Collisional energy loss of a heavy quark in a semiquark-gluon plasma

According to the previous discussions, it is possible to introduce a classical background field $A_{0}^{\rm cl}$ to describe the nontrivial Polyakov loop in the deconfining phase transition for $SU(N)$ gauge theories. The background field is assumed to be constant in spacetime and given by a diagonal matrix in color space,

$$(A_{0}^{\rm cl})_{ab}=\frac{1}{g}{\cal Q}^{a}\delta_{ab}\,.$$

(11)

Furthermore, it satisfies the traceless condition $\sum_{a=1}^{N}{\cal Q}^{a}=0$ with $a,b=1,\cdots,N$.

The Wilson line in the temporal direction is defined as

$${\bf L}={\cal P}\,{\rm exp}\Big{(}ig\int^{\beta}_{0}A^{\rm cl}_{0}\,d\tau\Big{)}\,,$$

(12)

where ${\cal P}$ denotes the time ordering, $\tau$ is the imaginary time and $\beta\equiv 1/T$ is the inverse temperature. Accordingly, the gauge invariant Polyakov loop takes the following form:

$$\ell=\frac{1}{N}{\rm Tr}\,{\bf L}\,.$$

(13)

One can also define higher loops, $(1/N){\rm Tr}\,{\bf L}^{n}$, and there are $N-1$ independent loops for $SU(N)$.

After taking into account the classical background field, the effective potential in general $SU(N)$ gauge theories can be computed in the perturbation theory. The one-loop result is

$${\cal V}_{\rm pt}=\frac{2\pi^{2}T^{4}}{3}\sum_{ab}{\cal P}^{ab,ba}B_{4}(|{\sf q}^{ab}|)\,.$$

(14)

The dependence on the background field ${\cal Q}$ is given by the fourth Bernoulli polynomials $B_{4}(x)=x^{2}(1-x)^{2}-1/30$. For later use, we also need the second Bernoulli polynomials $B_{2}(x)=x^{2}-x+1/6$. Notice that the argument $x$ should be understood as $x-[x]$ with $[x]$ the largest integer less than $x$, which is nothing but the modulo function. In the above equation, $a$ and $b$ run from $1$ to $N$ when summing over the color indices. In addition, we define ${\sf q}^{ab}={\sf q}^{a}-{\sf q}^{b}$ with ${\sf q}^{a}\equiv{\cal Q}^{a}/(2\pi T)$ and the projection operator $\mathcal{P}^{ab,cd}$ in the double line basis reads

$${\cal P}^{ab,cd}={\cal P}^{ab}_{dc}=\delta^{a}_{d}\delta^{b}_{c}-\frac{1}{N}\delta^{ab}\delta_{cd}\,.$$

(15)

It is more convenient to perform the calculation in the double line basis when a background field is present. In this basis, color indices in the fundamental representation are denoted by $a,b,\cdots=1,2,\cdots,N$, while in the adjoint representation, they are denoted by a pair of fundamental indices, $ab$. The generators of the fundamental representation are given by $(t^{ab})_{cd}=\frac{1}{\sqrt{2}}\mathcal{P}^{ab}_{cd}$, normalized as ${\rm Tr}\,(t^{ab}t^{cd})=\frac{1}{2}{\cal P}^{ab,cd}$. More details about the double line basis can be found in Refs. [35, 36].

The equation of motion for the background field based on Eq. (14) indicates that the system is always in a fully deconfined phase with vanishing background field, and no phase transition could happen. It can also be shown that after including the two-loop perturbative corrections [37, 38, 39, 40, 41], one still has ${\cal Q}=0$ as the vacuum. To drive the system to confinement, a nonperturbative contribution needs to be added to the effective potential. It is expected to play an important role near the critical temperature where the confined vacuum is favored. One possible way to generate such a contribution is to include a mass scale in the dispersion relation for gauge bosons, $\omega_{k}=\sqrt{k^{2}+M^{2}}$, then expand the resulting effective potential in the high temperature limit $M\ll T$. The leading order term in the expansion is nothing but the perturbative ${\cal V}_{\rm pt}$ as given in Eq. (14), and the next-to-leading order term appears as the desired nonperturbative contribution in the effective potential which takes the following form [8]:

$${\cal V}_{\rm npt}=\frac{M^{2}T^{2}}{2}\sum_{ab}{\cal P}^{ab,ba}{B_{2}(|{\sf q}^{ab}|)}\,.$$

