Doing the $l_{\perp}$ integration, assuming that $\rho_{\perp}\ll
k_{\perp}\ll\mu_{\perp}$, while at the same time taking $\rho_{\perp}\gg
l^{-},\,m_{D}$ yields

$\hat{q}(\rho_{\perp};l^{-})_{\text{subtr}}=\alpha_{s}C_{A}T\bigg{\\{}\underbrace{m_{D}^{2}\ln\left(\frac{\rho_{\perp}^{2}}{m_{D}^{2}}\right)}_{\mathrm{HO}}\,-\underbrace{l^{-2}\ln\left(1+\frac{m_{D}^{2}}{l^{-2}}\right)-m_{D}^{2}\ln\left(1+\frac{l^{-2}}{m_{D}^{2}}\right)}_{l^{-}\,\text{--
dependent}}\bigg{\\}}.$ (5.39)

####  5.4.1 HOA Contribution to Strict Single Scattering

Let us for the moment only concentrate on the first term. The UV logarithmic
divergence, characterised by $\rho_{\perp}$ signals the fact that there lies
another region of phase space bordering this one that could also give rise to
double logarithmic corrections to $\delta\hat{q}(\mu_{\perp})$. We will come
back to this issue. Applying the HOA to the first term above, we get

$\hat{q}_{0}(\rho_{\perp})=\alpha_{s}C_{A}Tm_{D}^{2}\ln\left(\frac{\rho_{\perp}^{2}}{m_{D}^{2}}\right)\to\hat{q}_{0},$
(5.40)

where we have inferred the value of $\hat{q}_{0}$ from the single-scattering
condition, $\rho_{\perp}\ll\mu_{\perp}$. This approximation is what has been
done to arrive at Eq. (5.14). To get the double logarithmic correction, we
then only need to do the $k^{+}$ and $k_{\perp}$ integrals. As was done in
Sec. 5.2, we instead do the integration in terms of the formation time, $\tau$

$\delta\hat{q}_{\text{semi}}^{\text{HO}}(\mu_{\perp})=\frac{\alpha_{s}C_{R}}{\pi}\hat{q}_{0}\int_{k^{+}_{\text{IR}}}^{\mu_{\perp}^{2}\tau_{\text{int}}}\frac{dk^{+}}{k^{+}}\left(1+2n_{\text{B}}\left(k^{+}\right)\right)\int_{k^{+}/\mu_{\perp}^{2}}^{\tau_{\text{int}}}\frac{d\tau}{\tau}$
(5.41)

The integration limits match those that come from integrating over regions $3$
and $4$ of Fig. 5.4151515Technically, the integration here assumes a generic
IR cutoff for the energy, satisfying $gT\ll k_{\text{IR}}\ll T$. Equating
$k_{\text{IR}}=\mu_{\perp}^{2}\tau_{\text{min}}$ would precisely match the
boundaries of regions $3$ and $4$ in Fig. 5.4. Our reason for keeping
$k_{\text{IR}}$ generic is that it will allow us to make cleaner contact with
the region, from which the classical corrections emerge in Sec. 5.8.. The
integration here can be dealt with in exactly the same way as for Eq. (5.20).
The result reads

$\displaystyle\delta\hat{q}_{\mathrm{semi}}^{\text{HO}}(\mu_{\perp})$
$\displaystyle=\frac{\alpha_{s}C_{R}}{2\pi}\hat{q}_{0}\bigg{[}\frac{4T\ln\left(\frac{\mu_{\perp}^{2}\tau_{\text{int}}}{k^{+}_{\text{IR}}e}\right)}{k^{+}_{\text{IR}}}+\left(\ln^{2}\left(\frac{\mu_{\perp}^{2}\tau_{\text{int}}}{\omega_{T}}\right)-2\gamma_{1}+\frac{\pi^{2}}{4}-\gamma_{E}^{2}\right)\bigg{]}$
$\displaystyle+\mathcal{O}\left(k^{+}_{\text{IR}},e^{-\mu_{\perp}\tau_{\text{int}}^{2}/T}\right),$
(5.42)

where $\gamma_{n}$ is the nth Stieltjes constant. The terms above depending on
the IR cutoff, $k^{+}_{\text{IR}}$ indeed suggest sensitivity to the region
giving rise to classical corrections [11]. We will elaborate on this further
in Sec. 5.8. For completeness, we specify the double logarithmic contribution

$\delta\hat{q}_{\mathrm{semi\,dlog}}^{\text{HO}}(\mu_{\perp})=\frac{\alpha_{s}C_{R}}{2\pi}\hat{q}_{0}\ln^{2}\frac{\mu_{\perp}^{2}\tau_{\text{int}}}{\omega_{T}}$
(5.43)

We now add this result to that from in the previous section, Eq. (5.27), the
double logarithmic contribution coming from integrating over the region for
which $\tau\geq\tau_{\text{int}}$

$\delta\hat{q}(\mu_{\perp})_{\text{dlog}}=\delta\hat{q}^{\text{few}}_{\text{dlog}}(\mu_{\perp})+\delta\hat{q}_{\mathrm{semi\,dlog}}^{\text{HO}}(\mu_{\perp})=\frac{\alpha_{s}C_{R}}{4\pi}\hat{q}_{0}\ln^{2}\frac{\mu_{\perp}^{4}}{\hat{q}_{0}\omega_{T}}.$
(5.44)

Upon adding the two contributions together, it is reassuring to observe the
disappearance of the intermediate cutoff, $\tau_{\text{int}}$. Moreover, with
respect to Eq. (5.17), the IR regulator $\tau_{\text{min}}$ has been replaced
with the thermal scale, $\omega_{T}$. This is one of the main results of this
thesis; by carefully including the contribution from the thermal scale, we
have bypassed many of the issues associated with $\tau_{\text{min}}$, which
were highlighted in Sec. 5.3. As we will demonstrate in the next section, Eq.
(5.44) turns out to be the dominant contribution to overall double logarithmic
correction to $\hat{q}(\mu_{\perp})$.

It is interesting to note that the correction above corresponds (up to the
prefactor of $\alpha_{s}C_{R}\hat{q}_{0}/\pi$) to the area of the sum of the
“1” and “3” regions from Fig. 5.4. This observation that neither region “2”
nor “4 ”contribute makes transparent the fact that, up to double-logarithmic
accuracy, the effect of a populated medium can be understood as cutting off
the phase space with a vertical, $k^{+}>\omega_{T}\sim T$ line instead of a
horizontal $\tau>\tau_{\text{min}}$ line.

Why does the thermal scale have such an effect? To start, we note that for
$k^{+}\ll T$

$2n_{\text{B}}(k^{+})\approx\frac{2T}{k^{+}}-1.$ (5.45)

Looking to Eq. (5.41), we note that in this limit, the part of the $k^{+}$
integrand giving rise to the logarithmic enhancement will be washed away; the
vacuum part – $1$ – cancels against the first quantum correction coming from
the expanded $n_{\text{B}}(k^{+})$161616Analogous cancellations of this kind
are known in the literature: they are discussed briefly and in general terms
in [295]. Similar cancellations also appear in the power corrections to Hard
Thermal Loops, which receive both vacuum and thermal contributions[296, 297].
These are separately IR-divergent but their sum is IR-finite.. This is exactly
what is exhibited in Eq. (5.4.1), where $k_{\text{IR}}^{+}$ only appears in
the power law term171717 Despite the presence of $k_{\text{IR}}^{+}$ in a
single logarithm in Eq. (5.4.1), that term can still be considered benign as
it multiplies a power of $k_{\text{IR}}^{+}$. As was discussed in Sec. 4.1,
corrections that depend on powers of the cutoff are unphysical and will
inevitably cancel against other power law corrections from other regions. We
explicitly show how this works with this power law correction in Sec. 5.8..

####  5.4.2 $l^{-}$ – Dependent Contrbution to Strict Single Scattering

We now turn to the calculation of the $l^{-}$ – dependent terms in Eq. (5.39).
As we will see, this $l^{-}$ dependence renders the integration a little more
challenging. Before beginning its evaluation, we use the delta function in Eq.
(5.36) to rewrite everything in terms of the formation time, i.e
$\tau=k^{+}/k_{\perp}^{2}=1/(2l^{-})$. The starting point is

$\displaystyle\mathcal{I}(\mu_{\perp})\equiv\int_{k^{+}_{\text{IR}}}^{\mu_{\perp}^{2}\tau_{\text{int}}}\frac{dk^{+}}{k^{+}}\left(\frac{1}{2}+n_{\text{B}}(k^{+})\right)\int_{k^{+}/\mu_{\perp}^{2}}^{\tau_{\text{int}}}\frac{d\tau}{\tau}\left[-\frac{1}{4\tau^{2}}\ln\left(1+4m_{D}^{2}\tau^{2}\right)-m_{D}^{2}\ln\left(1+\frac{1}{4m_{D}^{2}\tau^{2}}\right)\right].$
(5.46)

The $\tau$ integration yields

$\displaystyle\mathcal{I}(\mu_{\perp})=\int_{k^{+}_{\text{IR}}}^{\mu_{\perp}^{2}\tau_{\text{int}}}\frac{dk^{+}}{k^{+}}\left(\frac{1}{2}+n_{\text{B}}(k^{+})\right)\bigg{[}$
$\displaystyle\frac{m_{D}^{2}}{2}\mathrm{Li}_{2}\left(-\frac{\mu_{\perp}^{4}}{4k^{+2}m_{D}^{2}}\right)-\frac{m_{D}^{2}}{2}\ln\left(1+\frac{\mu_{\perp}^{4}}{4k^{+2}m_{D}^{2}}\right)$
$\displaystyle-\frac{\mu_{\perp}^{4}}{8k^{+2}}\ln\left(1+\frac{4k^{+2}m_{D}^{2}}{\mu_{\perp}^{4}}\right)+\mathcal{O}\left(\frac{1}{\tau_{\text{int}}^{2}}\right)\bigg{]},$
(5.47)

where $\mathrm{Li}_{2}$ is the dilogarithm and we have expanded for
$m_{D}\tau_{\text{int}}\gg 1$, recalling that $1/gT\ll\tau_{\text{int}}\ll
1/g^{2}T$. The vacuum ($1/2$) part needs to be integrated as is, without
further expansions: while close to the IR boundary we could exploit that
$\mu_{\perp}^{2}/(2k^{+}m_{D})\gg 1$, that would not be true close to the UV
boundary, where instead $\mu_{\perp}^{2}/(2k^{+}m_{D})\ll 1$. The integral is
however not problematic and yields

$\displaystyle\mathcal{I}_{\mathrm{vac}}(\mu_{\perp})=$
$\displaystyle-\frac{1}{6}m_{D}^{2}\bigg{[}\ln^{3}\left(\frac{\mu_{\perp}^{2}}{2k^{+}_{\text{IR}}m_{D}}\right)+\frac{3}{2}\ln^{2}\left(\frac{\mu_{\perp}^{2}}{2k^{+}_{\text{IR}}m_{D}}\right)+\frac{1}{4}\left(6+\pi^{2}\right)\ln\left(\frac{\mu_{\perp}^{2}}{2k^{+}_{\text{IR}}m_{D}}\right)+\frac{1}{8}\left(6+\pi^{2}\right)\bigg{]}$
$\displaystyle+\mathcal{O}\left(\frac{1}{\tau_{\text{int}}^{2}},(k^{+}_{\text{IR}})^{2}\right),$
(5.48)

where we have again expanded in the cutoffs. For the thermal part, we can on
the other hand expand for $\mu_{\perp}^{2}/(2k^{+}m_{D})\gg 1$, as the
effective UV cutoff introduced by Boltzmann suppression makes the region where
$\mu_{\perp}^{2}/(2k^{+}m_{D})\lesssim 1$ exponentially suppressed. Hence

$\displaystyle\mathcal{I}_{T}(\mu_{\perp})=-m_{D}^{2}\int_{k^{+}_{\text{IR}}}^{\infty}\frac{dk^{+}}{k^{+}}n_{\text{B}}(k^{+})\bigg{[}$
$\displaystyle\ln^{2}\frac{\mu_{\perp}^{2}}{2k^{+}m_{D}}+\ln\frac{\mu_{\perp}^{2}}{2k^{+}m_{D}}+\frac{1}{2}\left(1+\frac{\pi^{2}}{6}\right)\bigg{]}+\ldots,$
(5.49)

where the dots stand for higher-order terms in the expansions in the cutoffs
and for the exponentially suppressed term arising from approximating the UV
cutoff to infinity. This integral can be done with Eqs. (5.21), (5.24) and

$\int_{0}^{\infty}\frac{dk^{+}}{k^{+}}k^{+\epsilon}n_{\text{B}}(k^{+})\ln^{2}\frac{k^{+}}{T}=T^{\epsilon}\Gamma(\epsilon)\left[\zeta^{\prime\prime}(\epsilon)+2\psi(\epsilon)\zeta^{\prime}(\epsilon)+\zeta(\epsilon)\left(\psi(\epsilon)^{2}+\psi^{(1)}(\epsilon)\right)\right],$
(5.50)

where $\psi^{(1)}$ is the polygamma function of order 1. Using these three
integrals we find

$\displaystyle\mathcal{I}_{T}(\mu_{\perp})=$
$\displaystyle-\frac{m_{D}^{2}T\left(\ln^{2}\left(\frac{\mu_{\perp}^{2}}{2k^{+}_{\text{IR}}m_{D}}\right)-\ln\left(\frac{\mu_{\perp}^{2}}{2k^{+}_{\text{IR}}m_{D}}\right)+\frac{\pi^{2}}{12}+\frac{3}{2}\right)}{k^{+}_{\text{IR}}}$
$\displaystyle+\frac{m_{D}^{2}}{6}\bigg{\\{}\ln^{3}\left(\frac{\mu_{\perp}^{2}}{2k^{+}_{\text{IR}}m_{D}}\right)-\ln^{3}\left(\frac{\mu_{\perp}^{2}}{2m_{D}\omega_{T}}\right)+\frac{3}{2}\ln^{2}\left(\frac{\mu_{\perp}^{2}}{2k^{+}_{\text{IR}}m_{D}}\right)-\frac{3}{2}\ln^{2}\left(\frac{\mu_{\perp}^{2}}{2m_{D}\omega_{T}}\right)$
$\displaystyle+\frac{1}{4}\left(6+\pi^{2}\right)\ln\left(\frac{\mu_{\perp}^{2}}{2k^{+}_{\text{IR}}m_{D}}\right)+\ln\left(\frac{\mu_{\perp}^{2}}{2m_{D}\omega_{T}}\right)\left[6\gamma_{1}-\pi^{2}+3\gamma_{E}^{2}-\frac{3}{2}\right]$
$\displaystyle-6\gamma_{E}\gamma_{1}-3\left(-\gamma_{1}+\gamma_{2}+\frac{\pi^{2}}{8}\right)+\frac{1}{2}(3-4\gamma_{E})\gamma_{E}^{2}\bigg{\\}}+\ldots\,.$
(5.51)

Upon summing Eqs. (5.48) and (5.51) we see that all logarithmic dependence on
$k^{+}_{\text{IR}}$ vanishes, yielding

$\displaystyle\mathcal{I}(\mu_{\perp})=$
$\displaystyle-\frac{m_{D}^{2}T\left(\ln^{2}\left(\frac{\mu_{\perp}^{2}}{2k^{+}_{\text{IR}}m_{D}}\right)-\ln\left(\frac{\mu_{\perp}^{2}}{2k^{+}_{\text{IR}}m_{D}}\right)+\frac{\pi^{2}}{12}+\frac{3}{2}\right)}{k^{+}_{\text{IR}}}$
$\displaystyle+\frac{m_{D}^{2}}{6}\bigg{\\{}-\ln^{3}\left(\frac{\mu_{\perp}^{2}}{2m_{D}\omega_{T}}\right)-\frac{3}{2}\ln^{2}\left(\frac{\mu_{\perp}^{2}}{2m_{D}\omega_{T}}\right)+\ln\left(\frac{\mu_{\perp}^{2}}{2m_{D}\omega_{T}}\right)\left[6\gamma_{1}-\pi^{2}+3\gamma_{E}^{2}-\frac{3}{2}\right]$
$\displaystyle-6\gamma_{E}\gamma_{1}-3\left(-\gamma_{1}+\gamma_{2}\right)+\frac{1}{2}(3-4\gamma_{E})\gamma_{E}^{2}-\frac{1}{4}\left(3+2\pi^{2}\right)\bigg{\\}}+\ldots\,.$
(5.52)

Upon reinstating the proper prefactor we have

$\delta\hat{q}_{\mathrm{semi}}^{l^{-}-\mathrm{dep}}(\mu_{\perp})=\frac{2\alpha_{s}^{2}C_{R}C_{A}T}{\pi}\mathcal{I}(\mu_{\perp}),$
(5.53)

so that

$\delta\hat{q}_{\text{semi
dlog}}^{l^{-}-\mathrm{dep}}(\mu_{\perp})=-\frac{\alpha_{s}^{2}C_{R}C_{A}Tm_{D}^{2}}{3\pi}\ln^{3}\frac{e^{1/2}\mu_{\perp}^{2}}{2\omega_{T}m_{D}}.$
(5.54)

Let us compare this result to Eq. (5.44). Reinstating the Coulomb logarithm in
the latter, it becomes, up to its overall prefactor,

$\delta\hat{q}_{\mathrm{semi\,dlog}}^{\text{HO}}(\mu_{\perp})\propto\ln\frac{\rho_{\perp}^{2}}{m_{D}^{2}}\ln^{2}\frac{\mu_{\perp}^{2}}{\sqrt{\hat{q}_{0}\omega_{T}}}.$
(5.55)

Eq. (5.54) in parametrically smaller compared to
$\delta\hat{q}_{\mathrm{semi\,dlog}}^{\text{HO}}(\mu_{\perp})$. This can be
seen by noting that $\mu_{\perp}^{2}/\sqrt{\hat{q}_{0}\omega_{T}}$ is larger
by $1/g$ than $\mu_{\perp}^{2}/(m_{D}\omega_{T})$, which is instead comparable
to $\rho_{\perp}^{2}/m_{D}^{2}$, since $\rho_{\perp}\ll\mu_{\perp}$. We thus
emphasise that effects analysed in the semi-collinear evaluation, such as the
relaxation of the instantaneous approximation are not relevant at double-
logarithmic accuracy in the HOA.

###  5.5 Extension to $\mu_{\perp}>T$

Up until this point, we have taken $T\gg\mu_{\perp}\gg\sqrt{g}T$, the first
range where double-logarithm corrections appear. We now move on to consider
what happens if $\mu_{\perp}\gg T$ in terms of the structure of the relevant
portion of phase space.

If we start from Fig. 5.4 and proceed to increase $\mu_{\perp}$, line $(a)$
gets shifted to the right, making the original $(a)-(b)-(c)$ triangle larger.
As we argued previously, once
$\mu_{\perp}>\sqrt{\omega_{T}/\tau_{\text{min}}}\sim T$ the vertical
$k^{+}=\omega_{T}$ line intersects lines $(b)$ and $(c)$ rather than $(b)$ and
$(a)$. Hence, nothing would change for the part of the evaluation for
$\tau>\tau_{\text{int}}$, as given by Eq. (5.27). On the other hand, the
evaluation of the strict single scattering regime, for
$\tau<\tau_{\text{int}}$, needs to be amended to account for this. It must
also incorporate that for $\mu_{\perp}\gtrsim T$ we are including strict
single scatterings where $l_{\perp}$ is approaching or possibly exceeding the
$T$ scale. In principle, one would thus need to account for the fact that the
coefficient of the leading-log (harmonic oscillator) $\hat{q}_{0}$ is not
constant. As shown in [174] and summarized in Sec. 3.4, $\hat{q}_{0}(\rho\ll
T)$ and $\hat{q}_{0}(\rho\gg T)$ differ at leading-log by 15% for $N_{f}=3$
and $N_{c}=3$, so in a first approximation we may neglect this effect. We
leave a more precise evaluation of semi-collinear processes in this region to
future work.

12356

Figure 5.6: Structure of phase space once the cutoff on $\mu_{\perp}$ is
allowed to increase to values much larger than the temperature. Notably,
region “6” in dark green emerges. It cannot be well described by our setup due
to the associated small formation time.

Under this approximation, we can simply extend Eq. (5.44) to the present case.
This would however include the dark green triangle “6” in Fig. 5.6, which lies
under the $\tau=\tau_{\text{min}}$ line. It is not clear whether this sharp
line needs to be included, and only an evaluation like the one we show in
detail in the previous section, but extended to larger $\mu_{\perp}$ can
address the effect of the relaxation of the instantaneous approximation in
this setting. The expectation from the previous $\mu_{\perp}<T$ results is
that this effect should be subleading. We however choose to proceed
conservatively here and subtract that slice from our result. Moreover, parts
of this triangle lie below the $\tau k^{+}=1$ line, further motivating its
subtraction. In our double-logarithmic approximation, this corresponds again,
for the reasons explained at the end of Sec. 5.4.1, to subtracting off the
area of the dark green triangle. This leads to

$\displaystyle\delta\hat{q}(\mu_{\perp}\gg T)_{\mathrm{dlog}}$
$\displaystyle=\frac{\alpha_{s}C_{R}}{2\pi}\hat{q}_{0}(\rho_{\perp}\ll\mu_{\perp})\left[\frac{1}{2}\ln^{2}\left(\frac{\mu^{4}}{\hat{q}_{0}\omega_{T}}\right)-\ln^{2}\left(\frac{\mu_{\perp}^{2}\tau_{\text{min}}}{\omega_{T}}\right)\right]$
$\displaystyle=\frac{\alpha_{s}C_{R}}{2\pi}\hat{q}_{0}(\rho_{\perp}\ll\mu_{\perp})\left[\ln^{2}\left(\frac{\mu^{2}}{\hat{q}_{0}\tau_{\text{min}}}\right)-\frac{1}{2}\ln^{2}\left(\frac{\omega_{T}}{\hat{q}_{0}\tau_{\text{min}}^{2}}\right)\right].$
(5.56)

Unsurprisingly, the subtraction of the “6” triangle from the “1+3+6” triangle
on the first line is equal to that of the unshaded “2+5” triangle from the
original $(a)-(b)-(c)$ triangle on the second line. This implies that, for
$\mu_{\perp}>T$, our result is closer to the original one of [1].

