# Loop Quantum Gravity effects on electromagnetic properties
of charged leptons

J. P. S. Melo<EMAIL_ADDRESS>Centro Brasileiro de Pesquisas Físicas, Rua
Dr. Xavier Sigaud 150, Urca, CEP 22290-180, Rio de Janeiro, RJ, Brazil  M. J.
Neves<EMAIL_ADDRESS>Departamento de Física, Universidade Federal Rural do
Rio de Janeiro, BR 465-07, CEP 23890-971, Seropédica, RJ, Brazil  J. M. A.
Paixão<EMAIL_ADDRESS>Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier
Sigaud 150, Urca, CEP 22290-180, Rio de Janeiro, RJ, Brazil  J. A. Helayël-
Neto<EMAIL_ADDRESS>Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier
Sigaud 150, Urca, CEP 22290-180, Rio de Janeiro, RJ, Brazil

###### Abstract

The efforts in this contribution consist in reassessing a modified Dirac
equation that incorporates a $\gamma^{0}\gamma_{5}$-Lorentz-symmetry violating
(LSV) term induced as a Loop Quantum Gravity (LQG) effect. Originally, this
equation has been applied and considered as a good scenario for describing a
number of investigations on the flight time of cosmic photons and neutrinos,
which suggests that the speed of light in vacuum, in connection with the
geometry that describes a granular space-time, takes an energy-dependent form,
e.g., $v(E)=1\pm E/E_{\textrm{\tiny LSV}}$, with $E_{\textrm{\tiny
LSV}}\approx 6,5\times 10^{17}$ GeV for neutrinos. Once LQG provides a viable
way to consistently understand this picture, we pursue an analysis of this
effective Dirac equation to inspect some of its properties. These include: the
derivation of the modified fermionic propagator, attainment of the Gordon
decomposition of the vector current with minimal electromagnetic coupling to
obtain information on the form factors, examination of the non-relativistic
limit of the equation, evaluation of the spin- and velocity-dependent
corrections to the Coulomb potential due to LQG effects, and the modified
Hamiltonian in the low-relativistic regime. The study of the form factors may
open up paths to set up bounds on the LQG parameters from the precision
measurements of electromagnetic attributes of the charged leptons, such as
their respective electric and magnetic dipole moments.

##  I Introduction

The theoretical framework of Loop Quantum Gravity (LQG) is one of the most
promising ways to inspect the challenges of quantum gravity, which seeks to
understand the fundamental nature of the spacetime and gravity at the quantum
level [1, 2]. In LQG, the spacetime is considered to be discrete rather than
continuous, which is a departure from the smooth, continuous spacetime
described by general relativity. The theory represents the spacetime as a
network or quantum foam of interconnected loops. These loops are thought to be
the building blocks of space and time at the smallest scales. The key idea in
LQG is that quantities, such as area and volume, are quantized, meaning they
come in discrete, indivisible units. The theory uses mathematical structures
called spin networks and spin foams to describe the quantum properties of the
spacetime [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. LQG is still a
research area in progress and it has not yet been experimentally confirmed
once its principals predictions and phenomenological aspects become relevant
at the order of the Planck scale ($\ell_{P}\sim\\!10^{19}$ GeV) [16, 17, 18].
However, indirect ways to probe the soft effects of quantum gravity at larger
scales or through cosmological observations, e.g, some experiments and
observations related to cosmology [19, 20], high-energy astrophysics [21, 22,
23, 24], or precision measurements [25, 26] might provide insights into LQG
phenomenology.

The investigation on Lorentz-symmetry violation (LSV) is another significant
approach to uncover evidence in the pursuit of quantum gravity, once it is
also expected to happen at the Planck scale [21, 27, 28, 16, 29], where strong
space-time fluctuations are expected to occur. In this context, discussions
related to tests based on Modified Dispersion Relations (MDRs), using high-
energy photons available from astrophysical measurements, can contribute to
analyzes of possible deviations compared to standard dispersion relations,
indicating the presence of LSV or some effects of quantum gravity, such as LQG
[21, 27, 28]. Additionally, in this context, an important connection between
the LQG and LSV frameworks is presented in [16], where the modified Maxwell
equations point vacuum birefringence. Two other important works regarding the
MDRs, characterizing dispersion relations that could indicate the existence of
the LSV, can be found in Refs. [30, 17].

In the context of MDRs, as argued in [31] at the leading order of LSV, the
modified speed of a massless particle with energy $E$ can be expressed as
$v(E)\approx(1-\xi E)$, where $\xi$ refers to the LSV scale approximately of
the order of a fraction of the Planck scale. Consequently, the velocity
difference between two massless particles with different energies becomes
apparent whenever their energies are sufficiently high, and the discrepancies
in travel time accumulate over a considerable propagation distance. As a
result, cosmic photons and neutrinos, particularly those originating from
gamma-ray bursts (GRBs), and active galactic nuclei (AGNs) exhibiting
simultaneous high energy, provide a unique window for exploring LSV during
their journeys through the Universe [21]. Regarding the LSV exploration linked
to GRBs, a comprehensive summary is provided in [31] stating that: for
photons, without helicity dependence, we have
$v(E)=(1-E/E^{\gamma}_{\textrm{\tiny LSV}})$, with $E^{\gamma}_{\textrm{\tiny
LSV}}\approx 3.6\times 10^{17}$ GeV, indicating that photons are subluminal
and high energy photons propagate slower than low energy ones [32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42]; for neutrinos and anti-neutrinos, their speeds
are $v(E)\approx(1\mp E/E^{\nu}_{\textrm{\tiny LSV}})$, with
$E^{\nu}_{\textrm{\tiny LSV}}\approx 6.5\times 10^{17}$ GeV, indicating that
neutrinos and anti-neutrinos, or Majorana neutrinos with opposite helicities,
could be subluminal and superluminal with the sign of the velocity variations
depending on the helicities [41, 42, 43, 44, 45, 46, 47].

In [31] the authors find out that LQG is as a good theoretical candidate for
explaining the phenomenological suggestion of the speed variations of cosmic
photons and neutrinos picture consistently, and suggest that others observable
signals could also testify the predictions of LQG in the future. In order to
show that, they introduce LQG description that consists in assuming specifics
conditions and properties for the so-called weave states, which are
semiclassical states with a characteristic length $\mathcal{L}$ that describes
the discreteness of the spacetime represented by this state. When the de
Broglie wavelength $(\lambda)$ of a particle satisfies the condition
$\ell_{\textrm{\tiny P}}\ll\lambda$, in which $\ell_{\textrm{\tiny
P}}=1.61\times 10^{-35}$ m is the Planck length, and the characteristic length
$\mathcal{L}$ satisfies the condition $\ell_{\textrm{\tiny
P}}\ll\mathcal{L}\leq\lambda$, one can compute the weave state expectation of
the corresponding Hamiltonian of the particle and obtain an effective
Hamiltonian from which we can read off the modified dispersion relation and
the speed of such particle [17, 18]. Their results can be directly related to
observable phenomena and conversely the phenomenological analyses may be used
to examine the description of LQG based on wave states, leading towards better
understanding of the LQG framework and further progress of LQG theories.

Our efforts in this contribution consist in reassessing a modified Dirac
equation that incorporates a $\gamma^{0}\,\gamma_{5}$-LSV term induced as a
LQG effect by [31]. The paper is structured as follows: In Section II, we
present the modified Dirac equation discussed in [31]. We provide the action,
and the modified fermion propagator associated with this equation. The
dispersion relation is reevaluated, and additionally, we introduce the
correspondent group velocity. In Section III, we derive the positive and
negative energy solutions for this modified Dirac equation. The Section IV
focuses on obtaining the conserved current associated with the $U(1)$ gauge
symmetry of the model. Through minimal coupling with the electromagnetic (EM)
field, we demonstrate that this current aligns with the EM current within the
framework of modified Maxwell’s equations. In Section V, we obtain the Gordon
decomposition of the conserved current exploring its implications. Section VI
is devoted to obtaining spin- and velocity-dependent corrections to the
Coulomb potential due to the modified current structure of the model. The
Section VII involves deriving the non-relativistic limit of the modified Dirac
equation with minimal coupling. In Section VIII, we delve into the question of
renormalizability, concluding that the model is super-renormalizable. Finally,
Section IX encompasses our concluding considerations.

We adopt the natural system of units, in which $c=\hbar=1$, and we use the
Minkowski signature $(+,-,-,-)$.

##  II The modified Dirac equation from LQG

The modified Dirac equation emerging from the LQG effects in the context of
[31] is

$\displaystyle\left(i\gamma^{\mu}\partial_{\mu}-\dfrac{\hat{C}}{2\mathcal{L}}\,\gamma^{0}\gamma_{5}-m\right)\psi=0\;,$
(1)

where $\mathcal{L}$ is a characteristic length constraint by the condition
${\ell}_{\textrm{\tiny P}}\ll\mathcal{L}\leqslant\lambda$, in which
$\ell_{\textrm{\tiny P}}=1.61\times 10^{-35}$ m is the Planck length, and
$\lambda$ is the de Broglie wavelength of a particle. The $\hat{C}$-operator
is defined by

$\displaystyle\hat{C}=\dfrac{\kappa_{7}}{2}\left(\dfrac{\ell_{p}}{\mathcal{L}}\right)^{\\!\\!\Upsilon_{\\!f}}\\!\\!\ell_{p}^{2}\,\bm{\nabla}^{2}=\xi\,\bm{\nabla}^{2}\;,$
(2)

in which $\Upsilon_{f}$ is a phenomenological parameter from LQG, $\kappa_{7}$
is a magnitude parameter, and $\xi$ is the combination of these parameters
that has length dimension. Thus, it is convenient to define the quantity
$\vartheta:=\xi/(2\mathcal{L})$, with length dimension, and consequently, all
the known results of the fermion theory are recovered in the limit
$\vartheta\rightarrow 0$. The $\gamma^{\mu}$-matrices are the usual $4\times
4$ Dirac’s matrices that satisfy the relations
$\gamma^{\mu}\gamma^{\nu}=\eta^{\mu\nu}\,\mathds{1}-i\,2\Sigma^{\mu\nu}$, and
$\left\\{\gamma^{\mu},\gamma_{5}\right\\}=0$. The correspondent action from
Eq. (1) is

$\displaystyle{\cal
S}(\bar{\psi},\psi)=\\!\int\\!d^{4}x\,\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-\vartheta\,\gamma^{0}\gamma_{5}\,\bm{\nabla}^{2}-m\right)\psi\;,$
(3)

that, upon an integration by parts, we obtain

$\displaystyle{\cal
S}(\bar{\psi},\psi)\\!=\\!\int\\!d^{4}x\\!\left[\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}\\!-\\!m\right)\psi\\!+\\!\vartheta\,(\bm{\nabla}\bar{\psi})\\!\cdot\\!\gamma^{0}\gamma_{5}(\bm{\nabla}\psi)\right],$
(4)

where $\bar{\psi}:=\psi^{\dagger}\gamma^{0}$ is the adjoint field that
satisfies the equation

$\displaystyle
i\,\partial_{\mu}\bar{\psi}\,\gamma^{\mu}+\vartheta\,(\bm{\nabla}^{2}\bar{\psi})\,\gamma^{0}\gamma_{5}+m\,\bar{\psi}=0\;.$
(5)

Taking a plane wave solution $\psi(x)=u(p)\,e^{-ip\cdot x}$, the modified
Dirac equation in momentum space reads

$\displaystyle\left(\gamma^{\mu}p_{\mu}+\vartheta\,\bm{p}^{2}\gamma^{0}\gamma_{5}-m\right)u(p)=0\;,$
(6)

where $u(p)$ is an amplitude in the momentum space.