(16)

An alternative way to generate the above nonperturbative contribution is to embed a two-dimensional ghost field isotropically in four dimensions [9]. In this case, the mass scale in Eq. (16) should be considered as an upper limit for the transverse momentum of the embedded fields. It indicates that the ghost field is two-dimensional at short distances; however, for distances larger than the scale of confinement, it becomes fully four-dimensional field.

The combination of Eqs. (14) and (16) leads to an effective theory for studying the semi-QGP. One can derive the equation of motion for the classical background field in this theory, and a nonzero and temperature dependent background field arises. At low temperatures, the solution ${\sf q}^{a}=(N-2a+1)/(2N)$ minimizes the effective potential,¹¹1This is true under the straight line ansatz where the background fields ${\sf q}^{a}$ with $a=1,2,\cdots,N$ have constant spacing and automatically satisfy the traceless condition. which corresponds to the confining vacuum with vanishing Polyakov loop. When the temperature gets very large, ${\sf q}^{a}\sim M^{2}/T^{2}$ indicates a perturbative vacuum where the Polyakov loop approaches to unity. By requiring the phase transition to occur at the critical temperature $T_{d}$, the mass scale $M$, which is the only parameter in the effective theory can be determined. For the $SU(3)$ gauge theory, by parametrizing the background field as

$${\vec{\sf q}}=({\sf q},0,-{\sf q})\,,$$

(17)

the explicit $T$-dependence of the background field in the deconfined phase can be determined from its equation of motion as the following:

$${\sf q}=\frac{1}{36}\big{(}9-\sqrt{81-80(T_{d}/T)^{2}}\big{)}\,,$$

(18)

where $M/T_{d}=2\sqrt{10}\pi/9$ has been used. It is worth noting that besides the nontrivial behavior of the Polyakov loop in the deconfining phase transition, the above effective theory is also capable of qualitatively reproducing the lattice simulations on the thermodynamics for the $SU(N)$ gauge theories. Further improvements have been proposed, leading to various versions of the matrix models for deconfinement [42, 43, 44, 45].

In this work, we will investigate the influence of the background field on the heavy quark collisional energy loss when it passes through the hot and dense medium. For the hard scatterings, only thermal distributions of the medium partons are affected by the background field. The resulting Bose-Einstein distribution functions are given by [46, 5]

$$n(k_{0},{\sf q}^{ab})=\left\{\begin{aligned} &\frac{1}{e^{|k_{0}|/T-2\pi i{\sf q}^{ab}}-1}\equiv n_{+}^{ab}(k_{0})\quad\quad{\rm for}\quad k_{0}>0\\ &\frac{1}{e^{|k_{0}|/T+2\pi i{\sf q}^{ab}}-1}\equiv n_{-}^{ab}(k_{0})\quad\quad{\rm for}\quad k_{0}<0\\ \end{aligned}\right.\,.$$

(19)

On the other hand, for scatterings with soft gluon exchange, it becomes necessary to use the HTL resummed gluon propagator which regulates the infrared divergence. In the presence of a background field, the resummed propagator obtained from the Dyson-Schwinger equation gets modified through the ${\cal Q}$-dependent gluon self-energy.

Within the framework of perturbation theory, it was found that the HTL approximated gluon self-energy is not transverse due to the appearance of an anomalous term at nonzero ${\cal Q}$ [35, 47]. In addition, this anomalous term also results in an ill-defined gluon resummed propagator [48] as well as an unexpected discontinuity in the free energy at higher order in the coupling constant [49, 50]. These known issues further justify the necessity of employing an effective theory to properly incorporate the effect of a background field. After taking into account the contributions from the aforementioned two-dimensional ghosts, the anomalous term is completely canceled and the resulting gluon self-energy has the same Lorentz structure as its counterpart at ${\cal Q}=0$, and thus preserves the transversality [9].

