101
k/m
q 1.51.0 0.5 0.0 0.5 1.0 1.5
i
(k )/
()(k =
0)
0
/m
q|
/m
q= 0
/m
q1.67
( )
(csc) (S + P)
(S + P)adj (V + A)
(V + A) (V A)
(V A) (V + A)adj (V A)adj
0.0 0.5 1.0 1.5 2.0
/m
q1.5 1.0 0.5 0.0 0.5 1.0 1.5
i
( )/
()( = 0)
(QCD) ( ), 0
(QCD) csc, 0
Figure 6.6:The determination of the couplings of the LEM truncation at the transition scale Λ0
in terms of the couplings ¯λ(QCD)(σ-π),Λ
0 and ¯λ(QCD)(σ-π),Λ
0 from the RG flow in QCD is illustrated. Left panel:
Scale dependence of the (dimensionless) renormalized four-quark couplings in the zero-temperature limit for zero quark chemical potential (solid lines) and forµ/m¯q ≈1.67 (dashed lines), normalized to the scalar-pseudoscalar couplingλ(σ-π)(k= Λ0) at the transition scale. The black vertical line depicts a specific choice for the location of the transition scale, here at Λ0/m¯q≈1.67, at which the four-quark couplings are evaluated to serve as initial couplings of the LEM ansatz. Right panel: Four-quark couplings extracted at the transition scale Λ0 as functions of the quark chemical potential, normalized to the corresponding value of the scalar-pseudoscalar coupling in the vacuum limit, see main text for details.
and the behavior observed in Fig.6.1 rather resolves the effect of theµ-dependence of the critical scalekcr. In contrast to that, the four-quark couplings depicted in the right panel of Fig.6.6 are obtained at a fixed RG transition scale Λ0 and allow the comparison across different chemical potentials, thus revealing the actual dependence of the four-quark couplings on the quark chemical potential.
6.2 The equation of state of dense QCD matter
We now employ the QMD-model truncation (6.9), with the couplings determined by the RG flow of the four-quark couplings in QCD at higher scales, to compute the zero-temperature EOS
of isospin-symmetric QCD matter in an RG-consistent way. The effective potential U = Γ/V in the long-range limit k→0 is given by11
U[ ¯φ,{∆¯A}]] = 1
VΓΛ0[ ¯φ,{∆¯A}]−4L(Tk→0→0)(Λ0,φ¯2)−8Mk→0(T→0)(Λ0,φ¯2,|∆¯|2), (6.33) with
1
VΓΛ0[ ¯φ,{∆¯A}] =1
VΓΛ0[ ¯φ,{∆¯A}] + 4L(TΛ0→0)(Λ0,φ¯2)
µ=0+ 8MΛ(T0→0)(Λ0,φ¯2,|∆¯|2)
µ=0
+ 4µ2
∂µ2MΛ(T→0)
0 (Λ0,φ¯2,|∆¯|2)
µ=0
, (6.34)
where the term ∼ µ2 accounts for the renormalization of the diquark chemical potential again. Here, we have introduced the UV cutoff scale Λ0 for the “pre-initial” flow in order to distinguish from the UV scale Λ of the RG flow in QCD. This “pre-initial” flow realizes the RG consistency of the low-energy dynamics as described by the QMD truncation. In our QMD-model study, see Section6.1.2, the analysis of the Λ0-dependence of the zero-temperature EOS showed that the results for the pressure are converged for approximately Λ0/m¯q≈20, in particular in the regime of higher chemical potentials. This convergence indicates that the scale Λ0 is sufficiently large for the considered range of quark chemical potentials such that the condition µ/Λ01 holds. The latter implies that the condition (3.47) holds as well and the RG-consistency criterion is fulfilled. Cutoff artifacts and regularization scheme dependences can thus be considered removed at this scale. Here, based on theses finding, we shall set Λ0/m¯q = 20 in our computation of the EOS.
