††institutetext: Center for Theoretical Physics, Massachusetts Institute of
Technology, Cambridge, MA 02139, U.S.A.

# Radiative corrections for factorized jet observables in heavy ion collisions

Varun Vaidya<EMAIL_ADDRESS>
###### Abstract

I look at the renormalization of the medium structure function and a medium
induced jet function in a factorized cross section for jet substructure
observables in Heavy Ion collisions. This is based on the formalism developed
in Vaidya:2020lih , which uses an Open quantum system approach combined with
the Effective Field Theory(EFT) for forward scattering to derive a
factorization formula for jet observables which work as hard probes of a long
lived dilute Quark Gluon Plasma(QGP) medium. I show that the universal medium
structure function that captures the observable independent physics of the QGP
has both UV and rapidity anomalous dimensions that appear due to medium
induced Bremsstrahlung. The resulting Renormalization Group(RG) equations
correspond to the BFKL equation and the running of the QCD coupling
respectively. I present the first results for the numerical impact of
resummation using these RG equations on the mean free path of the jet in the
medium. I also briefly discuss the prospects of extending this formalism for a
short lived dense medium.

††preprint:  MIT–CTP 5310

##  1 Introduction

It is now widely accepted that heavy ion collisions are the laboratory for the
creation and study of the Quark Gluon Plasma medium. The high energy collision
of nuclei both at RHIC and the LHC creates sufficiently energetic partons that
can escape confinement from color neutral hadrons and give rise to a
strongly/weakly coupled soup of deconfined quarks and gluons known as the
Quark Gluon Plasma medium. Experimental evidence suggests that this medium
which exists for a very short time ($\sim 10$ fm/c) behaves as a near perfect
liquid in thermal equilibrium with very low viscosity. At the microscopic
level, we can think of this plasma as consisting of soft partons with typical
energy of the order of the temperature(T) of the medium which is usually much
lower than the center of mass energy of the initiating nuclear collision.
These stopping nuclear collisions which create the QGP are accompanied by hard
interactions which create highly energetic partons (E $\gg$ T) which
eventually form jets. These jets then have to traverse through a region of the
hot QGP as they evolve and hence they get modified in heavy ion collision,
compared with proton-proton collisions, due to the jet-medium interaction. The
modification of a jets in a medium compared to its vacuum evolution can shed
light on the properties of the medium, making them useful hard probes of the
Quark Gluon Plasma. A phenomenon that has been extensively studied in
literatureGyulassy:1993hr ; Wang:1994fx ; Baier:1994bd ; Baier:1996kr ;
Baier:1996sk ; Zakharov:1996fv ; Zakharov:1997uu ; Gyulassy:1999zd ;
Gyulassy:2000er ; Wiedemann:2000za ; Guo:2000nz ; Wang:2001ifa ; Arnold:2002ja
; Arnold:2002zm ; Salgado:2003gb ; Armesto:2003jh ; Majumder:2006wi ;
Majumder:2007zh ; Neufeld:2008fi ; Neufeld:2009ep  is that of Jet quenching,
which entails a suppression of particles with high transverse momenta in the
medium. This has also been recently observed in experiments at both
Relativistic Heavy Ion Collider (RHIC) Arsene:2004fa ; Back:2004je ;
Adams:2005dq ; Adcox:2004mh  and Large Hadron Collider (LHC) Aad:2010bu ;
Aamodt:2010jd ; Chatrchyan:2011sx . The suppression mechanism happens through
the mechanism of energy loss when jets travel through the hot medium. This can
happen either through a collision of the energetic partons in the jet with the
soft partons of the medium or through medium-induced radiation, but the latter
dominates at high energy. The key to understand jet quenching and jet
substructure modifications in heavy ion collisions is to understand how the
jet interacts with the expanding medium. There has been tremendous theoretical
effort to study the jet energy loss mechanism (see Refs. Mehtar-Tani:2013pia ;
Blaizot:2015lma ; Qin:2015srf ; Cao:2020wlm  for recent reviews). All these
attempts rely on a direct Feynman diagram calculation which is only valid in a
perturbative regime and the information about the medium is encoded in the
form of a transport co-efficient $\hat{q}$ which measures the average
transverse momentum gained by a parton per unit time. Given that the evolution
of the jet in the medium is a multi-scale problem with a hierarchy of scales,
a systematic Effective Field Theory(EFT) treatment is lacking. At the same
time, a complete analytical calculation which includes the hard interaction
which creates the jet along with the subsequent medium interactions is also
necessary for comparison with data. In Vaidya:2020lih , I proposed an EFT
approach using tools of Open Quantum systems and SCET(Soft Collinear Effective
Theory)(Bauer:2002aj ; Bauer:2003mga ; Manohar:2006nz ; Bauer:2000yr ;
Bauer:2001ct ; Bauer:2002nz ) for jet substructure observables in heavy ion
collisions. There are several advantages of adopting an EFT approach

  * •

It provides an explicit separation of physics at widely separated scales in
the form of a factorization formula for the the jet observables. In the
context of Heavy Ion collisions, it was shown in Vaidya:2020lih  that this
completely isolates the universal physics of the medium from the kinematic and
observable dependent physics of jet evolution.

  * •

It gives a gauge invariant operator definition for the factorized functions.
In particular, the physics of the medium was shown to be captured by a
correlator of Soft current computed in the background of the medium density
matrix. This definition was independent of the details of the medium density
matrix, with the assumption of homogeneity of the medium over certain scales.

  * •

The factorization does not rely on perturbation theory but only on a
separation of scales. So the factorization holds even when some of the
functions become non-perturbative. Thus, it gives a clear boundary between the
perturbatively calculable and non-calculable physics. This is in exact analogy
with factorization formulas for ep and pp experiments such as Deep Inelastic
scattering(DIS) and Drell-Yan where the universal physics of the proton is
defined by a Parton Distribution Function(PDF). This has a precise operator
definition and is non-perturbative while being factorized from process
dependent perturbative physics. This has the advantage that if some of the
factorized functions become non-perturbative, there is a possibility of
computing them from lattice/quantum computers.

  * •

Lastly, the separation of scales allows us to resum large logarithms in the
ratio of these scales via Renormalization group equation for the factorized
functions. This is systematically improvable and has already led to precise
predictions in pp and ep experiments.

The evolution of the jet in the medium usually depends on multiple scales
which can be broadly divided into three categories: Kinematic scales such as
such as the jet energy, thermal scales of the QGP , namely the temperature T
and the Debye screening mass $m_{D}\sim gT$ and the size or equivalently the
temporal extent of the medium. The second class of scales that appear due to
dynamical evolution of the system are the mean free path of the jet and the
formation time of the jet, which is the time scale over which the hard partons
created in the jet go on-shell. The third and final category are the
measurement scales imposed on the final state jet. The hierarchy between these
scales can vary widely depending on the details of the experiment. However, in
general the jet energy scale(Q) will be a hard scale much larger than all the
other scales. For sufficiently high temperature(T $\gg\Lambda_{QCD}$), we can
assume $g\ll 1$ and so that $m_{D}<<T$ which is what we will assume for the
rest of this paper. In current heavy ion collision experiments, the
temperature achieved lies in the range $150-500$ MeV, and may not always be a
perturbative scale. Thus, a fully weak coupling calculation may not be valid.
In this paper, I will stick to the case of a weakly coupled QGP for
simplicity. While this will enable me to do a perturbative calculation, it is
not a requirement for deriving a factorization of the physics at widely
separated scales. For $T\sim\Lambda_{QCD}$, some of the functions in the
factorization formula become non-perturbative and would then need to be
extracted from lattice/experiment.

At the same time, as a first simple case we will assume the size of the
medium(L) and the mean free path($\lambda_{\text{mfp}}$) to be much larger
than all other time scales in the problem. This may not always be a realistic
assumption, in particular the formation time of the jet($t_{F}$) would be
expected to be much larger than L in current experiments especially for high
energy jets. At the same time, a dense medium will lead to a small value for
$\lambda_{\text{mfp}}$ in which case the well known LPM(Landau-Pomeranchuk-
Migdal) effect (CaronHuot:2010bp ; Ke:2018jem ; Mehtar-Tani:2019ygg ;
Arnold:2015qya ; Arnold:2016kek ) will become important. However, the
assumption $t_{F}\ll L\sim\lambda_{\text{mfp}}$ greatly simplifies the
analysis of the system and gives us a taste of how factorization works out.
The more involved but phenomenologically relevant case
$t_{F}\geq\lambda_{\text{mfp}}\gg L$ will be considered in a companion paper
Varun . I will also argue why the more complicated cases will continue to have
the same form for factorization, albeit with some modifications in the
definitions of the functions so that most of the framework developed for the
simpler case can be carried over.

The existence of so many scales makes this a particularly rich system to
analyze. Due to the multiple scales involved in the problem, a powerful tool
to deal with large hierarchies of scales is Effective Field Theory (EFT). The
EFT that is extensively used for jet studies at high energy pp colliders is
Soft-Collinear Effective Theory (SCET) which provides a systematic approach
towards dealing with a multi-scale scattering problem. However, in heavy ion
collisions, there is the added complication of the presence of the medium
whose detailed time evolution cannot be easily described analytically. A way
to deal with our ignorance about the microscopic details of the medium is to
use an Open quantum systems approach (see Breuer:2002pc ; OQS  for review)
which works by tracing over the medium degrees of freedom and working out the
effective evolution of the reduced density matrix of the system(jet).

There are also formulations of SCET (known as SCETG) treating the Glauber
gluon, which is a type of mode appearing in forward scattering, as a
background field induced by the medium interacting with an energetic jet. By
making use of the collinear sector of the corresponding EFT, this formalism
has been used to address the question of jet quenching in the medium
Ovanesyan:2011kn ; Chien:2015hda ; Ovanesyan:2011xy ; Chien:2015vja ;
Kang:2014xsa . I will use a modern approach using a new EFT for forward
scattering that has been developed recently Rothstein:2016bsq  which also uses
the Glauber mode to write down contact operators between the soft and
collinear momentum degrees of freedom which is ideally suited for the
situation we want to study. As was shown in Vaidya:2020lih , this approach
gives a manifestly gauge invariant definition for the factorized functions. A
consequence of this, which I will show in this paper is that it captures
important radiative corrections which are otherwise absent in previous EFT
formulations.

The long term goal is to develop a theoretically robust formalism for
calculating jet substructure observables for both light parton and heavy quark
jets. For example, the bottom quark jets have been identified as an effective
probe of the QGP medium and will be experimentally studied at LHC, as well as
by the sPHENIX collaboration at RHIC. There has been recent work on computing
jet substructure observables for heavy quark jets in the context of proton-
proton collisions Lee:2019lge ; Makris:2018npl . The objective would then be
to compute the same observables in heavy ion collisions and study
modifications caused by the medium.

A first step was taken in Vaidya:2020cyi  which looked at the transverse
momentum spread of a single energetic quark as a function of the time of
propagation through the QGP medium. The multiple interactions of the quark
with the medium were resummed by solving a Lindblad master equation for the
quark jet. However for a realistic description of the system, we also need to
account for the initial hard interaction that creates the energetic quark
which is dressed with radiation from the subsequent parton shower which
accounts for any vacuum evolution of the jet along with any medium
interactions.

Combining SCET with Open quantum systems, a factorization formula for a
illustrative jet substructure observable taking into account the hard
interaction and subsequent parton shower was derived in Vaidya:2020lih  for an
idealized case of a long lived dilute medium. This paper introduced a
universal medium structure function for the Quark Gluon Plasma in analogy with
a PDF for hadron structure along with a medium induced jet function. Radiative
corrections corresponding to elastic collisions of the jet with the medium
were calculated in this paper and shown to be UV finite.

In this paper, I will present the results for radiative corrections from
medium induced radiation. I will show that these corrections induce rapidity
and UV divergences in the medium structure and jet functions which help
determine the Renormalization Group(RG) evolution of these functions. The
solution for these RG equations helps resum logarithms of widely separated
scales which form the dominant corrections compared to those from elastic
collisions. I will discuss the numerical impact of resummation on the mean
free path of the jet in the medium.

This paper is organized as follows. In Section 2 I review the details of the
observable and factorization formula derived in Vaidya:2020lih  and present
the corrections from elastic collisions of the jet partons with the medium. In
Section 3 I detail the one loop corrections from medium induced Brehmstrahlung
for the medium structure function and the medium jet function which also gives
their Renormalization Group(RG) equations. Section 4 gives analytical results
for the solution of RG evolution equations. Section 5 provides the solution of
the Lindblad type evolution equation which resums multiple incoherent
interactions of the jet with the medium with a RG evolved medium kernel.
Section 6 discusses the numerical impact of RG evolution on the mean free path
of the jet in the QGP medium. A summary and analysis of results along with
future directions in provided in Section 7.

##  2 Factorization for Jet substructure observables

In Vaidya:2020lih , I developed an EFT formalism for jet substructure
observables in a heavy ion collision environment, writing down a factorization
formula for an illustrative observable: The transverse momentum imbalance
between groomed dijets along with a cumulative jet mass measurement on each
jet. This observable was introduced to allow for a clean measurement while
countering the issue of jet selection bias. We want to consider final state
fat (jet radius R $\sim$ 1) dijet events produced in a heavy ion collision in
the background of a QGP medium. The jets are isolated using a suitable jet
algorithm such as anti-kT with jet radius $R\sim 1$. We examine the scenario
when the hard interaction creating the back to back jets happens at the
periphery of the heavy ion collision so that effectively only one jet passes
through the medium while the other evolves purely in vacuum.
Since, it is hard to give a reliable theory prediction for the distribution of
soft hadrons from the cooling QGP medium, we put a grooming on the jets. We
put in a simple energy cut-off sufficiently large to remove all partons at
energy T and lower. Given a hard scale Q $\sim 2E_{J}$, where $E_{J}$ is the
energy of the jet and an energy cut-off, $z_{c}E_{J}$, we work in the
hierarchy

$\displaystyle Q\sim z_{c}Q\gg T$ (1)

where T is the plasma temperature. The measurement we wish to impose is the
transverse momentum imbalance between the two jets and we want to to give
predictions for the regime $q_{T}\sim T$. We are going to assume a high
temperature weak coupling $g<<1$ scenario so that the Debye screening mass
$m_{D}$ is parametrically much smaller than the temperature.

While this constrains the radiation that lies outside the groomed jet, we
still need to ensure that the radiation that passes grooming and hence forms
the jet has a single hard subjet. This can be ensured by putting a cumulative
jet mass measurement $e_{i}$ on each groomed jet with the scaling $e_{i}\sim
T^{2}/Q^{2}$.

We wish to write down a factorization theorem within Soft Collinear Effective
Theory(SCET) which separates out functions depending on their scaling in
momentum space with $\lambda=q_{T}/Q\sim T/Q\sim\sqrt{e_{i}}$ as the expansion
parameter of our EFT. The dominant interaction of the jet with the medium
involves forward scattering of the jet in the medium and is mediated by the
Glauber mode. Using an open quantum systems approach combined with the EFT for
forward scattering developed in Rothstein:2016bsq , a factorization formula
for the reduced density matrix of the jet was derived in Vaidya:2020lih . This
factorization was derived under the assumption of a long lived dilute medium
meaning

  * •

The length of the medium L which is equivalent to the time of propagation of
the jet in the medium and the mean free path of the jet are larger than than
the formation time of the jet.

  * •

The medium is homogeneous over length scales probed by the jet.