The term $\vartheta\,\bm{p}^{2}$ appearing alongside the
$\gamma^{0}\gamma_{5}$-matrix in Eq. (6), is a CPT-violating contribution, as
argued in [48]. The CPT theorem states that CPT symmetry arises under
assumptions through the combination of Lorentz-symmetry and quantum-mechanical
assumptions. Then, if CPT symmetry is broken, one or more of the assumptions
necessary to prove the CPT theorem should not be valid. Once both Lorentz and
CPT invariance involve spacetime transformations, it is natural to suspect
that CPT violation implies Lorentz-symmetry breakdown. This statement was
rigorously proven in [49, 50, 51]. However, Lorentz-symmetry breaking does not
necessarily imply CPT violation.

To obtain the fermion propagator, it is convenient to define the matrix
$D=p_{\mu}\gamma^{\mu}+d_{\mu}\gamma^{\mu}\gamma_{5}-m\,\mathds{1}$ that acts
on the $u(p)$ amplitude in Eq. (6), where $d^{\mu}=(\vartheta\,\bm{p}^{2},{\bf
0})$ is a time-like vector whose the component depends on the ${\bm{p}}^{2}$.
Thereby, the fermion propagator is the inverse of the $D$-matrix:

$\displaystyle D^{-1}=$
$\displaystyle\;\Big{[}(p^{2}-m^{2}-d^{2})\left(m+p_{\mu}\gamma^{\mu}+d_{\mu}\gamma^{\mu}\gamma_{5}\right)-2(p\cdot
d)(d_{\mu}\gamma^{\mu}-p_{\mu}\gamma^{\mu}\gamma_{5})-2m\,(\,p^{\alpha}\,d^{\beta}-p^{\beta}\,d^{\alpha}\,)\,\varepsilon_{\alpha\beta\mu\nu}\,\Sigma^{\mu\nu}\Big{]}\,\Delta^{-1}\;,$
(7)

where the denominator $\Delta$ is given by

$\displaystyle\Delta=$ $\displaystyle\;(p^{2}-m^{2})^{2}-4(p\cdot
d)^{2}+2d^{2}(p^{2}+m^{2})+d^{4}\;.$ (8)

The dispersion relation of the particle is the propagator’s pole at
$\Delta=0$. Restricting these results to the case of the time-like $4$-vector
$d^{\mu}=(\vartheta\,\bm{p}^{2},{\bf 0})$, the fermion dispersion relation is
reduced to

$\displaystyle
4p^{2}\vartheta^{2}\bm{p}^{4}+\left(p^{2}-m^{2}-\vartheta^{2}\bm{p}^{4}+2E\vartheta\bm{p}^{2}\right)$
$\displaystyle\times\left(p^{2}-m^{2}-\vartheta^{2}\bm{p}^{4}-2E\vartheta\bm{p}^{2}\right)=0\;.$
(9)

The corresponding energy solutions of Eq. (9) are the positive, $+E_{\pm}({\bf
p})$, and negative, $-E_{\pm}({\bf p})$, energy eigenstates, where

$\displaystyle E_{\pm}({\bf
p})=\sqrt{\bm{p}^{2}+m^{2}+\vartheta^{2}\,\bm{p}^{4}\pm\vartheta(\bm{p}^{2})^{3/2}}\;,$
(10)

that confirm the results obtained in [31]. In the rest frame of the fermion,
we have no mass splitting

$\displaystyle E_{0}=m.$ (11)

This indicates that the ($\pm$) signals in Eq. (10) correspond to two energy
eigenstates of the same fermion.

The $i$-th component of the group velocity becomes

$\displaystyle\bm{v}_{i}=\dfrac{\partial E}{\partial\bm{p}_{i}}\;,$ (12)

splitting into time and space parts, and also taking the total differential of
Eq. (9), one can show the group velocity of the propagating wave is

$\displaystyle\bm{v}_{i}=\left[\dfrac{6\,(E^{2}-m^{2}-\bm{p}^{2})}{E^{2}-m^{2}-\bm{p}^{2}-\vartheta^{2}\,\bm{p}^{4}}+\vartheta^{2}\,\bm{p}^{2}-5\right]\dfrac{\bm{p}_{i}}{E}\;,$
(13)

which reduces to the usual case, $\bm{p}_{i}/E$, whether we turn off the LSV
terms. Substituting the positive energy solutions, one obtain

$\displaystyle\bm{v}_{\pm}=\left[\frac{1+\vartheta^{2}\,{\bf p}^{2}\pm
6\,\vartheta\,|{\bf
p}|}{\sqrt{\bm{p}^{2}+m^{2}+\vartheta^{2}\,\bm{p}^{4}\pm\vartheta(\bm{p}^{2})^{3/2}}}\right]\bm{p}\;.$
(14)

In both cases, with $(+)$ and $(-)$, even in the presence of LSV terms, the
expressions for the group velocities in Eq. (14) assume the condition
$|\bm{v}_{i}|<1$, indicating in this case that the causal structure of the
theory is maintained.

##  III The positive energy solutions for the modified Dirac equation

From the energy eigenstates, we can obtain the positive energy solutions for
the modified Dirac equation in the laboratory reference frame. We consider the
following plane wave solution for a free fermion $\psi$:

$\displaystyle\psi(x,\pm s)=u(p,\pm s)\,e^{-ip\cdot
x}=\begin{pmatrix}u_{a}(p,\pm s)\\\ u_{b}(p,\pm s)\end{pmatrix}e^{-ip\cdot
x}\;,$ (15)

in which the positive energy is described by the four-momentum fermion
$p^{\mu}=\left(E_{\pm},\bm{p}\right)$, in a laboratory reference frame, and
with spin projection $(\pm s)$. Using this definition, the equation field is

$\displaystyle\left(E_{\pm}\gamma^{0}-p_{i}\gamma^{i}-m+\vartheta\bm{p}^{2}\gamma^{0}\gamma_{5}\right)u(p,\pm
s)=0\;.$ (16)

In matrix form, this equation leads to the coupled system

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\left(E_{\pm}\\!-\\!m\right)\\!u_{a}(p,\pm
s)\\!+\\!\left(\vartheta\bm{p}^{2}\mathds{1}\\!-\\!\bm{\sigma}\\!\cdot\\!\bm{p}\right)\\!u_{b}(p,\pm
s)\\!=\\!0\;,$ (17a)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\left(E_{\pm}\\!+\\!m\right)\\!u_{b}(p,\pm
s)\\!+\\!\left(\vartheta\bm{p}^{2}\mathds{1}\\!-\\!\bm{\sigma}\\!\cdot\\!\bm{p}\right)\\!u_{a}(p,\pm
s)\\!=\\!0\;,$ (17b)

where $\mathds{1}$ is the $2\times 2$ identity. Thus, we obtain the relation

$\displaystyle u_{b}(p,\pm
s)=\dfrac{\bm{\sigma}\cdot\bm{p}-\vartheta\bm{p}^{2}\,\mathds{1}}{E_{\pm}+m}\,u_{a}(p,\pm
s)\;.$ (18)

Therefore, the general solution for a positive energy eigenstate in momentum
space is written as

$\displaystyle u(p,\pm s)=\begin{pmatrix}\mathds{1}\\\
\dfrac{\bm{\sigma}\cdot\bm{p}-\vartheta\,\bm{p}^{2}\,\mathds{1}}{E_{\pm}+m}\end{pmatrix}u_{a}(p,\pm
s)\;.$ (19)

As in the case of the usual Dirac equation, we construct the spin up solution
from the amplitude

$\displaystyle u_{a}(p,+s)=N_{+}\begin{pmatrix}1\\\ 0\end{pmatrix},$ (20)

that substituting in Eq. (18), the $u_{b}$-amplitude for $(+s)$ is

$\displaystyle u_{b}(p,+s)$
$\displaystyle=\dfrac{N_{+}}{E_{\pm}+m}\begin{pmatrix}p_{z}-\vartheta\bm{p}^{2}\\\
p_{x}+ip_{y}\end{pmatrix}\;.$ (21)

Thereby, the general solution has the form

$\displaystyle u(p,+s)=N_{+}\begin{pmatrix}1\\\ 0\\\
\dfrac{p_{z}-\vartheta\bm{p}^{2}}{E_{\pm}+m}\\\
\dfrac{p_{x}+ip_{y}}{E_{\pm}+m}\end{pmatrix}\;,$ (22)

where $N_{+}$ is a normalization constant. Using the normalization condition
$\bar{u}(p,+s)\,u(p,+s)=2m$, the $N_{+}$-constant is

$\displaystyle
N_{+}=\sqrt{\dfrac{2m(E_{\pm}+m)^{2}}{2m(E_{\pm}+m)+\vartheta(2p_{z}\pm|\bm{p}|)\bm{p}^{2}}}\;.$
(23)

The solution describing spin up is

$\displaystyle\psi(x,+s)=$
$\displaystyle\sqrt{\dfrac{2m(E_{\pm}+m)^{2}}{2m(E_{\pm}+m)+\vartheta(2p_{z}\pm|\bm{p}|)\bm{p}^{2}}}$
$\displaystyle\times\begin{pmatrix}1\\\ 0\\\
\dfrac{p_{z}-\vartheta\bm{p}^{2}}{E_{\pm}+m}\\\
\dfrac{p_{x}+ip_{y}}{E_{\pm}+m}\end{pmatrix}e^{-ip\cdot x}\,.$ (24)

For the case of the spin-down solution, the $u_{a}$-amplitude becomes

$\displaystyle u_{a}(p,-s)=N_{-}\begin{pmatrix}0\\\ 1\end{pmatrix},$ (25)

and thus, we obtain

$\displaystyle u_{b}(p,-s)$
$\displaystyle=\dfrac{N_{-}}{E_{\pm}+m}\begin{pmatrix}p_{x}-ip_{y}\\\
-p_{z}-\vartheta\bm{p}^{2}\end{pmatrix}\;.$ (26)

The general solution for $(-s)$ has the form

$\displaystyle u(p,-s)=N_{-}\begin{pmatrix}0\\\ 1\\\
\dfrac{p_{x}-ip_{y}}{E_{\pm}+m}\\\
\dfrac{-p_{z}-\vartheta\bm{p}^{2}}{E_{\pm}+m}\end{pmatrix},$ (27)

where $N_{-}$ is the correspondent normalization constant determined by the
condition $\bar{u}(p,-s)\,u(p,-s)=2m$. Thereby, we obtain

$\displaystyle
N_{-}=\sqrt{\dfrac{2m(E_{\pm}+m)^{2}}{2m(E_{\pm}+m)-\vartheta(2p_{z}\mp|\bm{p}|)\bm{p}^{2}}}\;,$
(28)

and the correspondent solution is given by

$\displaystyle\psi(x,-s)=$
$\displaystyle\sqrt{\dfrac{2m(E_{\pm}+m)^{2}}{2m(E_{\pm}+m)-\vartheta(2p_{z}\mp|\bm{p}|)\bm{p}^{2}}}$
$\displaystyle\times\begin{pmatrix}0\\\ 1\\\
\dfrac{p_{x}-ip_{y}}{E_{\pm}+m}\\\
\dfrac{-p_{z}-\vartheta\bm{p}^{2}}{E_{\pm}+m}\end{pmatrix}e^{-ip\cdot x}\,.$
(29)

The limit $\vartheta\rightarrow 0$ recovers all known results of the usual
Dirac equation. Notice that, the normalization constants are not equal by the
introduction of $\vartheta$-scale, i.e., $N_{+}\neq N_{-}$. Consequently, the
introduction of the LQG-scale breaks the symmetry in the Dirac equation
solutions, and still includes degenerate solutions $(\pm)$ in Eqs. (III) and
(III), respectively.