This can be better appreciated by noting that the vertical line at
$k^{+}=\mu_{\perp}^{2}\tau_{\text{min}}$ cuts the $(a)-(b)-(c)$ triangle into
two triangles of equal area. Hence, for
$\omega_{T}>\mu_{\perp}^{2}\tau_{\text{min}}$, i.e. $\mu_{\perp}\lesssim T$
our result (5.44) is less than half of the original double logarithm (5.17),
whereas for $\mu_{\perp}>\sqrt{\omega_{T}/\tau_{\text{min}}}$ our result
(5.56) is more than half of it.

###  5.6 Analysis of Neighbouring Regions

In the previous section, we showed that the dominant contribution to double
logarithmic corrections to $\hat{q}(\mu_{\perp})$ in a weakly coupled QGP are
given by the result Eq. (5.44). On the way to arriving at this result, we took
the HOA; the dependence on $\rho_{\perp}$, the UV cutoff for $l_{\perp}$ in
the first term of Eq. (5.39) signals sensitivity to a neighbouring region with
larger $l_{\perp}$, which could also potentially possess a double logarithmic
enhancement. In this section, we explore a couple of adjacent regions of phase
space and comment on whether or not they are relevant with respect to our
ultimate pursuit.

####  5.6.1 $l_{\perp}\sim\sqrt{g}T,\,l^{+}\sim l^{-}\sim gT$

Since we now have $L^{2}\gg m_{D}^{2}$, we can start by expanding
$G_{rr}^{--}(L)$, which will inevitably appear in our evaluation 181818This
can be understood as a consequence of the fact that in Fig. 5.2, the only non-
zero contributions are those with horizontal cuts going through the HTL
propagators, which produce either a $G_{>}^{--}(L)$ or $G_{<}^{--}(L)$. The
cut propagators can then always be traded for an $rr$ propagator, since
$l^{0}\ll T$ (see Eq. (4.55)).

$\displaystyle G_{rr}^{--}(L)$
$\displaystyle=\Big{(}\frac{1}{2}+n_{\text{B}}(l^{0})\Big{)}\Big{(}\rho_{\text{B}}^{L}(l)+\frac{l_{\perp}^{2}}{l^{2}}\rho_{\text{B}}^{T}(l)\Big{)}$
$\displaystyle\approx\frac{T}{l^{0}}\Big{(}\rho_{\text{B}}^{L}(l)+\frac{l_{\perp}^{2}}{l^{2}}\rho_{\text{B}}^{T}(l)\Big{)}.$

The expansion ($l^{0}\ll T$) is also valid for the scaling presented in the
previous section. However, looking in more detail at the spectral function, we
see that

$\displaystyle\rho^{L}_{\text{B}}(l)$
$\displaystyle=G_{R}^{L}(l)-G_{A}^{L}(l)$
$\displaystyle=\frac{i}{l^{2}+\Pi_{R}^{L}(l)}-\frac{i}{l^{2}+\Pi_{A}^{L}(l)}$
$\displaystyle\approx\frac{i}{l^{2}}-\frac{i\Pi_{R}^{L}(l)}{l^{4}}-\frac{i}{l^{2}}+\frac{i\Pi_{A}^{L}(l)}{l^{4}}$

where the causal HTL propagators have been defined in Eqs. (4.72),
(LABEL:htltrans). Let us now further unpack the causal self energies

$\displaystyle\Pi_{R/A}^{L}(l)$
$\displaystyle=m_{D}^{2}\Big{(}1-\frac{l^{0}\pm
i\varepsilon}{2l}\log\frac{l^{0}+l\pm i\varepsilon}{l^{0}-l\pm
i\varepsilon}\Big{)}$ $\displaystyle=m_{D}^{2}\Big{(}1-\frac{l^{0}\pm
i\varepsilon}{2l}\Big{(}\log|\frac{l^{0}+l}{l^{0}-l}\Big{|}+i\arctan\frac{\mp
2l\varepsilon}{L^{2}}\Big{)}\Big{)}$
$\displaystyle=m_{D}^{2}\Big{(}1-\frac{l^{0}}{2l}\Big{(}\log|\frac{l^{0}+l}{l^{0}-l}\Big{|}\mp
i\pi\Theta(-L^{2})\Big{)}\Big{)}$ (5.57)

where in going to the last line we have taken $\varepsilon\to 0$. When taking
the difference of the retarded and advanced self energies, only the imaginary
term survives and so

$\displaystyle\rho^{L}_{\text{B}}(l)$ $\displaystyle=\frac{m_{D}^{2}\pi
l^{0}}{l^{5}}\Theta(-L^{2})$ $\displaystyle\approx\frac{m_{D}^{2}\pi
l^{0}}{l_{\perp}^{5}}$ (5.58)

where we have used that $L$ is spacelike in going to the second line.
Similarly, the transverse spectral function can be computed to give191919In
this case, it is not only the imaginary term that survives when
$\Pi^{L}_{R}-\Pi^{L}_{A}$ computed. However, the surviving real part will be
proportional to $\delta(L^{2})\approx\delta(l_{\perp}^{2})$, which will always
be zero in this scaling.

$\displaystyle\rho_{\text{B}}^{T}(l)$ $\displaystyle=\frac{m_{D}^{2}\pi
l^{0}}{2lL^{4}}\Big{(}1-\frac{{l^{0}}^{2}}{l^{2}}\Big{)}\Theta(-L^{2})$ (5.59)
$\displaystyle\approx\frac{m_{D}^{2}\pi l^{0}}{2l_{\perp}^{5}}$ (5.60)

Next, we can go back to Sec. 5.1, add up the contributions from the three
diagrams and expand in $g$ in the usual coft scaling, $K\sim(T,gT,\sqrt{g}T)$,
while at the same time taking into account our new scaling for $L$

$\displaystyle\delta\mathcal{C}(k_{\perp})_{l_{\perp}\sim\sqrt{g}T}$
$\displaystyle=g^{4}C_{R}C_{A}\int\frac{dk^{+}}{2\pi}\int\frac{dl^{+}d^{2}l_{\perp}}{(2\pi)^{3}}\frac{1+n_{\text{B}}(k^{+})}{2k^{+}}\frac{4l_{\perp}^{2}}{k_{\perp}^{2}(\mathbf{k}+\mathbf{l})^{2}}G^{--}_{rr}(-L)$
$\displaystyle=g^{4}C_{R}C_{A}\int\frac{dk^{+}}{2\pi}\int\frac{dl^{+}d^{2}l_{\perp}}{(2\pi)^{3}}\frac{1+n_{\text{B}}(k^{+})}{2k^{+}}\frac{4l_{\perp}^{2}}{k_{\perp}^{2}(\mathbf{k}+\mathbf{l})^{2}}\frac{3Tm_{D}^{2}}{2l_{\perp}^{5}}.$
(5.61)

The integrand is independent of $l^{+}$ and so after the integration has been
done, the result will be proportional to a power of the cutoff on $l^{+}$. As
is always the case when one is computing integrals using the method of
regions, this cutoff term will be cancelled by the integration done in some
other adjacent region. However, we know that in the $L\sim(gT,gT,gT)$ scaling,
adopted in the previous section, there are no terms that depend on the $l^{+}$
cutoff. We can therefore conclude that this region is not adjacent to that
region, implying that we need not concern ourselves with it. We have checked
that something similar happens with the scaling
$(l^{+},l^{-},l_{\perp})\sim(\sqrt{g}T,gT,gT)$ and so we discard this region
as well.

####  5.6.2 $l_{\perp}\sim l^{+}\sim\sqrt{g}T,l^{-}\sim gT$

We proceed similarly to the last section by first calculating the expanded
spectral functions with this new scaling. The spectral functions read

$\displaystyle\rho^{L}_{\text{B}}(L)$ $\displaystyle=\frac{m_{D}^{2}\pi
l^{0}}{l^{5}}\Theta(-L^{2})$ $\displaystyle\approx\frac{m_{D}^{2}\pi
l^{+}}{({l^{+}}^{2}+l_{\perp}^{2})^{\frac{5}{2}}}$ (5.62)

and

$\displaystyle\rho_{\text{B}}^{T}(L)$ $\displaystyle=\frac{m_{D}^{2}\pi
l^{0}}{2lL^{4}}\Big{(}1-\frac{{l^{0}}^{2}}{l^{2}}\Big{)}\Theta(l^{2}-{l^{0}}^{2})$
$\displaystyle\approx\frac{m_{D}^{2}\pi
l^{+}}{2l_{\perp}^{2}({l^{+}}^{2}+l_{\perp}^{2})^{\frac{3}{2}}}.$ (5.63)

When using these to compute the scattering kernel, we see that this time, the
integrand does depend on $l^{+}$

$\displaystyle\delta\mathcal{C}(k_{\perp})_{l^{+}\sim l_{\perp}\sim\sqrt{g}T}$
$\displaystyle=g^{4}C_{R}C_{A}\int\frac{dk^{+}}{2\pi}\int\frac{dl^{+}d^{2}l_{\perp}}{(2\pi)^{3}}\frac{1+n_{\text{B}}(k^{+})}{2k^{+}}\frac{4l_{\perp}^{2}}{k_{\perp}^{2}(\mathbf{k}+\mathbf{l})^{2}}G^{--}_{rr}(-L)$
$\displaystyle=g^{4}C_{R}C_{A}\int\frac{dk^{+}}{2\pi}\int\frac{dl^{+}d^{2}l_{\perp}}{(2\pi)^{3}}\frac{1+n_{\text{B}}(k^{+})}{2k^{+}}\frac{4l_{\perp}^{2}}{k_{\perp}^{2}(\mathbf{k}+\mathbf{l})^{2}}\frac{3Tm_{D}^{2}\pi}{2({l^{+}}^{2}+l_{\perp}^{2})^{\frac{5}{2}}}$
$\displaystyle=2g^{4}C_{R}C_{A}\int\frac{dk^{+}}{2\pi}\int\frac{d^{2}l_{\perp}}{(2\pi)^{2}}\frac{1+n_{\text{B}}(k^{+})}{k^{+}}\frac{Tm_{D}^{2}}{k_{\perp}^{2}(\mathbf{k}+\mathbf{l})^{2}l_{\perp}^{2}}.$
(5.64)

It is not difficult to check that if we expand the expression above for
$k_{\perp}\gg l_{\perp}$, we recover Eq. (5.34), with the $l_{\perp}\gg
l^{-}\sim m_{D}$ limit taken there. The fact that these two expressions agree
in these limits implies that share a common boundary and are thus adjacent.

The $d^{2}l_{\perp}$ integration would, in this scaling run over the
$\mathbf{k}+\mathbf{l}\approx 0$ range, corresponding to a longer-lived
bremsstrahlung. Given this longer lifetime, we can expect multiple soft
scatterings at a rate of $g^{2}T$ to occur, in addition to a single, harder
scattering. Addressing this region properly would thus consist of dealing with
LPM resummation beyond the HOA. It is for this reason that we have been forced
to adopt the HOA and in doing so, are not able to compute the double
logarithmic contribution coming from this region. We discuss a pathway to
dealing with this issue in the next section.

###  5.7 Momentum Broadening Beyond the HOA

To briefly recap, up until this point, we have managed to compute double-
logarithmic corrections to $\hat{q}(\mu_{\perp})$, with the dominant
corrections coming from Eq. (5.44). In arriving at that expression, we
observed sensitivity to a neighbouring region (studied in Sec. 5.6.2) that
allows for both single and multiple scattering, which requires a more involved
treatment. We postponed dealing with this region by taking the HOA, inferring
the coefficient of $\hat{q}_{0}$ from the single-scattering consideration
$\mu_{\perp}\gg\rho_{\perp}$. In addition, we have had to impose by hand the
boundary $(b)$, which is intended to partition the single and multiple
scattering regimes. This is in contrast to the boundary separating the double-
logarithmic phase space from its classical counterpart, which, as we will
demonstrate, is clearly cut off by the scale $\omega_{\text{T}}$. In this
section, we present a way to proceed beyond these two hurdles, to self-
consistently determine $\delta\hat{q}(\mu_{\perp})$.

To this end, we will derive, starting from the formalism of [1, 220], an LPM
resummation equation that is not restricted to the HOA. By numerically solving
that equation and performing the integrations over the two logarithmic
variables $k^{+}$ and $\tau$ (or equivalently $k^{+}$ and $k_{\perp}$) one
would then see the emergence of multiple scatterings cutting off the double-
log and thus be able to determine how good of an approximation line $(b)$ is.
Let us then start from Eqs. (6), (7), (11) and (12) in [1] (see also (55)),
which construct a framework for resumming multiple interactions in the HOA and
in the large-$N_{c}$ limit. Combining (11) and (12) yields, in the notation of
[1]

$\displaystyle S(x_{\perp})=-\alpha_{s}C_{R}\mathrm{Re}$
$\displaystyle\int\frac{dk^{+}}{{k^{+}}^{3}}\int_{0}^{L_{M}}dz_{2}\int_{0}^{z_{2}}dz_{1}\nabla_{\mathbf{b}_{2}}\cdot\nabla_{\mathbf{b}_{1}}$
$\displaystyle\times\bigg{[}e^{-\hat{q}_{0\,R}x_{\perp}^{2}(L_{M}-z_{2}+z_{1})/4}G_{\text{HOA}}(\mathbf{b}_{2},z_{2};\mathbf{b}_{1},z_{1})-\mathrm{vac}\bigg{]}\bigg{|}^{\mathbf{b}_{2}=\mathbf{x}}_{\mathbf{b}_{2}=0}\bigg{|}^{\mathbf{b}_{1}=\mathbf{x}}_{\mathbf{b}_{1}=0},$
(5.65)

where we have already undone the large-$N_{c}$ approximation by replacing
$N_{c}/2$, the original large-$N_{c}$ limit of $C_{F}$, with $C_{R}$.
$\hat{q}_{0\,R}$, with $R=F,G$ denotes the specific partonic broadening
coefficients for a quark or gluon source. The $dk^{+}$ frequency integration
is understood over the positive frequencies of the radiated gluon. The
propagator $G_{\text{HOA}}$ is the Green’s function of the following
Schrödinger equation (see (55) there)

$\bigg{\\{}i\partial_{z}+\frac{\nabla^{2}_{\mathbf{b}}}{2k^{+}}+\frac{i}{4}\bigg{[}\hat{q}_{0R}x_{\perp}^{2}+\frac{\hat{q}_{0A}}{2}\big{(}b_{\perp}^{2}+(\mathbf{b}-\mathbf{x})^{2}-x_{\perp}^{2}\big{)}\bigg{]}\bigg{\\}}G_{\text{HOA}}(\mathbf{b},z;\mathbf{b}_{1},z_{1})=0,$
(5.66)

with

$G_{\text{HOA}}(\mathbf{b},z_{1};\mathbf{b}_{1},z_{1})=\delta^{(2)}(\mathbf{b}-\mathbf{b}_{1}).$
(5.67)

“vac” denotes the subtraction of the vacuum term $G_{0}$, i.e. the solution of
Eq. (5.71) with a vanishing $\hat{q}_{0}$. The double vertical bars at the end
of Eq. (5.65) signify that the expression in brackets should be understood as

$\displaystyle
e^{-\frac{\hat{q}_{0\,R}x_{\perp}^{2}}{4}(L_{M}-z_{2}+z_{1})}\bigg{[}$
$\displaystyle
G_{\text{HOA}}(\mathbf{x},z_{2};\mathbf{x},z_{1})+G_{\text{HOA}}(0,z_{2};0,z_{1})$
$\displaystyle-
G_{\text{HOA}}(\mathbf{x},z_{2};0,z_{1})-G_{\text{HOA}}(0,z_{2};\mathbf{x},z_{1})\bigg{]}-\mathrm{vac}.$
(5.68)

Physically, these four terms can be understood as the two positive, virtual
terms, where the gluon is emitted and reabsorbed within the amplitude (both
$\mathbf{b}=0$) or conjugate amplitude (both $\mathbf{b}=\mathbf{x}$) minus
the real terms, where the gluon is emitted on one side and absorbed on the
other side of the cut.

As a first step, we can identify their $S$-matrix element with our $\langle
W(x_{\perp})\rangle$.202020To this end, it suffices to note that the
broadening probability is given in our framework by the Fourier transform of
$\langle W(x_{\perp})\rangle$ and in theirs by that of $S$ — see their Eq.
(1). For a clear explanation of how the broadening probability is related to
$\mathcal{C}(k_{\perp})$, see App. A1 of [298]. Hence we can equate

$S(x_{\perp})=\exp[-L_{M}(\mathcal{C}(x_{\perp})+\delta\mathcal{C}(x_{\perp}))]\approx
e^{-L_{M}\mathcal{C}(x_{\perp})}(1-L_{M}\delta\mathcal{C}(x_{\perp})).$ (5.69)

Secondly, Eq. (5.65) is in the harmonic-oscillator approximation, i.e.

$\mathcal{C}_{R}(x_{\perp})\stackrel{{\scriptstyle{\text{HOA}}}}{{=}}\frac{\hat{q}_{0\,R}}{4}x_{\perp}^{2},$
(5.70)

We can then undo this approximation so that Eq. (5.66) becomes

$\bigg{\\{}i\partial_{z}+\frac{\nabla^{2}_{\mathbf{b}}-m_{\infty\,g}^{2}}{2k^{+}}+i\bigg{[}\mathcal{C}_{R}(x_{\perp})+\frac{1}{2}\big{(}\mathcal{C}_{A}(b_{\perp})+\mathcal{C}_{A}(|\mathbf{b}-\mathbf{x}|)-\mathcal{C}_{A}(x_{\perp})\big{)}\bigg{]}\bigg{\\}}G(\mathbf{b},z;\mathbf{b}_{1},z_{1})=0,$
(5.71)

where we have also introduced the gluon’s asymptotic mass, with
$m_{\infty\,g}^{2}=m_{D}^{2}/2$ 212121This mass term is necessary when
$\nabla^{2}_{b_{\perp}}$, the transverse momentum of the gluon, becomes of
order $g^{2}T^{2}$; it can be neglected in the deep LPM regime, where typical
transverse momenta are larger, $k_{\perp}^{2}\sim\sqrt{\hat{q}_{0}k^{+}}$,
with $k^{+}\gg T$.. To extract a leading-order determination of radiative
correction from this formalism, it should suffice to use, for
$\mathcal{C}_{F}(x_{\perp}),\,\mathcal{C}_{A}(x_{\perp})$ the Fourier
transform of the smooth kernel provided by Eq. (3.56).

Finally, we can also account for the effect of a populated medium by
considering the effects of stimulated emission and absorption, i.e $\int
dk^{+}\Theta(k^{+})\to\int dk^{+}(1/2+n_{\text{B}}(k^{+}))$. In a
longitudinally uniform medium, $G$ is only a function of $t\equiv z_{2}-z_{1}$
so that we can use our identification (5.69) to obtain, in the large-$L_{M}$
limit

$\displaystyle\delta\mathcal{C}(x_{\perp})=\alpha_{s}C_{R}\mathrm{Re}\int\frac{dk^{+}}{k^{+3}}\left(\frac{1}{2}+n_{\text{B}}(k^{+})\right)\int_{0}^{L_{M}}dt\,\nabla_{\mathbf{b}_{2}}\cdot\nabla_{\mathbf{b}_{1}}\bigg{[}$
$\displaystyle
e^{\mathcal{C}_{R}(x_{\perp})t}G(\mathbf{b}_{2},\mathbf{b}_{1};t)$
$\displaystyle-\mathrm{vac}\bigg{]}\bigg{|}^{\mathbf{b}_{2}=\mathbf{x}}_{\mathbf{b}_{2}=0}\bigg{|}^{\mathbf{b}_{1}=\mathbf{x}}_{\mathbf{b}_{1}=0}.$
(5.72)

We observe that the source-specific part of the scattering kernel in the
Hamiltonian (5.71) and the amplification factor
$e^{\mathcal{C}_{R}(x_{\perp})t}$ above can be eliminated by noting that if
$\tilde{G}(\mathbf{b},\mathbf{b}_{1};t)\equiv
e^{\mathcal{C}_{R}(x_{\perp})t}G(\mathbf{b},\mathbf{b}_{1};t)$ is a Green’s
function of the operator

$\bigg{\\{}i\partial_{t}+\frac{\nabla^{2}_{\mathbf{b}}-m_{\infty\,g}^{2}}{2k^{+}}+\frac{i}{2}\big{(}\mathcal{C}_{A}(b_{\perp})+\mathcal{C}_{A}(|\mathbf{b}-\mathbf{x}|)-\mathcal{C}_{A}(x_{\perp})\big{)}\bigg{\\}}\tilde{G}(\mathbf{b},\mathbf{b}_{1};t)=0,$
(5.73)

then $G(\mathbf{B},\mathbf{B}_{1};t)$ is a Green’s function of Eq. (5.71).
Hence

$\displaystyle\delta\mathcal{C}(x_{\perp})=\alpha_{s}C_{R}\mathrm{Re}\int\frac{dk^{+}}{k^{+3}}\left(\frac{1}{2}+n_{\text{B}}(k^{+})\right)\int_{0}^{L_{M}}dt\,\nabla_{\mathbf{b}_{2}}\cdot\nabla_{\mathbf{b}_{1}}\bigg{[}$
$\displaystyle\tilde{G}(\mathbf{b}_{2},\mathbf{b}_{1};t)$
$\displaystyle-\mathrm{vac}\bigg{]}\bigg{|}^{\mathbf{b}_{2}=\mathbf{x}}_{\mathbf{b}_{2}=0}\bigg{|}^{\mathbf{b}_{1}=\mathbf{x}}_{\mathbf{b}_{1}=0}.$
(5.74)

This reformulation makes transparent the fact that the Hamiltonian only
contains the purely non-abelian $\mathcal{C}_{A}$. That is because before or
after both emission vertices, there are only the two source lines for the hard
jet parton and one-gluon exchanges between them are resummed into
$\exp\big(-\mathcal{C}_{R}(x_{\perp})(L_{M}-t)\big{missing})$. In the time
region between the two emission vertices, the hard jet parton and conjugate
hard jet parton lines are no longer a color singlet, with corresponding color
factor $C_{R}$, but rather an octet, with color factor $C_{R}-C_{A}/2$. This
corresponds to the $\mathcal{C}_{R}(x_{\perp})-\mathcal{C}_{A}(x_{\perp})/2$
combination in Eq. (5.71). But we should remove the overall
$\exp\big(-\mathcal{C}_{R}(x_{\perp})L_{M}\big{missing})$ damping, which, as
per our dictionary (5.69), leads to the outright disappearance of the
$\mathcal{C}_{R}(x_{\perp})$ part, corresponding to the fact that those
exchanges would happen also in the absence of the radiated gluon. Up to the
statistical factors and thermal masses, Eqs. (5.73) and (5.74) agree with
[220].

