# Finite-temperature gluon spectral functions from \(N_f=2+1+1\) lattice QCD

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## Abstract

We investigate gluon correlation functions and spectral functions at finite temperature in Landau gauge on lattice QCD ensembles with \(N_f=2+1+1\) dynamical twisted-mass quarks flavors, generated by the tmfT collaboration. They cover a temperature range from \(0.8\le T/T_C\le 4\) using the fixed-scale approach. Our study of spectral properties is based on a novel Bayesian approach for the extraction of non-positive-definite spectral functions. For each binned spatial momentum we take into account the gluon correlation functions at all available discrete imaginary frequencies. Clear indications for the existence of a well defined quasi-particle peak are obtained. Due to a relatively small number of imaginary frequencies available, we focus on the momentum and temperature dependence of the position of this spectral feature. The corresponding dispersion relation reveals different in-medium masses for longitudinal and transversal gluons at high temperatures, qualitatively consistent with weak coupling expectations.

## 1 Introduction

Understanding the evolution of strongly interacting matter in a heavy-ion collision is one of the most demanding tasks in current theoretical physics [1]. Not only do we need to describe the real-time dynamics of matter in the high-temperature phase of QCD, the quark–gluon plasma, but also its transition to the low-temperature domain of hadrons seen and measured in experiment. In particular around the chiral cross-over transition, estimated on the lattice [2, 3, 4, 5] to occur at \(T_c=155\pm 9\) MeV, it is vital to uncover how the breaking of chiral symmetry and the onset of confinement proceed and how they affect the relevant degrees of freedom in the system.

Transport and thermal properties of strongly interacting matter have been extracted from the correlation functions of mesons, i.e. hadronic observables. For computations based on lattice simulations see e.g. [6, 7, 8, 9, 10, 11, 12, 13, 60]. On the other hand such real-time properties can also be accessed via the spectral functions of the fundamental constituents of QCD, gluons and quarks; see e.g. [14, 19, 20, 21, 22, 23, 24, 25].

In the present study we focus on the gluonic sector. The study of gluon correlation functions in gauge-fixed QCD has garnered interest for quite some time, both with lattice simulations and with functional approaches, for finite-temperature results see e.g. [26, 27, 28, 29, 30, 31] and [32, 33, 34, 35, 36, 37, 38, 39], respectively. The extraction of the corresponding spectral functions is, however, hampered by the fact that gauge-fixed gluon spectra contain non-positive-definite contributions; see e.g. [40]. This in turn defies standard approaches, based on Bayesian inference, such as the Maximum Entropy Method (MEM) [41]. In turn direct extensions [22, 42], modifications of the prior [20], as well as of the data part, such as the introduction of shift functions [21], have been applied in the literature. Over the last few years progress has been made in developing new Bayesian approaches independent from the MEM [43], which recently have also been generalized to non-positive-definite spectra [14]. Some non-Bayesian approaches, such as the Backus–Gilbert method [44] or the Sumudu transformation [45] also allow for the treatment of spectra with negative contributions.

Investigating the spectral properties of gluons serves several complementary purposes; first and foremost it provides a direct and intuitive handle on phenomena, such as the generation of a mass gap in the context of confinement [40, 46] as well as the emergence of thermal masses related to Debye screening.

Secondly gluon spectral functions play a vital role in the self-consistent computation of transport coefficients in functional approaches to QCD [19, 21], such as the functional renormalization group or Dyson–Schwinger approaches. If the gluon spectral function is known, it may serve as input to a closed set of real-time evolution equations for quark and gluon degrees of freedom, from which relevant quantities, such as energy-momentum correlation functions may be computed. Thus, in turn, transport properties become accessible.

From a practical point of view, we are also interested in using lattice gluon spectral functions to validate phenomenological models used in the description of heavy-ion collisions. Some of these rely on a quasi-particle picture for the fundamental constituents of strongly interacting matter, which at high temperature is matched to resummed perturbative predictions from hard-thermal loops. One example in this regard is the parton–hadron string dynamics model [47, 48, 49]. Elucidating the non-perturbative behavior of the gluon spectral function may therefore lead to more refined approximations and a better understanding of the validity of currently used model assumptions.

Single particle properties in QCD are conceptually more difficult to capture than those of e.g. mesons. Quarks and gluons represent color charged fields and thus their correlation functions are not gauge invariant. Hence gauge fixing becomes necessary and one has to carefully understand which of the observed properties is truly physical and which depends on the choice of gauge. Dismissing altogether the study of gauge dependent correlators, however, is a too narrow point of view, as they may still contain gauge independent information. One example is the extraction of the heavy-quark potential from Wilson-line correlators in Coulomb gauge [50, 51, 52]. In that case the gauge independent spectral feature encoding the potential is embedded in a gauge dependent background, which may be cleanly separated.

Inverting Eq. (9) using the simulated lattice correlator data, in order to obtain the spectral functions, represents a well known ill-posed problem, which we will attack via the use of Bayesian inference as laid out in detail in the next section.

## 2 Numerical methods

### 2.1 Lattice simulations and gauge fixing

*P*) contains all plaquettes and the sum (

*R*) all planar rectangles.

An economic procedure dealing with the \(N_f=2+1+1\) case consists in the choice \(a m_{0,l} = a m_{0,h} =\frac{1}{2 \kappa } -4 \) with a common hopping parameter. Tuning to maximal twist means tuning \(\kappa = \kappa _\mathrm{crit}(\beta )\). The critical \(\kappa \) corresponds to the vanishing of the PCAC light-quark mass \(m_\mathrm{PCAC}\) and is determined as a function of \(\beta \) at zero temperature [54].

