Returning CP-Observables to The Frames They Belong

Jona Ackerschott1, Rahool Kumar Barman2, Dorival Gonçalves2,

Theo Heimel1, and Tilman Plehn1

1 Institut für Theoretische Physik, Universität Heidelberg, Germany

2 Department of Physics, Oklahoma State University, Stillwater, USA

August 28, 2024

## Abstract

Optimal kinematic observables are often defined in specific frames and then
approximated at the reconstruction level. We show how multi-dimensional
unfolding methods allow us to reconstruct these observables in their proper
rest frame and in a probabilistically faithful way. We illustrate our approach
with a measurement of a CP-phase in the top Yukawa coupling. Our method makes
use of key advantages of generative unfolding, but as a constructed observable
it fits into standard LHC analysis frameworks.

###### Contents

  1. 1 Introduction
  2. 2 Reconstructing observables by unfolding
    1. 2.1 Generative unfolding
    2. 2.2 Periodic splines
    3. 2.3 Phase space parametrization
  3. 3 CP-phase from Higgs-top production
    1. 3.1 CP-observables
    2. 3.2 Unfolding-based analysis
    3. 3.3 Results
  4. 4 Outlook

##  1 Introduction

With the LHC continuing its success story of precision hadron collider
physics, the size and complexity of the datasets of the upcoming Run 3 and HL-
LHC are challenging the existing analysis methodology [1, 2, 3]. At the same
time, the goal of LHC physics has moved from model-based searches for physics
beyond the Standard Model (SM) to a comprehensive analysis of all its data,
based on consistent analysis frameworks like the Standard Model effective
theory [4, 5, 6].

The first step in any global analysis based on the fundamental principles of
QFT is to determine the underlying symmetries, which are required to construct
the effective Lagrangian. The, arguably, most interesting symmetry in the SM
is $CP$, linked to cosmology through the Sakharov conditions for baryogenesis
[7], and potentially realized in an extended Higgs sector [8]. In the language
of effective theory, $CP$-violation in the Higgs coupling to vector bosons is
loop-suppressed and arises at dimension six [9, 10, 11, 12, 13]. In contrast,
$CP$-violation in Higgs couplings to fermions can appear at dimension four
[14], making a $CP$-phase in the top Yukawa coupling the most sensitive link
between baryogenesis and LHC physics [15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32].

Obviously, we do not want to leave the test of fundamental Lagrangian
symmetries to a global analysis [33, 34] with limited control over
experimental and theoretical uncertainties [35, 36, 37, 38, 39], including
systematics from parton densities [40]. Instead, we should use dedicated
(optimal) observables to target one fundamental symmetry at a time [41, 42,
12, 30, 29, 43]. In the Higgs-gauge sector, the optimal observable is the
azimuthal angle between the two tagging jets in weak boson fusion. For
associated top–Higgs production, the azimuthal angle between a charged lepton
from one top decay and the down quark from the other plays a similar role.
Accurately extracting it faces the challenge of identifying the corresponding
decay jet. Another powerful observable probing the Higgs-top interaction is
the Collins–Soper angle [44, 23, 28, 29]. Again, the challenge is to map it
onto the observed final state after particle decays, parton shower, and
detector effects.

Both of these observables illustrate the common problem that an optimal or
ideal kinematic correlation is usually not defined on the reconstructed final
state. So while an optimal observable provides full sensitivity without the
need to consider additional phase space correlations, we pay a prize in its
reconstruction.

The standard inference approach for such kinematic correlations is to
approximate them at the reconstruction level. For this approximation, we can
use a directly observable correlation at the reconstruction level or rely on
some kind of algorithm. The approximation is unlikely to be optimal. An
improved approach would be to encode the observable in a learned mapping, for
instance, through neural networks. Fundamentally different and, in principle,
optimal alternatives are simulation-based inference [45, 46] or the matrix
element method [47, 48, 49, 50], but they come at a significant numerical cost
and are hard to re-interpret for other measurements.

For cases where an optimal observable is defined in some kinematic frame, we
propose a simplified unfolding approach, where we unfold the reconstruction-
level events to the appropriate reference frame, and then construct the
optimal observable for the down-stream task. Unfolding or reconstructing
events beyond the immediately available detector output is a long-standing
problem [51, 52, 53, 54], undergoing transformative progress through modern
machine learning (ML) [55, 56, 57, 58, 59, 60, 61, 62, 63]. One key
observation is that forward and backward simulations are completely symmetric
when we interpret them as sampling from conditional probabilities [64, 65].
This motivates ML-unfolding through probabilistic inverse simulations [59, 60,
62], which allows us to reconstruct observables or parameters defined at any
level of our forward simulations, for instance, unfolding detector effects,
parton shower, particle decays [61], all the way to measuring fundamental
parameters [66].

This generative unfolding technique allows us to just reconstruct key
observables, which have the advantage that they can be used in the standard
analysis frameworks of ATLAS and CMS, but with a performance increase from the
full set of kinematic correlations learned through the unfolding. To guarantee
stable network predictions and to be able to quantitatively extract the
training-induced network uncertainties, we use the Bayesian version [67] of
the conditional normalizing flows [68, 69], for which the likelihood losses
should lead to well-calibrated results. Eventually, this kind of analysis can
serve as a simple starting point for ML-unfolding, as it can be expanded
through additional observables step by step.

In this paper, we use $CP$-violation through a complex top Yukawa coupling to
show how ML-unfolding techniques can construct and numerically encode
observables in the reference frame where they are defined. In Sec. 2, we first
describe our neural network architecture, the physics task, and the treatment
of phase space. In Sec. 3, we introduce our reference process and discuss our
results and potential generalization errors. Finally, Sec. 4 is reserved for
summary and outlook.

##  2 Reconstructing observables by unfolding

In this study, we propose to use statistical unfolding through inverse
simulation [59, 60] to construct kinematic observables in a specific partonic
reference frame. While we are making use of unfolding techniques in
constructing a given observable, the precision, control, and model dependence
of the unfolding is not a limiting factor for our analysis. Instead, we treat
the so-defined observable like any other kinematics construction.