These assumptions implied that the dominant contribution to the cross section
was when the partons created in the jet went on-shell before interacting with
the medium and successive interactions of the jet with the medium were
incoherent. Upto quadratic order in the Glauber Hamiltonian expansion which is
equivalent to a single interaction of the jet with the medium, we can write a
factorized formula for the trace over then reduced density matrix of the jet
as a function of the time of propagation in the medium(t) with the jet
substructure measurement $\mathcal{M}$ described above.

$\displaystyle\text{Tr}[\rho(t)\mathcal{M}]\equiv\Sigma(e_{n},\vec{q}_{T},t)$
$\displaystyle=$ $\displaystyle
V\times\Bigg{[}\Sigma^{(0)}(e_{n},\vec{q}_{T})+t|C_{G}(Q)|^{2}\Sigma^{(0)}(e_{n},\vec{q}_{T})\otimes_{q_{T}}\int
d^{2}k_{\perp}S_{G}^{AB}(k_{\perp})\mathcal{J}_{n,\text{Med}}^{AB}\Bigg{]}+O(H_{G}^{3})+..$

V is the 4 d volume factor. $\Sigma^{(0)}(e_{n},\vec{q}_{T})$ is the vacuum
density matrix which captures the evolution of the jet in a vacuum background,
which is further factorized in terms of vacuum soft(S) and
jet($\mathcal{J}_{i}^{\perp}$) functions.

$\displaystyle\Sigma^{(0)}(q_{T},e_{n},e_{\bar{n}})=H(Q,\mu)S(\vec{q}_{T};\mu,\nu)\otimes_{q_{T}}\mathcal{J}^{\perp}_{n}(e_{n},Q,z_{c},\vec{q}_{T};\mu,\nu)\otimes_{q_{T}}\mathcal{J}^{\perp}_{\bar{n}}(e_{\bar{n}},Q,z_{c},\vec{q}_{T};\mu,\nu)$

where $\otimes_{q_{T}}$ indicates a convolution in $\vec{q}_{T}$.
$\vec{q}_{T}$ is the transverse momentum imbalance between the two jets (n and
$\bar{n}$) with $e_{n}$ and $e_{\bar{n}}$ as the cumulative jet mass
measurement respectively. H(Q) is the hard function which also includes the
born level term. The factorized functions are defined as follows

$\displaystyle S(\vec{q}_{T})=\frac{1}{N_{R}}\text{tr}\langle
X_{S}|\mathcal{T}\Big{\\{}e^{-i\int_{0}^{\infty}dt^{\prime}H_{S}(t^{\prime})}S_{\bar{n}}^{\dagger}S_{n}(0)\Big{\\}}\mathcal{\bar{T}}\Big{\\{}e^{-i\int_{0}^{\infty}dt^{\prime}H_{S}(t^{\prime})}S_{n}^{\dagger}S_{\bar{n}}(0)\Big{\\}}\delta^{2}(\vec{q}_{T}-\mathcal{P}_{\perp})|X_{S}\rangle$

where $S_{i}$ is a Soft Wilson line in the fundamental representation

$\displaystyle
S_{n}^{(r)}(x)=\mathcal{P}\exp\Bigg{[}ig\int_{\\-\infty}^{0}dsn\cdot
A_{s}^{B}(x+sn)T^{B}_{(r)}\Bigg{]}$ (3)

$H_{S}$ is the soft Hamiltonian which is equivalent to the full QCD
Hamiltonian. The vacuum quark jet function is defined as

$\displaystyle\mathcal{J}_{n}^{\perp}(e,Q)$ $\displaystyle=$
$\displaystyle\frac{(2\pi)^{3}}{N_{c}}\text{tr}\langle
X_{n}|\mathcal{T}\Big{\\{}e^{-i\int_{0}^{\infty}dt^{\prime}H_{n}(t^{\prime})}\bar{\chi}_{n}(0)\Big{\\}}|0\rangle$
$\displaystyle\langle
0|\mathcal{\bar{T}}\Big{\\{}e^{-i\int_{0}^{\infty}dt^{\prime}H_{n}(t^{\prime})}\frac{\not{\bar{n}}}{2}\chi_{n}\Big{\\}}\delta(Q-\mathcal{P}^{-})\delta^{2}(\mathcal{P}^{\perp})\Theta(e_{n}-\mathcal{E}_{\in
n,\text{gr}})\delta^{2}(\vec{q}_{T}-\mathcal{P}^{\perp}_{\not\in
n,\text{gr}})|X_{n}\rangle$

with $\in n,\text{gr}$ refers to collinear partons that pass grooming.
$n\equiv(1,0,0,1)$ refers to the direction of the jet. A similar definition
holds for the $\bar{n}\equiv(1,0,0,-1)$ jet. $H_{n}$ is the collinear
Hamiltonian of SCET. The interaction with the medium introduces three new
functions, namely the Glauber hard co-efficient $C_{G}(\mu)$, medium structure
function $S_{G}^{AB}$, and the medium jet function
$\mathcal{J}^{AB}_{n,\text{Med}}$. This is under the assumption that only the
n jet see the medium. The function $C_{G}(\mu)$ is simply
$8\pi\alpha_{s}(\mu)$ with a natural scale $\mu\sim Q$. The medium jet
function is the difference of a real and virtual medium jet function written
in impact parameter ($\vec{b}$) space which is conjugate to $\vec{q}_{T}$.

$\displaystyle\mathcal{J}^{AB}_{n,\text{Med}}=\mathcal{J}_{n,R}^{AB}-\mathcal{J}_{n,V}^{AB}$
(4) $\displaystyle\mathcal{J}_{n,R}^{AB}(e_{n},\vec{b},\vec{k}_{\perp})$
$\displaystyle=$
$\displaystyle\frac{1}{\mathcal{J}^{\perp}_{n}(e_{n},b)k_{\perp}^{2}}\frac{1}{[p_{\tilde{X}}^{-}]^{2}}\int\frac{d^{2}q_{n}e^{-i\vec{q}_{n}\cdot\vec{b}}}{(2\pi)^{2}}\langle
p_{\tilde{X}}-k,\bar{X}_{n}|\delta^{2}(\mathcal{P}_{\perp})\delta(Q-p^{-})\bar{\chi}_{n}(0)\frac{\not{\bar{n}}}{2}|0\rangle$
(5) $\displaystyle\times$ $\displaystyle\langle
0|\chi_{n}(0)|p_{\tilde{X}}-k,\bar{X}_{n}\rangle\langle\tilde{X}_{n}|O_{n}^{A}(0)|p_{\tilde{X}}-k\rangle\langle
p_{\tilde{X}}-k|O_{n}^{B}(0)|\tilde{X}_{n}\rangle\mathcal{M}_{X_{n}}$

This describes the real interaction of the medium with a single parton of the
jet at time. $p_{\tilde{X}}-k$ is the momenta of the single collinear
intermediate parton state which interacts with the medium $\tilde{X}_{n}$,
while $|X_{n}\rangle$ is a complete set of collinear final states. There is an
implicit integral over the phase space of all states. ($X_{n},\tilde{X}_{n}$).
$\mathcal{M}_{X_{n}}$ is the transverse momentum and jet mass measurement
imposed on the final state $X_{n},\tilde{X}_{n}\rangle$

$\displaystyle\mathcal{M}=\Theta(e_{n}-\mathcal{E}_{\in\text{gr}})\delta^{2}(\vec{q}_{n}-\mathcal{P}^{\perp}_{\notin\text{gr}}-\vec{k}_{\perp})$
(6) $\displaystyle\mathcal{J}^{AB}_{n,V}(b)$ $\displaystyle=$
$\displaystyle\frac{1}{k_{\perp}^{2}\mathcal{J}^{\perp}_{n}(e_{n},b)}\frac{1}{[p_{\tilde{Y}}^{-}]^{2}}\int\frac{d^{2}q_{n}e^{-i\vec{q}_{n}\cdot\vec{b}}}{(2\pi)^{2}}\langle
p_{\tilde{Y}}+k|O_{n}^{A}(0)|\tilde{Y}_{n}\rangle\langle\tilde{Y}_{n}|O_{n}^{B}(0)|p_{\tilde{Y}}+k\rangle$
(7) $\displaystyle\langle
p_{\tilde{Y}}+k,\bar{X}_{n}|\chi_{n}(0)\frac{\not{\bar{n}}}{2}|0\rangle\langle
0|\bar{\chi}_{n}(0)\delta^{2}(\mathcal{P}_{\perp})\delta^{2}(Q-\mathcal{P}^{-})\mathcal{M}_{V}|p_{\tilde{Y}}+k,\bar{X}_{n}\rangle$

This describes the virtual interaction of a jet
parton($|\tilde{Y}_{n}\rangle$) with the medium, again considering one parton
at a time. The measurement $\mathcal{M}_{V}$ in this case is identical to that
in Eq.6 but acts on the final states $|X_{n}\rangle,\tilde{Y}_{n}$.
$\mathcal{J}^{\perp}_{n}(e_{n},b)$ is the vacuum jet function in impact
parameter space. Both the vacuum and medium jet function are defined in terms
of the dressed quark field $\chi_{n}$.

$\displaystyle\chi_{n}=W_{n}^{\dagger}\xi_{n},\ \ \ \
W_{n}(x)=\mathcal{P}\exp\left(ig\int_{\infty}^{0}ds\bar{n}\cdot
A_{n}(x+\bar{n}s)\right)$ (8)

The medium jet function also has a gauge invariant quark and gluon collinear
current

$\displaystyle
O_{n}^{A}=\bar{\chi_{n}}T^{A}\frac{\not{\bar{n}}}{2}\chi_{n}+\frac{i}{2}f^{ACD}\mathcal{B}_{n\perp\mu}^{C}\frac{\bar{n}}{2}\cdot(\mathcal{P}+\mathcal{P}^{\dagger})\mathcal{B}_{n\perp}^{D\mu}$
$\displaystyle\mathcal{B}_{n\perp}^{\mu}=\mathcal{B}_{n\perp}^{\mu
A}T^{A}=\frac{1}{g}\Big{[}W_{n}^{\dagger}iD_{n\perp}^{\mu}W_{n}\Big{]}$ (9)

The medium structure function is defined as

$\displaystyle
S_{G}^{AB}(k_{\perp})=\int\frac{dk^{-}}{2\pi}\frac{1}{k_{\perp}^{2}}D^{AB}(k_{\perp},k^{-})$
(10)

where

$\displaystyle D_{>}^{AB}(k)=\int d^{4}xe^{-ik\cdot x}\langle
X_{S}|O_{S}^{A}(x)\rho O_{S}^{B}(0)|X_{S}\rangle$ (11)

is the correlator of Soft operators in the QGP background density matrix
$\rho$. The Soft operator $O_{S}^{A}$ is a sum over gauge invariant Quark and
Gluon soft currents defined in Rothstein:2016bsq

$\displaystyle
O_{S}^{A}=\bar{\psi}_{S}^{n}T^{A}\frac{\not{n}}{2}\psi_{S}^{n}+\frac{i}{2}f^{ACD}\mathcal{B}_{S\perp\mu}^{C}\frac{n}{2}\cdot(\mathcal{P}+\mathcal{P}^{\dagger})\mathcal{B}_{S\perp}^{n\mu
D}$ (12)

with

$\displaystyle\psi_{s}^{n}=S_{n}^{\dagger}\psi_{s},\ \ \ \
\mathcal{B}_{S\perp\mu}^{n}=\mathcal{B}_{S\perp\mu}^{nA}T^{A}=\frac{1}{g}\Big{[}S_{n}^{\dagger}iD_{S_{\perp}}^{\mu}S_{n}\Big{]}$
(13)

defined in terms of the Soft Wilson line (Eq.3) The one loop results and the
corresponding renormalization group equations for the vacuum soft and jet
functions were derived in Vaidya:2020lih . For the medium structure function
and the medium induced jet function there are two types of radiative
corrections at one loop:

  1. 1.

Elastic collisions of the jet partons with the medium which were also computed
in Vaidya:2020lih . These corrections are UV finite and but IR sensitive.
Hence they do not contribute to the renormalization of these functions.

  2. 2.

Medium induced Brehmstrahlung which will be considered in detail in this paper
in the next section.

##  3 One loop results for medium induced functions

In this section we will look at the complete set of radiative corrections upto
Next-to-Leading Order for our medium soft and jet function. To compute the
result for the Soft function, we need to assume a form for the density matrix
of the medium. A relevant choice is a thermal density matrix

$\displaystyle\rho=\frac{e^{-\beta H_{S}}}{\text{Tr}\Big{[}e^{-\beta
H_{s}}\Big{]}}$ (14)

The tree level result for the Soft function along with partial one loop
results for the medium jet function corresponding to elastic collisions with
the medium were presented in Vaidya:2020lih . Here we will evaluate the one
loop corrections for the medium Soft function as well as the remainder of the
one loop medium jet function which correspond to medium induced radiation.

###  3.1 Medium Jet function

The medium jet function is defined in Eq.4 as the difference between the
virtual and real interaction with the medium. The tree level result is

$\displaystyle\mathcal{J}_{n,\text{Med}}^{AB,(0)}(\vec{k}_{\perp},b,e_{n})=\frac{e^{-i\vec{k}_{\perp}\cdot\vec{b}}-1}{k_{\perp}^{2}}$
(15)

The medium induced jet function encodes the modification of the jet due to the
interaction with the medium. The complete set of corrections for this function
at one loop involves:

  * •

Elastic collisions of the jet partons with the medium.

  * •

Medium induced radiation.

The details of the one loop computation for the elastic collisions are
computed in Vaidya:2020lih .

