# Fermilab Booster beam emittances from
quadrupole modes measured by BPMs††thanks: This manuscript has been authored
by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the
U.S. Department of Energy, Office of Science, Office of High Energy Physics.

C. Y. Tan<EMAIL_ADDRESS>M. Balcewicz  Fermi National Accelerator Laboratory
Batavia  IL60510  USA

###### Abstract

The measurement of beam emittances by extracting the quadrupole mode signal
from a 4 plate beam position monitor (BPM) was published at least 40 years
ago. Unfortunately, in practice, this method suffers from poor signal to noise
ratio and requires a lot of tuning to extract out the emittances. In this
paper, an improved method where multiple BPMs are used together with better
mathematical analysis is described. The BPM derived emittances are then
compared with those measured by the Ion Profile Monitor (IPM). Surprisingly,
the BPM measured emittances behave very well and are more realistic than those
measured by the IPM.

##  1 Introduction

The measurement of beam emittances by extracting the quadrupole mode signal
from a 4 plate beam position monitors (BPM) was published at least 40 years
ago [1, 2]. However, the quadrupole signal is very small when compared to the
dipole signal and so, in practice, this method suffers from poor signal to
noise [3]. This means that to actually get the emittance, a lot of tuning is
required. We decided a revisit of this method because the Booster Ionization
Profile Monitors (IPMs) have idiosyncrasies that are very puzzling [4]. For
example, the beam current measured by the IPM is not conserved. It also shows
an emittance growth during the Booster ramp that is unexplained [5].

Our contribution to the improvement of this method are the use of multiple
BPMs which then gives us a better way for emittance extraction. When we
compared the emittances extracted with our method to those measured by the
IPMs in the Booster, we found that the BPM measured emittances were more
realistic than those from the IPMs.

##  2 Theory

This section only contains an abridged version of the theory. A full
derivation can be found in Ref. [6] which follows Miller [1].

The cross section of the 4 plate BPM is shown in Fig. 1. The image current
density, $J_{w}$, that is induced by a pencil current, $I_{b}$, at
$(r_{b},\theta_{b})$ is given by

$\displaystyle J_{w}(r_{b},\theta_{b};b,\phi_{w})=$ (1)
$\displaystyle\quad-\frac{I_{b}(r_{b},\theta_{b})}{2\pi
a}\left[1+2\sum_{n=1}^{\infty}\left(\frac{r_{b}}{a}\right)^{n}\cos
n\left(\phi_{w}-\theta_{b}\right)\right].$

We can integrate the above to obtain the current on each plate $R$(ight),
$L$(eft), $T$(op) and $B$(ottom). For example the current on the $R$(ight)
plate, $I_{R}$, is simply $I_{R}=\int_{-\phi_{0}/2}^{\phi_{0}/2}d\phi\;aJ_{w}$
to give

$\displaystyle I_{R}$
$\displaystyle=-\frac{I_{b}(x_{b},y_{b})}{2\pi}\left\\{\phi_{0}+2\left[2\left(\frac{x_{b}}{a}\right)\sin\frac{\phi_{0}}{2}+\right.\right.$
(2)
$\displaystyle\quad\left.\left.\left(\frac{x^{2}_{b}-y^{2}_{b}}{a^{2}}\right)\sin\phi_{0}\right]\right\\}.$

where we have only kept terms lower than $(r_{b}/a)^{3}$. We can obtain
similar equations for $I_{L}$, $I_{T}$ and $I_{B}$.

Figure 1: This cartoon shows the line current $I_{b}$ at $(r_{b},\theta_{b})$
inducing a current density on the $R$(ight) pickup plate. All the plates have
radius $a$ and subtends an angle $\phi_{0}$. The current density at
$(a,\phi_{w})$ is $J_{w}$.

And if the transverse distribution of the beam is a bi-gaussian distribution
centred at $(\bar{x},\bar{y})$ with standard deviations $\sigma_{x}$ and
$\sigma_{y}$ in the $x$ and $y$ directions respectively, then the normalized
gaussian distribution function is

$\rho(x,y)=\frac{1}{2\pi\sigma_{x}\sigma_{y}}\exp\left[-\frac{(x-\bar{x})^{2}}{2\sigma_{x}^{2}}\right]\exp\left[-\frac{(y-\bar{y})^{2}}{2\sigma_{y}^{2}}\right].$
(3)

With the above density function, the new current distribution on the $R$(ight)
plate is

$\displaystyle R=\int_{-\infty}^{\infty}dx\;dy\;\rho I_{R},$
$\displaystyle=-\frac{I_{b}}{2\pi}\left\\{\phi_{0}+2\left[2\left(\frac{\bar{x}}{a}\right)\sin\frac{\phi_{0}}{2}+\right.\right.$
(4)
$\displaystyle\quad\left.\left.\left(\frac{\sigma_{x}^{2}-\sigma_{y}^{2}}{a^{2}}+\frac{\bar{x}^{2}-\bar{y}^{2}}{a^{2}}\right)\sin\phi_{0}\right]\right\\}.$

The remaining plates will also have a similar current distributions $L$, $T$,
$B$.

Finally, we can take the appropriate sum and difference combinations of $R$,
$L$, $T$ and $B$ to create the dipole modes, $d_{x}$ and $d_{y}$ and
quadrupole mode $q$

$\left.\begin{aligned}
d_{x}&=\frac{R-L}{R+L+T+B}=\frac{\sin\frac{\phi_{0}}{2}}{\frac{\phi_{0}}{2}}\left(\frac{\bar{x}}{a}\right)\\\
d_{y}&=\frac{T-B}{R+L+T+B}=\frac{\sin\frac{\phi_{0}}{2}}{\frac{\phi_{0}}{2}}\left(\frac{\bar{y}}{a}\right)\\\
q&=\frac{2\sin\phi_{0}}{\phi_{0}}\left(\frac{\sigma_{x}^{2}-\sigma_{y}^{2}}{a^{2}}+\frac{\bar{x}^{2}-\bar{y}^{2}}{a^{2}}\right)\end{aligned}\quad\right\\}.$
(5)

$q$ can be written in terms of $d_{x}$ and $d_{y}$ to give

$q-(d_{x}^{2}-d_{y}^{2})\phi_{0}\cot\frac{\phi_{0}}{2}=\frac{2\sin\phi_{0}}{\phi_{0}}\left(\frac{\sigma_{x}^{2}-\sigma_{y}^{2}}{a^{2}}\right).$
(6)

The above can be written In terms of lattice functions by using the following
relationship

$\sigma_{x,y}^{2}=\frac{\beta_{x,y}\epsilon_{x,y}}{\pi}+(D_{x,y}\sigma_{p})^{2}$
(7)

where $\epsilon_{x,y}$ are the emittances, $\beta_{x,y}$ are the beta
functions, $D_{x,y}$ are the dispersions of the beam in the $x$ and $y$
directions respectively; and $\sigma^{2}_{p}=\langle(dp/p)^{2}\rangle$ is the
standard deviation of the relative momentum spread of the beam.

Thus, Eq. (6) becomes

$\Delta_{q}\equiv\frac{2}{a^{2}}\frac{\sin\phi_{0}}{\phi_{0}}\left[\frac{\beta_{x}\epsilon_{x}}{\pi}-\frac{\beta_{y}\epsilon_{y}}{\pi}+(D_{x}\sigma_{p})^{2}\right]$
(8)

where $\Delta_{q}$ is as defined above and we have set $D_{y}=0$ because the
vertical dispersion is small in Booster.

We can write down a matrix equation using the above from the measurements of
$1$ to $n$ BPMs or $1$ to $n$ samples from $m$ BPMs

$\begin{pmatrix}\beta_{x}(1)&-\beta_{y}(1)&D_{x}^{2}(1)\\\
\beta_{x}(2)&-\beta_{y}(2)&D_{x}^{2}(2)\\\ \vdots&\vdots&\vdots\\\
\beta_{x}(j)&-\beta_{y}(j)&D_{x}^{2}(j)\\\ \vdots&\vdots&\vdots\\\
\beta_{x}(n)&-\beta_{y}(n)&D_{x}^{2}(n)\end{pmatrix}\begin{pmatrix}\epsilon_{x}/\pi\\\
\epsilon_{y}/\pi\\\
\sigma_{p}^{2}\end{pmatrix}=\frac{a^{2}\phi_{0}}{2\sin\phi_{0}}\begin{pmatrix}\Delta_{q}(1)\\\
\Delta_{q}(2)\\\ \vdots\\\ \Delta_{q}(j)\\\ \vdots\\\
\Delta_{q}(n)\end{pmatrix}$ (9)

where the $j$th row comes from the $j$th pickup or the $j$th sample. If $n>3$
then we have a non-square matrix on the lhs, which means that we have an over-
determined set of equations. This non-square matrix is easily inverted using
SVD methods. See for example, Mathematica’s PseudoInverse[] function.

###  2.1 Data Manipulations

If we naïvely perform the inversion in Eq. (9) to obtain the emittances, we
will find that we will get garbage. The keys for getting sensible emittances
are discussed below.

  1. 1.

Theoretically, the gain on each BPM plate is required to be the same for Eq.
(9) to work. Therefore, we have to correct the signal from the BPM plates to
ensure that this condition is satisfied. Note: The choice of gain is critical
for ensuring the emittance solutions are real and not complex. Details about
our method is discussed in Ref. [6].

  2. 2.

Any DC offsets from each BPM plate when there is no beam.

  3. 3.

The most important observation of Eq. (9) is that on the rhs, there is the
difference $\beta_{x}\epsilon_{x}/\pi-\beta_{y}\epsilon_{y}/\pi$. This means
that if we had a round beam and $\beta_{x}\approx\beta_{y}$ then this
difference is close to zero. So, in this configuration, it may not be possible
to actually extract out the emittances. The ideal situation for extracting out
the emittances would be to have one of the $\beta$’s be a lot larger than the
other, i.e. either $\beta_{x}\gg\beta_{y}$ or $\beta_{x}\ll\beta_{y}$.

For the Booster, the BPMs in the S locations are good for extracting
$\epsilon_{x}$ because
$\beta_{x}($30\text{\,}\mathrm{m}$)\gg\beta_{y}($5\text{\,}\mathrm{m}$)$ while
the BPMs at the L locations are good for extracting $\epsilon_{y}$ because
$\beta_{x}($7\text{\,}\mathrm{m}$)\ll\beta_{y}($20\text{\,}\mathrm{m}$)$.111We
are using the symbols “$\gg$” and “$\ll$” loosely here. Therefore, Eq. (9)
must contain data from both L and S BPMs to get good $\epsilon_{x,y}$
solutions. In this paper, we have paired the data from S01 and L01, S02 and
L02, etc. for solving Eq. (9).

###  2.2 Systematic Error Removal

Figure 2: The systematic error of the emittances calculated with Eq. (9) for
BPM pairs (L15, S15) to (L23, S23).

We will find that even after applying the keys in the previous section, the
emittances found from Eq. (9) for every BPM pair have different, but
reproducible, systematic errors. The systematic errors of each BPM pair (L15,
S15) to (L23, S23) at three different beam intensities at 6, 10 and 15
turns222The Linac beam is “stacked” in Booster in units of turns. Higher turns
means higher intensity in Booster. can be seen in Fig. 2. For each BPM pair,
we have collected data at each intensity thrice to show that the curves are
reproducible.

Figure 3: The horizontal and vertical emittance growth w.r.t. 6 turns for BPM
pairs (L19, S19) to (L21, S21) for 10, 12, 15 and 17 turns are shown here.

Since the systematic error for each BPM pair is reproducible, we can simply
correct it by defining the 6 turn curve in Fig. 2 to be the reference. We can
then take the difference between any $n$ turn curve with the 6 turn curve to
obtain the emittance growth w.r.t. 6 turns emittance. A sample of the
horizontal and vertical emittance growth w.r.t. 6 turns for BPM pairs (L19,
S19) to (L21, S21) for 10, 12, 15 and 17 turns are shown in Fig. 3.

From this example, although the curves from each BPM pair are not the same, we
can identify some common features. This leads us to make the hypothesis that
the differences are due to underlying random noise that can be averaged out if
our system is ergodic. This means that we can average over the emittance from
all the BPM pairs (space average) which should be the same as the averaged
emittance at each BPM pair (time average).

##  3 Comparison to IPM emittance

The application of the ergodic hypothesis was the last step in the analysis of
our method. At this point, we have to do some verification. One way to do this
is to compare the BPM and IPM emittance measurements at different beam
intensities.

Figure 4: The vertical emittance growth w.r.t. 6 turns measured by the IPM
(left) and the BPMs (right) for 8, 10, 12, 14, and 16 turns. The dotted line
is where the beam crosses transition. Note: the IPM measured emittance is too
large; the shaded regions in the right plot are the statistical 1 sigma spread
of the emittance at each intensity. Sectors 5, 14 and 22 were ignored because
of bad BPM data.

For the first test, shown in Fig. 4, we will look at the vertical emittance at
different intensities because the vertical IPM is the reliable channel.
Unfortunately, when we took the data, the IPM emittance results are at least a
factor of 4 – 10 times too large and a linear growth which gets more steep as
the intensity increases. We have not found any mechanism, including ecloud,
that can cause this type of growth. On the other hand, the BPM emittance
growth is a lot more realistic. The beam has an initial growth then flattens.
A second growth happens after transition.

For the second test, we will look at the horizontal emittance. On a particular
occasion (11 Jul 2023), the horizontal IPM functioned and returned data that
looked reasonable except that it is too big by at least a factor of 4. Due to
the lack of horizontal IPM data, we can only compare the IPM emittance growth
taken on 11 Jul 2023 to the the BPM emittance growth data taken on 25 Mar 2024
in Fig. 5. Even though the data was taken about 7 months apart, they look
similar, with the emittance oscillating after transition.

Figure 5: The horizontal emittance growth w.r.t. 6 turns measured by the IPM
(left) and the BPMs (right) are shown here. The IPM emittance is about 4 times
too large. Unfortunately, we can only compare data taken on different dates.

