$\displaystyle\approx\,p+\frac{m_{D}^{2}}{P}+\frac{m_{D}^{4}}{32p^{3}}\left[3-2\ln\frac{8p^{2}}{m_{D}^{2}}\right]+\frac{m_{D}^{6}}{128p^{5}}\left[2\ln^{2}\frac{8p^{2}}{m_{D}^{2}}-10\ln\frac{8p^{2}}{m_{D}^{2}}+7\right]\
.$ (499b)

In addition to the pole contributions coming from time like domain
$\omega^{2}>p^{2}$ above the light cone, there is also a discontinuous part
corresponding to Landau damping coming from space like domain
$\omega^{2}<p^{2}$ due to the presence of logarithmic terms in (494) and
(495).

###  9.11 Spectral Representation of Gauge Boson Propagator

The effective propagator of a gauge boson in presence of thermal medium can be
written from (371) as

${D_{\mu\nu}=-\frac{\xi}{P^{4}}P_{\mu}P_{\nu}-\frac{1}{P^{2}+\Pi_{T}}A_{\mu\nu}-\frac{1}{P^{2}+\Pi_{L}}B_{\mu\nu}}\,,$
(500)

where

$\displaystyle P^{2}+\Pi_{L}$
$\displaystyle=\omega^{2}-p^{2}+\frac{(\omega^{2}-p^{2})}{p^{2}}\
m_{D}^{2}\,\left[1-\frac{\omega}{2p}\ln{\frac{\omega+p}{\omega-p}}\right]\,,$
(501a) $\displaystyle P^{2}+\Pi_{T}$
$\displaystyle=\omega^{2}-p^{2}-\frac{m_{D}^{2}}{2}\frac{\omega^{2}}{p^{2}}\left[1+\frac{p^{2}-\omega^{2}}{2\omega
p}\ln{\frac{\omega+p}{\omega-p}}\right]\,.$ (501b)

As discussed in subsec. 9.10 that $P^{2}+\Pi_{L}=0$ has solutions at
$\omega=\pm\omega_{L}$ and $P^{2}+\Pi_{T}=0$ has solutions at
$\omega=\pm\omega_{T}$. Both also have a cut part due to space like momentum
$\omega^{2}<p^{2}$. In-medium spectral function corresponding to the effective
photon propagator in (500) will have both pole and cut contribution as

$\displaystyle\rho_{L}(\omega,p)=\rho_{L}^{\textrm{pole}}(\omega,p)+\rho_{L}^{\textrm{cut}}(\omega,p)\,,$
(502a)
$\displaystyle\rho_{T}(\omega,p)=\rho_{T}^{\textrm{pole}}(\omega,p)+\rho_{T}^{\textrm{cut}}(\omega,p)\,.$
(502b)

The pole part of the longitudinal spectral function can be obtained using
(696) as

$\displaystyle\rho^{\textrm{pole}}_{L}(\omega,p)$ $\displaystyle=$
$\displaystyle\lim_{\epsilon\rightarrow
0}\frac{1}{\pi}{\textrm{Im}}\left.\frac{1}{P^{2}+\Pi_{L}}\right|_{\omega\rightarrow\omega+i\epsilon}$
(503) $\displaystyle=$
$\displaystyle\frac{\delta(\omega-\omega_{L})}{\left|\frac{d}{d\omega}(P^{2}+\Pi_{L})\right|_{\omega=\omega_{L}}}+\frac{\delta(\omega+\omega_{L})}{\left|\frac{d}{d\omega}(P^{2}+\Pi_{L})\right|_{\omega=-\omega_{L}}}\,$
$\displaystyle=$
$\displaystyle\frac{\omega}{p^{2}+m_{D}^{2}-\omega^{2}}\left[\delta(\omega-\omega_{L})+\delta(\omega+\omega_{L})\right].$

Similarly, the pole part corresponding to the transverse spectral function can
be obtained using (696) as

$\displaystyle\rho^{\textrm{pole}}_{T}(\omega,p)$ $\displaystyle=$
$\displaystyle\lim_{\epsilon\rightarrow
0}\frac{1}{\pi}{\textrm{Im}}\left.\frac{1}{P^{2}+\Pi_{T}}\right|_{\omega\rightarrow\omega+i\epsilon}$
(504) $\displaystyle=$
$\displaystyle\frac{\delta(\omega-\omega_{T})}{\left|\frac{d}{d\omega}(P^{2}+\Pi_{T})\right|_{\omega=\omega_{T}}}+\frac{\delta(\omega+\omega_{T})}{\left|\frac{d}{d\omega}(P^{2}+\Pi_{T})\right|_{\omega=-\omega_{T}}}\,$
$\displaystyle=$
$\displaystyle\frac{\omega(\omega^{2}-p^{2})}{m_{D}^{2}\omega^{2}+p^{2}(\omega^{2}-p^{2})}\left[\delta(\omega-\omega_{T})+\delta(\omega+\omega_{T})\right].$

For $\omega^{2}<p^{2}$, there is a discontinuity in
$\ln\frac{\omega+p}{\omega-p}$ as $\ln{y}=\ln\left|y\right|-i\pi\ ,$ which
leads to the spectral function, $\rho^{\textrm{cut}}_{L,T}(\omega,p)$,
corresponding to the discontinuity in $P^{2}+\Pi_{L,T}$ . The cut contribution
to the longitudinal spectral function can be obtained using (694) as

$\displaystyle\rho^{\textrm{cut}}_{L}(\omega,p)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi i}\textrm{Disc}\frac{1}{P^{2}+\Pi_{L}}$ (505)
$\displaystyle=$ $\displaystyle\frac{1}{\pi}\lim_{\epsilon\rightarrow
0}\textrm{Im}\left.\frac{1}{{P^{2}+\Pi_{L}}}\right|_{{\omega\rightarrow\omega+i\epsilon}\atop{\omega<p}}$
$\displaystyle=$
$\displaystyle\\!\\!\frac{(1-x^{2})x\,m_{D}^{2}\,\Theta(1-x^{2})/2}{\left[p^{2}(x^{2}-1)+(x^{2}-1)m_{D}^{2}-(x^{2}-1)x\,\frac{m_{D}^{2}}{2}\ln\left|\frac{x+1}{x-1}\right|\right]^{2}+\left[\pi(x^{2}-1)x\,\frac{m_{D}^{2}}{2}\right]^{2}}$
$\displaystyle=$ $\displaystyle\beta_{L}(x)\Theta(1-x^{2})\,,$

where $x=\omega/p$. Similarly, the cut part of the transverse spectral
function can be obtained using (694) as

$\displaystyle\rho^{\textrm{cut}}_{T}(\omega,p)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi i}\textrm{Disc}\frac{1}{P^{2}+\Pi_{T}}$ (506)
$\displaystyle=$ $\displaystyle\frac{1}{\pi}\lim_{\epsilon\rightarrow
0}\textrm{Im}\left.\frac{1}{{P^{2}+\Pi_{T}}}\right|_{{\omega\rightarrow\omega+i\epsilon}\atop{\omega<p}}$
$\displaystyle=$
$\displaystyle\\!\\!\frac{(x^{2}-1)x\,m_{D}^{2}\,\Theta(1-x^{2})/4}{\left[p^{2}(x^{2}-1)-\frac{m_{D}^{2}}{2}\left(x^{2}-\frac{(x^{2}-1)x}{2}\,\ln\left|\frac{x+1}{x-1}\right|\right)\right]^{2}+\left[\pi(x^{2}-1)x\,\frac{m_{D}^{2}}{4}\right]^{2}}$
$\displaystyle=$ $\displaystyle\beta_{T}(x)\Theta(1-x^{2})\,.$

##  10 Quantum Chromodynamics (QCD)

QCD is the theory of strong interaction which is a non-abelian $SU(3)$ gauge
theory. It describes the quarks and gluons in the similar way as QED does for
electrons and photons. The major difference between the two theories is that
QCD contains three colour charges in fundamental representation. So, a quark
can be represented by a vector with three colour states. The gauge field known
as gluon mediates the interaction of colour charges. The special unitary group
$SU(3)$ has $3^{2}-1=8$ generators, the number of charge mediating particles,
corresponding to eight gluons in QCD.

###  10.1 QCD Lagrangian

The QCD Lagrangian pbpal ; Peskin  is written as

${\cal
L}_{\textrm{QCD}}={\bar{\psi}}_{ab}(i\not{\partial}-m_{q})\psi_{ab}-g({\bar{\psi}}_{ab}\lambda_{i}\psi_{ab})\not{A}^{i}-\frac{1}{4}G_{\mu\nu}^{i}G^{\mu\nu}_{i}+{\cal
L}_{\textrm{gf}}+{\cal L}_{\textrm{gh}}\,.$ (507)

Figure 30: The QCD vertices. (a) quark-gluon interaction, (b) three gluon
interaction, (c) four gluon interaction and (d) ghost-gluon interaction.

We note that $\psi_{ab}$ is quark spinor with colour indices $a=(\textrm{r, \,
g,\, b})$ and $b$ indicates the flavour index. The Lagrangian in (507) is
quite similar to QED Lagrangian in (416), with the differences that the
electromagnetic field strength tensor $F^{\mu\nu}$, is replaced by the gluonic
field strength tensor $G^{i}_{\mu\nu}$ and an another set of indices has crept
in. The first term represents the non-interacting quarks with current mass
$m_{q}$. The second term represents quark-gluon interaction with QCD coupling
$g$ and shown in Fig. 30(a). The Gell-Mann matrices $\lambda_{i}$ were not
there in QED. $\lambda_{i}$ matrices change the colour of the interacting
particles. These matrices are traceless and obey the commutation relation
$[\lambda_{i},\,\lambda_{j}]=i\varepsilon^{ijk}\lambda_{k}$ with normalisation
relation ${\textrm{Tr}}\lambda_{i}\lambda_{j}=2\delta_{ij}$,
$\varepsilon^{ijk}$ is the structure constant of the group which is number. It
is also totally antisymmetric and vanishes if two of the indices become same.
Now, $G^{i}_{\mu\nu}$ indicates the gluon fields which is invariant under
gauge transformation and can be defined as

$G_{i}^{\mu\nu}=F_{i}^{\mu\nu}+g\varepsilon_{ijk}A_{j}^{\mu}A_{k}^{\nu}\,,$
(508)

where first term is the QED field tensor and the second term represents the
self interaction of gluons. In contrast to QED, the mediator gluons have
colour charge which enable them to interact among themselves. Now the gluonic
part of the Lagrangian can be written as

${\cal
L}_{G}=-\frac{1}{4}G_{i\mu\nu}{G^{\mu\nu}_{i}}=-\frac{1}{4}F_{i\mu\nu}{F^{\mu\nu}_{i}}+g\varepsilon_{ijk}A_{i\mu}A_{j\nu}\partial^{\mu}A_{k}^{\nu}-\frac{1}{4}g^{2}\varepsilon_{ijk}\varepsilon_{ilm}A^{\mu}_{j}A^{\nu}_{k}A_{l\mu}A_{m\nu}\,.$
(509)

The first term in (509) describes the Lagrangian for eight non-interacting,
massless spin 1 gluon fields. The second and third term, respectively, in
(509) describe the self interactions of gluon fields. This produces three- and
four-point vertices in perturbation theory, displayed in Fig. 30(b) and Fig.
30(c), respectively. The last two features have no analogue in QED, as photons
do not self interact.

Finally, the gauge fixing and ghost terms Peskin  are, respectively, given as

$\displaystyle{\cal L}_{\textrm{gf}}$
$\displaystyle=-\frac{1}{2\xi}(\partial_{\mu}A^{\mu}_{a})^{2}\,,$ (510a)
$\displaystyle{\cal L}_{\textrm{gh}}$
$\displaystyle=-{\bar{C}}_{i}\partial^{2}C_{i}-g\varepsilon_{ijk}{\bar{C}}_{i}\partial_{\mu}(A^{\mu}_{j}C_{k})\,,$
(510b)

where $C$ is the ghost field which are Grassmann variable. The gauge fixing
Lagrangian in (510a) is required to eliminate the unphysical degrees of
freedom present in the system. The ghost Lagrangian in covariant gauge is
given in (510b) which depends on the choice of the gauge fixing term. The
first term in (510b) represents free ghost fields whereas the second term
indicates interaction between ghost and gluon as shown in Fig. 30(d)

We note that the previous discussion on QED in sec. 9.9 in HTL approximation
would provide a good starting point for QCD, since these two theories follow
the similar formalism. We know that QCD is a non-Abelian $SU(3)$ gauge theory
whereas QED is a $U(1)$ gauge theory. Thus, the generalization of QED results
to QCD would mostly involve group-theoretical factors as we will see below.

###  10.2 One-loop Gluon Self-energy in HTL Approximation

We calculate the one-loop gluon self-energy Le_Bellac ; Weldon:1982aq ; VVK ;
KK  and the relevant diagrams are given in Fig. 31. The contribution of the
first diagram (tadpole) in HTL approximation can be written as

$\Pi_{\mu\nu}^{(a)}(P)=-3C_{A}g^{2}\delta_{\mu\nu}\int\frac{d^{4}K}{(2\pi)^{4}}\Delta_{B}(K)\,,$
(511)

where $g$ is the strong or QCD coupling constant, $C_{A}\,(=3)$ is the group
factor and $\Delta_{B}$ is the bosonic part of the propagator.

Figure 31: One-loop gluon self-energy diagrams.

The contribution of the second diagram (gluon loop) in HTL approximation is
obtained as

$\Pi_{\mu\nu}^{(b)}(P)=g^{2}C_{A}\int\frac{d^{4}K}{(2\pi)^{4}}5K_{\mu}K_{\nu}\Delta_{B}(K)\Delta_{B}(K+P)+g^{2}C_{A}\delta_{\mu\nu}\int\frac{d^{4}K}{(2\pi)^{4}}\Delta_{B}(K)\,.$
(512)

The contribution of the third diagram (ghost loop) can be obtained as

$\Pi_{\mu\nu}^{(c)}(P)=-g^{2}C_{A}\int\frac{d^{4}K}{(2\pi)^{4}}K_{\mu}K_{\nu}\Delta_{B}(K)\Delta_{B}(K+P)\,.$
(513)

Combining these three diagrams, one can write

$\displaystyle\Pi_{\mu\nu}^{(a+b+c)}(P)$ $\displaystyle=$ $\displaystyle
4g^{2}C_{A}\int\frac{d^{4}K}{(2\pi)^{4}}K_{\mu}K_{\nu}\Delta_{B}(K)\Delta_{B}(K+P)-2g^{2}C_{A}\delta_{\mu\nu}\int\frac{d^{4}K}{(2\pi)^{4}}\Delta_{B}(K)$
(514) $\displaystyle\approx$ $\displaystyle
g^{2}C_{A}\int\frac{d^{4}K}{(2\pi)^{4}}\left[4K_{\mu}K_{\nu}-2K^{2}\delta_{\mu\nu}\right]\Delta_{B}(K)\Delta_{B}(K+P)\,,$

where we have used in the last step as $\Delta_{B}(K+P)=1/(K+P)^{2}\approx
1/K^{2}$.

Now the contribution of the fourth diagram (quark loop) can be obtained
similar to QED in (484) as

$\Pi_{\mu\nu}^{(d)}(P)=-g^{2}\frac{N_{f}}{2}\int\frac{d^{4}K}{(2\pi)^{4}}\left[8K_{\mu}K_{\nu}-4K^{2}\delta_{\mu\nu}\right]\Delta_{F}(K)\Delta_{F}(K+P)\,,$
(515)

where $N_{f}$ is the number of quark flavour. We note that one can go from
fermionic loop to bosonic loop by the following transformations:

$\displaystyle\int\frac{d^{4}K}{(2\pi)^{4}}\Delta_{F}(K)$
$\displaystyle=-\frac{1}{2}\int\frac{d^{4}K}{(2\pi)^{4}}\Delta_{B}(K)\,,$
(516a)
$\displaystyle\int\frac{d^{4}K}{(2\pi)^{4}}\Delta_{F}(K)\Delta_{F}(K+P)$
$\displaystyle=-\frac{1}{2}\int\frac{d^{4}K}{(2\pi)^{4}}\Delta_{B}(K)\Delta_{B}(K+P)\,.$
(516b)

Using (516b), one can write (515) as

$\Pi_{\mu\nu}^{(d)}(P)=g^{2}\frac{N_{f}}{2}\int\frac{d^{4}K}{(2\pi)^{4}}\left[4K_{\mu}K_{\nu}-2K^{2}\delta_{\mu\nu}\right]\Delta_{B}(K)\Delta_{B}(K+P)\,,$
(517)

Now the contributions of all four diagrams in gluon self energy can be written
by combining (514) and (517) as

$\Pi_{\mu\nu}^{(a+b+c+d)}(P)=g^{2}\left(C_{A}+\frac{N_{f}}{2}\right)\int\frac{d^{4}K}{(2\pi)^{4}}\left[4K_{\mu}K_{\nu}-2K^{2}\delta_{\mu\nu}\right]\Delta_{B}(K)\Delta_{B}(K+P)\,.$
(518)

The photon self-energy can be written from (484) as

$\Pi_{\mu\nu}(P)=-e^{2}\int\frac{d^{4}K}{(2\pi)^{4}}\left[8K_{\mu}K_{\nu}-4K^{2}\delta_{\mu\nu}\right]\Delta_{F}(K)\Delta_{F}(K+P)\,,$
(519)

Using (516b), one gets

$\Pi_{\mu\nu}(P)=e^{2}\int\frac{d^{4}K}{(2\pi)^{4}}\left[4K_{\mu}K_{\nu}-2K^{2}\delta_{\mu\nu}\right]\Delta_{B}(K)\Delta_{B}(K+P)\,,$
(520)

Now the only difference between (518) and (520) is the overall factor. From
QED to QCD or photon to gluon, one changes

$e^{2}\Rightarrow g^{2}\left(C_{A}+\frac{N_{f}}{2}\right)\,.$ (521)

With this one can transform QED Debye mass in (489) to QCD Debye mass as

$\frac{e^{2}T^{2}}{3}\Rightarrow\frac{g^{2}T^{2}}{3}\left(C_{A}+\frac{N_{f}}{2}\right)=m_{D}^{2}\,:\,\,\,{\mbox{
\, \, QCD Debye mass}}.$ (522)

Therefore, we note that the HTL self-energies of photon given in (493), (494)
and (495) will be same for gluons with the replacement of Debye mass
$m_{D}^{2}$ for QCD as given in (522). We further note that in the IR limit
the gluon transverse self-energy in leading order in HTL vanishes implying
there is no magnetic screening in QCD and provides no magnetic screening mass
for gluons. This is a nonperturbative effect which can not be calculated
perturbatively Linde1 ; Linde2 . However, the longitudinal component of gluon
self-energy in the IR limit becomes $m_{D}^{2}$ which acts as a IR regulator
in the static electric scale.

We also note that the collective excitations of gluons is same as photons
which are displayed in Fig. 29.

###  10.3 One-loop Quark Self-energy in HTL Approximation

The evaluation of quark self-energy at one-loop order is even simpler. This is
because there is only one diagram which is similar to Fig. 26 where the
internal photon line is to be replaced by gluon line. The one loop quark self-
energy can be written as

$\displaystyle\Sigma(P)\delta_{ij}$ $\displaystyle=$ $\displaystyle
T\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{\\{K\\}}\,\,\,(-ig\gamma_{\mu}(\lambda^{a})_{ik})\frac{iK\\!\\!\\!\\!/}{K^{2}}(-ig\gamma^{\mu}(\lambda^{a})_{kj})\frac{i}{(P-K)^{2}}$
(523) $\displaystyle=$ $\displaystyle
g^{2}C_{F}\delta_{ij}T\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{\\{K\\}}\gamma_{\mu}\frac{K\\!\\!\\!\\!/}{K^{2}}\gamma^{\mu}\frac{1}{(P-K)^{2}}\,,$

where we have used the identity
$(\lambda^{a}\lambda^{a})_{ij}=\frac{Nc^{2}-1}{2N_{c}}\delta_{ij}=C_{F}\delta_{ij}$
with $N_{c}=3$. After performing the frequency sum and $k$-integration as done
in subsec. 9.6, the quark self-energy becomes

$\displaystyle\Sigma(P)$ $\displaystyle=$
$\displaystyle\frac{g^{2}T^{2}}{8}C_{F}\int\frac{d\Omega}{4\pi}\frac{\hat{\not{K}}}{P\cdot\hat{K}}.$
(524)

The electron self-energy is obtained in (470)

$\displaystyle\Sigma(P)$ $\displaystyle=$
$\displaystyle\frac{e^{2}T^{2}}{8}\int\frac{d\Omega}{4\pi}\frac{\hat{\not{K}}}{P\cdot\hat{K}}.$
(525)

The only difference between (525) and (524) is overall group factor as
$C_{F}=1$ for $U(1)$ gauge theory whereas $C_{F}=4/3$ for $SU(3)$ gauge
theory. So one obtains QCD results for quark by generalizing the QED results
of electron by replacing

$e^{2}\Rightarrow g^{2}C_{F}\,.$ (526)

With this one can transform electron thermal mass in (458) to quark thermal
mass as

$\frac{e^{2}T^{2}}{8}\Rightarrow\frac{g^{2}T^{2}}{8}C_{F}=\frac{g^{2}T^{2}}{8}\times\frac{4}{3}=\frac{g^{2}T^{2}}{6}=m_{\textrm{th}}^{2}:\,\,{\mbox{
\, \, quark thermal mass in QCD}}.$ (527)

Now the expression for electron self-energy given in (470) and (471) will be
same for quark self-energy with the replacement of electron thermal mass
($e^{2}T^{2}/8$) by quark thermal mass ($g^{2}T^{2}/6$) as given in (527).
Also the structure constants appearing in the general structure of fermion
self-energy and propagator need to be replaced by quark thermal mass to study
the collective excitations of quark in thermal medium, which are same as
electrons displayed in Fig. 27.