$\displaystyle\mathcal{J}_{n,\text{Med},\text{Elastic}}^{AB(1)}$
$\displaystyle=$
$\displaystyle\frac{\frac{1}{2}\delta^{AB}}{k_{\perp}^{2}}\left(Q(b)+G(b)\right)$
(16)

where the result is written in terms of Quark and Gluon contributions

$\displaystyle Q(b)$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}C_{F}e^{-i\vec{k}_{\perp}\cdot\vec{b}}}{2\pi}\Bigg{[}\int_{z_{c}}^{1-z_{c}}dzp_{gq}(z)\Big{\\{}\Theta(z_{c}-\frac{y}{e_{n}+y})\textrm{ln}\left(\frac{-B(z)}{e_{n}(1-z)}\right)-\Theta(z_{c}-\frac{e_{n}}{e_{n}+y})\textrm{ln}\frac{B(z)e_{n}}{yM}\Big{\\}}$
(17) $\displaystyle+$
$\displaystyle\Theta(\frac{e_{n}}{e_{n}+y}-z_{c})\Theta(\frac{y}{e_{n}+y}-z_{c})\Big{\\{}\int_{z_{c}}^{\frac{e_{n}}{e_{n}+y}}dzp_{gq}(z)\textrm{ln}\left(\frac{-B(z)}{e_{n}(1-z)}\right)-\int_{\frac{e_{n}}{e_{n}+y}}^{1-z_{c}}dzp_{gq}(z)\textrm{ln}\frac{B(z)e_{n}}{yM}\Big{\\}}\Bigg{]}$
$\displaystyle+$
$\displaystyle\frac{\alpha_{s}C_{F}}{2\pi}\left(e^{-i\vec{k}_{\perp}\cdot\vec{b}}-1\right)\int_{1-z_{c}}^{1}p_{gq}(z)\textrm{ln}\frac{m_{D}^{2}b^{2}e^{2\gamma_{E}}(1-z)}{4}$

The Gluon operator contribution G(b) is given by the difference $R_{g}-V_{g}$,
where

$\displaystyle R_{g}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}N_{c}e^{-i\vec{k}_{\perp}\cdot\vec{b}}}{2\pi}\Bigg{[}\int_{z_{c}}^{1-z_{c}}dzp_{gq}(z)\Big{\\{}\Theta(z_{c}-\frac{y}{e_{n}+y})\textrm{ln}\frac{A(z)}{M}-\Theta(z_{c}-\frac{e_{n}}{e_{n}+y})\textrm{ln}\left(\frac{-A(z)}{y(1-z)}\right)\Big{\\}}$
(18) $\displaystyle-$
$\displaystyle\Theta(\frac{e_{n}}{e_{n}+y}-z_{c})\Theta(\frac{y}{e_{n}+y}-z_{c})\Big{\\{}\int_{z_{c}}^{\frac{y}{e_{n}+y}}dzp_{gq}(z)\textrm{ln}\left(\frac{-A(z)}{y(1-z)}\right)-\int_{\frac{y}{e_{n}+y}}^{1-z_{c}}dzp_{gq}(z)\textrm{ln}\frac{A(z)}{M}\Big{\\}}\Bigg{]}$
$\displaystyle-$
$\displaystyle\frac{\alpha_{s}N_{c}}{2\pi}e^{-i\vec{k}_{\perp}\cdot\vec{b}}\int_{1-z_{c}}^{1}p_{gq}(z)\textrm{ln}\frac{m_{D}^{2}b^{2}e^{2\gamma_{E}}(1-z)}{4}$

written in terms of

$\displaystyle p_{gq}(z)=\frac{1+(1-z)^{2}}{z},\ \ \
y=\frac{4k_{\perp}^{2}}{Q^{2}},\ \ \ M=\frac{4m_{D}^{2}}{Q^{2}},\ \
A(z)=e_{n}z-(1-z)y,\ \ B(z)=zy-e_{n}(1-z)$

and $V_{g}$ is obtained by simply evaluating $R_{g}$ at $k_{\perp}=0$. The
integrals over z can be done exactly analytically but are not too illuminating
and hence we refrain from presenting them here. $k_{\perp}$ is the transverse
momentum exchanged with between the jet and the medium. The contributions Q(b)
and G(b) go to 0 as $k_{\perp}$ goes to 0 so that it is a purely medium
induced result. As we see here, this result in UV finite but is sensitive to
the IR scale $m_{D}$ in the form of single logarithms
$\textrm{ln}Q^{2}e_{n}/m_{D}^{2}\sim\textrm{ln}T^{2}/m_{D}^{2}\sim\textrm{ln}1/g^{2}$.
For the case $g<<1$, ideally we would like to resum these logarithms. In
principle, this can be achieved by matching the jet function to an EFT at the
scale $m_{D}$, which is something we leave for the future. As we will see, the
dominant radiative corrections actually come from medium induced radiation and
we will focus on their resummation and numerical analysis in this paper.

We will now compute in detail, the corrections for medium induced
Bremsstrahlung. We can separately consider the corrections for
$\mathcal{J}^{AB}_{n,R}$ and $\mathcal{J}_{n,V}^{AB}$ and evaluate the result
first in transverse momentum $\vec{q}_{Tn}$ space before moving to the impact
parameter ($\vec{b}$) space. We will begin with $\mathcal{J}^{AB}_{n,R}$

####  3.1.1 $\mathcal{J}^{AB}_{n,R}$

There are both real and virtual diagrams. The complete set of real diagrams is
given in Fig. 1 and correspond to a Glauber exchange on each side of the cut.
The real diagrams are all UV finite due to the imposed measurement on the jet
and hence we can work in 4 dimensions.

Figure 1: Real gluon emission corrections for $\mathcal{J}^{AB}_{n,R}$. k is
the momentum transferred to the jet from the medium. The $\otimes$ vertex
refers to the hard vertex which creates a high energy quark that goes on-shell
before interacting with the medium. The red dot is the Glauber collinear
current and contains both quark and gluon interaction with the medium.

  * •

Diagram (a) involves a the contribution from the Wilson line and an insertion
from the collinear Lagrangian. This evaluates to

$\displaystyle G_{a}^{R}$ $\displaystyle=$ $\displaystyle
2N_{c}\delta^{AB}g^{2}\nu^{\eta}\int
dp^{+}\int\frac{d^{4}q}{(2\pi)^{3}}\delta^{+}(p^{+}-\frac{(\vec{q}_{\perp}+\vec{k}_{\perp})^{2}}{Q-q^{-}})\delta^{+}(q^{2}-m_{D}^{2})\frac{Q|q^{-}|^{-\eta}}{q^{-}[(p^{+}+q^{+})Q-\vec{k}_{\perp}^{2}]}\mathcal{M}$

where the measurement function is

$\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle\Big{[}\Theta(Q-q^{-}-Qz_{c})\Theta(q^{-}-Qz_{c})\Theta(e_{n}-\frac{4}{Q^{2}}(p+q)^{2})\delta^{2}(\vec{q}_{Tn}-\vec{k}_{\perp})$
$\displaystyle+$
$\displaystyle\Theta(Qz_{c}-q^{-})\delta^{2}(\vec{q}_{Tn}-\vec{k}_{\perp}-\vec{q}_{\perp})+\Theta(Qz_{c}-(Q-q^{-}))\delta^{2}(\vec{q}_{Tn}+\vec{q}_{\perp}-\vec{k}_{\perp})\Big{]}$

p is the final state quark momentum while q is the gluon momentum. We then
have three contributions from the measurement functions. Among these, only the
second one where q fails the grooming condition leads to a divergence while
the other contributions are all finite. The divergence in the second term is a
rapidity divergence as $q^{-}\rightarrow 0$ which is regulated by our rapidity
regulator $\nu$ and will give the dominant contribution. We will concentrate
on extracting this contribution to the anomalous dimension of the medium jet
function.

We can isolate the rapidity pole ignoring finite corrections,

$\displaystyle G_{a,\eta}^{R}$ $\displaystyle=$ $\displaystyle
N_{c}\delta^{AB}g^{2}(\nu/Q)^{\eta}\Bigg{[}\int_{0}^{z_{c}}\frac{dzz^{-1-\eta}}{(2\pi)^{3}}\Bigg{]}\int
d^{2}q_{\perp}\frac{1}{[(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}+m_{D}^{2}]}\delta^{2}(\vec{q}_{Tn}-\vec{q}_{\perp})$
(19) $\displaystyle=$ $\displaystyle-
N_{c}\delta^{AB}g^{2}(\nu/[Qz_{c}])^{\eta}\frac{1}{\eta}\frac{1}{(2\pi)^{3}}\int
d^{2}q_{\perp}\frac{1}{[(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}+m_{D}^{2}]}\delta^{2}(\vec{q}_{Tn}-\vec{q}_{\perp})$

  * •

The next diagram in shown in Fig. 1(b) which is similar to Fig. 1(a) and gives
us

$\displaystyle G_{b}^{R}$ $\displaystyle=$ $\displaystyle
4\delta^{AB}N_{c}g^{2}\nu^{\eta}\int
d^{4}p\delta^{+}(p^{2})\int\frac{d^{4}q}{(2\pi)^{d-1}}\delta(Q-p^{-}-q^{-})\delta^{+}(q^{2}-m_{D}^{2})\delta^{2}(q_{\perp}+p_{\perp}+\vec{k}_{\perp})$
(20) $\displaystyle\times$
$\displaystyle\frac{(p^{-})^{2}|q^{-}|^{-\eta}}{q^{-}((p+k)^{2})}\mathcal{M}$

which once again has a rapidity divergence which gives the same contribution
as that from diagram Fig. 1(a), which we can isolate,

$\displaystyle G_{b,\eta}^{R}$ $\displaystyle=$ $\displaystyle-
N_{c}\delta^{AB}g^{2}(\nu/[Qz_{c}])^{\eta}\frac{1}{\eta}\frac{1}{(2\pi)^{3}}\int
d^{2}q_{\perp}\frac{1}{[(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}+m_{D}^{2}]}\delta^{2}(\vec{q}_{Tn}-\vec{q}_{\perp})$
(21)

  * •

Diagrams in Fig. 1 (c) and (d) evaluate to 0.

  * •

We now consider the diagram (e) including the mirror diagram

$\displaystyle G_{e}^{R}$ $\displaystyle=$
$\displaystyle\frac{g^{2}N_{c}}{4}\nu^{\eta}\int\frac{d^{2}q_{\perp}dq^{-}}{(2\pi)^{3}q^{-}}\frac{Q|q^{-}|^{-\eta}}{Q(\frac{(\vec{q}_{\perp}+\vec{k}_{\perp})^{2}}{Q-q^{-}}+\frac{\vec{q}_{\perp}^{2}+m^{2}}{q^{-}})-\vec{k}_{\perp}^{2}}\frac{1}{q^{-}(-\frac{(\vec{q}_{\perp}+\vec{k}_{\perp})^{2}}{Q-q^{-}})-(\vec{q}_{\perp}+\vec{k}_{\perp})^{2}-m^{2}}$
$\displaystyle\Bigg{[}\frac{-8(\vec{q}_{\perp}+\vec{k}_{\perp})\cdot\vec{q}_{\perp}}{q^{-}}-4\frac{(\vec{q}_{\perp}+\vec{k}_{\perp})^{2}}{Q-q^{-}}-4\vec{k}_{\perp}\cdot\frac{(\vec{q}_{\perp}+\vec{k}_{\perp})}{Q}-4q^{-}\frac{(\vec{q}_{\perp}+\vec{k}_{\perp})^{2}}{(Q-q^{-})^{2}}-4\vec{q}_{\perp}\cdot\frac{(\vec{q}_{\perp}+\vec{k}_{\perp})}{Q-q^{-}}\Bigg{]}\mathcal{M}$

Only the first term in the square brackets contributes to the rapidity
divergence

$\displaystyle G_{e,\eta}^{R}$ $\displaystyle=$
$\displaystyle-2g^{2}N_{c}(\nu/Q)^{\eta}\int_{0}^{z_{c}}\frac{dz}{z^{1+\eta}(2\pi)^{3}}\int
d^{2}q_{\perp}\frac{1}{(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}+m^{2}}\frac{1}{(\vec{q}_{\perp})^{2}+m^{2}}$
$\displaystyle\times$
$\displaystyle\Bigg{[}(\vec{q}_{\perp})\cdot(\vec{q}_{\perp}-\vec{k}_{\perp})\Bigg{]}\delta^{2}(\vec{q}_{Tn}-\vec{q}_{\perp})$
$\displaystyle=$ $\displaystyle
2g^{2}N_{c}(\nu/[Qz_{c}])^{\eta}\frac{1}{\eta}\frac{1}{(2\pi)^{3}}\int
d^{2}q_{\perp}\frac{\vec{q}_{\perp}\cdot(\vec{q}_{\perp}-\vec{k}_{\perp})}{[(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}+m^{2}][(\vec{q}_{\perp})^{2}+m^{2}]}\delta^{2}(\vec{q}_{Tn}-\vec{q}_{\perp})$

  * •

We can similarly compute the diagram Fig. 1(f), which gives us

$\displaystyle G_{f}^{R}$ $\displaystyle=$
$\displaystyle-\nu^{\eta}\frac{g^{2}}{4}N_{c}\int\frac{d^{2}q_{\perp}dq^{-}}{q^{-}(2\pi)^{3}}\frac{Q-q^{-}}{(Q-q^{-})(-\frac{q_{\perp}^{2}+m^{2}}{q^{-}})-\vec{q}_{\perp}^{2}}\frac{|q^{-}|^{\eta}}{q^{-}(-\frac{(\vec{q}_{\perp}+\vec{k}_{\perp})^{2}}{Q-q^{-}})-(\vec{q}_{\perp}+\vec{k}_{\perp})^{2}-m^{2}}$
(23) $\displaystyle\times$
$\displaystyle\Bigg{[}\frac{8(\vec{q}_{\perp}+\vec{k}_{\perp})\cdot\vec{q}_{\perp}}{q^{-}}+8(\vec{q}_{\perp}+\vec{k}_{\perp})\cdot\frac{(\vec{q}_{\perp}}{Q-q^{-}}+4q^{-}\frac{(\vec{q}_{\perp}+\vec{k}_{\perp})}{(Q-q^{-})^{2}}\cdot\vec{q}_{\perp}\Bigg{]}$

The rapidity divergence only appears again only in the first term in the
square brackets which gives us an identical result to diagram(e)

$\displaystyle G_{f,2}^{R,\eta}$ $\displaystyle=$ $\displaystyle
2g^{2}N_{c}(\nu/[Qz_{c}])^{\eta}\frac{1}{\eta}\frac{1}{(2\pi)^{3}}\int
d^{2}q_{\perp}\frac{\vec{q}_{\perp}\cdot(\vec{q}_{\perp}-\vec{k}_{\perp})}{[(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}+m^{2}][(\vec{q}_{\perp})^{2}+m^{2}]}\delta^{2}(\vec{q}_{Tn}-\vec{q}_{\perp})$

  * •

Next we consider the diagrams purely from Lagrangian insertions namely
diagrams (g),(h) and (i), none of which have a rapidity divergence

  * •

Finally we have the real diagram (j) which gives us

$\displaystyle G_{j}^{R}$ $\displaystyle=$
$\displaystyle-g^{2}\nu^{\eta}N_{c}\delta^{AB}\int
dp^{+}\delta(p^{+}-\frac{(\vec{q}_{\perp}+\vec{k}_{\perp})^{2}}{Q-q^{-}})\int\frac{d^{4}q}{(2\pi)^{3}}\delta(q^{2}-m^{2})\frac{p^{-}|q^{-}|^{-\eta}}{[(q+k)^{2}-m^{2}]^{2}}$
(24) $\displaystyle\times$
$\displaystyle(\vec{q}_{\perp}+\vec{k}_{\perp})^{2}\left(-4\frac{q^{-}}{p^{-}}-4-2\left(\frac{q^{-}}{p^{-}}\right)^{2}\right)\mathcal{M}$

Putting in the measurement, we see that this piece also has a rapidity
divergence, which we can isolate here

$\displaystyle G_{j,\eta}^{R}$ $\displaystyle=$
$\displaystyle-2g^{2}N_{c}\delta^{AB}\left(\frac{\nu}{Qz_{c}}\right)^{\eta}\frac{1}{\eta}\frac{1}{(2\pi)^{3}}\int
d^{2}q_{\perp}\frac{1}{\vec{q}_{\perp}^{2}+m^{2}}\delta^{2}(\vec{q}_{Tn}-\vec{q}_{\perp})$
(25)

We therefore see that the real diagrams do not have any UV divergence but do
contribute to the rapidity divergence. The rapidity divergent pieces can be
collected together to give

$\displaystyle G_{R}^{\eta}$ $\displaystyle=$
$\displaystyle-\frac{2N_{c}g^{2}}{(2\pi)^{3}}\left(\frac{\nu}{Qz_{c}}\right)^{\eta}\frac{1}{\eta}\int\frac{d^{2}q_{\perp}\delta^{AB}\delta^{2}(\vec{q}_{Tn}-\vec{q}_{\perp})}{k_{\perp}^{2}}\Bigg{\\{}\frac{1}{(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}}+\frac{1}{\vec{q}_{\perp}^{2}}-\frac{2\vec{q}_{\perp}\cdot(\vec{q}_{\perp}-\vec{k}_{\perp})}{q_{\perp}^{2}(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}}\Bigg{\\}}$
(26) $\displaystyle=$
$\displaystyle-\frac{2N_{c}g^{2}}{(2\pi)^{3}}\left(\frac{\nu}{Qz_{c}}\right)^{\eta}\frac{1}{\eta}\int\frac{d^{2}q_{\perp}\delta^{AB}\delta^{2}(\vec{q}_{Tn}-\vec{q}_{\perp})}{[q_{\perp}^{2}+m_{D}^{2}][(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}+m_{D}^{2}]}$

The natural scale is $\nu\sim Q\sim Qz_{c}$.