##  4 CONCLUSION

We have improved the extraction of emittance from the BPM quadrupole mode by
using multiple BPMs and with better data analysis. The compromise is that we
do not measure absolute emittance but the emittance growth w.r.t. some
reference. We have found that the emittance growth extracted from the BPM
quadrupole mode looks more sensible than those measured by the IPMs. As part
of making this method easy to use and without any expert tuning, we have
written a python GUI to interact with the user and a C++ program backend that
can calculate Booster’s emittance from real time BPM data. If this method
proves to be as robust as we believe it to be, it opens up the ability of
measuring emittances in nearly all rings that have 4 plate BPMs.

##  5 ACKNOWLEDGMENTS

We would like to thank S. Chaurize, K. Triplett and J. Kuharik of the
PS/Booster group who helped us take data. J. Diamond of the Controls group, R.
Santucci and D. Steincamp of the Instrumentation group who helped us access
the BPM frontends. V. Kapin of PS/Physics and R. Thurman-Keup of
Instrumentation who helped us get the raw IPM data.

## References

  * [1] R. H. Miller et al., “Nonintercepting emittance monitor”, SLAC, USA, Rep. SLAC-PUB-3186, Aug. 1983.
  * [2] G. Nassibian, “The measurement of the multipole coefficients of a cylindrical charge distribution”, CERN, Geneva, Switzerland, Rep. CERN-SI-Note-EL-70-13, Dec. 1970.
  * [3] S. J. Russell and B. E. Carlsten, “Measuring emittance using beam position monitors”, in _Proc. 15th IEEE Part. Accel. Conf._ , Washington, DC, USA, May 1993, pp. 2537–2539.
doi:10.1109/PAC.1993.309381

  * [4] C. Bhat, “Booster IPM puzzle”, https://beamdocs.fnal.gov/cgi-bin/sso/ShowDocument?docid=9264.
  * [5] V. Shiltsev, “On Booster intensity and IPM diagnostics (analysis of the 2019 Booster beam study S09 data)”,
https://beamdocs.fnal.gov/cgi-bin/sso/ShowDocument?docid=7997.

  * [6] C. Y. Tan, “Booster beam emittances from quadrupole modes measured by BPMs”, https://beamdocs.fnal.gov/cgi-bin/sso/ShowDocument?docid=9916.


# On-shell effective field theory and
quantum transport for hard photons

Marc Comadran1,2,3<EMAIL_ADDRESS>Cristina Manuel1,2
<EMAIL_ADDRESS>1Instituto de Ciencias del Espacio (ICE, CSIC)
C. Can Magrans s.n., 08193 Cerdanyola del Vallès, Spain and
2Institut d’Estudis Espacials de Catalunya (IEEC)
08860 Castelldefels (Barcelona), Spain and  3Departament de Física Quàntica i
Astrofísica and Institut de Ciències del Cosmos, Universitat de Barcelona
(IEEC-UB), Martí i Franqués 1, 08028 Barcelona, Spain

###### Abstract

We develop an effective field theory for the description of high energetic or
hard photons, the on-shell effective theory (OSEFT). The OSEFT describes the
so called eikonal or semi-classical optical limit, allowing for corrections
organized in a systematic expansion on inverse powers of the photon energy. We
derive the OSEFT from the Maxwell Lagrangian, and study its different
properties, such as the gauge symmetry and reparametrization invariance. The
theory can be finally formulated in terms of a gauge invariant vector gauge
field, without the need to introduce gauge-fixing. We then use the OSEFT to
compute corrections to the Wigner photon function, and derive its associated
side jump effect from reparametrization invariance. Finally, we discuss how to
properly define the Stokes parameters from transport theory once quantum
effects are considered, so as to preserve their well-defined properties under
Lorentz transformations.

###### Contents

  1. I Introduction
  2. II On-shell effective field theory for photons
    1. II.1 Vacuum propagator and dispersion relation
    2. II.2 Effective gauge field for almost on-shell photons
    3. II.3 OSEFT gauge transformations
    4. II.4 Polarization vectors
    5. II.5 Reparametrization invariance
    6. II.6 Going backward to the full theory variables
  3. III Quantum kinetic theory for photons from the OSEFT
    1. III.1 Wigner function for photons
    2. III.2 Side jumps from reparametrization invariance
  4. IV Stokes parameters with quantum corrections
  5. V Discussion
  6. A Operator notation and identities
  7. B RI transformations

##  I Introduction

Effective field theories are one of the most useful and advantageous tools in
physics. They rely on the idea that in order to describe some phenomena at a
given energy scale it is enough to identify the degrees of freedom that
operate at that scale, and uncover the Lagrangian that governs their dynamics,
exploiting the symmetries of the problem. The effective Lagrangian is then
organized in operators of increasing dimension over powers of the high energy
scale. A different set of effective field theories have been proposed to
describe a wide variety of physical phenomena, see the excellent reviews
Manohar:1996cq ; Manohar:2018aog ; Becher:2014oda  to discuss the most
relevant technical details and most used effective field theories for vacuum
physics.

In this manuscript we will focus on the so called on-shell effective field
theory (OSEFT) Manuel:2014dza ; Manuel:2016wqs ; Manuel:2016cit ;
Carignano:2018gqt ; Carignano:2019zsh ; Manuel:2021oah ; Comadran:2021pkv ,
which was first developed to describe energetic chiral fermions, with the main
aim to describe quantum corrections to the classical transport equations
Manuel:2014dza . The OSEFT also proved to be useful to compute power
corrections to photon self-energy diagrams in high temperature $T$ plasmas
Manuel:2016wqs , or mass corrections to the same amplitudes Comadran:2021pkv ,
taking profit of the natural hierarchy of energy scales that appear in these
systems.

Our focus in this manuscript is to describe the OSEFT associated to high
energetic or hard photons. This presents different challenges, among them, a
proper description respectful with the gauge symmetry of the photon degrees of
freedom. We first present the complete OSEFT associated to an electromagnetic
field, as being derived from the Maxwell Lagrangian. As we will discuss, the
leading order Lagrangian in a high energy expansion describes the so called
eikonal or optical limit, while higher order terms in such an expansion would
describe quantum corrections to the eikonal limit. Notably, the effective
field theory can be formulated without imposing any gauge fixing condition.
The non-physical components of the vector gauge field potential can be
eliminated, after integrating out and using local field redefinitions.
Remarkably, we will show that the final OSEFT Lagrangian only contains one
vector degree of freedom and demonstrate its invariance under gauge
transformations that respect the energy scaling of the effective photon field.
By studying the reparametrization invariance (RI) of the theory, we also check
that the OSEFT for photons is respectful with the Lorentz symmetry.

Then, we also consider a many-body system, such as a thermal plasma. By
employing the OSEFT for photons and well-established thermal field theory
techniques, we derive the quantum kinetic equations obeyed by the photon
Wigner function. Also, we show that the OSEFT allows us to systematically
compute quantum corrections associated to the photon Wigner function.

The polarization space components of the photon Wigner function can be related
to the so called Stokes parameters jackson . While the Stokes parameters are
not Lorentz invariant, some ratios of them are Cocke ; Han:1997zz . For
example, the percentage of circular polarization of a system is expected to be
a Lorentz invariant. In this work, we will see that when quantum corrections
are taken into consideration, the classical definition of the polarization
ratio is no longer Lorentz invariant. This is related to the fact that in
quantum kinetic theory, Lorentz transformations acquire non trivial
modifications when quantum effects are taken into account Chen:2014cla . The
unusual transformation properties have meaningful physical implications, for
instance the so called side jump effect Chen:2015gta ; Stone:2015kla . Hence,
we propose how to generalize the definition of the Stokes parameters, so as to
preserve the Lorentz invariance of the polarization ratios when small quantum
effects are taken into account. As we will discuss, the modification is
relevant when the frame of reference of observation is not at rest with the
thermal ensemble where the photon radiation is produced. For instance, this
could be realized in many astrophysical and cosmological settings, where those
conditions are usually met.

This manuscript is organized as follows, in Sec.II we develop the OSEFT
associated to highly energetic photons. In Sec.II.1 we derive the OSEFT vacuum
propagator and the dispersion relation, also, in Sec.II.2 we construct an
effective gauge field for almost on-shell photons and in Sec.II.3 we discuss
how the gauge symmetry is realized in the OSEFT. Then, in Sec.II.4 we
introduce a polarization basis in the effective field theory, while Sec.II.5
is devoted to the study of the RI of the theory. In Sec.II.6 we explain the
relation between the full theory and OSEFT variables. Then we employ the OSEFT
in Sec.III to develop a quantum kinetic theory for hard photons. Using those
results, in Sec.III.1 we construct the photon Wigner function while in
Sec.III.2 a derivation of the side jump effect for photons from first
principles is presented. In addition, in Sec.IV we discuss the Lorentz
transformation properties of the Stokes parameters when small quantum effects
are taken into account, from a quantum kinetic theory perspective. We present
our conclusions in Sec.V. Finally, we elaborate in App.A a simplified operator
notation used throughout the paper and explain some operator identities used
in this work. In App.B we give general RI transformations for the effective
photons fields which are too lengthy to include in the main text.

We use natural units $c=\hbar=k_{B}=1$, metric conventions
$\text{diag}(g^{\mu\nu})=(1,-1,-1,-1)$ and the normalization
$\epsilon^{0123}=1$ for the Levi-Civita tensor. Sometimes we will use the
shortened notations $A^{(\mu}B^{\nu)}=A^{\mu}B^{\nu}+A^{\nu}B^{\mu}$ and
$A^{[\mu}B^{\nu]}=A^{\mu}B^{\nu}-A^{\nu}B^{\mu}$. We employ boldface letters
for three dimensional vectors e.g
$A^{\mu}=(A^{0},A^{i})=(A^{0},\boldsymbol{A})$.

##  II On-shell effective field theory for photons

In this section we develop the OSEFT associated to hard or energetic photons.
It fully corresponds to an effective field theory treatment of the so called
eikonal or optical limit. The eikonal approximation is considered as a sort of
semi-classical approach, valid when the wavelength of the photon is much
shorter than any other length scale in the problem. The OSEFT allows us to
study corrections to the pure classical term as a series of operators of
increasing dimension over powers of the photon energy, which is the inverse of
the photon wavelength. These will represent quantum corrections to the semi-
classical picture, an explicit example of which will be presented in the
manuscript.

Our starting point is the Lagrangian describing the propagation of free
electromagnetic fields 111It is possible to generalize this procedure for
electromagnetic field propagation in a medium characterized by a refractive
index, as long as the space of variation of the index is much larger than the
photon wavelength. We will leave this case for future studies.

$\mathcal{L}=-\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}\ .$ (1)

Here $F^{\mu\nu}(x)=\partial^{\mu}A^{\nu}(x)-\partial^{\nu}A^{\mu}(x)$ is the
electromagnetic field strength tensor, and $A^{\mu}(x)$ is the vector gauge
field potential. The momentum of an almost on-shell photon in the frame
characterized by the time-like vector $u^{\mu}$ (satisfying $u^{2}=1$), can be
decomposed into on-shell and residual parts as follows

$q^{\mu}=p^{\mu}+k^{\mu}=Ev^{\mu}+k^{\mu}\ ,$ (2)

being $E=p\cdot u$ the energy of the photon in that frame, $v^{\mu}$ a light-
like vector ($v^{2}=0$) and $k^{\mu}$ the so called residual momentum. The
above decomposition assumes $k^{\mu}\ll Ev^{\mu}$, so that one can identify
the photon energy $E$ as the hard scale. We define another light-like vector
$\tilde{v}^{\mu}$ ($\tilde{v}^{2}=0$) satisfying $v\cdot\tilde{v}=2$, such
that

$u^{\mu}=\frac{v^{\mu}+\tilde{v}^{\mu}}{2}\ .$ (3)

Moreover, it will be useful to introduce an additional space-like vector

$n^{\mu}=\frac{\tilde{v}^{\mu}-v^{\mu}}{2}\ ,$ (4)

which satisfies $n^{2}=-1$. Then, the momentum splitting of Eq. (2) is
performed at the Lagrangian level

$\mathcal{L}_{E,v}=-\dfrac{1}{4}\left(\partial^{\mu}A^{\nu}_{v}-\partial^{\nu}A^{\mu}_{v}\right)^{2}\
,$ (5)

and we factor out the hard momenta of the vector gauge field potential as

$A^{\mu}_{v}(x)=e^{-iEv\cdot x}\xi^{\mu}(x)+e^{iEv\cdot
x}\xi^{\mu{\dagger}}(x)\ .$ (6)

The above decomposition then assumes that the field $\xi^{\mu}(x)$ only has a
dependence on the residual momenta. In terms of this field we can rewrite the
Lagrangian of Eq. (5) as

$\mathcal{L}_{E,v}=\dfrac{1}{2}\xi^{\mu{\dagger}}\left(g_{\mu\nu}\square-\partial_{\mu}\partial_{\nu}-2iEg_{\mu\nu}(v\cdot\partial)+E^{2}v_{\mu}v_{\nu}+iE(v_{\mu}\partial_{\nu}+\partial_{\mu}v_{\nu})\right)\xi^{\nu}+\text{h.c.}\
,$ (7)

where the oscillating terms $\sim e^{\pm 2iEv\cdot x}$ have been dropped,
also, we use $\square=\partial^{\mu}\partial_{\mu}$ and h.c. stands for
hermitian conjugate. However, we can as well construct an artificial covariant
derivative by introducing

$\mathcal{D}^{\mu}\equiv\partial^{\mu}-iEv^{\mu}\ ,$ (8)

and rewrite the Lagrangian in the compact form

$\mathcal{L}_{E,v}=\dfrac{1}{2}\xi^{\mu{\dagger}}\left(g_{\mu\nu}\mathcal{D}^{2}-\mathcal{D}_{\mu}\mathcal{D}_{\nu}\right)\xi^{\nu}+\text{h.c}\
,$ (9)

with
$\mathcal{D}^{2}=\mathcal{D}^{\mu}\mathcal{D}_{\mu}=\square-2iE(v\cdot\partial)$.
The $\xi^{\mu}(x)$ field contains both physical and nonphysical components. At
the classical level, this last can be eliminated for instance by imposing
gauge fixing conditions. Below we will show that, under the above assumptions,
the nonphysical components can be eliminated from the Lagrangian using
effective field theory techniques, with no need of gauge-fixing.

