###### Abstract

Within the finite-time-path out-of-equilibrium quantum field theory (QFT), we
calculate direct photon emission from early stages of heavy ion collisions,
from a narrow window, in which uncertainty relations are still important and
they provide a new mechanism for production of photons. The basic difference
with respect to earlier calculations, leading to diverging results, is that we
use renormalized QED of quarks and photons. Our result is a finite
contribution that is consistent with uncertainty relations.

###### keywords:

out-of-equilibrium quantum field theory; direct photons; dimensional
renormalization; finite-time-path formalism

3 4 44 Received: 15 July 2020; Accepted: 7 October 2020; Published: 20 October
2020 yes yes Direct Photons from Hot Quark Matter in Renormalized Finite-Time-
Path QED Ivan Dadić 1,∗, Dubravko Klabučar 2 and Domagoj Kuić 1 Ivan Dadić ,
Dubravko Klabučar and Domagoj Kuić Correspondence<EMAIL_ADDRESS>
##  1 Introduction

Heavy ion collisions (HIC) result in many-particle final states that carry a
lot of information, which is not easy to decode Baym:2016wox ;
Pasechnik:2016wkt ; David:2019wpt , including, for example, the recently
emerged ‘direct photon puzzle’ David:2019wpt . Various attempts to describe
HIC by theoretical means include increasingly involved approaches, such as
S-matrix QFT and equilibrium, as well as out-of-equilibrium QFT
Schwinger:1960qe ; Keldysh:1964ud ; KadanoffBook ; Danielewicz:1982kk ;
Chou:1984es ; Rammer:1986zz ; Landsman:1986uw ; Calzetta:1986cq ; Niemi:1987pm
; Remler:1990 ; LeBellacBook ; Brown:1998zx ; Blaizot:2001nr . Finally, the
fast evolving nature of HIC requires the finite-time-path description.
Pertinent calculations bv ; bvhs ; devega ; bvs0 ; wbvl ; wb ; ng ; bv68 ;
bvhep  of production of photons from the early stage of HIC, and the heated
discussion afterwards Arleo:2004gn , with the criticism on infinite energy
that is released in photon yield, indicate that the subject is far from
settled.

This calculation is performed in the finite-time-path (FTP) out-of-equilibrium
QFT. FTP is a variation of out-of-equilibrium QFT with propagators defined
through a finite-time contour. Close to our approach are the Dynamical
Renormalization Group approach by Boyanovsky and collaborators bv ; bvhs , and
Millington’s and Pilaftsis’ formulation Millington:2012pf ; Millington:2013qpa
of non-equilibrium thermal field theory. Specific to our approach is the use
of the retarded-advanced ($R$-$A$) basis, where the Keldysh ($K$) propagator
($D_{K}$) is also separated into its advanced and retarded parts ($D_{K,A}$
and $D_{K,R}\,,$ respectively). The formalism Dadic:1999bp ; Dadic:2002wv ;
Dadic:2009zz ; Dadic:2019lmm  is equivalent to the approach of Boyanovsky and
De Vega bv ; nevertheless, in the evaluation of production of direct photons,
the difference is that we are calculating in the energy-momentum
representation and avoid early approximations and simplifications.

Only having perturbative QED interactions, photons are “clean” probes of the
quark-gluon plasma (QGP), regardless of whether it is in the regime of
nonperturbative or perturbative QCD. For temperatures that are not much higher
than the (pseudo)critical temperature $T_{c}$ of the crossover transition, QGP
is still strongly coupled for sure Baym:2016wox ; Pasechnik:2016wkt . However,
there are claims Paquet:2015lta  that, in photon production calculations, one
can rely on perturbative QCD corrections. These claims are supported by
lattice around and even below $T\sim 1.3T_{c}$ Ghiglieri:2016tvj . Even in
references that stress the nonperturbative character of QCD significantly
above $T_{c}$, QGP begins approaching its perturbative regime beyond $T\sim
2T_{c}$ Ding:2015ona , i.e., above (250–300) MeV. Moreover, observables that
are dominated at high $T$ by quark rather than gluon contributions seem to
approach perturbative behavior earlier Ding:2015ona , and the fermionic sector
is weakly coupled for $T>300$ MeV Haque:2014rua . Thus, in the very early,
pre-equilibrium phase, where temperature $T$ is not even defined, but the
average energy per particle is high enough, i.e., higher than $k_{B}\times
300$ MeV, one can assume that the asymptotic QCD regime is reached, and
asymptotic freedom makes quarks move around quasi-free. Subsequently, in the
lowest approximation, one can neglect the dressing of quark-photon vertices by
QCD interactions. In particular, the fully dressed quark-photon vertex in the
vacuum polarization diagram of photons is replaced by the free quark-photon
vertex $\,e_{\rm q}\,\gamma_{\mu}\,$, as in Figure 1.

In any case, in the high-energy phase, various flavors of quarks and
antiquarks will be present and the energy of the heavy-ion collisions
determine the number of active quark flavors. With expansion and cooling, the
average energy per particle drops continuously until the phase in which quarks
became confined, and the system turns into hadron matter, where strong
interactions dominate. Finally further expansion produces interparticle
distances bigger than the range of strong interactions, and the system decays
to individual particles.

One expects the production of highest energy photons in the early stage of
HIC. The mechanism that is discussed in this paper only produces photons in
this stage.

The photons, which do not interact strongly with quarks and gluons, escape
relatively easily, carrying the valuable information on early stage of HIC. In
this stage, the uncertainty relations allow large energy uncertainty producing
fast oscillations.

The Dyson–Schwinger equation for photon $D_{K,R}$ propagator requires
renormalization Giambagi:1972 ; Thooft:1972 ; Ashmore:1972 ; Cicuta:1972 ;
Wilson:1973 ; Kislinger:1976 ; Donoghue:1983 ; Johansson:1986 ; Kobes:1989 ;
Keil:1989 ; LeBellac:1990 ; Elmfors:1992 ; Eijck:1996 ; Chapman:1997 ;
Nakkagawa:1997 ; Baacke:1998 ; Esposito:1998 ; Knoll:2002 ; Jakovac:2005 ;
Arrizabalaga:2007 ; Blaizot:2007 ; Blaizot:2015 ; RyderBook  of divergent
$\Pi_{R}$ and $\Pi_{A}$ vacuum polarizations. The related problems emerging
are an additional energy-not-conserving vertex and regularized $\Pi_{R}$ not
vanishing as $|p_{0}|\rightarrow\infty$ (potentially breaking causality). They
are solved in full analogy to the case of out-of-equilibrium $\lambda\phi^{3}$
field theory, as discussed in Dadic:2019lmm . The solution involves: energy
integrations performed, while $d<4$, subtractions in $\Pi_{F}$, and
“reparation” of causality in the products, like $D_{R}\Pi_{R}$ or
$\Pi_{R}D_{R}$ and analogously for $\Pi_{A}$.

Our final result is finite. In particular, the contributions that contain
initial distribution functions of quarks and/or antiquarks are finite. The
calculation is straightforward, but one should notice how potentially pinching
term turns into the contribution linear in time $t$.

Prospects for further development are discussed.

Figure 1: The lowest-order correction to the vacuum polarization
$\Pi^{\rho\sigma}$ of photons, due to quarks of the flavor $q$.

##  2 Direct Photon Production

At the time $t$, number density of detected photons of the polarization $e$
$(e=e_{0},e_{1},e_{2},e_{3})$ and momentum $\vec{p}$ is:

$\displaystyle\langle\,N_{\vec{p},e,t}\,\rangle\,=\,(2\pi)^{3}{d{\cal
N}_{e}\over d^{3}x\,d^{3}p}\,,$ (1)

where $d{\cal N}_{e}$ is the number of photons of the polarization $e$ inside
the differential volumes $d^{3}x$ and $d^{3}p$ of the coordinate and momentum
spaces.

