Paper Title: TASI Lectures on Physics for Machine Learning.







Quantum Non-Demolition Measurements and Leggett-Garg inequality
    Stefano Gherardini
    August 5, 2024
===============================================================





        0.75
        § ABSTRACT 
            These notes are based on lectures I gave at TASI 2024 on Physics for Machine Learning. The focus is on neural network theory, organized according to network expressivity, statistics, and dynamics. I present classic results such as the universal approximation theorem and neural network / Gaussian process correspondence, and also more recent results such as the neural tangent kernel, feature learning with the maximal update parameterization, and Kolmogorov-Arnold networks. The exposition on neural network theory emphasizes a field theoretic perspective familiar to theoretical physicists. I elaborate on connections between the two, including a neural network approach to field theory.
        
        









§ INTRODUCTION




Computer science (CS) is still in its infancy.  “What!?” you incredulously exclaim, thinking back to the earliest computer scientists you can think of, perhaps Lovelace or Babbage or Turing. Since its early days, CS has steadily progressed, from cracking the Enigma code in the 40's, to personal computers in the 70's and 80's, the internet in the 90's, powerful computers-in-your-pocket in the 2000's, and finally to some version of artificial intelligence in the last decade; see <cit.> for a thorough account. It shows no signs of stopping, and one might wager that future historians will not speak of decades of progress in CS, but centuries. Computer science is still in its infancy.

It is 2024, what have we seen recently? A plethora of “experimental" breakthroughs in machine learning (ML) that constitute some of the most exciting developments in CS to date. Reinforcement learning algorithms have become world-class at Go <cit.> and Chess <cit.>, merely by playing against themselves. Diffusion models <cit.> can generate images of people that don't exist, and yet you might never know. Large language models (e.g. GPT-3 <cit.> literature, or popular accounts <cit.>) can code better than you — or me, at least —  and they can even write compelling bedtime stories for your children. These forms of artificial intelligence have already started a trillion dollar industry, and some have speculated that we've already seen sparks <cit.>  of AGI, or human-level intelligence.

We've also started to see significant advances in the sciences. For instance, AlphaFold <cit.> has achieved state-of-the-art performance for predicting the structure of folded proteins, and neural networks (NNs) give state-of-the-art models for complex objects, from quantum many-body ground states <cit.> to metrics <cit.> on the Calabi-Yau manifolds of string theory. Lack-of-rigor is a natural concern, but in certain cases ML techniques can be made rigorous or interpretable at a level that would satisfy a formal theoretical physicist or pure mathematician; see, e.g., <cit.> and references therein for examples in string theory, algebraic geometry, differential geometry, and low-dimensional topology, including uses of ML theory for science.
Of course, this is just scratching the surface, see, e.g., <cit.> for a smattering of results across physics and the physical sciences.

With this in mind, we have some explaining to do. There are a large number of impressive ML results in both pop culture and the sciences, but experiments have significantly outpaced theory over the last decade. An analogy from physics might be useful, given the original audience. The current situation in ML might be reminiscent of the 1960's in particle physics: some essentials of the theory were known, e.g. quantum electrodynamics (QED), but a deluge of newly discovered hadrons left theorists scratching their heads and demanded an organizing principle to explain the new phenomena. This was one of the motivating factors for the development of string theory, but the right theory in the end is the quark model, as described by quantum chromodynamics (QCD). It's tempting to speculate that recent results in ML are waiting for someone to discover a foundational theory, akin to QCD for the strong interactions. As theoretical physicists, we hope for this sort of mechanistic understanding. However, it might be that ML and intelligence are so complex that sociology is a better analogy, and it would be a fool's errand to expect such a precise theory. My personal guess is that detailed theory will pay off in certain areas of ML — perhaps related to sparsity in NNs, architecture design, or the topics of these lectures — but not in all. Nevertheless, given the experimental progress, one must try.


These lectures were given at TASI 2024 <cit.> to Ph.D. students in theoretical high energy physics. I was given the title “Physics4ML", which means that I'm trying to keep ML-for-Physics to a minimum. I take a physics perspective, but acknowledge that this is an enormous field to which many communities have contributed.

The central idea of the lectures is that understanding NNs is essential to understanding ML, a story I built on three pillars: expressivity, statistics, and dynamics of NNs, one for each of the lectures I gave. The content builds on many ideas explained to me over the last five years by friends, acknowledged below, who pioneered many of the developments, as well as a few of my own results. Throughout, I take a decidedly field-theoretic lens to suit the audience, and also because I think it's a useful way to understand NNs. 










 §.§ The Setup.

Understanding ML at the very least means understanding neural networks. A neural network is a function

    ϕ_θ:^d →

with parameters θ. We've chosen outputs in  because, channeling Coleman, scalars already exhibit the essentials. We'll use the lingo

    Input:      x ∈^d 
    Output:     ϕ_θ(x) ∈
    Network:     ϕ_θ∈(^d, ) 
    Data:     ,

where the data  depends on the problem, but involves at least a subset of ^d, potentially paired 
with labels y∈.

With this minimal background, let's ask our central question:
What does a NN predict?
For any fixed value of θ, the answer is clear: ϕ_θ(x). However, the answer is complicated by issues of both dynamics and statistics. 

First, dynamics. In ML, parameters are updated to solve problems and we really have trajectories in 

    Parameter Space:     θ(t) ∈^| θ |
    Output Space:     ϕ_θ(t)(x) ∈
    Function Space:     ϕ_θ(t)∈(^d, ).

governed by some learning dynamics determined by the optimization algorithm and the nature of the learning problem. For instance, in supervised learning we have data

    = {(x_α, y_α) ∈^d ×}_α=1^||,

and a loss function 

    [ϕ_θ] = ∑_α=1^||ℓ(ϕ_θ(x_α), y_α),

where ℓ is a loss function such as ℓ_MSE = (ϕ_θ(x_α)-y_α)^2. One may optimize θ by gradient descent 

    dθ_i/dt = -∇_θ_i[ϕ_θ],

or other algorithms, e.g., classics like stochastic gradient descent (SGD) <cit.> or Adam <cit.>, or a more recent technique such as Energy Conserving Descent <cit.>. Throughout, t is training time of the learning algorithm unless otherwise noted.

Second, statistics. When a NN is initialized on your computer, the parameters θ are initialized as draws

    θ∼ P(θ)

from a distribution P(θ), where  means “drawn from" in this context.
Different draws of θ will give different functions ϕ_θ, and a priori we have no reason to prefer one over another. The prediction ϕ_θ(x) therefore can't be fundamental! Instead, what is fundamental is the average prediction and second moment or variance:

    [ϕ_θ(x)]     = ∫ dθ P(θ)  ϕ_θ(x) 
    [ϕ_θ(x) ϕ_θ(y)]     = ∫ dθ  P(θ)   ϕ_θ(x) ϕ_θ(y),

as well as the higher moments. Expectations are across different initializations. Since we're physicists, we henceforth replace [·]=· and we remember this is a statistical expectation value. It's useful to put this in our language:

    G^(1)(x)     = ϕ_θ(x)
    
        G^(2)(x, y)     = ϕ_θ(x) ϕ_θ(y),

the mean prediction and second moment are just the one-point and two-point correlation functions of the statistical ensemble of neural networks. Apparently ML has something to do with field theory. 

Putting the dynamics and statistics together, we have an ensemble of initial θ-values, each of which is the starting point of a trajectory θ(t), and therefore we have an ensemble of trajectories. We choose to think of θ(t) drawn as 

    θ(t) ∼ P(θ(t)),

a density on parameters that depends on the training time and yields time-dependent correlators 

    G_t^(1)(x)     = ϕ_θ(x)_t 
    
        G_t^(2)(x, y)     = ϕ_θ(x) ϕ_θ(y)_t,

where the subscript t indicates time-dependence and the expectation is with respect to P(θ(t)). Of course, assuming that learning is helping, we wish to take t→∞ and are interested in
G_∞^(1)(x) = mean prediction of ∞-number of NNs as t→∞.
Remarkably, we will see that in a certain supervised setting there is an exact analytic solution for this quantity.

There is one more pillar beyond dynamics and statistics that I need to introduce: expressivity. Let's make the point by seeing a failure mode. Consider neural networks of the following functional form, or architecture:

    ϕ_θ(x) = θ· x.

The one-point and two-point functions of these are analytically solvable, but going that far defeats the purpose: this is a linear model, and learning schemes involving it are linear regression, which will fail on general problems. We say that model is not expressive enough to account for the data. Conversely, we wish instead to choose expressive architectures that can model the data, which essentially requires that the architecture is complex enough to approximate anything. Of course, such architectures must be non-linear.

We now have the three pillars on which we'll build our understanding of neural networks, and associated questions:

    
  * Expressivity. How powerful is the NN?
    
  * Statistics. What is the NN ensemble?
    
  * Dynamics. How does it evolve?

We will approach the topics in this order, building up theory that takes us through the last few years through a physicist's lens. 

A physicist's lens means a few things. First, it means a physicist's tools, including:


  * Field Theory as alluded to above.

  * Landscape Dynamics from loss functions on θ-space.

  * Symmetries of individual NNs and their ensembles.
 
A physicist's lens also means we will try to find the appropriate balance of rigor and intuition. We'd like mechanisms and toy models, but we won't try to prove everything at a level that would satisfy a mathematician. This befits the current situation in ML, where there are an enormous number of empirical results that need an O(1) theoretical understanding, not rigorous proof.

Henceforth, we drop the subscript θ in ϕ_θ(x), and the reader should recall that the network depends on parameters.






§ EXPRESSIVITY OF NEURAL NETWORKS



Neural networks are big functions composed out of many simpler functions according to the choice of architecture. The compositions give more flexibility, prompting
How powerful is a NN?
Ultimately this is a question for mathematics, as it's a question about functions.  Less colloquially, what we mean by the power of a NN is its ability to approximate any function.



 §.§ Universal Approximation Theorem


The Universal Approximation Theorem (UAT) is the first result in this direction. It states that a neural network with a single hidden layer can approximate any continuous function on a compact domain to arbitrary accuracy. More precisely, the origin version of the theorem due to Cybenko is

Let f: ^d → be a continuous function on a compact set K⊂^d. Then for any ϵ>0 there exists a neural network with a single hidden layer of the form

    ϕ(x) = ∑_i=1^N ∑_j=1^d w_i^(1)σ(w_ij^(0)  x_j + b^(0)_i)+b^(1),

θ = {w_ij^(0), w_i^(1), b^(0)_i, b^(1)},
where σ:→ is a non-polynomial non-linear activation function, such that

    sup_x∈ K |f(x) - ϕ(x)| < ϵ.


The architecture in Eq. (<ref>) is known by many names, including the perceptron, a fully-connected network, or a feedforward network. The parameter N is known as the width. It may be generalized to include a depth dimension L encoding the number of compositions of affine transformations and non-linearities. Such an architecture is known as a multi-layer perceptron (MLP) or a deep feedforward network.

The UAT is a powerful result, but it has some limitations. First, though the error ϵ does get better with N, it doesn't say how many neurons are needed. Second, it doesn't say how to train the network: though there is a point θ^* in parameter space that is a good approximation to any f, existence doesn't imply that we can find it, which is a question of learning dynamics.

To ask the obvious,
Why does this UAT work? 
In Cybenko's original work, he focused on the case that σ was the sigmoid function 

    σ(x) = 1/1+e^-x,

which allows us to get a picture of what's happening. The sigmoids appear in the network function of the form 

    σ(w_i^(0)· x + b_i^(0))

which approximates a shifted step function as w^(0)→∞. A linear combination can turn it into a bump with approximately compact support that gets scaled by w^(1). These bumps can then be put together to approximate any function; see, e.g., Fig. <ref>.

Cybenko's version of the UAT was just the first, and there are many generalizations, including to deeper networks, to other activation functions, and to other domains.  See <cit.> for the original paper and <cit.> for a generalization to the case of multiple hidden layers.





    


 §.§ Kolmogorov-Arnold Theorem


The UAT essentially says that any function can be approximated by a neural network, i.e. a complicated function comprised out of many building blocks. It is natural to wonder whether there are other theorems to this effect, and whether they inspire other neural network constructions.

One such result is due to Kolmogorov <cit.> and Arnold <cit.>, who showed that any multivariate continuous function can be represented (exactly, not approximated) as a sum of continuous one-dimensional functions:


Let f:[0,1]^n→ be an arbitrary multivariate continuous function. Then it has the representation 

    f(x_1,…,x_n) = ∑_q=0^2nΦ_q(∑_p=1^n ϕ_q,p(x_p))

with continuous one-dimensional functions Φ_q and ϕ_q,p.


This result is perhaps less intuitive than the UAT. If every multivariate function is to be represented by univariate functions plus addition, what then of f(x,y)=xy? This is just a pedagogical example since

    f(x,y) = xy = e^log x + log y,

but of course the general theorem is very non-trivial.

A notable feature of (<ref>) is that it is a sum of functions of functions, which one might take to inspire a neural network definition. However, one aspect that is a little unusual is that ϕ_q,p depends on both p and q, and therefore lives on the connection between x_p and x^(1)_q:= ∑_p=1^n ϕ_q,p(x_p). If this is a neural network, it's one with activations ϕ_q,p on the edges, rather than on the nodes. Furthermore, for distinct values of p,q, the functions ϕ_q,p are independent in general.

In fact, this idea led to the recent proposal of Kolmogorov-Arnold networks (KAN) <cit.>, a new architecture in which activation functions are on the edges, as motivated by KART, and are learned. Much like how MLPs may be formed out of layers that repeat the structure in the single-layer UAT, KANs may be formed into layers to repeat the structure of KART. See Figure <ref> for a detailed comparison with MLPs, including a depiction of the architectures and how they are formed out of layers motivated by their respective mathematical theorems. The KAN implementation provided in <cit.> includes visualizations of the learned activation functions, which can lead to interpretable can architectures that can be mapped directly onto symbolic formulae.




§ STATISTICS OF NEURAL NETWORKS



Let's try to understand neural networks at initialization. 
 For this, firing up your computer once is not enough, since one initialization give you one draw θ∼ P(θ) and therefore one random neural network. Instead, we want to understand the statistics of the networks:
 What characterizes the stats of the NN ensemble?
 One aspect of this is encoded in the moments, or n-pt functions,

    G^(n)(x_1,…,x_n) = ϕ(x_1)…ϕ(x_n),

which may be obtained from a partition function as 

    Z[J]    = e^∫ d^dx J(x) ϕ(x)
    
    G^(n)(x_1,…,x_n)    = (δ/δ J(x_1)…δ/δ J(x_n) Z[J])|_J=0,

where J(x) is a source.
This expectation · is intentionally not specified here to allow for flexibility. For instance, using the expectation in the introduction we have

    Z[J]     = ∫ dθ P(θ) e^∫ d^dx  J(x) ϕ(x),

reminding the reader that the NN ϕ(x) depends on θ. The partition function integrates over the density of network parameters. But as physicists we're much more familiar with function space densities according to 

    Z[J]     = ∫ϕ e^-S[ϕ] e^∫ J(x) ϕ(x),

the Feynman path integral that determines the correlators from an action S[ϕ] that defines a density on functions.

Since starting a neural network requires specifying the data (ϕ,P(θ)), the parameter space partition function (<ref>) and associated parameter space calculation of correlators is always available to us. Given that mathematical data, one might ask
What is the action S[ϕ] associated to (ϕ,P(θ))?
When this question can be answered, it opens a second way of studying or understanding the theory. The parameter-space and function-space descriptions should be thought of as a duality.



 §.§ NNGP Correspondence


Having raised the question of the action S[ϕ] associated to the network data (ϕ,P(θ)), we can turn to a classic result of Neal <cit.>.

For simplicity, we again consider a single-layer fully connected network of width N, with the so-called biases turned off for simplicity:

    ϕ(x) = 1/√(N)∑_i=1^N ∑_j=1^d w_i^(1)σ(w_ij^(0) x_j),

where the set of network parameters is θ = {w_ij^(0), w_i^(1)} independently and identically distributed (i.i.d.). 

    w_ij^(0)∼ P(w^(0))      w_i^(1)∼ P(w^(1)).
 
Under this assumption, we see
ØThe network is a sum of N i.i.d. functions.
This is a function version of the Central Limit Theorem, generalizing the review in Appendix <ref>, and gives us the Neural Network / Gaussian Process (NNGP) correspondence, 
NNGP Correspondencein the N →∞ limit, ϕ is drawn from a Gaussian Process (GP), 

    lim_N→∞   ϕ(x) ∼(μ(x), K(x,y)),

with mean and covariance (or kernel) μ(x) and K(x,y).
By the CLT, exp(-S[ϕ]) is Gaussian and therefore
S[ϕ] is quadratic in networks. Now this really feels like physics, since the infinite neural network is drawn from a Gaussian density on functions, which defines a generalized free field theory.

We will address generality of the NNGP correspondence momentarily, but let's first get a feel for how to do computations.
To facilitate then, we take P(w^(1)) to have zero mean and finite variance, 

    w^(1)=0     w^(1)w^(1) = μ_2,
 which causes the one-point function to vanish
G^(1)(x) = 0. Following Williams <cit.>, we compute the two-point function in parameter space (with Einstein summation)

    G^(2)(x,y) 
                = 1/Nw_i^(1)σ(w_ij^(0) x_j)  w_k^(1)σ(w_kl^(0) y_l)
       1/N w_i^(1) w_k^(1)σ(w_ij^(0) x_j) σ(w_kl^(0) y_l)
        = μ_2/Nσ(w_ij^(0) x_j) σ(w_il^(0) y_l),

where the last equality follows from the ones being i.i.d., w_i^(1)w_k^(1) = μ_2 δ_ik. The sum over i gives us N copies of the same function, leaving us with 

    G^(2)(x,y) = μ_2  σ(w_ij^(0) x_j) σ(w_il^(0) y_l),

where we emphasize there is now 
no summation on i. This is an exact-in-N two-point function that now requires only on the computation of the quantity in bra-kets. One may try to evaluate it exactly by doing the integral over w^(0). If it can't be done, Monte Carlo estimates may be obtained from M samples of w^(0)∼ P(w^(0)) as

    G^(2)(x,y) ≃μ_2/M∑_samples^M σ(w_ij^(0) x_j) σ(w_il^(0) y_l).

In typical NN settings, parameter densities are easy to sample for convenience, allowing for easy computation of the estimate. If the density is more complicated, one may always resort to Markov chains, e.g. as in lattice field theory.

With this computation in hand, we have the defining data of this NNGP,

    lim_N→∞   ϕ(x) ∼(0, G^(2)(x,y)).

The associated action is 

    S[ϕ] = ∫ d^dx d^dy ϕ(x)  G^(2)(x,y)^-1 ϕ(y),

where 

    ∫ d^dy   G^(2)(x,y)^-1 G^(2)(y,z) = δ^(d)(x-z).

defines the inverse two-point function. In fact, this allows us to determine the action of any NNGP with μ(x)=G^(1)(x)=0, by computing the G^(2) in parameter space and inverting it.



So certain large neural networks are function draws from generalized free field theories. But at this point you might be asking yourself
How general is the NNGP correspondence?
Neal's result — that infinite-width single-layer feedforward NNs are drawn from GP — stood on its own for many years, perhaps (I am guessing) due to focus on non-NN ML techniques in the 90's and early 2000's during a so-called AI Winter. As NNs succeeded on many tasks in the 2010's after AlexNet <cit.>, however, many asked whether architecture X has a hyperparameter N such that the network is drawn from a Gaussian Process as N→∞. Before listing such X's, let's rhetorically ask
Didn't Neal's result essentially follow from summing N i.i.d. random functions? Maybe NNs do this all the time?
In fact, that is the case. Architectures admitting an NNGP limit include

    
  * Deep Fully Connected Networks, N=width,
    
  * Convolutional Neural Networks, N=channels,
    
  * Attention Networks, N=heads,

and many more. See, e.g., <cit.> and references therein.



 §.§ Non-Gaussian Processes


If the GP limit exists due to the CLT, then violating any of the assumptions of the CLT should introduce non-Gaussianities, which are interactions in field theory. From Appendix <ref>, we see that the CLT is violated by finite-N corrections and breaking statistical independence. See <cit.> for a systematic treatment of independence breaking, and derivation of NN actions from correlators.

We wish to see the N-dependence of the connected 4-pt function. William's technique for computing G^(2) extends to any correlator. To avoid a proliferation of indices, we will compute it using the notation

    ϕ(x) = ∑_i w_i φ_i(x)

where w_i is distributed as w^(1) was in the single layer case, and φ_i(x) are i.i.d. neurons of any architecture. The four-point function is 

    G^(4)   = ϕ(x)ϕ(y)ϕ(z)ϕ(w)
       = ∑_i,j,k,lw_i w_j w_k w_lφ_i(x)φ_j(y)φ_k(z)φ_l(w)
       = ∑_i w_i^4φ_i(x)φ_i(y)φ_i(z)φ_i(w)
               + ∑_i≠ jw_i^2w_j^2φ_i(x)φ_i(y)φ_j(z)φ_j(w) + perms.

One can see that you have to be careful with indices. The connected 4-pt function is <cit.>

    G^(4)_c(x_,y,z,w) = G^(4)(x,y,z,w) - (G^(2)(x,y)G^(2)(z,w) + perms),

and watching indices carefully we obtain 

    G^(4)_c(x_,y,z,w) = 1/N(μ_4  φ_i(x)φ_i(y)φ_i(z)φ_i(w)
     - μ_2^2  (φ_i(x)φ_i(y)φ_i(z)φ_i(w) + perms)),

with no Einstein summation on i.
We see that the connected 4-pt function is non-zero at finite-N, signalling interactions. We will see that in some examples G^(4)_c can be computed exactly.




 §.§ Symmetries


We have gotten some control over the statistics of the ensemble of neural networks. It natural 
at this point to ask 
Is there structure in the ensemble?
By this I mean properties that the ensemble realizes that an individual network might not see. We will return to this question broadly in Section <ref>, but for now we will focus on one type of structure: symmetries.

To allow for symmetries at both input and output, in this section we consider networks

    ϕ: ^d →^D.
 
with D-dimensional output. Sometimes the indices will be implicit.

A classic example is equivariant neural networks <cit.>.
We say that ϕ is G-equivariant with respect to a group G if 

    ρ_D(g) ϕ(x) = ϕ(ρ_d(g) x)     ∀ g ∈ G,

where 

    ρ_d ∈Mat(^d)    ρ_D ∈Mat(^D)

are matrix representations of G on ^D and ^d, respectively. The network is invariant if ρ_D = 1, the trivial representation.
Equivariance is a powerful constraint on the network that may be implemented in a variety of ways. For problems that have appropriate symmetries, building them into the network can improve the speed and performance of learning <cit.>, e.g., at the level of scaling laws <cit.>. For instance, in Phiala Shanahan's lectures you'll learn about SU(N)-equivariant neural networks from her work <cit.>, which are natural in lattice field theory due to invariance of the action under gauge transformations.

But this is the statistics section, so 
we're interested in symmetries that arise in ensembles of neural networks, which leave the statistical ensemble invariant. In field theory, we call them global symmetries. Let the network transform under a group action as 

    ϕ↦ϕ_g,      g ∈ G.

We say that ensemble of networks has a global symmetry group G if the partition function is invariant,

    Z_g[J] = Z[J],     ∀ g ∈ G.

At the level of expectations, this is 

    e^∫ d^d x J(x) ϕ_g(x)  =e^∫ d^d x J(x) ϕ(x)    ∀ g ∈ G,

where one can put indices on ϕ and J as required by D. By a network redefinition on the LHS, this may be cast as having a symmetry if · is invariant. 
In the usual path integral this is the statement of invariant action S[ϕ] and measure ϕ. In parameter space, the redefinition may be instituted <cit.> by absorbing g into a redefinition of parameters as θ↦θ_g, with symmetry arising when 

    ∫ dθ_g P(θ_g) e^∫ d^d x J(x)ϕ_θ(x) = ∫ dθ P(θ) e^∫ d^d x J(x)ϕ_θ(x),

i.e. the parameter density and measure must be invariant. We will give a simple example realizing this mechanism in a moment.

It is most natural to transform the input or output of the network. Our mechanism allows for symmetries of both types, which are analogs of spacetime and internal symmetries, respectively. It may also be interesting to study symmetries of intermediate layers, if one wishes to impose symmetries on learned representations. Equivariance fits into this picture because it turns a transformation at input into a transformation at output. The ensemble of equivariant NNs is invariant under ρ_d action on the input if the partition function is invariant under the induced ρ_D action on the output.


Example.  Consider any architecture of the form 

    ϕ(x) = ∑_i w_i φ_i(x),      w_i ∼ P(w)  even,

for any neuron φ, which could itself be a deep network. The _2 action 
ϕ↦ -ϕ may be absorbed into parameters w_g = -w_i with dw P(w) invariant by evenness, which is a global symmetry provided that the domain is invariant. This theory has only even correlators. 

Less trivial examples will be presented below, when we present specific networks and compute correlators.



 §.§ Examples


In the statistics of neural networks, we have covered three topics: Gaussian limits, non-Gaussian corrections from violating CLT assumptions, and symmetries. We will present this essential data in some examples and then discuss similarities and differences. For canonical cases in ML, like networks with ReLU activation or similar, see <cit.> and references therein.

Gauss-net.
The architecture and parameter densities are 

    ϕ(x) = w^(1)_iexp(w^(0)_ij x_j + b^(0)_i)/√(exp[2(σ_b_0^2 + σ^2_w_0/x^2 )]),

for parameters drawn i.i.d. from 

    w^(0)∼𝒩(0,σ^2_W_0/),     w^(1)∼𝒩(0, σ^2_w_1/2N),     b^(0)∼𝒩(0, σ^2_b_0).

The two-point function is

    G^(2)(x_1, x_2) = σ_w_1^2/2 e^-1/2σ_w_0^2 (x_1-x_2)^2

and we see that the theory has a correlation length 

    ξ = √(/σ_w_0^2).

The connected four-point function is

    G^(4)_c =   1/4 Nσ_w_1^4 [ 3e^4σ_b_0^2 e^-σ_w_0^2/2d∑_i x_i^2 -2(x_1 x_2 +  6  perms)
       - (e^-1/2dσ_w_0^2
        ((x_1-x_4)^2+(x_2-x_3)^2) +  2  perms 
        ) ].

We see that the 2-pt function is translation invariant, but not the 4-pt function.


Euclidean-Invariant Nets. 
We are interested in constructing input layers that are sufficient to ensure invariance under SO(d) and translations, i.e., under the Euclidean group. Consider an input layer of the form 

    ℓ_i(x) = F( w^(0)_i) cos(∑_j w^(0)_ij x_j + b^(0)_i),     i ∈1,…,N,

where the sum has been made explicit since i's are not summed over, with 

    w^(0)_ij∼ P(w^(0)_ij)      b^(0)_i ∼Unif[-π,π].

It is easy to see that x_i→ R_ij x_j for R∈ SO(d) can be absorbed into a redefinition of the w^(0)_ij's, and the partition function is invariant under this action provided that F( w^(0)_i) and P(w^(0)_ij) are invariant. Furthermore, arbitrary translations x_j → x_j+ϵ_j give a term w^(0)_ijϵ_j that can be absorbed into a redefinition of the b^(0)_i that leaves the theory invariant. See <cit.> for the 2-pt and 4-pt functions.

We emphasize two nice features:


  * Larger Euclidean Nets. Any network that builds on ℓ without reusing its parameters is Euclidean-invariant.

  * Spectrum Shaping. In computing G^(2)(p), b_i gets evaluated on p, and F may be chosen to shape the power spectrum (momentum space propagator) arbitrarily.

We refer the reader to <cit.> for general calculations of ℓ-correlators and to the specializations below for simple single-layer theories. We turn the ℓ_i(x) input layer into a scalar NN by

    ϕ(x) = ∑ w^(1)_i ℓ_i(x)

with drawn w^(1)_i ∼ P(w^(1)).

Cos-net.
The architecture is defined by specifying (<ref>) to F=1 

    ϕ(x) =  w^(1)_i cos( w^(0)_ij x_j + b^(0)_i),

and specific parameter densities

    w^(1)∼(0,σ_w_1^2/N)      w^(0)∼(0,σ_w_0^2/d)      b^(0)∼Unif[-π,π].

The two-point function is equivalent to that of Gauss-net

    G^(2)(x,y) = σ_w_1^2/2 e^-1/2σ_w_0^2 (x_1-x_2)^2.

The connected four-point function is 

    G^(4)|_c =    1/8 Nσ_w_1^4 [ 3  (e^-1/2dσ_w_0^2
       (x_1+x_2-x_3-x_4)^2+  2 perms) 
       -(2 e^-1/2dσ_b^2
       ((x_1-x_4)^2+(x_2-x_3)^2)+  2 perms) ].

The 2-pt function and 4-pt function are fully Euclidean invariant.

Equipped with both Cos-net and Gauss-net, we'd like to compare the two:


  * Symmetry. Cos-net is Euclidean-invariant ∀ N, by construction, while Gauss-net enjoys this symmetry only when N→∞; in that limit it only depends on G^(2)(x,y).

  * Large-N Duality. As N→∞, both theories are drawn from the same Gaussian Process, i.e., with the same G^(2)(x,y).


Scalar-net.
Here is one final example you might like, specializing (<ref>)

    ϕ(x) =  √(2 vol(B_Λ^d)/(2π)^d σ_w_1^2)1/√( w^(0)_i^2 + m^2)  w^(1)_i cos( w^(0)_ij x_j + b^(0)_i),

and specific parameter densities

    w^(1)∼(0,σ_w_1^2/N)      w^(0)∼Unif(B_Λ^d)      b^(0)∼Unif[-π,π],

where B_Λ^d is a d-ball of radius Λ. The theory is translation invariant by construction, and so we compute the power spectrum of the two-point function G^(2)(x-y) to be 

    G^(2)(p) = 1/p^2+m^2.

We see that we have a realization of the free scalar field theory in d Euclidean dimensions. See Sec. <ref> for an extension to ϕ^4 theory.
    



§ DYNAMICS OF NEURAL NETWORKS



Having covered expressivity and statistics, we turn to dynamics. Focusing on the most elementary NN dynamics, we ask
How does a NN evolve under gradient descent? First we will study a simplification known as the neural tangent kernel (NTK), and then will use it in the case of MSE loss to solve a model exactly. We'll discuss drawbacks of the NTK, and then improve upon them with a scaling analysis that ensures feature learning.

We will study the dynamics of supervised learning with gradient  descent, with data 

    = {(x_α,y_α)}_α=1^||,

and loss function

    [ϕ] = 1/||∑_α=1^||ℓ(ϕ(x_α), y_α),

We optimize the network parameters θ by gradient descent,

    dθ_i/dt = -η∇_θ_i[ϕ].

It is also convenient to define 

    Δ(x) = - δℓ(ϕ(x),y)/δϕ(x),

where y is to be understood as the label associated to x, which yields 

    dθ_i/dt= η/||∑_α=1^||Δ(x_α) ∂ϕ(x_α)/∂θ_i.

as another form of the gradient descent equation, by chain rule. Δ(x) is the natural object of gradient descent in function space.

We use Einstein summation throughout this section unless stated otherwise (which will happen).



 §.§ Neural Tangent Kernel


We now arrive at a classic dynamical result in ML theory, the neural tangent kernel <cit.>. We train the network by gradient descent 

    dϕ(x)/dt   = ∂ϕ(x)/∂θ_idθ_i/dt
       = η/||∑_α=1^||Δ(x_α)  Θ(x,x_α),

where 

    Θ(x,x_α) = ∂ϕ(x)/∂θ_i∂ϕ(x_α)/∂θ_i

is the neural tangent kernel (NTK).

Given the short derivation, this is clearly a fundamental object, but it seems terrible to work with, because it is:


  * Parameter-dependent. For modern NNs, there are billions of parameters to sum over.

  * Time-dependent. Of course, the learning trajectory is θ(t), and therefore the NTK time evolves.

  * Stochastic. Since θ(t) begins at θ(0) sampled at initialization, the NTK inherits the randomness.

It's also non-local, communicating information about the loss at train points x_α to the test point x. In summary, the NTK is unwieldy.

The reason it is a classic result, however, is that it simplifies in the N→∞ limit. In this limit, neural network training is in the so-called

    Lazy regime:     |θ(t) - θ(0)| ≪ 1,

i.e. the number of parameters is large but the evolution keeps them in a local neighborhood. In such a regime the network is approximately a linear-in-parameters model <cit.>

    lim_N→∞ϕ(x)≃ϕ_lin(x) := ϕ_θ_0(x) + (θ-θ_0)_i ∂ϕ(x)/∂θ_i|_θ_0,

and we have 

    lim_N→∞Θ(x,x') ≃Θ(x,x')|_θ_0.

That is, the infinite-width NTK is the NTK at initialization, provided that the network evolves as a linear model. Furthermore, in the same limit the law of large numbers often allows a sum to be replaced by an expectation value, e.g., 

    lim_N→∞Θ(x,x')|_θ_0 = β_θ(x,x') =: Θ̅(x,x'),

for computable β(x,x'), yielding network dynamics governed by 

    dϕ(x)/dt= -η/||∑_α=1^||δ l(ϕ(x_α),y_α)/δϕ(x_α)  Θ̅(x,x_α),

where Θ̅ is the so-called frozen NTK, a kernel that may be computed at initialization and fixed once-and-for-all. This is a dramatic simplification of the dynamics.

However, you should also complain.
ComplaintThe dynamics in (<ref>) simply interpolates between information at train points x_α and test point x,
according to a fixed function Θ̅. This isn't “learning" in  the usual NN sense, and there are zero parameters. In particular, since the NN only affects (<ref>) through Θ̅, which is fixed, nothing happening dynamically in the NN is affecting the evolution. We say that in this limit the NN does not learn features in the hidden dimensions (intermediate layers), since their non-trivial evolution would cause the NTK to evolve.

Example. 
Let's compute the frozen NTK for a single-layer network, to get the idea. The architecture is 

    ϕ(x) = 1/√(N)∑_i=1^N ∑_j=1^d w_i^(1)σ(w_ij^(0) x_j).

We make the sums explicit here because one is very important.
The NTK is 

    Θ(x,x')    = ∑_i∂ϕ(x)/∂ w_i^(1)∂ϕ(x')/∂ w_i^(1) + ∑_ij∂ϕ(x)/∂ w_ij^(0)∂ϕ(x')/∂ w_ij^(0)
       = 1/N∑_i=1^N (∑_j,l=1^d σ(w_ij^(0)x_j)σ(w_il^(0)x'_l)
            + ∑_j=1^d x_j x'_j  w_i^(1)w_i^(1) σ'(w_ij^(0) x_j) σ'(w_ij^(0)x'_j) ) 
       =: 1/N∑_i β_i(x,x').

If you squint a little, you'll see that the i-sum is a sum over the same type of object, β_i(x,x'), whose i dependence comes from all these i.i.d. parameter draws in the i-direction. By the law of large numbers, we have that in the N→∞ limit

    Θ̅(x,x') = β_i(x,x'),

with no sum on i.
We emphasize 
ØThe NTK in the N→∞ limit is deterministic (parameter-independent), depending only on P(θ).
Sometimes, the expectation may be computed exactly, and one knows the NTK that governs the dynamics once and for all.



 §.§ An Exactly Solvable Model


Let us consider a special case of frozen-NTK dynamics with MSE loss,

    ℓ(ϕ(x),y) = 1/2(ϕ(x)-y)^2.

Then the dynamics (<ref>) becomes 

    dϕ(x)/dt = -η/||∑_α=1^|| (ϕ(x_α)-y_α) Θ̅(x,x_α).

The solution to this ODE is 

    ϕ_t(x) = ϕ_0(x) + 1/||Θ̅(x,x_α) Θ̅(x_α, x_β)^-1(1-e^-ηΘ̅t)_βγ(y_γ-ϕ_0(x_γ)),

where computational difficulty is that Θ̅(x,x_α) is a ||× || matrix and takes O(||^3) time to invert. 
The solution defines a trajectory through function space from ϕ_0 to ϕ_∞. The converged network is 

    ϕ_∞(x) = ϕ_0(x) +Θ̅(x,x_α) Θ̅(x_α, x_β)^-1(y_β-ϕ_0(x_β)).

This is known as kernel regression, a classic technique in ML. 
In general kernel regression, one chooses the kernel. In our case, gradient descent training in the N→∞ limit is kernel regression, with respect to a specific kernel determined by the NN, the NTK Θ̅. 

On train points we have memorization

    ϕ_∞(x_α) = y_α    ∀α.

On test points x, the converged network is performing an interpolation, communicating residuals R_β on train points β through a fixed kernel Θ̅ to test points x. The prediction depends on ϕ_0, but may be averaged over to obtain 

    μ_∞(x) := ϕ_∞(x) = Θ̅(x,x_α) Θ̅(x_α, x_β)^-1y_β,

provided that ϕ_0=0, as in many initializations for the parameters.
Let's put some English on the remarkable facts,

    
  * μ_∞(x) is the mean prediction of an ∞ number of ∞-wide NNs trained to ∞ time.
    
  * If ϕ_0 is drawn from a GP, then ϕ_∞ is as well. The mean is precisely μ_∞(x), see <cit.> for the two-point function and covariance.




 §.§ Feature Learning


The frozen NTK is a tractable toy model, but it has a major drawback: it does not learn features.
In this section we perform a more general study of learning dynamics, with a focus on choosing wise N-scaling such that features are non-trivially learned during gradient descent. To do this we will engineer three properties: that features (pre-activations) are finite at initialization, predictions evolve in finite time, and features evolve in finite time. 

This section is denser than the others, so I encourage the reader to remember that we are aiming to achieve these three principles if they become overwhelmed with details. Throughout, I am following lecture notes of Pehlevan and Bordelon <cit.>  with some changes in notation to match the rest of the lectures; see those lectures for more details, <cit.> for original literature from those lectures that I utilize, and <cit.> for related work on feature learning.

We study a deep feedforward network with L layers and width N, which in all is a map

    ϕ:^D →

(note the input dimension D; d is reserved for below) defined recursively as 

    ϕ(x)    = 1/γ_0 N^dz^(L)(x) 
     
    z^(L)(x)    = 1/N^a_L w_i^(L) σ(z_i^(L-1)(x))
    
    z^(ℓ)_i(x)    = 1/N^a_ℓ W^(ℓ)_ijσ(z_j^(ℓ-1)(x)) 
    
    z_i^(1)(x)    = 1/N^a_1√(D) W^(1)_ijx_j

where Einstein summation is implied throughout this section   (unless stated otherwise) and all Latin indices run from {1,…,N} except in the j-index in the first layer, when they are {1,…,D}. The parameters are drawn 

    w_i^(ℓ)∼(0,1/N^b_L)      W^(ℓ)_ij∼(0,1/N^b_ℓ).

We scale the learning rate as 

    η = η_0 γ_0^2 N^2d-c

with γ_0,η_0 O(1)  constants. We use a parameterization that will be convenient where d has already been introduced but c is a new parameter. For notational brevity we will sometimes use a Greek index subscript in place of inputs, e.g. z^(ℓ)_α:= z^(ℓ)(x_α). The z's are known as the pre-activations, as they are the inputs to the activation functions σ.

We have a standard MLP but have parameterized our ignorance of N-scaling, governed by parameters (a_ℓ,b_ℓ,c,d). We will use this freedom to set some reasonable goals:

    
  * Finite Initialization Pre-activations. z^(ℓ)∼ O_N(1)   ∀ l.
    
  * Learning in Finite Time. dϕ(x)/dt∼ O_N(1).
    
  * Feature Learning in Finite Time. dz^(ℓ)/dt ∼ O_N(1)    ∀ l.

These constraints have a one-parameter family of solutions, which becomes completely fixed under an additional learning rate assumption. We take each constraint in turn.

Finite Pre-activations. Since the weights are zero mean,  trivially

    z^(ℓ)_α = 0     ∀ l.

We must also compute the covariance. For the first pre-activation, we have 

    z_iα^(1)z_jβ^(1) = 1/D N^2a_1w^(1)_imw^(1)_jn  x_mαx_nβ = δ_ij1/DN^2a_1+b_1 x_mαx_mβ.

A similar calculation for higher layers l  gives

    z_iα^(ℓ)z_jβ^(ℓ)   = 1/ N^2a_ℓw^(1)_imw^(1)_jn σ(z^(ℓ-1)_mα)σ(z^(ℓ-1)_mβ)
       = δ_ij1/N^2a_ℓ+b_ℓ-11/Nσ(z^(ℓ-1)_mα)σ(z^(ℓ-1)_mβ)
       = δ_ij1/N^2a_ℓ+b_ℓ-1Φ^(ℓ-1)_αβ,

where 

    Φ^(ℓ)_αβ := 1/Nσ(z^(ℓ)_mα)σ(z^(ℓ)_mβ)

is a feature kernel. We are constructing a proof-by-induction that the pre-activations are O_N(1), so at this stage we may assume that the pre-activations z^(ℓ-1)∼ O_N(1) and therefore Φ^(ℓ-1)∼ O_N(1) since it is the average of N O_N(1) quantities σ(z^(ℓ-1)_mα). With this in hand, 

    2a_1+b_1=0      2a_ℓ + b_ℓ = 1   ∀ l > 1,

ensures that the pre-activations z^(ℓ) are O_N(1), as is empirically required for well-behaved training.

As an aside: in the N→∞ limit feature kernels asymptote to deterministic objects, akin to the frozen NTK behavior we have already seen, and intermediate layer pre-activations z_α^(ℓ) are also Gaussian distributed. Therefore in that limit, the statistics of a randomly initialized neural network is described by a sequence of generalized free field theories where correlations are propagated down the network according to a recursion relation; see, e.g., <cit.>.

Learning in Finite Time. Similar to our NTK derivation, we have 

    dϕ(x)/dt = η/||∑_α=1^||Δ(x_α) ∂ϕ(x_α)/∂θ_i∂ϕ(x)/∂θ_i,

We have that 

    dϕ(x)/dt∼ O_N(1)   ↔   γ_0^2 N^2d/N^c∂ϕ(x_α)/∂θ_i∂ϕ(x)/∂θ_i∼ O_N(1).

A short calculation involving a chain rule, keeping track of the different types of layers, and tedium yields

    η_0γ_0^2 N^2d/N^c∂ϕ(x_α)/∂θ_i∂ϕ(x)/∂θ_i   = 1/N^c[ 1/N^2 a_L - 1Φ_μν^(L-1)
       + ∑_ℓ=2^L-11/N^2 a_ℓ - 1 G_α x^(ℓ)Φ_α x^(ℓ-1) + 1/N^2 a_1 G_α x^(1)Φ_α x^(0)]

where 

    G_μν^(ℓ) = 1/N g^(ℓ)_iμ  g^(ℓ)_iν,
          g^(ℓ)_iμ = √(N) ∂ z^(L)_μ/∂ z^(ℓ)_iμ

arises naturally when doing a chain rule through the layers and x in the subscript is a stand-in for an arbitrary input x. Some additional work <cit.> shows that g, G ∼ O_N(1). Since we've shown that the feature kernels are also O_N(1), then if 

    2a_1 + c = 0,      2a_ℓ + c = 1    ∀ l > 1,

we have that dϕ(x)/dt∼ O_N(1), i.e., predictions evolve in finite time, and therefore learning happens in finite time.

Features Evolve in Finite Time. 
Finally, we need the features to evolve. For the first layer we compute

    dz_iμ^(1)/dt   =1/N^a_1√(D)dW_ij/dt x_jμ 
     = 1/N^2a_1+c-d+1/2η_0γ_0/||∑_ν=1^||Δ_ν g_iν^(1)Φ_μν^(0),
 
where the only N-dependence arises from the exponent. Given (<ref>), 

    d=1/2

is required to have  dz^(1)/dt ∼ O_N(1), a constraint that in fact persists for all dz^(ℓ)/dt. We note that we have
No feature learningif d< 1/2,
which applies in particular to the NTK regime d=0.

Summarizing the Constraints. Putting it all together, by enforcing finite pre-activations and that predictions and features evolve in finite time, we have constraints given by

    2a_1 + b_1    = 0 
    
        2a_ℓ + b_ℓ   = 1   ∀ l > 1 
    
        2a_1 + c    = 0 
    
        2a_ℓ + c    = 1   ∀ l > 1 
    
        d    = 1/2.

These constraints are solves by a one-parameter family depending on a∈

    (a_ℓ, b_ℓ, c_ℓ)    = (a,1-2a,1-2a)   ∀ l > 1 
     
    (a_1, b_1, c_1)     = (a-1/2, 1-2a, 1-2a).

If one makes the addition demands that η∼ O_N(N^2d-c) is O_N(1), so that you don't have to change the learning rate as you scale up, then c=1 is fixed and there is a unique solution. This scaling is known as the maximal update parameterization, or μ P <cit.>.







§ NN-FT CORRESPONDENCE



Understanding the statistics and dynamics of NNs has led us naturally to objects that we are used to from field theory. The idea has been to understand ML theory, but one can also ask the converse, whether ML theory gives new insights into field theory. With that in mind, we ask
What is a field theory?
At the very least, a field theory needs 
    
  * Fields, functions from an appropriate function space, or sections of an appropriate bundle, more generally. 
    
  * Correlation Functions of fields, here expressed as scalars
    
    G^(n)(x_1,…,x_n) = ϕ(x_1)…ϕ(x_n).


You might already be wanting to add more beyond these minimal requirements – we'll discuss that in a second. For now, we have 
Answera FT is an ensemble of functions with a way to compute their correlators.
In the Euclidean case, when the expectation is a statistical expectation, one my say
Euclidean Answera FT is a statistical ensemble of functions.
Our minimal requirements get us a partition function

    Z[J] = e^∫ d^dx J(x)ϕ(x)

that we can use to compute correlators, where at this stage we are agnostic about the definition of ·. In normal field theory, the · is defined by the Feynman path integral 

    Z[J] = ∫ϕ  e^-S[ϕ] + ∫ d^dx J(x)ϕ(x),

which requires specifying an action S[ϕ] that determines a density on functions exp(-S[ϕ]). But that's not the data we specify when we specify a NN. The NN data (ϕ_θ, P(θ)) instead defines 

    Z[J] = ∫ dθ P(θ) e^∫ d^dx J(x)ϕ_θ(x).

These are two different ways of defining a field theory, and indeed given (ϕ_θ,P(θ)) one can try to work out the associated action, in which case we have dual description of the same field theory, as in the NNGP correspondence.
The parameter space description is already quite useful, though, as it enables the computation of correlation functions even if the action isn't known. In certain cases it enables the computation of exact correlators in interacting  theories.

Okay, you get it, this is a different way to do field theory. Now I'll let you complain about my definition. You're asking 
Shouldn't my definition of field theory include X?
I'm writing this before I give the lecture, and my guess is you already asked about a set of X's, e.g. 

    X ∈{Quantum ,
         Lagrangian,
         Symmetries,
         Locality, …}.

The problem is that with any such X, there's usually some community of physicists that doesn't care. For instance, not all statistical field theories are Wick rotations of quantum theories; not all field theories have a known Lagrangian; not all field theories have symmetry; not all field theories are local. So I'm going to stick with my definition, because at a minimum I want fields and correlators.

Instead, if your X isn't included in the definition of field theory, it becomes an engineering problem. Whether you're defining your specific theory by S[ϕ], (ϕ_θ,P(θ)), or something else, you can ask
Can I engineer my defining data to get FT + X?
For X=Symmetries you've already seen ways to do this at the level of actions in QFT1 and at the level of (ϕ_θ,P(θ)) in these lectures.


For a current account of recent progress in NN-FT, see the rather long Introduction of <cit.> and associated references, as well as results.



 §.§ Quantum Field Theory


We've been on Euclidean space the whole time[This can be relaxed, see e.g. for a recent paper defining an equivariant network in Lorentzian signature <cit.>.], so it's natural to wonder in what sense these field theories are quantum. In a course on field theory, we first learn to canonically quantize and then at some later point learn about Wick rotation, and how it can define Euclidean correlators. The theory is manifestly quantum.

But given a Euclidean theory, can it be continued to a well-behaved quantum theory in Lorentzian signature, e.g. with unitary time evolution and a Hilbert space without negative norm states? If we have a nice-enough local action, it's possible, but what if we don't have an action? We ask:
Given Euclidean correlators, can the theory be continued to a well-behaved Lorentzian quantum theory?
This is a central question in axiomatic quantum field theory, and the answer is that it depends on the properties of the correlators. The Osterwalder-Schrader (OS) theorem <cit.> gives a set of conditions on the Euclidean correlators that ensure that the theory can be continued to a unitary Lorentzian theory that satisfies the Wightman axioms.
The conditions of the theorem include 


  * Euclidean Invariance. The correlators are invariant under the Euclidean group, which after continuation to Lorentzian signature becomes the Poincaré group.

  * Permutation Invariance of the correlators G^(n)(x_1,…,x_n) under any permutation of the x_1,…,x_n.

  * Reflection Positivity. Having time in Lorentzian signature requires picking a Euclidean time direction τ. Let R(x) be the reflection of x in the τ=0 plane. Then reflection positivity requires that

    G^(2n)(x_1,…,x_n,R(x_1),…,R(x_n)) ≥ 0.

Technically, this is necessary but not sufficient. An accessible elaboration can be found in notes <cit.> from a previous TASI.

  * Cluster Decomposition occurs when the connected correlators vanish when any points are infinitely far apart.

If all of these are satisfied, then the pair (ϕ_θ, P(θ)) that defines the NN-FT actually defines a neural network quantum field theory <cit.>.
In NN-FT, permutation invariance is essentially automatic and Euclidean invariance may be engineered as described in Section <ref>. Cluster decomposition and reflection positivity hold in some examples <cit.>, but systematizing their construction is an important direction for future work.

There is at least one straightforward way to obtain an interacting NN-QFT. Notably, if (ϕ_θ, P(θ)) is a NNGP that satisfies the OS axioms (this is much easier <cit.>) with Gaussian partition function

    Z_G[J] = ∫ dθ P(θ) e^∫ d^dx J(x)ϕ_θ(x),

then one may insert an operator associated to any local potential V(ϕ), which deforms the action in the expected way and the NN-FT to 

    Z[J]    = ∫ dθ P(θ) e^∫ d^dx V(ϕ_θ(x)) e^∫ d^dx J(x)ϕ_θ(x)
       =: ∫ dθP̃(θ) e^∫ d^dx J(x)ϕ_θ(x),

where the architecture equation ϕ_θ(x) lets us sub out the abstract expression for a concrete function of parameters, defining a new density on parameters P̃(θ) in the process. The interactions in V(ϕ) break Gaussianity of the NNGP that was ensured by a CLT. This means a CLT assumption must be violated: it is the breaking of statistical independence in P̃(θ). The theory Z[J] defined by (ϕ_θ, P̃(θ)) is an interacting NN-QFT, since local potentials that deformed Gaussian QFTs still satisfy reflection positivity and cluster decomposition.



 §.§ ϕ^4 Theory


With this discussion of operator insertions, it's clear now how to get ϕ^4 theory. We just insert the operator

    e^∫ d^dx ϕ_θ(x)^4

into the partition function associated to the free scalar of Section <ref>; this operator with the architecture (<ref>) technically requires an IR cutoff, though other architectures realizing the free scalar may not. The operator insertion deforms the parameter densities in (<ref>) and breaks their statistical independence, explaining the origin of interactions in the NN-QFT. See <cit.> for a thorough presentation.



 §.§ Open Questions


In this section we've discussed a neural network approach to field theory in which the partition function is an integral over probability densities of parameters, and the fields (networks) are functions of these parameters. We have summarized some essential results, but there are many outstanding questions:


  * Reflection Positivity. Is there a systematic way to engineer NN-FTs (ϕ_θ, P(θ)) that satisfy reflection positivity?

  * Cluster Decomposition. Can  systematically understand the conditions under which cluster decomposition holds in NN-FT?

  * Engineering Actions. How generally can we define an NN-FT that is equivalent to a field theory with a fixed action?

  * Locality in the action can be realized at infinite-N, but is there a fundamental obstruction at finite-N?