The bare light-quark (\(\mu _l\)) twisted-mass parameter (in the first doublet) and the two bare heavy-quark twisted-mass parameters \(\mu _\sigma \) and \(\mu _\delta \) (in the second doublet) also need to be tuned (as functions of \(\beta \)) at zero temperature to stay on a line of constant physics, defined by the “pion mass” and by matching masses of hadrons containing strange and charm quarks. For light hadrons this has been performed for the first time for \(\beta =1.90\) and \(\beta =1.95\) in Ref. [54].

For a more detailed description of the simulation setup see Refs. [54, 55].

Properties of the three sets of finite-temperature ensembles used in our study, among them the deconfinement cross-over temperature \(T_{\mathrm {deconf}}\) (defined by the Polyakov loop susceptibility)

ETMC ens. (\(T=0\)) | A60.24 | B55.32 | D45.32 |

tmfT ens. (\(T\ne 0\)) | A370 | B370 | D370 |

\(\beta \) | 1.90 | 1.95 | 2.10 |

\(a \, [\text {fm}]\) | 0.0936 | 0.0823 | 0.0646 |

\(m_\pi \, [\text {MeV}]\) | 364(15) | 372(17) | 369(15) |

\(T_{\mathrm {deconf}}\, [\text {MeV}]\) | 202(3)(0) | 201(6)(0) | 193(13)(2) |

\(N_\tau =N_{q_4}\,\mathrm{range}\) | 4–14 | 10–14 | 4–20 |

### 2.2 Bayesian spectral reconstruction

*O*(1000) bins, while as shown in Table 1 the number of available correlator points ranges over \(N_{q_4}\in [4\ldots 20]\). Hence inverting a discretized Eq. (9)

In the present case of gluon spectra the severity of the ill-posedness of the inverse problem is worsened by the non-positivity of the gluon spectrum. Note that even if the sum rule (10) is implemented in the reconstruction this additional difficulty is not cured. A formal analysis of this issue is currently work in progress [59].

*I*) is proportional to the product of two terms

*I*to influence both terms on the right hand side. The first \(P[D|\rho ,I]=\mathrm{exp}[-L]\) refers to the likelihood probability, which in our case is related to the \(\chi ^2\) fitting functional. The likelihood

*L*measures the quadratic distance between the correlator points corresponding to the test function \(\rho \) and the simulated data \(D_i\)

*L*by itself possesses \(N_\omega -N_{q_4}\) flat directions, which in the Bayesian approach are regularized by introducing the prior probability \(P[\rho |I]=\mathrm{exp}[\alpha S (\omega )]\).

This second term encodes further information we possess about the spectrum, beyond the simulation data, which may take the form of a smoothness condition, a sum rule or in the case of hadronic spectra refer to positive-definiteness. Prior information enters in two ways: on the one hand the functional form of *S* itself encodes part of that information, on the other hand *S*[*m*] conventionally depends on a function \(m(\omega )\) called the default model. By definition *m* corresponds to the correct spectrum in the absence of data, i.e. it represents the unique extremum of *S*. Since Eq. (10) may not be exact at finite lattice spacing we refrain from strictly enforcing the sum rule in the following. As we, however, expect that the area under the spectrum will be close to zero and wish to use an otherwise unbiased default model we choose the function \(m(\omega )\) to vanish identically.

*m*can both take on the value zero, one uses a different measure for deviation between the default model and the spectrum \(r_l=|\rho _l-m_l|/h_l\). The function \(h_l\), absent in the standard BR regulator, corresponds to an additional default-model-like function, which encodes the confidence we have in \(m_l\). \(S_\mathrm{BR}^g\) does not require us to choose a decomposition a priori, since the role of \(m(\omega )\) is unchanged. Furthermore it imprints the form of the default model relatively weakly onto the end results, as its curvature in the region where \(\rho \) differs significantly from

*m*is smaller than in the \(S_{SJ}\) or the quadratic prior (as discussed in [14]).

*m*and of

*h*contribute to the systematic uncertainties of the reconstructed spectrum. Thus their values need to be varied to ascertain, which parts of the spectrum are fixed predominantly by the correlator data. Note also that a hyper-parameter \(\alpha \) has been introduced in the definition of the prior probability, taking into account that we may weight the influence of data and prior information independently from each other. The analytic form of \(S_\mathrm{BR}^g\) allows us to integrate \(\alpha \) out in a straight-forward fashion, assuming full ignorance about its values \(P[\alpha ]=1\)

For the reconstructions to be presented in Sect. 4 we deploy the generalized BR method on a frequency grid \(\omega \in [10^{-3},100]\) GeV divided in \(N_\omega =2000\) bins. To ensure that the convolution in (9) evaluated over such a relatively large frequency interval does not suffer from numerical precision losses, the computations are carried out using 512 bit arithmetic. In order to further improve the stability of the numerical optimization task, we deploy the following prescription for the kernel: \(K(q_4,\omega )=2\mathrm{ArcTan}(\omega )\omega /(q_4^2+\omega ^2)\). This means that instead of \(\rho \) itself, we reconstruct the function \(\rho (\omega )/\mathrm{ArcTan}(\omega )\). To plot and investigate the spectra we then divide out the arctan term. This rewriting of Eq. (9) enforces that the spectrum \(\rho (\omega )\) vanishes at \(\omega =0\). In addition by using the lattice momenta \(q_4\) in the kernel we take into account that the UV region of the Matsubara frequencies on the lattice is affected by the finite lattice spacing, making the reconstruction algorithm converge more quickly.