##  IV The $U(1)$ conserved current

Consider $\widehat{{\cal O}}$ a operator that acts of generic mathematical
objects $A$ and $B$, such as scalars, vectors and spinors. Throughout the
text, we assume the definitions $A\overleftarrow{\widehat{{\cal
O}}}B=(\widehat{{\cal O}}A)B$ and $A\overleftrightarrow{\widehat{{\cal
O}}}B\equiv(\widehat{{\cal O}}A)B-A(\widehat{{\cal O}}B)$. Then, assuming that
the transform under global $U(1)$ for the matter field $(\psi)$

$\displaystyle\psi\,\longmapsto\,\psi^{\prime}=e^{-i\alpha}\,\psi\;,$ (30)

the fermion action in Eq. (3) is invariant. On the other hand, if we introduce
a local $U(1)$ transformation for an infinitesimal $\alpha$-parameter, the
fermion action in Eq. (3) at first order in $\alpha$ is

$\displaystyle\mathcal{S}\,\longmapsto\,\mathcal{S}^{\prime}=$
$\displaystyle\;\mathcal{S}-\int
d^{4}x\,(\partial_{\mu}\alpha)\,\bar{\psi}\gamma^{\mu}\psi$
$\displaystyle\;+i\vartheta\int
d^{4}x\,(\bm{\nabla}\alpha)\cdot\bar{\psi}\gamma^{0}\gamma_{5}\vecev{\bm{\nabla}}\psi\;.$
(31)

The variation of Eq. (IV) with respect to $\alpha$-parameter yields the
$4$-current

$\displaystyle
J^{\mu}=(J^{0},J^{i})=\Big{(}\bar{\psi}\gamma^{0}\psi,\bar{\psi}\gamma^{i}\psi-i\vartheta\bar{\psi}\gamma^{0}\gamma_{5}\vecev{\partial}^{i}\psi\Big{)}\;,$
(32)

that satisfies the continuity equation $\partial_{\mu}J^{\mu}=0$. The minimal
coupling with the electromagnetic field is introduced substituting the
derivative operator by the covariant derivative operator
$\partial_{\mu}\longmapsto D_{\mu}=\partial_{\mu}+ie\,A_{\mu}$. The new action
for the QED is

$\displaystyle S=$ $\displaystyle\;\int
d^{4}x\bigg{[}-\dfrac{1}{4}\,F_{\mu\nu}F^{\mu\nu}-eA_{\mu}\bar{\psi}\gamma^{\mu}\psi$
$\displaystyle\;-i\vartheta
e\bm{A}\cdot\bar{\psi}\gamma^{0}\gamma_{5}\vecev{\bm{\nabla}}\psi+\vartheta
e^{2}\bm{A}^{2}\,\bar{\psi}\gamma^{0}\gamma_{5}\psi$
$\displaystyle\;+\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-\vartheta\,\gamma^{0}\gamma_{5}\,\bm{\nabla}^{2}-m\right)\psi\bigg{]}\,,$
(33)

which includes a new quartic coupling involving the quadratic vector potential
with the fermions fields. Since the Bianchi identity is the usual expression,
one can verify that magnetic Gauss and Faraday-Lenz laws are unchanged. Also
the electric Gauss law is protected, once the new current does not affect the
electrical charge density

$\displaystyle\bm{\nabla}\cdot\bm{E}=-e\,\psi^{\dagger}\,\psi\;.$ (34)

Just the Ampère-Maxwell law is modified by the spatial current density

$\bm{\nabla}\times\bm{B}=\partial_{t}\bm{E}\\!+\\!e\bar{\psi}\bm{\gamma}\psi\\!-\\!2\vartheta
e^{2}\bm{A}\bar{\psi}\gamma^{0}\gamma_{5}\psi\\!+\\!ie\vartheta\bar{\psi}\gamma^{0}\gamma_{5}\vecev{\bm{\nabla}}\psi\;.$
(35)

The displacement current on the right-hand side of Eq. (35) coincides with the
spatial part of $U(1)$ current in Eq. (32). Thereby, we can conclude that the
conserved $U(1)$ and the electromagnetic current is the same in this approach.

Substituting the solutions given by Eq. (19) in the spatial current density of
Eq. (32), we obtain the result in the momentum space

$\displaystyle\bm{J}=$
$\displaystyle\;\dfrac{1}{E_{\pm}+m}\,u^{\dagger}_{a}(p^{\prime})\Bigg{\\{}\bigg{[}1-\dfrac{\vartheta^{2}}{2}(\bm{q}^{2}+\bm{\ell}^{2})\bigg{]}\bm{\ell}-(\bm{q}\times\bm{\sigma})$
$\displaystyle\;-\dfrac{\vartheta}{2}\,\bm{q}^{2}\,\bm{\sigma}-\vartheta\bigg{(}\dfrac{\bm{\ell}^{2}}{2}\,\delta_{ij}-\bm{\ell}_{i}\,\bm{\ell}_{j}\bigg{)}\bm{\sigma}_{j}\Bigg{\\}}u_{a}(p)\;,$
(36)

where we have omitted the spins projections $(\pm s)$, and also have defined
the photon’s transfer 3-momentum $\bm{q}=\bm{p}^{\prime}-\bm{p}$, and the
total 3-momentum $\bm{\ell}=\bm{p}+\bm{p}^{\prime}$. Since the term
proportional to $\vartheta^{2}$ is too small, the only contribution of a
granular space-time in the current comes from a helicity-type interaction with
respect to the transfer momentum $\bm{q}$, and a projection of the spin on
$\bm{\ell}$-direction.

##  V The Gordon decomposition

In this Section, we introduce the Gordon decomposition procedure for the
modified Dirac equation, in which the interpretation of the current coupled to
the electromagnetic field is an important key to understand the form factors,
as well as the electron’s magnetic and electric dipole momentum. In order, we
must emphasize that the conserved current of the model given by Eq. (32) has
the interesting aspect of temporal and spatial sectors have different
interpretations. This fact suggests that the usual procedure for deducing
Gordon’s identity can no longer be applied at least to the spatial part of Eq.
(32), which differs form the usual case by the LSV terms. Thereby, we have two
Gordon identities, the first for the time component of Eq. (32), and the
second one for the spatial part. For the current’s time-component, we go back
to the Eq. (1), whose Dirac conjugate equation for $\bar{\psi}(x)$ is given by
Eq. (5), and combining Eq. (1) with Eq. (5), we multiply by
$\gamma^{\mu}\,\psi(x)$ at the right-hand side of Eq. (5), and also multiply
by $\bar{\psi}(x)\,\gamma^{0}$ at the left-hand side. By subtracting these
equations, we get

$\displaystyle
J^{0}=-\dfrac{i}{2m}\bar{\psi}\vecev{\partial}_{t}\psi-\dfrac{1}{m}\partial_{\mu}(\bar{\psi}\Sigma^{\mu
0}\psi)-\dfrac{\vartheta}{2m}\bar{\psi}\gamma_{5}\vecev{\bm{\nabla}}^{2}\psi\;,$
(37)

which is the Gordon identity for $J^{0}$-component of the conserved current.

For the spatial part of the conserved current, we repeat the procedure above
by multiplying by
${\bm{\gamma}}\,\psi+i\vartheta\,(\bm{\nabla})\gamma^{0}\gamma_{5}\psi-i\vartheta\,\gamma^{0}\gamma_{5}(\bm{\nabla}\psi)$
on the right-hand side of Eq. (5), and also multiplying by
$\bar{\psi}\,{\bm{\gamma}}+i\vartheta\,(\bm{\nabla}\bar{\psi})\gamma^{0}\gamma_{5}-i\vartheta\,\bar{\psi}\gamma^{0}\gamma_{5}\bm{\nabla}$
the left-hand side, with the LSV terms being necessary due to the new
structure of the current. Therefore, the Gordon identity for $J^{i}$ becomes
as given below:

$\displaystyle J^{i}\\!$ $\displaystyle=$
$\displaystyle\\!\dfrac{i}{2m}\,\bar{\psi}\partial^{i}\psi-\dfrac{1}{m}\,\partial_{\mu}(\bar{\psi}\,\Sigma^{\mu
i}\psi)+\dfrac{\vartheta}{2m}\,(\partial_{t}\bar{\psi})\gamma_{5}\partial^{i}\psi$
(38)
$\displaystyle-\dfrac{2i\vartheta}{m}\,(\bm{\nabla}\bar{\psi})\,\Sigma^{0i}\gamma_{5}\cdot(\bm{\nabla}\psi)+\dfrac{i\vartheta^{2}}{2m}\,\partial^{i}(\bar{\psi}\vecev{\bm{\nabla}}^{2}\psi)\;.$

The structure of both the components of the current as given in Eqs. (37) and
(38) shall be applied in the next Section, where we shall be calculating the
interparticle electron-electron ($e^{-}e^{-}$) non-relativistic potential
mediated by a photon exchange by means of an elastic scattering. The role of
the components of the currents will become clear after the expression for the
$e^{-}e^{-}$ potential will be written down with the spin and velocity
corrections on the Coulombian term.

From the action in Eq. (IV), the three-vertex coupling of the leptons current
with the EM field is described by

$\displaystyle S_{int}^{(3)}=-e\int d^{4}x\Big{[}\,\bar{\psi}\gamma^{\mu}\psi
A_{\mu}+i\vartheta\bm{A}\cdot\bar{\psi}\gamma^{0}\gamma_{5}\vecev{\bm{\nabla}}\psi\,\Big{]}\;,$
(39)

which, in the momentum space, reads as

$\displaystyle S_{int}^{(3)}=$
$\displaystyle\;-e\int\dfrac{d^{4}p^{\prime}}{(2\pi)^{4}}\frac{d^{4}p}{(2\pi)^{4}}\Big{[}\,\bar{u}(p^{\prime})\gamma^{\mu}u(p)A_{\mu}(q)$
$\displaystyle\;-\vartheta\,\bar{u}(p^{\prime})\gamma^{0}\gamma_{5}u(p)(\bm{p}^{\prime}+\bm{p})\cdot\bm{A}(q)\,\Big{]}\;.$
(40)

Using the Gordon identities, the time-component for the coupling is

$\displaystyle\bar{u}$ $\displaystyle(p^{\prime})\gamma^{0}u(p)A_{0}(q)=$
$\displaystyle=\bar{u}(p^{\prime})\bigg{[}\,\dfrac{\ell^{0}}{2m}\\!-\\!\dfrac{iq_{\mu}}{m}\,\Sigma^{\mu
0}\\!-\vartheta\,\dfrac{\bm{\ell}\cdot\bm{q}}{2m}\gamma_{5}\,\bigg{]}u(p)A_{0}(q)\;,$
(41)

where $q^{\mu}=p^{\prime\mu}-p^{\mu}$ and $\ell=p^{\prime\mu}+p^{\mu}$ are the
photon’s transfer and total $4$-momentums, respectively. This coupling is
interpreted as the interaction energy between the charge density and the
scalar potential $A^{0}\equiv V$. In configuration space, the $\vartheta$-term
in Eq. (V) for the electrostatic case becomes

$\displaystyle-\vartheta\,\dfrac{\bm{\ell}\cdot\bm{q}}{2m}\,V\,\gamma_{5}=-\frac{i\vartheta}{2m}(\bm{\ell}\cdot\bm{E})\,\gamma_{5}\;,$
(42)

where $\bm{q}\rightarrow-i\bm{\nabla}$, and ${\bf E}=-\bm{\nabla}V$. Thus, the
electric dipole momentum emerges with $\bm{\ell}\cdot\bm{E}$, and the ratio
$(\vartheta/2m)$ modulates the LQG contribution to the electron’s electric
dipole momentum.

