# Signature of the quantum critical point on the witness of the non-
Markovianity and the entanglement in spin-1/2 chain

Z. Saghafi  S. Samadi Rezaei  P. Azizi  E. Hosseini Lapasar  S. Mahdavifar
Department of Physics, University of Guilan, 41335-1914, Rasht, Iran

###### Abstract

A thermodynamic limit chain of spin-1/2 particles with $XX$ and three-spin
interactions (TSI) is considered. Using the fermionization technique, the
Hamiltonian of the chain with periodic boundary conditions is exactly
diagonalized. In the ground-state phase diagram of the chain system, a quantum
critical point separates the Luttinger liquid and chiral phases. Selecting
one-spin as an open quantum system, the rest of the spins play the role of its
environment. By choosing different initial states, we studied the dynamics of
the entanglement between the open quantum system and its environment. In the
initial state, the state of the open quantum system is a superposition of up
and down states and the environment is polarized. The revival of the
entanglement is observed in the whole range of interactions as an indicator of
non-Markovian dynamics. Our exact results revealed that non-Markovian behavior
is independent of the initial state. By tuning the initial state, the open
quantum system will be completely entangled with the environment at a special
time called entanglement-time, $t_{E}$. The value of the mentioned
entanglement-time decreased by increasing TSI. The signature of a quantum
critical point is confirmed by investigating the long-time average of the
entanglement and using the trace-distance as a witness of non-Markovianity.

##  I Introduction

Open quantum system dynamics has raised a great deal of interest both
theoretically and experimentally Eberly07 ; Yu09 ; Werlang09 ; Mazzola10 ;
Amico04 ; Horodecki09 . Using the theory of open quantum systems, the
dynamical behavior of the system which interacts with its environment is
classified into two categories: Markovian and non-Markovian Breuer16 . The
Markovian process involves the flow of information from the system to the
environment. On the other hand, the process by which part of the information
is returned from the environment to the system is called non-Markovian
dynamics. Describing the memory effects in the non-Markovian process is a
fundamental subject in the theory of open systems Breuer2 . Investigating
dynamics of the open quantum system was initially allocated to the Markovian
behavior, it has recently expanded to non-Markovian behaviors Piilo08 ;
Madsen11 ; Liu11 ; Huelga12 ; Barnes12 ; Addis14 ; Orieux15 . Systems that are
in contact with different non-Markovian environments show the revival of
entanglementBellomo07 ; Franco12 ; Franco13 . It is essential to show the
process of the revival of entanglement to introduce a general measurement of a
non-Markovianity degree in open quantum systems Breuer09 . Note that in
Markovian dynamics, the concurrence decreases exponentially Santos06 .
There are several ways to identify non-Markovianity via witnesses. A witness
of non-Markovianity is a quantity which will be vanished for all Markovian
dynamics Chru12 ; Chru14 . We know that a non-zero value of a witness of non-
Markovianity shows a non-Markovian dynamics. The witnesses of non-Markovianity
can be classified into two categories: the witnesses based on monotonicity
under completely positive maps and the witnesses based on monotonicity under
local completely positive maps Rivas14 . The first category includes the
following methods: Trace distance and the BLP quantifier Breuer09 ; Laine10 ,
Fidelity Uhlmann76 ; Jozsa94 , Quantum relative entropies Lindblad75 , Quantum
Fisher information Holevo82 ; Braunstein94 , Capacity measure Bylicka13 ,
Bloch volume measure Lorenzo13  and the second one contains Entanglement
Plenio7 , Quantum mutual information Luo12 , and Quantum discord Zurek00 ;
Ollivier1 ; Modi12 .

Theoretically, in a recent work, the dynamics of a qubit coupled to a spin
chain environment has been studied Apollaro11 . The environment is described
by a XY model in the transverse magnetic field. The dynamics of the qubit is
Markovian at a special point. Two regions that are separated by this point are
non-Markovian and lead to two completely different dynamical behaviors. It has
been shown that the contribution of energy density is responsible for the non-
Markovian effects. This result in an infinite environment is also correct.
From an experimental point of view, a non-Markovianity-assisted high-fidelity
refined Deutsh-Jozsa algorithm is implemented with a solid spin in a
diamondDong18 . Especially, a non-Markovian quantum process is observed by
measuring the non-Markovianity of the spin system. The control of the degree
of non-Markovianity in the dynamics of a nitrogen-vacancy center electron spin
is also demonstrated experimentally Hass18 . They have shown that, by changing
the population of the nitrogen spin, the non-Markovianity of the electron
spin’s dynamics is tuneable. In addition, using a randomized set of central
radio-frequency fields, a non-Markovian environment for a single nuclear
magnetic resonance qubit is effectively realized Khurana19 . The study of the
role of different interactions in inducing non-Morkovianity has been going on
in recent yearsLorenzo11 ; Znidaric11 ; Haikka12 ; Dubertrand18 ; Saghafi19-1
; Saghafi19-2 ; Motamedifar19 ; Chakraborty19 ; Roy19 ; Matern19 ; Roos19 ;
Kyaw20 .

Recently, systems involving multiple interactions, such as three-spin
interaction, four-spin interaction, etc., are significant by various parts of
physics Titvinidze03 ; Lou04 ; Lou06 ; Taras08 ; Tahvili17 ; Shadman18 ;
Moshfegh19 ; Yin20 ; Hu20 . A wide range of spin-1/2 Hamiltonians can be
created in different configurations of an optical lattice Kuklov03 . One type
of multiple-spin interaction is a three-spin interaction (TSI) which can be
represented by a triangular configuration Duan03 . TSI has been proposed with
various Hamiltonians, including the following

$\displaystyle{\cal{H}}_{TSI}^{(1)}$ $\displaystyle=$ $\displaystyle
J^{\prime}\sum_{j=1}^{N}(\vec{S}_{j}~{}.~{}\vec{S}_{j+2})_{xy}~{}{S}_{j+1}^{z},$
$\displaystyle{\cal{H}}_{TSI}^{(2)}$ $\displaystyle=$ $\displaystyle
J^{\prime}\sum_{j=1}^{N}[(\vec{S}_{j}~{}\times~{}\vec{S}_{j+2})~{}.~{}\hat{k}]~{}{S}_{j+1}^{z}.$
(1)

Thermodynamic properties and quantum phase transition for both interactions
with the Heisenberg spin-1/2 XX model have been investigated Titvinidze03 ;
Lou04 . There is a significant difference between the results of these two
models. Some of the differences express: The model that containing
${\cal{H}}_{TSI}^{(1)}$, shows that it has spontaneous magnetization in its
ground state, while the second model has zero magnetization in its ground
state. In the study of the specific heat as a function of $T/J$, it is
observed that the first model has a two-peak structure, while the second model
has a single-peak structure.

The dynamics of entanglement and dynamical transition from the Markovian to
the non-Markovian regime for a one-dimensional spin-1/2 XX model with
${\cal{H}}_{TSI}^{(1)}$ has been studied Mahmoudi17 . In this work, we focus
on the dynamical behavior for the system involving ${\cal{H}}_{TSI}^{(2)}$ to
explore the interplay between the critical phenomena, quantum correlations,
and non-Markovianity. One spin is selected from the chain and it plays the
role of the open system, while the rest of the chain constitutes its
environment. A class of initial states is considered, where the environment is
polarized and the one-spin system is a superposition of up and down states.
Using the fermionization approach, Hamiltonian is diagonalized and exact
expression for the time-dependent entanglement between the one-spin system and
its environment is obtained. Revivals phenomenon of entanglement is showed
that the dynamics is non-Markovian in presence of the TSI interaction. A
similar result is obtained by calculating the trace distance. For initial
states where one-spin is also polarized, the open quantum system will be
completely entangled with the environment at a special time called
entanglement-time, $t_{E}$. Entanglement time decreases by increasing TSI. On
the other hand, we also calculate the long-time average of the entanglement
and will find a clear signature of the quantum critical point.

The paper is structured as follows. In section II, first, the model is
introduced, then using the fermionization technique, the Hamiltonian is
diagonalized and analytical results are given. In section III, the dynamics of
entanglement between one-spin and its environment is studied. Results on the
trace distance are reported in section IV. Finally, we conclude all of our
results in section V.

##  II THE MODEL

The Hamiltonian of a spin-1/2 XX model with ${\cal{H}}_{TSI}^{(2)}$ is given
by

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle-J\sum_{j=1}^{N}({S}_{j}^{x}{S}_{j+1}^{x}+{S}_{j}^{y}{S}_{j+1}^{y})$
(2) $\displaystyle-$ $\displaystyle
J^{\prime}\sum_{j=1}^{N}({S}_{j}^{x}{S}_{j+2}^{y}-{S}_{j}^{y}{S}_{j+2}^{x})~{}{S}_{j+1}^{z},$

where $S_{j}$ is the spin-1/2 operator on the $j$-th site, $J$ and
$J^{\prime}$ are respectively the exchange coupling between the spins on the
nearest neighbors sites and on the three nearest neighbors sites.
$J^{\prime}=0$ represents the isotropic XX model. The TSI breaks
$\pi/2$-rotation symmetry along the $z$-axis. Since the dynamics of the system
is governed by Hamiltonian, so we expect remarkable changes in the dynamics
that will occur.

The Hamiltonian can be exactly diagonalized by the fermionization approach. In
the first step, using the Jordan-Wigner transformation

$\displaystyle{S}_{j}^{+}$ $\displaystyle=$ $\displaystyle
a_{j}^{\dagger}\exp(i\pi\sum_{l<j}a_{l}^{\dagger}a_{l}),$ (3)
$\displaystyle{S}_{j}^{-}$ $\displaystyle=$ $\displaystyle
a_{j}\exp(-i\pi\sum_{l<j}a_{l}^{\dagger}a_{l}),$ (4) $\displaystyle\
{S}_{j}^{z}$ $\displaystyle=$ $\displaystyle
a_{j}^{\dagger}a_{j}-\frac{1}{2},$ (5)

the Hamiltonian is mapped onto a 1D model of noninteracting spinless fermions

$\displaystyle{\cal H}_{f}$ $\displaystyle=$
$\displaystyle\frac{-J}{2}\sum_{j=1}^{N}(a^{{\dagger}}_{j}a_{j+1}+a^{{\dagger}}_{j+1}a_{j})$
(6) $\displaystyle+$ $\displaystyle
i\frac{J^{\prime}}{2}\sum_{j=1}^{N}(a^{{\dagger}}_{j}a_{j+2}-a^{{\dagger}}_{j+2}a_{j}).$

In the next step, performing a Fourier transformation into the momentum space
as
$a^{{\dagger}}_{j}=\frac{1}{\sqrt{N}}\sum^{N}_{j=1}e^{ikj}a^{{\dagger}}_{k}$,
the diagonalized Hamiltonian is given by

$\displaystyle{\cal H}_{f}=\sum_{k}\varepsilon(k)a_{k}^{\dagger}a_{k},$ (7)

where $\varepsilon(k)$ is the dispersion relation

$\displaystyle\varepsilon(k)$ $\displaystyle=$
$\displaystyle-J(\cos(k)-\frac{\alpha}{2}\sin(2k)),$ (8)

with $\alpha=\frac{J^{\prime}}{J}$. The ground state phase diagram is well
knownLou04 . System is in the gapless Luttinger liquid phase in the region
$\alpha\leq\alpha_{c}=1$ and shows chiral ordering with gapless excitations in
the region $\alpha>\alpha_{c}=1$.