For a medium that is isotropic in the azimuthal direction, we can use the
reflection symmetry $\mathbf{b}\to\mathbf{x}-\mathbf{b}$ of Eq. (5.73) to
simplify Eq. (5.74) into

$\displaystyle\delta\mathcal{C}(x_{\perp})=-2\alpha_{s}C_{R}\mathrm{Re}\int\frac{dk^{+}}{k^{+3}}\left(\frac{1}{2}+n_{\text{B}}(k^{+})\right)\int_{0}^{L_{M}}dt\,\nabla_{\mathbf{b}_{2}}\cdot\nabla_{\mathbf{b}_{1}}\bigg{[}$
$\displaystyle\tilde{G}(\mathbf{b}_{2},\mathbf{b}_{1};t)$
$\displaystyle-\mathrm{vac}\bigg{]}\bigg{|}^{\mathbf{b}_{2}=\mathbf{x},\mathbf{b}_{1}=0}_{\mathbf{b}_{2}=0,\mathbf{b}_{1}=0}.$
(5.75)

Eq. (5.73) is, together with Eq. (5.75), the main result of this section. We
anticipate that its solution would allow for a much better understanding of
how the double logarithm is cut off by the transition from single to multiple
scatterings as a function of the energy $k^{+}$ of the radiated gluon.
Furthermore, Eq. (5.73) goes beyond the HOA and would thus allow us to
properly deal with the region (discussed in Sec. 5.6.2) where
$\mathbf{l}+\mathbf{k}$ become small and in turn develop sensitivity to
multiple scatterings. However, Eq. (5.73) is not easy to solve, as it would
require generalizing the methods of [159, 198, 299, 14] to the extra
dependence on $x_{\perp}$ of $\tilde{G}$. As a first step, one could consider
using the improved opacity expansion, already mentioned in Sec. 3.7 to capture
the qualitative aspects of the transition from the HO approximation to
including rarer harder scatterings.

We shall leave the full or approximate solution of Eq. (5.73) to future work.
We continue this section by providing a non-trivial consistency check, namely
that the single-scattering term in Eqs. (5.73) and (5.75) agrees with Eq.
(5.12). The calculation here proceeds in exactly the same way as for the
single-scattering rate, derived in Sec. 3.5. Following the same steps, we
arrive at

$\displaystyle\delta\mathcal{C}(x_{\perp})^{N=1}=2\alpha_{s}C_{R}\int\frac{dk^{+}}{k^{+}}$
$\displaystyle\left(1+2n_{\text{B}}(k^{+})\right)\int_{\mathbf{p}}\int_{\mathbf{l}}\,\mathcal{C}_{A}(l_{\perp})\frac{(1-e^{i\mathbf{x}\cdot\mathbf{p}})(1+e^{i\mathbf{x}\cdot\mathbf{l}})}{p_{\perp}^{2}+m_{\infty\,g}^{2}}$
$\displaystyle\times\bigg{[}\frac{p_{\perp}^{2}}{p_{\perp}^{2}+m_{\infty\,g}^{2}}-\frac{\mathbf{p}\cdot(\mathbf{p}+\mathbf{l})}{(\mathbf{p}+\mathbf{l})^{2}+m_{\infty\,g}^{2}}\bigg{]}.$
(5.76)

We take a moment here to compare the equation above with Eq. (3.66), with the
$L_{M}\to\infty$ limit taken there. Even though the latter equation is a
radiation splitting rate, we can immediately understand that it resembles the
former, which computes corrections to the scattering kernel due to radiative
processes. However, the product
$(1-e^{i\mathbf{x}\cdot\mathbf{p}})(1+e^{i\mathbf{x}\cdot\mathbf{l}})$ above
makes manifest that the above equation includes real, probability-conserving
as well as virtual processes in comparison to Eq. (3.66), which only contains
real processes. To make contact with Eq. (5.12), we need to extract the real-
process contribution,
$\delta\mathcal{C}(k_{\perp})^{\mathrm{real}}_{\mathrm{single}}$. This just
corresponds to Fourier-transforming $\mathbf{x}$ into $\mathbf{k}$ and only
keeping terms proportional to $\exp(i\mathbf{x}\cdot\mathbf{p})$ and
$\exp(i\mathbf{x}\cdot(\mathbf{p}+\mathbf{l}))$, yielding

$\displaystyle\delta\mathcal{C}(k_{\perp})^{\mathrm{real}}_{\mathrm{single}}=$
$\displaystyle 4\alpha_{s}C_{R}\int
dk^{+}\frac{\frac{1}{2}+n_{\text{B}}(k^{+})}{k^{+}}\int_{\mathbf{l}}\mathcal{C}_{g}(l_{\perp})\left[\frac{\mathbf{k}}{k_{\perp}^{2}+m_{\infty\,g}^{2}}-\frac{\mathbf{k}+\mathbf{l}}{(\mathbf{k}{+}\mathbf{l})^{2}+m_{\infty\,g}^{2}}\right]^{2},$
(5.77)

which, in the $m_{\infty\,g}\to 0$ and
($1/2+n_{\text{B}}(k^{+})\to\Theta(k^{+})$) limits agree with Eq. (5.12). In
App. C.3 we compute the aforementioned virtual contributions diagrammatically
and in doing so, check that they are consistent with Eq. (5.76).

###  5.8 Connection to the Classical Contribution

Figure 5.7: The contour discussed in the main text for the evaluation of the
retarded part of the soft contribution to $\delta\mathcal{C}(k_{\perp})$. The
corresponding advanced part is not shown. The cross at the origin is Matsubara
zero mode, hence the deformation of the (red) integration contour there, which
gives rise to the Euclidean contribution.

When computing double logarithmic corrections to $\hat{q}(\mu_{\perp})$ in
Sec. 5.4, we observed corrections depending on a power law of the IR energy
cutoff, $k^{+}_{\text{IR}}$. Power law corrections of this kind will always
cancel against those coming from a neighbouring region of phase space and in
some sense, can be disregarded. In this instance however, we take advantage of
this fact to further clarify how the region of phase space from which the
quantum corrections emerge is connected to the region giving rise to the
classical corrections. In particular, we set out to show how the first term in
Eq. (5.4.1) is also produced (but with an opposite sign) in the calculation of
Caron-Huot [11]. We do not show how power law corrections from either Eq.
(5.26) or Eq. (5.52) cancel against their classical counterparts.

But how should this work in practice? Well, diagrams such as those in Fig. 5.2
are precisely those evaluated in [11] with $K\sim L\sim gT$. On that side, the
calculation should produce a $\sim 1/k^{+}_{\text{IR}}$ term. There,
$k^{+}_{\text{IR}}$ is a UV cutoff, which can thus be safely sent to infinity,
resulting in the absence of such corrections from [11].

As we have discussed in Sec. 4.3, based on ideas from the same paper, the
classical contribution is dominated by the Matsubara zero mode, which can be
mapped to EQCD. In terms of doing the $k^{+}$ integration in the 4D theory,
this consists of making the replacement $1/2+n_{\text{B}}(k^{+})\to T/k^{+}$
before closing the integration contour, $\mathcal{A}$ at a radius,
$k^{+}_{\text{IR}}$ between the zero and first Matsubara modes. This is
moreover consistent with our calculation, in the sense that $gT\ll
k^{+}_{\text{IR}}\ll T$. Any function that falls to zero faster than $1/k^{+}$
on this arc will only give rise to inverse powers of $k^{+}_{\text{IR}}$,
which could be safely neglected in the derivation [11]. The contour of
interest is depicted in Fig. 5.7.

Rather than starting from Caron-Huot’s calculation, we instead begin from
around Eq. (F.49) [294], which provides the NLO calculation of the
longitudinal momentum diffusion constant,
$\hat{q}_{L}(\mu_{\perp})\equiv\langle q_{z}^{2}\rangle/L_{M}$, also
conveniently carried out in Coulomb gauge. To make the connection with its
transverse counterpart, one can multiply by $(k_{\perp}/k^{+})^{2}$ under the
integral sign222222 See [18] for more details.. This yields

$\displaystyle\delta\hat{q}\Big{|}_{\mathrm{soft}}^{\mathrm{arcs}}(\mu_{\perp})\\!\\!$
$\displaystyle=$ $\displaystyle
2g^{4}C_{R}C_{A}T\\!\int_{\mathcal{A}_{R}}\\!\frac{dk^{+}}{2\pi}\\!\int^{\mu_{\perp}}\\!\frac{d^{2}k_{\perp}}{(2\pi)^{2}}\\!\int_{L}$
$\displaystyle\times\bigg{\\{}\frac{k_{\perp}^{2}}{k^{+2}}G^{--}_{rr}(L)\pi\delta(l^{-})\left(\frac{\mathbf{k}}{k_{\perp}^{2}+m_{\infty}^{2}}-\frac{\mathbf{k}+\mathbf{l}}{(\mathbf{k}+\mathbf{l})^{2}+m_{\infty}^{2}}\right)^{2}+\mathcal{O}\left(\frac{1}{k^{+3}}\right)\bigg{\\}}+\mathcal{A}_{A}.$

We can now perform some of the $L$ integrations using the reduction to EQCD
discussed in Sec. 4.3, finding

$\displaystyle\delta\hat{q}\Big{|}_{\mathrm{soft}}^{\mathrm{arcs}}(\mu_{\perp})\\!\\!$
$\displaystyle=$ $\displaystyle
g^{4}C_{R}C_{A}T^{2}\\!\int_{\mathcal{A}_{R}}\\!\frac{dk^{+}}{2\pi}\\!\int^{\mu_{\perp}}\\!\frac{d^{2}k_{\perp}}{(2\pi)^{2}}\\!\int_{\mathbf{l}}$
$\displaystyle\times\bigg{\\{}\frac{k_{\perp}^{2}}{k^{+2}}\frac{m_{D}^{2}}{l_{\perp}^{2}(l_{\perp}^{2}+m_{D}^{2})}\left(\frac{\mathbf{k}}{k_{\perp}^{2}+m_{\infty}^{2}}-\frac{\mathbf{k}+\mathbf{l}}{(\mathbf{k}+\mathbf{l})^{2}+m_{\infty}^{2}}\right)^{2}+\mathcal{O}\left(\frac{1}{k^{+3}}\right)\bigg{\\}}+\mathcal{A}_{A}.$

If we take the $l_{\perp},m_{D}\ll k_{\perp}$ approximation, consistently with
our treatment across the $k^{+}_{\text{IR}}$ boundary, we find

$\displaystyle\delta\hat{q}\Big{|}_{\mathrm{soft}}^{\mathrm{arcs}}(\mu_{\perp})\\!\\!$
$\displaystyle=$ $\displaystyle
g^{4}C_{R}C_{A}T^{2}\\!\int_{\mathcal{A}_{R}}\\!\frac{dk^{+}}{2\pi}\\!\int^{\mu_{\perp}}\\!\frac{d^{2}k_{\perp}}{(2\pi)^{2}}\\!\int_{\mathbf{l}}\frac{1}{k^{+2}k_{\perp}^{2}}\frac{m_{D}^{2}}{l_{\perp}^{2}+m_{D}^{2}}+\mathcal{A}_{A}.$
(5.80)

We regulate the $l_{\perp}$ integration using the HOA consistently with what
we did in Sec. 5.4.1, we find

$\displaystyle\delta\hat{q}\Big{|}_{\mathrm{soft}}^{\mathrm{arcs\,HO}}(\mu_{\perp})\\!\\!$
$\displaystyle=$ $\displaystyle
g^{2}C_{R}T\hat{q}_{0}\\!\int_{\mathcal{A}_{R}}\\!\frac{dk^{+}}{2\pi}\\!\int^{\mu_{\perp}}\\!\frac{d^{2}k_{\perp}}{(2\pi)^{2}}\frac{1}{k^{+2}k_{\perp}^{2}}+\mathcal{A}_{A}.$
(5.81)

We can then rewrite the $k_{\perp}$ integration as a $\tau$ one, with the same
boundaries of Eq. (5.41), i.e.

$\displaystyle\delta\hat{q}\Big{|}_{\mathrm{soft}}^{\mathrm{arcs\,HO}}(\mu_{\perp})\\!\\!$
$\displaystyle=$ $\displaystyle
g^{2}C_{R}T\hat{q}_{0}\\!\int_{\mathcal{A}_{R}}\\!\frac{dk^{+}}{2\pi}\\!\int_{k^{+}/\mu_{\perp}^{2}}^{\tau_{\text{int}}}\\!\frac{d\tau}{4\pi}\frac{1}{k^{+2}\tau}+\mathcal{A}_{A}.$
(5.82)

The contribution on the arc can be carried out as follows, understanding the
arc to be at constant $|k^{+}|=k^{+}_{\text{IR}}$. Changing variables to
$k^{+}=k^{+}_{\text{IR}}e^{it}$ then gives

$\int_{\mathcal{A}_{R}}\\!\frac{dk^{+}}{2\pi}\frac{1}{k^{+2}}\ln\left(\frac{\mu_{\perp}^{2}\tau_{\text{int}}}{k^{+}}\right)=\frac{i}{2\pi
k^{+}_{\text{IR}}}\int_{\pi}^{0}dte^{-it}\bigg{[}\ln\left(\frac{\mu_{\perp}^{2}\tau_{\text{int}}}{k^{+}_{\text{IR}}}\right)-it\bigg{]}=\frac{-1}{\pi
k^{+}_{\text{IR}}}\bigg{[}\ln\left(\frac{\mu_{\perp}^{2}\tau_{\text{int}}}{k^{+}_{\text{IR}}}\right)-1-i\frac{\pi}{2}\bigg{]}.$
(5.83)

The imaginary part cancels against an opposite one from $\mathcal{C}_{A}$. We
then have

$\displaystyle\delta\hat{q}\Big{|}_{\mathrm{soft}}^{\mathrm{arcs\,HO}}(\mu_{\perp})\\!\\!=$
$\displaystyle 2\frac{g^{2}C_{R}T^{2}\hat{q}_{0}}{4\pi}\frac{-1}{\pi
k^{+}_{\text{IR}}}\bigg{[}\ln\left(\frac{\mu_{\perp}^{2}\tau_{\text{int}}}{k^{+}_{\text{IR}}}\right)-1\bigg{]}$
$\displaystyle=$ $\displaystyle-2\frac{\alpha_{s}C_{R}T\hat{q}_{0}}{\pi
k^{+}_{\text{IR}}}\bigg{[}\ln\left(\frac{\mu_{\perp}^{2}\tau_{\text{int}}}{k^{+}_{\text{IR}}}\right)-1\bigg{]}.$
(5.84)

This is precisely opposite to the $T/k^{+}_{\text{IR}}$-proportional part of
Eq. (5.4.1), as we set out to show.

We provide a final comment with respect to how the double-logarithmic phase
space is cut off. The $k^{+}$ integration is clearly cut off around the scale
$\omega_{T}$ because below that scale, there is the cancellation between the
vacuum term and the first quantum correction, as was explained explicitly
around (…), which renders the $k^{+}$ integrand $\sim 1/{k^{+}}^{2}$. But can
one understand how the $\tau$ integration transitions to power law behaviour?
Well, we can understand from the calculation above that there is no analogous
scale to $\omega_{T}$ since a $1/\tau$ integrand persists for $k^{+}\ll T$.
Instead, $\omega_{T}$ cuts off the $\tau$ logarithm in a somewhat subtle way;
in approaching the $k^{+}_{\text{IR}}$ vertical line from the right, the
$\tau$ integration gives rise to single logs that are always multiplied by
powers of $k^{+}_{\text{IR}}$. As we have just shown, these single logs cancel
against identical terms, which are produced from approaching the
$k^{+}_{\text{IR}}$ vertical line from the left.

##  Chapter 6 Quantum Corrections to Forward Scattering

This chapter maintains the theme of computing higher order corrections to
quantities that serve as input to jet energy loss calculations. In particular,
we work towards the calculation of the $\mathcal{O}(g^{2})$ corrections to the
asymptotic mass, $m_{\infty}$ with two goals in mind:

  * •

Perform the matching between perturbative finite-temperature QCD and
perturbative EQCD in order to cancel EFT divergences coming from the EQCD
side. As was discussed at the end of Sec. 4.4.3, this matching procedure will
rid the non-perturbative determination of the classical corrections to
$m_{\infty}$ of any UV divergences. Recalling that111 The $Z_{g},\,Z_{f}$
operators in Eqs. (6.1), (6.2) are strictly speaking not equal to each other –
they are equal up to an overall colour factor. See, for instance, Eq. (4.108).
In what follows, we compute for a jet particle in colour representation, $R$.

$\displaystyle m_{\infty\,q}^{2}$ $\displaystyle=g^{2}C_{F}(Z_{g}+Z_{f}),$
(6.1) $\displaystyle m_{\infty\,g}^{2}$
$\displaystyle=g^{2}(C_{A}Z_{g}+2T_{F}N_{F}Z_{f}),$ (6.2)

we remind the reader that the operator $Z_{f}$ is not as sensitive as $Z_{g}$
to classical corrections, meaning that we need only complete this matching for
the $Z_{g}$ operator. This calculation makes up the content of Secs. 6.3 and
6.4 and is furthermore presented in [300]. In the latter, we take some time to
introduce the terminology from [15, 16] in order to shed some more light on
the context of the matching calculation.

  * •

Complete the $\mathcal{O}(g^{2})$ evaluation of $Z_{g}$. We begin this
endeavour in Sec. 6.5.

Before embarking on these goals, in Sec. 6.1, we introduce the various two-
loop diagrams that arise at NLO and argue why the majority of them are zero in
Feynman gauge, which we adopt throughout this chapter. In Sec. 6.2, we proceed
to demonstrate a further cancellation, which leaves us only needing to concern
ourselves with diagram $(c)$ from Fig. 6.1.

Figure 6.1: Diagrams relevant to the leading order and next-to-leading order
evaluation of $Z_{g}$. Eq. (4.108). External grey blobs denote $F^{-\perp}$
insertions whereas internal grey blobs denote self energies. The solid lines
represent Wilson lines and the curly lines a gauge boson.

###  6.1 Calculation Setup

We dive straight into the computation of the various diagrams, listed in Fig.
6.1. While in theory, the vacuum result should be subtracted from each
diagram, we neglect this term throughout the chapter, as it vanishes in DR
anyway.

Diagram $(a)$ just gives the LO contribution to $Z_{g}$, which we rewrite here
for convenience [279, 274]

$Z_{g}^{(a)}=\frac{T^{2}}{6}.$ (6.3)

We now proceed to the arduous task of computing all of the NLO diagrams.

####  6.1.1 Diagram $(c)$

In Sec. 4.4.2, we computed the EQCD counterpart of diagram $(a)$. Diagram
$(c)$ has the same structure, albeit with an extra self-energy insertion. We
can thus pick up from Eq. (4.111), writing

$\displaystyle Z_{g}^{(c)}$
$\displaystyle=-2\int_{0}^{\infty}dx^{+}\int_{K}x^{+}e^{-ik^{-}x^{+}}(k^{-2}G_{>}^{xx}(K)-2k^{x}k^{-}G_{>}^{x-}(K)+{k^{x}}^{2}G_{>}^{--}(K))$
$\displaystyle=2\int_{K}\frac{1}{(k^{-}-i\varepsilon)^{2}}(k^{-2}G_{>}^{xx}(K)-2k^{x}k^{-}G_{>}^{x-}(K)+{k^{x}}^{2}G_{>}^{--}(K))$
(6.4)
$\displaystyle=2\int_{K}\frac{1+n_{\text{B}}(k^{0})}{(k^{-}-i\varepsilon)^{2}}(k^{-2}(G_{R}^{xx}(K)-G_{A}^{xx}(K))-2k^{x}k^{-}(G_{R}^{x-}(K)-G_{A}^{x-}(K))$
$\displaystyle+{k^{x}}^{2}(G_{R}^{--}(K)-G_{A}^{--}(K)))$
$\displaystyle=2\int_{K}\frac{1+n_{\text{B}}(k^{0})}{(k^{-}-i\varepsilon)^{2}}(F_{R}(K)-F_{A}(K))$
(6.5)

where, in contrast to Eq. (4.111),

$\displaystyle F_{R/A}(K)$ $\displaystyle=\frac{i}{(K^{2}+i\varepsilon
k^{0})^{2}}\big{(}k^{-2}\Pi^{xx}_{R/A}(K)-2k^{x}k^{-}(\Pi^{0x}_{R/A}(K)-\Pi^{zx}_{R/A}(K))$
$\displaystyle+{k^{x}}^{2}(\Pi^{00}_{R/A}(K)+\Pi^{zz}_{R/A}(K)-2\Pi^{z0}_{R/A}(K))\big{)}.$
(6.6)

It is then useful to decompose the self-energies in terms of their
longitudinal and transverse components

$\Pi^{R/A}_{\mu\nu}(K)=P^{L}_{\mu\nu}(K)\Pi^{R/A}_{L}(K)+P^{T}_{\mu\nu}(K)\Pi^{R/A}_{T}(K)$
(6.7)

where $P^{L}_{\mu\nu}$ and $P^{T}_{\mu\nu}$ are defined in Eqs. (4.79) and
(4.80) respectively. Making use of those same equations then leads to

$\displaystyle F_{R/A}(K)=$ $\displaystyle=\frac{i}{(K^{2}+i\varepsilon
k^{0})^{2}}\bigg{[}k^{-2}\bigg{(}\Pi_{L}^{R}(K)\frac{k_{\perp}^{2}}{k^{2}}\bigg{)}$
$\displaystyle-2k^{x}k^{-}\bigg{(}\Pi_{L}^{R/A}(K)\Big{(}\frac{k^{x}k^{z}}{q^{2}}-\frac{k^{x}k^{-}}{K^{2}}\Big{)}-\Pi_{T}^{R/A}(K)\frac{k^{x}k^{z}}{q^{2}}\bigg{)}$
$\displaystyle+{k^{x}}^{2}\bigg{(}\Pi_{L}^{R/A}(K)\Big{(}1-\frac{k_{\perp}^{2}}{k^{2}}+\frac{k^{2}-2k^{0}k^{z}+{k^{z}}^{2}}{K^{2}}\Big{)}+\Pi_{T}^{R/A}(K)\frac{k_{\perp}^{2}}{k^{2}}\bigg{)}\bigg{]}$
$\displaystyle=\frac{i}{(K^{2}+i\varepsilon
k^{0})^{2}}\bigg{[}k^{-2}\Pi_{T}^{R/A}(K)+\frac{{k^{x}}^{2}K^{2}}{k^{2}}(\Pi_{L}^{R/A}(K)-\Pi_{T}^{R/A}(K))\bigg{]},$
(6.8)

which in turn results in222Technically, as the $K$ integration is done in DR
we should make the replacement $2k^{-2}\rightarrow(D-2)k^{-2}$ where
$D=4-2\epsilon$. Once we get to the stage where we are actually evaluating
divergent integrals, we will be careful to rewrite everything in the
appropriate $d$-dimensional form.

