Thermal Heavy Quark Self-Energy from Euclidean Correlators

TL;DR

This paper links the real self-energy of a single heavy quark to Euclidean electric field correlators along a Polyakov line, enabling lattice studies without analytical continuation, and compares Minkowski and Euclidean calculations.

Contribution

It identifies a real self-energy term for open heavy flavor and expresses it via Euclidean correlators, expanding the understanding of heavy quark self-energies.

Findings

The self-energy term can be directly computed from Euclidean correlators.

Minkowski and Euclidean correlators are consistent at NLO.

The self-energy for open heavy flavor differs from that of quarkonium.

Abstract

Brambilla, Escobedo, Soto, and Vairo have derived an effective description of quarkonium with two parameters; a momentum diffusion term and a real self-energy term. We point out that there is a similar real self-energy term for a single open heavy flavor and that it can be expressed directly in terms of Euclidean electric field correlators along a Polyakov line. This quantity can be directly studied on the lattice without the need for analytical continuation. We show that Minkowski-space calculations of this correlator correspond with the known NLO Euclidean value of the relevant electric field two-point function and that it differs from the real self-energy term for quarkonium.

Equations

$$\kappa =\frac{g^2}{6N_c}\mathrm{Re}{\int }_{-\infty }^{\infty }ds⟨\mathrm{T}{E}^{a,i}(s,0)E^{a,i}(0,0)⟩,$$

$$\gamma =\frac{g^2}{6N_{\text{c}}}\text{Im}{\int }_{-\infty }^{\infty }ds⟨\text{T}{E}^{a,i}(s,0)E^{a,i}(0,0)⟩.$$

$${\gamma }_{\mathrm{adj}}=\frac{g^2}{6N_{\text{c}}}\text{Im}{\int }_{-\infty }^{\infty }ds⟨\text{T}{E}^{a,i}(s,0)U(s,0{)}_{ab}{E}^{b,i}(0,0)⟩,$$

$$\gamma$$

$$\times$$

$${\int }_{-\infty }^{\infty }dt{e}^{i\omega t}\int d^3\mathbf{x}⟨\left[{\hat{\mathcal{J}}}^{\mu }(t,\mathbf{x}),{\hat{\mathcal{J}}}^{\nu }(0,0)\right]⟩,$$

$$G$$

$$=-\sum _{i=1}^{3-2ϵ}\frac{⟨\text{Re}\text{Tr}\left[U(\beta ,\tau )g_{\text{B}}{E}_i(\tau ,0)U(\tau ,0)g_{\text{B}}{E}_i(0,0)\right]⟩}{3⟨\text{Re}\text{Tr}\left[U(\beta ,0)\right]⟩},$$

$$\begin{array}{rl}\text{Im}{G}_R^{AB}(\omega =0)=& \text{Im}{\int }_0^{\infty }dt{G}^{AB}(t)\\ =& -{\int }_0^{\beta }d\tau G_E^{AB}(\tau )=-{\tilde{G}}_E({\omega }_n=0).\end{array}$$

$$\text{Im}{\int }_0^{\infty }dt\sum _{n,m}\frac{2}{Z}{A}_{mn}{B}_{nm}{e}^{-\frac{\beta }{2}(E_n+E_m)}$$

$$\times \sinh \left(\frac{\beta (E_n-E_m)}{2}\right)e^{-i(E_n-E_m)t}$$

$$=-\sum _{n,m}\frac{2}{Z}\mathrm{Re}\left[A_{mn}{B}_{nm}\right]\frac{e^{-\frac{\beta }{2}(E_n+E_m)}}{E_n-E_m}\sinh \left(\frac{\beta (E_n-E_m)}{2}\right),$$

$${\int }_0^{\beta }d\tau \sum _{n,m}\frac{1}{Z}{A}_{mn}{B}_{nm}{e}^{-\beta E_m}{e}^{-(E_n-E_m)\tau }$$

$$=\sum _{n,m}\frac{2}{Z}{A}_{mn}{B}_{nm}\frac{e^{-\frac{\beta }{2}(E_n+E_m)}}{E_n-E_m}\sinh \left(\frac{\beta (E_n-E_m)}{2}\right).$$

$${\gamma }_{\mathrm{fund}}=-{\int }_0^{\beta }d\tau G_{\text{E}}^{\text{HQ}}(\tau ),$$