Since the zero-area sum rule Eq. (10) is derived from the RG running at asymptotically large frequencies not present on a lattice, it is only estabilished in the continuum limit. Thus we here refrain from using it to further constrain the reconstruction. As the sum rule emerges in the continuum limit, we, however, set our initial default model \(m(\omega )\) to zero, while the confidence function \(h(\omega )\) is set to unity.

The robustness and reliability of the spectral reconstruction is ascertained in the following way. There exist two intertwined sources of uncertainty in our approach: on the one hand statistical uncertainty arises from the finite precision inherent in our lattice simulation correlators. On the other hand, due to the fact that only a finite number of datapoints are available, we incur a systematic uncertainty from the necessity to choose the default-model functions \(m(\omega )\) and \(h(\omega )\). How strongly the end result suffers from either statistical or systematic uncertainty can be explicitly checked by performing a Jackknife reweighting, where the spectral reconstruction is performed multiple times on a subset of the available simulation data for the former or by varying the values of the default models for the latter.

In Sect. 4 we will deploy a ten-bin Jackknife. It shows that the available signal to noise ratio in the lattice correlators is sufficiently high for statistical uncertainty to play only a minor role. Instead, as is probed by varying the value of the constant default model between \(m=\{-2,0,2\}\) and varying the confidence function \(h=\{1,2\}\), we find that systematic uncertainty dominates the error budget which will be presented in the sections below as transparent error-bands on the reconstructed spectra. Changing the default model by an absolute value of two might at first appear as a rather small change, but due to the relatively large range of frequencies up to \(\omega _\mathrm{max}=100\) GeV it represent a significant change in the area covered by the function. It is the area under the default model to which the final reconstruction result is susceptible, so that if reconstructed features remain unchanged even after changing the default-model area significantly, we can consider them as robustly encoded in the underlying correlator data.

## 3 Correlation functions

This section is devoted to a presentation and discussion of the computed gluon correlation functions in Landau gauge on \(N_f=2+1+1\) tmfT lattices. All figures below that show correlator data include statistical error estimates. Since the statistics of the tmfT ensembles is relatively high and the gluon propagator in imaginary frequencies does not show an exponential falloff, its signal to noise ratio is good enough so that our errorbars are mostly smaller than the point size used for plotting.

*D*370 ensemble (Table 2), which is both closest to the continuum and spans the broadest temperature range. Selected results from the other ensembles at different \(\beta \) values will be considered where it provides additional qualitative insight (Table 1).

Grid sizes and temperatures in the D370 ensembles used for the computation of the correlation functions below. \(N_{\mathrm{meas}}\) refers to the number of available correlator measurements

| 4 | 6 | 8 | 10 | 11 | 12 | 14 | 16 | 18 | 20 |
---|---|---|---|---|---|---|---|---|---|---|

| 762 | 508 | 381 | 305 | 277 | 254 | 218 | 191 | 170 | 152 |

\(N_s\) | 32 | 32 | 32 | 32 | 32 | 32 | 32 | 32 | 40 | 48 |

\(N_{\mathrm {meas}}\) | 310 | 400 | 120 | 410 | 420 | 380 | 790 | 610 | 590 | 280 |

We will connect to the current literature by carrying out Gribov–Stingl fits to the momentum space correlators and comparing with published results in both quenched QCD, as well as with more recent \(N_f=2\) dynamical QCD simulations. For future reference we also provide a visualization of the zero Matsubara frequency correlators \(D(0,|\mathbf {q}|^2)\), which in the literature have been used to benchmark functional computations in both Yang–Mills and \(N_f=2\) QCD.

*T*.

Best fit parameters and their uncertainty for the Gribov–Stingl fits, applied to the longitudinal (top five rows) and the transversal (bottom five rows) correlator at \(\beta =2.10\)

\(\text {Stingl fits}\) \(T/T_C\) | 3.95 | 2.63 | 1.98 | 1.58 | 1.44 | 1.32 | 1.13 | 0.99 | 0.88 | 0.79 |
---|---|---|---|---|---|---|---|---|---|---|

\(D_L\) – \(d/a^2\) | 0.25(2) | 0.30(2) | 0.48(3) | 0.68(4) | 0.75(5) | 0.82(6) | 0.92(6) | 0.96(7) | 1.25(10) | 1.36(9) |

\(D_L\) – \(r^2 a^2\) | 0.87(19) | 0.60(9) | 0.34(4) | 0.23(3) | 0.20(2) | 0.17(2) | 0.14(2) | 0.12(2) | 0.096(10) | 0.086(7) |

\(D_L\) – \(c / a^2\) | 1132 (82) | 946(47) | 642(28) | 478(26) | 436(24) | 408(25) | 366(22) | 349(22) | 281(18) | 262(15) |

\(D_L\) – \(b a^2\) | 0.88(2) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

\(D_L\) – \(\chi ^2/\mathrm{d.o.f.}\) | 0.78 | 0.74 | 0.75 | 0.83 | 0.85 | 0.84 | 0.86 | 0.86 | 0.90 | 0.91 |

