Determination of the temperature dependence of the up- down-quark mass in QCD
aa r X i v : . [ h e p - ph ] S e p Determination of the temperature dependence of the up- down-quark mass in QCD
C. A. Dominguez and L. A. Hernandez Centre for Theoretical & Mathematical Physics, and Department of Physics,University of Cape Town, Rondebosch 7700, South Africa (Dated: September 27, 2018)The temperature dependence of the sum of the QCD up- and down-quark masses, ( m u + m d ) andthe pion decay constant, f π , are determined from two thermal finite energy QCD sum rules forthe pseudoscalar-current correlator. This quark-mass remains mostly constant for temperatureswell below the critical temperature for deconfinement/chiral-symmetry restoration. As this criticaltemperature is approached, the quark-mass increases sharply with increasing temperature. Thisincrease is far more pronounced if the temperature dependence of the pion mass (determined inde-pendently from other methods) is taken into account. The behavior of f π ( T ) is consistent with theexpectation from chiral symmetry, i.e. that it should follow the thermal dependence of the quarkcondensate, independently of the quark mass. PACS numbers: 12.38.Aw, 12.38.Lg, 12.38.Mh, 25.75.Nq
The method of QCD sum rules (QCDSR) [1] is a wellestablished technique to obtain results in QCD analyti-cally. In particular, it has been widely used to determinethe values of all quark masses [2]-[3], except for the top-quark. This is achieved e.g. in the light-quark sectorby considering QCD sum rules for the pseudoscalar cur-rent correlator, proportional to the square of the quarkmasses. Current precision matches that from e.g. latticeQCD (LQCD)[3]. The extension of QCDSR to finite tem-perature was first proposed in [4], and subsequently usedover the years in a plethora of applications. Of partic-ular relevance are the thermal QCDSR results obtainedin the light-quark axial-vector [5], and vector channel [6],which will be used here. The most appropriate correla-tion function in the determination of quark masses is thepseudoscalar current correlator ψ ( q ) = i Z d x e iqx < | T ( ∂ µ A µ ( x ) ∂ ν A † ν (0)) | >, (1)with ∂ µ A µ ( x ) = m ud : d ( x ) i γ u ( x ) : , (2)and the definition m ud ≡ ( m u + md ) ≃
10 MeV , (3)where m u,d are the quark masses in the M S -scheme at ascale µ = 2 GeV [2]-[3], and u ( x ), d ( x ), the correspondingquark fields. The numerical value of these quark massesat T = 0 is irrelevant, as we shall only determine ra-tios. This has been the standard procedure in thermalQCDSR, always at one-loop order, ever since their in-troduction [4]. The relation between the QCD and thehadronic representation of current correlators is obtainedby invoking Cauchy’s theorem in the complex square-energy plane, Fig.1, which leads to the finite energy sum rules (FESR) [1]-[2] Z s ds π Im ψ ( s ) | HAD = − πi I C ( | s | ) ds ψ ( s ) | QCD , (4) Z s dss π Im ψ ( s ) | HAD + 12 πi I C ( | s | ) dss ψ ( s ) | QCD = ψ (0) , (5)where ψ (0) = Residue [ ψ ( s ) /s ] s =0 . (6)The radius of the contour, s , in Fig.1 is large enoughfor QCD to be valid on the circle. Information on thehadronic spectral function on the left hand side of Eq.(4)allows to determine the quark masses entering the con-tour integral. Current precision determinations of quarkmasses require the introduction of integration kernels onboth sides of Eq.(4). These kernels are used to enhanceor quench hadronic contributions, depending on the inte-gration region, and on the quality of the hadronic infor-mation available. They also deal with the issue of poten-tial quark-hadron duality violations, as QCD is not validon the positive real axis in the resonance region. Thiswill be of no concern here, as we are going to determineonly ratios, e.g. m ud ( T ) /m ud (0), to leading order in thehadronic and the QCD sectors. This has been so far thestandard approach in thermal FESR.To this order, the QCD expression of the pseudoscalarcorrelator, Eq.(1), is ψ ( q ) | QCD = m ud (cid:26) − π q ln (cid:18) − q µ (cid:19) + m ud h ¯ qq i q − q h α s π G i + O (cid:18) O q (cid:19)(cid:27) , (7) Re(s)Im(s)
FIG. 1: Integration contour in the complex s-plane. The dis-continuity across the real axis brings in the hadronic spectralfunction, while integration around the circle involves the QCDcorrelator. The radius of the circle is s , the onset of QCD. where h ¯ qq i = ( − ± from [7], and h α s π G i =0 . ± .
012 GeV from [8]. The gluon- and quark-condensate contributions in Eq.(7) are, respectively, oneand two orders of magnitude smaller than the leadingperturbative QCD term. Furthermore, at finite temper-ature both condensates decrease with increasing T , sothat they can be safely ignored in the sequel.The QCD spectral function at finite T , obtained fromthe Dolan-Jackiw formalism [9], in the rest frame of themedium ( q = ω − | q | → ω ) is given byIm ψ ( q , T ) | QCD = 38 π m ud ( T ) ω [1 − n F ( ω/ T )] , (8)where n F ( x ) = (1 + e x ) − is the Fermi thermal factor.At finite temperature there is in principle an additionalcontribution [4] from a cut centred at the origin in thecomplex energy ω -plane with extension −| q | ≤ ω ≤ + | q | .In the rest frame of the thermal medium ( | q | → δ ( ω ),depending on the correlator. If present, this so-calledQCD scattering term is proportional to the squared tem-perature times the delta function δ ( ω ). In the case ofthe pseudoscalar correlator, Eq.(1), this term is absent.This is due to the overall factor of q in the perturba-tive QCD term in Eq.(7), which prevents the formationof a delta function δ ( ω ). A non-vanishing QCD scat-tering term enters in e.g. the correlator of vector andaxial-vector currents, thus differentiating them from thepseudoscalar correlator. Such a term also appears in thehadronic representation of a current correlator, and itinvolves hadron loops. For instance, in the case of thevector current correlator, the hadronic scattering term isdue to a two-pion loop. In the hadronic sector of thepseudoscalar correlator the scattering term is due to aphase-space suppressed two-loop three-pion contribution, which is negligible in comparison with the pion-pole termIm ψ ( q , T ) HAD = 2 π f π ( T ) M π ( T ) δ ( q − M π ) , (9)where f π = 92 . ± .