From the spatial part, the current density is

$\displaystyle\bar{u}(p^{\prime})\big{[}$
$\displaystyle{\bm{\gamma}}-\vartheta\bm{\ell}\gamma^{0}\gamma_{5}\big{]}u(p)\cdot\bm{A}(q)=$
$\displaystyle=$
$\displaystyle\;\bar{u}(p^{\prime})\bigg{[}\dfrac{\bm{\ell}}{2m}-\dfrac{iq_{\mu}}{m}\Sigma^{\mu
i}+\dfrac{\vartheta q^{0}\bm{\ell}}{2m}\gamma_{5}$
$\displaystyle\;-\dfrac{i\vartheta(\bm{\ell}^{2}\\!-\\!\bm{q}^{2})}{2m}\Sigma^{0i}\gamma_{5}\\!-\\!\dfrac{\vartheta^{2}(\bm{q}\\!\cdot\\!\bm{\ell})\bm{q}}{2m}\bigg{]}u(p)\\!\cdot\\!\bm{A}(q)\;.$
(43)

In coordinate space, the last term can be eliminated by the Coulomb gauge,
$\bm{\nabla}\cdot{\bf A}=0$.

An important aspect arising from the Gordon decomposition in Eq. (V) is the
contribution to the electron’s electric dipole moment (EDM). Specifically, it
is associated with the term proportional to $\Sigma^{0i}\gamma_{5}$. This term
defines an interaction action for the EDM in momentum space such that

$\displaystyle{\cal S}_{\textrm{\tiny
EDM}}=\int\dfrac{d^{4}p^{\prime}}{(2\pi)^{4}}\frac{d^{4}p}{(2\pi)^{4}}\,\bar{u}(p^{\prime})\,d_{e}\,\bm{\Sigma}\cdot\bm{E}(q)\,u(p)\;,$
(44)

where ${\bf E}(q)$ is the momentum space version of the electric field,

$\displaystyle\bm{\Sigma}=\begin{pmatrix}\bm{\sigma}&0\\\
0&\bm{\sigma}\end{pmatrix}\;,$ (45)

and the EDM is

$\displaystyle d_{e}=\dfrac{\vartheta
e}{8m}\left(\dfrac{\bm{\ell}^{2}\\!-\\!\bm{q}^{2}}{q_{0}}\right)=\dfrac{\vartheta
e}{2m}\left(\dfrac{\bm{p}^{\prime}\cdot\bm{p}}{q_{0}}\right)\;.$ (46)

The corresponding Lagrangian density is

$\displaystyle\mathcal{L}_{\textrm{\tiny
EDM}}=\bar{u}(p^{\prime})\,d_{e}\,\bm{\Sigma}\cdot\bm{E}(q)\,u(p)\;.$ (47)

Figure 1: Upper bounds on the $d_{e}$ as function of the $\alpha$-angle. In
this plot, we fix the momenta and the energy at the MeV QED scale :
$|\bm{p}|=|\bm{p}^{\prime}|=q_{0}=1$ MeV.

The most recent experimental results for the EDM, obtained by using electrons
confined inside molecular ions subjected to a huge intramolecular electric
field, provide an upper bound of $d_{e}<4.1\times 10^{-30}$ $e\,\cdot$ cm
[52]. We can use this result to estimate a bound on the $\vartheta$-parameter.
In order, the Eq. (46) can be written as

$\displaystyle
d_{e}=\frac{\vartheta\,e}{2m}\left(\dfrac{|\bm{p}||\bm{p}^{\prime}|\cos\alpha}{q_{0}}\right)\leq
4.1\times 10^{-30}\,\,\mbox{e}\cdot\mbox{cm}\;,$ (48)

where $\alpha$ is the angle between $\bm{p}$ and $\bm{p}^{\prime}$. From the
Eqs. (1) and (2), the $\vartheta$-parameter is defined by

$\displaystyle\vartheta=\frac{\kappa_{7}}{4{\cal
L}}\left(\frac{\ell_{P}}{{\cal
L}}\right)^{\\!\\!\Upsilon_{\\!f}}\ell_{P}^{2}\;,$ (49)

and from ref. [31] comes that $\kappa_{7}=37.5$, $\Upsilon_{f}=-1$, and
$\ell_{p}=1.61\times 10^{-33}$ cm. Considering a energy scale at
$|\bm{p}|=|\bm{p}^{\prime}|=q_{0}=1$ MeV, and the electron mass of $m=0.5$
MeV, we can illustrate our $d_{e}$ values as a function of the $\alpha$-angle
in the plot of the FIG. 1. Since all $d_{e}$ values in FIG. 1 are on the order
of $10^{-33}\;e\cdot$ cm, the EDM predicted by this LQG model falls within the
current experimental limit of $d_{e}<4.1\times 10^{-30}\;e\cdot$ cm [52].

##  VI Electromagnetic Interaction Between Fermions

In this section, we aim to elucidate the implications of the preceding
electromagnetic current Gordon decomposition structure by examining its
influence on the Coulomb potential. To achieve this, we analyze the
interaction between two fermions, as described by the solutions in Eqs. (22)
and (27), in a tree-level elastic scattering considering the possibility of
spin-flip. The amplitude of this process can be characterized by the Feynman
diagram depicted in FIG. 2.

In this tree-level elastic scattering, the two fermions, each with charges
$e_{1}$ and $e_{2}$, possess incoming $4$-momenta denoted as
${p_{1}}^{\mu}=(E_{1},\bm{p}_{1})$ and ${p_{2}}^{\mu}=(E_{1},\bm{p}_{1})$, and
outgoing $4$-momenta denoted as
${p^{\prime}_{1}}^{\mu}=(E^{\prime}_{1},\bm{p}^{\prime}_{1})$ and
${p^{\prime}_{2}}^{\mu}=(E^{\prime}_{2},\bm{p}^{\prime}_{2})$, respectively.
Furthermore, it is convenient for us to work in the center of mass reference
frame, with the parameterization in terms of the following independent
momenta: the transferred momentum,
$q=p^{\prime}_{1}-p_{1}=-(p^{\prime}_{2}-p_{2})$, and the average momentum,
$\frac{\ell}{2}=\frac{(p^{\prime}_{1}+p_{1})}{2}=\frac{(p^{\prime}_{2}+p_{2})}{2}$.
Through the conservation of energy and momentum, we have that $q^{0}=0$ and
$\bm{q}\cdot\bm{\ell}=0$ to simplify the amplitude of FIG. 2.

Figure 2: An $e^{-}e^{-}$ tree-level elastic scattering involves incoming
$4$-momenta $p_{1}$ and $p_{2}$, and outgoing $4$-momenta $p^{\prime}_{1}$ and
$p^{\prime}_{2}$, mediated by a photon that transfers $4$-momentum
$p^{\prime}_{1}-p_{1}=-(p^{\prime}_{2}-p_{2})$ during the process.

In order to include spin and velocity corrections to the potential, we shall
employ the method described in [53, 54]. The prescription is that the
potential, $V$, can be obtained from the first Born approximation, i.e., by
performing the Fourier integral of the non-relativistic (NR) amplitude,
$\mathcal{M}_{\tiny{\textrm{NR}}}$, with respect to the transferred momentum
$\bm{q}$,

$\displaystyle
V(\bm{x})=-\int\dfrac{d^{3}\bm{q}}{(2\pi)^{3}}\mathcal{M}_{\tiny{\textrm{NR}}}e^{i\bm{q}\cdot\bm{x}},$
(50)

where $\mathcal{M}_{\tiny{\textrm{NR}}}$ is related to the relativistic
Feynman amplitude, $\mathcal{M}$, according to the conventions of [55],
through

$\displaystyle\mathcal{M}_{\tiny{\textrm{NR}}}=\dfrac{1}{\sqrt{2E_{1}}}\dfrac{1}{\sqrt{2E^{\prime}_{1}}}\dfrac{1}{\sqrt{2E_{2}}}\dfrac{1}{\sqrt{2E^{\prime}_{2}}}\mathcal{M}.$
(51)

The amplitude $\mathcal{M}$ is obtained by applying Feynman rules,
incorporating the fields, interaction vertices, and propagators. In the case
to be considered here, it can be rewritten as $\mathcal{M}\sim
J_{(1)}\langle\textrm{prop}\rangle J_{(2)}$, where $J_{(1)}$ and $J_{(2)}$
represent the sources/currents, and $\langle\textrm{prop}\rangle$ denotes the
propagator, with the possible Lorentz indices of the representations being
omitted. Therefore, taking the photon propagator in the Feynman gauge,

$\displaystyle\langle A_{\mu}A_{\nu}\rangle=i\dfrac{\eta_{\mu\nu}}{q^{2}},$
(52)

for the process in FIG. 2 we have the amplitude

$\displaystyle\\!\\!\\!\\!\\!i\mathcal{M}$
$\displaystyle=-e_{1}e_{2}J^{\mu}_{(1)}\langle A_{\mu}A_{\nu}\rangle
J^{\mu}_{(2)}$
$\displaystyle=-e_{1}e_{2}\bar{u}(p^{\prime}_{1})\gamma^{\mu}u(p_{1})\langle
A_{\mu}A_{\nu}\rangle\bar{u}(p^{\prime}_{2})\gamma^{\nu}u(p_{2}),$ (53)

with $J^{\mu}$ here denoting the vector current of the model given by Eq. (32)
in the momentum space. Then, utilizing the continuity equation
$q_{\mu}J^{\mu}=0$ and $q^{0}=0$, in the NR limit yields

$\displaystyle\mathcal{M}_{\tiny{\textrm{NR}}}$
$\displaystyle=-\dfrac{e_{1}e_{2}}{\bm{q}^{2}}\dfrac{J^{\mu}_{(1)}J_{(2)\mu}}{4E_{1}E_{2}}$
$\displaystyle=-\dfrac{e_{1}e_{2}}{\bm{q}^{2}}\dfrac{J^{0}_{(1)}J^{0}_{(2)}-\bm{J}_{(1)}\cdot\bm{J}_{(2)}}{4E_{1}E_{2}}.$
(54)

Expressing both $J^{0}_{(1)}$ and $\bm{J}_{(1)}$ in terms of their respective
Gordon decompositions and positive energy solutions, Eqs. (22) and (27), in
the NR limit,

$\displaystyle u_{\textrm{\tiny
NR}}(p_{1})\approx\sqrt{2m}\begin{pmatrix}\chi_{1}\\\\[5.69046pt]
\dfrac{\bm{\sigma}\cdot\bm{p}_{1}-\vartheta\bm{p}^{2}_{1}}{2m}\chi_{1}\end{pmatrix}$
(55)

and

$\displaystyle\bar{u}_{\textrm{\tiny
NR}}(p^{\prime}_{1})\approx\sqrt{2m}\left({\chi^{\prime}_{1}}^{\dagger},{\chi^{\prime}_{1}}^{\dagger}\dfrac{\bm{\sigma}\cdot\bm{p}^{\prime}_{1}-\vartheta{\bm{p}^{\prime}_{1}}^{2}}{2m}\right),$
(56)

with spin up and down respectively given by

$\displaystyle\chi_{1}=\begin{pmatrix}1\\\
0\end{pmatrix}\;\;\textrm{and}\;\;\chi_{1}=\begin{pmatrix}0\\\ 1\end{pmatrix}$
(57)

and the same for Dirac’s conjugate solution with

$\displaystyle{\chi^{\prime}_{1}}^{\dagger}=\left(1,0\right)\;\;\textrm{and}\;\;{\chi^{\prime}_{1}}^{\dagger}=\left(0,1\right),$
(58)

together the conditions of elastic scattering, we obtain

$\displaystyle
J^{0}_{(1)}\\!=\\!\left[2m_{1}\\!+\\!\dfrac{1}{8m_{1}}(\bm{\ell}^{2}\\!+\\!7\bm{q}^{2})\right]\\!\delta_{1}\\!+\\!\dfrac{3}{4m_{1}}i(\bm{q}\\!\times\\!\bm{\ell})\\!\cdot\\!\langle\bm{S}_{1}\rangle$
(59)

and

$\displaystyle\bm{J}_{(1)}\\!=\\!\left[\\!1\\!+\\!\dfrac{(\bm{q}^{2}\\!-\\!\bm{\ell}^{2})}{16m_{1}^{2}}\right]\\!\bm{\ell}\delta_{1}\\!-\\!2i\varepsilon_{ijk}\bm{q}_{j}\langle{\bm{S}_{1}}_{k}\rangle\\!+\\!\vartheta(\bm{\ell}^{2}\\!-\\!\bm{q}^{2})\langle{\bm{S}_{1}}\rangle,$
(60)

where we defined the quantities

$\displaystyle{\chi^{\prime}_{1}}^{\dagger}\chi_{1}=\delta_{1},$ (61)
$\displaystyle{\chi^{\prime}_{1}}^{\dagger}\dfrac{\bm{\sigma}}{2}\chi_{1}=\langle\bm{S}_{1}\rangle.$
(62)