Therefore, we learn about the collective excitations in a QCD plasma from the
acquired knowledge of QED plasma excitations by replacing the QED Debye mass
and electron thermal mass by QCD Debye mass in (522) and quark thermal mass in
(527), respectively, in photon self-energy, effective photon propagator,
electron self-energy and effective electron propagator.

##  11 Subtleties in Finite Temperature Field Theory

Let us start by introducing the parametric scales appearing in finite
temperature field theory. The periodicity or anti-periodicity over Euclidean
time introduces a scale present in the non-interacting theory where momentum,
$p\sim 2\pi T$, known as hard scale and the bosonic zero modes do not acquire
any scale in the non-interacting theory. As we have seen in previous sections
that interaction introduces softer scales corresponding to collective
excitations in the thermal medium. In particular, scalar fields and the
electric component of gauge fields are screened in Debye scale where momentum,
$p\sim gT$. On the other hand, the magnetic component of gauge field is
screened only non-perturbatively Linde1 ; Linde2  at the scale where momentum,
$p\sim g^{2}T$, known as the ultra-soft scale or non-perturbative magnetic
scale.

There is an expansion parameter related to bosonic fluctuations with momentum
(or mass) scale $p$ of the form

$\epsilon_{\textrm{b}}\sim\frac{1}{\pi}g^{2}(2\pi T)n_{B}(p)=\frac{g^{2}(2\pi
T)}{\pi(e^{p/T}-1)}\,\,\stackrel{{\scriptstyle p<T}}{{\sim}}\,\
\frac{g^{2}(2\pi T)T}{\pi p}\,.$ (528)

Thus, for the hard scale: $p\sim 2\pi T$, the expansion parameter becomes a
series in $\sim g^{2}/\pi^{2}$, the even power of $g$ like $T=0$ case. For
soft (electric) scale, $p\sim gT$, the series becomes $\sim g/\pi$. For ultra-
soft (magnetic) scale $\sim g^{2}T$, there is no perturbative series at all
Linde1 ; Linde2  and it has to be determined non-perturbatively. Therefore, at
very high temperature compared to any intrinsic mass scale of a given theory
and the coupling $g$ is less than unity, there appears a hierarchy of momentum
(mass) scales in the system and there are three distinct scales: hard, soft
(electric) and ultra-soft (magnetic).

In naive perturbation theory, both static and dynamic quantities can be
computed by expanding in coupling constant around the free theory. This works
in hard scale regime that uses free propagators and vertices, and the
contribution appears in even power of coupling ($g^{2n}$) as discussed above.
However, the naive application of perturbation theory would in most cases
result in infrared777singularities from both electric and magnetic sectors.
and/or collinear singularities, and some cases gauge dependent results. It is
to be noted that the infrared problems are associated with bosonic excitations
but not with fermionic excitations as the fermionic expansion parameter
remains finite for $p<T$. These in turn signal sensitivity to soft region
($p\sim gT$) of the phase space, where naive perturbation theory breaks down.
This breakdown corresponds to the emergence of collective effects, arising
from the dynamics of the thermal medium as discussed in previous sections. In
the soft (electric) scale $\sim gT$, for which perturbation theory in
principle works, does not exist in non-interacting theory but needs to be
generated. This means that the perturbation theory needs to be resummed or re-
organised. This is done through the effective field theory like the hard
thermal loop (HTL) resummation BP ; BP2 ; BP_90 ; TW ; FT ; Barton  techniques
and thereby HTL perturbation theory (HTLpt) andersen1 ; andersen2 ; 3loopglue1
; ASS ; ALSS1 ; HBAMSS ; HAMSS .

##  12 Hard Thermal Loop (HTL) Resummation and HTL Perturbation Theory

###  12.1 HTL Resummation

As discussed in sec. 11 that the naive perturbation theory suffers from
infrared and/or collinear singularities and gauge dependence results of some
quantities. This is because certain classes of diagrams were not considered in
naive perturbation theory which are higher order in the loop expansion that
contribute to the same order in the coupling constant as the one loop diagram
BP . These diagrams can be identified through the scale separation as
discussed below.

Considering the loop momenta to be hard ($\sim 2\pi T$), the amplitude for
higher-order loop diagrams BP  can be written through power counting as

$\sim\frac{g^{2}T^{2}}{P^{2}}\times{\mbox{Tree level amplitude}}\,,$ (529)

where $P$ is the external momentum. If the external momentum is hard, $P\sim
2\pi T$, then the amplitude is suppressed by $g^{2}$ of its tree level
amplitude. If the external momentum is soft, $P\sim gT$ then the amplitude
becomes equivalent to tree level amplitude. This indicates that diagrams of
higher order in loop expansion contribute to same order in coupling as the
one-loop by distinguishing the hard ($\sim 2\pi T$ ) arising from loop momenta
and soft scale ($\sim gT$) from external momenta. Therefore, one needs to take
into account those diagrams if the external momentum is sensitive to the
soft(electric) scale. The effective theory built around hard thermal loops
resums those diagrams. This is as illustrated below:

  1. 1.

One can resum those HTL diagrams in geometrical series through the effective
propagators and vertices in one loop as done in subsections 7.2 and 8.4. They
are related by the Ward-Takahashi identity in QED and by the Slanov-Taylor
identity in QCD.

  2. 2.

At the same time medium effects: viz., the electric screening mass, the
thermal mass, the collective behaviour of quasiparticles and the Landau
damping, are taken into account due to resummations. The effective $N$-point
functions can be used in perturbation theory leading to an effective
perturbation theory know as HTL perturbation theory (HTLpt) which will be
discussed in subsec 12.2. This effective perturbation theory leads to gauge
independent results and also complete in certain order of the coupling.

  3. 3.

In scalar field theory the infrared problems are cured due to appropriate
resummations which take into account the presence of the electric screening
(Debye) mass of order $gT$. In gauge theories like QED and QCD, the IR
singularities are improved due to electric scale. But there exists also
another sort of infrared problems associated with static magnetic fields.
Static magnetic fields are not screened at leading order in HTL because the
1-loop transverse photon/gluon self-energy vanishes in all gauges in the
infrared limit. Up to order $g^{5}$, the quantities can be calculated using
HTLpt that takes into account the screening of the chromoelectric scale but
breaks down at $g^{6}$ order due to the absence of magnetic screening Linde1 ;
Linde2 .

Now we write down the HTL improved Lagrangian in non-Abelian gauge theory
(QCD) BP2 ; BP_90 ; TW ; FT  as

${\cal
L}_{\tiny{\textrm{HTL}}}=im_{\textrm{th}}^{2}\bar{\psi}\gamma^{\mu}\left\langle\frac{y_{\mu}}{y\cdot\\!D}\right\rangle_{y}\psi-\frac{1}{2}m_{D}^{2}{\rm
Tr}\left(G_{\mu\alpha}\left\langle\frac{y^{\alpha}y_{\beta}}{(y\cdot\\!D)^{2}}\right\rangle_{y}G^{\mu\beta}\right)\,,$
(530)

where $G$ is the gluon field strength, $D$ is the covariant derivative,
$y^{\mu}=(1,{\bm{\hat{y}}})$ is a light-like four-vector with
${\bm{\hat{y}}}$= three-dimensional unit vector, and angular braces represent
the average over the directions specified by ${\bm{\hat{y}}}$. The overall
trace in the second term in (530) is for group indices. The two parameters
$m_{D}$ and $m_{\textrm{th}}$ are, respectively, the Debye screening mass and
the thermal quark mass which take into account the screening effects. The
Lagrangian is non-local and gauge symmetric, which forces the presence of the
covariant derivative in the denominator of (530), which makes it also non-
linear. When expanded in powers of gauge field, (530) generates an infinite
series of non-local self-energy and vertex corrections (viz., HTL $N$-point
functions). These $N$-point functions are related by Slanov-Taylor identity.
Note that since diagrams with external ghost legs do not produce hard thermal
loops, the ordinary ghost-gluon vertex remains the same. For details we refer
to the review article in Ref. Mike .

###  12.2 HTL perturbation Theory (HTLpt)

QCD Lagrangian density in Minkowski space can be written from (507) as

${\cal L}_{\rm
QCD}={\bar{\psi}}_{ab}i\not{\partial}\psi_{ab}-g({\bar{\psi}}_{ab}\lambda_{i}\psi_{ab})\not{A}^{i}-\frac{1}{4}G_{\mu\nu}^{i}G^{\mu\nu}_{i}+{\cal
L}_{\textrm{gf}}+{\cal L}_{\textrm{gh}}+\Delta{\cal L}_{\rm QCD}\,,$ (531)

where the counterterm $\Delta{\cal L}_{\rm QCD}$ is necessary to cancel the
ultraviolet (UV) divergences in perturbative calculations. HTLpt is a
reorganization of thermal QCD perturbation theory. The HTLpt Lagrangian
density andersen1 ; andersen2 ; 3loopglue1 ; ASS ; ALSS1 ; HBAMSS ; HAMSS  can
be written as

${\cal L}_{\tiny\rm{HTLpt}}=\left.({\cal L}_{\rm{QCD}}+(1-\delta){\cal L}_{\rm
HTL})\right|_{g\rightarrow\sqrt{\delta}g}+\Delta{\cal L}_{\rm HTL}$ (532)

where $\Delta{\cal L}_{\rm HTL}$ is additional counterterm needed to cancel
the UV divergences generated in HTLpt. The HTL improved Lagrangian, ${\cal
L}_{\rm HTL}$ is given in (530). HTLpt is defined by considering $\delta$ as a
formal expansion parameter. By adding the HTL improvement term in (532) to the
QCD Lagrangian in (531), HTLpt consistently shifts the perturbative expansion
from an ideal gas of massless particles, to a gas of massive quasiparticles
which are the more apt physical degrees of freedom at high temperature and
chemical potential. It is worth here to mention that the HTLpt Lagrangian
(532) becomes the QCD Lagrangian in (531) if one puts $\delta=1$.

Physical quantities are computed in HTLpt through expansion in powers of
$\delta$, terminating at some specified order, and then putting $\delta=1$. As
mentioned before, this signifies a rearrangement of the perturbation series in
which the screening effects through $m_{D}^{2}$ and $m_{\textrm{th}}^{2}$
terms in (532) have been considered to all orders but then systematically
subtracted out at higher orders in perturbation theory by the $\delta
m_{D}^{2}$ and $\delta m_{\textrm{th}}^{2}$ terms in (532). One usually
expands to orders $\delta^{0}$, $\delta^{1}$, $\delta^{2}$, respectively, for
computing leading order (LO), next-to-leading order (NLO), and next-to-next-
leading order (NNLO) results. Note that HTLpt is gauge invariant order-by-
order in the $\delta$-expansion and, consequently, the results obtained would
be gauge independent.

If the $\delta$-expansion could be computed to all orders the results would
not depend on $m_{D}$ and $m_{\textrm{th}}$ when one puts $\delta=1$. However,
any termination of the $\delta$-expansion generates $m_{D}$ and
$m_{\textrm{th}}$ dependent results. Therefore, a prescription is needed to
determine $m_{D}$ and $m_{\textrm{th}}$ as a function of $T$, $\mu$ and
$\alpha_{s}$. There are several prescriptions and some of them had been
discussed in andersen3  at zero chemical potential. The HTL perturbation
expansion produces UV divergences. In QCD perturbation theory,
renormalizability restricts the UV divergences in such a way that they can be
eliminated by the counterterm Lagrangian $\Delta{\cal L}_{\rm QCD}$. Usually
the renormalization of HTLpt can be considered by adding a counterterm
Lagrangian $\Delta{\cal L}_{\rm HTL}$ in (532). However, there is no such
proof yet that the HTLpt is renormalizable, so the general structure of the UV
divergences remain unknown. The most optimistic scenario is that HTLpt would
be renormalizable, such that the UV divergences in the physical observables
can all be eliminated using proper counterterms.

The HTLpt has been used to study the various physical quantities relevant for
understanding the properties of QGP, viz., the thermodynamic properties
andersen1 ; ABM ; HM ; CMT ; CMT1 ; CMT2 ; HMT ; ABM1 ; andersen2 ; HMS ;
najmul2qns ; 3loopglue1 ; ASS ; ALSS ; ALSS1 ; HBAMSS ; HAMSS ; andersen3 ;
andersen4 , dilepton production rate LT ; BPS ; BPY ; GHMT ; GM3 ; JG ;
Ghiglieri1 , photon production rate KLS ; BNNR ; AGKZ ; AMY ; AMY1 ; PT ;
GHKMT , single quark and quark-antiquark potentials MTC ; MTC1 ; CMT4 ; CMRT ;
LPRT ; DYS ; DGMS ; THKP , fermion damping rate RDP ; PPS , photon damping
rate MHT ; AD , gluon damping rate BP1 ; BP3  and parton energy-loss BT ; BT1
; TG ; CMT5 ; GMT .

###  12.3 One-loop Quark Free Energy in HTLpt

In thermal field theory the partition function is defined by a functional
determinant of the inverse propagator and by which the quark part of the free
energy density in one-loop order at the leading order in the
$\delta$-expansion can be written from Fig. 32 as

$\displaystyle F^{\textrm{1-loop}}_{q}$ $\displaystyle=$ $\displaystyle-
N_{c}N_{f}\int\frac{d^{4}P}{(2\pi)^{4}}\leavevmode\nobreak\
\ln\left({\mbox{det}}\left[S^{*-1}(P)\right]\right),$ (533)

Figure 32: one-loop quark contribution to free energy density.

where $P\equiv(p_{0},\bm{\vec{p}})$ is the four momentum with
$p=|\bm{\vec{p}}|$, $N_{c}$ is number of colour and $N_{f}$ is number of
flavour. For ideal gas of quarks the
$\textrm{det}\left[S^{*-1}(P)\right]=P^{4}$ as obtained in (283) and the free
energy density reads as

$\displaystyle F_{q}^{\textrm{ideal}}$ $\displaystyle=$
$\displaystyle-2N_{c}N_{f}\int\frac{d^{4}P}{(2\pi)^{4}}\leavevmode\nobreak\
\ln\left(P^{2}\right)$ (534) $\displaystyle=$
$\displaystyle-2N_{c}N_{f}\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{\\{P\\}}\ln(P^{2})$
$\displaystyle=$
$\displaystyle-\frac{7\pi^{2}T^{4}}{180}N_{c}N_{f}\left(1+\frac{120}{7}\hat{\mu}^{2}+\frac{240}{7}\hat{\mu}^{4}\right)\,,$

where $\hat{\mu}=\mu/2\pi T$.

Now, the effective quark propagator is given in (325) as

$\displaystyle S^{\star-1}(P)$ $\displaystyle=$
$\displaystyle{P\\!\\!\\!\\!/-\Sigma(P)}$ (535) $\displaystyle=$
$\displaystyle\frac{1}{2}(\gamma_{0}+{\vec{\gamma}}\cdot\bm{\hat{p}}){\cal
D}_{+}+\frac{1}{2}(\gamma_{0}-{\vec{\gamma}}\cdot\bm{\hat{p}}){\cal D}_{-}$
$\displaystyle=$ $\displaystyle\gamma_{0}p_{0}\,{\cal
C}-{\vec{\gamma}}\cdot\bm{\vec{p}}\,{\cal D}\,,$

where

$\displaystyle{\cal C}$ $\displaystyle=\frac{1}{2p_{0}}\left({\cal
D}_{+}+{\cal D}_{-}\right)\,,$ (536a) $\displaystyle{\cal D}$
$\displaystyle=\frac{1}{2p}\left({\cal D}_{-}-{\cal D}_{+}\right)\,.$ (536b)

Using (320) one can have

$\displaystyle{\cal C}$ $\displaystyle=1+{\cal A}+\frac{\cal B}{p_{0}}=1-{\cal
A}^{\prime}\,,$ (537a) $\displaystyle{\cal D}$ $\displaystyle=1+{\cal A}\,.$
(537b)

where

${\cal A}^{\prime}=-{\cal A}-\frac{\cal B}{p_{0}}\,.$ (538)

From (457) we get structure constant ${\cal A}$ as

$\displaystyle{\cal A}(p_{0},p)$ $\displaystyle=$
$\displaystyle-\frac{m^{2}_{\textrm{th}}}{p^{2}}\int\frac{d\Omega}{4\pi}\frac{{\bm{{\vec{p}}\cdot{\hat{k}}}}}{P\cdot\hat{K}}=\frac{m^{2}_{\textrm{th}}}{p^{2}}\left[1-{\cal
T}_{P}\right]\,,$ (539)

where

${\cal
T}_{P}=\int\frac{d\Omega}{4\pi}\frac{p_{0}}{p_{0}-\bm{{\vec{p}}\cdot{\hat{k}}}}\,,$
(540)

and in presence of chemical potential $\mu$ the quark thermal mass of (458)
can be written as

$m^{2}_{\textrm{th}}=\frac{g^{2}T^{2}C_{F}}{8}\left(1+4{\hat{\mu}}^{2}\right).$
(541)

Using (539) and (466) one can write (538) as

${\cal
A}^{\prime}=\frac{m^{2}_{\textrm{th}}}{p^{2}_{0}}\int\frac{d\Omega}{4\pi}\frac{p_{0}}{p_{0}-\bm{{\vec{p}}\cdot{\hat{k}}}}=\frac{m^{2}_{\textrm{th}}}{p^{2}_{0}}{\cal
T}_{P}\ .$ (542)

Similarly, one obtains

$\displaystyle{\cal C}$
$\displaystyle=1-\frac{m^{2}_{\textrm{th}}}{p^{2}_{0}}{\cal T}_{P}\,,$ (543a)
$\displaystyle{\cal D}$
$\displaystyle=1+\frac{m^{2}_{\textrm{th}}}{p^{2}}\left[1-{\cal T}_{P}\right]\
\,.$ (543b)

Now we calculate the $\textrm{det}(S^{*-1})$:

$\displaystyle\textrm{det}({S^{*}}^{-1})$ $\displaystyle=$
$\displaystyle\textrm{det}\left(\gamma_{0}p_{0}\,{\cal
C}-{\vec{\gamma}}\cdot\bm{\vec{p}}\,{\cal D}\right)$ (548) $\displaystyle=$
$\displaystyle\textrm{det }\left\\{\left[{\begin{array}[]{cc}p_{0}{\cal
C}&0\\\ 0&-p_{0}{\cal C}\\\ \end{array}}\right]+{\cal
D}\left[{\begin{array}[]{cc}0&\bm{{\vec{\sigma}}\cdot{\vec{p}}}\\\
-\bm{{\vec{\sigma}}\cdot{\vec{p}}}&0\\\ \end{array}}\right]\right\\}$
$\displaystyle=$ $\displaystyle\textrm{det
}\left[{\begin{array}[]{cc}p_{0}{\cal C}&{\cal
D}\,(\bm{{\vec{\sigma}}\cdot{\vec{p}}})\\\ -{\cal
D}\,(\bm{{\vec{\sigma}}\cdot{\vec{p}}})&-p_{0}{\cal C}\\\
\end{array}}\right]\,.$ (551)

We know

$\displaystyle\bm{{\vec{\sigma}}\cdot{\vec{p}}}$ $\displaystyle=$
$\displaystyle\sigma_{x}p_{x}+\sigma_{y}p_{y}+\sigma_{z}p_{z}$ (558)
$\displaystyle=$ $\displaystyle\left[{\begin{array}[]{cc}0&p_{x}\\\ p_{x}&0\\\
\end{array}}\right]+\left[{\begin{array}[]{cc}0&ip_{y}\\\ -ip_{y}&0\\\
\end{array}}\right]+\left[{\begin{array}[]{cc}p_{z}&0\\\ 0&-p_{z}\\\
\end{array}}\right]$ $\displaystyle=$
$\displaystyle\left[{\begin{array}[]{cc}p_{z}&p_{x}+ip_{y}\\\
p_{x}-ip_{y}&-p_{z}\\\ \end{array}}\right]\,.$ (561)

Using (561) in (551) one can write

$\displaystyle\textrm{det}(S^{*-1})$ $\displaystyle=$
$\displaystyle\textrm{det }\left[{\begin{array}[]{cccc}p_{0}\,{\cal
C}&0&-{\cal D}p_{z}&-{\cal D}(p_{x}+ip_{y})\\\ 0&p_{0}\,{\cal C}&-{\cal
D}(p_{x}-ip_{y})&{\cal D}p_{z}\\\ {\cal D}p_{z}&{\cal
D}(p_{x}+ip_{y})&-p_{0}\,{\cal C}&0\\\ {\cal D}(p_{x}-ip_{y})&-{\cal
D}p_{z}&0&-p_{0}\,{\cal C}\\\ \end{array}}\right]$ (566) $\displaystyle=$
$\displaystyle\left\\{p_{0}{\cal C}\left|{\begin{array}[]{ccc}p_{0}\,{\cal
C}&-{\cal D}(p_{x}-ip_{y})&{\cal D}p_{z}\\\ {\cal
D}(p_{x}+ip_{y})&-p_{0}\,{\cal C}&0\\\ -{\cal D}p_{z}&0&-p_{0}\,{\cal C}\\\
\end{array}}\right|-{\cal D}p_{z}\left|{\begin{array}[]{cccc}0&p_{0}\,{\cal
C}&{\cal D}p_{z}\\\ {\cal D}p_{z}&{\cal D}(p_{x}+ip_{y})&0\\\ {\cal
D}(p_{x}-ip_{y})&-{\cal D}p_{z}&-p_{0}\,{\cal C}\\\
\end{array}}\right|\right.$ $\displaystyle\left.+{\cal
D}(p_{x}+ip_{y})\left|{\begin{array}[]{cccc}0&p_{0}\,{\cal C}&-{\cal
D}(p_{x}-ip_{y})\\\ {\cal D}p_{z}&{\cal D}(p_{x}+ip_{y})&-p_{0}\,{\cal C}\\\
{\cal D}(p_{x}-ip_{y})&-{\cal D}p_{z}&0\\\ \end{array}}\right|\right\\}$
$\displaystyle=$ $\displaystyle p_{0}^{2}{\cal C}^{4}-p_{0}^{2}p^{2}{\cal
C}^{2}{\cal D}^{2}-p_{0}^{2}p_{z}^{2}{\cal C}^{2}{\cal
D}^{2}+p_{z}^{2}p^{2}{\cal D}^{4}-p_{0}^{2}(p_{x}^{2}+p_{y}^{2}){\cal
C}^{2}{\cal D}^{2}+(p_{x}^{2}+p_{y}^{2})p^{2}{\cal D}^{2}$ $\displaystyle=$
$\displaystyle\left(p_{0}^{2}{\cal C}^{2}-p^{2}{\cal D}^{2}\right)^{2}\,.$
(578)