Based on the Dyson-Schwinger equation, the HTL resummed gluon propagator in the effective theory was first computed in [48] where the covariant gauge was used. For our purpose, we carry out a similar calculation in the temporal axial gauge and the $N^{2}-N$ off diagonal components in color space are given by

$$\Delta_{ij}^{ab,cd}(Q,{\vec{\sf{q}}})\xlongequal[]{a\neq b}\delta^{ad}\delta^{bc}\left[\frac{1}{\omega^{2}-q^{2}-{({\cal M}_{D}^{2})}^{ab}({\vec{\sf{q}}})\Pi_{T}({\hat{\omega}})}A_{ij}+\frac{1}{\omega^{2}-{({\cal M}_{D}^{2})}^{ab}({\vec{\sf{q}}})\Pi_{L}({\hat{\omega}})}B_{ij}\right]\,,$$

(20)

where the ${\cal Q}$-modified screening mass can be written as

$${({\cal M}_{D}^{2})}^{ab}({\vec{\sf{q}}})=\left[\frac{3}{N}\sum_{e=1}^{N}\bigg{(}B_{2}(|{\sf q}^{ae}|)+B_{2}(|{\sf q}^{eb}|)\bigg{)}+\frac{3M^{2}}{4\pi^{2}T^{2}}\right]m_{D}^{2}\,,$$

(21)

with $a\neq b$ and the contribution $\sim M^{2}$ comes from the two-dimensional ghost field. As compared to Eq. (7), it turns out that the influence of the background field on the transverse and longitudinal gluon self-energy amounts to the same modification on the screening mass. It is simply a ${\cal Q}$-dependent multiplication as given by the terms in the square bracket in Eq. (21).

In addition, the diagonal components of the HTL resummed gluon propagator, after multiplying by the projection operator and summing over the color indices, read

$$\sum_{ab}{\cal P}^{aa,bb}\Delta_{ij}^{aa,bb}(Q,{\vec{\sf{q}}})=\sum_{\sigma=1}^{N-1}\bigg{[}\frac{1}{\omega^{2}-q^{2}-{({\cal M}_{D}^{2})}^{\sigma}({\vec{\sf{q}}})\Pi_{T}({\hat{\omega}})}A_{ij}+\frac{1}{\omega^{2}-{({\cal M}_{D}^{2})}^{\sigma}({\vec{\sf{q}}})\Pi_{L}({\hat{\omega}})}B_{ij}\bigg{]}\,.$$

(22)

Similarly, the influence of a nonzero background field is fully encoded in the modified screening masses which can be written as

$${({\cal M}_{D}^{2})}^{\sigma}({\vec{\sf{q}}})=\left(1+\frac{3M^{2}}{4\pi^{2}T^{2}}+\frac{6}{N}\lambda^{\sigma}\right)m_{D}^{2}\,,$$

(23)

where terms in the bracket represent the background field modifications and $\lambda^{\sigma}$ with $\sigma=1,2,\cdots,N-1$ denote a set of ${\cal Q}$-dependent functions. For $SU(3)$ gauge theory, the explicit form of $\lambda^{\sigma}$ can be obtained as

$${\vec{\lambda}}=(3{\sf q}^{2}-3{\sf q},\,9{\sf q}^{2}-5{\sf q})\,.$$

(24)

In fact, only the color sum $\sum_{abcd}{\cal P}^{ab,cd}\Delta_{ij}^{ab,cd}$ is relevant for studying the soft scattering processes. Combining the above results of the diagonal and off diagonal contributions, we arrive at the following compact form:

$$\sum_{abcd}{\cal P}^{ab,cd}\Delta_{ij}^{ab,cd}(Q,{\vec{\sf{q}}})=\sum_{ab/\sigma}\bigg{[}\frac{1}{\omega^{2}-q^{2}-{({\cal M}_{D}^{2})}^{ab/\sigma}({\vec{\sf{q}}})\Pi_{T}({\hat{\omega}})}A_{ij}+\frac{1}{\omega^{2}-{({\cal M}_{D}^{2})}^{ab/\sigma}({\vec{\sf{q}}})\Pi_{L}({\hat{\omega}})}B_{ij}\bigg{]}\,,$$

