Determination of the temperature dependence of the up- down-quark mass in QCD

# Determination of the temperature dependence of the up- down-quark mass in QCD

C. A. Dominguez    L. A. Hernandez Centre for Theoretical & Mathematical Physics, and Department of Physics, University of Cape Town, Rondebosch 7700, South Africa
July 26, 2019
###### Abstract

The temperature dependence of the sum of the QCD up- and down-quark masses, and the pion decay constant, , are determined from two thermal finite energy QCD sum rules for the pseudoscalar-current correlator. This quark-mass remains mostly constant for temperatures well below the critical temperature for deconfinement/chiral-symmetry restoration. As this critical temperature is approached, the quark-mass increases sharply with increasing temperature. This increase is far more pronounced if the temperature dependence of the pion mass (determined independently from other methods) is taken into account. The behavior of is consistent with the expectation from chiral symmetry, i.e. that it should follow the thermal dependence of the quark condensate, independently of the quark mass.

###### pacs:
12.38.Aw, 12.38.Lg, 12.38.Mh, 25.75.Nq

The method of QCD sum rules (QCDSR) QCDSR1 () is a well established technique to obtain results in QCD analytically. In particular, it has been widely used to determine the values of all quark masses reviewqmass ()-FLAG (), except for the top-quark. This is achieved e.g. in the light-quark sector by considering QCD sum rules for the pseudoscalar current correlator, proportional to the square of the quark masses. Current precision matches that from e.g. lattice QCD (LQCD)FLAG (). The extension of QCDSR to finite temperature was first proposed in BS (), and subsequently used over the years in a plethora of applications. Of particular relevance are the thermal QCDSR results obtained in the light-quark axial-vector axial (), and vector channel vector (), which will be used here. The most appropriate correlation function in the determination of quark masses is the pseudoscalar current correlator

 ψ5(q2)=i∫d4xeiqx<0|T(∂μAμ(x)∂νA†ν(0))|0>, (1)

with

 ∂μAμ(x)=mud:¯¯¯d(x)iγ5u(x):, (2)

and the definition

 mud≡(mu+md)≃10MeV, (3)

where are the quark masses in the -scheme at a scale reviewqmass ()-FLAG (), and , , the corresponding quark fields. The numerical value of these quark masses at is irrelevant, as we shall only determine ratios. This has been the standard procedure in thermal QCDSR, always at one-loop order, ever since their introduction BS (). The relation between the QCD and the hadronic representation of current correlators is obtained by invoking Cauchy’s theorem in the complex square-energy plane, Fig.1, which leads to the finite energy sum rules (FESR) QCDSR1 ()-reviewqmass ()

 ∫s00dss1πImψ5(s)|HAD + 12πi∮C(|s0|)dssψ5(s)|QCD (5) = ψ5(0),

where

 ψ5(0)=Residue[ψ5(s)/s]s=0. (6)

The radius of the contour, , in Fig.1 is large enough for QCD to be valid on the circle. Information on the hadronic spectral function on the left hand side of Eq.(4) allows to determine the quark masses entering the contour integral. Current precision determinations of quark masses require the introduction of integration kernels on both sides of Eq.(4). These kernels are used to enhance or quench hadronic contributions, depending on the integration region, and on the quality of the hadronic information available. They also deal with the issue of potential quark-hadron duality violations, as QCD is not valid on the positive real axis in the resonance region. This will be of no concern here, as we are going to determine only ratios, e.g. , to leading order in the hadronic and the QCD sectors. This has been so far the standard approach in thermal FESR.

To this order, the QCD expression of the pseudoscalar correlator, Eq.(1), is

 ψ5(q2)|QCD=m2ud{−38π2q2ln(−q2μ2) (7) + mud⟨¯qq⟩q2−18q2⟨αsπG2⟩+O(O6q4)},

where from GMOR (), and from GG (). The gluon- and quark-condensate contributions in Eq.(7) are, respectively, one and two orders of magnitude smaller than the leading perturbative QCD term. Furthermore, at finite temperature both condensates decrease with increasing , so that they can be safely ignored in the sequel.
The QCD spectral function at finite , obtained from the Dolan-Jackiw formalism DJ (), in the rest frame of the medium is given by

