Quark masses current

Abstract. Low-momentum quark determinant and effective action in the presence of current quark mass and external flavor fields is derived. The results of the calculations of various correlators are briefly presented. We conclude that, this approach is a reliable tool for the hadron physics, especially including strange quarks. 

In detyB we observe the competition between current mass m and overlapping matrix element a p2R 3. With typical instanton sizes p 1/3 fm and inter-instanton distances R 1/to, a is of the order of the strange current quark mass, ms = 150 MeV. So in this case it is very important to take properly into account the current quark mass. 

Within this approach it was proposed so called improved effective action which is more properly takes into account current quark masses and satisfies axial-anomaly low energy theorems also beyond the chiral limit (M.M. Musakhanov, 1999) at least at 0(m). 

In the present work we refine the calculations Diow and derive the QCD low-energy effective action not only with an account of current quark masses but also other external V = v + 075 + s + >75 fields, where v = a = 7v and a are vector and axial fields, s... 

In Eq. (37) soft external and a fields, carrying momentum q p l. were assumed. Then, they are present inside of the form-factor F in above mentioned form. If v, a external fields are flavor matrices then form-factor F also becomes matrix Nf x Nf. So, we get the partition function Z[m,V], where W are multi-quark interaction terms in the presence of current quark mass m and external fields V. 

The saddle-point equation leads to the momentum dependent dynamical quark mass Mf(k) = MfF2(k). Mf here is a function of current mass mf (M.M. Musakhanov, 2002). It was found that that M[m] is a decreasing function and for the strange quark with ms = 0.15 GeV Ms 0.5 Mu>d. This result in a good correspondence with (P. Pobylitsa, 1989), where another method was completely applied - direct sum is of planar diagrams. 

Here To = y 11/ is proportional to the unit matrix in flavor space. The quark field ip now contains a third component in flavor space, the strange quark, and consequently the mass matrix rh, see Eq. (4), is equally enlarged by the current strange quark mass, ms, which can in general be different from up and down quark masses. This interaction consists of a U(3)l x U(3)ft-syrnmetric 4-point interaction and a 7 Hooft-type 6-point interaction which breaks the UA (1) symmetry. 

Figure 2. Solutions of the chiral gap and the diquark gap A for the Gaussian model form-factor in the chiral limit (left panel) and for finite current quark mass mo = 2.41 MeV (right panel) at different temperatures T = 0,20,40 MeV for symmetric quark matter and G2 = 0.75 Gi.

We compare results in the chiral limit (mo = 0) with those for finite current quark mass mo = 2.41 MeV and observe that the diquark gap is not sensitive to the presence of the current quark mass, which holds for all form-factors However, the choice of the form-factor influences the critical values of the phase transition as displayed in the quark matter phase diagram (/j,q — T plane) of Fig. 2, see also Fig. 1. A softer form-factor in momentum space gives lower critical values for Tc and at the borders of chiral symmetry restoration and diquark condensation. 

The U-, d-, and s-quark masses are estimates of so-called current-quark masses, in a mass-independent subtraction scheme such as MS at a scale /r 2 GeV. The c-and 6-quark masses are the running masses in the MS scheme. For the 6-quark we also quote the IS mass. These can be different from the heavy quark masses obtained in potential models. 

We see that B is the extension of Lee-Bardeen s matrix B, taking into account the presence of the external fields V and with an account of the quark current mass m without making expansion over current mass m and also extended to a few flavors case. 

Figure 12. Kepler period versus the rotational mass for purely hadronic stars as well as hybrid stars. The following core compositions are considered i) nucleons and leptons (dotted line) ii) nucleons, hyperons, and leptons (dashed line) in) hadrons, quarks, and leptons (solid line). The shaded area represents the current range of observed data.

The value of the maximum mass of neutron stars obtained according to our analysis appears rather robust with respect to the uncertainties of the nuclear and the quark matter EOS. Therefore, the experimental observation of a very heavy (M > 1.6M ) neutron star, as claimed recently by some groups [41] (M ss 2.2 M ), if confirmed, would suggest that either serious problems are present for the current theoretical modelling of the high-density phase of nuclear matter, or that the assumptions about the phase transition between hadron and quark phase are substantially wrong. In both cases, one can expect a well defined hint on the high density nuclear matter EOS. 

The effect was first mentioned by Goldhaber and Sternheimer [109], who speculated on the fine and hyperfine structure of exotic atoms consisting of an atomic nucleus and an fl . The splitting pattern would provide a measurement of the quadrupole moment as well as the magnetic moment of the ft . In ref. [109], a quadrupole moment Q = fm was assumed for numerical illustration, on the grounds that the has a mass comparable to that of the deuteron and hence might also have Q of the same order of magnitude. In fact much smaller values of Q are obtained from current quark models [108,110]. 

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