(25)

where $\sum_{ab/\sigma}=\sum_{ab}+\sum_{\sigma}$ denotes a sum over both diagonal and off diagonal gluons and for general $SU(N)$, $a,b=1,2,\cdots,N$ with $a\neq b$ and $\sigma=1,2,\cdots,N-1$. For vanishing background field, it reduces to²²2Temperature independent contributions $\sim M^{2}$ in Eqs. (21) and (23) are negligible in the high temperature limit where the background field vanishes.

$$\sum_{abcd}{\cal P}^{ab,cd}\Delta_{ij}^{ab,cd}(Q,{\vec{\sf{q}}})\xlongequal[]{{\cal Q}=0}(N^{2}-1)\Big{[}\frac{1}{\omega^{2}-q^{2}-m_{D}^{2}\Pi_{T}({\hat{\omega}})}A^{ij}+\frac{1}{\omega^{2}-m_{D}^{2}\Pi_{L}({\hat{\omega}})}B^{ij}\Big{]}\,,$$

(26)

which indicates that the $N^{2}-1$ gluons have equal contribution to the above result. For nonzero background field, on the other hand, both the $N^{2}-N$ off diagonal gluons and $N-1$ diagonal gluons acquire different ${\cal Q}$-dependent modifications, and thus become distinguishable by their associated screening masses. Notice that according to our parametrization Eq. (17), there are only two different screening masses for the six off diagonal gluons in $SU(3)$ gauge theory.

In this subsection, we calculate the hard contributions to the collisional energy loss in the background field. As already mentioned before, the influence of a nonzero ${\cal Q}$ is reflected in the distributions of the thermal partons as given in Eq. (19). A special feature of such a ${\cal Q}$-dependent distribution is that the gluons are denoted by a pair of the color indices, $ab$. Accordingly, the color structures of the squared matrix element $|\mathcal{M}|^{2}$ need to be re-computed in the double line basis. We list the corresponding Feynman rules in the axial gauge in Fig. 2.

It is found that for nonvanishing background field, the energy loss $-dE/dx$ due to quark-gluon scattering can be obtained from Eqs. (3) and (4) by the replacement,

	$\displaystyle 4\int\frac{d^{3}{\bf k}}{(2\pi)^{3}}\frac{n(k)}{k}\int\frac{d^{3}{\bf{k}}^{\prime}}{(2\pi)^{3}}\frac{1-n(k^{\prime})}{k^{\prime}}\rightarrow$
	$\displaystyle\sum_{abcdef}\frac{1}{6}f^{ab,cd,ef}f^{ba,dc,fe}\int\frac{d^{3}{\bf k}}{(2\pi)^{3}}\frac{\big{(}n_{+}^{ab}(k)+n_{-}^{ab}(k)\big{)}/2}{k}\int\frac{d^{3}{\bf{k}}^{\prime}}{(2\pi)^{3}}\frac{1-\big{(}n_{+}^{cd}(k^{\prime})+n_{-}^{cd}(k^{\prime})\big{)}/2}{k^{\prime}}\,,$

where $4$ is the color factor in zero background field. Analogous to the ${\cal Q}=0$ case, the above form indicates that terms involving the product $n_{\pm}^{ab}({k})n_{\pm}^{cd}(k^{\prime})$ don’t contribute to the energy loss. This can be seen by interchanges of the momenta $\bf{k}\leftrightarrow\bf{k}^{\prime}$ and the color indices $ab\leftrightarrow cd$. Thus, for nonzero background field, we arrive at the following expressions for the hard contributions to $-dE/dx$:

	$\displaystyle-\bigg{(}\frac{dE}{dx}\bigg{)}_{\mathrm{BF,hard}}^{Qg(t)}$	$\displaystyle=$	$\displaystyle\frac{2\pi g^{4}}{v}{\sum_{a,b=1}^{3}}{\bigg{(}1-\frac{1}{3}\delta^{ab}\bigg{)}}\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}k}\frac{n_{+}^{ab}(k)+n_{-}^{ab}(k)}{2}\int\frac{d^{3}\mathbf{{k}^{\prime}}}{(2\pi)^{3}{k}^{\prime}}\delta(\omega-\mathbf{v\cdot q})$		(28)
		$\displaystyle\times$	$\displaystyle\frac{\omega}{(\omega^{2}-q^{2})^{2}}\left[(k-\mathbf{v\cdot k})^{2}+\frac{1-v^{2}}{2}(\omega^{2}-q^{2})\right]\theta(q-q^{*})\,,$

	$\displaystyle-\bigg{(}\frac{dE}{dx}\bigg{)}_{\mathrm{BF,hard}}^{Qg(s+u)}$	$\displaystyle=$	$\displaystyle\frac{\pi g^{4}}{4v}\sum_{a,b=1}^{3}\bigg{(}1-\frac{1}{3}\delta^{ab}\bigg{)}\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}k}\frac{n_{+}^{ab}(k)+n_{-}^{ab}(k)}{2}$		(29)
		$\displaystyle\times$	$\displaystyle\int\frac{d^{3}\mathbf{{k}^{\prime}}}{(2\pi)^{3}{k}^{\prime}}\delta(\omega-\mathbf{v\cdot q})\frac{\omega(1-v^{2})^{2}}{(k-\mathbf{v\cdot k})^{2}}\,.$

The energy loss due to the $s$- and $u$-channel scatterings can be obtained analytically, which reads

$$-\bigg{(}\frac{dE}{dx}\bigg{)}_{\mathrm{BF,hard}}^{Qg(s+u)}=\alpha_{s}^{2}\pi T^{2}{\sum_{ab}}{\bigg{(}1-\frac{1}{3}\delta^{ab}\bigg{)}}B_{2}(|{\sf q}^{ab}|)f_{1}(v)=\frac{\alpha_{s}^{2}T^{2}}{\pi}\sum_{n=1}^{\infty}\frac{|{\rm Tr}\,{\bf L}^{n}|^{2}-1}{n^{2}}f_{1}(v)\,,$$

(30)

where

$\displaystyle f_{1}(v)$

$\displaystyle=$

$\displaystyle\frac{1}{v}-\frac{1-v^{2}}{2v^{2}}\ln\frac{1+v}{1-v}\,.$

(31)

To get the above result, we used

$$\sum_{ab}\int_{0}^{\infty}k\,{n_{\pm}^{ab}(k)}dk=T^{2}\sum_{n=1}^{\infty}\frac{|{\rm Tr}\,{\bf L}^{n}|^{2}}{n^{2}}=\pi^{2}T^{2}\sum_{ab}B_{2}(|{\sf q}^{ab}|)\,.$$

(32)

Our results show that for the $s$- and $u$-channel scatterings, the modification of a background field is simply given by the factor $\sum_{ab}(1-\delta^{ab}/3)B_{2}(|{\sf q}^{ab}|)$. In the case of $SU(3)$, it equals $4(1-3{\sf q})^{2}/3$, which reduces to $4/3$ for vanishing background field as expected. However, integrals appearing in the $t$-channel scattering turn out to be more involved. To proceed further, we make the following change of the integral variables:

	$\displaystyle\frac{1}{2(2\pi)^{2}}\int\frac{d^{3}\mathbf{k}}{k}\int\frac{d^{3}\mathbf{k}^{\prime}}{k^{\prime}}$	$\displaystyle\to$	$\displaystyle\int_{\frac{1+v}{2}{q^{*}}}^{\infty}dk\int_{q^{*}}^{\frac{2k}{1+v}}qdq\int_{-vq}^{vq}d\omega+\int_{\frac{1+v}{2}{q^{*}}}^{\infty}dk\int_{\frac{2k}{1+v}}^{\frac{2k}{1-v}}qdq\int_{q-2k}^{vq}d\omega$		(33)
		$\displaystyle+$	$\displaystyle\int_{\frac{1-v}{2}{q^{*}}}^{\frac{1+v}{2}{q^{*}}}dk\int_{q^{*}}^{\frac{2k}{1-v}}qdq\int_{q-2k}^{vq}d\omega\,.$