Let us start by introducing the transverse projector to $v^{\mu}$ and
$\tilde{v}^{\mu}$

$\displaystyle
P_{\perp}^{\mu\nu}=g^{\mu\nu}-\dfrac{1}{2}(v^{\mu}\tilde{v}^{\nu}+\tilde{v}^{\mu}v^{\nu})\
,$ (10)

which obeys

$P_{\perp}^{\mu\nu}P_{\perp,\mu\rho}=P_{\perp,\rho}^{\nu}\ ,\quad
P_{\perp}^{2}=2\ .$ (11)

Note that in the rest frame, where $u^{\mu}=(1,\boldsymbol{0})$, one has
$v^{\mu}=(1,\boldsymbol{v})$, $\tilde{v}^{\mu}=(1,-\boldsymbol{v})$ and
$n^{\mu}=(0,-\boldsymbol{v})$, so that the transverse projector only has
spatial components in that frame and is perpendicular to $\boldsymbol{v}$.
Using the transverse projector, we can split the $\xi^{\mu}(x)$ field into
different components

$\xi^{\mu}(x)=\xi_{\perp}^{\mu}(x)+v^{\mu}\phi(x)+n^{\mu}\lambda(x)\ ,$ (12)

where each component is clearly identified

$\xi_{\perp}^{\mu}(x)=P_{\perp}^{\mu\nu}\xi_{\nu}(x)\
,\qquad\phi(x)=(u\cdot\tau)(x)\ ,\qquad\lambda(x)=(v\cdot\tau)(x)\ .$ (13)

The $\xi_{\perp}^{\mu}(x)$ field describes transverse degrees of freedom to
$v^{\mu}$ and $\tilde{v}^{\mu}$, while the remaining longitudinal and scalar
degrees of freedom are both described by the $\lambda(x)$ and $\phi(x)$
fields. There is a certain freedom in the decomposition of the non-transverse
part of Eq.(12). We have chosen the one that singles out the component that is
still transverse to the frame vector $u^{\mu}$, that is $\lambda(x)$ (since
$u\cdot n=0$). The suitability of this choice will be explained throughout the
paper. In terms of these components, the Lagrangian can be written as

$\displaystyle\mathcal{L}_{E,v}=\dfrac{1}{2}\bigg{\\{}\xi_{\perp}^{\mu{\dagger}}\left(g_{\mu\nu}\mathcal{D}^{2}-\partial_{\mu}\partial_{\nu}\right)\xi_{\perp}^{\nu}-\phi^{\dagger}(v\cdot\partial)^{2}\phi-\lambda^{\dagger}(\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2})\lambda$
$\displaystyle-\xi_{\perp}^{\mu{\dagger}}\partial_{\mu}(v\cdot\partial)\phi-\phi^{\dagger}(v\cdot\partial)\partial_{\mu}\xi_{\perp}^{\mu}-\xi_{\perp}^{\mu{\dagger}}\partial_{\mu}(n\cdot\mathcal{D})\lambda-\lambda^{\dagger}(n\cdot\mathcal{D})\partial_{\mu}\xi_{\perp}^{\mu}$
$\displaystyle+\phi^{\dagger}(\mathcal{D}^{2}-(v\cdot\partial)(n\cdot\mathcal{D}))\lambda+\lambda^{\dagger}(\mathcal{D}^{2}-(v\cdot\partial)(n\cdot\mathcal{D}))\phi\bigg{\\}}+\text{h.c}\
.$ (14)

Above, one can see that the operators accounting for the propagation of the
various components do not have the same power counting in energy $E$, for
instance $\mathcal{D}^{2}\sim E$ and $(n\cdot\mathcal{D})^{2}\sim E^{2}$.
Hence, we identify the $\lambda(x)$ field as the degree of freedom that can be
integrated out using its classical equation of motion

$\lambda(x)=-\frac{1}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\left[(n\cdot\mathcal{D})\partial_{\mu}\xi^{\mu}_{\perp}(x)-(\mathcal{D}^{2}-(v\cdot\partial)(n\cdot\mathcal{D}))\phi(x)\right]\
.$ (15)

The resulting Lagrangian may be written as

$\displaystyle\mathcal{L}_{E,v}=\dfrac{1}{2}\bigg{\\{}\xi_{\perp}^{\mu{\dagger}}\,\left(g_{\mu\nu}\mathcal{D}^{2}-\partial_{\mu}\partial_{\nu}\dfrac{\mathcal{D}^{2}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\right)\xi_{\perp}^{\nu}$
$\displaystyle-\xi_{\perp}^{\mu{\dagger}}\partial_{\mu}\dfrac{\mathcal{D}^{2}(u\cdot\mathcal{D})}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\phi-\phi^{\dagger}\dfrac{\mathcal{D}^{2}(u\cdot\mathcal{D})}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\partial_{\mu}\xi_{\perp}^{\mu}+\phi^{\dagger}\dfrac{\partial_{\perp}^{2}\mathcal{D}^{2}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\phi\bigg{\\}}+\text{h.c}\
,$ (16)

where we defined $\partial_{\perp}^{\mu}\equiv
P_{\perp}^{\mu\nu}\partial_{\nu}$. In the above Lagrangian and for the rest of
the paper, we will use a compact operator notation that we elaborate in App.
A. Subsequently, the above operators are expanded and organized in inverse
powers of the hard scale $(1/E)^{n}$, yielding an infinite series of
Lagrangians $\mathcal{L}_{E,v}^{(n)}$, each of them encompassing operators
that increase in dimension with the power of $n$. Specifically, using the
expansions

$\displaystyle\dfrac{\mathcal{D}^{2}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}=\dfrac{i(v\cdot\partial)}{E}+\mathcal{O}\left(\dfrac{1}{E^{2}}\right)\
,$ (17a)
$\displaystyle\dfrac{\mathcal{D}^{2}(u\cdot\mathcal{D})}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}=2(v\cdot\partial)+\dfrac{i}{E}\left(\partial_{\perp}^{2}-(v\cdot\partial)^{2}\right)+\mathcal{O}\left(\dfrac{1}{E^{2}}\right)\
,$ (17b)

we can write down the the first three orders

$\displaystyle\mathcal{L}_{E,v}^{(-1)}=-E\,\xi_{\perp}^{\mu{\dagger}}\,g_{\mu\nu}(iv\cdot\partial)\xi_{\perp}^{\nu}+\text{h.c}\
,$ (18a)
$\displaystyle\mathcal{L}_{E,v}^{(0)}=\dfrac{1}{2}\xi_{\perp}^{\mu{\dagger}}\,g_{\mu\nu}\square\xi_{\perp}^{\nu}-\xi_{\perp}^{\mu{\dagger}}\,\partial_{\mu}\left(v\cdot\partial\right)\phi-\phi^{\dagger}\left(v\cdot\partial\right)\partial_{\mu}\xi_{\perp}^{\mu}+\text{h.c}\
,$ (18b)
$\displaystyle\mathcal{L}_{E,v}^{(1)}=\dfrac{i}{E}\bigg{(}-\xi_{\perp}^{\mu{\dagger}}\,\partial_{\mu}(v\cdot\partial)\partial_{\nu}\xi_{\perp}^{\nu}-\dfrac{1}{2}\xi_{\perp}^{\mu{\dagger}}\,\partial_{\mu}\left(\partial_{\perp}^{2}-(v\cdot\partial)^{2}\right)\phi-\dfrac{1}{2}\phi^{\dagger}\,\left(\partial_{\perp}^{2}-(v\cdot\partial)^{2}\right)\partial_{\mu}\xi_{\perp}^{\mu}$
$\displaystyle+\phi^{\dagger}\,\partial_{\perp}^{2}(v\cdot\partial)\phi\bigg{)}+\text{h.c}\
.$ (18c)

Beyond the leading order, scalar and longitudinal degrees of freedom endure,
indicated by the presence of terms with the $\phi(x)$ field. However, we can
eliminate those terms through the application of local field redefinitions.
Indeed, the following transformation

$\xi^{\mu}_{\perp}(x)\ \longrightarrow\
\tau^{\mu}_{\perp}(x)=\xi_{\perp}^{\mu}(x)-\dfrac{i\partial_{\perp}^{\mu}}{E}\phi(x)-\dfrac{(v\cdot\partial+\tilde{v}\cdot\partial)\partial_{\perp}^{\mu}}{2E^{2}}\phi(x)\
,$ (19)

completely eliminates all terms with the $\phi(x)$ field from Eqs. (18b-18c).
We have demonstrated here a specific case of a much broader scenario. In
general, to eliminate the $\phi(x)$ field from the $n$-th order Lagrangian,
one needs to apply the following field redefinition

$\xi_{\perp}^{\mu}(x)\ \longrightarrow\
\tau^{\mu}_{\perp}(x)=\xi_{\perp}^{\mu}(x)-\dfrac{\partial_{\perp}^{\mu}}{u\cdot\mathcal{D}}\phi(x)\
,$ (20)

with the operators expanded up to $(n+1)$-th order in inverse powers of $E$.
In this way, the $\phi(x)$ field can be eliminated at all orders in the energy
expansion and the final Lagrangian can be written in terms of the locally
redefined field only as

$\begin{gathered}\mathcal{L}_{E,v}^{\prime}=\dfrac{1}{2}\tau_{\perp}^{\mu{\dagger}}\,\left(g_{\mu\nu}\mathcal{D}^{2}-\partial_{\mu}\partial_{\nu}\dfrac{\mathcal{D}^{2}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\right)\tau_{\perp}^{\nu}+\text{h.c}\
.\end{gathered}$ (21)

The final OSEFT Lagrangian only contains the vector degree of freedom
$\tau_{\perp}^{\mu}(x)$, corresponding to transverse, almost on-shell photons.
Note however, that we did not impose any particular gauge to derive it, in
fact, we demonstrate in Sec.II.3 that it enjoys a gauge symmetry. Notably, the
equivalence principle Haag:1958vt ; Coleman:1969sm ; Manohar:2018aog
guarantees that the on-shell quantities after the local field redefinition of
Eq.(20) remain unaffected.

Expanding now to the lowest orders we have

$\displaystyle\mathcal{L}_{E,v}^{,(-1)}=-E\tau_{\perp}^{\mu{\dagger}}\,g_{\mu\nu}(iv\cdot\partial)\tau_{\perp}^{\nu}+\text{h.c}\
,$ (22a)
$\displaystyle\mathcal{L}_{E,v}^{,(0)}=\dfrac{1}{2}\tau_{\perp}^{\mu{\dagger}}\,g_{\mu\nu}\square\tau_{\perp}^{\nu}+\text{h.c}\
,$ (22b)
$\displaystyle\mathcal{L}_{E,v}^{,(1)}=-\dfrac{1}{E}\tau_{\perp}^{\mu{\dagger}}\,\partial_{\mu}(iv\cdot\partial)\partial_{\nu}\tau_{\perp}^{\nu}+\text{h.c}\
.$ (22c)
$\displaystyle\mathcal{L}_{E,v}^{,(2)}=\dfrac{1}{2E^{2}}\tau_{\perp}^{\mu{\dagger}}\,\partial_{\mu}(\square-4(u\cdot\partial)(v\cdot\partial))\partial_{\nu}\tau_{\perp}^{\nu}+\text{h.c}\
.$ (22d)

The leading order Lagrangian describes the so called eikonal limit, while the
higher order Lagrangians capture the corrections to this limit. Let us discuss
how the above Lagrangian should be interpreted, as in an effective field
theory, one has to solve the theory order by order. For example, naively, the
equation of motion as derived from Eqs.(22a-22d) reads222We multiplied by an
overall factor of two for convenience.

$\left(-g_{\mu\nu}\left(2E\,iv\cdot\partial-\square\right)-\partial_{\mu}\left(\dfrac{2iv\cdot\partial}{E}-\dfrac{\square-4(u\cdot\partial)(v\cdot\partial)}{E^{2}}\right)\partial_{\nu}\right)\tau_{\perp}^{\nu}(x)=0\
.$ (23)

However, one can always use the equation of motion at a given order in $1/E$
as a constraint for the operators appearing at the next order. For instance,
the equation of motion at leading order
$i(v\cdot\partial)\tau_{\perp}^{\mu}(x)=0$ can be used to simplify the
operator
$\square\tau_{\perp}^{\mu}(x)=\partial_{\perp}^{2}\tau_{\perp}^{\mu}(x)+{\cal
O}(1/E)$ appearing at the next order. Performing this process at a fixed order
in $1/E$, one can always show without much difficulty that all structures
proportional to $\partial^{\mu}\partial^{\nu}$ vanish, so that the equation of
motion only contains the tensor structure $g^{\mu\nu}$ and can be formally
written as

$-\left(2E\,iv\cdot\partial-\square\right)\tau_{\perp}^{\mu}(x)=0\ ,$ (24)

or as $\mathcal{D}^{2}\tau_{\perp}^{\mu}(x)=0$, by employing the operator in
Eq.(8).