###  2.1 Generative unfolding

Generative unfolding is based on the observation that a forward simulation
from a parton-level event $x_{\text{part}}$ to a reco-level event
$x_{\text{reco}}$ just samples from an encoded conditional probability,

$\displaystyle r\sim\mathcal{N}(r)\;\stackrel{{\scriptstyle
x_{\text{part}}}}{{\longrightarrow}}\;x_{\text{reco}}\sim
p(x_{\text{reco}}|x_{\text{part}})\qquad\quad\text{(forward)}\;.$ (1)

This simulation can be trivially inverted on the same training data, so we can
unfold detector effects, initial-state jet radiation, or particle decays, by
sampling from the inverse conditional probability,

$\displaystyle r\sim\mathcal{N}(r)\;\stackrel{{\scriptstyle
x_{\text{reco}}}}{{\longrightarrow}}\;x_{\text{part}}\sim
p(x_{\text{part}}|x_{\text{reco}})\qquad\quad\text{(inverse)}\;.$ (2)

In both cases, the standard training relies on paired events
$\\{x_{\text{part}},x_{\text{reco}}\\}$. Obviously, this training dataset
leads to model dependence, which can be reduced by using iterative methods
[62]. The target phase space of the inverse simulation or unfolding can be
chosen flexibly, just unfolding detector effects [60, 62], but also jet
radiation [60], particle decays [61], or sampling right into model parameter
space using setups like BayesFlow [66]. Inference through conditional
normalizing flows is standard in many fields of physics [70, 71].

Our generative network encoding the conditional probability defined in Eq. (2)
is a conditional normalizing flow, specifically a conditional invertible
neural network (cINN) [70], trained with a likelihood loss to guarantee a
statistically correct and calibrated output. To link a batch of $B$ phase
space points $x_{i}$ to a Gaussian latent space $r_{i}$ with the condition
$c_{i}$, the likelihood loss reads

$\displaystyle\mathcal{L}_{\text{cINN}}=\sum_{i=1}^{B}\left(\frac{r_{i}(x_{i};c_{i})^{2}}{2}-\log\left|\frac{\partial
r_{i}(x_{i};c_{i})}{\partial x_{i}}\right|\right)\;.$ (3)

###  2.2 Periodic splines

$x_{2}$
---
$x_{3}$
---
$x_{4}$
---
$x_{5}$
---
$x_{6}$
---
$x_{1}$
---
$y_{5}$
---
$y_{4}$
---
$y_{3}$
---
$y_{6}$
---
$y_{1}$
---
$y_{2}$
---
Figure 1: Visualization of a modified coupling transformation for periodic
inputs. This transformation maps $K$ points $x_{i}$ to $K$ points $y_{i}$ on a
circle, while we use rational quadratics to interpolate between two points.
The modifications ensure that the $x_{i}$ and $y_{i}$ can be arbitrarily
skewed in relation to one another, and that the derivative at $x_{1}$ is
consistent with both adjacent rational quadratics.

The main part of our cINN is built from coupling layers, specifically rational
quadratic spline blocks [72], each followed by a random permutation. As we
will discuss in Sec. 2.3, some phase space directions are periodic and lead to
undesired boundary effects when we use these spline transformations. To
understand this problem in detail, let us consider a spline transformation

$\displaystyle g_{\theta}\mathrel{\mathop{\mathchar
58\relax}}\quad[-L,L]\rightarrow[-L,L]\;,$ (4)

with parameters $\theta$. This transformation is given by $K$ different
monotonic rational quadratics, parameterized by $K+1$ knot points
$(x_{k},y_{k})$ and $K+1$ derivatives. The boundaries are
$(x_{0},y_{0})=(-L,-L)$ and $(x_{K},y_{K})=(L,L)$.

For periodic inputs the conditions $g(x_{0})=y_{0}=-L$ and $g(x_{K})=y_{K}=L$
are unnecessarily restrictive and do not allow the network to map a
distribution onto or past the boundaries, to represent points on a circle. In
addition, we want $g^{\prime}(x_{0})=g^{\prime}(x_{K})$ for periodic inputs,
which is not necessarily true [73]. The first issue can be fixed by replacing
$g_{\theta}$ with

$\displaystyle\tilde{g}_{\theta}(x)=g_{\theta}(x)+g_{0}+2Lk\;,$ (5)

with an integer $k$ chosen such that $\tilde{g}(x)$ always lies within
$[-L,L]$, and a new parameter $g_{0}$ added to $\theta$. To solve the second
issue, we simply remove one of the derivative parameters from $\theta$ and set
$\tilde{g}_{\theta}^{\prime}(x_{K})=\tilde{g}_{\theta}^{\prime}(x_{0})$. The
resulting transformation $\tilde{g}_{\theta}$ is visualized in Fig. 1.

With these modifications, the number of parameters encoding the transformation
does not change. This means that we can use the same sub-network to determine
$\theta$ for, both, periodic and non-periodic inputs. In practice, we split
the input vector to the transformation into periodic and non-periodic inputs
and apply $g_{\theta}$ and $\tilde{g}_{\theta}$ separately to each part. This
also implies that we have to keep track of the permutations between coupling
blocks, to be able to determine the type (periodic or non-periodic) of each
input throughout the network. As a last detail, we use a uniform latent space
distribution instead of a Gaussian for periodic dimensions.

Bayesian neural networks allow us to efficiently control the training
stability and estimate training-related uncertainties. They extend standard
architectures for regression, classification, or generative networks to
distributions for each network weight, providing a key tool for explainable AI
for instance in fundamental physics applications. The uncertainty on the
output can then be extracted through sampling [67, 74, 75, 76, 77]. For
generative networks, the uncertainty can be defined for the underlying density
estimation [68, 69, 50, 3], for which the network learns an uncertainty map
over the target phase space. The critical aspect of Bayesian networks is how
to make them numerically viable and still retain all their promising features.
We use a variational approximation for the training, combined with independent
Gaussians for each network weight. Such Bayesian networks include an optimal
regularization, so they can outperform their deterministic counterparts with
limited extra numerical effort. As always, we emphasize that the underlying
approximations do not have to limit the expressivity of the networks when it
comes to the sampled uncertainties. Moreover, we can treat the formal bias in
the Gaussian widths as a hyperparameter, which needs to be adjusted and should
be checked for stability.

We use the standard Bayesian version of the cINN, as introduced in Ref. [69],
but with periodic splines. The network is implemented in PyTorch [78]. In
addition, we use the Adam [79] optimizer with a constant learning rate. The
hyper-parameters employed in our study are provided in Tab. 1.

Parameter | Value
---|---
Block type | periodic rational quadratic spline blocks
Number of bins | $10$
Block Period | $2\pi$
Block Domain (non-Periodic) | $[-5.0,5.0]\rightarrow[-5.0,5.0]$
Number of Blocks | 16
Layers per Block | 5
Units per Layer | 256
Weight Prior Type | Gaussian
Weight Prior $\log(\sigma^{2})$ | 1.0
Number of Epochs (Bayesian) | 100 (200)
Batch Size | 1024
Optimizer | Adam
Learning Rate | $2.0\times 10^{-4}$
Total number of training events |  $\sim$1.2M
Training/Testing split | $80\%/20\%$
Table 1: Setup and hyper-parameters of the Unfolding-cINN.