On the other hand, an NN-FT approach to CFT and to Grassmann fields is well underway and should appear in 2024.



§ RECAP AND OUTLOOK


Neural networks are the backbone of recent progress in ML. In these lectures we took the perspective that to understand ML, we must understand neural networks, and built our study around three pillars: the expressivity, statistics, and dynamics of neural networks.

Let's recap the essentials and then provide some outlook. In Section <ref> on expressivity, we asked “How powerful is a NN?" We presented the Universal Approximation Theorem (UAT) and gave a picture demonstrating how it works. We also presented the Kolmogorov-Arnold Representation Theorem (KART), which recently motivated a new architecture for deep learning where activations are learned and live on the edges. We wonder:
OutlookCybenko's UAT was around for over 20 years before the empirical breakthroughs of the 2010's. Existence of good approximators wasn't enough, we needed better compute and optimization schemes to find them dynamically.  However, given tremendous progress in dynamics, should we systematically return to the approximation theory literature, to help motivate new architectures, as e.g., in KART and KAN?

In Section <ref> on statistics, I reminded the reader that a randomly initialized NN is no more fundamental than a single role of the dice. It is a random function with parameters that is drawn from a statistical ensemble of NNs, and we asked “What characterizes the statistics of the NN ensemble?” I reviewed a classic result of Neal from the 90's that demonstrated that under common assumptions a width-N feedforward network is drawn from a Gaussian Process — a Gaussian density on functions — in the N→∞ limit. I explained that work from the last decade has demonstrated that this result generalizes to many different architectures, as a result of the Central Limit Theorem. Non-Gaussianities therefore arise by violating an assumption of the Central Limit Theorem, such as N→∞ or statistical independence, and by violating them weakly the non-Gaussianities can be made parametrically small. In field theory language, NNGPs are generalized free field theories and NN Non-Gaussian processes are interacting field theories. I also introduced a mechanism for global symmetries and exemplified many of the phenomena in the section.

In Section <ref> I presented some essential results from ML theory on NN dynamics, asking “How does a NN evolve under gradient descent?” When trained with full-batch gradient descent and a particular normalization, the network dynamics are governed by the Neural Tangent Kernel (NTK) that in general is an intractable object. However, in N→∞ limits it becomes a deterministic function and the NTK at initialization governs the dynamics for all time. An exactly solvable model with mean-squared error loss is presented that includes the mean prediction of an infinite number of infinitely wide neural networks trained to infinite time. The mean network prediction is equivalent to kernel regression, and the kernel communicates information from train points to test points in order to make predictions. However, from this description of the NTK a problem is already clear: nothing is being learned, and in particular late-time features in the hidden dimensions are in a local neighborhood of their initial values.  I showed how a detailed N-scaling analysis allows one to demand that network features and predictions update non-trivially, leading to richer learning regimes known as dynamical mean field theory or the maximal update parameterization.
OutlookThese recent theories of NN statistics and dynamics sometimes realize interesting toy models, but with known shortcomings from a learning perspective, though the N-scaling analysis in the feature learning section is promising. Surely there is still much to learn overall, e.g., we have said very little in these lectures about architecture design principles. For instance, some architectures, such as the Transformers <cit.> central to LLMs, are motivated by structural principles, whereas others are motivated by theoretical guarantees. 
 
 What are the missing principles that combines statistics, dynamics, and architecture design to achieve optimal learning?




Finally, in Section <ref> I explained how neural networks provide a new way to define a field theory, so that one might use ML theory for physics and not just physics for ML theory. This NN-FT correspondence is a rethink of field theory from an ML perspective, and I tried to state clearly about what is and isn't known about obtaining cherished field theory properties in that context. For example, I explained how by engineering a desired NNGP, such as the free scalar, one may do an operator insertion in the path integral and interpret the associated interactions as breaking the statistical independence necessary for the CLT. By such a mechanism, one may engineer ϕ^4 theory in an ensemble of infinite width neural networks.
OutlookCan NN-FTs motivate new interesting physics theories or provide useful tools for studying known theories?

Our discussion in this Section is directly correlated with the topics of these lectures. It reviews recent theoretical progress, but the careful reader surely has a sense that theory is significantly behind experiment and there is still much to do. It is a great time to enter  the field, and to that end I recommend carefully reading the literature that I've cited, as well as the book Geometric Deep Learning <cit.> that covers many current topics that will be natural to a physicist, including grids, groups, graphs, geodesics, and gauges. Many aspects of that book are related to principled architecture design, which complements the statistical and dynamical approach that I've taken in these lectures. I also recommend the book Deep Learning <cit.> for a more comprehensive introduction to the field.

I end with a final analogy between history and the current situation in ML. We are living in an ML era akin to the pre-PC era in classical computing. Current large language models (LLMs) or other deep networks are by now very powerful, but are the analogs of room-sized computers that can only be fully utilized by the privileged. Whereas harnessing the power of those computers in the 60's required access to government facilities or large laboratories, only billion dollar companies have sufficient funds and compute to train today's LLMs from scratch. Just as the PC revolution brought computing power to the masses in the 80's, a central question now is whether we can do the analog in ML by learning how to train equally powerful small models with limited resources. Doing so likely requires a deeper understanding of theory, especially with respect to sparse networks.



Acknowledgements: I would like to thank the students and organizers of TASI for an inspiring scientific environment and many excellent questions. I am grateful for the papers of and discussions with friends whose works have contributed significantly to these lectures, especially Yasaman Bahri, Cengiz Pehlevan, and Greg Yang.  I'd also like to thank my collaborators on related matters, including Mehmet Demirtas, Anindita Maiti, Fabian Ruehle, Matt Schwartz, and Keegan Stoner. I would like to thank Sam Frank, Yikun Jiang, and Sneh Pandya for correcting typos in a draft. Finally, thanks to GitHub Copilot for its help in writing these notes, including the correct auto-completion of equations and creation of  figures! Two-column landscape mode was chosen to facilitate boardwork, but I think it also makes a pleasant read, HT @ <cit.>. I am supported by the National Science Foundation under CAREER grant PHY-1848089 and
Cooperative Agreement PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental
Interactions).



Disclaimers: In an effort to post these notes shortly after TASI 2024, there are more typos and fewer references than is ideal. Other references and topics may be added in the future to round out the content that was presented in real-time. I also struck a playful tone throughout, because lectures are supposed to be fun, but may have overdone it at times. I welcome suggestions on the above, but updates will still aim for clarity and brevity, targeted at HET students.



§ CENTRAL LIMIT THEOREM


Let us recall a simple derivation of the Central Limit Theorem (CLT), in order to better understand the statistics of neural networks. Consider a sum of random variables

    ϕ = 1/√(N)∑_i=1^N X_i,

with X_i = 0. The moments μ_r and cumulants κ_r are determined by the moment generating function (partition function) Z[J] = e^Jϕ and cumulant generating function W[J] = log Z[J], respectively, as 

    μ_r     = (d/dJ)^r Z[J]|_J=0
    κ_r     = (d/dJ)^r W[J]|_J=0.

If the X_i are independent random variables, then the partition function factorizes 
Z_∑_i X_i[J] = ∏_i Z_X_i[J], and the cumulant generating function of the sum is the sum of the cumulant generating functions, yielding

    W_∑_i X_i[J] = ∑_i W_X_i[J] 
    κ_r^∑ X_i = ∑_i κ_r^X_i.
 
If the X_i are identically distributed, then the cumulants κ_r^X_i are the same for all i and we account for the 1/√(N) appropriately, we obtain

    κ_r^ϕ = κ_r^X_i/N^r/2-1.

This yields 

    lim_N→∞κ_r>2^ϕ = 0,

which is sufficient to show that ϕ is Gaussian in the large-N limit. In physics language, cumulants are connected correlators, and (<ref>) means that Gaussian (free) theories have no connected correlators.

In neural networks we will be interested in studying certain Gaussian limits. From this CLT derivation, we see two potential origins of non-Gaussianity:

    
  * 1/N-corrections from appearance in κ_r^ϕ.
    
  * Independence breaking since the proof relied on (<ref>).








utphys




 Paper Title: Final state interactions for high energy scattering off atomic electrons.










Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA



CALT-TH/2024-026



§ ABSTRACT 
    We consider the scattering of high energy leptons off bound atomic electrons focusing primarily on final state interactions 
    between the outgoing energetic electron, and the heavy residual charged “debris” in the final state. These effects are inherently absent from calculations for a free electron at rest. Coulomb exchanges are enhanced by the large number of electrons in the atomic debris, and are unsuppressed by non-relativistic velocities in the debris. We find that these exchanges can be resummed using operator methods, and cancel at the level of the cross section until at least O(α^3). Furthermore, we argue that both final and initial state Coulomb exchanges (enhanced by the number of electrons in the atom) do not affect the cross section until at least O(α^3). Transverse photon couplings  to non relativistic electrons are proportional to their small velocities, and rotational invariance suppresses their contribution to O(α^3).   Our results are relevant for precision experiments involving  neutrinos, electrons, positrons, and muons scattering off of atomic electrons in a fixed target. 



    



 Final state interactions for high energy scattering off atomic electrons
    Mark B. Wise
    August 5, 2024
========================================================================






§ INTRODUCTION  



Fixed target experiments play an important role in high energy physics at the intensity frontier. Examples include neutrino experiments <cit.>, searches for feebly interacting particles <cit.>, rare processes <cit.>, and precision measurements <cit.>. It is often implicitly assumed that atomic electrons can be treated as free and at rest when they participate in a scattering interaction with an incident high energy beam. This is a good leading approximation, however for high precision experiments sub-leading corrections must be understood theoretically. 

In recent work <cit.> we have studied the corrections due to atomic binding arising both from phase space, and the initial state (many-body) wavefunction of the atom. These corrections stem from the finite three-momentum of electrons in bound atomic orbitals, and the shifted kinematics from binding energies. Both of these effects enter at the same order and they must be treated simultaneously. They account for the non-perturbative bound-state dynamics i.e., from iterated Coulomb exchange in the initial state. 

Further corrections arise from perturbative photon exchange between “hard” leptons, and “soft” (i.e., non-relativistic) electrons and nuclei. These effects are inherently absent from calculations for a free electron at rest. For example, for neutrino scattering off an atomic electron, the outgoing ∼ GeV electron can exchange photons with the ionized “atomic debris”. Incident charged lepton scattering (e.g., μ e →μ e) involves both initial and final state photon exchange with atomic electrons or nuclei.

In this work, we focus mostly on perturbative final-state interactions involving the outgoing energetic electron (in <ref> we also comment on the “shake-up” of the initial state from Coulomb exchange with an incoming charged lepton). We specifically focus on ladder-graphs involving photon exchanges with the final state debris. If we track the charge of the debris, Q_B, and the charge of the electron, Q_e, separately then the (gauge invariant) class of corrections we consider are of order[We explicitly neglect “mixed” corrections such as those of order O(Q_BQ_e^3).] (Q_B Q_e)^n.  Although Q_B=+e, matrix elements involving the debris are enhanced for multi-electron atoms. This enhancement occurs for momentum transfers on the order of the atomic scale (i.e., the inverse Bohr radius) when the struck electron is able to see a partially (but not fully) screened atomic nucleus.  To match the order studied in Ref. <cit.>, requires working to order (Q_B Q_e)^2. 


These corrections are a necessary input for an extraction of hadronic vacuum polarization from μ e →μ e scattering <cit.> as is being pursued by the MuonE collaboration <cit.>. For neutrino electron scattering (relevant for neutrino flux normalization measurements <cit.>), we find these corrections are much smaller than the uncertainty from the 1-loop radiative corrections computed in Ref. <cit.> (roughly ∼ 0.5%).


We use a non-relativistic power counting for photon interactions with the atomic system, and standard perturbative counting for the highly relativistic electron in the final state. We find a number of interesting results. First, Coulomb corrections cancel in the differential cross section upon integration over the final state phase space of the atomic debris; this result holds to all orders in perturbation theory [i.e., resumming graphs of order (Q_B Q_e)^n] in the limit of small energy splittings of the debris. For hydrogen, the cancellation can be understood in terms of a final-state wavefunction, whereas for multi-electron atoms it stems from an operator-level identity. Second, sub-leading corrections due to a single transverse photon exchange in are also shown to cancel. Finally, a class of corrections which are sensitive to the energy splittings of the atomic debris and appear at order e^2 can be shown to have vanishing interference with the leading order matrix element. Taken together these results imply that final state interactions (as defined above) do not alter the predictions of Ref. <cit.>  up to and including O(Q_e^2Q_B^2) in our power counting. A similar result holds for interactions that involve the initial atomic state.


In everything that follows, we will omit factors of Q_e and Q_B and simply count powers of α. This includes factors of α that stem from bound-state dynamics i.e., that the typical velocity of an electron in an atom is O(α). We aim to work to an accuracy of O(α^2). 



The rest of the paper is organized as follows: In <ref> we discuss the case of scattering neutrinos from hydrogen. We find that Coulomb distortion of the final state cancels at the level of the cross section after integrating over the final-state proton's phase space. Next in <ref> we show how to generalize our analysis to multi-electron atoms where the corrections similarly cancel at the level of the cross section. We find an operator level expression for Coulomb photons, and also discuss sub-leading corrections such as transverse photon exchange with non-relativistic electrons in the final state debris. Transverse photons are discussed in <ref>. We find the non-relativistic nature of the electrons combined with rotational invariance of the initial atom implies that transverse photons first enter at O(α^3). In <ref> we describe how to apply our results at the level of a measurable cross section. We show that the relevant sum-rules that control cross sections (inclusive with respect to the debris)  are unaffected by Coulomb exchange through O(α^2). We discuss how this result applies both to neutrino electron scattering and muon electron scattering.  
Finally in <ref> we provide a summary and outlook.





§ HYDROGEN  


Let us first focus on neutrino scattering off hydrogen as the simplest example where only final state interaction are present, 


    ν(k) + (*0) →ν(k') + e^-(p') + p^+(h') .


The outgoing proton is heavy, and couples dominantly to Coulomb modes with couplings to transverse photons suppressed by |h'|/m_p∼α m_e/m_p. While discussing hydrogen, we will therefore focus exclusively on the exchange of Coulomb photons in the final state. 

Consider an outgoing scattering state |f⟩=|e^-(p') p^+(h')⟩ labeled by the asymptotic momentum of the electron and proton. Including the iterated Coulomb potential, the state |f⟩ satisfies a Lippmann-Schwinger equation 


    |f⟩ = |e^-(p')⟩_0 |p^+(h')⟩_0 + 1/E_f - Ĥ_0 - 0V̂|f⟩ ,


where |e^-(p')⟩_0|p^+(h')⟩_0 is the free-particle state. 
Here V̂ is the Coulomb interactions, Ĥ_0 is the free-Hamiltonian and E_f = √(p'^2+m_e^2) + √(h'^2+m_p^2) is the exact energy of |f⟩; the causal regulator (- 0) is fixed by demanding that |f⟩ be an out-state. 
The final state can be written in terms of its valence free-particle Fock states, as[Strictly speaking the Dirac wavefunction should be spin-dependent, but we suppress spin indices for simplicity. The answer is unaffected in the kinematic regime of interest <cit.> and <ref> is correct as written.]


    |f(p',h')⟩ = √(2E_e(p'))√(2 E_p(h'))   ∫^3 q/(2π)^3ϕ̃_p',h'(q)/√(2E_e(p'+q))√(2E_p(h'-q))|e^-(p'+q)⟩_0|p^+(h'-q)⟩_0 .



Note that when V̂=0, the scattering wavefunction is ϕ̃_p',h'(q)= (2π)^3 δ^(3)(q). The two-particle scattering state is normalized according to 


    ⟨f(p',h')|f(p,h)⟩ =  2   E_e(p) (2π)^3 δ^(3)(p-p') 
       × 2 E_p(h)(2π)^3 δ^(3)(h-h') .


The scattering state's wavefunction depends only on the asymptotic relative momentum f'≡p'-h' so that ϕ̃_p',h'(q) = ϕ̃_f'(q).  The scattering state wavefunction is normalized according to 


    ∫   ^3 q/(2π)^3ϕ̃_f'^*(q+f') ϕ̃_f(q+f)= (2π)^3 δ^(3)(f'-f) .


It is readily checked that <ref> can be substituted into <ref> to obtain <ref>. In coordinate space <ref> becomes, 


    ∫   ^3 x  ^ (f-f')·x ϕ_f'^*(x) ϕ_f(x)= (2π)^3 δ^(3)(f'-f) .




We similarly write the initial hydrogen atom in terms of free-particle valence states, 


    |(*0)⟩= ∫^3 p/(2π)^3   √(2M_H)/√(2E_e(p))√(2 E_p(p))
       ×ψ̃(p)|e^-(p)⟩_0 |p^+(-p)⟩_0 .


The bound state's wavefunction is normalized according to 


    ∫^3q/(2π)^3 |ψ̃(q)|^2=1 .


The relevant matrix element is 


    ℳ = f(p',h') ν(k')ℋ_W(0)ν(k) ,


where ℋ_W is the weak Hamiltonian density evaluated at the origin of spacetime. Using the above definitions, and _0⟨p^+(h'-q)|p^+(-p)⟩_0= 2 E_p (2π)^3 δ^(3)(q-(h'+p)), we arrive at 


    ℳ   = ∫^3 p/(2π)^3√(2M_H 2 E_p/2E_e)ϕ̃_p'-h'^*(p+h') ψ̃(p) 
       × M(p'+p+h',k';p,k) 
       ≃∫^3 p/(2π)^3√(2M_H 2 E_p/2E_e)ϕ̃_p'-h'^*(p+h') ψ̃(p) 
       × M(p',k';p,k) ,


where in the second line we have neglected the p+h' (both small) relative to p' (large); the error due to this approximation is very small ∼ |p|/|p'| <cit.>. The free-electron matrix element is defined as


    M(p',k';p,k)=  _0⟨ e^-(p')ν(k')| ℋ_W| e^-(p)ν(k)⟩_0 .


As mentioned above, in the absence of Coulomb exchanges ϕ̃_p'-h'(q) → (2π)^3 δ^(3)(q), and <ref> reduces to Eq. (14) of Ref. <cit.>.



 §.§ Perturbation theory  





It is instructive to compute the matrix element perturbatively. This both serves to illustrate the size of Coulomb corrections at the level of the amplitude and the connection between diagramatics and wavefunctions. Furthermore, by analyzing different regions of loop momenta we learn which characteristic scales dominate the Coulomb corrections. 

The O(α) correction involves an integral of the form (including a photon mass, μ_γ, to regulate IR divergences)


    I^(1)=∫^3 q/(2π)^3ψ̃(h'-q)  γ_0(p'+q+m) /2p'· q+q^2+ 0e^2/q^2-μ_γ^2+ 0 ,


where contraction against u̅(p') is implicit. 
The energy transfer is small, q_0 ≃ (2h'·q + q^2)/(2 m_p). We will suppress the causal regulators (+ 0) in what follows. 

Let us focus on a 1s orbital of hydrogen for concreteness,


    ψ̃_ 1s(h'-q) ≃8√(π)Λ^5/2/(|h'-q|^2 + Λ^2)^2 ,


where we have used the non-relativistic approximation and introduced Λ=1/a_0=α m_e with a_0 the Bohr radius. 

It is interesting to first understand the limit of large momentum transfers. These may occur in two qualitatively different regions of phase space. First consider the final state proton's momentum to be small |h'|∼Λ. Then, if the Coulomb photon mediates a large momentum transfer we must have a large initial momentum for the proton which is suppressed by the hydrogenic wavefunction; in this case we find a power suppression of ∼Λ^4/|q|^' 4. Alternatively, one can consider a large final state proton momentum with h'∼q. In this case the integral factorizes into a tree-level matrix element (for a proton at rest) multiplied by the wavefunction at the origin ψ_ 1s(x=0) (which is suppressed by powers of α). 

Hence the integral in <ref> is dominated by |q|∼Λ where one finds


    I^(1)∼αΛ^-3/2∼αψ̃_ 1s ,


which is O(α) relative to the leading expression.  We may therefore approximate I^(1) by expanding the integrand counting |h'|∼ |q|  ≪ E'_e, 


    I^(1)≃ e^2 ∫^3 q/(2π)^3 ψ̃_1s(h'-q)   1 /v'·q1/q^2+μ_γ^2 ,


where v'=p'/E'; this corresponds to the eikonal approximation. To get a sense of the size of the one-Coulomb-exchange correction, we may evaluate at the kinematic point h'=0. Then the integral is 



    I^(1)|_h'=0   ≃ e^2 ∫^3 q/(2π)^38√(π)Λ^5/2/(q^2 + Λ^2)^21 /v'·q1/q^2+μ_γ^2
        = α/|v'|[log( μ_γ^2 /Λ^2) + 1] ×ψ̃(h'=0)  .


As expected the correction is controlled by α/|v'|∼α and is naively much larger than the binding corrections computed in <cit.>. At the special kinematic point  h'=0 the correction is purely imaginary, but is in general complex[<Ref> has the form of a convolution and can be written in coordinate space ∫^3 x  ^-h'·x f(x) ψ_ 1s(x). If for h'=0 this is purely imaginary, then for non-zero h' it has both real and imaginary parts.] (with non-zero real and imaginary parts) for kinematics such that h'≠ 0. 


If we keep the next corrections from the electron's propagator we find terms suppressed by Λ/E' or Λ/m_p (coming from the energy transfer q_0=q^2/2m_p). As a concrete example, consider the correction proportional to q^2/2m_p. Expanding  1/(v'·q + q^2/2m_p)  leads to the correction, 


    δ I^(1)|_h'=0   ≃e^2 /2m_p∫^3 q/(2π)^38√(π)Λ^5/2/(q^2 + Λ^2)^2(1 /v'·q- 0)^2
       =α/|v'|^2(Λ/4m_p) ×ψ̃(h'=0) ,


which is suppressed by (Λ/m_p) relative to the leading result.[In the case of a multi-electron atom, the final state debris can contain electrons with kinetic energy T∼q^2/m_e. In this case the suppression is Λ/m_e ∼ O(α).] 


<Ref> is suppressed by the explicit perturbative factor of α, but is not suppressed by any hierarchy of scales. Coulomb corrections must therefore be controlled to at least two-loop order. As we discuss below, the Coulomb corrections actually exponentiate and cancel after integrating over the phase space of the outgoing proton. The exponentiation of Coulomb corrections can be seen transparently in coordinate space and we therefore return to the description in terms of final state wavefunctions. 




 §.§ Coordinate space representation  


For non-zero recoil momentum, h'≠ 0, we can obtain a useful representation of the integrals by noticing they have the form of a convolution. At n^ th-order in (e^2) we encounter integrals of the form,


    I^(n) =   ∫^3 x   E_f^*(n)(x) ^-h'·xψ(x) .


The function E_f^*(n) (x) is a series of nested convolutions. For example E_f^*(1) is given by, 


    E_f^*(1)(x)= e^2∫^3q_1/(2π)^3^q_1·x1/v'·q_11/|q_1|^2 ,


and E_f^*(2) by 


    E_f^*(2)(x)= e^4∫   ^3q_2/(2π)^3^q_2·x∫^3q_1/(2π)^31/v'·q_2
       ×1/|q_1-q_2|^21/v'·q_11/|q_1|^2 .


Inserting <ref> and <ref> into <ref>, writing ψ(x) = ∫^3p/(2π)^3 ^p·xψ̃(p), and carrying out the integral over ^3x one easily arrives at the momentum space representation of I^(1) and I^(2) [cf. <ref>]. 


Next, summing over n we find 


    ∑_n I^(n) =   ∫^3 x  ψ(x) ^-h'·x E_f^*(x)  ,


with 


    E_f^*(x)  = ∑_n E_f^*(n) (x) .


This resummation of diagrams is equivalent to solving the Lippmann-Schwinger equation and E_f(x). The function E_f(x) is the eikonal approximation to ϕ_f(x). For both relativistic and non-relativistic electrons, the eikonal wavefunction satisfies <cit.>


    [ v'·*∇ + V(x)] E_f(x) = 0 ,


with V(x) the Coulomb potential. When approximating ϕ_f with E_f,  <ref> hold up to corrections suppressed by[In the rest frame of the atom the momentum f≃p', and therefore p' labels the state.] Λ/|p'| as can be checked usign the method of stationary phase. The solution to <ref> exponentiates in coordinate space,


    E_f(x) = exp[-/|v'|∫_z^∞ s  V(|x_⊥|,s)] ≡^φ(x) ,


where |x_⊥| is the the spatial coordinate perpendicular to v', and z labels the coordinate direction parallel to v'.

<Ref> can be written in coordinate space rather than momentum space. One then finds, 


    ℳ=√(2M_H 2 E_p/2E_e)∫^3 x  ^-h'·x^-φ(x) M(-*∇) ψ(x) ,


where we have abbreviated M(p',k';p,k)= M(p) and transformed to coordinate space where the matrix element should be interpreted as an operator M(-*∇). 

Let us now consider the matrix element squared integrated over phase space. Importantly, in the m_p→∞ limit, the energy conserving delta function is independent of the proton's recoil momentum. Furthermore, corrections to the phase space due to a finite h' are suppressed by h'/E_e' and are negligible. We therefore have an unconstrained integral over h', 


    ∫^3 h'/(2π)^3 |ℳ|^2  =    ∫^3 h'/(2π)^3∫^3x ∫^3 y  ^h'·(y-x)
         ×^φ(y)^-φ(x)
       ×[ψ^*(y)  M^*(-←*∇_y)  M(-→*∇_x)ψ(x) ]  .
 

The integral over h' produces δ^(3)(x-y) and we find 


    ∫^3 h'/(2π)^3 |ℳ|^2     = ∫^3x 
          M(-*∇)ψ(x)^2
       = ∫^3p/(2π)^3 ψ̃(p)^2 × M(p)^2 .


This leads to a result which is identical to that obtained neglecting final state interactions. We see that the Coulomb corrections cancel only after integrating over final-state phase space. This result is independent of the details of the initial state wavefunction. 





§ MULTI-ELECTRON ATOMS  
 

The analysis for hydrogen is greatly simplified by the presence of a single recoiling proton in the final state. Inelastic excitations of the final state proton are gapped-out by large hadronic energy scales ∼ 300  MeV and can be safely neglected. By way of contrast when considering helium and other multi-electron atoms inelastic excitations must be included. The remnant system now contains a complex spectra of excitations with energy splittings on the order of ∼ 1  Ry= 12α^2 m_e and these are always energetically accessible. Furthermore, while the coupling of a proton to transverse photons is suppressed by 1/m_p, the coupling of electrons is only suppressed by 1/m_e, which necessitates the inclusion of both transverse and Coulomb modes. 

The addition of a single electron (i.e., going from hydrogen to helium)  therefore introduces a substantial increase in theoretical complexity. Nevertheless, as we will show below, the non-relativistic nature of the atomic debris allows us to simplify the analysis and retain theoretical control. Many of the properties derived above for hydrogen generalize readily to multi-electron atoms provided they are re-formulated at an operator level; for instance we will show that the exponentiation of Coulomb corrections persists.  In what follows we will first motivate our power counting, then analyze Coulomb photons neglecting energy splittings in the debris, and finally discuss sub-dominant corrections. 




 §.§ Power counting  



In this section we sketch our power counting, motivating the scaling of loop momentum, and the organizing principles behind our calculation. Following the analysis for hydrogen we take regions where |q|∼Λ∼α m_e to dominate all loop integrals. This guarantees that the atomic debris remains non-relativistic in both intermediate and final states. When considering photon exchange in the non-relativistic limit it is useful to separate different modes from one another. We will phrase our power counting in Coulomb gauge which simplifies the analysis and work in the rest frame of the atom. 

Non-relativistic particles have an unsurpressed coupling to Coulomb photons. The interaction strength between two particles with charges Q_1 and Q_2 with relative velocity β can be parameterized as Q_1 Q_2/β. For interactions among particles in the initial atom and/or atomic debris this coupling is non-perturbative, and the Coulomb interaction must be retained at leading order. We therefore include the intra-Coulomb interaction in the energies of atomic states which we will sum using completeness relations.  For Coulomb exchange with the “struck” electron we count Coulomb modes as O(α).  

The coupling of a particle to transverse photons are velocity suppressed in the non-relativistic limit. The atomic nucleus is sufficiently heavy that its velocity can be neglected entirely, while atomic electrons have velocities on the order of v_e∼α. We therefore count the exchange of transverse photons between the struck electron and the atomic debris as O(α v_e) ∼ O(α^2). 

Finally, in what follows we will often expand propagators making use of non-relativistic kinematics. For on-shell states |X⟩ coupled to a photon with momentum q,  we count energy splittings as Δ E_X ∼q^2/m_e ∼α |q|. Therefore we count powers of Δ E_X/|q| as O(α). 








 §.§ Analysis of Coulomb photons  


Let us first analyze the dominant final state interactions mediated by the exchange of Coulomb modes. In a multi-electron atom the wavefunction is replaced by the matrix element of the field operator with a state |B⟩. Let us begin from the in-out matrix element of the weak interaction Hamiltonian density (evaluated at x=0), 


    ν' e' B'ℋ_WAν =  u̅_k'γ_μ P_L u_k e' B'ψ̅Γ^μψA .


Next we insert a complete set of  in-states with electric charge Q=+e,


    e' B'ψ̅Γ^μψA = _B e' B'ψ̅BΓ^μBψA


The matrix element, e' B'ψ̅B, in the above equation involves in- and out-states and can be computed using standard techniques for scattering amplitudes. 


 
Notice that the matrix element involving ψ̅ is an “in-out” matrix element and so standard Feynman rules immediately apply. We now define 


    ψ_BA(p) ≡ _ inB(p)ψA(0)_ in ,


which is the analog of the hydrogenic wavefunction. 
 

A diagrammatic interpretation of e' B'ψ̅B is now clear. Drawing the highly energetic electron with a single line, and the atomic debris with a double line, we have 


    10pte' B'ψ̅B   =diagram_soft(20,6.5) 
        i1,i2o1,o2doublei1,b1,b2,o1plaini2,t1,t2,o2decor.shape=square,decor.size=10,decor.filled=emptyi2square,tension=0,shadedb2,t2,t1,b110pt   ,


where the white square is the insertion of ψ̅ and the shaded box includes all possible photon exchanges. In what follows we will expand in Δ E_X/|q|, but resumming an arbitrary number of Coulomb exchanges. 



  §.§.§ Leading order in Δ E_X/|q|


As shown in <ref>, for Coulomb exchanges <ref> can be reduced to an operator level expression.  When Δ E_X is neglected relative to |q| the only operator that appears[If energies of intermediate states are retained then the Hamiltonian also appears in an operator-level identity .]  is the Fourier transform of the the charge density at t=0, ρ̂(q). The atomic debris is non-relativistic, the energetic electron is ballistic (i.e., eikonal), and Coulomb modes are instantaneous; taken together this implies that only one time-ordering contributes to the matrix element.   Since ρ̂(q), commutes with itself for different arguments, i.e., [ρ̂(q), ρ̂(q')]=0, the sum over orders in perturbation theory ultimately exponentiates. 

The result is that the in-out matrix element in <ref> can be written as


    e'(p') B'(h')ψ̅B(h') = u̅(p') B'(h')Ŵ_v'B(h),


where, up to O(e^4), and evaluating operators at t=0, 


    Ŵ_v'=1 + e^2∫^3 q_1/(2π)^31/v'·q_1ρ̂(q_1)/q_1^2
       + e^4∫^3 q_1/(2π)^3^3 q_2/(2π)^31/v'·(q_1+q_2)ρ̂(q_2)/q_2^21/v'·q_1ρ̂(q_1)/q_1^2 
        +  …


The operator Ŵ_v' resembles a Wilson line, but is defined in terms of the operator ρ̂(q), as opposed to the photon field Â_μ. Matrix elements of Ŵ_v' may be thought of as resumming operator insertions of ρ̂, 


    6pt⟨Ŵ_v⟩ =  ∑_nwilson-line(30,7.5) 
        
        
        i1,b1,b2,b3,b4,b5,b6,o1i2,t1,t2,t3,t4,t5,t6,o2phantom,tension=4i1,b1,b2,b3,b4,b5,b6,o1dbl_dots,tension=0b1,t1dbl_dots,tension=0b2,t2dbl_dots,tension=0b5,t5dbl_dots,tension=0b6,t6phantomt3,v,b4label=…vplain,tension=4i2,t1,t2,t3,t4,t5,t6,o2label=⟨ρ̂(q_1) ρ̂(q_2)   … ρ̂(q_n-1) ρ̂(q_n)⟩b36pt    .



Here, the dotted lines refer to Coulomb modes (with propagators /q^2), and the solid line to the eikonalized electron.  See Ref. <cit.> for a similar construction in the context of Glauber gluons in QCD. 

The second integral in <ref> can be symmetrized between q_1 and q_2. Then, using standard eikonal identities, 


    1/2(1/v'·q_1 + 1/v'·q_2)= 1/2v'· (q_1+q_2)/(v'·q_1)(v'·q_2) ,


it is straightforward to show [in complete analogy with the eikonal wavefunction in <ref>] that <ref> exponentiates 


    Ŵ_v'   = exp[ e^2∫^3 q/(2π)^31/v'·q- 0ρ̂(q)/q^2+μ_γ^2]
       = exp[ - e^2∫_0^∞ s V̂(ŝv')  ] ,


where we have made the infrared regulator μ_γ explicit [cf. <ref>].  <Ref> is the operator level generalization of <ref> evaluated at x=0. We have introduced the Hermitian operator V̂ 


    V̂(x) = ∫^3 q/(2π)^3^q·xρ̂(q)/q^2+μ_γ^2 ,


which manifests that Ŵ_v' is unitary, Ŵ_v'^†Ŵ_v'=1. For a point-like particle (such as the proton), the matrix element of Ŵ_v' reduces to a standard Wilson line, expanded in the limit of small energy-transfer.


Upon summing over all states |B⟩⟨B| (which is the analog of folding with the wavefunction for hydrogen), we obtain the compact formula


    e' B'ψ̅Γ^μψA = u̅(p') B'(h')Ŵ_v'ψ̂A(0) .


Therefore, at the level of the cross section, we have the simple replacement <cit.>,


    Aψ̂^†δ(…) ψ̂A→Aψ̂^†Ŵ_v'^†δ(…) Ŵ_v'ψ̂A .


where the argument of the delta function includes the Hamiltonian, which does not commute with Ŵ_v'. This operator level replacement holds for arbitrary atoms. It relies only the non-relativistic dynamics of the atomic debris and on the scale separation between the energy of the outgoing electron, E'_e and the binding energies of the atoms,  ϵ_A∼ϵ_B. 

The results of this section apply to all orders in the Coulomb interaction in the limit of negligible energy transfer i.e., neglecting corrections of order Δ E_X/Λ. Phenomenologically (e.g., for an extraction of space-like hadronic vacuum polarization) we  expect working to O(α^2) to be sufficient; this corresponds to two Coulomb modes. At this order, however,  sub-leading corrections not captured in <ref> must also be considered. 




  §.§.§ Higher order in Δ E_X/|q| 


The leading effect arrises from corrections to the struck electron's propagator due to finite energy-level splittings, Δ E_X, in the debris. The contribution from the atomic-energy Δ E_X appears at O(αΔ E_X/|q|) ∼ O(α^2) and must therefore also be included. Using the expansion


    u'_0u'· q_1 +  0= -1 v' ·q_1 - 0+q^0  (v' ·q_1-0)^2 +…


where u_μ=γ(1,  v') is the four-velocity of the outgoing electron. 


For a single Coulomb exchange (i.e., at order e^2) the energy transfer is given by q_1^0=h^'0-h^0. The first correction [coming from the second term on the right-hand side of <ref>] gives rise to shift in <ref> that is proportional to 


    ⟨ B'(h')|[H,∫^3 q_1  (2 π)^3V̂(q_1)  (v' ·q_1- 0)^2]|B(h) ⟩ .


Note that q_1 in the integral of <ref> is actually fixed by momentum conservation. The operator in the matrix element above 


    O= [H,∫^3 q_1  (2 π)^3V̂( q_1)  (v'·q_1- 0)^2] ,


is anti-Hermitian [recall that V̂(q_1)^†=V̂(-q_1)] and therefore at linear order it's contribution to the cross section, which is proportional to

    Δσ∝∫^3 k_2  (2 π)^3∫^3 k_1  (2 π)^3⟨ A| â^†_k_2( O+ O^†)â_k_1 |A ⟩ ,

vanishes. This correction (at the amplitude level) was discussed in <ref> for the case h'=0 when the incoming atom is hydrogen so that  B and B' are a single proton.







 §.§ Transverse photons  


Having demonstrated the exponentiation of Coulomb modes, we now turn to transverse photons which enter first at O(α^2) in our power counting. Therefore, to match the size of the binding corrections considered in Ref. <cit.>, we will only study the exchange of a single transverse photon.  In Coulomb gauge a crucial difference between Coulomb and transverse modes appears in their propagators:


    D_00(q)= /q^2    vs.   D_ij(q)=-/q^2+ 0 [ δ_ij-q_i q_j/q^2] .


Transverse modes have poles in the complex q_0 plane and are propagating degrees of freedom. Coulomb modes are instantaneous, and non-propagating. 

When considering contributions from transverse photons it is therefore not possible to completely separate final states from initial states. We therefore organize our discussion along separate lines for final state interactions and interactions involving the initial state. 




  §.§.§ Exchange in the final state


Let us consider a contribution to <ref> from the exchange of a single transverse photon. The spatial photon propagator in Coulomb gauge we have


    D_ij= -/q_0^2 - q^2+ 0 [ δ_ij-q_i q_j/q^2] →/q^2- 0[ δ_ij] .


Where we have made use of current conservation, and an expansion in q_0/|q| in the second step. At this order (single transverse photon exchange) q_0 is explicitly fixed to be small by on-shell conditions from the atomic-debris side of the diagram.  

It is then clear from the structure of the loop amplitude that to this order in the expansion, transverse photons can be included by making the operator level replacement ρ̂→ρ̂ - v'·Ĵ. Counting non-relativistic matrix elements of the current operator ⟨Ĵ⟩∼ O(α)  then guarantees that transverse photons do not spoil the exponentiation at O(α^2). 



  §.§.§ Exchange with the initial state


There are also corrections that arise from the exchange of photons between the outgoing struck electron and the initial atom. Hard photons, with virtuality on the order of m_e^2 or larger, will factorize from bound state effects and are already included in existing calculations of QED corrections to ν e →ν e scattering with free electrons. 

Corrections involving soft photons may, however, be sensitive to atomic structure. Any transverse photon coupling from the atom to the debris is doubly suppressed by non-relativistic velocities and enters at O(α^3). We therefore focus on diagrams (shown below) in which the transverse photon couples to the outgoing ballistic electron. It is instructive to compare the one-loop diagrams involving the exchange of a Coulomb (dotted line) vs. a transverse (wavy line) photon,  

    coulomb-graph(16,16)
        tib,i1,i2,i3,i4,i5,itob,o1,o2,o3,o4,o5,otbdashesb,tdouble,tension=2i3,v1,v_hardphantomv_hard,o3doublev_hard,o1plain,tension=1.1v_hard,v2,otdbl_dots,tension=0,left=0.5,label=qv1,v2decor.shape=square,decor.size=10,decor.filled=emptyv_hardlabel=Ai3label=Bo1label=eottransverse-graph(16,16)
        tib,i1,i2,i3,i4,i5,itob,o1,o2,o3,o4,o5,otbdashesb,tdouble,tension=2i3,v1,v_hardphantomv_hard,o3doublev_hard,o1plain,tension=1.1v_hard,v2,otphoton,tension=0,left=0.5,label=qv1,v2decor.shape=square,decor.size=10,decor.filled=emptyv_hardlabel=Ai3label=Bo1label=eot20pt   .

We have drawn a dashed line where we insert a complete set of states in <ref>. 
The Coulomb graph vanishes, because its propagator supplies no poles in the complex q_0 plane, and the poles from the atom and the debris both lie on either the upper- or lower-half of the complex plane (see Appendix B of <cit.> for a more detailed discussion). A similar argument applies to Coulomb exchanges between the initial state atom and final state debris. Physically these results follow from the non-relativistic nature of the bound-state constituents, and the fact that the Coulomb interaction is instantaneous. 

The graph involving a transverse photon has additional poles from the photon propagator that invalidate the argument given above. Performing the integral over q_0 by closing  the contour in the upper half plane picks up only the photon's pole. This corresponds to an intermediate state that contains a photon [and contributes to the sum in <ref>]. 
However transverse photons couple proportional to a three momentum (i.e., Ĵ = ∑_i p̂_e,i/m_e)  and in the scattering amplitude this is integrated against the initial state multiparticle wave function.  For rotationally invariant targets this integral gives a contribution to the cross section that vanishes,[Strictly speaking, the expectation value of momentum can be proportional to the spin of the atom. For unpolarized targets this cancels upon spin averaging. ] because ⟨p̂_e ⟩≃0. Therefore, the contribution from transverse photons to the cross section can occur first at order e^2 ⟨p̂^2 ⟩ /m_e^2 ∼ O(α^3).









§ APPLICATIONS 
 


We now return to the phenomenological examples discussed in the introduction. We begin with neutrinos scattering from atomic electrons. This is convenient since in the previous sections we have already analyzed ν e →ν e scattering for atomic electrons in detail. In the absence of final-state interactions, the cross section[The same formula applies for the differential cross section σ /Π_e where Π_e is the phase space of the struck electron <cit.>.] for neutrino-atom scattering can be written as <cit.>, 


    σ   =  ∫ϵ∫^3 p/(2π)^3  S_A(ϵ,p)
       1/2 ω1/2E∫Π_e 1/2ω'δ( E-ϵ)1/2∑_ spins | M|^2  .


 Here M(k',p';k,p), is the free-particle matrix element for  ν(k) e(p) →ν(k') e(p') scattering and S_A(ϵ,p) is the non-relativistic spectral function of the target atom A, 


    S_A=∫^3 q/(2π)^3Aâ^†_qδ(ϵ_A+Ĥ-ϵ)  â_pA .


Binding corrections can be efficiently incorporated by expanding | M|^2 in ϵ and p <cit.>. 

To include final-state Coulomb exchange one simply makes the replacement S_A →S̃_A where 



    S̃_A=∫^3 q/(2π)^3Aâ^†_qŴ_v'^†δ(ϵ_A+Ĥ-ϵ) Ŵ_v'â_pA .


The same procedure (mentioned above) of expanding | M| in terms of p and ϵ still applies. We note that <ref> is reminiscent of the opperatorial definition of a parton distribution function. 


The unitarity of Ŵ_v' guarantees that corrections disappear after integrating over ϵ (which removes the delta function). The first term for which this is not true is the energy weighted sum ∫ϵ (-ϵ) × (…), which introduces the insertion of the Hamiltonian. The non-commutativity of Ĥ and Ŵ_v' then gives


    ∫ϵS̃_A(ϵ,p)    =  ∫^3 q/(2π)^3Aâ^†_qâ_pA ,
    ∫ϵ (-ϵ) S̃_A(ϵ,p)    =  ∫^3 q/(2π)^3Aâ^†_qŴ_v'^†  [Ŵ_v'â_p,Ĥ] A .


<Ref> generalizes the Koltun sum rule <cit.> to include final-state Coulomb interactions in the leading non-relativistic approximation. 

In our counting ϵ∼ O(α^2), and so when computing O(α^2) corrections to the cross section we can set Ŵ_v'=1 in <ref>. This demonstrates that Coulomb corrections do not alter the predictions of the cross section until at least O(α^3). 

For the same reason, when considering the sub-leading O(α^2) corrections we only need to interfere these contributions with the tree-level matrix element evaluated at p=0 and ϵ=0 (hereafter referred to as “1”). 
As argued in <ref> the contributions from final state transverse photon exchange, and corrections due to finite energy splitting in the debris do not contribute to the cross section at O(α^2).  We therefore conclude that all final-state interaction effects can be neglected at O(α^2) for observables that are inclusive with respect to the atomic debris (i.e., that have integrated over the debris' phase space). We note however, that corrections involving transverse photon exchange with the initial atom must still be evaluated for a complete O(α^2) analysis. 




 §.§ Neutrino flux normalization


One application of our results is for neutrino flux normalizations at long-baseline experiments such as DUNE and T2HK.  These experiments require the ν e →ν e cross section to be known with a precision on the order of (ideally at least a factor of ten smaller than) one percent. In reality these experiments measure hard outgoing electrons from atomic scattering. For example in the case of DUNE the relevant process is ν Ar→ν e B with B all the soft atomic debris. 

As discussed in <cit.>, atomic binding corrections are much smaller than power-counting would suggest due to accidental cancellations that come from the structure of the tree-level matrix element. Coulomb corrections vanish upon integrating over the final state debris, and transverse photon exchange vanishes at least until O(α^2).   We therefore conclude that all possible sources of QED corrections to ν A →ν e B scattering are under sufficient control (i.e., better than 1%) for neutrino flux normalization. 





 §.§ Extraction of hadronic vacuum polarization


Another process for which the results of this paper are relevant is the t-channel scattering of charged leptons off atomic electrons. We have in mind (in particular) muons scattering on electrons bound to carbon 


    μ(k)  ^12 C(0) →μ(k') e(p') B(h') .


This is relevant for the MuonE experiment <cit.> which aims to extract hadronic vacuum polarization at space-like kinematics. MuonE's data can potentially give an independent determination of hadronic vacuum polarization's contribution to the anomalous magnetic moment of the muon <cit.>. The theory demands of the experiment are a relative precision of 10^-5 <cit.>, which necessitates the inclusion of bound state effects. 

In Ref. <cit.> we have computed the shift in the shape of σ / t due to atomic binding effects. The results of this paper imply that final-state photon exchange between the struck electron and the atomic debris do not alter the conclusion of our previous work at O(α^2) (the working-order of Ref. <cit.>). 

Muon electron scattering, however, involves a charged muon which may also exchange photons with both the initial atom and the outgoing debris. The results of this paper are therefore not sufficient for a complete account of all of these effects, however we can comment immediately on a sub-class of diagrams. Let us first match the hard-photon mediated μ-e scattering vertex onto an effective four-Fermi contact operator, 𝒪̂. The operator's Wilson coefficient includes all short-distance radiative corrections.  

We are then interested in matrix elements of the form B'e'μ'𝒪̂Aμ. If we insert a complete set of states before the operator insertion, we can re-sum Coulomb exchange into an operator level expression involving Ŵ_v_μ^† where v_μ is the 3-velocity of the initial muon. We can resum Coulomb exchange between the final state leptons and the atomic debris as described above using the product of Ŵ_v_μ' and   Ŵ_v_e'. These operators commute with one-another since they only involve ρ̂(x) or equivalently ρ̂(q). Our experience with final state Coulomb exchanges suggests that for high energy muon atomic electron scattering, all the effects of soft Coulomb exchanges can be captured by making the replacement  


    â_p→Ŵ_v_μ'Ŵ_v_e'â_pŴ_v_μ^† ,


in the definition of the spectral function. The physical interpretation being that the Wilson-line before â_p “shakes up” the initial state. This leads to different sum rules than in the case of neutrino electron scattering, 


    ∫ϵS̃_A(ϵ,p) =  ∫^3 q/(2π)^3AŴ_v_μâ^†_qâ_pŴ_v_μ^†A .


We are interested in whether or not the operators Ŵ_v_μ and Ŵ_v_μ^† modify the sum rules that are relevant for our analysis. Corrections to the Koltun sum rule, and to the average kinetic energy are necessarily O(α^3) and can therefore be neglected in our O(α^2) analysis. 

 
The leading-order sum rule, ∫^3p/(2π)^3 ∫ϵS̃_A(ϵ,p) = Z, could in principle receive an O(α) or O(α^2) correction. Integrating over ^3 p, using ψ̂(0)= ∫^3 p/ (2π)^3 â_p and carrying out both integration over p and q we find


    ∫^3 p/(2π)^3∫ϵS̃_A(ϵ,p) = AŴ_v_μρ̂_e Ŵ_v_μ^†A .


where ρ̂_e=ψ̂^†ψ and the fields are evaluated at x=0. Then, since the electron charge density ρ̂_e commutes with Ŵ_v_μ we conclude that (provided <ref> holds) the modified spectral function satisfies the same sum rule as when Coulomb exchanges are ignored, 


    ∫^3 p/(2π)^3∫ϵS̃_A(ϵ,p) = ∫^3 p/(2π)^3∫ϵ  S_A(ϵ,p) = Z  ,


where Z is the atomic number of the state A. This is not surprising since Coulomb exchanges do not change the number of electrons in the atom. We therefore conclude that Coulomb exchanges in both the initial and final state do not interfere with the leading order amplitude for μ ^12 C→μ e B^+ scattering until at least O(α^3). 





§ CONCLUSIONS  
 

We have considered corrections to scattering cross sections that arise when a target electron is bound in an atom. The focus of the present paper is on final-state photon exchange and compliments our previous work on corrections due to the initial bound state <cit.>. 

We have studied which regions of loop momentum dominate virtual corrections, and find that photons with virtuality on the order of the atomic scale Λ∼α m_e dominate the integrals. We have organized these virtual corrections using a power counting scheme that makes use of the non-relativistic features of the atomic bound state, A, and final state debris, B^+. 

Within this power-counting, Coulomb exchanges provide the dominant leading correction (before summing over final states). We have studied these graphs in detail and find that they can be resummed in the leading non-relativistic approximation at an operator level. This includes arbitrary multi-particle intermediate states (e.g., ^12 C^3+ e^-e^-) with charge Q=+1.  Our work then provides an operator level definition of Coulomb corrections in the eikonal approximation for a system with small excitation energies. We find that the unitary nature of the resummed Coulomb corrections (they resemble a Wilson line) demonstrates that these corrections do not enter until at least O(α^3) for neutrino electron scattering. Our experience suggests that they likely do not enter for μ e →μ e scattering either (this second result relies on <ref> holding). 

We have also considered sub-leading corrections in the final state, such as transverse photon exchange (suppressed by non-relativistic velocities in the atom) and corrections to Coulomb exchanges from finite-energy splittings among different final states in the atomic debris. We find that both of these corrections (already down by α^2) do not interfere with the tree-level cross section after integrating over the final states of the debris, and therefore do not influence the cross section at O(α^2). We conclude that all effects necessary for a sub-percent prediction of the neutrino (atomic) electron cross section are under control. 


In the context of μ e →μ e scattering, the results of Ref. <cit.> and of the current work account for all of the leading “Z-enhanced” corrections to the cross section. These correspond to the class of diagrams that are O(Q_e^2 Q_B^2), with Z-enhancements in medium-heavy atoms; the correction to the cross section per electron scale parametrically as ∼ Z^4/3α^2. 



At O(e^4), there can be further corrections from final state interactions that stem from “mixed” corrections involving both Coulomb and transverse photons e.g., 


    diagram_mixed(20,6.5) 
        i1,i2o1,o2doublei1,b1,o1plaini2,t1,t2,t3,o2photon,tension=0,left=0.5t1,t3dbl_dots,tension=0t2,b1decor.shape=square,decor.size=10,decor.filled=emptyi210pt   .


These only receive Z-enhancements for Coulomb mode, but not for the transverse photon. Furthermore, loops with a transverse photon scale like α/(4π), whereas Coulomb modes scale like α/|v'| ∼α. We therefore expect that these “mixed” corrections scale as ∼ Z^2/3α^2/(4π) which is smaller than the purely kinematic binding corrections by[This estimate does not account for any large logarithms (or accidental numerical enhancements) that may appear in the mixed radiative corrections. ] Z^-2/3/(4π) ≈ 0.02. We therefore defer their evaluation to future work.





In summary, we have made further progress on understanding radiative corrections to leptonic interactions with atomic electrons. We find that photon exchanges (in both the initial and final state) between the atomic system and any high energy leptons cancel after integrating over the phase space of the atomic debris. The formalism for Coulomb corrections developed in this work is appropriate for systems with many low-lying energy levels that should be summed over. This is different from conventional treatments in terms of a static background potential. The results in this paper account for all maximally Z-enhanced corrections at O(α^2). The standard non-relativistic power-counting applied in this work should serve as a useful guide for future studies of atomic corrections to muon electron scattering for the MuonE experiment. 