Here, we are interested to study the dynamical behavior of a one-spin open
quantum system and understanding how the TSI changes the dynamical behavior of
the system. To study the dynamics and measuring the degree of non-
Markovianity, we use the trace distance, hence we need to calculate the
reduced density matrix. In the first step, it is necessary to determine the
initial state of the system at $t=0$. Then by using the time evolution
operator ($\hbar=1$ is considered) as

$\displaystyle U(t)=e^{-it\sum_{k}\varepsilon(k)a_{k}^{\dagger}a_{k}/\hbar},$
(9)

the physical state of the system at any time $t$ is obtained. The trace
distance which gives a measure of the distinguishability between two initial
quantum states ($\rho_{1}$ and $\rho_{2}$) can be expressed asBreuer09

$\displaystyle
D(\rho_{1},\rho_{2})=\frac{1}{2}\mathrm{tr}|\rho_{1}-\rho_{2}|,$ (10)

where
$|\rho_{1}-\rho_{2}|=\sqrt{(\rho_{1}-\rho_{2})^{\dagger}(\rho_{1}-\rho_{2})}$.
At any time, if the information is returned from the environment to the
system, the trace distance will increase; therefore, it is obviously related
to existence of the non-Markovianity. The witness of the non-Markovianity,
${\cal{N}}$, is defined as

$\displaystyle{\cal{N}}=max\int_{\sigma>0}\sigma(t,\rho_{1,2}(0))dt,$ (11)

where

$\displaystyle\sigma(t,\rho_{1,2}(0))=\frac{d}{dt}D(\rho_{1}(t),\rho_{2}(t)),$
(12)

is the rate of change of the trace distance at time $t$. This computational
method is common for calculating the non-Markovianity Breuer09 ; Lorenzo13-1 ;
Mahmoudi17 ; Saghafi19-1 . It is used in different models as: the damped
Janes-Cummings model Breuer09 , the spin-1/2 transverse field XY Heisenberg
chain model Saghafi19-1 , the extended cluster spin-1/2 XX chain model
Mahmoudi17  and the anisotropic spin-1/2 XY chain model Lorenzo13-1 . It is
important to note that if the trace distance increases, we can use the rate of
change for calculating the witness of non-Markovianity. Otherwise, we cannot
use the rate of change of the trace distance for calculating the witness of
non-Markovianity.

##  III One tangle

Figure 1:  The symbolic form of a one-spin open quantum system and its
environment.

In this section, we select one-spin of the chain as an open quantum system,
while the rest of the chain plays the role of the environment. A symbolic form
of the system is shown in Fig. 1. In fact, we concentrate on the entanglement
between a single spin located at site $m$ and the rest of the chain system.
For this purpose, we consider an initial state where the environment is
polarized but the state of the one-spin system is a superposition of up
($|\uparrow\rangle$) and down ($|\downarrow\rangle$) states as

$\displaystyle|\psi(t=0)\rangle$ $\displaystyle=$
$\displaystyle(\cos(\phi)|\uparrow\rangle+\sin(\phi)|\downarrow\rangle)_{S}$
(13) $\displaystyle\otimes$
$\displaystyle|\downarrow\downarrow.....\downarrow\rangle_{E},$

which is equivalent to

$\displaystyle|\psi(t=0)\rangle$ $\displaystyle=$
$\displaystyle(\cos(\phi)a_{m}^{{\dagger}}+\sin(\phi))~{}|0\rangle.$ (14)

in the fermionic form. $\phi$ is a phase factor and $|0\rangle$ denotes the
vacuum state. Applying the time evolution unitary operator, the time-dependent
physical state of the system is obtained as

$\displaystyle|\psi(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}(\sum_{k}e^{i(km-\varepsilon(k)t)}~{}\cos(\phi)a_{k}^{{\dagger}}$
(15) $\displaystyle+$ $\displaystyle\sqrt{N}\sin(\phi))~{}|0\rangle.$

The one-spin reduced density matrix, $\rho$, is defined as

$\displaystyle\rho=\left(\begin{array}[]{cc}\frac{1}{2}+\langle{S}_{m}^{z}\rangle&\langle{S}_{m}^{-}\rangle\\\
\langle{S}_{m}^{+}\rangle&\frac{1}{2}-\langle{S}_{m}^{z}\rangle\\\
\end{array}\right),$ (18)

where $\langle...\rangle$ denotes the expectation value on the time-dependent
physical state. Using the time-dependent state and fermionic operators, we
calculated the density matrix as:

$\displaystyle\rho=\left(\begin{array}[]{cc}A&B\\\ B^{*}&1-A\\\
\end{array}\right),$ (21)

where

$\displaystyle A(\phi,t)$ $\displaystyle=$
$\displaystyle{\cos}^{2}(\phi)~{}({f}^{2}(t)+{g}^{2}(t)),$ $\displaystyle
B(\phi,t)$ $\displaystyle=$
$\displaystyle{\sin}(\phi)~{}{\cos}(\phi)~{}({f}(t)-i{g}(t)),$ (22)

and

$\displaystyle f(t)$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{k}\cos(\varepsilon(k)t),$ $\displaystyle g(t)$
$\displaystyle=$ $\displaystyle\frac{1}{N}\sum_{k}\sin(\varepsilon(k)t).$ (23)

In the following, we concentrate on the entanglement between a single spin and
the rest of the chain. In this scenario, the amount of entanglement can be
measured by the so-called one-tangle Amico04 . It is defined as

$\displaystyle\tau$ $\displaystyle=$ $\displaystyle 4\det\rho,$
$\displaystyle=$ $\displaystyle
4~{}{\cos}^{4}(\phi)~{}({f}^{2}(t)+{g}^{2}(t))[1-({f}^{2}(t)+{g}^{2}(t))].$

At $t=0$, the one-spin system is not entangled with the rest of the chain
independent of the phase factor $\phi$. By passing the time, the non-zero
value of the entanglement is expected since, at a short time, low-lying
excited states contribute to the dynamics of the system. On the other hand,
the phase factor $\phi$ does not change the dynamical behavior of the system
and only plays the role of a positive constant.

Figure 2: (a) The dynamical behavior of the entanglement between the one-spin
open quantum system and its environment. (b) The entanglement-time with
respect to the TSI. (c) The long-time average of the entanglement with respect
to the TSI.

The results presented in Fig. 2 (a) provide information about the dynamical
behavior of the entanglement between the one-spin system and the rest of the
chain for three different values of the TSI. No entanglement is observed at
the initial time $t=0$. There is an upward trend at very short times
independent of the value of TSI. The entanglement increases and interestingly
the one-spin system will be completely entangled with its environment at a
special time $t_{E}$. It is noticeable that as soon as the time passes from
the entanglement-time, $t_{E}$, the entanglement decreases and reaches zero
for the first time. The mentioned dynamical behavior is repeated periodically
by passing time which is known as the revival of the entanglement. In fact,
such a revival is due to the reaction of the environment. Feedback of the
entanglement from the environment to the open quantum system is one of the
special characterizations of the non-Markovianity. We have to mention that the
entanglement between system and environment declines moderately and decays at
longer times. To find a deeper insight into the nature of the dynamical
behavior of entanglement between the one-spin system and its environment, we
have also investigated the entanglement-time $t_{E}$ with respect to the TSI.
Results are presented in Fig. 2 (b). The entanglement-time shows monotonic
decreasing behavior with respect to the TSI, which can be regarded as an
increasing effect of the TSI on the velocity of the entanglement transfer.

Figure 3: The witness of non-Markovianity ${N}$ as a function of TSI. Inset
shows results in the quantum critical region.

In addition, we have calculated the long-time average of the entanglement
defined as

$\displaystyle\tau_{av}=\lim_{T\longrightarrow\infty}\frac{1}{T}\int_{0}^{T}\tau(t)~{}dt.$
(25)

Results are presented in Fig. 2 (c) for different values of TSI. The signature
of the quantum critical point is also clearly seen in the behavior of
$\tau_{av}$. Three different regions are observed. A quantum critical region,
where the TSI is very close to the quantum critical point $\alpha_{c}=1$ and
two regions which are outside of the quantum critical region are distinguished
with values of $\alpha<\alpha_{c}$ and $\alpha>\alpha_{c}$. By increasing the
TSI from zero, the long-time average of the entanglement decreases slowly up
to the edge of the quantum critical region. As soon as the TSI enters into the
quantum critical region, $\tau_{av}$ increases rapidly and will be maximized
at the second edge of the quantum critical region. In the region,
$\alpha>\alpha_{c}$, it shows monotonic decreasing behavior.

##  IV MEASURE OF NON-MARKOVIANITY

In this section, we evaluate the trace distance as a measure of non-
Markovianity: we can anticipate similar results as those found for the
entanglement. In order to calculate the witness of non-Markovianity,
$\rho_{1}$ and $\rho_{2}$ must be determined. The two initial states are
selected as

$\displaystyle|\psi_{1}(t=0)\rangle$ $\displaystyle=$
$\displaystyle(\cos(\phi_{1})|\uparrow\rangle+\sin(\phi_{1})|\downarrow\rangle)_{S}\otimes(|\downarrow\downarrow.....\downarrow\rangle)_{E},$
$\displaystyle|\psi_{2}(t=0)\rangle$ $\displaystyle=$
$\displaystyle(\cos(\phi_{2})|\uparrow\rangle+\sin(\phi_{2})|\downarrow\rangle)_{S}\otimes(|\downarrow\downarrow.....\downarrow\rangle)_{E}.$

Using the time-dependent states and fermionic form, we calculated the
$\rho_{1}-\rho_{2}$ and finally the trace distance is obtained as

$\displaystyle D(\rho_{1},\rho_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathrm{tr}|\rho_{1}-\rho_{2}|,$ (27)
$\displaystyle=$ $\displaystyle\sqrt{A^{\prime 2}+|B^{\prime}|^{2}}$

where

$\displaystyle A^{\prime}$ $\displaystyle=$ $\displaystyle
A(\phi_{1},t)-A(\phi_{2},t),$ $\displaystyle B^{\prime}$ $\displaystyle=$
$\displaystyle B(\phi_{1},t)-B(\phi_{2},t).$ (28)

We want to determine whether the dynamical behavior of the one-spin system is
Markovian or non-Markovian. To reach this goal, the witness of non-
Markovianity ${\cal{N}}$ as a function of the TSI strength is calculated. It
should be noted that the time integral is extended over all intervals where
trace distance ($D(\rho_{1},\rho_{2})$) rises by increasing time. Choosing
different values of phases $\phi_{1}$ and $\phi_{2}$, we are looking to find
the maximum distance $|\rho_{1}-\rho_{2}|$, which is called the optimal state
pair. We have to mention that the optimal state pair leads to the maximal
possible backflow from the environment during the time evolution of states.
Results of the witness of non-Markovianity ${\cal{N}}$ are shown in Fig. 3. In
fact, the integral in Eq. (11) is calculated numerically for the time interval
from $0$ to $1000$ and a time step of $0.01$. It is clearly seen that the
dynamics of the system is non-Markovian in the whole range of considered two-
point and cluster interactions in complete agreement with the entanglement
results. In fact, by increasing TSI from zero, the degree of non-Markovianity
${\cal{N}}$ shows decreasing behavior. As soon as the system placed in the
quantum critical region, different behavior is signaled. As is seen in the
inset of Fig. 3, a cusp as a signature of the quantum critical point is
observed at $\alpha_{c}\sim 1$.