$\displaystyle Z_{g}^{(c)}$
$\displaystyle=i\int_{K}\frac{1+n_{\mathrm{B}}(k^{0})}{(k^{-}-i\varepsilon)^{2}}\bigg{[}\frac{2k^{-2}\Pi_{T}^{R}(K)}{(K^{2}+i\varepsilon
k^{0})^{2}}+\frac{k_{\perp}^{2}(\Pi_{L}^{R}(K)-\Pi_{T}^{R}(K))}{k^{2}(K^{2}+i\varepsilon
k^{0})}-\frac{2k^{-2}\Pi_{T}^{A}(K)}{(K^{2}-i\varepsilon k^{0})^{2}}$
$\displaystyle-\frac{k_{\perp}^{2}(\Pi_{L}^{A}(K)-\Pi_{T}^{A}(K))}{k^{2}(K^{2}-i\varepsilon
k^{0})}\bigg{]}.$ (6.9)

It is convenient to divide up the terms as follows

$Z_{g}^{(c)}\equiv Z_{g\,k^{-2}}^{(c)}+Z_{g\,\text{L}-\text{T}}^{(c)}.$ (6.10)

We find this to be a good stopping point, before matching the IR divergences
of the expression above to those coming from the EQCD calculation in Sec. 6.3.

Figure 6.2: Different topologies associated with diagram $(g)$ with momentum
flowing from left to right. The “1, 2” labels indicate the branch of the
Schwinger-Keldysh contour from which the vertices are being pulled.

####  6.1.2 Diagram $(g)$

For the computation of diagram $(g)$, working in the 1/2 basis there are two
possible thermal assignments, shown in Fig. 6.2. We start with the left
diagram

$\displaystyle Z_{g}^{(g)\,1}$
$\displaystyle=-\frac{2}{d_{A}}\int_{0}^{\infty}dx^{+}x^{+}\bigg{\langle}\Big{[}\partial^{-}A^{x,a}_{1}(x^{+})-\partial^{x}A^{-,a}_{1}(x^{+})\Big{]}gf^{abc}A^{-,b}_{1}(0)A^{x,c}_{1}(0)\bigg{\rangle}^{1}$
$\displaystyle=-\frac{2gf^{abc}}{d_{A}}\int_{0}^{\infty}dx^{+}x^{+}\int_{K}e^{-k^{-}x^{+}}\bigg{\langle}\Big{[}-ik^{-}A^{x,a}_{1}+k^{x}A^{-,a}_{1}\Big{]}A^{-,b}_{1}A^{x,c}_{1}(0)\bigg{\rangle}^{1},$
(6.11)

where the 1 subscripts on the gauge fields indicate that they are located on
the time-ordered branch of the Schwinger-Keldysh contour (see Fig. 4.1). The 1
superscript on the angled bracket indicates that the thermal expectation value
is to be evaluated with the three-gluon vertex of type 1. Upon evaluating the
expectation value, we get

$\displaystyle Z_{g}^{(g)\,1}$
$\displaystyle=-\frac{2gf^{abc}}{d_{A}}\int_{0}^{\infty}dx^{+}x^{+}\int_{K,Q}e^{-ik^{-}x^{+}}\;\left(gf^{abc}\;V_{\alpha\beta\gamma}(Q,K-Q,-K)\right)$
$\displaystyle\times\Big{[}\left(-\eta^{-\alpha}G_{F}(Q)\right)\left(-\eta^{x\beta}G_{F}(K-Q)\right)\left(-ik^{-}\left(-\eta^{x\gamma}G_{F}(K)\right)+ik^{x}\left(-\eta^{-\gamma}G_{F}(K)\right)\right)\Big{]}$
$\displaystyle=-2ig^{2}C_{A}\int_{0}^{\infty}dx^{+}x^{+}\int_{K,Q}e^{-ik^{-}x^{+}}\;\Big{[}G_{F}(Q)G_{F}(K-Q)k^{-}G_{F}(K)V_{+xx}(Q,K-Q,-K)$
$\displaystyle\hskip 5.0pt-
G_{F}(Q)G_{F}(K-Q)k^{x}G_{F}(K)V_{+x+}(Q,K-Q,-K)\Big{]}$ (6.12)

From Eq. (A.2), we see that the second term vanishes. Moreover,

$\int_{0}^{\infty}dx^{+}x^{+}e^{-ik^{-}x^{+}}=-\frac{1}{(k^{-}-i\varepsilon)^{2}},$
(6.13)

so that

$Z_{g}^{(g)\,1}=2ig^{2}C_{A}\int_{K,Q}G_{F}(Q)G_{F}(K-Q)G_{F}(K)\left(\frac{k^{-}q^{-}}{(k^{-}-i\varepsilon)^{2}}-2\right).$
(6.14)

Now, let us repeat the same procedure for the vertex assigned to the 2-branch
of the Schwinger-Keldysh contour (Fig.6.2-right)

$\displaystyle Z_{g}^{(g)\,2}$
$\displaystyle=-\frac{2}{d_{A}}\int_{0}^{\infty}dx^{+}x^{+}\bigg{\langle}\Big{[}\partial^{-}A^{x,c}_{1}(x^{+})-\partial^{x}A^{-,c}_{1}(x^{+})\Big{]}gf^{abc}A^{-,b}_{1}(0)A^{x,c}_{1}(0)\bigg{\rangle}^{2}$
$\displaystyle=-\frac{2}{d_{A}}\int_{0}^{\infty}dx^{+}x^{+}f^{abc}g^{2}\int_{K,Q}e^{-ik^{-}x^{+}}\left(-f^{abc}V_{\alpha\beta\gamma}(Q,K-Q,-K)\right)$
$\displaystyle\times\Big{[}\left(-\eta^{-\alpha}G_{<}(Q)\right)\left(-\eta^{x\beta}G_{<}(K-Q)\right)\left(-ik^{-}\left(-\eta^{x\gamma}G_{<}(K)\right)+ik^{x}\left(-\eta^{-\gamma}G_{<}(K)\right)\right)\Big{]}$
$\displaystyle=-2ig^{2}C_{A}\int_{K,Q}G_{<}(Q)G_{<}(K-Q)G_{>}(K)\left(\frac{k^{-}q^{-}}{(k^{-}-i\varepsilon)^{2}}-2\right).$
(6.15)

To get the full contribution to $Z_{g}^{(g)}$, we need to add Eqs. (6.14) and
Eq. (6.15) along with the mirror image of each diagram from Fig.6.2. While the
mirror of Fig.6.2-left gives the same contribution as Eq. (6.14), to get the
mirror of Fig.6.2-right, we should take Eq. (6.15) and make the replacement
$G_{<}(Q)G_{<}(K-Q)G_{>}(K)\rightarrow G_{>}(Q)G_{>}(K-Q)G_{<}(K)$. This
leaves us with

$\displaystyle Z^{(g)}_{g}$
$\displaystyle=2ig^{2}C_{A}\int_{K,Q}\left(\frac{q^{-}}{k^{-}-i\varepsilon}-2\right)\Big{[}2G_{F}(Q)G_{F}(K-Q)G_{F}(K)-G_{>}(Q)G_{>}(K-Q)G_{<}(K)$
$\displaystyle-G_{<}(Q)G_{<}(K-Q)G_{>}(K)\Big{]}$ (6.16)

In order to simplify things, we employ the amputated self-energy-like objects
defined in App. D.1, which allow us to rewrite Eq. (6.16) as

$\displaystyle Z^{(g)}_{g}$
$\displaystyle=2ig^{2}C_{A}\int_{K,Q}\left(\frac{q^{-}}{k^{-}-i\varepsilon}-2\right)\Big{[}\Big{(}2\widetilde{\Pi_{aa}}(K)+\widetilde{\Pi_{R}}(K)+\widetilde{\Pi_{A}}(K)\Big{)}G_{F}(K)$
$\displaystyle-\widetilde{\Pi_{>}}(K)G_{<}(K)-\widetilde{\Pi_{<}}(K)G_{>}(K)\Big{]}.$
(6.17)

In arriving at the above expression, one must invoke that the closed loops of
causality vanish (see Sec. 4.2.3). Specifically, this argument applies to the
terms containing $G_{R}(Q)G_{A}(K-Q),\,G_{A}(Q)G_{R}(K-Q)$. It then suffices
to use the KMS-like relations in App. D.1 to arrive at

$Z^{(g)}_{g}=4ig^{2}C_{A}\int_{K,Q}\left(\frac{q^{-}}{k^{-}-i\varepsilon}-2\right)\left(1+n_{\text{B}}(k^{0})\right)\left(\widetilde{\Pi_{R}}(K)G_{R}(K)-\widetilde{\Pi_{A}}(K)G_{A}(K)\right).$
(6.18)

We halt with the evaluation at this stage. Indeed, we will see that this
contribution cancels with the one coming from diagram $(f)$ in Sec. 6.2.2.

Figure 6.3: Different topologies associated with diagram $(f)$ with momentum
flowing from left to right. In all cases, we take the propagator attached to
the Wilson line (as opposed to the $F$ operator insertions) to have momentum
$K-Q$.

####  6.1.3 Diagram $(f)$

This is the only non-vanishing diagram where the gluons are being pulled from
the Wilson line itself. The corresponding contributions to this diagram are
listed in Fig. 6.3. We then have $8$ possible assignments overall, since each
of these $4$ subdiagrams can have either a $1$ or $2$ labelling for the three-
gluon vertex.

Since it is an $A^{-}$ field that is being pulled from the Wilson line (see
Eq. (4.109)), we can conclude that in each subdiagram, the only contribution
to the three-gluon vertex will be proportional to $\eta_{xx}$. We proceed to
first compute the contribution from $Z_{g}^{f\,I}$, pointing to Fig. 6.3 for
clarification about the labelling

$\displaystyle Z_{g}^{(f)\,I}$
$\displaystyle=\frac{-2}{d_{R}C_{R}}(ig)\int_{0}^{\infty}dx^{+}x^{+}\int_{-\infty}^{0}dy^{+}\bigg{[}\frac{if^{cba}}{4}\bigg{\langle}\partial^{-}A^{a,x}(x^{+})\partial^{-}A^{b,x}(0)A^{c,-}(y^{+})\bigg{\rangle}^{1}$
$\displaystyle+\frac{if^{acb}}{4}\bigg{\langle}\partial^{-}A^{a,x}(x^{+})\partial^{-}A^{b,x}(0)A^{c,-}(y^{+})\bigg{\rangle}^{2}\bigg{]}$
(6.19)

where the two factors in parentheses out in front come from the colour trace
and the Wilson line respectively. Then, Fourier transforming in a way that is
consistent with the labelling convention mentioned in Fig. 6.3, we find

$\displaystyle Z_{g}^{(f)\,I}$
$\displaystyle=-\frac{2ig}{d_{R}C_{R}}\int_{0}^{\infty}dx^{+}x^{+}\int_{-\infty}^{0}dy^{+}\int_{K,Q}k^{-}q^{-}e^{-ik^{-}x^{+}}e^{-i(q^{-}-k^{-})y^{+}}$
$\displaystyle\times\bigg{[}\frac{if^{cba}}{4}\bigg{\langle}A^{a,x}(K)A^{b,x}(Q)A^{c,-}(K-Q)\bigg{\rangle}^{1}+\frac{if^{acb}}{4}\bigg{\langle}A^{a,x}(K)A^{b,x}(Q)A^{c,-}(K-Q)\bigg{\rangle}^{2}\bigg{]}$
$\displaystyle=-\frac{2ig}{d_{R}C_{R}}\int_{K,Q}\frac{q^{-}}{(k^{-}-i\varepsilon)(k^{-}-q^{-}-i\varepsilon)}$
$\displaystyle\times\bigg{[}\frac{if^{cba}}{4}\bigg{\langle}A^{a,x}(K)A^{b,x}(Q)A^{c,-}(K-Q)\bigg{\rangle}^{1}+\frac{if^{acb}}{4}\bigg{\langle}A^{a,x}(K)A^{b,x}(Q)A^{c,-}(K-Q)\bigg{\rangle}^{2}\bigg{]}$
$\displaystyle=ig^{2}C_{A}\int_{K,Q}\frac{q^{-}(k^{-}+q^{-})}{(k^{-}-i\varepsilon)(k^{-}-q^{-}-i\varepsilon)}\Big{[}-G_{F}(Q)G_{F}(K)G_{F}(K-Q)$
$\displaystyle+G_{>}(Q)G_{<}(K)G_{>}(K-Q)\Big{]}.$ (6.20)

Note that the thermal assignments are separated by a minus: this is because,
in the second, we are pulling a 2 vertex from the action. We can immediately
then write down the expression for the contribution of the subdiagram with the
gluon instead attached to the Wilson line directly below

$\displaystyle Z_{g}^{(f)\,III}$
$\displaystyle=-ig^{2}C_{A}\int_{K,Q}\frac{q^{-}(k^{-}+q^{-})}{(k^{-}-i\varepsilon)(k^{-}-q^{-}-i\varepsilon)}\Big{[}-G_{F}(Q)G_{F}(K)G_{<}(K-Q)$
$\displaystyle+G_{>}(Q)G_{<}(K)G_{\bar{F}}(K-Q)\Big{]}.$ (6.21)

The minus sign out in front can be attributed to the fact that in this case,
the limits of the $y^{+}$ integral are switched. The thermal assignments are
of course different in this subdiagram, since the lower Wilson line is anti-
time-ordered. Combining the two subdiagrams then yields, after some algebra

$\displaystyle Z_{g}^{(f)\,I}+Z_{g}^{(f)\,III}$
$\displaystyle=-ig^{2}C_{A}\int_{K,Q}\frac{q^{-}(k^{-}+q^{-})}{(k^{-}-i\varepsilon)(k^{-}-q^{-}-i\varepsilon)}G_{R}(K-Q)\Big{[}G_{rr}(Q)G_{R}(K)$
$\displaystyle+G_{rr}(K)G_{A}(Q)+\frac{1}{2}\left(G_{R}(K)G_{R}(Q)+G_{A}(K)G_{A}(Q)\right)\Big{]}$
(6.22)

the last two terms are scaleless and therefore vanish in dimensional
regularisation.

Moving onto the subdiagram where the gluon is attached to the line between the
two $F$ insertions, we have

$\displaystyle Z_{g}^{(f)\,II}$
$\displaystyle=\frac{-2}{d_{R}C_{R}}(ig)\int_{0}^{\infty}dx^{+}x^{+}\int_{0}^{x^{+}}dy^{+}\int_{K,Q}k^{-}q^{-}e^{-ik^{-}x^{+}}e^{-i(q^{-}-k^{-})y^{+}}$
(6.23)
$\displaystyle\times\bigg{[}\frac{if^{cba}}{4}\bigg{\langle}A^{a,x}(K)A^{b,-}(K-Q)A^{c,x}(Q)\bigg{\rangle}^{1}+\frac{if^{bca}}{4}\bigg{\langle}A^{a,x}(K)A^{b,-}(K-Q)A^{c,x}(Q)\bigg{\rangle}^{2}\bigg{]}$
$\displaystyle=\frac{2g}{d_{R}C_{R}}\int_{K,Q}\frac{1}{q^{-}-k^{-}}\left(\frac{q^{-}}{k^{-}-i\varepsilon}-\frac{k^{-}}{q^{-}-i\varepsilon}\right)$
$\displaystyle\times\bigg{[}\frac{if^{cba}}{4}\bigg{\langle}A^{a,x}(K)A^{b,-}(K-Q)A^{c,x}(Q)\bigg{\rangle}^{1}+\frac{if^{bca}}{4}\bigg{\langle}A^{a,x}(K)A^{b,-}(K-Q)A^{c,x}(Q)\bigg{\rangle}^{2}\bigg{]}$
$\displaystyle=-ig^{2}C_{A}\int_{K,Q}(k^{-}+q^{-})\left(\frac{1}{k^{-}-i\varepsilon}+\frac{1}{q^{-}-i\varepsilon}\right)\Big{[}-G_{F}(Q)G_{F}(K)G_{F}(K-Q)$
$\displaystyle+G_{>}(Q)G_{<}(K)G_{>}(K-Q)\Big{]}.$ (6.24)

In this case, the corresponding diagram with the gluon attached below has the
same sign, due to relative minuses coming from both the colour trace and the
inversion of the limits on the $y^{+}$ integral. Thus,

$\displaystyle Z_{g}^{(f)\,IV}$
$\displaystyle=-ig^{2}C_{A}\int_{K,Q}(k^{-}+q^{-})\left(\frac{1}{k^{-}-i\varepsilon}+\frac{1}{q^{-}-i\varepsilon}\right)\Big{[}-G_{F}(Q)G_{F}(K)G_{<}(K-Q)$
$\displaystyle+G_{>}(Q)G_{<}(K)G_{\bar{F}}(K-Q)\Big{]}.$ (6.25)

Together these two add to give,

$\displaystyle Z_{g}^{(f)\,II}$
$\displaystyle+Z_{g}^{(f)\,IV}=ig^{2}C_{A}\int_{K,Q}(k^{-}+q^{-})\left(\frac{1}{k^{-}-i\varepsilon}+\frac{1}{q^{-}-i\varepsilon}\right)\Big{\\{}G_{A}(K-Q)\Big{[}\frac{1}{2}G_{A}(Q)G_{R}(K)$
$\displaystyle+2G_{rr}(K)G_{rr}(Q)+G_{rr}(K)G_{R}(Q)\Big{]}+2G_{rr}(K-Q)\Big{[}\frac{1}{2}G_{R}(K)G_{R}(Q)+G_{rr}(Q)G_{R}(K)$
$\displaystyle+G_{rr}(K)G_{A}(Q)\Big{]}\Big{\\}}$
$\displaystyle=ig^{2}C_{A}\int_{K,Q}\frac{k^{-}+q^{-}}{k^{-}-i\varepsilon}$
$\displaystyle\times\Big{\\{}4\left(\frac{1}{2}+n_{\text{B}}(k^{0})\right)\Big{[}G_{R}(K)\widetilde{\Pi_{R}}(K)-G_{A}(K)\widetilde{\Pi_{A}}(K)\Big{]}+2G_{R}(K)\widetilde{\Pi_{R}}(K)$
$\displaystyle-
G_{R}(K)\Big{[}G_{R}(K-Q)G_{rr}(Q)+\frac{1}{2}G_{R}(K-Q)G_{R}(Q)\Big{]}\Big{\\}}$
(6.26)

where we have made use of the “amputated self-energy” relations from App. D.1.
We come back to Eqs. (6.22) and (6.26) in Sec. 6.2.1.

####  6.1.4 Diagrams $(b),\,(d_{1}),\,(d_{2}),\,(e),\,(h),\,(i),\,(j)$

It suffices to write down the expression for diagram $(b)$

$\displaystyle Z_{g}^{(b)}$
$\displaystyle=-\frac{\mathcal{2C}(ig)^{2}}{d_{R}C_{F}}\int_{0}^{\infty}dx^{+}x^{+}\int_{0}^{x^{+}}dy^{+}\int_{0}^{y^{+}}dz^{+}\Bigl{\langle}\Big{(}\partial^{-}A^{x}_{a}(x^{+})-\partial^{x}A^{-}_{a}(x^{+})\Big{)}\Big{(}\partial^{-}A^{x}_{a}(0)-\partial^{x}A^{-}_{a}(0)\Big{)}\Bigr{\rangle}$
$\displaystyle\times\Bigl{\langle}A^{-}_{b}(y^{+})A^{-}_{b}(z^{+})\Bigr{\rangle},$
(6.27)

where $\mathcal{C}$ is some non-zero colour factor. In Feynman gauge, the
$\Bigl{\langle}A^{-}_{b}(y^{+})A^{-}_{b}(z^{+})\Bigr{\rangle}$ factor above
will inevitably produce an $\eta^{--}$, from which we can conclude that this
diagram vanishes. The other diagrams
$(b),\,(d_{1}),\,(d_{2}),\,(e),\,(h),\,(i),\,(j)$ then vanish for the same
reason as they only contain terms proportional to either
$\Bigl{\langle}A^{-}A^{-}\Bigr{\rangle}$ or
$\Bigl{\langle}A^{-}A^{x}\Bigr{\rangle}$, the latter of which is also zero in
Feynman gauge.

###  6.2 Proof of $Z_{g}^{(f)}+Z_{g}^{(g)}=0$

We provide two arguments, supporting the claim that in Feynman gauge, we need
only consider the contribution from diagram $(c)$ in Fig. 6.1.