§ ACKNOWLEDGEMENTS

Some of this work was done at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.  RP is supported by the Neutrino Theory Network under Award Number DEAC02-07CH11359. RP and MW are supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics under Award Number DE-SC0011632, and by the Walter Burke Institute for Theoretical Physics. 









§ EXPONENTIATION OF COULOMB CORRECTIONS  


In this section we derive <ref> by considering the perturbative series that arises from Coulomb exchange. We show that within a non-relativistic approximation, the Coulomb exchange leads to an expression involving a unitary operator that is closely related to a Wilson line . The main novelty of our derivation is the inclusion of arbitrary states of the atomic debris between subsequent insertions of the charge-density operator. In particular, we do not model Coulomb exchanges using a (classical) background potential, and instead include all possible non-relativistic excited states of the debris as intermediate states in the perturbative expansion. 

We find that only one time-ordering contributes (the uncrossed ladder graphs). After expanding the ballistic electron propagator to take advantage of the small energy transfers (fixed by the non-relativistic atomic debris) the expressions are independent of intermediate state energies at leading order. This allows the intermediate states to be removed via completeness, and a simple operator-level identity remains. We now discuss two different derivations, on in the time-domain and one in the frequency-domain. 



 §.§ Marks Way  


It is convenient to define,


    ∫^4x ^ Q· x⟨ B'(h')e(p')|ψ̅(x)|B(h) ⟩ = u̅I ,


where Q is a residual electron four momentum and our convention is that the final state electron has zero residual momentum. Factoring out the energy momentum conserving delta function we write, I(q_1)=(2π)^4δ(Q+(h'-h))J(q_1). Next, I (or J) can be expanded in a series I=I^(0)+I^(1)+ …  where the superscript is the number of Coulomb exchanges. The leading order piece is I^(0)=(2π)^4δ(Q+(h'-h)), or equivalently J^(0)=1 and for the one photon exchange in Coulomb gauge I^(1) we have, treating the electron as ballistic and using the appropriate vertex and propagator,


    I^(1)=e^2 u^'0 u'· Q∫^4x_1 ^ Q· x_1⟨ B'(h')|J_0(x_1)|B(h)⟩ .


Here u^μ is the electron four velocity (u^μ=γ(1,  v') in the rest frame of the atom) and we have used the fact that the constituents of the states are non-relativistic so matrix elements of the electromagnetic 3-vector current are suppressed. Translating the charge density operator J_0(x)→ J_0(x=0) to the origin of space-time and performing the integration over x_1 the above is equal to 


    I^(1)= u^'0 u'· Qe^2  Q^2 (2π)^4   δ(Q+(h'-h))
       ⟨ B'(h')|J_0(0)|B(h)⟩ .


Next we use 


    J_0(0)=∫^3q_1  (2π)^3J̃_0( q_1)  ,


and  ⟨ B'(h')|J̃_0( q)|B(h)⟩∝ (2π)^3δ( Q- q). Note that J̃_0( q_1) is also evaluated at t=0. It can be combined with the factor 1/ q_1^2 into the Fourier transform of the potential energy operator Ṽ. Finally since non relativistic energies  are small compared to momenta we can write u'· Q≃ -u'^0 v'· Q yielding


    J^(1)= e^2 ⟨ B'(h')|∫^3q_1  (2π)^3Ṽ( q_1) v'· q_1|B(h)⟩ .


At second order we have


    I^(2)= ∫^4 q_2  (2 π)^4 u^'0 u'· q_2+ 0 u^'0 u'· Q +0 e^2  q_2^2 e^2  | Q- q_2|^2      
       ∫^4 x∫^4 y ^- (Q-q_2)· x^- q_2· y
       ⟨ B'(h')|T{J_0(y)J_0(x)}|B(h)  ⟩ .
 

First imagine doing the q_2^0 integration (see <ref> for an explicit discussion of the method by contours). The ballistic electron propagator only has a pole in the lower half q_2^0 complex plane. Consequently in the time ordered product of the currents only the time ordering where y^0-x^0 >0 contributes and we can so we can replace[Since 1=θ(x^0-y^0)+θ(y^0-x^0) and the second theta function gives no contribution.] 


    T{J_0(x)J_0(y)}→ J_0(y)J_0(x)θ(y^0-x^0)→ J_0(y)J_0(x) ,

in the integral. Next insert a complete set of states |X_1(p_X_1)⟩ between the two currents and translate the currents to the origin of space-time.
Then perform the coordinate integrations. This gives  a product of two delta functions, (2π)^4δ(p_X_1-h'+q_2)(2π)^4δ(h-p_X_1-q_2+Q). They  can be rewritten as a product of  the overall four momentum conservation delta function and one that performs the q_2 integration setting q_2=h'-p_X_1.  This yields


    J^(2)≃_X_11 v'·( h'- h)1  v'·( h'- p_X_1)   e^2  | h'- p_X_1|^2e^2  | h- p_X_1|^2⟨ B'(h') | J_0(0)|X_1(p_X_1)⟩⟨ X_1(p_X_1)|J_0(0)|B(h)⟩ .


Next express the charge density operator at the origin and an integral over the three dimensional Fourier transform, and use the fact that matrix elements conserve momentum to reassemble terms into the Fourier transform of the potential operator Ṽ. At this point the sum over intermediate states can be done yielding

    J^(2)≃ e^4 1  v'· ( h'-  h)⟨ B'(h')|∫^3 q_1  (2π)^3 Ṽ( q_2)1  v'·q_2∫^3 q_1  (2π)^3 Ṽ(q_1) |B(h)⟩ .



Since the matrix element under the integral is proportional to δ^(3)([h-h']-[q_1+q_2] ) this can be rewritten as, cf. <ref>, 

    J^(2)=  e^4 ∫^3 q_1  (2π)^3    ∫^3 q_2  (2π)^3 1  v' · (q_1 +q_2)1  v'· q_2
       ⟨ B'(h')|Ṽ( q_2) V( q_1) |B(h)⟩ .

Obviously this procedure can be repeated to deduce expressions for all the J^(n).





 §.§ Ryan's way  


Let us repeat the above analysis with a different emphasis and method. Consider the n^ th-order in perturbation theory. Photon exchange between the electron line and the charged-debris |B⟩ leads to a conventional diagrammatic expansion (with fermion and photon propagators) and a “blob” on the bottom of the diagram with an incoming ionic system B and an outgoing system B'. This “blob” corresponds to a multi-current correlator, G(q_1,…, q_n) of time-ordered electromagnetic currents (specifically Ĵ_0=ρ̂ for Coulomb exchange), 


    G^(n)=∫{ x}^+∑ q_i · x_iB'T{Ĵ_0(x_1) …Ĵ_0(x_n)}B.


In analogy with the transition from I to J discussed above, it is convenient to factor out an overall momentum conserving delta function and introduce 𝒢. 


    G^(n)(q_1,…, q_n) =    (2π)^4 δ(Σ q- Q) 
         ×𝒢^(n)(Q; q_2,…, q_N)  .


where Q = h'-h  is enforced by the delta function. Notice that 𝒢^(n) is a function of n-1 variables, while Q is a parameter fixed by the states |B(h)⟩ and ⟨B'(h'|. 

Making use of the eikonal approximation on the highly energetic electron leg, and working in Coulomb gauge, we then find



    e' B'ψ̅B = u̅(p') [ 
    ⟨B'|B⟩ + e^21/q_1^2u'_0/u'·Q𝒢^(1)(Q) + e^4∫^4 q_2/(2π)^41/q_2^2u'_0/u'· q_21/|Q-q_2|^2u'_0/u'·  Q𝒢^(2)(Q; q_2) + …]  .


Extension to higher orders is straightforward: one appends an extra integral, an extra fermion and photon propagator, and uses 𝒢^(n)(Q; q_2,…, q_n). The term in square brackets in <ref> corresponds to J as defined beneath <ref>. 

It is useful to write Ĵ_0(0) in terms of its components which supply definite momentum transfer k. For example,


    𝒢^(1)(Q)= B'(h')Ĵ_0(0)B(h) = ∫^3 q_1/(2π)^3B'ρ̂(q_1)B ,


where ρ̂(q_1) = ∫^3x  ^-q_1·xĴ_0(x). 

We now proceed with a direct evaluation of the loop integrals by contour integration. Let us study 𝒢^(2)(Q; q_2) as an explicit example. Begin by explicitly time ordering the currents in 𝒢^(2). Inserting a complete set of states, re-writing Ĵ_0(0) in terms of ρ̂(q_1), and carrying out the remaining integral over ^4x we obtain (defining ω_2=q_2^0), 


    𝒢^(2)(Q; q_2)= ∫^3q_1/(2π)^3_X B'ρ̂(q_1)X-/ω_2-Δ E_X + 0Xρ̂(q_2)B+B'ρ̂(q_2)X-/ω_2 -Δ E_X'- 0 Xρ̂(q_1)B,




where Δ E_X = E_X - E_B and Δ E_X'=E_X-E_B'. This result is obtained by translating the currents to t=0 and integrating the exponentials. It essentially amounts to a derivation of time-ordered perturbation theory, and the two terms in <ref> can be interpreted as a crossed and uncrossed Feynman diagram.

Next notice that because 1/u'· q_2 ≡ 1/(u'· q_2 + 0) only has a pole in the lower-half plane. We may then close our contour in the upper-half plane which selects only the second term in <ref>. At higher orders in perturbation theory all time orderings with poles in the lower-half plane vanish after contour integration; these correspond to crossed ladders. It follows that only the uncrossed ladders contribute to the amplitude to any order in perturbation theory, in agreement with the time-domain analysis presented in <ref>. 



Expanding the eikonal propagators and dropping the (now on-shell) energy transfer Δ E_X gives, 


    [u'_0/u'· q_2 + 0]_ω_2=Δ E_X'≃1/-v'·q_2 + 0(1+O(Δ E_X'/|q|)) .


The propagator structure is then independent of Δ E_X, and the sums over intermediate states reduce to the identity by completeness. We are then left with <ref>, with the n^ th order contribution being given by 



    e^2n∫[^3q_1/(2π)^3…^3q_n/(2π)^3 ]1/v'·q_11/v'·(q_1+q_2)…1/v'· (∑_i q_i)∏_i ρ̂_i(q_i)/q_i^2 = e^2n/n![∫^3 q/(2π)^31/v'·q- 0ρ̂(q)/q^2]^n ,


where the equality follows from symmetrizing the eikonal propagators inside the curly braces (introducing the factor of 1/n!), and using the soft-photon identity <cit.>, 


    ∑_ permutations1/x_π(1)1/x_π(1) + x_π(2)…1/x_π(1) + x_π(2)+ … + x_π(n) = 1/x_11/x_2…1/x_n .











apsrev4-1




 Paper Title: Transition properties of Doubly Heavy Baryons.



 





        
Department of Physics, Government College, Daman-396210, 

U. T. of Dadra & Nagar Haveli and Daman & Diu, INDIA 

Transition properties of Doubly Heavy Baryons
    Kaushal ThakkarCorresponding Author: kaushal2physics@gmail.com
    August 5, 2024
==================================================================



§ ABSTRACT 
In this study, we have investigated the radiative and semileptonic decay of doubly heavy baryons. Our focus is to determine the static and dynamic properties such as ground state masses, magnetic moment, transition magnetic moment, radiative decay and heavy-to-heavy semileptonic decay rates including their corresponding branching fractions. The ground state masses are calculated by solving the six-dimensional hyperradial Schrödinger equation. The magnetic moments and transition magnetic moments for J^P=1/2^+ and J^P=3/2^+ baryons are also calculated. In addition, radiative M1 decay widths are computed from the transition magnetic moment. We have employed the Isgur-Wise function(IWF) to analyse the semileptonic decay widths of the doubly heavy baryons. The obtained results are compared with other theoretical predictions.





§ INTRODUCTION


All the ground state baryons with zero or one heavy quark have been well established experimentally <cit.>. Research on baryons containing two or more heavy charm or bottom quarks has gained interest in recent years. All the doubly heavy baryons with their quark content and their experimental status are shown in Table <ref>. Only two doubly charmed baryons have been experimentally confirmed<cit.>. The first observed doubly charmed baryon Ξ_cc^+(3520) was confirmed by SELEX collaboration<cit.>. Ξ_cc^++ was reported by LHCb Collaboration<cit.>. The spin-parity of both Ξ_cc^+ and Ξ_cc^++ are yet to be identified. A search for the doubly heavy Ξ_bc^0 baryon using its decay to the D_0pK^- final state was performed using proton-proton collision data by the LHCb experiment but no significant signal was found <cit.>. LHCb reported the first search for the Ω_bc^0 and a new search for the Ξ_bc^0 baryons in 2021. No significant excess was found for invariant predicted masses between 6.7 and 7.3 GeV/c^2<cit.>. A search for Ξ_cc^+(ccd) and Ω_cc^+(ccs) was done by LHCb Collaboration and only hints of signals were seen<cit.>. The experimental as well as theoretical data for masses and semileptonic decay and other properties of singly heavy baryons are available while there are no experimental data for doubly heavy baryons. 


    The properties of doubly heavy baryons have been investigated via different theoretical approaches such as Quark model (QM)<cit.>, Quark-diquark model<cit.>, Relativistic quark model (RQM)<cit.>, Non-relativistic quark model (NRQM)<cit.>, Light Front approach in diquark picture <cit.>, QCD sum rule (QCDSR)<cit.>, Heavy Diquark effective theory (HDiET)<cit.>, Bethe-Salpeter equation<cit.>, Lattice QCD (LQCD) <cit.>. The semileptonic decays of bottom baryons to charm baryons yield a significant source of knowledge on the internal structure of hadrons. The calculation of IWF yields insights into branching ratio, decay width, and the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix <cit.>.


     This paper is organised as follows: In section <ref>,
we have discussed the theoretical framework for the quark model to compute the ground state masses of doubly heavy baryons. The magnetic moments, transition magnetic moments and radiative decay widths for doubly heavy baryons are computed in section <ref>. In section <ref>, we have calculated the Isgur-wise function and the semileptonic decay width for heavy to heavy transition. The result is presented and discussed in section <ref>. The paper is summarised in section <ref>.





§ THEORETICAL FRAMEWORK


We have adopted the Hypercentral constituent quark model (HCQM) to study the doubly heavy baryons. We consider the doubly heavy baryon to be a bound state of two heavy and one light quark. The dynamics of three quarks can be described by Jacobi coordinates. The hyperspherical coordinates: hyperradius and hyperangle are described in terms of Jacobi coordinates<cit.>.

    ρ = 1/√(2)(𝐫_1 - 𝐫_2)


    λ = m_1 𝐫_1 + m_2 𝐫_2 - (m_1 + m_2) 𝐫_3/√(m_1^2 + m_2^2 + (m_1 + m_2)^2)

The kinetic energy operator in HCQM can be written as

    P_x^2/2m = -ħ^2/2m(∂^2/∂ x^2 + 5/x∂/∂ x + L^2(Ω)/x^2)

The model Hamiltonian for baryons can be expressed as

    H = P_x^2/2m + V(x)

The six-dimensional hyperradial Schrödinger equation can be written as

    [ d^2/dx^2 + 5/xd/dx - γ(γ + 4)/x^2] ψ_νγ(x)
    
    = -2m[E - V(x)]ψ_νγ(x)

Where ψ_νγ is the hyper-radial wave function. The potential is assumed to  depend only on the hyperradius and hence is a three-body potential since the hyperradius depends only on the coordinates of all the three quarks. The hyper Coulomb plus linear potential which is given as

    V(x) = τ/x + β x + V_0 + V_spin


    Where, τ = -2/3α_s is the hyper coulomb strength, the values of β and V_0 are fixed to get the ground state masses. V_spin is the spin dependent part given as <cit.>


    V_spin(x) =   -A/4α_s λ⃗_i ·λ⃗_j e^-x/x_0/x x_0^2∑_i<jσ⃗_i ·σ⃗_j/6 m_i m_j

Here, the parameter A and the regularisation parameter x_0 are considered as the hyperfine parameters of the model. λ_i,j are the SU(3) colour matrices, σ_i,j are the spin Pauli matrices, m_i,j are the constituent masses of two interacting quarks. α_s is the strong running coupling constant.
     We factor out the hyperangular part of three quark wavefunction which is given by hyperspherical harmonics. The hyperradial part of the wavefunction is evaluated by solving the Schrödinger equation. The hyper-coloumb trial radial wave function which is given by <cit.>


    ψ_νγ=[(ν - γ)! (2g)^6/(2ν + 5)(ν + γ + 4)!]^1/2 (2gx)^γ
    × e^-gx L^2γ + 4_ν-γ (2gx)


    Here, γ is the hyper angular quantum number and ω denotes the number of nodes of the spatial three-quark wave function. L^2γ + 4_ν-γ(2gx) is the associated Laguerre polynomial. The wavefunction parameter g and energy eigenvalues are obtained by applying the virial theorem. The masses of ground state doubly heavy baryons are calculated by summing the model quark masses (see Table <ref>), kinetic energy and potential energy.

    M_B = m_1 + m_2 + m_3 + ⟨ H ⟩

The computed ground state masses of doubly heavy baryons while comparing with others are given in Table <ref>.









§ MAGNETIC MOMENT AND RADIATIVE DECAY



 §.§ Effective quark masses and magnetic moment for doubly heavy baryons

Electromagnetic properties of the baryons are an important source of information on their internal structure. The magnetic moment of baryons are obtained in terms of its quarks spin-flavour wave function of the constituent quarks as, <cit.>

    μ_B = Σ_i ⟨ϕ_sf|μ_i|ϕ_sf⟩

where

    μ_i = e_iσ_i/2m_i^eff

where i = u,d,s,c,b; e_i and σ_i represents the charge and spin of constituting quarks of the baryonic state and |ϕ_sf⟩ represents the spin-flavour wave function of the respective baryonic state. The expressions for magnetic moments of J^P=1/2^+ and J^P=3/2^+ doubly heavy baryons are given in Table <ref>. Here, m_i the mass of i^th quark in the three body baryon is taken as an effective mass of the constituting quarks as their motions are governed by the three body force described through the Hamiltonian in Eqn.(<ref>). The baryon mass of the quarks may get modified due to its binding interactions with other two quarks. We account for this bound state effect by replacing the mass parameter m_i of Eqn. (<ref>)  by defining an effective mass to the bound quarks, m_i^eff given as <cit.>


    m_i^eff = m_i ( 1+⟨ H ⟩/∑_i m_i)


such that M_B = ∑_i=1^3 m_i^effwhere ⟨ H ⟩ = E + ⟨ V(x) ⟩. The calculated magnetic moments for doubly heavy baryons are listed and compared with other theoretical models in Table <ref>.











 §.§ Transition magnetic moment and radiative decay width

The transition magnetic moment for 3/2^+ →1/2^+ can be expressed as <cit.>

    μ_3/2^+ →1/2^+ = ∑_i ⟨ϕ_sf^3/2^+ | μ_iσ_i| ϕ_sf^1/2^+⟩

⟨ϕ_sf^3/2^+ | represent the spin flavour wave function of the quark composition for the respective  baryons with J^P=3/2^+ while |ϕ_sf^1/2^+⟩ represent the spin flavour wave function of the quark composition for the baryons J^P=1/2^+. To compute the transition magnetic moment (μ_3/2^+ →1/2^+), we take the geometric mean of effective quark masses of the constituent quarks of initial and final state baryons,

    m_i^eff = √(m_iB^*^effm_iB^eff)

 Here, m_iB^*^eff and m_iB^eff are the effective masses of the quarks constituting the baryonic states B^* and B respectively. Taking  into account the geometric mean of effective quark masses of the constituting quarks and the spin flavour wave functions of the baryonic states, the transition magnetic moments are computed using Eqn. (<ref>). The expressions for transition magnetic moments and the obtained transition magnetic moments of doubly heavy baryons are listed in Table <ref>. We can see that the results are in accordance with other theoretical predictions.

The radiative decay width can be expressed in terms of the radiative transition magnetic moment and photon momentum (k) as <cit.>

    Γ = α k^3/M_P^22/2J+1M_B/M_B^*μ^2(B^* → Bγ)

where μ^2(B^* → Bγ) is square of the transition magnetic moment, α=1/137, M_P is mass of proton = 0.938 GeV.  J and M_B^* are the spin and mass of the decaying baryon and M_B is the baryon mass of the final state. k is the photon momentum in the center-of-mass system of decaying baryon

    k = M^2_B^* - M^2_B/2M_B

Here, we ignore E2 amplitudes because of spherical symmetry of S-wave baryon spatial wave function and the M1 width of the decay B^* → Bγ has the form of Eqn. (<ref>). The calculated radiative decay widths are listed and compared in Table <ref>.










§ SEMILEPTONIC TRANSITION



 §.§ Form factors and Isgur-wise function:


One of the important topics in examining the features of doubly heavy baryons is their weak decay rates. The study of semileptonic decays of heavy hadrons allows for the determination of the CKM matrix elements. Other properties of semileptonic decays, such as the momentum dependence of transition form factors and exclusive decay rates are critical to our knowledge of heavy hadron structures. The Feynman diagram for b → c transition is shown in Fig <ref>.


    Our focus is to determine b → c transitions of the ground state of doubly heavy baryons. In the framework of Heavy quark effective theory(HQET),
    the heavy quark masses m_c, m_b ≫ Λ_QCD, Λ_QCD is the strong interaction scale. In the HQET, the total six form factors are reduced to one, which is represented by Isgur-wise function η.
    The remaining form factor is the function of the kinetic parameter ω.


    F_1(ω)=G_1(ω)=η(ω)



    F_2(ω) = F_3(ω) = G_2(ω) = G_3(ω) = 0


The Isgur-wise function η depends on ω which can be expressed as <cit.>


    η(ω) = exp(-3(ω - 1)m_cc^2/Λ_B^2)


where, ω = ν · ν' and ν, ν' are the four velocities of the initial and final state of doubly heavy baryons respectively. Λ_B is the size parameter that varies in range 2.5 ≤ Λ_B ≤ 3.5 GeV <cit.>.



 §.§ Differential decay widths

    At zero recoil point i.e. ω = 1, bb → bc and bc → cc becomes identical. The transversely polarised differential decay rate (Γ_T) and longitudinally polarised differential decay rate (Γ_L) neglecting the lepton masses, are given by,




























    dΓ_T/dω = G_F^2 |V_cb|^2 m_B'^3 /6π^3  q^2 ω√(ω^2 -1)η^2(ω)



    dΓ_L/dω = G_F^2 |V_cb|^2 m_B'^3 /24π^3×[(ω - 1)(m_B + m_B')^2 
    
    + (ω + 1)(m_B - m_B')^2]η^2(ω)


 where, q^2 is squared four-momentum transfer between the heavy baryons given as, q^2 = (p - p')^2 = m_B^2+m_B'^2-2m_Bm_B' where m_B and m_B' are masses of initial and final baryons, respectively. We have taken |V_cb| = 0.042. The total differential decay rate is given as,


    dΓ/dω = dΓ_T/dω + dΓ_L/dω



    Γ = ∫_1^ω_maxdΓ/dω dω

The total decay width is calculated by integrating the total differential decay rate from 1 to ω_max maximal recoil (q^2 = 0). The obtained values for ω_max for different transitions are shown in Table <ref>.


    ω_max = m_B^2 + m_B'^2/2m_B m_B'

















    Br = Γ×τ


    The branching ratio of doubly heavy baryons can be calculated using Eqn. (<ref>) where, τ is the lifetime of the initial baryon.





§ RESULTS AND DISCUSSIONS

    We have calculated the ground state masses of all the doubly heavy baryons using the parameters shown in Table <ref>. The calculated masses of ground state doubly heavy baryons are listed in Table <ref>. The mass difference between the up quark and down quark have been neglected in all other theoretical predictions shown in Table <ref>. In the present work, we have considered different quark masses for up and down quarks as, m_u = 0.330 GeV and m_d = 0.350 GeV (see Table <ref>). Our calculated masses of doubly heavy baryons are in agreement with the values obtained in Ref. <cit.>. The values obtained in Ref. <cit.> are smaller than our calculated masses.


    As shown in Table <ref>, the magnetic moments of doubly heavy baryons are almost matched with other models. The magnetic moment of Ξ^0*_bb predicted by the Ref. <cit.> has a negative value while all other theoretical approaches predicted including ours have  positive values.
    The transition magnetic moments of doubly heavy baryons are listed in  Table <ref>. As indicated in Table <ref>, we can see the good agreement of a computed transition magnetic moments with other predictions except the Ref.<cit.> which has relatively lower values.



    Comparing the radiative decay width with other models, we found that different approaches lead to different results as shown in Table <ref>. We can see that the radiative decay width is relatively large for Ξ_cc^++*→Ξ_cc^++γ and Ξ_cc^+*→Ξ_cc^+γ in the relativistic three quark model <cit.> while comparing with others. Our computed radiative decay width for Ω_bb^-*→Ω_bb^-γ transition is relatively lower than all other predictions.





     To calculate the semileptonic decay rate, we have considered m_bb = 2m_b = 9.9 GeV and for bc → cc transition, m_cc = 2m_c = 3.1 GeV in the Eqn. (<ref>). We have considered the size parameter Λ_B = 2.5 GeV <cit.>.
     The calculated decay rates of the baryons are listed and compared with other models in Table <ref>. The present results for the semileptonic decay width of doubly heavy baryons are close to the results predicted by the Ref.<cit.>. The predicted result of semileptonic decay for Ξ_bc^+ →Ξ_cc^++ lν̅_l by the Ref.<cit.> and Ref.<cit.> are in accordance with the present computed result.      It is found that the present computed decay width for Ξ_bb^0 →Ξ_bc^+ lν̅_l transition is lower compared to Ref.<cit.> and Ref.<cit.>.


     The total differential decay rate (dΓ/dω) can be written as a summation of transverse differential decay rate (dΓ_T/dω) and longitudinal decay rate (dΓ_L/dω) as indicated in Eqn.(<ref>). It is found that the contribution from the transverse decay (Γ_T) is relatively higher compared to the longitudinal decay (Γ_L) as shown in Table <ref>. We can see that almost 60% of contributions come from Γ_T while 40% of the contribution comes from Γ_L.


     The behaviour of the predicted differential decay rate for semileptonic decay of doubly heavy baryons with ω are shown in Figure <ref> to Figure <ref>. The peak value of differential decay rate (dΓ/dω)  for  Ξ_bb and Ω_bb baryons is found at ω≈1.01 while the peak value for Ξ_bc and Ω_bc baryons is found at ω≈1.06. The dΓ/dω of Ξ_bb and Ω_bb baryons gets saturated around ω≈1.06 while Ξ_bc and Ω_bc baryons are at peak value for ω≈1.06.








   The lifetimes of baryons have been studied in Ref. <cit.>. We have considered τ_Ξ_bb^0 = 0.52×10^-12 s, τ_Ξ_bb^- = 0.53 ×10^-12 s, τ_Ξ_bc^+ = 0.24×10^-12 s, τ_Ξ_bc^0 = 0.22×10^-12 s, τ_Ω_bb^- = 0.53 ×10^-12 s, τ_Ω_bc^0 = 0.18×10^-12 s as given in the Ref.<cit.>. We have calculated the branching ratios using the life time of doubly heavy baryons predicted by the Ref.<cit.>.
   While comparing our results for branching ratio with other theoretical predictions, we have computed branching ratio from their predicted decay width in corresponding model and life time mentioned in the Ref.<cit.>. The computed branching ratio for the Ω_bc^0 →Ω_cc^+lν̅_l semileptonic decay is 1.11% which is in agreement with the Ref.<cit.>.









§ CONCLUSIONS

 The ground state masses are calculated using Hypercentral Constituent Quark Model(hCQM). The magnetic moments of doubly heavy baryons are computed using the spin-flavour wave functions of the constituent quarks and their effective masses within the baryon. We have calculated the radiative M1 decay width from the obtained transition magnetic moment for 3/2^+ →1/2^+ transition.  The semileptonic decay rates for doubly heavy baryons are calculated after obtaining the Isgur-wise function. Also, the transverse and longitudinal components of the decay widths are calculated.




100
pdg2016C. Patrignani et al. (Particle Data Group), Chin. Phys. C 40(10), 100001 (2016). 
pdg2020P. A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020). 
pdg2022R. L. Workman et al. Review of particle physics. Prog. Theor. Exp. Phys.2022, 083C01 (2022). 
Mattson2002M. Mattson et al.(SELEX Collaboration), Phys. Rev. Lett. 89, 112001 (2002). 
Ocherashvili2005A. Ocherashvili et al. (SELEX Collaboration), Phys. Lett. B 628, 18 (2005). 
Aaij2017R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 119, 112001 (2017). 
Aaij2018R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 121, 162002 (2018). 
Aaij2019R. Aaij et al. (LHCb Collaboration), JHEP 10, 124 (2019). 
RAaij2020cR. Aaij et al. (LHCb  collaboration), Chin.  Phys.  C 44, 022001 (2020). 
RAaij2020R. Aaij et al. (LHCb collaboration), JHEP 02, 049 (2020). 
Aaij2020R. Aaij et al. (LHCb Collaboration),   JHEP 11, 095 (2020). 
Aaij2021R. Aaij et al. (LHCb Collaboration), Chin. Phys. C 45, 093002(2021). 
2020AaijR. Aaij et al. (LHCb collaboration), Sci. China Phys. Mech. Astron. 63, 221062 (2020). 
2021AaijR. Aaij et al. (LHCb collaboration), JHEP 12, 107 (2021). 
2021cAaijR. Aaij et al. (LHCb collaboration), Sci. China Phys. Mech. Astron. 64, 101062 (2021). 
Roberts2008W. Roberts and Muslema Pervin, arXiv:0711.2492 [nucl-th] (2008).
Farhadi2023M. Farhadi et al. Eur. Phys. J. A  59, 171 (2023). 
Ebert2004D. Ebert, R. N. Faustov, V. O. Galkin and A. P. Martynenko, Phys. Rev. D 70, 014018 (2004). 
Ghalenovi2022Z. Ghalenovi, C.P. Shen, M. Moazzen Sorkhi, Phys. Lett. B 834, 137405 (2022). 
zhao2018Z. X. Zhao, Eur. Phys. J. C 78, 756 (2018). 
Tousi2024M. S. Tousi, K. Azizi, Phys. Rev. D, 109(5), 054005(2024). 
shi2020Y. J. Shi, W. Wang, Z. X. Zhao, and U. G. Meiner, Eur. Phys. J. C 80, 398 (2020). 
Yu2019Q. X. Yu and X. H. Guo, Nucl. Phys. B 947, 114727 (2019). 
Brown2014Z. S. Brown, W. Detmold, S. Meinel, K. Orginos, Phys. Rev. D 90(9), 094507 (2014). 
Gershtein2000 S. S. Gershtein, V. V. Kiselev, A. K. Likhoded, and A. I. Onishchenko, Phys. Rev. D 62, 054021 (2000). 
Padmanath2019M. Padmanath, arXiv:1905.10168 (2019). 
Isgur1990N. Isgur, M.B. Wise, Phys. Lett. B 237, 527 (1990). 
Thakkar2017K. Thakkar, Z. Shah, A. K. Rai and  P. C. Vinodkumar, Nuclear Physics A 965, 57-73  (2017).
Thakkar2020K. Thakkar, Eur. Phys. J. C 80(10), 926 (2020). 
Garcilazo2007  H Garcilazo, J Vijande and A Valcarce, et al., J. Phys. G: Nucl. and Part. Phys. 34, 961 (2007).
Santopinto1998E. Santopinto, F. Lachello, M.M. Giannini, Eur. Phys. J. A 1, 307 (1998). 
Ferraris1995M. Ferraris, M.M. Giannini, M. Pizzo, E. Santopinto, L. Tiator, Phys. Lett. B 364, 231 (1995). 
Thakkar2011K. Thakkar, B. Patel, A. Majethiya, P.C. Vinodkumar, Pramana J. Phys. 77, 1053 (2011). 
majethiya2008 A. Majethiya, B. Patel, P. C. Vinodkumar, Eur. Phys. J. A, 42, 213 (2009); [Erratum]: Eur. Phys. J. A 38,307 (2008).
Bernotas2013 A. Bernotas, V. Šimonis, Phys. Rev. D 87, 074016 (2013). 
Wagner2000G. Wagner, A. J. Buchmann, A. Faessler, J. Phys. G 26, 267 (2000). 
Gadaria2016A. N. Gadaria, N. R. Soni, J. N. Pandya, DAE Symp. Nucl. Phys. 61, 698-699 (2016). 
shah2021 Z. Shah, A. Kakadiya, K. Gandhi, A. K. Rai, Universe, 7, 337 (2021). 
Hazra2021 A. Hazra, S. Rakshit, R. Dhir, Phys. Rev. D 104, 053002 (2021). 
Simonis2018 V. Šimonis, arXiv:1803.01809 (2018). 
Branz2010T. Branz, A. Faessler, T. Gutsche, M. A. Ivanov, J. G. Körner, V. E. Lyubovitskij, and B. Oexl, Phys. Rev. D 81, 114036 (2010). 
Lu2017Q. F. Lü, K. L. Wang, L. Y. Xiao and X. H. Zhong, Phys. Rev. D 96, 114006 (2017). 
Albertus2005C. Albertus, E. Hernández and J. Nieves, Phys. Rev. D 71, 014012 (2005).
Faessler2009A. Faessler et. el, Phys. Rev. D 80, 034025 (2009). 
Ebert2005D. Ebert, R. N. Faustov et al. Phys. Atom. Nuclei 68, 784-807 (2005). 
Rahmani2020S. Rahmani and H. hassanabadi, Eur. Phys. J. C 80, 312 (2020). 
Ivanov1997M. A. Ivanov, V. E. Lyubovitskij, J. G. Körner and P. Kroll, Phys. Rev. D 56, 348 (1997). 
Ghalenovi2023Z. Ghalenovi, M. Moazzen Sorkhi, Chin. Phys. C 47, 033105(2023). 
Wang2017W. Wang, F. S. Yu, Z.X. Zhao, Eur. Phys. J. C 77, 781 (2017). 
Karliner2014M. Karliner, J. L. Rosner, Phys. Rev. D 90, 094007 (2014). 
Albertus2007C. Albertus, E. Hernández, J. Nieves and J. M. Verde-Velasco, Eur. Phys. J. A 32, 183 (2007), [Erratum]: Eur. Phys. J. A 36, 119 (2008). 
Berezhnoy2018A. V. Berezhnoy, A. K. Likhoded, A. V. Luchinsky, Phys. Rev. D 98, 113004 (2018). 
Kiselev2002V. V. Kiselev and A. K. Likhoded, Phys. Usp. 45, 455 (2002) [Usp. Fiz. Nauk 172, 497 (2002)]. 
Likhoded2018A. K. Likhoded, A.V.Luchinsky, Physics of Atomic Nuclei, Vol. 81, 6 (2018). 
Cheng2019H. Cheng and F. Xu, Phys. Rev. D 99, 073006 (2019). 



 Paper Title: Hot and Dense QCD Shear Viscosity at (almost) NLO.


=1
















Λ_QCD
#1Eq. (<ref>)
 Re 
 Im 

Li_2
𝒞
𝒪
N_f
ff̅hc
ff̅h
fh
hc
fhc
𝒬_max
#1 [ID: #1] 
bremsstrahlung
↔
ν
d^3/(2π)^3 
d^3/2||   (2π)^3 
Π_00
Π_T
Tr 
12
22

P
m_ eff,g
m_ eff,f
m_eff,s
Q_max

μ__EQCD

ç
h
ȷj
x
p
q
k̨
v̌
ŭ
E
D
f
h
b
q_⊥
||
f

n̂
C_A
C_F
d_A
T_A
T_F
d_F
N_c 
N_c
m_F
n_B
n_q
𝐃
𝐫
𝐑
𝐱
𝐲
𝐪
α_eff
𝕀1
Π^μν
ρ
λ
𝐤
||
𝐩
𝐏

|q^0|
||
ℒ

C_R
∇
||
||
ℒ
𝒞
α_s
csch


m_D

semi

    𝒞#1Appendix <ref>#1Section <ref>#1Ref. <cit.>#1Fig. <ref>Institut für Kernphysik, Technische Universität Darmstadt
    
    Schlossgartenstraße 2, D-64289 Darmstadt, Germanyidanhoni@theorie.ikp.physik.tu-darmstadt.de,guy.moore@physik.tu-darmstadt.deThe next-to-leading order weak-coupling shear viscosity of QCD was computed 6 years ago.
     However, these results have never been applied at finite baryon chemical potential μ, even though intermediate-energy heavy ion collisions and merging neutron stars may explore the Quark-Gluon Plasma in a regime where baryon chemical potentials are large.
     Here, we extend the next-to-leading order shear viscosity calculations to finite μ, and we show that, while the convergence of the weak-coupling expansion is questionable for achievable plasmas, it is somewhat better at μ > T than at μ=0.
    
    Hot and Dense QCD Shear Viscosity at (almost) NLO
        Isabella Danhoni, Guy D. Moore
        August 5, 2024
    =================================================
    
    
    
    § INTRODUCTION
    
    
    The deconfined phase of hot and dense QCD is known as the quark-gluon plasma (QGP). Evidence shows that this phase was present in the early universe and that it has been recreated at microscopic scales in heavy ion collision experiments<cit.>.
    These results have sparked large interest in this subject over the past few decades, and experiments indicate that the QGP in such collisions is a strongly coupled fluid<cit.> with behavior close to that of a perfect fluid<cit.>.
    On the other hand, sufficiently early in the history of the early Universe, the running coupling should have been small enough that a perturbative treatment may be valid.
    For thermodynamical quantities, suitably resummed perturbative calculations work starting at temperatures a few times that of the chiral symmetry restoration transition<cit.>.
    
    
    
    
    
    In lower-energy collisions, quark-gluon plasma is generated with a significant net baryon number, such that the baryon chemical potential μ_B is several times higher than the temperature.
    This regime cannot be directly studied on the lattice due to the sign problem (or complex action problem).
    For a chemical potential μ_B≠ 0 the fermion determinant (or Monte Carlo action) becomes complex, which ruins the probabilistic interpretation of this quantity used in lattice QCD calculations<cit.>.
    For this reason, the high-density region of the QCD phase diagram is much less well understood than the μ_B=0 axis.
    As experimental efforts focus on bringing clarity to this region, such as RHIC-BES <cit.>, HADES<cit.> and CBM at
    FAIR<cit.>, the need for parallel theoretical development becomes clear.
    
    At high temperatures, we claim that the perturbative series should work better at a high chemical potential, such that μ_B > T > T_c, than in the case of μ_B=0<cit.>. This is a consequence of the fact that scattering from quarks is enhanced by a factor of μ_B^2 / T^2.
    The failure of perturbation theory in hot QCD occurs before the coupling becomes large, due to the strongly-coupled nature of soft gluons.
    These play a central role in the breakdown of the perturbative expansion for the shear viscosity <cit.>.
    But at high chemical potential, where quarks play a larger role, one expects the perturbative series to be better behaved.
    This means that perturbation theory has a better chance of working at high μ/T than at μ=0.
    Although the (μ,T) values which can be explored in experiment are unlikely to extend into the region where perturbation theory is directly predictive, the perturbative results can still be used in modeling the (μ,T) behavior of the shear viscosity phenomenologically, as was recently done using leading-log results in Ref. <cit.>.
    Essentially, perturbation theory serves as a limiting behavior so that η/s can be interpolated based on high-density perturbative, low-density hadron-resonance gas, and intermediate density data-driven estimates.
    
    Previously Arnold, Moore, and Yaffe have computed the shear viscosity at leading perturbative order in the absence of a chemical potential
    <cit.>, and Ghiglieri et al have extended this calculation to “almost” next-to-leading order (NLO) <cit.>.
    Separately, we have presented the shear viscosity at finite chemical potential at leading-log order <cit.>.
    The goal of this paper is to work to almost next-to-leading order at finite chemical potential, both to obtain the most reliable estimates for the viscosity in this regime that we can, and to test how perturbation theory converges in the high-μ regime.
    We start in Section <ref> by reviewing the kinetic theory and computational approach that Arnold, Moore, and Yaffe used in <cit.>.
    In Section <ref>, we review how to extend the collision operator to next-to-leading order and describe the computational tools required. 
    In Section <ref>, we show what changes must be made to the calculation to include chemical potentials.
    The analysis of our calculations and final results are shown in
    Section <ref>.
    We end in Section <ref> with a short discussion and summary.
    But we can present the main conclusions very briefly now.
    The shear viscosity, rescaled as η T/(ε + P), is a weak function of μ_q/T (with μ_q=μ_B/3 the chemical potential of a quark).
    The convergence of the perturbative series improves when
    μ_q^2 > N_c π^2 T^2, but the improvement is somewhat modest.
    For more complete details, see Section <ref>.
    
    
    
    
    
    § KINETIC THEORY
    
    In this section, we briefly summarize how shear viscosity is defined and how it can be computed to leading order (LO) in the Effective Kinetic Theory (EKT).
    Shear viscosity is a property relevant for systems with space-nonuniform flow velocity, meaning that the energy and momentum densities vary through space, but in a manner which is sufficiently smooth that the system is locally near equilibrium.
    In this case we can exchange the conserved densities T^μ 0 for the local 4-velocity and energy density, defined with the Landau-Lifshitz convention <cit.>:
    u^μ(x,t) is the timelike unit eigen(four)vector of T^μν and ε is its eigenvalue,
    T^μν u_ν = ε u^μ
    with[We use a mostly-positive metric tensor g_μν = Diag[-1,+1,+1,+1],
    and natural units, c=1, ħ = 1. Capital letters P,K are 4-vectors, lower case letters p⃗,k⃗ or p_i,k_i are the space components, p^0,k^0 are the time components, and p,k are the magnitudes of the space components.]u^μ u_μ = -1.
    For uniform u^μ,ε the system will be in equilibrium with
    T^μν = ε u^μ u^ν + P g^μν,
    where P the pressure is determined from ε by the equilibrium equation of state.
    
    However, for nonuniform u^μ the system will depart from equilibrium and T^μν will not follow its ideal form.
    Defining the projection operator into the local rest frame,
    
        Δ^μν≡ g^μν + u^μ u^μ ,
    
    the shear tensor is defined as
    
        σ^μν≡∇^μ u^ν + ∇^ν u^μ
            - 2/3Δ^μν∇_α u^α .
    
    For small and slowly varying σ_ij, linear response theory predicts that the stress tensor should change by an amount linear in the shear tensor:  in the local rest frame,
    
        T_ij = P g_ij - η σ_ij
    
    with P the pressure and η, the shear viscosity, defined as the coefficient of the linear response of T_ij to nonvanishing σ_ij.
    If the fluid flow also has a divergence θ = ∂_i u_i, then an additional term T_ij = … - ζθΔ_ij would be present, with ζ the bulk viscosity.
    Bulk viscosity requires a different analysis <cit.> which we will not attempt here.
    
    Many (not all) theories have a kinetic description, and this should include nonabelian gauge theories in the weak coupling regime.
    In this case one can give a coarse-grained description of the state of the system in terms of the occupancies of each field type at each point in time and phase space, f^a(p⃗,x⃗,t).
    In equilibrium and at lowest order in the coupling, we have
    
        f_0^a(p⃗,x⃗,t) = 1/exp(γβ ( p^0 - u_i p^i - μ_a ) ∓ 1 
            = 1/exp (β (-u_μ P^μ - μ_a)) ∓ 1.
    
    Here u^μ = (γ , γv⃗) is the local velocity 4-vector as before, γ is the associated relativistic gamma-factor, and β=β(x⃗,t) = 1/T is the local temperature.
    The sign ∓ 1 is - for bosons and + for fermions (the upper sign will always apply to bosons), and μ_a is the chemical potential for species a, which will be nonzero for quark species and opposite for a particle and its antiparticle.
    In this work, we will consider SU(3) gauge theory with  vectorlike quark species each with the same chemical potential.
    
    
    Shear viscosity, as well as other transport coefficients, is theoretically defined in a Kubo-type formulae relating them to the zero-frequency transport limit of the spectral functions of two-point correlators of the appropriate operators.
    Performing a perturbative evaluation at leading order using the diagrammatic approach based directly on these Kubo formulas would require complex resummations of numerous sub-diagrams to all orders.
    This has been done successfully for scalar field theory
    <cit.> and for QED for electrical conductivity
    <cit.> and shear viscosity <cit.>,
    and in each case the result proved to be the same as a kinetic description.
    In the case of full QCD such a diagrammatic calculation has never been successfully attempted, but physical arguments indicate that the result should again justify a kinetic description <cit.>.
    Therefore, we adopt the linearized version of the EKT developed in <cit.>.
    
    One starts as usual with a Boltzmann equation
    
        [ v⃗_p ·∂/∂x⃗] f^a(p⃗,x⃗,t) = - ^a[f],
    
    where f^a refers to the nonequilibrium form of the statistical function,
    which we linearize about the equilibrium form
    
        f^a(p⃗,x⃗) = f_0^a(p⃗, x⃗) + f_0(1 ± f_0)  f_1^a(p⃗,x⃗)  ,
    
    according to the Landau-Lifshitz conventions under which f_0 contributes all of the energy, momentum density, and net particle numbers in the system.
    The choice to normalize f_1 with a factor of f_0(1± f_0) will be convenient in what follows.
    For a more detailed description one can refer to <cit.> which developed the leading-log application of Boltzmann equations to the evaluation of the QCD shear viscosity,
    <cit.>, which extended the treatment to leading order and established most of our notation and techniques,
    and <cit.>, which extended the treatment to finite chemical potentials, albeit at leading-logarithmic order.
    We will save the discussion of the collision operator ^a[f] for the next section.
    
    The departure from equilibrium must first arise at linear order in σ_ij.
    Using rotational invariance, one can then write the departure from equilibrium as
    
        f_1^a(p⃗,x⃗) = β^2 X_ij(x⃗) χ^a_ij(p⃗) = β^2 X_ij(x⃗) I_ij(p⃗) χ^a(p)  .
    
    Here χ^a(p) is a species-dependent, pure scalar function of p which characterizes the departure from equilibrium in the presence of shear stress, and
    
        X_ij   ≡1/√(6)σ_ij
        
            I_ij(p⃗)    ≡√(3/2)( p̂_i p̂_j - 1/3δ_ij)  .
    
    The normalization is explained in Ref. <cit.> and is chosen so that I_ij is a properly normalized projection operator.
    Evaluating the space derivatives on the lefthand side of boltz
    one finds to lowest order,
    
        v⃗_p ·∂/∂x⃗
            f^a_0(p⃗,x⃗,t) = β^2 X_ij S^a_ij
    
    with
    
        S^a_ij(p⃗)  ≡ p^0   T   f_0^a(1± f_0^a) I_ij(p⃗)  .
    
    Using this and inserting chidef into the RHS, boltz reduces to
    
        β^2 X_ij  S^a_ij(p⃗) = ^a[β^2 X_ij χ^a_ij(p⃗)]
                
            S^a_ij(p⃗) = χ^a_ij(p⃗) ,
    
    where χ^a_ij(p⃗) means the result of factoring β^2 X_ij out of the collision operator.
    
    This equation is solved using a variational method.
    This strategy was used in <cit.> in leading log and leading order calculations. 
    First one defines an inner product on departures from equilibrium:
     
        (f,g) = β^3 ∑^_a∫_p⃗ f^a(p⃗)g^a(p⃗)
    
    It can be shown that the collision operator is Hermitian with respect to this inner product.
    We define
     
        𝒬[χ] = (χ_ij, S_ij)- 1/2(χ_ij,χ_ij)  ,
    
    which can be viewed as a functional over the space of χ^a(p) values.
    That is, the functional 𝒬 takes its maximum value when the Boltzmann equation is satisfied.
    Furthermore, its value at this maximum,
     
        =1/2(χ_ij, S_ij)=1/2 (χ_ij,χ_ij) = 1/2 ( S_ij,^-1 S_ij )  ,
     
    determines the viscosity:
     
        η = 2/15.
    
    As known from other variational problems, one can obtain highly accurate approximate results by performing a restricted extremization within a well-chosen finite dimensional subspace.
    For this purpose, we expand the χ(p) functions into a finite basis set:
    
        χ(p) =∑_m=1^N a_m ϕ^(m)(p) .
    
    We will use the variational Ansatz proposed
    [To our knowledge this variational Ansatz was first used in Ref. <cit.>, but it is not actually presented there; we thank Larry Yaffe for private communications on this point.]
    in Ref. <cit.>:
    
        ϕ^(m)=p(p/T)^m/(1+p/T)^N-1,  m=1,...,N  .
    
    The subspace spanned by this basis strictly increases as N is increased.
     
    
    
    
    § COLLISION OPERATORS FOR AN NLO EVALUATION
    
    
    At vanishing chemical potential, the perturbative framework suggests that the QGP predominantly comprises quasiparticle excitations with momenta of the order of the temperature.  
    Their dispersion and occupancy are corrected only at (g^2), which for an (g)-subleading calculation means that we can ignore corrections both to the lowest-order entropy density s and to the lowest-order particle dispersion relations (except for splitting, see below).
    Therefore NLO corrections will only be needed in the collision integral, which we turn to next.
    The LO calculation <cit.> requires two types of scattering processes, the  scatterings with all hard ((T)) external particles and  effective splitting processes between hard participants. 
    
    At NLO, all the scatterings and corrections to be added are the ones suppressed by a single power of g.  As shown in detail in <cit.>, there are only a
    few such (g) subleading effects, which can be described as follows. The rate of soft
     scattering is modified to be considered as an
    additional momentum-diffusion coefficient δq̂.  This along with an (g) correction to the medium corrections to dispersion provides an (g) shift in the  splitting rate.  Next, the  splitting rate must be corrected wherever one participant of the process is "soft", p ∼ gT, or the opening angle is less collinear.  Finally, the numerical implementation of the LO scattering kernel <cit.> already resums a small amount of these NLO effects, requiring a subtraction to δq̂ and the longitudinal momentum diffusion, δq̂_L. 
    This section will briefly review the collision operator at LO and NLO at vanishing chemical potential.
    
    
    
     §.§ Collision Integrals at LO
    
    
    The leading-order collision operator encodes the contribution of tree-level 
    scattering processes, with Hard-Loop resummed
    propagators in the soft-sensitive channels,
    as well as collinear, effective  processes resumming the effect
    of splitting caused by any number of soft scatterings.
    Both processes contribute to order g^4T to the collision operator; a subset of C_a^[f]
    is logarithmically enhanced, g^4Tln(1/g), due to the sensitivity
    to the soft scale gT.
    C_a^[f] and C_a^[f] are described in detail
    in <cit.>. Therefore, the collision term has the form:
    
        𝒞_a[f] = 𝒞_a^2↔ 2[f] + 𝒞_a^1↔ 2[f]
    
    
    
    The matrix elements that contribute to 2 ↔ 2 processes at leading order come from the diagrams shown in Fig. <ref>. In terms of the collision operator, these contributions can be written as:
    
        ^a[f](p⃗) =1/2∑_bcd∫_k⃗,p⃗,k⃗'⃗   |ℳ_abcd(P,K,P',K')|^2/2p^0  2k^0  2p'^0  2k'^0
            (2π)^4 δ^4(P+K-P'-K') 
           ×{ f^a(p⃗)f^b(k⃗)[1± f^c(p⃗'⃗)][1± f^d(k⃗'⃗)] 
           ×{ -  f^c(p⃗'⃗)f^d(k⃗'⃗)[1± f^a(p⃗)][1± f^b(k⃗)]}
    
    where the incoming/outgoing momenta p⃗,k⃗ and p⃗',k⃗' are all on shell, p^0=p. The squared matrix element 
    |ℳ_cd^ab| is hard thermal loop (HTL) resummed
    <cit.>
    and is summed over all spin polarizations and colors.  We write ∫_k⃗ = ∫d^3k⃗/(2π)^3 to simplify the notation. 
    