The number density is connected to the equal time limit of the $K$ (Keldysh)
component of the dressed photon propagator ${\cal D}_{\mu\nu}$ (for
simplicity, we use Feynman gauge):

$\displaystyle 2\langle\,N_{\vec{p},e,t}\,\rangle+1\,=\,-\,{\omega_{p}\over
2\pi}\lim_{t_{1}\rightarrow t}\int dp_{0}{\cal
D}_{K,t_{1},t,\mu\nu}(p)e^{\mu}e^{*\nu}\,.$ (2)

There are various definitions Blaizot:2001nr ; wbvl ; Garbrecht:2002pd  of
number density based on the product of creation and annihilation operators,
$a^{\dagger}$ and $a$. They are mostly equivalent, at least at low orders. Our
definition is adapted to the presently used formalism, as the dressed Keldysh
propagator ${\cal D}_{K}={\cal D}_{K,R}-{\cal D}_{K,A}$ is easily calculated
from perturbation expansion for matrix propagators (${\cal D}_{ij}$) as well
as from Dyson–Schwinger equation.

As expected, at the lowest order of the perturbation expansion, it is

$\displaystyle
2\langle\,N_{\vec{p},e,t}^{0}\,\rangle+1\,=\,2\,f_{e}(\vec{p})\,+\,1\,,$ (3)

where the function $f_{e}(\vec{p})$ is the initial ($t=0$) distribution
function for photons. However, they freely escape from the medium after they
are created in the collision, and do not accumulate in the quark medium
created by the collision of nucleons. Thus, we set $f_{e}(\vec{p})=0$ as the
initial condition. We are only interested in the photons from the early quark
phase.

The initial distributions of quarks ($n_{+}(\vec{p})$) and of antiquarks
($n_{-}(\vec{p})$) are the input functions for out-of-equilibrium field
theory, where they are independent. All of the initial distribution functions
($f_{\rm i}=f_{e},n_{\pm}$) have to satisfy the following conditions:
$\int\,d^{3}pf_{\rm i}(\vec{p})<\infty$, $\int
d^{3}p\,\omega_{\vec{p}}\,f_{\rm i}(\vec{p})<\infty$, saying, respectively,
that the probability and the average energy are finite. There is no condition
regarding analyticity.

In principle, highly anisotropic situations can be considered. Nevertheless,
in many applications, the basic assumption is that the system is close to
equilibrium and, in such cases, a natural choice for $n_{\pm}$ is the
Fermi–Dirac distribution function, which is isotropic:

$n_{\pm}(\vec{p})\,\to\,n_{\pm}(\omega_{p})\,=\,{\,1\over
e^{(\omega_{p}-\mu_{\pm})/T}+1\,}~{}.$ (4)

This was the choice of Wang and Boyanovsky bv , and we will also use it when
we compare our result to theirs.

Even for out-of-equilibrium situations, some phenomenologists speak loosely
about temperature $T$ and chemical potential $\mu$ as the characterization of
average single particle energy and particle number. We expect that, by fitting
the measured direct photon data, one can extract some knowledge regarding
$f(\vec{p})$ function.

In comparison with the cross-section, which is proportional to time derivative
of exclusive number of particles, the number of “detected” photons is
inclusive. It is the density number of photons (i.e., “gain minus loss”)
detected until the time $t$ (yield).

In order to calculate the first nontrivial order contributions to
$\langle\,N_{\vec{p},e,t}\,\rangle$ in (2), for ${\cal
D}_{K,t_{1},t_{2},\mu\nu}(p)$, we use Dyson–Schwinger Equations (5), where
time and momenta variables are suppressed for compactness, and where $*$
denotes the convolution product defined in Appendix A by Equation (17):

$\begin{array}[]{l}{\cal
D}_{\mu\nu,K}\,=\,{D}_{\mu\nu,K}+i[D_{\mu\rho,R}*\Pi^{\rho\sigma}_{K}*{D}_{\sigma\nu,A}+D_{\mu\rho,R}*\Pi^{\rho\sigma}_{R}*{D}_{\sigma\nu,K}+D_{\mu\rho,K}*\Pi^{\rho\sigma}_{A}*{D}_{\sigma\nu,A}]\,\,,\cr\\\
\Pi^{\rho\sigma}_{K}\,=\,-\Pi^{\rho\sigma}_{K,R}+\Pi^{\rho\sigma}_{K,A}\,\,,\end{array}$
(5)

where all vacuum polarizations $\Pi^{\rho\sigma}$ are of the lowest order (one
loop), as in Figure 1.

The expression (5) contains two types of problems: divergences Dadic:2019lmm
and vertices Dadic:1998yd ; Dadic:1999bp ; Dadic:2002wv ; Dadic:2009zz .

The divergences are contained in $\Pi^{\rho\sigma}_{R}$ and
$\Pi^{\rho\sigma}_{A}$. These divergences are descendants of the divergence of
the Feynman vacuum polarization $\Pi^{\rho\sigma}_{F}$. In the dimensional
regularization at $d<4$, they are regulated (i.e., finite) and can be
subtracted. Other vacuum polarizations $\Pi^{\rho\sigma}_{K}$ are expected to
be finite.

Vertices are of the following three types:

  1. 1.

Vertices with at least one outgoing retarded propagator or incoming advanced
propagator. Energy conservation is achieved by simple integration over the
energy of such propagator, by closing the contour from above for the retarded
and from below for the advanced propagator. This case includes the cases when
there are more than one such propagators connecting the same vertex. In this
case, the loop integrals should not diverge. The convolution product $*$
containing them turns into the usual algebraic product. Examples: the vertex
between the two-point functions $D_{\mu\rho,R}$ and $\Pi^{\rho\sigma}_{K,R}$;
between $\Pi^{\rho\sigma}_{K,A}$ and ${D}_{\sigma\nu,A}$; between
$\Pi^{\rho\sigma}_{R}$ and $D_{\mu\rho,K,R}$; between ${D}_{\sigma\nu,K,A}$
and $\Pi^{\rho\sigma}_{A}$, as they appear in the Dyson–Schwinger equation
terms
$D_{\mu\rho,R}*\Pi^{\rho\sigma}_{R}*{D}_{\sigma\nu,K}+D_{\mu\rho,K}*\Pi^{\rho\sigma}_{A}*{D}_{\sigma\nu,A}$.

  2. 2.

Vertices without any outgoing retarded propagator or incoming advanced
propagator. They are lower in time than neighboring vertices. Closing the
integration path always catches some singularities of the propagators. These
terms will not conserve energy, but they oscillate in time, with high
frequency. The examples are: the vertex between the two-point functions
$D_{\mu\rho,R}$ and $\Pi^{\rho\sigma}_{K,A}$; between $\Pi^{\rho\sigma}_{K,R}$
and ${D}_{\sigma\nu,A}$; between $\Pi^{\rho\sigma}_{R}$ and $D_{\mu\rho,K,A}$
; and, between ${D}_{\sigma\nu,K,R}$ and $\Pi^{\rho\sigma}_{A}$.

  3. 3.

Vertices with at least one outgoing retarded propagator, or at least one
incoming advanced propagator, but with two or more such propagators entering
the same vertex, where the corresponding loop integral diverges. These
vertices should conserve energy, but divergent integrals make them ill-
defined. At $d<4$, the loop integrals are regulated, and the usual closing of
the integration contour leads to energy conservation. This make them group 1.
vertices. Examples of such vertices are the ones between the two-point
functions $D_{\mu\rho,R}$ and $\Pi^{\rho\sigma}_{R}$, as well as
$\Pi^{\rho\sigma}_{A}$ and ${D}_{\sigma\nu,A}$. Additionally, in the case that
the mentioned vertex is connected with yet another one, where this connection
satisfies the condition for the group 1 vertices, the mentioned vertex
immediately belongs to the group 1.