The above integral intervals are determined to meet the constraint condition that $-1\leq\cos\theta\leq 1$, where $\theta$ denotes the angle between $\mathbf{k}$ and $\mathbf{k}^{\prime}$ and $\cos\theta=(k^{2}+(\omega+k)^{2}-q^{2})/(2k(k+w))$. Further constraints due to the two theta functions $\theta(q-q^{*})$ and $\theta(v^{2}q^{2}-\omega^{2})$ have also been taken into account in Eq. (33). The second theta function arises after averaging the integrand over the direction of ${\bf v}$, which is valid because the energy loss is independent of the incident direction of the heavy quark in an isotropic medium.

The cutoff $q^{*}$ was introduced to eliminate the infrared divergence associated with soft momentum transfer. In fact, in the limit $q^{*}\rightarrow 0$, only the first integral over $q$ in Eq. (33) becomes divergent, and thus needs to be regulated. Consequently, a simplified version of Eq. (33) can be obtained as

$$\frac{1}{2(2\pi)^{2}}\int\frac{d^{3}\mathbf{k}}{k}\int\frac{d^{3}\mathbf{k}^{\prime}}{k^{\prime}}\to\int_{0}^{\infty}dk\int_{q^{*}}^{\frac{2k}{1+v}}qdq\int_{-vq}^{vq}d\omega+\int_{0}^{\infty}dk\int_{\frac{2k}{1+v}}^{\frac{2k}{1-v}}qdq\int_{q-2k}^{vq}d\omega\,,$$

(34)

which coincides with the variable change as used in [18] and leads to an analytical form for the energy loss even for nonzero background field. The result can be expressed as

	$\displaystyle-\bigg{(}\frac{dE}{dx}\bigg{)}_{\mathrm{BF,hard}}^{Qg(t)}$	$\displaystyle=$	$\displaystyle\frac{2\alpha_{s}^{2}}{\pi}{\sum_{ab}}{\bigg{(}1-\frac{1}{3}\delta^{ab}\bigg{)}}\int_{0}^{\infty}kdk\,\frac{n_{+}^{ab}(k)+n_{-}^{ab}(k)}{2}\bigg{[}f_{2}(v)+f_{1}(v)$		(35)
		$\displaystyle\times$	$\displaystyle\Big{(}\ln(k/q^{*})-\frac{1}{2}\Big{)}\bigg{]}$
		$\displaystyle=$	$\displaystyle\frac{2\alpha_{s}^{2}T^{2}}{\pi}\sum_{n=1}^{\infty}\frac{|{\rm Tr}\,{\bf L}^{n}|^{2}-1}{n^{2}}\bigg{[}f_{1}(v)\bigg{(}\ln\frac{T}{q^{*}}+\frac{1}{2}-\gamma-\ln n\bigg{)}+f_{2}(v)\bigg{]}\,,$

where

	$\displaystyle f_{2}(v)$	$\displaystyle=$	$\displaystyle\Big{[}\frac{1}{2}-\frac{1}{2}\ln(1-v^{2})+\ln 2\Big{]}f_{1}(v)-\frac{1-v^{2}}{4v^{2}}\Big{[}{\rm Li}_{2}\big{(}\frac{1+v}{2}\big{)}$		(36)
		$\displaystyle-$	$\displaystyle{\rm Li}_{2}\big{(}\frac{1-v}{2}\big{)}+\frac{1}{2}\ln\frac{1-v^{2}}{4}\ln\frac{1+v}{1-v}\Big{]}-\frac{2}{3}v\,,$

and ${\rm Li}_{2}(x)$ is the polylogarithm. As we can see in the limit $q^{*}\rightarrow 0$, the above result has a logarithmic divergence $\sim\ln q^{*}$ and the background field modification on this divergent part is the same as that on the $s$- and $u$-channel contributions. On the other hand, for the finite part, besides the term $\sim k$ in the integral over $k$, there is also a term $\sim k\ln k$, which leads to a more involved ${\cal Q}$ dependence of the energy loss which cannot be described simply by the Bernoulli polynomial.