###  2.3 Phase space parametrization

The unfolding method introduced above is identical to full, high-dimensional
unfolding to the parton-level. However, in this application, we will only
target a small number of kinematic distributions. Moreover, the unfolding
network will then be used to define these distributions as part of the
standard LHC analysis chain. This application allows us to improve the
description of relevant phase space directions at the potential expense of
correlations which are not useful for the measurement. In our case, we will
guarantee the correct descriptions of the intermediate top-mass peaks through
the network architecture.

The simplest way of encoding LHC events at the parton-level is through the
components of the final-state 4-momenta. However, the corresponding redundant
degrees of freedom are not adapted to the production of intermediate on-shell
particles and it’s reduced phase space. One way to improve the performance of
generative networks is to add a maximum mean discrepancy (MMD) between a given
set of generated and truth distributions [80] in the loss function. Its main
advantage is that it only affects the target distribution and avoids an
unnecessarily large model dependence. The disadvantage is that the additional
loss term complicates the training and consequently limits the precision of
the network. For our INN architecture, the computation of an MMD loss requires
samples generated from the latent distribution, while the usual INN loss works
on latent-space samples.

In our case, where the dominant signal and background processes share
intermediate mass peaks, we can learn these features directly, through an
appropriate phase space parametrization. For top decays with 9 degrees of
freedom in the final state, a natural parametrization starts with the
corresponding top 4-momentum, and then adds the invariant $W$-mass and a set
of less sensitive angular observables,

$\displaystyle\left\\{\;m_{t},p_{T,t},\eta_{t},\phi_{t},m_{W},\eta^{t}_{W},\phi^{t}_{W},\eta^{W}_{\ell,u},\phi^{W}_{\ell,u}\;\right\\}\;.$
(6)

Here $m_{t(W)}$ indicates the reconstructed invariant mass of the
corresponding resonance. The superscripts $t$ and $W$ indicate the rest frame
where the observable is defined, otherwise we use the laboratory frame. The
indices $\ell$ and $u$ indicate the charged lepton and the up-type quark for
leptonic or hadronic $W$-decays.

A network trained on this parametrization will reproduce the invariant top and
$W$-mass distributions, but with drawbacks in the correlations of the hadronic
$W$-decay. To extract $CP$-information, we also want to give the network
access to the most important $CP$-observables, which we will discuss in detail
in Sec. 3.1. This means we will include the Collins-Soper angle
$\theta_{\textrm{CS}}$ [44, 23, 28, 29] and the angle between the charged
lepton and the down-quark $\Delta\phi_{\ell d}$. One such parametrization for
the entire $t\bar{t}$ system with 18 degrees of freedom is

$\displaystyle\Big{\\{}\;\vec{p}_{t\bar{t}},m_{t_{\ell}},|\vec{p}^{\text{CS}}_{t_{\ell}}|,\theta^{\text{CS}}_{t_{\ell}},\phi^{\text{CS}}_{t_{\ell}},m_{t_{h}},$
$\displaystyle\qquad\operatorname{sign}(\Delta\phi^{t\bar{t}}_{\ell\nu})\,m_{W_{\ell}},|\vec{p}^{t\bar{t}}_{\ell}|,\theta^{t\bar{t}}_{\ell},\phi^{t\bar{t}}_{\ell},|\vec{p}^{t\bar{t}}_{\nu}|,$
$\displaystyle\qquad\operatorname{sign}(\Delta\phi^{t\bar{t}}_{du})\,m_{W_{h}},|\vec{p}^{t\bar{t}}_{d}|,\theta^{t\bar{t}}_{d},\Delta\phi^{t\bar{t}}_{\ell
d},|\vec{p}^{t\bar{t}}_{u}|\;\Big{\\}}\;.$ (7)

The superscripts CS and $t\bar{t}$ indicate the Collins-Soper frame of the
$t\bar{t}$-system and the $t\bar{t}$ rest frame; the latter rotated such that
$\vec{p}^{t\bar{t}}_{t_{\ell}}$ points in the direction of the positive
$z$-axis. Also, $t_{\ell}$ and $t_{h}$ denote the leptonically and
hadronically decaying tops, while $u$ and $d$ denote the up- and down-quarks
from the $W$-decay. Using
$\operatorname{sign}(\Delta\phi^{t\bar{t}}_{AB})m_{W}$ as a phase space
direction makes it harder for the network to generate the $W$-peaks, but
solves the problem of quadratic phase space constraints.

We emphasize that the combination of generative unfolding with the phase space
parametrization of Eq. (7) is expected to introduce a bias in the unfolding.
However, for our application, we can ignore this bias given our choice of
signal channel and our choice of target observable. Moreover, a potential bias
will render the network-defined observable sub-optimal, but does not affect
its evaluation in a standard analysis.

##  3 CP-phase from Higgs-top production

The example we choose to illustrate unfolding as a way to define dedicated
observables is associated Higgs and top quark pair production

$\displaystyle pp\to
t\bar{t}h+\text{jets}\to(bu\bar{d})\;(\bar{b}\ell^{-}\bar{\nu})\;(\gamma\gamma)+\text{jets}\;,$
(8)

plus the charge-conjugated process. $CP$-violating BSM effects modifying the
top Yukawa coupling can be parametrized through the Lagrangian [81]

$\displaystyle\mathscr{L}\supset-\frac{m_{t}}{v}\kappa_{t}\bar{t}(\cos\alpha+i\gamma_{5}\sin\alpha)th\;,$
(9)

where $\alpha$ is the $CP$-violating phase, $\kappa_{t}$ the absolute value of
the top Yukawa coupling, and $v=246$ GeV the Higgs VEV. The SM-limit is
$\kappa_{t}=1$ and $\alpha=0$. Deviations from the SM will affect Higgs
production and the decay. While changes in the scalar Higgs decay will only
impact the total rate, we focus on kinematic effects in the production.