####  6.2.1 Proof from the Correlator

Let us start by looking at Eq. (4.69) and writing

$Z_{g}=-\frac{1}{d_{R}C_{R}}\int_{0}^{\infty}dx^{+}x^{+}\,\Tr\,\Bigl{\langle}U_{R}(-\infty;x^{+})F^{-\perp}(x^{+})\,U_{R}(x^{+};0)\,F^{-\perp}(0)U_{R}(0;-\infty)\Bigr{\rangle},$
(6.28)

where $F^{-\perp}F^{-\perp}$ is taken to mean $F^{-x}F^{-x}+F^{-y}F^{-y}$.
Continuing onwards, let us recast Eq. (4.69) as a double integral, using
translation invariance

$\displaystyle Z_{g}$
$\displaystyle=-\frac{1}{d_{R}C_{R}}\int_{0}^{\infty}dx^{+}\int_{0}^{x^{+}}dy^{+}\Tr\,\Bigl{\langle}U_{R}(-\infty;x^{+}+y^{+})F^{-\perp}(x^{+}+y^{+})\,U_{R}(x^{+}+y^{+};y^{+})\,F^{-\perp}(y^{+})$
$\displaystyle\times U_{R}(y^{+};-\infty)\Bigr{\rangle},$
$\displaystyle=-\frac{1}{d_{R}C_{R}}\int_{0}^{\infty}dx^{+}\int_{0}^{x^{+}/2}\\!dy^{+}\,\Tr\,\Bigl{\langle}U_{R}(-\infty;x^{+})F^{-\perp}(x^{+})\,U_{R}(x^{+};y^{+})\,F^{-\perp}(y^{+})U_{R}(y^{+};-\infty)\Bigr{\rangle},$
(6.29)

where we have shifted $x^{+}$ in going to the second line. We now rewrite
$Z_{g}$ as

$Z_{g}\equiv Z_{g}^{\perp\perp}+Z_{g}^{\perp-}+Z_{g}^{-\perp}+Z_{g}^{--},$
(6.30)

with the apex labelling to be elaborated on momentarily. By calling on a
method introduced in [282]: we rewrite

$F^{-\perp}=-\partial^{\perp}A^{-}+[D^{-},A^{\perp}].$ (6.31)

The equation of motion of the Wilson line (see Eq. (B.8)),
$D^{-}_{x^{+}}U(x^{+};y^{+})=0$ can then be used to rewrite the commutator as
a total derivative, i.e.

$U_{R}(a;x^{+})[D^{-},A^{\perp}(x^{+})]U_{R}(x^{+};b)=\frac{d}{dx^{+}}[U_{R}(a;x^{+})\,A^{\perp}(x^{+})\,U_{R}(x^{+};b)].$
(6.32)

This should shed light on the labelling in Eq. (6.30): the apexes there denote
whether the $-\partial^{\perp}A^{-}$ (⟂) or the $[D^{-},A^{\perp}]$ component
(-) of the field strength tensor has been taken.

Starting with the first term from Eq. (6.30),

$Z_{g}^{\perp\perp}=-\frac{1}{d_{R}C_{R}}\int_{0}^{\infty}dx^{+}x^{+}\,\Tr\,\Bigl{\langle}U_{R}(-\infty;x^{+})\partial^{\perp}A^{-}(x^{+})\,U_{R}(x^{+};0)\,\partial^{\perp}A^{-}(0)U_{R}(0;-\infty)\Bigr{\rangle}.$
(6.33)

Whereas for the second term, we have, employing Eq. (6.32)

$\displaystyle Z_{g}^{\perp-}$
$\displaystyle=\frac{1}{d_{R}C_{R}}\int_{0}^{\infty}dx^{+}\,\Tr\,\Bigl{\langle}U_{R}(-\infty;x^{+})\partial^{\perp}A^{-}(x^{+})\,U_{R}(x^{+};x^{+}/2)\,\Big{[}A^{\perp}(x^{+}/2)U_{R}(x^{+}/2;0)$
$\displaystyle-
U_{R}(x^{+}/2;0)A^{\perp}(0)\Big{]}U_{R}(0;-\infty)\Bigr{\rangle}.$ (6.34)

Translation invariance dictates that for the first term

$\displaystyle\Bigl{\langle}U_{R}(-\infty;x^{+})\partial^{\perp}A^{-}(x^{+})\,U_{R}(x^{+};x^{+}/2)\,A^{\perp}(x^{+}/2)U_{R}(x^{+}/2;-\infty)\Bigr{\rangle}=\Bigl{\langle}U_{R}(-\infty;x^{+}/2)$
$\displaystyle\times\partial^{\perp}A^{-}(x^{+}/2)\,U_{R}(x^{+}/2;0)\,A^{\perp}(0)U_{R}(0;-\infty)\Bigr{\rangle}.$
(6.35)

By aptly redefining the integration variable for this term one finds that

$\displaystyle
Z_{g}^{\perp-}=\frac{1}{d_{R}C_{R}}\int_{0}^{\infty}dx^{+}\,\Tr\,\Bigl{\langle}U_{R}(-\infty;x^{+})\partial^{\perp}A^{-}(x^{+})\,U_{R}(x^{+};0)\,A^{\perp}(0)U_{R}(0;-\infty)\Bigr{\rangle}.$
(6.36)

For its counterpart we have

$\displaystyle
Z_{g}^{-\perp}=-\frac{1}{d_{R}C_{R}}\int_{0}^{\infty}dy^{+}\,\Tr\,\Bigl{\langle}U_{R}(-\infty;2y^{+})A^{\perp}(2y^{+})\,U_{R}(2y^{+};y^{+})\,\partial^{\perp}A^{-}(y^{+})U_{R}(y^{+};-\infty)\Bigr{\rangle}.$
(6.37)

where we used the fact that, in a non-singular gauge, $A^{\perp}(\infty)=0$.
By applying translation invariance and relabeling $y^{+}$ into $x^{+}$ this
becomes

$\displaystyle
Z_{g}^{-\perp}=-\frac{1}{d_{R}C_{R}}\int_{0}^{\infty}dx^{+}\,\Tr\,\Bigl{\langle}U_{R}(-\infty;x^{+})A^{\perp}(x^{+})\,U_{R}(x^{+};0)\,\partial^{\perp}A^{-}(0)U_{R}(0;-\infty)\Bigr{\rangle}.$
(6.38)

Finally, for the term with both commutators, we find

$\displaystyle
Z_{g}^{--}=\frac{1}{d_{R}C_{R}}\,\Tr\,\Bigl{\langle}U_{R}(-\infty;0)A^{\perp}(0)\,A^{\perp}(0)U_{R}(0;-\infty)\Bigr{\rangle},$
(6.39)

where we have again set the boundary term at infinity to zero.

At this point, we recall that in Feynman gauge the – gluon propagator vanishes
and furthermore, that a three-gluon vertex with more than one - gluon does as
well. Let us then consider Eq. (6.33): at LO it must vanish in Feynman gauge.
At NLO it can only contribute to part of diagram $(c)$. Note however that the
Wilson lines cannot contribute on their own for the same reasons outlined in
Sec. (6.1.4). Moreover, there cannot be a contribution of the form of diagram
$(g)$ as the $gA^{-}A^{\perp}$ part of the covariant derivative has been
removed from the correlator through the previous manipulations.

Similar arguments can be safely applied to Eqs. (6.36) and (6.38): they also
have a vanishing $(f)$ contribution and they correspond to the mixed
$G^{x-}_{>}(K)$ terms in Eq. (6.4).

Eq. (6.39) does not vanish at LO: it corresponds precisely to diagram $(a)$.
At NLO it too contains a contribution to $(c)$, that proportional to
$G^{xx}_{>}(K)$ terms in Eq. (6.4). It also contains the only contribution
directly involving the Wilson lines, as either of them can source an $A^{-}$
gluon that can connect with a three-gluon vertex to the two $A^{\perp}$ ones.
However, that would give rise to the following colour structure

$\mathcal{C}=f^{abc}A^{\perp\,a}A^{\perp\,b}A^{-\,c},$ (6.40)

which vanishes, due to the symmetric (anti-symmetric) nature of
$A^{\perp\,a}A^{\perp\,b}$ ($f^{abc}$). Given that, through this
reorganisation, we only encounter non-zero contributions from diagram $(c)$,
we conclude that the respective contributions from diagrams $(f)$ and $(g)$
must cancel each other. We further back up this argument with a direct
diagrammatic proof in the next section.

####  6.2.2 Diagrammatic Proof

Coming back to Eq. (6.22), we note that by suitably relabelling momenta, we
can drastically simplify this expression: let us start by considering the
second term and relabeling $K-Q\rightarrow K,\\\ Q\rightarrow
Q-K\,,K\rightarrow Q$. This yields

$\displaystyle Z_{g}^{(f)\,I}+Z_{g}^{(f)\,III}$
$\displaystyle=-ig^{2}C_{A}\int_{K,Q}\Big{[}\frac{q^{-}(k^{-}+q^{-})}{(k^{-}-i\varepsilon)(k^{-}-q^{-}-i\varepsilon)}+\frac{(q^{-}-k^{-})(2q^{-}-k^{-})}{(q^{-}-i\varepsilon)(k^{-}-i\varepsilon)}\Big{]}$
$\displaystyle\times G_{R}(K-Q)G_{rr}(Q)G_{R}(K)$
$\displaystyle=-ig^{2}C_{A}\int_{K,Q}\Big{[}\frac{q^{-}(k^{-}+q^{-})}{(k^{-}-i\varepsilon)(k^{-}-q^{-}-i\varepsilon)}+\frac{2q^{-}}{(k^{-}-i\varepsilon)}\Big{]}$
$\displaystyle\times G_{R}(K-Q)G_{rr}(Q)G_{R}(K)$ (6.41)

where in going to the second line, we have used that we can throw away any
terms in the second term with a factor of $k^{-}$ in the numerator, since the
$k^{-}$ integral can then be closed in the upper half plane. We then perform
another relabelling of the momenta in the second term: $K-Q\rightarrow
K\,,\,Q\rightarrow-Q\,,\,K\rightarrow K-Q$, under which the multiplied
propagators are invariant

$\displaystyle Z_{g}^{(f)\,I}+Z_{g}^{(f)\,III}$
$\displaystyle=-ig^{2}C_{A}\int_{K,Q}\Big{[}\frac{q^{-}(k^{-}+q^{-})}{(k^{-}-i\varepsilon)(k^{-}-q^{-}-i\varepsilon)}-\frac{2q^{-}}{(k^{-}-q^{-}-i\varepsilon)}\Big{]}$
$\displaystyle\times G_{R}(K-Q)G_{rr}(Q)G_{R}(K)$
$\displaystyle=ig^{2}C_{A}\int_{K,Q}\frac{q^{-}}{k^{-}-i\varepsilon}G_{R}(K-Q)G_{rr}(Q)G_{R}(K).$
(6.42)

At this point, let us go back and look at the expression coming from the
evaluation of diagram $(g)$, Eq. (6.18) and relabel so that $Q\rightarrow
K-Q\,,\,K-Q\rightarrow Q\,,\,K\rightarrow K$, yielding

$Z^{(g)}_{g}=-4ig^{2}C_{A}\int_{K,Q}\frac{k^{-}+q^{-}}{k^{-}-i\varepsilon}\left(1+n_{\text{B}}(k^{0})\right)\left(\widetilde{\Pi_{R}}(K)G_{R}(K)-\widetilde{\Pi_{A}}(K)G_{A}(K)\right).$
(6.43)

which of course resembles what we have in Eq. (6.26). Let us then add both
contributions to see if they cancel

$\displaystyle Z_{g}^{(f)\,II}$
$\displaystyle+Z_{g}^{(f)\,V}+Z_{g}^{(g)}=ig^{2}C_{A}\int_{K,Q}\frac{k^{-}+q^{-}}{k^{-}-i\varepsilon}\Big{\\{}2\widetilde{\Pi_{A}}(K)G_{A}(K)$
$\displaystyle-
G_{R}(K)G_{R}(K-Q)G_{rr}(Q)-\frac{1}{2}G_{R}(K)G_{R}(K-Q)G_{R}(Q)\Big{\\}}.$
(6.44)

The first term above is advanced in $k^{-}$, so we can just close the
integration in the lower half plane to see that it vanishes. Likewise, for the
$k^{-}/(k^{-}-i\varepsilon)$ proportional part of the second term, we can
close the $k^{-}$ integration in the upper half plane to see that it does not
contribute. The third term will vanish after a suitable relabelling of the
momenta whereas the $q^{-}/(k^{-}-i\varepsilon)$ part of the second term
cancels against what we found in Eq. (6.42).

Alternatively, we could invoke the ideas from Sec. 4.3 since in the case of
the second and third term, the propagators are lightlike separated and are
thus retarded in both $k^{-}$ and $k^{+}$. The $1/(k^{-}-i\varepsilon)$ out in
front means that we cannot close the $k^{-}$ in the upper half plane without
picking up a contribution there. However, there are no such extra poles in
$k^{+}$ and so we can therefore close the $k^{+}$ integration in the upper
half-plane.

In any case, we undoubtedly conclude

$Z_{g}^{(f)}+Z_{g}^{(g)}=0.$ (6.45)

###  6.3 Matching from Full QCD

In this section, we compute zero-mode contributions from the gluonic part of
diagram $(c)$ and thus $Z_{g}$, before checking that they cancel with the
corresponding EQCD contributions in the next section. Throughout, for
consistency’s sake, we set up the integration so that the $K$ integration is
done in DR whereas the $Q$ integration is done in $4$ dimensions333We choose
this scheme to avoid having to consider the $d$-dimensional equivalents of the
one-loop self-energies, listed in App. D.2.. It is useful in this context to
think of the contributions to $Z_{g}$, in diagram $(c)$ as a double sum over
Matsubara modes, one sum coming from $k^{0}$ and the other coming from
$q^{0}$. Since EQCD is a theory of soft modes, it only accounts for the _zero
modes_ of the full theory. It is then natural to expect corresponding
divergences on the full QCD side to show up in the term of the double sum with
both zero modes. Indeed, this is _physically_ what happens. However, this
double zero mode term will give no contribution in DR, as it is the result of
a scaleless integral. Thus, _mathematically_ , what happens is that the
divergence gets shifted from the $n=0\,,\,n=0$ term to the $n=0\,,\,n\neq 0$
terms. We make a point to organise things so the divergence comes from the
zero mode associated with the momentum $K$.

We emphasise that we only need to do this matching for the gluonic
contribution to $Z_{g}^{(c)}$ as EQCD only describes the dynamics of the
gluonic zero modes.

####  6.3.1 $k^{-2}$ Piece

We begin by computing the $k^{-2}$ contribution, defined by Eqs. (6.9) and
(6.10). We moreover need the gluonic form of $\Pi_{T}^{R}(K)$, Eq. (D.12)

$\displaystyle Z^{(c)}_{g\,k^{-2}\,G}$
$\displaystyle=i\frac{g^{2}C_{A}}{8\pi^{2}}\int_{K}\int_{0}^{\infty}dq\frac{(1+n_{B}(k^{0}))}{k^{3}}n_{B}(q)\Bigg{[}\frac{1}{(K^{2}+i\varepsilon
k^{0})^{2}}\Bigg{(}8qk(k^{2}+k_{0}^{2})$
$\displaystyle-K^{2}(4q^{2}+3k^{2}+k_{0}^{2})\left(\ln\frac{k_{0}+k-2q+i\varepsilon}{k_{0}-k-2q+i\varepsilon}+\ln\frac{k_{0}-k+2q+i\varepsilon}{k_{0}+k+2q+i\varepsilon}\right)$
$\displaystyle-4qk_{0}K^{2}\left(\ln\frac{k_{0}+k+i\varepsilon}{k_{0}-k+i\varepsilon}+\ln\frac{k_{0}-k+2q+i\varepsilon}{k_{0}+k+2q+i\varepsilon}-\ln\frac{k_{0}+k-2q+i\varepsilon}{k_{0}-k-2q+i\varepsilon}\right)\Bigg{)}-\text{adv.}\Bigg{]}$
(6.46)

Let us then the write $k^{0}$ integral as a sum over the Matsubara modes. This
leaves us with

$\displaystyle Z^{(c)}_{g\,k^{-2}\,G}$
$\displaystyle=-\frac{g^{2}C_{A}T}{8\pi^{2}}\sum_{n}\int_{k}\int_{0}^{\infty}dq\frac{1}{k^{3}}n_{B}(q)\frac{1}{(-\omega_{n}^{2}-k^{2})^{2}}\Bigg{(}8qk(k^{2}-\omega_{n}^{2})$
$\displaystyle-(-\omega_{n}^{2}-k^{2})(4q^{2}+3k^{2}-\omega_{n}^{2})\left(\ln\frac{i\omega_{n}+k-2q}{i\omega_{n}-k-2q}+\ln\frac{i\omega_{n}-k+2q}{i\omega_{n}+k+2q}\right)$
$\displaystyle-4iq\omega_{n}(-\omega_{n}^{2}-k^{2})\left(\ln\frac{i\omega_{n}+k}{i\omega_{n}-k}+\ln\frac{i\omega_{n}-k+2q}{i\omega_{n}+k+2q}-\ln\frac{i\omega_{n}+k-2q}{i\omega_{n}-k-2q}\right)\Bigg{)}.$
(6.47)

Next, we extract the zero mode contribution

$Z^{(c)}_{g\,k^{-2}\,G\,n=0}=-\frac{g^{2}C_{A}T}{8\pi^{2}}\int_{k}\int_{0}^{\infty}dq\frac{1}{k^{3}}n_{B}(k)\frac{1}{k^{4}}\Bigg{(}8qk^{3}+2k^{2}(4q^{2}+3k^{2})\ln\left|\frac{2q-k}{k+2q}\right|\Bigg{)}.$
(6.48)

Before carrying out the integration in DR, it is first useful to rescale
$k\rightarrow kq$ so that the integrations factorise, i.e

$\displaystyle Z^{(c)}_{g\,k^{-2}\,G\,n=0}$
$\displaystyle=-\frac{g^{2}C_{A}T}{8\pi^{2}}\Big{(}\frac{\bar{\mu}^{2}e^{\gamma_{E}}}{4\pi}\Big{)}^{\epsilon}\Big{(}\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d}{2}\right)(2\pi)^{d}}\Big{)}\int_{0}^{\infty}dk\int_{0}^{\infty}dq\frac{k^{d-1}}{k^{5}}\frac{q^{d+2}}{q^{5}}n_{B}(q)\Bigg{(}8k$
$\displaystyle+2(4+3k^{2})\ln\left|\frac{2-k}{k+2}\right|\Bigg{)}$
$\displaystyle=\frac{g^{2}C_{A}T^{2}}{32\pi^{2}}\bigg{[}-\frac{3}{\epsilon}-\left(7+6\ln\frac{\bar{\mu}}{4T}\right)\bigg{]},$
(6.49)

where, as is specified in App. A, DR is carried out in $d=3-2\epsilon$
dimensions.

####  6.3.2 $\Pi_{L}-\Pi_{T}$ Piece

Let us start by writing the full contribution (i.e, that coming from all
modes) using Eq.s (6.9) and (D.14). This yields

$\displaystyle Z^{(c)}_{g\,L-T\,G}$
$\displaystyle=i\frac{g^{2}C_{A}}{16\pi^{2}}\int_{K}\int_{0}^{\infty}dq\frac{(1+n_{B}(k^{0}))}{k^{3}(k^{-}-i\varepsilon)^{2}}n_{B}(q)\Bigg{[}\frac{k_{\perp}^{2}}{k^{2}(K^{2}+i\varepsilon
k^{0})}\Bigg{(}-8qk(3k_{0}^{2}-k^{2})$
$\displaystyle+K^{2}(12q^{2}-k^{2}+3k_{0}^{2})\left(\ln\frac{k_{0}+k-2q+i\varepsilon}{k_{0}-k-2q+i\varepsilon}+\ln\frac{k_{0}-k+2q+i\varepsilon}{k_{0}+k+2q+i\varepsilon}\right)$
$\displaystyle+12qk_{0}K^{2}\left(\ln\frac{k_{0}+k+i\varepsilon}{k_{0}-k+i\varepsilon}+\ln\frac{k_{0}-k+2q+i\varepsilon}{k_{0}+k+2q+i\varepsilon}-\ln\frac{k_{0}+k-2q+i\varepsilon}{k_{0}-k-2q+i\varepsilon}\right)\Bigg{)}-\text{adv.}\Bigg{]}.$
(6.50)

After doing a Caron-Huot shift, i.e $k^{z}\rightarrow k^{z}+k^{0}$, we isolate
the zero mode contribution by replacing $\int\frac{dk^{0}}{2\pi}$ with
$iT\sum_{n}$ and then setting $\omega_{n}\rightarrow 0$. We are left with

$\displaystyle Z^{(c)}_{g\,L-T\,G\,n=0}$
$\displaystyle=-\frac{g^{2}C_{A}T}{16\pi^{2}}\int_{k}\int_{0}^{\infty}dq\frac{1}{k^{3}(k^{z}+i\varepsilon)^{2}}n_{B}(q)\frac{k_{\perp}^{2}}{k^{2}(-k^{2})}\Bigg{(}-8qk(-k^{2})$
$\displaystyle+(-k^{2})(12q^{2}-k^{2})\left(\ln\frac{k-2q}{-k-2q}+\ln\frac{-k+2q}{k+2q}\right)\Bigg{)}$
$\displaystyle=\frac{g^{2}C_{A}T}{16\pi^{2}}\int_{k}\int_{0}^{\infty}dq\frac{k_{\perp}^{2}}{k^{7}(k^{z}+i\varepsilon)^{2}}n_{B}(q)\Bigg{(}8qk^{3}-k^{2}(12q^{2}-k^{2})\left(2\ln|\frac{2q-k}{k+2q}\bigg{|}\right)\Bigg{)}.$
(6.51)

In contrast to the previous piece, the integrand depends on the polar angle,
$\theta$, which we deal with by making the substitution $x=\cos\theta$. The
integral then yields

$\displaystyle Z^{(c)}_{g\,L-T\,G\,n=0}$
$\displaystyle=\frac{g^{2}C_{A}T}{16\pi^{2}}\left(\frac{\bar{\mu}^{2}e^{\gamma_{E}}}{4\pi}\right)^{\epsilon}\left(\frac{2\pi^{\frac{d-1}{2}}}{\Gamma\left(\frac{d-1}{2}\right)(2\pi)^{d}}\right)\int_{0}^{\infty}dk\int_{-1}^{1}dx\int_{0}^{\infty}dq\,n_{B}(q)k^{d-8}\frac{(1-x^{2})^{\frac{d-1}{2}}}{x^{2}}$
$\displaystyle\Bigg{(}8qk^{3}-k^{2}(12q^{2}-k^{2})\left(2\ln|\frac{2q-k}{k+2q}\bigg{|}\right)\Bigg{)}$
$\displaystyle=\frac{g^{2}C_{A}T}{16\pi^{2}}\left(\frac{\bar{\mu}^{2}e^{\gamma_{E}}}{4\pi}\right)^{\epsilon}\left(\frac{2\pi^{\frac{d-1}{2}}}{\Gamma\left(\frac{d-1}{2}\right)(2\pi)^{d}}\right)\int_{0}^{\infty}dk\int_{-1}^{1}dx\int_{0}^{\infty}dqT^{d-2}\frac{q^{d-7}}{e^{q}-1}k^{d-8}\frac{(1-x^{2})^{\frac{d-1}{2}}}{x^{2}}$
$\displaystyle\Bigg{(}8q^{4}k^{3}-2k^{2}q^{2}(12q^{2}-k^{2}q^{2})\ln|\frac{2-k}{k+2}\bigg{|}\Bigg{)}$
$\displaystyle=\frac{g^{2}C_{A}T^{d-1}}{16\pi^{2}}\left(\frac{\bar{\mu}^{2}e^{\gamma_{E}}}{4\pi}\right)^{\epsilon}\left(\frac{2\pi^{\frac{d-1}{2}}}{\Gamma\left(\frac{d-1}{2}\right)(2\pi)^{d}}\right)\int_{0}^{\infty}dk\int_{-1}^{1}dx\int_{0}^{\infty}dq\frac{q^{d-3}}{e^{q}-1}k^{d-6}\frac{(1-x^{2})^{\frac{d-1}{2}}}{x^{2}}$
$\displaystyle\Bigg{(}8k-2(12-k^{2})\ln|\frac{2-k}{k+2}\bigg{|}\Bigg{)}.$
(6.52)

In the first line, we have dropped the $i\varepsilon$ piece since the $x$
dependent part of the integrand is a $\beta$ function, the integral of which
will converge for some $d$. Going from the second to the third line, we have
rescaled $q\rightarrow qk$ and $k\rightarrow k/T$. Upon doing the
integrations, we find

$Z^{(c)}_{g\,L-T\,G\,n=0}=\frac{g^{2}C_{A}T^{2}}{32\pi^{2}}\left(-\frac{1}{\epsilon}+4-\ln\frac{\bar{\mu}^{2}e^{\gamma_{E}}}{4T^{2}}-\psi\left(\frac{3}{2}\right)\right),$
(6.53)

where $\psi(x)$ is the digamma function. Having extracted these IR
divergences, we can now move on to identify their (UV) EQCD counterparts in
the next section.