These properties lead to the simplified version of Dyson–Schwinger equation at
the first order:

$\begin{array}[]{l}{\cal
D}_{\mu\nu,K}\,=\,{D}_{\mu\nu,K}+i\,[\,-D_{\mu\rho,R}\Pi^{\rho\sigma}_{K,R}*{D}_{\sigma\nu,A}+D_{\mu\rho,R}*\Pi^{\rho\sigma}_{K,A}{D}_{\sigma\nu,A}\cr\\\
\qquad\qquad-
D_{\mu\rho,R}\Pi^{\rho\sigma}_{R}*{D}_{\sigma\nu,K,A}+D_{\mu\rho,K,R}*\Pi^{\rho\sigma}_{A}{D}_{\sigma\nu,A}\cr\\\
\qquad\qquad+D_{\mu\rho,R}\Pi^{\rho\sigma}_{R}{D}_{\sigma\nu,K,R}-D_{\mu\rho,K,A}\Pi^{\rho\sigma}_{A}{D}_{\sigma\nu,A}\,].\end{array}$
(6)

Inserting this ${\cal D}_{\mu\nu,K}$ in Equation (2) and using the convolution
product defined in Appendix A, reveals that the one-loop contribution to the
photon number density is

$\begin{array}[]{l}\langle\,N^{1}_{\vec{p},t}\,\rangle\,=\,\sum_{\mu,\nu}g^{\mu\nu}{\omega_{p}\over
4\pi}i\int dp_{01}dp_{02}dp_{0}P_{t}(p_{0},{p_{01}+p_{02}\over 2}){i\over
2\pi}{e^{-it(p_{01}-p_{02}+i\epsilon)}-1\over p_{01}-p_{02}+i\epsilon}\cr\\\
\times[\,-D_{\mu\rho,R}(p_{01},\vec{p})\Pi^{\rho\sigma}_{K,R}(p_{01},\vec{p}){D}_{\sigma\nu,A}(p_{02},\vec{p})-D_{\mu\rho,R}(p_{01},\vec{p})\Pi^{\rho\sigma}_{R}(p_{01},\vec{p}){D}_{\sigma\nu,K,A}(p_{02},\vec{p})\cr\\\
\quad+D_{\mu\rho,R}(p_{01},\vec{p})\Pi^{\rho\sigma}_{K,A}(p_{02},\vec{p}){D}_{\sigma\nu,A}(p_{02},\vec{p})+D_{\mu\rho,K,R}(p_{01},\vec{p})\Pi^{\rho\sigma}_{A}(p_{02},\vec{p}){D}_{\sigma\nu,A}(p_{02},\vec{p})\cr\\\
\quad+D_{\mu\rho,R}(p_{01},\vec{p})\Pi^{\rho\sigma}_{R}(p_{01},\vec{p})D_{\sigma\nu,K,R}(p_{02},\vec{p})-D_{\mu\rho,K,A}(p_{01},\vec{p})\Pi^{\rho\sigma}_{A}(p_{02},\vec{p})D_{\sigma\nu,A}(p_{02},\vec{p})\,]~{}.\end{array}$
(7)

The last two terms ($RRR$ and $AAA$) vanish in the limit of equal time. The
function $P_{t}(p_{0},{p_{01}+p_{02}\over 2})$ is the projecting function
defined in the Appendix A.

In the rest of the text, we suppress the Lorentz indices.

Next, we perform equal time limit procedure. Equal time limit of the product
of two or more retarded functions vanishes, as one can, in that case, close
the integration contour $dp_{0}$ in a way to catch no singularity. Such a
product of advanced functions also vanishes. The diverging $\Pi_{R}$ and
$\Pi_{A}$ are represented by vacuum polarizations in the matrix representation
Dadic:2019lmm  as

$\begin{array}[]{l}\Pi_{j,k}=-\Pi_{K,R}+\Pi_{K,A}-\,k\,\Pi_{R}-\,j\,\Pi_{A}\qquad(i,j=1,2)\,\,,\cr\\\
\Pi_{11}=\Pi_{F}-\Pi_{n_{+},n_{-},R}+\Pi_{n_{+},n_{-},A}\,\,,\cr\\\
Re\Pi_{R(A),n_{\pm}=0}(p)=Re\Pi_{F}~{},\end{array}$ (8)

where $\Pi_{n_{+},n_{-},R(A)}$ are the corrections to $\Pi_{F}$ by finite
$n_{\pm}$.

Calculated at $d<4$ and regularized by subtraction of constant term, the
closing the integration path over $dp_{01}$ from above shows that the
subtraction term vanishes.

Integration over $dp_{0}$ removes (see Appendix A) the
$P_{t}(p_{0},p^{\prime}_{0})$ projector.

In the first two terms in the square bracket in Equation (7), one can close
the $dp_{02}$ integration path from above and catch only the singularities of
$D_{A}$ and $D_{K,A}$. The integration over $dp_{01}$ can be closed from below
with the singularities caught from $D_{R}$ and $D_{K,R}$ and from $\Pi_{R}$
and $\Pi_{K,R}$. Important to notice is that the poles of both, $D_{A}$ and
$D_{K,A}$, are equal $[\bar{p}_{02}]_{1(2)}=\pm\omega_{p}+i\epsilon$, for
$D_{R}$ and $D_{K,R}$ they are
$[\bar{p}_{01}]_{1(2)}=\pm\omega_{p}-i\epsilon$. Singularities of $\Pi_{R}$
and $\Pi_{K,R}$ may be complicated.

In the terms with only poles of propagators, the vertex factor requires a
limiting procedure

$\displaystyle\lim_{\epsilon\rightarrow
0}[{e^{-it(\bar{p}_{01}-\bar{p}_{02}+i\epsilon)}-1\over\bar{p}_{01}-\bar{p}_{02}+i\epsilon}]_{\bar{p}_{01}=\bar{p}_{02}*=\pm\omega_{p}-i\epsilon}=-it.$
(9)

The corresponding terms are growing linearly with time.

Similar, but complex conjugated, are the contributions of the last two terms
in Equation (7). Thus, we obtain

$\begin{array}[]{l}\int_{-\infty}^{\infty}dp_{0}\,{\cal
D}^{1}_{K,t}(p_{0},\vec{p})\,=\,-\,{i\pi t\over
2\omega_{p}^{2}}\sum_{\lambda=\pm}[\,-\Pi_{K,R}(\lambda\omega_{p},\vec{p})+\lambda\Pi_{R}(\lambda\omega_{p},\vec{p})(1+2f_{e}(\omega_{p}))\cr\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+\,\Pi_{K,A}(-\lambda\omega_{p},\vec{p})-\lambda\Pi_{A}(-\lambda\omega_{p},\vec{p})\left(1+2f_{e}(\omega_{p})\,\right)\,]\cr\\\
\qquad\qquad\qquad-\,{\pi\over
4\omega_{p}^{2}}\sum_{\lambda=\pm}{e^{-i2t\lambda\omega_{p}}-1\over\omega_{p}}\,[\,\lambda\Pi_{K,R}(\lambda\omega_{p},\vec{p})-\Pi_{R}(\lambda\omega_{p},\vec{p})(1+2f_{e}(\omega_{p}))\cr\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\,\lambda\Pi_{K,A}(-\lambda\omega_{p},\vec{p})-\Pi_{A}(-\lambda\omega_{p},\vec{p})(1+2f_{e}(\omega_{p}))]\cr\\\
+\,{1\over\omega_{p}}\sum_{\lambda=\pm}{\cal P}\int_{cut}{dp_{0}\over
p^{2}}{e^{-it(p_{0}-\lambda\omega_{p})}-1\over
p_{0}-\lambda\omega_{p}}\,[\lambda
Im\Pi_{K,R}(p_{0},\vec{p})+Im\Pi_{R}(p_{0},\vec{p})(1+2f_{e}(\omega_{p}))\,]\cr\\\
+{1\over\omega_{p}}\sum_{\lambda=\pm}{\cal P}\int_{cut}{dp_{0}\over
p^{2}}{e^{it(p_{0}-\lambda\omega_{p})}-1\over
p_{0}-\lambda\omega_{p}}\,[\,\lambda
Im\Pi_{K,A}(p_{0},\vec{p})-Im\Pi_{A}(p_{0},\vec{p})(1+2f_{e}(\omega_{p}))]~{},\end{array}$
(10)

where $\cal P$ denotes the principal value of the integrals.