The Lagrangian in Eq. (9) can be linked to the standard SMEFT framework used
for general LHC analyses at mass dimension six. In this case, we introduce two
Wilson coefficients to modify the top Yukawa [82, 83]

$\displaystyle\mathscr{L}$
$\displaystyle\supset\frac{f_{t}}{\Lambda^{2}}\mathcal{O}_{t}+\frac{\tilde{f}_{t}}{\Lambda^{2}}\tilde{\mathcal{O}}_{t}$
$\displaystyle\equiv\left(\phi^{\dagger}\phi-\frac{v^{2}}{2}\right)\left(\frac{f_{t}}{\Lambda^{2}}\left(\bar{q}_{L}t_{R}\tilde{\phi}+\tilde{\phi}^{\dagger}\bar{t}_{R}q_{L}\right)+i\frac{\tilde{f}_{t}}{\Lambda^{2}}\left(\bar{q}_{L}t_{R}\tilde{\phi}-\tilde{\phi}^{\dagger}\bar{t}_{R}q_{L}\right)\right)\;,$
(10)

where $\phi$ is the Higgs doublet, $\tilde{\phi}=i\sigma_{2}\phi^{*}$, and
$q_{L}$ the heavy quark doublet $(t_{L},b_{L})$. The parameters $\kappa_{t}$
and $\alpha$ in Eq. (9) can be computed as

$\displaystyle\kappa_{t}^{2}=\left(-1+\frac{v^{3}f_{t}}{\sqrt{2}m_{t}\Lambda^{2}}\right)^{2}+\left(\frac{v^{3}\tilde{f}_{t}}{\sqrt{2}m_{t}\Lambda^{2}}\right)^{2}\qquad\text{and}\qquad\tan\alpha=\dfrac{\tilde{f}_{t}}{f_{t}-\dfrac{\sqrt{2}m_{t}\Lambda^{2}}{v^{3}}}\;.$
(11)

We emphasize that the SMEFT description implicitly assumes that new physics
enters through higher-dimensional operators. In contrast, a $CP$-phase of the
top Yukawa can already arise as a dimension-4 modification of the SM-
Lagrangian, reflected by the scale combination $v^{3}/(m_{t}\Lambda^{2})$
appearing above.

###  3.1 CP-observables

There are, fundamentally, two ways of testing the $CP$-structure of the Higgs
Yukawa coupling introduced in Eq. (9): we can measure the angle $\alpha$ and
conclude from a significant deviation $\alpha\neq 0$ that $CP$ is violated in
the top Yukawa coupling, ideally using simulation-based inference [84, 46, 29,
30] or the matrix element method [50]. Alternatively, we can define an optimal
observable for $CP$-violation and test the actual symmetry [42, 12, 23].

#### Classical reconstruction

To search for $CP$-violation, spin correlations between the top and anti-top
quarks in $t\bar{t}h$ production are ideal, because the short top-lifetime
allows for a transfer of the top-polarization to the decay products prior to
hadronization or spin decorrelation [85]. The angular correlation between the
top-spin and the momenta of the top decay products is given by

$\displaystyle\frac{1}{\Gamma_{t}}\frac{d\Gamma}{d\cos\xi_{i}}=\frac{1}{2}\left(1+\beta_{i}P_{t}\cos\xi_{i}\right)\;,$
(12)

where $\xi_{i}$ is the angle between the top spin and the $i$-th particle in
the top quark rest frame, $P_{t}\in[0,1]$ is the polarization of the top
quark, and $\beta_{i}$ is the spin analyzing power of the $i$-th decay
product. Due to the left-handed nature of the weak interaction, the charged
lepton and $d$-quark display the largest spin analyzing power,

$\displaystyle\beta_{\ell^{+}}=\beta_{\bar{d}}=1\qquad\text{(to leading
order)}.$ (13)

While one cannot tag a $d$-jet, it is possible to find efficient proxies. A
practical solution is to select the softer of the two light-flavor jets in the
top rest frame. This choice gives a spin analyzing power for this jet as 50%
of that of the charged lepton [86, 29, 87]. Assuming that the softer $W$-decay
jet in the top rest frame comes from the $d$-quark, we can now construct
appropriate angular correlations to measure.

#### Linear CP-observables

The basis of optimal observables testing a symmetry are $U$-even or $U$-odd
observables, defined through their transformation properties on the incoming
and outgoing states,

$\displaystyle\mathcal{O}\left(U\ket{i}\to
U\ket{f}\right)=\pm\mathcal{O}\left(\ket{i}\to\ket{f}\right)\;,$ (14)

where in our case $U=CP$. Furthermore, a genuine $U$-odd observable is defined
as an observable which vanishes in a $U$-symmetric theory

$\displaystyle\bigl{\langle}\mathcal{O}\bigr{\rangle}_{\mathscr{L}=U\mathscr{L}U^{-1}}=0\;.$
(15)

The two definitions are related in that any $U$-odd observable is also a
genuine $U$-odd observable under the condition that the initial state and the
phase space are $U$-symmetric [88, 12], so the genuine $U$-odd property is
weaker.

Unfortunately, we cannot infer a $CP$-invariant theory from
$\langle\mathcal{O}\rangle$ of a $CP$-odd observable alone. While a
$\langle\mathcal{O}\rangle\neq 0$ always points to a $CP$-violating theory,
the result $\langle\mathcal{O}\rangle=0$ can appear in $CP$-symmetric and in
$CP$-violating theories. To further analyze this case, we can construct a
$CP$-odd observable that is also odd under the so-called naive time reversal
$\hat{T}$. Now, the expectation value of this observable is completely tied to
the $CP$-symmetry of the underlying theory [12].

$CP$-odd observables can be constructed either as $\hat{T}$-even scalar
products of two 4-momenta or as a $\hat{T}$-odd contraction of four
independent 4-momenta through the Levi-Civita tensor. For the
$t\bar{t}$-system of $t\bar{t}h$ production we can use two top momenta and
decay momenta

$\displaystyle\left\\{p_{b_{l}},p_{l},p_{\nu},p_{b_{h}},p_{u},p_{d}\right\\}\;.$
(16)

It is straightforward to construct the $C$-even, $P$-odd, and $\hat{T}$-odd
observable

$\displaystyle\mathcal{O}=\varepsilon_{\mu\nu\,\sigma\rho}p_{t_{h}}^{\mu}p_{t_{\ell}}^{\nu}p_{A}^{\rho}p_{B}^{\sigma}\;,$
(17)

with suitable top decay momenta $p_{A,B}$. We can use the $CP$-invariance of
the initial state and the phase space for $t\bar{t}h$ production to show that
its expectation value in Eq. (17) vanishes in the SM.