To express $J^{0}_{(2)}$ and $\bm{J}_{(2)}$, one simply needs to replace the
index $1$ with $2$ in Eqs. (59) and (60), as well as replace the momenta
$\bm{q}$ and $\bm{\ell}$ with $-\bm{q}$ and $-\bm{\ell}$, respectively. With
these result and still in the NR limit, one can write the expression for the
Coulomb potential in the form

$\displaystyle V(\bm{x})=$
$\displaystyle\;e_{1}e_{2}\bigg{\\{}\left[\dfrac{5}{16}\left(\dfrac{1}{m_{1}^{2}}+\dfrac{1}{m_{2}^{2}}\right)\delta_{1}\delta_{2}-\dfrac{2}{3}\dfrac{\langle{\bm{S}_{1}}\rangle\cdot\langle{\bm{S}_{2}}\rangle}{m_{1}m_{2}}+\dfrac{\vartheta}{m_{1}m_{2}}\Big{(}\langle{\bm{S}_{1}}\rangle\times\langle{\bm{S}_{2}}\rangle\Big{)}\cdot\bm{\nabla}\right]\delta^{(3)}(\bm{x})$
$\displaystyle\;+\dfrac{\delta_{1}\delta_{2}}{4\pi|\bm{x}|}\left[1+\dfrac{\bm{\ell}^{2}}{4m_{1}m_{2}}-\bm{\ell}^{2}\left(\dfrac{1}{m_{1}^{2}}+\dfrac{1}{m_{2}^{2}}\right)\right]-\dfrac{\vartheta}{4\pi|\bm{x}|}\dfrac{\bm{\ell}^{2}}{m_{1}m_{2}}\bm{\ell}\cdot\Big{(}\langle{\bm{S}_{2}}\rangle\delta_{1}-\langle{\bm{S}_{1}}\rangle\delta_{2}\Big{)}$
$\displaystyle\;+\dfrac{\bm{Q}_{ij}}{4\pi|\bm{x}|^{2}}\dfrac{\langle{\bm{S}_{1,i}}\rangle\langle{\bm{S}_{2,j}}\rangle}{m_{1}m_{2}}-\dfrac{\bm{L}}{4|\bm{x}|^{3}}\\!\cdot\\!\bigg{[}\delta_{1}\langle{\bm{S}_{2}}\rangle\left(\dfrac{3}{4m_{2}^{2}}+\dfrac{1}{m_{1}m_{2}}\right)+1\leftrightarrow
2\bigg{]}+\dfrac{\vartheta\bm{\ell}^{2}}{4\pi
m_{1}m_{2}|\bm{x}|^{3}}\bm{x}\cdot\Big{(}\langle{\bm{S}_{2}}\rangle\times\langle{\bm{S}_{1}}\rangle\Big{)}\bigg{\\}},$
(63)

where $\bm{L}\\!=\\!\bm{x}\\!\times\\!\frac{\bm{\ell}}{2}$ is the angular
momentum and
$\bm{Q}_{ij}\\!=\\!\delta_{ij}\\!-\\!3\frac{\bm{x}_{i}\bm{x}_{j}}{\bm{|}\bm{x}|^{2}}$.

As anticipated in Section V, the linear contributions in the $\vartheta$
parameter are explicitly calculated and we pay attention to the Coulomb-like
term in $\vartheta$ with dependence on the spin. Also, it is worth to
highlight the spin correction coming from the $\vartheta$-contribution to the
contact term.

In the case of the $e^{-}e^{-}$ scattering, $m_{1}=m_{2}=m_{e}$ and
$e_{1}=e_{2}=e$, with $m_{e}$ and $e$ being the electron mass and charge,
respectively. Then, in the situation of an $e^{-}e^{-}$ scattering with spin-
flip, both $\delta_{1}$ and $\delta_{2}$ are trivial. In this sense, the
potential given by Eq. (VI) assumes the form

$\displaystyle V(\bm{x})=$
$\displaystyle\;\dfrac{e^{2}}{4\pi|\bm{x}|^{2}}\left[\dfrac{\bm{Q}_{ij}\langle{\bm{S}_{1,i}}\rangle\langle{\bm{S}_{2,j}}\rangle\\!+\\!\vartheta\bm{\ell}^{2}\bm{\hat{x}}\\!\cdot\\!\Big{(}\\!\langle{\bm{S}_{2}}\rangle\\!\times\\!\langle{\bm{S}_{1}}\rangle\\!\Big{)}}{m_{e}^{2}}\right]$
$\displaystyle\;+\dfrac{\vartheta
e^{2}}{m_{e}^{2}}\Big{(}\langle{\bm{S}_{1}}\rangle\times\langle{\bm{S}_{2}}\rangle\Big{)}\cdot\bm{\nabla}\delta^{(3)}(\bm{x}).$
(64)

In the case where the $e^{-}e^{-}$ scattering does not have the spin-flip,
$\delta_{1}=\delta_{2}$ and
$\langle{\bm{S}_{1}}\rangle=\langle{\bm{S}_{2}}\rangle$, and the Eq. (VI)
reduces to

$\displaystyle V(\bm{x})=$
$\displaystyle\;\dfrac{e^{2}}{4\pi|\bm{x}|}\left(1-\dfrac{7\bm{\ell}^{2}}{4m_{e}^{2}}\right)\delta_{1}^{2}$
$\displaystyle\;+\dfrac{e^{2}}{m_{e}^{2}}\bigg{[}\left(\dfrac{5\delta_{1}\delta_{2}}{8}-\dfrac{2\langle{\bm{S}_{1}}\rangle\cdot\langle{\bm{S}_{2}}\rangle}{3}\right)\delta^{(3)}(\bm{x})$
$\displaystyle\;+\dfrac{\bm{Q}_{ij}\langle{\bm{S}_{1,i}}\rangle\langle{\bm{S}_{2,j}}\rangle}{4\pi|\bm{x}|^{2}}-\dfrac{7\delta_{1}}{8|\bm{x}|^{3}}\bm{L}\cdot\langle{\bm{S}_{2}}\rangle\bigg{]}\;.$
(65)

The contribution from LQG arises specifically in scenarios involving
scattering without spin-flip, as outlined in Eq. (VI). Notably, this
contribution does not take into account the typical Coulomb-like term. The
Coulombian interaction, being a gauge interaction, preserves the chirality of
the particles. Hence, it follows that, in the absence of spin-flip, as Eq.
(VI) shows, Coulomb potential comes out.

##  VII The coupling with the EM field and the non-relativistic Hamiltonian

The gauge principle applied to the Eq. (3) means to introduce the minimal
coupling with the EM field via the usual redefinition of the $4$-momentum
$p_{\mu}\longmapsto p_{\mu}-e\,A_{\mu}$ in which the modified Dirac Eq. (1) is
given by

$\displaystyle\left[\gamma^{\mu}\left(p_{\mu}\\!-\\!eA_{\mu}\right)+\vartheta\left(\bm{p}-e\bm{A}\right)^{2}\gamma^{0}\gamma_{5}\\!-\\!m\right]u(p)=0\;.$
(66)

Using the previous notation for the $u$-amplitude in the momentum space, i.e.,
$u=(u_{a}\;\;\;u_{b})^{t}$, the system of Eqs. in (66) is read as

$\displaystyle\\!\\!\\!\left(E\\!-\\!e\phi\\!-\\!m\right)\\!u_{a}\\!-\\!\left[\bm{\sigma}\\!\cdot\\!\left(\bm{p}\\!-\\!e\bm{A}\right)\\!-\\!\vartheta\left(\bm{p}\\!-\\!e\bm{A}\right)^{2}\right]\\!u_{b}\\!=\\!0,$
(67a)
$\displaystyle\\!\\!\\!\left(E\\!-\\!e\phi\\!+\\!m\right)\\!u_{b}\\!-\\!\left[\bm{\sigma}\\!\cdot\\!\left(\bm{p}\\!-\\!e\bm{A}\right)\\!-\\!\vartheta\left(\bm{p}\\!-\\!e\bm{A}\right)^{2}\right]\\!u_{a}\\!=\\!0.$
(67b)

From (67b), we have the relation

$u_{b}=\dfrac{\bm{\sigma}\cdot\left(\bm{p}-e\bm{A}\right)-\vartheta\left(\bm{p}-e\bm{A}\right)^{2}}{E-e\phi+m}\,u_{a}\;,$
(68)

that allows us to write the spinor $u(p)$ in the form

$\displaystyle u(p)=\begin{pmatrix}\mathds{1}\\\\[5.69046pt]
\dfrac{\bm{\sigma}\cdot\left(\bm{p}-e\bm{A}\right)-\vartheta\left(\bm{p}-e\bm{A}\right)^{2}}{E-e\phi+m}\end{pmatrix}u_{a}\;.$
(69)

In the non-relativistic regime, the linear momentum is $\bm{p}\ll m$, and the
dominant energy term is of the order of the rest energy of the particle, that
is, $E\approx m$. The kinetic and potential energy are much smaller than the
rest energy, that is, the approximation $e\phi\ll m$ is valid in the previous
relations. Therefore, the relations between the Eqs. (68) and (69) are
simplified as

$\displaystyle u_{b}\\!$ $\displaystyle\approx$
$\displaystyle\\!\dfrac{\bm{\sigma}\cdot\bm{\pi}-\vartheta\bm{\pi}^{2}}{2m}\,u_{a}\;,$
(70a) $\displaystyle u_{\textrm{\tiny NR}}(p)\\!$ $\displaystyle\approx$
$\displaystyle\\!\begin{pmatrix}\mathds{1}\\\\[5.69046pt]
\dfrac{\bm{\sigma}\cdot\bm{\pi}-\vartheta\bm{\pi}^{2}}{2m}\end{pmatrix}u_{a}\;,$
(70b)

where $\bm{\pi}\equiv\bm{p}-e\bm{A}$. Substituting Eq. (70a) in Eq. (67a), the
Eq. (67a) is so written in the form $E_{\textrm{\tiny NR}}u=(E-m)u=Hu$, in
which $E_{\textrm{\tiny NR}}$ is the energy of a particle, and $H$ is the
Hamiltonian operator in the NR limit:

$\displaystyle
H=e\phi+\dfrac{\left(\bm{\sigma}\cdot\bm{\pi}-\vartheta\bm{\pi}^{2}\right)^{2}}{2m}\,.$
(71)