It can also be shown that the hard contribution in Eq. (28) is always positive when varying $q^{*}$. However, Eq. (35) may lead to a negative energy loss at large $q^{*}$ because of the simplification used in Eq. (34). Therefore, Eq. (34) is only a good approximation when $q^{*}\ll T$. For an arbitrary cutoff, one should adopt Eq. (33) to change the integral variables for mathematical rigour. Thus, Eq. (28) can be written as

	$\displaystyle-\bigg{(}\frac{dE}{dx}\bigg{)}_{\mathrm{BF,hard}}^{Qg(t)}$	$\displaystyle=$	$\displaystyle\frac{3\alpha_{s}^{2}}{\pi v^{2}}{\sum_{ab}}{\bigg{(}1-\frac{1}{3}\delta^{ab}\bigg{)}}\iiint_{{\rm sum}}dk\,dq\,d\omega\frac{n_{+}^{ab}(k)+n_{-}^{ab}(k)}{2}\frac{\omega}{q^{2}}$		(37)
		$\displaystyle\times$	$\displaystyle\bigg{[}\frac{k^{2}+k\omega+\omega^{2}/4}{q^{2}}-\frac{1-v^{2}}{q^{2}-\omega^{2}}\frac{k^{2}+k\omega+q^{2}}{3}-\frac{v^{2}}{12}\bigg{]}\,,$

where we use the shorthand notation $\iiint_{{\rm sum}}dk\,dq\,d\omega$ to denote the sum of the three triple integrals as given in Eq. (33). Consequently, the integral over $k$ needs to be carried out numerically. Similar to the ${\cal Q}=0$ case discussed in [20], the energy loss based on Eq. (37) is always positive for an arbitrary cutoff and agrees with Eq. (35) in the small $q^{*}$ region.

The soft contributions to the collisional energy loss can be studied based on Eq. (10), where the influence of the background field amounts to a modification on the gluon propagator. In the double line basis, the gluon propagator is not diagonal in the color space and summing over the color indices becomes nontrivial. In order to explicitly show the color structures, we rewrite Eq. (10) as

	$\displaystyle-\bigg{(}\frac{dE}{dx}\bigg{)}_{\mathrm{BF,soft}}$	$\displaystyle=$	$\displaystyle\sum_{abcd}\frac{4\pi\alpha_{s}}{v}\frac{1}{2N}{\cal P}^{ab,cd}\,{\rm Im}\int\frac{d^{3}\mathbf{q}}{(2\pi)^{3}}(\mathbf{v}\cdot\mathbf{q})v^{i}\big{[}\Delta_{ij}^{ab,cd}(Q,{\sf q})$		(38)
		$\displaystyle-$	$\displaystyle{(\Delta_{0})}_{ij}^{ab,cd}(Q)\big{]}v^{j}\theta(q^{*}-q)\Big{|}_{\omega={\bf v}\cdot{\bf q}}\,.$

In the above equation, ${(\Delta_{0})}_{ij}^{ab,cd}(Q)$ is not the usual bare propagator in perturbation theory. It actually corresponds to the zero temperature limit of the resummed propagator in the effective theory and can be written as

$${(\Delta_{0})}_{ij}^{ab,cd}(Q)={\cal{P}}^{ab,cd}\bigg{[}\frac{1}{\omega^{2}-q^{2}-\frac{3g^{2}M^{2}}{4\pi^{2}}\Pi_{T}({\hat{\omega}})}A_{ij}+\frac{1}{\omega^{2}-\frac{3g^{2}M^{2}}{4\pi^{2}}\Pi_{L}({\hat{\omega}})}B_{ij}\bigg{]}\,.$$

(39)

The usual bare propagator can be recovered by setting $M^{2}=0$ in the above equation which has no contribution to the energy loss because of a vanishing imaginary part. However, the propagator as given in Eq. (39) leads to an extra contribution which as we will see, is important to ensure the cancellation of the cutoff dependence in the weak coupling limit although such a contribution is temperature independent.

