Quark masses

Figure 3 compares P(s) of full QCD with Nf = 3 flavors and quark mass ma = 0.05 to the RMT result. In the confinement as well as in the deconfinement phase we observe agreement with RMT up to very high [3 (not shown). The observation that P(s) is not influenced by the presence of dynamical quarks is expected from the results of Ref. (Fox and Kahn, 1964), which apply to the case of massless quarks. Our... 

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. 

The figure represents dynamical quark mass M(p) from a lattice simulation (P. Bowman et.al., 2004). Here solid curve from instantons vacuum model no fitting (D. Diakonov et.al., 1986). So, rescattering of massless quarks on an instanton vacuum leads to the dynamical quark mass M(p), which is perfectly confirmed by lattice calculations. 

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. 

Figure 3. The quasiparticle fit (solid line) of the lattice data [9] (open symbols) for the pressure in QCD with Nj = 2 light flavors. The full symbols, representing data with large quark masses, agree with the results for the pure SU(3) plasma (hatched band) for details see [10].

While the anomaly matching conditions are still in force at nonzero quark chemical potential [32] the persistent mass condition [50] ceases to be valid. Indeed a phase transition, as function of the strange quark mass, between the CEL and the 2SC phases occurs. 

It is quite likely to find dense quark matter inside compact stars like neutron stars. However, when we study the quark matter in compact stars, we need to take into account not only the charge and color neutrality of compact stars and but also the mass of the strange quark, which is not negligible at the intermediate density. By the neutrality condition and the strange quark mass, the quarks with different quantum numbers in general have different chemical potentials and different Fermi momenta. When the difference in the chemical potential becomes too large the Cooper-pairs breaks or other exotic phases like kaon condensation or crystalline phase is more preferred to the BCS phase. 

The Fermi sea of up and strange quarks is shown in Fig. 9. Because of the strange quarks mass, they have different Fermi momenta. Note that the Cooperpairing occurs for quarks with same but opposite momenta. Therefore, at least one of the pairing quarks should be excited away from the Fermi surface, costing some energy. Let us suppose that the Cooper-pair gap opens at p p between two Fermi surfaces, psF < p < pf. 

Although our result precludes breaking of vector symmetries at asymptotic density in the case of three exactly massless quarks [29], it does not necessarily apply to the case when the quark masses are allowed to be slightly non-zero. In that case the results depend on precisely how the limits of zero quark masses and asymptotic density are taken, as we discuss below. 

In [21] the authors investigate the effect of quark masses on the CFL phase. These calculations are done in the asymptotic limit, and are reliable for sufficiently small quark masses. When mu = rnd = m ms (unbroken SU(2) isospin, but explicitly broken SU(3)), one finds a kaon condensate. The critical value of rns at which the condensate forms is m to1/3 Aq 3, where Ao is the CFL gap (see, in particular, equation (8) of the first paper). As kaons transform as a doublet under isospin, the vector SU(2) symmetry is broken in seeming contradiction with our result. 

To investigate spontaneous symmetry breaking, one ordinarily has to start at finite volume and insert a source which explicitly breaks the symmetry. The source is removed only after the infinite volume limit is taken. We stress that the source does not have to be a quark mass (it could be a higher dimension operator), so one can investigate symmetry breaking even when the quark mass is exactly zero throughout the calculation. (To be precise, a quark mass does not explicitly violate vector symmetries, so it cannot play the role of the source in the thermodynamic limit needed here.)... 

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. 

In the framework of the bag model [5], or, in general, under the assumption that the strange quark mass is small [6], one can exclude the 2SC phase in the interior of compact stars when charge neutrality is considered. 

We have seen that the quark mass dependence of ferromagnetism should be important, while we have treated it as an input parameter. When we consider the realization of chiral symmetry in QCD, the quark mass should be dynamically generated as a result of the vacuum superconductivity qq pairs are condensed in the vacuum. We consider here SU(2)l x SU(2)r symmetry. Then Lagrangian should be globally invariant under the operation of any group element with constant parameters, except the symmetry-breaking term... 

One expects the diquark condensate to dominate the physics at densities beyond the deconfinement/chiral restoration transition and below the critical temperature. Various phases are possible. E.g., the so called 2-color superconductivity (2SC) phase allows for unpaired quarks of one color. There may also exist a color-flavor locked (CFL) phase [7] for not too large value of the strange quark mass ms, for 2A > m2s/fiq, cf. [8], where the color superconductivity... 

We will employ parametrisations of the nonlocal quark model which reproduce pion properties mass mn = 140 MeV, decay constant fn = 93 MeV and which have the same quark mass gap (7 = 0, /z = 0) = 330 MeV in the vacuum. The results for the Darameterization are taken from [23],... 

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. 

Elementary particles. Table of quarks (mass is not shown because quarks are never observed alone). 

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