Other O(a) improvement coefficients in Nf=4 simulations . 85

We have already discussed in section 5.5 that theO(a)improvement of the Schrödinger functional requires additional improvement terms on the boundaries. For the choice of the considered boundary conditions, there are two such coefficients: the weight of the time-like plaquettes Pattached to the boundaryw(P) =c_t(g₀)and the fermionic improvement coefficient ˜c_t. The 2-loop expression of c_t and the 1-loop expression for ˜c_t quoted in equations 5.31 and 5.32 are used to determine these parameters in our simulations.

Let us now devote a few more words to the determination of the Sheikholeslami-Wohlert improvement coefficientc_SW, which is crucial for theO(^a)improvement of the Wilson fermion action. This coefficient can be determined non-perturbatively

— utilizing similar considerations to those applied in the effort to compute c_A non-perturbatively. Namely, one first introduces an alternative definition of the quark mass

M(x₀,y₀) = m(x₀)−s(x₀)^m(^y0)−^m(^y0)

s(y₀)−s(y₀) ^(6.24)

=^r(^x0) −^s(^x0)^r(^y0)−^r(^y0)

s(y₀)−s(y₀)^{. (6.25)} In theO(a)improved theory that we are considering, this alternative quark mass definition differs frommonly inO(a²)terms. M(x₀,y₀)is defined in an analogue way, but with forward and backward correlation functions interchanged. A con-dition imposed for the determination of c_SW was the vanishing of the following difference (see Ref. [43])

10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 0.05

0.1 0.15 0.2 0.25 0.3

μ/Λ αSF

Nf= 4 2-loopβ- funct.

3-loopβ- funct.

Figure 6.3: The running coupling inN_f =4 obtained from the simulation data in [21], together with the perturbative determinations of the coupling in the Schrödinger functional renormalization scheme to 2- and 3-loops.

The originally published data had a mistake in the error analysis. We plot here the data from ref. [21] with the corrected error bars.

Thec_Aterm in the definitions ofmandm(cf. equations 6.13 and 6.16) is canceled in the above definition of the mass. Therefore, in spite of the problems we have experienced in the non-perturbative determination ofc_A, with a moderate com-putational effort it is possible to determinec_SWnon-perturbatively. This determi-nation is done in Refs. [95, 104] and the resulting interpolation formula is quoted in Table 2.1, together with the results of similar determinations for different num-bers of fermion flavors. This interpolating result is used for our simulations that follow.

6.4 Previous computation of the running coupling in N

= 4 Wilson QCD and how to improve it

The running of the QCD coupling in N_f = 4 theory has previously been com-puted non-perturbatively only within two collaborations. The results for the four flavors of staggered fermions are published in Ref. [105], whereas the result for the four flavors of Wilson fermions are published in Refs. [21, 95]. The latter is the only non-perturbative computation of this quantity that has been performed so far for the N_f =4 Wilson QCD. The final result for the Lambda parameter in

6.4 Previous computation of the running coupling in Nf=4 Wilson QCD

units of the hadronic scale L_maxobtained in Refs. [21, 95] reads⁴

ln(ΛLmax) = −2.294(153), at g²(^Lmax) =3.45, (6.27) while the result of the running SF coupling obtained in Refs. [21, 95] is shown in Figure 6.3. The importance of an even more precise non-perturbative deter-mination of the Λparameter in the N_f = 4 theory for the world average of this quantity has already been discussed in chapter 1. The first important step to-wards this result is the precise determination of the step scaling function in the N_f =4 theory. Our goal is to achieve this step by

• performing a cross-check of the results obtained in [21],

• improving the precision of the previously obtained SSF for roughly a fac-tor of two, while still maintaining control over the systematic errors and keeping them negligible.