Figure 4: The Energy spectrum as a function of the momentum for different
values of the TSI as: (a) In the absence of the TSI, $\alpha=0$, (b) In the
presence of the TSI but less than quantum critical point,
$\alpha=0.5<\alpha_{c}$, (c) In the presence of the TSI equal to the quantum
critical point $\alpha=1.0=\alpha_{c}$, (d) In the presence of the TSI but
larger than quantum critical point, $\alpha=1.5>\alpha_{c}$.

It is known that the most important quantity in determining the dynamical
behavior of the fermion systems is the energy spectrum $\varepsilon(k)$. One
should note that the ground state of the chain model corresponds to the
configuration where all states with $\varepsilon(k)<0$ are filled and
$\varepsilon(k)>0$ are empty. In the region $\alpha<\alpha_{c}=1$, there is
two-Fermi points at $k_{F}=\pm\frac{\pi}{2}$. Two additional Fermi points
appear in the region $\alpha>\alpha_{c}=1$ at
$k_{F}^{1}=\arcsin(\frac{1}{\alpha})$ and
$k_{F}^{2}=\pi-\arcsin(\frac{1}{\alpha})$. Since
$\left\langle\psi_{GS}|\psi(t)\right\rangle=0$, the ground state energy has no
effect in the dynamical behavior of the system (this is not general and
depends on the initial state). Thus, only excited states of the Hamiltonian
play important role in the dynamical behavior of the system. In the region
$\alpha\leq\alpha_{c}=1$, the ground state consists of all quasi-particles
with momentum $-\frac{\pi}{2}<k<\frac{\pi}{2}$. But in the region
$\alpha>\alpha_{c}=1$, it consists of quasi-particles with momentum
$-\frac{\pi}{2}<k<k_{F}^{1}$ and $\frac{\pi}{2}<k<k_{F}^{2}$. Low-lying
excited states of the chain model correspond to the configurations where in
addition to filled states with $\varepsilon(k)<0$, quasi-particles with
momentum value close to the Fermi points are also created.

Here, we focus on the dispersion relation Eq. (8) for different values of the
TSI. Results are presented in Fig. 4. In absence of the TSI, the asymmetry
between quasi-particles with positive and negative momentum is seen in Fig.
(8) (a). But, in presence of TSI (Fig. 4 (b-d)), the mentioned symmetry is
broken and at first, the sub-branch with positive momentum
($k>k_{F}=\frac{\pi}{2}$) will be filled. As soon as the TSI crosses the
quantum critical point, two additional Fermi points $k_{F}^{1(2)}$ are
induced. In this case, quasi-particles with momentum
$k_{F}^{1}<k<\frac{\pi}{2}$ will be also created in low-lying excited states
and their impact as a signal shows itself in quantities such as the long-time
average of the entanglement and the measure of the non-Markovianity.

##  V Conclusion

In this paper, we have studied the dynamical behavior of a one-spin open
quantum system in a one-dimensional spin-1/2 isotropic XX Heisenberg model
with TSI. The TSI breaks $\pi/2$-rotation along $Z$-axis symmetry. We have
shown that notable changes in the dynamics of the systems have occurred since
the dynamics of the system is governed by Hamiltonian. Calculating the witness
of non-Markovianity and the entanglement between the one-spin open quantum
system and its environment, we showed that the one-spin open quantum system
experiences no dynamical transition, since the one-way flow of information
from the system to the environment was never observed.

In the short-time dynamics of the entanglement between the one-spin and
environment, we found a special time where the open quantum system becomes
completely entangled with its environment. The entanglement-time shows
decreasing behavior with respect to the TSI. We also calculated the long-time
average of the entanglement and a clear signature of the quantum critical
point observed. The mentioned signal of the quantum critical point observed
also in the dynamical behavior of the measure of non-Markovianity.

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#  Almost relevant corrections for direct measurements of electron’s g-factor

Benjamin Koch<EMAIL_ADDRESS>Institut für Theoretische Physik, Technische
Universität Wien, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria
Pontificia Universidad Católica de Chile
Instituto de Física, Pontificia Universidad Católica de Chile,
Casilla 306, Santiago, Chile  Felipe Asenjo  Facultad de Ingeniería y
Ciencias, Universidad Adolfo Ibáñez, Santiago 7491169, Chile.  Sergio Hojman
Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo
Ibáñez, Santiago 7491169, Chile  Departamento de Física, Facultad de Ciencias,
Universidad de Chile, Santiago 7800003, Chile.  Centro de Recursos Educativos
Avanzados, CREA, Santiago 7500018, Chile.

###### Abstract

We revisit the observable used for the direct measurements of the electron’s
$g$ factor. This is done by considering the sub-leading effects of the large
magnetic background field and virtual Standard Model processes. We find
substantial corrections to the Landau levels of the electron. Implications for
the observed magnetic moment and the tension between direct and indirect
measurement are discussed.

Youtube: paperpitch

###### Contents

  1. I Introduction
    1. I.1 Electron $g-2$ in a nut-shell
    2. I.2 Idealized experimental setting
    3. I.3 Corrections to expect
  2. II Corrections from effective Lagrangian approach
    1. II.1 Hamiltonian from effective Lagrangian
    2. II.2 Between theory and experiment
    3. II.3 Further gauge invariant bilinear corrections
      1. II.3.1 $\sim 1$ contributions
      2. II.3.2 $\sim\gamma^{\mu}$ contributions
      3. II.3.3 $\sim\sigma^{\mu\nu}$ contributions
    4. II.4 Parity violating corrections
      1. II.4.1 $\sim\gamma^{5}$ contributions
      2. II.4.2 $\sim\gamma^{5}\gamma^{\mu}$ contributions
      3. II.4.3 Considering the electric field
    5. II.5 Combined corrections
    6. II.6 Anomalous magnetic moment
  3. III Discussion
    1. III.1 Indirect $g-2$ measurements
    2. III.2 Benchmark without parity violation
    3. III.3 Benchmark with parity violation
    4. III.4 What happened?
  4. IV Conclusion
  5. A Appendix
    1. A.1  Explicit solutions
    2. A.2 QED background corrections from Suszuki approach
    3. A.3 Parity violating SM processes

##  I Introduction

###  I.1 Electron $g-2$ in a nut-shell

Quantum electrodynamics (QED) is one of the cornerstones of our understanding
of the electron, its properties, and interactions Feynman:1949zx ;
Feynman:1950ir . This theory allows performing the interplay between
theoretical prediction and experimental confirmation to unprecedented
precision. The anomalous magnetic moment of the electron plays a pivotal role
in this ongoing success.

Luttinger and Schwinger made the first theoretical calculations suggesting
that the magnetic moment of the electron should differ from two
Schwinger:1948iu ; Luttinger:1948zz . Over the decades, these calculations
became more and more precise Aoyama:2007mn ; Aoyama:2012wj ; Aoyama:2017uqe ,
such that they currently include even tenth order in perturbation theory (for
a review see Aoyama:2019ryr ). For recent developments concerning the
anomalous magnetic moment of the muon see Aoyama:2020ynm .

Such an amazing effort on the theoretical side only makes sense, if the
experimental observations have a precision that is comparable to the
theoretical uncertainty. Since the first experimental observation of $g\neq 2$
Foley:1948zz ; Kusch:1948mvb , the measurements have been continuously
improved. Currently the most precise direct measurement of the electron’s $g$
factor go down to 13 significant digits Hanneke:2010au ; Hanneke:2008tm ;
Gabrielse:2006gg . These experiments, which will be discussed in more detail
below, use Penning traps. A comparison between theory and experiment at such
precision does not only allow for better testing of QED, but it also is an
excellent tool to test the Standard Model of particle physics (SM) and to to
constrain models that go beyond the Standard Model of particle physics (BSM)
Broggio:2014mna ; Essig:2013vha ; Endo:2012hp ; Giudice:2012ms ; Kahn:2016vjr
; Raggi:2015yfk .

These direct measurements can also be complemented by measurements of the fine
structure constant $\alpha$, which, by virtue of the theoretical predictions,
allow inferring the value of $g$ Bouchendira:2010es ; AmelinoCamelia:2010me .
Three years ago, such a comparison showed a significant tension of $2.4$
standard deviations between the directly measured and the inferred magnetic
moment Parker:2018vye . This triggered new hope for the search of physics
beyond the Standard Model. Currently, the most prominent examples of these
models, which are capable of resolving the tension, are AmelinoCamelia:2008qg
; Crivellin:2018qmi ; Davoudiasl:2018fbb ; Badziak:2019gaf ; Bauer:2019gfk ;
Endo:2019bcj ; Liu:2018xkx ; Hiller:2019mou ; Han:2018znu ; Dutta:2018fge .
However, last year a new experiment gave a discrepancy pointing in the
opposite direction, but with a smaller absolute value, such that one can not
speak of a significant tension any more Morel:2020dww .

In this paper, we revisit the observable which is used to determine the
electron’s anomalous magnetic moment in the most precise direct measurements
Hanneke:2010au ; Hanneke:2008tm ; Gabrielse:2006gg . The paper is organized as
follows. In the following subsection I.2, we give an idealized and thus
simplified summary of the key observable. We then motivate the study by
discussing and comparing the orders of magnitude involved in this problem I.3.
Section II uses an effective Lagrangian approach. It contains four parts in
which the path from Lagrangian corrections to changes in the observable is
described. In section III, these findings are then applied and discussed in
the context of a comparison between direct and indirect measurements of
$(g-2)$. A summary and conclusion is given in section IV. While preparing this
paper, we noticed the preprint Kim:2021ydg , which also considers the effect
of a finite magnetic field. It is, however, mainly discussing the implications
for the anomalous magnetic moment of the muon.