    
    Diagrams (D) and (E) are processes involving t (or u) channel fermion exchange.
    These processes have a logarithmic infrared divergence when computed using free propagators.
    To cut off this divergence, one can include the retarded HTL fermionic thermal self-energy, as was originally done in Ref. <cit.>.
    It was later shown in Ref. <cit.> that there is a simpler treatment which is also correct at the leading order.
    The most important feature of the HTL self-energy insertion is that it correctly reproduces the total scattering rate for this process; since the process is nearly isotropic, the detailed angle dependence is not relevant.
    A mass-like insertion, with the correctly chosen coefficient, reproduces this rate and is much simpler to use in calculation.
    Specifically for a soft fermion exchange in the t or u channel, one can simply use:
    
        u-s/t→u-s/tq^2/q^2 + ξ_q^2 m^2_∞,q
    
    where q is the exchanged momentum, ξ_q = e/2 ≃ 1.359, and m^2_∞,q is the medium-induced effective quark mass-squared for quarks, given by
    
        m^2_∞,q    = 2g^2∫d^3 p/2p^0(2π^3)[2 f_g(p) +  (f_q(p)+ f_q̅(p) )]
         
            = g^2 /4( T^2 + μ^2/π^2) = g^2/3( T^2 + μ^2/π^2).
    
    We have allowed ourselves to make this simplification – also because the cumulative effect of these diagrams is rather small and is exponentially suppressed ∝ e^-μ/T in the large μ/T region.
    
    On the other hand, diagrams (A),(B), and (C) represent processes involving t or u-channel gauge boson exchange that require the inclusion of the thermal
    gauge boson self-energy on the internal propagator to cut off the infrared sensitivity of these processes. The matrix element of interest is (s^2+u^2)/t^2 which can be written as:
    
        1/8D_μα^retD_νβ^adv(pγ^μkγ^ν)(p'γ^αk'γ^β)
    
    where D_μα^ret and D_μα^adv, are the retarded and advanced boson propagators, respectively.
    Following the same notation as <cit.>, the plasma frame frequency and momentum
    carried by the gauge boson propagator are denoted as ω and q. The retarded gauge boson propagator can be conveniently expressed in Coulomb gauge as:
    
        D_00^ret(ω,q)=-1/q^2 - ^ret(ω,q)
           D_ij^ret(ω,q)=δ_ij- q̂_iq̂_j/q^2 - ω^2 +^ret(ω,p)
           D_0i^ret(ω,q)=D_i0^ret(ω,q)=0 .
    
    The equilibrium transverse and longitudinal gauge boson self-energies in the HTL approximation are <cit.>
        ^ret(ω,)     =  ^2
        	{ω^2/2q^2 
        	+
        	    ω  ( q^2 -ω^2 )/4 q^3[
        		ln( q+ω/q-ω) - i  π]
        	}  , 
        
        Π^ret(ω,)     =  ^2
        	{
        	    1
        	    -
        	    ω/2q[
        		ln(q+ω/q-ω) - i  π]
        	}  ,
    
    where we have assumed |ω| < q,
    which is the only case of relevance. 
    Here ^2 is the squared Debye mass, which we will give for the case of nonzero μ,T in mDebye.
    The advanced propagators are the complex conjugates of the retarded ones.
    
    The remaining diagrams and the interference terms are all finite and can be computed directly. One should refer to <cit.> for more details on these calculations.
    
    The second type of contribution to leading order comes from 1 ↔ 2 nearly collinear processes. These can be approximated as strictly collinear, which only produces relative changes suppressed by at least one power of g.  Corrections at this order will be treated in the next subsection.
    The leading-order computation proceeds as follows
    <cit.>:
    
        𝒞_a^1↔ 2[f](p⃗) =    (2π)^3/2|p⃗ |^2ν_a∑_bc∫_0^∞ dp' dk δ(|p|-p'-k')γ^a_bc(p;p',k') 
           {f^a(p)[1± f^b(p')][1 ± f^c(k)]-f^b(p') f^c(k)[1 ± f^a(p)]} 
        
            +   (2π)^3/2|p⃗ |^2ν_a∑_bc∫_0^∞ dk dp' δ(|p|+k-p')γ^c_ab(p';p,k) 
           {f^a(p)f^b(k)[1 ± f^c(p')]-f^c(p') [1 ± f^a()][1 ± f^b(k)]},
    
    where γ_bc^a represents the splitting rate, and is given by <cit.>:
    
        γ_bc^a(p;p-k,k)= g^2d_R_bC_R_b/64π^4{[ p^4+k^4+(p-k)^4/p^3k^3(p-k)^3                        g ↔ gg;     p^2+(p-k)^2/p^2k^3(p-k)^2                        q ↔ qg;     p^2+(p-k)^2/p^3k^2(p-k)^2                        g ↔ qq;                               ]}×∫dh^2/(2π)^22 h·F_b,
    
    where h= p×k is a transverse, two-dimensional vector related to the transverse momentum picked up during the splitting process. And F_b can be obtained through the following integral equation:
    
        2=     iδ E(h)F_b(h) + ∫dq_⊥^2/(2π)^2C(q_⊥) {(C_R_b-C_A/2)[F_b(h)-F_b(F-kq_⊥)]
            C_A/2[F_b(h)-F_b(h+pq_⊥)] + C_A/2[F_b(h)-F_b(h-(p-k)q_⊥)]}.
    
    
    For the case of g ↔ q q̅, C_R_b - C_A/2 =  - C_A/2 multiplies the term with F_b(h-pq_⊥)
    rather than F_b(h-kq_⊥). Here C(q_⊥) represents the  leading-order transverse scattering kernel:
    
        C(q_⊥)=g^2T^2/q_⊥^2(q_⊥^2+^2),
    
    and δ E(h) is the energy difference between initial and final collinear particles, expanded to first nontrivial order in thermal masses and transverse momenta:
    
        δ E(h) = h^2/2pk(p-k)+m^2_∞,c/2k+m^2_∞,b/2(p-k) - m^2_∞,a/2p.
    
    We encountered m^2_∞,q already in thermalquarkmass.
    For gluons, m^2_∞,g = ^2/2.
    
    
    
    
     §.§ Next-to-Leading Order corrections
    
    
    We now focus on the next-to-leading order corrections.
    Treating kinetic theory to NLO was pioneered in
    Ref. <cit.> and applied to QCD shear viscosity in Ref. <cit.>.
    In this work, the authors observed that the analytically complex parts of the collision operator do not receive (g) corrections, which instead only modify collinear exchange, incorporated by changing γ^a_bc,
    and soft exchange, which can be modeled as momentum-diffusion and identity-changing processes.
    We briefly review the extended treatment that captures these O(g) corrections, and then we summarize it and present an explicit form of the collision operator at NLO.  
    
    
    
    
      §.§.§ Momentum diffusion
    
    
    The small momentum exchange part of diagrams (A), (B), (C) can be well approximated by momentum diffusion for the particles on each line.
    Beyond leading order, there are additional small-momentum-exchange effects which are only suppressed by (g) relative to LO, which must be incorporated.
    One such effect is the possibility for the gluon in diagrams (A), (B) to carry a soft momentum.
    Other effects involve the interference between multiple soft scatterings, soft gluon emission during scattering, and nonabelian interactions between scattering processes, such as those depicted in Figure <ref>.
    
    
    
    Because the momentum exchanges involved in these processes are small (that is the reason they can occur at (g) and not (g^2)), these terms can be adequately described in terms of Langevin dynamics, that is, in terms of momentum drag and longitudinal and transverse momentum diffusion.
    This is analyzed in the context of heavy quark dyamics in Ref. <cit.> and jet energy loss in Ref. <cit.>, and applied to shear viscosity in Ref. <cit.>.
    As shown there, the effective collision term due to soft scatterings, cut off at a scale μ_⊥, is
     
        (C_diff^ f_1)^a() =
            -1/2∂/∂ p^i q̂^ij_a    f_0^a(p) (1 ± f_0^a(p))
            ∂ f_1^a()/∂ p^j  +   ,
    
    where
    
        q̂^ij_a = q̂_a,Lp̂^i p̂^j
          +  12q̂_a (δ^ij - p̂^i p̂^j )
    
    represents the momentum diffusion parallel and perpendicular to the particle's momentum through q̂_L and q̂ respectively.
    
    Contracting with the departure from equilibrium and integrating over , the contribution to the inner product of Qdef takes the form
    
     (f_1, C^2 → 2_diff f_1)  ≡. (f_1, C^2 → 2_diff f_1)|_ loss
     +
     . (f_1, C^2 → 2_diff f_1)|_ gain  ,
     
     where
     . (f_1, C^2 → 2_ diff f_1) |_ loss
    = 1/2β^3 ∑_aν_a  ∫_  f_0^a(p)(1 ± f_0^a(p))  q̂^ij_a ∂ f_1^a() /∂ p^i∂ f_1^a() /∂ p^j   ,
    
    
    The gain collision term takes the form
    
        =    1/2∂/∂ p^i (
           f_0^a(p) (1 ± f_0^a(p))   ∑_bν_b ∫_^ij_ab (·̨̂)   f_0^b(k) (1 ± f_0^b(k))
           ∂ f_1^b()̨/∂ k^j )
        
        . (f_1, C^2 → 2_diff f_1)|_ gain 
        =      -β^3/2∑_abν_a ν_b ∫_
        f_0^a(p)(1 ± f_0^a(p)) f^b_0(k) (1 ± f_0^b(k))
        
           ×_ab^ij (·̨̂) ∂ f_1^a() /∂ p^i∂ f_1^b()̨/∂ k^j,
    
    where the angular function is
    _ab^ij(·̨̂)
    =  g^4 C_R_a C_R_b/d_A∫d^4Q/(2π)^4   |G_μν^R(Q) v_^μ v_^̨ν|^2     2πδ(v_· Q) 2πδ(v_·̨Q) q^i q^j   ,
    
    which is related to q̂^ij_a through
    q̂^ij_a =  ∑_b ν_b ∫_f̨_0^b(k) (1 ± f_0^b(k))  _ab^ij(·̨̂)   .
    
    Beyond leading order it turns out to be much easier to determine q̂^ij_a than ^ab_ij(·̨̂),
    since light-cone techniques work for the former but not the latter.
    Therefore, in practice we will not be able to compute this gain term.
    Since our best estimates based on leading-order analyses indicate that angular cancellations make this term small, we will instead drop it.
    This is why we generally refer to our results as “almost next-to-leading order.”
    
    Examining the expression for
    . (f_1, C_diff^f_1) |_ loss,
    it becomes evident that it involves a single light-like vector denoted as v_. 
    Using the causality and KMS properties of such light-like correlators <cit.>, these leading-order
    soft contributions to q̂ and q̂_L  can be evaluated
    in closed form <cit.>,
    
        q̂^a|_soft=g^2C_R_a T
        	^2/2πlnμ_⊥/,	    q̂_ L^a|_soft=g^2C_R_a T
        	^2/4πln√(2)μ_⊥/.
    
    In these expressions, gT≪μ_⊥≪ T is a cutoff on the q_⊥≡√(q^2 - ω^2)
    integration, which separates the soft from the hard scale. In the region where ω,q≳ T, and the bare matrix elements can be used to compute the hard contribution to the collision operator C^, there is a cancellation on this cutoff dependence. Since light-like separated points are effectively causally disconnected
    as far as the soft gauge fields are concerned,  q̂ and q̂_L have a simple form.
    Using the explicit form of the angular dependence of
    f_1^a() = β^2 χ_ij() X_ij,
    
    straightforward analysis shows that the loss term  reduces to
    
        . ( χ_ij, C^_lossshort χ_ij) |_ loss =
         	 β^3/2∑_a ν_a∫_ f^a_0(p)
         	 (1± f_0^a(p)) ×
        [(χ^a(p)')^2  q̂^a_ L|_soft
         	+
         	ℓ(ℓ+1)χ^a(p)^2/2p^2q̂^a|_soft]   ,
    
    which is the most useful form for evaluating the transport coefficients
    numerically.
    
    The next-to-leading order value of q̂^a has been computed <cit.>.
    This result can be simply included by adding it to the above results <cit.>:
      
        δq̂^a=g^4C_R_a C_A  T^2 /32π^2(3π^2+10-4ln 2)  .
    
    the shift included in the soft gluon exchange diffusion rate gives
     
        ( χ_ij, C^δq χ_ij) =
         	 β^3/2∑_aν_a δq^a
         	 ∫_   f^a_0(p)
         	[1± f_0^a(p)]
         	 ℓ(ℓ+1)χ^a(p)^2/2p^2 .
    
    
    As discussed in Ref. <cit.>, the precise procedure described in the last subsection for including HTL effects in the leading-order calculation also introduces some NLO effects in that calculation.
    In particular, the phase space and statistical functions for scattering from soft gluons incorporate some of the (g) effects which we have just described.
    Therefore, the leading-order procedure we use will not in fact reproduce qhatql, but will reproduce that expression plus some (g) additional terms.
    These contributions must then be subtracted from the (g) contribution we include here, in order to avoid overcounting such effects.
    Fortunately, since the effects arise from scattering off of gluons, they are unmodified by the inclusion of quark chemical potentials, and we can take over the analysis of
    Ref. <cit.>.
    The resulting counterterm can be written as
     
        ( χ_ij, C^_(g) finite g χ_ij)    =   d_AC_Ag^4 /32π^5 T∑_a T_R_a∫_0^∞ dp  p^2 f^a_0(p)[1± f_0^a(p)]
              ×{4.2695[(χ^a(p)')^2+(χ^g(0)')^2]
        	  +7.1769ℓ(ℓ+1)/2p^2[(χ^a(p))^2+(pχ^g(0)')^2]
              +
        	  18.0669 δ_ℓ2[χ^g(0)']^2
        	  },
    
    and the soft fermion exchange counterterm is given by
    
        ( χ_ij, C^_(g) finite q χ_ij) =     d_F ^2 g^4
        m_∞/32π^5T^2 9.95268 ∫_0^∞ dk k 
         f^q_0(k)	[1+ f_0^g(k)]
           ×[(χ^q(k)-χ^g(k))^2+(χ^q(0))^2] .
    
    One should refer to <cit.> for a detailed derivation of these counterterms.
    
    
    
    
      §.§.§ Collinear and semi-collinear contributions 
    
    
    The collinear splitting process, collop21 to deltaEeq, involves the same momentum diffusion effects we just described.
    And as we just described, the strength of momentum diffusion receives (g) corrections.
    Furthermore, deltaEeq shows that collinear splitting is sensitive to thermal effective masses.
    These also turn out to receive (g) corrections.
    Therefore we need to analyze how these two corrections impact the collinear splitting rate.
    To conduct the analysis, we will follow Ref. <cit.> and Fourier transform collop21 from transverse-momentum _⊥ to transverse impact parameter _⊥ space.
    This not only diagonalizes the collision kernel C(k_⊥), but also turns an integral equation into a differential equation.
    Furthermore,  the source on the left-hand side becomes a boundary condition at b=0
    and the desired final integral becomes a boundary
    value of the ODE solution.
    Hence, defining
    
        F() = ∫d^2h/(2π)^2 e^i· hF(h) ,
    
    we have
    
        ∫d^2h/(2π)^2 2  ·F(q)
        = ( 2 ∇_ b·F(0)) ,
    
    and split_int becomes
    
        -2iδ^2()   =   i/2pω(p-ω)(p(p-ω)m_∞ ω^2+
        p ω m_∞ p-ω^2-ω(p-ω)m_∞ p^2-∇_^2)F()
              +
        ('_R(|ω| b)-'_A(|ω| b)/2
        +'_A(| p| b)/2+'_A(| p-ω| b)/2)F(),
    
    with
    
        _R'(|ω| b) ≡∫d^2k_⊥/(2π)^2( 1 - e^i ω·k_⊥) _R(k_⊥) .
    
    As noted above, we need to incorporate the change in the collision operator,
    
        '_R  , LO+NLO(b) = _R'(b) + δ_R'(b),
    
    where the _⊥-space correction of _R is the unintegrated version of δq̂^a detailed in qhatnlo.
    The details appear in Appendix C of Ref. <cit.>.
    We must also incorporate a shift in the thermal masses:
    
        m^2_∞ a, LO+NLO    = m^2_∞ a +δ m^2_∞ a,     δ m^2_∞ q    = -gT/2π,    δ m^2_∞ g    = -gC_AT/2π.
    
    These shifts are incorporated into bspace by solving for their effects perturbatively, by treating F()
    formally as an expansion in powers of δ m_∞, δ;
    F() = F_0()+F_1()+…  to first order. For details on the boundary conditions and the numerical evaluation of these
    equations, one should refer to 
    <cit.>
    
    The above analysis considers (g) corrections to the regime in which the external states are collinear up to (g^2) corrections.
    This allowed for simplifications in the analysis, in particular in expressing deltaEeq in terms of the leading mass and transverse-momentum corrections.
    But this is not always the case.
    Integrating over opening angles, deltaEeq breaks down if the external momenta P,K obey P· K ∼ gT^2,
    and this regime represents an (g) part of the splitting processes considered.
    Similarly, for hard scattering, if the transfer momentum Q becomes nearly lightlike, our treatment also breaks down, and this again represents an (g) effect.
    Both regimes feature small angles, but not as small as in the collinear splitting processes; so we refer to them as semi-collinear.
    The analysis of this kinematic region was carried out in Ref. <cit.>, and we will follow the development there and in Ref. <cit.>.
    
    The good news is that, in treating this regime, the kinematic variable h becomes large, namely h≫ gT^2.
    The collision operator then becomes subdominant, ≪δ E, and one can perturb in it to first order.
    The lowest-order and first nontrivial-order solutions are
    
        F_b1(h)     = 2/iδ E(h)  , 
        F_b2(h)     = i/δ E(h)∫d^2 q_⊥/(2π)^2C̅(q_⊥) {( C_R_b - C_A/2)
               [F_b1(h) - F_b1(h-kq_⊥)] .
               
             . + C_A/2 [F_b1(h) - F_b1(h+pq_⊥)]
               + C_A/2 [ F_b1(h) - F_b1 (q-(p-k)q_⊥)]
               } .
    
    This is the same as treating the emission in the Bethe-Heitler limit.
    However, at large h, q^- = δ E ∼ gT ∼ q_⊥, which changes the kinematics of the scattering. Therefore,
    C̅(q_⊥) must be recomputed.  A more accurate form for
    C̅(q_⊥) in this regime is
    <cit.>
        C̅_NLO(q_⊥,δ E) =
          g^2 T ^2/(q_⊥^2 + δ E^2)(q_⊥^2 + δ E^2 + ^2)
               + 2 g^2 T δ E^2/q_⊥^2(q_⊥^2 + δ E^2) .
    
    Physically, this is a consequence of the
    splitting being induced by an elastic scattering or by the
    absorption of a soft on-shell particle, as observed in <cit.>. Therefore,
    the already-computed LO  contribution, corresponding to the small δ E
    limit, and the already-included LO  contribution <cit.> need to be subtracted as:
    
        δC̅(q_⊥,δ E) =
          g^2 T ^2/(q_⊥^2 + δ E^2)(q_⊥^2 + δ E^2 + ^2)
               - g^2 T ^2/q_⊥^2(q_⊥^2+^2) .
    
    The second term is the LO collinear form for C̅ (the small
    δ E limit of CLONLO).
    
    The semi-collinear contribution is found by substituting CNLO
    into Fseries and using it to evaluate split_int and finally
    collop21, which results in
    
        γ^a_bc|_semi(p;p-k,k)
              = g^2/32π^4{[ d_AC_Ap^4+k^4+(p-k)^4/p^3k^3(p-k)^3                               g↔ gg;       d_F p^2+(p-k)^2/p^2(p-k)^2k^3                              q↔ q g;       d_F (p-k)^2+k^2/(p-k)^2k^2p^3                              g↔ qq̅ ].
        ∫d^2h/(2π)^2∫d^2/(2π)^2δC̅(,δ E)
           ×[(C_R-C_A/2)(h/δ E(h)-
        h-kq_⊥/δ E(h-kq_⊥))^2+C_A/2(h/δ E(h)-
        h+pq_⊥/δ E(h+pq_⊥))^2.
           .+C_A/2(h/δ E(h)-
        h-(p-k)q_⊥/δ E(h-(p-k)q_⊥))^2],
    
    with (C_R-C_A/2) appearing on the h + pq_p term for
    g ↔ qq̅ processes.  This is then inserted into
    collop21, resulting in
    
        (f_1,𝒞^semif_1)
        	   ≡   2π/T^3∑_abc∫_0^∞ dp
                ∫_0^p dk γ^a_bc|_semi (p;p-k,k)
        	f^a_0(p)[1± f^b_0(k)][1± f^c_0(p-k)]
              ×[f_1^a()-f_1^b(k)-
        	f_1^c((p-k))]^2.
    
    
      §.§.§ Summary
    
    
    In summary, the corrections to be added to the strict leading-order contributions
    of the previous section produce a collision operator that is fully
    NLO.  At NLO, the collision term will be given as
    
        ( f_1, C_NLO  f_1 )
            =
            ( f_1, C_LO  f_1 ) +
            ( f_1, δ C   f_1 )  ,
    
    where
    
        ( f_1, δ C   f_1 )
          ≡   ( f_1, C^δq  f_1 )-
          ( f_1, C^_(g) finite  f_1 )		  +
          ( f_1, C^  f_1 )  +
          ( f_1, δ C^  f_1 ).
    
    Here, the first two terms are the contribution from the soft gluon exchange and its counterterm. The last two terms are the contribution from semi-collinear processes and the correction to the splitting rate, respectively. All of these corrections were carefully studied in <cit.>, and what was found is that the main difference between LO and NLO results arises from the soft gluon exchange. 
    
    
    
    § FROM ZERO TO NONZERO CHEMICAL POTENTIAL
    
    
    In this section, we delve into the differences between the description of the QGP at vanishing chemical potential, μ=0, and at high densities, μ>0.
    Temperature, density, and screening enter the calculation, and we must make sure that we include them correctly each place they both arise.
    The structure fits within the general effective kinetic framework of Ref. <cit.>, for which finite μ and T case is a special case.
    
    When introducing a chemical potential we must shift the equilibrium occupancies from their simple thermal values; we have already done this in f0.
    This change propagates into the Boltzmann equation and the definition of the extremization problem, Qdef, in a straightforward way.
    There is no need to change the basis of variational functions, but more terms may be needed because the most important momentum range is likely to be higher, p ∼μ rather than p ∼ T.
    
    The matrix elements depend on the chemical potential indirectly through the strength of screening effects.
    As shown in Ref. <cit.>, the form of the HTLs are identical for high density as for high temperature.
    The two effects simply combine in the Debye screening mass squared,
    
        ^2 = N_c/3 g^2T^2 + /6 g^2(T^2 + 3/π^2μ^2).
    
    The last term represents the increase in the screening due to the quark abundance induced by the chemical potential.
    The strength of quark screening is similarly enhanced; one uses the occupancies with chemical potentials in evaluating thermalquarkmass.
    The resulting expression is already written in that equation.
    
    In evaluating splitting rates, a pivotal role was played by the transverse scattering rate, described by the kernel C̅(_⊥).
    Remarkably, scattkernel still holds at finite chemical potential, provided one uses the adjusted value of the Debye mass squared.
    This is discussed in Ref. <cit.>.
    Essentially, the shifted ^2 value accounts for the change in the number of scattering “targets” in the medium and therefore correctly adjusts the strength of medium scattering which induces the splitting.
    Similarly, the calculation of δ E for splitting, deltaEeq, must use the form of the asymptotic masses which accounts for chemical potentials.
    
    Similarly, at NLO we must correctly account for the strength of scattering in the medium, but generally the calculations presented so far remain true after one does so.
    The light-cone methods which were used to derive qhatql and qhatnlo are based on integrating out the quarks and incorporating their effects in the HTLs or equivalently within the effective theory of EQCD.
    This proceeds the same with μ as without, except that the value of ^2 must be adjusted using mDebye.
    By similar reasoning, mshift and CLONLO apply as written, but using the corrected form of ^2.
    
    Next, let us briefly analyze the formal convergence of perturbation theory.
    At weak coupling ≪ 1 and μ = 0 hot QCD has three scales: the hard scale π T, the soft scale gT, and the ultrasoft scale g^2 T.
    At leading order the scale π T participates as the dominant propagating degrees of freedom, while the gT scale is relevant as the momentum-exchange scale for soft scattering relevant in both  and  processes.
    NLO corrections occur primarily because these gT scale degrees of freedom have occupancies f(q) ∼ 1/g and loop corrections are only suppressed by a single power of g.
    Therefore the structure of NLO (g) corrections relative to LO is often as a ratio g^2 T/.
    Including color and other numerical factors, these corrections often prove to be quite large.
    
    The inclusion of a chemical potential with μ > π T raises the hard scale from π T to μ, and it increases the Debye scale as one sees in mDebye:
    ∼ g μ rather than gT.
    Therefore the separation between the soft and ultrasoft scales is increased, and (g) corrections are expected to scale as g^2 T / gμ instead of g^2 T/gT.
    This underlies the expectation that NLO effects should be milder in the large-μ/T case.
    
    In practice, the largest NLO effect is the shift in the transverse momentum diffusion coefficient, qhatnlo.
    This effect is linear in , while the leading-order part is proportional to ^2.
    The ratio δq̂ / q̂∼ g^4  T^2/g^2 ^2 T is therefore of order g^2 T/, as anticipated above.
    For parametrically large μ/T, one indeed finds an improvement in the NLO convergence by a single factor of T/μ.
    In the next section we will see to what extent this really improves the convergence in practice.
    
    
    
    
    § RESULTS
    
    In the last sections, we have presented the ingredients necessary for the computation of shear viscosity at leading order, discussed the relevant next-to-leading-order corrections, and presented a good basis set for these calculations.
    Using this computational apparatus, we compute the shear viscosity as a function of μ/T at both leading order and (almost) next-to-leading order.
    
    
    
    In figure <ref> we plot the shear-viscosity to entropy density ratio η/s for LO and NLO, and in figure <ref> we
    show the kinematic viscosity, that is, the ratio of the shear viscosity to the enthalpy density, η T /(e + P ).
    [Raw data for all plots are included in the ancillary materials.]
    Note that the chemical potential μ we refer to is that for a quark, and we treat it to be the same for all quark species.
    The baryon number chemical potential is then 3 times larger.
    We have used the EQCD-inspired scale setting by choosing the MS renormalization point to be
    =2.7T, following the treatment from <cit.>, and we run the coupling using the 2-loop beta function and the value = 0.118 at the Z-pole.
    We assume that when  reaches a quark mass, one crosses the quark-mass threshold and the quark becomes active both in the medium and in the running of the coupling constant.
    So for temperatures below 2.7T=m_b we use =4 and for temperatures above 2.7T=m_b we use =5 (and similarly for the charm and top quarks).
    The plots clearly show that the difference between LO and NLO is smaller at large μ than at μ=0, particularly at high temperatures where the coupling is smaller and  is larger.
    
    
    
    
    Since one motivation to study high-density regimes comes from the expectation that perturbation theory should show better convergence than at μ=0, we present a more detailed comparison between LO and NLO results in Fig. <ref>.
    We plot the ratio between LO and NLO calculations as a function of the temperature for each fixed chemical potential.
    We reproduce the previous result <cit.> that the NLO corrections at μ=0 are large, more than a factor of 2, up to 10^3 GeV temperatures.
    Once one goes to higher densities, the NLO effects become less dominant, falling below a factor of 2 by T=500 MeV for μ/T = 6.
    This suggests that the expansion in g is better behaved at large μ/T.
    Nevertheless, NLO effects are rather important at all temperatures and chemical potentials we consider.
    
    
    
    
    
    
    
    
    
    
    § SUMMARY AND DISCUSSION
    
    
    We have extended the investigation of shear viscosity for hot QCD with chemical potentials to both leading and (almost) next-to-leading order.
    We find that the viscosity-to-entropy ratio η/s is larger at finite μ than at zero μ for all temperatures we consider.
    However, the kinematic viscosity η T/(e+P) is more weakly dependent on μ, falling below the μ=0 result except at NLO below T=1 GeV, where the μ/T dependence almost vanishes.
    We argued previously that, at a given temperature or coupling strength, the convergence of perturbation theory should be better in the large μ/T regime than for μ=0.
    Our results show that this is indeed the case, but the improvement is not dramatic.
    NLO effects are a factor of 2 at μ/T=0 for
    temperatures of hundreds of GeV,
    whereas for μ=6T they are a factor of 2 at about 500 MeV temperature.
    Perturbation theory does not converge at experimentally achievable combinations of (μ,T), but it comes closer at high μ than at small μ.
    
    It would be straightforward to apply the techniques developed here to the more physical case in which different quark species experience different chemical potentials.
    This could allow the development of an interpolating estimate for the viscosity based on experimental data, hadron-resonance modeling, and NLO perturbative input, similar to what was recently developed by one of us <cit.>.
    
    
    
    § ACKNOWLEDGEMENTS
    
    The authors acknowledge the support by the State of Hesse within the Research Cluster ELEMENTS (Project ID 500/10.006), and support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the CRC-TR 211 'Strong-interaction matter under extreme conditions'– project number 315477589 – TRR 211. The authors also thank Nicolas Wink, Jacopo Ghiglieri, and Kevin Ingles for instructive conversations.
    
    
    
    
    
    
    § SELF-ENERGIES BEYOND THE HARD THERMAL LOOP APPROXIMATION
    
    As discussed in <cit.>, the t-channel exchange process is subtle, since the use of vacuum matrix elements leads to logarithmic divergences in the “transport” cross-section and in the inverse viscosity.
    Baym et al already identified the correct procedure to fix this problem in the context of viscosity <cit.>; one should amend the matrix element through the inclusion of the thermal self-energy.
    At the leading-log level, relevant in Refs. <cit.>, it is sufficient to identify the energy scale where this occurs.
    At leading order it is necessary to really include the self-energy, at least within an approximation which is adequate in the formal kinematic regime (p,k) ∼ T but (q,ω) ∼ gT ≪ T.
    This is precisely the regime where the HTL approximation should be valid.
    
    At large chemical potentials we can try to do better.
    In this regime, the self-energy is dominated by fermionic loops.
    For these, the self-energy is gauge invariant and the full expression, not just the hard-loop limit, is known <cit.>.
    Specifically, the medium-induced part of the fermion-loop contribution to the gluonic self-energy is
    <cit.>
        = g^2  T^2/3H(ω,q),  = g^2  T^2/3(-1/2H(ω,q) + q^2/2(q^2-ω^2)G(ω,q))
    
    where H(ω,q) and G(ω,q) are, writing 2ω_+=ω+q and 2ω_-=ω-q:
    
        G(ω,q)   = 3/π^2T^2∫_0^∞ dk f(k)(4k+q^2-ω^2/2qln[(k+ω_-)(k-ω_+)/(k-ω_-)(k+ω_+)]) 
         H(ω,q)   = 3/π^2T^2∫_0^∞ dk f(k)(2k-4k^2+ω^2-q^2/4qln[(k+ω_-)(k-ω_+)/(k-ω_-)(k+ω_+)] 
           -2kω/qlnω_+/ω_--kω/qln[(k+ω_-)(k-ω_+)/(k-ω_-)(k+ω_+)]) 
         G(ω,q)   = 3/π^2T^2∫_0^∞ dk f(k)q^2-ω^2/2q[Θ(k+ω_+)Θ(-k-ω_-)-Θ(-k+ω_+)Θ(k-ω_-)] 
         H(ω,q)   = 3/π^2T^2∫_0^∞ dk f(k)[-(2k+ω)^2-q^2/4qΘ(k+ω_+)Θ(-k-ω_-)
           +(2k-ω)^2-q^2/4qΘ(k-ω_-)Θ(-k+ω_+)+2kω/qΘ(ω_+)Θ(-ω_-)]
    
    The references only present the real parts, but the imaginary parts are easily obtained by using the retarded prescription ω→ω+i0 and carefully interpreting the logarithms.
    The hard thermal loop approximation corresponds to expanding the k integrals in the approximation k ≫ (q,ω), in which case the functions simplify:
    
        H_HTL(ω/q) = 1 - ω/2qlnω_+/ω_-+iπω/2qΘ(ω_+)Θ(-ω_-)
           G_HTL(ω/q) = 1 .
    
    
    One concern regarding the AMY leading-order calculation <cit.> has always been the use of hard thermal loop self-energies, rather than full thermal self-energies.
    At high chemical potential, where the hard loop is dominated by fermions, the above self-energies accurately represent the correct loop screening effect at any energy and momentum, without the need for an HTL approximation.
    Therefore we can find out how well the HTL approximation works by comparing how the leading-order shear viscosity changes when we “improve” the screening effects by replacing the HTL approximation, GH-HTL, with the full self-energy, GH.
    
    We make this comparison in figure <ref>.
    The figure considers a single collision process, diagram (C), and computes the collision integral (χ_ij,χ_ij) for the single-parameter Ansatz.
    We then take the ratio of this collision rate for the case of the full self-energy over the collision rate using the HTL self-energy.
    The Debye screening strength is chosen to vary with μ as it would if the coupling strength g^2 is held fixed.
    
    The figure shows, first and foremost, that the effect of replacing the HTL self-energy with the full self-energy is not that large.
    Also, the HTL approximation actually becomes more reliable as the chemical potential is increased.
    This indicates that the use of the HTL approximation for the self-energies is justified and represents a smaller effect than the NLO (and, presumably, unknown NNLO) corrections.
    
    
    JHEP2
    
    § TESTS AND CHECKS 
    
    Since previous results were computed using different numerical integration techniques, it is convenient to compare our results with AMY leading order results. This is done in figure <ref>. The plot shows that there are small differences due to different integration methods and base choice, but they are compatible, especially at small values of , where one expects that the consequences of these effects should have a small role on the transport coefficient calculation.
    
    We also verified that a basis of size N=4 is enough to give accurate results, especially at high chemical potential, where the most important contribution comes from the diagram (C). This is shown in figure <ref>, where we compare the results given by a basis of size N=1 and N=4 to a basis of size N=6, which was considered to give accurate results for LL calculations.


 Paper Title: Axion-like Particle Effects on Photon Polarization in High-Energy Astrophysics.





§ INTRODUCTION


The Standard Model (SM) of particle physics provides a satisfactory theoretical baseline able to explain all known processes acting on elementary
particles. However, the SM cannot be considered as the ultimate theory of fundamental interactions since it does not really unify the electroweak and the strong interaction and does not contain gravity. Moreover, the SM describes neither dark matter nor dark energy, which together make up the majority of our Universe. Therefore, the SM represents a low-energy manifestation of a more fundamental theory describing and unifying all four fundamental interactions at the quantum level.

Several proposals have been presented with the aim of extending and completing the SM, such as
multidimensional Kaluza–Klein theories <cit.>, four-dimensional unsymmetrical models <cit.>, and superstring and superbrane theories <cit.>. A remarkable fact is that all these theories invariably predict the existence of axion-like particles (ALPs, for a review, see, e.g., <cit.>).

ALPs represent a generalization of the axion, the pseudo-Goldstone boson originating from the breakdown of the global Peccei–Quinn symmetry U(1)_ PQ, introduced to solve the strong CP problem (see, e.g., <cit.>). The axion is characterized by a strict relationship between its mass and its coupling to photons, and it interacts also with gluons and fermions so that the Peccei–Quinn mechanism can work. Instead, two aspects differentiate ALPs from axions: (i) the ALP mass m_a is unrelated to the two-photon-to-ALP coupling g_aγγ, and (ii) ALPs primarily interact with photons only, while other possible interactions can safely be neglected in the present context. ALPs are currently considered among the best candidates for being the long-sought dark matter <cit.>. Furthermore, ALPs produce two significant effects in the presence of external magnetic fields: (i) photon–ALP oscillations <cit.>, which are similar to the oscillations of massive neutrinos with different flavors with the difference that, in the ALP case, the external magnetic field is necessary for spin compensation between spin-0 ALPs and spin-1 photons, and (ii) the change of the polarization state of . Due to their faint interaction with photons, ALPs can hardly be detected over the limited distances of laboratory experiments. Instead, the astrophysical context allows us to overcome this restriction with photons that can originate at cosmological distances and can cross several magnetized media, where photon–ALP interaction can take place. Therefore, the astrophysical environment appears as the most promising way to study the ALP physics, especially in the high-energy (HE) and in the very-high-energy (VHE) band (see, e.g., <cit.> for a review). 

In particular, photon–ALP oscillations occur in different magnetized media, such as inside active galactic nuclei (AGN; see, e.g., <cit.> and Section <ref>). Specifically, photon–ALP oscillations explain the detection of photons with energies of up to ∼400   GeV originated from flat spectrum radio quasars (FSRQs, an AGN class), while conventional physics prevents any emission for energies above ∼20   GeV (see also Section <ref>): this fact represents the first hint at ALP existence <cit.>. Moreover, photon–ALP interaction solves the anomalous redshift dependence of the spectra of BL Lacs (another AGN class) <cit.>. In particular, within conventional physics, the values assumed by the BL Lac VHE-emitted spectral indexes decrease as the redshift grows, while no redshift dependence is expected, since cosmological effects are not present and selection biases are excluded, as discussed in <cit.>. The introduction of photon–ALP conversion inside the extragalactic space increases the Universe transparency <cit.>, partially avoiding the absorption of VHE photons due to their interaction with the extragalactic background light (EBL, see also Section <ref> ), and makes BL Lac VHE-emitted spectral indexes redshift independent. Since this is the only possibility according to the physical expectation, this result represents the second hint 
 at ALP existence <cit.>. Moreover, photon–ALP conversion inside the blazar jet, in the host galaxy, in the extragalactic space and in the Milky Way produces two features <cit.>: (i) spectral irregularities and (ii) photon excess in blazar spectra at VHE, which can be detected by current and planned observatories, such as ASTRI <cit.>, CTA <cit.>, GAMMA-400 <cit.>, HAWC <cit.>, HERD <cit.>, LHAASO <cit.>, and TAIGA-HiSCORE <cit.>. Furthermore, ALP-induced spectral irregularities produced inside the galaxy cluster turbulent magnetic field have been used to place constraints on the ALP parameter space <cit.>. In addition, photon–ALP interaction provokes modifications on stellar evolution <cit.>; it has been employed inside galaxy clusters as an explanation of the spectral distortions of the continuum thermal emission (T∼2  keV–8   keV) <cit.> and of the unexpected spectral line at 3.55   keV, viewed as dark matter decay into ALPs with their subsequent oscillations into photons <cit.>. ALPs have also been used in order to explain a blazar line-like feature <cit.>. The recent detection of the gamma-ray burst GRB 221009A by the LHAASO Collaboration <cit.> up to 18   TeV strongly challenges conventional physics, while the photon–ALP scenario naturally explains the VHE observation of this GRB, which represents the third hint and the firmest indication to date at ALP existence <cit.> (see also Section <ref> and <cit.>). Moreover, all three above-mentioned hints are derived with the same ALP parameters, which further strengthens the indication itself.

In this review, we concentrate on the other main effect of the photon–ALP interaction, i.e., the change of the polarization state of photons. This ALP effect has attracted less interest so far with respect to that producing spectral modifications. However, the existing and proposed polarimetric missions in both the X-ray band such as IXPE <cit.> (already operative), eXTP <cit.>, XL-Calibur <cit.>, NGXP <cit.>, XPP <cit.> and in the HE range like COSI <cit.> (approved to launch), e-ASTROGAM <cit.>, AMEGO <cit.> allow polarization studies to be carried out with unprecedented sensitivity. Therefore, these observatories represent a unique opportunity to perform additional studies on ALP physics. In particular, the ALP effects on the polarization of photons produced in GRBs have been studied in <cit.>. Attention to various ALP-induced polarization topics regarding different astrophysical sources, such as AGN, white dwarfs, and neutron stars, has been paid in <cit.>. Moreover, the photon–ALP interaction has been demonstrated as a unique way to measure the emitted photon polarization <cit.>. Quite recently, the impact of the photon–ALP interaction on the polarization of photons produced in the central region of typical galaxy clusters or generated at the jet base of generic blazars has been analyzed in <cit.>. Results obtained in <cit.> have been applied to concrete cases: (i) to Perseus and Coma concerning galaxy clusters, as shown in <cit.>, and (ii) to OJ 287, BL Lacertae, Markarian 501, and 1ES 0229+200 regarding blazars, as discussed in <cit.>, showing that, in all the cases, the initial photon polarization is strongly modified by the photon–ALP interaction with the production of features observable by present and planned missions <cit.>. The above-cited studies concerning blazars <cit.> and especially galaxy clusters <cit.> demonstrate that the measure of the polarization degree of photons produced by these astrophysical sources may provide us a new and an additional way to study ALPs and their effects. In fact, whenever the expectations of conventional physics about the photon polarization degree of astrophysical sources are in tension with possible future observations performed by missions in the X-ray band <cit.> and in the HE range <cit.>, this fact would represent a hint at ALP existence. This is especially true regarding galaxy clusters. As discussed below, photons diffusely produced in the central zone of galaxy clusters are expected to be unpolarized both in the X-ray and in the HE band within conventional physics, but the photon–ALP conversion makes them partially polarized, as shown in <cit.>. Therefore, a signal of partially polarized photons diffusely produced in a galaxy cluster would be a strong indication at ALP existence. Concerning blazars, we could achieve a similar conclusion if the detected polarization degree were extremely high, above the predictions of standard hadronic models (see below and <cit.>). In case no tension was found, the ALP parameter space (m_a, g_aγγ) could still be limited further. In this review, our aim is thus to describe the ALP-induced effects on photon polarization, which may be detected by missions in the X-ray band <cit.> and in the HE range <cit.> and which may provide us new ways to make progress in the ALP physics.

In particular, we introduce ALPs and the photon–ALP system in Section <ref>, and we illustrate polarization in Section <ref>. Section <ref> describes the astrophysical media crossed by the photon–ALP beam. In Section <ref>, the ALP-induced polarization effects are reported both in the X-ray and in the HE range for Perseus as an example for galaxy clusters and for OJ 287 as an example for blazars. In Section <ref>, the above-mentioned polarization effects are discussed along with their real detectability. Section <ref> is also devoted to describing possible future perspectives in the VHE range, where strong ALP signatures could be detected, along with a discussion about GRBs. Finally, we draw our conclusions in . Throughout this review, we employ the natural Lorentz–Heaviside (rationalized) units with ℏ = c = k_B = 1.



§ AXION-LIKE PARTICLES


ALPs are neutral, spin-zero pseudo-scalar bosons, whose primary interaction with photons (other interactions with fermions or gluons may exist but are subdominant and negligible in the present context) is described by the Lagrangian 

    ℒ_ ALP =  1/2 ∂^μ a  ∂_μ a - 1/2  m_a^2   a^2 -  1/4  g_aγγ  F_μνF̃^μν a  = 1/2 ∂^μ a  ∂_μ a - 1/2  m_a^2   a^2 + g_aγγ  E· B a,

where a denotes the ALP field, while E and B respectively represent the electric and magnetic components of the electromagnetic tensor F_μν, whose dual is F̃^μν.

Many bounds about the photon–ALP coupling g_a γγ and the ALP mass m_a have been derived in the literature <cit.>. The most reliable ALP limit is represented by g_a γγ < 0.66 × 10^- 10  GeV^- 1 for m_a < 0.02   eV at the 2 σ level from no detection of ALPs from the Sun <cit.>. Other bounds are more or less affected by the description of the astrophysical environment <cit.>.

Furthermore, the QED one-loop vacuum polarization effects, which become important in the presence of a strong external magnetic field, are accounted by the Heisenberg–Euler–Weisskopf (HEW) effective Lagrangian <cit.>

    ℒ_ HEW = 2 α^2/45 m_e^4 [ ( E^2 -  B^2 )^2 + 7 ( E· B)^2 ],

where α represents the fine-structure constant, while m_e is the electron mass.

We consider a photon–ALP beam of energy E propagating in the y-direction in a magnetized medium with an external magnetic field denoted by B and entering ), where E describes the photon electric field. Because of the off-diagonality of the mass matrix of the γ - a system, the propagation eigenstates differ from the interaction eigenstates, producing γ↔ a oscillations. This is similar to the oscillations of different flavor massive neutrinos, but in the case of the photon–ALP system, the external B field is essential to compensate for the spin mismatch between photons and ALPs. In particular, the functional expression of the photon–ALP coupling in Equation (<ref>) shows that a couples only with the component B_T of B transverse to the photon momentum k <cit.>. Furthermore, the short-wavelength approximation <cit.> holds in the present analysis since E ≫ m_a (we consider extremely light ALPs), so that the propagation of an unpolarized photon–ALP beam arising from ℒ_ ALP of Equation (<ref>) is described by the Von Neumann-like equation

    i d ρ (y)/d y = ρ (y)  ℳ^† ( E, y) - ℳ ( E, y)  ρ (y),

where ℳ (E,y) represents the photon–ALP mixing matrix, while ρ (y) is the polarization density matrix of the photon–ALP system. By defining ϕ, the angle that B_T forms with the z axis, ℳ reads as follows:

    ℳ (E,y) ≡(
    [        Δ_xx (E,y)        Δ_xz (E,y)  Δ_a γ(y)   sin ϕ;        Δ_zx (E,y)        Δ_zz (E,y)  Δ_a γ(y)   cos ϕ;  Δ_a γ(y)   sin ϕ Δ_ a γ(y)   cos ϕ         Δ_a a (E);                   ]),

with

    Δ_xx (E,y) ≡Δ_ (E,y)   cos^2  ϕ + Δ_∥ (E,y)   sin^2  ϕ,


    Δ_xz (E,y) = Δ_zx (E,y) ≡(Δ_∥ (E,y) - Δ_ (E,y) )  sin ϕ  cos ϕ,


    Δ_zz (E,y) ≡Δ_ (E,y)   sin^2  ϕ + Δ_∥ (E,y)   cos^2  ϕ,


    Δ_a γ(y) = 1/2g_aγγB_T(y),


    Δ_aa (E) = - m_a^2/2 E,

and

    Δ_ (E,y) = i/2  λ_γ (E,y) - ω^2_ pl(y)/2 E   + 2 α/45 π(B_T(y)/B_ cr)^2 E + ρ_ CMBE,


    Δ_∥ (E,y) = i/2  λ_γ (E,y) - ω^2_ pl(y)/2 E  + 7 α/90 π(B_T(y)/B_ cr)^2 E + ρ_ CMBE ,

[-15]where B_ cr≃ 4.41 × 10^13  G is the critical magnetic field and ρ_ CMB=0.522 × 10^-42.  describes the photon–ALP interaction, while Equation (<ref>) accounts for the ALP mass effect. The first term in Equations (<ref>) and (<ref>) describes photon absorption (e.g., due to the EBL <cit.>) with λ_γ being the γγ→ e^+e^- mean free path. In the second term of Equations (<ref>) and (<ref>), ω_ pl=(4 πα n_e / m_e)^1/2 represents the plasma frequency with n_e denoting the electron
number density. The third term in Equations (<ref>) and (<ref>) describes the QED one-loop vacuum polarization coming from ℒ_ HEW of Equation (<ref>), producing polarization variation and birefringence 
on the beam, and the fourth term expresses the photon dispersion on the CMB <cit.>. 

The solution of Equation (<ref>) can be expressed in terms of the transfer matrix 𝒰(E;y,y_0) of the photon–ALP system as

    ρ ( y ) = 𝒰(E; y, y_0 )  ρ_0  𝒰^†(E; y, y_0 ),

where ρ_0 is the density matrix at position y_0. The probability to find the photon–ALP beam in the final state ρ at position y starting in
the initial state ρ_0 at position y_0 reads as follows:

    P_ρ_0 →ρ (E,y) =  Tr[ρ 𝒰 (E; y, y_0)  ρ_0  𝒰^† (E; y, y_0) ],

with Tr ρ_0 =  Tr ρ =1 <cit.>.

We concentrate now on a photon–ALP system with the following characteristics: (i) fully polarized photons, (ii) no photon absorption (λ_γ→∞), (iii) homogeneous medium, and (iv) constant B field. Therefore, we can choose the z axis along the direction of B_T, resulting in ϕ=0 in Equation (<ref>). With these assumptions, the γ→ a conversion probability reads as follows:

    P_γ→ a (E, y) = (g_aγγB_T   l_ osc (E)/2π)^2  sin^2 (π (y-y_0)/l_ osc (E)),

with

    l_ osc (E) ≡2 π/[(Δ_zz (E) - Δ_aa (E) )^2 + 4  Δ_aγ^2 ]^1/2

being the photon–ALP beam oscillation length. The different energy dependence of the several Δ terms in Equation (<ref>) produces various regimes, which can be experienced by the photon–ALP beam. Hence, we can introduce the low-energy threshold E_L and the high-energy threshold E_H reading as follows:

    E_L ≡|m_a^2 - ω^2_ pl|/2 g_a γγ  B_T,

and 

    E_H ≡ g_a γγ  B_T [7 α/90 π(B_T/B_ cr)^2 + ρ_ CMB]^- 1,
 
respectively. In the energy interval E_L ≲ E ≲ E_H, the photon–ALP system lies in the strong-mixing regime, where the plasma, the ALP mass, the QED one-loop, and the photon dispersion on the CMB effects are all negligible with respect to the photon–ALP mixing term. In this case, P_γ→ a is maximal and energy independent and reduces to

    P_γ→ a (y) =  sin^2 ( g_aγγ B_T/2  (y-y_0) ).

Instead, both in the case E ≲ E_L, where the plasma contribution and/or the ALP mass term are prominent, and in the case E ≳ E_H, where the QED one-loop effect and/or the photon dispersion on the CMB dominate, the photon–ALP beam propagates in the weak-mixing regime, where P_γ→ a becomes energy dependent and progressively vanishes.

The usual situation of a generic photon initial polarization, structured magnetic fields, and non-homogeneous media represents a generalization of the above-considered case, which is reported here for the simplicity of the analytical expressions of the involved equations. However, in all our following calculations, we employ the appropriate photon polarization and the proper expressions of the magnetization, dispersion, and absorption properties of the crossed media.




§ POLARIZATION EFFECTS


The generalized polarization density matrix ρ associated with the photon–ALP beam can be expressed as 

    ρ (y) = ([ A_x (y); A_z (y);   a (y) ])
    ⊗([ A_x (y)   A_z (y)  a (y) ])^*,

where A_x (y) and A_z (y) represent the photon linear polarization amplitudes along the x and z axis, respectively, and a(y) is the ALP amplitude. Equation (<ref>) allows us to describe totally polarized (pure states), unpolarized, and partially polarized beams at once. In particular, pure photon states polarized in the x and z direction are expressed as

    ρ_x = (
    [ 1 0 0; 0 0 0; 0 0 0;   ]),         ρ_z = (
    [ 0 0 0; 0 1 0; 0 0 0;   ]),

respectively, and the ALP state as

    ρ_a = (
    [ 0 0 0; 0 0 0; 0 0 1;   ]),

whereas an unpolarized photon state is described by

    ρ_ unpol = 1/2(
    [ 1 0 0; 0 1 0; 0 0 0;   ]).

Partially 
 polarized photons are characterized by a polarization density matrix with an intermediate functional expression between Equations (<ref>) and (<ref>). 

The photonic part of the polarization density matrix of Equation (<ref>) can be expressed in terms of the Stokes parameters as <cit.>

    ρ_γ = 1/2(
    [  I+Q U-iV; U+iV  I-Q;      ]),

while the photon degree of linear polarization Π_L and the polarization angle χ are defined <cit.> as

    Π_L ≡(Q^2+U^2)^1/2/I = [ (ρ_11-ρ_22)^2+(ρ_12+ρ_21)^2]^1/2/ρ_11+ρ_22,

and

    χ≡1/2 arctan(U/Q)=1/2 arctan(ρ_12+ρ_21/ρ_11-ρ_22),

respectively, where ρ_ij with i,j=1,2 are the photon polarization density matrix elements. In the following Sections, we will show the behavior of the final Π_L, which turns out to be strongly modified by the photon–ALP interaction with respect to the initial photon degree of the linear polarization Π_L,0.