In the limit ${\cal Q}\rightarrow 0$, the resummed gluon propagation $\Delta_{ij}^{ab,cd}(Q,{\sf q})$ is also proportional to ${\cal P}^{ab,cd}$, and the sum over the color indices in Eq. (38) can be easily carried out, which leads to the following color factor:

$$\sum_{abcd}\frac{1}{2N}{\cal P}^{ab,cd}{\cal P}^{ab,cd}=C_{F}=\frac{4}{3}\,,$$

(40)

for $SU(3)$. On the other hand, when ${\cal Q}\neq 0$, based on the resummed gluon propagator as given in Sec. III.1, the following compact form can be found for the collisional energy loss due to soft scatterings,

	$\displaystyle-\bigg{(}\frac{dE}{dx}\bigg{)}_{\mathrm{BF,soft}}$	$\displaystyle=$	$\displaystyle\frac{\alpha_{s}}{12\pi v^{2}}\sum_{ab/\sigma/s}\mathrm{Im}\int^{+v}_{-v}d\hat{\omega}\frac{v^{2}-\hat{\omega}^{2}}{\hat{\omega}}\bigg{[}\frac{\hat{\omega}^{2}}{(1-\hat{\omega}^{2})^{2}}F^{ab/\sigma/s}_{T}(\hat{\omega},{\vec{\sf{q}}})$		(41)
		$\displaystyle\times$	$\displaystyle\ln\frac{(1-\hat{\omega}^{2})(q^{*})^{2}+F^{ab/\sigma/s}_{T}(\hat{\omega},{\vec{\sf{q}}})}{F^{ab/\sigma/s}_{T}(\hat{\omega},{\vec{\sf{q}}})}+\frac{1}{v^{2}-\hat{\omega}^{2}}F^{ab/\sigma/s}_{L}(\hat{\omega},{\vec{\sf{q}}})$
		$\displaystyle\times$	$\displaystyle\ln\frac{F^{ab/\sigma/s}_{L}(\hat{\omega},{\vec{\sf{q}}})-(q^{*})^{2}\hat{\omega}^{2}}{F^{ab/\sigma/s}_{L}(\hat{\omega},{\vec{\sf{q}}})}\bigg{]}\,,$

where we used $d^{3}\mathbf{q}=2\pi q^{2}dqd{\hat{\omega}}/v$ and the integral over $q$ has been done analytically. Furthermore, the shorthand notation of the color sums is defined as

$$\sum_{ab/\sigma/s}=\sum_{ab}+\sum_{\sigma}-\sum_{s}\,,$$

(42)

where the first two sums on the right-hand side have already appeared in Eq. (25). Notice the minus sign for the last sum which is related to the contribution associated with ${(\Delta_{0})}_{ij}^{ab,cd}(Q)$ in Eq. (38). In the high temperature limit where ${\cal Q}\rightarrow 0$, such a contribution becomes negligible as compared to the contributions from the first two sums. However, near the critical temperature, the three sums in Eq. (42) give comparable contributions to the energy loss. In addition, we have

$$F^{ab/\sigma/s}_{T/L}(\hat{\omega},{\vec{\sf{q}}})={({\cal M}_{D}^{2})}^{ab/\sigma/s}({\vec{\sf{q}}})\Pi_{T/L}(\hat{\omega})\,,$$

(43)

where $({\cal M}_{D}^{2})^{s}=3g^{2}M^{2}/(4\pi^{2})$ is independent on the color index $s$ and for $SU(N)$, $s=1,2,\cdots,N^{2}-1$. Therefore, the purpose of introducing such a trivial sum over $s$ is merely to make the form of Eq. (41) compact.