We also have virtual diagrams which are shown in Fig. 2 but these have already
been computed in Rothstein:2016bsq , so we will take the result from there
directly.

Figure 2: Virtual gluon diagrams for $\mathcal{J}^{AB}_{n,R}$.

The virtual diagrams again do not have any UV divergences but only rapidity
divergences.

$\displaystyle V_{R}^{\eta}$ $\displaystyle=$
$\displaystyle-\frac{N_{c}g^{2}}{(2\pi)^{3}}\left(\frac{\nu}{Q}\right)^{\eta}\int_{0}^{1}\frac{dz}{z^{1+\eta}}\int\frac{d^{2}q_{\perp}\delta^{AB}\delta^{2}(\vec{q}_{Tn}-\vec{k}_{\perp})}{q_{\perp}^{2}(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}}$
(27) $\displaystyle=$
$\displaystyle\frac{N_{c}g^{2}}{(2\pi)^{3}}\left(\frac{\nu}{Q}\right)^{\eta}\frac{1}{\eta}\int\frac{d^{2}q_{\perp}\delta^{AB}\delta^{2}(\vec{q}_{Tn}-\vec{k}_{\perp})}{q_{\perp}^{2}(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}}$

####  3.1.2 $\mathcal{J}_{n,V}^{AB}$

The set of diagrams to be computed for this piece remains the same, the only
difference is the measurement function, hence we can immediately write down
the rapidity divergent result

$\displaystyle\mathcal{J}_{n,V}^{AB}$ $\displaystyle=$
$\displaystyle\delta^{AB}\delta^{2}(\vec{q}_{Tn})\frac{2N_{c}g^{2}}{(2\pi)^{3}}\left(\frac{\nu}{Q}\right)^{\eta}\int_{0}^{1}\frac{dz}{z^{1+\eta}}\left(\frac{1}{q_{\perp}^{2}(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}}-\frac{1}{2q_{\perp}^{2}(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}}\right)$
(28)

Using this result along with Eqs. 26 and 27 in Eq.4, we can now write

$\displaystyle\nu\frac{d}{d\nu}\mathcal{J}_{n,\text{Med}}^{AB(1)}(\vec{k}_{\perp})=-\frac{\alpha_{s}N_{c}}{\pi^{2}}\int
d^{2}q_{\perp}\left(\frac{\mathcal{J}_{n,\text{Med}}^{AB(0)}(\vec{q}_{\perp})}{(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}}-\frac{k_{\perp}^{2}\mathcal{J}_{n,\text{Med}}^{AB(0)}(k_{\perp})}{2q_{\perp}^{2}(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}}\right)$
(29)

The result therefore gives us the final Rapidity RGE for the Jet function at
this perturbative order

$\displaystyle\nu\frac{d}{d\nu}J(\vec{k}_{\perp})=-\frac{\alpha_{s}N_{c}}{\pi^{2}}\int
d^{2}q_{\perp}\left(\frac{J(\vec{q}_{\perp})}{(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}}-\frac{k_{\perp}^{2}J(k_{\perp})}{2q_{\perp}^{2}(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}}\right)$
(30)

This is identical to the BFKL equation with a negative sign. The medium
induced jet function does not yield any UV anomalous dimension.

###  3.2 Medium Soft function

The medium soft function defined in Eq.10 is just the Wightman function in a
thermal bath. To compute the corrections upto NLO I will use the Realtime
formalism of thermal field theory. Operationally, this involves introducing
two copies of every field in our action dubbing them as type 1 and type 2
fields. The two types of fields only talk to each other via the kinetic term,
which allows type 1 to propagate into type 2 and vice versa. All the fields in
an interaction term have either a type 1 or type 2 field with opposite sign of
the coupling for the interaction term with type 2 fields. In my case, for
simplicity I assume that the thermal bath is only sourcing quarks so that I
only double the quark fields. This is sufficient to derive the RG equations
for the soft function. Including the gluons sourced by the thermal medium will
only change the boundary condition for the solution of the RG equations.
Doubling of the quark fields leads to 4 types of fermion propagators

$\displaystyle D_{11}(k)$ $\displaystyle=$
$\displaystyle\not{k}\Bigg{[}\frac{i}{k^{2}+i\epsilon}-2\pi\delta(k^{2})n_{F}(|k^{0}|)\Bigg{]},\
\ \ \
D_{22}(k)=\not{k}\Bigg{[}\frac{-i}{k^{2}-i\epsilon}-2\pi\delta(k^{2})n_{F}(|k^{0}|)\Bigg{]}$
$\displaystyle D_{21}(k)$ $\displaystyle=$ $\displaystyle
2\pi\not{k}\Bigg{[}\Theta(k^{0})\delta(k^{2})-\delta(k^{2})n_{F}(|k^{0}|)\Bigg{]},\
\ \ \
D_{12}(k)=2\pi\not{k}\Bigg{[}\Theta(-k^{0})\delta(k^{2})-\delta(k^{2})n_{F}(|k^{0}|)\Bigg{]}$

where $n_{F}(\omega)=\frac{1}{e^{\beta\omega}+1}$ is the Fermi distribution
function.

The tree level diagram is shown in Fig. 3 which gives us

Figure 3: The tree level medium structure function in a thermal bath is just
the advanced Wightman function. The black square is the Soft Glauber current
vertex.

$\displaystyle
S_{G}^{AB,(0)}=-\frac{2(2\pi)^{3}}{k_{\perp}^{2}}\int\frac{dk^{-}}{(2\pi)}\int\frac{d^{4}p}{(2\pi)^{3}}(p^{+})^{2}\Big{[}\delta^{-}(p^{2})-\delta(p^{2})n_{F}(|p^{0}|)\Big{]}$
$\displaystyle\Big{[}\delta^{+}((p+k)^{2})-\delta((p+k)^{2})n_{F}(|p^{0}+k^{0}|)\Big{]}\text{Tr}\Big{[}\frac{\not{n}}{2}\frac{\not{\bar{n}}}{2}\Big{]}\text{Tr}[T^{A}T^{B}]$
(31)

which evaluates to

$\displaystyle S_{G}^{AB,(0)}(\vec{k}_{\perp})$ $\displaystyle=$
$\displaystyle\frac{\delta^{AB}}{k_{\perp}^{2}}\int
d^{2}p_{\perp}\int_{0}^{\infty}dp^{+}\Bigg{\\{}n_{F}\left(|\frac{p^{+}}{2}+\frac{p_{\perp}^{2}}{2}|\right)\Big{[}1-n_{F}(|\frac{p^{+}}{2}+\frac{(\vec{p}_{\perp}+\vec{k}_{\perp})^{2}}{2p^{+}}|)\Big{]}$
$\displaystyle+$ $\displaystyle
n_{F}(|\frac{p^{+}}{2}+\frac{(\vec{p}_{\perp}+\vec{k}_{\perp})^{2}}{2p^{+}}|)\Big{[}1-n_{F}\left(|\frac{p^{+}}{2}+\frac{p_{\perp}^{2}}{2}|\right)\Big{]}\Bigg{\\}}$

We now turn to the one loop corrections. We can look at the real and virtual
graphs separately, where by real graphs we mean those where the gluon
propagator is of type $D_{12}$ or $D_{21}$. We will only compute the rapidity
divergences to establish consistency of the Rapidity anomalous dimension. The
UV divergence of the soft function can be obtained by consistency.

####  3.2.1 Real Soft Diagrams

All the real graphs are shown in Fig. 4. We also include the mirror diagrams
not shown explicitly in this figure.

Figure 4: "Real" gluon emission diagrams for the medium structure function.
The black square is the Soft Glauber vertex. The real diagrams are those where
the gluon propagator is of type $D_{12}$ or $D_{21}$, i.e, the non-thermal
piece of the propagator is on-shell.

  * •

Diagram (a) involves the soft Wilson line along with an insertion from the
Soft Hamiltonian which is identical to the full QCD Hamiltonian.

$\displaystyle S_{a}^{R}$ $\displaystyle=$
$\displaystyle-4N_{c}g^{2}\nu^{\eta/2}\frac{\delta^{AB}}{k_{\perp}^{2}}\int
dk^{-}\int d^{4}q\int
d^{4}p(p^{+})^{2}\Big{[}\delta^{-}(p^{2})-\delta(p^{2})n_{F}(|p^{0}|)\Big{]}\delta^{+}(q^{2}-m^{2})$
$\displaystyle\times$
$\displaystyle\Big{[}\delta^{+}((p+k-q)^{2})-\delta((p+k-q)^{2})n_{F}(|p^{0}+k^{0}-q^{0}|)\Big{]}\frac{p^{+}}{(p+k)^{2}}\frac{|q^{-}-q^{+}|^{-\eta/2}}{q^{+}}$

p is the quark momentum whole q is the gluon momentum. k is the Glauber
momentum exchanged with the jet. The overall minus sign comes from the Quark
loop. In our case, there is no $\bar{n}$ sector for the medium interaction and
hence the rapidity divergence only exists in the limit
$q^{-}\rightarrow\infty$ when the soft sector overlaps with the n collinear
sector. Hence we can simply use $|q^{-}|^{-\eta/2}$ as a rapidity regulator in
place of the complete $|q^{-}-q^{+}|^{\eta/2}$ which simplifies our
computation. If we only keep track of the pieces with rapidity divergence, we
end up with two terms

$\displaystyle S_{a,1}^{R,\eta}$ $\displaystyle=$ $\displaystyle
g^{2}N_{c}\frac{\delta^{AB}}{k_{\perp}^{2}}\int
d^{2}p_{\perp}\int_{0}^{\infty}dp^{+}\int\frac{d^{2}q_{\perp}}{(q_{\perp}^{2}+m_{D}^{2})^{\eta/2}}n_{F}(|\frac{p^{+}}{2}+\frac{p_{\perp}^{2}}{2p^{+}}|)$
(32) $\displaystyle\times$ $\displaystyle(\nu
p^{+})^{\eta/2}\int_{0}^{1}\frac{dz}{(2\pi)^{d-1}}\frac{(1-z)}{z(\vec{p}_{\perp}+\vec{k}_{\perp}-\vec{q}_{\perp})^{2}+(1-z)[q_{\perp}^{2}+m_{D}^{2}]-z(1-z)(\vec{p}_{\perp}+\vec{k}_{\perp})^{2}}\frac{1}{(z)^{1-\eta/2}}$
$\displaystyle\times$
$\displaystyle\left(1-n_{F}(|\frac{p^{+}(1-z)}{2}+\frac{(\vec{p}_{\perp}+\vec{k}_{\perp}-\vec{q}_{\perp})^{2}}{2p^{+}(1-z)}|)\right)$

and

$\displaystyle S_{a,2}^{R,\eta}$ $\displaystyle=$ $\displaystyle
g^{2}N_{c}\frac{\delta^{AB}}{k_{\perp}^{2}}\int
d^{2}p_{\perp}\int_{0}^{\infty}dp^{+}\int\frac{d^{2}q_{\perp}}{(q_{\perp}^{2}+m_{D}^{2})^{\eta/2}}\left(1-n_{F}(|\frac{p^{+}}{2}+\frac{p_{\perp}^{2}}{2p^{+}}|)\right)$
(33) $\displaystyle\times$ $\displaystyle(\nu
p^{+})^{\eta/2}\frac{\int_{0}^{\infty}dz}{(2\pi)^{d-1}}\frac{(1-z)}{z(\vec{p}_{\perp}+\vec{k}_{\perp}-\vec{q}_{\perp})^{2}+(1-z)[q_{\perp}^{2}+m_{D}^{2}]-z(1-z)(\vec{p}_{\perp}+\vec{k}_{\perp})^{2}}\frac{1}{(z)^{1-\eta/2}}$
$\displaystyle\times$ $\displaystyle
n_{F}(|\frac{p^{+}(1+z)}{2}+\frac{(\vec{p}_{\perp}+\vec{k}_{\perp}-\vec{q}_{\perp})^{2}}{2p^{+}(1+z)}|)$

Due to the presence of the Fermi distribution function, it is not possible to
do the integral over z analytically. However, if we are only interested in the
rapidity pole, we see that this happens in the limit $z\rightarrow 0$, which
allows us to isolate the rapidity pole

$\displaystyle S_{a}^{R,\eta}$ $\displaystyle=$ $\displaystyle
g^{2}N_{c}\frac{\delta^{AB}}{k_{\perp}^{2}}\frac{2}{\eta}\int
d^{2}p_{\perp}dp^{+}\int\frac{d^{2}q_{\perp}}{(2\pi)^{3}}\frac{1}{q_{\perp}^{2}+m_{D}^{2}}\left(\frac{\nu
p^{+}}{q_{\perp}^{2}+m_{D}^{2}}\right)^{\eta/2}$ (34)
$\displaystyle\Bigg{\\{}n_{F}(|\frac{p^{+}}{2}+\frac{p_{\perp}^{2}}{2p^{+}}|)\left(1-n_{F}(|\frac{p^{+}}{2}+\frac{(\vec{p}_{\perp}+\vec{k}_{\perp}-\vec{q}_{\perp})^{2}}{2p^{+}}|)\right)$
$\displaystyle+$
$\displaystyle(1-n_{F}(|\frac{p^{+}}{2}+\frac{p_{\perp}^{2}}{2p^{+}}|))n_{F}(|\frac{p^{+}}{2}+\frac{(\vec{p}_{\perp}+\vec{k}_{\perp}-\vec{q}_{\perp})^{2}}{2p^{+}}|)\Bigg{\\}}$

  * •

We next consider the contribution from Diagram (b)

$\displaystyle S_{b}^{R}$ $\displaystyle=$ $\displaystyle
4g^{2}N_{c}\nu^{\eta/2}\frac{\delta^{AB}}{k_{\perp}^{2}}\int
d^{4}p(p^{+})^{2}\left(\delta^{+}((p+k-q)^{2})-\delta((p+k-q)^{2})n_{F}(|p^{0}+k^{0}-q^{0}|)\right)$
(35) $\displaystyle\times$
$\displaystyle\left(\delta^{-}(p^{2})-\delta(p^{2})n_{F}(|p^{0}|)\right)\int
d^{4}q\delta^{+}(q^{2}-m_{D}^{2})\frac{p^{+}-q^{+}}{(p-q)^{2}}\frac{|q^{-}|^{-\eta/2}}{q^{+}}$

The rapidity divergence contribution is identical to case (a).

  * •

Diagrams (c) and (d) reduce to 0.