As shown by some theorems enunciated and demonstrated in <cit.>, in the absence of photon absorption, there exists a relationship between photon survival probability in the presence of photon–ALP interaction P_γ→γ and initial Π_L,0 (P_γ→ a=1-P_γ→γ due to the absence of photon absorption), and in particular, the following hold: 




  * In case of no photon absorption and a starting condition of only photons with an initial degree of linear polarization Π_L,0, the inequalities P_γ→ a≤ (1+Π_L,0)/2 and P_γ→γ≥ (1-Π_L,0)/2 hold. In case of initially unpolarized photons (Π_L,0=0), we observe P_γ→ a≤ 1/2 and P_γ→γ≥ 1/2.


  * In the previous conditions, Π_L,0 represents the measure of the overlap between the values assumed by P_γ→ a and P_γ→γ. In case of initially unpolarized photons (Π_L,0=0), P_γ→ a and P_γ→γ possess the common value of 1/2, at most.



We have employed the inequalities described in item 1 to verify the correctness of the behavior of P_γ→γ reported in several figures of this review (see the following sections).

As shown in <cit.>, the results presented in item 2 can be used to measure the initial Π_L,0 when only spectral data of an astrophysical source are known. The strategy works in the energy band in which the photon–ALP system lies in the weak mixing regime, where P_γ→ a and P_γ→γ are energy dependent and their assumed values satisfy the inequalities described in item 1. 

The procedure can be sketched as follows. We consider systems in an energy range where photon absorption is negligible. Starting from observational data of an astrophysical source with the observed spectrum Φ_ obs, we can infer P_γ→γ as 

    P_γ→γ=Φ_ obs/Φ_ em,

where Φ_ em represents the emitted spectrum, which can be either known or derivable from Φ_ obs (see <cit.> for more details). Furthermore, the absence of photon absorption by hypothesis ensures that P_γ→ a = 1 - P_γ→γ. Now, Π_L,0 can be obtained by measuring the interval of overlap between the values assumed by P_γ→γ and P_γ→ a.

In Figure <ref>, we show how the previous procedure works: in the left panels, an application in the optical-X-ray band is reported, while the right panels address a similar study but in the MeV range. We show how an observer would proceed to infer the initial Π_L,0 from spectral data. Note that cosmological effects are taken into account and E in Figure <ref> represents the photon energy, as observed from the Earth.



In the optical-X-ray band, we consider a high-frequency peaked blazar (HBL) placed inside a poor galaxy cluster at redshift z = 0.1 with an initial Π_L,0=0.3, while in the MeV range, we deal with an HBL but now placed inside a quite rich galaxy cluster at redshift z = 0.05 with Π_L,0=0.1. We assume the phenomenological emitted spectra derived in <cit.> and typical values concerning the magnetization properties of the astrophysical media crossed by the photon–ALP beam both in the optical-X-ray and in the MeV energy band.

In particular, in the optical-X-ray band, the photon–ALP interaction turns out to be negligible inside the poor galaxy cluster (see also Section <ref>), where the blazar is supposed to be located and inside the hosting elliptical galaxy, whose typical properties are described in Section <ref>. Instead, we have efficient photon–ALP conversion inside the jet of the blazar, whose characteristics are described in Section <ref>: For the magnetic field in the jet B^ jet, we assume a toroidal profile of Equation (<ref>) with magnetic field strength at the emission distance y_ em = 3 × 10^16  cm equal to B_0^ jet≡ B^ jet(y_ em)= 0.5   G, and for the electron number density n_e^ jet, we consider the profile expressed by Equation (<ref>), taking n_e,0^ jet≡ n_e^ jet(y_ em)=5 × 10^4   cm^-3. We hypothesize an efficient photon–ALP interaction inside the extragalactic space by assuming an extragalactic magnetic field
strength B_ ext = 1   nG and the B_ ext coherence properties described in Section <ref>. Lastly, the photon–ALP interaction turns out to be efficient also inside the Milky Way, where we adopt the Jansson and Farrar model <cit.> concerning the magnetic field B_ MW and that presented in <cit.> regarding the electron number density in the Galaxy n_ MW,   e (see also Section <ref>). For more details, we send the reader to <cit.>.



Instead, in the MeV range, we have an efficient photon–ALP conversion in the cluster since the blazar is supposed to be located inside a rich galaxy cluster, whose properties are reported in Section <ref>: we assume the cluster magnetic field B^ clu profile of Equation (<ref>), where n_e^ clu represents the cluster electron number density described by means of the β model expressed by Equation (<ref>). Concerning the parameters entering Equations (<ref>) and (<ref>), we assume k_L = 0.1   kpc^-1, k_H = 3   kpc^-1, q = -11/3, B^ clu_0 = 20   G, n_e,0^ clu = 0.1   cm^-3, η_ clu = 0.75, β_ clu = 2/3, a cluster core radius r_ core = 150   kpc, and a cluster radius of 1   Mpc (see Sections <ref> and <ref> for the definition of the parameters). In the present situation, we hypothesize a negligible photon–ALP conversion inside the extragalactic space by assuming B_ ext < 10^-15  G (see also Section <ref>). Concerning the photon–ALP beam propagation inside the blazar jet, host galaxy, and Milky Way, we consider the same models and parameter values assumed in the previous case (optical-X-ray band) apart from the jet magnetic field strength at the emission distance, which is now B_0^ jet≡ B^ jet(y_ em)= 0.1   G (see also Sections <ref>, <ref>, and <ref>). We send the reader to <cit.> for more details.



In both of the previous cases (optical-X-ray and MeV band), a typical realization of P_γ→γ is evaluated and reported in the top panels of Figure <ref>: we assume g_aγγ = 0.5 × 10^-11  GeV^-1 for both of the situations, but we take m_a = 5 × 10^-14  eV for the first blazar in the optical-X-ray band, while we deal with an alternative scenario by taking m_a = 2 × 10^-10  eV for the second blazar in the MeV range. The central panels of Figure <ref> show the observed binned spectra Φ_ obs (with a typical energy resolution <cit.>), simulated by multiplying the emitted spectra by P_γ→γ in the presence of photon–ALP interaction. In the bottom panels of Figure <ref>, we show P_γ→γ and P_γ→ a with P_γ→γ inferred from the simulated observed spectra Φ_ obs, reported in the central panels, by means of Equation (<ref>), where Φ_ em can be reconstructed by fitting the upper spectral bins in the central panels, since photon–ALP interaction produces an energy-dependent dimming in the observed spectra (see also <cit.>). We recall that P_γ→ a = 1 - P_γ→γ due to the absence of photon absorption. Finally, by employing the results presented in item 2, we can measure the initial Π_L,0, and we obtain Π_L,0 = 0.288 ± 0.016 in the X-ray band and Π_L,0 = 0.090 ± 0.018 in the MeV range. We observe that the inferred values are very close to the initially assumed ones, which demonstrates the power of the present method.

We want to stress that the above-described procedure is the only established possibility to measure the initial polarization of photons produced by astrophysical sources. Furthermore, this procedure can be employed in a wide energy range, where photon absorption is negligible, and only needs spectral observations of the source, so that flux-measuring observatories also become polarimeters.




§ PHOTON–ALP BEAM PROPAGATION IN DIFFERENT MEDIA


In the present section, we describe the main properties of the media crossed by the photon–ALP beam starting where photons are produced up to their arrival at the Earth: blazar, host galaxy, galaxy cluster, extragalactic space, and Milky Way. We concentrate mainly on the characteristics that are crucial for the photon–ALP system, such as the magnetization and absorption properties of the media, while we send the reader to the quoted papers for an exhaustive discussion.



 §.§ Blazars


Blazars are powerful astrophysical sources emitting in the whole electromagnetic spectrum, from radio waves up to the VHE band. They are a particular class of AGN that are basically extragalactic supermassive black holes (SMBHs) accreting matter from the surroundings. In these systems, two collimated relativistic jets can develop in opposite directions. In the accidental case where one of these jets turns out to be  in the Earth direction, the AGN is called blazar, while in other situations, the AGN shows a different phenomenology, as displayed in Figure <ref>, where the AGN structure is also shown. 



Blazars are divided into two sub-classes: flat spectrum radio quasars (FSRQs) and BL Lac objects (BL Lacs). FSRQs are powerful sources, where accretion from the disk surrounding the central SMBH is efficient. Furthermore, they show intense broad emission lines generated by clouds of gas in the so-called
broad-line region (BLR) encompassing the central SMBH and photoionized by the
ultraviolet photons produced by the disk. Moreover, FSRQs display absorption regions for VHE photons, which can interact with the optical/ultraviolet radiation emitted by the above-mentioned BLR and with the infrared radiation produced by the dusty torus (see Figure <ref>), preventing the observation of photons with energy above ∼20   GeV <cit.>. However, photons of up to ∼400   GeV have been observed from FSRQs <cit.>. Only ad hoc solutions have been proposed inside conventional physics, while photon–ALP oscillations provide a natural solution of the problem by partially preventing BLR absorption: this fact represents a hint at ALP existence <cit.>. Instead, BL Lacs are less powerful, and they show neither absorption regions nor significant emission lines due to the presence of a radiatively inefficient
accretion flow (RIAF) onto the central SMBH <cit.>.

In the following, we will concentrate on BL Lacs, where the emission region of photons is located at a distance of about y_ em= (10^16– 10^17)   cm from the central SMBH. Once produced at y_ em, photons propagate inside the BL Lac jet up to a distance y_ jet≃ 1   kpc, where they enter the host galaxy. During their path inside the jet, these photons can oscillate into ALPs inside the jet magnetic field B^ jet. The toroidal part of B^ jet, which is transverse to the jet axis <cit.>, turns out to be the dominant one at the distances under consideration, and its profile reads as follows:

    B^ jet ( y ) = B^ jet_0 (y_ em/y),

where B^ jet_0 denotes the magnetic field strength of the jet at y_ em. Another crucial information in order to evaluate photon–ALP conversion inside BL Lacs is the electron number density profile in the jet n_e^ jet, which, due to the conical shape of
the jet, is expected to be 

    n^ jet_e ( y ) = n^ jet_e,0(y_ em/y)^2,

where n^ jet_e,0 represents the electron number density at y_ em. Synchrotron Self Compton (SSC) diagnostics applied to blazar spectra suggests assuming n^ jet_e,0=5 × 10^4   cm^-3 <cit.>.

Two competing photon emission mechanisms have been proposed in order to explain blazar phenomenology: (i) the leptonic model (see, e.g., <cit.>) and (ii) the hadronic model (see, e.g., <cit.>). Both scenarios describe photon production in the optical up to the X-ray band as generated by electron synchrotron emission. At higher energies, leptonic models explain photon emission with inverse Compton scattering of synchrotron radiation or of external photons from the disk and/or clouds with the synchrotron producing relativistic electrons <cit.>. Instead, hadronic models attribute the high-energy photon generation to proton–synchrotron emission or photomeson production <cit.>. Hadronic models require higher values of B^ jet_0 and y_ em with respect to leptonic scenarios. Moreover, hadronic models predict a higher initial degree of linear polarization Π_L,0 with respect to leptonic scenarios, as shown in Figure <ref> for two particular BL Lacs.



The propagation of the photon–ALP beam is computed in the comoving frame of the jet so that a transformation E →γ E, with γ being the Lorentz factor, is required to pass to the fixed frames of the subsequently encountered regions. 

The values assumed by B^ jet_0, y_ em, γ and Π_L,0 affect the calculation of the photon–ALP beam propagation in the jet and will be discussed in Section <ref> for both leptonic and hadronic models. Then, the photon–ALP beam transfer matrix in the jet 𝒰_ jet can be evaluated by following the calculation reported in <cit.>.




 §.§ Host Galaxy


The effectiveness of the photon–ALP conversion inside the host galaxy depends on its nature and characteristics. Ideal galaxies in order to have an efficient photon–ALP conversion are spirals and even more starbursts. The reason lies in the high magnetic field strength inside the host B^ host, which can reach values of B^ host=𝒪(5-10)  μ G for spirals <cit.> and B^ host=𝒪(20-50)  μ G for starbursts <cit.>, and in the large coherence length L_ dom^ host of B^ host, which is around L_ dom^ host =𝒪(1-4)   kpc for both spiral and starburst galaxies outside the central region <cit.>. The magnetic field coherence can be modeled by employing a Kolmogorov-type turbulence power spectrum in a similar way as discussed in the following subsection concerning galaxy clusters (for more details about the parameter values, see also <cit.>). A possible example of photon–ALP conversion inside spirals or starbursts is represented by photons produced by GRBs, which are usually located in the central region of these galaxies <cit.>. 

In particular, the location of GRB 221009A—the brightest GRB to date—in the central region of a disk-like galaxy <cit.> allows for an efficient photon–ALP conversion inside the host, which is able to justify the detection by the LHAASO Collaboration of this GRB above 10   TeV <cit.>. The latter fact represents a strong hint at ALP existence, as demonstrated in <cit.> (see also Section <ref> and <cit.>).

Instead, blazars are often hosted inside elliptical galaxies, where B^ host is described by means of a domain-like model with the typical values B_ host≃ 5   G and L_ dom^ host≃ 150   pc <cit.>. In such a situation, the photon–ALP oscillation length turns out to be much larger than L_ dom^ host, resulting in an inefficient photon–ALP conversion, as shown in <cit.>. However, whatever the nature and the properties of the host galaxy are, we evaluate the associated photon–ALP beam transfer matrix 𝒰_ host.



 §.§ Galaxy Clusters


Galaxy clusters consist of 30 to more than 1000 galaxies with a total mass within the range (10^14–10^15)   M_⊙, making them the largest gravitationally bound structures in the Universe. Properties such as galactic content, density profile, shape, concentration, and symmetry allow galaxy clusters to be classified into three main classes: (i) regular clusters, (ii) intermediate clusters, and (iii) irregular clusters <cit.>. The approximate spherical symmetry and the satisfying knowledge of the density profile of regular clusters allow us to model their electron number density n_e^ clu and the linked magnetic field B^ clu carefully so that we can properly calculate the photon–ALP conversion. 

The strength of B^ clu is set to B^ clu = 𝒪 (1-10)   G by Faraday rotation measurements and synchrotron radio emission <cit.>. The morphology of B^ clu is turbulent and modeled with a Kolmogorov-type turbulence power spectrum M(k)∝ k^q, with index q=-11/3 and where k represents the wave number in the interval [k_L,k_H] with k_L = 2π/Λ_ max and k_H = 2π/Λ_ min being the minimal and maximal turbulence scales, respectively <cit.>. Consequently, the behavior of B^ clu can be modeled as <cit.>

    B^ clu(y)=ℬ( B_0^ clu,k,q,y ) ( n_e^ clu(y)/n_e,0^ clu)^η_ clu,

where ℬ represents the spectral function describing the Kolmogorov-type turbulence of the cluster magnetic field (for more details, see, e.g., <cit.>), η_ clu is a cluster parameter, and B_0^ clu and n_e,0^ clu are the magnetic field strength and the electron number density at the center of the cluster, respectively.

Several models are employed to describe the profile of n_e^ clu, such as the single β model typically used for non-cool-core (nCC) clusters, the double β model for cool-core (CC) clusters, or a modified version of them <cit.>. We will fix the behavior of n_e^ clu in Section <ref>, where we will concentrate on real situations.

Photons can be produced by some astrophysical source such as a blazar or a GRB and can pass through a galaxy cluster, or they can be diffusely generated in the central region of the cluster itself. In the latter situation, it is necessary to discuss their initial Π_L,0. In the X-ray band, photons are produced in the cluster central region via thermal Bremsstrahlung <cit.> with photons that turn out to be therefore unpolarized. In the HE range, several processes are proposed to explain possible photon emission, such as the cascade of VHE photons, inverse Compton scattering, and neutral pion decay (see, ). Even in the HE range, photons turn out to be effectively .

The transfer matrix 𝒰_ clu of the photon–ALP beam propagating from the central region of the cluster up to its virial radius can be evaluated by following the calculation reported in <cit.>. In the specific case in which photons are not diffusely produced in the central region of a rich cluster but are generated by another astrophysical source that does not lie inside a rich cluster—such a situation is frequent concerning blazars—the photon–ALP conversion in this region turns out to be inefficient.



 §.§ Extragalactic Space


The effectiveness of the photon–ALP conversion inside the extragalactic space is determined by the strength and morphology of the extragalactic magnetic field B_ ext. However, the properties of B_ ext are still poorly understood. While many configurations for B_ ext exist in the literature <cit.>, its limits are in the interval 10^- 7  nG≲ B_ ext≲ 1.7   nG on the scale of 𝒪 (1)   Mpc <cit.>.

The shape of B_ ext is modeled as a domain-like network, where B_ ext is homogeneous over a whole domain of size L_ dom^ ext, which is equal to its coherence length, and with B_ ext randomly and discontinuously changing its direction from one domain to the subsequent one, but maintaining approximately the same strength <cit.>. Since the exact orientation of B_ ext in each domain is unknown, the photon–ALP beam propagation becomes a stochastic process, and many realizations of it can be evaluated by varying the B_ ext orientations, in order to obtain the statistic properties of the system. However, only a single realization of the photon–ALP beam propagation process can be physically observed.

The domain-like model proposed for B_ ext is justified by several scenarios of outflows from primeval galaxies, further amplified by turbulence <cit.>. According to the latter scenarios, the usual values for B_ ext are B_ ext = 𝒪 (1)   nG and L_ dom^ ext = 𝒪 (1)   Mpc (for details, see <cit.>). In this fashion, we assume B_ ext = 1   nG and L_ dom^ ext in the interval (0.2–10)   Mpc with ⟨ L_ dom^ ext⟩ = 2   Mpc.

However, the commonly used discontinuous model for B_ ext produces unphysical results when the photon–ALP beam oscillation length l_ osc becomes smaller than L_ dom^ ext due to the photon dispersion on the CMB <cit.>, and consequently, the system turns out to be sensitive to the B_ ext substructure. In order to solve such a problem, we employ a more complex model developed in <cit.>, where the components of B_ ext change continuously, crossing from a domain to the subsequent one.

[-15]In the UV-X-ray and HE bands, photon absorption due to the interaction with EBL photons—we assume the conservative EBL model of Franceschini and Rodighiero <cit.>—is negligible, but it becomes substantial for E > 𝒪(100)   GeV <cit.>. Therefore, for , propagation in the extragalactic space produces additional photon–ALP oscillations when the photon–ALP conversion is efficient. Instead, for E > 𝒪(100)   GeV, where we have a large EBL absorption, the effect of the photon–ALP beam propagation in the extragalactic space is also to mitigate the EBL absorption of VHE photons. The reason lies in the fact that when photons become ALPs, they do not undergo EBL absorption so that the effective optical depth is reduced and more photons can reach the Earth.

In both of the previous situations, with large or negligible EBL absorption and in the case of both efficient or ineffective photon–ALP conversion, the transfer matrix 𝒰_ ext of the photon–ALP system in the extragalactic space can be calculated by following the analysis performed in <cit.>.



 §.§ Milky Way


Photon–ALP conversion in the Milky Way can be computed once the morphology and strength of the magnetic field B_ MW and the profile of the electron number density n_ MW,   e in the Galaxy are known.

Concerning B_ MW, we employ the model developed by Jansson and Farrar <cit.>, which includes a disk and a halo component, both parallel to the Galactic plane, and a poloidal `X-shaped' component at the galactic center. The most relevant component of B_ MW for the photon–ALP conversion turns out to be its regular part, to which we add for completeness the turbulent one developed in <cit.>. The alternative B_ MW model of Pshirkov et al. <cit.> does not substantially produce a different photon–ALP conversion in the Milky Way. Still, we prefer the model by Jansson and Farrar <cit.> since it determines the Galactic halo component with a higher level of accuracy. 

Regarding n_ MW,   e we employ the model presented in <cit.>, where the Galaxy is modeled with an extended thick disk standing for the warm interstellar medium, a thin disk accounting for the Galactic molecular ring, a Galactic Center disk, spiral arms and features for the Gum Nebula, Galactic Loop I, and the Local Bubble.

By following the procedure developed in <cit.>, we can evaluate the transfer matrix 𝒰_ MW of the photon–ALP system in the Milky Way.



 §.§ Overall Photon–ALP Beam Propagation


Once all the transfer matrices discussed in the previous subsections are known, we can combine them in order to calculate the total transfer matrix 𝒰 describing the photon–ALP beam propagation starting from the photon emission zone up to the Earth. When photons are diffusely generated in the central zone of a galaxy cluster, 𝒰 reads as follows:

    𝒰=𝒰_ MW 𝒰_ ext 𝒰_ clu,

while when photons are produced in the jet of a blazar, 𝒰 is expressed by

    𝒰=𝒰_ MW 𝒰_ ext 𝒰_ clu 𝒰_ host 𝒰_ jet.


In both of the previous cases, we can calculate the photon survival probability in the presence of photon–ALP interaction by specializing Equation (<ref>) as

    P_γ→γ  = ∑_i = x,z Tr[ρ_i  𝒰 ρ_ in 𝒰^†],

where ρ_x and ρ_z read from Equations (<ref>), and ρ_ in represents the beam initial polarization density matrix, which is specified in the following subsections for concrete cases, for which the initial photon degree of linear polarization Π_L,0 is known. Moreover, by recalling Equation (<ref>) with ρ_0 ≡ρ_ in, we obtain the final photon degree of the linear polarization Π_L and the polarization angle χ, thanks to Equations (<ref>) and (<ref>), respectively (see also <cit.>).




§ ALP EFFECTS ON PHOTON POLARIZATION


Hereafter, we show how photon–ALP interaction modifies the final degree of linear polarization Π_L and polarization angle χ for photons diffusely generated in the galaxy cluster central region or produced in the blazar jet. Here, we report two concrete examples: one for galaxy clusters, and specifically, we concentrate on Perseus, and one for BL Lacs and in particular, we focus on OJ 287. The latter two sources represent good observational targets to study ALP effects inside galaxy clusters <cit.> and blazars <cit.>, respectively. We send the interested reader to the above-quoted papers, where a deeper discussion is presented along with other concrete examples of both galaxy clusters and blazars <cit.>. We now describe the properties of Perseus and of OJ 287, which are crucial to evaluate the photon–ALP interaction effects on Π_L and χ.

Perseus is a deeply studied CC rich regular cluster located at redshift z = 0.01756 and represents the brightest galaxy cluster in the X-ray band. Its electron number density n_e^ clu can be expressed by

    n_e^ clu(y)=n_e,01^ clu( 1+y^2/r_ core,1^2)^-3/2β_ clu,1+ 
    
    n_e,02^ clu( 1+y^2/r_ core,2^2)^-3/2β_ clu,2,

with n_e,01^ clu=3.9 × 10^-2  cm^-3, r_ core,1=80   kpc, β_ clu,1=1.2, n_e,02^ clu=4.05 × 10^-3  cm^-3, r_ core,2=280   kpc, and β_ clu,2=0.58 <cit.>. The Perseus magnetic field B^ clu is described by Equation (<ref>) by taking B_0^ clu = 16   G, which represents an average estimate among those reported in the literature <cit.>. Due to the similarity of the stochastic properties of the Perseus B^ clu with those of the Coma cluster, we assume the following values regarding Perseus: q=-11/3, Λ_ min= 2   kpc and Λ_ max= 34   kpc <cit.>, and the average value η_ clu=0.5 entering Equation (<ref>). Finally, we take an initial degree of linear polarization Π_L,0=0 both in the UV-X-ray and in the HE band according to the discussion in Section <ref>. Then, the photon–ALP beam propagation can be evaluated as discussed in Section <ref>, and its transfer matrix reads from Equation (<ref>).

OJ 287 represents a typical low-frequency peaked blazar (LBL) located at redshift z=0.3056. Due to its high flux in both the X-ray and HE bands, OJ 287 turns out to be a favorable observational target for polarimetric studies in both energy ranges <cit.>. In the following, we consider the typical values regarding the LBL parameters: for the leptonic model, we take B^ jet_0 = 1   G, y_ em = 3 × 10^16  cm and γ = 10, while for the hadronic model, we assume B^ jet_0 = 20   G, y_ em = 10^17  cm and γ = 15 <cit.>. The profiles of B^ jet and n_e^ jet are represented by Equations (<ref>) and (<ref>), respectively. As discussed in Section <ref>, the initial Π_L,0 for OJ 287 is plotted in Figure <ref> for both the leptonic and the hadronic scenario. OJ 287 is hosted inside an elliptical galaxy with a resulting inefficient photon–ALP conversion in the host, as discussed in Section <ref>. Moreover, OJ 287 is not found within a rich galaxy cluster with a consequent negligible ALP effect also in this zone (see ). However, for completeness, we consider all the contributions for the evaluation of the total photon–ALP beam transfer matrix, which is expressed by Equation (<ref>). A remark regarding photons emitted by blazars should be kept in mind: due to the limited spatial resolution of real polarimeters, photons coming from different zones inside the transverse section of the blazar jet cannot be distinguished, and all photons are then collected together. Although the latter fact might in principle wash out the ALP-induced polarization features, the studies developed in <cit.> demonstrate that this is not the case in realistic situations. In order to be conservative, the magnetic field profile expressed by Equation (<ref>) and a propagation distance of 1   pc inside the jet are assumed. We send the reader to the papers quoted above <cit.> for a deeper discussion regarding the latter problem and the adopted strategy to take the polarimeter technological limitations into account. 

We can now show the final photon survival probability P_γ→γ, the corresponding photon degree of the linear polarization Π_L, and the polarization angle χ arising from photon–ALP oscillations inside the magnetized media analyzed in Section <ref> (blazar jet, host galaxy, galaxy cluster, extragalactic space, Milky Way) for the two above-discussed scenarios: (i) photons are diffusely generated in the central zone of a galaxy cluster—we take Perseus as an example, and (ii) photons are produced at the jet base of a BL Lac—we consider OJ 287 as an illustration. In order to assess the robustness of the results, several realizations of the photon–ALP beam propagation process are evaluated by considering different magnetic field configurations due to limited knowledge especially regarding B^ clu and B_ ext. The probability density function f_Π associated with the final Π_L can be computed by collecting all the values assumed by Π_L in the various realizations and gives us information about the most probable scenarios.

Regarding the ALP parameters, we assume g_aγγ=0.5 × 10^-11  GeV^-1 and the two ALP masses: (i) m_a ≲ 10^-14  eV and (ii) m_a = 10^-10  eV, which are within the CAST bound <cit.>. For m_a ≲ 10^-14  eV, the plasma term dominates over the ALP mass term for the chosen model parameters, while the opposite is true in the case m_a = 10^-10  eV. Results about P_γ→γ, Π_L and χ are obtained for photons produced at redshift z with emitted energy E and observed energy E_0=E/(1+z) in the two intervals: (i) UV-X-ray band and 

Note that, in the following figures, binned data about Π_L and χ are also shown in order to provide an idea of how a possible observation of ALP-induced features may look like. The binning procedure is based on the realistic energy resolution of current and/or planned polarimeters both in the X-ray band <cit.> and in the HE range <cit.> (see <cit.> for more details).



 §.§ UV-X-ray Band


In the UV-X-ray band, the photon–ALP beam propagates in the weak mixing regime due to the strength of the plasma and/or ALP mass term inside the mixing matrix of Equation (<ref>). As a result, P_γ→γ, Π_L and χ turn out to possess an energy-dependent behavior in the case of both diffuse photon production in the central region of the Perseus cluster and photon emission at the jet base of OJ 287.

In particular, in the case m_a = 10^-10  eV, the ALP mass term is so strong that it dominates over the photon–ALP mixing term. As a result, the photon–ALP conversion turns out to be inefficient in the UV-X-ray band for m_a = 10^-10  eV, apart from a moderate dimming of Π_L,0 for OJ 287 in the case of the hadronic emission model due to the high central value of B^ jet (see <cit.> for more details). This is the reason why we only consider the case m_a ≲ 10^-14  eV in this subsection. 

Regarding the Perseus cluster, we report P_γ→γ, Π_L and χ in the top subfigure of Figure <ref> and the corresponding f_Π associated with the final Π_L in the bottom subfigure (see <cit.> for more details). Similarly, concerning OJ 287, we show P_γ→γ, Π_L and χ in the top subfigure of Figure <ref> and the linked f_Π in the bottom subfigure, but we now consider the two possible emission mechanisms inside blazars (see Section <ref>): (i) the leptonic model in the left panels and (ii) the hadronic model in the right panels (see <cit.> for more details). In both cases, i.e., for both Perseus and OJ 287, we concentrate on two benchmark energies in the evaluation of f_Π: (i) E_0 = 1   keV and (ii) E_0 = 10   keV.




In the case considered here, i.e., g_aγγ=0.5 × 10^-11  GeV^-1, m_a ≲ 10^-14  eV, the energy dependence of P_γ→γ, Π_L, and χ in the top subfigures of Figures <ref> and <ref> is caused by the strength of the plasma term, which is not negligible with respect to the photon–ALP mixing term, especially inside the cluster in the case of Perseus and in the BL Lac jet regarding OJ 287. As a result, the photon–ALP beam propagates in the weak mixing regime, which protracts for a few energy decades due to the large variation with respect to the distance of B^ clu and n_e^ clu regarding Perseus and of B^ jet and n_e^ jet concerning OJ 287 (see  and <cit.> for details). In addition, the previous figures demonstrate that the final Π_L turns out to be heavily modified with respect to the initial one Π_L,0 for both Perseus and OJ 287. The binned data in the top Subfigures of Figures <ref> and <ref> concerning Π_L and χ show that the ALP-induced features can be detected by IXPE <cit.>, eXTP <cit.>, XL-Calibur <cit.>, NGXP <cit.>, and XPP <cit.> for E_0 ≳ 2   keV regarding Perseus and for E_0 ≳ 0.5   keV concerning OJ 287. Regarding OJ 287, the top subfigure of Figure <ref> shows that the hadronic model produces a larger energy variability of Π_L and χ with respect to the leptonic scenario due to the higher value of B^ jet_0 and the wider extension of the jet region.



The behavior of f_Π reported in the bottom subfigures of Figures <ref> and <ref> shows a broadening of the final Π_L with respect to the initial Π_L,0 for both Perseus and OJ 287. Perseus appears as a strong target to study ALP effects on photon polarization, since the final value Π_L = 0 is never the most probable expected result, while conventional physics predicts Π_L = Π_L,0 = 0. Instead, for OJ 287, the most probable final value of Π_L is not very different from the initial Π_L,0 apart from the case of the leptonic model at E_0 = 10   keV, where the most probable result turns out to be Π_L ≳ 0.8, which is extremely difficult to explain within conventional physics. For more details, we send the reader to the discussion developed in <cit.> concerning Perseus and to that presented in <cit.> regarding OJ 287.
















 §.§ High-Energy Band


In the HE band, both the scenario with ALP parameters g_aγγ=0.5 × 10^-11  GeV^-1, m_a ≲ 10^-14  eV and that with g_aγγ=0.5 × 10^-11  GeV^-1, m_a = 10^-10  eV produce sizable ALP effects on P_γ→γ, Π_L and χ in the case both of photon diffuse production in the central zone of the Perseus cluster and of photon emission at the jet base of OJ 287.

Concerning the Perseus cluster, we show the behavior of P_γ→γ, Π_L and χ in the top subfigure of Figure <ref> for the case g_aγγ=0.5 × 10^-11  GeV^-1, m_a ≲ 10^-14  eV and in the top subfigure of Figure
 <ref> for the case g_aγγ=0.5 × 10^-11  GeV^-1, m_a = 10^-10  eV. The associated f_Π are reported in the bottom subfigures of Figures <ref> and <ref>, respectively. Regarding OJ 287, we likewise report P_γ→γ, Π_L, and χ in the top subfigure of Figure <ref> for the case g_aγγ=0.5 × 10^-11  GeV^-1, m_a ≲ 10^-14  eV and in the top subfigure of Figure <ref> for the case g_aγγ=0.5 × 10^-11  GeV^-1,  m_a = 10^-10  eV. The corresponding f_Π are shown in the bottom subfigures of Figures <ref> and <ref>, respectively. For OJ 287, we assume (i) the leptonic emission model in the left panels of Figures <ref> and <ref> and (ii) the hadronic one in the right panels of Figures <ref> and <ref> (see Section <ref> and <cit.> for more details). Moreover, f_Π is evaluated for both Perseus and OJ 287 at the two benchmark energies: (i) E_0 = 300   keV and (ii) E_0 = 3   MeV.




We first consider the scenario g_aγγ=0.5 × 10^-11  GeV^-1, m_a ≲ 10^-14  eV. In the present situation, the photon–ALP beam propagates in the strong mixing regime, since the photon–ALP mixing term dominates over all other terms inside the mixing matrix of Equation (<ref>) (see Section <ref>). Consequently, P_γ→γ, Π_L, and χ are energy independent in almost all the considered energy band in the case both of photon production in the central region of Perseus, as shown by the top subfigure of Figure <ref>, and of photon generation in the jet of OJ 287, as reported in the top subfigure of Figure <ref>. We can also infer that, in all the considered cases, the final Π_L turns out to be strongly modified with respect to the initial Π_L,0. The binned data show a lower uncertainty than in the UV-X-ray band due to fact that the photon–ALP system lies in the strong mixing regime, which makes both Perseus and OJ 287 good observational targets for missions such as COSI <cit.>, e-ASTROGAM <cit.>, and AMEGO <cit.>. Moreover, regarding OJ 287, we do not observe substantial differences between the leptonic and the hadronic emission model.




As far as the behavior of f_Π is concerned, for both Perseus and OJ 287, the bottom subfigures of Figures <ref> and <ref> respectively show that the most probable final value of Π_L is larger than the initial one Π_L,0 with a substantial broadening. For both Perseus and OJ 287, the most promising energy range to study ALP effects on photon polarization appears around E_0 = 3   MeV, where the most probable value for Π_L is Π_L ≳ 0.8, which is larger than that predicted by conventional physics in all considered cases.





We now move to the scenario g_aγγ=0.5 × 10^-11  GeV^-1, m_a = 10^-10  eV. In this situation, the photon–ALP system lies in the weak mixing regime, since the ALP mass term turns out to be non-negligible with respect to the photon–ALP mixing term inside the mixing matrix of Equation (<ref>) (see Section <ref>). As a result, P_γ→γ, Π_L, and χ become energy dependent, as shown in the top subfigure of Figure <ref> regarding Perseus and in the top subfigure of Figure <ref> concerning OJ 287. Furthermore, we observe that the final Π_L turns out to be altered compared with the initial Π_L,0 especially in the case of Perseus and in that of the leptonic model for OJ 287. As already observed in the X-ray band for the case g_aγγ=0.5 × 10^-11  GeV^-1, m_a ≲ 10^-14  eV, the wide energy range, where the photon–ALP system lies in the weak mixing regime, is due to the high variation of the magnetic field and electron number density profiles in the crossed media (see also Sections <ref> and <ref> and <cit.> for more details). The binned data in the top subfigures of Figures <ref> and <ref> show that observatories like COSI <cit.>, e-ASTROGAM <cit.>, and AMEGO <cit.> may detect ALP-induced features on Π_L for E_0 ≳ 3   MeV concerning Perseus, and similar features for E_0 ≳ 0.2   MeV regarding OJ 287 in the case both of the leptonic and of the hadronic model.










The behavior of f_Π reported in the bottom subfigure of Figure <ref> for Perseus and in the bottom subfigure of Figure <ref> for OJ 287 shows a broadening of the final Π_L with respect to the initial Π_L,0 in all the cases. Moreover, the best energy interval for studies of ALP-induced features on Π_L appears to be E_0 ≳ 3   MeV concerning the Perseus cluster, where the most probable value for the final Π_L is Π_L ≳ 0.2, while conventional physics predicts Π_L = Π_L,0 = 0. Even OJ 287 represents a good observational target, since the most probable values of the final Π_L are larger than the initial ones, but in this case, conventional physics could justify moderately high values of Π_L albeit with some difficulty.



§ DISCUSSION AND FUTURE PERSPECTIVES


In Section <ref>, the ALP impact on the polarization of photons coming from Perseus has been reported as an example of ALP effects on galaxy clusters in the X-ray and in the HE band (for details, see <cit.>). Similarly, OJ 287 has been presented as an illustration of the ALP-induced features on photon polarization in BL Lacs (see <cit.> for details). Present and planned observatories in the X-ray band <cit.> and in the HE range <cit.> possess the capability to detect possible ALP signatures as those analyzed in Section <ref>, as shown by the realistic binned data reported therein. An analysis about ALP effects on photon polarization in the VHE band has been performed in <cit.>: we will briefly discuss this topic in Section <ref> along with its future crucial importance for the possible identification of an ALP signal.

The ALP-induced features on the final photon degree of linear polarization Π_L discussed in Section <ref> mainly originate inside the galaxy cluster central region and/or in the blazar jet, while the effect of other crossed media is subdominant, as shown . Specifically, a negligible photon–ALP conversion inside the extragalactic space (when B_ ext < 10^-15  G) does not substantially change the results apart from a smaller broadening of Π_L shown in the behavior of its associated probability density function f_Π <cit.>.

In particular, photons diffusely emitted in the central region of galaxy clusters are expected to be observed as unpolarized within conventional physics both in the X-ray and in the HE range (i.e., with Π_L = 0, see Section <ref>). Instead, the photon–ALP interaction model consistently predicts Π_L > 0 for them in both energy ranges. Therefore, galaxy clusters appear among the best observational targets for studies concerning ALP-induced effects on photon polarization. Specifically, a possible future detection of photons diffusely emitted from galaxy clusters with Π_L > 0 would represent a hint at ALP existence in addition to the already-existing ones <cit.>, since conventional physics along with other scenarios like Lorentz invariance violation <cit.> is unable to explain Π_L > 0 (see <cit.> for details). BL Lacs are also promising targets especially in the cases where the photon–ALP interaction model predicts Π_L > 0.8 (see Section <ref>), since, in this case, even the hadronic emission model can hardly justify such high polarization degrees within conventional physics (for details, see <cit.>). However, in case of lower Π_L values detected from BL Lacs, the exclusion of an explanation within conventional physics and the demonstration of the need for ALPs may be less robust. Note that the magnetic field in the jet B^ jet is expected to be quite turbulent, which implies a lower initial photon degree of linear polarization Π_L,0 especially within the leptonic emission model, as shown by a recent theoretical study <cit.> and confirmed by IXPE results <cit.>. Consequently, the final Π_L predicted by the photon–ALP interaction model is reduced in the cases where the initial Π_L,0 remains the most probable final value, as shown by the behavior of the corresponding f_Π in Section <ref>.

Other possible targets to evaluate ALP effects on photon polarization are represented by (i) white dwarfs, where ALP bounds can be derived from polarization measurements <cit.>; (ii) neutron stars, where ALPs produce polarization signatures detectable in the X-ray band <cit.>; and (iii) GRBs, where the photon–ALP conversion over cosmological distances strongly modifies the initial photon polarization degree, as shown in the pioneering study performed in <cit.>. We will focus our attention in Section <ref> on GRBs due to their importance also for the Universe transparency problem regarding the recent detection by the LHAASO Collaboration of GRB 221009A up to 18   TeV <cit.>, which is difficult to justify within conventional physics, but is instead explained by the photon–ALP interaction (see Section <ref> and <cit.> for details).

[-25]As far as the ALP parameter space (m_a, g_aγγ) is concerned, we have analyzed two cases: (i) m_a ≲ 10^-14  eV , g_aγγ = 0.5 × 10^-11  GeV^-1 and (ii) m_a = 10^-10  eV, . Both the previous possibilities are within the CAST bound <cit.>. The former ALP scenario, i.e., the case with m_a ≲ 10^-14  eV, is disfavored by some recent bounds <cit.>, but this possibility cannot be excluded, as discussed in <cit.>. Instead, the latter choice, i.e., the case with m_a = 10^-10  eV, meets all existing bounds and is compatible with the ALP parameters employed in previous studies, where ALP hints have been found <cit.>. This is the reason why the case m_a = 10^-10  eV, g_aγγ = 0.5 × 10^-11  GeV^-1 appears to be more promising. As discussed in <cit.>, other possibilities, such as an ALP with mass m_a = 10^-11  eV and coupling to photons g_aγγ= 0.5 × 10^-11  GeV^-1, will produce an intermediate behavior between the two cases considered in Section <ref> and can meet all current ALP bounds. Note that, while an increment (decrement) of the strength of g_aγγ implies a corresponding increase (decrease) of the ALP effects on the final Π_L and χ at a given energy, a variation in the value of m_a produces the modification of the energy threshold, starting from which the ALP-induced polarization features become relevant. In particular, as shown in Section <ref>, for m_a ≲ 10^-14  eV down to extremely low ALP masses, we do not expect modifications in the ALP-induced effects on Π_L and χ, since the ALP mass term turns out to be subdominant with respect to the plasma term in Equation (<ref>). For 10^-14  eV≲ m_a ≲ 10^-10  eV, the ALP mass term starts to become important and progressively overcomes the strength of the plasma term with a resulting translation at higher energies of the threshold above which the ALP-induced effects become sizable. For the above-mentioned ALP mass m_a = 10^-11  eV, substantial effects on Π_L and χ are expected starting from E_0 ≳𝒪(2-5)   keV, while, as shown in Section <ref>, for m_a = 10^-10  eV, the ALP impact on Π_L and χ is relevant for E_0 ≳𝒪(0.2-0.5)   MeV. Lastly, for m_a progressively increasing above 10^-10  eV, the ALP-induced effects on Π_L and χ will correspondingly appear at larger and larger energies up to disappear in the HE range and to become sizable only in the VHE band and beyond.

As far as polarization measurements are concerned, unfortunately, we do not presently have many observational data regarding the astrophysical sources considered in this review, i.e., galaxy clusters, blazars, and GRBs. In particular, polarization data in both the X-ray and HE bands regarding diffuse emission from galaxy clusters are substantially missing. Since, as discussed above, these latter astrophysical [-15]sources are extremely important for analyzing ALP-induced effects on photon polarization and for studying ALP physics in general, a dedicated observational campaign by IXPE <cit.> could provide vital information in this regard. Concerning blazars, we now have recent polarimetric data obtained in the X-ray band by IXPE (see, e.g., <cit.>), which promises to provide us with further important results in the near future. Regarding GRBs, some polarization measurements have been obtained in the X-ray band <cit.>. For all the astrophysical sources discussed in this review, further polarimetric data are expected from new missions proposed both in the X-ray band <cit.> and in the HE range <cit.>.






 §.§ Very High Energy Band


In the VHE band, we consider the case of a BL Lac emitting within the leptonic model and placed inside a rich galaxy cluster, as studied in <cit.>; therefore, this represents an alternative possibility with respect to those contemplated in Sections <ref> and <ref>. In the present situation, photon–ALP interaction inside the galaxy cluster is relevant, while the conversion inside the host is subdominant (see Section <ref> for more details). The total transfer matrix of the photon–ALP beam is represented by Equation (<ref>). In the following, two possibilities are evaluated: (i) a BL Lac with average properties (see below) is located at a redshift z=0.03; (ii) the same BL Lac is placed at z=0.4. The BL Lac redshift represents a crucial parameter to evaluate the photon–ALP system behavior in the VHE band: since, for E_0 ≳ 100   GeV, the absorption of VHE photons due to their interaction with the EBL is not negligible, as it occurs instead for lower energies, as in Sections <ref> and <ref> (see also Section <ref>), a larger redshift means a higher photon absorption.

In the present scenario, the profiles of the magnetic field in the jet B^ jet and of the electron number density n_e^ jet are provided by Equations (<ref>) and (<ref>), respectively, by taking the following parameter values: B^ jet_0 = 0.5   G, y_ em = 3 × 10^16  cm, n_e,0^ jet=5 × 10^4   cm^-3, and γ = 15. Photons are assumed to be produced within the leptonic model as unpolarized, i.e., with the initial degree of the linear polarization Π_L,0=0 (see <cit.> for more details). Moreover, we consider a CC galaxy cluster, hosting the BL Lac, with the electron number density profile

    n_e^ clu(y)=n_e,0^ clu( 1+y^2/r_ core^2)^-3/2β_ clu,

with the typical parameter values n_e,0^ clu=5 × 10^-2  cm^-3, r_ core= 100   kpc, and β_ clu=2/3 <cit.>. The cluster magnetic field B^ clu reads from Equation (<ref>) by taking the following typical parameter values: B_0^ clu=15   G, η_ clu=0.75, q=-11/3, k_L=0.2   kpc^-1, and k_H=3   kpc^-1 <cit.>. Furthermore, we assume an extragalactic magnetic field strength B_ ext < 10^-15  G, resulting in an inefficient photon–ALP conversion inside the extragalactic space. Concerning the ALP parameters, we take g_aγγ=0.5 × 10^-11  GeV^-1 and m_a = 10^-10  eV. 

For a BL Lac with the above-discussed properties, we show P_γ→γ and Π_L in the top subfigure of Figure <ref> and the corresponding f_Π associated with the final Π_L in the bottom subfigure. The left panels of Figure <ref> report the results for the BL Lac placed at z=0.03, while the right panels show the same results but for the BL Lac located at z=0.4 (see <cit.> for more details). 




The top subfigure of Figure <ref> shows that P_γ→γ and Π_L are energy dependent, since the photon–ALP system lies in the weak mixing regime due to the strength of the QED one-loop term, which is not negligible at high energies especially inside the BL Lac jet. Specifically, the final Π_L turns out to be strongly modified with respect to the initial one Π_L,0 in both cases of the BL Lac placed at z=0.03 and at z=0.4. Now, a new aspect must be considered in the VHE band with respect to the cases at lower energies discussed in Sections <ref> and <ref>: the top subfigure of Figure <ref> shows that P_γ→γ considerably decreases due to the EBL absorption starting from E_0 ≳ 3   TeV at z=0.03 and from E_0 ≳ 200   GeV at z=0.4. Correspondingly, Π_L starts to grow up to its limit value Π_L=1, which is reached at E_0 ≳ 30   TeV for z=0.03 and at E_0 ≳ 5   TeV for z=0.4, where photons are fully polarized and P_γ→γ≲ 10^-2. The different behavior with respect to the redshift is due to the fact that the EBL absorption grows as both the energy and the redshift increase. As deeply discussed in <cit.>, we have therefore found a peculiar feature: in the presence of photon–ALP interaction, photons, observed at energies where EBL absorption is very strong, are fully polarized. The full explanation of this important feature regarding Π_L, and shown in the top subfigure of Figure <ref>, is provided in <cit.>: in short, after photon–ALP interaction inside the source, we have both photons and ALPs entering the extragalactic space. Then, when EBL absorption is extremely high, all remaining photons are lost during their propagation in the extragalactic space, and we therefore collect only those photons that are reconverted back from ALPs in the Milky Way magnetic field B_ MW. Still, only the coherent component of B_ MW is efficient in producing photon–ALP conversion so that photons reconverted back from ALPs inside the Milky Way turn out to be fully polarized.

The behavior of f_Π in the bottom subfigure of Figure <ref> confirms the robustness of the previous discussion. In particular, upper panels deal with the case of small EBL absorption, while the lower ones describe the situation illustrated above, in which Π_L tends to its limit value Π_L=1 with a concomitant high EBL absorption. Note that the behavior of f_Π in the left panels is similar to that in the right ones, but different energies are considered (see the legend inside the panels) due to the above-described behavior of the EBL absorption as a function of energy and redshift. Since conventional physics cannot explain the scenario Π_L=1, a possible detection of fully polarized VHE photons not only would be an additional hint at ALPs in addition to those discussed in <cit.>, but also would represent the proof of ALP existence. Therefore, this ALP-induced feature identified in <cit.> producing fully polarized photons at VHE possesses a high discovery power. However, this possibility nowadays represents only a hope for the future, as currently known techniques cannot measure photon polarization in the VHE band but are limited to tens of GeV at most <cit.>.



 §.§ Gamma-Ray Bursts


ALP effects on the polarization of photons produced by GRBs have been analyzed in a pioneering study <cit.>, where photon–ALP conversion is evaluated inside the source, in the extragalactic space, inside possibly crossed galaxy clusters, and in the Milky Way within the tens of KeV-few MeV band. In particular, the benchmark energy E_0 = 100   keV and the ALP parameters g_aγγ= 10^-11  GeV^-1, m_a = 10^-13  eV are taken into account.

GRBs are extremely intense explosions originating at cosmological distances and are characterized by a prompt and an afterglow phase. By considering the typical parameters of the prompt phase—which is analyzed in <cit.>—regarding the magnetic field strength and the electron number density, the photon–ALP conversion inside the source turns out to be negligible. Regarding the magnetic field structure in extragalactic space, galaxy clusters, and Milky Way, domain-like models with different parameter values inside the several crossed media are assumed in <cit.>. In particular, by calling B_ ext, B_ clu, and B_ MW the magnetic field strengths in extragalactic space, galaxy clusters, and Milky Way, respectively, and L_ dom^ ext, L_ dom^ clu, and L_ dom^ MW the corresponding magnetic field coherence lengths, the following values are assumed: B_ ext=𝒪(1)   nG, B_ clu=𝒪(1)  μ G, B_ MW=𝒪(1-5)   G and , L_ dom^ clu=𝒪(10)   kpc, L_ dom^ MW=𝒪(10)   kpc (see <cit.> for more details).

For the chosen parameter values concerning both the photon–ALP system and the crossed media, the photon–ALP beam propagates in the strong mixing regime in all the previous zones within the considered energy band. Consequently, both P_γ→γ and Π_L turn out to be energy independent with Π_L which is modified with respect to the initial Π_L,0. Therefore, the only informative quantity in the present situation is represented by f_Π, which is shown in Figure <ref> for photon–ALP conversion inside the extragalactic space. In particular, the behavior of f_Π for different initial values of Π_L,0 is shown in the top subfigure of Figure <ref> for a GRB placed at redshift z=0.3 and in the bottom subfigure for a GRB located at z=2. Note that, in Figure <ref>, the initial degree of linear polarization is denoted by Π_0 and the photon–ALP coupling by g_aγ, but these quantities possess the same meaning of Π_L,0 and g_aγγ, respectively, which are used in the whole review.



Figure <ref> shows that the initial Π_L,0 (we recall the different notation here with ) is strongly modified and broadened in a similar way as it occurs in the systems considered in Sections <ref> and <ref>. In particular, the bottom subfigure shows that this effect increases with the redshift. Moreover, as shown in <cit.>, the photon–ALP beam propagation inside galaxy clusters and in the Milky Way further increases the broadening effect of the initial Π_L,0 to a point that, at high redshift, the most probable value of the final Π_L is no more the initial one, apart from the case of the initial Π_L,0≡Π_0 = 1.

These findings demonstrate that GRBs are very important astrophysical sources where ALP-induced effects on photon polarization can be studied, 
in addition to galaxy clusters and BL Lacs (see Sections <ref> and <ref>). The recent detection by the LHAASO Collaboration of photons with energy of up to 18   TeV produced by GRB 221009A <cit.>—which is difficult to justify in the context of conventional physics—shows the need to extend the analysis of the ALP impact on GRB polarization even at higher energy bands in a similar way as discussed in Sections <ref> and <ref>. Moreover, the explanation of the GRB 221009A detection up to 18   TeV within the photon–ALP interaction model developed in <cit.> demonstrates the importance of the photon–ALP conversion also inside the galaxy hosting the GRB. Therefore, an analysis similar to that performed in <cit.>, by considering the photon–ALP beam propagation also in the host galaxy and updated models for the photon–ALP conversion inside extragalactic space, galaxy clusters, and Milky Way, appears very promising in order to discover further ALP-induced features on photon polarization and even possible new hints at ALP existence or to improve constraints on the ALP parameter space.