In the $t\bar{t}$ center of mass frame, we can turn Eq. (17) into a triple
product, a standard form for $CP$-odd observables,

$\displaystyle\mathcal{O}$
$\displaystyle=2\,E_{t_{h}}\,\vec{p}_{t_{\ell}}\cdot(\vec{p}_{A}\times\vec{p}_{B})\;.$
(18)

However, it depends on the top 4-momentum, which are hard to determine
accurately. It can be modified by introducing the azimuthal angle difference
$\Delta\phi^{t\bar{t}}_{AB}=\phi_{A}^{t\bar{t}}-\phi_{B}^{t\bar{t}}$ in the
$t\bar{t}$ frame [23, 29],

$\displaystyle\Delta\phi_{AB}^{t\bar{t}}=\textrm{sgn}[\vec{p}_{t_{\ell}}\cdot(\vec{p}_{A}\times\vec{p}_{B})]\arccos\left[\frac{\vec{p}_{t_{\ell}}\times\vec{p}_{A}}{|\vec{p}_{t_{\ell}}\times\vec{p}_{A}|}\cdot\frac{\vec{p}_{t_{\ell}}\times\vec{p}_{B}}{|\vec{p}_{t_{\ell}}\times\vec{p}_{B}|}\right]\;,$
(19)

to give us

$\displaystyle\mathcal{O}=2p_{t}^{z}\,E_{t}\,p_{T,\,A}\,p_{T,\,B}\,\sin\Delta\phi^{t\bar{t}}_{AB}\;,$
(20)

where we choose $p_{t_{\ell}}=\\{E_{t},0,0,p_{t}^{z}\\}$ and
$p_{t_{h}}=\\{E_{t},0,0,-p_{t}^{z}\\}$. By construction, $\mathcal{O}$ and
$\Delta\phi^{t\bar{t}}_{AB}$ are sensitive to the linear interference terms in
the scattering cross section, and therefore sensitive to the sign of the
$CP$-phase.

Figure 2: Fisher information $I$ for the linear CP-observables
$\sin\Delta\phi_{AB}^{t\bar{t}}$ (blue) and $\mathcal{O}_{AB}$ (red), probing
the sensitivity to $CP$-violating phase $\alpha$ in $t\bar{t}h$ production.

These linear $CP$-observables can be constructed for various $\\{A,B\\}$
pairs, and their $CP$-sensitivity dependents on the spin-analyzing power of
the particles $A$ and $B$. We compute the Fisher information metric $I$ to
rank their $CP$-sensitivity, using MadMiner [46, 29]. The $\alpha$-dependent
component of $I$ is defined as

$\displaystyle I=\mathbb{E}\left[\frac{\partial\log
p(x|\kappa_{t},\alpha)}{\partial\alpha}\frac{\partial\log
p(x|\kappa_{t},\alpha)}{\partial\alpha}\right],$ (21)

where $p(x|\kappa_{t},\alpha)$ represents the likelihood of a phase space
configuration $x$ given the theory parameters $\kappa_{t}$ and $\alpha$.
$\mathbb{E}$ denotes the expectation value at the SM point,
$(\kappa_{t},\alpha)_{\text{SM}}=(1,0)$. In Fig. 2, we show the Fisher
information at parton-level associated with the linear $CP$-observables
$\mathcal{O}_{AB}$ in red and the Fisher information for
$\sin\Delta\phi_{AB}^{t\bar{t}}$ in blue.

First, we see that for all combinations $(A,B)$ the Fisher information in
$\mathcal{O}_{AB}$ is slightly larger than the Fisher information in
$\sin\Delta\phi_{AB}^{t\bar{t}}$, an effect of the momentum-dependent
prefactor in $\mathcal{O}_{AB}$. Among the various combinations $(A,B)$, the
combination of the lepton and the down-type quark is the most sensitive. This
corresponds to the maximal spin analyzing power for this pair. Next comes the
combination where either the charged lepton or the down quark is replaced by
the $b$-quark or the $W$-boson. In this case, the Fisher information is
suppressed by two powers of

$\displaystyle\beta_{b}=\beta_{W}\sim 0.4\;.$ (22)

The correlation between a pair of $b$-quarks or $W$-bosons is further
suppressed by another factor $\beta_{b,W}^{2}$.

#### Non-linear observables and Collins-Soper angle

For a given realization of $CP$-violation in an SM-like interaction vertex,
the $CP$-observable defined in the previous section is not guaranteed to be
the most powerful observable [10]. This is obvious for dimension-6 operators,
where the symmetry structure is often combined with a momentum dependence of
the interaction [12], and the two aspects can, in principle, be tested
independently. Comparing the two handles, $CP$-odd observables are only
sensitive to the interference between the SM-contribution and the
$CP$-violating matrix element, while observables testing the momentum
structure of the interaction vertex can be dominated by the new-physics-
squared contribution. For large $CP$-phases $\alpha$, the more promising
analysis strategy will use a general test of the structure of the top-Higgs
coupling. This motivates using a combination of dedicated $CP$-observables
with general interaction probes as an optimal search strategy.

Several observables have been evaluated as probes of the $CP$-phase $\alpha$
in Eq. (9) using $t\bar{t}h$ production [23, 28, 29]. They include the
pseudorapidity difference between the two tops and the azimuthal angle between
the two tops, the Higgs transverse momentum [81, 89], or the invariant mass of
the top and anti-top pair,

$\displaystyle\left\\{\;\Delta\eta_{t\bar{t}},\Delta\phi_{t\bar{t}},p_{T,h},m_{t\bar{t}}\;\right\\}\,.$
(23)

These standard observables can be supplemented with the projection angle [81,
89, 90]

$\displaystyle
b_{4}=\frac{p_{z,t}}{|\vec{p}_{t}|}\cdot\frac{p_{z,\bar{t}}}{|\vec{p}_{\bar{t}}|}\;.$
(24)

Finally, we can use the Collins-Soper angle $\theta_{\text{CS}}$ [44], the
angle between the top quark and the bisector of the incoming hadrons in the
$t\bar{t}$ center of mass frame. The original motivation for the Collins-Soper
angle was to define an observable for the Drell-Yan process
$pp\to\ell^{+}\ell^{-}$ that corresponds to the scattering angle.
Factorization arguments suggest the di-lepton rest frame, to minimize ISR-
effects and then study the angular correlations between the incoming quarks
and the outgoing leptons. In this frame the 3-momenta of the quarks and
leptons each define a plane, and in turn an azimuthal angle and a polar angle
between the two planes.