###  6.4 Matching from EQCD

Before completing the matching of divergences between full QCD and EQCD, we
take a moment to lay out some details of the perturbative EQCD (pEQCD)
calculation of $Z_{g}$ from [15, 16]. From the EQCD Lagrangian, Eq. (4.28),
one derives the EQCD analog of the Wilson line, Eq. (4.109)

$\tilde{U}_{R}(L;0)=\mathcal{P}\;\exp\left(ig_{\text{3d}}\int_{0}^{L}dz\left(A_{z}^{a}(z)+i\Phi^{a}(z)\right)T^{a}_{R}\right).$
(6.54)

In EQCD, since the field operators are classical, they commute, which allows
us to write

$\frac{1}{d_{R}C_{R}}\Tr\langle\tilde{U}_{R}(-\infty;L)\mathcal{O}^{a}(L)T_{R}^{a}\tilde{U}_{R}(L;0)\mathcal{O}^{b}(0)T_{R}^{b}\tilde{U}_{R}(0;-\infty)\rangle=\frac{T_{R}}{d_{R}C_{R}}\langle\mathcal{O}^{a}(L)\tilde{U}^{ab}_{A}(L;0)\mathcal{O}^{b}(0)\rangle.$
(6.55)

Thus, the EQCD equivalent of $Z_{g}$ reads, given rotational invariance in the
transverse plane

$Z_{g}^{3\text{d}}=-\frac{2T_{R}T}{d_{R}C_{R}}\int_{0}^{L_{\text{min}}}dL\,L\Bigl{\langle}\Big{(}F^{a}_{xz}(L)+i(D_{x}\Phi(L))^{a}\Big{)}\tilde{U}^{ab}_{A}(L;0)\Big{(}F^{b}_{xz}(0)+i(D_{x}\Phi(0))^{b}\Big{)}\Bigr{\rangle}$
(6.56)

In terms of the lattice computation, it is more convenient to work with the
quantities

$\displaystyle\langle EE\rangle$
$\displaystyle\equiv\frac{1}{2}\bigl{\langle}(D_{x}\Phi(L))^{a}\,\tilde{U}^{ab}_{A}(L;0)\,(D_{x}\Phi(0))^{b}\bigr{\rangle}\;,$
(6.57) $\displaystyle\langle BB\rangle$
$\displaystyle\equiv\frac{1}{2}\bigl{\langle}F_{xz}^{a}(L)\,\tilde{U}^{ab}_{A}(L;0)\,F_{xz}^{b}(0)\bigr{\rangle}\;,$
(6.58) $\displaystyle\langle EB\rangle$
$\displaystyle\equiv\frac{i}{2}\bigl{\langle}(D_{x}\Phi(L))^{a}\,\tilde{U}_{A}^{ab}(L;0)\,F_{xz}^{b}(0)\bigr{\rangle}$
(6.59)
$\displaystyle+\frac{i}{2}\bigl{\langle}F_{xz}^{a}(L)\,\tilde{U}_{A}^{ab}(L;0)\,(D_{x}\Phi(0))^{b}\bigr{\rangle}\;,$
(6.60)

which leaves us with

$\displaystyle Z_{g}^{3\text{d}}$
$\displaystyle=-\frac{4T_{R}T}{d_{R}C_{R}}\int_{0}^{L_{\text{min}}}dL\,L\Big{(}-\langle
EE\rangle+\langle BB\rangle+i\langle EB\rangle\Big{)}$
$\displaystyle=\frac{4T_{R}T}{d_{R}C_{R}}\int_{0}^{L_{\text{min}}}dL\,L\langle
FF\rangle,$ (6.61)

where we have implicitly defined $\langle FF\rangle=\langle EE\rangle-\langle
BB\rangle-i\langle EB\rangle$.

Figure 6.4:  Continuum-extrapolated EQCD lattice data on $\langle EE\rangle$
(left) and $\langle FF\rangle$ (right). Short dashes represent the long
$L$-tail whole short dashes denote the NLO pEQCD evaluation. $L^{3}$
compensates the $c_{0}/L^{3}$ divergence. Figure from [16].

In Fig. 6.4, the lattice EQCD evaluations of $\langle EE\rangle$ (left) and
$\langle FF\rangle$ (right) are shown. One cannot integrate up to arbitrarily
large values of $L$ on the lattice, which is made apparent by the rather large
error bars in that region. This IR region is addressed by a fitting ansatz,
intended to reproduce the expected exponential falloff from electrostatic and
magnetostatic screening. In the UV, where $m_{D}L\ll 1$, the lattice
determination agrees well with the pEQCD determination, as expected. However,
as one moves further into the deep UV, the lattice approach becomes
impractical due to associated discretisation effects and one must rely on a
perturbative evaluation. Such a perturbative evaluation does not, however,
come from pEQCD; as an IR EFT of full QCD, EQCD exhibits unphysical behaviour
for momenta $k\gg m_{D}$.

Because of the super-renormalisability of EQCD, in the UV, it is natural to
expect that

$\langle FF\rangle_{m_{D}L\ll
1}=\frac{c_{0}}{L^{3}}+\frac{c_{2}g^{2}_{\text{3d}}}{L^{2}}+\frac{c_{4}g^{4}_{\text{3d}}}{L}+...\,.$
(6.62)

The first and second terms above are divergent when inserted into Eq. (6.61).
The former would have come out of the calculation in Sec. 4.4.2 if we had
employed a regularisation other than DR. It moreover cancels against a power
law divergence coming from the LO calculation444Similarly, this divergence
would be present in, for instance, Eq.(6.3) if we had used a regularisation
other than DR. See Eq. (4.115).. The latter was computed in [16] and, being a
logarithmic divergence, does show up in DR.

Figure 6.5: EQCD equivalent of diagram $(c)$. The labelling is identical to
that from Fig. 6.1 except that here, curly lines represent spatial gauge
bosons, solid double lines are (adjoint) Wilson lines and solid lines are
adjoint scalars.

To arrive at a solution for $Z_{g}^{\text{3d}}$, accurate across all energy
scales, one can imagine stitching the perturbative, intermediate and IR
regions together

$\displaystyle Z_{g}^{\text{3d}}\big{|}^{\text{merge}}=$
$\displaystyle\frac{4T}{d_{A}}\bigg{\\{}\int_{0}^{L_{\text{min}}}dL\,L\bigg{[}\langle
FF\rangle_{\text{NLO}}-\frac{c_{0}}{L^{3}}-\frac{c_{2}g_{\text{3d}}^{2}}{L^{2}}\bigg{]}+\int_{L_{\text{min}}}^{L_{\text{max}}}dL\,L\bigg{[}\langle
FF\rangle_{\text{lat}}-\frac{c_{0}}{L^{3}}\bigg{]}$ $\displaystyle\hskip
14.22636pt+\int_{L_{\text{max}}}^{\infty}dL\,L\bigg{[}\langle
FF\rangle_{\text{tail}}-\frac{c_{0}}{L^{3}}\bigg{]}\bigg{\\}}.$ (6.63)

$L_{\text{min}}$ is taken to be $0.25/g_{3\text{d}}^{2}$, as can be seen in
Fig. 6.4.

Our goal in this section is essentially to check the validity of the NLO
subtraction in the first term above. In particular, such a logarithmic UV
divergence should cancel against the logarithmic IR divergence from the full
QCD calculation that we computed in the previous section. Moreover, from the
arguments of Sec. 6.2, we know that this UV divergence must come from the EQCD
analogue of diagram $(c)$, shown in Fig. 6.5. We first take the $\langle
BB\rangle$ contribution from Appendix A.3 of [16]555 At the time of writing,
an extra factor of $1/2$ is present in Eq. (6.64) compared to its counterpart
in [16], due to a typo identified in the original work. For the same reason,
an extra factor of $1/2$ is inserted in Eq. (6.68). (Eq. A.14)

$(c)^{BB}_{\text{div}}=\frac{1}{4}\int_{k}\frac{e^{iLk_{z}}}{k^{4}}(k_{\perp}^{2}+2k_{z}^{2})\delta^{ab}\Big{(}-\Pi^{T}_{A^{a}_{i}A^{b}_{j}\,\text{div}}(k)\Big{)}.$
(6.64)

where

$\displaystyle\Pi^{T}_{A^{a}_{i}A^{b}_{j}}(k)$
$\displaystyle=\delta^{ab}\frac{g^{2}_{\text{3d}}C_{A}}{16\pi}\left(-2m_{D}+\frac{14}{4}k+\frac{k^{2}+4m_{D}^{2}}{k}\arctan\frac{k}{2m_{D}}\right)$
$\displaystyle\stackrel{{\scriptstyle k\gg
m_{D}}}{{\approx}}\delta^{ab}\frac{g^{2}_{3d}C_{A}}{16\pi}\left(-3\pi
k\right)\equiv\Pi^{T}_{A^{a}_{i}A^{b}_{j}\,\text{div}}(k).$ (6.65)

We have expanded for $k\gg m_{D}$ in order to probe its UV, i.e, the region
where EQCD is no longer applicable. The loop integration, performed to arrive
at Eq. (6.65) was done in $3$ dimensions, whereas the $k$ integration in Eq.
(6.64) will be done in $d$ dimensions – this is consistent with the
integration procedure followed on the full QCD side. The second term in Eq.
(6.65), proportional to $2k_{z}^{2}$ should correspond to the $2k^{-2}$ term
computed in Sec. 6.3.1. Indeed, looking to Eq. (6.61), we find

$\displaystyle Z^{3d\,(c)}_{g\,k^{-2}\,G\,\text{div}}$
$\displaystyle=-\frac{4T}{d_{A}}\int_{0}^{L_{\text{min}}}dL\,L\,(c)^{BB}_{k^{-2}\,\text{div}}$
$\displaystyle=\frac{3g^{2}_{3d}C_{A}T}{32\pi^{2}}\Big{(}\frac{1}{\epsilon}-2+3\gamma_{E}+2\ln{L_{\text{min}}\bar{\mu}}+\psi\left(\frac{1}{2}\right)\Big{)}.$
(6.66)

Given that $g^{2}_{3d}\sim g^{2}T$, it is clear that the $1/\epsilon$ term
cancels the one from Eq. (6.49).

As for the $L-T$ piece, we need the contribution from the $\langle EE\rangle$
correlator. Again, we extract this from Eq. A.8 of [16] and expand towards the
UV to arrive at

$\displaystyle\Pi_{\Phi^{a}\Phi^{b}}(k)$
$\displaystyle=\delta^{ab}\frac{g^{2}_{3d}C_{A}m_{D}}{16\pi}\left(-4-\frac{6C_{F}-C_{A}}{6C_{A}}\frac{\lambda_{E}}{g^{2}_{3d}}-8\frac{k^{2}-m_{D}^{2}}{km_{D}}\arctan\frac{k}{m_{D}}\right)$
$\displaystyle\stackrel{{\scriptstyle k\gg
m_{D}}}{{\approx}}\delta^{ab}\frac{g^{2}_{3d}C_{A}}{16\pi}\left(-4\pi
k\right)\equiv\Pi_{\Phi^{a}\Phi^{b}\,\text{div}}(k).$ (6.67)

From Eq. $A.13$ of [16], we then write

$(c)^{EE}_{\text{div}}=\frac{1}{4}\int_{k}\frac{e^{iLk_{z}}}{k^{4}}k_{\perp}^{2}\delta^{ab}\Big{(}-\Pi^{T}_{\Phi^{a}\Phi^{b}\,\text{div}}(k)\Big{)}.$
(6.68)

In order to get the full $\Pi_{L}-\Pi_{T}$ EQCD contribution, we need to add
to this the first term in Eq. (6.64), proportional to $k_{\perp}^{2}$.
Inserting these into Eq. (6.61),

$\displaystyle Z^{3d\,(c)}_{g\,L-T\,G\,\text{div}}$
$\displaystyle=-\frac{4T}{d_{A}}\int_{0}^{L_{\text{min}}}dL\,L\,\left((c)^{BB}_{k_{\perp}^{2}\,\text{div}}-(c)^{EE}_{\text{div}}\right)$
$\displaystyle=\frac{g^{2}_{3d}C_{A}T}{32\pi^{2}}\bigg{(}\frac{1}{\epsilon}-3+3\gamma_{E}+2\ln{L_{\text{min}}\bar{\mu}}+\psi\left(-\frac{1}{2}\right)\bigg{)},$
(6.69)

which, reassuringly contains a $1/\epsilon$ pole that cancels exactly against
the one from Eq. (6.53). Having confirmed the validity of the subtraction in
Eq. (6.63), we proceed to write

$Z_{g}^{\mathrm{match}}=\frac{g^{2}C_{A}T^{2}}{4\pi^{2}}\bigg{[}\ln(2L_{\text{min}}T)+\gamma_{E}-\frac{3}{2}\bigg{]}\,,\qquad
Z_{g}^{\text{non-pert
class}}=Z_{g}^{\text{3d}}\big{|}^{\text{merge}}+Z_{g}^{\text{match}},$ (6.70)

where, from Eqs. (6.49), (6.53), (6.66) and (6.69)

$\displaystyle Z_{g}^{\text{match}}$
$\displaystyle=Z^{(c)}_{g\,k^{-2}\,G\,n=0}+Z^{\text{3d}\,(c)}_{g\,k^{-2}\,G\,\text{div}}+Z^{(c)}_{g\,L-T\,G\,n=0}+Z^{\text{3d}\,(c)}_{g\,L-T\,G\,\text{div}}$
$\displaystyle=\frac{g^{2}C_{A}T^{2}}{8\pi^{2}}\bigg{[}2\ln(2L_{\text{min}}T)+2\gamma_{E}-3\bigg{]}.$
(6.71)

Note that in computing $Z_{g}^{\text{non-pert class}}$, the $\ln
L_{\text{min}}$ above cancels against the $\int
dL\frac{c_{2}g_{\text{3d}}^{2}}{L}$ term in Eq. (6.63). For reasonable values
of $g_{\text{3d}}$, extracted from [13], we find that $Z_{g}^{\text{non-pert
class}}$ provides a negative contribution to $Z_{g}$. However, we should not
yet make any conclusive remarks about the convergence of $Z_{g}$ until all of
its $\mathcal{O}(g^{2})$ corrections have been obtained. We present the first
steps towards a full determination of the $\mathcal{O}(g^{2})$ corrections in
the next section.

###  6.5 Full Evaluation of Diagram $(c)$

We conclude this chapter by completing the evaluation of diagram $(c)$ – the
self-energy diagram. The computation is split as follows: the finite-
temperature fermionic and gluonic self-energies are given in Secs.6.6, 6.6.1
respectively and the vacuum fermionic and gluonic self-energies are presented
together in Sec. 6.6.2. We begin by computing the finite-temperature fermionic
contribution, which, naively should be free of any divergences. We discover
that such an expectation is not met and comment on sources of these
divergences where appropriate. As was done in Sec. 6.3, we further divide up
the finite-temperature self-energies into the $k^{-2}$ and $\Pi_{L}-\Pi_{T}$
pieces with the $Q$ integration performed in $4$ dimensions and the $K$
integration in $d$ dimensions. Many of the technical details associated with
the integration are left to App. D.3.

###  6.6 Finite-Temperature Fermionic Contribution

We pick up directly from Eq. (6.9), with the finite-temperature gluon and
fermion self-energies given in App. D.2. The contribution from the
$k^{-2}$-proportional part is computed in App. D.3.1 and reads

$\displaystyle Z_{g\,k^{-2}\,F}^{(c)}=$
$\displaystyle\frac{g^{2}T_{F}N_{F}T^{2}}{96\pi^{2}}\bigg{\\{}-3\left(4\ln(2)\ln\left(\frac{2}{A^{8}}\right)+2\ln(2)\right)+\frac{\pi^{2}}{6}-8\gamma_{E}\ln
2$ $\displaystyle+4.4713-\frac{96\ln 2}{\pi^{2}}+3.7933+1.70118\bigg{\\}}.$
(6.72)

From the absence of any $\epsilon$-dependent terms, it is evidently finite. We
have cross-checked this result by numerically performing the sum over the
Matusbara modes associated with $K$.

However, we find that the $\text{L}-\text{T}$ contribution, computed in App.
D.3.2 is

$\displaystyle Z_{g\,F\,\text{ L-T
}}^{(c)}=\frac{g^{2}T_{F}N_{F}T^{2}}{48\pi^{2}}\bigg{\\{}\frac{2}{\epsilon^{2}}+\frac{2\ln\frac{\bar{\mu}\exp(\gamma_{E}+1/2)}{4\pi
T}+2(\ln(2)-1)+2\ln\frac{\bar{\mu}A^{12}e^{-1/2}}{16\pi T}}{\epsilon}$
$\displaystyle+2\ln^{2}\frac{\bar{\mu}\exp(\gamma_{E}+1/2)}{4\pi
T}-4\gamma_{1}+\frac{\pi^{2}}{4}-2\gamma_{E}^{2}+\frac{3}{2}$
$\displaystyle+2(\ln(2)-1)\bigg{[}2\ln\frac{\bar{\mu}\exp(\gamma_{E}+1/2)}{4\pi
T}\bigg{]}$ $\displaystyle+2\ln^{2}\frac{\bar{\mu}A^{12}e^{-1/2}}{16\pi T}$
$\displaystyle+48\ln(A)(\gamma_{E}+\ln(2\pi)-6\ln(A))+\frac{12\zeta^{\prime\prime}(2)}{\pi^{2}}+\frac{1}{2}\text{Li}_{2}\left(\frac{2}{3}\right)+\frac{1}{4}\text{Li}_{2}\left(\frac{3}{4}\right)$
$\displaystyle-\frac{1}{2}-\frac{11}{2}\ln^{2}(2)-2\ln(\pi)\ln(4\pi)-2\gamma_{E}\left(\gamma_{E}+\ln\left(4\pi^{2}\right)\right)-\frac{1}{2}\ln(3)\coth^{-1}(7)$
$\displaystyle\underbrace{-0.438129}_{k<2q}+\underbrace{1.97382}_{k>2q}\bigg{\\}}.$
(6.73)

In contrast to the $k^{-2}$ piece, this contribution is divergent – the
$1/\epsilon^{2}$ pole signals the presence of a double logarithmic divergence.
We can understand that one of these divergences coming from the region where
$k>2q$, implying it is of a UV nature. From only the expression given above,
the origin of the other logarithmic divergence is not at all apparent.
However, by looking to, for instance, Eq. (D.53), we note the potential for a
logarithmic divergence in $k$ when $k^{0}-k\ll k^{0}+k$. This is precisely the
region of phase space that is sensitive to _collinear physics_ , where
$K^{\mu}\sim T$ with $K^{2}$ small. We thus identify the second logarithmic as
a collinear one. We further take this opportunity to hammer home the point
that these divergences _are not_ cancelled by opposing ones coming from the
soft sector, as EQCD is a theory of gluons only.

####  6.6.1 Finite-Temperature Gluonic Contribution

We start with the finite-temperature gluonic $k^{-2}$-proportional part, with
the details left to App. D.3.3. In this case, we must be mindful not to double
count the zero mode contribution already obtained in Eq. (6.49). In doing so,
we identify a finite contribution that comes from the non-zero Matsubara modes

$Z^{(c)}_{g\,k^{-2}\,G\,n\neq
0\,\text{finite}}=-\frac{g^{2}C_{A}T^{2}}{32\pi^{2}}\times 0.00380658,$ (6.74)

which we have obtained numerically. In addition, we isolate a divergent
contribution from the non-zero Matsubara modes, given by

$\displaystyle Z^{(c)}_{g\,k^{-2}\,G\,n\neq 0\,\text{div}}$
$\displaystyle=\frac{g^{2}C_{A}T^{2}}{48\pi^{2}}\left(\frac{1}{\epsilon}-2\ln\frac{4\pi
T}{A^{12}\bar{\mu}}-2.44147-2.44091\right).$ (6.75)

Unlike the finite-temperature fermionic $k^{-2}$ piece, the gluonic part is
divergent – one from the zero-mode and another from the non-zero modes. While
we have already discussed the former at length, the latter is of UV nature and
we comment further on how it is dealt with at the end of Sec. 6.6.2.