###  I.2 Idealized experimental setting

The motion of an electron in a real Penning trap has four eigenfrequencies,
which are the spin-, cyclotron-, axial-, and magnetron-frequency. For most
parts of this paper, it is, however, sufficient to consider the idealized
case, where electric field and magnetron effects are absent.

The magnetic field strength, used in this experiment, can be inferred from the
measured cyclotron frequency of about $\nu_{c}=\omega/(2\pi)\approx 149~{}$GHz
Hanneke:2010au

$|B|=\frac{2\pi\nu_{c}m}{e_{0}}\approx 5.3~{}T.$ (1)

Here, $e_{0}$ is the absolute value of the electron charge. The energy-
eigenvalues of this system are Brown:1985rh

$E_{n}^{\pm}=\frac{1}{2}\left(2n+1\pm
1\right)h\nu_{c}\pm\frac{a}{2}h\nu_{c}-\frac{1}{8}\frac{h^{2}\nu_{c}^{2}}{mc^{2}}\left(2n+1\pm
1\right)^{2},$ (2)

where the last term is the leading relativistic correction. The parameter of
interest in (2) is the anomalous magnetic moment

$a\equiv\frac{(g-2)}{2}.$ (3)

Measurements of transitions between these energies are used to determine the
value of $a$. It turns out that a particular ratio of such transition energies
cancels several effects of imperfections and is thus particularly suited for
this purpose Hanneke:2010au ; Hanneke:2008tm ; Gabrielse:2006gg

$a_{d}=\frac{E_{0}^{+}-E_{1}^{-}}{E_{1}^{+}-E_{0}^{+}+3h\delta/2}.$ (4)

The subscript of $a_{d}$ refers to “direct” measurement.

###  I.3 Corrections to expect

Quantum field theoretical descriptions are effective many-particle theories,
that consider excitations of all fields $(\psi,A_{\mu},\dots)$. This leads to
corrections to the quantum mechanical Dirac equation

$i\hbar\psi=H_{D}\psi.$ (5)

When discussing these corrections for the above experimental setting we have
three small dimensionless numbers at our disposal, which can be used to obtain
good estimates. In order to compare these quantities, we introduce a common
small expansion parameter $\epsilon$.

  * •

First, for the non-vanishing magnetic field in (1), there is

$\delta_{c}\equiv\frac{h\nu_{c}}{mc^{2}}\sim 10^{-9}\sim\epsilon^{4}.$ (6)

The Foldy-Wouthuysen transformation Foldy:1949wa  is essentially based on an
expansion in (6) since relativistic energy corrections to the rest mass are
given as series of the form $1+\delta_{c}+\delta_{c}^{2}+\delta_{c}^{3}\dots$.

  * •

Second, there is the fine-structure constant

$\alpha\sim 2\pi a\sim 0.007\sim\epsilon.$ (7)

Perturbation theory in QED is an expansion in (7). The famous anomalous
magnetic moment contributes to the energies with a term $\alpha\delta_{c}$.

  * •

The third dimensionless parameter is the ratio between the electron mass $m$
and and the mass of the $W$-boson

$\delta_{W}=\frac{m}{m_{W}}\sim 10^{-5}\sim\epsilon^{2.5}.$ (8)

The leading contribution to the Hamiltonian of these corrections appears at
quadratic order in $\delta_{W}$ and is induced by loop corrections. It is
suppressed by $\alpha\delta_{c}\delta_{W}^{2}$ or by $\alpha\delta_{W}^{2}$,
as shown in (72).

The above expansions are present in energy formulas like (2). To see the
relevance of the different terms in this expansion it is instructive to
compare them to the experimental precision of Hanneke:2008tm . In this
experiment “$a$” was determined by measuring a dimensionless ratio
$\sim(\Delta E/(mc^{2}\delta_{c}))$ between different transition energies
($\Delta E,\delta_{c}$) of the quantum mechanical system to a precision of
$2.8\cdot 10^{-13}$. This comparison of magnitudes is shown in figure 1.

Figure 1:  Orders of magnitude of the expansion parameters (6, 7, 8). The
precision reached by current experiments is indicated by the green bar. The
theoretical expansion order used to interpret the experimental results (2, 4)
is indicated by the red bar. The energies in this sketch are normalised to the
dimensionless ratio $\sim(\Delta E/(mc^{2}\delta_{c}))$.

Remembering that the interpretation of the experimental measurement considers
dimensionless corrections up to order $\sim\delta_{c}$, figure 1 gives some
expected and some surprising insight. One sees that sub-sub-leading order
relativistic corrections ($\sim\delta_{c}^{2}$), or loop corrections with
three external photon lines ($\alpha\delta_{c}^{2}$) are well beyond
experimental reach. On the other hand, figure 1 shows that loop corrections
with two external photon lines $\sim\alpha\delta_{c}$ and loop corrections
involving a weak interaction and a single external photon line
$\sim\alpha\delta_{W}$ can be expected to give relevant corrections to the
interpretation of experimental results. From the point of view of a combined
expansion in (6, 7, 8) there is no reason, why such terms should be absent.

We will discuss possible corrections to the energies and to the observable (2,
4) up to order $\alpha\delta_{c}^{2}$ in a systematic way. We address the
issue of the seemingly relevant corrections $\sim\alpha\delta_{c}$ and
$\alpha\delta_{W}^{2}$.

Note that there are further dimensionless scales. In particular, if one would
consider axial motion with the axial frequency $\nu_{z}$, there would be a
fourth small dimensionless parameter

$\delta_{z}\equiv\frac{h\nu_{z}}{mc^{2}}\sim 10^{-13}\sim\epsilon^{6}.$ (9)

This parameter is shortly mentioned in the discussion. There are, of course,
further higher orders in fine structure constant $\alpha$, which are not shown
in figure 1 since these corrections do not introduce new effective operators.

Note further that, in principle, this discussion of orders of magnitude also
applies to the anomalous magnetic moment of the muon, but there are two
crucial differences: First, the experimental observable is different from (4).
Second the muon is about two hundred times heavier than the electron, which
further suppresses $\delta_{c}$ by the same factor. Thus, most of the possible
corrections discussed for the electron are irrelevant for the muon.

##  II Corrections from effective Lagrangian approach

We address the problem of corrections to the anomalous magnetic moment of the
electron with an effective Lagrangian approach. This approach allows to obtain
the energy spectrum as a result of the solutions of the effective equations of
motion. Alternatively the energies can be read off from the spectral structure
of the exact electron propagator in the presence of a finite magnetic field,
using Schwinger‘s proper-time representation Ayala:2021lor . We chose the
former approach, because it allows to include non-QED effects straight
forwardly.

###  II.1 Hamiltonian from effective Lagrangian

In an effective field theory approach the anomalous magnetic moment of the
electron is given in terms of the effective Lagrangian Itzykson:1980rh

${\mathcal{L}}_{eff}=\bar{\psi}\left[\gamma^{\mu}(i\hbar\partial_{\mu}-eA_{\mu})-mc^{2}+a\frac{e\hbar}{4m}\sigma^{\mu\nu}F_{\mu\nu}+\dots\right]\psi.$
(10)

The anomalous contribution to the $g$-factor is encoded in the parameter
$(a)$. In a top-down approach the corresponding last term in (10) arises from
the sunset diagram with a single external photon line (see figure 2 with
$y=1$). In a bottom-up approach, one can also take $(a)$ as a phenomenological
parameter of the effective Lagrangian (10) and determine its value from an
experiment. For this purpose, one uses (10) to derive the corresponding
quantum mechanical Dirac equation

$\left[\gamma^{\mu}(i\hbar\partial_{\mu}-eA_{\mu})-mc^{2}+a\frac{e\hbar}{4m}\sigma^{\mu\nu}F_{\mu\nu}+\dots\right]\psi=0.$
(11)

For a constant external magnetic field, and $a=0$ the energy levels and
eigenfunctions of the system with a single electron are known exactly
Thaller:1992ji ; Suzuki:2005wra , with the eigenenergies given in (61). With
$a\neq 0$, the anomalous magnetic moment is described by an additional term,

$\Delta_{1}H=a\gamma^{0}\sigma^{\mu\nu}F_{\mu\nu},$ (12)

adding to the Dirac Hamiltonian. Since these are also eigenfunctions of the
operator $\Delta_{1}H$ the corresponding energy levels (61) only get modified
by an additive factor

$E_{n}^{\pm}=mc^{2}\sqrt{1+\frac{h\nu_{c}}{mc^{2}}(1+2n\pm
1)}\pm\frac{a}{2}h\nu_{c}.$ (13)

Since our aim is to study corrections to (2) we expand the energies (13) up to
powers of $\delta_{c}^{3}$

$E_{n}^{\pm}=mc^{2}+\frac{1}{2}\left(2n+1\pm
1\right)h\nu_{c}\pm\frac{a}{2}h\nu_{c}-\frac{1}{8}\frac{h^{2}\nu_{c}^{2}}{mc^{2}}\left(2n+1\pm
1\right)^{2}+\frac{1}{16}\frac{h^{3}\nu_{c}^{3}}{m^{2}c^{4}}\left(2n+1\pm
1\right)^{3}.$ (14)

One notes that this contains the constant mass term, the energy formula (2),
and the additional last term $\sim\delta_{c}^{3}\sim\epsilon^{12}$, which is
the type of correction we are interested in. Schematically one can write

$\displaystyle\frac{E_{n}^{\pm}}{mc^{2}\delta_{c}}$ $\displaystyle=$
$\displaystyle{\mathcal{O}}\left(\frac{1}{\delta_{c}}+1+\alpha+\delta_{c}+\delta_{c}^{2}+\dots\right)$
(15)

The contributions $\sim\delta_{c}$ lead to relative changes in the energy
differences of $\sim 10^{-9}$ and are thus an essential part of the current
measurements which have a precision of about $10^{-13}$. The corrections
$\sim\delta_{c}^{2}$ lead to relative changes in the energy differences of
$\sim 10^{-18}$, they will only be accessible to future experiments. One
realizes that the solution (13) does not give rise to $\alpha\delta_{c}$ or
$\alpha\delta_{W}^{2}$ terms. For these terms to appear one needs to consider
additional operators in the effective action (10).

In the following subsections, we will explore this possibility.

###  II.2 Between theory and experiment

There is a conceptual gap between the theoretical definition of the anomalous
magnetic moment and the observable which is used to measure this quantity.
From theory side, one defines $a$ in terms of the photon-electron vertex
function $\Gamma_{e\bar{e}\rho}(p,-p,q)$, evaluated at zero momentum transfer
Czarnecki:1995wq ; Heinemeyer:2004yq . From the experimental side, $a$ is
defined through equation (4) in terms energy eigenvalues of an electron in the
background of a finite magnetic field.