  * •

For the diagram (e), we have

$\displaystyle
S_{e}^{R}=g^{2}N_{c}\nu^{\eta/2}\frac{\delta^{AB}}{k_{\perp}^{2}}\int
d^{4}p\left(\delta^{-}(p^{2})-\delta(p^{2})n_{F}(|p^{0}|)\right)\left(\delta^{+}((p+k-q)^{2})-\delta((p+k-q)^{2})n_{F}(|p^{0}+k^{0}-q^{0}|)\right)$
(36) $\displaystyle\times$ $\displaystyle\int
d^{4}q\frac{\delta^{+}(q^{2}-m_{D}^{2})p^{+}}{(p+k)^{2}[(q-k)^{2}-m^{2}]}\Big{[}\frac{8p^{+}(p^{+}-q^{+})q_{\perp}\cdot(q_{\perp}-k_{\perp})|q^{-}|^{-\eta/2}}{q^{+}}$
$\displaystyle-$ $\displaystyle
4p^{+}(2q_{\perp}-k_{\perp})\cdot(p_{\perp}+k_{\perp}-q_{\perp})-4(p^{+}-q^{+})((p_{\perp}+k_{\perp})\cdot(q_{\perp}-k_{\perp})+p_{\perp}\cdot
q_{\perp})$ $\displaystyle+$ $\displaystyle
4p^{+}(2p_{\perp}+k_{\perp})\cdot(p_{\perp}+k_{\perp}-q_{\perp})-4p^{+}(p+k-q)\cdot(2p+k)+4(p^{+}-q^{+})p\cdot(p+k)\Big{]}$

Only the first term in the square brackets above has a rapidity divergence.
This yields a rapidity pole

$\displaystyle S_{e}^{R,\eta}$ $\displaystyle=$ $\displaystyle
2N_{c}g^{2}\frac{\delta^{AB}}{k_{\perp}^{2}}\int
d^{2}p_{\perp}dp^{+}\int\frac{d^{2}q_{\perp}}{(2\pi)^{3}}\frac{q_{\perp}\cdot(q_{\perp}-k_{\perp})}{(q_{\perp}^{2}+m_{D}^{2})((\vec{q}_{\perp}-\vec{k}_{\perp})^{2}+m_{D}^{2})}$
(37) $\displaystyle\left(\frac{\nu
p^{+}}{q_{\perp}^{2}+m_{D}^{2}}\right)^{\eta/2}\frac{2}{\eta}\Bigg{\\{}n_{F}(|\frac{p^{+}}{2}+\frac{p_{\perp}^{2}}{2p^{+}}|)\left(1-n_{F}(|\frac{p^{+}}{2}+\frac{(\vec{p}_{\perp}+\vec{k}_{\perp}-\vec{q}_{\perp})^{2}}{2p^{+}}|)\right)$
$\displaystyle+$
$\displaystyle(1-n_{F}(|\frac{p^{+}}{2}+\frac{p_{\perp}^{2}}{2p^{+}}|))n_{F}(|\frac{p^{+}}{2}+\frac{(\vec{p}_{\perp}+\vec{k}_{\perp}-\vec{q}_{\perp})^{2}}{2p^{+}}|)\Bigg{\\}}$

  * •

Diagram(f) gives the same contribution as diagram (e)

  * •

Diagram (g) only has insertions from the Soft Hamiltonian and evaluates to

$\displaystyle S_{g}^{R}$ $\displaystyle=$ $\displaystyle
g^{2}C_{F}\delta^{AB}\int
d^{2}p_{\perp}\frac{dp^{+}}{p^{+}}\int\frac{d^{4}q}{(2\pi)^{3}}\delta^{+}(q^{2}-m^{2})\frac{\Big{\\{}-2(p-q)^{2}p^{+}+4p\cdot(p-q)(p^{+}-q^{+})\Big{\\}}}{[(p-q)^{2}]^{2}}$
(38) $\displaystyle\times$
$\displaystyle\left(\Theta(-p^{+})-\text{sgn}(-p^{+})n_{F}(|p^{0}|)\right)\left(\Theta(p^{+}-q^{+})-\text{sgn}(p^{+}-q^{+})n_{F}(|p^{0}+k^{0}-q^{0}|)\right)$

This piece does not have a rapidity divergence and hence we will not compute
it further.

  * •

$\displaystyle S_{h}^{R}$ $\displaystyle=$ $\displaystyle-
C_{F}\frac{g^{2}}{4}\int
d^{2}p_{\perp}dp^{+}\int\frac{d^{2}q_{\perp}dq^{+}}{(2\pi)^{3}q^{+}(p^{+}-q^{+})}\frac{\left(4(p+k)^{2}(p^{+}-q^{+})-8p^{+}(p+k)\cdot(p+k-q)\right)}{[(p+k)^{2}]^{2}}$
(39) $\displaystyle\times$
$\displaystyle\left(\Theta(-p^{+})-\text{sgn}(-p^{+})n_{F}(|p^{0}|)\right)\left(\Theta(p^{+}-q^{+})-\text{sgn}(p^{+}-q^{+})n_{F}(|p^{0}+k^{0}-q^{0}|)\right)$

which is also rapidity finite.

  * •

Same is the case for (i) which has the form

$\displaystyle S_{i}^{R}$ $\displaystyle=$
$\displaystyle-2g^{2}\delta^{AB}\left(\frac{C_{F}}{2}-\frac{Nc}{4}\right)\int
d^{2}p_{\perp}\frac{dp^{+}}{p^{+}}\int\frac{d^{4}q\delta^{+}(q^{2}-m^{2})}{(p^{+}-q^{+})(2\pi)^{3}}\frac{\left(-2(p+k)^{2}q^{+}(p^{+}-q^{+})+2(p-q)^{2}p^{+}q^{+}\right)}{[(p+k)^{2}][(p-q)^{2}]}$
(40)
$\displaystyle\left(\Theta(-p^{+})-\text{sgn}(-p^{+})n_{F}(|p^{0}|)\right)\left(\Theta(p^{+}-q^{+})-\text{sgn}(p^{+}-q^{+})n_{F}(|p^{0}+k^{0}-q^{0}|)\right)$

  * •

We finally have (j) which does have a rapidity divergence

$\displaystyle
S_{j}^{R}=-g^{2}\nu^{\eta/2}\frac{N_{c}}{4}\frac{\delta^{AB}}{k_{\perp}^{2}}\int
d^{2}p_{\perp}\frac{dp^{+}}{p^{+}}\int\frac{d^{2}q_{\perp}dq^{+}}{(2\pi)^{3}(p^{+}-q^{+})}\frac{|\frac{q_{\perp}^{2}+m_{D}^{2}}{q^{+}}|^{-\eta/2}}{q^{+}[q^{+}(\frac{p_{\perp}^{2}}{p^{+}}-\frac{(\vec{p}_{\perp}+\vec{k}_{\perp}-\vec{q}_{\perp})^{2}}{p^{+}-q^{+}}-(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}-m^{2}]^{2}}$
(41)
$\displaystyle\left(\Theta(-p^{+})-\text{sgn}{-p^{+}}n_{F}(|p^{0}|)\right)\left(\Theta(p^{+}-q^{+})-\text{sgn}(p^{+}-q^{+})n_{F}(|p^{0}+k^{0}-q^{0}|)\right)$
$\displaystyle\Big{\\{}-8q^{+}p^{+}(q_{\perp}-k_{\perp})\cdot(p_{\perp}+k_{\perp}-q_{\perp})+8p^{+}(p^{+}-q^{+})(q_{\perp}-k_{\perp})\cdot(q_{\perp}-k_{\perp})$
$\displaystyle+$ $\displaystyle
8(q^{+})^{2}(p_{\perp}\cdot(p_{\perp}+k_{\perp}-q_{\perp})-p\cdot(p+k-q))-8q^{+}(p^{+}-q^{+})p_{\perp}\cdot(q_{\perp}-k_{\perp})\Big{\\}}$

As before, we can isolate the rapidity divergence

$\displaystyle S_{j}^{R,\eta}$ $\displaystyle=$ $\displaystyle
2g^{2}N_{c}\frac{\delta^{AB}}{k_{\perp}^{2}}\Bigg{\\{}\int
d^{2}p_{\perp}dp^{+}\int\frac{d^{2}q_{\perp}}{(2\pi)^{d-1}}n_{F}(|\frac{p^{+}}{2}+\frac{p_{\perp}^{2}}{2p^{+}}|)\left(1-n_{F}(|\frac{p^{+}}{2}+\frac{(\vec{p}_{\perp}+\vec{k}_{\perp}-\vec{q}_{\perp})^{2}}{2p^{+}}|)\right)$
(42) $\displaystyle+$ $\displaystyle\int
d^{2}p_{\perp}dp^{+}(1-n_{F}(|\frac{p^{+}}{2}+\frac{p_{\perp}^{2}}{2p^{+}}|))n_{F}(|\frac{p^{+}}{2}+\frac{(\vec{p}_{\perp}+\vec{k}_{\perp}-\vec{q}_{\perp})^{2}}{2p^{+}}|)\Bigg{\\}}$
$\displaystyle\times$ $\displaystyle\left(\frac{\nu
p^{+}}{q_{\perp}^{2}+m_{D}^{2}}\right)^{\eta/2}\frac{2}{\eta}\frac{1}{(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}+m_{D}^{2}}$

We can combine all the rapidity pole pieces above and make the shift
$\vec{q}_{\perp}\rightarrow-\vec{q}_{\perp}+\vec{k}_{\perp}$ to write

$\displaystyle
S^{R,\eta}=\delta^{AB}\frac{\alpha_{s}N_{c}}{\pi^{2}k_{\perp}^{2}}\int
d^{2}p_{\perp}dp^{+}\int d^{2}q_{\perp}\left(\frac{\nu
p^{+}}{(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}+m_{D}^{2}}\right)^{\eta/2}\frac{2}{\eta}\frac{k_{\perp}^{2}}{q_{\perp}^{2}(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}}$
$\displaystyle\Bigg{\\{}n_{F}(|\frac{p^{+}}{2}+\frac{p_{\perp}^{2}}{2p^{+}}|)\Big{[}1-n_{F}\left(\frac{p^{+}}{2}+\frac{(\vec{p}_{\perp}+\vec{q}_{\perp})^{2}}{2p^{+}}|\right)\Big{]}+n_{F}(|\frac{p^{+}}{2}+\frac{(\vec{p}_{\perp}+\vec{q}_{\perp})^{2}}{2p^{+}}|)\Big{[}1-n_{F}\left(\frac{p^{+}}{2}+\frac{\vec{p}_{\perp}^{2}}{2p^{+}}|\right)\Big{]}\Bigg{\\}}$

so that the result is now written in terms of the tree level soft function.

####  3.2.2 Virtual Soft diagrams

The virtual diagrams are the one where the gluon propagator is of type
$D_{11}$, $D_{22}$. All the possible diagrams in this category are shown in
Fig. 5

Figure 5: "Virtual" gluon emission for medium structure function. The gluon
propagator is of type $D_{11}$ or $D_{22}$ meaning the non-thermal piece of
the propagator is just the zero temperature Feynman propagator.

As for the real diagrams, we concentrate on isolating the rapidity divergences
in each diagram.

  * •

Diagram (a) evaluates to

$\displaystyle S_{a}^{V}=\frac{N_{c}}{4}\nu^{\eta/2}16g^{2}\int
d^{4}pp^{+}(\delta^{-}(p^{2})-\delta(p^{2})n_{F}(|p^{0}|))\int\frac{d^{d}q}{(2\pi)^{d-1}}\frac{|q^{-}|^{-\eta/2}}{q^{+}}p^{+}\Big{\\{}\frac{i(p^{+}-q^{+})}{(p+k-q)^{2}}$
$\displaystyle-(p^{+}-q^{+})\delta((p+k-q)^{2})n_{F}(|p^{0}+k^{0}-q^{0})\Big{\\}}\Big{\\{}\delta^{+}((p+k)^{2})-\delta((p+k)^{2})n_{F}(|p^{0}+k^{0}|)\Big{\\}}$

This is now proportional to the tree level result and gives us a rapidity
divergence

$\displaystyle S_{a}^{V,\eta}$ $\displaystyle=$ $\displaystyle g^{2}N_{c}\int
d^{2}p_{\perp}\int_{0}^{\infty}dp^{+}\Bigg{\\{}n_{F}(|p^{0}|))\Big{\\{}1-n_{F}(|p^{0}+k^{0}|)\Big{\\}}+(1-n_{F}(|p^{0}|))n_{F}(|p^{0}+k^{0}|)\Bigg{\\}}$
(44) $\displaystyle\times$
$\displaystyle\int\frac{d^{2}q_{\perp}}{(2\pi)^{d-1}}\frac{1}{q_{\perp}^{2}+m_{D}^{2}}\left(\frac{\nu
p^{+}}{q_{\perp}^{2}+m_{D}^{2}}\right)^{\eta/2}\frac{2}{\eta}$

where $p^{0}=\frac{p^{+}}{2}+\frac{p_{\perp}^{2}}{2p^{+}}$ and
$p^{0}+k^{0}=\frac{p^{+}}{2}+\frac{(\vec{p}_{\perp}+\vec{k}_{\perp})^{2}}{2p^{+}}$
enforcing on-shell conditions.

  * •

Diagram (b) evaluates to

$\displaystyle S_{b}^{V}$ $\displaystyle=$
$\displaystyle\frac{N_{c}}{4}\nu^{\eta/2}16g^{2}\int
d^{4}pp^{+}(\delta^{-}(p^{2})-\delta(p^{2})n_{F}(|p^{0}|))\int\frac{d^{d}q}{(2\pi)^{d-1}}\frac{i(p^{+}+q^{+})}{(p+q)^{2}}\frac{1}{q^{2}-m_{D}^{2}+i\epsilon}$
(45) $\displaystyle\times$
$\displaystyle\frac{|q^{-}|^{-\eta/2}}{q^{+}}p^{+}\Big{\\{}\delta^{+}((p+k)^{2})-\delta((p+k)^{2})n_{F}(|p^{0}+k^{0}|)\Big{\\}}$

Evaluating this tells us that the rapidity divergence is identical to that of
diagram (a).

  * •

Diagram (c) has the same trace structure as the corresponding real diagram and
hence we can use that result directly

$\displaystyle
S_{c}^{V}=g^{2}N_{c}\nu^{\eta/2}\frac{\delta^{AB}}{k_{\perp}^{2}}\int
d^{4}p\left(\delta^{-}(p^{2})-\delta(p^{2})n_{F}(|p^{0}|)\right)\left(\delta^{+}((p+k)^{2})-\delta((p+k)^{2})n_{F}(|p^{0}+k^{0}|)\right)$
(46) $\displaystyle\times$ $\displaystyle\int
d^{4}q\frac{|q^{-}|^{-\eta/2}}{[q^{2}-m^{2}]\left((p^{+}-q^{+})(\frac{(\vec{p}_{\perp}+\vec{k}_{\perp})^{2}}{p^{+}}-q^{-})-(\vec{p}_{\perp}+\vec{k}_{\perp}-\vec{q}_{\perp})^{2}\right)}$
$\displaystyle\frac{1}{[q^{+}(q^{-}-\frac{(\vec{p}_{\perp}+\vec{k}_{\perp})^{2}}{p^{+}}+\frac{p_{\perp}^{2}}{p^{+}})-(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}-m_{D}^{2}]}\Big{[}\frac{8p^{+}(p^{+}-q^{+})q_{\perp}\cdot(q_{\perp}-k_{\perp})}{q^{+}}$
$\displaystyle-4p^{+}(2q_{\perp}-k_{\perp})\cdot(p_{\perp}+k_{\perp}-q_{\perp})-4(p^{+}-q^{+})((p_{\perp}+k_{\perp})\cdot(q_{\perp}-k_{\perp})+p_{\perp}\cdot
q_{\perp})$ $\displaystyle+$ $\displaystyle
4p^{+}(2p_{\perp}+k_{\perp})\cdot(p_{\perp}+k_{\perp}-q_{\perp})-4p^{+}(p+k-q)\cdot(2p+k)+4(p^{+}-q^{+})p\cdot(p+k)\Big{]}$

We can now isolate the rapidity divergence

$\displaystyle S_{c}^{V,\eta}$ $\displaystyle=$
$\displaystyle-2g^{2}N_{c}\frac{\delta^{AB}}{k_{\perp}^{2}}\int
d^{2}p_{\perp}\int_{0}^{\infty}dp^{+}\Bigg{\\{}n_{F}(|p^{0}|))\Big{\\{}1-n_{F}(|p^{0}+k^{0}|)\Big{\\}}+(1-n_{F}(|p^{0}|))n_{F}(|p^{0}+k^{0}|)\Bigg{\\}}$
(47) $\displaystyle\times$
$\displaystyle\int\frac{d^{2}q_{\perp}}{(2\pi)^{3}}\frac{q_{\perp}\cdot(q_{\perp}-k_{\perp})}{[(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}+m_{D}^{2}][q_{\perp}^{2}+m_{D}^{2}]}\left(\frac{\nu
p^{+}}{q_{\perp}^{2}+m_{D}^{2}}\right)^{\eta/2}\frac{2}{\eta}$

  * •

Diagrams (d) and (e) are simply wave-function renormalization for the quark
and hence do not have any rapidity divergence.

