Low-energy effective degrees of freedom

As strong-interaction matter in the zero-temperature limit is governed by spontaneous symme-try breaking associated with the formation of condensates, the four-quark couplings develop a singularity at a finite critical scalek_cr(µ). In order to integrate out the remaining fluctuations in the low-energy regime, we employ an LEM ansatz to resolve the effective bosonic degrees of freedom and thus to account for the formation of the corresponding condensates. The relevant bosonic degrees of freedom are identified from first principles in QCD by making use of the Fierz-complete RG flow of the four-quark couplings as obtained from the ansatz (5.1) discussed in Section5.1. In this approach, the aspect ofFierz completeness is essential in order to determine the low-energy effective degrees of freedom in an unconstrained and unbiased manner. The RG flow is initiated at the UV cutoff scale Λ = 10 GeV. This large value ensures that the conditionµ/Λ1 holds true for the considered range of values of the quark chemical potential to avoid any cutoff and regularization scheme dependences. The four-quark couplings are initially set to zero and are therefore dynamically generated by the gluodynamics as soon as the RG scale kis lowered and fluctuations are integrated out, see our discussion in Section5.2.

The scale dependence of the strong coupling is given by the one-loop running (5.9) which has the advantage of being universal and not depending on details of the gauge fixing process.

Extracted from experimental measurements at the τ mass scale M_τ = 1.78 GeV, the gauge coupling is found to assume the value αs(Mτ) = 0.330±0.014 [38]. Evolved to higher scales on the assumption of the one-loop running, this value corresponds to αs(Λ)≈0.176±0.004

6.1 low-energy dynamics 155 at Λ = 10 GeV, which shall serve as the initial UV value of the strong coupling in our computa-tions. The scale dependence of the gauge coupling as obtained from a computation at one-loop order is completely sufficient for our purpose in this study. Well before theLandaupole (LP) of the running gauge coupling atkLP≈248 MeV shows any effect, the RG flow of the four-quark couplings shall be stopped at a transition scale Λ0 = 450−600 MeV in order to continue with the LEM ansatz to integrate out the remaining low-energy dynamics. The transition scale might be considered as the defined upper boundary of the low-energy sector. The figure in the left panel of Fig.5.1in Section5.1 shows that the running gauge coupling at one-loop order differs only marginally at the relevant scalesk≥450 MeV in comparison to, e.g., αQCD taken from Refs. [392, 393]. Only for scales k.800 MeV, the running gauge coupling in one-loop approximation assumes slightly larger values. In Section5.3, we have tested the influence of using the scale-dependent strong couplingαYM as obtained from Eq. (5.7) withηAtaken from Refs. [392, 393] forN_f = 0, i.e., in pure YM theory. This coupling is similarly increased at the scales 450 MeV≤k.1 GeV, in fact even stronger than the gauge coupling at one-loop order, cf. the right panel of Fig. 5.1. Still, we have found very little influence on the dominance pattern of the four-quark couplings. Our findings suggest that the relative strengths of these couplings are predominantly determined by the dynamics within the quark sector. Moreover, see our discussion below in Section6.1.3, the values of the four-quark couplings, which are extracted from the RG flow at the transition scale Λ0 in order to determine the couplings of the LEM ansatz, are rescaled such that the constituent quark mass is obtained in the vacuum limit. This procedure fixes the scale of the low-energy dynamics and suppresses possible influences which might originate from details of the scale dependence of the strong coupling.