$$G_{\mathrm{E\,LO}}^{\mathrm{HQ}}(\tau)=-\frac{g^{2}C_{F}}{3}\sumint\limits_{K}e^{ik_{n}\tau}\frac{(D-1)k_{n}^{2}+k^{2}}{k_{n}^{2}+k^{2}},$$

$$G_{\mathrm{E}\mathrm{LO}}^{\mathrm{HQ}}(\tau )=g^2{C}_F{\pi }^2{T}^4\left[\frac{{\cos }^2(\pi \tau T)}{{\sin }^4(\pi \tau T)}+\frac{1}{3{\sin }^2(\pi \tau T)}\right].$$

$$G_{\mathrm{E}\mathrm{LO}}^{\mathrm{HQ}}(\tau ,T=0)=\frac{g^2{C}_F}{{\pi }^2{\tau }^4}.$$

$${\gamma }_{\mathrm{fund}}=$$

$$+2{\int }_{\beta /2}^{\infty }d\tau G_{\mathrm{E}\mathrm{LO}}^{\mathrm{HQ}}(\tau ,T=0)=0+\mathcal{O}(g^4),$$

$${\gamma }_{\mathrm{fund}}^{\mathrm{LO}}=$$

$$=$$

$${\mathcal{I}}_1,{\tilde{\mathcal{I}}}_1=$$

$${\mathcal{I}}_2,{\tilde{\mathcal{I}}}_2=$$

$${\sigma }_f(T)$$

$${\mathcal{I}}_1=$$

$${\mathcal{I}}_2=$$

$${\mathcal{I}}_a\propto \frac{1}{{\beta }^{2d-3}}\to {\tilde{\mathcal{I}}}_a={\mathcal{I}}_a\times (4^{2-d}-1).$$

$${\gamma }_{\mathrm{fund}}^{\mathrm{LO}}=-2{\alpha }_{\mathrm{s}}^2{T}^3\zeta (3)C_F{N}_{\text{f}}.$$

$${\gamma }_{\mathrm{adj}}^{\mathrm{LO}}=-2{\alpha }_{\mathrm{s}}^2{T}^3\zeta (3)C_F\left(\frac{4}{3}{N}_{\text{c}}+N_{\text{f}}\right).$$

$${\gamma }_{\mathrm{NLO}}=\frac{g^2{C}_F}{3}{\int }_k\frac{k^2}{k^2+m_D^2}=\frac{{\alpha }_{\mathrm{s}}{C}_F{m}_D^3}{3},$$

$${\gamma }_{\mathrm{adj}}^{(\mathrm{j})}=$$

$$\displaystyle\times\frac{if^{acb}f^{abc}}{(\vec{p}+\vec{q})^{2}}\bigg{[}q^{0}p^{0}(q^{0}-p^{0})G^{11}_{ik}(P)G^{11}_{ki}(Q)$$

$$\displaystyle\hskip 14.22636pt+2ip^{0}\hat{q}^{i}\hat{q}^{j}G^{11}_{ij}(P)-2iq^{0}\hat{p}^{i}\hat{p}^{j}G^{11}_{ij}(Q)\bigg{]},$$

$${\gamma }_{\mathrm{fund}}^{(\mathrm{j})}=$$

$$\displaystyle\times\int_{Q}\int_{P}e^{iq^{0}t}e^{-i(p^{0}+q^{0})t^{\prime}}\frac{if^{acb}f^{abc}}{(\vec{p}+\vec{q})^{2}}\bigg{[}q^{0}p^{0}(q^{0}-p^{0})$$

Full text

quarkonium, quark-gluon plasma, electric–field correlator

††preprint: CERN-TH-2019-025

I Introduction

Quarkonium (bound heavy quark-antiquark states) are an intriguing probe of the quark-gluon plasma Andronic et al. (2016). Originally proposed by Matsui and Satz Matsui and Satz (1986), the suppression of quarkonia has remained an active topic of experimental Adare et al. (2015); Adamczyk et al. (2014a, b); Abelev et al. (2014); Adam et al. (2017); Khachatryan et al. (2017) and theoretical Rapp and Du (2017); Brambilla et al. (2008, 2011); Mocsy et al. (2013); Burnier et al. (2015); Brambilla et al. (2017, 2018) investigation ever since. The central idea is that a thermal medium tends to break up quarkonium bound states; one then investigates how strong this effect is expected to be theoretically, and how much such states are suppressed experimentally. Recently it has become clear that, at the highest collision energies, charmonium experiences important recombination effects from the many open charm quarks in the plasma Young and Shuryak (2010); Zhao and Rapp (2011); Blaizot et al. (2016).