§ CONCLUSIONS


In this review, we have described the ALP-induced effects on photon polarization, which can provide a different and complementary way to study the ALP physics with respect to the analysis of the ALP consequences on astrophysical spectra. As mentioned above, there exist three hints at ALP existence linked to the ALP effects on astrophysical spectra, with the most recent and strongest indication coming from the detection by the LHAASO Collaboration of GRB 221009A up to 18   TeV <cit.>, which can hardly be justified within conventional physics but is naturally explained by photon–ALP interaction <cit.> (see also <cit.>). In the near future, we expect that new spectral data from observatories such as ASTRI <cit.>, CTA <cit.>, GAMMA-400 <cit.>, HAWC <cit.>, HERD <cit.>, LHAASO <cit.>, and TAIGA-HiSCORE <cit.> will be able to increase our knowledge of the ALP properties.

The study of the ALP effects on the polarization of photons generated by astrophysical sources such as galaxy clusters, blazars, GRBs, neutron stars, and white dwarfs can add further information by shedding light on the ALP parameter space with current spectral hints suggesting m_a = 𝒪(10^- 10)   eV and g_a γγ = 𝒪(10^- 11)   GeV^- 1. As discussed in this review, galaxy clusters represent very promising observational targets in this respect. In fact, according to conventional physics, photons are emitted unpolarized in the central region of galaxy clusters both in the X-ray and in the HE band, resulting in an expected zero polarization degree for the observed photons, while the photon–ALP interaction makes them partially polarized with ALP parameters in the same ballpark of those producing spectral effects. Therefore, a possible future signal of partially polarized photons diffusely produced inside galaxy clusters would be a further strong hint at ALP existence. A negative signal would instead be useful to constrain the ALP parameter space further. In any case, as discussed in this review, future observations of the photon polarization degree from other astrophysical sources like blazars and GRBs are also very promising. Missions such as IXPE <cit.> (already operative), eXTP <cit.>, XL-Calibur <cit.>, NGXP <cit.>, XPP <cit.> in the X-ray band and like COSI <cit.>, e-ASTROGAM <cit.>, and AMEGO <cit.> in the HE range may test our findings about ALP effects on photon polarization. 

Finally, an ALP direct detection could be achieved by laboratory experiments such as ALPS II <cit.>, IAXO <cit.> and STAX <cit.>, thanks to the techniques developed by Avignone and collaborators <cit.> and possibly with the ABRACADABRA experiment <cit.>.



 




























The work of the author is supported by a contribution through Grant No. ASI-INAF 2023-17-HH.0 and by the INAF Mini Grant `High-energy astrophysics and axion-like particles', PI: Giorgio Galanti.

Data used in this paper can be asked from the authors of the quoted papers. 










The author thanks Giacomo Bonnoli, Silvano Molendi, and Lara Nava for the discussions and all collaborators in this field, especially Enrico Costa, Alessandro De Angelis, Marco Roncadelli, and Fabrizio Tavecchio.

The author declares no conflict of interest. 



-0cm


References














999

kaluzaklein1 Chang, S.; Tazawa, S.; Yamaguchi, M. Axion model in extra dimensions with TeV scale gravity. Phys. Rev. D 2000, 61, 084005. [http://doi.org/10.1103/PhysRevD.61.084005CrossRef]

kaluzaklein2 Dienes, K.R.; Dudas, E.; Gherghetta, T. Invisible axions and large-radius compactifications. Phys. Rev. D 2000, 62, 105023. [http://dx.doi.org/10.1103/PhysRevD.62.105023CrossRef]

turok Turok, N. Almost-Goldstone Bosons from Extra-Dimensional Gauge Theories. Phys. Rev. Lett. 1996, 76, 1015. [http://dx.doi.org/10.1103/PhysRevLett.76.1015CrossRef] [http://www.ncbi.nlm.nih.gov/pubmed/10061612PubMed]

susy1 Corianò, C.; Irges, N. Windows over a new low energy axion. Phys. Lett. B 2007, 651, 298–305. [http://dx.doi.org/10.1016/j.physletb.2007.06.022CrossRef]

susy2 Corianò, C.; Irges, N.; Morelli, S. Stückelberg axions and the effective action of anomalous abelian models 1. A unitarity analysis of the Higgs-axion mixing. J. High Energy Phys. 2007, 2007, 008. [http://dx.doi.org/10.1088/1126-6708/2007/07/008CrossRef]

susy3 Baer, H.; Krami, S.; Sekmen, S.; Summy, H. Dark matter allowed scenarios for Yukawa-unified SO(10) SUSY GUTs. J. High Energy Phys. 2008, 2008, 056. [http://dx.doi.org/10.1088/1126-6708/2008/03/056CrossRef]

susy4 Baer, H.; Summy, H. SO(10) SUSY GUTs, the gravitino problem, non-thermal leptogenesis and axino dark matter. Phys. Lett. B 2008, 666, 5–9. [http://dx.doi.org/10.1016/j.physletb.2008.06.072CrossRef]

susy5 Baer, H.; Haider, M.; Kraml, S.; Sekmen, S.; Summy, H.  Cosmological consequences of Yukawa-unified SUSY with mixed axion/axino cold and warm dark matter. J. Cosmol. Astropart. Phys. 2009, 2009, 002. [http://dx.doi.org/10.1088/1475-7516/2009/02/002CrossRef]

string1 Witten, E. Some properties of O(32) superstrings. Phys. Lett. B 1984, 149, 351–356. [http://dx.doi.org/10.1016/0370-2693(84)90422-2CrossRef]

string2 Conlon, J.P. The QCD axion and moduli stabilisation. J. High Energy Phys. 2006, 2006, 078. [http://dx.doi.org/10.1088/1126-6708/2006/05/078CrossRef]

string3 Svrcek, P.; Witten, E. Axions in string theory. J. High Energy Phys. 2006, 2006, 051. [http://dx.doi.org/10.1088/1126-6708/2006/06/051CrossRef]

string4 Conlon, J.P. Seeing an Invisible Axion in the Supersymmetric Particle Spectrum. Phys. Rev. Lett. 2006, 97, 261802. [http://dx.doi.org/10.1103/PhysRevLett.97.261802CrossRef] [http://www.ncbi.nlm.nih.gov/pubmed/17280417PubMed]

string5 Choi, K.-S.; Kim, I.-W.; Kim, J.E. String compactification, QCD axion and axion–photon–photon coupling. J. High Energy Phys. 2007, 2007, 116. [http://dx.doi.org/10.1088/1126-6708/2007/03/116CrossRef]

axiverse Arvanitaki, A.; Dimopoulos, S.; Dubovsky, S.; Kaloper, N.; March-Russell, J. String axiverse. Phys. Rev. D 2010, 81, 123530. [http://dx.doi.org/10.1103/PhysRevD.81.123530CrossRef]

abk2010 Acharya, B.S.; Bobkov, K.; Kumar, P. An M theory solution to the strong CP-problem, and constraints on the axiverse. J. High Energy Phys. 2010, 11, 105. [http://dx.doi.org/10.1007/JHEP11(2010)105CrossRef]

cicoli2012 Cicoli, M.; Goodsell, M.; Ringwald, A. The type IIB string axiverse and its low-energy phenomenology. J. High Energy Phys. 2012, 10, 146. [http://dx.doi.org/10.1007/JHEP10(2012)146CrossRef]

alp1 Jaeckel, J.; Ringwald, A. The Low-Energy Frontier of Particle Physics. Ann. Rev. Nucl. Part. Sci. 2010, 60, 405–437. [http://dx.doi.org/10.1146/annurev.nucl.012809.104433CrossRef]

alp2 Ringwald, A. Exploring the role of axions and other WISPs in the dark universe. Phys. Dark Univ. 2012, 1, 116–135. [http://dx.doi.org/10.1016/j.dark.2012.10.008CrossRef]

axionrev1 Kim, J.H. Light pseudoscalars, particle physics and cosmology. Phys. Rep. 1987, 150, 1–177. [http://dx.doi.org/10.1016/0370-1573(87)90017-2CrossRef]

axionrev2 Cheng, H.Y. The strong CP problem revisited. Phys. Rep. 1988, 158, 1–89. [http://dx.doi.org/10.1016/0370-1573(88)90135-4CrossRef]

axionrev3 Kim, J.E.; Carosi, G. Axions and the strong CP problem. Rev. Mod. Phys. 2010, 82, 557. [http://dx.doi.org/10.1103/RevModPhys.82.557CrossRef]

axionrev4 Marsch, D.J.E. Axion cosmology. Phys. Rep. 2016, 643, 1. [http://dx.doi.org/10.1016/j.physrep.2016.06.005CrossRef]

preskill Preskill, J.; Wise, M.B.; Wilczek, F. Cosmology of the invisible axion. Phys. Lett. B 1983, 120, 127–132. [http://dx.doi.org/10.1016/0370-2693(83)90637-8CrossRef]

abbott Abbott, L.F.; Sikivie, P. A cosmological bound on the invisible axion. Phys. Lett. B 1983, 120, 133–136. [http://dx.doi.org/10.1016/0370-2693(83)90638-XCrossRef]

dine Dine, M.; Fischler, W. The not-so-harmless axion. Phys. Lett. B 1983, 120, 137–141. [http://dx.doi.org/10.1016/0370-2693(83)90639-1CrossRef]

arias2012 Arias, P.; Cadamuro, D.; Goodsell, M.; Jaeckel, J.; Redondo, J.; Ringwald, A. WISPy cold dark matter. J. Cosmol. Astropart. Phys. 2012, 2012, 013. [http://dx.doi.org/10.1088/1475-7516/2012/06/013CrossRef]

mpz Maiani, L.; Petronzio, R.; Zavattini, E. Effects of nearly massless, spin-zero particles on light propagation in a magnetic field. Phys. Let. B 1986, 175, 359–363. [http://dx.doi.org/10.1016/0370-2693(86)90869-5CrossRef]

raffeltstodolsky Raffelt, G.G.; Stodolsky, L. Mixing of the photon with low-mass particles. Phys. Rev. D 1988, 37, 1237. [http://dx.doi.org/10.1103/PhysRevD.37.1237CrossRef] [http://www.ncbi.nlm.nih.gov/pubmed/9958803PubMed]

irastorzaredondo Irastorza, I.G.; Redondo, J. New experimental approaches in the search for axion-like particles. Prog. Part. Nucl. Phys. 2018, 102, 89–159. [http://dx.doi.org/10.1016/j.ppnp.2018.05.003CrossRef]

grRew Galanti, G.; Roncadelli, M. Axion-like Particles Implications for High-Energy Astrophysics. Universe 2022, 8, 253. [http://dx.doi.org/10.3390/universe8050253CrossRef]

trg2015 Tavecchio, F.; Roncadelli, M.; Galanti, G. Photons to axion-like particles conversion in Active Galactic Nuclei. Phys. Lett. B 2015, 744, 375–379. [http://dx.doi.org/10.1016/j.physletb.2015.04.017CrossRef]

trgb2012 Tavecchio, F.; Roncadelli, M.; Galanti, G.; Bonnoli, G. Evidence for an axion-like particle from PKS 1222+216? Phys. Rev. D 2012, 86, 085036. [http://dx.doi.org/10.1103/PhysRevD.86.085036CrossRef]

grdb Galanti, G.; Roncadelli, M.; De Angelis, A.; Bignami, G.F. Hint at an axion-like particle from the redshift dependence of blazar spectra. Mon. Not. Astron. R. Soc. 2020, 493, 1553. [http://dx.doi.org/10.1093/mnras/stz3410CrossRef]

drm2007 De Angelis, A.; Roncadelli, M.; Mansutti, O. Evidence for a new light spin-zero boson from cosmological gamma-ray propagation? Phys. Rev. D 2007, 76, 121301. [http://dx.doi.org/10.1103/PhysRevD.76.121301CrossRef]

dgr2011 De Angelis, A.; Galanti, G.; Roncadelli, M. Relevance of axionlike particles for very-high-energy astrophysics. Phys. Rev. D 2011, 84, 105030; Erratum in Phys. Rev. D 2013, 87, 109903. [http://dx.doi.org/10.1103/PhysRevD.84.105030CrossRef]

grExt Galanti, G.; Roncadelli, M. 
Extragalactic photon–axion-like particle oscillations up to 1000 TeV.  J. High Energy Astrophys. 2018, 20, 1–17. [http://dx.doi.org/10.1016/j.jheap.2018.07.002CrossRef]

dgr2013 De Angelis, A.; Galanti, G.; Roncadelli, M. Transparency of the Universe to gamma rays. Mon. Not. Astron. R. Soc. 2013, 432, 3245. [http://dx.doi.org/10.1093/mnras/stt684CrossRef]

franceschinirodighiero Franceschini, A.; Rodighiero, G. The extragalactic background light revisited and the cosmic photon-photon opacity. Astron. Astrophys. 2017, 603, A34. [http://dx.doi.org/10.1051/0004-6361/201629684CrossRef]

gptr Galanti, G.; Piccinini, F.; Roncadelli, M.; Tavecchio, F. Estimating γγ absorption for ultrahigh-energy photons with lepton and hadron production. Phys. Rev. D 2020, 102, 123004. [http://dx.doi.org/10.1103/PhysRevD.102.123004CrossRef]

gtre2019 Galanti, G.; Tavecchio, F.; Roncadelli, M.; Evoli, C. Blazar VHE spectral alterations induced by photon–ALP oscillations. Mon. Not. Astron. R. Soc. 2019, 487, 123. [http://dx.doi.org/10.1093/mnras/stz1144CrossRef]

gtl2020 Galanti, G.; Tavecchio, F.; Landoni, M. Fundamental physics with blazar spectra: A critical appraisal. Mon. Not. Astron. R. Soc. 2020, 491, 5268. [http://dx.doi.org/10.1093/mnras/stz3411CrossRef]

astri Vercellone, S.; Bigongiari, C.; Burtovoi, A.; Cardillo, M.; Catalano, O.; Franceschini, A.; Lombardi, S.; Nava, L.; Pintore, F.; Stamerra, A.; et al. ASTRI Mini-Array core science at the Observatorio del Teide. J. High Energy Astrophys. 2022, 35, 1–42. [http://dx.doi.org/10.1016/j.jheap.2022.05.005CrossRef]

cta  Available online: <https://www.cta-observatory.org/> (accessed on 10 June 2024).

g400 Egorov, A.E.; Topchiev, N.P.; Galper, A.M.; Dalkarov, O.D.; Leonov, A.A.; Suchkov, S.I.; Yurkin, Y.T. Dark matter searches by the planned gamma-ray telescope GAMMA-400. J. Cosmol. Astropart. Phys. 2020, 11, 049. [http://dx.doi.org/10.1088/1475-7516/2020/11/049CrossRef]

hawc Available online: <https://www.hawc-observatory.org/> (accessed on 10 June 2024).

herd Huang, X.; Lamperstorfer, A.S.; Tsai, Y.L.S.; Xu, M.; Yuan, Q.; Chang, J.; Dong, Y.W.; Hu, B.L.; Lü, J.G.; Wang, L.; et al. Perspective of monochromatic gamma-ray line detection with the High Energy cosmic-Radiation Detection (HERD) facility onboard China's space station. Astropart. Phys. 2016, 78, 35–42. [http://dx.doi.org/10.1016/j.astropartphys.2016.02.003CrossRef]

LHAASOsens  Cao, Z.; della Volpe, D.; Liu, S.; Bi, X.; Chen, Y.; D'Ettorre Piazzoli, B.; Feng, L.; Jia, H.; Li, Z.; Ma, X.; et al. The Large High Altitude Air Shower Observatory (LHAASO) Science Book (2021 Edition). Chin. Phys. C 2022,  46, 035001–035007.

desy Available online: <https://taiga-experiment.info/taiga-hiscore/> (accessed on 10 June 2024).

fermi2016 Ajello, M. et al. [Fermi-LAT collaboration]. Search for spectral irregularities due to photon–axionlike-particle oscillations with the Fermi Large Area Telescope. Phys. Rev. Lett. 2016, 116, 161101. [http://dx.doi.org/10.1103/PhysRevLett.116.161101CrossRef] [http://www.ncbi.nlm.nih.gov/pubmed/27152783PubMed]

CTAfund Abdalla, H.; Abe, H.; Acero, F.; Acharyya, A.; Adam, R.; Agudo, I.; Aguirre-Santaella, A.; Alfaro, R.; Alfaro, J.; Alispach, C.; et al. Sensitivity of the Cherenkov Telescope Array for probing cosmology and fundamental physics with gamma-ray propagation. J. Cosmol. Astropart. Phys. 2021, 2021, 048. [http://dx.doi.org/10.1088/1475-7516/2021/02/048CrossRef]

straniero Ayala, A.; Domínguez, I.; Giannotti, M.; Mirizzi, A.; Straniero, O. Revisiting the Bound on Axion-Photon Coupling from Globular Clusters. Phys. Rev. Lett. 2014, 113, 191302. [http://dx.doi.org/10.1103/PhysRevLett.113.191302CrossRef] [http://www.ncbi.nlm.nih.gov/pubmed/25415896PubMed]

thermal Conlon, J.P.; Marsh, M.C.D.; Powell, A.J. Galaxy cluster thermal x-ray spectra constrain axionlike particles. Phys. Rev. D 2016, 93, 123526. [http://dx.doi.org/10.1103/PhysRevD.93.123526CrossRef]

DMdecay Jaeckel, J.; Redondo, J.; Ringwald, A. 3.55 keV hint for decaying axionlike particle dark matter. Phys. Rev. D 2014, 89, 103511. [http://dx.doi.org/10.1103/PhysRevD.89.103511CrossRef]

wang Zhou, J.; Wang, Z.; Huang, F.; Chen, L. A possible blazar spectral irregularity case caused by photon-axionlike-particle oscillations. J. Cosmol. Astropart. Phys. 2021, 2021, 007. [http://dx.doi.org/10.1088/1475-7516/2021/08/007CrossRef]

LHAASO LHAASO Collaboration. GCN Circular n. 32677. 2022. Available online: <https://gcn.gsfc.nasa.gov/gcn3/32677.gcn3> (accessed on 10 June 2024).

LHAASOspectrumHigh Cao, Z. et al. [The LHAASO Collaboration]. Very high-energy gamma-ray emission beyond 10 TeV from GRB 221009A. Sci. Adv. 2023, 9, eadj2778.

gnrtb2023 Galanti, G.; Nava, L.; Roncadelli, M.; Tavecchio, F.; Bonnoli, G. Observability of the Very-High-Energy Emission from GRB 221009A. Phys. Rev. Lett. 2023, 131, 251001. [http://dx.doi.org/10.1103/PhysRevLett.131.251001CrossRef] [http://www.ncbi.nlm.nih.gov/pubmed/38181368PubMed]

ALPinGRB2 Galanti, G.; Roncadelli, M.; Tavecchio, F. Assessment of ALP scenarios for GRB 221009A. arXiv 2022, arXiv:2211.06935.

ixpe Weisskopf, M.C.; Soffitta, P.; Baldini, L.; Ramsey, B.D.; O’Dell, S.L.; Romani, R.W.; Matt, G.; Deininger, W.D.; Baumgartner, W.H.; Bellazzini, R.; et al. Imaging X-ray Polarimetry Explorer: Prelaunch. J. Astron. Telesc. Instrum. Syst. 2022, 8, 026002. [http://dx.doi.org/10.1117/1.JATIS.8.2.026002CrossRef]

extp  Zhang, S.; Santangelo, A.; Feroci, M.; Xu, Y.; Lu, F.; Chen, Y.; Feng, H.; Zhang, S.; Brandt, S.; Hernanz, M.; et al. The enhanced X-ray Timing and Polarimetry mission–eXTP. Sci. China Phys. Mech. Astron. 2019, 62, 29502. [http://dx.doi.org/10.1007/s11433-018-9309-2CrossRef]

xcalibur Abarr, Q.; Awaki, H.; Baring, M.G.; Bose, R.; De Geronimo, G.; Dowkontt, P.; Errando, M.; Guarino, V.; Hattori, K.; Hayashida, K.;  et al. XL-Calibur—A second-generation balloon-borne hard X-ray polarimetry mission. Astropart. Phys. 2021, 126, 102529. [http://dx.doi.org/10.1016/j.astropartphys.2020.102529CrossRef]

ngxp Soffitta, P.; Bucciantini, N.; Churazov, E.; Costa, E.; Dovciak, M.; Feng, H.; Heyl, J.; Ingram, A.; Jahoda, K.; Kaaret, P.;  et al. A polarized view of the hot and violent universe. Exp. Astron. 2021, 51, 1109–1141. [http://dx.doi.org/10.1007/s10686-021-09722-yCrossRef]

xpp Jahoda, K.; Krawczynski, H.; Kislat, F.; Marshall, H.; Okajima, T.; Agudo, I.; Angelini, L.; Bachetti, M.; Baldini, L.; Baring, M.; et al. The X-ray Polarization Probe mission concept. arXiv 2019, arXiv:1907.10190.

cosi Yang, C.-Y.; Lowell, A.; Zoglauer, A.; Tomsick, J.; Chiu, J.-L.; Kierans, C.; Sleator, C.; Boggs, S.; Chang, H.-K.; Jean, P.; et al. The polarimetric performance of the Compton Spectrometer and Imager (COSI). Proc. SPIE 2018, 10699, 642.

eastrogam1 De Angelis, A.; Tatischeff, V.; Tavani, M.; Oberlack, U.; Grenier, I.; Hanlon, L.; Walter, R.; Argan, A.; von Ballmoos, P.; Bulgarelli, A.; et al. The e-ASTROGAM mission. Exp. Astron. 2017, 44, 25–82. [http://dx.doi.org/10.1007/s10686-017-9533-6CrossRef]

eastrogam2 Tatischeff, V. et al. [e-ASTROGAM Collaboration]. The e-ASTROGAM gamma-ray space observatory for the multimessenger astronomy of the 2030s. Proc. SPIE 2018, 10699, 106992J.

amego Kierans, C.A. et al. [AMEGO Collaboration]. AMEGO: Exploring the Extreme Multimessenger Universe. Proc. SPIE 2020, 11444, 1144431.

bassan Bassan, N.; Mirizzi, A.; Roncadelli, M. Axion-like particle effects on the polarization of cosmic high-energy gamma sources. J. Cosmol. Astropart. Phys. 2010, 2010, 010. [http://dx.doi.org/10.1088/1475-7516/2010/05/010CrossRef]

ALPpol1 Jain, P.; Panda, S.; Sarala, S. Electromagnetic polarization effects due to axion-photon mixing. Phys. Rev. D 2002, 66, 085007. [http://dx.doi.org/10.1103/PhysRevD.66.085007CrossRef]

ALPpol2 Agarwal, N.; Kamal, A.; Jain, P. Alignments in quasar polarizations: Pseudoscalar-photon mixing in the presence of correlated magnetic fields. Phys. Rev. D 2011, 83, 065014. [http://dx.doi.org/10.1103/PhysRevD.83.065014CrossRef]

ALPpolMWD Gill, R.; Heyl, J.S. Constraining the photon-axion coupling constant with magnetic white dwarfs. Phys. Rev. D 2011, 84, 085001. [http://dx.doi.org/10.1103/PhysRevD.84.085001CrossRef]

ALPpol3 Payez, A.; Cudell, J.R.; Hutsemekers, D. Can axionlike particles explain the alignments of the polarizations of light from quasars? Phys. Rev. D 2011, 84, 085029. [http://dx.doi.org/10.1103/PhysRevD.84.085029CrossRef]

ALPpol4 Agarwal, N.; Aluri, P.K.; Jain, P.; Khanna, U.; Tiwari, P. A complete 3D numerical study of the effects of pseudoscalar–photon mixing on quasar polarizations. Eur. Phys. J. C 2012, 72, 1928. [http://dx.doi.org/10.1140/epjc/s10052-012-1928-yCrossRef]

ALPpol5 Perna, R.; Ho, W.C.G.; Verde, L.; Adelsberg, M. Jimenez, R. Signatures of photon-axion conversion in the thermal spectra and polarization of neutron stars. Astrophys. J. 2012, 748, 116. [http://dx.doi.org/10.1088/0004-637X/748/2/116CrossRef]

day Day, F.; Krippendorf, S. Searching for axion-like particles with X-ray polarimeters. Galaxies 2018, 6, 45. [http://dx.doi.org/10.3390/galaxies6020045CrossRef]

galantiTheorems Galanti, G. Photon-ALP interaction as a measure of initial photon polarization. Phys. Rev. D 2022, 105, 083022. [http://dx.doi.org/10.1103/PhysRevD.105.083022CrossRef]

galantiPol Galanti, G. Photon-ALP oscillations inducing modifications to photon polarization. Phys. Rev. D 2023, 107, 043006. [http://dx.doi.org/10.1103/PhysRevD.107.043006CrossRef]

grtcClu Galanti, G.; Roncadelli, M.; Tavecchio, F.; Costa, E. ALP induced polarization effects on photons from galaxy clusters. Phys. Rev. D 2023, 107, 103007. [http://dx.doi.org/10.1103/PhysRevD.107.103007CrossRef]

grtPolBla Galanti, G.; Roncadelli, M.; Tavecchio, F. ALP-induced polarization effects on photons from blazars. Phys. Rev. D 2023, 108, 083017. [http://dx.doi.org/10.1103/PhysRevD.108.083017CrossRef]

cast CAST Collaboration. New CAST limit on the axion–photon interaction. Nat. Phys. 2017, 13, 584–590. [http://dx.doi.org/10.1038/nphys4109CrossRef]

payez2015 Payez, A.; Evoli, C.; Fischer, T.; Giannotti, M.; Mirizzi, A.; Ringwald, A. Revisiting the SN1987A gamma-ray limit on ultralight axion-like particles. J. Cosmol. Astropart. Phys. 2015, 2015, 006. [http://dx.doi.org/10.1088/1475-7516/2015/02/006CrossRef]

berg Berg, M.; Conlon, J.P.; Day, F.; Jennings, N.; Krippendorf, S.; Powell, A.J.; Rummel, M. Constraints on axion-like particles from X-ray observations of NGC1275. Astrophys. J. 2017, 847, 101. [http://dx.doi.org/10.3847/1538-4357/aa8b16CrossRef]

conlonLim Conlon,  J.P.; Day, F.; Jennings, N.; Krippendorf, S.; Rummel, M. Constraints on axion-like particles from non-observation of spectral modulations for X-ray point sources. J. Cosmol. Astropart. Phys. 2017, 2017, 005. [http://dx.doi.org/10.1088/1475-7516/2017/07/005CrossRef]

meyer2020 Meyer, M. et al. [Fermi-LAT Collaboration]. Search for Axionlike-Particle-Induced Prompt γ-Ray Emission from Extragalactic Core-Collapse Supernovae with the Fermi Large Area Telescope. Phys. Rev. Lett. 2020, 124, 231101; Erratum in Phys. Rev. Lett. 2020, 125, 119901. [http://dx.doi.org/10.1103/PhysRevLett.124.231101CrossRef]

limFabian Reynolds, C.S.; Marsh, M.C.D.; Russell, H.R.; Fabian, A.C.; Smith, R.; Tombesi, F.; Veilleux, S. Astrophysical limits on very light axion-like particles from Chandra grating spectroscopy of NGC 1275. Astrophys. J. 2020, 890, 59. [http://dx.doi.org/10.3847/1538-4357/ab6a0cCrossRef]

limJulia Sisk-Reynés, J.; Matthews, J.H.; Reynolds, C.S.; Russell, H.R.; Smith, R.N.; Marsh, M.C.D. New constraints on light axion-like particles using Chandra transmission grating spectroscopy of the powerful cluster-hosted quasar H1821+643. Mon. Not. Astron. R. Soc. 2022, 510, 1264. [http://dx.doi.org/10.1093/mnras/stab3464CrossRef]

limKripp Schallmoser, S.; Krippendorf, S.; Chadha-Day, F.; Weller, J. Updated Bounds on Axion-Like Particles from X-ray Observations. Mon. Not. Astron. R. Soc. 2022, 514, 329. [http://dx.doi.org/10.1093/mnras/stac1224CrossRef]

limRey2 Matthews, J.H.; Reynolds, C.S.; Marsh, M.C.D.; Sisk-Reynés, J.; Rodman, P.E. How do Magnetic Field Models Affect Astrophysical Limits on Light Axion-like Particles? An X-ray Case Study with NGC 1275. Astrophys. J. 2022, 930, 90. [http://dx.doi.org/10.3847/1538-4357/ac5625CrossRef]

mwd Dessert, C.; Dunsky, D.; Safdi, B.R. Upper limit on the axion-photon coupling from magnetic white dwarf polarization. Phys. Rev. D 2022, 105, 103034. [http://dx.doi.org/10.1103/PhysRevD.105.103034CrossRef]

hew1 Heisenberg, W.; Euler, H. Folgerungen aus der diracschen theorie des positrons. Z. Phys. 1936, 98, 714–732. [http://dx.doi.org/10.1007/BF01343663CrossRef]

hew2 Weisskopf, V.S. Über die elektrodynamik des vakuums auf grund des quanten-theorie des elektrons. K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 1936, 14, 6.

hew3 Schwinger, J. On Gauge Invariance and Vacuum Polarization. Phys. Rev. 1951, 82, 664. [http://dx.doi.org/10.1103/PhysRev.82.664CrossRef]

raffelt2015 Dobrynina, A.; Kartavtsev, A.; Raffelt, G. Photon-photon dispersion of TeV gamma rays and its role for photon-ALP conversion. Phys. Rev. D 2015, 91, 083003; Erratum in Phys. Rev. D 2015, 91, 109902. [http://dx.doi.org/10.1103/PhysRevD.91.083003CrossRef]

poltheor1 Kosowsky, A. Cosmic Microwave Background Polarization. Ann. Phys. 1996, 246, 49. [http://dx.doi.org/10.1006/aphy.1996.0020CrossRef]

poltheor2 Rybicki, G.B.; Lightman, A.P. Radiative Processes in Astrophysics; Wiley: New York, NY, USA, 1979.

blazarSeq Ghisellini, G.; Righi, C.; Costamante, L.; Tavecchio, F. The Fermi blazar sequence. Mon. Not. Astron. R. Soc. 2017, 469, 255. [http://dx.doi.org/10.1093/mnras/stx806CrossRef]

jansonfarrar1 Jansson, R.; Farrar, G.R. A New Model of the Galactic Magnetic Field. Astrophys. J. 2012, 757, 14. [http://dx.doi.org/10.1088/0004-637X/757/1/14CrossRef]

jansonfarrar2 Jansson, R.; Farrar, G.R. The galactic magnetic field. Astrophys. J. 2012, 761, L11. [http://dx.doi.org/10.1088/2041-8205/761/1/L11CrossRef]

yaomanchesterwang2017 Yao, J.M.; Manchester, R.N.; Wang, N. A New Electron-density Model for Estimation of Pulsar and FRB Distances. Astrophys. J. 2017, 835, 29. [http://dx.doi.org/10.3847/1538-4357/835/1/29CrossRef]

optical 
Roming, P.W.A.; Kennedy, T.E.; Mason, K.O.; Nousek, J.A.; Ahr, L.; Bingham, R.E.; Broos, P.S.; Carter, M.J.; Hancock, B.K.; Huckle, H.E.; et al. The Swift Ultra-Violet/Optical Telescope. Space Sci. Rev. 2005, 120, 95. [http://dx.doi.org/10.1007/s11214-005-5095-4CrossRef]

swift Burrows, D.N.; Hill, J.E.; Nousek, J.; Kennea, J.A.; Wells, A.; Osborne, J.P.; Abbey, A.F.; Beardmore, A.; Mukerjee, K.; Short, A.D.T.;  et al. The Swift X-ray Telescope. Space Sci. Rev. 2005, 120, 165. [http://dx.doi.org/10.1007/s11214-005-5097-2CrossRef]

fermiSens Available online: <https://www.slac.stanford.edu/exp/glast/groups/canda/lat_Performance.htm> (accessed on 10 June 2024).

AGNfig1 Urry, C.M.; Padovani, P. Unified schemes for radio-loud active galactic nuclei. Pub. Astron. Soc. Pacific 1995, 107, 803. [http://dx.doi.org/10.1086/133630CrossRef]

AGNfig2 Collmar, W. News from CGRO-COMPTEL. Mem. Soc. Astron. Ital. 2002, 73, 99.

AGNfig3 Torres, D.F.; Anchordoqui, L.A. Astrophysical origins of ultrahigh energy cosmic rays. Rep. Prog. Phys. 2004, 67, 1663. [http://dx.doi.org/10.1088/0034-4885/67/9/R03CrossRef]

FSRQabs1 Liu, T.H.; Bai, J.M. Absorption of 10–200 GeV gamma rays by radiation from broad-line regions in blazars. Astrophys. J. 2006, 653, 1089. [http://dx.doi.org/10.1086/509097CrossRef]

FSRQabs2 Tavecchio, F.; Mazin, D. Intrinsic absorption in 3C 279 at GeV–TeV energies and consequences for estimates of the extragalactic background light. Mon. Not. Astron. R. Soc. 2009, 392, L40. [http://dx.doi.org/10.1111/j.1745-3933.2008.00584.xCrossRef]

FSRQabs3 Poutanen, J.; Stern, B. GeV Breaks in Blazars as a Result of Gamma-ray Absorption within the Broad-Line Region. Astrophys. J. 2010, 717, L118. [http://dx.doi.org/10.1088/2041-8205/717/2/L118CrossRef]

gltcTorus Galanti, G.; Landoni, M.; Tavecchio, F.; Covino, S. Probing the absorption of gamma-rays by IR radiation from the dusty torus in FSRQs with the Cherenkov telescope array. Mon. Not. Astron. R. Soc. 2020, 495, 3463. [http://dx.doi.org/10.1093/mnras/staa1391CrossRef]

FSRQvhe1 Albert, J.; Aliu, E.; Anderhub, H.; Antonelli, L.A.; Antoranz, P.; Backes, M.; Baixeras, C.; Barrio, J.A.; Bartko, H.; Bastieri, D.; et al. Very-High-Energy Gamma Rays from a Distant Quasar: How Transparent Is the Universe? Science 2008, 320, 1752. [http://dx.doi.org/10.1126/science.1157087CrossRef]

FSRQvhe2 Aleksić, J.; Antonelli, L.A.; Antoranz, P.; Backes, M.; Barrio, J.A.; Bastieri, D.; Becerra González, J.; Bednarek, W.; Berdyugin, A.; Berger, K.; et al. MAGIC Discovery of Very High Energy Emission from the FSRQ PKS 1222+21. Astrophys. J. 2011, 730, L8. [http://dx.doi.org/10.1088/2041-8205/730/1/L8CrossRef]

FSRQvhe3 Wagner, S.J.; Behera, B. Hess Observations of Quasars. In Proceedings of the 10th HEAD Meeting, BAAS, Honolulu, HI, USA, 5 July 2010; Volume 41.

riaf1 Ghisellini, G.; Maraschi, L.; Tavecchio, F. The Fermi blazars’ divide. Mon. Not. Astron. R. Soc. 2009, 396, L105–L109. [http://dx.doi.org/10.1111/j.1745-3933.2009.00673.xCrossRef]

riaf2 Righi, C.; Tavecchio, F.; Inoue, S. Neutrino emission from BL Lac objects: The role of radiatively inefficient accretion flows. Mon. Not. Astron. R. Soc. 2019, 483, L127–L131. [http://dx.doi.org/10.1093/mnrasl/sly231CrossRef]

bbr1984 Begelman, M.C.; Blandford, R.D.; Rees, M.J. Theory of extragalactic radio sources. Rev. Mod. Phys. 1984, 56, 255–351. [http://dx.doi.org/10.1103/RevModPhys.56.255CrossRef]

ghisellini2009 Ghisellini, G.; Tavecchio, F. Canonical high-power blazars. Mon. Not. Astron. R. Soc. 2009, 397, 985–1002. [http://dx.doi.org/10.1111/j.1365-2966.2009.15007.xCrossRef]

pudritz2011 Pudritz, R.E.; Hardcastle, M.J.; Gabuzda, D.C. Magnetic Fields in Astrophysical Jets: From Launch to Termination. Space Sci. Rev. 2012, 169, 27–72. [http://dx.doi.org/10.1007/s11214-012-9895-zCrossRef]

tavecchio2010 Tavecchio, F.; Ghisellini, G.; Ghirlanda, G.; Foschini, L.; Maraschi, L. TeV BL Lac objects at the dawn of the Fermi era. Mon. Not. Astron. R. Soc. 2010, 401, 1570–1586. [http://dx.doi.org/10.1111/j.1365-2966.2009.15784.xCrossRef]

Maraschi92 Maraschi, L.; Ghisellini, G.; Celotti, A. A jet model for the gamma-ray emitting blazar 3C 279. Astrophys. J. 1992, 397, L5–L9. [http://dx.doi.org/10.1086/186531CrossRef]

Sikora94 Sikora, M.; Begelman, M.G.; Rees, M.J. Comptonization of diffuse ambient radiation by a relativistic jet: The source of gamma rays from blazars? Astrophys. J. 1994, 421, 153–162. [http://dx.doi.org/10.1086/173633CrossRef]

ssc1 Bloom, S.D.; Marscher, A.P. An analysis of the synchrotron self-compton model for the multi–wave band spectra of blazars. Astrophys. J. 1996, 461, 657. [http://dx.doi.org/10.1086/177092CrossRef]

mannheim1 Mannheim, K.  The proton blazar. Astron. Astrophys. 1993, 269, 67–76.

mannheim2 Mannheim, K. TeV gamma-rays from proton blazars. Space Sci. Rev. 1996, 75, 331–340. [http://dx.doi.org/10.1007/BF00195043CrossRef]

Muecke2003 Mücke, A.; Protheroe, R.J.; Engel, R.; Rachen, J.P.; Stanev, T. BL Lac objects in the synchrotron proton blazar model. Astropart. Phys. 2003, 18, 593. [http://dx.doi.org/10.1016/S0927-6505(02)00185-8CrossRef]

zhangBott Zhang, H.; Böttcher, M. X-ray and Gamma-Ray Polarization in Leptonic and Hadronic Jet Models of Blazars. Astrophys. J. 2013, 774, 18. [http://dx.doi.org/10.1088/0004-637X/774/1/18CrossRef]

SpiralBrev Beck, R. Magnetic fields in spiral galaxies. Astron. Astrophys. Rev. 2016, 24, 4. [http://dx.doi.org/10.1007/s00159-015-0084-4CrossRef]

Fletcher2010 Fletcher, A. Magnetic Fields in Nearby Galaxies. The Dynamic Interstellar Medium: A Celebration of the Canadian Galactic Plane Survey. Astron. Soc. Pac. Conf. Ser. 2011, 438, 197.

Thompson2006 Thompson, T.A.; Quataert, E.; Waxman, E.; Murray, N.;  Martin, C.L. Magnetic Fields in Starburst Galaxies and the Origin of the FIR-Radio Correlation. Astrophys. J. 2006, 645, 186. [http://dx.doi.org/10.1086/504035CrossRef]

LopezRodriguez2021 Lopez-Rodriguez, E.; Guerra, J.A.; Asgari-Targhi, M.; Schmelz, J.T. The Strength and Structure of the Magnetic Field in the Galactic Outflow of Messier 82. Astrophys. J. 2021, 914, 24. [http://dx.doi.org/10.3847/1538-4357/abf934CrossRef]

GRBposition Blanchard, P.K.; Berger, E.; Fong, W. The Offset and Host Light Distributions of Long Gamma-Ray Bursts: A New View from HST Observations of Swift Bursts. Astrophys. J. 2016, 817, 144. [http://dx.doi.org/10.3847/0004-637X/817/2/144CrossRef]

GRBposition2 Lyman, J.D.; Levan, A.J.; Tanvir, N.R.; Fynbo, J.P.U.; McGuire, J.T.W.; Perley, D.A.; Angus, C.R.; Bloom, J.S.; Conselice, C.J.; Fruchter, A.S.;  et al. The host galaxies and explosion sites of long-duration gamma-ray bursts: Hubble Space Telescope near-infrared imaging. Mon. Not. Astron. R. Soc. 2017, 467, 1795–1817. [http://dx.doi.org/10.1093/mnras/stx220CrossRef]

GRB221009Ahost Levan, A.J.; Lamb, G.P.; Schneider, B.; Hjorth, J.; Zafar, T.; de Ugarte Postigo, A.; Sargent, B.; Mullally, S.E.; Izzo, L.; d’Avanzo, P.;  et al. The First JWST Spectrum of a GRB Afterglow: No Bright Supernova in Observations of the Brightest GRB of all Time, GRB 221009A. Astrophys. J. Lett. 2023, 946, L28. [http://dx.doi.org/10.3847/2041-8213/acc2c1CrossRef]

moss1996 Moss, D.; Shukurov, A. Turbulence and Magnetic Fields in Elliptical Galaxies. Mon. Not. Astron. R. Soc. 1996, 279, 229–239. [http://dx.doi.org/10.1093/mnras/279.1.229CrossRef]

GalCluClass Bahcall, N.A. Clusters and superclusters of galaxies. In Formation of Structure in the Universe; Dekel, A., Ostriker, J.P., Eds.; Cambridge University Press: Cambridge, UK, 1999.

cluB1 Govoni, F.; Feretti, L. Magnetic fields in clusters of galaxies. Int. J. Mod. Phys. D 2004, 13, 1549–1594. [http://dx.doi.org/10.1142/S0218271804005080CrossRef]

cluB2 Feretti, L.; Giovannini, G.; Govoni, F.; Murgia, M. Clusters of galaxies: Observational properties of the diffuse radio emission. Astron. Astrophys. Rev. 2012, 20, 54. [http://dx.doi.org/10.1007/s00159-012-0054-zCrossRef]

cluFeretti Bonafede, A.; Feretti, L.; Murgia, M.; Govoni, F.; Giovannini, G.; Dallacasa, D.; Dolag, K.; Taylor, G.B. The Coma cluster magnetic field from Faraday rotation measures. Astron. Astrophys. 2010, 513, A30. [http://dx.doi.org/10.1051/0004-6361/200913696CrossRef]

clu2 Kuchar, P.; Enßlin, T.A. Magnetic power spectra from Faraday rotation maps. Astron. Astrophys. 2011, 529, A13. [http://dx.doi.org/10.1051/0004-6361/200913918CrossRef]

meyerKolm Meyer, M.; Montanino, D.; Conrad, J. On detecting oscillations of gamma rays into axion-like particles in turbulent and coherent magnetic fields. J. Cosmol. Astropart. Phys. 2014, 2014, 003. [http://dx.doi.org/10.1088/1475-7516/2014/09/003CrossRef]

cluValues Hudson, D.S.; Mittal, R.; Reiprich, T.H.; Nulsen, P.E.J.; Andernach, H.; Sarazin, C.L. What is a cool-core cluster? A detailed analysis of the cores of the X-ray flux-limited HIFLUGCS cluster sample. Astron. Astrophys. 2010, 513, A37. [http://dx.doi.org/10.1051/0004-6361/200912377CrossRef]

mitchell1979 Mitchell, R.J.; Dickens, R.J.; Bell Burnell, J.S.; Culhane, J.L. The X-ray spectra of clusters of galaxies and their relationship to other cluster properties. Mon. Not. Astron. R. Soc. 1979, 189, 329. [http://dx.doi.org/10.1093/mnras/189.2.329CrossRef]

cluGammaEm1 Houston, B.P.; Wolfendale, A.W.; Young, E.C.M. Cosmic gamma rays from galaxy clusters. Phys, J. G Nucl. Phys. 1984, 10, L147–L150. [http://dx.doi.org/10.1088/0305-4616/10/6/007CrossRef]

cluGammaEm2 Dennison, B. Formation of radio halos in clusters of galaxies from cosmic-ray protons. Astrophys. J. 1980, 239, L93–L96. [http://dx.doi.org/10.1086/183300CrossRef]

cluGammaEm3 Colafrancesco, S.; Marchegiani, P. Warming rays in cluster cool cores. Astron. Astrophys. 2008, 484, 51–65. [http://dx.doi.org/10.1051/0004-6361:20078428CrossRef]

cluGammaEm4 Timokhin, A.N.; Aharonian, F.A.; Neronov, A.Y. On the non-thermal high energy radiation of galaxy clusters. Astron. Astrophys. 2004, 417, 391. [http://dx.doi.org/10.1051/0004-6361:20040004CrossRef]

polarRev Lei, F.; Dean, A.J.; Hills, G.L. Compton Polarimetry in Gamma-Ray Astronomy. Space Sci. Rev. 1997, 82, 309. [http://dx.doi.org/10.1023/A:1005027107614CrossRef]

FermiPol Giomi, M.; Bühler, R.; Sgrò, C.; Longo, F.; Atwood, W.B.;  on behalf of the Fermi LAT Collaboration. Estimate of the Fermi large area telescope sensitivity to gamma-ray polarization. AIP Conf. Proc. 2017, 1792, 0022.

kronberg1994 Kronberg, P.P. Extragalactic magnetic fields. Rept. Prog. Phys. 1994, 57, 325–382. [http://dx.doi.org/10.1088/0034-4885/57/4/001CrossRef]

grassorubinstein Grasso, D.; Rubinstein, H.R. Magnetic fields in the early Universe. Phys. Rep. 2001, 348, 163–266. [http://dx.doi.org/10.1016/S0370-1573(00)00110-1CrossRef]

wanglai Wang, C.; Lai, D. Axion-Photon Propagation in Magnetized Universe. J. Cosmol. Astropart. Phys. 2016, 6, 006. [http://dx.doi.org/10.1088/1475-7516/2016/06/006CrossRef]

neronovvovk Neronov, A.; Vovk, I. Evidence for Strong Extragalactic Magnetic Fields from Fermi Observations of TeV Blazars. Science 2010, 328, 73. [http://dx.doi.org/10.1126/science.1184192CrossRef]

durrerneronov Durrer, R.; Neronov, A. Cosmological magnetic fields: Their generation, evolution and observation. Astron. Astrophys. Rev. 2013, 21, 62. [http://dx.doi.org/10.1007/s00159-013-0062-7CrossRef]

upbbext Pshirkov, M.S.; Tinyakov, P.G.; Urban, F.R. New Limits on Extragalactic Magnetic Fields from Rotation Measures. Phys. Rev. Lett. 2016, 116, 191302. [http://dx.doi.org/10.1103/PhysRevLett.116.191302CrossRef]

reessetti1968 Rees, M.J.; Setti, G. Model for the Evolution of Extended Radio Sources. Nature 1968, 219, 127. [http://dx.doi.org/10.1038/219127a0CrossRef]

hoyle1969 Hoyle, F. Magnetic Fields and Highly Condensed Objects. Nature 1969, 223, 936. [http://dx.doi.org/10.1038/223936a0CrossRef]

kronbergleschhopp1999 Kronberg, P.P.; Lesch, H.; Hopp, U. Magnetization of the Intergalactic Medium by Primeval Galaxies. Astrophys. J. 1999, 511, 56. [http://dx.doi.org/10.1086/306662CrossRef]

furlanettoloeb2001 Furlanetto, S.; Loeb, A. Intergalactic Magnetic Fields from Quasar Outflows. Astrophys. J. 2001, 556, 619–634. [http://dx.doi.org/10.1086/321630CrossRef]

grSM Galanti, G.; Roncadelli, M. Behavior of axionlike particles in smoothed out domainlike magnetic fields. Phys. Rev. D 2018, 98, 043018. [http://dx.doi.org/10.1103/PhysRevD.98.043018CrossRef]



BMWturb Beck, M.C.; Beck, A.M.; Beck, R.; Dolag, K.; Strong, A.W.;  Nielaba, P. New constraints on modelling the random magnetic field of the MW. J. Cosmol. Astropart. Phys. 2016, 2016, 056. [http://dx.doi.org/10.1088/1475-7516/2016/05/056CrossRef]

pshirkov2011 Pshirkov, M.S.; Tinyakov, P.G.; Kronberg, P.P.; Newton-McGee, K.J. Deriving the Global Structure of the Galactic Magnetic Field from Faraday Rotation Measures of Extragalactic Sources. Astrophys. J. 2011, 738, 192. [http://dx.doi.org/10.1088/0004-637X/738/2/192CrossRef]

nePerseus Churazov, E.; Forman, W.; Jones, C.; Böhringer, H. XMM-Newton Observations of the Perseus Cluster. I. The Temperature and Surface Brightness Structure. Astrophys. J. 2003, 590, 225. [http://dx.doi.org/10.1086/374923CrossRef]

BpersHigh Taylor, G.B.; Gugliucci, N.E.; Fabian, A.C.; Sanders, J.S.; Gentile, G.;  Allen, S.W. Magnetic fields in the centre of the Perseus cluster. Mon. Not. Astron. R. Soc. 2006, 368, 1500–1506. [http://dx.doi.org/10.1111/j.1365-2966.2006.10244.xCrossRef]

BpersLow Aleksić, J. et al. [MAGIC Collaboration]. Constraining cosmic rays and magnetic fields in the Perseus galaxy cluster with TeV observations by the MAGIC telescopes. Astron. Astrophys. 2012, 541, A99. [http://dx.doi.org/10.1051/0004-6361/201118502CrossRef]

LeptHadrBott Böttcher, M.; Reimer, A.; Sweeney, K.; Prakash, A. Leptonic and Hadronic Modeling of Fermi-detected Blazars. Astrophys. J. 2013, 768, 54. [http://dx.doi.org/10.1088/0004-637X/768/1/54CrossRef]

LIVpol Kostelecký, V.A.; Mewes, M. Electrodynamics with Lorentz-violating operators of arbitrary dimension. Phys. Rev. D 2009, 80, 015020. [http://dx.doi.org/10.1103/PhysRevD.80.015020CrossRef]

blazarPolNew Zhang, H.; Böttcher, M.; Liodakis, I. Revisiting High-Energy Polarization from Leptonic and Hadronic Blazar Scenarios. arXiv 2024, arXiv:2404.12475.

IXPEmrk421 Di Gesu, L.; Donnarumma, I.; Tavecchio, F.; Agudo, I.; Barnounin, T.; Cibrario, N.; Di Lalla, N.; Di Marco, A.; Escudero, J.; Errando, M.; et al. The X-ray Polarization View of Mrk 421 in an Average Flux State as Observed by the Imaging X-ray Polarimetry Explorer. Astrophys. J. Lett. 2022, 938, L7. [http://dx.doi.org/10.3847/2041-8213/ac913aCrossRef]

IXPEmrk501 Liodakis, I.; Marscher, A.P.; Agudo, I.; Berdyugin, A.V.; Bernardos, M.I.; Bonnoli, G.; Borman, G.A.; Casadio, C.; Casanova, V.; Cavazzuti, E.; et al. Polarized blazar X-rays imply particle acceleration in shocks. Nature 2022, 611, 677. [http://dx.doi.org/10.1038/s41586-022-05338-0CrossRef]

IXPEblLacertae Middei, R.; Liodakis, I.; Perri, M.; Puccetti, S.; Cavazzuti, E.; Di Gesu, L.; Ehlert, S.R.; Madejski, G.; Marscher, A.P.; Marshall, H.L.;  et al. X-ray Polarization Observations of BL Lacertae. Astrophys. J. Lett. 2023, 942, L10. [http://dx.doi.org/10.3847/2041-8213/aca281CrossRef]

IXPEblLacertae2 Peirson, A.L.; Negro, M.; Liodakis, I.; Middei, R.; Kim, D.E.; Marscher, A.P.; Marshall, H.L.; Pacciani, L.; Romani, R.W.; Wu, K.; et al. X-ray Polarization of BL Lacertae in Outburst. Astrophys. J. Lett. 2023, 948, L25. [http://dx.doi.org/10.3847/2041-8213/acd242CrossRef]

IXPEpg1553 Middei, R.; Perri, M.; Puccetti, S.; Liodakis, I.; Di Gesu, L.; Marscher, A.P.; Cavero, N.R.; Tavecchio, F.; Donnarumma, I.; Laurenti, M.; et al. IXPE and Multiwavelength Observations of Blazar PG 1553+113 Reveal an Orphan Optical Polarization Swing. Astrophys. J. Lett. 2023, 953, L28. [http://dx.doi.org/10.3847/2041-8213/acec3eCrossRef]

IXPE1ES0229 Ehlert, S.R.; Liodakis, I.; Middei, R.; Marscher, A.P.; Tavecchio, F.; Agudo, I.; Kouch, P.M.; Lindfors, E.; Nilsson, K.; Myserlis, I.; et al. X-ray Polarization of the BL Lacertae Type Blazar 1ES 0229+200. Astrophys. J. 2023, 959, 61. [http://dx.doi.org/10.3847/1538-4357/ad05c4CrossRef]

IXPEmrk421-2 Di Gesu, L.; Marshall, H.L.; Ehlert, S.R.; Kim, D.E.; Donnarumma, I.; Tavecchio, F.; Liodakis, I.; Kiehlmann, S.; Agudo, I.; Jorstad, S.G.; et al. Discovery of X-ray polarization angle rotation in the jet from blazar Mrk 421. Nature Astronomy 2023, 7, 1245. [http://dx.doi.org/10.1038/s41550-023-02032-7CrossRef]

IXPE1ES1959 Errando, M.; Liodakis, I.; Marscher, A.P.; Marshall, H.L.; Middei, R.; Negro, M.; Peirson, A.L.; Perri, M.; Puccetti, S.; Rabinowitz, P.L.; et al. Detection of X-ray Polarization from the Blazar 1ES 1959+650 with the Imaging X-ray Polarimetry Explorer. Astrophys. J. 2024, 963, 5. [http://dx.doi.org/10.3847/1538-4357/ad1ce4CrossRef]

IXPEPKS2155 Kouch, P.M.; Liodakis, I.; Middei, R.; Kim, D.E.; Tavecchio, F.; Marscher, A.P.; Marshall, H.L.; Ehlert, S.R.; Di Gesu, L.; Jorstad, S.G.; et al. IXPE observation of PKS 2155-304 reveals the most highly
polarized blazar. arXiv 2024, arXiv:2406.01693.