In Fig.6.1, the “hierarchy” of the (dimensionless) renormalized four-quark couplings along the zero-temperature axis is shown as a function of the quark chemical potential. Note that the finite-density region in the zero-temperature limit is governed by spontaneous symmetry breaking, implying that the four-quark couplings always develop a singularity at a critical scalek_cr=k_cr(µ) which itself depends on the quark chemical potential. In order to determine the relative strengths, the couplings are evaluated at a scale slightly higher than the particular critical scale at a given value of the chemical potential. To be specific, the couplings are evaluated at k/k0 =kcr(µ)/k0+ 0.0039, with the critical scalekcr(µ= 0)≡k0 ≈256 MeV in the vacuum limit. Moreover, the values of the couplings are normalized to the couplingλ_(σ-π) of the scalar-pseudoscalar interaction channel at zero quark chemical potential and at the corresponding scalek/k0=kcr(0)/k0+ 0.0039. The relative strengths show a clear dominance of the scalar-pseudoscalar coupling at smaller chemical potentials and of the CSC coupling at higher chemical potentials, indicating the formation of a chiral and a diquark condensate, respectively. Note that the magnitude of the CSC coupling is equal to the magnitude of the (S+P)^adj− -coupling as a consequence of the U_A(1)-symmetric RG flow. Here, we do not take into account UA(1)-violating boundary conditions at the UV cutoff scale Λ as the effect is found to be negligible at the scale Λ0 at which the RG flow of the four-quark couplings is paused in order to continue with the LEM ansatz for the low-energy regime.² Entering the

2 Comparing Figs.5.6and5.9in Chapter5suggests that explicitUA(1)-breaking initial conditions even amplifies the dominances of the scalar-pseudoscalar coupling at small chemical potential and of the CSC coupling at high chemical potential by suppressing the remaining couplings, in particular the (S+P)^adj₋ -coupling. The impact on the absolute values of the scalar-pseudoscalar and the CSC coupling is surprisingly small.

0.0 0.5 1.0 1.5 2.0 2.5

/k

0 1.0

0.5 0.0 0.5 1.0 1.5

i

( )/

()

( = 0)

( - ) (csc)

(S + P)

(S + P)^adj (V + A)

(V + A) (V A)

(V A) (V + A)^adj (V A)^adj

Figure 6.1:Values of the various (dimensionless) renormalized couplings atk/k0=kcr(µ)/k0+ 0.0039 (with the critical scalekcr(µ= 0)≡k0 ≈256 MeV in the vacuum limit) as functions of the quark chemical potential at zero temperature, illustrating the “hierarchy” of the four-quark couplings in terms of their relative strength along the zero-temperature finite-density axis. The finite-density region at zero temperature is governed by spontaneous symmetry breaking, implying aµ-dependent critical scale k_cr(µ) at which the four-quark couplings develop a singularity. In order to evaluate the relative strengths of the four-quark couplings, the RG flow is stopped slightly above the particular critical scale at the given quark chemical potential. The values of the couplings are normalized to the couplingλ_(σ-π)of the scalar-pseudoscalar interaction channel at zero quark chemical potential and k/k₀=k_cr(0)/k₀+ 0.0039, see main text for further details.

regime governed by spontaneous symmetry breaking, however, the U_A(1) symmetry is broken simultaneously. Therefore, it is legitimate to consider an ansatz for the low-energy sector which explicitly breaksUA(1) invariance.

At this point, a comment is in order to put the dominance pattern observed in Fig.6.1 into context with the ones studied earlier, see, e.g., Fig.5.6 in Section5.3. Here, the dominances appear to change from the scalar-pseudoscalar to the CSC channel already at the rather small quark chemical potential µ_χ/k₀ ≈ 0.5. Note, however, that in our present calculation the critical scalek₀ ≈256 MeV in the vacuum limit is significantly smaller compared to the critical scales obtained in our previous studies. As the critical scale sets the scale for the low-energy observables, the small value of k₀ implies that, e.g., the corresponding constituent quark mass is smaller than the typically assumed value. For instance, the mean-field computation in the one-channel approximation discussed in Section 4.1.3, which was used to fix the scale in theFierz-complete NJL model, relates the constituent quark mass m_q = 0.3 GeV to the valuek0≈484 MeV of the critical scale in the vacuum limit (for the regularization scheme also presently applied), which is almost twice as large as the critical scale obtained here. On account of these deviating low-energy scales, a direct comparison of the four-quark couplings as a function of the quark chemical potential is not meaningful. This observation indicates, however, that the four-quark couplings as extracted from the RG flow in QCD in order to determine the couplings of the LEM truncation must be appropriately rescaled to adjust the ansatz to the low-energy scales as given by, e.g., the constituent quark mass, see our discussion