We use the symmetries of the one-loop vacuum polarizations to simplify the
above expression

$\begin{array}[]{l}\Pi_{K,R}(-|p_{0}|,\vec{p})=-\Pi^{*}_{K.R}(|p_{0}|,\vec{p})=\Pi^{*}_{K,A}(-|p_{0}|,\vec{p})=-\Pi_{K,A}(|p_{0}|,\vec{p}),\cr\\\
\Pi_{R}(-|p_{0}|,\vec{p})=\Pi^{*}_{R}(|p_{0}|,\vec{p})=\Pi^{*}_{A}(-|p_{0}|,\vec{p})=\Pi_{A}(|p_{0}|,\vec{p})~{}.\end{array}$
(11)

Our final result for the one-loop contribution to the photon number density is

$\begin{array}[]{l}\langle\,N^{1}_{\vec{p},t}\,\rangle\,=\,{\omega_{p}\over
4\pi}\int_{-\infty}^{\infty}dp_{0}\,{\cal D}^{1}_{K,t}(p_{0},\vec{p})\,=\cr\\\
{t\over
2\omega_{p}}[-Im\Pi_{K,R}(\omega_{p},\vec{p})+Im\Pi_{R}(\omega_{p},\vec{p})(1+2f_{e}(\omega_{p}))]\cr\\\
-{1\over
4\omega_{p}^{2}}(1+2f_{e}(\omega_{p}))[(1-\cos{2t\omega_{p}})Re\Pi_{R}(\omega_{p},\vec{p})+\sin{2t\omega_{p}}Im\Pi_{R}(\omega_{p},\vec{p})]\cr\\\
+{2\over\pi^{2}\omega_{p}}\,{\cal
P}\int_{p_{0}>0,~{}cut}{dp_{0}\over(p_{0}^{2}-\omega_{p}^{2})^{2}}[p_{0}(1-\cos
tp_{0}\cos t\omega_{p})-\omega_{p}\sin tp_{0}\sin t\omega_{p}]\cr\\\
\times(1+2f_{e}(\omega_{p}))Im\Pi_{R}(p_{0},\vec{p})~{}.\end{array}$ (12)

In Appendix C, we give the results on the vacuum polarizations that are needed
in Equation (12). To evaluate it numerically, we, of course, also need the
early-time distribution functions. They are presently still unknown, but we
are presently considering several Ansätze proposed in literature as physically
motivated “educated guesses”.

##  3 Discussion of the Results

Thus, we end up with four sorts of terms. One of them is just the initial
particle distribution, i.e., Equation (3). As we explain after Equation (3),
this “zeroth order term” is not contained in Equation (12).

###  3.1 Energy Conserving Terms

The terms appearing in the first square bracket in (12) are the terms from
propagator poles satisfying $\bar{p}_{01}=\bar{p}_{02}^{*}$. These terms grow
linearly with time (see Equation (9)). At large times, these terms would be
dominating. They have the other desired properties:

  1. (1)

They conserve energy.

  2. (2)

They vanish for the distribution functions satisfying detailed balance
principle. Indeed, the bracket from (59) in Appendix C is the defect of
detailed balance in all of the channels:

$\begin{array}[]{l}\Theta(p_{0}-q_{0})\Theta(q_{0})[n_{+}(\omega_{p-q})n_{-}(\omega_{q})(1+f(\omega_{p}))(1-n_{+}(\omega_{p-q}))(1-n_{-}(\omega_{q}))f(\omega_{p})]\cr\\\
+\Theta(q_{0}-p_{0})\Theta(-q_{0})[n_{-}(\omega_{p-q})n_{+}(\omega_{q})(1+f(\omega_{p}))(1-n_{-}(\omega_{p-q}))(1-n_{+}(\omega_{q}))f(\omega_{p})]\cr\\\
+\Theta(q_{0}-p_{0})\Theta(q_{0})[(1-n_{-}(\omega_{p-q}))n_{-}(\omega_{q})(1+f(\omega_{p}))-n_{-}(\omega_{p-q})(1-n_{-}(\omega_{q}))f(\omega_{p})]\cr\\\
+\Theta(p_{0}-q_{0})\Theta(-q_{0})[(1-n_{+}(\omega_{p-q}))n_{+}(\omega_{q})f(\omega_{p})\cr\\\
-n_{+}(\omega_{p-q})(1-n_{+}(\omega_{q}))(1+f(\omega_{p}))(1-n_{+}(\omega_{p-q}))n_{+}(\omega_{q})f(\omega_{p})]\,].\end{array}$
(13)

In Equation (13), unpolarized photons are assumed; hence, no subscripts appear
on their distribution functions $f(\omega_{p})$.

  3. (3)

They are proportional to the lowest order Collision integral. Nevertheless,
their contribution vanishes, owing to the kinematical limitations. (Otherwise,
these terms would correspond to the contribution from the usual $S$-matrix
formalism.)

###  3.2 Term Containing $Re\Pi_{R}$

The second term in Equation (12) is the term from propagator poles satisfying
$\bar{p}_{01}=-\bar{p}_{02}^{*}$. Energy is not conserved in this term. The
contribution containing $Re\Pi_{R}$ requires the renormalization of finite-
time-path out-of-equilibrium $\phi^{3}$ QFT Dadic:2019lmm . The important
points are: The vertices in the products $D_{R}*\Pi_{R}$ and $\Pi_{A}*D_{A}$
should conserve energy, but, owing to the divergences in $\Pi_{R}$ and
$\Pi_{A}$, the energy-delta function appears only if we integrate over
intermediate energy while keeping $d<4$. Subsequently, thanks to properties of
the convolution product Dadic:1999bp , we obtain $D_{R}*\Pi_{R}=D_{R}\Pi_{R}$
and $\Pi_{A}*{D}_{A}=\Pi_{A}{D}_{A}$.

The connection between vacuum polarizations Dadic:2019lmm , together with
symmetry relations (11) gives for the vacuum parts $\Pi_{F}$ (where
$n_{\pm}=0$):

$\begin{array}[]{l}\Pi_{j,k}={1\over
2}[-\Pi_{K,R}+\Pi_{K,A}-(-1)^{k}\Pi_{R}-(-1)^{j}\Pi_{A}]~{},\qquad(j,k=1,2)~{},\cr\\\
Re\Pi_{R,n_{\pm}=0}=Re\Pi_{A,n_{\pm}=0}=Re\Pi_{F}~{},\end{array}$ (14)

where for the second line we have used the symmetries (11). Now, we obtain the
renormalized value for
$Re\Pi_{ren,R,f=0}(q_{0},\vec{q})=Re\Pi_{R,f=0}(q_{0},\vec{q})-CT$, where $CT$
denotes the counter-term given by the $1/(4-d)$ term in Equation (67) or
Equation (68) in Appendix C.

The quantity $Re\Pi_{ren,R,f=0}(q_{0},\vec{q})$, in spite of having the label
$R$, is not a true retarded function, as it does not vanish when
$|q_{0}|\rightarrow\infty$. This creates a causality problem, which is
repaired Dadic:2019lmm  by considering the composite objects
$D_{R}\Pi_{ren,R}$ and $\Pi_{ren,A}{D}_{A}$, which are retarded and advanced
functions, respectively.

###  3.3 Cut Contributions

These are the terms from propagator poles and singularities of vacuum
polarizations (principal-value contributions in Equation (12)). They are also
oscillating.

While, in the first term, we may identify kinetic energy $\bar{p}_{0}$ as
$\pm\omega_{p}$, in second and third term it does not make sense as there are
two different values for $\bar{p}_{0}$ for each term.

At short times, all of the terms are important, as they make sure that the
uncertainty relations between energy and time are satisfied.

###  3.4 Comparison to the Wang-Boyanovsky Result

The above expression should be compared to the Wang–Boyanovsky expression wb :

$\displaystyle E\,{d{\cal N}(t)\over d^{3}p\,d^{3}x}={1\over
4\pi^{4}}\int_{-\infty}^{\infty}dp_{0}{1-\cos(p_{0}-E)t\over(p_{0}-E)^{2}}R(p_{0}),$
(15)

where $R(p_{0})$ (in their notation, which is $f=n_{+}$ and ${\bar{f}}=n_{-}$
in our notation (4)) is

$\begin{array}[]{l}R(p_{0})={20\pi^{2}\alpha\over
3}\int{d^{3}k\over(2\pi)^{3}}\cr\\\
\left([1-(\hat{p}\hat{k})(\hat{p}\hat{q})][f(k)[1-\bar{f}(q)]\delta(p_{0}-k+q)+[1-f(k)]\bar{f}(q)]\delta(p_{0}+k-q)\right.\cr\\\
\left.+[1+(\hat{p}\hat{k})(\hat{p}\hat{q})]f(k)\bar{f}(q)\delta(p_{0}-k-q)+[1-f(k)][1-\bar{f}(q)]\delta(p_{0}+k+q)\right),\end{array}$
(16)

and where $k=|\vec{k}|$, $\hat{k}={\vec{k}\over k}$, $\hat{p}={\vec{p}\over
E}$, $\vec{q}=\vec{p}-\vec{k}$, $q=|\vec{q}|$, and $\hat{q}=\vec{q}/q$.