  * •

Diagram (f) evaluates to

$\displaystyle S_{f}^{V}$ $\displaystyle=$ $\displaystyle
4g^{2}\text{Tr}[T^{C}T^{A}T^{C}T^{B}]\int
d^{2}p_{\perp}dp^{+}\left(\Theta(-p^{+})-\text{sgn}(-p^{+})n_{F}(|p^{0}|)\right)\left(\Theta(p^{+})-\text{sgn}(p^{+})n_{F}(|p^{0}+k^{0}|)\right)$
$\displaystyle\times$
$\displaystyle\int\frac{d^{4}q}{(p^{+})^{2}}\frac{1}{[q^{2}-m^{2}][(p+k-q)^{2}][(p-q)^{2}]}4\Big{[}(p^{+}-q^{+})^{2}p\cdot(p+k)$
$\displaystyle+$
$\displaystyle(p^{+})^{2}(p-q)\cdot(p+k-q)-p^{+}(p^{+}-q^{+})[(p+k)\cdot(p+k-q)+p\cdot(p-q)]\Big{]}$

which gives a UV divergence but does not lead to a rapidity divergence.

  * •

We also have wavefunction renormalization for the Glauber gluon which is shown
in fig (g), (h) which will contribute to only the UV divergence.

We can combine the Rapidity divergent pieces here to give

$\displaystyle S_{V}^{\eta}$ $\displaystyle=$ $\displaystyle
g^{2}N_{c}\frac{\delta^{AB}}{k_{\perp}^{2}}\int
d^{2}p_{\perp}\int_{0}^{\infty}dp^{+}\Bigg{\\{}n_{F}(|p^{0}|))\Big{\\{}1-n_{F}(|p^{0}+k^{0}|)\Big{\\}}+(1-n_{F}(|p^{0}|))n_{F}(|p^{0}+k^{0}|)\Bigg{\\}}$
(48) $\displaystyle\times$ $\displaystyle(\nu
p^{+})^{\eta/2}\int\frac{d^{2}q_{\perp}}{(2\pi)^{3}}\left(\frac{-2q_{\perp}\cdot(q_{\perp}-k_{\perp})}{[q_{\perp}^{2}+m_{D}^{2}]^{1+\eta/2}[(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}+m_{D}^{2}]}+\frac{2}{[q_{\perp}^{2}+m_{D}^{2}]^{1+\eta/2}}\right)\frac{2}{\eta}$

Since we have neglected any contribution except the one with the rapidity
pole, it is not obvious what is the natural scale for $\nu$. This will be, in
general hard to answer by a direct calculation due to the presence of the
Fermi distributions function. However, we know that the soft function does not
know about the scale of the transverse momentum measurement $\vec{q}_{T}$, but
only depends on $k_{\perp}$ and on T due to the presence of the density
matrix. Hence, we conjecture that the natural scale for the soft function will
be $\vec{k}_{\perp}\sim T$. However, we have an additional scale $m_{D}$ which
is hierarchically separated from T and the tree level Soft function does
provide support for $k_{\perp}$ all the way down to $m_{D}$. Hence a more
rigorous treatment requires a further factorization of the scale T from the
scale $m_{D}$ which can be done by matching the current EFT to an EFT that
lives at $k_{\perp}\sim m_{D}$. This is beyond the scope of the current paper,
but will be taken up in the future. For now, for the purposes of numerical
analysis, we will assume that the scale for the Soft function is $k_{\perp}$.
For the phenomenologically interesting case of $m_{D}\sim T$, there is a
single scaling for the soft function but in that case, we expect the rapidity
evolution to have a different form, where we incorporate the exact form of the
mass dependent gluon propagator instead of simply using $m_{D}$ as an IR
regulator.

####  3.2.3 Soft function anomalous dimension

We can now combine the results of the real and virtual diagrams( Eqs. 3.2.1
and 48) to write the rapidity anomalous dimension for the Soft function at
this order in perturbation theory.

$\displaystyle\nu\frac{d}{d\nu}S(\vec{k}_{\perp})=\frac{\alpha_{s}N_{c}}{\pi^{2}}\int
d^{2}q_{\perp}\left(\frac{S(\vec{q}_{\perp})}{(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}}-\frac{k_{\perp}^{2}S(k_{\perp})}{2q_{\perp}^{2}(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}}\right)$
(49)

This is just the BFKL equation and the resulting rapidity anomalous dimension
is equal and opposite to the medium jet function so that we have a powerful
check on the consistency of soft-collinear factorization. From explicit
calculation we already know that the jet function has no UV divergence and
hence we can infer the UV anomalous dimension of the Soft function by
consistency of factorization knowing that the hard function is just
$\alpha^{2}(\mu)$.

$\displaystyle\mu\frac{d}{d\mu}S(\vec{k}_{\perp})$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}\beta_{0}}{\pi}S(k_{\perp})$ (50)

where $\beta_{0}=11/3C_{A}-C_{F}n_{F}T_{F}$, which has a value of 23/3 if we
assume 5 active quark flavors.

##  4 Resummation

We can now solve the renormalization group equations in both $\mu,\nu$. The
factorization of Hard-Soft-Collinear modes has completely separated out the
physics at the scale Q from the IR scales $q_{T}$, $T$ and $k_{\perp}$. The
solutions of the RG equations allow us to resum large logarithms in
$Q/k_{\perp}$ thereby increasing the precision of the prediction. We run the
jet function in $\nu$ and the Soft function in $\mu$ as shown in Fig. 6.

Figure 6: Path in $\mu$, $\nu$ space for Renormalization Group evolution of
the medium structure function S and the medium induced jet function J.

The rapidity RGE is the same as the BFKL equation and we first turn towards
solving this equation.

###  4.1 Solving the BFKL equation

We follow the procedure outlined in Kovchegov:2012mbw . We evolve the jet
function using the BFKL equation from its natural scale $\nu\sim Q\sim Qz_{c}$
to the natural rapidity scale of the soft function $\nu\sim k_{\perp}$.

$\displaystyle\nu\frac{d}{d\nu}J(\vec{k}_{\perp})=-\frac{\alpha_{s}N_{c}}{\pi^{2}}\int
d^{2}q_{\perp}\left(\frac{J(\vec{q}_{\perp})}{(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}}-\frac{k_{\perp}^{2}J(k_{\perp})}{2q_{\perp}^{2}(\vec{q}_{\perp}-\vec{k}_{\perp})^{2}}\right)$
(51)

We define the BFKL kernel as

$\displaystyle\int
d^{2}q_{\perp}K_{BFKL}(q,k)J(q_{\perp})=\frac{1}{\pi}\int\frac{d^{2}q_{\perp}}{(\vec{k}_{\perp}-\vec{q}_{\perp})^{2}}\Bigg{[}J(q_{\perp})-\frac{k_{\perp}^{2}}{2q_{\perp}^{2}}J(k_{\perp})\Bigg{]}$
(52)

So $K_{BFKL}$ defines for us the matrix in $q_{\perp}$ space which needs to be
diagonalized. To do that, we need to find the eigenvalues and eigenfunctions
of this matrix. The BFKL kernel is conformally invariant and its
Eigenfunctions are $f(\vec{k}_{\perp})=k_{\perp}^{2\gamma-1}e^{in\phi_{k}}$
with n being an integer and $\phi_{k}$ is the azimuthal angle for
$\vec{k}_{\perp}$. If we plug in this ansatz in the equation above and do some
manipulations,

$\displaystyle\int
d^{2}q_{\perp}K_{BFKL}(k,q)q_{\perp}^{2(\gamma-1)}e^{in\phi_{q}}=\chi(n,\gamma)k_{\perp}^{2(\gamma-1)}e^{in\phi_{k}}$
(53)

where

$\displaystyle\chi(n,\gamma)=\int_{0}^{\infty}dt\Bigg{[}\frac{1}{2\pi}\int_{0}^{2\pi}\frac{d\phi
t^{\gamma-1}e^{in\phi}}{1+t-2\sqrt{t}\cos(\phi)}-\frac{1}{t}\left(\frac{1}{|t-1|}-\frac{1}{\sqrt{4t^{2}+1}}\right)\Bigg{]}$
(54)

is the eigenvalue for the BFKL kernel which can be written as

$\displaystyle\chi(n,\gamma)=2\psi(1)-\psi\left(\gamma+\frac{|n|}{2}\right)-\psi\left(1-\gamma+\frac{|n|}{2}\right)$
(55)

where $\psi$ is the derivative of the logarithm of the $\Gamma$ function, also
known as the Polygamma function.

$\displaystyle\psi(z)=\frac{d}{dz}\textrm{ln}\Gamma(z)$ (56)

The result for the eigenvalue is valid for $0<Re(\gamma)<1$.
Given this result, we can now expand out our jet function in terms of these
Eigenfunctions

$\displaystyle
J(k_{\perp},\nu)=\sum_{n=-\infty}^{\infty}\int_{a-\infty}^{a+i\infty}\frac{d\gamma}{2\pi
i}C_{n,\gamma}(\nu)k_{\perp}^{2(\gamma-1)}e^{in\phi_{k}}$ (57)

We now plug this in our Rapidity Renormalization group equation

$\displaystyle\nu\frac{d}{d\nu}J(k_{\perp},\nu)$ $\displaystyle=$
$\displaystyle\sum_{n=-\infty}^{\infty}\int_{a-i\infty}^{a+i\infty}\frac{d\gamma}{2\pi
i}k_{\perp}^{2(\gamma-1)}e^{in\phi}\Bigg{[}\nu\frac{d}{d\nu}C_{n,\gamma}(\nu)\Bigg{]}$
$\displaystyle=$
$\displaystyle-\frac{\alpha_{s}N_{c}}{\pi}\sum_{n=-\infty}^{\infty}\int_{a-i\infty}^{a+i\infty}\frac{d\gamma}{2\pi
i}\chi(n,\gamma)C_{n,\gamma}(\nu)k_{\perp}^{2(\gamma-1)}e^{in\phi}$

Given that we have an expansion in a complete set of basis functions, we can
write

$\displaystyle\nu\frac{d}{d\nu}C_{n,\gamma}(\mu,\nu)=-\frac{\alpha_{s}(\mu)N_{c}}{\pi}\chi(n,\gamma)C_{n,\gamma}(\nu)$
(58)

so that

$\displaystyle
C_{n,\gamma}(\nu_{f})=C_{n,\gamma}(\mu,\nu_{0})e^{-\frac{\alpha_{s}(\mu)N_{c}}{\pi}\chi_{n,\gamma}\textrm{ln}\frac{\nu_{f}}{\nu_{0}}}$
(59)

For the jet function, $\nu_{0}\sim Q\sim Qz_{cut}$, so that our final solution
looks like

$\displaystyle
J(\mu,\nu_{f},k_{\perp})=\sum_{n=-\infty}^{\infty}\int_{a-i\infty}^{a+i\infty}\frac{d\gamma}{2\pi
i}k_{\perp}^{2(\gamma-1)}e^{in\phi}C_{n,\gamma}(\mu,Q)e^{-\frac{\alpha_{s}(\mu)N_{c}}{\pi}\chi_{n,\gamma}\textrm{ln}\frac{\nu_{f}}{Q}}$
(60)

We also know that at $\nu_{f}=Q$, the result reduces to the tree level result
for the jet function which in transverse momentum space looks like

$\displaystyle
J^{(0)}(\vec{k}_{\perp},\vec{q}_{Tn})=\frac{\delta^{2}(\vec{q}_{Tn}-\vec{k}_{\perp})}{k_{\perp}^{2}}-\frac{\delta(\vec{q}_{Tn})}{k_{\perp}^{2}}$
(61)

We therefore have the boundary condition

$\displaystyle\frac{\delta^{2}(\vec{q}_{Tn}-\vec{k}_{\perp})}{k_{\perp}^{2}}-\frac{\delta^{2}(\vec{q}_{Tn})}{k_{\perp}^{2}}=\sum_{n=-\infty}^{\infty}\int_{a-i\infty}^{a+i\infty}\frac{d\gamma}{2\pi
i}k_{\perp}^{2(\gamma-1)}e^{in\phi}C_{n,\gamma}(\mu,Q)$ (62)

which needs to be solved to fix $C_{n,\gamma}(Q)$.
We can first multiply both sides by $e^{-im\phi_{k}}k_{\perp}^{2\alpha^{*}-1}$
which is one element of the expansion basis and integrate over $\int
d^{2}\vec{k}_{\perp}$. We now us the orthogonality of the basis functions

$\displaystyle\int
d^{2}\vec{k}_{\perp}e^{-im\phi_{k}}k_{\perp}^{2(\alpha^{*}-1)}k_{\perp}^{2(\gamma-1)}e^{in\phi_{k}}=2\pi\delta_{m,n}\int
dre^{(2\alpha_{R}+2\gamma_{R}-2)r}e^{i(-2\alpha_{I}+2\gamma_{I})r}$ (63)

To enforce orthogonality, we need to choose a single value for the real parts
of $\alpha$ and $\gamma$ which now tells us that $\alpha_{R}=\gamma_{R}=1/2$
which then gives us

$\displaystyle\int
d^{2}\vec{k}_{\perp}e^{-im\phi_{k}}k_{\perp}^{2(\alpha^{*}-1)}k_{\perp}^{2(\gamma-1)}e^{in\phi_{k}}=2\pi^{2}\delta_{m,n}\delta(\alpha_{I}-\gamma_{I})$
(64)

We therefore have

$\displaystyle\pi C_{m,\alpha}$ $\displaystyle=$ $\displaystyle
e^{-im\phi_{q}}\frac{q_{tn}^{2(\alpha^{*}-1)}}{q_{Tn}^{2}}-2\pi\delta^{2}(\vec{q}_{Tn})\delta_{m,0}\int
dk_{\perp}\frac{k_{\perp}^{2(\alpha^{*}-1)}}{k_{\perp}}$ (65)

The second term here essentially regulates the singularity as
$k_{\perp}\rightarrow 0$. We can now write our result for our RG evolved jet
function as

$\displaystyle J(\mu,\nu_{f},k_{\perp})$ $\displaystyle=$
$\displaystyle\sum_{n=-\infty}^{\infty}\int_{1/2-i\infty}^{1/2+i\infty}\frac{d\gamma}{2\pi^{2}i}k_{\perp}^{2(\gamma-1)}e^{in\phi_{k}}$
(66) $\displaystyle\times$
$\displaystyle\Bigg{[}e^{-in\phi_{q}}\frac{q_{Tn}^{2(\gamma^{*}-1)}}{q_{Tn}^{2}}-2\pi\delta^{2}(\vec{q}_{Tn})\delta_{n,0}\int
dl_{\perp}\frac{l_{\perp}^{2(\alpha^{*}-1)}}{l_{\perp}}\Bigg{]}e^{-\frac{\alpha_{s}(\mu)N_{c}}{\pi}\chi_{n,\gamma}\textrm{ln}\frac{\nu_{f}}{Q}}$