6.1 low-energy dynamics 157 below in Section6.1.3. Alternatively, we may adapt the value of the strong coupling at the

initial scale Λ to adapt the low-energy scales accordingly.³

As indicated by the “hierarchy” of the four-quark couplings in Fig.6.1, toward the IR the dynamics of the quark sector is dominated by either the scalar-pseudoscalar interaction channel or the CSC interaction channel. Since the gluon sector is expected to decouple from the matter sector at lower momentum scales, see, e.g., Refs. [186, 192, 193, 195, 211, 380, 381, 432], and all the remaining four-quark interactions become insignificant in contrast to the dominant channels, we assume the effective action at the transition scale Λ0 to be well described by the (purely fermionic (F)) form

Γ^(QCD)_Λ0 ≡S_LEM^(F) =^Z

x

ψ¯ i∂/−iµγ₀ψ+1

2λ¯_(σ-π)L_(σ-π)+ 1

2¯λ_cscL_csc

, (6.1)

at leading order of the derivative expansion. The effective action Γ^(QCD)_Λ0 at the transition scalek= Λ0 as obtained from the RG flow in QCD at higher scales thus provides the initial form of the LEM truncation. We emphasize again that Λ0 should not be confused withk_cr(µ).

In particular, Λ0 is chosen to beµ-independent in order to simplify scale fixing. Note that in this spirit the renormalized four-quark couplings of Γ^(QCD)_Λ0 become the initial bare couplings from the perspective of the LEM ansatz. However, the pointlike approximation of the four-quark interactions does not allow us to access the regime governed by spontaneous symmetry breaking as the formation of bound states requires to resolve the momentum dependence of the associated vertices. In order to resolve part of the momentum structure, we introduce auxiliary bosonic fields by means of a Hubbard-Stratonovich transformation of the purely fermionic actionS_LEM^(F) given by Eq. (6.1). On the level of the path integral, as discussed in Section4.1.3, this amounts to inserting the identity

1 N

Z

DφD∆D∆^∗ e⁻ R

x

n₁

2m¯²_(σ-π)φ²+ ¯m²_csc∆A∆^∗_A

o

= 1, (6.2)

with appropriate normalizationN. The auxiliary fields can be considered as composites of two quark fields. Phenomenologically, the scalar fieldsφ^T= (σ, ~π^T) carry the quantum numbers of theσ meson,σ ∼( ¯ψψ), and the pions,~π ∼( ¯ψ~τ γ5ψ), respectively. Here, theτi’s are thePauli matrices which couple the quark spinorsψin flavor space. The sigma and the pion field do not carry an internal charge, e.g., color, flavor or baryon number. The complex-valued scalar fields ∆A carry the quantum numbers of diquark states, ∆A∼( ¯ψγ5τ2T^Aψ^C). As introduced in Section2.2, the latter corresponds to a J^P = 0⁺ state, with the color index A referring only to the antisymmetric color generatorsT^Ain the fundamental representation. By shifting the auxiliary fields according to

σ →σ+ i¯h_(σ-π)

¯ m²_(σ-π)

ψψ¯ , ∆A→∆A+ ¯h_csc

¯ m²_csc

i ¯ψγ5τ2T^Aψ^C, (6.3)

3 For QCD in the chiral limit, there is in principle only one parameter, e.g., ΛQCD, and the values of all physical quantities are universal in units of this parameter.

π_i→π_i− ¯h_(σ-π)

¯ m²_(σ-π)

ψγ¯ ₅τ_iψ, ∆^∗A→∆^∗A+ h¯_csc

¯ m²_csc

i ¯ψ^Cγ₅τ₂T^Aψ , (6.4)

and choosing the parameters to fulfill the relations⁴ λ¯_(σ-π)= ¯h²_(σ-π)

