Quark Masses

and have been calculated at one loop level[86]. The results are listed in Table 4.3.

More reliable renormalization constants have to be determined non-perturbatively. In the vector and axial-vector channel the validity of the Ward identities, Eq.(2.19), can be used as a criterion to determine the renormalization constantsZ_V, Z_A and b_V. Using the Schr¨odinger functional scheme, L¨uscher et. al.[90] calculated these factors near zero quark mass at various couplings for 0 < g² ≤ 1 and reached an error of less than 1%. Fitting these data leads to the parameterizations

Z_V = 1−0.7663g²+ 0.0488g⁴

1−0.6369g² (4.10)

Z_A = 1−0.8496g²+ 0.0610g⁴

1−0.7332g² (4.11)

b_V = 1−0.6518g²−0.1226g⁴

1−0.8467g² . (4.12)

In these channels, the non-perturbatively determined renormalization constants are used from now on, whereas in the other channels we rely on the values obtained from the TI perturbation theory presented above. A complete list of these perturbatively and non-perturbatively determined constants for the particular couplings and quark masses can be found in Appendix B. As can be seen there, they differ at most by 2% in those channels where both values are available. This can be used also as a guideline for error estimates in the other channels, where non-perturbative values are not available.

4.3 Quark Masses

As explained in Section 2.3.1, the quark mass can be calculated in two ways: One may make use of the vanishing pion mass in the chiral limit to compute the additive renormalization term which appears due to the explicit chiral symmetry breaking of the Wilson action. The other one utilizes the axial Ward identity. Their connection and a possible temperature dependence in this observable is discussed in this section.

The first way to compute the additive renormalization is to define the chiral limit at the quark mass value where the pion mass vanishes. Extrapolating the pion mass to this regime by utilizing the GMOR relation, Eq.(1.19),

m²_π(m) ∝ m_b (4.13)

defines the appropriate subtraction from the quark mass m_b(a) ≡ m−m_c with m_c = 1/2κ_c(a) at given m = 1/2κ. This is obviously valid only below the chiral symmetry restoration temperature. In the high temperature regime, the quark mass has to be defined differently. Here one can make use of the axial Ward identity. From this one can determine a quark mass (see Sections 2.3.1 and 2.3.2),

2m_{AW I}(a) = P

xh∇4(A₄(x, τ) +ac_A∇4P(x, τ))P^†(0)i P

xhP(x, τ)P^†(0)i , (4.14)

0.00 0.01 0.02 0.03 0.04 0.05

0.0 0.1 0.2 0.3 0.4 0.5

am_AWI

zT,τT N_σN_τ d 32 16 z 32 16 τ 16 16 z 16 16 τ

Figure 4.1: Quark mass forβ = 6.136 andκ= 0.1346 for different directions and different volumes

in which the discretization effects are shifted to full order O(a²)¹, if the improvement constant c_A for the SW-action is chosen appropriately (see Eq.(2.40)). Here only the fourth component of the axial current is used which further enhances the overlap with the pion state. Eq.(4.14) is an operator identity and should be, apart from possible lattice artefacts, valid for all distances. That makes it possible to extract the mass from both directions, temporal and spatial, where for the spatial direction the sum over the so called funny space, which consists of the two space directions x and y and the time direction τ, is taken instead. As shown in Fig. 4.1, for 0.6Tc as an example, they indeed coincide in the large distance regime, independent of the volume and direction. It has also been checked that the same holds true aboveT_c. The plateau in the spatial direction is in any case longer than the temporal one and will be used from now on to determine the quark mass. Also note the large signal to noise ratio in the figure which is typical and allows a reliable determination of the quark mass.

In Table 4.5 the results in the hadronic phase are listed.² They show a significant volume dependence but this effect is only about 4%. Therefore, neglecting this effect and using only the 32³×16 lattice, where Eq.(4.14) has been evaluated for different quark masses, one can extrapolate to the chiral limit, i.e., the κ parameter where the quark mass vanishes.

This leads to 0.135586(58) at 0.9Tc and to 0.135772(4) at 0.6Tcwhich is in good agreement with the definition ofκ_cfrom Eq.(4.13) and the interpolated values from [40]. Hence, below T_c no temperature effect can be oberved in this quantity.

Both definitions are related to the renormalized quark mass m_R with m_R = Z_A(1 +b_Aam_b)

Z_P(1 +b_Pam_b)m_{AW I} (4.15)

= Z_m(1 +b_mam_b)m_b . (4.16)

This leads to the relation

mAW I = Z{1 + [bm+ (bP −bA)]amb}mb (4.17)

1As pointed out in [91], one has to take care of the discretizing the derivative. Therefore their improved lattice derivative with errors ofO(a⁴) is used for this equation from now on.

2The necessary operators have not been computed for all parameters listed in Table 4.1, because of the limited programming memory of the APE machines.