Recently Brambilla, Escobedo, Soto, and Vairo have used potential non-relativistic QCD (pNRQCD) Pineda and Soto (1998); Brambilla et al. (2000, 2005) at second order in the multipole expansion to rigorously derive in Brambilla et al. (2017) an open quantum system effective description for quarkonium evolution in a quark-gluon plasma for $m\gg 1/a_{0}\gg T$, where $m$ is the heavy quark mass and $a_{0}\sim 1/(m\alpha_{\mathrm{s}})$ the Bohr radius. Their description depends on two (in principle nonperturbative) parameters describing the interaction of the thermal medium with heavy quarks. One parameter is the well-known heavy-quark momentum-diffusion coefficient Casalderrey-Solana and Teaney (2006)

[TABLE]

where $E^{a,i}$ is the color electric field, $s$ is the Minkowski time value, both $E$ fields are at the same space coordinate $\mathbf{0}$, $N_{c}=3$ is the number of colors, $\mathrm{T}$ is the time ordering symbol, and a Wilson line is implicitly included in the time-ordered correlator indicated, i.e., in the notation of Brambilla et al. (2017, 2018) the $\vec{E}$ field has been redefined as $\vec{E}\to\Omega\vec{E}\Omega^{\dagger}$, with $\Omega\equiv\mathrm{P}\exp\,-ig\int_{-\infty}^{t}dsA^{0}(s,\mathbf{0})$. This quantity has been extensively investigated in the literature, both in weak-coupling QCD Svetitsky (1988); Moore and Teaney (2005); Caron-Huot and Moore (2008a, b), effective models van Hees et al. (2008), holographic dual theories Casalderrey-Solana and Teaney (2006); Herzog et al. (2006), and via analytical continuation from lattice data Caron-Huot et al. (2009); Ding et al. (2011); Francis et al. (2011); Banerjee et al. (2012); Francis et al. (2015).

The second parameter is a real non-dissipative plasma effect which induces a mass shift in the heavy-quark bound states. At lowest order in pNRQCD, the shift is $\delta m=\frac{3}{2}a_{0}^{2}\gamma$ Brambilla et al. (2008), where $\gamma$ is the following electric-field correlator

[TABLE]

This correlator has received less attention in the literature. In this paper we show that there are two distinct operators, depending on the Wilson lines connecting the electric fields; Brambilla et al. (2008)

[TABLE]

with $U(s,0)_{ab}$ an adjoint-representation111To fix clearly the conventions, we choose $D_{\mu}=\partial_{\mu}-igA_{\mu}$, so that $U(a,b)\equiv\mathrm{P}\exp\,ig\int_{b}^{a}dsA^{0}(s,\mathbf{0})$ is the Wilson line in either representation. Wilson line, and

[TABLE]

with $E^{i}=E^{a,i}T^{a}$, the Wilson lines $U$ in the fundamental representation and P is the path-ordering symbol, indicating that fields are contour-ordered along the Wilson lines; this equals time ordering for the $E$ fields but not for the $U$ operators.

The different Wilson line structures reflect the different time evolution of the two objects. The former corresponds to the real part of the heavy quarkonium singlet self-energy. Before (after) the pNRQCD dipole vertex at $t=0$ ($t=s$) the bound state is a color singlet and thus only the adjoint Wilson line connecting the two $E$ fields exists in Eq.(3). Conversely, Eq. (4) is linked to the real part of the open heavy-quark self-energy: the Wilson lines are always fundamental and reflect the fact that the heavy quark can interact with the environment at all times.