The kinetic term yields the operator

$\left(\bm{\sigma}\\!\cdot\\!\bm{\pi}\\!-\\!\vartheta\bm{\pi}^{2}\right)^{2}\\!=\\!\left(\bm{\sigma}\\!\cdot\\!\bm{\pi}\right)^{2}\\!+\\!\vartheta^{2}\bm{\pi}^{4}\\!-\\!\vartheta\bm{\pi}^{2}\left(\bm{\sigma}\cdot\bm{\pi}\right)\\!-\\!\vartheta\left(\bm{\sigma}\\!\cdot\\!\bm{\pi}\right)\bm{\pi}^{2},$
(72)

in which new terms as $\bm{\pi}^{4}$ and
$\bm{\pi}^{2}\left(\bm{\sigma}\cdot\bm{\pi}\right)$ emerge with the
$\vartheta$-parameter. We use the following relations

$\displaystyle\left(\bm{\pi}\cdot\bm{\sigma}\right)^{2}\\!$ $\displaystyle=$
$\displaystyle\\!\left(\bm{p}-e\bm{A}\right)^{2}-e\bm{\sigma}\cdot\bm{B}\;,$
(73a) $\displaystyle\bm{\pi}^{2}\left(\bm{\sigma}\cdot\bm{\pi}\right)\\!$
$\displaystyle=$
$\displaystyle\\!\left(\bm{\sigma}\cdot\bm{\pi}\right)\bm{\pi}^{2}+e\bm{\sigma}\cdot\partial_{t}\bm{E}$
(73b)
$\displaystyle+\,2e^{2}\bm{\sigma}\cdot\left[\bm{A}\times\left(\bm{p}\times\bm{A}\right)\right]\;,$

for a magnetic field $({\bf B})$, in which $\bm{B}=\bm{\nabla}\times\bm{A}$.
Therefore, the Hamiltonian operator is

$\displaystyle H=$
$\displaystyle\;e\phi+\dfrac{\left(\bm{p}-e\bm{A}\right)^{2}}{2m}-\dfrac{e}{2m}\bm{\sigma}\cdot\bm{B}$
$\displaystyle\;-\dfrac{\vartheta}{m}\left(\bm{\sigma}\cdot\bm{\pi}\right)\left(\bm{p}-e\bm{A}\right)^{2}-\dfrac{\vartheta\,e}{2m}\bm{\sigma}\cdot\partial_{t}\bm{E}$
$\displaystyle\;-\dfrac{\vartheta
e^{2}}{m}\,\bm{\sigma}\cdot\left[\bm{A}\times\left(\bm{p}\times\bm{A}\right)\right]+\dfrac{\vartheta^{2}}{2m}\bm{\pi}^{4}\;.$
(74)

In the particular case of a weak and homogeneous magnetic field, the vector
potential is $\bm{A}=(\bm{B}\times\bm{x})/2$, and we can neglect quadratic
terms in $\bm{A}$. Thereby, we can approximate the terms

$\displaystyle\bm{\pi}^{2}\approx\bm{p}^{2}-e(\bm{B}\cdot\bm{L})\;,$ (75a)
$\displaystyle\bm{\pi}^{4}\approx\bm{p}^{4}-2e\bm{p}^{2}\left(\bm{B}\cdot\bm{L}\right)\;,$
(75b)
$\displaystyle\bm{\sigma}\cdot\left[\bm{A}\times\left(\bm{p}\times\bm{A}\right)\right]\approx
0\;,$ (75c)
$\displaystyle\bm{\sigma}\cdot(\bm{p}-e\bm{A})(\bm{p}-e\bm{A})^{2}\approx-e\bm{p}^{2}\bm{B}\cdot(\bm{r}\times\bm{S})$
$\displaystyle+\,\frac{1}{2}(\bm{S}\cdot\bm{p})\left(\bm{p}^{2}-e\bm{B}\cdot\bm{L}\right)\;,$
(75d)

where we have used that $\bm{S}=\bm{\sigma}/2$. Under these conditions, the
Hamiltonian operator is

$\displaystyle H=$
$\displaystyle\;e\phi+\dfrac{\bm{p}^{2}}{2m}\Big{[}1+2\vartheta\big{(}e\bm{B}\cdot(\bm{x}\times\bm{S})-2\bm{S}\cdot\bm{p}\big{)}$
$\displaystyle\;+\vartheta^{2}\left(\bm{p}^{2}-2\bm{B}\cdot\bm{L}\right)\Big{]}-\dfrac{e}{2m}\left(\bm{L}+2\bm{S}\right)\cdot\bm{B}$
$\displaystyle\;-\dfrac{\vartheta\,e}{m}\,\bm{S}\cdot\partial_{t}\bm{E}+\dfrac{2\vartheta\,e}{m}\left(\bm{B}\cdot\bm{L}\right)\left(\bm{S}\cdot\bm{p}\right).$
(76)

The term $\bm{S}\cdot\partial_{t}\bm{E}$ is projection of the electric field
on the spin direction. It emerges from the Faraday law, and expresses the
electric dipole momentum variable with the time. In the case of a
electrostatic field, as the Coulomb electric field, this term is null. The
other terms have contribution of the uniform magnetic field, of the linear
momentum, and also of angular momentum $({\bf L})$ for the NR particle.

##  VIII Renormalizability

The presence of a $\bm{\nabla}^{2}$ in the modified Dirac action introduces a
higher-order space derivative. Besides gauge symmetry, causality, and
renormalizability, that are fundamental ingredients of a consistent quantum
field theory can be checked in this model. From the discussions in Section II
and IV, respectively, we see that there are configurations in which tachyon
modes are avoided, and that $U(1)$ gauge symmetry is ensured if minimal
coupling with the EM field is introduced . The missing point, regarding
renormalizability, is a relevant discussion and let us take into consideration
the fact that the modified fermion propagator given by Eq. (7) becomes more
convergent with respect to the usual case of QED. Thus, we conclude that the
model is super-renormalizable. This is the matter discussed in the present
Section.

The ultraviolet behavior of the modified fermion propagator in the Eq. (7)
goes with $\sim 1/p^{2}$. Thereby, we obtain that the superficial degree of
divergence of the primitively divergent diagrams is given by

$\displaystyle\delta_{\textrm{\tiny
graph}}=4-E_{\gamma}-E_{\psi}-V_{\textrm{\tiny QED}}\;,$ (77)

where $E_{\gamma}$ and $E_{\psi}$ are the number of external photonic and the
fermionic lines, respectively, and $V_{\textrm{\tiny QED}}$ is the usual QED
vertex with two fermionic lines, one photonic line, and without momentum.
Since that $V_{\textrm{\tiny QED}}$ appears with a negative coefficient, the
conclusion is in favor to a super-renormalizability model from the point of
view of a quantum field theory. This unexpected result is consequence of the
behavior of the modified fermion propagator given by Eq. (7) in the
ultraviolet regime.

##  IX Concluding remarks

In this paper, we study the solutions of a modified Dirac equation by the
effects of the LQG, and it coupling with the EM field through the $U(1)$ gauge
symmetry. The modification of the Dirac action is due to presence of a length
scale $(\vartheta)$ introduced by the LQG, where the Dirac kinetic term
contains the higher order derivative term
$\vartheta\,\bar{\psi}\gamma^{0}\gamma_{5}\,\bm{\nabla}^{2}\psi$. We obtain
the plane solutions in the momentum space associated with the positive and
negative energies for the modified Dirac operator. These solutions express an
asymmetry in these solutions due to Lorentz symmetry breaking induced by the
$\vartheta$-length scale. This is the consequence of the asymmetry for the
frequencies associated with the fermion dispersion relations. The group
velocity for fermions is showed for the positive energy solution. The fermion
propagator in this LSV scenario is obtained in the momentum space. In the
ultraviolet regime, the propagator has an improved behavior in which it goes
to zero with the squared inverse of the momentum $(\sim 1/k^{2})$.

The $U(1)$ global symmetry in the LQG Dirac action motivates us to calculate
the correspondent conserved current through the Noether theorem. The term
$\vartheta\,\bar{\psi}\gamma^{0}\gamma_{5}\,\bm{\nabla}^{2}\psi$ with the
Laplacian operator implies that the spatial-component for the conserved
current is altered with new terms that depend on the
${\bm{\vartheta}}$-parameter, whereas that time-component is unaltered. The
next step is so to investigate the $U(1)$ local symmetry of this theory. The
minimal coupling with the electromagnetic (EM) field preserves the $U(1)$
local transformation, that leads us to a QED emerged from the LQG. Thereby, we
study the coupling of the conserved current with the EM field through the
Gordon identity. New terms emerge in the current that can be interpreted as
contributions to the lepton’s electric dipole momentum due exclusively to the
${\bm{\vartheta}}$-length scale. Afterwards, we obtain the non-relativistic
limit in the Dirac equation, where the Hamiltonian for the NR particle
acquires terms that depend on the spin and orbital angular momentum projected
on the uniform magnetic field, and also the projection on the particle linear
momentum. A new term that depends on the interaction of the electric dipole
momentum emerges in the NR limit. All these new effects are removed whenever
$\vartheta\rightarrow 0$.

Also, to render clearer the terms that appear after carrying out the Gordon
decomposition of the electric magnetic current, we have devoted special
attention to the calculation of the $e^{-}e^{-}$ interparticle potential and
the linear corrections in the $\vartheta$ parameter shows us how LQG effects
correct the usual spin- and velocity-dependent contributions present in the
potential.

To end up, we briefly investigate the power-counting renormalizability of the
model. Since the fermion propagator exhibits a more convergent ultraviolet
behavior, $D^{-1}\sim 1/k^{2}$, the model becomes super-renormalizable. Thus,
this conclusion motivates us to investigate new effects of this modified QED,
such as the corrections to the $g-2$ factor for the electron and muon, which
are decisive precision measurements in QED. Furthermore, the study of the
scalar sector and of the EM modified by the LQG, as described in the Ref.
[31], are interesting challenges for a forthcoming project. As a perspective,
it is worth noting that one can explore the effects of the Hamiltonian in Eq.
(VII) on spectral lines, e.g., of the Hydrogen atom.

##  X Acknowledgements

J.P.S.M. and J.M.A.P. express their gratitude to CNPq-Brazil for granting
their PhD Fellowships.

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# Peeking into the $\theta$ vacuum

Ryuichiro Kitano  High Energy Accelerator Research Organization (KEK), Tsukuba
305-0801, Japan  Graduate University for Advanced Studies (SOKENDAI), Tsukuba
305-0801, Japan  Ryutaro Matsudo  High Energy Accelerator Research
Organization (KEK), Tsukuba 305-0801, Japan  Norikazu Yamada  High Energy
Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan  Graduate
University for Advanced Studies (SOKENDAI), Tsukuba 305-0801, Japan  Masahito
Yamazaki  Kavli Institute for the Physics and Mathematics of the Universe
(WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan

###### Abstract

We propose a subvolume method to study the $\theta$ dependence of the free
energy density of the four-dimensional $\mathrm{SU}$($N$) Yang-Mills theory on
the lattice. As an attempt, the method is first applied to $\mathrm{SU}$(2)
Yang-Mills theory at $T=1.2\,T_{c}$ to understand the systematics of the
method. We then proceed to the calculation of the vacuum energy density and
obtain the $\theta$ dependence qualitatively different from the high
temperature case. The numerical results combined with the theoretical
requirements provide the evidence for the spontaneous CP violation at
$\theta=\pi$, which is in accordance with the large $N$ prediction and
indicates that the similarity between 4d SU($N$) and 2d CP${}^{N\\!-\\!1}$
theories does not hold for $N$=2.

##  I Introduction

The $\theta$ parameter of the 4d Yang-Mills theory controls relative weights
of different topological sectors in the path integral. Despite long history,
it still remains as a challenging problem to identify the effect of the
$\theta$ parameter on the non-perturbative dynamics of the theory.