A similar computation, contained in App. D.3.4 shows that the finite-
temperature gluonic $\text{L}-\text{T}$ reads

$\displaystyle Z_{g\,\text{G L-T
}}^{(c)}=\frac{g^{2}C_{A}T^{2}}{48\pi^{2}}\bigg{\\{}\frac{2}{\epsilon^{2}}+\frac{2\ln\frac{\bar{\mu}\exp(\gamma_{E}+1/2)}{4\pi
T}+2(\ln(2)-1)+2\ln\frac{\bar{\mu}A^{12}e^{-1/2}}{8\pi T}}{\epsilon}$
$\displaystyle+2\ln^{2}\frac{\bar{\mu}\exp(\gamma_{E}+1/2)}{4\pi
T}-4\gamma_{1}+\frac{\pi^{2}}{4}-2\gamma_{E}^{2}+\frac{3}{2}$
$\displaystyle+2(\ln(2)-1)\bigg{[}2\ln\frac{\bar{\mu}\exp(\gamma_{E}+1/2)}{4\pi
T}\bigg{]}$ $\displaystyle+2\ln^{2}\frac{\bar{\mu}A^{12}e^{-1/2}}{8\pi T}$
$\displaystyle+48\ln(A)(\gamma_{E}+\ln(2\pi)-6\ln(A))+\frac{12\zeta^{\prime\prime}(2)}{\pi^{2}}+\frac{1}{2}\text{Li}_{2}\left(\frac{2}{3}\right)+\frac{1}{4}\text{Li}_{2}\left(\frac{3}{4}\right)$
$\displaystyle-\frac{1}{2}-\frac{3}{2}\ln^{2}(2)-2\ln(\pi)\ln(4\pi)-2\gamma_{E}\left(\gamma_{E}+\ln\left(4\pi^{2}\right)\right)-\frac{1}{2}\ln(3)\coth^{-1}(7)$
$\displaystyle+\underbrace{0.145277}_{k<2q}\underbrace{-5.13832}_{k>2q}\bigg{\\}}.$
(6.76)

As was the case in Eq. (6.73), there is a $1/\epsilon^{2}$ pole; by adding Eq.
(6.73) and Eq. (6.76) we find the divergent terms are proportional to
$m_{D}^{2}$, signifying that the divergences present in both contributions
have the same origin.

####  6.6.2 Vacuum Contribution

Another piece, which was not needed in the matching calculation is given by
the vacuum self-energy inserted between the two thermal propagators. The
vacuum self-energy (composed of both the fermionic and gluonic pieces) is
given in Eq. (D.19). Upon plugging it into Eq. (6.9), we find

$Z_{g\text{
vac}}^{(c)}=i\frac{\alpha_{s}}{2\pi}\int_{K}[1+n_{\mathrm{B}}(k^{0})]\bigg{[}\frac{-\ln\frac{k^{2}-(k^{0}+i\varepsilon)^{2}}{\bar{\mu}^{2}}\bigg{(}\frac{5}{3}C_{A}-\frac{4}{3}T_{F}N_{F}\bigg{)}+\frac{31}{9}C_{A}-\frac{20}{9}T_{F}N_{F}}{(K^{2}+i\varepsilon
k^{0})}-\text{adv}\bigg{]}.$ (6.77)

To proceed, we rewrite the frequency integration as a Matsubara sum, i.e

$Z_{g\text{
vac}}^{(c)}=T\sum_{n}\frac{\alpha_{s}}{2\pi}\int_{k}\frac{-\ln\frac{k^{2}+\omega_{n}^{2}}{\bar{\mu}^{2}}\bigg{(}\frac{5}{3}C_{A}-\frac{4}{3}T_{F}N_{F}\bigg{)}+\frac{31}{9}C_{A}-\frac{20}{9}T_{F}N_{F}}{k^{2}+\omega_{n}^{2}}.$
(6.78)

This yields

$\displaystyle Z_{g\text{ vac}}^{(c)}=$
$\displaystyle\frac{g^{2}C_{A}T^{2}}{864\pi^{2}}\bigg{[}30\ln\frac{A^{12}\bar{\mu}}{4\pi
T}+31\bigg{]}-\frac{g^{2}T_{F}N_{F}T^{2}}{216\pi^{2}}\bigg{[}6\ln\frac{A^{12}\bar{\mu}}{4\pi
T}+5\bigg{]}$ $\displaystyle=$
$\displaystyle\frac{g^{2}T^{2}}{48\pi^{2}}\bigg{[}\bigg{(}\frac{5}{3}C_{A}-\frac{4}{3}T_{F}N_{F}\bigg{)}\ln\frac{A^{12}\bar{\mu}}{4\pi
T}+\frac{31}{18}C_{A}-\frac{10}{9}T_{F}N_{F}\bigg{]}.$ (6.79)

It is finite as the $\overline{MS}$ subtraction has already been performed in
Eq. (6.77). If we instead keep the $1/\epsilon$ poles in Eq. (D.18), we find

$Z_{g\text{ vac
pole}}^{(c)}=T\sum_{n}\frac{\alpha_{s}}{2\pi}\int_{k}\frac{\frac{1}{\epsilon}\bigg{(}\frac{5}{3}C_{A}-\frac{4}{3}T_{F}N_{F}\bigg{)}}{k^{2}+\omega_{n}^{2}}=\frac{g^{2}T^{2}}{96\pi^{2}\epsilon}\bigg{(}\frac{5}{3}C_{A}-\frac{4}{3}T_{F}N_{F}\bigg{)}.$
(6.80)

Upon adding this to Eq. (6.75), we see that they match the $\beta$ function
$C_{A},\,T_{F}N_{F}$ coefficients in (2.13). We therefore understand that
these UV divergences are taken care of by the usual UV zero-temperature
renormalisation procedure.

###  6.7 Recap and Discussion of Divergences

With that, we conclude our evaluation of diagram $(c)$ and therefore the
$\mathcal{O}(g^{2})$ corrections to $Z_{g}$. Let us briefly lay out our
findings:

  * •

In Sec. 6.3, we extracted $Z_{g}$’s (IR) zero-mode divergences. These
divergences are expected and are in fact needed as they cancel against UV
divergences coming from EQCD at NLO, as we showed in Sec. 6.4. This is one of
the main results of the chapter.

  * •

We also identified what we call UV, vacuum divergences coming from Eqs.
(6.75), (6.80). We have confirmed that these are subtracted off in the zero-
temperature renormalisation.

  * •

Lastly, we have found double-logarithmic UV, collinear divergences,
contributing to both the gluonic and fermionic $\text{L}-\text{T}$ pieces in
Eqs. (6.73) and (6.76). In contrast to what has been discussed above, these
divergences signify that our treatment here is incomplete. As they come from a
region of phase space where collinear physics dominates, it is likely that
some form of LPM resummation will be needed to cure these divergences. Further
investigation on this matter is underway.

As a closing remark, we emphasise that, without a full determination of the
$\mathcal{O}(g^{2})$ corrections to $Z_{g}$, the arbitrariness of how the
classical corrections are separated from the quantum, $\mathcal{O}(g^{2})$
corrections prevents us from making concrete inferences regarding the
numerical size of the non-perturbative evaluation and furthermore, the
convergence of the perturbative expansion for $Z_{g}$.

##  Chapter 7 Conclusion and Outlook

A common theme that has permeated throughout this thesis is the computation of
higher-order corrections to $\hat{q}$ and $m_{\infty}$, both of which serve as
central inputs to jet energy loss calculations. In particular, the
advancements contained in this work are tied to the exploration of quantum
corrections at $\mathcal{O}(g^{2})$ to these quantities and how they are
related to classical corrections, present at $\mathcal{O}(g)$.

In Ch. 5, we calculated double-logarithmic corrections to $\hat{q}$ in the
setting of a weakly coupled QGP. We investigated the consequences of taking
the thermal scale seriously in this context and provided clarification,
previously lacking in the literature, on how the logarithmically enhanced
region of phase space is connected to the classical one. Additionally, we were
able to show how the inclusion of semi-collinear processes allows one to go
beyond the instantaneous approximation made in the original works [1, 2].

Our result for $\delta\hat{q}(\mu_{\perp})$ at leading double logarithmic
accuracy is given by Eq. (5.43) and was obtained in the harmonic oscillator
approximation. The necessity of implementing this approximation comes from the
need to neglect a neighbouring region of phase space, where both single and
multiple scattering can occur; a concrete understanding of this region is
lacking in the sense that we do not know the form of the transverse scattering
kernel there. In the future, we hope that this issue can be addressed by
solving Eq. (5.75). While solving this equation exactly would likely prove to
be a formidable task, a first step could be to solve it approximately by using
the improved opacity expansion [190, 195, 196, 197]. This could also serve to
provide a more satisfying cutoff for the single-scattering regime in the
direction of multiple scattering – boundary $(b)$.

We are eager to perform a similar computation and analysis of the single
logarithmic corrections to $\hat{q}$. Indeed, next to leading logarithm
corrections are often needed to fairly assess the quantitative impact of their
leading logarithm counterparts – see [161] for the collinear splitting rate
and [124] for QGP transport coefficients. Nevertheless, it does not make sense
to move forward in this direction until all of the double logarithmic
corrections have been computed.

We could also imagine applying what we have learned to some of the recent
developments in jet energy loss theory, mentioned in Sec. 3.7. In more detail,
it would be interesting to investigate the impact of the thermal scale in the
context of double gluon emission [165, 207, 208, 209, 210, 211, 212, 213, 214,
215, 216, 217], given that the same _universal_ double logarithms also arise
in that context.

Another fruitful direction could be to study the resummation of these
logarithmic corrections, similarly to what was done in [220, 221, 222, 223].
These authors have performed a resummation according to original triangle
shown in Fig. 5.3, evolving $\hat{q}$ from some initial timescale $\tau_{0}$
to longer timescales by resumming longer-lived quantum fluctuations. Our
evaluation of the $k^{+}\sim T$ and smaller $\tau$ regions could then be
incorporated by evolving $\hat{q}$ through these regions, with the result then
being passed on as an initial condition to the resummation equations given in
[222, 223].

In Ch. 6, we set out to complete the evaluation of $\mathcal{O}(g^{2})$
corrections to $Z_{g}$ by computing the contribution from the thermal scale.
We were able to show that the unphysical UV EQCD divergences, identified in
[16], cancel against IR divergences on the full QCD side, thereby providing a
finite result to the non-perturbative evaluation of the classical corrections,
also calculated in [16]. When computing the rest of the corrections at
$\mathcal{O}(g^{2})$, we encountered two other kinds of divergences. The first
kind comes from both the fermion and gluon vacuum contribution as well as the
$k^{-2}$ finite-temperature gluonic piece. These are UV, vacuum divergences
and are appropriately dealt with through the renormalisation of the zero-
temperature theory.

The set of divergences, present in both the finite-temperature gluonic and
fermionic “$\text{L}-\text{T}$” pieces are more troublesome. They have a
collinear origin, i.e, they come from a region of phase space where
$K^{\mu}\sim T$ but with $K^{2}$ much smaller. The next course of action in
this project is to correctly treat the region from which they emerge, which
will likely involve having to take LPM resummation into account. Following
this, we plan to compute the $\mathcal{O}(g^{2})$ corrections to $Z_{f}$,
which will complete the $\mathcal{O}(g^{2})$ determination of the asymptotic
masses. Since $Z_{f}$ is not as sensitive to classical corrections, this
should be, in principle, a more straightforward task.

With all of the $\mathcal{O}(g^{2})$ to $\hat{q}$ and $m_{\infty}$ in hand, a
more long-term goal would be to assess the quantitative impact of these
corrections on the in-medium splitting rate, as was done for the classical
corrections to $\hat{q}$ in [13, 14]. This would then pave the way for more
phenomenologically-based studies [301, 302], which we originally motivated the
need for in the Introduction.

##  Appendix A Conventions

We take here a moment to specify our notation, which is intended to be
particularly suited for studying jet energy loss. We adopt the use of natural
units ($\hbar=c=k_{B}=1$) and work in the mostly minus metric, $(+,-,-,-)$.

We write four-momentums with capital letters, $Q=(p^{0},\vec{\mathbf{q}})$,
the modulus of three-momentums with lower case letters, $q$ and the modulus of
transverse vectors as $q_{\perp}$. Position vectors are denoted similarly. It
will often be convenient to use light-cone coordinates such that
$Q=(q^{+},q^{-},\mathbf{q})$ with the following convention

$\displaystyle q^{+}$
$\displaystyle\equiv\frac{q^{0}+q^{z}}{2}=\bar{v}_{\mu}q^{\mu},$
$\displaystyle q^{-}$ $\displaystyle\equiv q^{0}-q^{z}=v_{\mu}q^{\mu},$
$\displaystyle q\cdot k$
$\displaystyle=q^{+}k^{-}+q^{-}k^{+}-\mathbf{q}\cdot\mathbf{k},$

where we have defined the two light-like reference vectors as

$\displaystyle\bar{v}^{\mu}$ $\displaystyle\equiv\frac{1}{2}(1,0,0,-1),$
$\displaystyle v^{\mu}$ $\displaystyle\equiv(1,0,0,1).$

We shall often deal with scalings where $q^{-}\ll q^{+}$, which then implies
$q^{0}\approx q^{z}\approx q^{+}$. For the sake of brevity, we introduce the
following shorthand for integration over position and momenta in various
dimensions

$\displaystyle\int_{Q}$ $\displaystyle\equiv\int\frac{d^{4}Q}{(2\pi)^{4}}$
$\displaystyle\int_{q}$ $\displaystyle\equiv\int\frac{d^{3}q}{(2\pi)^{3}}$
$\displaystyle\int_{\mathbf{q}}$
$\displaystyle\equiv\int\frac{d^{2}q_{\perp}}{(2\pi)^{2}}$
$\displaystyle\int_{\mathbf{x}}$ $\displaystyle\equiv\int d^{2}x_{\perp}$

Our convention for the covariant derivative is consistent with that of
Schwartz [23]

$D_{\mu}=\partial_{\mu}-igA_{\mu},$ (A.1)

which in turn fixes the three-gluon vertex to be

$\vbox{\hbox{\includegraphics[width=99.58464pt,height=85.35826pt]{three_gluon.pdf}}}\qquad\qquad\begin{aligned}
=gf^{abc}V^{\mu\nu\rho},\end{aligned}$ (A.2)

where

$V^{\mu\nu\rho}=\eta^{\mu\nu}(K-P)^{\rho}+\eta^{\nu\rho}(P-Q)^{\mu}+\eta^{\rho\mu}(Q-K)^{\nu}.$
(A.3)

Finallly, we mention our convention for dimensional regularisation

$\int_{q}=\Big{(}\frac{\bar{\mu}^{2}e^{\gamma_{E}}}{4\pi}\Big{)}^{\epsilon}\int\frac{d^{d}q}{(2\pi)^{d}},$
(A.4)

where we will often regularise integrals in $d=3-2\epsilon$ dimensions.

##  Appendix B Wilson Line Details

A Wilson line is a bi-local field that in general depends on two spacetime
points

$\mathcal{V}(Y,Z)=\mathcal{P}\exp\Big(ig\int_{s_{Y}}^{s_{Z}}ds\frac{dX^{\mu}}{ds}A_{\mu}(X(s))\Big{missing}),$
(B.1)

where $A_{\mu}=A_{\mu}^{a}T^{a}$ is a (non-abelian) Lie-algebra-valued gauge
(gluon) field and $\mathcal{P}$ is the path-ordering operator. $\mathcal{P}$
ensures that fields evaluated earlier on the path from $Y$ to $Z$ are placed
to the right with respect to fields that are evaluated later on the
path111Beware that in the more standard definition found in the literature,
$\mathcal{P}$ orders colour matrices but not the fields, which are instead
time-ordered.. As well as transforming covariantly under gauge transformation,
they naturally incorporate the resummation of gluons. We can expand the
expression above by first writing

$\mathbf{\mathcal{F}}(s)=\frac{dX^{\mu}}{ds}A_{\mu},$ (B.2)

where the bold font is meant to emphasis that these objects are matrices.
Then, incorporating the usual definition of the matrix exponential
expansion222The expression Eq. (B.3) can be adapted to the case of a conjugate
Wilson line by sending $i\to-i$ and enforcing instead an anti-path-ordering
prescription.

$\mathcal{V}(Y,Z)=\sum_{n=0}^{\infty}\left(\frac{ig}{n!}\right)^{n}\int_{s_{Y}}^{s_{Z}}ds_{1}\int_{s_{Y}}^{s_{Z}}ds_{2}...\int_{s_{Y}}^{s_{Z}}ds_{n}\mathcal{P}\Big{[}\mathbf{\mathcal{F}}(s_{1})\mathbf{\mathcal{F}}(s_{2})...\mathbf{\mathcal{F}}(s_{n})\Big{]}$
(B.3)

The above integration is done over an $n$-dimensional hypercube but one can
divide this region into $n!$ sub-regions. To give an illustrative example,
consider $n=2,\,s_{Y}=0,\,s_{Z}=1$

$\int_{0}^{1}ds_{1}\int_{0}^{1}ds_{2}\mathcal{P}\Big{[}\mathbf{\mathcal{F}}(s_{1})\mathbf{\mathcal{F}}(s_{2})\Big{]}=\int_{0}^{1}ds_{1}\int_{0}^{s_{1}}ds_{2}\mathbf{\mathcal{F}}(s_{1})\mathbf{\mathcal{F}}(s_{2})+\int_{0}^{1}ds_{2}\int_{0}^{s_{2}}ds_{1}\mathbf{\mathcal{F}}(s_{2})\mathbf{\mathcal{F}}(s_{1}).$
(B.4)

At this point, $s_{1},\,s_{2}$ are just dummy variables, so we can write

$\int_{0}^{1}ds_{1}\int_{0}^{1}ds_{2}\mathcal{P}\Big{[}\mathbf{\mathcal{F}}(s_{1})\mathbf{\mathcal{F}}(s_{2})\Big{]}=2\int_{0}^{1}ds_{1}\int_{0}^{s_{1}}ds_{2}\mathbf{\mathcal{F}}(s_{1})\mathbf{\mathcal{F}}(s_{2}),$
(B.5)

where such a procedure can be suitably generalised to the the $n$-dimensional
case. In this way, one can eliminate the path-ordering rewrite Eq. (B.3) as

$\mathcal{V}(Y,Z)=\sum_{n=0}^{\infty}\left(ig\right)^{n}\int_{s_{Y}}^{s_{Z}}ds_{1}\int_{s_{Y}}^{s_{1}}ds_{2}...\int_{s_{Y}}^{s_{n-1}}ds_{n}\mathbf{\mathcal{F}}(s_{1})\mathbf{\mathcal{F}}(s_{2})...\mathbf{\mathcal{F}}(s_{n}).$
(B.6)

In practice, it is this formula that we make use of to calculate Wilson lines
in this thesis.

Furthermore, let $s_{Y}=s_{0},\,s_{Z}=s$ so that $Y=X(s_{0})\,Z=X(s)$. Then,
by taking a derivative with respect to $s$

$\frac{dx^{\mu}}{ds}\mathcal{V}(x(s),x(s_{0}))=ig\frac{dx^{\mu}}{ds}A_{\mu}(x(s))\mathcal{V}(x(s),x(s_{0})),$
(B.7)

we can derive an equation of motion for the Wilson line, which reads

$\frac{dx^{\mu}}{ds}\Big{(}\frac{\partial}{\partial
x^{\mu}}-igA_{\mu}(x(s))\Big{)}\mathcal{V}(s,s_{0})=0=\frac{dx^{\mu}}{ds}D_{\mu}\mathcal{V}(s,s_{0}),$
(B.8)

where we have used the definition of the covariant derivative, Eq. (A.1).

##  Appendix C $\hat{q}(\mu_{\perp})$ Computation

###  C.1 Side rails in Coulomb Gauge

Figure C.1: Diagrams obtained from expanding the Wilson loop to the first non-
trivial order, where at least one of the gluon legs are attached to a side
rail. In the main text, we argue that none of these diagrams are relevant for
the computations performed in this thesis.

Back in Sec. 4.4.1, we showed how to compute the leading order soft
contribution to $\hat{q}(\mu_{\perp})$ using the Wilson loop defined in [287].
While carrying out that computation, we only needed to consider the diagrams
in Fig. 4.7 where gluons are attached to the horizontal Wilson lines. In this
Appendix, we argue why other diagrams, some of which are shown in Fig. C.1 are
not relevant for such that calculation, as well as the subsequent ones carried
out in Ch. 5.

We start with the top left diagram from Fig. C.1. Using Eqs. (4.91), (4.92),
we have

$\displaystyle(a)$
$\displaystyle=-g^{2}C_{R}\int_{0}^{1}ds\int_{-\frac{L_{M}}{2}}^{\frac{L_{M}}{2}}dx^{+}x_{\perp}\langle
G_{>}^{\perp-}(-\frac{L_{M}}{2}-x^{+},0,y_{\perp}s)\rangle$
$\displaystyle=-g^{2}C_{R}\int_{0}^{1}ds\int_{-\frac{L_{M}}{2}}^{\frac{L_{M}}{2}}dx^{+}\int_{K}e^{ik^{-}(\frac{L_{M}}{2}+x^{+})}e^{i\mathbf{k}\cdot\mathbf{x}s}x_{\perp}\cdot\langle
G_{>}^{\perp-}(K)\rangle,$ (C.1)

where a sum over the $\perp=x,\,y$ directions is implied. We recognise that
$x^{+}$ integration can be rewritten in terms of a delta function, i.e

$(a)=-g^{2}C_{R}\int_{0}^{1}ds\int_{K}\delta(k^{-})e^{ik^{-}\frac{L_{M}}{2}}e^{i\mathbf{k}\cdot\mathbf{x}s}x_{\perp}\cdot\langle
G_{>}^{\perp-}(K)\rangle.$ (C.2)

Then, upon doing the $k^{-}$ integration, $L_{M}$ will vanish from the entire
integrand, implying that this type of diagram will not produce a linear factor
of $L_{M}$. Therefore, according to Eq. (4.94), it will furthermore not
contribute to $\mathcal{C}(x_{\perp})$.