In this paper we show that $\gamma_{\mathrm{adj}}$, which was calculated to LO in Ref. Brambilla et al. (2008), differs from $\gamma_{\mathrm{fund}}$, whose LO computation we present here for the first time. We also show that the latter object can be determined via lattice QCD much more easily than the coefficient $\kappa$, as it maps directly to a Euclidean expression, without the need of analytic continuation or spectral function reconstruction. Therefore we will focus in this paper on the coefficient $\gamma_{\mathrm{fund}}$. We plan to investigate the Euclidean mapping of $\gamma_{\mathrm{adj}}$ in a follow-up paper Eller et al. , together with the physical interpretation of $\gamma_{\mathrm{fund}}$. This follow-up paper will also discuss the difference between $\kappa_{\mathrm{adj}}$ and $\kappa_{\mathrm{fund}}$, which are also distinct objects.

In the next section we will show how to analytically continue Eq. (4) to Euclidean time. This leads to a time-integral of a correlator of electric fields along a Polyakov loop. In the remainder of the paper we check this derivation by showing that the next-to-leading order (NLO) value of $\gamma_{\mathrm{fund}}$, derived by Minkowski space methods, correctly corresponds to the appropriate integral moment of the known NLO Euclidean correlator from Ref. Burnier et al. (2010). This is a nice check of our continuation.

II Analytic Continuation of Electric-Field Correlator

The analytic continuation of the electric-field correlator Eq. (4) is not trivially doable. However, we don’t consider this correlator directly, but instead use the heavy-quark current-current correlator

[TABLE]

which Eq. (4) originates from. Here $\mathcal{\hat{J}}^{\mu}\equiv\hat{\bar{\psi}}\gamma^{\mu}\hat{\psi}$ is the heavy-quark current and $\hat{\psi}$ is the heavy-quark field operator. The definition of the heavy-quark momentum-diffusion coefficient $\kappa$ as the spectral function of Eq. (5) in the $M_{\text{kin}}\rightarrow\infty$ limit is derived in Ref. Caron-Huot et al. (2009). Its authors also show that in the heavy-quark limit the current-current correlator can be analytically continued to Euclidean time leading to the Euclidean color–electric correlator

[TABLE]

where the minus sign emerges from different factors of $i$ in the definition of the color–electric field in real and imaginary time. For the details of the continuation we refer the reader to the original paper Caron-Huot et al. (2009).

In order to analytically continue $\gamma_{\mathrm{fund}}$, we need to relate the imaginary part of Eq. (5) with its analytic continuation Eq. (6). Therefore we want to remind the reader of the relation between the imaginary part of a two-point function of two hermitian operators $A,B$ in real time with the zero Matsubara frequency limit of the corresponding Euclidean correlator, such that

[TABLE]

The two-point functions are defined in the usual way, $G^{AB}(t)=\text{Tr}\left\{\hat{\rho}\left[A(t),B(0)\right]\right\}$ and $G^{AB}_{E}(\tau)=\text{Tr}\left\{\hat{\rho}A(-i\tau)B(0)\right\}$ with $\hat{\rho}\equiv\frac{1}{Z}e^{-\beta H}$ the finite-temperature equilibrium density matrix. (Note that $\int_{0}^{\infty}dtG^{AB}(t)$ is purely imaginary, since the commutator of Hermitian operators gives twice the imaginary part. Nevertheless, we take the imaginary part explicitly because the finite-frequency transform contains real and imaginary parts.)

We insert two complete sets of energy eigenstates in the definition of the two-point function such that the LHS of Eq. (7) becomes

[TABLE]

where we used the notation $A_{nm}=\left\langle n\left|A(t=0)\right|m\right\rangle$.

Using the same procedure on the RHS of Eq. (7) yields

[TABLE]

So as long as $A_{mn}B_{nm}$ has no imaginary part, which happens if $A$ and $B$ are Hermitian operators or if $A=B^{\dagger}$, Eq. (7) is true. From this we conclude that the analytic continuation of $\gamma_{\mathrm{fund}}$ is

[TABLE]

One of the main results of this paper is therefore that the thermal effects on $\gamma_{\mathrm{fund}}$ can be determined by a nonperturbative calculation using the vacuum-subtracted Euclidean color–electric correlator on the lattice.

