Exploring the Tan contact term in Yang-Mills theory

Reliably computing the free energy in a gauge theory like QCD is a challenging and resource-demanding endeavor. As an alternative, we explore here the possibility to obtain the associated thermodynamic anomaly by exploiting its relation to the Tan contact. Optimally, this would reduce the determination of the free energy to a high-precision calculation of two-point correlators. We study this possibility using the lattice and functional methods and compare them to the expected behavior for the SU(2) Yang-Mills case.


I. INTRODUCTION
Thermodynamic observables including, e.g., density correlations are among the most prominent observables that provide information about the phase structure of heavy ion collisions. Their computations rely on access to the bulk thermodynamic information of the system. This information is encoded in the free energy, whose determination at finite temperature and density is of prime interest. At large densities functional approaches such as the Functional Renormalization Group (FRG) and Dyson-Schwinger equations (DSEs) circumvent the eminent sign problem that at present prevents lattice simulations in this regime. However, while the part of correlations and thermodynamics that comes from the matter fluctuations does not pose problems in functional approaches, the access to the thermodynamics of gauge fluctuations poses a formidable challenge beyond perturbation theory. At its root it is related to the relevant momentum-scale running of the thermodynamic potentials such as the free energy. Even though by now functional methods have reached high quantitative precision, such a computation still remains a demanding calculation in terms of resources. This asks for computational approaches that reduce the computational effort.
Such an alternative may be provided by a computation in terms of the Tan contact [1,2], for a discussion in Yang-Mills theory see [3]. Essentially, it boils down to the idea, rehearsed in section II, that the thermodynamic part could be encoded in a simple way in the high-momentum behavior of the two-point correlation functions. These are much simpler to determine reliably. While the extracted part still needs some processing to obtain the free energy, drastic features, e. g. phase transitions, should manifest themselves already directly in the unprocessed data. The aim of the present work is to explore exactly this possibility.
To this end, we use the Landau-gauge gluon propagator of SU(2) Yang-Mills theory at finite temperature. This theory undergoes a second-order phase transition at a well established critical temperature. Furthermore, its free energy is known quite well. It is thus an ideal testbed for a new method. We use for this purpose the gluon propagator as obtained using lattice methods, the functional renormalization group, and Dyson-Schwinger equations. The results are shown in section III.
The Tan contact term is related in a straightforward way to the thermodynamic anomaly. This is exploited in section IV, where in a proof-of-principle style the anomaly is determined. There it will also be discussed what further steps are required to make this a quantitatively competitive approach.
In fact, the results show that interesting features of the thermodynamics are already manifest for the unprocessed data. These results are encouraging in that this is possible and may be a promising future avenue, as is concluded in section V.

II. SETUP
The basic idea of how to transfer the Tan contact term formalism [1,2] from solid state physics and ultracold atoms, e.g. [4][5][6][7][8], to particle physics has been outlined in [3]. It essentially boils down to that the high-momentum behavior of a propagator D(T, p 0 , p), i. e. at momenta p ≫ Λ YM , depending on both the temperature T and the momentum p, should behave essentially as where C(T ) is the Tan contact term, which is the only source of temperature dependence. D 0 is the vacuum propagator and Z is a total normalisation of the propagator. Here it is chosen such that Z D 0 (µ 2 ) = 1/µ 2 , i.e. we choose a temperature-independent renormalization scheme. By construction, the Tan contact term satisfies C(T = 0) = 0 if the fit form (1) describes the propagator perfectly. Note also that the Tan contact in (1) is not RG invariant, it runs with twice the anomalous dimension of the propagator. An RG-invariant form is easily achieved by multiplication with the wave function renormalization squared. As we concentrate on the comparison between functional approaches and the lattice, this is not important for us.
In the present work we consider the gluon propagator. As we are interested in high energies, we set D 0 to be the one-loop resummed propagator D 0 (p 2 , µ 2 ) = 1 which entails Z = 1. The quantity ω also involves the coupling g. To accommodate for different renormalization prescriptions, we fit ω 2 to the zero-temperature propagator for the different methods rather than to use some prescribed value. This approach describes the gluon propagator above 2 GeV at zero temperature for all methods at the 1-2% level. In this regime also the propagator from all methods coincide at this level of precision.
In addition, the thermal gluon propagator splits into a longitudinal chromoelectric one and a transverse chromomagnetic one with respect to the four-velocity of the heat-bath. Accordingly, we use (1) independently for both, thus computing a chromoelectric and chromomagnetic Tan contact, C L (T ) and C ⊥ (T ) respectively. To be in the asymptotic regime, we use only data above | p| > 2 GeV and the zeroth Matsubara frequency, though at these energies the approximation D(T, p 2 0 , p 2 ) ≈ D(T, 0, p 2 0 + p 2 ) holds well anyway [9]. For the lattice case, we use the data from [10] with some additional statistics and two additional lattice discretizations at T /T c = 0.9 and T /T c = 1.1 with an 8 × 40 3 lattice. For the zero-temperature form (2) data from [11] are used. This entails statistical errors on the fit parameters Z and ω, which were propagated to the fit of C(T ). While ω = 0.82 +0.04 −0.03 is essentially βindependent, Z was interpolated for different β values by Z 0 1.50 +0.06 −0.03 (ln β) −1.11 +0.01 −0.04 , where Z 0 is the arbitrarily chosen renormalization prescription at zero temperature at fixed µ = 2 GeV. In all cases fits where done along spatial diagonals, which are least affected by discretization effects at high momenta [11].
For the FRG, we use the results from [12]. The vacuum results yield Z = 2.69Z 0 and ω = 0.795 at µ = 2 GeV. The value for ω agrees well with the lattice result.
We also extract the Tan contact term from DSE results. Although they are obtained from a much simpler truncation than the FRG results, the high momentum behavior is determined sufficiently well to extract the relevant information as shown below. Details of the DSE calculations can be found in Appendix A. Their fit parameters are Z = 1.78Z 0 and ω = 0.752, again in good agreement to the other methods.
The temperatures are taken from the respective works as well, i.e. we did not additionally try to fix any scales independently. This yields agreement of the spatialdiagonal lattice data and the DSE and FRG results from 2 GeV up to 12 GeV at the percent level and thus for the whole range of relevant momenta in this work.

III. RESULTS
At finite temperature, we find that the fits work at the few percent level well in all cases. However, results from the lattice for the two temperatures T /T c = 0.9 and T /T c = 1.1 with ten times more statistics reveal that (1) is insufficient if at that level of statistics a sub-percent fit is desired. Rather, C(T ) needs then to be replaced by some extended form, e. g. C(T ) + p 2 D(T ). A similar result is obtained in the FRG and DSE cases. However, for the present purpose, and without a major effort for creating more statistics for the lattice, we contend ourselves here with fits at the 2-3% level, which at low statistics is also the statistical accuracy of the lattice results, allowing for agreement within errors. Note that for the continuum results the fit stops to work above roughly T /T c ≈ 5. This is expected, as when T becomes larger, eventually screening effects will propagate to larger momenta which are not included in the fit ansatz (1).
The results are shown in Fig. 1. First of all, it is visible that the general agreement between lattice and functional methods is satisfactory, except for the DSE in the transverse case at high temperatures. Also, at this level of statistics no statistically significant dependency on lattice parameters is visible. Then, there are a number of visible trends which are quite different for the transverse and the longitudinal Tan contact term.
The probably most significant one is the difference between the transverse one and the longitudinal one at high temperatures. The transverse one starts to rise from essentially zero somewhere around t = T /T c ≈ 0.8 for the lattice data, levels off shortly after t 1, and stays constant up to t ≈ 3. There is no significant change happening at the phase transition. The functional results switch on smoothly, but follow the same trend. However, above t 2.5, the functional methods yield again a slow rise of the Tan contact term.
The longitudinal one is quite different. Up to t ≈ 1, the lattice results are compatible with zero. There is a slight systematic, though not statistically satisfactory trend to non-zero values above t = 1. However, at large temperatures the Tan contact term rises quicker than quadratically with temperature. Except for the smoothing of the transition, this behavior is also seen in the functional results, this time with no particular impact at t 3.
In comparison to the low-momentum behavior [10,12], this provides a consistent picture. There, also the transverse propagator shows no substantial impact of the phase transition, while the longitudinal one seems to do so. At the same time, the impact at high temperatures is also stronger for the longitudinal one.
This leads us to the following picture: The transverse sector carries non-trivial thermodynamic behavior, which is sensitive to the interactions which create a strongly-  interacting phase above the phase transition for a range of a few T c . The bulk thermodynamics is manifested in the longitudinal degrees of freedom, including both the phase transition and the Stefan-Boltzmann trend at high temperatures.

IV. THE ANOMALY FROM THE TAN CONTACT TERM
While the Tan contact term in solid state physics and ultracold atoms encodes the thermodynamics, it is in itself not yet equivalent to a thermodynamic potential. However, it is linked to the thermodynamic anomaly A(T ), see e.g. [13], wherein β(g) is the β-function and g(T ) the temperaturedependent running coupling evaluated at the temperature. An analogous derivation in Yang-Mills theory faces several intricacies. First of all this concerns the unphysical nature of gluon fields in comparison to that in solid state and ultracold atomic systems. This leads us to negative norm states in the Fock space as well as the occurrence of ghost fields. Accordingly, a Yang-Mills analogue of the relation (3) will involve C ⊥ , C L and C ghost and the respective β-functions β ⊥ , β L , β ghost as well additional normalisation factors. The latter differ in the stronglycorrelated low temperature regime with T T c . Being short of the full resolution of the different ingredients of the Yang-Mills relation we here discuss the chromomagnetic and chromoelectric parts of this relation. They are given by where we will use the same β-function for chromomagnetic and chromoelectric parts and take the normalisation factors to unity. Note that the left-hand side of (4) are related to an observable, the thermodynamic or trace anomaly in Yang-Mills theory. Thus, scheme-dependencies on the righthand side need to cancel, implying that the Tan contact term is scheme-dependent. In addition, the miniMOM or Taylor scheme [14,15] employed in the calculation of the gluon propagators yields a multi-valued β-function and its precise determination in lattice calculations requires high statistics. While the former can be remedied by using the temperature-dependent correct branch, the latter precludes us yet from a full determination within each method separately. Also, as will be seen, the Tan contact term needs to be determined at much higher precision in the low-temperature domain.
However, as a proof-of-principle, we will use here an analytic, temperature-independent coupling motivated by analytic perturbation theory, taking Λ 2 c = 1.21 GeV 2 for the cutoff momentum and Λ 2 YM = 0.81 GeV 2 for the scale. Note that this will necessarily upset the overall scale of the result, as we do not use matched schemes.
The results are shown in figure 2. The lattice results, albeit with large errors, are consistent with the temperature dependence of the anomaly, showing a peak around the phase transition, and a slow decrease towards large temperatures. At low temperatures, where already the Tan contact term is compatible with zero within the er-  rors, so is necessarily the anomaly. Moreover, we deduce from figure 2 that the overall normalisation of (4) is non-trivial as the trace anomaly A YM in Yang-Mills obeys A YM 3, see e.g. [16], while A ⊥/L 10 3 . Both functional results show a quite similar behavior at high temperatures, but tend to have the peak at far too low temperatures. This is likely partly because this temperature regime is in the deep infrared, where the β-function is not dominated by its perturbative behavior. Here, a determination of the β-function in a consistent scheme would likely cure these problems.
Nonetheless, the anomaly shows qualitatively the expected behavior, indicating that the Tan contact term may indeed be a suitable approach to obtain thermodynamic information from propagators.

V. CONCLUSIONS
We have extracted for the first time the Tan contact term for Yang-Mills theory from the gluon propagator. We see that known thermodynamic features, the phase transition, the asymptotic Stefan-Boltzmann behavior, and the strongly-interacting liquid behavior imprint themselves qualitatively in the Tan contact term. We also see that the various effects distribute themselves among the transverse and longitudinal degrees of freedom differently. While the strong-interaction regime above the phase transition seems to be encoded in the chromomagnetic sector, the critical and bulk behavior seems to be carried by the chromoelectric sector. This agrees with observations in the infrared [10]. It has also been shown that it is, in principle, possible to use the Tan contact term to determine the anomaly and thus thermodynamic bulk properties.
The obvious steps to be taken from here are to improve statistics and systematics on the lattice and to compare to further results from other sources, e. g. hard-thermal loop calculations or results from dimensionally-reduced calculations [17,18]. Another issue are contributions from the ghost, which at first sight seems to be inert to temperature [10,12,19,20]. For a reconstruction of the thermodynamic potential in full it is required to find the correct normalisation, a suitable scheme and sufficient precision to determine the anomaly. Finally, an extension to finite density is of high interest. Here, also QCDlike theories without sign problem, e. g. 2-color QCD or G 2 -QCD, could be interesting testing grounds. The truncation used for the calculation of the propagators from their DSEs is described in the following. The equations that were solved are the ones for the ghost and gluon propagators truncated to one-loop without tad-poles. The only remaining higher n-point functions are the ghost-gluon and three-gluon vertices. The former is taken as bare, which is within the context of the present work sufficient since we are only interested in the highmomentum behavior. The deviation from a bare vertex is known to be a bump around 1 GeV which falls off quickly [12,20,21].
The three-gluon vertex plays a crucial role for the gluon propagator. It is not only quantitatively relevant, but also the existence of a solution for the gluon propagator depends strongly on its properties. Here, the following model adapted from Ref. [22] was used for dressing the tree-level tensor, The momentum p 2 is (p 2 + q 2 + (p + q) 2 )/2 and p and q are four-momenta. G and Z T are the ghost and the transverse gluon dressing functions, respectively. The first term in the parentheses determines the IR behavior of the vertex, the second the UV behavior. The term in front of the parentheses accounts for missing perturbative higher loop contributions relevant for the resummed one-loop behavior [23][24][25]. The model contains two scales which are fixed as Λ s = Λ 3g = 0.741 GeV. The integral kernels for the Dyson-Schwinger equations can be found, e.g., in Ref. [26]. Here they were derived with DoFun [27][28][29], and the equations were solved with CrasyDSE [30]. Quadratic divergences in the gluon propagator DSE were renormalized via second renormalization conditions chosen as the value of the propagators at zero momentum [25,31,32]. For the employed truncation this leaves some ambiguity how to select these conditions, but at the scales of relevance here it is expected that such effects are subleading.
The overall scale was set for the lowest calculated temperature by matching the UV tail to FRG results. The relative scales for the other temperatures were set by matching the perturbative couplings.
The resulting dressing functions for the gluon propagators are shown in Fig. 3 for selected temperatures. Clearly, the present truncation cannot capture the IR behavior but reproduces the momentum and temperature dependencies qualitatively. Both, in the FRG and the DSE computation of the propagators the computation of the non-trivial A 0 -background has not been taken into account. This background A 0 = 0 is the equation of motion and is directly linked to the vanishing of the Polyakov loop in the confining phase, [33][34][35][36][37], for perturbative computations within the background see [38].
In [12] it has been argued that this should lead to deviations of the chromo-electric propagators in functional approaches from the chromo-electric lattice propagators (as they are computed on different a background) for temperatures with 0.5 T c T 1.