 Imψ5(q2,T)|QCD=38πm2ud(T)ω2[1−2nF(ω/2T)], (8)

where is the Fermi thermal factor. At finite temperature there is in principle an additional contribution BS () from a cut centred at the origin in the complex energy -plane with extension . In the rest frame of the thermal medium (), this cut can either lead to a vanishing contribution to a spectral function, or to a delta function of the energy, , depending on the correlator. If present, this so-called QCD scattering term is proportional to the squared temperature times the delta function . In the case of the pseudoscalar correlator, Eq.(1), this term is absent. This is due to the overall factor of in the perturbative QCD term in Eq.(7), which prevents the formation of a delta function . A non-vanishing QCD scattering term enters in e.g. the correlator of vector and axial-vector currents, thus differentiating them from the pseudoscalar correlator. Such a term also appears in the hadronic representation of a current correlator, and it involves hadron loops. For instance, in the case of the vector current correlator, the hadronic scattering term is due to a two-pion loop. In the hadronic sector of the pseudoscalar correlator the scattering term is due to a phase-space suppressed two-loop three-pion contribution, which is negligible in comparison with the pion-pole term

where PDG2014 ().

Corrections to this relation arise from the radial excitations of the pion, e.g. , and . In the chiral symmetry limit, these states are not Goldstone bosons, so that their decay constants vanish in this limit. In the real world these decay constants are at the level of a few . Their contribution to the pseudoscalar correlator is only meaningful in precision determinations of the light quark masses from QCD FESR reviewqmass (). Furthermore, we find that the value of at from the FESR is below these resonances, whose already large width will grow even larger with increasing temperature. Together with the well established fact that decreases monotonically with increasing , these states can be safely neglected here. We also notice that at the end we shall divide all thermal results by their values.

The two FESR, Eqs.(3)-(4), at finite become

 2f2π(T)M4π(T)=3m2ud(T)8π2∫s0(T)0s[1−2nF(√s2T)]ds,
 2f2π(T)M2π(T) = −2mud(T)⟨¯qq⟩(T)+38π2m2ud(T) (11) × ∫s0(T)0[1−2nF(√s2T)]ds.

Equation (11) is the thermal Gell-Mann-Oakes-Renner relation incorporating a higher order QCD quark-mass correction, . While at this correction is normally neglected GMOR (), at finite temperature this cannot be done, as it is of the same order in the quark mass as the right-hand-side of Eq.(10). As done in deriving Eq.(10), hadronic corrections due to radial excitations of the pion have been neglected in Eq.(11).
The thermal quark condensate is the order parameter of chiral-symmetry restoration, i.e. the phase transition between a Nambu-Goldstone to a Wigner-Weyl realization of . On the other hand, is a phenomenological parameter signalling quark deconfinement BS (). One expects these two parameters to be related if the two phase transitions take place at a similar critical temperature. In fact, the relation

 s0(T)s0(0)≃[⟨¯qq⟩(T)⟨¯qq⟩(0)]2/3, (12)

was suggested long ago from FESR in the axial-vector channel CAD0 (), and confirmed soon after by a more detailed analysis Gatto (). Notice that while Eq.(12) only involves ratios, one still matches the dimensions of the individual parameters. Using current information this relation was reconfirmed in axial (), using thermal FESR for the vector-current correlator (independent of the pseudoscalar correlator, Eq.(1)). Further support for the relation, Eq.(12), is provided by LQCD results LQCD2 (). We do not expect this relation to be valid very close to the critical temperature, , as we are using the thermal quark condensate for finite quark masses, which is non-vanishing close to . Using this result on as input in the FESR, Eqs. (10)-(11), together with LQCD results for for finite quark masses LQCD (), and independent determinations of MPIT (), we can determine the ratios and . We expect the latter ratio to be close to . This is because in the Nambu-Goldstone realization of chiral symmetry the pion mass vanishes as the quark mass

 M2π=Bmq, (13)

while the pion decay constant vanishes as the quark condensate

 f2π=1B⟨¯qq⟩ (14)

with a constant CHPT (). This expectation is confirmed by the results from the FESR as discussed next.
The LQCD results for the thermal quark condensate LQCD () are shown in Fig.2 (solid circles), together with our fit to these points (continuous curve). Results from the FESR, Eqs.(10)-(11), are shown in Figs. 3 and 4, respectively. The thermal quark mass is essentially constant at low temperatures, rising sharply at . This rise is far more pronounced for the case of a -dependent pion mass (obtained from MPIT ()). Beyond the FESR cease to have real solutions, as approaches zero. Figure 4 confirms the expectation from chiral symmetry that should be independent of the thermal behaviour of the pion mass, and follow instead the behavior of the quark condensate, Eq.(14).
The temperature behaviour of the quark mass determined here is consistent with the expectation that at the critical temperature for deconfinement the free quarks would acquire a constituent mass, much bigger than the small QCD mass.

Acknowledgements

This work was supported in part by the National Research Foundation (South Africa).

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