The effective action ΓΛ0 of our QMD-model truncation, which by itself amounts to the initial effective action at the scale Λ0 in the vacuum limit, is given by
1
VΓΛ0[ ¯φ,{∆¯A}] = 1 2
1 λ¯(QCD)(σ-π),Λ
0
φ¯2−1 2
1
¯λ(QCD)csc,Λ
0
|∆¯|2, (6.35)
with the values ¯λ(QCD)(σ-π),Λ
0 and ¯λ(QCD)csc,Λ
0 determined by the RG flow of the four-quark couplings in QCD at higher scales as described in Section6.1.3. Note that the effective action at the scale Λ0 is given by the expression (6.35) alone only in the vacuum limit and receives otherwise additional contributions in form of cutoff corrections as entailed by the RG-consistency criterion.
With the effective potentialU at hand, the ground state values of the fields ¯φ and ¯∆A are determined by the global minimum of the effective potential. The pressure P is then obtained from the latter as follows:
P =−Uφ¯gs,{∆¯A}gs+Uφ¯gs,{∆¯A}gs
µ=0, (6.36)
where we have normalized the pressure with respect to the pressure in the vacuum limit.
In Fig. 6.7, we show our results (light-red band) for the zero-temperature EOS of
isospin-11 Recall that the scalar fieldφhas been redefined to absorb theYukawacoupling ¯h(σ-π).
6.2 the equation of state of dense qcd matter 175
100 101 102
n
B/n
0 00.2 0.4 0.6 0.8 1.0
P/P
SBFRG, approx.: no diquark gapFRG Chiral EFT N2LO/N3LO pQCD (Nf= 3)
Figure 6.7:PressureP of symmetric nuclear matter normalized by theStefan-Boltzmann pressure of the free quark gasPSB as a function of the baryon number densitynB/n0 in units of the nuclear saturation density as obtained from chiral EFT, FRG, including results from an approximation without taking into account a diquark gap (FRG, approx.: no diquark gap), and perturbative QCD (pQCD), see main text for details.
symmetric QCD matter at intermediate densities. The EOS is given in terms of the pressure as a function of the baryon number densitynB, where the latter is readily obtained from the pressure according to
nB = 1 3
∂P(µ)
∂µ , (6.37)
cf. the thermodynamic relations (2.23) introduced in Section2.1.1. In Fig.6.7, the density is given in units of the nuclear saturation densityn0 and the pressure is normalized by the Stefan-Boltzmann pressurePSB of the free quark gas. To estimate the uncertainties arising from the presence of the scale Λ0 describing the “transition” in the effective degrees of freedom, we vary this scale from Λ0= 450. . .600 MeV. Together with the uncertainty in the RG flow of the four-quark couplings due to fixing of the strong coupling within the experimental error, i.e., the uncertainty related to the values of ¯λ(QCD)(σ-π),Λ
0 and ¯λ(QCD)csc,Λ
0 , this gives rise to the light-red band. Within the band, we show three representative EOSs associated with Λ0 = 450,500,600 MeV depicted by the solid, dashed and dotted red line, respectively. The
extent of the light-red band at high densities is set by the constraintµ≤Λ0.
In Fig. 6.7, we have included results for the EOS in the low-density regime obtained from computations based on chiral EFT interactions. For this regime, chiral EFT represents a powerful framework to describe the nuclear dynamics and interactions within a systematic expansion based on nucleons and pions as the low-energy degrees of freedom [89,90]. Substantial progress has been achieved in recent years in deriving new nuclear forces and computing the EOS microscopically based on nucleon-nucleon (NN), three-nucleon (3N) and four-nucleon (4N) interactions derived within chiral EFT [91–93, 281, 283, 440–444]. In particular, an efficient framework was presented in Ref. [93] to compute the energy of nuclear matter at
zero temperature within many-body perturbation theory (MBPT) up to high orders in the many-body expansion and for general proton fractions. It allows to include all contributions from two- and many-body forces up to N3LO and to explore the connection of properties of matter and nuclei [445]. Moreover, in Ref. [92], a set of NN and 3N interactions was fitted to few-body observables, where all derived interactions led to good saturation properties without adjustment of free parameters. In particular, one interaction of this set was found to also correctly predict the ground state energies of medium-mass nuclei up to 100Sn [446, 447].