Without ISR the $z$-axis of the so-defined CS-frame is trivially given by the
parton and hadron directions. Including ISR, we instead define this $z$-axis
as halving the angle between one of the hadrons and the reverse direction of
the other hadron. The Collins-Soper angle can be used to measure the
polarization of the intermediate gauge boson, the weak mixing angle [91], or
the (Lorentz) structure of the interaction vertices. The Collins-Soper angle
can also be used to probe the structure of the Higgs-photon coupling [92, 93]
and to boost new physics searches in $h^{*}\to ZZ$, $Zh$, and $t\bar{t}Z$
channels [94, 95, 96, 97]. Finally, it can be generalized to $t\bar{t}$
production, where it is constructed for the top momentum in the $t\bar{t}$
rest frame [98, 99, 23]. While the Collins-Soper angle has no specific
sensitivity to $CP$-violation, we view it as the Swiss Army knife of coupling
tests.

All above-mentioned kinematic observables are sensitive to the new-physics-
squared terms, proportional to $\sin^{2}\alpha$ or $\cos^{2}\alpha$, in the
$t\bar{t}h$ rate, with no sensitivity to the sign of the CP-phase. From Ref.
[29], we know the relative sensitivity of these observables to probe the
Higgs-top $CP$-structure through a modified Fisher information metric,
accounting for non-linear effects. The top-five observables with the highest
Fisher information for $\alpha$ are (symbolically written)

$\displaystyle\Delta\eta_{t\bar{t}}>\theta_{\text{CS}}>b_{4}>\Delta\phi_{t\bar{t}}>p_{T,h}\;.$
(25)

We show the parton-level distributions for the four most sensitive observables
in the semileptonic $t\bar{t}h$ channel for the SM value $\alpha=0$ and
$\alpha=\pi/4,\pi/2$ at the LHC with $\sqrt{s}=14$ TeV in Fig. 3. Different
values of $\alpha$ lead to distinctly different profiles in the distributions.

As alluded to above, the technical challenge and a limitation to the
optimality of a given analysis is to construct the different observables in
their respective kinematic frames. Considering their strong sensitivity on
$\alpha$, we include the leading observables in the phase space
parametrization given in Eq. (7) to target this problem directly.

Figure 3: Distributions for the CS-angle $\theta_{\text{CS}}$,
$\Delta\eta_{t_{\ell}t_{h}}$, $b_{4}$, and $\Delta\phi_{t_{\ell}t_{h}}$, all
at parton-level for semileptonic $t\bar{t}h$ production. The distributions are
shown for SM (green), $\alpha=\pi/4$ (blue) and $\alpha=\pi/2$ (red).

###  3.2 Unfolding-based analysis

The standard challenge for every LHC analysis is to extract a small signal
from a large (continuum) background. For our simple study, we show how we can
avoid modeling this step. The generative unfolding trained on $t\bar{t}h$
events gives us the probability $p(x_{\text{part}}|x_{\text{reco}},S)$ that a
parton-level signal event $x_{\text{part}}$, corresponds to an assumed signal
event $x_{\text{reco}}$ at reconstruction level. What we are ultimately
interested in, however, is a model parameter $\alpha$, which could be a mass,
a $CP$-phase, or any other continuous theory parameter, which affects our
signal distribution. Since we do not know if a particular reco-level event
$x_{\text{reco}}$ is signal or background, we only care about the full
probability $p(\alpha|x_{\text{reco}})$ of our model parameter, given some
reco-level event $x_{\text{reco}}$ which is either signal or background. Since
$\alpha$ does not change the background, this probability can be split into
the distribution $p(\alpha|x_{\text{part}})$, where $x_{\text{part}}$ is a
parton-level signal event, and the probability
$p(x_{\text{part}}|x_{\text{reco}})$ of $x_{\text{part}}$ given
$x_{\text{reco}}$:

$\displaystyle p(\alpha|x_{\text{reco}})=\int
p(\alpha|x_{\text{part}})p(x_{\text{part}}|x_{\text{reco}})\operatorname{d\\!}{x}.$
(26)

The challenge is to compute $p(x_{\text{part}}|x_{\text{reco}})$ from our
unfolding result $p(x_{\text{part}}|x_{\text{reco}},S)$. Using the definition
of conditional probabilities we can write

$\displaystyle p(x_{\text{part}}|x_{\text{reco}})$
$\displaystyle=\sum_{T\in\\{S,B\\}}p(x_{\text{part}}|x_{\text{reco}},T)p(T|x_{\text{reco}})$
$\displaystyle=p(x_{\text{part}}|x_{\text{reco}},S)p(S|x_{\text{reco}})+p(x_{\text{part}}|x_{\text{reco}},B)(1-p(S|x_{\text{reco}}))\;,$
(27)

where the probabilities of $x_{\text{reco}}$ being a signal or background
event, $p(T|x_{\text{reco}})$, can be encoded in a trained classifier. Let us
consider for a moment what the probability
$p(x_{\text{part}}|x_{\text{reco}},B)$ tells us. We are interested in signal
events $x_{\text{part}}$, i.e. events that are affected by $\alpha$. By
definition, background events $x_{\text{reco}}$ cannot give us any
information, beyond prior knowledge, about $x_{\text{part}}$. For this reason,
we can drop $x_{\text{reco}}$ and write
$p(x_{\text{part}}|x_{\text{reco}},B)=p(x_{\text{part}})$, where
$p(x_{\text{part}})$ is only constrained through prior knowledge. This
includes our model assumptions as well as phase-space constraints due to a
finite center-of-mass energy. We can now write

$\displaystyle
p(x_{\text{part}}|x_{\text{reco}})=p(x_{\text{part}}|x_{\text{reco}},S)p(S|x_{\text{reco}})+p(x_{\text{part}})(1-p(S|x_{\text{reco}})).$
(28)

What Eq. (28) shows, is that we can limit our unfolding model to extracting
$p(x_{\text{part}}|x_{\text{reco}},S)$ and still include background events
into our analysis later, without changing our model.

As given in Eq. (8), we study $pp\to t_{h}\bar{t}_{\ell}h$ production with
$h\to\gamma\gamma$ at the HL-LHC. The dominant background is continuum
$t\bar{t}\gamma\gamma$ production, subdominant contributions arise from the
process $Wb\bar{b}(h\to\gamma\gamma)$. We use MadGraph5_aMC@NLO [100] with
NNPDF2.3QED [101] to generate signal events at leading order with
$\sqrt{s}=14\leavevmode\nobreak\ $TeV. Signal events are simulated without
kinematic cuts using the HC_NLO_X0 UFO model [102, 103]. Parton showering and
hadronization effects are simulated using Pythia 8 [104]. The detector
response is simulated with Delphes 3 [105], using the default ATLAS HL-LHC
card [106, 107].