Using (543a) and (543b), we get

$\displaystyle\textrm{det}(S^{*-1})$ $\displaystyle=$
$\displaystyle\left[\left\\{p_{0}-\frac{m^{2}_{\textrm{th}}}{p_{0}}{\cal
T}_{P}\right\\}^{2}-\left\\{p+\frac{m^{2}_{\textrm{th}}}{p}\left(1-{\cal
T}_{P}\right)\right\\}^{2}\right]^{2}$ (579) $\displaystyle=$
$\displaystyle\left[A_{0}^{2}-A_{S}^{2}\right]^{2}\,,$

where

$\displaystyle A_{0}$
$\displaystyle=p_{0}-\frac{m^{2}_{\textrm{th}}}{p_{0}}{\cal T}_{P}\,,$ (580a)
$\displaystyle A_{S}$
$\displaystyle=p+\frac{m^{2}_{\textrm{th}}}{p}\left(1-{\cal T}_{P}\right)\,.$
(580b)

Combining (533) and (579), the one-loop quark free energy density in HTL
approximation can be written as

$\displaystyle F^{\textrm{1-loop}}_{q}$ $\displaystyle=$ $\displaystyle-
N_{c}N_{f}\int\frac{d^{4}P}{(2\pi)^{4}}\leavevmode\nobreak\
\ln\left(A_{0}^{2}-A_{S}^{2}\right)^{2}=-2N_{c}N_{f}\int\frac{d^{4}P}{(2\pi)^{4}}\leavevmode\nobreak\
\ln\left(A_{0}^{2}-A_{S}^{2}\right)$ (581) $\displaystyle=$
$\displaystyle-2N_{c}N_{f}\int\frac{d^{4}P}{(2\pi)^{4}}\leavevmode\nobreak\
\ln\left(P^{2}\right)-2N_{c}N_{f}\int\frac{d^{4}P}{(2\pi)^{4}}\leavevmode\nobreak\
\ln\left(\frac{A_{0}^{2}-A_{S}^{2}}{P^{2}}\right)\,.$

Now the argument of the logarithm in the second term in (581) can be
simplified using (580a) and (580b) as

$\displaystyle\frac{(A^{2}_{0}-A^{2}_{S})}{P^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{P^{2}}\left[p_{0}^{2}-2m^{2}_{\textrm{th}}{\cal
T}_{P}+\frac{m^{4}_{\textrm{th}}}{p_{0}^{2}}{\cal
T}_{P}^{2}-p^{2}-2m^{2}_{\textrm{th}}(1-{\cal
T}_{P})-\frac{m^{4}_{\textrm{th}}}{p^{2}}(1-{\cal T}_{P})^{2}\right]$ (582)
$\displaystyle=$ $\displaystyle\frac{1}{P^{2}}\left[P^{2}-2p_{0}^{2}{\cal
A}^{\prime}+p_{0}^{2}{\cal A^{\prime}}^{2}-2p^{2}{\cal A}-p^{2}{\cal
A}^{2}\right]$ $\displaystyle=$ $\displaystyle
1+\left(\frac{\mathcal{A}^{\prime}(\mathcal{A}^{\prime}-2)p_{0}^{2}-\mathcal{A}(\mathcal{A}+2)p^{2}}{P^{2}}\right)\,,$

where in the second line we have used (539) and (542). In the high temperature
approximation, the logarithmic term in (581) can be expanded in a series of
coupling constants $g$ and then keeping terms up to $\mathcal{O}(g^{4})$ one
obtains

$\displaystyle\ln\left[\frac{(A^{2}_{0}-A^{2}_{S})}{P^{2}}\right]$
$\displaystyle=$ $\displaystyle\frac{\mathcal{A}^{\prime
2}p_{0}^{2}-\mathcal{A}^{2}p^{2}-2\mathcal{A}^{\prime}p_{0}^{2}-2\mathcal{A}p^{2}}{P^{2}}-2\frac{\left(\mathcal{A}^{\prime}p_{0}^{2}+\mathcal{A}p^{2}\right)^{2}}{P^{4}}+\mathcal{O}(g^{6})\,$
(583) $\displaystyle=$ $\displaystyle\frac{\mathcal{A}^{\prime
2}p_{0}^{2}-\mathcal{A}^{2}p^{2}}{P^{2}}-2\frac{\mathcal{A}^{\prime}p_{0}^{2}+\mathcal{A}p^{2}}{P^{2}}-2\frac{(\mathcal{A}^{\prime}p_{0}^{2}+\mathcal{A}p^{2})^{2}}{P^{4}}+\mathcal{O}(g^{6})\,.$

Using (539) and (542), we can obtain

$\displaystyle(\mathcal{A}^{\prime}p_{0}^{2}+\mathcal{A}p^{2})$
$\displaystyle=m_{\textrm{th}}^{2},$ (584a) $\displaystyle(\mathcal{A}^{\prime
2}p_{0}^{2}-\mathcal{A}^{2}p^{2})$
$\displaystyle=m_{\textrm{th}}^{4}\left[\frac{{\cal
T}_{P}^{2}}{p_{0}^{2}}-\frac{\left(1-{\cal
T}_{P}\right)^{2}}{p^{2}}\right]\,.$ (584b)

Now (583) becomes

$\displaystyle\ln\left[\frac{(A^{2}_{0}-A^{2}_{S})}{P^{2}}\right]$
$\displaystyle=$
$\displaystyle-\frac{2m^{2}_{\textrm{th}}}{P^{2}}+m^{4}_{\textrm{th}}\left[\frac{{\cal
T}_{P}^{2}}{p_{0}^{2}P^{2}}-\frac{2}{P^{4}}-\frac{1}{p^{2}P^{2}}+\frac{{2\cal
T}_{P}}{p^{2}P^{2}}-\frac{{\cal T}_{P}^{2}}{p^{2}P^{2}}\right]\,.$ (585)

Using (585) in (581), the one-loop free energy density up to
$\mathcal{O}(g^{4})$ can be written as

$\displaystyle F^{\textrm{1-loop}}_{q}$ $\displaystyle=$ $\displaystyle
N_{c}N_{f}\Bigg{[}-2\int\frac{d^{4}P}{(2\pi)^{4}}\leavevmode\nobreak\
\ln\left(P^{2}\right)+4m_{\textrm{th}}^{2}\int\frac{d^{4}P}{(2\pi)^{4}}\frac{1}{P^{2}}$
(586) $\displaystyle-$ $\displaystyle
m_{\textrm{th}}^{4}\int\frac{d^{4}P}{(2\pi)^{4}}\left(\frac{2{\cal
T}_{P}^{2}}{p_{0}^{2}P^{2}}-\frac{4}{P^{4}}-\frac{2}{p^{2}P^{2}}-\frac{2{\cal
T}_{P}^{2}}{p^{2}P^{2}}+\frac{4{\cal T}_{P}}{p^{2}P^{2}}\right)\Bigg{]}$
$\displaystyle=$ $\displaystyle
N_{c}N_{f}\Bigg{[}-2\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{\\{P\\}}\ln\left(P^{2}\right)+4m_{\textrm{th}}^{2}\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{\\{P\\}}\frac{1}{P^{2}}$
$\displaystyle-\
m_{\textrm{th}}^{4}\left(\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{\\{P\\}}\frac{2{\cal
T}_{P}^{2}}{p_{0}^{2}P^{2}}-\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{\\{P\\}}\frac{4}{P^{4}}-\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{\\{P\\}}\frac{2}{p^{2}P^{2}}-\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{\\{P\\}}\frac{2{\cal
T}_{P}^{2}}{p^{2}P^{2}}+\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{\\{P\\}}\frac{4{\cal
T}_{P}}{p^{2}P^{2}}\right)\Biggr{]}\,.$

Using the sum-integrals of (687a) to (687g) given in appendix A.1, the one-
loop quark free energy density becomes

$\displaystyle F^{\textrm{1-loop}}_{q}$ $\displaystyle=$ $\displaystyle
N_{c}N_{f}\Bigg{[}-\frac{7\pi^{2}T^{4}}{180}\left(1+\frac{120}{7}\hat{\mu}^{2}+\frac{240}{7}\hat{\mu}^{4}\right)$
(587)
$\displaystyle+\frac{m_{\textrm{th}}^{2}T^{2}}{6}\left(\frac{\Lambda}{4\pi
T}\right)^{2\epsilon}\left(1+12\hat{\mu}^{2}+2\epsilon[1+12\hat{\mu}^{2}+12\aleph(1,z)]\right)$
$\displaystyle+4m_{\textrm{th}}^{4}\left(1+\frac{1-2\Delta_{3}+\Delta_{4}^{\prime\prime}-\Delta_{3}^{\prime\prime}}{2-d}\right)\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{\\{P\\}}\frac{1}{P^{4}}\Bigg{]}\,.$

Substituting (688a) to (688c) and (687c), we get leading order quark free
energy density HMS  in thermal medium as

$\displaystyle F^{\textrm{1-loop}}_{q}$ $\displaystyle=$ $\displaystyle
N_{c}N_{f}\Bigg{[}-\frac{7\pi^{2}T^{4}}{180}\left(1+\frac{120}{7}\hat{\mu}^{2}+\frac{240}{7}\hat{\mu}^{4}\right)$
(588)
$\displaystyle+\frac{m_{\textrm{th}}^{2}T^{2}}{6}\left(\frac{\Lambda}{4\pi
T}\right)^{2\epsilon}\left(1+12\hat{\mu}^{2}+2\epsilon[1+12\hat{\mu}^{2}+12\aleph(1,z)]\right)$
$\displaystyle+4m_{\textrm{th}}^{4}\left[\left(\frac{\pi^{2}}{3}-2\right)\epsilon\right]\frac{1}{\left(4\pi\right)^{2}}\left(\frac{\Lambda}{4\pi
T}\right)^{2\epsilon}\Bigg{[}\frac{1}{\epsilon}-\aleph(z)\Bigg{]}$
$\displaystyle{=\atop\epsilon\rightarrow 0}$ $\displaystyle
N_{c}N_{f}\Bigg{[}-\frac{7\pi^{2}T^{4}}{180}\left(1+\frac{120}{7}\hat{\mu}^{2}+\frac{240}{7}\hat{\mu}^{4}\right)+\frac{m_{\textrm{th}}^{2}T^{2}}{6}\left(1+12\hat{\mu}^{2}\right)+\frac{m_{\textrm{th}}^{4}}{12\pi^{2}}\left(\pi^{2}-6\right)\Bigg{]}$
$\displaystyle=$ $\displaystyle
N_{c}N_{f}\Bigg{[}-\frac{7\pi^{2}T^{4}}{180}\left(1+\frac{120\hat{\mu}^{2}}{7}+\frac{240\hat{\mu}^{4}}{7}\right)+\frac{g^{2}C_{F}T^{4}}{48}\left(1+4\hat{\mu}^{2}\right)\left(1+12\hat{\mu}^{2}\right)$
$\displaystyle+\,\frac{g^{4}C_{F}^{2}T^{4}}{768\pi^{2}}\left(1+4\hat{\mu}^{2}\right)^{2}\left(\pi^{2}-6\right)\Bigg{]}.$

###  12.4 One-loop Gluon Free Energy in HTLpt

Similar to photon partition function in QED in subsec 9.5, the QCD partition
function for a gluon can be written in Euclidean space-time as

${\cal Z}_{g}={\cal Z}{\cal Z}^{\textrm{gh}},\leavevmode\nobreak\
\leavevmode\nobreak\ {\cal
Z}=N_{\xi}\prod_{n,\bm{p}}\sqrt{\frac{\pi^{D}}{\textsf{det}D_{\mu\nu,E}^{-1}}},\leavevmode\nobreak\
\leavevmode\nobreak\ {\cal Z}^{\textrm{gh}}=\prod_{n,\bm{p}}P_{E}^{2},$ (589)

where the product over $n$ is for the discrete bosonic Matsubara frequencies
($\omega_{n}=2\pi n\beta;\,\,n=0,1,2,\cdots$) due to Euclidean time whereas
that of $\bm{p}$ is for the spatial momentum. $D_{\mu\nu,E}^{-1}$ is the
inverse gauge boson propagator in Euclidean space,
$P_{E}^{2}=\omega_{n}^{2}+p^{2}$ is the square of the Euclidean four-momentum
and $D$ is the space-time dimension of the theory. As discussed in subsec. 9.5
the normalization $N_{\xi}=1/(\pi^{D}\xi)^{1/2}$ arises due to the
introduction of the Gaussian integral at every location of position while
averaging over the gauge condition function with a width $\xi$, the gauge
fixing parameter. Gluon free energy can now be written from Fig. 33 as

$F_{g}=-(N_{c}^{2}-1)\frac{T}{V}\ln{\cal
Z}_{g}=(N_{c}^{2}-1)\left[\frac{1}{2}\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P_{E}}\leavevmode\nobreak\
\ln\Big{[}\textsf{det}\left(D_{\mu\nu,E}^{-1}(P_{E})\right)\Big{]}-\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P_{E}}\leavevmode\nobreak\
\ln P_{E}^{2}+\frac{1}{2}\ln\xi\right]\,.$ (590)

where $\sum\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P_{E}}$ is a bosonic sum-
integral and the gauge dependence will explicitly cancel as can be seen later.

Figure 33: One-loop gluon and ghost contribution to free energy density.

For an ideal case
$\textsf{det}\left(D_{\mu\nu,E}^{-1}(P)\right)=(P_{E}^{2})^{4}/\xi$ as
obtained in (435) and hence the free energy for $(N_{c}^{2}-1)$ massless spin
one gluons yields as

$\displaystyle
F_{g}^{\textrm{ideal}}=(N_{c}^{2}-1)\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P_{E}}\leavevmode\nobreak\
\ln P_{E}^{2}=-(N_{c}^{2}-1)\frac{\pi^{2}T^{4}}{45},$ (591)

where $P_{E}$ is the four-momentum in Euclidean space and can be written as
$P^{2}=p_{0}^{2}+p^{2}.$

In presence of thermal background medium one can have

$\displaystyle\textsf{det}\left(D_{\mu\nu,E}^{-1}(P_{E})\right)=\frac{P_{E}^{2}}{\xi}\left(-P_{E}^{2}+\Pi_{T}\right)^{2}\left(-P_{E}^{2}+\Pi_{L}\right),$
(592)

with four eigenvalues; respectively $P_{E}^{2}$, $(-P_{E}^{2}+\Pi_{L})$ and
two fold degenerate $(-P_{E}^{2}+\Pi_{T})$. Here $\Pi_{T}$ and $\Pi_{L}$ are
the transverse and longitudinal part of the gluon self-energy in thermal
medium. Using (592) in (590), one gets 1-loop gluon free energy at the leading
order in $\delta$-expansion as

$\displaystyle F_{g}$ $\displaystyle=$
$\displaystyle(N_{c}^{2}-1)\left[\frac{1}{2}\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P_{E}}\,\,\ln\left(1-\frac{\Pi_{L}}{P_{E}^{2}}\right)+\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P_{E}}\,\,\ln\left(P_{E}^{2}-\Pi_{T}\right)\right],$
(593)

where a term $i\pi$ has been neglected as that would make the free energy
complex. Now transforming into Minkowski space $(P_{E}^{2}\rightarrow-P^{2})$
we have

$\displaystyle F_{g}$ $\displaystyle=$
$\displaystyle(N_{c}^{2}-1)\left[\frac{1}{2}\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\,\,\ln\left(1+\frac{\Pi_{L}}{P^{2}}\right)+\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\,\,\ln\left(P^{2}+\Pi_{T}\right)\right]=(N_{c}^{2}-1)\left[F_{g}^{L}+F_{g}^{T}\right]\,,$
(594)

where $F_{g}^{L}$ and $F_{g}^{T}$ are, respectively, the longitudinal and
transverse part of the gluon free energy and are given as

$\displaystyle F_{g}^{L}$
$\displaystyle=\frac{1}{2}\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\,\,\ln\left(1+\frac{\Pi_{L}}{P^{2}}\right)\,,$
(595a) $\displaystyle F_{g}^{T}$
$\displaystyle=\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\,\,\ln\left(P^{2}+\Pi_{T}\right)\,.$
(595b)

The longitudinal and transverse part of the gluon self-energy can,
respectively, be written from (493) and (495) as

$\displaystyle\Pi_{L}$
$\displaystyle=\frac{m_{D}^{2}P^{2}}{p^{2}}\left(1-{\cal T}_{P}\right)\,,$
(596a) $\displaystyle\Pi_{T}$
$\displaystyle=-\frac{m_{D}^{2}}{2p^{2}}\left(p_{0}^{2}-P^{2}{\cal
T}_{P}\right)\ \,,$ (596b)

where ${\cal T}_{P}$ is given in (540) and the QCD Debye mass in presence of
quark chemical potential is given as

$m_{D}^{2}=\frac{g^{2}T^{2}}{3}\left[C_{A}+\frac{N_{f}}{2}(1+12{\hat{\mu}}^{2})\right]\,.$
(597)

Now expanding the logarithm in high temperature approximation as

$\displaystyle F_{g}^{L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\,\,\ln\left(1+\frac{\Pi_{L}}{P^{2}}\right)=\frac{1}{2}\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\left(\frac{\Pi_{L}}{P^{2}}-\frac{\Pi^{2}_{L}}{2P^{4}}\right)\,$
(598) $\displaystyle=$
$\displaystyle\frac{m_{D}^{2}}{2}\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\frac{1}{p^{2}}-\frac{m_{D}^{2}}{2}\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\frac{{\cal
T}_{P}}{p^{2}}-m_{D}^{4}\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\left[\frac{1}{4p^{4}}-\frac{{\cal
T}_{P}}{2p^{4}}+\frac{{\cal T}_{P}^{2}}{4p^{4}}\right]$

and

$\displaystyle F_{g}^{T}$ $\displaystyle=$
$\displaystyle\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\,\,\ln\left(P^{2}+\Pi_{T}\right)=\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\,\,\ln(P^{2})+\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\,\,\ln\left(1+\frac{\Pi_{T}}{P^{2}}\right)=\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\,\,\ln(P^{2})+\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\,\,\left(\frac{\Pi_{T}}{P^{2}}-\frac{\Pi^{2}_{T}}{2P^{4}}\right)\,$
(599) $\displaystyle=$
$\displaystyle\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\,\,\ln(P^{2})-m_{D}^{2}\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\,\,\frac{p_{0}^{2}}{2p^{2}P^{2}}+m_{D}^{2}\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\,\,\frac{{\cal
T}_{P}}{2p^{2}}-m_{D}^{4}\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\left[\frac{p_{0}^{4}}{8p^{4}P^{4}}-\frac{p_{0}^{2}{\cal
T}_{P}}{4p^{4}P^{2}}+\frac{{\cal T}_{P}^{2}}{8p^{4}}\right]\,,$

where we have kept terms up to ${\mathcal{O}}[m_{D}^{4}]$. The hard
contribution to gluon free energy can be obtained combining (598) and (599)
with (594) as

$\displaystyle F_{g}^{\textrm{hard}}$ $\displaystyle=$
$\displaystyle(N_{c}^{2}-1)\left(F_{g}^{L}+F_{g}^{T}\right)=(N_{c}^{2}-1)\left[\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\,\,\ln(P^{2})-\frac{m_{D}^{2}}{2}\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\,\frac{1}{P^{2}}\right.$
(600)
$\displaystyle\left.-\frac{m_{D}^{4}}{8}\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\left[\frac{1}{P^{4}}+\frac{2}{p^{2}P^{2}}-\frac{2{\cal
T}_{P}}{p^{2}P^{2}}-\frac{6{\cal T}_{P}}{p^{4}}+\frac{3{\cal
T}_{P}^{2}}{p^{4}}\right]\right]\,.$

The bosonic sum integrals have been obtained in (691a) to (691f) in appendix
A.2. Using them in (600) one gets the hard contribution to gluon free energy
as

$\displaystyle F_{g}^{\textrm{hard}}$ $\displaystyle=$
$\displaystyle(N_{c}^{2}-1)\\!\\!\left[-\frac{\pi^{2}T^{4}}{45}+\frac{m_{D}^{2}T^{2}}{24}\\!\\!\left(\frac{\Lambda}{4\pi
T}\right)^{2\epsilon}\\!\\!\\!\\!\\!\left(1+\mathcal{O}[\epsilon]\right)\right.$
(601)
$\displaystyle\left.-\frac{m_{D}^{4}}{128\pi^{2}}\\!\\!\left(\frac{\Lambda}{4\pi
T}\right)^{2\epsilon}\\!\\!\left(\frac{1}{\epsilon}+2\gamma_{E}+\frac{2\pi^{2}}{3}-7\right)\right].$

For the soft contribution the only important term in the integral is
$p_{0}=0$. Putting $p_{0}=0$, we get from (596a) and (596b), respectively, as

$\displaystyle\Pi_{L}$ $\displaystyle=-m_{D}^{2}\,,$ (602a)
$\displaystyle\Pi_{T}$ $\displaystyle=0\,,$ (602b)

where the longitudinal mode provides the electric screening through the Debye
mass $m_{D}$ but the transverse mode does not contribute in the soft scale
which provides no magnetic screening in HTL.