Moving on to the rest of the diagrams from Fig. C.1, we note that their sum is
analogous to the sum of the three diagrams from Fig. 4.7, but in this case
with all gluons attached to the side rails

$\mathcal{C}(x_{\perp})_{\text{side
rails}}=-\frac{g^{2}C_{R}}{L_{M}}\int_{1}^{0}ds\int_{1}^{0}dt\int\frac{d^{4}k}{(2\pi)^{4}}e^{-i\mathbf{k}\cdot\mathbf{x}(s+t-1)}(1-e^{-ik^{-}L_{M}})x_{\perp}G^{\perp\perp}_{>}(k)x_{\perp}.$
(C.3)

The relevant part that we must pay attention to here is

$\lim_{L_{M}\to\infty}\frac{1}{L_{M}}\int
dk^{-}(1-e^{-ik^{-}L_{M}})G^{\perp\perp}_{>}(k).$ (C.4)

where we have added the limit explicitly because of its particular importance
here. For reasons that will soon become clear, it is convenient to divide the
integrand by ${k^{-}}^{2}$. Then, by making a change of variable
$z=k^{-}L_{M}$ and noting that $G^{\perp\perp}_{>}(z/L_{M})$ is regular as
$L_{M}\to\infty$ (in the Coulomb gauge) we can write

$\displaystyle\lim_{L_{M}\to\infty}\frac{1}{L^{2}}\int
dz\frac{1-e^{-iz}}{\frac{z^{2}}{L_{M}^{2}}}G^{\perp\perp}_{rr}(\frac{z}{L_{M}})=$
$\displaystyle\int dz\frac{1-e^{-iz}}{z^{2}}G^{\perp\perp}_{>}(0)$
$\displaystyle=G^{\perp\perp}_{>}(0)\int dz\frac{1-\cos(z)}{z^{2}}$
$\displaystyle=\pi G^{\perp\perp}_{>}(0)$ (C.5)

where the final integral over $z$ can be calculated using residue
methods111One can use a principal value prescription to show that the odd part
of the integrand vanishes since $\frac{\sin(z)}{z^{2}}$ is divergent as $z\to
0$.. We have just shown that (now it is clear why we divided the integrand by
${k^{-}}^{2}$) Eq. C.4 is indeed proportional to the integral of
${k^{-}}^{2}\delta(k^{-})$, which indeed justifies why we need not worry about
these kinds of diagrams, where gluons are attached to the side rails. This
clarifies why, throughout the main body of the thesis, the only diagrams that
are included are ones where gluons are attached to the horizontal,
$\mathcal{W}$ Wilson lines.

###  C.2 Hard Semi-Collinear Subtraction

In this Appendix, we will show how the second term in Eq. (5.37) is
automatically taken into account if we add the hard contribution to
$\hat{q}(\mu_{\perp})$ from [174]. Indeed, that paper computes the
contribution to $\hat{q}(\mu_{\perp})$ for $\sqrt{ET}\gg k_{\perp}\gg gT$ at
leading order, i.e. through elastic Coulomb scatterings with the light quarks
and gluons of the medium. The incoming and outgoing momenta for such
scatterers are named $P_{2}$ and $P_{2}-Q$ there. We can then identify $P_{2}$
with our $L$ and $Q$ with $-K$. In order to obtain the contribution to
$\mathcal{C}(k_{\perp})$ and thence $\hat{q}(\mu_{\perp})$, [174] integrates
over all values and orientations of $p_{2}$. In so doing, it ends up including
the slice where $p_{2}\sim gT$, $k_{\perp}\sim\sqrt{g}T$, which is precisely
the semi-collinear scaling we investigated in Sec. 5.4. As explained there, we
need to subtract this limit of [174], so as to avoid double countings.

Our starting point is Eq. (3.8) of [174], where we have specialised to the
case of scattering off a soft gluon ($n_{\text{B}}(l^{0})\approx T/l^{0}$) and
undone a couple of auxiliary integrals (see (3.7) there), as well as applied
the dictionary just described. We then have

$\displaystyle\delta\mathcal{C}(k_{\perp})^{\text{hard}}$
$\displaystyle=\frac{2g^{4}C_{R}C_{A}}{k_{\perp}^{4}}\int\frac{dk^{0}}{2\pi}\int\frac{dk^{z}}{2\pi}\int\frac{d^{4}L}{(2\pi)^{4}}(l^{0}-l^{z})^{2}\frac{T}{l^{0}}(1+n_{\text{B}}(l^{0}+k^{0}))$
$\displaystyle\times
2\pi\delta(k^{0}-k^{z})2\pi\delta((L+K)^{2})2\theta(l^{0}+k^{0})2\theta(l^{0})2\pi\delta(L^{2}).$
(C.6)

For a soft gluon $L\sim gT$ we can simplify the expression above as

$\displaystyle\delta\mathcal{C}(k_{\perp})^{\text{hard}}$
$\displaystyle=\frac{2g^{4}C_{R}C_{A}}{k_{\perp}^{4}}\int\frac{dk^{+}}{2\pi}\int\frac{d^{4}L}{(2\pi)^{4}}\frac{Tl^{-2}}{l^{+}+l^{-}/2}(1+n_{\text{B}}(k^{+}))$
$\displaystyle\times
2\pi\delta(2k^{+}l^{-}-k_{\perp}^{2})2\theta(k^{+})2\theta(l^{+}+l^{-}/2)2\pi\delta(L^{2}).$
(C.7)

We can set up the $\delta$ function to fix $l^{+}$, i. e.

$\displaystyle\delta\mathcal{C}(k_{\perp})^{\text{hard}}$
$\displaystyle=\frac{2g^{4}C_{R}C_{A}}{k_{\perp}^{4}}\int\frac{dk^{+}}{2\pi}\int\frac{d^{4}L}{(2\pi)^{4}}\frac{Tl^{-2}}{l^{+}+l^{-}/2}(1+n_{\text{B}}(k^{+}))\frac{\theta(k^{+})}{k^{+}}$
$\displaystyle\times
2\pi\delta\left(l^{-}-\frac{k_{\perp}^{2}}{2k^{+}}\right)\frac{\theta(l^{+}+l^{-}/2)}{\absolutevalue{l^{-}}}2\pi\delta\left(l^{+}-\frac{l_{\perp}^{2}}{2l^{-}}\right).$
(C.8)

This finally yields

$\delta\mathcal{C}(k_{\perp})^{\text{hard}}=\frac{g^{2}C_{R}}{\pi
k_{\perp}^{4}}\int\frac{dk^{+}}{k^{+}}(1+n_{\text{B}}(k^{+}))\theta(k^{+})\int\frac{d^{2}l_{\perp}}{(2\pi)^{2}}\frac{g^{2}C_{A}T\;2l^{-2}}{l_{\perp}^{2}+l^{-2}}\bigg{|}_{l^{-}=\frac{k_{\perp}^{2}}{2k^{+}}}.$
(C.9)

It is encouraging to see the resemblance of this result to that of the second
term of Eq. (5.37) when plugged in Eq. (5.36). However, to get the two results
to match exactly, we need to compute the same result with same integral with
instead $L\to L-K$ and $L+K\to L$ keeping $L$ soft, that is, a soft outgoing
gluon scatterer. This second integral will give the negative $k^{+}$
contribution, and the integrals’ sum will indeed yield

$\delta\mathcal{C}(k_{\perp})_{\mathrm{semi}}^{\text{hard}}=\frac{g^{2}C_{R}}{\pi
k_{\perp}^{4}}\int\frac{dk^{+}}{k^{+}}(1+n_{\text{B}}(k^{+}))\hat{q}_{\text{\cite[cite]{[\@@bibref{Number}{Arnold:2008vd}{}{}]}}}\left(\rho_{\perp};\frac{k_{\perp}^{2}}{2k^{+}}\right).$
(C.10)

Figure C.2:  Diagrams whose cuts correspond to virtual processes.

###  C.3 Partially Virtual Diagrams

We start from the first diagram in Fig. C.2. As was the case for the real II
and X diagrams, the $C_{R}^{2}$-proportional, abelian part of this diagram
vanishes with the counterparts, not shown in the figure, where the coft gluon
does not straddle the soft one. Accounting for this and for the symmetric
diagrams with the coft gluon attached to the other Wilson line we have

$\delta\mathcal{C}(l_{\perp})^{\mathrm{virt}}_{\widehat{\mathrm{T}}}=-g^{4}C_{R}C_{A}\int\frac{d^{4}K}{(2\pi)^{4}}\int\frac{dl^{+}}{2\pi}\frac{1}{(k^{-}+i\epsilon)^{2}}G_{>}^{--}(L)G_{rr}^{--}(K)\Big{|}_{l^{-}=0},$
(C.11)

where we have labeled this diagram $\widehat{\mathrm{T}}$ following its
topology. We note that the Wilson line integration forces $l^{-}=0$,
differently from the real processes. In this case too we defer the evaluation
of this expression until after the second diagram has been evaluated. Its
contribution, together with that of its symmetric counterpart with two
vertices on the bottom Wilson line, is222At first glance, it is not clear why
we cannot have any other real-time assignments for this diagram. Indeed, one
could imagine the assignment $G_{F}(L)G_{>}(K+L)G_{>}(-L)$. We have checked
that this leaves us with an expression that, at first order in the collinear
expansion, is odd in $k^{+}$ and thus integrates to zero.

$\displaystyle\delta\mathcal{C}(x_{\perp})^{\mathrm{virt}}_{{\mathrm{Y}}}=$
$\displaystyle\frac{g^{4}C_{R}C_{A}}{2}\int\frac{d^{4}K}{(2\pi)^{4}}\int\frac{d^{4}L}{(2\pi)^{4}}\frac{-i}{k^{-}-i\epsilon}e^{i\mathbf{l}\cdot\mathbf{x}}G_{>}^{\beta-}(L)2\pi\delta(l^{-})$
$\displaystyle\times\big{(}G_{F}^{-\delta}(K)G_{F}^{-\alpha}(K{+}L)-G_{\bar{F}}^{-\delta}(K)G_{\bar{F}}^{-\alpha}(K{+}L)\big{)}(g_{\delta\beta}(K{-}L)+g_{\beta\alpha}(2L{+}K)_{\delta}-g_{\alpha\delta}(2K{+}L)_{\beta}).$
(C.12)

We can use the analogue of Eq. (5.10) for the anti-time-ordered propagator,
i.e.

$G_{\bar{F}}(K)=-\frac{1}{2}\Big{(}G_{R}(K)+G_{A}(K)\Big{)}+G_{rr}(K)$ (C.13)

Using these definitions, we get

$\displaystyle\delta\mathcal{C}(l_{\perp})^{\mathrm{virt}}_{{\mathrm{Y}}}=$
$\displaystyle
g^{4}C_{R}C_{A}\int\frac{d^{4}K}{(2\pi)^{4}}\int\frac{dl^{+}dl^{-}}{2\pi}\frac{i}{k^{-}-i\epsilon}G_{>}^{\beta-}(L)\delta(l^{-})$
$\displaystyle\times\big{(}G_{rr}^{-\delta}(K)G_{R}^{-\alpha}(K+L)+G_{R}^{-\delta}(K)G_{rr}^{-\alpha}(K+L)\big{)}(g_{\delta\beta}(K{-}L)+g_{\beta\alpha}(2L{+}K)_{\delta}-g_{\alpha\delta}(2K{+}L)_{\beta}).$
(C.14)

The virtual contribution is given by the sum of the two, that is

$\delta\mathcal{C}(l_{\perp})^{\mathrm{virt}}=\delta\mathcal{C}(l_{\perp})^{\mathrm{virt}}_{\widehat{\mathrm{T}}}+\delta\mathcal{C}(l_{\perp})^{\mathrm{virt}}_{{\mathrm{Y}}}.$
(C.15)

In this case we do not enforce a semi-collinear scaling for the coft gluon,
since, as we shall show, there is no double-logarithmic contribution.
Furthermore, the Wilson line integrations set $l^{-}=0$, thus making the
interaction with the medium instantaneous. We rather assume $l^{+}\sim
l_{\perp}\sim k_{\perp}\sim gT\ll k^{+}$, leading to

$\displaystyle\delta\mathcal{C}(l_{\perp})^{\mathrm{virt}}=$ $\displaystyle
2g^{4}C_{R}C_{A}\int\frac{dl^{+}}{2\pi}\int\frac{dk^{+}d^{2}k_{\perp}}{(2\pi)^{3}}\frac{\frac{1}{2}+n_{\text{B}}(k^{+})}{k^{+}}\frac{G^{--}_{>}(l^{+},l_{\perp})}{k_{\perp}^{2}+m_{\infty\,g}^{2}}$
$\displaystyle\times\bigg{[}\frac{k_{\perp}^{2}+\mathbf{k}\cdot\mathbf{l}}{(\mathbf{k}+\mathbf{l})^{2}+m_{\infty\,g}^{2}}-\frac{k_{\perp}^{2}}{k_{\perp}^{2}+m_{\infty
g}^{2}}\bigg{]}.$ (C.16)

The asymptotic masses at the denominator have been included since we assumed
$k_{\perp}\sim gT$. They arise from the $k^{+}\sim T$, $K^{2}\sim g^{2}T^{2}$
limit of the HTL propagators in Sec. 4.2.4. The $l^{+}$ integration can be
carried out through the mapping to EQCD, as was demonstrated in Sec. 4.4.1. It
leads to

$\displaystyle\delta\mathcal{C}(l_{\perp})^{\mathrm{virt}}=$ $\displaystyle
2g^{4}C_{R}C_{A}\int\frac{dk^{+}d^{2}k_{\perp}}{(2\pi)^{3}}\frac{\frac{1}{2}+n_{\text{B}}(k^{+})}{k^{+}}\frac{Tm_{D}^{2}}{l_{\perp}^{2}(l_{\perp}^{2}+m_{D}^{2})}\frac{1}{k_{\perp}^{2}+m_{\infty\,g}^{2}}$
$\displaystyle\times\bigg{[}\frac{k_{\perp}^{2}+\mathbf{k}\cdot\mathbf{l}}{(\mathbf{k}+\mathbf{l})^{2}+m_{\infty\,g}^{2}}-\frac{k_{\perp}^{2}}{k_{\perp}^{2}+m_{\infty\,g}^{2}}\bigg{]}.$
(C.17)

We recognise that this expression is proportional to the leading-order,
adjoint soft scattering kernel
$\mathcal{C}(l_{\perp})^{\mathrm{LO}}_{\mathrm{soft}}$, Eq. (3.24). From this
expression and its associated $\hat{q}$ contribution one can see that the
$|\mathbf{k}+\mathbf{l}|\gg l_{\perp}$ single-scattering region is free of
double-logarithmic enhancements, as found in [1]. Furthermore, for
$k_{\perp}\sim l_{\perp}\sim gT$, which we have used for our derivation, the
lifetime of the virtual coft fluctuation is long, so that multiple soft
scatterings can occur within it. Hence, Eq. (C.17) should just be considered
as the $\mathcal{N}=1$ term in the opacity expansion of the virtual
contribution to Eq. (5.75). Indeed, we have checked that Eq. (C.17) can also
be obtained from the virtual terms in Eq. (5.76), i.e. those proportional to
$\exp(i\mathbf{x}\cdot\mathbf{l})$, further confirming the soundness of Eqs.
(5.73) and (5.75).

##  Appendix D $Z_{g}$ Computation

This appendix contains some extra details relevant for the calculation in Ch.
6.

###  D.1 Amputated Self-Energies

It is pointed out in [166] that using KMS relations [263, 264], one can show
that amputated self-energies of thermal propagators obey

$\displaystyle\Pi^{aa}(K)$
$\displaystyle=\left(\frac{1}{2}+n_{\text{B}}(k^{0})\right)\Big{[}\Pi^{R}(K)-\Pi^{A}(K)\Big{]}$
(D.1) $\displaystyle\Pi^{>}(K)$
$\displaystyle=\left(1+n_{\text{B}}(k^{0})\right)\Big{[}\Pi^{R}(K)-\Pi^{A}(K)\Big{]}$
(D.2) $\displaystyle\Pi^{<}(K)$
$\displaystyle=n_{\text{B}}(k^{0})\Big{[}\Pi^{R}(K)-\Pi^{A}(K)\Big{]},$ (D.3)

where

$\displaystyle\Pi^{R}(K)$ $\displaystyle=\Pi^{ar}$ (D.4)
$\displaystyle\Pi^{A}(K)$ $\displaystyle=\Pi^{ra}.$ (D.5)

We then define the amputated self-energy-like objects111Note that the $1/4$ in
Eq. (D.11) comes from the extra factor associated with the $aaa$ vertex from
the _r/a_ basis. In particular, see Fig. 4.3.

$\displaystyle\widetilde{\Pi^{T}}(Q)$ $\displaystyle\equiv
G^{F}(K)G^{F}(Q-K),$ (D.6) $\displaystyle\widetilde{\Pi^{<}}(Q)$
$\displaystyle\equiv G^{<}(K)G^{<}(Q-K),$ (D.7)
$\displaystyle\widetilde{\Pi^{>}}(Q)$ $\displaystyle\equiv
G^{>}(K)G^{>}(Q-K),$ (D.8) $\displaystyle\widetilde{\Pi^{R}}(Q)$
$\displaystyle\equiv G^{rr}(K)G^{R}(Q-K)+G^{rr}(Q-K)G^{R}(K),$ (D.9)
$\displaystyle\widetilde{\Pi^{A}}(Q)$ $\displaystyle\equiv
G^{rr}(K)G^{A}(Q-K)+G^{rr}(Q-K)G^{A}(K),$ (D.10)
$\displaystyle\widetilde{\Pi^{aa}}(Q)$ $\displaystyle\equiv
G^{rr}(K)G^{rr}(Q-K)+\frac{1}{4}G^{R}(Q-K)G^{R}(K)+\frac{1}{4}D^{A}(Q-K)D^{A}(K).$
(D.11)

Strictly speaking, there should be a $K$ integral on the right-hand side of
the equations above. However, this would then greatly muddle the notation in
and around Eq. (6.18), for instance, where these identities are employed. In
the end, it does not matter as we only use these relations to help combine the
different assignments; in doing the actual loop integration, we always rewrite
everything in terms of the propagators.

###  D.2 One-Loop Self Energies

In this appendix, we list some finite-temperature one-loop self energies that
are needed for the computation in Ch. 6. We essentially copy them from Chapter
$8$ of [281] but rewrite them here in such a way that is more suited to our
computation.

To start, the gluonic part of the retarded, transverse self-energy reads

$\displaystyle\Pi_{T}^{R}(K)_{G}$
$\displaystyle=\frac{g^{2}C_{A}}{16\pi^{2}k^{3}}\int_{0}^{\infty}dq\,n_{\mathrm{B}}(q)\bigg{[}8qk\left(k^{2}+k_{0}^{2}\right)$
$\displaystyle-K^{2}\left(4q^{2}+4k^{2}+K^{2}\right)\left(\ln\left(\frac{k^{0}+k-2q+i\varepsilon}{k^{0}-k-2q+i\varepsilon}\right)+\ln\left(\frac{k^{0}-k+2q+i\varepsilon}{k^{0}+k+2q+i\varepsilon}\right)\right)$
$\displaystyle-4qk^{0}K^{2}\left(2\ln\left(\frac{k^{0}+k+i\varepsilon}{k^{0}-k+i\varepsilon}\right)+\ln\left(\frac{k^{0}-k+2q+i\varepsilon}{k^{0}+k+2q+i\varepsilon}\right)-\ln\left(\frac{k^{0}+k-2q+i\varepsilon}{k^{0}-k-2q+i\varepsilon}\right)\right)\bigg{]}.$
(D.12)

The finite-temperature, gluonic part of the longitudinal retarded self-energy
is instead

$\displaystyle\Pi_{L}^{R}(K)_{G}=$
$\displaystyle-\frac{g^{2}C_{A}K^{2}}{8\pi^{2}k^{3}}\int_{0}^{\infty}dq\,n_{\mathrm{B}}(q)\bigg{[}8qk$
$\displaystyle-\left(4q^{2}-2k^{2}+k_{0}^{2}\right)\bigg{[}\ln\frac{k^{0}+k-2q+i\varepsilon}{k^{0}-k-2q+i\varepsilon}+\ln\frac{k^{0}-k+2q+i\varepsilon}{k^{0}+k+2q+i\varepsilon}\bigg{]}$
$\displaystyle-4qk^{0}\bigg{[}2\ln\frac{k^{0}+k+i\varepsilon}{k^{0}-k+i\varepsilon}+\ln\frac{k^{0}-k+2q+i\varepsilon}{k^{0}+k+2q+i\varepsilon}-\ln\frac{k^{0}+k-2q+i\varepsilon}{k^{0}-k-2q+i\varepsilon}\bigg{]}\bigg{]}.$
(D.13)

It is also worthwhile to have the difference of these two quantities on hand

$\displaystyle\Pi_{L}^{R}(K)_{G}-\Pi_{T}^{R}(K)_{G}=$
$\displaystyle\frac{g^{2}C_{A}}{16\pi^{2}k^{3}}\int_{0}^{\infty}dq\,n_{\mathrm{B}}(q)\bigg{[}-8qk(3k_{0}^{2}-k^{2})$
$\displaystyle+K^{2}\left(12q^{2}-k^{2}+3k_{0}^{2}\right)\bigg{[}\ln\frac{k^{0}+k-2q+i\varepsilon}{k^{0}-k-2q+i\varepsilon}+\ln\frac{k^{0}-k+2q+i\varepsilon}{k^{0}+k+2q+i\varepsilon}\bigg{]}$
$\displaystyle+12qk^{0}K^{2}\bigg{[}2\ln\frac{k^{0}+k+i\varepsilon}{k^{0}-k+i\varepsilon}+\ln\frac{k^{0}-k+2q+i\varepsilon}{k^{0}+k+2q+i\varepsilon}-\ln\frac{k^{0}+k-2q+i\varepsilon}{k^{0}-k-2q+i\varepsilon}\bigg{]}.\bigg{]}.$
(D.14)

The fermionic part of the transverse self-energy is then

$\displaystyle\Pi^{R}_{T}(K)_{F}=$
$\displaystyle\frac{g^{2}T_{F}N_{F}}{8\pi^{2}k^{3}}\int_{0}^{\infty}dq\,n_{\mathrm{F}}(q)\bigg{[}8qk\left(k^{2}+k_{0}^{2}\right)$