These are different things, which only agree in certain limits and
approximations. One way to discuss this issue is in terms of the effective
Lagrangian (10) and the resulting field equations, as sketched in the previous
subsection. The photon-electron vertex function
$\Gamma_{e\bar{e}\rho}(p,-p,q)$ evaluated at zero momentum gives rise to the
effective Hamiltonian contribution (12). This contribution is thus an
unambiguous SM prediction, which is experimentally tested by measuring
transitions between energy eigenvalues of (11) and plugging them into relation
(4). This test is correct as long as additional contributions to the effective
Lagrangian, symbolized by the dots in relations (10, 11) are negligible or at
least give negligible contributions to the energy eigenvalues in (4).

Higher order corrections to the effective field equations (11) which are not
included in the standard analysis because they do not arise from the photon-
electron vertex function $\Gamma_{e\bar{e}\rho}(p,-p,q)$ are a consequence of
three different aspects:

  * •

The experiments are performed at finite magnetic field, wich means that the
assumption of zero momentum transfer is only approximately true. Since such
corrections are typically quadratic in the momentum transfer, one can expect
that they lead to $\sim\delta_{c}^{2}$ corrections in (15).

  * •

When allowing contributions to second or higher order in the finite external
magnetic field, (11) will contain additional terms which have a field and
Lorentz structure, which is not captured by (12). Such corrections can already
appear at order $\sim\alpha\delta_{c}$ in (15).

  * •

Additional, non QED, fields can also lead to operators with a different
structure than (12) and which are thus not part of the standard analysis in
the literature. If the additional field is a weak gauge boson, one expects
$\sim\alpha\delta_{W}^{2}$ corrections to (15). If the additional field is
some hypothetical beyond the Standard Model field, one expects a correction
suppressed by the coupling and by the mass of this field.

In the following discussion we will estimate these effects for the Standard
Model of particle physics, but some of the findings can be useful for studies
that go beyond this model.

###  II.3 Further gauge invariant bilinear corrections

In a top-down approach, further corrections to the effective action (10) arise
from irreducible Feynman diagrams with an arbitrary number of external legs.
We are interested in the cases with two external fermion lines and
$y$-external photon lines, as shown in figure 2.

Figure 2: Feynman diagram for coupling to $y=3$ external photons.

For $y=1$, the only contribution is the one given in (10). Loop corrections
induce numerous terms with $y>1$. In an effective field theory approach we
study a class of possible terms in the extended Lagrangian that meet the
following criteria:

  * •

they are bi-linear in the Dirac field $\bar{\psi}{\mathcal{O}}\psi$

  * •

they form a Lorentz scalar

  * •

the field content of the operator
${\mathcal{O}}={\mathcal{O}}(A_{\mu},F^{\mu\nu})$ only contains gauge-
invariant contributions of the external electromagnetic field which are either
the field strength tensor $F^{\mu\nu}$, or the covariant derivative operator
$D_{\mu}$, thus ${\mathcal{O}}={\mathcal{O}}(D_{\mu},F^{\mu\nu})$.

  * •

the field content of ${\mathcal{O}}$ is only up to second power in the field
and has at least one contribution from $F^{\mu\nu}$

Since the operator ${\mathcal{O}}$ forms a $4\times 4$ matrix in spinor-space,
it is useful to use
$\\{1,\gamma^{\mu},\sigma^{\mu\nu},\gamma^{5}\gamma^{\mu},\gamma^{5}\\}$ as
complete, five-dimensional basis for the classification of these operators.
While the first three basis elements are naturally induced in QED, the last
two elements require virtual non-QED contributions such as weak interactions
or BSM physics.

####  II.3.1 $\sim 1$ contributions

The term one can add to the effective Lagrangian (10), which is proportional
to the identity matrix, is

$\displaystyle\Delta_{1,FF}{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\alpha\frac{\xi_{1,FF}}{m^{3}c^{6}}\;\bar{\psi}F_{\mu\nu}F^{\mu\nu}\psi,$
(16)

where $\xi_{i}$ is the effective dimensionless coupling parameter, which is
expected to be of order one. The contributions of these operators to the
energy spectrum can be estimated by using $|\phi_{n}^{\pm}\rangle$, the
eigenfunctions of (11), which are given in (60). These eigenfunctions are
normalized as $\langle\phi_{n}^{\pm}|\phi_{n}^{\pm}\rangle=1$. Since small
terms like (16) only lead to small modifications of these wave-functions one
can apply perturbation theory to estimate to which extend these modifications
shift the energy levels $E_{n}^{\pm}$. These shifts are labeled
$\Delta_{\dots}E_{n}^{\pm}$. One finds

$\displaystyle\Delta_{1,FF}E_{n}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{\xi_{1,FF}\alpha}{m^{3}c^{6}}\langle\phi_{n}^{\pm}|\gamma^{0}F_{\mu\nu}F^{\mu\nu}|\phi_{n}^{\pm}\rangle$
$\displaystyle=$
$\displaystyle-\xi_{1,FF}\alpha\frac{h^{2}\nu^{2}}{2mc^{2}\sqrt{1+\frac{h\nu}{mc^{2}}(2n+1\pm
1)}}$ $\displaystyle\approx$
$\displaystyle\xi_{1,FF}\alpha\left(-\frac{h^{2}\nu_{c}^{2}}{mc^{2}}+\frac{h^{3}\nu_{c}^{3}(2n+1\pm
1)}{m^{2}c^{4}}\right).$ (18)

In terms of the series in figure 1, these contributions are of the order of

$\Delta_{1,FF}E_{n}^{\pm}/(mc^{2}\delta_{c})\sim\xi_{1,FF}{\mathcal{O}}(\alpha\delta_{c}+\alpha\delta_{c}^{2}).$
(19)

Note that these corrections are also dependent on the other quantum number
$l$, as long as $l\leq n$. The above relation is for $l=0$.

Note further that corrections of this type can also appear due to finite
temperature effects, but these are so small that they do not interfere with
our discussion.

####  II.3.2 $\sim\gamma^{\mu}$ contributions

When including one $\gamma^{\mu}$, the first option is to consider one
$D_{\alpha}$ and one $F^{\mu\nu}$. Since
$D_{\alpha}\gamma^{\alpha}F^{\mu}_{\mu}=0$, the only possibility, which is
linear in the field strength is

$\displaystyle\Delta_{\gamma,DF}{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\frac{\xi_{\gamma,DF}}{m^{2}c^{4}}\;\bar{\psi}D_{\alpha}\gamma^{\beta}F^{\alpha}_{\beta}\psi\neq
0.$ (20)

However, when integrating the corresponding contribution to the energy, one
finds that it vanishes

$\displaystyle\Delta_{\gamma,DF}E_{n}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{\xi_{1,DF}}{m^{2}c^{4}}\langle\phi_{n}^{\pm}|\gamma^{0}D_{\alpha}\gamma^{\beta}F^{\alpha}_{\beta}|\phi_{n}^{\pm}\rangle$
$\displaystyle=$ $\displaystyle 0.$

Second-order in $F^{\mu\nu}$, there are two non-vanishing contributions

$\displaystyle\Delta_{\gamma,DFF1}{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\alpha\frac{\xi_{\gamma,DFF1}}{m^{4}c^{8}}\;\bar{\psi}D_{\alpha}\gamma^{\alpha}F_{\mu\nu}F^{\mu\nu}\psi$
(22)

with

$\displaystyle\Delta_{\gamma,DFF1}E^{\pm}_{n}$ $\displaystyle=$
$\displaystyle\xi_{\gamma,DFF1}\alpha\frac{h^{2}\nu^{2}}{mc^{2}\sqrt{1+h\nu(1+2n\pm
1)}}$ (23) $\displaystyle\approx$
$\displaystyle\xi_{\gamma,DFF1}\alpha\left(\frac{2h^{2}\nu^{2}}{mc^{2}}-\frac{h^{3}\nu^{3}(1+2n\pm
1)}{m^{2}c^{4}}\right)$ (24)

and

$\displaystyle\Delta_{\gamma,DFF2}{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\alpha\frac{\xi_{\gamma,DFF2}}{m^{4}c^{8}}\;\bar{\psi}D_{\alpha}\gamma^{\beta}F_{\beta\nu}F^{\alpha\nu}\psi$
(25)

with

$\displaystyle\Delta_{\gamma,DFF2}E^{\pm}_{n}$ $\displaystyle=$
$\displaystyle-\Delta_{\gamma,DFF2}\alpha\frac{h^{3}\nu^{3}(1+2n\pm
1)}{m^{2}c^{4}\sqrt{1+h\nu(1+2n\pm 1)}}$ (26) $\displaystyle\approx$
$\displaystyle-\xi_{\gamma,DFF2}\alpha\frac{h^{3}\nu^{3}(1+2n\pm
1)}{m^{2}c^{4}}.$ (27)

In terms of the series in figure 1, these contributions are of the order of

$\displaystyle\Delta_{\gamma,DFF1}E_{n}^{\pm}/(mc^{2}\delta_{c})$
$\displaystyle\sim$
$\displaystyle\xi_{\gamma,DFF1}{\mathcal{O}}(\alpha\delta_{c}+\alpha\delta_{c}^{2})$
(28) $\displaystyle\Delta_{\gamma,DFF2}E_{n}^{\pm}/(mc^{2}\delta_{c})$
$\displaystyle\sim$
$\displaystyle\xi_{\gamma,DFF2}{\mathcal{O}}(\alpha\delta_{c}^{2}).$ (29)

####  II.3.3 $\sim\sigma^{\mu\nu}$ contributions

The contribution, which is first-order in $~{}F^{\mu\nu}$, is already
considered by the coupling in (11). Second-order contributions such as $\sim
F^{\alpha}_{\beta}\sigma^{\beta\mu}F_{\mu\alpha}$ are identically zero. A
possible third-order contribution is

$\displaystyle\Delta_{\sigma,FFF1}{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\alpha\frac{\xi_{\sigma,FFF1}}{m^{5}c^{10}}\;\bar{\psi}\sigma^{\mu\nu}F_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\psi$
(30)

with

$\displaystyle\Delta_{\sigma,FFF1}E^{\pm}_{n}$ $\displaystyle=$
$\displaystyle\pm\xi_{\sigma,FFF1}\alpha\frac{4h^{3}\nu^{3}}{m^{2}c^{4}}$ (31)

Similarly, there is

$\displaystyle\Delta_{\sigma,FFF2}{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\alpha\frac{\xi_{\sigma,FFF2}}{m^{5}c^{10}}\;\bar{\psi}\sigma^{\mu\nu}F_{\mu\alpha}F_{\nu\beta}F^{\alpha\beta}\psi$
(32)

with

$\displaystyle\Delta_{\sigma,FFF2}E^{\pm}_{n}$ $\displaystyle=$
$\displaystyle\pm\xi_{\sigma,FFF2}\alpha\frac{2h^{3}\nu^{3}}{m^{2}c^{4}}.$
(33)