###  II.1 Vacuum propagator and dispersion relation

The OSEFT Green function in vacuum, that we define as
$\mathcal{G}_{\perp}^{\mu\nu}(x,y)=\braket{0}{\mathcal{T}\tau_{\perp}^{\mu{\dagger}}(x)\tau_{\perp}^{\nu}(y)}{0}$,
where $\mathcal{T}$ denotes time-ordering, obeys

$\mathcal{D}^{2}_{x}\,\mathcal{G}_{\perp}^{\mu\nu}(x,y)=P_{\perp}^{\mu\nu}\delta(x-y)\
,$ (25)

as can be deduced from Eq.(24). The above equation can be easily inverted in
(residual) momentum space, yielding to

$\mathcal{G}_{\perp}^{\mu\nu}(k)=\dfrac{-P_{\perp}^{\mu\nu}}{2E(v\cdot
k)+k^{2}+i0^{+}}\ .$ (26)

The dispersion equation is $2E(v\cdot k)+k^{2}=0$, which has to be solved
order by order in the energy expansion (see for instance Refs.Manuel:2014dza ;
Manuel:2016wqs ; Manuel:2016cit ). Doing so up to order $1/E$, one gets

$2E(v\cdot k)+k_{\perp}^{2}-\dfrac{(\tilde{v}\cdot k)k_{\perp}^{2}}{2E}=0\ ,$
(27)

so that in the rest frame, using the notation $k^{\mu}=(k_{0},\boldsymbol{k})$
for the residual momentum, the denominator of the propagator can be written as
$2E(k_{0}-\mathrm{f}(\boldsymbol{k}))+i0^{+}$ where
$\mathrm{f}(\boldsymbol{k})$ is the dispersion relation of the OSEFT, given by

$\displaystyle\mathrm{f}(\boldsymbol{k})=\boldsymbol{v}\cdot\boldsymbol{k}+\dfrac{\boldsymbol{k}_{\perp}^{2}}{2E}-\dfrac{(\boldsymbol{v}\cdot\boldsymbol{k})\boldsymbol{k}_{\perp}^{2}}{2E^{2}}+\mathcal{O}\left(\dfrac{1}{E^{3}}\right)\
,$ (28)

where we used the fact that in the rest frame one has
$k_{\perp}^{2}=-\boldsymbol{k}_{\perp}^{2}$ and $\tilde{v}\cdot k\approx
2(\boldsymbol{v}\cdot\boldsymbol{k})$.

###  II.2 Effective gauge field for almost on-shell photons

It is interesting to relate our starting effective field $\xi^{\mu}(x)$ with
the locally redefined field of the OSEFT Lagrangian $\tau_{\perp}^{\mu}(x)$.
In order to do so, we first insert the equation of motion of the $\lambda(x)$
field into Eq.(12), in addition, by writing
$\partial_{\perp}^{\mu}=\partial^{\mu}-v^{\mu}(u\cdot\partial)-n^{\mu}(v\cdot\partial)$
in the field redefinition of Eq.(20) and using the operator identity of
Eq.(94), we find that an effective gauge field for almost on-shell photons is

$\displaystyle\xi^{\mu}(x)=\left(g^{\mu}_{\nu}-n^{\mu}\partial_{\nu}\frac{n\cdot\mathcal{D}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\right)\tau_{\perp}^{\nu}(x)+\dfrac{\mathcal{D}^{\mu}}{u\cdot\mathcal{D}}\phi(x)\
.$ (29)

Actually, it is relatively easy to derive again the OSEFT Lagrangian in very
few steps, by directly plugging the above relation into Eq.(9) and noting that
all terms with the $\phi(x)$ field vanish due to
$\left(g_{\mu\nu}\mathcal{D}^{2}-\mathcal{D}_{\mu}\mathcal{D}_{\nu}\right)\mathcal{D}^{\mu}=0$.

The above expression is quite useful in computations, for instance, in Sec.III
we will use it to construct the photon Wigner function in a semi-classical
approximation, furthermore, it will help to understand the physical picture
underlying the OSEFT. This last question will be elaborated throughout the
manuscript, in particular in the next section, where we discuss how the gauge
symmetry is realized in the effective field theory.

###  II.3 OSEFT gauge transformations

In effective field theories, where there is a well defined hierarchy of
scales, it is common to separate the gauge transformations for each sector of
the theory according to their scale, respecting such a separation. One
particular example of this fact occurs in SCET, where one talks on a gauge
symmetry associated to the hard (or collinear) gluon fields, and another one
associated to the soft gluon fields Becher:2014oda . An additional multipole
expansion of the different fields might be needed to respect the energy
separation Beneke:2002ni . An analogous situation is expected in the OSEFT for
photons. In this work, we only consider the hard sector of the theory,
ignoring the soft gauge fields, as they do not interact. Further, we are not
considering the interaction with matter particles neither. All of them could
be incorporated, following the same SCET techniques. Here we will discuss the
gauge symmetry associated to hard photons. The Lagrangian of Eq.(5) enjoys the
gauge symmetry

$A^{\mu}_{v}(x)\longrightarrow A^{\mu}_{v}(x)+\partial^{\mu}\theta_{v}(x)\ ,$
(30)

for an arbitrary function $\theta_{v}(x)$ that respects the energy scaling of
the gauge field. Hence, the OSEFT Lagrangian of Eq.(21) should also posses a
gauge symmetry. By writing $\theta_{v}(x)=e^{-iEv\cdot x}\eta(x)+\text{h.c}$,
we can deduce how the different components of the photon field transform under
a gauge transformation. Explicitly, one then finds that the OSEFT vector gauge
field transforms as

$\xi^{\mu}(x)\longrightarrow\xi^{\mu}(x)+\mathcal{D}^{\mu}\eta(x)\ ,$ (31)

and thus

$\displaystyle\xi_{\perp}^{\mu}(x)\longrightarrow\xi_{\perp}^{\mu}(x)+\partial^{\mu}_{\perp}\eta(x)\
,$ (32a)
$\displaystyle\phi(x)\longrightarrow\phi(x)+(u\cdot\mathcal{D})\eta(x)\ ,$
(32b)
$\displaystyle\lambda(x)\longrightarrow\lambda(x)+(v\cdot\partial)\eta(x)\ .$
(32c)

Then, we can show that the final OSEFT Lagrangian is gauge invariant. To prove
it, we note that the locally redefined field is itself invariant under the
above set of gauge transformations. Indeed,

$\tau_{\perp}^{\mu}(x)\longrightarrow\xi_{\perp}^{\mu}(x)+\partial^{\mu}_{\perp}\eta(x)-\dfrac{\partial_{\perp}^{\mu}}{u\cdot\mathcal{D}}\left[\phi(x)+(u\cdot\mathcal{D})\eta(x)\right]=\tau_{\perp}^{\mu}(x)\
.$ (33)

Let us conclude this section with the following observation. Taking the four
divergence in Eq.(6) and resorting to Eq.(29) one can show that

$\partial_{\mu}A^{\mu}_{v}=\dfrac{\mathcal{D}^{2}}{(u\cdot\mathcal{D})}\phi(x)e^{-iEv\cdot
x}+\text{h.c}\ .$ (34)

Thus, if we had not carried out the local field redefinition and rather we
would have fixed the gauge as $\phi(x)=0$ in the initial Lagrangian, that
would imply that the gauge field obeys $u_{\mu}A_{v}^{\mu}=0$ and
$\partial_{\mu}A^{\mu}_{v}=0$. In conclusion, the resulting framework in the
OSEFT if we impose that $\phi(x)=0$ is equivalent to the Coulomb gauge.
However, let us stress that in this work we perform a local field redefinition
to eliminate the non-physical $\phi(x)$ component of the gauge field from the
Lagrangian, which allows us to work without choosing any particular gauge.

###  II.4 Polarization vectors

The field $\tau_{\perp}^{\mu}(x)$ is transverse to $v^{\mu}$ and
$\tilde{v}^{\mu}$. Thus, in the rest frame, it only has spatial components and
is transverse to $\boldsymbol{v}$. It is suggesting then to introduce
polarization vectors in the effective field theory.
Let us define a linear polarization basis, as $\\{e_{i}^{\mu}\\}$ with
$i=\\{1,2\\}$, satisfying $(e_{i}^{\mu})^{*}=e_{i}^{\mu}$, also

$v\cdot e_{i}=\tilde{v}\cdot e_{i}=0\ ,\quad e_{i}\cdot
e_{j}=g_{ij}=-\delta_{ij}\ ,$ (35)

so that they are unitary and space-like. We shall, however, work with circular
polarization vectors, that we introduce as $\\{e_{h}^{\mu}\\}$ with
$h=\\{+,-\\}$, obeying the properties

$v\cdot e_{h}=\tilde{v}\cdot e_{h}=0\ ,\quad e_{h}^{*}\cdot
e_{h^{\prime}}=-\delta_{hh^{\prime}}\ .$ (36)

Their relation with the linear polarization vectors is

$e^{\mu}_{h}=\dfrac{1}{\sqrt{2}}(e_{1}^{\mu}+ih\,e_{2}^{\mu})\ ,$ (37)

so that $(e_{h}^{\mu})^{*}=e_{-h}^{\mu}$. Note that both the linear and
circular polarization vectors are also transverse to $u^{\mu}$ and $n^{\mu}$.
We can relate the polarization vectors with the transverse projector and spin
tensor (introduced below) as

$\displaystyle
P_{\perp}^{\mu\nu}=g^{\mu\nu}-\dfrac{1}{2}(v^{\mu}\tilde{v}^{\nu}+\tilde{v}^{\mu}v^{\nu})=-e^{*\mu}_{+}e^{\nu}_{+}-e^{*\mu}_{-}e^{\nu}_{-}\
,$ (38a) $\displaystyle
S_{\perp}^{\mu\nu}=i\epsilon^{\mu\nu\alpha\beta}v_{\alpha}u_{\beta}=e^{*\mu}_{+}e^{\nu}_{+}-e^{*\mu}_{-}e^{\nu}_{-}\
.$ (38b)

The spin tensor obeys $S_{\perp}^{2}=-2$, but it is not a good projector,
since $S_{\perp}^{\mu\nu}S_{\perp,\mu\rho}=-P_{\perp,\rho}^{\nu}$. Thus, it is
useful to introduce the right ($h=+$) and left ($h=-$) projectors

$\displaystyle
P_{h}^{\mu\nu}=\dfrac{1}{2}(P_{\perp}^{\mu\nu}-hS_{\perp}^{\mu\nu})=-e^{*\mu}_{h}e^{\nu}_{h}\
,$ (39)

satisfying the properties

$\displaystyle P_{\pm}^{2}=0\ ,\quad P_{\pm}^{\mu\nu}P_{\mp,\mu\nu}=1\ ,\quad
P_{\pm}^{\mu\nu}P_{\pm,\mu\rho}=0\ ,\quad
P_{\pm}^{\mu\nu}P_{\mp,\mu\rho}=-P_{\mp,\rho}^{\nu}\ .$ (40)

Now, the field $\tau_{\perp}^{\mu}(x)$ can be decomposed in the circular
polarization basis as

$\tau_{\perp}^{\mu}(x)=\sum_{h=\pm}e_{h}^{\mu}\tau_{h}(x)\ .$ (41)

Hence, the components $\tau_{h}(x)=-(e_{h}^{*}\cdot\tau_{\perp})(x)$
correspond to right $(h=+)$ and left $(h=-)$ handed circularly polarized
photons respectively.

###  II.5 Reparametrization invariance

In general, an effective field theory should remain invariant under
infinitesimal transformations of the parameters used to describe the theory.
The invariance under these types of transformations is called
reparametrization invariance (RI) and it was first discussed in the context of
heavy quark effective field theory (HQEFT) Luke:1992cs , and later on
generalized for massless degrees of freedom in the context of soft collinear
effective field theory (SCFT) Manohar:2002fd .

RI is the symmetry associated with the ambiguity of the decomposition of the
full momentum $q^{\mu}$ in Eq. (2). A small shift in the velocity $v^{\mu}$
could be reabsorbed in the definition of the residual momentum $k^{\mu}$,
while the physics should remain unchanged. On the other hand, the explicit
choices of the vectors $v^{\mu}$ and ${\tilde{v}}^{\mu}$ seem to imply an
apparent breaking of the Lorentz symmetry. Checking the RI of the theory
ultimately confirms that Lorentz symmetry is respected in the effective field
theory.

RI has been extensively studied also in the OSEFT for fermions in
Carignano:2018gqt ; Manuel:2021oah . We will discuss how to generalize these
concepts for the photon fields here. The main idea is that the field
$A_{v}^{\mu}(x)$ should not change under infinitesimal transformations of the
parameters used to construct the effective field theory. Then, under a RI
transformation one should have

$A_{v}^{\mu}(x)\overset{\Lambda}{\longrightarrow}A_{v^{\prime}}^{\mu}(x)=A_{v}^{\mu}(x)\
,$ (42)

where we introduced the label $\Lambda=\\{\text{I, II, III}\\}$ for the three
types of RI transformations Manohar:2002fd . Thus, the OSEFT Lagrangian should
enjoy the following symmetries

$\text{(I)}:\begin{cases}v^{\mu}\rightarrow v^{\mu}+\Delta_{\perp}^{\mu}\\\
\tilde{v}^{\mu}\rightarrow\tilde{v}^{\mu}\end{cases}\
,\quad\text{(II)}:\begin{cases}v^{\mu}\rightarrow v^{\mu}\\\
\tilde{v}^{\mu}\rightarrow\tilde{v}^{\mu}+\tilde{\Delta}_{\perp}^{\mu}\end{cases}\
,\quad\text{(III)}:\begin{cases}v^{\mu}\rightarrow(1+\alpha)v^{\mu}\\\
\tilde{v}^{\mu}\rightarrow(1-\alpha)\tilde{v}^{\mu}\end{cases}\ ,$ (43)

where $\\{\Delta_{\perp}^{\mu},\tilde{\Delta}_{\perp}^{\mu},\alpha\\}$ are
five infinitesimal parameters, satisfying

$\Delta_{\perp}\cdot
v=\Delta_{\perp}\cdot\tilde{v}=\tilde{\Delta}_{\perp}\cdot
v=\tilde{\Delta}_{\perp}\cdot\tilde{v}=0\ .$ (44)

In Tab.(I-II) we show the transformation rules for the projected fields and
other relevant quantities of the effective field theory respectively.
Polarization vectors also change under RI transformations, as can be seen in
the last row of Tab.(II), so that they preserve their transversality to
$v^{\mu}$ and $\tilde{v}^{\mu}$.

|  Type (I) |  Type (II) |  Type (III)
---|---|---|---
$\xi_{\perp}^{\mu}$ |  $\xi_{\perp}^{\mu}-\Delta_{\perp}^{\mu}\phi+\frac{1}{2}\Delta_{\perp}^{\mu}\lambda-\frac{1}{2}v^{\mu}(\Delta_{\perp}\cdot\tau)$ |  $\xi_{\perp}^{\mu}-\frac{1}{2}v^{\mu}(\tilde{\Delta}_{\perp}\cdot\tau)-\frac{1}{2}\tilde{\Delta}_{\perp}^{\mu}\lambda$ |  $\xi_{\perp}^{\mu}$
$\phi$ |  $\phi+\frac{1}{2}(\Delta_{\perp}\cdot\tau)$ |  $\phi+\frac{1}{2}(\tilde{\Delta}_{\perp}\cdot\tau)$ |  $(1-\alpha)\phi+\alpha\lambda$
$\lambda$ |  $\lambda+\Delta_{\perp}\cdot\tau$ |  $\lambda$ |  $(1+\alpha)\lambda$
Table 1: Transformation rules for the projected fields
$\tau_{\perp}^{\mu}(x)$, $\lambda(x)$ and $\phi(x)$. We dropped the space time
arguments to enlighten the notation.