Despite a number of common features, there are significant differences: Within
the one-loop order approximation, our result is exact, while Refs. wb ; ng
ignore a few terms, and make further approximations and simplifications.

In our case, the emitted photons have undefined energy in accordance with the
Heisenberg uncertainty principle, similar to the findings of Millington and
Pilaftsis in the context of a simple self-interacting theory of a real massive
scalar field Millington:2012pf . For Equation (15), Wang and Boyanovsky claim
wb  that the photons are on mass shell, i.e., $p_{0}=\omega_{p}$. As the
average of $\omega_{p}$ diverges Arleo:2004gn , it means infinite energy
emitted in photons. In our case, the infinities are subtracted from
$\Pi_{R(A)}$. It is evident that other terms are not “dangerous”.

Our result allows for the distribution functions for $q$ and $\bar{q}$ to be
determined phenomenologically, i.e., not determined by $T,\mu$.

##  4  Conclusions

In this paper, we have calculated the production of photons from the early
stage of quark gluon plasma, within the finite-time-path out-of-equilibrium
QFT. The renormalization of mass divergence is analogously performed to the
method that was developed for renormalization of $\lambda\phi^{3}$ out-of-
equilibrium QFT Dadic:2019lmm . The result is finite. The contribution in
Equation (12) oscillates (similarly as in Ref. Millington:2012pf ) in a way
that is consistent with uncertainty relations between energy and time (except
for energy conserving term, which vanishes for kinematical reasons). It is
oscillating with the period $\propto 1/\omega_{p}$. Thus, after a few periods
it will be dominated by other, higher order terms, in a lower energy-density
medium, but still in pre-equilibrium. Nevertheless, it is important, as this
period may provide the highest energy photons.

The result (12) inspires further investigation that is necessary to compare
and predict the production of photons by numerical calculations at the lowest
order and develop methods in order to obtain higher order contributions. We
are in the process of performing the numerical analysis (to be published
separately) aiming to investigate the following aspects:

  1. 1.

Equation (12) contains renormalized $Re\Pi_{R}(\vec{p})$ linearly, whereas in
usual $S$-matrix calculations it appears quadratically in higher orders of the
perturbation expansion. Thus, Equation (12). at least in principle, offers a
possibility, albeit challenging, to extract some information about
$Re\Pi_{R}(\vec{p})$ from experiment.

  2. 2.

One should distinguish the direct photon stage from the later stage in which
the energy uncertainties are much smaller, but higher order perturbation
contributions become more important and even start to dominate (the damping
phase).

  3. 3.

The early-time distributions of quarks ($n_{+}$) and antiquarks ($n_{-}$) are
still unknown, and one should consider two very different situations: (a) the
quarks are distributed isotropically and the probing functions could be taken
as a thermalized Fermi–Dirac form like in Ref. wb . Or (b) the initial
distribution of quarks may reflect the early stage distribution of nucleons.
Some testing of Ansätze for the $n_{\pm}$ distributions will be necessary
before reaching the final conclusion on the importance of the presented
mechanism and its result (12), but we hope this will contribute to resolving
the direct photon puzzle David:2019wpt .

Other developments may follow in the more ambitious direction of
renormalization of the full out-of-equilibrium QED (and QCD).

Conceptualization, I.D. and D.K. (Dubravko Klabučar); Formal analysis, I.D.;
Investigation, I.D., D.K. (Dubravko Klabučar) and D.K. (Domagoj Kuić);
Methodology, I.D. and D.K. (Dubravko Klabučar); Software, D.K. (Domagoj Kuić);
Supervision, I.D. and D.K. (Dubravko Klabučar); Validation, I.D., D.K.
(Dubravko Klabučar) and D.K. (Domagoj Kuić); Writing—original draft, I.D.;
Writing—review & editing, I.D., D.K. (Dubravko Klabučar) and D.K. (Domagoj
Kuić). All authors have read and agreed to the published version of the
manuscript.

Domagoj Kuić acknowledges partial support of the European Regional Development
Fund—the Competitiveness and Cohesion Operational Programme: KK.01.1.1.06—RBI
TWIN SIN, as well as the Croatian Science Foundation (HrZZ) Project No.
IP-2016-06-3347. Dubravko Klabučar thanks for partial support to COST Actions
CA15213 THOR and CA16214 PHAROS.

###### Acknowledgements.

I. Dadić acknowledges useful discussions with R. Baier in the early stages of
this paper. The authors declare no conflict of interest. yes

##  Appendix A Convolution Product

In the next step, we perform the convolution products and equal time limit.

The convolution product is defined as

$\displaystyle C=A*B\Leftrightarrow
C(x,y)=\int_{0}^{\infty}dzA(x,z)B(z,y)~{}.$ (17)

In terms of Wigner transforms of projected functions Dadic:1999bp , it becomes

$\displaystyle C_{X_{0}}(p_{0},\vec{p})=\int
dp_{01}dp_{02}\,P_{X_{0}}(p_{0},{p_{01}+p_{02}\over 2})\,{1\over
2\pi}{ie^{-iX_{0}(p_{01}-p_{02}+i\epsilon)}\over
p_{01}-p_{02}+i\epsilon}\,A_{\infty}(p_{01},\vec{p})\,B_{\infty}(p_{02},\vec{p})~{},$
(18)

where

$\displaystyle P_{X_{0}}(p_{0},p^{\prime}_{0})={1\over
2\pi}\Theta(X_{0})\int_{-2X_{0}}^{2X_{0}}ds_{0}e^{is_{0}(p_{0}-p^{\prime}_{0})}={1\over\pi}\Theta(X_{0}){\sin\left(2X_{0}(p_{0}-p^{\prime}_{0})\right)\over
p_{0}-p^{\prime}_{0}},$ (19)

and

$\displaystyle
e^{-is_{0}p^{\prime}_{0}}\Theta(X_{0})\Theta(2X_{0}+s_{0})\Theta(2X_{0}-s_{0})=\int
dp_{0}e^{-is_{0}p_{0}}P_{X_{0}}(p_{0},p^{\prime}_{0}).$ (20)

We have defined Dadic:1999bp  the following properties: (1) the function of
$p_{0}$ is analytic above (below) the real axis, (2) the function goes to zero
as $|p_{0}|$ approaches infinity in the upper (lower) semi-plane. The choice
above (below) and upper (lower) refers to Retarded (Advanced) functions.

In the following cases the product simplifies even further. These cases are:
1\. If $A$ is advanced function and $B$ advanced, retarded, or even constant
function. 2\. If $B$ is retarded function and $A$ advanced, retarded, or even
constant function. Then the product becomes:

$\displaystyle C_{X_{0}}(p_{0},\vec{p})=\int
dp_{01}P_{X_{0}}(p_{0},p_{01})A_{\infty}(p_{01},\vec{p})B_{\infty}(p_{01},\vec{p}).$
(21)

Note that when $X_{0}\to\infty$, (19) becomes the Dirac $\delta$-function, and
the convolution (21) reduces to the algebraic product: $\quad
C_{\infty}(p_{0},\vec{p})\,=\,A_{\infty}(p_{0},\vec{p})\,B_{\infty}(p_{0},\vec{p})$.

However, we omit the subscript ∞ throughout this paper.

##  Appendix B QED—Propagators

The propagators in the covariant gauge and in the $R,A$ basis are:

$\begin{array}[]{l}D_{\mu\nu,R}(p)=[g_{\mu\nu}-(1-a){p_{\mu}p_{\nu}\over
p^{2}+2ip_{0}\epsilon}]\Delta_{R}(p),\cr\\\
D_{\mu\nu,A}(p)=[g_{\mu\nu}-(1-a){p_{\mu}p_{\nu}\over
p^{2}-2ip_{0}\epsilon}]\Delta_{A}(p)\cr\\\ \Delta_{R(A)}(p)={i\over p^{2}\pm
2ip_{0}\epsilon}~{}.\end{array}$ (22)

It would be interesting to obtain $D_{{\mu\nu},K}(p)$ as well as the number
operator for different gauges. For simplicity we set $a=1$, obtaining Feynman
gauge.