Next, we select events containing exactly two photons, two $b$-tagged jets,
one lepton, and at least two light-flavored jets. The individual particles in
the final state are required to satisfy the acceptance cuts

$\displaystyle p_{T,b}$ $\displaystyle>25\leavevmode\nobreak\ \text{GeV}\;,$
$\displaystyle\quad p_{T,j}$ $\displaystyle>25\leavevmode\nobreak\
\text{GeV}\;,$ $\displaystyle\quad p_{T,\ell}$
$\displaystyle>15\leavevmode\nobreak\ \text{GeV}\;,$ $\displaystyle\quad
p_{T,\gamma}$ $\displaystyle>15\leavevmode\nobreak\ \text{GeV}\;,$
$\displaystyle|\eta_{b}|$ $\displaystyle<4\;,$ $\displaystyle\quad|\eta_{j}|$
$\displaystyle<5\;,$ $\displaystyle\quad|\eta_{\ell}|$ $\displaystyle<4\;,$
$\displaystyle\quad|\eta_{\gamma}|$ $\displaystyle<4\;.$ (29)

At the parton-level, the signal phase space involves eight final state
particles; following Sec. 2.3 it requires 22 parameters if we are assuming the
Higgs is fully and uniquely reconstructed. The training dataset involves an
event-wise pairing of parton and detector level events with up to six light-
flavored jets, satisfying the selection cuts in Eq. (29). While the event at
the reconstruction level requires additional degrees of freedom for jet
radiation, the number of degrees of freedom is reduced by the neutrino. An
additional challenge is the combinatorics of the $b$-jets and light-flavor
jets.

###  3.3 Results

Figure 4: Jet combinatorics — cINN-generated distributions for $m_{t_{\ell}}$,
$m_{t_{h}}$, $m_{ht_{h}}$ and $m_{t_{\ell}t_{h}}$ in the SM. Unfolded
distributions are shown as solid lines, parton-level truth as dashed lines.
The training data set either does not include ISR (green) or up to six ISR
jets (red).

##### Jet combinatorics

The first results from unfolding $t\bar{t}h$ SM-events are presented in Fig.
4. We train the unfolding network on SM-events and also apply it to SM-events.
First, we examine the robustness of the network to unfold a variable number of
jets to the parton-level. For our lepton-hadron reference process in Eq. (8)
two light-flavor jets come from the hadronic top decay, while additional jets
arise from QCD jet radiation. The unfolding network has to reconstruct the two
hard jets at the parton level from a variable number of jets at the detector
level [60].

To evaluate the unfolding performance, we examine four invariant masses:
$m_{t_{\ell}}$, $m_{t_{h}}$, $m_{ht_{h}}$, and $m_{t_{\ell}t_{h}}$. We train
one network on SM events without ISR and one network on events with up to six
light-flavored jets. The corresponding cINN-generated distributions are shown
as solid lines in Fig. 4. The parton-level truth is displayed as dashed lines.
We find that unfolded distributions generated by both networks are in good
agreement with the parton-level truth in the bulk of the phase space. Despite
the added combinatorial ambiguity, the performance of both networks is largely
comparable. We also show the uncertainties from the Bayesian setup,
represented as $1\sigma$ error bands. They test the stability of the unfolding
network similar to an ensemble of networks. It is important to observe that
the truth distributions remain within these error bands.

Figure 5: Reconstructing dedicated observables — cINN-generated distributions
and distributions based on classical reconstruction [29] for
$\theta_{\text{CS}}$, $\Delta\eta_{t_{\ell}t_{h}}$, $b_{4}$ and
$\Delta\phi_{t_{\ell}t_{h}}$ for SM events. The secondary panels show the bin-
wise agreement between the cINN-generated distributions and the parton-level
truth.

##### Reconstructing dedicated observables

For Fig. 5 we train the unfolding network on SM events with up to six light-
flavor jets. We compare cINN-generated events at the parton-level and in the
appropriate rest frame with events from a classical reconstruction for four
particularly interesting observables from Sec. 3.1: $\theta_{\textrm{CS}}$,
$\Delta\eta_{t\bar{t}}$, $b_{4}$, and $\Delta\phi_{t\bar{t}}$. For comparison,
we display the parton-level truth as dashed lines. In the ratio we observe
that the generated distributions agree with the truth within a few percent.
Slightly larger deviations in the tails are due to limited training
statistics.

Figure 6: Model dependence — cINN-generated distributions for
$\theta_{\text{CS}}$, $\Delta\eta_{t_{\ell}t_{h}}$, $b_{4}$, and
$\Delta\phi_{t_{\ell}t_{h}}$. Upper two rows: unfolding of SM events using
three different networks, trained on data with $\alpha=-\pi/4,0,\pi/4$. Lower
two rows: unfolding of events with $\alpha=-\pi/4,0,\pi/4$, with a network
trained on SM events.

The conventional approach to complex kinematic correlations in the
semileptonic $t\bar{t}$ system relies on a complex reconstruction algorithm,
with a significant loss of information due to missing correlations [29]. We
show the reconstructed distributions from the classical reconstruction
strategy developed in Ref. [29] as dotted lines. Comparing these distributions
to the cINN-unfolded version, we see that at least for a network trained and
tested on SM events, the improvement from generative unfolding is striking.

Figure 7: Sensitivity — Left: cINN-generated distributions for
$\theta_{\text{CS}}$ from unfolding events with two $\alpha$ values. These
generated distributions are compared to the distributions obtained from
classical reconstruction methods, as described in Ref. [29], and the
respective truth. Right: To quantify the sensitivity of the cINN, as shown
here for $\theta_{\text{CS}}$, we compute the reduced $\chi^{2}$-value between
the distributions ($\sim$120k events and 64 bins) for both $\alpha$ values,
using the Poisson errors of the bin counts. We do this for the cINN-generated
(red), classically reconstructed (green), and truth distributions (blue) of
$\theta_{\text{CS}}$, $\Delta\eta_{t_{\ell}t_{h}}$, $b_{4}$, and
$\Delta\phi_{t_{\ell}t_{h}}$. In the bottom panel, we also show the ratio of
the reduced $\chi^{2}$-values of the cINN and the classical reconstruction to
the truth. Uncertainties on cINN-generated values are obtained from the
Bayesian setup.