\(D_T\) – \(d/a^2\) | 0.28(1) | 0.35(2) | 0.52(3) | 0.68(5) | 0.75(6) | 0.80(8) | 0.87(8) | 0.91(9) | 0.99(4) | 0.85(7) |

\(D_T\) – \(r^2 a^2\) | 0.21(2) | 0.26(2) | 0.19(2) | 0.15(2) | 0.13(2) | 0.12(2) | 0.11(2) | 0.11(2) | 0.11(3) | 0.12(1) |

\(D_T\) – \(c / a^2\) | 806(20) | 698(23) | 523(27) | 427(26) | 394(29) | 374(27) | 348(27) | 336(30) | 324(9) | 357(23) |

\(D_T\) – \(b a^2\) | 0.411(4) | 0.12(2) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

\(D_T\) – \(\chi ^2/\mathrm{d.o.f.}\) | 0.80 | 0.84 | 0.84 | 0.87 | 0.89 | 0.88 | 0.88 | 0.90 | 0.89 | 0.88 |

### 3.1 Zero-Matsubara frequency correlators

*n*is set to unity, corresponding to the ’quasi-particle’ scenario, as detailed below. The asymptotic behavior of the correlators for large momenta is known perturbatively and exhibits powers of logarithms [40, 46], which we here neglect. In turn this simple description is expected to fail eventually as we move to higher momenta.

The individual parameters inform us about vital aspects of the physics encoded in the correlator. Most easily this is seen for \(b=0\) and \(d=0\), where the form is equivalent to \(D^\mathrm{GS}_{L/T}(q)=\frac{\kappa _1}{(q^2+r^2)^2}\), which encodes a pair of complex-conjugate poles that may be associated with a stable quasi-particle. The dynamically generated mass of such an excitation is then related to the value of the parameter \(m=r\). The presence of the parameter \(b\ne 0\), the so-called Gribov–Zwanziger term, introduces a modification to the pole structure, i.e. destabilizing the quasi-particle, while the \(d\ne 0\) term introduces the possibility for a genuine zero of the analytically continued propagator when the two terms in the numerator cancel. Since here we deploy a fixed-scale approach, the renormalization of the correlators remains unchanged between different temperatures. Otherwise, since the correlator renormalizes multiplicatively, the factor *c* would experience direct contributions from changes in scale.

To determine the parameter values of Eq. (31) we fit the unrenormalized longitudinal and transversal correlators \(D_{L/T}(0,|\mathbf {q}|)\) on the D370 ensembles in the range above \(|q|\sim 0.6\) GeV up to 6 GeV. Except for the highest in the longitudinal and the two highest temperatures in the transversal sector, the fits show a clear preference for a vanishing \(b=0\). The resulting best fit values are listed in Table 3, the data and corresponding fit functions are plotted in Fig. 1. With consistent \(\chi ^2/d.o.f.\) values smaller than unity we find that the Gribov–Stingl form works well in the momentum regime considered here. The fit continues to describe the longitudinal data well even up to \(|q|\sim 10\) GeV, while it seems to deviate from the transversal correlator sizably above \(|q|\sim 8\) GeV. One possible reason may be an earlier emergence of the logarithmic corrections to the rational form in the transversal sector.

Already by eye significant differences in the temperature dependence of the infrared behavior of the longitudinal and transversal correlator are visible in Fig. 1. The former shows a much stronger change to smaller values with increasing *T* compared to the latter, manifest also in the fitted \(r^2\) and *c* values. Since the \(q_4=0\) correlator at vanishing spatial momentum (in that order of limits) provides insight on the inverse screening mass of the theory, screening in the longitudinal (electric) sector, as expected, is more efficient than in the transversal (magnetic) sector. We will come back to the determination of the quasi-particle masses in the context of Bayesian spectral reconstructions in the following sections.

*r*is also consistent with the results obtained from quenched QCD in [18].

The genuine phase transition present in *SU*(3) manifests itself in a change of behavior in all longitudinal fit parameters around \(T=T_C\), while for \(N_f=2+1+1\) the cross-over does not appear to induce a similar feature. On the other hand our \(N_f=2+1+1\) data stops shortly below the deconfinement cross-over transition temperature and we may just not be able to observe a similar change in the parameter behavior without extending the ensembles to smaller temperatures. The transversal sector shows less variation in the parameters, as expected from a naive inspection by eye of the correlators themselves. Neither the quenched data, nor our \(N_f=2+1+1\) data shows significant changes around \(T=T_C\).

*c*, as a renormalization of the mass can affect

*r*. A reanalysis of the temperature dependence using \(N_f=2\) data and appropriately renormalized correlators may thus be illuminating.

As last item in this subsection we plot for completeness in Fig. 3 the full spatial momentum dependence of the longitudinal and transversal correlators at the different available values of \(\beta =2.10,1.95,1.90\) (top, middle, bottom, respectively). They illustrate the approach of the correlators to their \(T=0\) behavior at large momenta such that for \(|\mathbf {q}|^2\gg T^2\) they take on the same values and at the highest momenta shown are virtually indistinguishable.