After integrating out, the transformations in Tab.(I) with the presence of the
$\lambda(x)$ field have to be modified. We derive and discuss the general
expressions for those transformations, i.e valid at any order in $1/E$, in
App. B. Remarkably, it can also be shown that the OSEFT Lagrangian, obtained
after the local field redefinition to eliminate the $\phi(x)$ field, is RI
invariant. In order to prove it, one just needs to know how the field
$\tau_{\perp}^{\mu}(x)$, introduced in Eq.(20), changes under each type of
transformation. We also give the general form of these last transformations in
App. B.

Hence, using Eqs.(103-105), it is possible to check that

$\delta_{(\text{I})}\mathcal{L}_{E,v}=\delta_{(\text{II})}\mathcal{L}_{E,v}=\delta_{(\text{III})}\mathcal{L}_{E,v}=0\
,$ (45)

or in other words, the OSEFT Lagrangian is RI invariant. It is worth
mentioning that all non-transverse structures in these last transformations
are not necessary to proof the RI invariance of the Lagrangian, as those
pieces always vanish when contracted with transverse tensors. However, they
would be necessary to derive the transformation rules for other quantities,
such as currents. The general transformations for $\tau_{\perp}^{\mu}(x)$ are
quite simple when expanded in powers of $1/E$. Precisely, keeping terms up to
first order in the energy expansion we find

$\displaystyle\tau_{\perp}^{\mu}\overset{\text{(I)}}{\longrightarrow}\tau_{\perp}^{\mu}-\dfrac{i\partial_{\perp}^{\mu}}{2E}(\Delta_{\perp}\cdot\tau_{\perp})-\dfrac{i\Delta_{\perp}^{\mu}}{2E}(\partial\cdot\tau_{\perp})-\dfrac{\tilde{v}^{\mu}}{2}(\Delta_{\perp}\cdot\tau_{\perp})\
,$ (46a)
$\displaystyle\tau_{\perp}^{\mu}\overset{\text{(II)}}{\longrightarrow}\tau_{\perp}^{\mu}-\dfrac{i\partial_{\perp}^{\mu}}{2E}(\tilde{\Delta}_{\perp}\cdot\tau_{\perp})+\dfrac{i\tilde{\Delta}_{\perp}^{\mu}}{2E}(\partial\cdot\tau_{\perp})-\frac{v^{\mu}}{2}(\tilde{\Delta}_{\perp}\cdot\tau_{\perp})\
,$ (46b)
$\displaystyle\tau_{\perp}^{\mu}\overset{\text{(III)}\,}{\longrightarrow}\tau_{\perp}^{\mu}\
.$ (46c)

Employing the expressions above, we derive in Sec.(III) the transformation
rules for the distribution function for photons at $1/E$ accuracy, in
particular using type (II) transformations one can dervie the so called side
jump effect.

|  Type (I) |  Type (II) |  Type (III)
---|---|---|---
$E$ |  $E$ |  $E$ |  $(1-\alpha)E$
$\partial^{\mu}$ |  $\partial^{\mu}+iE\Delta_{\perp}^{\mu}$ |  $\partial^{\mu}$ |  $\partial^{\mu}$
$\partial_{\perp}^{\mu}$ |  $\partial_{\perp}^{\mu}-\frac{1}{2}(\Delta_{\perp}\cdot\partial)\tilde{v}^{\mu}-\frac{1}{2}(\tilde{v}\cdot\mathcal{D})\Delta_{\perp}^{\mu}$ |  $\partial_{\perp}^{\mu}-\frac{1}{2}(\tilde{\Delta}_{\perp}\cdot\partial)v^{\mu}-\dfrac{1}{2}(v\cdot\partial)\tilde{\Delta}_{\perp}^{\mu}$ |  $\partial_{\perp}^{\mu}$
$\mathcal{D}^{\mu}$ |  $\mathcal{D}^{\mu}$ |  $\mathcal{D}^{\mu}$ |  $\mathcal{D}^{\mu}$
$u\cdot\mathcal{D}$ |  $u\cdot\mathcal{D}+\frac{1}{2}(\Delta_{\perp}\cdot\partial)$ |  $u\cdot\mathcal{D}+\frac{1}{2}(\tilde{\Delta}_{\perp}\cdot\partial)$ |  $u\cdot\mathcal{D}-\alpha(n\cdot\mathcal{D})$
$n\cdot\mathcal{D}$ |  $n\cdot\mathcal{D}-\frac{1}{2}(\Delta_{\perp}\cdot\partial)$ |  $n\cdot\mathcal{D}+\frac{1}{2}(\tilde{\Delta}_{\perp}\cdot\partial)$ |  $n\cdot\mathcal{D}-\alpha(u\cdot\mathcal{D})$
$\partial_{\perp}^{2}$ |  $\partial_{\perp}^{2}-(\Delta_{\perp}\cdot\partial)(\tilde{v}\cdot\mathcal{D})$ |  $\partial_{\perp}^{2}-(\tilde{\Delta}_{\perp}\cdot\partial)(v\cdot\partial)$ |  $\partial_{\perp}^{2}$
$e_{h}$ |  $e_{h}-\frac{1}{2}(e_{h}\cdot\Delta_{\perp})\tilde{v}^{\mu}$ |  $e_{h}-\frac{1}{2}(e_{h}\cdot\tilde{\Delta}_{\perp})v^{\mu}$ |  $e_{h}$
Table 2: Transformation rules for the operators of the effective field theory
and the OSEFT polarization vectors. The transformation rules for the negative
energy sector of the theory can be esasily recovered after replacing
$E\rightarrow-E$ and $h\rightarrow-h$.

###  II.6 Going backward to the full theory variables

When constructing the OSEFT, we decomposed the photon momentum $q^{\mu}$ as in
Eq.(2), introducing the effective field theory variables $E,v^{\mu}$ and
$k^{\mu}$. Any quantity computed from the OSEFT depends on these variables.
However, it is desirable to be able to re-express the results in terms of the
full theory momentum $q^{\mu}$.

This is a well established process which has been extensively discussed in the
literature of the OSEFT, see e.g Refs.Manuel:2016wqs ; Manuel:2016cit ;
Carignano:2018gqt ; Carignano:2019zsh ; Manuel:2021oah , so we just recall
here the relevant expressions needed for this work. First, we will need the
expression of the on-shell velocity $v^{\mu}_{q}$ in the effective field
theory, which reads

$v^{\mu}_{q}=v^{\mu}+\dfrac{k_{\perp}^{\mu}}{E}-\dfrac{(\tilde{v}\cdot
k)k_{\perp}^{\mu}}{2E^{2}}-\dfrac{k_{\perp}^{2}n^{\mu}}{2E^{2}}+{\cal
O}\left(\frac{1}{E^{3}}\right)\ .$ (47)

It follows that $\tilde{v}^{\mu}_{q}=2u^{\mu}-v_{q}^{\mu}$, while the frame
vector $u^{\mu}$ does not change when moving back to the full theory. Also, it
is useful to define a space-like vector in the full theory, such that

$\displaystyle
n^{\mu}_{q}=\dfrac{\tilde{v}^{\mu}_{q}-v^{\mu}_{q}}{2}=n^{\mu}-\dfrac{k_{\perp}^{\mu}}{E}+\dfrac{(\tilde{v}\cdot
k)k_{\perp}^{\mu}}{2E^{2}}+\dfrac{k_{\perp}^{2}n^{\mu}}{2E^{2}}+{\cal
O}\left(\frac{1}{E^{3}}\right)\ .$ (48)

Let us introduce the polarization vectors of the full theory in the circular
basis as $\\{e_{q,h}^{\mu}\\}$ with the suffix $(q)$ indicating that they are
functions of momentum and $h=\\{+,-\\}$. We can relate them with the OSEFT
polarization vectors $\\{e_{h}^{\mu}\\}$ using the following trick. By
requiring that at each order in $1/E$

$u\cdot e_{q,h}=v_{q}\cdot e_{q,h}=\tilde{v}_{q}\cdot e_{q,h}=0\ ,\quad
e_{q,h}\cdot e_{q,h^{\prime}}=-\delta_{hh^{\prime}}\ ,$ (49)

holds, it is not difficult to realize that

$e_{q,h}^{\mu}=e_{h}^{\mu}-\dfrac{e_{h}\cdot
k_{\perp}}{E}n^{\mu}+\dfrac{(e_{h}\cdot
k_{\perp})}{2E^{2}}\left(k_{\perp}^{\mu}+(\tilde{v}\cdot
k)n^{\mu}\right)+{\cal O}\left(\frac{1}{E^{3}}\right)\ .$ (50)

Then, we can introduce the transverse projector and spin tensor of the full
theory as

$\displaystyle
P_{\perp,q}^{\mu\nu}=g^{\mu\nu}-\dfrac{1}{2}(v^{\mu}_{q}\tilde{v}^{\nu}_{q}+\tilde{v}^{\mu}_{q}v^{\nu}_{q})=-e_{q,+}^{*\mu}e_{q,+}^{\nu}-e_{q,-}^{*\mu}e_{q,-}^{\nu}\
,$ (51a) $\displaystyle
S_{\perp,q}^{\mu\nu}=\dfrac{i\epsilon^{\mu\nu\alpha\beta}q_{\alpha}u_{\beta}}{u\cdot
q}=e_{q,+}^{*\mu}e_{q,+}^{\nu}-e_{q,-}^{*\mu}e_{q,-}^{\nu}\ ,$ (51b)

respectively. In the definitions above, $q^{\mu}=(E_{q},\boldsymbol{q})$ is
the on-shell momentum (with $E_{q}=u\cdot q$), also
$v^{\mu}_{q}=(1,\boldsymbol{q}/E_{q})$ and
$\tilde{v}^{\mu}_{q}=(1,-\boldsymbol{q}/E_{q})$. They can also be expressed in
terms of the OSEFT variables, either by using Eq.(47) or Eq.(50). For
instance, we can write

$\displaystyle
P_{\perp,q}^{\mu\nu}=P_{\perp}^{\mu\nu}-\dfrac{1}{E}n^{(\mu}k_{\perp}^{\nu)}+\dfrac{(\tilde{v}\cdot
k)}{2E^{2}}n^{(\mu}k_{\perp}^{\nu)}+\dfrac{1}{E^{2}}(k_{\perp}^{\mu}k_{\perp}^{\nu}+k_{\perp}^{2}n^{\mu}n^{\nu})+{\cal
O}\left(\frac{1}{E^{3}}\right)\ ,$ (52a) $\displaystyle
S_{\perp,q}^{\mu\nu}=S_{\perp}^{\mu\nu}+\dfrac{1}{E}n^{[\mu}S_{\perp}^{\nu]\alpha}k_{\alpha}-\dfrac{1}{2E^{2}}k_{\perp}^{[\mu}S_{\perp}^{\nu]\alpha}k_{\alpha}-\dfrac{(\tilde{v}\cdot
k)}{2E^{2}}n^{[\mu}S_{\perp}^{\nu]\alpha}k_{\alpha}+{\cal
O}\left(\frac{1}{E^{3}}\right)\ .$ (52b)

##  III Quantum kinetic theory for photons from the OSEFT

In the previous section we derived the OSEFT associated to an energetic photon
field. Here we will consider a many body system, such as a plasma,
characterized by a temperature $T$. Then, one should consider that there are
many electromagnetic fields with energy scales of the order or much larger
than $T$, which then admit an OSEFT description. The electromagnetic fields
with typical energy scales lower than $T$ can be then treated as classical
gauge fields. This can be justified by the fact that the Bose-Einstein
distribution function is well approximated by a classical field distribution
function at low energies.

In this section we derive photon quantum kinetic equations from the effective
field theory developed in Sec. II, that is, we assume that we are describing
the high energetic photons in the system. To this aim, apart from the OSEFT
for photons we employ the Schwinger-Keldysh formalism of thermal field theory
Kapusta:2006pm ; Bellac:2011kqa ; Laine:2016hma , see also
Refs.Ghiglieri:2020dpq ; Mustafa:2022got  for recent reviews on the subject.

Let us emphasize that in the OSEFT, we have performed a local field
redefinition to eliminate the non-physical component of the photon field from
the Lagrangian (c.f Eq.(20)). Notably, as the equivalence theorem was extended
to finite density and then to general thermodynamic observables in
Ref.Furnstahl:2000we , doing a similar reasoning as in Sec.II allows us to
work with the locally redefined field with no need to worry about affecting
any on-shell quantity.