GRBpol1 Kole, M.; De Angelis, N.; Berlato, F.; Burgess, J.M.; Gauvin, N.; Greiner, J.; Hajdas, W.; Li, H.C.; Li, Z.H.; Pollo, A.;  et al. The POLAR gamma-ray burst polarization catalog. Astron. Astrophys. 2020, 644, A124. [http://dx.doi.org/10.1051/0004-6361/202037915CrossRef]

GRBpol2 Gill, R.; Kole, M.; Granot, J. GRB Polarization: A Unique Probe of GRB Physics. Galaxies 2021, 9, 82. [http://dx.doi.org/10.3390/galaxies9040082CrossRef]

GRBpol3 Chattopadhyay, T.; Gupta, S.; Iyyani, S.; Saraogi, D.; Sharma, V.; Tsvetkova, A.; Ratheesh, A.; Gupta, R.; Mithun, N.P.S.; Vaishnava, C.S.; et al. Hard X-ray Polarization Catalog for a Five-year Sample of Gamma-Ray Bursts Using AstroSat CZT Imager. Astrophys. J. 2022, 936, 12. [http://dx.doi.org/10.3847/1538-4357/ac82efCrossRef]

IXPE-GRB221009A Negro, M.; Di Lalla, N.; Omodei, N.; Veres, P.; Silvestri, S.; Manfreda, A.; Burns, E.; Baldini, L.; Costa, E.; Ehlert, S.R.;  et al. The IXPE View of GRB 221009A. Astrophys. J. Lett. 2023, 946, L21. [http://dx.doi.org/10.3847/2041-8213/acba17CrossRef]

polLimit Depaola, G.O.; Kozameh, C.N.; Tiglio, M.H. A method to determine the polarization of high energy gamma rays. Astropart. Phys. 1999, 10, 175. [http://dx.doi.org/10.1016/S0927-6505(98)00054-1CrossRef]

alps2 Bähre, R.; Döbrich, B.; Dreyling-Eschweiler, J.; Ghazaryan, S.; Hodajerdi, R.; Horns, D.; Januschek, F.; Knabbe, E.-A.; Lindner, A.; Notz, D.; et al. Any light particle search II—Technical design report. J. Instrum. 2013, 8, T09001. [http://dx.doi.org/10.1088/1748-0221/8/09/T09001CrossRef]





iaxo Armengaud, E.; Attié, D.; Basso, S.; Brun, P.; Bykovskiy, N.; Carmona, J.M.; Castel, J.F.; Cebrián, S.; Cicoli, M.; Civitani, M.; et al. Physics potential of the international axion observatory (IAXO). J. Cosmol. Astropart. Phys. 2019, 2019, 047. [http://dx.doi.org/10.1088/1475-7516/2019/06/047CrossRef]

stax Capparelli, L.M.; Cavoto, G.; Ferretti, J.; Giazotto, F.; Polosa, A.D.; Spagnolo, P. Axion-like particle searches with sub-THz photons. Phys. Dark Univ. 2016, 12, 37. [http://dx.doi.org/10.1016/j.dark.2016.01.003CrossRef]

avignone1 Avignone, F.T., III. Potential for large germanium detector arrays for solar-axion searches utilizing the axio-electric effect for detection. Phys. Rev. D 2009, 79, 035015. [http://dx.doi.org/10.1103/PhysRevD.79.035015CrossRef]

avignone2 Avignone, F.T., III; Crewick, R.J.; Nussinov, S.  Can large scintillators be used for solar-axion searches to test the cosmological axion–photon oscillation proposal? Phys. Lett. B 2009, 681, 122–124. [http://dx.doi.org/10.1016/j.physletb.2009.09.011CrossRef]

avignone3 Avignone, F.T., III; Crewick, R.J.; Nussinov, S. The experimental challenge of detecting solar axion-like particles to test the cosmological ALP-photon oscillation hypotheses. Astropart. Phys. 2011, 34, 640–642. [http://dx.doi.org/10.1016/j.astropartphys.2010.12.012CrossRef]

abracadabra Kahn, Y.; Safdi, B.R.; Thaler, J. Broadband and resonant approaches to axion dark matter detection. Phys. Rev. Lett. 2016, 117, 141801. [http://dx.doi.org/10.1103/PhysRevLett.117.141801CrossRef]






























 Paper Title: Mass mixing between QCD axions.





 
lihaijun@itp.ac.cn
Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
 

yfzhou@itp.ac.cn
Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China
International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China

ITP-24-166



§ ABSTRACT 

We introduce a novel level crossing in the mass mixing between two QCD axions, one canonical QCD axion and one Z_𝒩 QCD axion.
The level crossing can take place at the QCD phase transition critical temperature or slightly before it, depending on the ratio of the axion decay constants ∼1.69.
The cosmological evolution of the mass eigenvalues in these two cases is similar, however, the transition of axion energy density is completely different.
Finally, we estimate the relic density of the QCD axion dark matter.
This level crossing may also have some cosmological implications.





Mass mixing between QCD axions
    Yu-Feng Zhou
    August 5, 2024
==============================






Introduction.—
The QCD axions are attractive candidates for the cold dark matter (DM).
The canonical QCD axion was predicted by the Peccei-Quinn (PQ) mechanism <cit.> to solve the strong CP problem in the Standard Model (SM) <cit.>.
It obtains a tiny mass from the QCD non-perturbative effects <cit.>.
As the DM candidate, it can be non-thermally produced in the early Universe through the misalignment mechanism <cit.>.
At high cosmic temperatures, the QCD axion is massless and obtains a non-zero mass during the QCD phase transition, the axion oscillation when its mass is comparable to the Hubble parameter explains the observed DM abundance.
See e.g. Ref. <cit.> for a recent review.
The reduced-mass QCD axion, Z_𝒩 axion <cit.>, can also solve the strong CP problem with 𝒩⩾3 <cit.> and account for the DM through the trapped plus kinetic misalignment mechanism <cit.>.
Here 𝒩 is an odd number, the 𝒩 mirror worlds that are nonlinearly realized by the axion field under a Z_𝒩 symmetry can coexist in Nature.
Due to the suppressed non-perturbative effects on axion potential from the 𝒩 degenerate QCD groups, the Z_𝒩 axion mass is exponentially suppressed at the QCD phase transition.
To avoid confusion, the following “QCD axion" only stands for the canonical QCD axion.

The mass mixing in the multiple axions model <cit.> has attracted extensive attention in recent years.[For instance, the mass mixing between the QCD axion and axionlike particle (ALP) <cit.> or sterile axion <cit.>, and the mixing between the Z_𝒩 axion and ALP <cit.>, etc.]
Considering the non-zero mass mixing between these axions, the cosmological evolution process of the mass eigenvalues called the level crossing can take place and induce the adiabatic transition of the axion energy density, which is similar to the MSW effect <cit.> in the neutrino oscillations.
The level crossing has some interesting cosmological implications, such as the modification of the axion relic density and isocurvature perturbations <cit.>, the domain walls formation <cit.>, the gravitational waves emission and primordial black holes formation <cit.>, and also the dark energy composition <cit.>, etc.

In this letter, we investigate the mass mixing between two QCD axions, one QCD axion and one Z_𝒩 axion.
We find that the level crossing can take place at the QCD phase transition critical temperature or slightly before it, depending on the relation of the axion decay constants.
The conditions for level crossing to occur in these two cases are discussed in detail.
The cosmological evolution of the mass eigenvalues in these two cases is similar, while the transition of energy density between the axions is different.
Finally, we briefly estimate the relic density of the QCD axion and Z_𝒩 axion DM through the misalignment mechanism.
To our knowledge, this is the first time to investigate the effect of level crossing in the mass mixing with a multiple QCD axions model.

The model.—
Here we consider a minimal multiple QCD axions model, one QCD axion ϕ and one Z_𝒩 axion φ, with the low-energy effective Lagrangian

    ℒ   ⊃12∑_ϕ, φ∂^μΦ∂_μΦ-m_a^2(T) f_a^2 [1-cos(ϕf_a)]
       -m_𝒩^2(T) F_a^2𝒩^2[1-cos(ϕf_a+𝒩φF_a)]  ,

where m_a(T) and m_𝒩(T) are the temperature-dependent QCD axion and Z_𝒩 axion masses, respectively, f_a and F_a are the QCD axion and Z_𝒩 axion decay constants.
The QCD axion mass m_a(T) can be described by

    m_π f_πf_a√(z)1+z  ,     T≤ T_ QCD
    m_π f_πf_a√(z)1+z(TT_ QCD)^-b  ,     T> T_ QCD
 
where m_π and f_π are the mass and decay constant of the pion, respectively, z≡ m_u/m_d≃0.48 is the ratio of the up to down quark masses, T_ QCD≃150  MeV is the QCD phase transition critical temperature, and b≃4.08 is an index taken from the dilute instanton gas approximation.
The first term in Eq. (<ref>) corresponds to the zero-temperature QCD axion mass m_a,0.
While the Z_𝒩 axion mass m_𝒩(T) can be described by

    m_π f_π√(π) F_a√(1-z1+z)𝒩^3/4z^𝒩/2  ,    T≤ T_ QCD
    m_π f_πF_a√(z1-z^2)  ,    T_ QCD < T ≤T_ QCDγ
    m_π f_πF_a√(z1-z^2)(γ TT_ QCD)^-b  ,    T>T_ QCDγ

where γ∈(0,1) is a temperature parameter.
The first term in Eq. (<ref>) also corresponds to the zero-temperature Z_𝒩 axion mass m_𝒩,0, which is suddenly exponentially suppressed at T_ QCD due to the Z_𝒩 symmetry.
Then the equations of motion of ϕ and φ are given by

    ϕ̈   +3Hϕ̇+m_a^2(T) f_asin(ϕf_a)
       +m_𝒩^2(T) F_a^2𝒩^2 f_asin(ϕf_a+𝒩φF_a)=0  ,


    φ̈+3Hφ̇+m_𝒩^2(T) F_a𝒩sin(ϕf_a+𝒩φF_a)=0   ,

where H(T) is the Hubble parameter, and the dots represent derivatives with respect to the physical time t.
Diagonalizing the mass mixing matrix

    𝐌^2=
    (
    [ m_a^2(T)+m_𝒩^2(T) F_a^2𝒩^2 f_a^2                 m_𝒩^2(T) F_a𝒩f_a;                 m_𝒩^2(T) F_a𝒩f_a                         m_𝒩^2(T) ])  ,

we derive the heavy (a_h) and light (a_l) mass eigenstates

    (
    [ a_h; a_l ])
    =
    (
    [  cosα  sinα; -sinα  cosα ])
    (
    [ ϕ; φ ])  ,
 
with the corresponding mass eigenvalues m_h,l(T)

    m_h,l^2(T)   =12[m_a^2(T)+m_𝒩^2(T)+ m_𝒩^2(T) F_a^2𝒩^2 f_a^2]
       ±12𝒩^2 f_a^2[-4 𝒩^4 m_a^2(T) m_𝒩^2(T) f_a^4
       +((m_a^2(T)+m_𝒩^2(T))𝒩^2 f_a^2
       +m_𝒩^2(T) F_a^2)^2]^1/2  ,
 
where α is the mass mixing angle

    cos^2α=12(1+m_𝒩^2(T)-m_a^2(T)-m_𝒩^2(T) F_a^2𝒩^2 f_a^2m_h^2(T)-m_l^2(T))  .
 

The so-called axion level crossing is that when considering the interaction between two axion fields, there is a nontrivial mass mixing between their temperature-dependent mass eigenvalues m_h,l(T) that as functions of the cosmic temperature T.
It can take place when the difference of m_h^2(T)-m_l^2(T) gets a minimum value.




Case I: Level crossing at T_ QCD.—
We first consider a case that the zero-temperature QCD axion mass m_a,0 is smaller than the second term in Eq. (<ref>) (we can define it as the mass m_𝒩,π)

    m_π f_πf_a√(z)1+z<m_π f_πF_a√(z1-z^2)  ,

it can be characterized as

    ζ≡F_af_a<√(1+z1-z)≃ 1.69   ,

where we have defined the ratio ζ.
On the other hand, in order for the level crossing to occur in this case, the zero-temperature QCD axion mass should be larger than the zero-temperature Z_𝒩 axion mass m_𝒩,0, we have

    m_π f_πf_a√(z)1+z>m_π f_π√(π) F_a√(1-z1+z)𝒩^3/4z^𝒩/2  ,

it can also be characterized as

    ζ   >1√(π)√((1-z)(1+z)^3z^2)𝒩^3/4z^𝒩/2
       ≃ 1.24×0.48^𝒩/2𝒩^3/4  ,

which depends on the value of 𝒩.
Then we show in Fig. <ref> an illustration of level crossing in this case.
We have set f_a=10^12  GeV, F_a=10^11.5  GeV, 𝒩=17, and γ=0.25.
The red and blue lines correspond to the temperature-dependent mass eigenvalues m_h(T) and m_l(T), respectively.
Note that the level crossing occurs at the QCD phase transition critical temperature

    T_× = T_ QCD  .
 
Let us discuss the cosmological evolution of the mass eigenvalues in this case.
At high temperatures, the heavy mass eigenvalue m_h(T) corresponds to the Z_𝒩 axion, while the light one m_l(T) corresponds to the QCD axion.
They will approach to each other at T_ QCD and then move away from each other.
When T < T_ QCD, the m_h(T) represents the QCD axion, and the m_l(T) represents the Z_𝒩 axion.
For a more intuitive description, they can be expressed as

    m_h(T)⇒
    m_a(T)  ,    T< T_×
    
    m_𝒩(T)  ,    T> T_×


    m_l(T)⇒
    m_𝒩(T)  ,    T< T_×
    
    m_a(T)  .    T> T_×
 
The above discussion is focused on the Case I that level crossing occurs at T_ QCD.

To compare with another case discussed below, we show the condition for level crossing to occur in the Case I by using Eqs. (<ref>) and (<ref>).
See Fig. <ref> with the red shadow region in the {𝒩,  ζ} plane.
The blank regions represent that no level crossing occurs.







Case II: Level crossing slightly before T_ QCD.—
Next we consider another case that the zero-temperature QCD axion mass is larger than m_𝒩,π, we have

    ζ>√(1+z1-z)≃ 1.69   .

We show an illustration of this case in Fig. <ref>.
We have set f_a=10^12  GeV, F_a=10^12.5  GeV, 𝒩=13, and γ=0.25.
Different from the Case I, here the level crossing will take place slightly before the temperature T_ QCD.
By solving d(m_h^2(T)-m_l^2(T))/dT=0 at T_ QCD < T < T_ QCD/γ, we obtain the level crossing temperature

    T_×=T_ QCD(√(1-z1+z)ζ)^1/b(1-ζ^2𝒩^2)^-1/(2b)  ,

which is slightly larger than T_ QCD.
The QCD axion mass at T_× is given by

    m_a(T_×)≃ m_𝒩,π√(1-ζ^2𝒩^2)  .

Note that ζ<𝒩 should be satisfied in Eq. (<ref>).
This level crossing will last for a parametric duration

    Δ T_×=|1cosα(T_×)dcosα(T_×)dT|^-1  ,
 
where α(T) is the mass mixing angle given by Eq. (<ref>). 
This is the Case II.
The temperature-dependent behavior of the mass eigenvalues is similar to the Case I.
However, the transition of energy density between them is completely different, which will be discussed later.
In fact, the Case I can transform into the Case II, depending on the value of the ratio ζ∼1.69. 

Then we show in Fig. <ref> (the blue shadow region) the condition for level crossing to occur in the Case II by using Eq. (<ref>) and ζ<𝒩.
Additionally, we also note that at T_ QCD/γ the QCD axion mass should be smaller than the Z_𝒩 axion mass   

    m_π f_πf_a√(z)1+zγ^b<m_π f_πF_a√(z1-z^2)  ,

it can be characterized as

    ζ<√(1+z1-z)γ^-b≃ 1.69  γ^-b  ,

which depends on the value of γ.
Since for the small γ, the constraint of ζ<𝒩 is more stringent than Eq. (<ref>), we do not show it in Fig. <ref>.

Axion relic density.—
Here we briefly estimate the relic density of the axion DM in the Cases I and II through the misalignment mechanism.
The pre-inflationary scenario is considered, in which the PQ symmetry is spontaneously broken during inflation. 

We first discuss the Case I.
As an example, we focus our attention on the Z_𝒩 axion DM.
We begin with the QCD axion field, which is frozen at high temperatures with an arbitrary initial misalignment angle θ_1,a, and starts to oscillate at T_1,a.
The oscillation temperature T_1 is given by m(T)=3H(T) with the Hubble parameter

    H(T)=√(π^2 g_*(T)90)T^2m_ Pl  ,
 
where m_ Pl is the reduced Planck mass, and g_* is the number of effective degrees of freedom of the energy density.
In the following, the subscript “x" stands for the physical quantity at T_x or corresponding to x.
The initial energy density in the QCD axion field is

    ρ_a,1=12m_a,1^2 f_a^2 θ_1,a^2  .

At T_ QCD<T<T_1,a, the QCD axion energy density is adiabatic invariant with the comoving number N_a ≡ρ_a a^3 /m_a, where a is the scale factor.
Then we have the QCD axion energy density just before T_ QCD as

    ρ_a,  QCD=12 m_a,0 m_a,1 f_a^2 θ_1,a^2 (a_1,aa_ QCD)^3   .

In this case, the QCD axion is trapped around θ_ tr≡θ_a(T_ QCD)∼π until T_ QCD with the initial axion velocity θ̇_ tr≡θ̇_a(T_ QCD).
At T=T_ QCD, the light mass eigenvalue m_l(T) will comprise the Z_𝒩 axion, the axion mass is suddenly suppressed and the true minimum ∼ 0 develops.
Then we have the mean velocity of the Z_𝒩 axion

    √(⟨θ̇_ tr^2⟩)=1√(2)ζ√(m_a,0 m_a,1)θ_1,a(a_1,aa_ QCD)^3/2  .

Note that the Z_𝒩 axion energy density at T_ QCD is non-adiabatic, which just after T_ QCD is given by

    ρ_𝒩,  tr=12 F_a^2 θ̇_ tr^2+2m_𝒩^2 F_a^2𝒩^2  ,

where we consider a case that the axion has a large mean velocity.
Then the Z_𝒩 axion will start to oscillate at T_2, which is defined as the temperature when the kinetic energy is equal to the barrier height.
At T_2<T<T_ QCD, the Z_𝒩 axion energy density is conserved with the comoving PQ charge q_ kin=θ̇_𝒩 a^3, and we have the scale factor at T_2 as a_2=(𝒩θ̇_ tr/(2m_𝒩))^1/3a_ QCD.
Then at T_0<T<T_2, the adiabatic approximation is valid, and we have the Z_𝒩 axion energy density at present T_0 as

    ρ_𝒩,0 = m_𝒩,0 F_a^2 θ̇_ tr𝒩(a_ QCDa_0)^3  .

Substituting the mean velocity, it reads

    ρ_𝒩,0 = C m_𝒩,0√(m_a,0 m_a,1)θ_1,a F_a^2√(2)𝒩ζ(√(a_1,a a_ QCD)a_0)^3  ,

where C≃2 is a constant.
 

 

Next we discuss the Case II, focusing on the QCD axion DM.
We begin with the Z_𝒩 axion field, it starts to oscillate at T_1,𝒩 with the initial misalignment angle θ_1,𝒩, and the initial energy density is given by

    ρ_𝒩,1=12m_𝒩,1^2 F_a^2 θ_1,𝒩^2  .

At T_×<T<T_1,𝒩, the Z_𝒩 axion energy density is adiabatic invariant, and at T_× we have 

    ρ_𝒩,×=12 m_𝒩,π m_𝒩,1 F_a^2 θ_1,𝒩^2 (a_1,𝒩a_×)^3   .

At T=T_×, the level crossing occurs, the heavy mass eigenvalue m_h(T) will comprise the QCD axion and the energy density ρ_𝒩,× is transferred to the QCD axion ρ_a,×.
To ensure this axion energy transition is adiabatic at T_×, we need

    Δ t_×≫max[2πm_l(T_×),  2πm_h(T_×)-m_l(T_×)]  ,

where the time Δ t_× corresponding to the temperature in Eq. (<ref>).
We also check it in Fig. <ref> with the parameter set shown in Fig. <ref>.
Then at T_0<T<T_×, the adiabatic approximation is valid again, and we have the present QCD axion energy density 

    ρ_a,0=12 m_a,0 m_𝒩,1 F_a^2 θ_1,𝒩^2 (a_1,𝒩a_0)^3   .

Note that we only show two examples above to estimate the relic density of the Z_𝒩 axion and QCD axion DM in the Cases I and II, respectively, one can also calculate others in a similar way. 
   
Conclusion.—
In summary, we have investigated the mass mixing between the QCD axion and Z_𝒩 axion, and introduced a novel level crossing.
In the mixing, we find that the level crossing can take place at the QCD phase transition critical temperature T_ QCD (Case I) or slightly before T_ QCD (Case II), depending on the ratio of the axion decay constants ζ∼1.69.
The conditions for level crossing to occur in the Cases I and II are discussed in detail.
In the Case I, the zero-temperature QCD axion mass m_a,0 should be smaller than the mass m_𝒩,π and larger than the zero-temperature Z_𝒩 axion mass m_𝒩,0, which can be characterized as 1.24×0.48^𝒩/2𝒩^3/4<ζ<1.69.
While in the Case II, the zero-temperature QCD axion mass m_a,0 should be larger than m_𝒩,π and the ratio should be ζ<𝒩, which can also be characterized as 1.69<ζ<𝒩.
Note that in this case we also have a relatively weak constraint ζ<1.69  γ^-b, which will be stringent for the large value of γ.

In these two cases, the cosmological evolution of the mass eigenvalues is similar, however, the transition of energy density between the axions is different.
In the Case I, the axion energy transition at the level crossing is non-adiabatic (T_ QCD), while in the Case II it is considered to be adiabatic (T_×).
Finally, we briefly estimate the relic density of the axion DM through the misalignment mechanism, focusing on the Z_𝒩 axion and QCD axion DM in the Cases I and II, respectively.
The level crossing in this scenario may also have some other interesting cosmological implications.

Acknowledgments.—
This work was supported by the CAS Project for Young Scientists in Basic Research YSBR-006, the National Key R&D Program of China (Grant No. 2017YFA0402204), and the National Natural Science Foundation of China (Grants No. 11821505, No. 11825506, and No. 12047503).




 Paper Title: Probing effective operators in single top quark production in association with a lepton-neutrino pair.



	
	

		August 5, 2024

	
	

	

		Probing effective operators in single top quark production in association with a lepton-neutrino pair
	

	
	

	

		Reza Goldouzian[Rgoldouz@nd.edu] and Michael D. Hildreth[Mikeh@nd.edu]  

		
 
		Department of Physics, 225 Nieuwland Science Hall,University of Notre Dame, Notre Dame, IN 46556, USA
	

	
	

	§ ABSTRACT 
		
  We study single top quark production in association with a lepton-neutrino pair at the LHC  within the framework of the Standard Model Effective Field Theory (SMEFT). We focus on relevant two-quark-two-lepton operators (3lq, 1lequ, and 3lequ). It is known that these operators  have tiny  effects on the inclusive cross section of standard model tW production and are thus typically ignored in the SMEFT searches. However, we show that by employing smart observables, such as m_T2, the pp→tℓν process is significantly sensitive to these operators. We set the most stringent limit on the coupling strength of the 3lq operator by reinterpreting the results of a search for new phenomena  with two opposite-charge leptons, jets and missing transverse momentum at √(s) = 13 TeV performed by the ATLAS collaboration. The limits derived on the 1lequ and 3lequ operators are  comparable to those obtained  from probing EFT effects in pp→ttℓν and pp→ttℓℓ processes at the LHC. Consequently, we propose to include the effects of these three operators on the pp→tℓν process in future global SMEFT analyses to increase the sensitivity and to reduce possible degeneracies.  
	
	





	

§ INTRODUCTION

All direct searches at the large hadron collider (LHC)  have so far been unsuccessful in finding clear indications of physics beyond the standard model (BSM). This might mean that the BSM energy scale (Λ) is above what can be directly probed at the LHC. If Λ is well separated from the electroweak scale, heavy states can be integrated out and then cast into extensions of the SM Lagrangian by higher dimensional operators (O) <cit.>, i.e. operators of dimension D>4:

    L=L_SM + ∑_D,iC_i^D/Λ^D-4O_i^D ,

The dimensionless effective couplings of new physics, called Wilson coefficients (WC), are denoted by C_i^D. 
The structure of the operators is entirely dictated  by the field content and symmetry restrictions. Writing L including the SM gauge symmetries and particle content results in the  SM Effective Field Theory (SMEFT) representation <cit.>.
The SMEFT is a consistent model-independent framework for systematically parameterising and predicting deviations from the SM. Furthermore, 
the SMEFT is a renormalizable theory and its predictions can be improved by adding terms of higher order quantum electrodynamic (QCD) and electroweak corrections <cit.>. 

For dimension D=6, there are 59 independent operators with minimal flavour violation and Baryon number conservation <cit.>. 
Each operator may affect multiple SM physics processes in different parts of the available phase space.
In order to explore the SMEFT parameter space, it is necessary to discover regions sensitive to SMEFT effects for different types of processes
and include them within a coherent global SMEFT analysis.   
In recent years, the SMEFT has been probed in different sectors by measurements of quantities related to electroweak bosons, the Higgs boson and the top quark by performing global fits <cit.>. 
All these global fits are performed on a few hand-picked unfolded kinematic  distributions provided by the experiments.

Recently, the CMS Collaboration has performed  the most global detector-level SMEFT search to date in the production of one or more top quarks with additional leptons, jets, and b jets  in proton-proton (pp) collisions <cit.>.
Constraints are set on 26 WCs which affect the pp→ttX and  pp→tqX (X= W, Z and Higgs bosons and q=light jet) processes.
They have also included non-Z/W/H mediated ttℓℓ and ttℓν processes for probing the two-quark-two-lepton operators in multilepton final states <cit.>.  In this analysis there are two important improvements with respect to the previous version <cit.>:  first, including 10 more sensitive operators in the global fit, and second, employing smarter observables in various regions of the phase space. Consequently,  a more realistic global fit is performed and the 2σ profiled confidence intervals are improved by factors of approximately 2 to 6, depending on the WC.  
Therefore,  recognizing  sensitive operators in any available phase space and including them in the global fit is a crucial task <cit.>.

In this work, we focus on the SMEFT effects on single top quark production in association with a W boson at the LHC. 
We show that the number of sensitive operators increases if we extend the pp→tW process to include pp→tlν.  
We propose a unique phase space for probing the extended  operators and examine its sensitivity by reinterpreting the results of a search for  new phenomena  with two opposite-charge leptons, jets and missing transverse momentum performed by the ATLAS Collaboration <cit.>. 




§ SMEFT OPERATORS


In this section, we review the SMEFT operators that affect  single top quark production in association with a lepton-neutrino pair in pp collisions. 
Eight operators  can affect the pp→tlν process at leading order (LO); these operators are listed in Table <ref> together with corresponding WCs <cit.>. The operators are classified into two categories: operators that involve two quarks and gauge bosons, and operators that involve two quarks and two leptons. 

 



In the SM, the pp→tlν  process involves tW production with the leptonic decays of the W boson.  The   3φ qij, φ udij,  dWij, and uWij operators modify the SM Wtb vertex and thus affect the tW production<cit.>.  The 3φ qij operator has the same  structure as the SM Wtb vertex and only rescales the Wtb coupling without affecting any kinematic distribution.
In the m_b=0 limit, the φ udij and dWij operators do not interfere with the SM amplitude and only contribute at order 1/Λ^4 to the cross section.  The  uWij operator involves the right-handed top quark and interferes with the SM amplitudes. The φ udij,  dWij, and uWij operators affect the  W boson polarization fractions in top quark decays and are well constrained via the W polarization measurements by the CMS and ATLAS Collaborations <cit.>.
These operators can also affect the pp→ttX and pp→tqX processes <cit.>.
The uGij operator  contributes to tW production via modification of the SM top-gluon vertex <cit.>.  This operator is also well constrained using tt events at the LHC <cit.>.

The  3lqijkl,  1lequijkl, and 3lequijkl operators do not have a W boson  and directly connect a top-bottom quark pair to a lepton-neutrino pair <cit.>. 
The  3lqijkl,  1lequijkl, and 3lequijkl  operators have vector-, scalar-, and tensor-like Lorentz structure, respectively.
It is worth mentioning that these operators are of great interest for new physics searches because of the observed anomalies in B meson decays <cit.>.  They can   represent  the low energy  effects of heavy new physics particles such as W', Z', leptoquarks, etc. which could be solutions that can describe the “B anomalies"<cit.>. These two-quark-two-lepton operators with a top quark leg also affect the ttℓℓ and ttℓν processes and are probed in three lepton final states by the CMS collaboration <cit.>.





§ SIGNAL SAMPLE SIMULATION

We aim to study dimension-six EFT effects on  tlν production at the LHC.
Representative Feynman diagrams for  pp→tlν process in the presence of the EFT vertices are shown in Figure. <ref>.

The signal sample is modeled at leading order using the Madgraph@NLO Monte Carlo event generator in the five-flavour scheme  <cit.>.
The SMEFT Lagrangian is included using the dim6top model described in Ref. <cit.>.
Events are generated using the  NNPDF3.0 parton distribution function (PDF) set <cit.> and  are passed to PYTHIA8 <cit.> for showering and hadronization. 

In the dim6top model, only operators involving one or more top quarks are provided with no limitations on the lepton flavors.  
In this study we assume the same couplings for all generations of leptons  (e.g.  = = ). 
All eight operators mentioned in Table  <ref>  are included simultaneously in the signal sample generation.
Only diagrams with one EFT vertex are included together with the SM diagrams. 
The signal sample is generated at a non-zero point of eight-dimensional WC phase space.
In order to predict the EFT effects at arbitrary points of the WC phase space,  we use the Madgraph@NLO event reweighting technique <cit.>.
By knowing the fact that the cross section of the pp→tlν process depends  quadratically on the WCs and by having  enough event weights,  we can determine the eight-dimensional quadratic function of the cross section in any region of the phase space or observable bin <cit.>.   





§ SENSITIVE  OBSERVABLES


In Figure <ref>,  relative SMEFT  contributions to the  pp→tlν inclusive cross section are shown for individual WCs.
It can be seen that the 3φ qij,  uWij and uGij operators, with considerable interference with the SM,  have the largest effects.
The dWij operator, with no interference with the SM,  has the second largest effect.
Finally, the φ udij and  two-lepton-two-quark operators have very small effects on the inclusive pp→tlν cross section.






In Ref. <cit.>, the authors constrained the two-quark-two-lepton WCs  by reinterpreting the   unfolded differential cross sections  of the pp→tW→tlν process measured by the ATLAS collaboration <cit.>.
Although the invariant mass of the ℓℓ b system (M(ℓℓ b)) and  the transverse mass of the ℓℓνν b system (M_T(ℓℓνν b)) show better  separation power between the  EFT and SM events compared to the inclusive cross section,  the 95% CL limits obtained from these two distributions are much looser than those set by the CMS collaboration by analyzing the  ttℓℓ and ttℓν processes (see table <ref> for an exact comparison) <cit.>.

There is an important difference between the tlν production via the  SM and two-quark-two-lepton operators: the SM process involves on-shell W boson production while the EFT process does not. Therefore, one can discriminate between the SM and EFT events by reconstructing  the mass of the lν system. 
In the dilepton final state, the measured missing transverse momentum vector (p_T^miss) is associated to the sum of the two neutrinos which causes ambiguity in the reconstruction of the lν system.  
A well known variable which can be used to extract information about the mother particle's mass in  events where two massive particles each decay to a detected and an undetected object is m_T2 <cit.>.  
The m_T2 variable is defined as:
 
 
 
    (  p_T,1,  p_T,2,  p_T^miss) = min_ q_T,1 +  q_T,2 =   p_T^miss
       {max [ 
    m_T(  p_T,1,  q_T,1 ),
    m_T(  p_T,2,  q_T,2
    )
     ]
    } ,


where p_T and q_T are the transverse momenta of the detected  and undetected particles, and
m_T is the transverse mass 



    m_( p_T, q_T )=√(2| p_T||  q_T |(1-cos(Δϕ(  p ,  q)))).


The minimisation is performed over all the possible decompositions of p_T^miss  =  q_T,1 +  q_T,2.
The  distribution has an endpoint at the W boson mass for the SM tW events,  while signal events are expected to populate the tails of   distribution.
The m_T2 variable is  used widely in searches for supersymmetric particles <cit.> and top quark mass measurements <cit.>. 





In Figure <ref>,  distributions of the M(ℓℓ b), M_T(ℓℓνν b), and m_T2 observables are shown for the pp→tlν process in the SM and SMEFT scenarios. Events  with an opposite-sign lepton pair (, , and ) and a b-quark  with p_T>20 GeV and |η|<2.4 are selected at the generator level. We also require events to have  |  p_T^miss|>40 GeV to be closer to the selection used in Ref. <cit.>. We only show the effects related to non-zero values of the  two-quark-two-lepton WCs.
In the tail of the m_T2 distribution, EFT variations are enhanced significantly compared to the M(ℓℓ b) and M_T(ℓℓνν b) distributions. This proves that the   pp→tlν process could be sensitive to the 3lqijkl,  1lequijkl, and 3lequijkl operators if one chooses a smart observable.





§ EXPERIMENTAL INPUTS

In order to evaluate the power of the m_T2 observable for constraining the two-quark-two-lepton operators via tlν production, we reinterpret the results of a search  for top squarks in events with two leptons, b-jets and missing transverse energy at √(s)=13 TeV performed by the ATLAS Collaboration  with the full Run-2 data corresponding to 139 fb^-1 <cit.>.
Dedicated event selections are optimised in the ATLAS search to probe various decay modes of the top squark. 
One of these event selections, called the “two-body  selection”, targets the two-body top squark decay into an on-shell top quark and the lightest neutralino. The estimated SM yields in the signal region defined in the “two-body selection” are reported in multiple bins of the m_T2 distribution in the range of [100,inf) GeV.
This signal region is proper for probing the  pp→tlν process  and the m_T2  variable that is our variable of interest in this study.  Therefore, we use the published results in Ref.  <cit.> in the “two-body  selection” for constraining the two-quark-two-lepton operators.

In the  “two-body  selection”,  events are accepted if they have two opposite-sign leptons (, , and ) with the leading lepton  p_T(ℓ 1) larger than 25 GeV and the subleading lepton p_T(ℓ 2) larger than 20 GeV.  
Electrons (muons) are required to be in the |η|<2.47 (2.40) region. Electrons in the transition region between the 
barrel and endcap electromagnetic calorimeters of the ATLAS detector, 1.37<|η|<1.52, are excluded. 
To remove leptons from Drell–Yan and low-mass resonances,  dilepton events with an invariant mass m_ll<20  are rejected. 
 Events with same flavor (SF) lepton pairs with 71.2<m_ll<111.2 GeV are also rejected to reduce the Z boson background.
Jets are reconstructed using the anti-k_t algorithm <cit.> with a radius parameter of 0.4 and are selected if they have p_T>20 GeV and |η|<2.8.  Selected jets are tagged as b-jets  using the  MV2C10 boosted decision tree algorithm with 77% efficiency <cit.>. Events should have at least one reconstructed b-tagged jet.
Linked to the missing transverse momentum, the object-based missing transverse momentum’s  significance (E_T^misssignificance) is used  to discriminate
between the events in which the reconstructed E_T^miss originates from weakly interacting particles and the
events in which the E_T^miss is related to the resolution and inefficiencies in particle measurements. 
The E_T^misssignificance is computed by considering the expected resolution and mismeasurement likelihood of
all the objects that enter the E_T^miss  reconstruction, as detailed in Ref. <cit.>.
Another variable employed in this search is the azimuthal angle between the  p_T^miss and  the vectorial sum of  p_T^miss and the leptons' transverse momentum vectors,  denoted as Δϕ_ boost. 
Selected events are required to have  E_T^miss significance larger than 12, Δϕ_ boost<1.5, and finally m_T2 greater than 110 GeV.
The two-body selection requirements are summarized in Table <ref>. 








To reinterpret the results of the ATLAS search, simulated signal events are passed to Delphes 3 for modeling the response of the ATLAS detector <cit.>.
The “two-body selection” requirements are applied to the  Delphes reconstructed objects. 
It is not trivial to extract precise  object-based E_T^miss significance from the   Delphes reconstructed objects.
Therefore, we approximate the object-based E_T^miss significance by an event-based E_T^miss significance (S) as:

    S=E_T^miss/√(H)_T

where H_T  is the sum of the transverse momenta of visible particles  <cit.>.  
The cut value for the event-based E_T^miss significance was chosen to obtain a similar SM prediction for the pp→tW process in the tt validation region reported by the ATLAS collaboration (see Table 7 in Ref <cit.>).  The  tt validation region has exactly the same selection requirements as the “two-body selection” except for the m_T2 variable which is restricted to [100, 110] GeV. 
We found that cutting on the event-based E_T^miss significance at 9.6 GeV^1/2 for our Delphes simulated sample could reproduce the reported ATLAS prediction for the pp→tW process in the  tt validation region. Therefore we replace the cut  of E_T^miss significance >12 GeV  with the   S>9.6 GeV^1/2 requirement in the “two-body selection”.




In Figure <ref>,  the m_T2 distribution predicted by  the ATLAS collaboration for the SM  pp→tW process in the signal region is compared to the SMEFT predictions via the Delphes simulated sample. There is an acceptable agreement between our simulations at the SM point of WC phase space and the ATLAS predictions.  Although the ATLAS prediction for the SM tW process  in the tail of the m_T2 distribution has large statistical uncertainties, the agreement between our prediction and the ATLAS prediction for the SM tW yield in the last bin of the m_T2 distribution, which is the most sensitive bin, is very good. 
 In addition to the SM prediction, SMEFT predictions for the pp→tlν process in the presence of the two-quark-two-lepton operators are shown. 
 The ratio plots in Figures <ref> and <ref> show similar deviations with respect to the SM prediction at the generator and reconstructed levels.  This confirms that the “two-body selection” is proper for probing the two-quark-two-lepton operators at the reconstruction level.




§ RESULTS

No excess in data over the SM prediction is observed  by the ATLAS Collaboration <cit.>. The results are presented in terms of model independent 95% confidence levels (CL) on the number of signal events using the CLs method <cit.>.   
The upper limits are derived for the  different flavor and same flavor signal regions combined. 
Multiple upper limits are calculated using event yields from the following cuts on the lower edge of the range for the  m_T2 variable, with the upper end of the range at infinity: m_T2> 110, 120, 140, 160,  180, 200, or 220 GeV.  
We use these upper limits to constraint the two-quark-two-lepton WCs affecting tℓν production. 






It is more sensitive to include both the yield and shape variations in multiple bins of the m_T2 distribution because of the SMEFT effects.
However, because the SMEFT signal shapes are different from the top squark signal shape we are constrained to consider on the event yield variation in one cumulative bin in order to use the model dependent limits in Ref  <cit.>.
Based on the reported expected limits, we found that the tightest constraints are obtained in the range where m_T2 ∈ [160, ∞] GeV. In Figure <ref> the difference between the SMEFT  and  SM yield predictions for the pp→tℓν process in the range m_T2∈ [160, ∞] GeV are shown as a function of individual two-quark-two-lepton WCs. The 95% CL observed and expected  upper limits on the number of signal events in the same range of m_T2, reported by the ATLAS Collaboration,  are also shown. 
The constraints on the two-quark-two-lepton  WCs are determined by finding where the quadratic curves cross the 95% CL upper limit lines. In Table <ref>, the 95% CL upper limits obtained in this analysis  by using the m_T2 observable are listed. 
In addition, our results are compared to the limits obtained from analyzing  the M_T(ℓℓνν b) observable <cit.> and to those extracted from events in the three lepton final states  <cit.>. 
The obtained limit on the  operator is the most stringent reported on this WC to date.
Constraints on the   and      couplings are comparable to those obtained in the three lepton final state. 
Therefore, combining  dilepton  and three lepton final states in a global fit can improve the results on top of the individual constraints.                         








§ SUMMARY

In this work, we have investigated the possibility of probing the top quark related two-quark-two-lepton SMEFT operators in the   production of a single top quark in association with a lepton-neutrino pair at the LHC. We showed that the pp→tlν process is very sensitive to the 3lq, 1lequ, and 3lequ operators in the tail of the m_T2 distribution.  To make a realistic evaluation of the  power of the proposed phase space for constraining these SMEFT operators, we reinterpreted the results of an ATLAS search in a very similar phase space.
We set strong 95%  confidence level limits on the Wilson coefficients of the mentioned operators.  
The obtained bound on the  operator is around four times tighter than the latest reported bounds to date.
Consequently, we advocate for including the m_T2 variable as analyzed in this paper in future SMEFT searches and global SMFFT fits as a probe of this specific corner of phase space.





§ ACKNOWLEDGMENTS

R.G. would like to thank Hamzeh Khanpour for useful input. 
	
	tocsectionReferences
	JHEP
	
	
	
	
	
	


 Paper Title: The Smile of Cheshire Cat At High Density.










=1ex plus0.5ex minus0.2ex

AŁLØO
⟨⟩

    3pt1pt∼1pt<
    3pt1pt∼1pt>
    ≤[ m_N^+m_N^-m_π( [ )trm_πm_ρMeVm_N_+m_N_-m_p^-m_p^+≤( [ )trF_π^2m_πeMeVχ̅ω̅^0-hls EFT_ bsHLS-hls Skyrmion_ sHLS Tr∂ 𝔸
    
     
    The Smile of Cheshire Cat  At High Density 0.5cm
    
    Mannque Rho[ e-mail:mannque.rho@ipht.fr]Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique, 
     91191, Gif-sur-Yvette, France  0.3cm
     (August 5, 2024)
     ABSTRACT  Baryons in finite nuclei, nuclear matter and dense compact-star matter are described in terms of Cheshire Cat for  QCD. A potential conceptual link, admittedly short in mathematical rigor, between their manifestations is  made by what's called Cheshire Cat Principle. Put in terms taken  “dual" to QCD variables, going to very high density exposes quantum Hall droplets – or pancakes –  at which the dilaton-limit fixed point with g_A→ 1, f_π→ f_χ– where f_π and f_χ are respectively the pion and dilaton decay constants – and the baryon parity-doubling are reached. This scenario suggests a thus-far totally unexplored structure of dual baryonic matter at high density which does neither require nor  rule out  (rapid) first-order phase transitions from hadrons to quarks in the core of compact stars on the verge of gravitational collapse.  
     
     1.0cm
    
    
    
    § INTRODUCTION
    
    With the advent of gravity-wave observations with neutron-rich compact stars that are being – and will be – discovered, a thus-far more or less unexplored domain of physics is opening up,  resurrecting the old question raised more than a half a century ago: What is the structure of dense baryonic matter stable against, but on the verge of gravitational collapse into black a black hole? Of course general relativity enters with its own multifacets, but what has been baffling in modern physics  is how are the baryons crunched into such high density in the standard model (SM) of particle physics? This question arises in specific terms of what is the equation of state (EoS) of the matter that figures in the TOV (Tolman-Oppenheimer-Volcoff)  equation to describe the properties of massive compact (predominantly neutron) stars stable against gravitational collapse? This question becomes highly topical as the on-going – and up-coming – gravitational wave measurements are to provide ultimately precision data on the properties of such massive stars.
    
    This question brings back the long-standing issue on the basic structure of nucleon in the fundamental theory of strong interactions  (now belonging to  the Standard Model),  widely discussed several decades ago, as a model of QCD,  the  MIT bag model.  There was a big dispute in early 1970's between  Stony Brook, Long Island and the nuclear and particle theorists at MIT in Cambridge where the “MIT Bag"  was conceived <cit.> on the structure of baryons in heavy nuclei.  Taking  the well-studied heavy nucleus ^208Pb as a specific example, the MIT theory argued that  this nucleus should be more properly described in terms of 3× 208=624  quarks instead of 208 nucleons given that the MIT bag for the single nucleon was to have the confinement size  R_conf∼ 1 fm. This would depict the nucleon bags more or less close packed in heavy nuclei, giving the picture of the nucleons resembling  “grapefruits in a salad bowl." A nucleon has the size of radius ∼ 1 fm and a nucleon in the MIT bag model contains N_c=3 quarks  confined in the bag with a confinement radius ∼ 1 fm. This structure was considered,  generally at the time,  in nuclear community and particularly by the Stony Brook theorists,   “glaringly" at odds with the then generally accepted description of  nucleons in the Pb nucleus treated in the shell model with the nucleons pictured as quasi-fermions moving in  an average potential.  The nucleon was considered as a small object of radius much less than 1 fm  surrounded by meson cloud with energy much larger than the nucleon mass m_N ∼ 1  GeV,  in consistency with the old Yukawa theory of nuclear forces with strong self-energy processes with the pion field – and subsequently heavier meson fields –  bringing down the nucleon mass to its known measured value by self-energy corrections.  Following the notion of bag-confinement  in QCD,  while I was spending a year at the Nuclear Theory Group with  Gerry Brown, he and I proposed “The Little Bag Model" <cit.> to contrary the Big MIT bag.  The little bag was depicted as confining three quarks,  massless due to large density restoring chiral symmetry within the bag.  During the whole  period of my stay I was witnessing  the daily on-going disputes taking place at Stony Brook with its Little Bag (LB) and MIT with its big Bag (BB)[Gerry used to commute, quite frequently, between Stony Brook and Cambridge giving talks, generating heated arguments with the local protagonists of the BB. Amusingly each time he returned from MIT, his little bag got a little bit bigger, visibly persuaded by the local theorists. But it shrunk back after a little while,  dissuaded by us in the Stony Brook group. At his office door Gerry had hung two brown bags, one a big (brown shopping)  bag and another a small (lunch) bag contained inside the big one, clearly indicating his ambivalence.]. This – what one might call –“dichotomy" between the Stony Brook LB and the MIT BB becomes more striking at higher densities probed in compact stars as illustrated in Fig. <ref> copied from  <cit.> where it was used to argue persuasively against rapid first-order phase transitions from baryons to quarks inside massive compact stars.
    
    
    
    The LB was constructed for the nucleons with three colored quarks coupled to the small-mass pions, recognized as the Nambu-Goldstone (NG) bosons resulting from spontaneous breaking of the nearly perfect chiral symmetry. In proceeding with the LB, the quark-pion coupling at the bag boundary was constrained to be consistent  with chiral symmetry with the quarks confined inside the bag and the pions clouded outside.   
    With a suitable coupling boundary condition imposed at a given “confinement" radius R_bag,  the “Chiral Bag Model" (CBM) <cit.> was applied with some success to certain class of phenomena of nuclear dynamics <cit.>.   But there was however an obvious technical difficulty  due to its too big a size of  the self-energy corrections required to bring down the mass to near its physical value.  
    
    
    The way out of this stumbling block came with what one could  say a striking “tongue-in-cheek" idea proposed by Nadkarni, Nielsen and Zahed  in 1985 <cit.>, inspired from the Cheshire Cat in  Lewis Carroll's  1865 fable  “Alice in the Wonderland."  The  baryon charge that leaks out from the bag is linked to the  smile that disappears from the face of the Cheshire Cat in the Wonderland.
    
    
    
    What's  magical in the Cheshire Cat solution  was  that the nucleon could be described as a soliton, nowadays referred to as skyrmion, discovered by Skyrme in early 1960s <cit.>. What happens is that the baryon charge  B=1 for the nucleons can leak out of the LBs into the pion cloud residing outside of the bag. The crucial point, elaborated below, is that this leakage is caused by a quantum anomaly lodged at the bag boundary that makes the baryon charge disappear from the bag. This anomaly, associated with a gauge symmetry,  turns out to make the  bag radius R_bag, or equivalently confinement radius R_conf, an unphysical quantity. This is the emergence of what is referred to as “Cheshire Cat Principle (CCP)" and the associated mechanism  “Cheshire Cat Mechanism (CCM)."  As will be mentioned below there could be an alternative approach with the quark-pion coupling made in the whole space invoking no boundary conditions. This approach, referred to as “Cloudy Bag Model" <cit.>, could be considered as a variant of CCP without bag boundary conditions and hence involves no leakage of baryon charge. In this model the baryon charge is strictly confined inside the bag. If the bag is taken sufficiently large,  it could be formulated as a particular gauge-fixed version as described in Sec. <ref>.
    
    
    I first recount, qualitatively,  what took place when the CCM was introduced into the LB and how the resulting CC model fared with Nature in nuclear processes. I then describe how  a conceptually new development took place in the efforts to access the properties of superdense neutron matter relevant to massive compact stars currently being observed in gravity-wave and associated measurements. 
    
    
    
    § THE CHESHIRE CAT (CC)
    
    Reconciling  the “grapefruits-in-the-salad (GFF)" dilemma in heavy nuclei, say, ^208Pb in the LB scenario turns out to come from how  topology figures in the baryon structure: It turns out that the BBs in ^208Pb can be shrunk to the LBs relying on that that baryons are  topological solitons. The nucleons, proton and neutron, belonging to the baryon flavor  symmetry SU (N_f ) with N_f=2, can be considered as skyrmions in the large N_c limit  of QCD <cit.>  where N_c is the number of colors, thereby resurrecting Skyrme's idea left dormant for ∼ 2 decades. The outcome was that the whole baryon octet, not just proton and neutron, can be related to the meson octet via topology.  This is stated mathematically in terms of  the homotopy group π_3 (SU(N_f))=Z.  But it holds only if N_f >1,  not for N_f=1. There is no-go theorem for  a single flavor, i.e., π_3 (U(1))=0, so there can be no flavor-singlet  skyrmion coming from the flavor singlet meson, η^'.  Somewhat surprisingly this is due to the color symmetry of  QCD <cit.>. However  the flavor singlet soliton can actually be present topologically but not as a skyrmion. We will come back to this issue which is one of what could constitute a surprising new development in nuclear physics. It leads to a fractional quantum Hall droplet involving Chern-Simons topology. It  seems to play no significant role at low density, but, I suggest,  can figure importantly at high density.  
    
    
    In addressing hadronic interactions in nuclear and astrophysical systems, a true Cheshire Cat Mechanism posits that “the bosonic and fermionic degrees of freedom  can  in no way be independently constrained at the boundary" <cit.>.  This would  imply that the boundary terms should be  necessarily determined by locality, renormalizability and hermiticity, plus appropriate symmetries of QCD. This means that the bag wall, therefore the confinement radius, cannot be a physically relevant quantity. 
    