Thus, one realizes that in the spirit of the expansion in figure 1, all these
contributions are of order

$\Delta_{\sigma\dots}E_{n}^{\pm}/(mc^{2}\delta_{c})\sim\xi_{\sigma\dots}{\mathcal{O}}(\alpha\delta_{c}^{2}).$
(34)

###  II.4 Parity violating corrections

Once one leaves the realm of QED, there is also the possibility of forming
Lorentz invariants with $\gamma^{5},\gamma^{\mu}\gamma^{5}$ and through
contractions with $\epsilon^{\mu\nu\alpha\beta}$

####  II.4.1 $\sim\gamma^{5}$ contributions

One might consider mass contributions which have the form

$\displaystyle\Delta_{\gamma^{5}}{\mathcal{L}}$ $\displaystyle=$
$\displaystyle
mc^{2}\xi_{\gamma^{5}}\alpha\delta_{W}^{2}\bar{\psi}\gamma^{5}\psi.$ (35)

They do not give corrections to the energy

$\displaystyle\Delta_{\gamma^{5}}E_{n}^{\pm}$ $\displaystyle=$ $\displaystyle
0.$ (36)

####  II.4.2 $\sim\gamma^{5}\gamma^{\mu}$ contributions

Parity violating contributions to the minimal coupling can, for example, be
introduced by virtual weak interactions

$\displaystyle\Delta_{\gamma^{5}\gamma^{\mu}}{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\xi_{\gamma^{5}\gamma^{\mu}}\alpha\delta_{W}^{2}\bar{\psi}\gamma^{5}\gamma^{\mu}D_{\mu}\psi.$
(37)

Two examples are calculated in the appendix A.3. Even though these terms come
with the expected prefactor, they do not lead to corrections to the energy
levels

$\displaystyle\Delta_{\gamma^{5}\gamma^{\mu}}E_{n}^{\pm}$ $\displaystyle=$
$\displaystyle 0.$ (38)

Similarly, there is the possibility of

$\displaystyle\Delta_{\gamma^{5}\gamma^{\mu}FF}{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\alpha\frac{\xi_{\gamma^{5}\gamma^{\mu}FF\delta_{W}^{2}}}{m^{3}c^{6}}\bar{\psi}\gamma^{5}\gamma^{\alpha}D_{\alpha}F^{\mu\nu}F_{\mu\nu}\psi,$
(39)

also leading to vanishing energy corrections

$\displaystyle\Delta_{\gamma^{5}\gamma^{\mu}FF}E_{n}^{\pm}$ $\displaystyle=$
$\displaystyle 0.$ (40)

Notably, all parity-violating contributions to the energy vanish at the
considered order.

####  II.4.3 Considering the electric field

If one considers an electric field, e.g. the field confining the electrons in
$z$-direction, the solutions of equation (11) will contain additional quantum
numbers, like $n_{z}$ referring to this degree of freedom.

$\psi=\psi^{\pm}_{n,n_{z},l,l_{z}}\quad.$ (41)

The strength of the electric field will depend on $z$, and it will be related
to the dimensionless parameter (9) and the energy contribution of the electric
oscillations will be

$E_{n,n_{z}}^{\pm}-E_{n,n_{z}=0}^{\pm}=n_{z}mc^{2}\delta_{z}+{\mathcal{O}}(\delta_{z}^{2}).$
(42)

In order to avoid such complications, let us consider a constant electric
field $\vec{E}=\hat{z}(mc^{2})^{2}\delta_{z}$. Thus, as long as we consider
only eigenstates with the same quantum number $n_{z}$ (e.g. $n_{z}=0$), we can
reuse the eigenstates of the previous sections

$\psi^{\pm}_{n,l}=\psi^{\pm}_{n,n_{z}=0,l,l_{z}=0}.$ (43)

The discussion of the preceding sections remains valid in a perturbative
sense, even though the electromagnetic stress tensor now reads

$F_{\mu\nu}=(mc^{2})^{2}\left(\begin{array}[]{cccc}0&0&0&\delta_{z}\\\
0&0&\delta_{c}&0\\\ 0&-\delta_{c}&0&0\\\ -\delta_{z}&0&0&0\end{array}\right).$
(44)

The existence of the electric field $\sim(mc^{2})^{2}\delta_{z}$ does not
alter the differences between energy levels of the state (43). This
invisibility becomes spoiled due to parity-violating contributions, analogous
to the ones discussed above. One obtains, for example, an effective coupling

$\Delta_{\sigma\tilde{F}}{\mathcal{L}}=\alpha\frac{\xi_{\sigma\tilde{F}}\delta_{W}^{2}}{mc^{2}}\sigma^{\mu\nu}\tilde{F}_{\mu\nu},$
(45)

where the dual electromagentic tensor is given by
$\tilde{F}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}F^{\alpha\beta}$.
This coupling contributes to the energy levels with

$\Delta_{\sigma\tilde{F}}E_{n}^{\pm}=\pm
mc^{2}\alpha\delta_{W}^{2}\xi_{\sigma\tilde{F}}\delta_{z}.$ (46)

###  II.5 Combined corrections

As a result of the previous two subsections, we have obtained eight
corrections (II.3.1, 23, 26, 31, 33, 36, 38, 40) of the standard energy
formula (2). Up to order $\delta_{c}^{3}$ the expansion of the standard energy
formula reads (14). Within the same expansion, the combined correction arising
from (II.3.1, 23, 26, 31, 33, 36, 38, 40) reads

$\displaystyle\frac{\Delta E_{n}^{\pm}}{mc^{2}\delta_{c}}$ $\displaystyle=$
$\displaystyle\alpha\delta_{c}^{1}(\xi_{\gamma^{\mu}FF1}-\xi_{1,FF})\pm\alpha\frac{\delta_{W}^{2}\delta_{z}}{\delta_{c}}\xi_{\sigma\tilde{F}}$
$\displaystyle+\alpha\delta_{c}^{2}(\xi_{IdFF}(2n\pm
1+1)-2n(\xi_{\gamma^{\mu}FF1}+\xi_{\gamma^{\mu}FF2})-\xi_{\gamma^{\mu}FF1}-\xi_{\gamma^{\mu}FF2}$
$\displaystyle\quad\quad\mp(\xi_{\gamma^{\mu}FF1}+\xi_{\gamma^{\mu}FF1}-2(\xi_{\sigma
FFF1}+\xi_{\sigma FFF2})))+\dots$

In equation (68) a theoretical prediction for the proportionality factor of
the leading coefficient of (II.5) is given
$(\xi_{\gamma^{\mu}FF1}-\xi_{1,FF})\approx 0.02$.

###  II.6 Anomalous magnetic moment

Experimentally, the value of $a_{d}$ is obtained from measuring transition
energies and applying relation (4). This formula is valid for energies
expanded up to $\delta_{c}^{2}$. The value inferred from the direct Penning
trap experiment by using this relation is Hanneke:2008tm

$a_{d}=0.001\,159\,652\,180\,73\,(28).$ (48)

If one wants to include the relativistic $\delta_{c}^{3}$ energy corrections
(14) one has to adopt the formula (4) to

$a_{d}=\frac{E_{0}^{+}-E_{1}^{-}}{E_{1}^{+}-E_{0}+\frac{3}{2}mc^{2}\delta_{c}-\frac{7}{2}mc^{2}\delta_{c}^{2}}.$
(49)

If one includes now, on top of the relativistic improvement (14), the combined
corrections (II.5), the actual energies read

$\tilde{E}_{n}^{\pm}=E_{n}^{\pm}+\Delta E_{n}^{\pm}.$ (50)

Using these energies in the relativistically improved formula (49) one finds
the inferred direct anomalous moment $a_{d}$ (the l.h.s. of (49)) differs from
the actual magnetic moment $a_{a}$ (the one appearing in (10)). One finds

$\displaystyle a_{d}$ $\displaystyle=$ $\displaystyle
a_{a}+\frac{2\alpha\delta_{W}^{2}\delta_{z}}{\delta_{c}}\xi_{\sigma\tilde{F}}-2\alpha\delta_{c}^{2}\left(a_{a}(\xi_{IdFF}-\xi_{\gamma^{\mu}FF1}-\xi_{\gamma^{\mu}FF2})-2(2\xi_{\sigma
FFF1}+\xi_{\sigma FFF2})\right)+{\mathcal{O}}(\epsilon^{11}).$ (51)

The above relation used a combined expansion in the small parameter $\epsilon$
(6, 7, 8). This relation can be inverted to give the actual value of the
anomalous magnetic moment $a_{a}$ as a function of
$a_{a}=a_{a}(a_{d},\xi_{i})$

$\displaystyle a_{a}$ $\displaystyle=$ $\displaystyle
a_{d}-\frac{2\alpha\delta_{W}^{2}\delta_{z}}{\delta_{c}}\xi_{\sigma\tilde{F}}+2\alpha\delta_{c}^{2}\left(a_{d}(\xi_{IdFF}-\xi_{\gamma^{\mu}FF1}-\xi_{\gamma^{\mu}FF2})-2(2\xi_{\sigma
FFF1}+\xi_{\sigma FFF2})\right).$ (52)

It is now the task of theory to predict the precise values of $\xi_{i}$ and
the task of experiment to determine and/or constrain these values. One way of
doing this is via a comparison with indirect $g-2$ measurements.

##  III Discussion

###  III.1 Indirect $g-2$ measurements

Theoretical calculations determine a relation between the value of $(g-2)$ and
the fine structure constant up to fifth-order in $\alpha$

$\displaystyle a(\alpha)$ $\displaystyle=$ $\displaystyle
C_{2}\left(\frac{\alpha}{\pi}\right)+C_{4}\left(\frac{\alpha}{\pi}\right)^{2}+C_{6}\left(\frac{\alpha}{\pi}\right)^{3}+C_{8}\left(\frac{\alpha}{\pi}\right)^{4}$
$\displaystyle+C_{10}\left(\frac{\alpha}{\pi}\right)^{5}+a_{\mu\tau}+a_{had}+a_{weak},$

where the constants $C_{i}$ are given in Gabrielse:2006gg  (consider the
erratum). Thus, an independent measurement of $\alpha$ (and the lepton
masses), also allows to infer a value for $g-2$, and vice versa.

In recent years this has been done by Parker et al. Parker:2018vye  and Morel
et al. Morel:2020dww . They obtained $\alpha$-induced values

$\displaystyle a_{P}(\alpha)$ $\displaystyle=$ $\displaystyle
0.001\,159\,652\,181\,610\,(230)$ (54) $\displaystyle a_{M}(\alpha)$
$\displaystyle=$ $\displaystyle 0.001\,159\,652\,180\,252\,(95).$ (55)

In contrast to this, the “direct” measurement gave (48). This situation is
shown in figure 3, where we plot the value of $a(\alpha)$ relative to the
value $a_{d}$, which was inferred from the direct measurement.