The soft contribution from longitudinal part can be written from (594) as

$\displaystyle F_{g}^{\textrm{soft}}$ $\displaystyle=$
$\displaystyle\frac{(N_{c}^{2}-1)}{2}\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{P}\,\,\ln(p^{2}+m^{2}_{D})$
(603) $\displaystyle=$
$\displaystyle-(N_{c}^{2}-1)\frac{m_{D}^{3}T}{12\pi}\left(\frac{\Lambda}{2m_{D}}\right)^{2\epsilon}\left[1+\frac{8}{3}\epsilon\right]\,.$

Now total gluon free energy in 1-loop can be written as

$F_{g}^{\textrm{1-loop}}=F_{g}^{\textrm{hard}}+F_{g}^{\textrm{soft}}+\Delta_{0}{\cal
E}_{0}\,,$ (604)

where the HTL leading order vacuum counter term ASS  is given as

$\Delta_{0}{\cal E}_{0}=\frac{(N_{c}^{2}-1)m_{D}^{4}}{128\pi^{2}\epsilon}\,.$
(605)

Substituting (601), (603) and (605) in (604), one gets leading order gluon
free energy ASS  as

$\displaystyle F_{g}^{\textrm{1-loop}}$ $\displaystyle=$
$\displaystyle(N_{c}^{2}-1)\left[-\frac{\pi^{2}T^{4}}{45}+\frac{m_{D}^{2}T^{2}}{24}\left(\frac{\Lambda}{4\pi
T}\right)^{2\epsilon}\left(1+\mathcal{O}[\epsilon]\right)-\frac{m_{D}^{3}T}{12\pi}\left(\frac{\Lambda}{2m_{D}}\right)^{2\epsilon}\left(1+\mathcal{O}[\epsilon]\right)\right.$
(606)
$\displaystyle\left.-\frac{m_{D}^{4}}{128\pi^{2}}\left(\frac{\Lambda}{4\pi
T}\right)^{2\epsilon}\left(\frac{1}{\epsilon}+2\gamma_{E}+\frac{2\pi^{2}}{3}-7\right)\right]+\frac{(N_{c}^{2}-1)m_{D}^{4}}{128\pi^{2}\epsilon}$
$\displaystyle{=\atop\epsilon\rightarrow 0}$
$\displaystyle(N_{c}^{2}-1)\left[-\frac{\pi^{2}T^{4}}{45}+\frac{m_{D}^{2}T^{2}}{24}-\frac{m_{D}^{3}T}{12\pi}-\frac{m_{D}^{4}}{128\pi^{2}}\left(2\ln\left(\frac{\Lambda}{4\pi
T}\right)+2\gamma_{E}+\frac{2\pi^{2}}{3}-7\right)\right.$
$\displaystyle\left.-\frac{m_{D}^{4}}{128\pi^{2}\epsilon}\right]+\frac{(N_{c}^{2}-1)m_{D}^{4}}{128\pi^{2}\epsilon}$
$\displaystyle=$
$\displaystyle-(N_{c}^{2}-1)\frac{\pi^{2}T^{4}}{45}\left[1-\frac{15}{2}{\hat{m}}_{D}^{2}+30{\hat{m}}_{D}^{3}+\frac{45}{8}{\hat{m}}_{D}^{4}\left(2\ln\left(\frac{\hat{\Lambda}}{2}\right)+2\gamma_{E}+\frac{2\pi^{2}}{3}-7\right)\right]$
$\displaystyle=$ $\displaystyle
F_{g}^{\textrm{ideal}}\left[1-\frac{15}{2}{\hat{m}}_{D}^{2}+30{\hat{m}}_{D}^{3}+\frac{45}{8}{\hat{m}}_{D}^{4}\left(2\ln\left(\frac{\hat{\Lambda}}{2}\right)+2\gamma_{E}+\frac{2\pi^{2}}{3}-7\right)\right]\,,\,$

where

$\displaystyle{\hat{m}}_{D}$ $\displaystyle=\frac{m_{D}}{2\pi T}\,,$ (607a)
$\displaystyle{\hat{\Lambda}}$ $\displaystyle=\frac{\Lambda}{2\pi T}\,.$
(607b)

###  12.5 Leading Order (LO) Thermodynamics of QGP in HTLpt

The leading order free energy density of quarks and gluons above the
deconfinement temperature is defined as

$F^{\textrm{LO}}=F_{q}^{\textrm{1-loop}}+F_{g}^{\textrm{1-loop}}\,,$ (608)

where one-loop quark and gluon free energy density are, respectively, given in
(588) and (606). Using them the free energy at leading order in the
$\delta$-expansion becomes

$\displaystyle F^{\textrm{LO}}$ $\displaystyle=$ $\displaystyle-
d_{A}\frac{\pi^{2}T^{4}}{45}\left[1+\frac{7}{4}\frac{d_{F}}{d_{A}}\left(1+\frac{120}{7}\hat{\mu}^{2}+\frac{240}{7}\hat{\mu}^{4}\right)-30\frac{d_{F}}{d_{A}}(1+12{\hat{\mu}}^{2}){\hat{m}}^{2}_{\textrm{th}}\right.$
$\displaystyle\left.-\frac{15}{2}{\hat{m}}_{D}^{2}+30{\hat{m}}_{D}^{3}-60\frac{d_{F}}{d_{A}}(\pi^{2}-6){\hat{m}}^{4}_{\textrm{th}}+\frac{45}{8}{\hat{m}}_{D}^{4}\left(2\ln\left(\frac{\hat{\Lambda}}{2}\right)+2\gamma_{E}+\frac{2\pi^{2}}{3}-7\right)\right],$
(609)

where $d_{F}=N_{c}N_{f}$, $d_{A}=N_{c}^{2}-1$ and
${\hat{m}}^{2}_{\textrm{th}}={m}^{2}_{\textrm{th}}/2\pi T$. The leading order
pressure is given by

${\cal P}^{\textrm{LO}}=-F^{\textrm{LO}}\,.$ (610)

###  12.6 Next-to-leading Order (NLO) Thermodynamics of QGP in HTLpt

The NLO free energy density from two-loop HTLpt has been obtained complete
analytically in Ref. HMS ; najmul2qns  as

$\displaystyle F^{\rm NLO}$ $\displaystyle=$ $\displaystyle F^{\rm 2-loop}=$
(611) $\displaystyle-$ $\displaystyle d_{A}{\pi^{2}T^{4}\over
45}\Bigg{\\{}1+{7\over 4}{d_{F}\over
d_{A}}\left(1+\frac{120}{7}\hat{\mu}^{2}+\frac{240}{7}\hat{\mu}^{4}\right)-15\hat{m}_{D}^{3}-{45\over
4}\left(\log\hat{\Lambda\over 2}-{7\over 2}+\gamma_{E}+{\pi^{2}\over
3}\right)\hat{m}_{D}^{4}$ $\displaystyle+$ $\displaystyle 60{d_{F}\over
d_{A}}\left(\pi^{2}-6\right)\hat{m}^{4}_{\textrm{th}}+{\alpha_{s}\over\pi}\Bigg{[}-{5\over
4}\left(c_{A}+{5\over 2}s_{F}\left(1+\frac{72}{5}\
\hat{\mu}^{2}+\frac{144}{5}\ \hat{\mu}^{4}\right)\right)$ $\displaystyle+$
$\displaystyle
15\left(c_{A}+s_{F}(1+12\hat{\mu}^{2})\right)\hat{m}_{D}-{55\over
4}\left\\{c_{A}\left(\log{\hat{\Lambda}\over 2}-{36\over
11}\log\hat{m}_{D}-2.001\right)\right.$ $\displaystyle+$
$\displaystyle\left.{4\over 11}s_{F}\left[\left(\log{\hat{\Lambda}\over
2}-2.337\right)+(24-18\zeta(3))\left(\log{\hat{\Lambda}\over
2}-15.662\right)\hat{\mu}^{2}+120\left(\zeta(5)-\zeta(3)\right)\right.\right.$
$\displaystyle\times$ $\displaystyle\left.\left.\left(\log{\hat{\Lambda}\over
2}-1.0811\right)\hat{\mu}^{4}+{\cal
O}\left(\hat{\mu}^{6}\right)\right]\\!\\!\right\\}\hat{m}_{D}^{2}-45\,s_{F}\left\\{\log{\hat{\Lambda}\over
2}+2.198-44.953\hat{\mu}^{2}\right.$ $\displaystyle-$
$\displaystyle\left.\left(288\ln{\frac{\hat{\Lambda}}{2}}+19.836\right)\hat{\mu}^{4}+{\cal
O}\left(\hat{\mu}^{6}\right)\right\\}\hat{m}^{2}_{\textrm{th}}+{165\over
2}\left\\{c_{A}\left(\log{\hat{\Lambda}\over 2}+{5\over
22}+\gamma_{E}\right)\right.$ $\displaystyle-$ $\displaystyle\left.{4\over
11}s_{F}\left(\log{\hat{\Lambda}\over 2}-{1\over 2}+\gamma_{E}+2\ln
2-7\zeta(3)\hat{\mu}^{2}+31\zeta(5)\hat{\mu}^{4}+{\cal
O}\left(\hat{\mu}^{6}\right)\right)\right\\}\hat{m}_{D}^{3}$ $\displaystyle+$
$\displaystyle
15s_{F}\left(2\frac{\zeta^{\prime}(-1)}{\zeta(-1)}+2\ln\hat{m}_{D}\right)\left[(24-18\zeta(3))\hat{\mu}^{2}+120(\zeta(5)-\zeta(3))\hat{\mu}^{4}+{\cal
O}\left(\hat{\mu}^{6}\right)\right]\hat{m}_{D}^{3}$ $\displaystyle+$
$\displaystyle
180\,s_{F}\hat{m}_{D}\hat{m}^{2}_{\textrm{th}}\Bigg{]}\Bigg{\\}}\;,$

where $c_{A}=N_{c}$ and $s_{F}=N_{f}/2$. The NLO pressure is given by

${\cal P}^{\textrm{NLO}}=-F^{\textrm{NLO}}\,.$ (612)

###  12.7 Next-to-next-leading Order (NNLO) Thermodynamics of QGP in HTLpt

The NNLO free energy density from three-loop HTLpt has been obtained complete
analytically in Ref. HBAMSS ; HAMSS  as

$F^{\rm NNLO}=F_{q}^{\rm 3-loop}+F_{g}^{\rm 3-loop},$ (613)

where the 3-loop quark contribution is given. HBAMSS ; HAMSS  as

$\displaystyle F_{q}^{\rm 3-loop}$ $\displaystyle=$
$\displaystyle-\frac{d_{A}\pi^{2}T^{4}}{45}\Bigg{[}\\!\\!\Bigg{[}\frac{7}{4}\frac{d_{F}}{d_{A}}\left(1+\frac{120}{7}\hat{\mu}^{2}+\frac{240}{7}\hat{\mu}^{4}\right)-\frac{s_{F}\alpha_{s}}{\pi}\bigg{[}\frac{5}{8}\left(1+12\hat{\mu}^{2}\right)\left(5+12\hat{\mu}^{2}\right)$
(614)
$\displaystyle-\frac{15}{2}\left(1+12\hat{\mu}^{2}\right)\hat{m}_{D}-\frac{15}{2}\bigg{(}2\ln{\frac{\hat{\Lambda}}{2}-1-\aleph(z)}\Big{)}\hat{m}_{D}^{3}+90\hat{m}_{\textrm{th}}^{2}\hat{m}_{D}\bigg{]}$
$\displaystyle+$ $\displaystyle
s_{2F}\left(\frac{\alpha_{s}}{\pi}\right)^{2}\bigg{[}\frac{15}{64}\bigg{\\{}35-32\left(1-12\hat{\mu}^{2}\right)\frac{\zeta^{\prime}(-1)}{\zeta(-1)}+472\hat{\mu}^{2}+1328\hat{\mu}^{4}$
$\displaystyle+64\Big{(}-36i\hat{\mu}\aleph(2,z)+6(1+8\hat{\mu}^{2})\aleph(1,z)+3i\hat{\mu}(1+4\hat{\mu}^{2})\aleph(0,z)\Big{)}\bigg{\\}}$
$\displaystyle-$
$\displaystyle\frac{45}{2}\hat{m}_{D}\left(1+12\hat{\mu}^{2}\right)\bigg{]}+\left(\frac{s_{F}\alpha_{s}}{\pi}\right)^{2}\left[\frac{5}{4\hat{m}_{D}}\left(1+12\hat{\mu}^{2}\right)^{2}+30\left(1+12\hat{\mu}^{2}\right)\frac{\hat{m}_{\textrm{th}}^{2}}{\hat{m}_{D}}\right.$
$\displaystyle+$
$\displaystyle\left.\frac{25}{12}\Bigg{\\{}\left(1+\frac{72}{5}\hat{\mu}^{2}+\frac{144}{5}\hat{\mu}^{4}\right)\ln\frac{\hat{\Lambda}}{2}+\frac{1}{20}\left(1+168\hat{\mu}^{2}+2064\hat{\mu}^{4}\right)+\frac{3}{5}\left(1+12\hat{\mu}^{2}\right)^{2}\gamma_{E}\right.$
$\displaystyle\left.-\frac{8}{5}(1+12\hat{\mu}^{2})\frac{\zeta^{\prime}(-1)}{\zeta(-1)}-\frac{34}{25}\frac{\zeta^{\prime}(-3)}{\zeta(-3)}\right.-\frac{72}{5}\Big{[}8\aleph(3,z)+3\aleph(3,2z)-12\hat{\mu}^{2}\aleph(1,2z)$
$\displaystyle+12i\hat{\mu}\,(\aleph(2,z)+\aleph(2,2z))-\left.i\hat{\mu}(1+12\hat{\mu}^{2})\,\aleph(0,z)-2(1+8\hat{\mu}^{2})\aleph(1,z)\Big{]}\Bigg{\\}}\right.$
$\displaystyle-\left.\frac{15}{2}\Bigg{\\{}\left(1+12\hat{\mu}^{2}\right)\left(2\ln\frac{\hat{\Lambda}}{2}-1-\aleph(z)\right)\Bigg{\\}}\hat{m}_{D}\right]$
$\displaystyle+$
$\displaystyle\left(\frac{c_{A}\alpha_{s}}{3\pi}\right)\left(\frac{s_{F}\alpha_{s}}{\pi}\right)\Bigg{[}\frac{15}{2\hat{m}_{D}}\left(1+12\hat{\mu}^{2}\right)-\frac{235}{16}\Bigg{\\{}\bigg{(}1+\frac{792}{47}\hat{\mu}^{2}+\frac{1584}{47}\hat{\mu}^{4}\bigg{)}\ln\frac{\hat{\Lambda}}{2}$
$\displaystyle-\frac{144}{47}\left(1+12\hat{\mu}^{2}\right)\ln\hat{m}_{D}+\frac{319}{940}\left(1+\frac{2040}{319}\hat{\mu}^{2}+\frac{38640}{319}\hat{\mu}^{4}\right)-\frac{24\gamma_{E}}{47}\left(1+12\mu^{2}\right)$
$\displaystyle-\frac{44}{47}\left(1+\frac{156}{11}\hat{\mu}^{2}\right)\frac{\zeta^{\prime}(-1)}{\zeta(-1)}-\frac{268}{235}\frac{\zeta^{\prime}(-3)}{\zeta(-3)}-\frac{72}{47}\Big{[}4i\hat{\mu}\aleph(0,z)$
$\displaystyle+\left(5-92\hat{\mu}^{2}\right)\aleph(1,z)+144i\hat{\mu}\aleph(2,z)+52\aleph(3,z)\Big{]}\Bigg{\\}}+90\frac{\hat{m}_{\textrm{th}}^{2}}{\hat{m}_{D}}$
$\displaystyle+\frac{315}{4}\Bigg{\\{}\left(1+\frac{132}{7}\hat{\mu}^{2}\right)\ln\frac{\hat{\Lambda}}{2}+\frac{11}{7}\left(1+12\hat{\mu}^{2}\right)\gamma_{E}+\frac{9}{14}\left(1+\frac{132}{9}\hat{\mu}^{2}\right)$
$\displaystyle+\frac{2}{7}\aleph(z)\Bigg{\\}}\hat{m}_{D}\Bigg{]}\Bigg{]}\\!\\!\Bigg{]}\,,$

whereas the free energy density up to three-loop pure glue case has been
calculated in 3loopglue1 ; ASS  and read as

$\displaystyle F_{g}^{\rm 3-loop}$ $\displaystyle=$
$\displaystyle-\frac{d_{A}\pi^{2}T^{4}}{45}\Bigg{[}\\!\\!\Bigg{[}1-\frac{15}{4}\hat{m}_{D}^{3}+\frac{c_{A}\alpha_{s}}{3\pi}\Bigg{[}-\frac{15}{4}+\frac{45}{2}\hat{m}_{D}-\frac{135}{2}\hat{m}_{D}^{2}-\frac{495}{4}\left(\ln\frac{\hat{\Lambda}_{g}}{2}+\frac{5}{22}+\gamma_{E}\right)\hat{m}_{D}^{3}\Bigg{]}$
(615) $\displaystyle+$
$\displaystyle\left(\frac{c_{A}\alpha_{s}}{3\pi}\right)^{2}\Bigg{[}\frac{45}{4\hat{m}_{D}}-\frac{165}{8}\left(\ln\frac{\hat{\Lambda}_{g}}{2}-\frac{72}{11}\ln\hat{m}_{D}-\frac{84}{55}-\frac{6}{11}\gamma_{E}-\frac{74}{11}\frac{\zeta^{\prime}(-1)}{\zeta(-1)}+\frac{19}{11}\frac{\zeta^{\prime}(-3)}{\zeta(-3)}\right)$
$\displaystyle+$
$\displaystyle\frac{1485}{4}\left(\ln\frac{\hat{\Lambda}_{g}}{2}-\frac{79}{44}+\gamma_{E}+\ln
2-\frac{\pi^{2}}{11}\right)\hat{m}_{D}\Bigg{]}\Bigg{]}\\!\\!\Bigg{]}.$

It is important to note that chemical potential dependence also appears in
pure glue diagrams from the internal quark loop in effective gluon propagators
and effective vertices. This chemical potential$(\mu)$ dependence are present
within Debye mass$(m_{D})$. Besides the choice of the renormalization scales,
the analytic result does not contain any free fit parameters and the result is
also gauge-invariant. The NNLO pressure is given by

${\cal P}^{\textrm{NNLO}}=-F^{\textrm{NNLO}}\,.$ (616)

The higher orders thermodynamical quantities HBAMSS ; HAMSS  of hot and dense
matter, such as, the entropy density, the equation of state, the speed of
sound, the interaction measure or the trace anomaly and various
susceptibilities associated with conserved density fluctuations can be
computed using NNLO free energy density and pressure. The equation of state is
a generic quantity of a hot and dense many particle system and is required to
investigate the expansion dynamics of hot and dense matter by using
hydrodynamics. The obtained results on thermodynamic quantities HBAMSS ; HAMSS
are very good agreement with lattice results within error down to 200 MeV
temperature. These calculations certainly have huge impact on the
thermodynamics of QCD matter at finite temperature and chemical potential that
agree quite well with data from lattice QCD, a first principle calculation. On
the other hand the higher order thermodynamic quantities and various order
quark number susceptibilities (QNS) are of huge interest to both theorists and
experimentalists, for understanding the phase diagram of QCD.

Apart from QCD thermodynamics andersen1 ; ABM ; HM ; CMT ; CMT1 ; CMT2 ; HMT ;
ABM1 ; andersen2 ; HMS ; najmul2qns ; 3loopglue1 ; ASS ; ALSS ; ALSS1 ; HBAMSS
; HAMSS ; andersen3 ; andersen4 , if readers are interested in application to
other physical quantities related to QGP created in heavy-ion collisions can
go through the following extensive list of references: for dilepton production
rate LT ; BPS ; BPY ; GHMT ; GM3 ; JG ; Ghiglieri1 , photon production rate
KLS ; BNNR ; AGKZ ; AMY ; AMY1 ; PT ; GHKMT , single quark and quark-antiquark
potentials MTC ; MTC1 ; CMT4 ; CMRT ; LPRT ; DYS ; DGMS ; THKP , fermion
damping rate RDP ; PPS , photon damping rate MHT ; AD , gluon damping rate BP1
; BP3  and parton energy-loss BT ; BT1 ; TG ; CMT5 ; GMT .

##  13 Thermal Medium with Non-perturbative Effects

Finite temperature QCD has been applied to study the properties of a QGP,
which is believed to have existed in the early Universe, just a few
microsecond after the big bang and in the fireball created in high energy
relativistic heavy-ion collisions at RHIC in BNL and at LHC in CERN. Lattice
QCD (lQCD) provides a first-principles-based method that can take into account
the non-perturbative effects of QCD. lQCD has been applied to investigate the
behaviour of QCD near the critical temperature $T_{c}$, where hadronic matter
undergoes a phase transition to the deconfined QGP phase. Beside lQCD also
perturbation theory has been used to investigate the phenomenologically
relevant properties of QGP. In contrast to lQCD computations the purterbative
method is able to deal with dynamical quantities, a finite baryon density and
non-equilibrium situations. To perturbatively understand the properties of QGP
one needs to have very good understanding of the different collective
excitations appear due to the presence of a thermal bath. There are three
types of collective excitations which are associated with different thermal
scales, They are (i) the energy (or hard) scale $T$, (ii) the electric scale
$gT$, and (iii) the magnetic scale $g^{2}T$. In the literature the hard and
electric scales are well studied, but not the magnetic scale since it is
related to the difficult non-perturbative physics of confinement.

Based on the HTL resummations BP2 ; BP ; TW ; FT ; BP1 , a reorganization of
finite-temperature and chemical potential perturbation theory known as HTL
perturbation theory (HTLpt) has been discussed in sec. 12.2. HTLpt deals with
the hard scale $T$ and the electric scale $gT$ as the soft scale. This has
been widely applied to compute various physical quantities associated with QGP
by using HTL resummed propagators and vertices. HTLpt works well at a
temperature of approximately $2T_{c}$ and above, where $T_{c}\sim 160$ MeV is
the critical temperature for the QGP phase transition. Near $T_{c}$, the
running coupling $g$ is moderately high and the QGP could therefore be
completely non-perturbative in the vicinity of $T_{c}$.