To further clarify the need for vacuum subtraction, let us look at the Euclidean color–electric correlator at leading order (LO), $\mathcal{O}(g^{2})$. It is obtained trivially by connecting the two chromoelectric fields with a gluon propagator, yielding Caron-Huot et al. (2009); Burnier et al. (2010)

[TABLE]

where \sumint\limits_{K}\equiv T\sum_{k_{n}}\int_{k}, $\int_{k}\equiv\int d^{d}k/(2\pi)^{d}$ with $D=d+1$ the dimension of spacetime and $k_{n}$ the bosonic Matsubara frequency. One could immediately perform the $\tau$ integration of Eq. (8), obtaining $\beta\delta_{k_{n}}$, at which point the $\int_{k}$ integral would vanish in dimensional regularization (DR). So would, in this scheme, the vacuum contribution, where the $\tau$ integrations runs from $-\infty$ to $+\infty$ and the Matsubara sum is replaced with an integral over a continuous Euclidean frequency $k_{4}$. However, to better illustrate the need for vacuum subtraction in other schemes, such as the lattice, let us instead perform first the Matsubara sum and then the $\int_{k}$, which gives Caron-Huot et al. (2009)

[TABLE]

The integration of this object over the compactified time direction does not converge, as the integrand diverges as $\tau^{-4}$ as $\tau\to 0$ and as $(\beta-\tau)^{-4}$ as $\tau\to\beta$. But this divergence is ultraviolet, as it comes about when the two $E$ fields are brought together. It is thus equal to the behavior observed in vacuum, which can be easily obtained from the $k_{4}$ integration, leading to

[TABLE]

Hence, vacuum subtraction in a non-DR scheme takes the form

[TABLE]

where we have exploited the symmetry of the thermal contribution at $\tau=\beta/2$ and that of the vacuum at $\tau=0$. It is precisely a subtraction of this kind that would need to be performed on the lattice: for all $\tau<\beta/2$ values, one computes the difference between the correlator on the thermal lattice and the vacuum lattice, and one then subtracts the integral of the vacuum contribution over $\tau>\beta/2$. In practice, due to the noisy denominator in Eq. (6), it may be impossible to subtract $\gamma_{\mathrm{fund}}$ at very low temperature on the lattice; in practice a subtraction at a temperature where thermal effects are expected to be small should be sufficient.

At small separation, where the vacuum and thermal correlators diverge but the difference stays finite, it may be difficult to extract the difference with good statistical power. However, we believe that, while the individual short-distance values are sensitive to even small amounts of gradient flow Narayanan and Neuberger (2006); Lüscher (2010a, b), the difference should not be. This is supported by existing analytical studies Eller and Moore (2018), and it would be useful to investigate this issue further. We also refer to Christensen and Laine (2016) for the perturbative $\mathcal{O}(\alpha_{\mathrm{s}})$ renormalization of Eq. (6) in the lattice scheme.

III NLO Integration of the Euclidean correlator

In this section, we validate our result in a perturbative calculation at next–to–leading order: we first present the imaginary-time integration of the correlator in Eq. (6), to be followed by the corresponding real-time counterpart. The Euclidean color–electric correlator was calculated in perturbation theory up to next–to–leading order in Ref. Burnier et al. (2010) and the contributing diagrams are shown in Fig. 1.

Since we are interested in the thermal contributions to $\gamma_{\mathrm{fund}}$, we subtract the vacuum contribution of the correlator during the calculation. As previously highlighted, dimensional regularization does that automatically, so in the following we will not keep track of scale-free contributions that vanish in any $D$.

Using the integral expression of $G_{\mathrm{E\,NLO}}^{\mathrm{HQ}}(\tau)$ obtained in Ref. (Burnier et al., 2010) from the diagrams above, we notice that the only $\tau$ dependence is in the Fourier transform. After applying the Kronecker delta $\beta\delta_{k_{n}}$ arising from the $\tau$ integration, we obtain that222We thank Viljami Leino for the discovery of an issue in our evaluation of Eq. (13), then reflected in Eq. (19), in a previous version of this article. We also thank M. Á. Escobedo, N. Brambilla, J. Soto and A. Vairo for comments on this revised version.