Figure 3:  Results for anomalous magnetic moment $(a_{P/M}(\alpha)-a_{d})\cdot
10^{13}$ plotted as a function of time. The blue point is the direct
measurement Hanneke:2008tm , the red and orange points are the indirect
measurements Parker:2018vye ; Morel:2020dww .

Since the latter measurement of $\alpha$ has much smaller error bars, there
seems to be no significant tension between $a_{d}$ and $a_{M}(\alpha)$. The
(non)-tension between the direct measurement and the indirect measurements can
be quantified by

$\frac{a_{P/M}(\alpha)-a_{d}}{\sqrt{\delta a_{d}^{2}+\delta
a^{2}_{P/M}(\alpha)}}=\left\\{\begin{array}[]{cc}t_{i,P}=+2.4\\\
t_{i,M}=-1.6\end{array}\right..$ (56)

The values of $t_{i,P/M}$ are measured in standard deviations. Here the sign
indicates whether $a(\alpha)$ is smaller or bigger than $a_{d}$.

In the following subsections, we will explore a comparison between $a_{d}$ and
$a(\alpha)$ in the light of the difference between $a_{d}$ and $a_{a}$ (52).
In this comparison we will identify $a(\alpha)$ with $a_{a}$. This
identification assumes implicitly that there are no corrections to the
measurements of the fine-structure constant $\alpha$.

###  III.2 Benchmark without parity violation

Let’s first consider a benchmark without parity violation by setting
$\xi_{\sigma\tilde{F}}=0$. The result in section II suggests correcting the
interpretation of the direct measurement by using the value (51):
$a_{d}\rightarrow a_{a}=a_{a}(\xi_{i})$ and the error induced by this
correction.

As the first benchmark, let’s treat all parity conserving contributions with
the same numerical factor $\xi_{i}$. For this benchmark, one can study how the
comparison between direct measurement and indirect measurement depends on the
parameter $\xi_{i}$. If one assumes these corrections come without an
increased error bar, one can plot $a_{a}(\xi_{i})$ relative to $a_{d}$ and in
comparison with the indirect results $a_{P/M}-a_{d}$. This is shown in figure
4.

Figure 4:  The red line shows $(a_{M}-a_{d})$, with the error bars of $a_{M}$
Morel:2020dww . The orange line shows $(a_{P}-a_{d})$, with the error bars of
$a_{P}$ Parker:2018vye . The blue region are the error bars of $a_{d}$
Hanneke:2008tm . The black line shows $(a_{a}(\xi_{i})-a_{d})$, with the error
bars of $a_{d}$.

From figure 4 one sees that the parity conserving corrections only have a
relevant effect on the comparison between direct and indirect observation, if
the dimensionless parameters $\xi_{i}$ are of the order of $10^{6}$. However,
the prefactors of the effective operators are chosen such that within the SM
the parameters $\xi_{i,\dots}$ are expected to be of order one. Thus, figure 4
means that there are six orders of magnitude missing in the sensibility before
one can appreciate the effect of these SM contributions.

This finding can be compared with the QED corrections for finite momentum
transfer. The one-loop QED form factor for finite photon momentum squared
$q^{2}$ is given by

$\displaystyle F_{2}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2\pi}\int_{0}^{1}dxdydz\;\delta(x+y+z-1)\frac{2m^{2}z(1-z)}{m^{2}(1-z)^{2}-q^{2}xy}$
$\displaystyle=$
$\displaystyle\frac{\alpha}{2\pi}\left(1-\frac{q^{2}}{6m^{2}}\right).$

If one evaluates the correction due to the second term with the momentum scale
given by the magnetic background field $q^{2}\approx(h\nu_{c})^{2}$, one
obtains

$F_{2}\approx\frac{\alpha}{2\pi}\left(1-\frac{\delta_{c}^{2}}{6}\right).$ (58)

Thus, one also finds that finite momentum effects are suppressed by
$\alpha\delta_{c}^{2}\approx 10\cdot 10^{-20}$, which is also six orders of
magnitude below the current experimental precision.

From a phenomenological perspective (e.g. for BSM models) one has to demand
that $|a_{a}-a_{d}|\leq 10\cdot 10^{-13}$ which constrains the $\xi_{i}$
parameter of this benchmark to

$\xi_{i}\leq 1\cdot 10^{6}.$ (59)

The parameters $\xi_{i}$ are thus useful mediators between BSM models and the
precision measurement of $a_{d}$.

###  III.3 Benchmark with parity violation

In a second benchmark, we set all parity conserving corrections to zero and
only retain $\xi_{\sigma\tilde{F}}$. Figure 5 shows how the comparison between
direct and indirect measurements is affected by the remaining parameter.

Figure 5:  The red line shows $(a_{M}-a_{d})$, with the error bars of $a_{M}$
Morel:2020dww . The orange line shows $(a_{P}-a_{d})$, with the error bars of
$a_{P}$ Parker:2018vye . The blue region are the error bars of $a_{d}$
Hanneke:2008tm . The black line shows $(a_{a}(\xi_{i})-a_{d})$, with the error
bars of $a_{d}$.

One notes that a natural $\xi_{\sigma\tilde{F}}={\mathcal{O}}(1)$ is three
orders of magnitude too small for observation under the current simplifying
assumptions.

###  III.4 What happened?

In subsection I.3 it has been shown that background field effects
$(\sim\alpha\delta_{c})$ and effects of the weak interaction
$(\sim\alpha\delta_{W}^{2})$ can be expected to give important contributions
to the energy levels $E_{n}^{\pm}/(mc^{2}\delta_{c})$, as depicted in figure
1. These energy levels are used to determine $a_{d}$. However, when studying
the actual comparison between direct and indirect measurements, the effect of
these corrections was rather negligible, as shown in figure 4. The
disappearance of these seemingly substantial contributions happens for
different reasons:

  * •

The parity-violating contributions, arising from virtual weak interactions,
introduce new types of effective couplings (35, 37, 39). These couplings do,
however, not contribute to the leading-order corrections of the energy levels
$E_{n}^{\pm}/(mc^{2}\delta_{c})$. This is due to the particular off-diagonal
nature of the $\gamma^{0}\cdot\gamma^{5}$ matrix.

  * •

The parity conserving corrections (16, 20,22, 25, 30, 32) do all contribute to
$E_{n}^{\pm}/(mc^{2}\delta_{c})$ and they do so with corrections
$\sim\alpha\delta_{c}^{1}$ and $\sim\alpha\delta_{c}^{2}$. The problem is that
all the the $\sim\alpha\delta_{c}^{1}$ corrections are independent of the
quantum numbers $(\pm,\;n)$. Thus, when evaluating energy differences in (49),
only the much smaller $\sim\alpha\delta_{c}^{2}$ corrections survive. It
remains to be seen whether larger corrections $\sim\alpha\delta_{c}^{1}$ can
be noted by a different type of experimental setup.

  * •

The largest corrections arise from the parity-violating contributions (45).
Under the simplifying assumptions used above, they are still three orders of
magnitude below the sensibility threshold. Concering this result, one has to
consider two additional aspects. On the on hand, one could imagine an
experimental setup with an increased electric field, which would then reach
the observability limit. On the other hand, one must consider that the
assumption of a homogenous and constant electric field is not realistic. In a
more realistic scenario, it is likely that contributions from positive and
negative electric fields cancel each other.

##  IV Conclusion

Penning trap experiments allow for a precise direct measurement of the
electron’s anomalous magnetic moment $a=(g-2)/2$. This is done by relating
observed transition energies between Landau levels, as shown in equation (4).

In this paper, we studied the impact of parametrized corrections to the
effective Lagrangian (10) on the electron’s anomalous g factor, deduced from
the observable (49). It is found that the presence of these parameters leads
to a difference between the deduced anomalous magnetic moment $a_{d}$ and the
actual magnetic moment $a_{a}$ given by (51). This generic result is then
studied for two benchmarks.

It is shown that there are QED-loop-induced parity conserving corrections to
the energies $E_{n}^{\pm}$ which are of the order of $\alpha\delta_{c}^{2}$.
Such corrections are, in principle, numerically very relevant since $\Delta
a\sim\Delta E/\delta_{c}\sim\alpha\delta_{c}\sim 10^{-11}$. However, these
leading corrections do not depend on the quantum numbers $(n,\,\pm)$ and thus
they get canceled in the actual observable (49), where only energy differences
are considered. The remaining sub-leading effects $\Delta
a_{d}\sim\delta_{c}^{2}$ are, at the current stage, not observable. They are
about six orders of magnitude smaller than the experimental precision.

Most parity-violating corrections to the energy vanish unless one considers
effects of the electric field. An example for such a contribution is given in
relation (45). Whether this scenario has the potential of studying parity
violation with $g-2$ experiments has to be investigated with a detailed study
considering a spatial variation of the electric field.

While this paper has its focus on SM contributions, the relations (II.5) are
perfectely suited to test and constrain BSM effects.

## Acknowledgements

We thank R. Schoeffbeck, J. Suzuki, A. Rebhan, and F. Hummel for their
comments and C. Kat for his efforts to improve our pronunciation of “Foldy-
Wouthuysen”.

##  Appendix A Appendix

###  A.1  Explicit solutions

The exact solutions of (11) are well known Suzuki:2005wra . Since we made
massive use of the particle eigenstates $\psi_{n=(0,1),\,l=0}^{\pm}$, these
functions in symmetric gauge are given explicitly

$\displaystyle\psi_{0,0}^{+}$ $\displaystyle=$
$\displaystyle\left(\frac{\rho}{2\sqrt{2\pi}}\sqrt{1+\frac{m}{\sqrt{m^{2}+\rho^{2}}}},0,0,\frac{i\rho^{3}(x+iy)}{4\sqrt{2\pi}\sqrt{m(\sqrt{m^{2}+\rho^{2}}+m)+\rho^{2}}}\right)e^{-\frac{1}{8}\rho^{2}(x^{2}+y^{2})-it\sqrt{m^{2}+\rho^{2}}}$
(60) $\displaystyle\psi_{0,0}^{-}$ $\displaystyle=$
$\displaystyle\left(0,\frac{\rho}{2\sqrt{\pi}},0,0\right)e^{-\frac{1}{8}\rho^{2}(x^{2}+y^{2})-itm}$
$\displaystyle\psi_{1,0}^{+}$ $\displaystyle=$
$\displaystyle\left(\frac{\rho^{2}(x+iy)}{4\sqrt{2\pi}}\sqrt{1+\frac{m}{\sqrt{m^{2}+\rho^{2}}}},0,0,\frac{i\rho^{4}(x+iy)^{2}}{8\sqrt{2\pi}\sqrt{m(\sqrt{m^{2}+2\rho^{2}}+m)+2\rho^{2}}}\right)e^{-\frac{1}{8}\rho^{2}(x^{2}+y^{2})-it\sqrt{m^{2}+2\rho^{2}}}$
$\displaystyle\psi_{1,0}^{-}$ $\displaystyle=$
$\displaystyle\left(\frac{\rho^{2}(\sqrt{m^{2}+\rho^{2}}+m)(x+iy)}{4\sqrt{2\pi}\sqrt{m(\sqrt{m^{2}+2\rho^{2}}+m)+2\rho^{2}}},0,0\frac{-i\rho^{2}}{2\sqrt{2\pi}\sqrt{m(\sqrt{m^{2}+2\rho^{2}}+m)+2\rho^{2}}}\right)e^{-\frac{1}{8}\rho^{2}(x^{2}+y^{2})-it\sqrt{m^{2}+\rho^{2}}},$

where the abbreviation $\rho^{2}=2mh\nu$ and $c\equiv 1$ was used.