The main object in the Schwinger-Keldysh formalism is the Green function,
which is expressed as a matrix in the complex time path contour. Similarly, in
the OSEFT, one can define for positive energy photons

$\boldsymbol{\cal G}^{\mu\nu}_{\perp}(x,y)=\begin{pmatrix}{\cal
G}^{c,\mu\nu}_{\perp}(x,y)&{\cal G}^{<,\mu\nu}_{\perp}(x,y)\\\ {\cal
G}^{>,\mu\nu}_{\perp}(x,y)&{\cal
G}^{a,\mu\nu}_{\perp}(x,y)\end{pmatrix}=\begin{pmatrix}\braket{\mathcal{T}_{C}\,\tau_{\perp}^{\mu}(x)\tau_{\perp}^{\nu{\dagger}}(y)}&\braket{\tau_{\perp}^{\nu{\dagger}}(y)\tau_{\perp}^{\mu}(x)}\\\
\braket{\tau_{\perp}^{\mu}(x)\tau_{\perp}^{\nu{\dagger}}(y)}&\braket{\widetilde{\mathcal{T}}_{C}\,\tau_{\perp}^{\mu}(x)\tau_{\perp}^{\nu{\dagger}}(y)}\end{pmatrix}\
,$ (53)

where $\mathcal{T}_{C}$ and $\widetilde{\mathcal{T}}_{C}$ denote time and
anti-time ordering along the complex time contour respectively, while
$\braket{\ldots}$ denotes thermal average over an ensemble of states.
Analogous definitions can be done for the negative energy sector of the
theory. Indeed, we can also define

$\boldsymbol{\widetilde{\cal
G}}^{\mu\nu}_{\perp}(x,y)=\begin{pmatrix}\widetilde{\cal
G}^{c,\mu\nu}_{\perp}(x,y)&\widetilde{\cal G}^{<,\mu\nu}_{\perp}(x,y)\\\
\widetilde{\cal G}^{>,\mu\nu}_{\perp}(x,y)&\widetilde{\cal
G}^{a,\mu\nu}_{\perp}(x,y)\end{pmatrix}=\begin{pmatrix}\braket{\mathcal{T}_{C}\,\tau_{\perp}^{\mu{\dagger}}(x)\tau_{\perp}^{\nu}(y)}&\braket{\tau_{\perp}^{\nu}(y)\tau_{\perp}^{\mu{\dagger}}(x)}\\\
\braket{\tau_{\perp}^{\mu{\dagger}}(x)\tau_{\perp}^{\nu}(y)}&\braket{\widetilde{\mathcal{T}}_{C}\,\tau_{\perp}^{\mu{\dagger}}(x)\tau_{\perp}^{\nu}(y)}\end{pmatrix}\
.$ (54)

Our interest is in the lesser (or greater) components of the Green functions,
as these are related with the photon phase-space distribution function after a
Wigner transform. From their definition, it follows that the lesser and
greater components of each sector of the theory are related by

$\displaystyle{\cal G}^{<,\mu\nu}_{\perp}(x,y)=\widetilde{\cal
G}^{>,\nu\mu}_{\perp}(y,x)\ ,$ (55)

and satisfy the hermiticity property

$\displaystyle({\cal G}^{<,\mu\nu}_{\perp}(x,y))^{*}=\widetilde{\cal
G}^{>,\mu\nu}_{\perp}(x,y)={\cal G}^{<,\nu\mu}_{\perp}(y,x)\ .$ (56)

Let us focus on the positive energy sector of the theory. We define the OSEFT
Wigner function for positive energy photons as

$\displaystyle{\cal G}^{<,\mu\nu}_{\perp}(X,k)=\int d^{4}s\,e^{ik\cdot s}{\cal
G}^{<,\mu\nu}_{\perp}(x,y)\ ,$ (57)

where $X^{\mu}=(x^{\mu}+y^{\mu})/2$ and $s^{\mu}=x^{\mu}-y^{\mu}$ are the
central and relative coordinate respectively, while $k^{\mu}$ is the residual
momentum. We can build kinetic equations to the desired order in $1/E$ by
adding and subtracting the Wigner transformed equations of motion. Precisely,
taking into account Eq.(25), we can define

$\displaystyle\left(I_{\pm}\right)^{\mu\nu}=-\dfrac{1}{2}\int
d^{4}s\,e^{ik\cdot
s}\left(\mathcal{D}^{2}_{x}\pm\mathcal{D}^{2}_{y}\right)\,{\cal
G}^{<,\mu\nu}_{\perp}(x,y)=0\ .$

Performing the Wigner transform, the dispersion
$\left(I_{+}\right)^{\mu\nu}=0$ and transport $\left(I_{-}\right)^{\mu\nu}=0$
equations give

$\displaystyle\left(2E(v\cdot k)+k^{2}-\dfrac{\partial^{2}}{4}\right)\,{\cal
G}_{\perp}^{<,\mu\nu}(X,k)=0\ ,$ (58a)
$\displaystyle\left(E\,iv\cdot\partial+ik\cdot\partial\right){\cal
G}^{<,\mu\nu}_{\perp}(X,k)=0\ ,$ (58b)

respectively. Above and for the rest of the paper, we will use the shortened
notation $\partial^{\mu}=\partial_{X}^{\mu}$ when all derivatives in the
expressions are respect to the center coordinate. The kinetic equations can
also be written in the following form, obtained after solving the theory order
by order in $1/E$ (see the remarks in Sec.II)

$\displaystyle\left(2E(v\cdot
k)+k_{\perp}^{2}-\dfrac{\partial_{\perp}^{2}}{4}-\dfrac{(\tilde{v}\cdot
k)k_{\perp}^{2}}{2E}+\dfrac{(\tilde{v}\cdot
k)\partial_{\perp}^{2}}{8E}+\dfrac{(k_{\perp}\cdot\partial)(\tilde{v}\cdot\partial)}{4E}\right)\,{\cal
G}_{\perp}^{<,\mu\nu}(X,k)=0\ ,$ (59a)
$\displaystyle\left(E\,iv\cdot\partial+i(k_{\perp}\cdot\partial)-\dfrac{ik_{\perp}^{2}(\tilde{v}\cdot\partial)}{4E}+\dfrac{i(\tilde{v}\cdot\partial)\partial_{\perp}^{2}}{16E}-\dfrac{i(\tilde{v}\cdot
k)(k_{\perp}\cdot\partial)}{2E}\right){\cal G}^{<,\mu\nu}_{\perp}(X,k)=0\ .$
(59b)

In addition, one should complement the kinetic equations with the following
constraints which, by construction, are obeyed at any order in $1/E$

$\displaystyle v_{\mu}{\cal G}^{<,\mu\nu}_{\perp}=\tilde{v}_{\mu}{\cal
G}^{<,\mu\nu}_{\perp}=v_{\nu}{\cal G}^{<,\mu\nu}_{\perp}=\tilde{v}_{\nu}{\cal
G}^{<,\mu\nu}_{\perp}=0\ .$ (60)

Analogous kinetic equations and constraints can be derived for the negative
energy sector, just by replacing $E\rightarrow-E$ and ${\cal
G}\rightarrow\widetilde{\cal G}$. The above constraints suggest that the OSEFT
Wigner function can be decomposed into the polarization basis introduced in
Sec.II.3 as

$\displaystyle{\cal
G}^{<,\mu\nu}_{\perp}(X,k)=\sum_{h,h^{\prime}=\pm}e^{\mu}_{h}e^{*\nu}_{h^{\prime}}\,{\cal
G}^{<,hh^{\prime}}(X,k)\ .$ (61)

In the remaining part of the paper we will assume that there is no
polarization mixing in the photon ensemble, and thus write

$\displaystyle{\cal
G}^{<,\mu\nu}_{\perp}(X,k)=\sum_{h=\pm}e^{\mu}_{h}e^{*\nu}_{h}\,{\cal
G}^{<,h}(X,k)\ .$ (62)

The polarization space components are defined as the Wigner transform of the
corresponding Green function as

$\displaystyle{\cal G}^{<,h}(X,k)=\int d^{4}s\,e^{ik\cdot s}{\cal
G}^{<,h}(x,y)\ ,$ (63)

where ${\cal G}^{<,h}(x,y)=\braket{\tau_{h}^{\dagger}(y)\tau_{h}(x)}$. After
the Wigner transform, their general structure is

$\displaystyle{\cal G}^{<,h}(X,k)=2\pi\delta\big{(}K_{h}\big{)}f^{h}(X,k)\ ,$
(64)

where we denote with $f^{h}(X,k)$ the off-shell distribution function for
right/left circular polarized photons of positive energy, while $K_{h}$ is the
function that governs the dispersion relation, given by the expression inside
the parenthesis in Eq.(58a). Again, similar definitions can be done for the
negative energy sector, e.g $\widetilde{\cal
G}^{<,h}(X,k)=2\pi\delta\big{(}\widetilde{K}_{h}\big{)}\widetilde{f}^{h}(X,k)$.

###  III.1 Wigner function for photons

Transport equations associated to the photon Wigner function were studied in
the literature long ago, see for example Vasak:1987um . The photon Wigner
function can be determined in several ways, for instance by using the Fourier
decomposition of the vector field gauge potential or by directly solving the
quantum kinetic equations Huang:2020kik ; Hattori:2020gqh ; Lin:2021mvw . Here
we present yet an alternative derivation using the OSEFT developed in Sec. II.

The lesser component of the Wigner function for photons is defined as

$G^{<,\mu\nu}(X,q)=\int d^{4}s\,e^{iq\cdot s}\braket{A^{\nu}(y)A^{\mu}(x)}\ .$
(65)

Then, plugging the ansatz for the photon field of Eq.(6) onto the above
equation and using the momentum decomposition $q^{\mu}=\pm Ev^{\mu}+k^{\mu}$
for the positive/negative energy sector of the theory respectively, one can
write

$G^{<,\mu\nu}(X,q)=\int d^{4}s\,e^{ik\cdot
s}\left\\{\braket{\xi^{\nu{\dagger}}(y)\xi^{\mu}(x)}+\braket{\xi^{\nu}(y)\xi^{\mu{\dagger}}(x)}\right\\}\
,$ (66)

where we used that

$\braket{\xi^{\nu}(y)\xi^{\mu}(x)}=\braket{\xi^{\nu{\dagger}}(y)\xi^{\mu{\dagger}}(x)}=0\
,$ (67)

which is equivalent to impose causality preserving commutation relations
between creation and annihilation operators. Then, for free photons we can
directly use Eq.(29) to write the Wigner function as follows

$\begin{gathered}G^{<,\mu\nu}(X,q)=\int d^{4}s\,e^{ik\cdot
s}\bigg{\\{}\left(\mathcal{O}^{\mu}_{\
\alpha}\right)_{x}\left(\mathcal{O}^{\nu}_{\ \beta}\right)_{y}^{*}{\cal
G}^{<,\alpha\beta}_{\perp}(x,y)+\left(\mathcal{O}^{\mu}_{\
\alpha}\right)_{x}^{*}\left(\mathcal{O}^{\nu}_{\
\beta}\right)_{y}\widetilde{\cal G}^{<,\alpha\beta}_{\perp}(x,y)\bigg{\\}}\
,\end{gathered}$ (68)

where ${\cal G}^{<,\alpha\beta}_{\perp}(x,y)$ and $\widetilde{\cal
G}^{<,\alpha\beta}_{\perp}(x,y)$ are the OSEFT Green functions for positive
and negative energies introduced in Eqs.(53-54) respectively, and we used the
shortened notation

$\mathcal{O}^{\mu}_{\
\nu}=g^{\mu}_{\nu}-n^{\mu}\partial_{\nu}\frac{n\cdot\mathcal{D}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\
.$ (69)

Note that in Eq.(68), we assumed that

$\braket{\phi^{\dagger}(y)\tau_{\perp}^{\mu}(x)}=\braket{\tau_{\perp}^{\mu{\dagger}}(y)\phi(x)}=\braket{\phi^{\dagger}(y)\phi(x)}=0\
,$ (70)

which can be justified by noting that in the Lagrangian of Eq.(21) there are
no terms that couple those fields. Then, using the expansion
$\mathcal{O}^{\mu}_{\ \nu}\approx
g^{\mu}_{\nu}-\frac{i}{E}n^{\mu}\partial_{\nu}$ and decomposing the OSEFT
Wigner functions onto the circular polarization basis, we reach to

$\begin{gathered}G^{<,\mu\nu}(X,q)=\sum_{h={\pm}}\int d^{4}s\,e^{ik\cdot
s}\bigg{\\{}\left(e^{\mu}_{h}e_{h}^{*\nu}+\dfrac{i}{E}\left(e^{\mu}_{h}n^{\nu}(e_{h}^{*}\cdot\partial_{y})-n^{\mu}e_{h}^{*\nu}(e_{h}\cdot\partial_{x})\right)\right){\cal
G}^{<,h}(x,y)\\\
+\left(e^{*\mu}_{h}e_{h}^{\nu}-\dfrac{i}{E}\left(e^{*\mu}_{h}n^{\nu}(e_{h}\cdot\partial_{y})-n^{\mu}e_{h}^{\nu}(e_{h}^{*}\cdot\partial_{x})\right)\right)\widetilde{\cal
G}^{<,h}(x,y)\bigg{\\}}\ .\end{gathered}$ (71)

By writing
$\partial_{x}^{\mu}=\frac{1}{2}\partial_{X}^{\mu}+\partial_{s}^{\mu}$ and
$\partial_{y}^{\mu}=\frac{1}{2}\partial_{X}^{\mu}-\partial_{s}^{\mu}$ above,
we can easily perform the Wigner transform, as $\partial_{s}^{\mu}\rightarrow-
ik^{\mu}$ after integrating by parts. The result may be written as

$\displaystyle G^{<,\mu\nu}(X,q)=\sum_{h=\pm}\left(\Pi^{\mu\nu}_{h}(k){\cal
G}^{<,h}(X,k)+\widetilde{\Pi}^{\mu\nu}_{h}(k)\widetilde{\cal
G}^{<,h}(X,k)\right)\ ,$ (72)

where we defined the tensor

$\displaystyle\Pi^{\mu\nu}_{h}(k)=e^{\mu}_{h}e^{*\nu}_{h}-\dfrac{1}{E}\left(e^{\mu}_{h}n^{\nu}(e_{h}^{*}\cdot
k)+n^{\mu}e^{*\nu}_{h}(e_{h}\cdot
k)\right)+\dfrac{i}{2E}\left(e^{\mu}_{h}n^{\nu}(e_{h}^{*}\cdot\partial)-n^{\mu}e^{*\nu}_{h}(e_{h}\cdot\partial)\right)\
,$ (73)

while $\widetilde{\Pi}^{\mu\nu}_{h}(k)$ can be obtained after replacing
$E\rightarrow-E$ and $h\rightarrow-h$ in the above expression. Note that in
the above tensor we are reproducing the expansion of the full theory
polarization vectors of Eq.(50). The tensor $\Pi_{h}^{\mu\nu}(k)$ can also be
written in terms of the transverse projector and the spin tensor

$\displaystyle\Pi^{\mu\nu}_{h}(k)=-\dfrac{1}{2}\left(P_{\perp}^{\mu\nu}+hS_{\perp}^{\mu\nu}-\dfrac{1}{E}n^{(\mu}k^{\nu)}_{\perp}+\dfrac{h}{E}n^{[\mu}S_{\perp}^{\nu]\alpha}k_{\alpha}-\dfrac{i}{2E}n^{[\mu}\partial^{\nu]}_{\perp}+\dfrac{ih}{2E}n^{(\mu}S_{\perp}^{\nu)\alpha}\partial_{\alpha}\right)\
,$ (74)

after resorting to Eq.(39). Above, it can be easily seen that we are
reproducing the expansions of the full theory transverse projector and spin
tensor of Eqs.(52a-52b) up to first order in $1/E$, additionally, we produce
quantum corrections to the Wigner function.