In Fig. 6.7, we show the results for the pressure of symmetric nuclear matter up to twice nuclear density based on the set of interactions of Ref. [92] (individual blue lines) as well as the interactions up to N3LO fitted to the empirical saturation point of Ref. [93] (blue bands).
The uncertainty bands at N2LO (light-blue band) and N3LO (dark-blue band) have been determined following the strategy of Ref. [448] and represent the combined uncertainties based on the results at the two cutoff scales Λ = 450 and 500 MeV (see also Ref. [93]). We observe that the results for the pressure as obtained from our FRG analysis at intermediate densities are remarkably consistent with those obtained from the many-body framework based on chiral EFT interactions at lower densities. However, our present approximation does not allow us to reliably compute the pressure at densities smaller than nB .3n0 and is not capable to resolve the exact position of any chiral transition or crossover, since we do not observe a clear dominance pattern in the spectrum of the four-quark couplings in this regime. Moreover, in order to resolve this regime more reliably, the incorporation of fermionic six-point functions associated with baryon dynamics is expected to be necessary.
In the regime of very high densities, the EOS can be calculated using perturbative meth-ods [302–306] owing to the fact that the dynamics is dominated by modes with momenta
|p| ∼µ. This effectively renders the QCD coupling g2s/4π small. Although the ground state is expected to be governed by diquark condensation [98,99,102,103,110], calculations which do not include condensation effects are reliable, provided that the chemical potential is much larger than the scale set by the diquark gap. With respect to this high-density limit, we find that the results from our FRG approach at intermediate densities are also found to be consistent with those from perturbative QCD calculations (light-green band) [306]. We indicate, however, that these results have been obtained in a perturbative calculation at order O(gs4) for isospin-symmetric nuclear matter including the strange quark. In this calculation with massless up and down quarks, the chemical potential of the strange quark has been set to zero. Still, the consideration of three flavors in contrast to our two-flavor computation may restrict a direct comparison.
In our RG study, the gluon-induced four-quark interactions serve as proxies for the various order parameters. The analysis of their RG flows indeed indicate that the ground state is governed by spontaneous symmetry breaking, even at high densities. This can be effectively described by a transition in the relevant degrees of freedom at a finite scale. In order to make contact with perturbative calculations, we drop the running of the four-quark interactions and consider an FRG computation restricted to the running of the quark and gluon wavefunction renormalization factors at leading order in the derivative expansion. From the latter, the dressed quark and gluon propagators are obtained which are then used to compute the pressure.12 In
12 Details of this computation are not part of this thesis and can be found in Ref. [449].
6.2 the equation of state of dense qcd matter 177
1 2 3 4 5 6 7 8 9 10
n
B/n
0 100101 102 103
P
[MeV fm3]FRG, approx.: no diquark gapFRG Chiral EFT N2LO/N3LO DBHF (Bonn A)LEM
NL ,NL DDD3C DD-FKVR KVOR
Figure 6.8:Pressure of symmetric nuclear matter as obtained from chiral EFT, FRG, and perturbative QCD (pQCD), as in Fig.6.7, in comparison with different models (see main text and also Ref. [450]).
this case, the RG flow of the pressure can be followed from high-energy scales down to the deep IR limit without encountering any pairing instabilities as associated with spontaneous symmetry breaking. In Fig.6.7, we show our results for the pressure from this calculation (orange band). We observe very good agreement with recent perturbative calculations [306].