    The principal idea for what the Cheshire Cat  is all about can be glimpsed in an amusing pedestrian presentation given in 1987 Les Houches  meeting <cit.> where it is stated that   “The Cheshire Cat point of view is to postulate that the bag itself of the hybrid bag model does not exist, in analogy to the non-existence of the Cheshire Cat in the Lewis Caroll fable. Only the grin of the Cheshire Cat and the formalism of the bag exist <cit.>."  The application of the idea to nuclear physics up to year ∼ 2000 is partly summarized in <cit.>. Here I will recount some of what has transpired since then, going into the regime of nuclear physics that has not yet been explored.
    
    
    
     §.§ The chiral bag
    
    The gist of the basic point can be illustrated with  a (1+1)-dimensional CC involving an “infinite hotel." . 
    
    The chiral bag action consists of three components 
    
    S=S_V+S_V̅+S_∂ V
    where the first term is of the bag of volume V containing N_c=3  massless quarks ψ for the baryon, the second is the outside of the bag of volume V̅ populated by mesons (massless pions π– for now,  later heavier mesons will be brought in) and the third is the boundary connecting the inside to the outside. 
    
    What enters in the first and second terms of the  action is known classically.  What crucially matters is the boundary term which makes the Cheshire Cat Mechanism (CCM) operative. As it can be seen easiest for the issue, we proceed in 1 space dimension focussing on a single quark but it applies to all N_c of them.  The argument given for (1+1) dimensions applies equally to 3d for the flavor number ≥  2. The subtlety for N_f=1, a totally new element, will be postponed to later.
    
    
    With the appropriate constraints, e.g.,    chiral symmetries etc.,  imposed in the action (<ref>), the problem arises in the boundary term because there is a quantum anomaly on the vector current that triggers the leakage of the baryon charge confined in V into V̅. In <cit.>, this scenario is, amusingly, depicted as a “jailed" quark trying to escape through the jail boundary exploiting the “infinite hotel," the anomaly.  The consistency of the theory requires the anomaly generated at the bag side be cancelled by the pion field in V̅ wherever the bag boundary might be located and this is done by the pion field “turning topological" and picking up the leaked charge.  This action is independent of the bag location  R_bag/con and this renders the bag/confinement size physically irrelevant. The baryon charge, carried by the quarks confined inside the bag, disappears into the the pion field, with the latter carrying the soliton topology that leaks out. The baryon charge therefore is the smile of the Cheshire Cat. 
    
    
    
     §.§ The   Smile of CC is topological
    
    In (1+1)d, there is an exact bosonization of the fermion quark to a boson so the baryon charge B  is 1 independently of the confinement wall R_bag. Neither the charge in the bag nor the charge in the pion field separately  is physical while the total is. When the bag is shrunk to zero radius, the baryon charge remains in the smile of the cat.  Given the exact bosonization in (1+1) dimensions, it is possible to see that not just the baryon charge but also certain other  conserved quantities remain independent of the  R_bag.  But this is not feasible in (1+3) dimensions.
    
    How this phenomenon occurs is nicely illustrated in <cit.>. Consider a massless quark ψ swimming in attempt to escape from the “jail" to the positive direction on top of the Dirac sea. It gets blocked when it reaches R_bag. This is because the symmetries do  not allow it. Since it cannot turn back on the surface,  it will  plunge into the sea  but the Dirac sea is completely occupied, so  it tries wiggle itself out below the Dirac surface. This is where the quantum anomaly kicks in at the boundary, allowing the quark to escape exploiting  the “infinite hotel."  The fermion number is conserved, so the quark disappears into the “smile" of the CC.  What takes place there can be  precisely described thanks to the exact bosonization that exists in (1+1) dimensions. 
    
    
    
     §.§ How CC works in real world
    
    Now what about in (1+3) dimensions where the exact bosonization is not known to exist?  The early efforts to address this issue in nuclear physics are reviewed in <cit.>.
    
    It was first seen in <cit.> at the pion'a magic chiral angle θ=π/2 that exactly half of the baryon charge B=1/2 leaks out of the bag which then is picked up by a half-skyrmion. This magic-angle phenomenon holding trivially – thanks to the topology involved – for the baryon charge in both 1 and 3 space dimensions has a very interesting phenomenon particularly at high density. We will return to it below. 
    
    It was shown by Goldstone and Jaffe <cit.> using the multiple reflection method that  the leaking baryon charge from the bag in (1+3) dimension is  precisely picked up at any chiral angle by the fractionalized skyrmion charge in the exterior of the bag so that the total baryon charge remains intact. As in the (1+1) dimensions what is in action is the anomaly. The leaking charge which of course depends on the bag radius R_bag is precisely carried by the pion field (skyrmion) also in (1+3)d. This must happen because the baryon charge is a  topologically conserved quantity and it is the gauge invariance that tackles the anomaly that does the job. 
    
    Now the question is: What about other physical quantities that are not subject to topology?
    
    
    
     §.§ CC Mechanism (CCM) as gauge invariance in action:
     Flavor-singlet axial charge of the proton
    
    There is no exact bosonizaton known in (1+3))d, so it is not very likely that apart from what's involved with topology, one can expect the CCP to hold for  all physical observables.  Surprisingly there are certain properties that are not directly controlled by topology but seem however to come rather close to having the CCP work. Let me describe one case where while topology is not evidently involved but there is an anomaly present at the boundary.  Here  a conserved charge does play a role, namely, the flavor-singlet axial coupling constant (FSAC) g_A^(0) that involves the U_A (1) anomaly.  With the modern progress toward “first-principles" approaches to nuclear structure, calculations could be done a lot more accurately but even at this early stage of development with the technique which is more or less crude it becomes surprisngly clear that intricate details, not obviously topologically connected, need to be as fully treated as possible to see how the CCP can be arrived at. The FSAC, described below, gives an excellent illustration of this case.
    
    
    
      §.§.§  The η^' from the U_A(1) anomaly
    
    One prominent illustration of what could be involved in the working of the CCP is the case of the flavor-singlet axial charge g_A^(0) of the proton measured in the EMC at CERN <cit.>– erroneously, as it turned out, linked to the proton spin crisis <cit.>.  It is an extremely interesting case where the leakage of color from the bag and the flavor-singlet η^' excitation outside can be described by a Cheshire Cat mechanism involving the U_A(1) anomaly <cit.>.  
    
    
    First let me first give a simple description of the η^' properties combining the U_A(1) anomaly with the CCP.  It illustrates  what the CCP states at low energy: “hadronic processes do not discriminate between QCD degrees of freedom (quarks, gluons) on one hand and meson degrees of freedom (pions, vector mesons etc.) on the other, provided all the necessary quantum effects are properly taken into account <cit.>." The basic idea is to consider a slowly-fluctuating  η^' configuration in the vacuum. Over a small volume of size V,  applying the CCP allows one then to describe η^' excitation in terms of QCD variables or effective bosonized degrees of freedom. As in the case of the baryon charge, the presence of η^' at the boundary makes the color charge of the quarks to leak-out of V <cit.>. This symmetry-breaking term which involves the η^' coupling to the Chern-Simons current K_5^μ
    
    K_5^μ=ϵ^μναβ (A_ν^a G^a_αβ-2/3 f^abcgA_ν^a A_α^bA_β^c), 
     
    where A_μ^a is the gluon field, 
    needs to be cancelled by quantum effects. The boundary term reproduces the Witten-Veneziano relation <cit.> relating the  η^' mass to the topological susceptibility χ <cit.>.  
    
    That the properties of the η^' emerge correctly from the CC is an important point for the g_A^(0) treated below.  
    
    
    
      §.§.§  A chiral bag with  η^'
    
    We are now equipped to address the FSAC g_A^(0) problem <cit.>. The chiral bag model we will treat is given pictorially in Fig. <ref>.  
    
    
    The action contains three components as in (<ref>), now in,  in (1+3)d  
    
    S=S_V+S_V̅+S_∂ V.
    The first term is the usual QCD action. The other two terms are
    
    
    
    
    S_Ṽ   =   f^2/4∫_Ṽ d^4x(_μ U^†^μ U +1/4N_f
    m^2_η^'( lnU- lnU^†)^2
    )
       +    S_WZW,
    
    S_ V   =   1/2∫_ V dΣ^μ{(n_μψ̅ U^γ_5ψ)
    +i g^2/16π^2K_5_μ ( ln U^†- ln U)}
    where N_f=3 is the number of flavors and 
    
    U   =   e^iη^'/f_0e^2iπ/f,
    
    f_0   ≡   √(N_f/2) f. 
    Here π stands for the octet pseudo-scalar mesons and η^' for the singlet pseudoscalar.
    It should be mentioned at this point in anticipation of what's to come later that hidden local symmetry bosons (ρ,ω), in particular, via the homogeneous Wess-Zumino term <cit.>,  will figure importantly at high density.[It will be seen that unlike the pseudo-scalar octet mesons π which give rise to all the octet baryons as skyrmions, the flavor singlet η^' does not give a flavor singlet skyrmion baryon B^(0). It will lead to a fractional quantum Hall droplet which will figure at high density.] In the present problem, they are not important.
    
    
    
    
    The matrix element of the FSAC gotten from the model (<ref>) consists of terms from the bagged degrees freedom and the η^' satisfying the U_A(1) anomaly. As detailed in <cit.>, there are (1) the contribution from the bagged quark, a^0_Bq, with the baryon charge leakage properly taken into account, (2) the η^' contribution a^0_η, (3) the contribution of static gluons due to quark source a^0_G,stat, and (4) the gluon Casimir a^0_G,vac.  The results obtained (with available parameters) for bag radius 0 < R < 1.5 fm are plotted together with  the sum a^0_total in Fig. <ref>. The shaded area corresponds to the EMC result for g_A^(0).
    It shows  that  the CC-ism is quite sensitive to the detailed account of various contributions. It involves significant cancellations for the  CCP to to work out. As far as I am aware this is the only calculation in the literature that shows how the CC feature is captured numerically for quantities that are not fully topological. It would be theoretically challenging and meaningful to repeat this calculation in a more rigorous treatment. 
     
    
    
    
     §.§ CC Mechanism for finite nuclei and nuclear matter
    
    The CCP states that when the bag radius is shrunk to zero, the CC smile remains and physics should not depend on the “confinement radius" in the sense implied in the MIT model. Therefore nuclei should be describable in terms of the  smiles of the multiple CCs with the bags shrunk to points. Let me remark only briefly on this issue which deserves a lot more space than available.
    
    Nuclear physics in terms entirely of skyrmions is currently becoming a new frontier in the field. It could be considered as  of a “first-principles" approach in nuclear physics on par with χ EFT in QCD at low energy <cit.>  in the sense of Weinberg's “Folk Theorem" of QCD <cit.>. There is a great deal of developments going on  and expected to come. How this picture works has been extensively studied with the Skyrme model with only the quartic derivative term (referred in the literature as Skyrme term) with some remarkable success in various observables <cit.>.  It is now well established that the Skyrme quartic term alone would fail to give nuclear binding energies correctly. It gives too big a binding energy and some of the ground-state properties rather poorly. This defect can be cured to certain extent by the role of heavy vector mesons incorporated into the mesonic sector <cit.> and holographic dual models with infinite tower of vector mesons <cit.>. But as will become clearer in what follows, there is a great deal of what degrees of freedom – and what subtleties – that need to be figured out.
    
    The early effort to access dense matter, particularly compact star matter, is found in <cit.>.  Only very recently are some challenging efforts  made in mathematical rigor in the skyrmion approach to  dense matter in compact stars etc. starting  to appear in the literature <cit.>.  For instance, conformal symmetry found to play a significant role in the CCP – and treated further below – is yet to be incorporated in these efforts and would certainly impact on some of the topics covered below. An example of what might be in store is given below. 
    
    
    
      §.§.§  Precocious pseudo-conformailty
    
    There is a precocious onset of conformality which begs to be clarified and is presumably connected to the CC smiles at high density.   It could be a macroscopic rendition of what's taking place in the microscopically fluctuating behavior as one encounters in the sound speed in the interior of compact stars appearing above the skyrmion-half-skyrmion  transition at n∼ n_1/2.   This is observed in a crystal-lattice simulation of the skyrmion matter above the density n_1/2 which takes place roughly at n_1/2 3 n_0 where skyrmions transform to half-skyrmions. It is one of the phenomena in skyrmion matter that remains quite mysterious and intriguing and not understood, but it is believed to be one of the essential features in probing topology in CCM. Let me make a comment,  somewhat detailed, on this matter. 
     
    The half-skyrmion phase in the skyrmion-crystal simulation of dense baryonic matter is, it seems most likely,  in a state that can be described  entirely by mean fields,  largely undistorted by strong nuclear interactions. This phenomenon, discussed in detail already in <cit.>, and given a brief reminder below in Sect. <ref>, is found in Landau-Fermi liquid fixed point theory where the β function for the quasiparticle interactions is suppressed. This striking feature was first found in the Skyrme model  with the Atiyah-Manton ansatz in <cit.>.  Here we illustrate the result from <cit.>.
    
    Write the chiral field U as U(x⃗) = ϕ_0(x,  y,  z ) + i ϕ^j_π(x,  y,  z )τ^j  with the Pauli matrix τ^j and j=1,2,3. Including ρ and ω, we write the fields placed in the lattice size L as ϕ_η,  L(x⃗ ) with η =0, π, ρ, ω and normalize them with respect to their maximum values denoted ϕ_η,L, max for given L.  It can be shown, as in <cit.>, that in the half-skyrmion phase with L L_1/2 where  L_1/2≃2.9 fm, the field configurations are invariant under scaling in density as the lattice is scaled from L_1 to L_2ϕ_η, L_1(L_1t⃗ )/ϕ_η, L_1,  max= ϕ_η, L_2(L_2t⃗ )/ϕ_η, L_2,  max. 
    Since other fields are quite similar, we only show in  Fig. <ref> the case of ϕ_0,π for  ϕ_0,π (t, 0,0) vs. t with t≡ x/L.  What is seen there is that denisty-scale invariance sets in for L L_1/2. One can see that the field is independent of density in the half-skyrmion phase with L L_1/2  whereas for the skyrmion phase with lower density with L >  L_1/2, it is appreciably dependent on density.
    
    
    What does this imply for the energy density?
    
    The energy density for the skyrmion matter put on the lattice of lattice size L can be written as
    
        ϵ = E/A/V(=L^3) = 1/L^3∫^L_0 d^3x ∑_n, m c_n,m  f_n,m( ∇⃗_x, ϕ_η, L(x⃗ )) ,
    
    where c_n,m is the coefficient of f_n,m which is the function of ∇⃗_x and ϕ_η, L(x⃗ ) having nth power of ∇⃗_x and mth power ϕ_η, L(x⃗ ) with ∇_x, j = ∂/∂ x^j. One can reduce it to <cit.>
        ϵ
        
        = ∑_n, m(1/L)^n (ϕ_η, L,  max)^m  A_n, m ,
    
     where A_n, m is a constant independent of the lattice size L.
    
     Calculating the energy density (<ref>) in skyrmion-crystal simulations involves field configurations satisfying their equations of motion. Hence (<ref>) is a mean field expression. It captures all essential dynamics in terms of the mean fields of each degrees of freedom involved, with residual interactions suppressed. This is in the same spirit as the Landau Fermi-liquid fixed approximation that underlies GnEFT.  The density dependence lies, apart from the (1/L)^n factor, in the maximum field configuration (ϕ_η, L,  max)^m. This implies that in the half-skyrmion phase,  considered to set in at high density, the mean-field structure dominates. This agrees with the lore that at high density – and in the large N_c limit, the skyrmion crystal picture becomes valid  in QCD. In clear contrast, however, as one can see in Fig. <ref>, the mean-filed structure breaks down in the lower-density phase with L > L_1/2. Taking the RMF approximation to be more or less equivalent to Fermi-liquid fixed point theory at some high density difficult to pin down numerically, one can take the breakdown of the mean-field structure  as perhaps a signal for non-Fermi liquid structure. 
     
    In short, while it is far from quantitative and short in rigor, it is intriguing that the precocious pseudo-conformality seems to set in the half-skyrmion property (<ref>) in GnEFT as  in the sound velocity in compact stars <cit.> and also in the long-standing puzzle of the quenched g_A in nuclei <cit.>. Whether or not these features survive in  “first-principles" calculations or in Nature remains to be seen. They are not yet ruled out. 
    
    
    
     §.§ CC as a gauge symmetry
    
     To reiterate what has been developed up to here and to be developed further,  the CCP implies that hadronic processes do not discriminate between QCD degrees of freedom (quarks, gluons) on one hand and meson degrees of freedom (pions, vector mesons etc.) on the other, provided all the necessary quantum effects are properly taken into account. 
     An appealing suggestion for what may be involved is an action of  gauge symmetry <cit.>. As a specific case is the MIT bag radius R_bag associated with confinement. By introducing the bag radius expressed in terms of  a collective field as  a gauge degree of freedom, a given value of R_bag in the sense of the MIT bag with confined quarks can be taken as a gauge-fixing as in the cloudy bag <cit.>. With chiral as well other symmetric couplings to appropriate degrees freedom π, vector mesons etc. without the space divided into inside and outside but suitably fitted to data, a model in the class of cloudy bag (with no boundary conditions) could potentially be formulated as a gauge-fixed CC gauge theory. Of course it would be more powerful if it could be not just for the confinement size but in a broader sense as suggested in <cit.>.  It would be interesting to see how far one can proceed in confronting nature.
     
    
    
    § Η^' AND FRACTIONAL QUANTUM HALL DROPLET
    
    
    
     §.§ CC for quantum Hall (QH) droplet
    
    We have seen that the flavor singlet meson η^' played a crucial role for the color symmetry for baryons and  for the FSAC g_A^(0).   In Sect. <ref> the properties of η^' were derived  by  CCP. Now the question is why there is no skyrmion baryon coming via CC from the η^' as there are  octet skyrmions coming the octet mesons π?  There is no description of flavor-singlet  baryon B^(0) in the baryon multiplet whereas the η^' figures as a singlet in the multiplet of the octet π. This may be  due to the role of the color symmetry in QCD <cit.>. It is surprising how this is implied by the homotopy group  π_3 (U(1))=0 as opposed to π_3(SU(2))=π_3 (S^3)=Z. 
    
    This however does not mean that there is no topological baryonic object coming from the η^'.
    
    It gas been recently pointed out by Komargodski <cit.>  that for N_f=1  the chiral effective theory is dominated by the axial U(1) anomaly, and  the topological charge cannot be identified. It was noted that the presence of stable superselection rules in the
    QCD  vacuum (instanton tunneling between vacua with different Chern-Simons number) implies  the existence of 1(time) +2 (space) dimensional domain walls. These walls connect vacua with different Chern-Simons number and are observed to be stable at large N_c. When these sheets are finite dimensional with a boundary, they are found to 
    carry massless edge excitations with baryon quantum numbers. They  are identified with fast spinning baryons. These sheets, unconnected to skyrmions,  are  described by
    a topological field theory through a level-rank duality argument, much like in the fractional quantum Hall (FQH) effect <cit.>. The
     baryons are analogous to the gapless edge excitations in quantum Hall (QH) droplets. 
     
     
     
     Most remarkably this quantum Hall droplet (variably called  pita or pancake) can be obtained by the CCP <cit.>.  A chiral bag with a single-quark species of charge b  (a fermion number) confined to a 1+2 dimensional annulus is found to leak the fermion charge in a way different from the skyrmion falling into an “infinite hotel" <cit.> . When the bag radius is shrunk to zero,  the “smile" of the cat  is left with spinning gapless  quarks running  luminally, accounting for the edge modes and their chirality <cit.>. A current transverse to the smile is shown to appear, embodying the Callan-Harvey anomaly out-flow <cit.>, well-known to string theorists. This transverse current is shown to be quite analogous to the Hall current typical of the QH effect through the emergence of an effective U(1)gauge field, say, Chern-Simons field. This U(1)  gauge field lives in the disc enclosed by the Cheshire cat  smile, and is described by a purely topological field theory in 1+2 dimensions. The quantum numbers of this baryon as a QH droplet follow readily from the chiral bag construction. The droplet outside the bag is illustrated  in Fig. <ref>. 
    
    
    The QH droplet is found to be captured by the  action 
    
    S_1+2= b^2/4π∫_1+2 AdA
    where A_μ is the abelian Chern-Simons field <cit.>. It will be noted, as elaborated later,  that this U(1) field is an  emergent  field dual to the ω field in HLS theory. What results for N_f=1 is a topological object but quite different from skyrmions for N_f≥ 2.  In constructing the CC model Fig. <ref>, why the anomaly out-flow takes place instead of the “infinite hotel" mechanism taken for the skyrmion. The mathematical constraint π_3 (U(1) is the singlet in the flavor space, with no other quantum numbers involved. That the QH droplet/pancake is not visible at low energy in nuclear processes could be because it's unstable or high-lying in energy. In QCD it is the color that could be responsible for the flavor singlet B^0 not to join the octet skyrmion baryons coming from the octet π <cit.>.
    
    
    
     §.§ Tale of two hotels
    
    In <cit.> a chiral bag model was constructed to prevent the color charge  from leaking from the bag following the CCP.  A boundary term was added to the chiral bag to seal  the leaking charge. This boundary term can be readily obtained by noting that for y-independent fields, Ref. <cit.>  describes the  outside of the bag as a line segment in 1+1 dimensions with
     b/2π∫_1+1Adθ= b/2π∫_1+1θ F- b/2π∫_BA_0θ
    after an integration by parts, clearly showing the leaking of the  b-charge through the boundary. To seal the leak, the  inside of the bag has to be supplemented
    by the opposite boundary term
     b/2π∫_B nAη^'/f_η≡ - b/2π∫_B  ϵ^μνn_ν A_μη^'/f_η
    with n^ν the spatial normal to the bag boundary, after enforcing covariance on the 2-form.
    This is exactly the surface term suggested in the Cheshire cat construction.  The present arguments illustrate the subtle relationship between the chiral bag in <cit.> for the baryon charge as well as  <cit.> for the color charge and the present chiral bag for the baryon as a FQH droplet. In the former the  b-charge is absolutely confined, while in the latter the  b-charge is allowed to flow transversely, both making use of a Chern-Simons term. This is the tale of two hotels: the infinite hotel in our world for the confined anomaly, and the finite hotel in the other world for the flowing anomaly. This tale is highly relevant for our formulation of nuclear and astrophysical processes involving hadron-quark continuity. For instance, the role of  the η^' for the color charge conservation is responsible for the Cheshire Cat mechanism for the  tiny flavor singlet axial charge for the proton g_A^(0) described in Fig. <ref>. Furthermore since the η^' is expected to become light at high density, it could have a strong impact on the stiffness of the equation of state in compact-star matter required for the observed massive 2 M_⊙ stars as discussed in <cit.>. This possibility has not  been considered up to date.
    
    
    Now the question is: Are the two hotels connected and if so how?
    
    
    
    
     §.§ Going from skyrmions to quantum Hall droplet
    
    Several interesting questions are raised at this point. First can the skyrmions in the N_f=2 baryons, say, p and n, be made to approach the QH droplet structure  by tweaking parameters of the chiral Lagrangian? 
    
    This issue is addressed by Karasik  in two ways <cit.>. 
    
    First is to start with the two-flavor skyrmions in the presence of η^' field with zero quark masses m_u=m_d=0 in the large N_c limit. The mass m_d is then continuously increased by fiat from 0 to ∞ while keeping m_u at zero. The tweaking makes the theory continuously move to  a chiral compact boson living on a boundary, leading to a Chern-Simons theory. The process is quite intricate, involving a certain amount of involved arguments and reproduces the pancake structure of <cit.>. It also predicts the spin of B^0 to be N_c/2, the same as the decuplet Δ. Of course this by itself is not a controllable operation in reality but it shows how the QH droplet is connected to the skyrmions.
    
    The second way to see the appearance of the QH droplet is to couple the HLS vector mesons <cit.> to the pions and the η^'. This way, as I will describe below, offers a possibility – the only way known (to the author) – to probe the QH droplet by going to high density.
    
    Concisely stated, the key observation Karasik makes is that the cusp singularity in the potential V_η^' in the effective Lagrangian for η^' generated as gluon fields jump from one vacuum to another
    
    V_η^'≈1/2 f_π^2 m_η^'^2   min_k∈ Z (η^' - 2π k)^2 +⋯
    can be eliminated by the vector mesons in hidden local symmetry <cit.>  in the effective theory which play the role of Seiberg dual to the gluons <cit.>.  This leads to the fractional quantum Hall structure on the η^' sheet as a configuration carrying a baryon charge that interpolates between η^'=0 and η^'=2π captured by a U(1)_N_c Chern-Simons theory <cit.>. This structure is also indicated in holographic dual QCD <cit.>. 
    
    The power of exploiting hidden local symmetry (HLS) and hidden scale symmetry (HSS) together with the notion – up to date lacking rigorous formulation – of vector dominance (VD) is that there is a possibility of probing the QH droplet, more precisely, fractional QH sheet structure via an effective field theory suitably applicable to dense nuclear matter relevant to massive compact stars.  What figures is the homogenous Wess-Zumino term gauge-invariant with the global symmetries and with time reversal symmetry that acts on the field U, U↔  U^†.  Limited to N_f=2 for convenience, the 5-d Wess-Zumino term does not enter.  Remarkably it is the ω field in HLS that will play the role of the Chern-Simons field A_μ in (<ref>).
    
    
    To discuss how this can be achieved, we need a brief discussion of the effective field theory generalized from Weinberg's χEFT <cit.> to apply to density n 3n_0 relevant to compact-star matter. We called it GnEFT and was mentioned above..
    
    
    
      §.§.§  Generalized nuclear EFT (GnEFT)
    
    It is now well established that the χEFT anchored on Weinberg's “Folk Theorem" – call it standard (S)χEFT – works quantitatively well up to baryon density 2n_0 where n_0≈ 0.16 fm^-2  is the nuclear equilibrium density. The central density of massive compact-star matter measured in astrophysical observations could reach ∼ (5-7)n_0. To access this density regime, however, degrees of freedom higher in scale than what figures in  SχEFT need to be introduced. Holographic dual approaches incorporate infinite towers of vector mesons. In GnEFT, the next most relevant degrees of freedom  taken into account are the HLS vectors V_μ=(ρ_μ,ω_μ) and the hidden scale symmetry dilaton D that we will represent by the “conformal compensator field" χ=f_χ e^D/f_χ. To this GnEFT,   is added the flavor singlet η^'.[The mixing between the octet scalar η and the flavor singlet η^' will be ignored.] What figures importantly in this approach is the degree of freedom associated with the spontaneous breaking of scale symmetry hidden in QCD. It is not usually considered explicitly but appears in SχEFT at some low order in chiral perturbation theory.
    
    There is a long history – and with controversy – on the scalar dilaton resulting from the broken scale symmetry that we will choose to avoid to get into details. For our problem, we  adopt the most recent approach called “Genuine Dilaton (GD)" by Crewther (with Tunstall) <cit.> and/or “QCD-Conformal Dilaton (QCD-CD)" by Zwicky <cit.>, both of which are anchored on assuming an IR fixed-point in QCD with N_f 3.[The controversy has to do with whether or not there can be an IR fixed point for N_f ≪ 8 in QCD, away from the conformal window, being discussed re:  dilatonic Higgs in particle theory. Some current developments are at odds with the long-standing controversy. The work discussed in this article is in favor of the recent  proposals <cit.> as a possibility for an emergence phenomenon in nuclear correlations.]   Applied to our case where scale symmetry is mist likely  “emergent" by strong baryonic correlations,  these two approaches are essentially the same except for certain fine details we are not concerned with. The IR fixed point we are concerned with is characterized by the presence of the Nambu-Goldstone bosons π and χ with  nonzero decay constants, f_π≠ 0 and f_χ≠ 0, thereby accommodating massive matter fields such as baryons ψ, vector mesons V etc. The GnEFT was formulated to apply, at least semi-quantitatively,  from normal nuclear matter to compact-star matter. 
    
    The effective Lagrangian that leads to the GnEFT will be denoted L_ψ, HS,χ  with HS standing for hidden symmetries, both local and scale,  and  with the η^' coupled in to account for the U_A(1) anomaly, it will be written L_ψ, HS,χ,η^'.  Now the key ingredient among the quantities that enter into L_ψ, HS,χ is topological as befits the CCP. It is the topology change from skyrmions to half-skyrmions assumed to take place at the density n_1/2∼ (2-3) n_0. It encodes a wide-range of density-dependent parameters in the Lagrangian connected logically to the topological structure consistent with the CCP <cit.>.  This many-nucleon approach is to encompass the  SχEFT and go up in density scale to the compact-star regime by implementing Landau Fermi-liquid (LFL) fixed point approximation in a generalized density functional (GDF) approach à la Hohenberg-Kohn theorem. In our notation, this is referred to as pseudo-conformal model, concisely  summarized  <cit.>.  There one finds that its prediction, with essentially no free parameters, encounters, so far, no serious inconsistency with observables from normal nuclear matter densities to the densities measured in the recent gravity-wave and associated data in compact-star matter[Just for comparison, see the most up-to-date analyses <cit.>.].
    
    
    
      §.§.§  From chiral EFT to Kohn-Sham density functional 
     and the dilaton-limit fixed-point (DLFP)
    
    It may be somewhat nontrivial to those unfamiliar to how a generalized density functional can be gotten from  χEFT with hidden symmetries minimally implemented. This follows from the work of Ref. <cit.> based on the strategy given in <cit.> on the formulation of renormalization-group approach to many-fermions on the Fermi surface.
    In fact one arrives at it by a simple formulation in terms of the Landau Fermi-liquid fixed-point interactions given by BR scaling <cit.>. This GDF formulation anchored on Hohenberg-Kohn theorem invoked above  is most likely closely connected to what is discussed in the fractional Quantum Hall phenomenon in condensed matter physics.   The Kohn-Sham density functional theory, widely applied in nuclear physics, turns to have a powerful impact in the fractional quantum Hall effect studied in mesoscopic devices as shown in <cit.>. It would be interesting to further elucidate the possible conceptual link between the two fields.  
    
    In approaching the FQH droplet  B^0,  L_ψ, HS,χ,η^' is treated in the mean-field which corresponds to the  LFL fixed-point approximation. Next    
    one redefines the field Σ=f_π/f_χ Uχ with U=e^iπ/f_π  and take the limit Tr (Σ^†Σ)→ 0. In order to avoid the singularities that  develope  in the limiting process, one is required to put what are referred to as the “dilaton-limit fixed-point (DLFP)" constraints <cit.>[In this reference the notion of DLFP was first formulated in terms of quark model but it has been verified to equally apply to quasi-nucleons as in GnEFT.]       f^∗_π→  f^∗_χ,   g^∗_ρ NN=g^∗_ρ g^∗_ COR→ 0, 
           g^∗_A→ 1, m^∗_N→ m_0∝ f_χ^∗
    where g^∗_ρ is the HLS gauge coupling in medium in RG analysis <cit.> and  g^∗_ COR stands for standard (many-body) nuclear correlation effects in dense matter. The ∗ stands for density dependence encoded in the condensates of the vacuum modified by the medium. 
    
    There are several remarkably powerful implications in the constraints (<ref>), some of which, it should be mentioned, have not yet been  given solid theoretical support. The first is that the DLFT indicates an approach to the IR fixed point à la  GD (and CD-QCD) scenario that our approach espouses. Indeed f^∗_π→ f^∗_χ deviates strongly from the conformal window scenario (relevant to dilaton Higgs model)  f_π^2/f_d^2  ≪ 1 <cit.>. What's directly relevant to our approach to the FQH droplet is the decoupling of the iso-vector vector mesons in HLS   before reaching the vector manifestation fixed point g^∗_ρ→ 0 at which the ρ becomes the genuine gauge field.  Furthermore g^∗_A→ 1 is linked to the long-standing quenching of g_A in nuclear matter <cit.> and the relation m_N^∗≈ f^∗_χ follows from the emergence of parity-doubling symmetry in dense medium responsible for  the sound speed in the core of compact stars approaching precociously the pseudo-conformal sound velocity v^2_pcs≈ 1/3 in the core of compact stars <cit.>.
    
    
    
      §.§.§  The role of “homogeneous"/“hidden" Wess-Zumino term
    
     Now when the HLS fields are “integrated in," apart from the 5-d Wess-Zumino-Witten term which is absent for N_f=2 that we are concerned with, there is a 4-d action that has the homogeneous solution to the  Wess-Zumino anomaly equation, called homogeneous[Called “hidden" by Karasik when combined with the VD.] WZ term <cit.>. This action, conserving parity and charge conjugation but violating intrinsic parity, contains four terms including an external field constructed with Maurer-Cartan 1-forms. 
     
     What's most significant for the issue concerned here is that the vector mesons in HLS are most likely Seiberg-dual gluons <cit.> and in the DLFP,  the iso-vector mesons decouple before reaching the IR fixed point as seen in (<ref>). That leaves the ω meson. As argued by Karasik <cit.> in terms of the domain wall with lim_z→ -∞η^' (z)=0 and  lim_z→ +∞η^' (z)=2π, one  then arrives at the (1+2) domain theory
     L_ domain-wall= N_c/4πω dω
    which is lodged in the bulk of the sheet. This reproduces the CC result (<ref>) relating the ω field to the Chern-Simons field A_μ. The  VD (vector dominance) emerges in the theory fixing the arbitrary parameters in the hWZ of HLS <cit.> to hidden(h) WZ <cit.> .
    
    What we have here is just one sheet. There is an ongoing work of sheet structure in  dense matter in the field of skyrmion crystals. There are mathematical treatments of multi-sheet structure actively studied in the field <cit.> and it is much too early to make contact with Nature. In fact such a sheet structure is already seen in skyrmion-crystal structure at finite density <cit.> and speculated in compact-star physics where fractionally charged quasi-baryons play the role of deconfined quarks in the interior of compact stars <cit.>.
    
    In the DLFP limit, the ρ mesons decouple. In Nature, the DLFP density can be  pinned down neither by theory nor by experiments. It is likely at a lower density than that of the vector manifestation. The question then arises what happens if the iso-vector mesons are not sufficiently decoupled. It seems plausible that one would then encounter non-abelian Chern-Simons droplets before reaching the abelian Chern-Simons droplet. In fact this is what one finds in the Cheshire Cat description <cit.>. 
    For arbitrary N_f the spin and statistics arguments do not change as they are solely fixed by N_c. However, the leaking flavor currents lead to a U(N_f) flavor-valued emergent gauge field A_μ. The CCP applies  mutatis mutandis. 
    The  emergent non-Abelian Chern-Simons action  would then be
    N_c/4π∫_1+2 Tr( A  d A+2/3 A^3).
    
    This suggests a wide range of stacked sheets of Chern-Simons droplets/pancakes before reaching the truly deconfined quarks which may not survive the gravitational collapse. As in condensed matter systems, the phase structure as viewed in this scenario could be much richer than what can be seen in QCD variables. It could be that ultimately which way is superior will depend on which is more efficient in capturing the vast variety of physics.
    
    
    
    § SOME COMMENTS AND OPEN ISSUES
    
    Even at the far-from-rigorous level that was adopted in this article, there are several issues that beg to be further elaborated. Let me mention a few of them here for possible future development. In the present status of poor understanding of highly dense matter, it would be premature to rule out any of  them.
    
    ∙ Multifaceted role of ω meson in nuclear matter
    
    In the vacuum the ω meson  is with the ρ meson in hidden local symmetry. But in nuclei, it starts deviating from U(3) in controlling the nuclear forces while the ρ meson decouples from the dense matter before the vector manifestation fixed point (VMFP) <cit.> is arrived at. It provides the crucial repulsion to counter the attraction due to the scalar (dilaton)-meson exchange as the density approaches the dilaton-limit-fixed point (DLFP). It gets transformed via Vector Dominance to the abelian Chern-Simons field. How this feature gets manifested in terms of QCD variables would be important for the physics of compact-star matter. This is an open issue that has not been addressed up to date.
    
    ∙ Half-skymion structure
    
    The half-skyrmion structure figures crucially in the framework of GnEFT in controlling the baryonic matter at high density (as reviewed in <cit.>). Although it does not appear explicitly, a single skyrmion may actually be a bound two half-skyrmions by a hidden monopole as suggested in <cit.>. One would wonder whether  the monopole is suppressed at high density, so that one can consider deconfined half-skyrmions interacting scale-invariantly as seen in skyrmion crystal configurations in Fig. <ref>. In condensed matter physics in lower dimensions, deconfined quantum critical phenomena could involve such deconfined half-skyrmions <cit.>[More recent developments do away in condensed matter physics with the deconfined half-skyrmions, going to local effective field theories directly without changing  essential physics.]. Also in CCM, the magic angle that involves half-skymions with the other half shared by the vector meson ρ provides an extremely simple account to the electromagnetic form factors of the proton at high momentum transfers <cit.>. It is not clear how to transform the half-skyrmions to deconfined quarks.
    
    ∙ Stacks of FQH droplets
    
    As mentioned, there are indications for the formation of the sheets of half-skyrmions in the skyrmion crystal simulations <cit.> at increasing density. The discussion in Sect. <ref> suggested the formation of a QH droplet with J=N_c/3 as density is increased  toward the DLFP. It seems that there can be a stack of  QH droplets mixed with, possibly unstable, half-skyrmion sheets before reaching the DLFP. That may give rise to something like the domain-wall structure with deconfined quarks on the sheets (or droplets) and confinement in the bulk <cit.>.  This could lead to the precocious pseudo-conformal EoS <cit.> resembling what's considered as evidence of quark-matter core in massive compact stars <cit.>.
    
    
    
     §.§ Acknowledgments
    
    
    I am grateful for the invitation by Su Houng Lee to write this article for Elsevier.
    
    
    
    
    
    
    
    
    
    
    
    
    42MITbag K. Johnson,
    “The M.I.T. bag model,”
    Acta Phys. Polon. B 6 (1975), 865
    MIT-CTP-494.
    
    LB G. E. Brown and M. Rho,
    “The Little Bag,” Phys. Lett. B 82 , 177 (1979).
    
    weise L. Brandes and W. Weise,
    “Constraints on phase transitions in neutron star matter,”
    Symmetry 16 (2024) no.1, 111.
    
    CBM
    G. E. Brown and M. Rho,
    “The Chiral Bag,”
    Comments Nucl. Part. Phys. 18  no.1, 1 (1988).
    
    
    CBMapplied
    G. E. Brown, M. Rho and V. Vento,
    “Little Bag dynamics,”
    Phys. Lett. B 84, 383-388 (1979).
    
    
    CCP S. Nadkarni, H. B. Nielsen and I. Zahed,
    “Bosonization relations as bag boundary conditions,”
    Nucl. Phys. B 253, 308-322 (1985); S. Nadkarni and I. Zahed,
    “Nonabelian Cheshire Cat bag models in (1+1)-dimensions,”
    Nucl. Phys. B 263 (1986), 23-36 (1985).
    
    TCCM S. Nadkarni and H.B. Nielsen, “PARTIAL BOSONIZATION: The formalism of Cheshire cat bag models," Nucl. Phys. B263, 1 (1986).
    
    
    skyrme T. H. R. Skyrme,
    “A nonlinear field theory,”
    Proc. Roy. Soc. Lond. A 260 (1961), 127-138;  `A unified field theory of mesons and baryons,”
    Nucl. Phys. 31 (1962), 556-569.
    
    CloudyBM A. W. Thomas,
    “Chiral symmetry and the bag model: A new starting point for nuclear physics,”
    Adv. Nucl. Phys. 13, 1-137 (1984).
    
    Witten-qcd E. Witten, “Current algebra, baryons, and quark Confinement,”
    Nucl. Phys. B 223, 433-444 (1983).
    
    noB0 J. Estévez, F. J. Llanes-Estrada, V. Martínez-Fernández and A. Pastor-Gutiérrez,
    “Lightest flavor-singlet qqq baryons as witnesses to color,”
    Phys. Rev. D 102, no.11, 114032 (2020).
    
    nielsen-wirzba H. B. Nielsen and A. Wirzba,
    “The Cheshire  Cat Principle applied to hybrid bag models," in Les Houches Workshop on Theoretical Physics, March 24-April 2, 1987 (Springer Proceedings in Physics. V 26, (1988)) 
    
    
    MRCCP M. Rho,
    “The Cheshire Cat Hadrons Revisited,”
    [arXiv:hep-ph/0206003 [hep-ph]] revision and addendum to "Cheshire Cat Hadrons," Phys. Repts.  240, 1-141 (1994).
    
    RGB M. Rho, A. S. Goldhaber and G. E. Brown,
    “Topological soliton bag model for Baryons,”
    Phys. Rev. Lett. 51, 747-750 (1983).
    
    
    GJ J. Goldstone and R. L. Jaffe,
    “The baryon number in chiral bag models,”
    Phys. Rev. Lett. 51, 1518 (1983).
    
    
    protonspin B. W. Filippone and X. D. Ji,
    “The spin structure of the nucleon,”
    Adv. Nucl. Phys. 26, 1 (2001).
    
    EL E. Leader,
    “The non-existence of the proton spin crisis,”
    PoS SPIN2018, 123 (2018).
    
    cheshireeta H. B. Nielsen, M. Rho, A. Wirzba and I. Zahed,
    “The tale of the eta-prime from the cheshire cat principle,”
    Phys. Lett. B 281, 345-350 (1992).
    
    colorleakage H. B. Nielsen, M. Rho, A. Wirzba and I. Zahed,
    “Color anomaly in a hybrid bag model,”
    Phys. Lett. B 269, 389-393 (1991).
    
    Witten E. Witten,
    “Current algebra theorems for the U(1) Goldstone boson,”
    Nucl. Phys. B 156, 269-283 (1979).
    
    Veneziano G. Veneziano,
    “U(1) without instantons,”
    Nucl. Phys. B 159, 213-224 (1979).
    
    
    FSAC H. J. Lee, D. P. Min, B. Y. Park, M. Rho and V. Vento,
    “The Proton spin in the chiral bag model: Casimir contribution and Cheshire cat principle,”
    Nucl. Phys. A 657, 75-94 (1999). 
    
    HLS M. Harada and K. Yamawaki,
    “Hidden local symmetry at loop: A New perspective of composite gauge boson and chiral phase transition,”
    Phys. Rept. 381, 1-233 (2003).
    
    
      vankolck H. W. Hammer, S. König and U. van Kolck,
    “Nuclear effective field theory: status and perspectives,”
    Rev. Mod. Phys. 92, no.2, 025004 (2020).
    
    WeinbergEFT S. Weinberg,
    “What is quantum field theory, and what did we think it is?,”
    [arXiv:hep-th/9702027 [hep-th]].  
    
    
    
     
    
    battye R. A. Battye, N. S. Manton, P. M. Sutcliffe and S. W. Wood,
    “Light nuclei of even mass number in the Skyrme model,”
    Phys. Rev. C 80, 034323 (2009).
    
    manton N. S. Manton,
    “Skyrmions – A Theory of Nuclei,”
    World Scientific, 2022.
    
    sutcliffe C. Naya and P. Sutcliffe,
    “Skyrmions in models with pions and rho mesons,”
    JHEP 05, 174 (2018).
    
    
    HRYY  D. K. Hong, M. Rho, H. U. Yee and P. Yi,
      “Chiral dynamics of baryons from string theory,”
      Phys. Rev. D  76, 061901 (2007)
     
      [hep-th/0701276 [HEP-TH]].
    
    SS H. Hata, T. Sakai, S. Sugimoto and S. Yamato,
      “Baryons from instantons in holographic QCD,”
      Prog. Theor. Phys.  117, 1157 (2007)
     
      [hep-th/0701280 [HEP-TH]].
      
    park-vento B.-Y. Park and V. Vento, “Skyrmion approach to fininte density and temperature" in  The Multifaceted Skyrmion  (World Scientific, 2017) ed. by M. Rho and I. Zahed.  
    
    adametal C. Adam, C. Naya and A. Wereszczyński,
    “Carbon-12 in the generalized Skyrme model,”
    [arXiv:2401.08778 [nucl-th]];
    C. Adam, A. Garcia Martin-Caro, M. Huidobro and A. Wereszczynski,
    “Skyrme Crystals, Nuclear Matter and Compact Stars,”
    Symmetry 15, no.4, 899 (2023); 
    
    
    
    
    C. Adam, A. García Martín-Caro, M. Huidobro, R. Vázquez and A. Wereszczynski,
    “Kaon condensation in skyrmion matter and compact stars,”
    Phys. Rev. D 107, no.7, 074007 (2023);
    C. Adam, A. García Martín-Caro, M. Huidobro, R. Vázquez and A. Wereszczynski,
    “A new consistent neutron star equation of state from a generalized Skyrme model,”
    Phys. Lett. B 811, 135928 (2020).
    PKLMR W. G. Paeng, T. T. S. Kuo, H. K. Lee, Y. L. Ma and M. Rho,
    “Scale-invariant hidden local symmetry, topology change, and dense baryonic matter. II.,”
    Phys. Rev. D 96, no.1, 014031 (2017).
    
    
    atiyah-manton-skyrmion B. Y. Park, D. P. Min, M. Rho and V. Vento,
      “Atiyah-Manton approach to skyrmion matter,”
      Nucl. Phys. A  707, 381 (2002).
      
    
    shao-rho L. Q. Shao and M. Rho,
    “Corrections to Landau Fermi-liquid fixed-point approximation in nonlinear bosonized theory: An application to g_A^L in nuclei,”
    Phys. Rev. C 110, no.1, 015204 (2024).
    
    smooth-bosonization P. H. Damgaard, H. B. Nielsen and R. Sollacher,
    “Smooth bosonization: The Cheshire cat revisited,”
    Nucl. Phys. B 385, 227-250 (1992).    
      
    
    ZOHAR
      Z. Komargodski,
      “Baryons as quantum Hall droplets,”
      arXiv:1812.09253 [hep-th].
      
      CMHALL 
      D. Tong,
      “Lectures on the Quantum Hall Effect,”
      arXiv:1606.06687 [hep-th].
      
     MNRZ Y. L. Ma, M. A. Nowak, M. Rho and I. Zahed,
    “Baryon as a Quantum Hall Droplet and the quark-hadron duality,”
    Phys. Rev. Lett. 123, 172301 (2019).
      
      CALLAN
      C. G. Callan, Jr. and J. A. Harvey,
      “Anomalies and fermion zero modes on strings and domain walls,”
      Nucl. Phys. B  250, 427 (1985).
      
      
    PCMprediction M. Rho,
    “Dense baryonic matter predicted in “pseudo-conformal model”,”
    Symmetry 15, no.6, 1271 (2023)
    [arXiv:2305.04715 [nucl-th]].
    
      
     karasik A. Karasik,
    “Vector dominance, one flavored baryons, and QCD domain walls from the `hidden' Wess-Zumino term,”
    SciPost Phys. 10, no.6, 138 (2021); “From skyrmions to one-flavored baryons and beyond," Symmetry 2022, 14, 2347.
    
    sdual Z. Komargodski,
    “Vector mesons and an interpretation of Seiberg duality,”
    JHEP 02, 019 (2011).
     
     dualqcd F. Bigazzi, A. L. Cotrone and A. Olzi,
    “Hall droplet sheets in holographic QCD,”
    JHEP 02, 194 (2023).
    
    Crewther
    R. J. Crewther,
    “Genuine dilatons in gauge theories,”
    Universe 6, no.7, 96 (2020);  
    R. J. Crewther and L. C. Tunstall,
    “Δ I=1/2 rule for kaon decays derived from QCD infrared fixed point,”
    Phys. Rev. D 91,  034016 (2015).
    
    Zwicky
    R. Zwicky,
    “QCD with an infrared fixed point and a dilaton,”
    [arXiv:2312.13761 [hep-ph]]. 
    
    
    MRreview 
    Y. L. Ma and M. Rho,
    “Towards the hadron–quark continuity via a topology change in compact stars,”
    Prog. Part. Nucl. Phys. 113, 103791 (2020);
    “Manifestation of hidden symmetries in baryonic matter: from finite nuclei to neutron stars,”
    Mod. Phys. Lett. A 36,  2130012 (2021).
    FR B. Friman and M. Rho,
    “From chiral Lagrangians to Landau Fermi liquid theory of nuclear matter,”
    Nucl. Phys. A 606, 303-319 (1996).
    
    shankar R. Shankar,
    “Renormalization group approach to interacting fermions,”
    Rev. Mod. Phys. 66, 129-192 (1994).
    
    hu-jain Y. Hu and J.K. Jain, “Kohn-Sham theory of fractional quantum Hall effect," Phys. Rev. Lett.  123, 176802 (2019).
    
    lattimer N. Rutherford, M. Mendes, I. Svensson, A. Schwenk, A. L. Watts, K. Hebeler, J. Keller, C. Prescod-Weinstein, D. Choudhury and G. Raaijmakers, et al.“Constraining the dense matter equation of state with new NICER mass-radius measurements and new chiral effective field theory inputs,”
    [arXiv:2407.06790 [astro-ph.HE]].
    
    dlfp-Bira S. R. Beane and U. van Kolck,
    “The dilated chiral quark model,”
    Phys. Lett. B 328, 137-142 (1994) [arXiv:hep-ph/9401218 [hep-ph]].
    
    
    appelquist T. Appelquist, J. Ingoldby and M. Piai,
    “Dilaton Effective Field Theory,”
    Universe 9, no.1, 10 (2023).
    
    gA* Y. L. Ma and M. Rho,
    “Quenched g_A in nuclei and emergent scale symmetry in baryonic matter,”
    Phys. Rev. Lett. 125, no.14, 142501 (2020)
    
    
    canfora E.g. among many in the literature,  F. Canfora, E. Delgado and L. Urrutia,
    “Ordered patterns of (3+1)-dimensional hadronic gauged solitons in the low-energy limit of Quantum Chromodynamics at a finite baryon density, Their magnetic fields and novel BPS bounds,” Symmetry 16, no.5, 518 (2024);  M. Eto, K. Nishimura and M. Nitta,
    “How baryons appear in low-energy QCD: Domain-wall Skyrmion phase in strong magnetic fields,”
    [arXiv:2304.02940 [hep-ph]].
    
    paeng-park-vento B. Y. Park, W. G. Paeng and V. Vento,
    “The inhomogeneous phase of dense skyrmion matter,”
    Nucl. Phys. A 989, 231-245 (2019) [arXiv:1904.04483 [hep-ph]].
    
    MR-fractional M. Rho,
    “Fractionalized quasiparticles in dense baryonic matter,”
    [arXiv:2004.09082 [nucl-th]]; 
    “Probing fractional Quantum Hall sheets in dense baryonic matter,”
    [arXiv:2211.14890 [nucl-th]].
    
    cho Y. M. Cho, K. Kimm, J. H. Yoon and P. Zhang,
    “New topological structures of Skyrme theory: Baryon number and monopole number,”
    Eur. Phys. J. C 77, no.2, 88 (2017); P. Zhang, K. Kimm, L. Zou and Y. M. Cho,
    “Re-interpretation of Skyrme theory: New topological structures,”
    [arXiv:1704.05975 [hep-th]].
    
    QCP T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M. P. A. Fisher,
    “Deconfined quantum critical Point,” Science 303, no.5663, 1490-1494 (2004).
    
    magic G. E. Brown, M. Rho and W. Weise,
    “Phenomenological sizes of confinement regions in baryons,”
    Z. Phys. A 331, 139-149 (1988).
    
    domain-wall T. Sulejmanpasic, H. Shao, A. Sandvik and M. Unsal,
    “Confinement in the bulk, deconfinement on the wall: infrared equivalence between compactified QCD and quantum magnets,”
    Phys. Rev. Lett. 119, no.9, 091601 (2017).
    
    evidence-core E. Annala, T. Gorda, A. Kurkela, J. Nättilä and A. Vuorinen,
    “Evidence for quark-matter cores in massive neutron stars,”
    Nature Phys. 16, no.9, 907-910 (2020). ] ] ]