Now we would like to re-express the Wigner function in terms of the full
theory variables. At this expansion order we can associate

$\displaystyle{\cal
G}^{<,h}(X,k)=2\pi\delta(K_{h})f^{h}(X,k)\,\longrightarrow\,4\pi\delta(q^{2})\theta(u\cdot
q)f^{h}(X,q)\ ,$ (75a) $\displaystyle\widetilde{\cal
G}^{<,h}(X,k)=2\pi\delta(\widetilde{K}_{h})\widetilde{f}^{h}(X,k)\,\longrightarrow\,-4\pi\delta(q^{2})\theta(-u\cdot
q)f^{h}(X,q)\ ,$ (75b)

where $f^{h}(X,q)$ is the off-shell distribution for right/left handed
circularly polarized photons of the full theory. If, for instance, one
considers that the photon ensemble is at thermal equilibrium and that there is
no CP violating effect, so that one can assume $f^{+}(X,q)=f^{-}(X,q)\equiv
f_{\text{eq}}(q_{0})$, the distribution function takes the form
$f_{\text{eq}}(q_{0})=(e^{q_{0}/T}-1)^{-1}$ in the rest frame of the medium,
being $T$ the equilibrium temperature. In that scenario, one has

$f_{\text{eq}}(q_{0})=\begin{cases}n_{B}(\boldsymbol{q})\ ,&\text{if}\quad
q_{0}=|\boldsymbol{q}|\\\ -[1+n_{B}(\boldsymbol{q})]\ ,&\text{if}\quad
q_{0}=-|\boldsymbol{q}|\end{cases}\ ,$ (76)

where $n_{B}(\boldsymbol{q})=(e^{|\boldsymbol{q}|/T}-1)^{-1}$ is the Bose-
Einstein distribution function. The tensors $\Pi^{\mu\nu}_{h}(k)$ and
$\widetilde{\Pi}^{\mu\nu}_{h}(k)$ both translate to $\Pi^{\mu\nu}_{h}(q)$,
which may be written either in terms of the polarization vectors or the
transverse projector and spin tensor as (see Sec.(II.6))

$\displaystyle\Pi^{\mu\nu}_{h}(q)=e^{\mu}_{q,h}e^{*\nu}_{q,h}+\dfrac{i}{2E_{q}}\left(e^{\mu}_{q,h}n^{\nu}_{q}(e_{q,h}^{*}\cdot\partial)-n^{\mu}_{q}e^{*\nu}_{q,h}(e_{q,h}\cdot\partial)\right)\
,$ (77a)
$\displaystyle\Pi^{\mu\nu}_{h}(q)=-\dfrac{1}{2}\left(P_{\perp,q}^{\mu\nu}+hS_{\perp,q}^{\mu\nu}-\dfrac{i}{2E_{q}}n_{q}^{[\mu}\partial^{\nu]}_{\perp,q}+\dfrac{ih}{2E_{q}}n_{q}^{(\mu}S_{\perp,q}^{\nu)\alpha}\partial_{\alpha}\right)\
,$ (77b)

where we recall that $n^{\mu}_{q}=u^{\mu}-v^{\mu}_{q}$, being
$v^{\mu}_{q}=q^{\mu}/E_{q}$ the photon on-shell velocity. Then, after moving
back to the full theory variables, one can write the Wigner function for
photons at $1/E_{q}$ accuracy as

$G^{<,\mu\nu}(X,q)=4\pi\sum_{h=\pm}\Pi^{\mu\nu}_{h}(q)\delta(q^{2})\text{sgn}(u\cdot
q)f^{h}(X,q)\ ,$ (78)

with $\text{sgn}(x)$ denoting the sign function. The kinetic equations obeyed
by the photon Wigner function can be deduced from Eqs.(58a-58b). In terms of
the full theory variables they read

$\displaystyle q^{2}\,G^{<,\mu\nu}(X,q)=0\ ,$ (79a)
$\displaystyle\left(q\cdot\partial\right)G^{<,\mu\nu}(X,q)=0\ .$ (79b)

In Eq. (79a) we dropped a piece proportional to $\sim\partial^{2}$ to be
consistent with the gradient expansion assumed here. We note that at this
expansion order the following constraints are satisfied

$\left(\dfrac{1}{2}\partial_{\mu}-iq_{\mu}\right)\Pi^{\mu\nu}_{h}(q)=\left(\dfrac{1}{2}\partial_{\nu}+iq_{\nu}\right)\Pi^{\mu\nu}_{h}(q)=0\
,$ (80)

and also

$u_{\mu}\Pi^{\mu\nu}_{h}(q)=u_{\nu}\Pi^{\mu\nu}_{h}(q)=0\ ,$ (81)

so that the Wigner function obeys the Coulomb gauge-fixing conditions. We note
that the Wigner function of Eq.(78) coincides exactly with that encountered in
Refs.Huang:2020kik ; Hattori:2020gqh ; Lin:2021mvw . However, let us remark
that we did not impose any gauge fixing condition to achieve this result,
which means that the Wigner function associated to the high-energy modes, when
computed in a semi-classical approach, is a gauge invariant quantity, provided
that the gauge transformations are respectful with the separation of energy
scales

###  III.2 Side jumps from reparametrization invariance

In a semi-classical approach to describe massless chiral fermions the Lorentz
transformations need to be modified in the presence of small quantum effects
in order to preserve the frame independence of the theory Chen:2014cla . This
is related to the fact that the total angular momentum of a relativistic
spinning particle is ambiguous, because the definition of the spin part is
non-unique. The issue is resolved by imposing the condition
$u_{\mu}S_{\perp,q}^{\mu\nu}=q_{\mu}S_{\perp,q}^{\mu\nu}=0$, so that the spin
tensor is uniquely fixed in the inertial frame, and is given then by Eq.(51b).
Consequently, when moving from frame $u^{\mu}$ to $u^{\prime\mu}$ the particle
position also changes, so that the total angular momentum is conserved. The
shift on the particle position when changing between inertial frames is the so
called side jump effect Chen:2014cla ; Chen:2015gta . In the context of chiral
kinetic theory, describing a system of massless fermions, this effect is
manifested by the fact that the fermion distribution function is no longer a
Lorentz scalar.

In chiral kinetic theory, the side jump effect can naturally be derived from
OSEFT, and it is linked with the reparametrization invariance of the theory.
This was checked for massless fermions in Carignano:2018gqt ;
Carignano:2019zsh . As we will see, the same applies to photons. Note that
while it has been widely accepted that this side jump effect would affect also
other massless spinning particles Huang:2018aly , and not only fermions, we
are however unaware of any explicit proof of this side jump effect from
quantum field theory.

In order to derive the side jump effect, we first need to know the RI
transformations of the polarization space components of the Wigner function.
Those can be derived from

${\cal G}^{<,h}(X,k)=e_{h,\mu}^{*}e_{h,\nu}\int d^{4}s\,e^{ik\cdot
s}\braket{\tau_{\perp}^{{\dagger}\nu}(y)\tau_{\perp}^{\mu}(x)}\ ,$ (82)

by employing the transformation rules of Eqs.(46a-46c). Under a type (II)
transformation one finds

$\displaystyle\delta_{(\text{II})}{\cal
G}^{<,h}(X,k)=\dfrac{i}{2E}e_{h,\mu}^{*}e_{h,\nu}\int d^{4}s\,e^{ik\cdot
s}\bigg{(}(\partial_{x}+\partial_{y})^{\mu}\tilde{\Delta}_{\perp}^{\nu}-\tilde{\Delta}_{\perp}^{\mu}(\partial_{x}+\partial_{y})^{\nu}\bigg{)}{\cal
G}^{<,h}(x,y)=$ $\displaystyle\qquad\qquad\qquad\
=\dfrac{ih}{2E}S_{\perp}^{\mu\nu}\tilde{\Delta}_{\perp,\nu}\partial_{\mu}{\cal
G}^{<,h}(X,k)\ .$ (83)

The contribution from the negative energy sector,
$\delta_{(\text{II})}\widetilde{\cal G}^{<,h}(X,k)$, is the same as above, as
can be seen by replacing $E\rightarrow-E$ and $h\rightarrow-h$. After adding
the positive and negative energy sector contribution, we can deduce the
transformation rule for the photon distribution function under a type (II)
transformation, which reads

$\displaystyle
f^{h}(X,q)\overset{\text{(II)}}{\longrightarrow}f^{h}(X,q)+\dfrac{ih}{2E_{q}}S_{\perp,q}^{\mu\nu}\tilde{\Delta}_{\perp,\nu}\partial_{\mu}f^{h}(X,q)\
,$ (84)

after moving back to the full theory variables, which is the expected side
jump effect. Note that Eq. (84) gives the infinitesimal change of the
distribution function, where one has to take into account that
$\tilde{\Delta}_{\perp}^{\mu}/2=u^{\prime\mu}-u^{\mu}$.

Doing a similar reasoning as above, one can show that at $1/E_{q}$ accuracy
$S^{\mu\nu}_{\perp,q}$ and $f^{h}(X,q)$ are both invariant under a type (I)
and a type (III) RI transformation, so that there is no side jump effect in
those cases.

##  IV Stokes parameters with quantum corrections

By projecting the Wigner function of Eq.(78) onto the circular polarization
basis, we can build a naive photon current associated to every helicity state.
Precisely, in the absence of polarization mixing in the photon ensemble, we
can define

$j^{h,\mu}(X)=\int\dfrac{d^{4}q}{(2\pi)^{4}}q^{\mu}G^{h}(X,q)\ ,\quad h=\pm\
,$ (85)

where we dropped the lesser symbol $(<)$ from the Wigner function to enlighten
the notation. Also, we introduced

$G^{h}(X,q)=4\pi\delta(q^{2})\text{sgn}(u\cdot q)f^{h}(X,q)\ .$ (86)

The polarization space components of the Wigner function (or the zeroth
component of the photon current), can be directly related to the Stokes
parameters jackson ; Beneke:2010eg . Then, with this setting, the naive Stokes
parameters matrix can be defined as

$\rho(X)=\begin{pmatrix}j^{+}_{0}(X)&0\\\
0&j^{-}_{0}(X)\end{pmatrix}=\begin{pmatrix}j^{I}_{0}(X)-j^{V}_{0}(X)&0\\\
0&j^{I}_{0}(X)+j^{V}_{0}(X)\end{pmatrix}\ ,$ (87)

where $j^{I}_{0}=(j_{0}^{+}+j_{0}^{-})/2$ and
$j^{V}_{0}=-(j_{0}^{+}-j_{0}^{-})/2$ give the values of the intensity and
degree of polarization of the photon ensemble, respectively. It is well-known
that the Stokes parameters are not Lorentz invariant, but the percentage of
circulation polarization, obtained as the ratio of the degree of circular
polarization over the intensity, i.e $j^{V}_{0}/j^{I}_{0}$, is usually
regarded in the literature as a Lorentz invariant Cocke ; Han:1997zz .

However, in a semi-classical approach to photon propagation based on quantum
kinetic theory, as soon as quantum corrections are considered, and due to the
side jump effect discussed in Sec. III.2 this is no longer the case.
Modification of the definition of the Stokes parameters is then needed in
order to restore the Lorentz invariance of the polarization ratios.

The above issue is related to the fact that the naive current of Eq.(88) does
not transform as a Lorentz vector when quantum corrections are taken into
account, because the photon distribution function is no longer a Lorentz
scalar. A solution was found in Ref.Chen:2014cla , by including a
magnetization contribution to the naive helicity current one can restore its
frame independence in the collisionless limit. Indeed, one can define a frame
independent photon current in the collisionless limit as

$J^{h,\mu}(X)=\int\dfrac{d^{4}q}{(2\pi)^{4}}\left(q^{\mu}-ihS^{\mu\nu}_{\perp,q}\partial_{\nu}\right)G^{h}(X,q)\
.$ (88)

From the above current, we can define new Stokes parameters as
$J^{I}_{0}=(J_{0}^{+}+J_{0}^{-})/2$ and $J^{V}_{0}=-(J_{0}^{+}-J_{0}^{-})/2$,
so that the Lorentz invariance of the polarization ratio $J^{V}_{0}/J_{0}^{I}$
is restored. Hence, we can write the intensity and the degree of polarization
of the photon ensemble as

$\displaystyle
J^{I}_{0}(X)=\int\dfrac{d^{4}q}{(2\pi)^{3}}2\delta(q^{2})\text{sgn}(u\cdot
q)\left(q_{0}f^{I}(X,q)+\dfrac{\boldsymbol{u}\times\boldsymbol{q}}{u\cdot
q}\cdot\boldsymbol{\nabla}f^{V}(X,q)\right)\ ,$ (89) $\displaystyle
J^{V}_{0}(X)=\int\dfrac{d^{4}q}{(2\pi)^{3}}2\delta(q^{2})\text{sgn}(u\cdot
q)\left(q_{0}f^{V}(X,q)+\dfrac{\boldsymbol{u}\times\boldsymbol{q}}{u\cdot
q}\cdot\boldsymbol{\nabla}f^{I}(X,q)\right)\ ,$ (90)

respectively, where the we have defined

$\displaystyle f^{I}(X,q)=\dfrac{f^{+}(X,q)+f^{-}(X,q)}{2}\ ,$ (91)
$\displaystyle f^{V}(X,q)=-\dfrac{f^{+}(X,q)-f^{-}(X,q)}{2}\ .$ (92)

Note that in the rest frame of the medium ($\boldsymbol{u}=\boldsymbol{0}$) or
in the frames satisfying $\boldsymbol{u}\times\boldsymbol{q}=0$, the naive
polarization percentage $j^{V}_{0}/j^{I}_{0}$ is still a LI quantity, even in
the presence of small quantum effects.