The width of this band illustrates the uncertainty arising from a variation of the regularization scheme and a variation of the running gauge coupling within the experimental error bars at the τ mass scale [38]. Following the pressure toward smaller densities, we observe that our results for the intermediate-density and high-density regime are consistent in the sense that a naive extrapolation of our results at intermediate densities “flows” into the results for perturbative QCD at (very) high densities. However, our findings make apparent that toward the regime of intermediate densities condensation effects eventually become essential.
In Fig. 6.8, we compare our results with different models. These include relativistic mean-field calculations, such as NLρ and NLρδ [451], DD, D3C and DD-F [452] as well as KVR and KVOR [453] (see also Ref. [450]). In addition, we show results of Dirac-Brueckner Hartree-Fock calculations (DBHF) [454] and from a ‘conventional’ LEM, see Ref. [115] and the third example in Section6.1.2.13 At densities up to around twice nuclear saturation density, the different models are compatible with the chiral EFT uncertainty bands at N2LO (but not all at N3LO). At higher densities, however, the pressure obtained from most models is found to be significantly higher than our FRG results. Considering the ‘conventional’ LEM calculation, the values are significantly increased as compared to our present results and tend to overestimate the pressure toward lower densities as well as toward higher densities. In particular, the pressure appears to be inconsistent with perturbative QCD computations at asymptotically high densities, which is also true for the results obtained from most of the other models shown in Fig. 6.8. This comes as no surprise as in conventional LEM studies the applicability in
13 The ‘conventional’ LEM refers to the QMD model with UV cutoff Λ = Λ0= 600 MeV, see Section6.1.2for details.
100 101 102
n
B/n
0 00.1 0.2 0.3 0.4 0.5
c
2 sFRG, approx.: no diquark gapFRG Chiral EFT N2LO/N3LO pQCD (Nf= 3)
10 15 20
0.40 0.41 0.42 0.43
0= 450 MeV 0= 500 MeV
0= 600 MeV
estimated peak position
Figure 6.9: The speed of sound squared c2s as a function of the baryon number density nB/n0 in units of the nuclear saturation density as derived from the pressure shown in Fig.6.7. The inset shows the estimated position of the maximum of the speed of sound.
terms of the range of external parameters such as the quark chemical potential is typically limited. Toward higher densities, the condition µ/Λ 1 becomes violated and the model begins to resolve cutoff artifacts and regularization scheme dependencies.
Let us now turn to the speed of sound which is given by c2s = ∂P
∂ = 1 µ
∂P/∂µ
∂2P/∂µ2 , (6.38)
where we have used the relation= 3µnB−P for the energy density in the zero-temperature limit.14 In Fig. 6.9, we present our results for the speed of sound squaredc2s as derived from the pressure shown in Fig. 6.7. The light-red band, corresponding to the one in Fig. 6.7, is associated with the results from our FRG approach taking diquark condensation into account.
The band describes again the uncertainty estimate obtained from varying the “transition”
scale Λ0 and a variation of the running gauge coupling within the experimental error bars. Its extent at high densities is set by the constraintµ≤Λ0as before. Also, we show once more three representative computations associated with the transition scales Λ0 = 450,500,600 MeV depicted by the solid, dashed and dotted red line, respectively. Toward lower densities, the obtained speed of sound is consistent with the N3LO derived from chiral EFT interactions, while the N2LO uncertainty increases rapidly at densities around twice nuclear saturation density and would constrain only very large values for the speed of sound. Our results for the speed of sound exceed the non-interacting limit c2s = 1/3 (indicated by the dashed gray line in Fig. 6.9), which is expected to be approached from below at asymptotically high densities according to perturbative QCD studies [302–306]. Thus, our findings at intermediate densities suggest that the speed of sound assumes a maximum. In order to estimate the maximal value
14 The pressure as a function of the quark chemical potential was interpolated withChebyshev polynomials in order to numerically compute the second derivative with respect to the chemical potential.
6.3 conclusions 179