Initial densities for two transversal polarizations $e_{1,t}$, $e_{2,t}$
(linear or circular, perpendicular to $(|\vec{p}|,\vec{p})$ and mutually
perpendicular) are given as $f_{1}(\vec{p})$, $f_{2}(\vec{p})$. They could be
joined by density $f_{l}(\vec{p})$ for “unphysical’ longitudinal polarization
$e_{l}=(0,\vec{p}/|\vec{p}|)$ and density $f_{0}(\vec{p})$ for timelike
polarization $e_{0}=(1,0)$. As the longitudinal and timelike densities do not
evolve with time one can fix them in a various ways. In particular one can set
them equal $f_{l}(\vec{p})=f_{0}(\vec{p})$, or vanishing
$f_{l}(\vec{p})=f_{0}(\vec{p})=0$. This is included in the definition of
gauge. The vectors defined above, form a new basis in four vector space:
$e_{0},e_{1},e_{2},e_{3}=e_{l}$ and $g_{\mu,\nu}$ is easily transformed to
this basis:

$g_{ij}=\sum_{\mu,\nu}g_{\mu\nu}\,e_{i}^{*\mu}e_{j}^{\nu}$,

$g_{\mu\nu}=\sum_{i,j}g_{ij}\,e^{*i}_{\mu}e^{j}_{\nu}$,

$g_{00}=1$, $g_{11}=g_{22}=g_{33}=-1$.

Then:

$\begin{array}[]{l}D_{\mu\nu,R}(p)=g_{\mu\nu}\,\Delta_{R}(p),\cr\\\
D_{\mu\nu,A}(p)=g_{\mu\nu}\,\Delta_{A}(p),\cr\\\
D_{{\mu\nu,K}}(p)=2\pi\delta(p^{2})\sum_{i,j}g_{ij}e^{*i}_{\mu}e^{j}_{\nu}[1+2f_{i}(\omega_{p})]\cr\\\
=D_{{\mu\nu},K,R}(p)-D_{{\mu\nu},K,A}(p),\cr\\\
D_{{\mu\nu},K,R}(p)=-D_{{\mu\nu},K,A}(-p)\cr\\\
=-{p_{0}\over\omega_{p}}\sum_{i,j}g_{ij}\,e^{*i}_{\mu}\,e^{j}_{\nu}[1+2f_{i}(\omega_{p})]\,\Delta_{R}(p),\cr\\\
\omega_{p}=|\vec{p}|.\end{array}$ (23)

One needs spinor propagator (we omit the label ∞, it is understood that
whenever the time label is omitted it should be ∞)

$\begin{array}[]{l}S_{R}(p)\,=\,(p\hbox to0.0pt{\hss/\kern
1.0pt}+m)\,G_{R}(p,m)~{},\cr\\\ S_{A}(p)\,=\,(p\hbox to0.0pt{\hss/\kern
1.0pt}+m)\,G_{A}(p,m)~{},\cr\\\ G_{R(A)}(p,m)\,=\,{-i\over p^{2}-m^{2}\pm
2ip_{0}\epsilon}~{},\cr\\\ S_{K}(p)=(P\hbox to0.0pt{\hss/\kern
1.0pt}+{mp_{0}\over\omega_{p}})\,2\pi\delta(p^{2}-m^{2})\,[1-2n(\omega_{p})]=S_{K,R}(p)-S_{K,A}(p),\cr\\\
S_{K,R}(p)=-[1-2n(\omega_{p})]\,(P\hbox to0.0pt{\hss/\kern
1.0pt}+{mp_{0}\over\omega_{p}})\,G_{R}(p,m),\cr\\\
S_{K,A}(p)=-[1-2n(\omega_{p})]\,(P\hbox to0.0pt{\hss/\kern
1.0pt}+{mp_{0}\over\omega_{p}})\,G_{A}(p,m),\cr\\\ p\hbox to0.0pt{\hss/\kern
1.0pt}=\gamma^{\mu}p_{\mu},~{}p=(p_{0},\vec{p}),~{}P\hbox to0.0pt{\hss/\kern
1.0pt}=\gamma^{\mu}P_{\mu},~{}P=(\omega_{p},{p_{0}\over\omega_{p}}\vec{p}),~{}~{}\omega_{p}=\sqrt{\vec{p}^{2}+m^{2}\,}.\end{array}$
(24)

$n(\omega_{p})$ is initial fermion distribution function. However, initial
fermion and antifermion distribution functions can in general be different.
For unequal distributions one has

$\displaystyle
n(p_{0},\omega_{p},\vec{p})=\Theta(p_{0})n_{+}(\omega_{p},\vec{p})+\Theta(-p_{0})n_{-}(\omega_{p},-\vec{p})~{}.$
(25)

Now we decompose $K$-propagator into it‘s retarded and advanced part

$\begin{array}[]{l}S_{K}(p,m)\,=\,S_{K,R}(p,m)-S_{K,A}(p,m)~{},\cr\\\
S_{K,R}(p,m)\,=\,-\,G_{R}(p,m)\,L(p_{0},\vec{p})~{},\cr\\\
S_{K,A}(p,m)\,=\,-\,G_{A}(p,m)\,L(p_{0},\vec{p})~{},\end{array}$ (26)

where

$\begin{array}[]{l}L(p_{0},\vec{p})\,=\,[1-2n_{+}(\omega_{p},\vec{p})]{p_{0}+\omega_{p}\over
2\omega_{p}}~{}2m\Lambda_{+}(\omega_{p},\vec{p})\cr\\\
+[1-2n_{-}(\omega_{p},-\vec{p})]{p_{0}-\omega_{p}\over
2\omega_{p}}~{}2m\Lambda_{-}(\omega_{p},-\vec{p})~{},\cr\\\
\Lambda_{+}(\omega_{p},\vec{p})={\gamma_{0}\omega_{p}-\vec{\gamma}\vec{p}+m\over
2m}~{},\cr\\\
\Lambda_{-}(\omega_{p},\vec{p})={-\gamma_{0}\omega_{p}+\vec{\gamma}\vec{p}+m\over
2m}~{},\end{array}$ (27)

where the projectors $\Lambda_{\pm}$ satisfy
$\Lambda_{\pm}(\omega_{p},\vec{p})^{2}=\Lambda_{\pm}(\omega_{p},\vec{p})$,
while $\Lambda_{+}(\omega_{p},\vec{p})\Lambda_{-}(\omega_{p},\vec{p})=0$,
$\Lambda_{-}(\omega_{p},\vec{p})\Lambda_{+}(\omega_{p},\vec{p})=0$, and
$\Lambda_{-}(\omega_{p},\vec{p})+\Lambda_{+}(\omega_{p},\vec{p})=1$.

The above result can be rewritten as:

$\begin{array}[]{l}S_{K,R(A)}(p)=[1-2{\bar{n}}(\omega_{p})]{i(P\hbox
to0.0pt{\hss/\kern 1.0pt}+mp_{0}/\omega_{p})\over p^{2}-m^{2}\pm
2ip_{0}\epsilon}-2n_{\Delta}(\omega_{p}){i(p\hbox to0.0pt{\hss/\kern
1.0pt}+m)\over p^{2}-m^{2}\pm 2ip_{0}\epsilon},\cr\\\
\bar{n}(\omega_{p})={n_{+}(\omega_{p})+n_{-}(\omega_{p})\over
2},~{}~{}n_{\Delta}(\omega_{p})={n_{+}(\omega_{p})-n_{-}(\omega_{p})\over
2}~{},\end{array}$ (28)

We can also obtain $S_{11}$,

$\begin{array}[]{lll}S_{11}(p)&=&{1-2{\bar{n}}(\omega_{p})\over 2}{i(P\hbox
to0.0pt{\hss/\kern 1.0pt}+mp_{0}/\omega_{p})\over
p^{2}-m^{2}+2ip_{0}\epsilon}+{1-2n_{\Delta}(\omega_{p})\over 2}{i(p\hbox
to0.0pt{\hss/\kern 1.0pt}+m)\over p^{2}-m^{2}+2ip_{0}\epsilon}\cr\\\
&-&{1-2{\bar{n}}(\omega_{p})\over 2}{i(P\hbox to0.0pt{\hss/\kern
1.0pt}+mp_{0}/\omega_{p})\over
p^{2}-m^{2}-2ip_{0}\epsilon}+{1+2n_{\Delta}(\omega_{p})\over 2}{i(p\hbox
to0.0pt{\hss/\kern 1.0pt}+m)\over p^{2}-m^{2}-2ip_{0}\epsilon}~{}.\end{array}$
(29)

These propagators satisfy the following properties under inversion:

$S_{R}(-p)=-\bar{S}_{A}(p),~{}~{}S_{K,R}(-p)=-\bar{S}_{K,A}(p)$, where
$\bar{S}$ is the propagator of antifermion (i.e., with the the replacement of
$n_{+}$ by $n_{-}$).