###  A.2 QED background corrections from Suszuki approach

In the effective Lagrangian approach (10), $\xi_{i}$ are phenomenological
parameters that have to be determined by experiment. In a full QFT approach,
with non-vanishing background fields, these parameters can be predicted. Below
we will sketch how such a calculation relates to the above results.

Complementary to using the effective action (10), one can also calculate the
loop corrections to the different energies $E_{n}^{\pm}$ directly in the
underlying quantum field theory. Usually, this is done to leading order in the
magnetic field. For a comparison with the results obtained in the previous
sections, this has to be done up to second order in $B$. While the complete
calculation of these QED corrections goes beyond the scope of this paper, one
can use the known shift in the ground state energy to get an estimate of the
uncertainty. The finite ground state corrections, which are proportional to
$B^{2}$, are given by Suzuki Suzuki:2005wra . Strong field effects in a
broader sense are discussed in Dunne:2008kc .

Suzuki uses the exact QM solution for the energy levels of the electron
Suzuki:2005wra

$E_{n}^{\pm}=mc^{2}\sqrt{1+\frac{h\nu_{c}}{mc^{2}}(1+2n\pm 1)},$ (61)

Note that when comparing the above formula with the energies given in
Itzykson:1980rh ; Suzuki:2005wra  one needs to do the identification
$n\rightarrow n+1$. The Energies (61) are identical to (13) for $a=0$.

The actual (QFT corrected) energies are then labeled
${\mathcal{E}}_{n}^{\pm}$, they can be expressed as corrections to (61). One
finds

$\Delta E_{n}^{\pm}\equiv{\mathcal{E}}_{n}^{\pm}-E_{n}^{\pm}=\int d^{3}x\int
d^{3}x^{\prime}\bar{\psi}_{E,n,\pm}(x){\mathcal{M}}(E_{n}^{\pm},\vec{x},\vec{x}^{\prime})\psi_{E,n,\pm}(x^{\prime}),$
(62)

where ${\mathcal{M}}$ is the mass operator and $\psi_{E,n,\pm}$ are the
eigenfunctions for the QM eigenvalues (61). The calculation is tedious, but
for a small magnetic field $h\nu_{c}\ll mc^{2}$, the result can be written as
an expansion

$\Delta
E_{n}^{\pm}=\frac{a}{2}h\nu_{c}\left(1+\frac{h\nu_{c}}{mc^{2}}(b_{0}^{\pm}+b_{1}^{\pm}n)\right)+{\mathcal{O}}\left(\frac{h\nu_{c}}{mc^{2}}\right)^{3}.$
(63)

In the recent paper of Kim at al. Kim:2021ydg , this type of expansion is
given for the ground state energy and the coefficients are calculated
explicitly. However, the above formula allows also for $n\neq 0$. These energy
contributions do, in principle, depend on the quantum numbers $(n,\pm)$, since
the eigenfunctions $\psi_{n,\pm}$ depend on these quantum numbers. This
dependence on $n$ does not have to be linear, but since (4) only considers
$n=(0,1)$ a distinction with different powers of $n$ is not necessary. Here,
$a,b_{i}^{\pm}$ are numerical coefficients that can be obtained in a loop
expansion

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2\pi}+{\mathcal{O}}(\alpha^{2})$ (64)
$\displaystyle b_{i}^{\pm}$ $\displaystyle=$ $\displaystyle
b_{i,0}^{\pm}+\alpha\cdot b_{i,1}^{\pm}+{\mathcal{O}}(\alpha^{2}).$ (65)

The coefficients of $a$ are very well studied to high order, while the
coefficients $b_{i}^{\pm}$ are hard to find in the literature. Their order of
magnitude can be estimated from the $\sim B^{2}$ correction to the ground
state energy given in Suzuki:2005wra . Using the leading order expansion in
$\epsilon$ and minimal subtraction, this ground state correction is

$\Delta E_{0}^{-}|_{B^{2}}=\alpha
B^{2}\frac{1259}{16800}\frac{e^{2}\hbar^{2}}{m^{3}c^{2}\pi}+{\mathcal{O}}\left(\epsilon^{8}\right).$
(66)

From this, one can read off the numerical value of the first coefficient of
(65)

$b_{0}^{-}=\frac{1159}{4200}\approx 0.3+{\mathcal{O}}(\alpha).$ (67)

From equation (II.5) and the preceding discussion, one can expect that the
coefficients which are proportional to $n$, vanish at order $\delta_{c}$
$b_{b_{1}}^{\pm}=0$. Further, one can extract from a comparison between (67)
and (II.5) a theoretical prediction for a combination of the parameters
$\xi_{\gamma^{\mu}FF1}$ and $\xi_{idFF}$

$\xi_{\gamma^{\mu}FF1}-\xi_{1,FF}=\frac{b_{0}-}{4\pi}\approx 0.02.$ (68)

###  A.3 Parity violating SM processes

There are, SM processes that can contribute to parity violation in low energy
QED processes, see for example Dittmaier:1997dx ; Jones:2004yd ;
Zykunov:2005md ; Du:2019evk .

To quantify the magnitude of similar effects in the context of the observable
(49) one needs to consider the weak process shown in figure 6.

Figure 6: Feynman diagram weak sub-process

The prefactor of the Feynman amplitude of this process gives the magnitude one
should expect for the parameters $\xi_{\gamma^{5}\dots}$. As discussed in the
previous section, the contribution, relevant for the $g-2$ observable is
$\xi_{\gamma^{5}\gamma^{\mu}}$. The amplitude shown in figure 6 can be
calculated with the software packages based on Denner:1991kt ; Hahn:2000kx ;
Hahn:2016ebn . It reads

$\displaystyle A$ $\displaystyle=$ $\displaystyle\frac{2i\pi^{3}\alpha
e}{\sin^{2}(\theta_{W})}\left(-2(-2B_{0}(m^{2},0,m_{W}^{2})+B_{0}(0,m_{W}^{2},m_{W}^{2})-2m_{W}^{2}C_{0}(0,m^{2},m^{2},m_{W}^{2},m_{W}^{2},0)\right.$
$\displaystyle\left.+2m^{2}C_{0}(0,m^{2},m^{2},m_{W}^{2},mW^{2},0)-DC_{00}(m^{2},0,m^{2},0,m_{W}^{2},m_{W}^{2}))+2C_{00}(m^{2},0,m^{2},0,m_{W}^{2},m_{W}^{2})\right.$
$\displaystyle\left.+3m^{2}C_{1}(m^{2},0,m^{2},0,m_{W}^{2},m_{W}^{2})\right.\cdot\bar{\psi}_{2}\gamma^{\mu}\gamma^{7}\psi_{1}$
$\displaystyle
m\bar{\psi}_{2}\gamma^{7}\psi_{1}\cdot(D-2)p_{2}^{\mu}((D-8)C_{1}(m^{2},0,m^{2},0,m_{W}^{2},m_{W}^{2})2(D-2)C_{12}(m^{2},0,m^{2},0,m_{W}^{2},m_{W}^{2})))$
$\displaystyle
m\bar{\psi}_{2}\gamma^{6}\psi_{1}\cdot((D-2)p_{1}^{\mu}(C_{1}(m^{2},0,m^{2},0,m_{W}^{2},m_{W}^{2})+2C_{11}(m^{2},0,m^{2},0,m_{W}^{2},m_{W}^{2}))$
$\displaystyle
p_{2}^{\mu}((D-8)C_{1}(m^{2},0,m^{2},0,m_{W}^{2},m_{W}^{2})+2(D-2)C_{12}(m^{2},0,m^{2}0,m_{W}^{2},m_{W}^{2})))$
$\displaystyle+6m^{2}C_{1}(m^{2},0,m^{2},0,m_{W}^{2},m_{W}^{2})\cdot\bar{\psi}_{2}\gamma^{\mu}\gamma^{6}\psi_{1}),$

where $D=4-\tilde{\epsilon}$ is the spacetime dimension of dimensional
regularization, $\gamma^{6}=(1+\gamma^{5})/2$, $\gamma^{7}=(1-\gamma^{5})/2$,
and $C_{ij}$ are the Passarino-Veltman loop integral functions which are
evaluated with Denner:2005nn ; Patel:2015tea . To reduce the resulting
expression to the parity-violating terms, which are relevant for our analysis,
we use the following simplifications

  * •

retain only terms proportional to
$\bar{\psi}_{2}\gamma^{\mu}\gamma^{5}\psi_{1}$

  * •

subtract divergent terms $\sim 1/\tilde{\epsilon}$

  * •

perform an expansion in $\delta_{W}^{2}$ in which
$A\sim\delta_{W}^{0}+\delta_{W}^{2}+\delta_{W}^{4}$. Since the order
$\delta_{W}^{0}$ is controlled by the dimensional regularization parameters
($\tilde{\epsilon}$, $\mu$) and since the order $\delta^{4}_{W}$ is
negligible, we keep the $\sim\delta_{W}^{2}$ terms

  * •

set $D\rightarrow 4$.

The result is then

$A|_{WW}=-i\frac{5e^{3}m^{2}}{96\pi^{2}m_{W}^{2}\sin^{2}(\theta_{W})}\bar{\psi}_{2}\gamma^{\mu}\gamma^{5}\psi_{1}.$
(70)

The same procedure can be repeated with the triangle diagram with a single
Z-boson in the loop giving

$A|_{Z}=i\frac{e^{3}m^{2}(6\log(m_{Z}^{2}/m^{2})-1)(4\sin^{2}(\theta_{W})-1)}{192\pi^{2}m_{Z}^{2}\sin^{2}(\theta_{W})\cos^{2}(\theta_{W})}\bar{\psi}_{2}\gamma^{\mu}\gamma^{5}\psi_{1}.$
(71)

Thus one can conclude that parity-violating corrections to the QED couplings
are indeed suppressed by

$\sim\alpha\frac{m^{2}}{m_{W/Z}^{2}}.$ (72)

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