###  4.2  Running the soft function

We can run the soft function in $\mu$ using the beta function

$\displaystyle S(k_{\perp},\nu\sim k_{\perp},\mu\sim Q)$ $\displaystyle=$
$\displaystyle
S(k_{\perp},\nu=k_{\perp},\mu=k_{\perp})e^{\int_{k_{\perp}}^{Q}d\textrm{ln}\mu\frac{\alpha_{s}(\mu)\beta_{0}}{\pi}}$
(67) $\displaystyle=$ $\displaystyle
S(k_{\perp},\nu=k_{\perp},\mu=k_{\perp})\frac{\alpha_{s}^{2}(k_{\perp})}{\alpha_{s}^{2}(Q)}$

##  5 The master equation

We can now go back and look at our factorized reduced density matrix and try
to derive the master equation for jet evolution. From Eq.2, we can write the
trace over the reduced density matrix for the jet upto quadratic order in the
Glauber interaction as

$\displaystyle\Sigma(t,\vec{q}_{T})=\Sigma^{(0)}(\vec{q}_{T})+\Sigma^{(1)}(\vec{q}_{T})$
(68)

where we have

$\displaystyle\Sigma^{(1)}(\vec{q}_{T})=t|C_{qq}|^{2}(Q)\Sigma^{(0)}(Q,z_{c},\vec{q}_{T})\otimes_{q_{T}}\int\frac{d^{4}k}{k_{\perp}^{4}(2\pi)^{4}}S^{AB}(k)\mathcal{J}_{n,\text{Med}}^{AB}(Q,z_{c},\vec{q}_{T},\vec{k}_{\perp})$
(69)

$C_{qq}=8\pi\alpha_{s}(Q)$ is the Hard function. We can use the RGE solutions
described in the previous section to evolve medium induced soft and jet
functions in rapidity and virtuality so that

$\displaystyle\Sigma^{(1)}(\vec{q}_{T})=t|C_{qq}|^{2}(Q)\Sigma^{(0)}(Q,z_{c},\vec{q}_{T})\otimes_{q_{T}}K_{\text{Med}}(\vec{q}_{T})$
(70)

with

$\displaystyle
K_{\text{Med}}(p_{\perp})=(N_{c}^{2}-1)\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}S_{G}^{\text{resum}}(k_{\perp})J^{\text{resum}}(Q,z_{c},\vec{p}_{\perp},k_{\perp})$
(71)

where the resummed jet and soft functions are given in Eq. 66 and Eq. 67. Eq.
68 describes the evolution of the density matrix over a time scale t. We can
write the evolution equation in a suggestive form by going to impact parameter
space( $\vec{b}$) which is conjugate to $\vec{q}_{T}$,

$\displaystyle\Sigma(t,\vec{b})=\Sigma^{(0)}(\vec{b})\Big{[}1+\frac{t}{\lambda_{\text{mfp}}(\vec{b})}\Big{]}$
(72)

where

$\displaystyle\lambda^{-1}_{\text{mfp}}(b)=|C_{qq}|^{2}K_{\text{Med}}(\vec{b})$
(73)

can be thought of as the inverse mean free path of the jet in the medium.
While deriving this form ,we have assumed that t, which represents the time of
evolution of the jet in the medium or equivalently the size of the medium L,
is much greater than all the time scales of the jet, including the formation
time of the jet $t_{F}\sim Q/q_{T}^{2}$. If the medium is very dilute then the
mean free path(mfp) can be large compared to t in which case Eq.72 is a good
enough approximation.

The second case is when the medium is dense enough so that the mfp is
comparable to the medium size then it becomes necessary to resum higher powers
of $t/\lambda_{\text{mfp}}$. These higher order corrections correspond to
multiple interactions of the jet with the medium. We are still working in a
hierarchy when $t\sim\lambda_{\text{mfp}}>>t_{F}$ so that these multiple
interactions are incoherent and hence Markovian in nature. We can therefore
write a Lindblad equation by taking the limit $t\rightarrow 0$, where we
assume that a single time step $t-0=\delta t$ of the evolution is much smaller
than the medium size, at the same time being much greater than $t_{F}$. We can
first relate the density matrix with the cross section that we wish to
compute,

$\displaystyle
P(\vec{q}_{T})\equiv\frac{d\sigma(t)}{d^{2}\vec{q}_{T}}=\mathcal{N}\frac{\Sigma(t)}{V}$

where $\mathcal{N}$ is a kinematical normalization factor. This leads to our
master equation

$\displaystyle\partial_{t}P(\vec{q}_{T})(t)=P(\vec{q}_{T})\otimes_{q_{T}}K_{\text{Med}}(\vec{q}_{T})$
(74)

The semi classical equation can now be solved by going to impact parameter
space (conjugate to $\vec{q}_{T}$)

$\displaystyle\frac{d\sigma(t)}{d^{2}\vec{q}_{T}}(e_{n},t)=\int
d^{2}\vec{b}e^{i\vec{b}\cdot\vec{q}_{T}}[V(\vec{b},e_{n})]e^{(K_{\text{Med}}(\vec{b}))t}$
(75)

where V(b) is the Inverse Fourier Transform of the vacuum cross section.

The third case is when $\lambda_{\text{mfp}}$ is much smaller than L but is
comparable to the formation time of the jet $t_{F}$. In this case we can
imagine higher order corrections of the form
$tt_{F}/\lambda^{2}_{\text{mfp}}\sim t/\lambda_{\text{mfp}}$ which are now
equally important and hence need to be accounted for. This correction is one
in which quantum coherence is maintained over successive interactions of the
jet with the medium and is well known in literature as the Landau-Pomeranchuk-
Migdal(LPM) Effect. These corrections are beyond the scope of the paper but
will be considered in detail in an upcoming paper Varun . However, we can
still give some arguments for the the factorization in this case. The
important point to note is that this effect is important for high energy
radiation(since $t_{F}\propto E$) and hence only affects the collinear jet
function and not the soft function. Therefore, the factorization of the soft
physics from the collinear should still hold with the Soft function remaining
unchanged. This also means that the BFKL evolution which is a property of the
soft function also holds and by consistency of factorization, so does the RG
evolution of the jet function. I conjecture that the coherence effects will
merely modify the natural rapidity scale for the jet function from its current
value of Q to a smaller value involving the formation time of the jet.

##  6 Numerical results for the medium kernel

We can now look at the numerical impact of resummation for the medium kernel.
We have from Eq. 71

$\displaystyle
K_{\text{Med}}(p_{\perp})=(N_{c}^{2}-1)\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}S_{G}^{\text{resum}}(k_{\perp})J^{\text{resum}}(Q,z_{c},\vec{p}_{\perp},k_{\perp})$
(76)

As pointed out in Eq.73, we can interpret this quantity(upto a factor of
$|C_{qq}|^{2}$) as the inverse mean free path and look at the impact of
resummation at specific values of $p_{\perp}$, in this case we look at
$p_{\perp}\sim T$.

The expression for the resumed Soft function (Eq.67) is straightforward to
implement numerically. However, the jet function (Eq.66) has a complicated
solution which involves an infinite sum. For a non zero value of $p_{\perp}$,
we can write the complete solution for the resummed jet function (Eq.66) as

$\displaystyle
J(\mu,\nu_{f},k_{\perp})=\frac{1}{\pi}\sum_{n=-\infty}^{\infty}\int_{1/2-i\infty}^{1/2+i\infty}\frac{d\gamma}{2\pi
i}k_{\perp}^{2(\gamma-1)}e^{in\phi_{k}}e^{-in\phi_{q}}\frac{q_{Tn}^{2(\gamma^{*}-1)}}{q_{Tn}^{2}}e^{-\frac{\alpha_{s}(\mu)N_{c}}{\pi}\chi_{n,\gamma}\textrm{ln}\frac{\nu_{f}}{Q}}$
(77)

Following Kovchegov:2012mbw , can resort to a saddle point approximation in
three regimes namely $k_{\perp}<<p_{\perp}$, $k_{\perp}\sim p_{\perp}$ and
$k_{\perp}>>p_{\perp}$ and interpolate smoothly between the three
approximations. We consider each regime in turn ,

  * •

$p_{\perp}\sim k_{\perp}$
We can do an approximation for the $\gamma$ integral, using the saddle point
approximation when $\textrm{ln}Q/\nu_{f}\sim\textrm{ln}Q/k_{\perp}$ is large
which is true in our case. Defining

$\displaystyle\bar{\alpha}_{s}\equiv\frac{\alpha_{s}N_{c}}{\pi}$ (78)

We can write the solution as

$\displaystyle
J(\mu,\nu_{f},k_{\perp},\vec{q}_{T_{n}})=\frac{1}{2\pi^{2}k_{\perp}q_{Tn}}\sqrt{\frac{\pi}{14\zeta(3)\bar{\alpha}_{s}Y}}e^{(\alpha_{p}-1)Y-\frac{\textrm{ln}^{2}(k_{\perp}/q_{Tn})}{14\zeta(3)\bar{\alpha}_{s}Y}}$
(79)

with

$\displaystyle\alpha_{p}-1=\frac{4\alpha_{s}N_{c}}{\pi}\textrm{ln}2,\ \ \ \
Y=\textrm{ln}\frac{Q}{\nu_{f}}$ (80)

  * •

$k_{\perp}>>q_{Tn}$
This is known as the double logarithmic approximation and we can also do a
saddle point approximation here. In this case we have

$\displaystyle
J(k_{\perp},q_{Tn})\Big{|}_{k_{\perp}>>q_{Tn}}\approx\frac{1}{2\pi^{3/2}q_{Tn}^{2}k_{\perp}^{2}}\frac{(\bar{\alpha}_{s}Y)^{1/4}}{\textrm{ln}^{3/4}\left(k_{\perp}^{2}/q_{Tn}^{2}\right)}e^{2\sqrt{\bar{\alpha}_{s}Y\textrm{ln}(k_{\perp}^{2}/q_{Tn}^{2})}}$
(81)

The contribution from this region should be the most suppressed compared to
the other two regimes.

  * •

$k_{\perp}<<q_{Tn}\sim T$
Since the integral in $\gamma_{I}$ is symmetric under the interchange
$k_{\perp}\leftrightarrow p_{\perp}$, we can again do a saddle point
approximation

$\displaystyle
J(k_{\perp},q_{Tn})\Big{|}_{k_{\perp}<<q_{Tn}}\approx\frac{1}{2\pi^{3/2}q_{Tn}^{4}}\frac{(\bar{\alpha}_{s}Y)^{1/4}}{\textrm{ln}^{3/4}\left(q_{Tn}^{2}/k_{\perp}^{2}\right)}e^{2\sqrt{\bar{\alpha}_{s}Y\textrm{ln}(q_{Tn}^{2}/k_{\perp}^{2})}}$
(82)

Since this is multiplied by the soft function that has a $1/k_{\perp}^{2}$
singularity cut-off by the gluon mass, this region will give the most
contribution.

Once we have the approximation for the jet function in the three regimes, we
can suitably interpolate between the three.

We choose a fixed perturbative $q_{Tn}\sim T~{}15$GeV with a value of
$\alpha_{s}=0.15$ at this scale and a hard scale $Q=100GeV$. The value of
$m_{D}\sim gT$ is of the same order as the temperature for this choice of
scales. This is different from the hierarchy that we have assumed where
$m_{D}$ is much smaller than the scale of the temperature for simplicity of
analysis. A separation of these scales can be achieved but only at very high
temperatures which are beyond the reach of current colliders. Hence, for the
present purposes, we assume an unrealistic small value of $m_{D}=3GeV$ which
is sufficiently separated from the scale of the temperature. The scale $m_{D}$
in our case therefore only serves the purpose of an IR regulator.

A more realistic analysis for the case $m_{D}\sim T$ would require us to keep
track of all factors of $m_{D}$. This will not change the UV structure of the
factorization but is likely to alter the rapidity evolution equation from the
current conformally invariant BFKL equation to a mass dependent one. I will
leave the derivation of this more phenomenologically relevant case as an added
complication to be incorporated in the theoretical framework.

We can now do a piecewise interpolation between our three saddle point
approximations to generate a single curve for our resummed jet function valid
at all $k_{\perp}$. Fig. 7 shows the approximations in the three regimes for
the jet function along with an interpolating curve.

Figure 7: Approximate solutions for the resummed jet function in three regimes
of $k_{\perp}$ using saddle point approximations. The black dotted curve shows
a piecewise interpolation between the three approximations that we use to
perform a numerical analysis.

In order to show the impact of the radiative corrections, I will give the
ratio of the resummed kernel to the tree level result. At tree level for a
non-zero $p_{\perp}$, the jet function has a simple form

$\displaystyle
J^{(0)}(k_{\perp},p_{\perp})=\frac{\delta^{2}(\vec{p}_{\perp}-\vec{k}_{\perp})}{k_{\perp}^{2}}$
(83)

while the Soft function is

$\displaystyle S_{G}^{(0)}(\vec{k}_{\perp})$ $\displaystyle=$
$\displaystyle\frac{1}{k_{\perp}^{2}}\int
d^{2}p_{\perp}\int_{0}^{\infty}dp^{+}\Bigg{\\{}n_{F}\left(|\frac{p^{+}}{2}+\frac{p_{\perp}^{2}}{2}|\right)\Big{[}1-n_{F}(|\frac{p^{+}}{2}+\frac{(\vec{p}_{\perp}+\vec{k}_{\perp})^{2}}{2p^{+}}|)\Big{]}$
(84) $\displaystyle+$ $\displaystyle
n_{F}(|\frac{p^{+}}{2}+\frac{(\vec{p}_{\perp}+\vec{k}_{\perp})^{2}}{2p^{+}}|)\Big{[}1-n_{F}\left(|\frac{p^{+}}{2}+\frac{p_{\perp}^{2}}{2}|\right)\Big{]}\Bigg{\\}}$
$\displaystyle\equiv$ $\displaystyle\frac{1}{k_{\perp}^{2}}D(k_{\perp})$

so that

$\displaystyle K^{(0)}_{\text{Med}}(q_{\perp})$ $\displaystyle=$
$\displaystyle\frac{N_{C}^{2}-1}{q_{\perp}^{4}}D(q_{\perp})$ (85)

At the same time

$\displaystyle K^{(1)}_{\text{Med}}(q_{\perp})$ $\displaystyle=$
$\displaystyle(N_{C}^{2}-1)\int\frac{d^{2}k_{\perp}}{k_{\perp}^{2}+m_{D}^{2}}D(k_{\perp})\frac{\alpha^{2}_{s}(k_{\perp})}{\alpha_{s}(Q)^{2}}J^{\text{Resum}}(k_{\perp},q_{\perp})$
(86)

If we look at $D(k_{\perp})$, then this is exactly the same quantity as
$\widehat{\mathcal{W}}(k_{\perp})$ which was evaluated in Vaidya:2020cyi .
From the numerical result we see that this quantity has a very mild dependence
on $k_{\perp}$ and we can practically assume it to be a constant over the
entire range of $k_{\perp}$ . Since we are looking at the ratio of quantities,
we can therefore eliminate this factor altogether so that

$\displaystyle R(q_{\perp})$ $\displaystyle=$
$\displaystyle\frac{K^{(1)}_{\text{Med}}(q_{\perp})}{K^{(0)}_{\text{Med}}(q_{\perp})}=q_{\perp}^{4}\int\frac{d^{2}k_{\perp}}{k_{\perp}^{2}+m_{D}^{2}}\Bigg{[}\frac{\alpha_{s}(k_{\perp})}{\alpha_{s}(Q)}\Bigg{]}^{2}J^{\text{Resum}}(k_{\perp},q_{\perp})$
(87)

From this expression we see that there two separate corrections to the medium
kernel and we can explore the impact of each separately.