Then,

$\begin{array}[]{l}\langle\,1-2N_{\pm,\pm\vec{p}}(t)\,\rangle\,=\,{\omega_{p}\over
m\pi}\int dp_{0}{1\over 4}Tr_{\pm}[S_{K,R,t}(p)-S_{K,A,t}(p)]\gamma_{0}.\cr\\\
={\omega_{p}\over m\pi}\int dp_{0}{1\over
4}Tr\Lambda_{\pm}(\omega_{p},\pm\vec{p})[S_{K,R,t}(p)-S_{K,A,t}(p)]\gamma_{0}.\end{array}$
(30)

where the subscript “+” indicates that the trace $Tr$ is taken over the
fermion degrees of freedom and “$-$” over the antifermion degrees of freedom.
When acting on momentum & spin eigenstates of fermions,
$|+,\vec{p},s\,\rangle$, and of antifermions, $|-,-\vec{p},s\,\rangle$ (both
normalized to $\langle\,\pm,\pm\vec{p},s|\pm,\pm\vec{p},s\,\rangle\,=\,1/m$),
the projectors $\Lambda_{\pm}$ satisfy

$\Lambda_{+}(\omega_{p},\vec{p})|+,\vec{p},s\,\rangle\,=\,|+,\vec{p},s\,\rangle\quad$
and
$\quad\Lambda_{-}(\omega_{p},\vec{p})|+,\vec{p},s\,\rangle\,=\,0~{},\quad\quad$
while

$\Lambda_{+}(\omega_{p},\vec{p})|-,-\vec{p},s\,\rangle\,=\,0\quad$ and
$\quad\Lambda_{-}(\omega_{p},\vec{p})|-,-\vec{p},s\,\rangle\,=\,|-,-\vec{p},s\,\rangle$.

##  Appendix C One-Loop Vacuum Polarizations

In this appendix, the vacuum polarizations are calculated as contributions of
a single quark flavor q with the charge $e_{\rm q}=C_{\rm q}\,e$, where
$C_{\rm q}$ can take the values $C_{\rm q}=\pm 1/3,\pm 2/3$. ($e$ is the
electron charge.) Each flavor has its initial distribution functions
$n_{\pm}(\omega_{p})$, and mass $m_{\rm q}$. In the present paper, we used the
mass symbol $m$ without the flavor subscript q because our analysis pertains
to a single flavor. To obtain the full result, one has to sum over all active
quark flavors.

One obtains vacuum polarizations easily as one knows the perturbation
expansion for matrix propagators. Here we use $\bar{S}$ as a symbol for anti-
fermion propagator. It differs from the fermion propagator by the fact that
the roles of $n_{+}$ and $n_{-}$ are interchanged. For all one-loop vacuum
polarizations in Equation (31), $\Pi^{1}=\Pi_{A},\Pi_{R},\Pi_{K}$ (the label
$\infty$ is omitted for simplicity after (31)),

$\displaystyle\Pi_{t}(p_{0},\vec{p})=\int_{-\infty}^{\infty}dp^{\prime}_{0}P_{t}(p_{0},p_{0}^{\prime})\Pi_{\infty}(p_{0}^{\prime},\vec{p})~{},$
(31) $\displaystyle\Pi_{A}(p_{0},\vec{p})={ig^{2}\over
4}\int{dq_{0}d^{d-1}q\over(2\pi)^{d}}[S_{A}(p_{0}-q_{0},\vec{p}-\vec{q},m)\bar{S}_{K}(q_{0},\vec{q},m)$
(32)
$\displaystyle+S_{K}(p_{0}-q_{0},\vec{p}-\vec{q},m)\bar{S}_{A}(q_{0},\vec{q},m)]~{},$
(33) $\displaystyle\Pi_{R}(p_{0},\vec{p})=-{ig^{2}\over
4}\int{dq_{0}d^{d-1}q\over(2\pi)^{d}}[S_{R}(p_{0}-q_{0},\vec{p}-\vec{q},m)\bar{S}_{K}(q_{0},\vec{q},m)$
(34)
$\displaystyle+S_{K}(p_{0}-q_{0},\vec{p}-\vec{q},m)\bar{S}_{R}(q_{0},\vec{q},m)]~{},$
(35) $\displaystyle Im~{}\Pi_{A}(p_{0},\vec{p})=-{g^{2}\pi\over
4}\int{d^{d-1}q\over(2\pi)^{d}}\int
dq_{0}\delta(q_{0}^{2}-\omega_{q}^{2})\delta((p_{0}-q_{0})^{2}-\omega_{p-q}^{2})$
(36) $\displaystyle\times sign[q_{0}(p_{0}-q_{0})][Tr\gamma^{0}(p\hbox
to0.0pt{\hss/\kern 1.0pt}-q\hbox to0.0pt{\hss/\kern 1.0pt}+m)\gamma^{0}(q\hbox
to0.0pt{\hss/\kern
1.0pt}+m)]_{|q_{0}|=\omega_{q},~{}|p_{0}-q_{0}|=\omega_{p-q}}$ (37)
$\displaystyle[\Theta(p_{0}-q_{0})(1-2n_{+}(\omega_{p-q}))-(1-\Theta(p_{0}-q_{0}))(1-2n_{-}(\omega_{p-q}))$
(38)
$\displaystyle+\Theta(q_{0})(1-2n_{-}(\omega_{q}))-(1-\Theta(q_{0}))(1-2n_{+}(\omega_{q}))]~{},$
(39) $\displaystyle
Im~{}\Pi_{A}(p_{0},\vec{p})=-Im~{}\Pi_{R}(p_{0},\vec{p})=-Im~{}\Pi_{A}(-p_{0},\vec{p})=Im~{}\Pi_{R}(-p_{0},\vec{p})~{},$
(40) $\displaystyle
Re~{}\Pi_{A}(p_{0},\vec{p})=Re~{}\Pi_{R}(p_{0},\vec{p})~{},$ (41)
$\displaystyle\Pi_{K}(p_{0},\vec{p})=-\Pi_{K,R}(p_{0},\vec{p})+\Pi_{K,A}(p_{0},\vec{p})~{},$
(42) $\displaystyle\Pi_{K,R}(p_{0},\vec{p})=$ (43) $\displaystyle{-ig^{2}\over
2}\int{dq_{0}d^{d-1}q\over(2\pi)^{d}}[S_{K,R}(p_{0}-q_{0},\vec{p}-\vec{q},m)\bar{S}_{K,R}(q_{0},\vec{q},m)$
(44)
$\displaystyle+S_{R}(p_{0}-q_{0},\vec{p}-\vec{q},m)\bar{S}_{R}(q_{0},\vec{q},m)]~{},$
(45) $\displaystyle\Pi_{K,A}(p_{0},\vec{p})=$ (46) $\displaystyle{ig^{2}\over
2}\int{dq_{0}d^{d-1}q\over(2\pi)^{d}}[S_{K,A}(p_{0}-q_{0},\vec{p}-\vec{q},m)\bar{S}_{K,A}(q_{0},\vec{q},m)$
(47)
$\displaystyle+S_{A}(p_{0}-q_{0},\vec{p}-\vec{q},m)\bar{S}_{A}(q_{0},\vec{q},m)]$
(48) $\displaystyle
Re~{}\Pi_{K,A}(p_{0},\vec{p})=Re~{}\Pi_{K,R}(p_{0},\vec{p}),$ (49)
$\displaystyle Im~{}\Pi_{K,A}(p_{0},\vec{p})$ (50)
$\displaystyle=-{g^{2}\pi\over 4}\int{d^{d-1}q\over(2\pi)^{d-1}}\int
dq_{0}\delta(q_{0}^{2}-\omega_{q}^{2})\delta((p_{0}-q_{0})^{2}-\omega_{p-q}^{2})$
(51) $\displaystyle\times sign[q_{0}(p_{0}-q_{0})][Tr\gamma^{0}(p\hbox
to0.0pt{\hss/\kern 1.0pt}-q\hbox to0.0pt{\hss/\kern 1.0pt}+m)\gamma^{0}(q\hbox
to0.0pt{\hss/\kern
1.0pt}+m)]_{|q_{0}|=\omega_{q},~{}|p_{0}-q_{0}|=\omega_{p-q}}$ (52)
$\displaystyle[[\Theta(p_{0}-q_{0})(1-2n_{+}(\omega_{p-q}))-(1-\Theta(p_{0}-q_{0}))(1-2n_{-}(\omega_{p-q}))]$
(53)
$\displaystyle\times[\Theta(q_{0})(1-2n_{-}(\omega_{q}))-(1-\Theta(q_{0}))(1-2n_{+}(\omega_{q}))]+1]~{},$
(54) $\displaystyle
Im~{}\Pi_{K,A}(p_{0},\vec{p})=-Im~{}\Pi_{K,R}(p_{0},\vec{p}),$ (55)
$\displaystyle\omega_{q}^{2}=m^{2}+\vec{q}^{2},~{}~{}~{}\omega_{p-q}^{2}=m^{2}+(\vec{p}-\vec{q})^{2}$
(56)
$\displaystyle\delta(q_{0}^{2}-\omega_{q}^{2})\delta((p_{0}-q_{0})^{2}-\omega_{p-q}^{2})$
(57)
$\displaystyle=\sum_{\lambda,\lambda^{\prime}=\pm}{\delta(q_{0}-\lambda\omega_{q})\delta(p_{0}-q_{0}-\lambda^{\prime}\omega_{p-q})\over
4\omega_{q}\omega_{p-q}}~{},$ (58)