##### Model dependence

After observing the significant improvement through our new method for SM
events, we need to test how model-dependent the network training is. In the
upper panels of Fig. 6, we reconstruct the usual set of key observables for SM
events, but with three different networks, trained on events generated with
the $CP$-angles $\alpha=-\pi/4,0,\pi/4$. We adopt the BSM values
$\alpha=\pm\pi/4$ here, as these choices closely align with the current
experimental limits [108, 109]. From Fig. 6 we expect, for instance, the
network trained on events with $\alpha=\pi/4$ to be biased towards a broader
$\theta_{\text{CS}}$ distribution, a wider rapidity difference
$\Delta\eta_{t_{\ell},t_{h}}$, and a flatter $b_{4}$ distribution. In the
different panels we see a slight bias, especially in the secondary panels. But
the bias remains at the order of 10%, at most 20%, much below the change in
the corresponding distributions from varying $\alpha$. On the other hand, this
bias is significantly larger than the uncertainty band, which indicates that
this model dependence can be reduced through the proposed iterative method of
Ref. [62]. The corresponding study is beyond the scope of this paper, because
it balances a reduced bias of the unfolding with less statistics, an aspect
which we do not include in this proof-of-principle study.

In the lower panels of Fig. 6 we test the model dependence the other way
around, by unfolding data for different $\alpha=-\pi/4,0,\pi/4$ using a
network trained on SM events. The figure of merit are the ratios of cINN-
unfolded and respective truth distributions, shown in the secondary panels.
This situation is closer to the reality of a measurement, where we infer
$\alpha$ by comparing the distribution extracted from data to different
simulated hypotheses. As before, we see a slight bias, now towards the SM
structure of a more narrow $\theta_{\text{CS}}$ distribution, a narrow
rapidity difference $\Delta\eta_{t_{\ell},t_{h}}$, and a steeper $b_{4}$
distribution. Also, as before, the effect of the bias is much smaller than the
effect of $\alpha$ on the data, leaving us optimistic that we can use the
cINN-unfolded distribution to measure $\alpha$.

##### Sensitivity

Finally, in Fig. 7, we apply the generative unfolding to SM and $\alpha=\pi/4$
events. The unfolding network is trained on SM events. As a baseline
comparison, we also show the same two curves for classical reconstruction of
$\theta_{\text{CS}}$, following Ref. [29] as dotted lines in the left panel of
Fig. 7. As mentioned earlier, generative unfolding leads to a major
improvement over classical reconstruction. The difference in the two unfolded
kinematic distributions, shown in solid lines, illustrates the reach of an
analysis based on the kinematic distribution. To showcase the improvement in
new physics sensitivity, we calculate the reduced $\chi^{2}$ values for
$\theta_{\text{CS}}$, $\Delta\eta_{t_{\ell}t_{h}}$, $b_{4}$, and
$\Delta\Phi_{t_{\ell}t_{h}}$ between the SM and $\alpha=\pi/4$ hypotheses,
using the Poisson errors of the bin counts. The reduced $\chi^{2}$ values are
computed with $\sim$120k events and 64 bins, for three scenarios: parton-level
truth (blue), classical reconstruction from Ref. [29] (green), and the cINN-
based generative model trained on SM events (red). A higher $\chi^{2}$ value
indicates a greater sensitivity to new physics.

The results show that the unfolding setup leads to an enhancement in
sensitivity compared to the classical reconstruction strategy. This indicates
that the generative unfolding approach is effective in extracting more
information from the kinematic distributions, thereby improving the analysis’
capability to detect and explore new physics phenomena. We further observe
that the network is slightly more consistent in reproducing the sensitivity
relations of the true observable distributions than the classical
reconstruction. The latter performs well on some observables, but quite bad
for others. Especially surprising is the classical sensitivity on
$\theta_{\text{CS}}$, given that the reconstruction here is far from the
actual CS-angle.

##  4 Outlook

Unfolding is one of the most exciting development of analysis preservation and
publication at the LHC. Modern machine learning makes it possible to unfold
high-dimensional distributions, covering all correlations without binning.
Generative unfolding defines this unfolding in a statistically consistent
manner. However, using unfolded data is a challenge for the ATLAS and CMS
analysis chains, especially in controlling and estimating uncertainties.

We investigated a simpler application of the unfolding technique, the
extraction of a kinematic observable in a specific partonic reference frame.
It solves the dilemma that on the one hand an optimal observable requires no
complex correlations, but on the other hand such an observable is, typically,
hard to reconstruct. In this case the generated kinematic distribution can be
used like any other observable; the unfolding network is nothing but a
kinematic reconstruction algorithm.

The perfect examples for a challenging kinematic correlation are the Collins-
Soper angle or the optimal $CP$-observables in $t\bar{t}h$ production. They
allow us to measure a $CP$-phase in the top Yukawa coupling, a cosmologically
relevant parameter entering an LHC signature at dimension four and at leading
order. We have shown that unfolding allows us to extract the leading
observables for such a $CP$-phase $\alpha$, with the help of an appropriate
phase space parametrization. While such a parametrization might shape the
unfolded kinematic distribution, this effect can be controlled through
calibration.

First, we have shown that the cINN-unfolding can solve the combinatorics of
$W$-decay jets vs QCD jet radiation. Second, the unfolded distributions of SM
events, with a network trained on SM events, show excellent agreement with the
parton-level truth. Potential differences are covered by the uncertainty
estimate from the Bayesian network. Third, we have tested the model dependence
in two different ways — unfolding SM event using networks trained on events
with different amounts of $CP$-violation and unfolding events with
$CP$-violation using a network trained on SM events. For the former, we have
found that there exists a small, but significant model dependence, which can
be removed through Bayesian iterative improvements. For the latter, the
unfolded distributions do not perfectly reproduce the respective truth, but
the bias is much smaller than the kinematic effect of the $CP$-angle.

All these tests have motivated a comparison of the reach of the HL-LHC for the
$CP$-angle $\alpha$, based on classical reconstruction methods and on cINN-
unfolded distributions. The generative unfolding approach effectively extracts
more information from kinematic distributions, enhancing sensitivity to new
physics phenomena. This highlights the importance of advanced machine learning
techniques, such as cINNs, for the HL-LHC.

While this study is clearly not the last word on this analysis technique, we
consider the outcome promising enough for an experimental study, with a proper
treatment of statistical limitations, continuum backgrounds, calibration, and
iterative improvements of the unfolding network.

## Acknowledgements

RKB and DG thank the U.S. Department of Energy for financial support, under
grant number DE-SC0016013. Some computing for this project was performed at
the High Performance Computing Center at Oklahoma State University, supported
in part through the National Science Foundation grant OAC-1531128. TH is
funded by the Carl-Zeiss-Stiftung through the project Model-Based AI: Physical
Models and Deep Learning for Imaging and Cancer Treatment. This research is
supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under grant 396021762 – TRR 257: Particle Physics Phenomenology
after the Higgs Discovery and through Germany’s Excellence Strategy EXC 2181/1
– 390900948 (the Heidelberg STRUCTURES Excellence Cluster).

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