02 MeV [10]. Corrections to this
FIG. 2: The thermal quark condensate normalized to its valueat T = 0 from [14] (solid circles), and our fit (solid line). Thephenomenological deconfinement parameter, s ( T ) /s (0), isexpected to follow the quark condensate behaviour [5], exceptpossibly very close to T c . relation arise from the radial excitations of the pion, e.g. π (1300), and π (1800). In the chiral SU (2) × SU (2) sym-metry limit, these states are not Goldstone bosons, sothat their decay constants vanish in this limit. In thereal world these decay constants are at the level of afew M eV . Their contribution to the pseudoscalar cor-relator is only meaningful in precision determinations ofthe light quark masses from QCD FESR [2]. Further-more, we find that the value of s at T = 0 from theFESR is below these resonances, whose already largewidth (Γ ≃ −
600 MeV) will grow even larger with in-creasing temperature. Together with the well establishedfact that s ( T ) decreases monotonically with increasing T , these states can be safely neglected here. We also no-tice that at the end we shall divide all thermal results bytheir T = 0 values.The two FESR, Eqs.(3)-(4), at finite T become2 f π ( T ) M π ( T ) = 3 m ud ( T )8 π Z s ( T )0 s (cid:20) − n F (cid:18) √ s T (cid:19)(cid:21) ds, (10)2 f π ( T ) M π ( T ) = − m ud ( T ) h ¯ qq i ( T ) + 38 π m ud ( T ) × Z s ( T )0 (cid:20) − n F (cid:18) √ s T (cid:19)(cid:21) ds. (11)Equation (11) is the thermal Gell-Mann-Oakes-Rennerrelation incorporating a higher order QCD quark-mass FIG. 3: The ratio of the quark masses m ud ( T ) /m ud (0) as afunction of T from the FESR Eqs.(10)-(11). Curve (a) is fora T -dependent pion mass from [15], and curve (b) is for aconstant pion mass.FIG. 4: The ratio of the pion decay constant f π ( T ) /f π (0) asa function of T from the FESR Eqs.(10)-(11). Curve (a) isfor a T -dependent pion mass from [15], and curve (b) is for aconstant pion mass. correction, O ( m ud ). While at T = 0 this correction isnormally neglected [7], at finite temperature this cannotbe done, as it is of the same order in the quark massas the right-hand-side of Eq.(10). As done in derivingEq.(10), hadronic corrections due to radial excitations ofthe pion have been neglected in Eq.(11).The thermal quark condensate is the order parameter ofchiral-symmetry restoration, i.e. the phase transition be-tween a Nambu-Goldstone to a Wigner-Weyl realizationof SU (2) × SU (2). On the other hand, s ( T ) is a phe-nomenological parameter signalling quark deconfinement[4]. One expects these two parameters to be related ifthe two phase transitions take place at a similar criticaltemperature. In fact, the relation s ( T ) s (0) ≃ (cid:20) h ¯ qq i ( T ) h ¯ qq i (0) (cid:21) / , (12) was suggested long ago from FESR in the axial-vectorchannel [11], and confirmed soon after by a more de-tailed analysis [12]. Notice that while Eq.(12) only in-volves ratios, one still matches the dimensions of the in-dividual parameters. Using current information this re-lation was reconfirmed in [5], using thermal FESR forthe vector-current correlator (independent of the pseu-doscalar correlator, Eq.(1)). Further support for the re-lation, Eq.(12), is provided by LQCD results [13]. Wedo not expect this relation to be valid very close to thecritical temperature, T c , as we are using the thermalquark condensate for finite quark masses, which is non-vanishing close to T c . Using this result on s ( T ) /s (0) asinput in the FESR, Eqs. (10)-(11), together with LQCDresults for h ¯ qq i ( T ) for finite quark masses [14], and inde-pendent determinations of M π ( T ) [15], we can determinethe ratios m ud ( T ) /m ud (0) and f π ( T ) /f π (0). We expectthe latter ratio to be close to h ¯ qq i ( T ) / h ¯ qq i (0). This is be-cause in the Nambu-Goldstone realization of chiral sym-metry the pion mass vanishes as the quark mass M π = B m q , (13)while the pion decay constant vanishes as the quark con-densate f π = 1 B h ¯ qq i (14)with B a constant [16]. This expectation is confirmed bythe results from the FESR as discussed next.The LQCD results for the thermal quark condensate [14]are shown in Fig.2 (solid circles), together with our fit tothese 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 lowtemperatures, rising sharply at T ≃
150 MeV. This riseis far more pronounced for the case of a T -dependent pionmass (obtained from [15]). Beyond T ≃
170 MeV theFESR cease to have real solutions, as s ( T ) approacheszero. Figure 4 confirms the expectation from chiral sym-metry that f π ( T ) /f π (0) should be independent of thethermal behaviour of the pion mass, and follow insteadthe behavior of the quark condensate, Eq.(14).The temperature behaviour of the quark mass deter-mined here is consistent with the expectation that at thecritical temperature for deconfinement the free quarkswould acquire a constituent mass, much bigger than thesmall QCD mass. Acknowledgements
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