$\displaystyle
Im\Pi_{K,R}(\omega_{p},\vec{p})+Im\Pi_{R}(\omega_{p},\vec{p})(1+2f(\omega_{p}))$
(59) $\displaystyle={g^{2}\pi\over 4}\int{d^{d-1}q\over(2\pi)^{d-1}}\int
dq_{0}\delta(q_{0}^{2}-\omega_{q}^{2})\delta((p_{0}-q_{0})^{2}-\omega_{p-q}^{2})$
(60) $\displaystyle\times sign[q_{0}(p_{0}-q_{0})][Tr\gamma^{0}(p\hbox
to0.0pt{\hss/\kern 1.0pt}-q\hbox to0.0pt{\hss/\kern 1.0pt}+m)\gamma^{0}(q\hbox
to0.0pt{\hss/\kern
1.0pt}+m)]_{|q_{0}|=\omega_{q},~{}|p_{0}-q_{0}|=\omega_{p-q}}$ (61)
$\displaystyle\left[\,\\{\,[\,\Theta(p_{0}-q_{0})(1-2n_{+}(\omega_{p-q}))-(1-\Theta(p_{0}-q_{0}))(1-2n_{-}(\omega_{p-q}))]\right.$
(62)
$\displaystyle\times[\,\Theta(q_{0})(1-2n_{-}(\omega_{q}))-(1-\Theta(q_{0}))(1-2n_{+}(\omega_{q}))]+1\,\\}$
(63)
$\displaystyle-[\,\Theta(p_{0}-q_{0})(1-2n_{+}(\omega_{p-q}))-(1-\Theta(p_{0}-q_{0}))(1-2n_{-}(\omega_{p-q}))$
(64)
$\displaystyle+\,\Theta(q_{0})(1-2n_{-}(\omega_{q}))-(1-\Theta(q_{0}))(1-2n_{+}(\omega_{q}))](1+2f(\omega_{p}))\,]\,.$
(65)

At $|q_{0}|=\omega_{q}$ and $|p_{0}-q_{0}|=\omega_{p-q}$, the bracket
containing distribution functions vanishes when the functions satisfy detailed
balance condition in all channels:

$\begin{array}[]{l}Im\Pi_{K,R}(\omega_{p},\vec{p})+Im\Pi_{R}(\omega_{p},\vec{p})(1+2f(\omega_{p}))\cr\\\
={g^{2}\pi\over 4}\int{d^{d-1}q\over(2\pi)^{d-1}}\int
dq_{0}\delta(q_{0}^{2}-\omega_{q}^{2})\delta((p_{0}-q_{0})^{2}-\omega_{p-q}^{2})\cr\\\
\times sign[q_{0}(p_{0}-q_{0})][Tr\gamma^{0}(p\hbox to0.0pt{\hss/\kern
1.0pt}-q\hbox to0.0pt{\hss/\kern 1.0pt}+m)\gamma^{0}(q\hbox to0.0pt{\hss/\kern
1.0pt}+m)]_{|q_{0}|=\omega_{q},~{}|p_{0}-q_{0}|=\omega_{p-q}}\cr\\\
\times[4\Theta(p_{0}-q_{0})\Theta(q_{0})[n_{+}(\omega_{p-q})n_{-}(\omega_{q})(1+f(\omega_{p}))(1-n_{+}(\omega_{p-q}))(1-n_{-}(\omega_{q}))f(\omega_{p})]\cr\\\
+4\Theta(q_{0}-p_{0})\Theta(-q_{0})[n_{-}(\omega_{p-q})n_{+}(\omega_{q})(1+f(\omega_{p}))(1-n_{-}(\omega_{p-q}))(1-n_{+}(\omega_{q}))f(\omega_{p})]\cr\\\
+4\Theta(q_{0}-p_{0})\Theta(q_{0})[(1-n_{-}(\omega_{p-q}))n_{-}(\omega_{q})(1+f(\omega_{p}))-n_{-}(\omega_{p-q})(1-n_{-}(\omega_{q}))f(\omega_{p})]\cr\\\
+4\Theta(p_{0}-q_{0})\Theta(-q_{0})[(1-n_{+}(\omega_{p-q}))n_{+}(\omega_{q})f(\omega_{p})\cr\\\
-n_{+}(\omega_{p-q})(1-n_{+}(\omega_{q}))(1+f(\omega_{p}))(1-n_{+}(\omega_{p-q}))n_{+}(\omega_{q})f(\omega_{p})]].\end{array}$
(66)

Notice that in this section of appendix, we have assumed that the initial
photon distribution does not depend on the photon polarization $e$. Thus, we
wrote $f(\omega_{p})$ instead of $f_{e}(\omega_{p})$.

Regularized $\Pi_{F}$

For the dimensionally regularized $\Pi_{F}$, $\kappa\equiv 4-d>0$, the result
is causal, as $|q_{0}|\rightarrow\infty$ implies
$|\Pi_{\mu\nu,F,d}(q)|\rightarrow 0$. The Feynman component ($n_{\pm}=0$) of
the vacuum polarization, expanded around small $\kappa$, can be written as
RyderBook

$\displaystyle\Pi_{\mu\nu,F}(q)={e^{2}\over
2\pi^{2}}\,(q_{\mu}q_{\nu}-q^{2}g_{\mu\nu})\,[\,{1\over
3\kappa}\,-\,{\gamma_{\rm E}\over
6}-\int_{0}^{1}dz\,z(1-z)\,\ln{q^{2}z(1-z)-m^{2}\over 4\pi\mu^{2}}]~{}.$ (67)

For small $q^{2}$ Dadic:2019lmm , this becomes

$\displaystyle\Pi_{\mu\nu,F}(q)={e^{2}\over
6\pi^{2}}(q_{\mu}q_{\nu}-q^{2}g_{\mu\nu})({1\over\kappa}+{q^{2}\over
10m^{2}}+...)$ (68) $\displaystyle={e^{2}\over
6\pi^{2}}(q_{\mu}q_{\nu}-q^{2}g_{\mu\nu}){1\over\kappa}\,+\,{\rm finite}~{}.$
(69)

We see that the finite part of $\Pi_{\mu\nu,F}(q)$ is not vanishing when
$|q_{0}|\rightarrow\infty$. This implies that also “retarded” and ”advanced”
part of it do not satisfy this requirement. To repair causality, we turned to
the composite operators pointed out below Equation (14), namely
$D_{R}\Pi_{ren,R}$ and $\Pi_{ren,A}{D}_{A}$ which satisfy this requirement.
($D_{R}$ and $D_{A}$ yield the $1/\omega_{p}^{2}$ suppression of the second
square bracket in Equation (12).)

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