For the special value $\theta=\pi$ the Lagrangian has CP symmetry, and we can
ask whether or not the CP symmetry is spontaneously broken. In the large $N$
limit tHooft:1973alw  spontaneous CP violation at $\theta=\pi$ was
demonstrated in Refs. Witten:1980sp ; tHooft:1981bkw ; Witten:1998uka . For
finite $N$ a mixed anomaly between the CP symmetry and the $\mathbb{Z}_{N}$
center symmetry shows that the CP symmetry in the confining phase has to be
broken Gaiotto:2017yup . A similar conclusion was derived by studying
restoration of the equivalence of local observables in $\mathrm{SU}$($N$) and
$\mathrm{SU}$($N$)/$\mathbb{Z}_{N}$ gauge theories in the infinite volume
limit Kitano:2017jng . (See also Refs. Azcoiti:2003ai ; Yamazaki:2017dra ;
Wan:2018zql .) While these theoretical developments have narrowed down
possible scenarios, an explicit nonperturbative calculation is necessary to
unambiguously settle the fate of the CP symmetry at $\theta=\pi$. Any direct
numerical simulation at $\theta=\pi$, however, has been difficult due to the
notorious sign problem 111For recent related efforts towards direct
simulations, see, for example, Refs. Hirasawa:2020bnl ; Gattringer:2020mbf ..

In Ref. Kitano:2020mfk  three of the authors of the present paper studied the
vacuum energy density of the 4d $\mathrm{SU}$(2) Yang-Mills theory around
$\theta=0$ by lattice numerical simulations. The case of $\mathrm{SU}$(2)
gauge group is of particular interest since $N=2$ is farthest away from the
large $N$ limit: there is a well-known parallel between 2d CP1 and 4d
$\mathrm{SU}$(2) model 222See Refs. Yamazaki:2017ulc ; Yamazaki:2017dra  for
more precise connections between the two., and the known vacuum at
$\theta=\pi$ in the former CP1  alludes to the appearance of gapless theory in
the latter. By observing that the first two numerical coefficients in the
$\theta$ expansion obey the large $N$ scaling 333See Refs. Lucini:2001ej ;
DelDebbio:2002xa ; Bonati:2016tvi  for large $N$ scaling of the first two
coefficients in the $\theta$ expansion in the SU($N$) gauge theory. For SU(2)
theory the first coefficient in the $\theta$ expansion, the topological
susceptibility $\chi$, was estimated in Refs. deForcrand:1997esx ;
Alles:1997qe ; DeGrand:1997gu ; Lucini:2001ej ; Berg:2017tqu ., it was
inferred in Ref. Kitano:2020mfk  that the 4d $\mathrm{SU}$(2) Yang-Mills
theory at $\theta=\pi$ has spontaneous CP breaking, contrary to the naive
expectation from the 2d CP1 model.

In this work, we develop a subvolume method to explore the $\theta$ dependence
of the free energy _without any series expansion in $\theta$_ and apply to 4d
$\mathrm{SU}$(2) theory. We find that it indeed works and show an evidence of
spontaneous CP violation at $\theta=\pi$ at zero temperature, in consistency
with the results of Ref. Kitano:2020mfk . The subvolume method here is
inspired by Ref. Luscher:1978rn  and similar to that introduced in Ref.
KeithHynes:2008rw  to study 2d CP${}^{N\\!-\\!1}$ model.

##  II Subvolume Method and Lattice Set Up

In what follows, the subvolume method is described. After generating a number
of gauge configurations at $\theta=0$ and implementing the $n_{\rm APE}$ steps
of APE smearing Albanese:1987ds , the topological charge density $q(x)$ is
calculated with the five-loop improved topological charge operator
deForcrand:1997esx  on the lattice. Defining the subvolume topological charge
$Q_{\rm sub}$,

$\displaystyle Q_{\rm sub}=\displaystyle\sum_{x\in V_{\rm sub}}q(x)\ ,$ (1)

we calculate the free energy density for the subvolume as Luscher:1978rn ;
KeithHynes:2008rw ,

$\displaystyle f_{\rm sub}(\theta)=\frac{-1}{V_{\rm
sub}}\ln\frac{Z(\theta)}{Z(0)}=\frac{-1}{V_{\rm
sub}}\ln\,\langle\,\cos\left(\theta\,Q_{\rm sub}\right)\,\rangle\ ,$ (2)
$\displaystyle Z(\theta)=\int\\!\\!{\cal D}U\,e^{-S_{g}+i\theta Q_{\rm sub}}\
,$ (3)

where $U$ denotes the link variable and $S_{g}$ the gauge action. Note that
the expectation value $\langle\cdots\rangle$ is estimated on the $\theta=0$
configurations. The free energy density is then obtained as the infinite
volume limit of $f_{\rm sub}(\theta)$:

$\displaystyle f(\theta)=\lim_{V_{\rm sub}\to\infty}f_{\rm sub}(\theta)\ .$
(4)

The goal of this study is to answer whether the spontaneous CP violation does
occur at $\theta=\pi$. Thus, the order parameter should be useful and is
calculated through

$\displaystyle\frac{d\,f(\theta)}{d\,\theta}=\lim_{V_{\rm
sub}\rightarrow\infty}\frac{d\,f_{\rm sub}(\theta)}{d\,\theta}\ ,$ (5)
$\displaystyle\frac{d\,f_{\rm sub}(\theta)}{d\,\theta}=\frac{1}{V_{\rm
sub}}\frac{\left\langle Q_{\rm sub}\,\sin(\theta Q_{\rm
sub})\right\rangle}{\langle\,\cos(\theta Q_{\rm sub})\,\rangle}\,.$ (6)

$f$ and $df/d\theta$ are evaluated separately and later used to make a
consistency check.

Some cautions are in order. First, the size of the subvolume $V_{\rm sub}$
cannot be arbitrary. Obviously it has to be large enough to cover the typical
correlation length of the system, which is considered to be around
$1/(aT_{c})=9.50$ Giudice:2017dor  in the lattice unit. We thus restrict the
size of the subvolume to $V_{\rm sub}\geq 10^{4}$. Suppose that $V_{\rm sub}$
is large enough but $V_{\rm sub}\ll V_{\rm full}$, where $V_{\rm full}$
denotes the full volume. Then, the resulting $f(\theta)$ is expected to be
independent of $V_{\rm full}$, and $f_{\rm sub}$ would shows a scaling
behavior as a function of $V_{\rm sub}$. As $V_{\rm sub}$ grows and approaches
$V_{\rm full}$, $Q_{\rm sub}$ becomes close to an integer global $Q$, and
$f_{\rm sub}$ starts to converge a full volume result, in which we are not
interested as it becomes clear later. The critical goal in this method is to
identify the scaling behavior from which the infinite volume limit is
extracted.

Secondly, the observables (2) and (6) contain the expectation value
$\langle\cos(\theta Q_{\rm sub})\rangle$. Either observable becomes
incalculable when $\langle\cos(\theta Q_{\rm sub})\rangle$ fluctuates across
zero. This is the sign problem in this method and sets the $\theta$ dependent
upper limit on the size of $V_{\rm sub}$. Thus, another crucial point is
whether the scaling behavior is realized before $V_{\rm sub}$ reaches the
upper limit. We will return to this point in the discussion section.

The free energy density of 4d $\mathrm{SU}$(2) Yang-Mills theory is calculated
at high and zero temperatures below. We employ the configurations generated in
our previous work at $T=0$ Kitano:2020mfk , while the ensemble corresponding
to $T=1.2\,T_{c}$ is newly generated at the lattice coupling $\beta=1.975$,
the same as the $T=0$ case. The simulation parameters are summarized in Tab.
1.

$N_{S}^{3}\times N_{T}$ | $T/T_{c}$ | [${n_{\rm APE}}_{\rm min},\ {n_{\rm APE}}_{\rm max}$] | statistics | $a^{4}\chi\times 10^{5}$
---|---|---|---|---
$24^{3}\times 8$ | 1.21 | [25, 45] | 10,000 | 1.35(5)
$24^{3}\times 48$ | 0 | [20, 40] | 68,000 | 2.54(3)
Table 1:  The lattice parameters of the ensembles. The tree-level Symanzik
improved action Weisz:1982zw  is adopted with the periodic boundary conditions
in all four directions. The value $1/(aT_{c})=9.50$ Giudice:2017dor  gives the
value of $T/T_{c}$ in the table, where $T_{c}$ is the critical temperature at
$\theta=0$.

The topological charge density measured at each smearing step is uniformly
shifted as $q(x)\to q(x)+\epsilon$ at every configurations so that the global
topological charge $Q=\sum_{x\in V_{\rm full}}q(x)$ takes the integer closest
to the original value. The calculation of $Q_{\rm sub}$ is carried out every
five smearing steps. For the APE smearing, we take $\alpha=0.6$ in the
notation of Ref. Alexandrou:2017hqw .

Topological observables on the lattice can be distorted by topological lumps
originating from lattice artifacts. One can take away those by the smearing
procedure, but at the same time the smearing may deform physical topological
excitations, too. We studied this point in detail in Ref. Kitano:2020mfk , and
developed the procedure to restore the physical information. The procedure
consists of the extrapolation of the observables to the zero smearing limit by
fitting over a suitable interval of the smearing steps. The fit range is fixed
in advance by examining the response of the global topological charge to the
smearing as done in Ref. Kitano:2020mfk . The resulting fit ranges and the
topological susceptibilities $\chi=\langle\,Q^{2}\,\rangle/V_{\rm full}$ thus
determined using the global topological charge $Q$ are shown in Tab. 1 for
later use.

The $\theta$-dependence is explored in the range of $\theta=k\,\pi/10$ with
$k\in[1,\,20]$. Each configuration is separated by ten Hybrid Monte Carlo
(HMC) trajectories. In the following analysis, statistical errors are
estimated by the single-elimination jack-knife method with the bin size of 500
and 100 configurations for zero and high temperatures, respectively. We mainly
show the analysis of $f(\theta)$, but $df(\theta)/d\theta$ is analyzed in
parallel with similar quality.

##  III Testing the method at High Temperature

We first apply the subvolume method to the calculation of the free energy
density above $T_{c}$, where the instanton prediction,
$f(\theta)\sim\chi(1-\cos\theta)$ 'tHooft:1976fv ; Callan:1977gz ;
Gross:1980br , is believed to be valid and numerically supported for
$\mathrm{SU}$($N$) with $N\geq 3$ Bonati:2013tt ; Frison:2016vuc  as well.
Using the translational invariance, a single-sized subvolume $V_{\rm
sub}/a^{4}=l^{3}\times N_{T}$ with $l=10,\ 12,\ \cdots,\ 24$ is taken from 64
places per a configuration, and the results are averaged.

The $l$ dependence of $f_{\rm sub}$ is shown in Fig. 1.

Figure 1:  The subvolume dependence of $f_{\rm sub}(\theta)$ and the
extrapolation to the infinite volume limit.

It is found that in general the measured $l$ dependence is not constant and
the leading correction linear in $1/l$ is inevitable. The linear dependence is
seen for $\theta=\pi/2$ for $l\leq 22$, whereas it ends around $l=20$ for
$\theta=\pi$ and it is hard to determine the linear region for
$\theta=3\pi/2$. Thus, it turns out to be difficult to identify the scaling
region unambiguously especially when $\theta$ is large, and hence we give up
the precise determination. Instead, we choose three fit ranges in each
extrapolation and try to estimate the potential size of the systematic
uncertainty due to the ambiguity of the scaling region. Three fit ranges,
$l\in[12,16]$, $[14,18]$ and $[16,20]$, are examined when fitting to the
expected scaling behavior

$\displaystyle f_{\rm sub}(\theta)=f(\theta)+\frac{a^{-1}\,s(\theta)}{l}\,,$
(7)

where $s(\theta)$ denotes the surface tension of the nonzero $\theta$ domain
and $a$ the lattice spacing. All fits performed in this analysis yield
$\chi^{2}/\mbox{dof}<3$. It is interesting to see that the relative relation
$f_{\rm sub}(3\pi/2)>f_{\rm sub}(\pi)$ at small $l$ flips toward the large $l$
limit and $f(\theta)$ ends up with non-monotonic function. Since the data of
$f_{\rm sub}$ smoothly (but sometimes rapidly) connect the full volume
($l^{3}\times 6=24^{3}\times 6$) results from the above, the extrapolation
fitting the data near the full volume tends to be smaller than that fitting
the data far from the full volume. As a result, the discrepancy, i.e. the
potential size of the systematic uncertainty, turns out to be larger at larger
$\theta$.