4.3. Quark Masses 71

β T /Tc N_σ³×Nτ κ m_{AW I}(a) m_MS(¯µ)[MeV]

16³×16 0.13460 0.0315(1) 101.7(3) 16³×16 0.13540 0.0101(2) 30.7(6) 32³×16 0.13300 0.0771(1) 233.2(3) 6.136 0.55 32³×16 0.13400 0.0494(1) 148.5(3) 32³×16 0.13460 0.0326(1) 98.4(3) 32³×16 0.13495 0.0226(1) 67.4(3) 32³×16 0.13540 0.0097(2) 29.4(6) 16³×16 0.13450 0.0337(1) 176(1) 32³×16 0.13300 0.0761(2) 389(1) 32³×16 0.13400 0.0476(2) 245(1) 6.499 0.93 32³×16 0.13460 0.0302(2) 158(1) 32³×16 0.13531 0.0094(3) 48(2) 32³×16 0.13540 0.0066(3) 34(2) 48³×16 0.13300 0.0758(1) 387(1) 48³×16 0.13460 0.0301(2) 157(1)

Table 4.5: Quark masses as obtained from axial Ward identity belowT_c in the MS scheme at ¯µ≈2GeV.

β T /T_c κ N_σ³×N_τ m_{AW I}(a) m_MS(¯µ)[MeV]

7.457 6.0 0.13390 32³×8 0.01045(3) 182.9(44) 7.457 3.0 0.13390 64³×16 0.003545(2) 62.0(4) 7.457 3.0 0.13390 32³×16 0.003715(17) 65.0(0) 7.192 3.0 0.13440 48³×12 0.00373(7) 46.8(8) 6.872 3.0 0.13495 32³×8 0.0095(1) 79.7(8) 7.192 1.5 0.13440 64³×24 0.00172(3) 21.6(3) 6.872 1.5 0.13495 64³×16 0.002155(7) 18.1(5) 6.640 1.5 0.13536 48³×12 0.003115(17) 19.4(10) 6.640 1.5 0.13525 48³×12 0.00635(3) 39.6(19) 6.499 1.25 0.13558 48³×12 0.0030(1) 15.6(5)

Table 4.6: Quark masses aboveT_c from the axial Ward identity.

with Z = Z_mZ_P/Z_A which is valid up to order a². The coefficients b_m, b_P −b_A and Z are calculated non-perturbatively by Guagnelli et. al.[92]. Their equivalence in the limit g, a → 0 becomes obvious by using the perturbative expansion Z = 1 +O(g²), b_m =−0.5 +O(g²) and Eq.(4.9).

The great advantage of defining the quark mass through the current quark mass is its validity also above Tc. The results are listed in Table 4.6. Now one can utilize Eq.(4.17) to calculate the critical quark mass also above T_c from only a single current quark mass.

The result is summarized in Fig. 4.2. The zero temperature points are taken from [40], where a 16³×32 lattice has been used and the critical hopping parameter was determined from the vanishing of the axial Ward identity. The solid line is a smooth interpolation between them to guide the eye. The simulatedκ values, used here, are marked with grey

0.1330 0.1335 0.1340 0.1345 0.1350 0.1355 0.1360

6.4 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 β κ_c(β)

V κ

T/T_c 0.0 1.2 1.5 3.0 6.0 Sim

Figure 4.2: Critical hopping parameter versus β obtained from Eq.(4.17) for different temperatures and couplings.

crosses. Obvious is a shift to largerκ values for higher temperatures at constantβ. This difference decreases for largerβ as it should become vanish in the free theory. To exclude that this is due to possible large O(am_q) effects, two different κ values with the same temperature at β = 6.640 have been used to determine κc. As one can see, the critical hopping parameter obtained from them agree with each other. The same holds true for possible volume effects. For β = 7.457, two different volumes with otherwise equivalent parameters have been used and no significant difference could be observed. A unique interpretation of this result is, however, not possible with the present data. The first reason can be a temperature effect onκ_c. Another interpretation is a thermal quark mass, i.e., quark gets heavier with increasing temperature and the chiral limit, as determined below T_c, remains unchanged above T_c. Also a cut-off effect proportional to N_τ⁻² is a possible explanation for the observed effect.

In the case of degenerate quark mass QCD has only two independent parameters: the coupling constant g and m_q. Both have to be renormalized and depend therefore on the scale µ and the renormalization scheme. To compare the different quark masses, obtained at the different scales 1/a, one has to rescale them. Imposing a mass independent renormalization scheme like MS, the quark mass is only a function of the scaleµand the coupling g. As for the coupling, introduced in Section 2.5, the running of the mass is described by a renormalization group equation. For the renormalized quark mass mR it is³

µ∂m_R

∂µ = τ(g_R)m_R. (4.18)

The renormalization group functionτ(g_R) has the perturbative expansion

τ(g_R) = −g²_R{d₀ + d₁ g_R² +· · · }, (4.19)

3Note that the functionτ has nothing to do with the Euclidian timeτ used earlier.