Given the uncertainty involved in the lattice computation of dynamical
quantities and also HTLpt near $T_{c}$, it is always desirable to formulate an
alternative approach to consider non-perturbative effects which can be dealt
in a similar way as done in HTLpt. There are some approaches available in the
literature: one such approach is a semi-empirical way to include non-
perturbative effects by considering a gluon condensate within the Green
functions in momentum space lavelle ; markus ; MST . The gluon condensate has
a substantial effect on the equation of state of QCD matter, in contrast to
the quark condensate. In subsec. 13.1 the quark propagation in QGP with gluon
condensate will be discussed. Another approach nansu1 ; BHMS  would be to
consider the non-perturbative physics involved in the QCD magnetic scale. This
is taken into account through the non-perturbative magnetic screening scale
within the Gribov-Zwanziger (GZ) action Gribov1 ; zwanziger1 . The inclusion
of magnetic scale regulates the magnetic infrared (IR) behaviour of QCD, the
physics associated with it is completely non-perturbative. The gluon
propagator with the GZ action is IR regulated which mimics confinement. This
also makes the calculations more compatible with the results of lQCD. In
subsec. 13.2 the quark propagation in QGP with GZ action will be discussed.

###  13.1 Quark Propagation in QGP with Gluon Condensate

####  13.1.1 Quark self-energy

Now in the rest frame of the heat bath, $u^{\mu}=(1,0,0,0)$, the most general
ansatz for fermionic self-energy reads from (313) as

$\displaystyle\Sigma(L)$ $\displaystyle=$ $\displaystyle-{\cal
A}(\omega,l)L\\!\\!\\!\\!/-{\cal B}(\omega,l)\gamma_{0},$ (617)

with structure functions

$\displaystyle{\cal A}(\omega,l)$
$\displaystyle=\frac{1}{4l^{2}}\left({\rm{Tr}}\left[\Sigma
L\\!\\!\\!\\!/\,\right]-\omega{\rm{Tr}}\left[\Sigma\gamma_{0}\right]\right),$
(618a) $\displaystyle{\cal B}(\omega,l)$
$\displaystyle=\frac{1}{4l^{2}}\left(L^{2}{\rm{Tr}}\left[\Sigma\gamma_{0}\right]-\omega{\rm{Tr}}\left[\Sigma
L\\!\\!\\!\\!/\,\right]\right),$ (618b)

where the four momentum of fermion is $L\equiv(\omega,\bm{\vec{l}})$ with
$l=|\bm{\vec{l}}|$.

Figure 34: Quark self-energy containing gluon condensate.

The lowest order interaction of a quark with gluon condensate is given by
self-energy diagram in Fig. 34. One can write the quark self-energy following
Fig. 34 as

$\displaystyle\Sigma(L)\delta_{il}$ $\displaystyle=$ $\displaystyle
T\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{K}\,\,\left(-ig\gamma^{\mu}\lambda^{a}_{ij}\right)\left(\frac{i\not{Q}\delta_{jk}}{Q^{2}}\right)\left(-ig\gamma^{\nu}\lambda^{b}_{kl}\right)i{\tilde{D}}_{\mu\nu}^{ab}(K)\delta_{ab}$
$\displaystyle\Sigma(L)$ $\displaystyle=$
$\displaystyle\frac{4}{3}g^{2}T\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{K}\,\,{\tilde{D}}_{\mu\nu}(K)\gamma^{\mu}\frac{\not{Q}}{Q^{2}}\gamma^{\nu}\,$
(619)

where $Q=L-K$, $\sum\\!\\!\\!\\!\\!\\!\\!\int\limits_{K}$ is a bosonic sum-
integral,
$\lambda_{ij}^{a}\lambda_{jl}^{a}=\frac{(N_{c}^{2}-1)}{2N_{c}}\delta_{il}=\frac{4}{3}\delta_{il}$
with $N_{c}=3$ and ${\tilde{D}}_{\mu\nu}$ is the non-perturbative gluon
propagator containing gluon condensate. We will consider purely non-
perturbative input from lQCD as parametrized by temperature dependent gluon
condensates. The most general ansatz for the non-perturbative gluon propagator
at finite temperature can be written as

${\tilde{D}}_{\mu\nu}(K)={\tilde{D}}_{L}(k_{0},k)P^{L}_{\mu\nu}+{\tilde{D}}_{T}(k_{0},k)P^{T}_{\mu\nu}\,,$
(620)

where the longitudinal and transverse projectors are given by

$\displaystyle P^{L}_{\mu\nu}$
$\displaystyle=\frac{K_{\mu}K_{\nu}}{K^{2}}-\eta^{\mu\nu}-P^{T}_{\mu\nu}\,,$
(621a) $\displaystyle P^{T}_{\mu 0}$ $\displaystyle=0\,,$ (621b)
$\displaystyle P^{T}_{ij}$
$\displaystyle=\delta_{ij}-\frac{k_{i}k_{j}}{k^{2}}\,.$ (621c)

In order to relate the propagator in (620) to the gluon condensate one can
follow the zero temperature calculation lavelle  and expand the quark
propagator in (619) for small loop momenta. Then considering terms which are
only bilinear in $K$, one can relate the gluon condensates with the moments of
the gluon propagator. Following this one can obtain markus  the structure
functions in (618a) and (618b) as

$\displaystyle{\cal A}$
$\displaystyle=-\frac{4}{3}g^{2}\frac{1}{L^{6}}T\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{K}\,\,\left[\frac{1}{K^{2}}\left\\{-\omega^{2}k_{0}^{4}+\frac{22}{3}\omega^{2}k_{0}^{2}k^{2}-\frac{1}{3}l^{2}k_{0}^{4}+\frac{34}{15}l^{2}k_{0}^{2}k^{2}+\frac{5}{3}\omega^{2}k^{4}-\frac{1}{3}l^{2}k^{4}\right\\}{\tilde{D}}_{L}(k_{0},k)\right.$
$\displaystyle\left.+\left\\{-2\omega^{2}k_{0}^{2}-\frac{2}{3}l^{2}k_{0}^{2}-2\omega^{2}k^{2}+\frac{2}{5}l^{2}k^{2}\right\\}{\tilde{D}}_{T}(k_{0},k)\right]\,,$
(622a) $\displaystyle{\cal B}$
$\displaystyle=-\frac{4}{3}g^{2}\frac{l_{0}}{L^{6}}T\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{K}\,\,\left[\frac{1}{K^{2}}\left\\{-\frac{8}{3}l^{2}k_{0}^{4}-\left(\frac{40}{3}\omega^{2}+\frac{104}{15}l^{2}\right)k_{0}^{2}k^{2}-\frac{8}{3}\omega^{2}k^{4}\right\\}{\tilde{D}}_{L}(k_{0},k)\right.$
$\displaystyle\left.-\left\\{\frac{16}{3}l^{2}k_{0}^{2}+\frac{16}{15}l^{2}k^{2}\right\\}{\tilde{D}}_{T}(k_{0},k)\right]\,.$
(622b)

Assuming the temperature scale to be large, $T>>l$ and one can set $k_{0}=2\pi
inT=0$ under the plane wave approximation. The above two equations become

$\displaystyle{\cal A}(\omega,l)$
$\displaystyle=-\frac{4}{3}g^{2}\frac{1}{L^{6}}T\int\frac{d^{3}k}{(2\pi)^{3}}\left[\left(\frac{1}{3}\omega^{2}-\frac{5}{3}l^{2}\right)k^{2}{\tilde{D}}_{L}(0,k)+\left(-2\omega^{2}+\frac{2}{5}l^{2}\right)k^{2}{\tilde{D}}_{T}(0,k)\right]\,,$
(623a) $\displaystyle{\cal B}(\omega,l)$
$\displaystyle=-\frac{4}{3}g^{2}\frac{\omega}{L^{6}}T\int\frac{d^{3}k}{(2\pi)^{3}}\left[\frac{8}{3}\omega^{2}k^{2}{\tilde{D}}_{L}(0,k)-\frac{16}{15}l^{2}k^{2}{\tilde{D}}_{T}(0,k)\right]\,.$
(623b)

The moments of the longitudinal and the transverse gluon propagators,
respectively, in (623a) and (623b) can be related to the chromoelectric and
the chromomagnetic condensates as

$\displaystyle\langle E^{2}\rangle_{T}$
$\displaystyle=8T\int\frac{d^{3}k}{(2\pi)^{3}}k^{2}{\tilde{D}}_{L}(0,k)+{\cal
O}(g)\,,$ (624a) $\displaystyle\langle B^{2}\rangle_{T}$
$\displaystyle=-16T\int\frac{d^{3}k}{(2\pi)^{3}}k^{2}{\tilde{D}}_{T}(0,k)+{\cal
O}(g)\,.$ (624b)

Using (624a) and (624b) in (623a) and (623b), one can write the structure
functions as

$\displaystyle{\cal A}(\omega,l)$
$\displaystyle=-\frac{1}{6}g^{2}\frac{1}{L^{6}}\left[\left(\frac{1}{3}\omega^{2}-\frac{5}{3}l^{2}\right)\langle
E^{2}\rangle_{T}+\left(\frac{1}{5}l^{2}-\omega^{2}\right)\langle
B^{2}\rangle_{T}\right]\,,$ (625a) $\displaystyle{\cal B}(\omega,l)$
$\displaystyle=-\frac{4}{9}g^{2}\frac{\omega}{L^{6}}\left[\omega^{2}\langle
E^{2}\rangle_{T}+\frac{1}{5}l^{2}\langle B^{2}\rangle_{T}\right]\,,$ (625b)

where $g^{2}=4\pi\alpha_{s}$. These condensates can be obtained in terms of
the spacelike ($\Delta_{\sigma}$) and timelike ($\Delta_{\tau}$) plaquette
expectation values computed on a lattice boyd  in Minkowski space as markus

$\displaystyle\frac{\alpha_{s}}{\pi}\langle E^{2}\rangle_{T}$
$\displaystyle=\frac{4}{11}T^{4}\Delta_{\tau}-\frac{2}{11}\langle
G^{2}\rangle_{T=0}\ ,$ (626a) $\displaystyle\frac{\alpha_{s}}{\pi}\langle
B^{2}\rangle_{T}$
$\displaystyle=-\frac{4}{11}T^{4}\Delta_{\sigma}+\frac{2}{11}\langle
G^{2}\rangle_{T=0}\ .$ (626b)

The plaquette expectation values are related to the gluon condensate above
$T_{c}$ as boyd ; leut

$\langle G^{2}\rangle_{T}=\langle G^{2}\rangle_{T=0}-\Delta T^{4}\ ,$ (627)

where $\Delta=\Delta_{\sigma}+\Delta_{\tau}$ and $\langle
G^{2}\rangle_{T=0}=(2.5\pm 1.0)T_{c}^{4}$.

####  13.1.2 Quark propagator and dispersion

Figure 35: Effective quark propagator containing gluon condensate.

The effective quark propagator containing gluon condensate follows from
diagram in Fig. 35. In helicity representation, the effective quark propagator
is given in (326) as

$S^{*}(L)=\frac{1}{\not{L}-\Sigma(L)}=\frac{\gamma_{0}-\vec{\gamma}\cdot\bm{\hat{l}}}{2{\cal
D}_{+}(\omega,l)}+\frac{\gamma_{0}+\vec{\gamma}\cdot\bm{\hat{l}}}{2{\cal
D}_{-}(\omega,l)}\ ,$ (628)

where

${\cal D}_{\pm}(\omega,l)=(\omega\mp l)(1+{\cal A})+{\cal B},$ (629)

and the expressions for $\cal A$ and $\cal B$ are obtained in terms of the
chromoelectric and the chromomagnetic condensates in (625a) and (625b).

The dispersion relation of a quark interacting with the thermal gluon
condensate is obtained by the poles of ${\cal D}_{\pm}(\omega,l)=0$ in (629).
The functions ${\cal A}$ and ${\cal B}$ have been determined by using (626a)
and (626b), where the plaquette expectation values are taken from the lattice
calculations of Ref. boyd .

Figure 36: Quark dispersion in presence of gluon condensate. These figures are
taken from Ref. MST .

${\cal D}_{+}(\omega,l)=0$ has two ploes at $\omega=\omega^{+}(l)$ and
$\omega=-\omega^{-}(l)$ whereas ${\cal D}_{-}(\omega,l)=0$ has two ploes at
$\omega=\omega^{-}(l)$ and $\omega=-\omega^{+}(l)$. In Fig. 36 we have
displayed the dispersion relation of a quark having momentum $l$ for
$T=1.1T_{c}$ (left panel) and $2T_{c}$ (right panel), respectively. Only
positive solutions of ${\cal D}_{\pm}(L)=0$ have been displayed in Fig. 36.
The mode with energy $\omega^{+}$ describes the in-medium propagation of a
particle excitation. It is a Dirac spinors which is a eigenstate of
$(\gamma_{0}-{\vec{\gamma}}\cdot\bm{\hat{l}})$ with chirality to helicity
ratio $+1$. Also there is a new long wavelength mode known as plasmino with
energy $\omega^{-}$. It is also a Dirac spinor and is a eigenstate of
$(\gamma_{0}+{\vec{\gamma}}\cdot\bm{\hat{l}})$ with chirality to helicity
ratio $-1$. Both branches are situated in the time like domain (i.e., above
the free dispersion relation $\omega=l$), and they begin from a common
effective mass which is obtained in the $l\rightarrow 0$ limit as markus

$\omega^{+}(0)=\omega^{-}(0)=m_{\rm{eff}}=\left[\frac{2\pi\alpha_{s}}{3}\left(\langle{E}^{2}\rangle_{T}+\langle{B}^{2}\rangle_{T}\right)\right]^{1/4}\
,$ (630)

which is given by $m_{\rm{eff}}\approx 1.15$ $T$ and found to be independent
of $g$. For small momenta $l\rightarrow 0$, the dispersion relation behaves
markus  like

$\omega^{\pm}=m_{\rm{eff}}\pm c_{2}l,$ (631)

where

$c_{2}=\frac{3}{4}\frac{\langle{E}^{2}\rangle_{T}}{\langle{E}^{2}\rangle_{T}+\langle{B}^{2}\rangle_{T}}\,.$
(632)

It is to be noted that because of the opposite slopes of two branches,
$\omega^{-}$ branch has a minimum at low momenta then it rapidly approaches
the free dispersion for large momenta, indicating a purely long wavelength
mode. This minimum leads to Van Hove singularities in soft dilepton rate MST
akin to HTL case BPY ; KMT  and will be discussed in subsec 13.3.2. On the
other hand, the $\omega^{+}$ mode at large momenta is given by markus

$\omega^{+}=l+c_{1},$ (633)

where

$c_{1}=\left[\frac{2\pi}{9}\alpha_{s}\left(\langle E^{2}\rangle_{T}+\langle
B^{2}\rangle_{T}\right)\right]^{1/4}.$ (634)

The dispersion relation of a quark interacting with the in-medium gluon
condensate is similar to that obtained from the HTL resummed quark propagator
displayed in Fig. 27. It is important to note that the dispersion relations
with the HTL approximation and gluon condensate, respectively, exhibit similar
features which is the general consequence of the presence of the heat bath.

####  13.1.3 Spectral representation of the quark propagator

The spectral functions, $\rho^{\pm}(\omega,l)$, corresponding to the effective
propagator in (628) can be obtained following subsec. 9.8 or (696) in appendix
A.3 as MST

$\rho^{\pm}(\omega,l)=R^{\pm}(\omega,l)\delta\left(\omega-\omega^{\pm}\right)+R^{\mp}(-\omega,l)\delta\left(\omega+\omega^{\mp}\right)\
,$ (635)

where

$R^{\pm}=\left|\frac{\left(\omega^{2}-l^{2}\right)^{3}}{C^{\pm}}\right|$ (636)

with

$\displaystyle C^{\pm}$ $\displaystyle=$ $\displaystyle-\left[\left(1+{\cal
A}\right)\left(\omega^{2}-l^{2}\right)^{3}\ +\ \frac{{\cal
B}\left(\omega^{2}-l^{2}\right)^{3}}{\omega}\ +\ 6\,\omega(\omega\mp
l)(\omega^{2}-l^{2})^{2}\right.$ (637) $\displaystyle\left.+\
\frac{g^{2}}{3}\omega(\omega\mp
l)\left(\frac{5}{3}\langle{E}^{2}\rangle_{T}-\langle{B}^{2}\rangle_{T}\right)\
-\ \frac{8}{9}g^{2}\omega^{2}\langle{E}^{2}\rangle_{T}\right].$

The spectral functions in (635) has only contribution from the poles of the
effective propagator. The solutions are collective quark modes with energy
$\omega^{\pm}$. Since the effective quark propagator (628) does not have an
imaginary part coming from the quark self-energy, the spectral functions do
not have a contribution from discontinuities or Landau cut.

###  13.2 Quark Propagation in QGP with Gribov-Zwanziger Action

####  13.2.1 Gribov-Zwanziger action and its consequences

Gribov showed in 1978 Gribov1  that in a non-Abelian gauge theory, fixing the
divergence of the potential does not commute with the gauge fixing.
Unfortunately, the solutions of the differential equations, which specify the
gauge fixing with vanishing divergence, can have several copies (Gribov
copies) or none at all. This is known as Gribov ambiguity. To resolve this
ambiguity, the domain of functional integral has to be restricted within a
fundamental modular region, bounded by Gribov horizon. Following this in 1989
Zwanziger zwanziger1  derived a local, renormalizable action for non-Abelian
gauge theories which fulfills the idea of restriction. He also showed that by
introducing this GZ action the divergences may be absorbed by suitable field
and coupling constant renormalization.

The GZ action is given by NV

$\displaystyle S_{\textrm{GZ}}$ $\displaystyle=$ $\displaystyle
S_{0}+S_{\gamma\textrm{G}};$ (638) $\displaystyle S_{0}$ $\displaystyle=$
$\displaystyle S_{\textrm{YM}}+S_{\textrm{gf}}+\int
d^{D}x\left({\bar{\phi}}^{ac}_{\mu}\partial_{\nu}D^{ab}_{\nu}\phi^{bc}_{\mu}-{\bar{\omega}}^{ac}_{\mu}\partial_{\nu}D^{ab}_{\nu}\omega^{bc}_{\mu}\right)+\Delta
S_{0};$ (639) $\displaystyle S_{\gamma\textrm{G}}$ $\displaystyle=$
$\displaystyle\gamma_{G}^{2}\int d^{D}x\ g\
f^{abc}A_{\mu}^{a}\left(\phi^{bc}_{\mu}+{\bar{\phi}}^{bc}_{\mu}\right)+\Delta
S_{\gamma};$ (640)

where $(\phi^{bc}_{\mu}+{\bar{\phi}}^{b}c_{\mu})$ and
$(\omega^{bc}_{\mu}+{\bar{\omega}}^{bc}_{\mu})$ are a pair of complex
conjugate bosonic and Grassmann fields respectively, introduced due to
localization of the GZ action. $S_{\textrm{YM}}$ and $S_{\textrm{gf}}$ are the
normal Yang-Mills and the gauge fixing term of the action and $D$ is the
dimension of the theory. $\Delta S_{0}$ and $\Delta S_{\gamma}$ are the
corresponding counterterms of the $\gamma_{G}$ independent and dependent parts
of GZ action. $\gamma_{G}$ is called the Gribov parameter. In reality,
$\gamma_{G}$ is computed self-consistently using a one-loop888Equation (641)
is a one-loop result. In the vacuum, the two-loop result has been computed
Gracey:2005cx  and the form of Gribov propagator in (643) remains unaffected.
Only $\gamma_{G}$ itself is changed to take into account the two-loop
correction. It is expected that this would be valid also at finite
temperature. gap equation and at asymptotically high temperatures it becomes
nansu1 ; zwanziger2 ; nansu2

$\displaystyle\gamma_{G}=\frac{D-1}{D}\frac{N_{c}}{4\sqrt{2}\pi}g^{2}T,$ (641)

where $N_{c}$ is the number of colors. The one-loop running strong coupling,
$g^{2}=4\pi\alpha_{s}$, is

$\displaystyle
g^{2}(T)=\frac{48\pi^{2}}{(33-2N_{f})\ln\left(\frac{Q^{2}_{0}}{\Lambda_{0}^{2}}\right)},$
(642)

where $Q_{0}$ is the renormalization scale, which is usually chosen to be
$2\pi T$ unless specified and $N_{f}$ is the number of quark flavors. The
scale $\Lambda_{0}$ is fixed by requiring that $\alpha_{s}$(1.5 GeV) = 0.326,
which is obtained from lattice calculations alphas_lat . For one-loop running,
this procedure gives $\Lambda_{0}=176$ MeV.

We know that gluons have an important role in confinement. In the GZ action
Gribov1 ; zwanziger1  the confinement is expected to be governed kinematically
with the gluon propagator in covariant gauge Gribov1 ; zwanziger1

$\displaystyle
D^{\mu\nu}(P)=-\left[\eta^{\mu\nu}-(1-\xi)\frac{P^{\mu}P^{\nu}}{P^{2}}\right]\frac{P^{2}}{P^{4}+\gamma_{G}^{4}}\,,$
(643)

where the four-momentum $P=(p_{0},\bm{\vec{p}})$ and $\xi$ is the gauge
parameter. The term $\gamma_{G}$ in the denominator in (643) shifts the poles
of the gluon propagator off the energy axis and there are no asymptotic gluon
modes exist. For maintaining the consistency of the theory, obviously these
unphysical poles should not appear in gauge-invariant quantities. This
indicates that the gluons are unphysical excitations. In reality, this means
that the addition of the Gribov parameter yields the effective confinement of
gluons.

