Static quark-antiquark pairs at finite temperature
In a framework that makes close contact with modern effective field theories for non-relativistic bound states at zero temperature, we study the real-time evolution of a static quark-antiquark pair in a medium of gluons and light quarks at finite temperature. For temperatures ranging from values larger to smaller than the inverse distance of the quark and antiquark, , and at short distances, we derive the potential between the two static sources, and calculate their energy and thermal decay width. Two mechanisms contribute to the thermal decay width: the imaginary part of the gluon self energy induced by the Landau damping phenomenon, and the quark-antiquark color singlet to color octet thermal break up. Parametrically, the first mechanism dominates for temperatures such that the Debye mass is larger than the binding energy, while the latter, which we quantify here for the first time, dominates for temperatures such that the Debye mass is smaller than the binding energy. If the Debye mass is of the same order as , our results are in agreement with a recent calculation of the static Wilson loop at finite temperature. For temperatures smaller than , we find new contributions to the potential, both real and imaginary, which may be relevant to understand the onset of heavy quarkonium dissociation in a thermal medium.
The study of heavy quark-antiquark pairs in a thermal medium at temperature has received a lot of attention since it was suggested that quarkonium dissociation due to color screening may be a striking signature of the quark-gluon plasma formation (1). Based on this idea and the assumption that medium effects can be understood in terms of a temperature dependent potential, the problem of quarkonium dissociation has been addressed in terms of potential models with screened temperature-dependent potentials over the past 20 years (see e.g. Refs. (2); (3); (4) for some representative works). A derivation from QCD of the in-medium quarkonium potential has not appeared in the literature so far and expectedly not all medium effects can be incorporated into a potential. A first step towards a QCD derivation of the quarkonium potential at finite temperature has been a recent calculation (5) of the static Wilson loop in the imaginary-time formalism at order . After analytical continuation to real time, the calculation shows a real part, which is a screened Coulomb potential, and an imaginary part that may be traced back to the scattering of particles in the medium carrying momenta of order with space-like gluons, a phenomenon also known as Landau damping. Some applications can be found in (6); (7). First principle calculations of quarkonium properties at finite temperature include calculations of Euclidean correlation functions in lattice QCD and the reconstruction of the corresponding spectral functions using the Maximum Entropy Method. At present, however, a reliable determination of the quarkonium spectral functions from the lattice data appears very difficult due to statistical errors and lattice discretization effects (see discussion in Ref. (8) and references therein).
In this work, we will study static quark-antiquark pairs in a thermal bath in real-time formalism (see e.g. (9)) and in a framework that makes close contact with effective field theories (EFTs) for non-relativistic bound states at (10). In this framework, we will address the problem of defining and deriving the potential between the two static sources, and we will calculate their energy and thermal decay width. We will describe the system for temperatures that range from larger to smaller values with respect to the inverse distance of the quark and antiquark, . In some range, we will agree with previous findings; for temperatures lower than the inverse distance of the quark and antiquark, we will find new contributions to the potential, both real and imaginary, with a non trivial analytical structure. In particular, we will point out the existence of a new type of process that contributes to the quark-antiquark thermal decay width besides the Landau damping.
We will deal with static quarks only. The static case is relevant also for the study of bound states made of quarks with a large but finite mass , like quarkonia, in a thermal medium. Quarkonium is expected to exist in the medium if the temperature and the other thermodynamical scales are much lower than . In this situation, one may consistently integrate out the mass from QCD and expand order by order in . The leading order of the expansion corresponds to QCD with a static quark and a static antiquark. Higher-order corrections in may be systematically included in the framework of non-relativistic QCD (NRQCD) (11).