Let us emphasize that the frame independence of the current of Eq.(88) is only
valid under the assumption of a collisionless medium. Nevertheless, under
certain circumstances this program can also be generalized in the presence of
collisions Chen:2015gta , so that further modification of the Stokes
parameters would be required in that scenario.

##  V Discussion

In this manuscript we have fully developed the OSEFT associated to photons,
generalizing previous work carried out for chiral fermions. Assuming a high
energy expansion, we derived the Lagrangian associated to the high energy or
hard photons from the free Maxwell Lagrangian. By splitting the vector gauge
field into different components, we identified the degree of freedom that can
be integrated out from the Lagrangian using its classical equation of motion,
given in Eq.(15). Subsequently, we showed that the remaining non-physical
degree of freedom can be eliminated employing the local field redefinition of
Eq.(20). After this last step, we obtained an effective field theory that only
contains a transverse vector gauge field (cf. Eq. (21)).

Generally, in the context of effective field theories, it is common to split
the fields into different components, according to their energy scale (e.g
into hard and soft parts). In a gauge theory then one expects a gauge symmetry
associated to each sector of the theory which is respectful with the energy
scaling of such decomposition. In this work, a gauge symmetry associated to
the hard part of the photon field has been presented, and we demonstrated that
the OSEFT Lagrangian enjoys that symmetry. We have also proven the RI of the
theory, which basically means that the OSEFT is respectful with the Lorentz
symmetry. Using the RI transformations, a first principles derivation of the
so called side jump effect for photons has been provided (see Eq.(84)).

Since at high energies, or when the photon wavelength is much shorter than any
length scale in the system, electromagnetic waves can be accurately studied
under the so called eikonal limit, the OSEFT seems a suitable tool to study
how the physics beyond the eikonal approximation is corrected. For instance,
we used the OSEFT and the Schwinger-Keldysh formalism of thermal field theory
to construct a quantum kinetic theory for photons. We have shown how to use
the OSEFT to systematically compute quantum corrections to the semi-classical
photon Wigner function and we explicitly computed the leading order quantum
correction (see Eqs.77a-77b and Eq.(78)). In addition, we derived the quantum
kinetic equations obeyed by the photon Wigner function in the collisionless
limit, given by Eqs.(79a-79b). Our results agree with those found in
Refs.Huang:2020kik ; Hattori:2020gqh ; Lin:2021mvw . However, as in the
effective field theory approach we separate the gauge field into physical and
non-physical components, in OSEFT computations one can clearly identify which
modes contribute at each order in the energy expansion, thus providing
valuable insight.

We have then elaborated on the proper definition of the Stokes parameters as
deduced from quantum kinetic theory. At the classical level, the percentage of
circular polarization in a system is a Lorentz invariant quantity. However, we
have shown that as soon as quantum corrections are considered, the definition
of the Stokes parameters needs to be modified in order to preserve the Lorentz
invariance of the percentage of circular polarization. We have given such a
definition in Eqs.(89-90) for the intensity and degree of circular
polarization. Let us recall that we assumed the absence of polarization mixing
in the photon ensemble. It would be interesting to generalize the discussion
in the presence of such polarization modes, as they are known to be generated
in relevant cosmological physical scenarios.

Finally, let us mention that the OSEFT for photons could be used also for the
computation of power corrections to different sort of Feynman diagrams, for
example, at high temperature and/or density. One could extend the study we
carry out by allowing interactions of the hard photons with either hard and
soft fermions, and proceed with the same methods for the study of the effects
of these interactions, as it was done for the contributiuon of hard fermions
in Manuel:2016wqs ; Manuel:2016cit .

Acknowledgements

We thank Joan Soto and Stefano Carignano for discussions. This work was
supported by Ministerio de Ciencia, Investigación y Universidades (Spain)
MCIN/AEI/10.13039/501100011033/ FEDER, UE, under the project
PID2022-139427NB-I00, by Generalitat de Catalunya by the project 2021-SGR-171
(Catalonia). This work was also partly supported by the Spanish program Unidad
de Excelencia Maria de Maeztu CEX2020-001058-M.

##  Appendix A Operator notation and identities

In this manuscript we use a simplified operator notation in order to shorten
the length of the expressions. When we write one over a differential operator
we mean the inverse of the differential operator in the denominator. To
compactify even more the notation, we often write fractions of different
differential operators. For example it should be understood the following

$\frac{n\cdot\mathcal{D}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\partial_{\mu}\xi_{\perp}^{\mu}\longrightarrow\frac{1}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}(n\cdot\mathcal{D})\partial_{\mu}\xi_{\perp}^{\mu}\
.$ (93)

Also, in most of the derivations presented in this work, we used the fact that
some combinations of operators, which appear acting on the $\phi(x)$ field,
vanish identically. For instance, when deriving Eq.(29) we used the fact that

$\dfrac{n\cdot\mathcal{D}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\dfrac{\partial_{\perp}^{2}}{u\cdot\mathcal{D}}-\dfrac{\mathcal{D}^{2}-(v\cdot\partial)(n\cdot\mathcal{D})}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}+\dfrac{v\cdot\partial}{u\cdot\mathcal{D}}=0\
.$ (94)

Similarly, one can show other operator identities which are relevant in our
derivations

$\displaystyle\dfrac{u\cdot\mathcal{D}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\dfrac{\mathcal{D}^{2}}{n\cdot\mathcal{D}}-\dfrac{\mathcal{D}^{2}-(v\cdot\partial)(n\cdot\mathcal{D})}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}-\dfrac{v\cdot\partial}{n\cdot\mathcal{D}}=0\
,$ (95a)
$\displaystyle\dfrac{(u\cdot\mathcal{D})(n\cdot\mathcal{D})(\mathcal{D}^{2}-(v\cdot\partial)(n\cdot\mathcal{D}))}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}-(\mathcal{D}^{2}-(v\cdot\partial)(u\cdot\mathcal{D}))+\dfrac{\mathcal{D}^{2}\partial_{\perp}^{2}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}=0\
.$ (95b)

##  Appendix B RI transformations

As we explained in Sec.(II.5), the transformation rules are modified after
integrating out the $\lambda(x)$ field. Precisely, for the transverse field we
find

$\displaystyle\xi^{\mu}_{\perp}\overset{\text{(I)}}{\longrightarrow}\xi^{\mu}_{\perp}-\dfrac{\Delta_{\perp}^{\mu}}{2}\left(1+\dfrac{(u\cdot\mathcal{D})(n\cdot\mathcal{D})}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\right)\phi-\dfrac{\Delta_{\perp}^{\mu}}{2}\dfrac{(n\cdot\mathcal{D})(\partial\cdot\xi_{\perp})}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}-\frac{v^{\mu}}{2}(\Delta_{\perp}\cdot\xi)-n^{\mu}\delta_{(\text{I})}\lambda\
,$ (96)
$\displaystyle\xi^{\mu}_{\perp}\overset{\text{(II)}}{\longrightarrow}\xi^{\mu}_{\perp}+\dfrac{\tilde{\Delta}_{\perp}^{\mu}}{2}\left(\frac{(n\cdot\mathcal{D})(\partial\cdot\xi_{\perp})}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}-\frac{\mathcal{D}^{2}-(v\cdot\partial)(n\cdot\mathcal{D})}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\phi\right)-\frac{v^{\mu}}{2}(\tilde{\Delta}_{\perp}\cdot\xi)-n^{\mu}\delta_{(\text{II})}\lambda\
,$ (97)
$\displaystyle\xi^{\mu}_{\perp}\overset{\text{(III)}\,}{\longrightarrow}\xi^{\mu}_{\perp}-\alpha\,n^{\mu}\left(\frac{(n\cdot\mathcal{D})(\partial\cdot\xi_{\perp})}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}-\frac{\mathcal{D}^{2}-(v\cdot\partial)(n\cdot\mathcal{D})}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\phi\right)-n^{\mu}\delta_{(\text{III})}\lambda\
.$ (98)

In the above transformations, we defined the quantities
$\delta_{(\Lambda)}\lambda$ for $\Lambda=\\{\text{I, II, III}\\}$ given by

$\displaystyle\delta_{(\text{I})}\lambda=\dfrac{1}{2}\dfrac{\mathcal{D}^{2}+2(n\cdot\mathcal{D})^{2}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}(\Delta_{\perp}\cdot\xi_{\perp})+\dfrac{1}{2}\dfrac{\mathcal{D}^{2}}{(\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2})^{2}}(\Delta_{\perp}\cdot\partial_{\perp})(\partial\cdot\xi_{\perp})$
$\displaystyle\
\qquad+\dfrac{1}{2}\left(\dfrac{(n\cdot\mathcal{D})(\mathcal{D}^{2}-(v\cdot\partial)(n\cdot\mathcal{D}))}{(\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2})^{2}}+\dfrac{v\cdot\partial}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\right)(\Delta_{\perp}\cdot\partial)\phi\
,$ (99)
$\displaystyle\delta_{(\text{II})}\lambda=\dfrac{1}{2}\dfrac{\mathcal{D}^{2}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}(\tilde{\Delta}_{\perp}\cdot\xi_{\perp})-\dfrac{1}{2}\dfrac{\mathcal{D}^{2}}{(\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2})^{2}}(\tilde{\Delta}_{\perp}\cdot\partial_{\perp})(\partial\cdot\xi_{\perp})$
$\displaystyle\
\qquad-\dfrac{1}{2}\left(\dfrac{(n\cdot\mathcal{D})(\mathcal{D}^{2}-(v\cdot\partial)(n\cdot\mathcal{D}))}{(\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2})^{2}}+\dfrac{v\cdot\partial}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\right)(\tilde{\Delta}_{\perp}\cdot\partial)\phi\
,$ (100)
$\displaystyle\delta_{(\text{III})}\lambda=-\alpha\left(\frac{n\cdot\mathcal{D}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}(\partial\cdot\xi_{\perp})-\frac{\mathcal{D}^{2}-(v\cdot\partial)(n\cdot\mathcal{D})}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\phi\right)+\alpha\dfrac{\mathcal{D}^{2}(u\cdot\mathcal{D})}{(\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2})^{2}}(\partial\cdot\xi_{\perp})$
$\displaystyle\
\qquad+\alpha\left(\dfrac{(u\cdot\mathcal{D})(n\cdot\mathcal{D})(\mathcal{D}^{2}-(v\cdot\partial)(n\cdot\mathcal{D}))}{(\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2})^{2}}-\dfrac{\mathcal{D}^{2}-(v\cdot\partial)(u\cdot\mathcal{D})}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\right)\phi\
,$ (101)

which account for the infinitesimal variations of the operators on the
equation of motion of $\lambda(x)$ (c.f Eq.(15)). For the $\phi(x)$ field,
only the transformation rule under type (III) needs to be modified

$\phi\overset{\text{(III)}\,}{\longrightarrow}(1-\alpha)\phi-\alpha\left(\frac{n\cdot\mathcal{D}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}(\partial\cdot\xi_{\perp})-\frac{\mathcal{D}^{2}-(v\cdot\partial)(n\cdot\mathcal{D})}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}\phi\right)\
.$ (102)

Remarkably, using Eqs.(96-98) and the corresponding transformations for the
$\phi(x)$ field, one can show that the Lagrangian of Eq.(II), obtained after
integrating out the hard field $\lambda(x)$, is RI invariant. Finally, let us
conclude by writing down the general transformations for the locally redefined
field of Eq.(20), they read

$\displaystyle\tau_{\perp}^{\mu}\overset{\text{(I)}}{\longrightarrow}\tau_{\perp}^{\mu}-\dfrac{\partial_{\perp}^{\mu}}{2(u\cdot\mathcal{D})}(\Delta_{\perp}\cdot\tau_{\perp})-\dfrac{\Delta_{\perp}^{\mu}}{2}\dfrac{n\cdot\mathcal{D}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}(\partial\cdot\tau_{\perp})-\dfrac{v^{\mu}}{2}(\Delta_{\perp}\cdot\tau_{\perp})$
$\displaystyle-\dfrac{n^{\mu}}{2}\bigg{(}\dfrac{\mathcal{D}^{2}+2(n\cdot\mathcal{D})^{2}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}(\Delta_{\perp}\cdot\tau_{\perp})+\dfrac{\mathcal{D}^{2}}{(\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2})^{2}}(\Delta_{\perp}\cdot\partial)(\partial\cdot\tau_{\perp})\bigg{)}\
,$ (103)
$\displaystyle\tau_{\perp}^{\mu}\overset{\text{(II)}}{\longrightarrow}\tau_{\perp}^{\mu}-\dfrac{\partial_{\perp}^{\mu}}{2(u\cdot\mathcal{D})}(\tilde{\Delta}_{\perp}\cdot\tau_{\perp})+\dfrac{\tilde{\Delta}_{\perp}^{\mu}}{2}\dfrac{n\cdot\mathcal{D}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}(\partial\cdot\tau_{\perp})-\frac{v^{\mu}}{2}(\tilde{\Delta}_{\perp}\cdot\tau_{\perp})\
,$
$\displaystyle-\dfrac{n^{\mu}}{2}\bigg{(}\dfrac{\mathcal{D}^{2}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}(\tilde{\Delta}_{\perp}\cdot\tau_{\perp})-\dfrac{\mathcal{D}^{2}}{(\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2})^{2}}(\tilde{\Delta}_{\perp}\cdot\partial)(\partial\cdot\tau_{\perp})\bigg{)}\
,$ (104)
$\displaystyle\tau_{\perp}^{\mu}\overset{\text{(III)}}{\longrightarrow}\tau_{\perp}^{\mu}+\alpha\dfrac{\partial_{\perp}^{\mu}}{u\cdot\mathcal{D}}\dfrac{n\cdot\mathcal{D}}{\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2}}(\partial\cdot\tau_{\perp})+\alpha\,n^{\mu}\dfrac{\mathcal{D}^{2}(u\cdot\mathcal{D})}{(\mathcal{D}^{2}+(n\cdot\mathcal{D})^{2})^{2}}(\partial\cdot\tau_{\perp})\
.$ (105)

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