[TABLE]

where we have introduced these sum integrals

[TABLE]

where $K^{2}=k_{n}^{2}+\mathbf{k}^{2}$. The notation $\left\{Q\right\}$ represents the fermionic Matsubara frequencies $q_{n}=\pi T(2n+1)$ for the fermionic sum integrals, denoted by $\tilde{\mathcal{I}}$. These are directly related to the bosonic ones via

[TABLE]

The bosonic integrals are easily evaluated as

[TABLE]

where we have used a Feynman parameter for $\mathcal{I}_{2}$. The fermionic counterparts, obtained from Eq. (16), are

[TABLE]

Putting everything together, we find

[TABLE]

This results differs from the real-time calculation of Eq. (3) with its adjoint Wilson line given in Ref. Brambilla et al. (2008), which found

[TABLE]

To our knowledge there is no real-time determination of Eq. (4) in the literature; let us thus present it briefly. We find Coulomb gauge to be a good choice: in this gauge the $A_{0}A_{0}$ retarded bare propagator is $G_{00}^{R}(q^{0},q)=i/q^{2}$ and equals the advanced one. Thus, the spectral density vanishes, making the off-diagonal entries of the propagator matrix in the “12” formalism of real-time perturbation theory vanish. Moreover, the lack of frequency dependence of the diagonal elements makes many diagrams vanish in DR. We label the real-time graphs as in Fig. 1, though the Wilson lines now start and end at $t=-\infty$. The only non-vanishing diagrams in this gauge are then (i), (j) and (k). Of these, (i) and (k) do not source any gluons from the Wilson lines: they thus contribute equally to $\gamma_{\mathrm{adj}}$ and $\gamma_{\mathrm{fund}}$, as the color trace gives the same result. In this gauge, any difference between the two can thus only arise from diagram (j) and its equivalent for $\gamma_{\mathrm{adj}}$. We show in App. A how the Coulomb gauge evaluation of the contribution of this diagram to Eqs. (3) and (4) yields the difference between Eqs. (20) and (19),333Ref. Brambilla et al. (2008) obtained Eq. (20) in the temporal axial gauge $A_{0}=0$. We have also checked that the sum of diagrams (i), (j) and (k) in the expansion of $\gamma_{\mathrm{adj}}$ in Coulomb gauge reproduces Eq. (20). thus confirming the correctness of our analytical continuation, Eq. (8), to first non-trivial order in perturbation theory.

Interestingly, it is also possible to go beyond this order. Our Euclidean analysis has so far used unresummed perturbation theory, which is appropriate when all momenta are of order $T$ and the Matsubara frequency $k_{n}$ is nonzero. Unlike $\kappa$, which receives a contribution from the $gT$ scale at LO, $\gamma_{\mathrm{fund}}$ does not. Inspection of Eqs. (8) and (9) shows that $k\sim gT$ contributes to $\gamma_{\mathrm{fund}}$ at $\mathcal{O}(g^{5})$. This contribution is easily obtained by replacing Eq. (9) with its resummed version. Since the $\tau$ integration forces $k_{n}=0$, it suffices to use Electrostatic QCD (EQCD) Braaten (1995); Braaten and Nieto (1995, 1996); Kajantie et al. (1996, 1997), where ${\bf E}\approx-i\bm{\nabla}A_{0}$. Then the temporal component of the gauge field gets Debye-screened, yielding

[TABLE]

where $m_{D}^{2}=g^{2}T^{2}(N_{\text{c}}/3+N_{\text{f}}/6)$ is the Debye mass. Eq. (21) agrees with Eq. (87) of Brambilla et al. (2008), recalling that $\mathrm{Re}\,\delta V_{s}(r)_{11}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Brambilla:2008cx}{\@@citephrase{(}}{\@@citephrase{)}}}}}=\gamma_{\mathrm{adj}}\,r^{2}/2$. As the Wilson lines do not contribute, this $\mathcal{O}(g^{5})$ contribution $\gamma_{\mathrm{NLO}}$ is identical for $\gamma_{\mathrm{fund}}$ and $\gamma_{\mathrm{adj}}$ and is the only NLO contribution to both $\gamma$; to the best of our knowledge, this was not observed in the previous literature.