Bound states at finite temperature are systems characterized by many energy scales. There are the thermodynamical scales that describe the motion of the particles in the thermal bath: the temperature scale (we will not distinguish between and multiple of ), the Debye mass , which is the scale of the screening of the chromoelectric interactions, and lower energy scales. In the weak-coupling regime, which we will assume throughout this work, one has . Moreover, there are the scales typical of the bound state. In the case of a system of two static sources, the scales may be identified with the inverse of the quark-antiquark distance and the static potential, the first being much larger than the second. We will also assume that is much larger than the typical hadronic scale , i.e. we will concentrate on the short-distance part of the potential. This may be the part of the potential relevant for the lowest quarkonium resonances like the or the , which are the most tightly bound states. Thermodynamical and bound-state scales get entangled and different hierarchies are possible. Bound states are expected to dissolve in the bath at temperatures such that is larger than the typical inverse size of the bound state. Hence we will concentrate on the situation (i.e. or ), and distinguish between the two cases and .
The paper is organized as follows. Sections II and III are introductory: they deal with QCD with static sources, which we call static QCD for short, but do not include bound states. In Sec. II, we write the quark and gluon propagators in static QCD at finite . In Sec. III, we summarize one-loop finite contributions in static QCD that are relevant to the present work. In Sec. IV, we introduce the relevant EFTs and calculate the static potential in a situation where the inverse distance between the static quark and antiquark is larger than the temperature of the thermal bath: . When is as small as the binding energy, we also calculate the leading thermal contribution to the static energy and the decay width. In Sec. V, we provide an alternative derivation of the potential in perturbative QCD. In Sec. VI, we calculate the static potential in the situation . In Sec. VII, we summarize and discuss our results and list some possible developments.
Ii Static QCD at finite
We consider here QCD with a static quark and antiquark; in particular, we write the quark, antiquark and the gluon propagators at finite . To simplify the notation, we will drop the color indices from the propagators. Throughout the paper, the complex time contour for the evaluation of the real-time thermal expectation values goes from a real initial time to a real final time , from to , from to and from to . The propagators will be given with this conventional choice of contour. Furthermore, the following notations will be used. We indicate thermal averages as
where is the Hamiltonian of the system. We also define
ii.1 Quark propagator
In order to show the behaviour of static sources in a thermal bath, it may be useful to consider first a quark (or antiquark) with a large but finite mass , , and then perform the limit.
We define the propagators
where is the Pauli spinor field that annihilates the fermion (in the following, the Pauli spinor field that creates the antifermion will be denoted ). The free propagators,
satisfy the equations (in momentum space: ):
where we have neglected corrections of order or smaller: they will eventually vanish in the limit.
If the heavy quarks are part of the thermal bath, they satisfy the Kubo–Martin–Schwinger relation:
From the equal-time canonical commutation relation it follows the sum rule
The spectral density is given by
and the free propagator,
is given in momentum space by
In the static limit , the propagators simplify because for . Moreover, we may get rid of the explicit mass dependence by means of the field redefinition , which amounts to change to in the expressions for the propagators and the spectral density; they read now
The free static propagator is the same as at zero temperature. On the other hand, if we would have assumed from the beginning that , i.e. that there is no backward propagation of a static quark (in agreement with the Kubo–Martin–Schwinger formula in the limit) then, together with the equations of motion , (obtained after removing via field redefinitions) and the sum rule (10), we would have obtained Eqs. (16), (18) and (19).
The real-time free static propagator for the quark reads
and for the antiquark
The main observation here is that, since the component vanishes, the static quark (antiquark) fields labeled “2” never enter in any physical amplitude, i.e. any amplitude that has the physical fields, labeled “1”, as initial and final states. Hence, when considering physical amplitudes, the static fields “2” decouple and may be ignored.
The propagator may be written in a diagonal form as
Throughout the paper, we will use bold-face letters to indicate matrices in the real-time formalism.
ii.2 Gluon propagator
The gluon propagator in the real-time formalism can be written as (9)
Gluons being bosonic fields, the Kubo–Martin–Schwinger relation reads
from which it follows that
is the spectral density.
We may express also in terms of the retarded and advanced propagators and :
In the free case, in Coulomb gauge, the longitudinal and transverse propagators have the following expressions (12):
where . Note that the longitudinal part of the gluon propagator in Coulomb gauge does not depend on the temperature. The temperature enters only the transverse part, which splits in the sum of a piece and a thermal one.