####  13.2.2 Quark self-energy

In the high-temperature limit one can calculate the quark self-energy $\Sigma$
using the modified gluon propagator given in (643) as nansu1 ; BHMS

$\displaystyle\Sigma(P)$ $\displaystyle=$ $\displaystyle
C_{F}T\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{\\{K\\}}\\!\\!(-ig\gamma_{\mu})\left(\frac{i\not{K}}{K^{2}}\right)(-ig\gamma_{\nu})(iD^{\mu\nu}(P-K))\approx
g^{2}C_{F}\sum_{\pm}\int\limits_{0}^{\infty}\frac{dk}{2\pi^{2}}k^{2}\int\frac{d\Omega}{4\pi}$
(644)
$\displaystyle\times\frac{\tilde{n}_{\pm}(k,\gamma_{G})}{4E_{\pm}^{0}}\left[\frac{\gamma_{0}+\bm{\hat{k}}\cdot\vec{\gamma}}{\omega+k-E_{\pm}^{0}+\frac{\bm{\vec{p}}\cdot{\vec{k}}}{E_{\pm}^{0}}}+\frac{\gamma_{0}-\bm{\hat{k}}\cdot\vec{\gamma}}{\omega-k+E_{\pm}^{0}-\frac{\bm{\vec{p}}\cdot{\vec{k}}}{E_{\pm}^{0}}}\right],$

where Casimir factor $C_{F}=4/3$,
$\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{\\{K\\}}$ is a fermionic sum-
integral and

$\displaystyle\tilde{n}_{\pm}(k,\gamma_{G})$ $\displaystyle\equiv$
$\displaystyle n_{B}\\!\left(\sqrt{k^{2}\pm i\gamma_{G}^{2}}\right)+n_{F}(k),$
$\displaystyle E_{\pm}^{0}$ $\displaystyle=$ $\displaystyle\sqrt{k^{2}\pm
i\gamma_{G}^{2}}\ ,$ (645)

with $n_{B}$ and $n_{F}$ are Bose-Einstein and Fermi-Dirac distribution
functions, respectively. In presence of the Gribov term the modified thermal
quark mass can also be obtained as nansu1

$\displaystyle
m_{q}^{2}(\gamma_{G})=\frac{g^{2}C_{F}}{4\pi^{2}}\sum_{\pm}\int\limits_{0}^{\infty}dk\,\frac{k^{2}}{E_{\pm}^{0}}\,\tilde{n}_{\pm}(k,\gamma_{G}).$
(646)

####  13.2.3 Quark propagator and dispersion

The effective quark propagator is an important ingredient for computing
various properties of a hot and dense QGP using (semi-)perturbative methods, .
Using the modified quark self-energy given in (644), it would now be
convenient to obtain the effective quark propagator with the Gribov term. The
resummed quark propagator in (325) can now be rearranged as

$\displaystyle S^{\star-1}(P)$ $\displaystyle=$
$\displaystyle{P\\!\\!\\!\\!/-\Sigma(P)}$ (647) $\displaystyle=$
$\displaystyle\frac{1}{2}(\gamma_{0}+{\vec{\gamma}}\cdot\bm{\hat{p}}){\cal
D}_{+}+\frac{1}{2}(\gamma_{0}-{\vec{\gamma}}\cdot\bm{\hat{p}}){\cal D}_{-}$
$\displaystyle=$ $\displaystyle\gamma_{0}\,{\cal
A}_{0}-{\vec{\gamma}}\cdot\bm{\vec{p}}\,{\cal A}_{s}\,,$

where

$\displaystyle{\cal A}_{0}$ $\displaystyle=\frac{1}{2}\left({\cal D}_{+}+{\cal
D}_{-}\right)\,,$ (648a) $\displaystyle{\cal A}_{s}$
$\displaystyle=\frac{1}{2}\left({\cal D}_{-}-{\cal D}_{+}\right)\,.$ (648b)

${\cal A}_{0}$ and ${\cal A}_{s}$ are obtained within the HTL approximation as
nansu1 ; BHMS

$\displaystyle{\cal A}_{0}(\omega,p)$
$\displaystyle=\omega-\frac{2g^{2}C_{F}}{(2\pi)^{2}}\sum_{\pm}\int
dk\,k\,\tilde{n}_{\pm}(k,\gamma_{G})\left[Q_{0}(\tilde{\omega}_{1}^{\pm},p)+Q_{0}(\tilde{\omega}_{2}^{\pm},p)\right],$
(649a) $\displaystyle{\cal A}_{s}(\omega,p)$
$\displaystyle=p+\frac{2g^{2}C_{F}}{(2\pi)^{2}}\sum_{\pm}\int
dk\,k\,\tilde{n}_{\pm}(k,\gamma_{G})\left[Q_{1}(\tilde{\omega}_{1}^{\pm},p)+Q_{1}(\tilde{\omega}_{2}^{\pm},p)\right].$
(649b)

The shifted frequencies are defined here as $\tilde{\omega}_{1}^{\pm}\equiv
E_{\pm}^{0}(\omega+k-E_{\pm}^{0})/k$ and $\tilde{\omega}_{2}^{\pm}\equiv
E_{\pm}^{0}(\omega-k+E_{\pm}^{0})/k$. The Legendre functions of the second
kind, $Q_{0}$ and $Q_{1}$, are given as

$\displaystyle Q_{0}(\omega,p)$
$\displaystyle=Q_{0}\left(\frac{\omega}{p}\right)\equiv\frac{1}{2p}\ln\frac{\frac{\omega}{p}+1}{\frac{\omega}{p}-1},$
(650a) $\displaystyle Q_{1}(\omega,p)$
$\displaystyle=Q_{1}\left(\frac{\omega}{p}\right)\equiv\frac{1}{p}\left[1-\omega
Q_{0}\left(\frac{\omega}{p}\right)\right].$ (650b)

Following (326) the effective quark propagator in helicity representation can
also be written as

$\displaystyle S^{*}(P)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{(\gamma_{0}-{\vec{\gamma}}\cdot\bm{\hat{p}})}{{\cal
D}_{+}}+\frac{1}{2}\frac{(\gamma_{0}+{\vec{\gamma}}\cdot\bm{\hat{p}})}{{\cal
D}_{-}},$ (651)

where ${\cal D}_{\pm}$ are obtained as

$\displaystyle{\cal D}_{+}(\omega,p,\gamma_{G})$ $\displaystyle={\cal
A}_{0}(\omega,p)-{\cal
A}_{s}(\omega,p)=\omega-p-\frac{2g^{2}C_{F}}{(2\pi)^{2}}\sum_{\pm}\int
dkk\tilde{n}_{\pm}(k,\gamma_{G})$ $\displaystyle\hskip
56.9055pt\times\left[Q_{0}(\tilde{\omega}_{1}^{\pm},p)+Q_{1}(\tilde{\omega}_{1}^{\pm},p)+Q_{0}(\tilde{\omega}_{2}^{\pm},p)+Q_{1}(\tilde{\omega}_{2}^{\pm},p)\right],$
(652a) $\displaystyle{\cal D}_{-}(\omega,p,\gamma_{G})$ $\displaystyle={\cal
A}_{0}(\omega,p)+{\cal
A}_{s}(\omega,p)=\omega+p-\frac{2g^{2}C_{F}}{(2\pi)^{2}}\sum_{\pm}\int
dkk\tilde{n}_{\pm}(k,\gamma_{G})$ $\displaystyle\hskip
56.9055pt\times\left[Q_{0}(\tilde{\omega}_{1}^{\pm},p)-Q_{1}(\tilde{\omega}_{1}^{\pm},p)+Q_{0}(\tilde{\omega}_{2}^{\pm},p)-Q_{1}(\tilde{\omega}_{2}^{\pm},p)\right].$
(652b)

Figure 37: Plot of the dispersion relations for different values of
$\gamma_{G}$. In the parenthesis, the first one represents a collective
excitation mode whereas the second one is the corresponding energy of that
mode. These figures are taken from Ref. BHMS .

The zeros of ${\cal D}_{\pm}(\omega,p,\gamma_{G})$ correspond to the
dispersion relations for the collective excitations in the non-perturbative
medium. In Fig. 37 the dispersion relations are displayed for three values of
$\gamma_{G}$. For HTL case when $\gamma_{G}=0$, one gets two massive
quasiparticle modes. One is a normal quark mode $q_{+}$ with energy
$\omega_{+}$ and another one is a long wavelength plasmino mode $q_{-}$ with
energy $\omega_{-}$. They are displayed in Fig. 27 also in the left panel of
Fig. 37. The $q_{-}$ mode has a minimum and then it quickly approaches to the
non-interacting massless mode in the high-momentum limit. The minimum in
$q_{-}$ mode (plasmino mode) leads to Van Hove singularities in soft dilepton
production rate BHMS  which will be discussed in subsec 13.3.3. In presence of
the $\gamma_{G}$, there appears a new massless spacelike mode $q_{G}$ with
energy $\omega_{G}$, in addition to the two massive modes, $q_{+}$ and $q_{-}$
nansu1  as shown in the middle and in the right panel of Fig. 37. This new
spacelike massless mode $q_{G}$ in spacelike domain is due to the inclusion of
the magnetic scale through the GZ action. It becomes lightlike at large
momentum as can also be seen from the middle and the right panel of Fig. 37.
The existence of this extra spacelike mode could affect lattice calculations
of the dilepton rate because the recent lQCD results kitazawa1 ; kitazawa2
considered that there were only two poles of the in-medium propagator leading
to a quark mode and a plasmino mode motivated by the HTL approximation.

It is also to be noted that the slope of the dispersion curve for the new
massless spacelike mode $q_{G}$ exceeds unity in some domain of momentum. This
indicates that the group velocity, $d\omega_{G}/dp$, of the new mode is
superluminal, and then it approaches to the light cone ($d\omega/dp=1$) from
above as shown in Fig. 38. Since the mode is spacelike, there is no causality
problem but could be termed as anomalous dispersion because the presence
$\gamma_{G}$ converts the Landau damping in the spacelike domain into
amplification of a massless spacelike dispersive mode.

Figure 38: Plot of the group velocity for different values of $\gamma_{G}$.
The group velocity for the space like Gribov mode $d\omega_{G}/dp$ becomes
superluminal, as can be seen from both plots.

####  13.2.4 Spectral representation of the quark propagator

In absence of Gribov parameter ($\gamma_{G}=0$), i.e., in the HTL
approximation apart from poles the propagator contains a discontinuity in
complex plane originating from the logarithmic terms in (652a) and (652b) due
to spacelike momentum $\omega^{2}<p^{2}$. The HTL spectral function contains
contributions from two collective excitations and the Landau cut as discussed
in subsec. 9.8. On the other hand, for $\gamma_{G}\neq 0$ there are three
collective excitations $q_{+}$, $q_{-}$ and $q_{G}$, and no Landau cut
contributions in the complex plane due to the fact that the poles come in
complex-conjugate pairs and ultimately cancel out. It seems that the Landau
cut contribution in spacelike domain for $\gamma_{G}=0$ is converted into a
new massless spacelike dispersive mode in presence of magnetic scale
($\gamma_{G}\neq 0$). Since there is no Landau cut contribution, the spectral
representation of the quark propagator ${\cal D}_{\pm}^{-1}$ for
$\gamma_{G}\neq 0$ has only pole contributions and obtained following subsec.
9.8 or (696) in appendix A.3 as BHMS

$\displaystyle\rho_{\pm}^{G}(\omega,p)=\frac{\omega^{2}-p^{2}}{2m_{q}^{2}(\gamma_{G})}\left[\delta(\omega\mp\omega_{+})+\delta(\omega\pm\omega_{-})+\delta(\omega\pm\omega_{G})\right],$
(653)

where ${\cal D}_{+}$ has poles at $\omega_{+}$, $-\omega_{-}$, and
$-\omega_{G}$ and ${\cal D}_{-}$ has poles at $\omega_{-}$, $-\omega_{+}$, and
$\omega_{G}$ with a prefactor, $(\omega^{2}-p^{2})/2m_{q}^{2}(\gamma_{G})$, as
the residue.

####  13.2.5 Quark-Photon vertex

The quark-photon three-point vertex can be obtained BHMS  by using the Ward-
Takahashi identity 999This procedure only constrains the longitudinal part of
the vertex function. as

$\displaystyle(P_{1}-P_{2})_{\mu}\Gamma^{\mu}(P_{1},P_{2})=S^{-1}(P_{1})-S^{-1}(P_{2})\
.$ (654)

The temporal and spatial parts of the modified effective vertex can be written
as

$\displaystyle\Gamma^{0}$ $\displaystyle=a_{G}\leavevmode\nobreak\
\gamma^{0}+b_{G}\leavevmode\nobreak\ \bm{\gamma\cdot\hat{p}},$ (655a)
$\displaystyle\Gamma^{i}$ $\displaystyle=c_{G}\leavevmode\nobreak\
\gamma^{i}+b_{G}\leavevmode\nobreak\
\hat{p}^{i}\gamma_{0}+d_{G}\leavevmode\nobreak\
\hat{p}^{i}\left(\bm{\gamma\cdot\hat{p}}\right),$ (655b)

where the coefficients are given by

$\displaystyle a_{G}$
$\displaystyle=1-\frac{2g^{2}C_{F}}{(2\pi)^{2}}\sum_{\pm}\int
dk\,k\,\tilde{n}_{\pm}(k,\gamma_{G})\frac{1}{\omega_{1}-\omega_{2}}\left[\delta
Q_{01}^{\pm}+\delta Q_{02}^{\pm}\right],$ (656a) $\displaystyle b_{G}$
$\displaystyle=-\frac{2g^{2}C_{F}}{(2\pi)^{2}}\sum_{\pm}\int
dk\,k\,\tilde{n}_{\pm}(k,\gamma_{G})\frac{1}{\omega_{1}-\omega_{2}}\left[\delta
Q_{11}^{\pm}+\delta Q_{12}^{\pm}\right],$ (656b) $\displaystyle c_{G}$
$\displaystyle=1+\frac{2g^{2}C_{F}}{(2\pi)^{2}}\sum_{\pm}\int
dk\,k\,\tilde{n}_{\pm}(k,\gamma_{G})\frac{1}{3(\omega_{1}-\omega_{2})}\left[\delta
Q_{01}^{\pm}+\delta Q_{02}^{\pm}-\delta Q_{21}^{\pm}-\delta
Q_{22}^{\pm}\right],$ (656c) $\displaystyle d_{G}$
$\displaystyle=\frac{2g^{2}C_{F}}{(2\pi)^{2}}\sum_{\pm}\int
dk\,k\,\tilde{n}_{\pm}(k,\gamma_{G})\frac{1}{\omega_{1}-\omega_{2}}\left[\delta
Q_{21}^{\pm}+\delta Q_{22}^{\pm}\right],$ (656d)

with

$\displaystyle\delta Q_{n1}^{\pm}$
$\displaystyle=Q_{n}(\tilde{\omega}_{11}^{\pm},p)-Q_{n}(\tilde{\omega}_{21}^{\pm},p){\rm{\leavevmode\nobreak\
for\leavevmode\nobreak\ }}n=0,1,2\,\,,$ (657a)
$\displaystyle\omega_{m1}^{\pm}$
$\displaystyle=E_{\pm}^{0}(\omega_{m}+k-E_{\pm}^{0})/k{\rm{\leavevmode\nobreak\
for\leavevmode\nobreak\ }}m=1,2\,\,,$ (657b) $\displaystyle\omega_{m2}^{\pm}$
$\displaystyle=E_{\pm}^{0}(\omega_{m}-k+E_{\pm}^{0})/k{\rm{\leavevmode\nobreak\
for\leavevmode\nobreak\ }}m=1,2\,\,$ (657c)

Similarly, the quark-photon four-point function can be obtained from the
following generalized Ward-Takahashi identity

$\displaystyle
P_{\mu}\Gamma^{\mu\nu}(-P_{1},P_{1};-P_{2},P_{2})=\Gamma^{\nu}(P_{1}-P_{2},-P_{1};P_{2})-\Gamma^{\nu}(-P_{1}-P_{2},P_{1};P_{2})\
.$ (658)

###  13.3 Dilepton Production Rate from QGP

Thermal dileptons ($q{\bar{q}}\rightarrow\gamma^{*}\rightarrow l^{+}l^{-}$,
where $q$ and $\bar{q}$ are (anti)quark, $\gamma^{*}$ is virtual photon and
$l^{+}l^{-}$ are lepton pair) emitted from the fireball in ultrarelativistic
heavy ion collisions might serve as a promising signature Ruuskanen:1991au
for the QGP formation in such collisions. In contrast to hadronic signals
dileptons and photons carry direct information about the early phase of the
fireball, since they do not interact with the surrounding medium after their
production. Therefore, they can be used as a direct probe for the QGP.
Unfortunately there is a huge background coming from hadronic decays. Hence it
would be desirable to have some specific features in the dilepton spectrum
which could signal the presence of deconfined matter. Indeed perturbative
calculations BPY ; KMT  have shown distinct structures (van Hove peaks VH ; AM
, gaps) in the production rate of low mass dileptons caused by non-trivial in-
medium quark dispersion relations. In the following subsec 13.3.1 we briefly
discuss the dilepton production rate from a thermal medium.

####  13.3.1  Dilepton rate in presence of thermal medium

The dilepton multiplicity per unit space-time volume is given Weldon:1990iw
as

$\displaystyle\frac{dN}{d^{4}X}$ $\displaystyle=$ $\displaystyle 2\pi
e^{2}e^{-\beta
p_{0}}L_{\mu\nu}\rho^{\mu\nu}\frac{d^{3}\bm{\vec{q}}_{1}}{(2\pi)^{3}E_{1}}\frac{d^{3}\bm{\vec{q}}_{2}}{(2\pi)^{3}E_{2}},$
(659)

where where $e$ is the electromagnetic coupling, $\bm{\vec{q}}_{i}$ and
$E_{i}$ with $i=1,2$ are three momentum and energy of the lepton pairs. The
photonic tensor or the electromagnetic spectral function in thermal medium can
be written as

$\displaystyle\rho^{\mu\nu}(p_{0},\bm{\vec{p}})$ $\displaystyle=$
$\displaystyle-\frac{1}{\pi}\frac{e^{\beta p_{0}}}{e^{\beta
p_{0}}-1}\textrm{Im}\left[D^{\mu\nu}(p_{0},\bm{\vec{p}})\right]\equiv-\frac{1}{\pi}\frac{e^{\beta
p_{0}}}{e^{\beta p_{0}}-1}\leavevmode\nobreak\
\frac{1}{P^{4}}\textrm{Im}\left[\Pi^{\mu\nu}(p_{0},\bm{\vec{p}})\right],$
(660)

where $`\textrm{Im}$’ stands for imaginary part, $\Pi^{\mu\nu}$ is the two
point current-current correlation function or the self-energy of photon and
$D^{\mu\nu}$ represents the photon propagator. Here we have used the relation
Weldon:1990iw

$\displaystyle
D^{\mu\nu}(p_{0},\bm{\vec{p}})=\frac{1}{P^{4}}\Pi^{\mu\nu}(p_{0},\bm{\vec{p}})\,,$
(661)

where $P\equiv(p_{0},\bm{\vec{p}})$ is the four momenta of the photon.

Also the leptonic tensor in terms of Dirac spinors is given by

$\displaystyle L_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum\limits_{\mathrm{spins}}\mathrm{Tr}\left[\bar{u}(Q_{2})\gamma_{\mu}v(Q_{1})\bar{v}(Q_{1})\gamma_{\nu}u(Q_{2})\right]$
(662) $\displaystyle=$ $\displaystyle
Q_{1\mu}Q_{2\nu}+Q_{1\nu}Q_{2\mu}-(Q_{1}\cdot Q_{2}+m_{l}^{2})g_{\mu\nu},$

where $Q_{i}\equiv(q_{0},\bm{\vec{q}}_{i})$ is the four momentum of the $i$-th
lepton and $m_{l}$ is the mass of the lepton.

Now inserting $\int d^{4}P\,\delta^{4}(Q_{1}+Q_{2}-P)=1$, one can write the
dilepton multiplicity from (659) as

$\displaystyle\frac{dN}{d^{4}X}\ $ $\displaystyle=$ $\displaystyle 2\pi
e^{2}e^{-\beta p_{0}}\int
d^{4}P\,\delta^{4}(Q_{1}+Q_{2}-P)L_{\mu\nu}\rho^{\mu\nu}\frac{d^{3}\bm{\vec{q}}_{1}}{(2\pi)^{3}E_{1}}\frac{d^{3}\bm{\vec{q}}_{2}}{(2\pi)^{3}E_{2}}.$
(663)

Using the identity

$\displaystyle\int\frac{d^{3}\bm{\vec{q}}_{1}}{E_{1}}\frac{d^{3}\bm{\vec{q}}_{2}}{E_{2}}\delta^{4}(Q_{1}+Q_{2}-P)\
L_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{3}\left(1+\frac{2m_{l}^{2}}{P^{2}}\right)\sqrt{\left(1-\frac{4m_{l}^{2}}{P^{2}}\right)}\left(P_{\mu}P_{\nu}-P^{2}g_{\mu\nu}\right)$
(664) $\displaystyle=$
$\displaystyle\frac{2\pi}{3}F_{1}(m_{l},P^{2})\left(P_{\mu}P_{\nu}-P^{2}g_{\mu\nu}\right),$

the dilepton production rate in (663) comes out to be

$\displaystyle\frac{dN}{d^{4}Xd^{4}P}=\frac{dR}{d^{4}P}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{12\pi^{4}}\frac{n_{B}(p_{0})}{P^{2}}F_{1}(m_{l},P^{2})\
\textrm{Im}\left[\Pi^{\mu}_{\mu}(p_{0},\bm{\vec{p}})\right]$
$\displaystyle\frac{dR}{d^{4}P}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{12\pi^{4}}\frac{n_{B}(p_{0})}{P^{2}}F_{1}(m_{l},P^{2})\
\frac{1}{2i}{\textrm{Disc}}\left[\Pi^{\mu}_{\mu}(p_{0},\bm{\vec{p}})\right],$
(665)

where $n_{B}(p_{0})=(e^{p_{0}/T}-1)^{-1}$ and $e^{2}=4\pi\alpha$, $\alpha$ is
the electromagnetic coupling constant. We have also used the transversality
condition $P_{\mu}\Pi^{\mu\nu}=0$. The invariant mass of the lepton pair is
defined as $M^{2}\equiv
P^{2}(=p_{0}^{2}-|\bm{\vec{p}}|^{2}=\omega^{2}-|\bm{\vec{p}}|^{2})$. We note
that for massless lepton ($m_{l}=0$) $F_{1}(m_{l},P^{2})=1$.