IV Discussion and conclusions

In this paper we considered the coefficient $\gamma_{\mathrm{adj}}$, introduced by Brambilla, Escobedo, Soto and Vairo Brambilla et al. (2017) and defined in Eq. (3), which describes the in-medium mass shift to heavy quarkonium for $m\gg 1/a_{0}\gg T$, and $\gamma_{\mathrm{fund}}$ (Eq. (4)). The latter object can be seen as the fluctuation counterpart to $\kappa$, the momentum-diffusion coefficient for a single heavy quark. We have shown that $\gamma_{\mathrm{fund}}$ and $\gamma_{\mathrm{adj}}$ are not equal; this was done by presenting an explicit calculation for $\gamma_{\mathrm{fund}}$, which shows how it differs from $\gamma_{\mathrm{adj}}$ at the first non-trivial order in perturbation theory, $\mathcal{O}(g^{4}T^{3})$. Physically, the difference between the two can be understood as follows: $\gamma_{\mathrm{fund}}$ is related to the propagation of a single heavy quark, which can interact with the medium at any time, whereas $\gamma_{\mathrm{adj}}$ describes a $Q\bar{Q}$ pair, which is a medium-blind singlet before (after) the first (last) $E$ field insertion. Indeed, our explicit evaluation, presented in App. A, shows how the difference arises from the Wilson lines before/after the $E$ fields.444

The computation of $\gamma_{\mathrm{adj}}$ in Brambilla et al. (2008) was performed in the $A^{0}=0$ gauge, where the contribution of the Wilson lines vanishes and one would naively expect $\gamma_{\mathrm{adj}}$ and $\gamma_{\mathrm{fund}}$ to be equal. However, more care is needed when temporal Wilson lines stretch to $t=-\infty$ in the $A^{0}=0$ gauge; we plan to return to the issue of gauge invariance in this singular gauge in Eller et al. . We also observe that, at zero temperature, the NLO result for $g^{2}/(6N_{\text{c}})\langle E^{a}(t)U_{ab}(t,0)E^{b}(0)\rangle$ in Eidemuller and Jamin (1998) and that for $g^{2}/(3N_{\text{c}})\left\langle\text{Tr}\left[\text{T}~{}U(-\infty,t)E^{i}(t)U(t,0)E^{i}(0)U(0,-\infty)\right]\right\rangle$, which can be extracted from the $T=0$ limit of Burnier et al. (2010), do not agree, further confirming the different nature of the two operators.

We have furthermore shown that $\gamma_{\mathrm{fund}}$ can be re-expressed in terms of a Euclidean correlation function, (6), which is highly amenable to a lattice determination. With the vacuum contributions removed, the time integral of the correlator, (8), should not suffer from divergences and the computational cost should be reasonable if smoothing techniques like gradient flow are employed. We confirmed that the LO results for $\gamma_{\mathrm{fund}}$, evaluated via real-time techniques, agree with the Euclidean time-integration of the results of Ref. Burnier et al. (2010), which is a nontrivial check on our derivation of the Euclidean continuation. We also obtained the NLO correction to $\gamma_{\mathrm{fund}}$ and $\gamma_{\mathrm{adj}}$ in Eq. (21).

The physical interpretation of $\gamma_{\mathrm{fund}}$ is at the moment however not completely clear to us; we plan to return to this issue in a follow-up publication, Eller et al. , where we also intend to address the issue of the Euclidean counterpart to $\gamma_{\mathrm{adj}}$. Similarly, we can define $\kappa_{\mathrm{fund}}$ as the real rather than imaginary part of Eq. (4), and $\kappa_{\mathrm{adj}}$ as the real part of Eq. (3). The former is relevant for the medium interactions of open heavy quarks, while the latter is relevant in quarkonium physics. Perturbative results show that they agree up to order $g^{5}T^{3}$ Caron-Huot and Moore (2008a, b); Brambilla et al. (2008), but there is no reason why this should persist to all orders. We plan to touch this issue as well in Eller et al. , together with that of gauge invariance discussed in footnote 4.

Acknowledgements.

We thank the Technische Universität Darmstadt and its Institut für Kernphysik, where this work was conducted and where JG was hosted during the early phase of this work. This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project number 315477589 – TRR 211.

Note added

As we were finalizing this paper, we became aware of the preprint “Transport coefficients from in medium quarkonium dynamics” Brambilla et al. (2019) by N. Brambilla, M. Á. Escobedo, A. Vairo and P. Vander Griend. It proposes a way to determine $\gamma$ from the quarkonium spectral function reconstructed from lattice QCD Kim et al. (2018). We thank the authors for sharing their results with us prior to publication and for discussion.