The Lagrangian of QCD with a static quark, a static antiquark and massless quark fields is
where , , and . The free static quark propagator is given by Eq. (20), the free static antiquark propagator by Eq. (21) and the free gluon propagator (in Coulomb gauge) by Eqs. (36) and (LABEL:D0ij). Note that transverse gluons do not couple directly to static quarks.
Iii One-loop finite contributions in static QCD
Throughout this work, we will assume that , ; this enables us to evaluate thermal properties of QCD in the weak-coupling regime. In this section, we consider one-loop thermal contributions to the static quark propagator, quark-gluon vertices and gluon propagator. When the loop momenta and energies are taken at the scale and the external momenta are much lower, so that we may expand with respect to them, these correspond to the hard thermal loop (HTL) contributions (13).
The one-loop contributions to the static-quark self energy, to the static-quark longitudinal-gluon vertex and to the static-quark transverse-gluon vertex are displayed in Figs. 1, 2 and 3 respectively. It is convenient to fix the Coulomb gauge. In that gauge, longitudinal gluons do not depend on the temperature (see Eq. (36)) and the above diagrams do not give thermal contributions. Moreover, if evaluated in dimensional regularization they vanish after expanding in the external momenta. Throughout this work, we will adopt the Coulomb gauge unless stated otherwise.
Momenta and energies of order contribute to the gluon self-energy diagrams. Since only longitudinal gluons couple to static quarks, we will focus on the longitudinal part of the polarization tensor. This will be the only component of the gluon polarization tensor relevant to the paper. Diagrams contributing to the thermal part of the longitudinal gluon polarization tensor in real-time formalism at one-loop order are shown in Fig. 4. In Sec. III.1, we will give a general expression, and in Secs. III.2, III.3, III.4 we will expand it in some relevant limits.
iii.1 The longitudinal gluon polarization tensor
where “R” stands for retarded, “A” for advanced, “F” labels the contribution coming from the loops of massless quarks (first diagram of Fig. 4) and “G” labels the contribution from the second, third and fourth diagram of Fig. 4. is the number of colors and . In the context of the imaginary-time formalism, Eqs. (42) and (43) can be found also in textbooks like (15). The original derivation of (43) is in (16).
The retarded and advanced gluon self energies contribute to the retarded and advanced gluon propagators. From the retarded and advanced gluon propagators we may derive the full propagator, the spectral density and finally all components of the matrix of the real-time gluon propagator along the lines of Sec. II.2. In the following, we study Eqs. (39)-(43) in different kinematical limits.
iii.2 The longitudinal gluon polarization tensor for
iii.3 The longitudinal gluon polarization tensor for
If we assume that all components of the external four-momentum are much smaller than the loop momentum , then we may expand and in . At leading order, we obtain the well-known HTL expression for the longitudinal gluon polarization tensor, which may be found, for instance, in (9):
where is the Debye mass:
The resummed longitudinal gluon propagator
The longitudinal polarization tensor induces corrections to the longitudinal gluon propagator:
Since contains a real and an imaginary part, also acquires a real and an imaginary part.
If the typical momentum transfer is of the order of the Debye mass, , then the series in (49) needs to be resummed:
The resummed longitudinal propagator depends on and has a real and an imaginary part. The Debye mass plays the role of a screening mass for longitudinal gluons whose momenta are such that and . A study of the resummed gluon propagator in the real-time formalism may be found in (17).
The role of the screening mass can be made more evident if we assume further that and expand Eq. (50) in up to order ; then we obtain
where the “” and “” signs refer to the retarded and advanced propagators respectively. The corresponding spectral density is
Following Sec. II.2 and expanding in also the Bose factor, , at leading order in , we obtain that the resummed HTL longitudinal propagator in the real-time formalism is
iii.4 The longitudinal gluon polarization tensor for
The result is real and does not depend on . Moreover, only the gluonic part of the polarization tensor contributes in this limit and at this order. Higher-order real corrections are suppressed by , while higher-order imaginary corrections are exponentially suppressed.
Iv Bound states for
Starting from this section, we shall address bound states made of a static quark and antiquark in QCD at finite . Bound states introduce extra scales in the dynamics, besides and , that we have to account for. The most relevant one is the distance between the quark and the antiquark. Throughout the paper, we will assume that . We will further assume that also the binding energy of the quark-antiquark static pair is larger than .