The results thus obtained are then extrapolated to $n_{\rm APE}=0$ at each
value of $\theta$ with the fit range shown in Tab. 1. In the extrapolation,
the linear fit goes well with $\chi^{2}/\mbox{dof}<3$. The stability against
small shifts of the fit range is seen in Fig. 2.

Figure 2:  The linear extrapolation of $f(\theta)$ to $n_{\rm APE}=0$, where
$f(\theta)$ is obtained by the fit with $l\in[14,18]$.

Finally, the free energy density obtained with three fit ranges are shown in
Fig. 3 together with the full volume result (filled squares), where
$f(\theta)$ is normalized by the topological susceptibility in Tab. 1.

Figure 3:  The $\theta$ dependence of $f(\theta)$ at $T=1.2\,T_{c}$. The
results obtained with different fit ranges in Fig. 1, $l\in[12,16]$, $[14,18]$
and $[16,20]$, are shown in triangle-up, circle and triangle-down,
respectively.

The prediction from the dilute instanton gas approximation, $1-\cos(\theta)$,
is shown by the dashed curve. The function, $\theta^{2}/2$, is also shown as
the solid curve for comparison. Taking into account the uncertainty arising
from the ambiguity of the scaling region, the numerical results are consistent
with the instanton prediction. Note that non-monotonic behavior of $f(\theta)$
seems robust at high temperature but is far from obvious before the
extrapolations, as the surface tension term in Eq. (7) is monotonic.

$f(\theta)$ can also be obtained from the numerical integration of
$df(\theta)/d\theta$ as shown by the dotted curves. The agreement with those
curves supports that the two nontrivial extrapolations included in the whole
analysis do not pick up accidental fluctuations and are stable.

The result with full volume is found to well agree with the instanton
prediction. One may think that this is the simplest way to obtain $f(\theta)$.
However, we will see that it does not work at $T=0$. From the test, assuming
that the instanton prediction is valid at high temperature, we learn that the
scaling behavior of $f_{\rm sub}$ would be linear and the region showing such
a behavior starts around the dynamical length scale ($\sim 1/(aT_{c})$).

##  IV Applying to Zero Temperature

Next we apply the subvolume method to calculate the vacuum energy density.
This time the subvolume is defined by $V_{\rm sub}/a^{4}=l^{4}$ with
$l=10,12,\ \cdots,\ 24$ and taken from 512 places per configuration. The $l$
dependence of $f_{\rm sub}(\theta)$ is shown in Fig. 4 as before.

Figure 4:  The linear extrapolation of $f_{\rm sub}(\theta)$ to the infinite
volume limit.

Due to the sign problem in this method, some results at large $\theta$ and
large $l$ could not be calculated. But the available data show linear
behavior. Following the previous analysis, three fit ranges of $l\in[10,14]$,
$[12,16]$ and $[14,18]$ are taken in the fit to (7) to estimate the systematic
uncertainty. Contrary to the high temperature case, $f(\theta)$ turns out to
be stable against the variation of the fit range, and does not show any sign
of the flip, indicating monotonic behaviors of $f(\theta)$ as a function of
$\theta$.

The linear extrapolation to $n_{\rm APE}=0$ is carried out with the fit range
shown in Tab. 1, and the fit is found to work well with $\chi^{2}/$dof $<3$ as
shown in Fig. 5.

Figure 5:  The linear extrapolation of $a^{4}f(\theta)$ to $n_{\rm APE}=0$ for
the data obtained with $l\in[12,16]$.

The stability against shift of the fit range is also confirmed.

Finally, the resulting $f(\theta)$ and $df(\theta)/d\theta$ are shown in Fig.
6 together with the predictions from the large $N$ ($\theta^{2}/2$) and the
instanton calculus ($1-\cos\theta$).

Figure 6:  $f(\theta)$ (top) and $df(\theta)/d\theta$ (bottom). The results
obtained with different fit ranges in Fig. 4, $l\in[10,14]$, $[12,16]$ and
$[14,18]$, are shown in triangle-up, circle and triangle-down, respectively.

The stability of the two extrapolations during the analysis is confirmed as
$f(\theta)$ and $df(\theta)/d\theta$ well agree with the dotted curves. While
the full volume calculation works only in the vicinity of $\theta=0$, the
subvolume method succeeds to calculate, at least, to $\theta\sim\pi$. There
are crucial differences from the high temperature case. First, the different
choices of the fit range in $l$ yield consistent results, and hence the
potential systematic error from the ambiguity of the scaling region seems to
be under control. Second, $f(\theta)$ is a monotonically increasing function,
at least, to $\theta\sim\pi$, and the direct calculation of
$df(\theta)/d\theta$ clearly shows $d\,f(\theta)/d\theta|_{\theta=\pi}\neq 0$.
Since $d\,f(\theta)/d\theta=-i\langle q(x)\rangle$ is CP odd, we conclude that
CP is spontaneously broken at $\theta=\pi$ in the vacuum of the 4d
$\mathrm{SU}$(2) Yang-Mills theory 444See also Refs. Unsal:2012zj  and
Unsal:2020yeh  for analytic discussions. and that there is a phase transition
to recover the CP symmetry at some finite temperature. In other words, it is
found that the 4d SU(2) Yang-Mills theory is in the large-$N$ class unlike the
2d CP1 model 555There is a logical possibility that there is a phase
transition at some $\theta$ below $\pi$, which the subvolume method could not
detect. In that case, the CP symmetry may be left unbroken at $\theta=\pi$. In
any case, the fact that the free energy does not show the $2\pi$ periodicity
indicates that there are multiple branches in the vacuum structure as in the
case of the large $N$ limit. We thank Yuya Tanizaki for discussion on this
point..

##  V Discussion

The symmetry of $\mathrm{SU}$($N$) gauge theories indicates
$f(\theta)=f(-\theta)$ and $f(\theta)=f(\theta+2\pi)$. In the subvolume
method, $f(\theta)=f(-\theta)$ is automatic from (2) but the
$2\pi$-periodicity is not seen in $f(\theta)$ shown in Fig. 6.

The subvolume method is equivalent to modifying the value of $\theta$ inside
the subvolume. If the difference of $\theta$ is a multiple of $2\pi$ and the
calculation respects the $2\pi$-periodicity, the free energy would scale as
the surface area of the subvolume when the subvolume is large enough. The lack
of $2\pi$ periodicity in the free energy density should thus be interpreted as
the presence of a meta-stable vacuum for a fixed value of $\theta$ (except for
$\theta=\pi$ where two vacua interchanged by CP are degenerate and stable).
Thus, we expect that the meta-stable vacuum should eventually decay into the
stable one by the creation of a dynamical domain wall that attaches to the
interface.

The absence of the decay of the domain into the domain-wall in the lattice
calculation has an analog in the calculation of the static potential 666The
similar reasoning is found for 2d CP${}^{N\\!-\\!1}$ -model in
KeithHynes:2008rw .. The static potential is calculated by inserting a Wilson
loop, and should show the string breaking for configurations with light
dynamical quarks when the two test charges are distant enough. But it does not
occur, at least, within naive methods, and the resulting potential sticks to
the original branch even after passing the transition point. The probable
reason is that the overlap between the original state with two static charges
and another lower energy state with two mesons are extremely small. We infer
that the same happens to the calculation of $f(\theta)$ for $\theta>\pi$ and
the first order phase transition is missing 777We expect that the subvolume
method can capture second order transitions in principle because meta-stable
states do not exist.. It is clearly interesting to directly see the formation
of the domain wall on the lattice though it would not be straightforward.

We have mentioned the $\theta$ dependent upper limit on the size of $V_{\rm
sub}$ in sec. II. We examine the relation between the limit and
$\theta\langle|Q_{\rm sub}|\rangle$ at $T=0$. Figure 7 shows
$\theta\langle|Q_{\rm sub}|\rangle$ as a function of $1/l$, where we have used
the approximate relation $\langle|Q_{\rm sub}|\rangle=(l^{4}/V_{\rm
full})^{1/2}\langle|Q|\rangle$ and the measured value of $\langle|Q|\rangle$
at $T=0$ by ignoring the corrections to the relation of $O(1/l)$ and
$O(1/(\chi V_{\rm sub}))$.

Figure 7:  Subvolume dependence of $\theta\langle\,|Q_{\rm sub}|\,\rangle$.
The open symbols represent the calculations which succeeded while the filled
ones represent those which failed.

In the figure, the filled symbols represent the points where the calculations
are failed due to the sign problem. It is seen that the upper limit indeed
decreases with $\theta$. Numerical investigation suggests that the upper limit
on the subvolume scales as $1/\theta$.

##  VI Summary

We developed the subvolume method, which enables us to extract the $\theta$
dependence of the free energy density in 4d Yang-Mills theory not restricted
to $\theta\sim 0$. At high temperature, the method yields $\theta$ dependence
consistent with the instanton prediction, as expected, within the large
uncertainty due to the ambiguity of the scaling region. To fix this ambiguity,
it is necessary to go to larger lattices. On the other hand, at zero
temperature the sign problem arises instead, but still $f(\theta)$ could be
calculated to $\theta\sim\pi$ with the systematic uncertainty under control.
Combining the numerical result with the theoretical requirement leads to the
conclusion that the vacuum of 4d $\mathrm{SU}$(2) Yang-Mills theory undergoes
spontaneous CP violation at $\theta=\pi$ as large $N$ theory does. Although
the overlap problem prohibits the domain-wall from being formed, it is
interesting to learn that such a object actually exists in the Yang-Mills
theory Luscher:1978rn .

We have tested the stability of the results by exchanging the order of
extrapolations and obtained the consistent results with enlarged
uncertainties.

In order to promote this study to the quantitative level, it is necessary to
perform lattice simulations with larger volumes and finer lattice spacings.
Further studies will be presented in the forthcoming paper InProgress . It is
a fascinating question to ask if our method is applicable for other questions
with sign problems, such as gauge theories with finite values of the chemical
potential.

While numerical results are not accurate past $\theta\sim 3\pi/2$, there are
indications that the derivative $df(\theta)/d\theta$ decreases past
$\theta=\pi$, and becomes smaller near $\theta\sim 2\pi$. This is consistent
with the expectation Yamazaki:2017ulc  that there are two meta-stable branches
of the $\mathrm{SU}$(2) theory, each of which has $4\pi$ periodicity.

## Acknowledgments

This work is based in part on the Bridge++ code Ueda:2014rya  and is supported
in part by JSPS KAKENHI Grant-in-Aid for Scientific Research (Nos. 19H00689
[RK, NY, MY], 18K03662 [NY], 19K03820, 20H05850, 20H05860 [MY]) and MEXT
KAKENHI Grant-in-Aid for Scientific Research on Innovative Areas (No. 18H05542
[RK]). Numerical computation in this work was carried out in part on the
Oakforest-PACS and Cygnus under Multidisciplinary Cooperative Research Program
(No. 17a15) in Center for Computational Sciences, University of Tsukuba;
Fujitsu PRIMERGY CX600M1/CX1640M1 (Oakforest-PACS) in the Information
Technology Center, the University of Tokyo.

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