Appendix A Explicit real-time computation of diagram (j)

In the adjoint case, diagram (j) is shown in Fig. 2. It contributes to

[TABLE]

where we have rewritten the integral over negative and positive times of the contour-ordered operator as twice the positive-time integral of the forward Wightman operator. Thus, the three fields sourced by the operator, $E(t)$, $A^{0}(t^{\prime})$ and $E(0)$, are naturally time-ordered and thus of type “1” in the “12” formalism of real-time perturbation theory. In Coulomb gauge the $A^{0}(t^{\prime})$ field can only connect to another $A^{0}$ field, which has furthermore to be of type “1” as well, due to the diagonal nature of the bare temporal propagator matrix. Hence, the three-gluon vertex has to be of type “1”, so that the propagators of the transverse gluons have to be of type “11”, i.e., time-ordered. Indeed, the second line of Eq. (22) is the contribution with two transverse gluons sourced by the two $E$ fields, as depicted in Fig. 2, while on the final line they source one transverse and one temporal gluon. Finally, $\int_{P}=\int d^{D}P/(2\pi)^{D}$ is the Minkowski $D$-dimensional integral.

In the fundamental case one has instead the configurations shown in Fig. 3. They give

[TABLE]

where, as shown in the figure, there are now two possible “12” assignments for the fields sourced by the operator: the $E$ fields are always of type “1”, while the $A_{0}$ gluon is “1” if it comes from $U(0,-\infty)$ or $U(s,0)$, “2” if from $U(-\infty,s)$. For reasons which will become clearer soon, we have rewritten this last Wilson line as $U(-\infty,s)=U(-\infty,\infty)U(\infty,s)$, with $U(-\infty,\infty)$ thus of type “2” and $U(\infty,s)$ of type “1”, hence the $\int_{t}^{\infty}dt^{\prime}$ contribution.555In a covariant gauge the contribution of $\int_{t}^{\infty}dt^{\prime}$ vanishes, as expected from the unitarity of the Wilson lines. In Coulomb gauge one needs anyway to consider $U(-\infty,s)$ as $U(-\infty,s+\delta^{+})U(\delta^{+},s)$, with $\delta^{+}$ arbitrarily small and positive. This avoids the appearance of ill-defined $\theta(0)$ contributions arising from the time integrations of the bare temporal propagators, which are instantaneous in time. The second line describes the diagrams with two transverse gluons, where we have used the definition $G^{>}=G^{21}$. For these diagrams, as shown in Fig. 3, we have two assignments contributing to each of the three $dt^{\prime}$ integrations. The relative sign between the two, encoded in the sign functions $\epsilon(t^{\prime})\epsilon(t-t^{\prime})$, arises from the combination of a minus sign from the different color ordering — in the $0<t^{\prime}<t$ region — together with another minus sign from the opposite direction of the Wilson lines. The final line encodes the contribution of graphs with a single transverse gluon, for which the lower “2” contour does not contribute. The overall factor of $1/2$ in front of Eq. (23) with respect to Eq. (22) arises from color tracing in the different cases.

If we take the difference between Eq. (23) and (22) we obtain $\Delta\gamma\equiv\gamma_{\mathrm{fund}}-\gamma_{\mathrm{adj}}$. It reads

[TABLE]

so that the structure of the time integrations simplifies greatly; hence our choice of introducing the $\int_{t}^{\infty}dt^{\prime}$ contribution. Upon using $G_{ij}(P)=(\delta_{ij}-\hat{p}_{i}\hat{p}_{j})G_{T}(P)$ we find

[TABLE]

$G_{T}^{>}(Q)=(\theta(q^{0})+n_{\mathrm{B}}(|q^{0}|))2\pi\delta(Q^{2})$ is purely real and thus does not contribute to $\Delta\gamma$. $G^{11}_{T}(Q)=i\mathbf{P}1/(q_{0}^{2}-q^{2})+(1/2+n_{\mathrm{B}}(|q^{0}|))2\pi\delta(Q^{2})$ has both real and imaginary parts, with $\mathbf{P}$ a principal-value prescription and $n_{\mathrm{B}}$ the Bose–Einstein distribution, so that

[TABLE]

where we have also used the $p\leftrightarrow q$ symmetry of the integrand. The final integration has been carried out in DR, showing that Eq. (26) is equal to the difference between Eqs. (19) and (20), as we set out to prove.