First, we deal with the situation where the inverse distance of the two static sources is much larger than the temperature: . Under this condition, we may integrate out from static QCD at order by order in . The EFT that we obtain is potential non-relativistic QCD (pNRQCD) in the static limit (18); (19), whose Lagrangian can be written as
The fields and are static quark-antiquark singlet and octet fields respectively, is the chromoelectric field: , and . The trace is over the color indeces. The matching coefficients , , , are at leading order: , , and . Gluon fields are multipole expanded and depend only on the center of mass coordinate; they scale with the low-energy scales (, , , , …) that are still dynamical in the EFT. The dots in the last line stand for higher-order terms in the multipole expansion.
iv.1 Singlet and octet propagators
The free real-time singlet and octet static propagators at finite are similar to the free static quark propagator (20), although singlet and octet are bosons:
iv.2 Non-thermal part of the singlet static potential
The contribution to the singlet static potential coming from the scale can be read from the Lagrangian (55); it is just the Coulomb potential, which in real-time formalism reads
where is a series in : at leading order, . Starting from order , is infrared divergent; these divergences, which appear at zero temperature, have been considered elsewhere (20) and will not matter here.
The matrix in (58) is such that
Together with Eq. (22), this guarantees that
i.e. that, like at , the sum of all insertions of a potential exchange between a free quark and antiquark amounts to the propagator (56).
The singlet static potential does not get only contributions from the scale , but, if the next relevant scales are the thermal ones, it will also get thermal contributions. These will be the subject of the following sections.
iv.3 Chromoelectric correlator
In the paper, it will become necessary to calculate the chromoelectric correlator where is a Wilson line in the adjoint representation connecting the points and by a straight line. Such a correlator enters each time we consider diagrams with two chromoelectric dipole insertions. In the real-time formalism, the chromoelectric correlator is a matrix in the field indices “1” and “2”. We shall write it as
where, at zeroth order in , is the free gluon propagator. In Coulomb gauge, the free gluon propagator has been given in Eqs. (36) and (LABEL:D0ij); since the chromoelectric correlator is a gauge invariant quantity the choice of the gauge does not matter. At zeroth order in , the thermal part of is
Note that for Eq. (62) gives the thermal part of the gluon condensate in the weak-coupling regime:
iv.4 Thermal corrections to the singlet static potential
We calculate now the leading thermal contributions to the static potential assuming for definiteness that and are the next relevant scales after , i.e. that the binding energy is much smaller than . We will further assume that all other thermodynamical scales are much smaller than the binding energy, so that we can ignore them. We recall that from an EFT point of view, only energy scales larger than the binding energy contribute to the potential, which is the matching coefficient entering the Schrödinger equation of the bound state, while all energy scales contribute to physical observables like, for instance, the static energy (10). We will comment on the impact of degrees of freedom with energies and momenta of the order of the binding energy in Secs. IV.5 and IV.6.
The calculation proceeds in two steps. The first step will be performed in Sec. IV.4.1. It consists in integrating out from the pNRQCD Lagrangian (55) modes of energy and momentum of order . This modifies pNRQCD into a new EFT where only modes with energy and momentum lower than the temperature are dynamical. For our purposes, it only matters that the pNRQCD Lagrangian gets additional contributions in the singlet and in the Yang–Mills sectors. In the first sector, it is an additional contribution to the singlet static potential. In the second one, the additional contribution corresponds to the HTL Lagrangian (22).
The second step, which will be performed in Sec. IV.4.2, consists in integrating out from the previous EFT modes of energy and momentum of order . The resulting EFT will only have degrees of freedom that are dynamical at energy and momentum scales lower than the Debye mass. In the singlet sector, the EFT gets modified by a further additional contribution to the static potential.
In summary, if both and are much larger than the binding energy, the real-time Coulomb potential (58), gets two type of corrections . The first one comes from the scale and the other one from the scale .
Contributions from the scale
The leading thermal correction is induced by the dipole terms