### 3.2 Finite-Matsubara frequency correlators

We continue with an inspection of the finite Matsubara frequency correlators, several representative ones we have plotted in the panels of Fig. 4. The top row contains the longitudinal correlators, while the bottom row those of the transversal sector. In each panel there are \(N_\tau /2\) curves corresponding to the resolved imaginary frequencies on the corresponding lattices vs. spatial momentum. A clear ordering in imaginary frequencies is present at all temperatures, with the values at higher \(q_4\) being smaller than those at lower imaginary frequencies. Only at \(q_4=0\) is the value at \(|q|\rightarrow 0\) well defined, which is why only the top most data include this point.

The first observation to be made is that indeed at vanishing Matsubara frequency both longitudinal and transversal correlators show clear distinguishable values at different temperatures. For the former the \(q_4=0\) values are strictly ordered by temperature, decreasing in value as *T* increases, while for the latter the values first seem to rise below \(T_C\) and then decrease above \(T_C\). As expected from our previous inspection of the \(q_4=0\) correlators, temperature effects in the transverse sector are milder than in the longitudinal sector.

While not surprising, it is important to note that, for finite imaginary frequencies, in particular above the first Matsubara frequency \(\omega _1=2\pi T\), the datapoints all but collapse onto a single curve, which represents essentially \(T=0\) physics. This phenomenon is even more pronounced at higher momenta, where temperature effects are naturally suppressed by the momentum scale. In anticipation of the extraction of gluon spectral functions this is a stark reminder of the difficulties involved. In essence we will attempt to extract from two or maximally three datapoints, which are temperature sensitive, the full in-medium modification of the gluon spectrum, a challenging proposition. To succeed, it then becomes necessary to determine the minute changes in the datapoints at higher \(q_4\) with very good precision. On the other hand a significant improvement of the spectral reconstruction would result if the regime between the zeroth and first Matsubara frequency, where most of the temperature effects are hidden, could be resolved (first attempts in this direction have been reported in [58]).

At \(T=0\) the conceptual restriction of a finite spacing of imaginary frequencies is non-existent and they may be as finely resolved as computationally feasible. In addition at \(T=0\) the Euclidean correlation functions naturally exhibit an *O*(4) symmetry, which may be used to obtain the correlator values at finite imaginary frequencies by evaluating the correlator at zero imaginary frequencies, while appropriately shifting the finite spatial momentum \(D(q_4,|\mathbf {q}|)\approx D(0,\sqrt{q_4^2+|\mathbf {q}|^2})\). At finite temperature compactified imaginary time starts to play a special role and *O*(4) invariance is not exact anymore. It may nevertheless provide a useful approximation and using spline interpolations along spatial momenta it has been verified in continuum computations. The *O*(4) scaling assumption there applies with less than 10% error up to the first Matsubara frequency and with even less error at the higher frequencies. In turn one may also attempt to access the regime between the zeroth and first Matsubara frequency, which promises improvements in the spectral reconstruction. This experience in the continuum has subsequently motivated the use of the *O*(4) ansatz in lattice studies; see e.g. [20].

### 3.3 The *O*(4) scaling ansatz on the lattice

*O*(4) symmetry is already broken by the finite lattice spacing and the finite box. Therefore we set out here to investigate the validity of the

*O*(4) scaling assumption for the longitudinal (top row) and transversal (bottom row) correlator in Fig. 6. In the left panels we set up a spline interpolation \(\tilde{D}_{L/T}\) (solid yellow line) of the correlator \(D_{L/T}\) for \(\beta =2.10\) at \(T=254\) MeV at vanishing imaginary frequencies (topmost points) along spatial momenta \(|\mathbf {q}|\). The orange and red curves then correspond to this interpolation, evaluated according to \(\tilde{D}_{L/T} (0,\sqrt{q_4^2+|\mathbf {q}|^2})\) at the first three finite available imaginary frequencies. We find that at \(q_4\approx 2 \pi T\) the

*O*(4) ansatz works acceptably well, starting from \(|\mathbf {q}|\approx 1.5\) GeV\(^2\). As expected from experience with continuum computations the

*O*(4) scaling does not work as well for the transversal part at higher imaginary frequencies and starts to deviate from the simulated data already at around \(|\mathbf {q}|=3\) GeV.

Different from the continuum, the finite lattice spacing manifests itself in the breaking of rotational symmetry, which affects the edge of the Brillouin zone most severely. It is then exactly at high spatial momenta where we see deviations from the *O*(4) ansatz appearing. In the right panels we use \(\tilde{D}_{L/T}\) to evaluate the correlator along imaginary frequencies. We find that it provides a smooth curve, which for the four lowest finite imaginary frequencies, available on the lattice, lies quite close to the actual datapoints, while it starts to deviate again towards the edge of the Brillouin zone.

In conclusion, since for the spectral reconstruction very precise correlator data is required and we find that systematic uncertainties due to finite lattice spacing artifacts are manifest in the application of the *O*(4) ansatz to finite-temperature lattice gluon correlators, we do not deploy it further in this study. Instead we will only use the actual computed correlator values along discrete imaginary frequencies for the spectral reconstructions in the next section.

## 4 Reconstructed spectral functions

### 4.1 Low-temperature spectral functions

We start with the longitudinal (left) and transversal (right) spectra at the lowest available temperature \(T=0.152\) GeV plotted in Fig. 7 for different spatial momenta. Lighter colors correspond to higher momenta. These reconstructions correspond to the most robust ones, since the underlying correlators are resolved with the largest number \(N_{q_4}/2=10\) of imaginary frequencies in our ensemble. At the lowest momenta available, the spectrum exhibits a characteristic peak–trough structure. A well defined lowest lying positive peak is followed by a negative valley, which approaches the *x*-axis from below. The amplitude of both structures diminishes as one increases momenta. Consistently the trough in the transverse spectrum is more strongly pronounced than in the longitudinal sector. Their position is clearly correlated with spatial momentum, which we will study quantitatively in the next section.