¯

m²_(σ-π), 2(−¯λ_csc) = ¯h²_csc

¯

m²_csc , (6.5)

we obtain the partially bosonized action S_LEM =^Z

x

ψ¯i∂/−iµγ₀+ i¯h_(σ-π)(σ+ iγ₅τ_iπ_i)ψ+1

2m¯²_(σ-π)φ²+ ¯m²_csc∆^∗A∆A

+ i¯h_cscψ¯^Cγ₅τ₂∆AT^Aψ+ ¯ψγ₅τ₂∆^∗_AT^Aψ^C

. (6.6)

The four-quark interactions have been replaced by Yukawa-type interactions between the quark fields and the scalar fields φand between the quark fields and the diquark fields ∆A

and ∆^∗_A, associated with theYukawacouplings ¯h_(σ-π) and ¯h_csc, respectively. More precisely, the auxiliary bosonic fields mediate the four-quark interactions which effectively resolve part of the momentum dependence of these interactions. The access to the momentum structure now allows us to integrate out the remaining low-energy dynamics in the IR regime governed by spontaneous symmetry breaking and to explicitly study the formation of condensates.

In our approach presented here, we employ the partial bosonized version in its instant form and compute the effective action in a one-loop approximation where only the purely fermionic loops are taken into account. To be more specific, we neglect the RG running of the wavefunction renormalizations of the meson and diquark fields and set them to zero, i.e., we shall drop terms of the following form in our computation of the effective action:

Z

x

1

2Z_⊥^(φ)∇φ~ ²+1

2Z_k^(φ)(∂_τφ)²+Z_⊥^(∆)(|∆|²)|∇~∆|²+Z_k^(∆)(|∆|²)|∂_τ∆|²

+ 2µZ_µ^(∆)(|∆|²)(∆∂_τ∆^∗−∆^∗∂_τ∆), (6.7) where ∆^∗O∆≡^P_A∆^∗_AO∆A and|O∆|² ≡^P_A|O∆A|² with Obeing some operator acting on the diquark fields. In general, such terms are dynamically generated due to quantum effects, even if only purely fermionic loops are taken into account. Note that the fields in Eq. (6.7) denote the corresponding classical fields associated with the quantum fields appearing in the classical action (6.6). As before, in line with the approximations introduced above, we neglect corrections to the wavefunction renormalization factors of the quark fields (as well as to the quark chemical potential) and also the RG runnings of the Yukawa-type couplings. The latter can thus be absorbed into the fields by appropriate redefinitions, leading to the mappings

h¯_(σ-π)φ→φ , ¯hcsc∆A→∆A, ¯hcsc∆^∗A→∆^∗A. (6.8)

4 Note that the four-quark coupling ¯λcsc always assumes negative values in the RG flow, see, e.g., Fig.6.1, implying that the mass parameter ¯m²cscis positive - if we assume theYukawacoupling ¯hcscto be positive as well - as it should be for the identity (6.2) to be well-defined.

6.1 low-energy dynamics 159 This allows us to recast the initial form of the LEM truncation into a formulation in terms

of the four-quark couplings again which directly connects to the effective action Γ^(QCD)_Λ0 as obtained from the QCD flow at the transition scale Λ0:

Γ^(LEM)_Λ0 =^Z

x

ψ¯ i∂/−iµγ0+ i (σ+ iγ5τiπi)ψ+1 2

¯ 1

λ_(σ-π)φ²−1 2

¯1

λ_csc∆^∗A∆A

+ i ¯ψ^Cγ₅τ₂∆AT^Aψ+ i ¯ψγ₅τ₂∆^∗AT^Aψ^C

. (6.9)

With this ansatz we can eventually integrate out the remaining fluctuations in the low-energy regime. From the effective action determined in the one-loop approximation, we can then derive the Ginzburg-Landau-type effective potential for the bosonic fields which allows a straightforward analysis of the ground-state properties. The formation of a chiral or a diquark condensate corresponds to the chiral fieldφ or the diquark fields ∆A acquiring a non-zero ground-state expectation value, respectively. The details of this computation are discussed in the next section before we proceed to combine the LEM truncation (6.9) with the RG flow in QCD at higher scales.

Im Dokument Phase Structure and Equation of State of Dense Strong-Interaction Matter (Seite 164-169)