The (665) is the familiar result most widely used for the dilepton emission
rate from a thermal medium. It must be emphasized that this relation is valid
only to ${\cal O}(e^{2})$ since it does not account for the possible
reinteractions of the virtual photon on its way out of the thermal bath. The
possibility of emission of more than one photon has also been neglected here.
However, the expression is true to all orders in strong interaction.

####  13.3.2 Dilepton production rate from QGP with gluon condensate

In this subsection we calculate the effect of an in-medium gluon condensates,
as discussed in subsec 13.1, on the production rate of lepton pairs from QGP
MST . This effect can be included by using effective propagators, $S^{*}$, as
given in (628) containing the gluon condensate for the exchanged quarks in
photon self-energy in Fig. 39. The photon self energy in Fig. 39 can now be
written as

$\Pi^{\mu\nu}(P)=-3\times
2\times\frac{5}{9}e^{2}T\sum_{\\{k_{0}\\}}\int\frac{d^{3}k}{(2\pi)^{3}}{{\rm
Tr}}\left[S^{*}(K)\gamma^{\mu}S^{*}(Q)\gamma^{\nu}\right],$ (666)

where $\sum_{\\{k_{0}\\}}$ is the frequency sum over fermionic Matsubara
frequency, $K$ and $Q=P-K$ are the fermionic loop four-momenta. We have
considered only massless $u$ and $d$ quarks and the total electric charge of
two flavours is
$(\frac{2}{3})^{2}e^{2}+(\frac{1}{3})^{2}e^{2}=\frac{5}{9}e^{2}$, the factor
$2$ is for antiquarks and the color factor of quark is $3$.

Figure 39: One-loop photon self-energy with effective quark propagators
containing gluon condensate. This figure is taken from Ref. MST .

Substitution of (628) in (666) and performing the traces one gets

$\displaystyle{\Pi}_{\mu}^{\mu}(P)$ $\displaystyle=$
$\displaystyle-\frac{10}{3}e^{2}T\sum_{\\{k_{0}\\}}\int\frac{d^{3}k}{(2\pi)^{3}}\left[\frac{1}{D_{+}(K)}\left(\frac{1-\bm{\hat{k}\cdot\hat{q}}}{D_{+}(Q)}+\frac{1+\bm{\hat{k}\cdot\hat{q}}}{D_{-}(Q)}\right)\right.$
(667)
$\displaystyle+\left.\frac{1}{D_{-}(K)}\left(\frac{1+\bm{\hat{k}\cdot\hat{q}}}{D_{+}(Q)}+\frac{1-\bm{\hat{k}\cdot\hat{q}}}{D_{-}(Q)}\right)\right]\
.$

Now according to (665) one needs to compute the imaginary or discontinuity
part of $\Pi^{\mu}_{\mu}(P)$. The discontinuity can be obtained by using the
Braaten-Pisarski-Yuan (BPY) prescription BPY  obtained in (708) in appendix
A.3 as

$\displaystyle\textrm{Im}\,T\sum_{k_{0}}F_{1}(k_{0})F_{2}(q_{0})$
$\displaystyle=$ $\displaystyle\frac{1}{2i}\textmd{Disc\leavevmode\nobreak\
}T\sum_{k_{0}}F_{1}(k_{0})F_{2}(q_{0})$ (668) $\displaystyle=$
$\displaystyle\pi(1-e^{\beta\omega})\int d\omega_{1}\int
d\omega_{2}\leavevmode\nobreak\ n_{F}(\omega_{1})n_{F}(\omega_{2})$
$\displaystyle\times\ \delta(\omega-\omega_{1}-\omega_{2})\
\rho_{1}(\omega_{1})\rho_{2}(\omega_{2}),$

where $\delta(\omega-\omega_{1}-\omega_{2})$ is the energy conserving
$\delta$-function, $n_{F}$ is the Fermi-Dirac distribution function and
$\rho_{1}$ and $\rho_{2}$ are the spectral functions corresponding to the
functions $F_{1}$ and $F_{2}$.

Now using (668) one can write the imaginary part of $\Pi^{\mu}_{\mu}$ as

$\displaystyle\textrm{Im}\,\Pi_{\mu}^{\mu}(P)$ $\displaystyle=$
$\displaystyle\frac{10\pi}{3}e^{2}\left(e^{E/T}-1\right)\int\frac{d^{3}k}{(2\pi)^{3}}\int_{-\infty}^{\infty}d\omega\int_{-\infty}^{\infty}d\omega^{\prime}\
$ (669)
$\displaystyle\times\delta\left(E-\omega-\omega^{\prime}\right)n_{F}(\omega)n_{F}(\omega^{\prime})$
$\displaystyle\times\left[\left(1+\bm{\hat{q}\cdot\hat{k}}\right)\left\\{\rho^{+}\left(\omega,k\right)\rho^{-}\left(\omega^{\prime},q\right)+\rho^{-}\left(\omega,k\right)\rho^{+}\left(\omega^{\prime},q\right)\right\\}\right.$
$\displaystyle\left.+\left(1-\bm{\hat{q}\cdot\hat{k}}\right)\left\\{\rho^{+}\left(\omega,k\right)\rho^{+}\left(\omega^{\prime},q\right)+\rho^{-}\left(\omega,k\right)\rho^{-}\left(\omega^{\prime},q\right)\right\\}\right],$

where $\rho^{\pm}$ are the spectral functions corresponding to $1/{\cal
D}_{\pm}(L)$ and obtained in (635). Inserting (635) into (669) and performing
the $\omega$-integrations by exploiting the delta functions of the spectral
functions, one finds ($x=\bm{\hat{p}\cdot\hat{k}}$)

$\displaystyle\textrm{Im}\,\Pi_{\mu}^{\mu}(P)$ $\displaystyle=$
$\displaystyle\frac{5}{6\pi}e^{2}\left(e^{E/T}-1\right)\int_{0}^{\infty}dk\,k^{2}\int_{-1}^{+1}dx$
(670) $\displaystyle\ \ \ \times\left[\left(1\
+\bm{\hat{q}\cdot\hat{k}}\right)A\ +\ \left(1\
-\bm{\hat{q}\cdot\hat{k}}\right)B\right],\ $

where

$\displaystyle A$ $\displaystyle=$ $\displaystyle
n_{F}\left(\omega^{+}(k)\right)n_{F}\left(\omega^{-}(q)\right)R_{+}\left(\omega^{+}(k),k\right)R_{-}\left(\omega^{-}(q),q\right)\delta\left(E-\omega^{+}(k)-\omega^{-}(q)\right)$
$\displaystyle+$ $\displaystyle
n_{F}\left(-\omega^{-}(k)\right)n_{F}\left(\omega^{-}(q)\right)R_{-}\left(\omega^{-}(k),k\right)R_{-}\left(\omega^{-}(q),q\right)\delta\left(E+\omega^{-}(k)-\omega^{-}(q)\right)$
$\displaystyle+$ $\displaystyle
n_{F}\left(\omega^{+}(k)\right)n_{F}\left(-\omega^{+}(q)\right)R_{+}\left(\omega^{+}(k),k\right)R_{+}\left(\omega^{+}(q),q\right)\delta\left(E-\omega^{+}(k)+\omega^{+}(q)\right)$
$\displaystyle+$ $\displaystyle
n_{F}\left(-\omega^{-}(k)\right)n_{F}\left(-\omega^{+}(q)\right)R_{-}\left(\omega^{-}(k),k\right)R_{+}\left(\omega^{+}(q),q\right)\delta\left(E+\omega^{-}(k)+\omega^{+}(q)\right)$
$\displaystyle+$ $\displaystyle
n_{F}\left(\omega^{-}(k)\right)n_{F}\left(\omega^{+}(q)\right)R_{-}\left(\omega^{-}(k),k\right)R_{+}\left(\omega^{+}(q),q\right)\delta\left(E-\omega^{-}(k)-\omega^{+}(q)\right)$
$\displaystyle+$ $\displaystyle
n_{F}\left(-\omega^{+}(k)\right)n_{F}\left(\omega^{+}(q)\right)R_{+}\left(\omega^{+}(k),k\right)R_{+}\left(\omega^{+}(q),q\right)\delta\left(E+\omega^{+}(k)-\omega^{+}(q)\right)$
$\displaystyle+$ $\displaystyle
n_{F}\left(\omega^{-}(k)\right)n_{F}\left(-\omega^{-}(q)\right)R_{-}\left(\omega^{-}(k),k\right)R_{-}\left(\omega^{-}(q),q\right)\delta\left(E-\omega^{-}(k)+\omega^{-}(q)\right)$
$\displaystyle+$ $\displaystyle
n_{F}\left(-\omega^{+}(k)\right)n_{F}\left(-\omega^{-}(q)\right)R_{+}\left(\omega^{+}(k),k\right)R_{-}\left(\omega^{-}(q),q\right)\delta\left(E+\omega^{+}(k)+\omega^{-}(q)\right),$

and

$\displaystyle B$ $\displaystyle=$ $\displaystyle
n_{F}\left(\omega^{+}(k)\right)n_{F}\left(\omega^{+}(q)\right)R_{+}\left(\omega^{+}(k),k\right)R_{+}\left(\omega^{+}(q),q\right)\delta\left(E-\omega^{+}(k)-\omega^{+}(q)\right)$
$\displaystyle+$ $\displaystyle
n_{F}\left(-\omega^{-}(k)\right)n_{F}\left(\omega^{+}(q)\right)R_{-}\left(\omega^{-}(k),k\right)R_{+}\left(\omega^{+}(q),q\right)\delta\left(E+\omega^{-}(k)-\omega^{+}(q)\right)$
$\displaystyle+$ $\displaystyle
n_{F}\left(\omega^{+}(k)\right)n_{F}\left(-\omega^{-}(q)\right)R_{+}\left(\omega^{+}(k),k\right)R_{-}\left(\omega^{-}(q),q\right)\delta\left(E-\omega^{+}(k)+\omega^{-}(q)\right)$
$\displaystyle+$ $\displaystyle
n_{F}\left(-\omega^{-}(k)\right)n_{F}\left(-\omega^{-}(q)\right)R_{-}\left(\omega^{-}(k),k\right)R_{-}\left(\omega^{-}(q),q\right)\delta\left(E+\omega^{-}(k)+\omega^{-}(q)\right)$
$\displaystyle+$ $\displaystyle
n_{F}\left(\omega^{-}(k)\right)n_{F}\left(\omega^{-}(q)\right)R_{-}\left(\omega^{-}(k),k\right)R_{-}\left(\omega^{-}(q),q\right)\delta\left(E-\omega^{-}(k)-\omega^{-}(q)\right)$
$\displaystyle+$ $\displaystyle
n_{F}\left(-\omega^{+}(k)\right)n_{F}\left(\omega^{-}(q)\right)R_{+}\left(\omega^{+}(k),k\right)R_{-}\left(\omega^{-}(q),q\right)\delta\left(E+\omega^{+}(k)-\omega^{-}(q)\right)$
$\displaystyle+$ $\displaystyle
n_{F}\left(\omega^{-}(k)\right)n_{F}\left(-\omega^{+}(q)\right)R_{-}\left(\omega^{-}(k),k\right)R_{+}\left(\omega^{+}(q),q\right)\delta\left(E-\omega^{-}(k)+\omega^{+}(q)\right)$
$\displaystyle+$ $\displaystyle
n_{F}\left(-\omega^{+}(k)\right)n_{F}\left(-\omega^{+}(q)\right)R_{+}\left(\omega^{+}(k),k\right)R_{+}\left(\omega^{+}(q),q\right)\delta\left(E+\omega^{+}(k)+\omega^{+}(q)\right).$

Changing the integration variable from $x$ to
$q=|{\vec{p}}-{\vec{k}}|=\sqrt{p^{2}+k^{2}-2pkx}$ the dilepton production rate
in (665) with massless leptons can be written as

$\displaystyle\frac{dN}{d^{4}Xd^{4}P}$ $\displaystyle=$
$\displaystyle\frac{5}{18\pi^{4}}\frac{\alpha^{2}}{M^{2}}\frac{1}{p}\left[\int_{0}^{p}dk\int_{p-k}^{p+k}dq+\int_{p}^{\infty}dk\int_{k-p}^{p+k}dq\right]$
(673)
$\displaystyle\times\left[\left(p^{2}-(k-q)^{2}\right)A+\left((k+q)^{2}-p^{2}\right)B\right].$

where the invariant mass of the dilepton is $M^{2}\equiv
P^{2}(=p_{0}^{2}-|\bm{\vec{p}}|^{2}=E^{2}-|\bm{\vec{p}}|^{2})$, where $E$ is
the photon energy.

Now one can perform the $q$-integration by means of the remaining
$\delta$-functions in $A$ and $B$ leading to

$\displaystyle\frac{dN}{d^{4}Xd^{4}P}$ $\displaystyle=$
$\displaystyle\frac{5}{18\pi^{4}}\frac{\alpha^{2}}{M^{2}}\frac{1}{p}\int_{0}^{\infty}dk\left[\left(p^{2}-(k-q_{s})^{2}\right)\left(A_{1}+A_{2}+A_{3}+A_{5}+A_{6}+A_{7}\right)\right.$
(674)
$\displaystyle\left.+\left((k+q_{s})^{2}-p^{2}\right)\left(B_{1}+B_{2}+B_{3}+B_{5}+B_{6}+B_{7}\right)\right]_{|p-k|\leq
q_{s}\leq p+k}\>,$

where the $q_{s}$ determined by the various $\delta$-functions in (LABEL:ca)
and (LABEL:cb) can assume two different values in the case of the plasmino
branch due to the presence of the minimum and

$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle
n_{F}\left(\omega^{+}(k)\right)n_{F}\left(\omega^{-}(q_{s})\right)R_{+}\left(\omega^{+}(k),k\right)\frac{R_{-}\left(\omega^{-}(q_{s}),q_{s}\right)}{|d\omega^{-}(q)/dq|_{q_{s}}},$
$\displaystyle A_{2}$ $\displaystyle=$ $\displaystyle
n_{F}\left(-\omega^{-}(k)\right)n_{F}\left(\omega^{-}(q_{s})\right)R_{-}\left(\omega^{-}(k),k\right)\frac{R_{-}\left(\omega^{-}(q_{s}),q_{s}\right)}{|d\omega^{-}(q)/dq|_{q_{s}}},$
$\displaystyle A_{3}$ $\displaystyle=$ $\displaystyle
n_{F}\left(\omega^{+}(k)\right)n_{F}\left(-\omega^{+}(q_{s})\right)R_{+}\left(\omega^{+}(k),k\right)\frac{R_{+}\left(\omega^{+}(q_{s}),q_{s}\right)}{|d\omega^{+}(q)/dq|_{q_{s}}},$
$\displaystyle A_{5}$ $\displaystyle=$ $\displaystyle
n_{F}\left(\omega^{-}(k)\right)n_{F}\left(\omega^{+}(q_{s})\right)R_{-}\left(\omega^{-}(k),k\right)\frac{R_{+}\left(\omega^{+}(q_{s}),q_{s}\right)}{|d\omega^{+}(q)/dq|_{q_{s}}},$
$\displaystyle A_{6}$ $\displaystyle=$ $\displaystyle
n_{F}\left(-\omega^{+}(k)\right)n_{F}\left(\omega^{+}(q_{s})\right)R_{+}\left(\omega^{+}(k),k\right)\frac{R_{+}\left(\omega^{+}(q_{s}),q_{s}\right)}{|d\omega^{+}(q)/dq|_{q_{s}}},$
$\displaystyle A_{7}$ $\displaystyle=$ $\displaystyle
n_{F}\left(\omega^{-}(k)\right)n_{F}\left(-\omega^{-}(q_{s})\right)R_{-}\left(\omega^{-}(k),k\right)\frac{R_{-}\left(\omega^{-}(q_{s}),q_{s}\right)}{|d\omega^{-}(q)/dq|_{q_{s}}},$
$\displaystyle B_{1}$ $\displaystyle=$ $\displaystyle
n_{F}\left(\omega^{+}(k)\right)n_{F}\left(\omega^{+}(q_{s})\right)R_{+}\left(\omega^{+}(k),k\right)\frac{R_{+}\left(\omega^{+}(q_{s}),q_{s}\right)}{|d\omega^{+}(q)/dq|_{q_{s}}},$
$\displaystyle B_{2}$ $\displaystyle=$ $\displaystyle
n_{F}\left(-\omega^{-}(k)\right)n_{F}\left(\omega^{+}(q_{s})\right)R_{-}\left(\omega^{-}(k),k\right)\frac{R_{+}\left(\omega^{+}(q_{s}),q_{s}\right)}{|d\omega^{+}(q)/dq|_{q_{s}}},$
$\displaystyle B_{3}$ $\displaystyle=$ $\displaystyle
n_{F}\left(\omega^{+}(k)\right)n_{F}\left(-\omega^{-}(q_{s})\right)R_{+}\left(\omega^{+}(k),k\right)\frac{R_{-}\left(\omega^{-}(q_{s}),q_{s}\right)}{|d\omega^{-}(q)/dq|_{q_{s}}},$
$\displaystyle B_{5}$ $\displaystyle=$ $\displaystyle
n_{F}\left(\omega^{-}(k)\right)n_{F}\left(\omega^{-}(q_{s})\right)R_{-}\left(\omega^{-}(k),k\right)\frac{R_{-}\left(\omega^{-}(q_{s}),q_{s}\right)}{|d\omega^{-}(q)/dq|_{q_{s}}},$
$\displaystyle B_{6}$ $\displaystyle=$ $\displaystyle
n_{F}\left(-\omega^{+}(k)\right)n_{F}\left(\omega^{-}(q_{s})\right)R_{+}\left(\omega^{+}(k),k\right)\frac{R_{-}\left(\omega^{-}(q_{s}),q_{s}\right)}{|d\omega^{-}(q)/dq|_{q_{s}}},$
$\displaystyle B_{7}$ $\displaystyle=$ $\displaystyle
n_{F}\left(\omega^{-}(k)\right)n_{F}\left(-\omega^{+}(q_{s})\right)R_{-}\left(\omega^{-}(k),k\right)\frac{R_{+}\left(\omega^{+}(q_{s}),q_{s}\right)}{|d\omega^{+}(q)/dq|_{q_{s}}},$

The group velocity factors in (LABEL:cabs) follow from the dispersion
relation, ${\cal D}_{\pm}(L)=0$, of (629) as

$\frac{d\omega^{\pm}(l)}{dl}=\pm\frac{F^{\pm}\left(\omega^{\pm}(l),{\cal
A},{\cal B},l\right)}{G^{\pm}\left(\omega^{\pm}(l),{\cal A},{\cal
B},l\right)}\ ,$ (676)

where

$\displaystyle F^{\pm}$ $\displaystyle=$ $\displaystyle\left(1+{\cal
A}\right)^{2}\left({\omega^{\pm}}^{2}(l)-l^{2}\right)^{3}\ \mp\ 6{\cal
B}\left({\omega^{\pm}}^{2}(l)-l^{2}\right)^{2}l\ \mp\ \frac{g^{2}}{6}{\cal
B}\left(\frac{2}{3}\langle{E}^{2}\rangle_{T}-\frac{2}{5}\langle{B}^{2}\rangle_{T}\right)l$
$\displaystyle\pm\ \frac{8}{45}g^{2}\left(1+{\cal
A}\right)l\omega^{\pm}(l)\langle{B}^{2}\rangle_{T}\ \ ,$ $\displaystyle
G^{\pm}$ $\displaystyle=$ $\displaystyle-\left(1+{\cal A}\right)C_{\pm}\ ,$
(677)

and ${\cal A}$, ${\cal B}$, and $C_{\pm}$ are given in (625a), (625b) and
(637), respectively. As we will see that the group velocity leads to a
characteristic feature of the dilepton rate. In (674) we have dropped terms
$A_{4}$, $A_{8}$, $B_{4}$ and $B_{8}$ as the corresponding $\delta$-functions
in (LABEL:cb) can never be satisfied by virtue of energy conservation since
$\omega^{\pm}$ is always positive. Now, one can perform the $k$-integration in
(674) numerically, and we find that the terms, which satisfy the energy
conservation, correspond to various physical processes involving two
quasiparticles with different momentum $k$ and $q$.