  * •

Soft function running
We see that the correction from the running in $\mu$ is simply to replace the
coupling at the hard scale Q with that at the scale $k_{\perp}$ which is
sensible since the momentum transfer during the interaction of the jet with
the medium is just $k_{\perp}$. $k_{\perp}$ is free to range all the way down
to the scale $m_{D}$, where the coupling gets stronger, at the same time this
region is supported by the singularity $\sim 1/(k_{\perp}^{2}+m_{D}^{2})$
which is cutoff at the scale $m_{D}$ so that effectively we get an enhancement
$\sim\alpha^{2}(m_{D})/\alpha^{2}(Q)\sim 5$ compared to the tree level result.

  * •

BFKL resummation
The solution to the BFKL equation resums the large rapidity logarithm
$\textrm{ln}Q/k_{\perp}$ along with logarithms of $q_{\perp}/k_{\perp}$ as
seen in the various approximations presented earlier in this section. The
singularity in the form of a factor $1/(k_{\perp}^{2}+m_{D}^{2})$ from the
soft function gets cut-off at the scale $q_{\perp}$ when convolved with the
tree level Jet function. However, when we include radiative corrections, the
BFKL resummed jet function supports this singularity all the way down to the
cut-off scale $m_{D}$. We therefore have additional logarithmic enhancement
due to the presence of this singularity. We can judge the impact of this by
temporarily ignoring the effect of the Soft function running. The curve marked
$R_{\text{BFKL}}$ in Fig. 8 presents this enhancement over a small range of
values of $q_{\perp}$ about a central $q_{\perp}\sim T=15$ GeV.

The same figure also shows the complete enhancement for $K_{\text{Med}}$ due
to both soft and Jet resummation. This suggests that there is significant
reduction in the mean free path due to enhancement of the interaction with the
medium induced by radiative corrections. In light of the discussion in the
previous section, this means that due to the radiative corrections, the medium
effectively appears dense which would enhance the importance of the LPM
effect.

We also have UV finite $m_{D}$ dependent radiative corrections due to elastic
collisions presented in Section 2. These induce logarithms
$\sim\textrm{ln}q_{T}/m_{D}\sim\textrm{ln}1/g$ which are sub-leading
corrections compared to the large logarithm $\textrm{ln}Q/k_{\perp}$ induced
by BFKL evolution as well as running of the soft function which is why we have
not included them in the numerical analysis. Indeed, in the phenomenologically
relevant case of $m_{D}\sim T$, these corrections are quite small.

Figure 8: Enhancement for the medium Kernel due to resummed radiative
corrections as a function of transverse momentum compared to the tree level
result. The curve marked $R_{\text{BFKL}}$ estimates the impact of BFKL
resummation in conjunction with the forward scattering singularity. The curve
marked R also includes the running of the Soft function.

##  7 Summary and Outlook

In this paper I have worked out the renormalization for the medium structure
function and the medium jet function which appear in the factorization formula
for jet substructure observables in heavy ion collisions. This factorization
formula was derived in Vaidya:2020lih  under the following assumptions

  * •

The size of the medium or the time of propagation of the jet in the medium is
larger than the formation time of the jet.

  * •

The medium is dilute or more precisely, the mean free path of the jet in the
medium in comparable to or larger than the size of the medium.

  * •

The medium is homogeneous over the length scales probed by the jet.

  * •

The medium temperature T is high enough for the gluon mass $m_{D}$ to be much
smaller T.

The first three of these conditions then imply that the dominant radiative
corrections are those when the jet partons created in the hard interaction go
on shell between successive interactions with the medium. This formalism
provides a separation of physics at different scales in terms of matrix
elements of manifestly gauge invariant operators. The physics of the medium is
captured by the observable independent medium structure function which is
defined as a gauge invariant correlator of soft currents in the background of
the medium density matrix. In a sense, this function can be thought of as the
"Parton Distribution Function" of the QGP that describes the probabililty of
sourcing partons from and subsequently sinking them in the medium. The
observable dependent medium induced jet function describes the evolution of
high energy partons taking into account elastic collisions with the medium as
well as medium induced radiation.

The radiative corrections for elastic collisions were worked out in
Vaidya:2020lih  and given in section 2. These corrections are UV finite and do
not lead to any renormalization for the medium soft function and the medium
induced jet function. In this paper, I detail the computation of the other
type of corrections at one loop, namely medium induced radiation. All the
divergences in the factorized functions are induced by the medium induced
Bremsstrahlung in the form of UV and rapidity anomalous dimensions. In
particular, the universal soft medium structure function has a rapidity
anomalous dimension which leads to an Renormalization Group equation identical
to the BFKL equation. The UV anomalous dimension induces the running of the
QCD coupling. The medium jet function only has rapidity anomalous dimension
which is equal and opposite the soft function which is a powerful check on the
consistency of factorization. I look at the numerical impact of resummation on
the mean free path of a high energy jet comparing it to the tree level result
and show that the radiative corrections give significant corrections due to a
combination of resummation and the forward scattering singularity which is
cut-off by the medium induced gluon mass. The contribution from the medium
induced radiation dominates over that from elastic collisions and effectively
leads to a smaller mean free path for the jet in the medium.

This is the first consistent factorization formalism for jet substructure
observables in heavy ion collisions. This is still an idealized case as
evidenced by the assumptions that have gone into this derivation but is an
important step towards addressing a realistic scenario. In particular, we
would like to consider the following phenomenologically relevant cases of
increasing complexity

  1. 1.

The medium exists for a short time comparable to or shorter than the formation
time of the jet but the medium is dilute. This means that the mean free path
is large compared to the medium size and only a single interaction of the jet
with the medium is relevant. In this case the jet partons wont go on-shell
before they interact with the medium. This would mean a hard interaction that
creates the jet and the forward scattering from the medium can have
interference diagrams. Since the formation time for Soft radiation is small,
we conjecture that the Soft function remains unchanged and only the jet
function will get modified. However, due to consistency of factorization the
rapidity RG for the jet function will also remain unchanged, albeit with a
modified natural scale for the rapidity.

  2. 2.

A more complicated case would be that the mean free path can be comparable to
or smaller than the formation time of the jet which in turn is comparable to
the size of the medium. This would mean that apart from hard-medium
interaction interference, multiple interactions of the jet with the medium are
important and there is quantum inteference between successive interactions(LPM
effect). Again, this will only impact the jet function which by RG consistency
should still have a BFKL type evolution. This entails defining and computing
jet functions at higher orders in the Glauber Hamiltonian and summing the
series leading to possibly an AMY (Arnold:2002ja ) type formalism, albeit with
a BFKL resummation implemented at each order.

  3. 3.

The Debye mass scale $m_{D}$ is comparable to the scale of the temperature.
This is relatively straightforward to address in that we need to keep the
complete $m_{D}$ dependence which will likely lead to a massive version of the
BFKL equation.

Many of these interesting open questions will be addressed in an upcoming
paper Varun .

###### Acknowledgements.

I thank Iain Stewart and Yuri Kovchegov for useful discussions on several
conceptual aspects of this project. This work is supported by the Office of
Nuclear Physics of the U.S. Department of Energy under Contract DE-SC0011090
and Department of Physics, Massachusetts Institute of Technology.

## References

  * (1) V. Vaidya, [arXiv:2010.00028 [hep-ph]].
  * (2) M. Gyulassy and X. n. Wang, Nucl. Phys. B 420, 583 (1994) [nucl-th/9306003].
  * (3) X. N. Wang, M. Gyulassy and M. Plumer, Phys. Rev. D 51, 3436 (1995) [hep-ph/9408344].
  * (4) R. Baier, Y. L. Dokshitzer, S. Peigne and D. Schiff, Phys. Lett. B 345, 277 (1995) [hep-ph/9411409].
  * (5) R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne and D. Schiff, Nucl. Phys. B 483, 291 (1997) [hep-ph/9607355].
  * (6) R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne and D. Schiff, Nucl. Phys. B 484, 265 (1997) [hep-ph/9608322].
  * (7) B. G. Zakharov, JETP Lett. 63, 952 (1996) [hep-ph/9607440].
  * (8) B. G. Zakharov, JETP Lett. 65, 615 (1997) [hep-ph/9704255].
  * (9) M. Gyulassy, P. Levai and I. Vitev, Nucl. Phys. B 571, 197 (2000) [hep-ph/9907461].
  * (10) M. Gyulassy, P. Levai and I. Vitev, Nucl. Phys. B 594, 371 (2001) [nucl-th/0006010].
  * (11) U. A. Wiedemann, Nucl. Phys. B 588, 303 (2000) [hep-ph/0005129].
  * (12) X. f. Guo and X. N. Wang, Phys. Rev. Lett. 85, 3591 (2000) [hep-ph/0005044].
  * (13) X. N. Wang and X. f. Guo, Nucl. Phys. A 696, 788 (2001) [hep-ph/0102230].
  * (14) P. B. Arnold, G. D. Moore and L. G. Yaffe, JHEP 0206, 030 (2002) [hep-ph/0204343].
  * (15) P. B. Arnold, G. D. Moore and L. G. Yaffe, JHEP 0301, 030 (2003) [hep-ph/0209353].
  * (16) C. A. Salgado and U. A. Wiedemann, Phys. Rev. D 68, 014008 (2003) [hep-ph/0302184].
  * (17) N. Armesto, C. A. Salgado and U. A. Wiedemann, Phys. Rev. D 69, 114003 (2004) [hep-ph/0312106].
  * (18) A. Majumder, B. Müller and S. A. Bass, Phys. Rev. Lett. 99, 042301 (2007) [arXiv:hep-ph/0611135 [hep-ph]].
  * (19) A. Majumder, B. Müller and X. Wang, Phys. Rev. Lett. 99, 192301 (2007) [arXiv:hep-ph/0703082 [hep-ph]].
  * (20) R. Neufeld, B. Müller and J. Ruppert, Phys. Rev. C 78, 041901 (2008) [arXiv:0802.2254 [hep-ph]].
  * (21) R. Neufeld and B. Müller, Phys. Rev. Lett. 103, 042301 (2009) [arXiv:0902.2950 [nucl-th]].
  * (22) I. Arsene et al. [BRAHMS Collaboration], Nucl. Phys. A 757, 1 (2005) [nucl-ex/0410020].
  * (23) B. B. Back et al., Nucl. Phys. A 757, 28 (2005) [nucl-ex/0410022].
  * (24) J. Adams et al. [STAR Collaboration], Nucl. Phys. A 757, 102 (2005) [nucl-ex/0501009].
  * (25) K. Adcox et al. [PHENIX Collaboration], Nucl. Phys. A 757, 184 (2005) [nucl-ex/0410003].
  * (26) G. Aad et al. [ATLAS Collaboration], Phys. Rev. Lett. 105, 252303 (2010) [arXiv:1011.6182 [hep-ex]].
  * (27) K. Aamodt et al. [ALICE Collaboration], Phys. Lett. B 696, 30 (2011) [arXiv:1012.1004 [nucl-ex]].
  * (28) S. Chatrchyan et al. [CMS Collaboration], Phys. Rev. C 84, 024906 (2011) [arXiv:1102.1957 [nucl-ex]].
  * (29) Y. Mehtar-Tani, J. G. Milhano and K. Tywoniuk, Int. J. Mod. Phys. A 28, 1340013 (2013) [arXiv:1302.2579 [hep-ph]].
  * (30) J. P. Blaizot and Y. Mehtar-Tani, Int. J. Mod. Phys. E 24, no. 11, 1530012 (2015) [arXiv:1503.05958 [hep-ph]].
  * (31) G. Y. Qin and X. N. Wang, Int. J. Mod. Phys. E 24, no. 11, 1530014 (2015) [arXiv:1511.00790 [hep-ph]].
  * (32) S. Cao and X. N. Wang, arXiv:2002.04028 [hep-ph].
  * (33) G. Ovanesyan and I. Vitev, Phys. Lett. B 706, 371 (2012) [arXiv:1109.5619 [hep-ph]].
  * (34) Y. T. Chien and I. Vitev, JHEP 1605, 023 (2016) [arXiv:1509.07257 [hep-ph]].
  * (35) G. Ovanesyan and I. Vitev, JHEP 1106, 080 (2011) [arXiv:1103.1074 [hep-ph]].
  * (36) Y. T. Chien, A. Emerman, Z. B. Kang, G. Ovanesyan and I. Vitev, Phys. Rev. D 93, no. 7, 074030 (2016) [arXiv:1509.02936 [hep-ph]].
  * (37) Z. B. Kang, R. Lashof-Regas, G. Ovanesyan, P. Saad and I. Vitev, Phys. Rev. Lett. 114, no. 9, 092002 (2015) [arXiv:1405.2612 [hep-ph]].
  * (38) I. Z. Rothstein and I. W. Stewart, JHEP 08, 025 (2016) [arXiv:1601.04695 [hep-ph]].
  * (39) S. Caron-Huot and C. Gale, Phys. Rev. C 82, 064902 (2010) [arXiv:1006.2379 [hep-ph]].
  * (40) W. Ke, Y. Xu and S. A. Bass, Phys. Rev. C 100, no.6, 064911 (2019) [arXiv:1810.08177 [nucl-th]].
  * (41) Y. Mehtar-Tani and K. Tywoniuk, arXiv:1910.02032 [hep-ph].
  * (42) P. Arnold and S. Iqbal, JHEP 1504, 070 (2015) Erratum: [JHEP 1609, 072 (2016)] [arXiv:1501.04964 [hep-ph]].
  * (43) P. Arnold, H. C. Chang and S. Iqbal, JHEP 1609, 078 (2016) [arXiv:1605.07624 [hep-ph]].
  * (44) H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, (2002).
  * (45) G. Schaller, Open Quantum Systems far from Equilibrium, Lecture Notes in Physics 881, Springer (2014).
  * (46) V. Vaidya, In preparation
  * (47) C. Lee, P. Shrivastava and V. Vaidya, JHEP 1909, 045 (2019) [arXiv:1901.09095 [hep-ph]].
  * (48) Y. Makris and V. Vaidya, JHEP 1810, 019 (2018) [arXiv:1807.09805 [hep-ph]].
  * (49) V. Vaidya and X. Yao, [arXiv:2004.11403 [hep-ph]].
  * (50) C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. D 67, 071502 (2003) [arXiv:hep-ph/0211069 [hep-ph]].
  * (51) C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. D 68, 034021 (2003) [arXiv:hep-ph/0303156 [hep-ph]].
  * (52) A. V. Manohar and I. W. Stewart, Phys. Rev. D 76, 074002 (2007) [arXiv:hep-ph/0605001 [hep-ph]].
  * (53) C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, Phys. Rev. D 63, 114020 (2001) [arXiv:hep-ph/0011336 [hep-ph]].
  * (54) C. W. Bauer and I. W. Stewart, Phys. Lett. B 516, 134-142 (2001) [arXiv:hep-ph/0107001 [hep-ph]].
  * (55) C. W. Bauer, S. Fleming, D. Pirjol, I. Z. Rothstein and I. W. Stewart, Phys. Rev. D 66, 014017 (2002) [arXiv:hep-ph/0202088 [hep-ph]].
  * (56) Y. V. Kovchegov and E. Levin, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 33, 1-350 (2012) doi:10.1017/CBO9781139022187