Qualitatively the observed structures agree with expectations from continuum computations, which predict that the gluon spectrum at large frequencies will be negative and will asymptote to zero [40, 46]. Due to the ill-posed nature of the reconstruction task, the reconstructed spectra, however, show artificial oscillations around the *x*-axis with a monotonously decreasing amplitude, which both precludes both a sensible quantitative comparison to the asymptotic form and to the continuum zero-area sum rule. We note that close to the frequency origin the spectral function may start out flat or even with a small negative slope, related to a possible flattening off of the correlator close to \(q_4=0\) [59].

*T*, we need to make sure that what we observe is genuine in-medium physics and not simply a methods artifact.

To this end we carry out the following crosscheck: We artificially reduce the input data (for a similar test in the context of non-relativistic spectral functions see [11]). At the lowest available temperature, i.e. the \(\beta =2.10\) ensemble with \(N_{q_4}=20\), we discard from the imaginary frequency correlator every 2nd, 3rd or 4th datapoint and feed these thinned out datasets into the reconstruction algorithm. This construction corresponds to carrying out the reconstruction on the *reconstructed correlator* (see also [60]), i.e. the correlator, which ensues if the low-temperature spectrum was present in a system at high temperatures.

In Fig. 9 we plot the reconstructed spectra based on the thinned out correlators at the smallest spatial momentum \(|\mathbf {q}|=0.4\) GeV. The reconstructions are carried out for a single choice of default model \(m=1,h=1\), since we are interested here simply in identifying possible systematic trends induced by the deterioration of the dataset. And as expected, the lower the number of input datapoints, the less structure the reconstructions contain, in particular the strength of the negative trough is strongly dependent on the \(N_{q_4}\) used. On the other hand the position of the lowest lying peak is quite robust in the transversal sector and only for \(N_{q_4}/2=3\) deviates strongly from all other reconstructions. In the longitudinal sector, reducing the underlying correlator data points already leads to a trend of shifting the peak position to lower frequencies when using half the original number.

Keeping these systematics in mind we can now proceed to an investigation of the in-medium modification of the gluon spectra.

### 4.2 Temperature dependence of the spectral functions

We continue by inspecting the outcome of the spectral reconstructions, based on actual in-medium correlators, as shown in Fig. 10. The left panel contains longitudinal, the right panel the transversal ones. In the upper row reconstructions are obtained at the lowest available spatial momentum \(q\approx 0.6\) GeV on the \(\beta =2.10\) lattices, while in the bottom row we show the results for intermediate \(q\approx 1.68\) GeV. The temperature range covered lies between \(T=0.152\ldots 0.381\) GeV and the eight temperatures shown here correspond to those at which the default-model dependence was mild enough for a robust determination of the positive peak feature.

As can be expected from the discussion of the temperature dependence of the correlators, the longitudinal spectra show significantly stronger changes with increasing temperature than the transversal ones. At the same time the in-medium modification (in absolute terms) at low momenta appears as pronounced as at higher momenta. Qualitatively the changes are nonetheless similar in both the longitudinal and the transverse sectors. The lowest lying peak broadens, moves to higher frequencies and diminishes in height. Still at \(T=0.381\) GeV we find clear indications for the presence of such a quasi-particle structure also in the longitudinal sector. For larger temperatures the statistical significance of the results is insufficient and we have refrained from presenting them. The observed behavior is consistent with an increase in the mass of a gluon quasi-particle and thus signals an increase in the strength of screening affecting the interactions mediated by it. At the same time the depth of the negative trough at low momenta appears to weaken. However, its width also seems to broaden and its minimum shifts to higher frequencies.

*x*-axis one would find that the systematic uncertainties make the result compatible with no negative trough present at all, even though the mean value of the reconstructed spectra always shows a regime where it falls below zero. We hence expect that with improved statistics and an increased number of datapoints \(N_{q_4}\) the trough would eventually be recovered.

Our interest, however, lies mainly in determining the properties of the low lying positive peak, since e.g. its position can be interpreted as representing the in-medium dispersion relation \(\omega ^0_{L/T}(|\mathbf {q}|)\) of a gluon quasi-particle, which in turn may become part of a dynamical model of the quark–gluon plasma. The peak width would inform us about the lifetime of such a quasi-particle excitation. However, due to the small number of available correlator datapoints, the observed values of the width are still dominated by reconstruction uncertainties.

Before embarking on a quantitative investigation of the quasi-particle peak, we need to ascertain whether the changes observed here are genuine in-medium effects. It is here that we can utilize the reconstructions based on the sparsened low-temperature datasets, as shown for the longitudinal (top row) and transversal (bottom row) sector in Fig. 11.

In the left column we contrast the actual in-medium reconstructions (solid colored lines) at \(T=0.152\) GeV, \(T=0.305\) GeV and \(T=0.508\) GeV with those from sparsened low-temperature datasets at \(N_{q_4}=20,10,5\) (gray dashed). What is important here is to observe that at \(T=0.305\) GeV (green curve) the in-medium results show a weakened but nevertheless present negative trough. It is shallower than what can be explained simply by the degradation of the spectral reconstruction itself, visible in the gray short dashed curve. The \(T=0.508\) GeV result already carries quite large error-bands, so that no significant difference regarding the negative trough compared to the sparsened data result can be found.

In the right column of Fig. 11 we now turn to the changes observed in the peak position, defined here naively via the topmost point of the reconstructed spectrum. We already saw that the effect of truncating the correlator is a shift to lower frequencies, while the in-medium spectra show a shift to higher frequencies. This difference is quantified here using the colored points, corresponding to the peak positions in the actual \(T>0\) reconstructions, and the red triangles, which denote the position obtained in the sparsened reconstructions.

We conclude that both the observation of a diminishing trough depth and the shift of the positive peak position to higher frequencies with increasing temperature are genuine in-medium effects, which go significantly beyond the uncertainties introduced by the reduced number of datapoints.

### 4.3 In-medium gluon dispersion relation

Having inspected the behavior of the reconstructed spectra qualitatively, we proceed to quantitatively determine position of the first positive peak \(\omega _{L/T}^0\). In the left panel of Fig. 12 we show the peak positions, again defined naively via its topmost point, plotted against spatial momentum for those eight temperatures at which the default-model dependence was mild enough for a robust determination. Both the peak position and the momenta are rescaled by the temperature, which allows a straight-forward comparison of non-trivial differences in the behavior of \(\omega _{L/T}^0\) at different temperatures. The errorbars arise from the variation of the results among changing both \(m(\omega )\) and \(h(\omega )\) as described in Sect. 2.2.

*y*-axis at a non-zero value and that above \(|\mathbf {q}|/T\approx 6\) GeV all of them exhibit an identical behavior within the relatively large systematic errorbars. As the number of correlator points reduces with increasing temperature, the size of the combined systematic and statistical errorbars increases concurrently. In turn we are not able to distinguish differences among the peak positions in the deconfined phase at low momenta. On the other hand in the confined phase, i.e. for the lowest two temperatures, the peak position seems to flatten off at a value above that in the deconfined phase. This difference is probed more quantitatively in the right panel of Fig. 12, where the peak positions for \(T=0.152\) GeV\(<T_c\) and \(T=0.381\) GeV \(>T_c\) are plotted together with a modified free-theory fit, \(\omega _{L/T}^0(|\mathbf {q}|)=A\sqrt{B^2+|\mathbf {q}|^2}\). This simple fit ansatz manages to retrace the values reasonably well and leads to significantly different intercepts, i.e. quasi-particle masses, between the lowest and highest temperature both in the longitudinal sector,

*g*is small and the electric scale of Debye screening \(\sim g T\) is expected to be well separated from the non-perturbative magnetic sector, \(\sim g^2 T\). The corresponding magnetic in-medium mass therefore will be smaller than its electric Debye counterpart.

For comparison purposes we also display (red solid curve) a recent lattice QCD determination of the Debye mass with \(N_f=2+1\) flavors of light HISQ quarks [13, 61], which at \(T=0.381\) GeV coincidentally agrees within errors with the HTL value of \(m_D\) evaluated for four massless flavors at the scale \(\mu =2\pi T\). Even at \(T=0.381\) GeV though, the Debye mass result is consistently smaller than what is observed as quasi-particle mass directly from the gluon spectra. In addition we have included for the longitudinal sector previous estimates [33] for the electric screening mass computed in a Dyson–Schwinger approach including \(N_f=2+1+1\) quark flavors. What we find is that at low temperatures their continuum values differ from our finite lattice spacing value by around 13%, with the functional result lying below the lattice one. At high temperatures the Dyson–Schwinger result agrees very well with the HTL Debye mass estimate and thus also lies below the gluon spectral function result.

## 5 Conclusion

We have presented the first computation of finite-temperature gluon correlation and spectral functions in Landau gauge on full QCD ensembles with \(N_f=2\,+1+1\) flavors of dynamical quarks, generated by the tmfT collaboration and gauge fixed using the cuLGT library on GPU’s. Based on these data we both carried out a Gribov–Stingl fit analysis of the correlators themselves and a Bayesian investigation of the corresponding gluon spectral functions. It is the first Bayesian study in this context, which is independent of the assumption of *O*(4) invariance, containing a systematic error budget. Spectral function reconstructions were performed with a novel Bayesian approach, which generalizes the recent BR method to arbitrary, i.e. non-positive-definite functions.

The outcome of the Gribov–Stingl fits is collected in Table 3 and the corresponding best fit curves plotted in Fig. 1. As was expected from perturbative computations at high temperature, the longitudinal correlators show a much stronger dependence on temperature than the transversal ones. We found that the temperature dependence of e.g. the quasi-particle mass parameter *r* in the fits shows a monotonous increase, which is qualitatively compatible with previous results obtained in quenched QCD. Since previous studies in \(N_f=2\) full QCD used a fixed box approach and did not provide Gribov–Stingl fits on the renormalized propagators, no conclusive comparison could be made.

We further found (see Fig. 5) that, for a fixed momentum, at imaginary frequencies above the first Matsubara frequency \(q_4\approx 2\pi T\) the correlator values lie already very close to their \(T\approx 0\) behavior, while at \(q_4=0\) significant differences between the correlators are manifest. At \(q_4=0\) the correlator will however suffer most severely from the inherent finite extent of the Euclidean axis in standard lattice simulations. We have checked (see Fig. 6) that while interpolating the correlators using the assumption of *O*(4) scaling works reasonably well at small \(q_4\), it degrades towards the boundaries of the Brillouin zone. The interpolation is found to work better on the longitudinal correlators than on the transversal ones but is globally not sufficiently precise to be deployed for the determination of spectral functions.