The dilepton production rate for $\bm{\vec{p}}=0$ is obtained by setting
$\bm{\vec{q}}=-\bm{\vec{k}}$ in (670) as

$\displaystyle\frac{dN}{d^{4}Xd^{4}P}(\bm{\vec{p}}=0)$ $\displaystyle=$
$\displaystyle\frac{20}{9\pi^{4}}\frac{\alpha^{2}}{M^{2}}\int_{0}^{\infty}dk\,k^{2}\left[n_{F}^{2}\left(\omega^{+}(k)\right)R_{+}^{2}\left(\omega^{+}(k)\right)\delta\left(E-2\omega^{+}(k)\right)\right.$
(678) $\displaystyle+$
$\displaystyle\left.2n_{F}\left(\omega^{+}(k)\right)n_{F}\left(-\omega^{-}(k)\right)R_{+}\left(\omega^{+}(k)\right)R_{-}\left(\omega^{-}(k)\right)\delta\left(E-\omega^{+}(k)+\omega^{-}(k)\right)\right.$
$\displaystyle+$
$\displaystyle\left.2n_{F}\left(\omega^{-}(k)\right)n_{F}\left(-\omega^{+}(k)\right)R_{+}\left(\omega^{+}(k)\right)R_{-}\left(\omega^{-}(k)\right)\delta\left(E+\omega^{+}(k)-\omega^{-}(k)\right)\right.$
$\displaystyle+$
$\displaystyle\left.n_{F}^{2}\left(\omega^{-}(k)\right)R_{-}^{2}\left(\omega^{-}(k)\right)\delta\left(E-2\omega^{-}(k)\right)\right]\
.$

First we would like to discuss the dilepton production from a QGP at momentum
$\bm{\vec{p}=0}$ of the virtual photon. The different terms in (678)
correspond to various physical processes involving two quasiparticles ${\rm
q}^{+}$ and ${\rm q}^{-}$ with same momentum $k$. The first term represents
the annihilation process ${\rm q}^{+}\bar{\rm q}^{+}\rightarrow\gamma^{*}$.
The second term corresponds to ${\rm q}^{+}\rightarrow{\rm q}^{-}\gamma^{*}$,
a decay process from a ${\rm q}^{+}$ mode to a plasmino plus a virtual photon.
Energy conservation does not allow the process given by the third term (${\rm
q}^{-}\rightarrow\bar{\rm q}^{+}\gamma^{*}$). Finally, the fourth term
corresponds to a process, ${\rm q}^{-}\bar{\rm q}^{-}\rightarrow\gamma^{*}$,
i.e. annihilation of plasmino modes.

The $k$-integration in (678) can be performed using the standard delta
function identity

$\displaystyle\delta(f(x))$ $\displaystyle=$
$\displaystyle\sum_{i}\frac{\delta(x-x_{i})}{\mid\\!f^{\prime}(x)\\!\mid_{x=x_{i}}},$
(679)

where $x_{i}$ are the solutions of $f(x_{i})=0$. After performing the
$k$-integration in (678) the expression for the dilepton rate at
$\bm{\vec{p}}=0$ becomes

$\displaystyle\frac{dN}{d^{4}Xd^{4}P}(\bm{\vec{p}}=0)$ $\displaystyle=$
$\displaystyle\frac{20}{9\pi^{4}}\frac{\alpha^{2}}{M^{2}}\sum_{k_{s}}k_{s}^{2}\left[n_{F}^{2}\left(\omega^{+}(k_{s})\right)R_{+}^{2}\left(\omega^{+}(k_{s})\right)\frac{1}{2}\left|\frac{d\omega^{+}(k)}{{\rm
d}k}\right|_{k_{s}}^{-1}\right.$ (680) $\displaystyle+$
$\displaystyle\left.2n_{F}\left(\omega^{+}(k_{s})\right)n_{F}\left(-\omega^{-}(k_{s})\right)R_{+}\left(\omega^{+}(k_{s})\right)R_{-}\left(\omega^{-}(k_{s})\right)\left|\frac{d\left(\omega^{+}(k)-\omega^{-}(k)\right)}{dk}\right|_{k_{s}}^{-1}\right.$
$\displaystyle+$
$\displaystyle\left.2n_{F}\left(\omega^{-}(k_{s})\right)n_{F}\left(-\omega^{+}(k_{s})\right)R_{+}\left(\omega^{+}(k_{s})\right)R_{-}\left(\omega^{-}(k_{s})\right)\left|\frac{d\left(\omega^{-}(k)-\omega^{+}(k)\right)}{dk}\right|_{k_{s}}^{-1}\right.$
$\displaystyle+$
$\displaystyle\left.n_{F}^{2}\left(\omega^{-}(k_{s})\right)R_{-}^{2}\left(\omega^{-}(k_{s})\right)\frac{1}{2}\left|\frac{d\omega^{-}(k)}{dk}\right|_{k_{s}}^{-1}\right]\
.$

Figure 40: The dilepton rate from QGP with gluon condensate at virtual photon
momentum $\bm{\vec{p}}=0$. This figure is taken from Ref. MST .

The static differential rate of the aforementioned processes are displayed in
Fig. 40 for $T=1.1T_{c}$ (solid line), $2T_{c}$ (dashed curve) and $4T_{c}$
(dotted curve). Similar to the hard thermal loop case BPY  the differential
rate in the presence of a gluon condensate also shows peaks (van Hove
singularities 101010A van Hove peak VH ; AM  appears where the density of
states diverges due to the vanishing group velocity.) at different invariant
masses of the virtual photon. Now we discuss the contributions to the rate
from each process in detail. The channel, ${\rm q}^{+}\rightarrow{\rm
q}^{-}\gamma^{*}$, opens up at $M=0$. This process continues up to the first
peak appears due to the vanishing group velocity $dE/dk=0$ at the maximum
$E=M=\omega^{+}(k)-\omega^{-}(k)$, since the density of states is inversely
proportional to the group velocity. The ${\rm q}^{+}\rightarrow{\rm
q}^{-}\gamma^{*}$ channel terminates at the peak, after which there is a gap
because neither of the other processes is possible in this invariant mass
regime. The size of the gap depends on the temperature. For $T=1.1T_{c}$ it
ranges from $M=1.01T_{c}$ to 2.07$T_{c}$, for $T=2T_{c}$ from $M=1.83T_{c}$ to
3.73$T_{c}$, and for $T=4T_{c}$ from $M=2.14T_{c}$ to 8.76$T_{c}$.

The process, ${\rm q}^{-}\bar{\rm q}^{-}\rightarrow\gamma^{*}$, starts at an
energy which is twice the energy of the minimum of the plasmino branch,
$E=M=2\omega_{-}(k_{min})$. The diverging density of states at that point
again causes a van Hove singularity BPY ; VH ; AM . This process continues
with increasing $M$ but falls off very fast due to two reasons: i) as $M$
increases the high energy plasmino modes come into the game and the
corresponding square of the residue $R_{-}^{2}(\omega^{-}(k),k)$, to which the
rate is proportional, becomes very small since it is proportional to
$({\omega^{-}}^{2}(k)-k^{2})^{6}$, and ii) with increasing $M$ the density of
states decreases gradually.

At $M=E=2\omega_{+}(k)\geq 2m_{\rm{eff}}$, the process, ${\rm q}^{+}\bar{\rm
q}^{+}\rightarrow\gamma^{*}$, shows up. As $M$ increases, the contribution
from this process grows and dominates over the plasmino annihilation process,
resulting in a dip in the dilepton rate. For large $M$ this annihilation
process is solely responsible for the dilepton rate, which approaches the Born
contribution (${\rm q}\bar{\rm q}$ annihilation of massless quarks) there BPY
; CFR :

$\frac{dN^{\rm{Born}}}{d^{4}Xd^{4}P}(\bm{\vec{p}}=0)=\frac{5}{18\pi^{4}}{\alpha^{2}}e^{-E/T}\
.$ (681)

The reason for this is that for high energy quarks the effective propagator
reduces to the bare one and the contribution to the dilepton rate comes from
hard loop momenta in Fig. 40.

Figure 41: The dilepton rate from QGP with gluon condensate at virtual photon
momentum $\bm{\vec{p}}=T_{c}$. This figure is taken from Ref. MST .

Next, we turn our attention to the dilepton rate at non-zero virtual photon
momentum. The corresponding rate is given in (674). The processes
corresponding to terms $A_{2}$, $A_{3}$, $A_{6}$, $A_{7}$, $B_{6}$ and
$B_{7}$, namely transitions within a branch and transitions from the lower to
the upper branch, do not contribute to the rate, because they are forbidden
for timelike photons decaying into dileptons due to energy conservation SMH .
The processes corresponding to $A_{1}$ and $A_{5}$ indicate annihilation
between a quark (${\rm q}^{+}$) and a plasmino mode (${\rm q}^{-}$) with
different momentum to a virtual photon with energy $E$, which were absent at
$\bm{\vec{p}}=0$. The process given by $B_{1}$ is the annihilation between a
quark and antiquark (${\rm q}^{+}(k)\bar{\rm q}^{+}(q)\rightarrow\gamma^{*}$),
whereas $B_{5}$ corresponds to the annihilation (${\rm q}^{-}(k)\bar{\rm
q}^{-}(q)\rightarrow\gamma^{*}$) between two plasmino modes. The term $B_{2}$
corresponds to the decay process, ${\rm q}^{+}(q)\rightarrow{\rm
q}^{-}(k)\gamma^{*}$, whereas $B_{3}$ corresponds to ${\rm
q}_{+}(k)\rightarrow{\rm q}^{-}(q)\gamma^{*}$. The differential rate involving
these processes are displayed in Fig. 41 for virtual photon momentum $p=T_{c}$
at different temperatures, namely $T=1.1T_{c}$ (solid line), $2T_{c}$ (dashed
line), and $4T_{c}$ (dotted line).

####  13.3.3 Dilepton production rate from QGP with Gribov-Zwanziger action

Figure 42: One-loop photon self-energy diagram (left) and tadpole diagram
(right) with the effective propagators and vertices with GZ action. These
diagrams are taken from Ref. BHMS .

The general features of non-perturbative GZ action have been discussed in
subsec 13.2. Now, in this subsec we want to compute the dilepton production
rate BHMS  with the GZ action from QGP. At one-loop order, the dilepton
production rate is associated with photon self-energy and tadpole diagrams as
shown in Fig. 42. The contributions to the one-loop photon self-energy can be
written from the two diagrams in Fig. 42 as

$\displaystyle\Pi_{\mu}^{\mu}(Q)$ $\displaystyle=$
$\displaystyle-\frac{10}{3}e^{2}T\sum_{\\{p_{0}\\}}\int\frac{d^{3}p}{(2\pi)^{3}}\biggl{\\{}\textmd{Tr}\biggl{[}S^{*}(P)\leavevmode\nobreak\
\Gamma_{\mu}(K,Q,-P)\leavevmode\nobreak\ S^{*}(K)\leavevmode\nobreak\
\Gamma^{\mu}(-K,-Q,P)\biggr{]}$ (682) $\displaystyle+$
$\displaystyle\textmd{Tr}\biggl{[}S^{*}(P)\leavevmode\nobreak\
\Gamma_{\mu}^{\mu}(-P,P;-Q,Q)\biggr{]}\biggr{\\}},$

where $K=P-Q$, $S^{*}$ is the effective quark propagator as given in (651),
$\Gamma^{\mu}$ is the quark-photon vertex as given in (654) and
$\Gamma^{\mu}_{\mu}$ are four-point quark-photon vertex as given in (658). The
second term in (682) is due to the tadpole diagram shown in Fig. 42 which
eventually does not contribute as $\Gamma_{\mu}^{\mu}=0$. However, the tadpole
diagram is essential to satisfy the Ward-Takahashi identity
$Q_{\mu}\Pi^{\mu\nu}(Q)=0$ and thus the gauge invariance and charge
conservation in the system.

Using the $N$-point functions and performing traces, one obtains the photon
self-energy with photon three momentum, $\bm{\vec{q}}=0$ as

$\displaystyle\Pi_{\mu}^{\mu}(\bm{\vec{q}}=0)$ $\displaystyle=$
$\displaystyle-\frac{10}{3}e^{2}T\sum_{p_{0}}\int\frac{d^{3}p}{(2\pi)^{3}}$
(683) $\displaystyle\times\Biggl{[}\left\\{\frac{(a_{G}+b_{G})^{2}}{{\cal
D}_{+}(\omega_{1},p,\gamma_{G}){\cal
D}_{-}(\omega_{2},p,\gamma_{G})}+\frac{(a_{G}-b_{G})^{2}}{{\cal
D}_{-}(\omega_{1},p,\gamma_{G}){\cal
D}_{+}(\omega_{2},p,\gamma_{G})}\right\\}$
$\displaystyle-\left\\{\frac{(c_{G}+b_{G}+d_{G})^{2}}{{\cal
D}_{+}(\omega_{1},p,\gamma_{G}){\cal
D}_{-}(\omega_{2},p,\gamma_{G})}+\frac{(c_{G}-b_{G}+d_{G})^{2}}{{\cal
D}_{-}(\omega_{1},p,\gamma_{G}){\cal
D}_{+}(\omega_{2},p,\gamma_{G})}\right\\}$
$\displaystyle-2c_{G}^{2}\left\\{\frac{1}{{\cal
D}_{+}(\omega_{1},p,\gamma_{G}){\cal
D}_{+}(\omega_{2},p,\gamma_{G})}+\frac{1}{{\cal
D}_{-}(\omega_{1},p,\gamma_{G}){\cal
D}_{-}(\omega_{2},p,\gamma_{G})}\right\\}\Biggr{]},$

where ${\cal D}_{\pm}$ are given, respectively, in (652a) and (652b) whereas
$a_{G}$, $b_{G}$, $c_{G}$ and $d_{G}$ are given, respectively, in (656a),
(656b), (656c) and (656d).

Now using the BPY prescription BPY  given in (668) or in (708) in appendix A.3
we first find out the imaginary or discontinuous part of the (683) and then
performing some more algebra, we write down the dilepton production rate with
massless leptons following (665) as

$\displaystyle\frac{dR}{d\omega d^{3}q}(\bm{\vec{q}}=0)$ $\displaystyle=$
$\displaystyle\frac{20\alpha^{2}}{9\pi^{4}}\frac{1}{\omega^{2}}\int\limits_{0}^{\infty}p^{2}dp\int\limits_{-\infty}^{\infty}d\omega_{1}\int\limits_{-\infty}^{\infty}d\omega_{2}n_{F}(\omega_{1})n_{F}(\omega_{2})\delta(\omega-\omega_{1}-\omega_{2})$
(684)
$\displaystyle\Bigg{[}4\left(1-\frac{\omega_{1}^{2}-\omega_{2}^{2}}{2p\,\omega}\right)^{2}\rho_{+}^{G}(\omega_{1},p)\rho_{-}^{G}(\omega_{2},p)$
$\displaystyle+\left(1+\frac{\omega_{1}^{2}+\omega_{2}^{2}-2p^{2}-2m_{q}^{2}(\gamma_{G})}{2p\,\omega}\right)^{2}\rho_{+}^{G}(\omega_{1},p)\rho_{+}^{G}(\omega_{2},p)$
$\displaystyle+\left(1-\frac{\omega_{1}^{2}+\omega_{2}^{2}-2p^{2}-2m_{q}^{2}(\gamma_{G})}{2p\,\omega}\right)^{2}\rho_{-}^{G}(\omega_{1},p)\rho_{-}^{G}(\omega_{2},p)\Bigg{]},$

where $\omega$ is the photon energy and $\rho^{G}_{\pm}$ are the spectral
functions in presence of Gribov term given in (653). Using (653) one can
obtain the dilepton production rate as

$\displaystyle\frac{dR}{d\omega d^{3}q}\Big{|}^{pp}({\vec{q}}=0)$
$\displaystyle=$
$\displaystyle\frac{20\alpha^{2}}{9\pi^{4}}\frac{1}{\omega^{2}}\int\limits_{0}^{\infty}p^{2}\,dp\times$
(685) $\displaystyle\Biggl{[}\delta(\omega-2\omega_{+})\
n_{F}^{2}(\omega_{+})\left(\frac{\omega_{+}^{2}-p^{2}}{2m_{q}^{2}(\gamma_{G})}\right)^{2}\left\\{1+\frac{\omega_{+}^{2}-p^{2}-m_{q}^{2}(\gamma_{G})}{p\leavevmode\nobreak\
\omega}\right\\}^{2}$ $\displaystyle+\leavevmode\nobreak\
\delta(\omega-2\omega_{-})\
n_{F}^{2}(\omega_{-})\left(\frac{\omega_{-}^{2}-p^{2}}{2m_{q}^{2}(\gamma_{G})}\right)^{2}\left\\{1-\frac{\omega_{-}^{2}-p^{2}-m_{q}^{2}(\gamma_{G})}{p\leavevmode\nobreak\
\omega}\right\\}^{2}$ $\displaystyle+\leavevmode\nobreak\
\delta(\omega-2\omega_{G})\
n_{F}^{2}(\omega_{G})\left(\frac{\omega_{G}^{2}-p^{2}}{2m_{q}^{2}(\gamma_{G})}\right)^{2}\left\\{1-\frac{\omega_{G}^{2}-p^{2}-m_{q}^{2}(\gamma_{G})}{p\leavevmode\nobreak\
\omega}\right\\}^{2}$ $\displaystyle+4\ \delta(\omega-\omega_{+}-\omega_{-})\
n_{F}(\omega_{+})\
n_{F}(\omega_{-})\left(\frac{\omega_{+}^{2}-p^{2}}{2m_{q}^{2}(\gamma_{G})}\right)\left(\frac{\omega_{-}^{2}-p^{2}}{2m_{q}^{2}(\gamma_{G})}\right)$
$\displaystyle\times\left\\{1-\frac{\omega_{+}^{2}-\omega_{-}^{2}}{2p\,\omega}\right\\}^{2}$
$\displaystyle+\delta(\omega-\omega_{+}+\omega_{-})\
n_{F}(\omega_{+})n_{F}(-\omega_{-})\left(\frac{\omega_{+}^{2}-p^{2}}{2m_{q}^{2}(\gamma_{G})}\right)\left(\frac{\omega_{-}^{2}-p^{2}}{2m_{q}^{2}(\gamma_{G})}\right)$
$\displaystyle\times\left\\{1+\frac{\omega_{+}^{2}+\omega_{-}^{2}-2p^{2}-2m_{q}^{2}(\gamma_{G})}{2p\,\omega}\right\\}^{2}\leavevmode\nobreak\
\Biggr{]}.$

The momentum integration in (685) can be performed using the standard delta
function identity given in (679). Now, inspecting the arguments of the various
energy conserving $\delta$-functions in (685) one can understand the physical
processes originating from the poles of the propagator. The first three terms
in (685) correspond to the annihilation processes of
$q_{+}{\bar{q}}_{+}\rightarrow\gamma^{*}$,
$q_{-}{\bar{q}}_{-}\rightarrow\gamma^{*}$, and
$q_{G}{\bar{q}}_{G}\rightarrow\gamma^{*}$, respectively. The fourth term
corresponds to the annihilation of $q_{+}{\bar{q}}_{-}\rightarrow\gamma^{*}$.
On the other hand, the fifth term corresponds to a process, $q_{+}\rightarrow
q_{-}\gamma*$, where a $q_{+}$ mode makes a transition to a $q_{-}$ mode along
with a virtual photon. These processes involve soft quark modes
($q_{+},\,q_{-}$, and $q_{G}$ and their antiparticles) which originate by
cutting the self-energy diagram in Fig. 42 through the internal lines without
a “blob”. The virtual photon, $\gamma^{*}$, in all these five processes decays
to lepton pair and can be visualized from the dispersion plot as displayed in
the Fig. 43.

Figure 43: Various dilepton processes originate from the in-medium dispersion
of quasiparticles with Gribov term are displayed. This figure is taken from
Ref. BHMS . Figure 44: The dilepton production rates corresponding to
quasiparticle processes in Fig. 43. This figure is taken from Ref. BHMS .
Figure 45: Comparison of dilepton production rates involving various
quasiparticle modes with and without inclusion of $\gamma_{G}$. This figure is
taken from Ref. BHMS .

The contribution of various individual processes to the dilepton production
rate in presence of the Gribov term are displayed in the Fig. 44. The
transition process, $q_{+}\rightarrow q_{-}\gamma*$, begins at the energy
$\omega=0$ and ends up with a van-Hove peak where all of the transitions from
$q_{+}$ branch are directed towards the minimum of the $q_{-}$ branch. The
annihilation process involving the massless spacelike Gribov modes,
$q_{G}{\bar{q}}_{G}\rightarrow\gamma^{*}$, also starts at $\omega=0$ and
falls-off very quickly. The annihilation of the two plasmino modes,
$q_{-}{\bar{q}}_{-}\rightarrow\gamma^{*}$, opens up with again a van-Hove peak
at $\omega=2\times$ the minimum energy of the plasmino mode. The contribution
of this process decreases exponentially. At $\omega=2m_{q}(\gamma_{G})$, the
annihilation processes involving usual quark modes,
$q_{+}{\bar{q}}_{+}\rightarrow\gamma^{*}$, and that of a quark and a plasmino
mode, $q_{+}{\bar{q}}_{-}\rightarrow\gamma^{*}$, begin. However, the former
one ($q_{+}{\bar{q}}_{+}\rightarrow\gamma^{*}$) grows with the energy and
would converge to the usual Born rate (leading order perturbative rate) CFR
at high mass whereas the later one ($q_{+}{\bar{q}}_{-}\rightarrow\gamma^{*}$)
initially grows at a very fast rate, but then decreases slowly and finally
drops very quickly. The behavior of the latter process can easily be
understood from the dispersion properties of quark and plasmino mode. Summing
up, the total contribution of all theses five processes is displayed in Fig.
45. This is compared with the similar dispersive contribution when
$\gamma_{G}=0$ BPY , comprising processes $q_{+}\rightarrow q_{-}\gamma*$,
$q_{+}{\bar{q}}_{+}\rightarrow\gamma^{*}$,
$q_{-}{\bar{q}}_{-}\rightarrow\gamma^{*}$ and
$q_{+}{\bar{q}}_{-}\rightarrow\gamma^{*}$. We note that when $\gamma_{G}=0$,
the dilepton rate contains both van-Hove peaks and an energy gap BPY . In
presence of the Gribov term ($\gamma_{G}\neq 0$), the van-Hove peaks remain,
but the energy gap disappears due to the annihilation of new massless Gribov
modes, $q_{G}{\bar{q}}_{G}\rightarrow\gamma^{*}$. This new contribution could
be important for low mass dilepton spectra.

Figure 46: Comparison of various dilepton production rates from the deconfined
matter. This figure is taken from Ref. BHMS .

In Fig. 46 we compare the rates obtained using various approximations: