In-medium effects on the gauge anomalous dimension

In our study, the anomalous dimension η_A determines the scale dependence of the gauge coupling gs, see Eq. (5.7) and our discussion in Section 5.1. It receives contributions from the gluonic sector as well as from quark fluctuations. For our computations presented in this chapter thus far, we have taken the anomalous dimension η_A as external input from Refs. [392, 393], where these contributions were calculated for all scales and temperatures.

However, the quark fluctuations which contribute to the anomalous dimension are modified at finite density. We briefly discuss in the following some aspects of in-medium effects on the anomalous dimension ηA and the resulting implications for the finite-temperature phase boundary. In order to estimate the influence of such effects, we adopt the following form for the anomalous dimension:

ηA=η_A^YM+ ∆ηA, (5.11)

where the quark contribution ∆η_A is added to the anomalous dimension η^YM_A as obtained in pure YM theory. This approximation has been applied earlier in Refs. [186,187, 193,397], and has also been used in Dyson-Schwinger studies, see, e.g., Refs. [188,426–428]. The pure YM anomalous dimension η^YM_A is again taken from Refs. [392, 393]. Following Refs. [193,429,430], the quark contribution to the gluon anomalous dimension is given by:

∆ηA= −Z_A⁻¹ 3(N_c²−1)V

∂

∂p² p=0

tr











δ^abP_µν^⊥ ·

undefined

1 undefined

1

p, b, ⌫

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¯ g_s

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









, (5.12)

with the transversal projection operatorP_µν^⊥ defined in Eq. (5.3) and the four-dimensional space-time volume V. The diagram represents the expression

undefined

1 undefined

1

p, a, µ

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¯

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δ δA^b_ν(p)

1 2STr

( ∂tRk

Γ⁽²⁾_k +R_k )!

A=0¯ ψ=ψ=0

, (5.13)

where p is the external momentum and the crossed circle depicts the regulator insertion.

As the quark propagators depend on the quark chemical potential, we can thus incorporate in-mediums effects on the gauge anomalous dimension η_A with this contribution. Only the class of diagrams associated with fermionic propagators of equal sign structure in their µ-dependence contributes in Eq. (5.12), corresponding to the blue labels in Fig. 5.2. For the computation of the quark contribution ∆η_A, we employ the three-dimensional Litim regulator (3.31) introduced in Section 3.2 to regularize the fermion loop (5.13).¹¹ In the vacuum limit, the quark contribution (5.12) reduces to the perturbative one-loop result

11 The application of covariant regularization schemes to computations beyond the leading order of the derivative expansion is possible but very difficult due to the non-analyticity at theFermi surface in the zero-temperature limit and is not considered here.

5.3 phase diagram and symmetry breaking patterns 141

∆ηA = g_s²/(6π²). However, employing a different three-dimensional regularization scheme introduces deviating relations of scales as opposed to the covariant regularization scheme used otherwise. We therefore emphasize that we here only aim for a qualitative analysis of in-medium effects on the quark contribution ∆ηA. Moreover, note that the applicability of the approximation (5.12) relies on a mild momentum dependence of the quark contribution, we refer to Ref. [193] for a detailed discussion. Such a mild momentum dependence is guaranteed as long as the quark propagator remains gapped as entailed by, e.g., a finite RG scalek, a thermal mass or non-zero quark masses.

At high temperatures, the quark contribution to the gauge anomalous dimension is sup-pressed as the quarks decouple because of their thermal mass. Consequently, the scale dependence of the gauge coupling is only very mildly modified by the quark chemical potential.

At large RG scales k, the running gauge coupling remains unchanged and approaches the vacuum limit asT /k1 andµ/k1. For increasing quark chemical potential, however, the quark contribution becomes significant in the regime whereµ > T. The gauge coupling starts to increase at larger scales and the location of the maximum of the gauge coupling, cf. Fig.5.1, is more and more dominated by the scale of the chemical potential. For small temperatures and high quark chemical potential, the maximum is located neark∼µ, while its maximum value increases for greater chemical potentials. For the RG scale kapproaching zero at finite temperature, the quarks decouple and the scale dependence of the gauge couplings remains unaffected by the quark chemical potential, i.e., the gauge couplings agree with the running as obtained in pure YM theory.

In Fig.5.7, the finite-temperature phase boundary resulting from a computation based on the running gauge coupling with the quark contribution (5.11) and (5.12) is shown in comparison to the former results shown in Fig.5.4, obtained with the strong couplingα_QCD determined from Eq. (5.7) with the anomalous dimension ηA for Nf = 2 taken from Refs. [392, 393].

We first neglect the dependence on the quark chemical potential by setting µ = 0 in the expression (5.12) in order to estimate the influence of the different three-dimensional scheme used to regularize this particular fermionic loop. The corresponding phase boundary is given by the yellow line in Fig.5.7labeled α_QCD|_∆η

A(µ=0). The phase boundary agrees quite well with the former results depicted by the red line. Only at large quark chemical potential a deviation of approximately 5% is observed. This may be traced back to the fact that the gauge couplingα_QCD from Refs. [392,393] incorporates higher-order quark fluctuation effects as compared to Eq. (5.12), resulting in a decrease in the running gauge coupling owing to fermionic screening. As a consequence, the running gauge coupling determined from Eq. (5.11) with (5.12) is generally stronger. However, the scale fixing procedure anchoring our computation at the critical temperatureTcr(µ= 0) = 132 MeV entails a smaller initial UV value for theα_QCD|_∆η

A(µ=0) coupling and thus partially counteracts the stronger running. As a result, the gauge couplingα_QCD still starts to increase at higher scales and is thus capable of driving the quark sector to criticality even at temperatures slightly above the critical temperature associated with the coupling α_QCD|_∆η

A(µ=0) at a given finite quark chemical potential. In summary, the influence of the three-dimensional regularization scheme appears to be rather mild which, however, might be a consequence of the applied scale fixing procedure.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

/T

0.0 0.2 0.4 0.6 0.8 1.0

T/ T

QCD QCD| A( ) QCD| A( = 0)

Figure 5.7: Phase boundary associated with the spontaneous breakdown of at least one of the fundamental symmetries as obtained from theFierz-complete ansatz (5.1) including dynamic gauge fields. The scale dependence of the gauge couplinggsis computed from Eq. (5.7) with the anomalous dimension ηA for Nf = 2 either taken from Refs. [392, 393] (red line denoted by αQCD) or given by the approximation (5.11), with the quark contribution (5.12) evaluated at zero quark chemical potential (yellow line denoted by αQCD|∆η_A(µ=0)) or at finite quark chemical (green line denoted by αQCD|∆η_A(µ)), see main text for details.

The finite-temperature phase boundary as determined with aµ-dependent quark contribution to the gauge anomalous dimension is depicted by the green line in Fig.5.7labeledα_QCD|_∆ηA(µ). The in-medium effects on the running gauge coupling tend to further increase the critical temperature while the effect is stronger for larger chemical potentials. At µ/T0 = 4.4, the critical temperature is increased by almost 40% compared to the results computed with the coupling α_QCD. An intriguing result is that the dominances along each phase boundary in Fig. 5.7 remain completely unaffected. The solid lines indicate again a dominance of the scalar-pseudoscalar coupling, the dashed lines a CSC dominance and the dotted lines a transition region characterized by a “mixed” dominance pattern. As the differences between the underlying computations concern the gauge sector, this finding is another indicator that the dominance pattern is determined by the dynamics in the quark sector.

At this point, let us emphasize once more that the results of the computation with the quark contribution (5.12) has to be taken with care and that our analysis of such in-medium effects is only qualitative. Foremost, the three-dimensional regularization scheme used in the computation of the quark contribution implies a different relation of scales and therefore the comparability to the computation based on a covariant regularization scheme is limited. As discussed in Sections 3.2 and 4.2.4, three-dimensional regularization schemes lack locality in the temporal direction and potentially amplify effects which are associated with this temporal direction such as the influence of the temperature or the quark chemical potential, see also our discussion of the curvature of the finite-temperature phase boundary at zero quark chemical potential in Section 5.3.2. Thus, the increased critical temperature resulting from a computation with the couplingαQCD|_∆ηA(µ) might be overestimated. Furthermore, the applied scale fixing procedure leads to a critical scale kcr/T0 ≈3.5 in the vacuum limit as compared

5.3 phase diagram and symmetry breaking patterns 143 tokcr/T0 ≈2.6 obtained in a computation with the couplingαQCD. The different approaches

will therefore lead to different values for the low-energy observables, with the consequence that a direct quantitative comparison is limited. Nonetheless, qualitative comparisons are still considered meaningful.