Our low-temperature reconstructed spectral functions (see Fig. 7) both in the longitudinal and transversal sector, show clear signs of positivity violation at high frequencies. In general we find one well defined positive peak structure at low frequencies, followed by a negative trough at higher \(\omega \). At higher frequencies, the spectrum approaches the frequency axis from below, in qualitative agreement to the continuum asymptotic behavior. Due to the imperfections of the reconstruction process at high frequencies, the spectra however begin to artificially oscillate around the \(\omega \)-axis with a diminishing amplitude.

At higher temperatures (see Fig. 10) the spectral features change such that the positive peak structure broadens, moves to higher frequencies and shrinks, while the negative trough becomes more and more shallow and also moves to higher frequencies. In order to check whether these changes are actual in-medium effects, we carried out a systematic crosscheck based on reconstructions performed on sparsened low-temperature imaginary frequency datasets. It revealed (see Figs. 9 and 11) that the observed modifications of the peak and trough can indeed be attributed to in-medium physics. That is, while the overall form of the reconstructed spectra became more and more featureless, as the number of datapoints was reduced, the position of the lowest lying peak remained relatively stable down to \(N_{q_4}/2=4\), moving slightly towards lower values. On the other hand the actual \(T>0\) reconstructions showed a clear behavior of tending to larger values of \(\omega ^0_{L/T}\).

Since a well pronounced positive peak at low frequencies was identified in all reconstructed spectra, we use its tip as a naive definition of a quasi-particle dispersion relation. The corresponding values plotted against spatial momentum (see Fig. 12) are compatible with a finite intercept at \(|\mathbf {q}|=0\). Such an intercept, determined from a modified free-theory fit, can give a first rough estimate of the quasi-particle mass in the longitudinal and transversal sector (see (33) and (35)). We find that below \(T_c\) the values of this mass appear to be consistent with each other for longitudinal and transversal gluons, while above the deconfinement transition a clear separation of values emerges. This difference is qualitatively consistent with the weak coupling expectation that the magnetic mass should be parametrically smaller than the Debye mass, which screens the electric fields. Comparing with previous results on QCD screening properties, either from the Debye mass extracted from the heavy-quark potential or a direct computation of the electric screening mass in a \(N_f=2+1+1\) Dyson–Schwinger approach, shows values that are consistently smaller than our lattice estimates, the difference being around 13% at low temperatures and close to 28% at the highest temperatures investigated.

Our findings are encouraging: it appears to be possible to reconstruct characteristic features of gluon spectral functions from Landau-gauge lattice QCD correlators with a relatively small number of available frequency points, since the statistics of the ensembles is high. The position of the lowest lying peak structure is one example. To connect to the perturbative high momentum regime, where the signal to noise ratio in the correlators is still weak, will require increasing the statistics further. The quasi-particle peak width on the other hand demands simulations with a significantly larger number of temporal lattice points, i.e. a smaller lattice spacing. Connecting our lattice results on gluon spectra to e.g. the PHSD framework therefore needs to be postponed to future studies. Performing the full continuum extrapolation on the correlators and subsequent reconstructions has to be attempted in a future study as the current ensembles are not tuned for this purpose and thus feature a too coarse temperature resolution.

Already with the currently available data we may attempt to use the reconstructed spectra in a self-consistent computation for transport coefficients in full QCD, which is work in progress.

We are confident that with a further increase of the statistics on the tmfT ensembles and subsequently on the computed correlators the determination of the quasi-particle dispersion relation can be brought to a more robust quantitative level, in particular that it will become possible to resolve more clearly the temperature dependence of its intercept at \(|\mathbf {q}|=0\).

One open problem left is to estimate the influence of the number of light-quark species and of the light-quark mass on the properties of the gluon spectral functions. Considering the ongoing work on thermodynamics for \(N_f=2\,+1+1\) flavors, we will be able in the near future to investigate the case of more realistic light-quark masses (pion masses of close to 200 MeV) for \(N_f=2\,+1+1\) flavors along the lines of the present paper. For a sensible comparison with \(N_f=2\) simulations one would have to return to the previous tmfT datasets and carry out a careful reanalysis of the Gribov–Stingl fits taking into account the renormalization of the quasi-particle masses and widths.

## Notes

### Acknowledgements

E.-M. I. and A. T. are grateful to the members of the tmfT collaboration for years of common work, in particular we thank Florian Burger who has run most of the simulations for the \(N_f=2+1+1\) project. We thank Michel Müller-Preussker and Maria-Paola Lombardo for numerous discussions and continuous interest in gauge-fixed correlation functions and real-time approaches. Landau-gauge fixing and generation of part of the tmfT configurations have been performed on the “Lomonosov” supercomputer of Moscow State University and on the “HybriLIT” cluster of JINR. We are grateful to the MSU Supercomputer Center and the HybriLIT team for extensive computational resources and responsive support. We also thank A. Cyrol for careful reading of the manuscript. This work is supported by EMMI, the grants ERC-AdG-290623, BMBF 05P12VHCTG. It is part of and supported by the DFG Collaborative Research Centre “SFB 1225 (ISOQUANT)”.

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