For the computation of the coupling, the massless SF scheme is achieved by tun-ing the PCAC mass (defined in section 6.1) to zero. The value of the hopptun-ing parameterκ for which the current quark mass vanishes, κc (critical κ), has to be tuned explicitly. We could re-use the entire tuning procedure done in [21, 95], but for the additional ensembles that we add to the following computation we had to perform this procedure ourselves. The starting hint and some more details of the PCAC mass tuning are discussed in appendix 5. Nevertheless, we would like to discuss here one important point regarding the tuning procedure. Namely, in order to achieve the wanted improvement in precision, and at the same time maintain control over the systematics resulting from the tuning of the quark mass to zero, we have to reinvestigate the bounds on the PCAC quark mass m₁. The criterion for defining a massless scheme in [21] was to achieve |^m1L| ≤ 0.005 in the tuning of the quark mass. It is not a priori clear whether this criterion would suffice with the reduced uncertainty of the SSF that we are aiming at. We will therefore discuss the derivation of the bound on|^m1L|in the following section.

6.4.1 Tuning criteria for the PCAC mass

In order to estimate the systematic error in the SSF coming from the inexact tun-ing of the PCAC mass to zero, one needs to determine the dependence of the lattice SSF on the quark mass m₁. The derivative of Σ with respect to z = ^m1L

4The error on ln(ΛLmax)quoted in ref. [21] does not coincide with the error given in eq. 6.27.

Value quoted in eq. 6.27 is the corrected value, obtained after a mistake in the previously used error propagation code is removed.

β g¯² am₁ g¯² am₁ 5.0 3.932(39) 0.03752(13) 3.638(34) 0.00037(14)

Table 6.1: The renormalized coupling results for two L/a = 8 simulations ob-tained in Ref. [21] at the same value of the inverse bare couplingβ. The data is used to estimate the numerical derivative in eq. 6.30.

reads

∂g²(2L)

∂z |_g²_(L)=u =Φ(a/L)^g4(^L), z= ^Lm1. (6.28) The authors of [19] argue thatΦis a slowly varying function ofa/Land give the estimate of its universal part from perturbation theory

Φ(0) = 0.00957N_f. (6.29)

We perform here an additional non-perturbative estimation of the derivative 6.28.

For that purpose, we use the two sets ofL/a =8 simulations from Ref. [21] and obtain

∂g²(2L)

∂z |_β=5.0,L/a=8 =Φ(a/L)g⁴(L) = 0.98(17), z= Lm₁, (6.30) at the highest value of the renormalized coupling that has been considered in Ref.

[21]. The data taken from [21] for the purpose of this estimation are given in Table 6.1. We show the two discussed estimates ofΦ(a/L) in Figure 6.4. To be on a safe side, we take for the new universal estimate ofΦ(a/L)the non-perturbative value

Φ(a/L) = 0.08 at L/a =8, (6.31)

coming from the upper bound depicted with the dashed blue line in Figure 6.4.

Due to the mentioned weak dependence of Φ on a/L, we use this value in the following estimations of the limit to|Lm₁|for all lattice extents considered.

Coming to the reduction of the statistical error in SSF, let us recall that the typ-ical statisttyp-ical precision of the SSF from Monte Carlo simulations reads

Δ(g⁻²) = ¹

g⁴Δ(g²). (6.32)

We require the systematic error coming from the mismatch in the tuning of the

6.4 Previous computation of the running coupling in Nf=4 Wilson QCD per-turbative(continuum) value for s/g⁴ given in 6.29. The red point at g² ≈ 3.9 comes from the non-perturbative estimate in eq. 6.30 at L/a =8.

PCAC mass to be at mostone thirdof the statistical error in the SSF Δ^sys(g(2L))≤ ¹

3 g⁴(L)Δ^stat(g⁻²(2L)). (6.33) Using the non-perturbatively obtained value from eq. 6.31 we estimate the sys-tematic error to be

Δ^sys(^g(2L)) = ^∂g2(2L)

∂z |^z| ≈0.08g⁴(^L)|^z|, z =^m1a×^L

a. (6.34) Finally, we obtain the limit onm₁L, such that the condition 6.33 is satisfied

|^m1a| ≤ ^a L ×¹

3× ¹

0.08 ×Δ^stat(^g−2(2L)). (6.35) In Appendix 6, we quote the statistical precision for the latices L/a = 6, 8 from Ref [21]. We can see that the tuning of the PCAC mass performed in [21] already satisfies the bound 6.35, even if we were to reduce the statistical error by a factor of two in the simulations we plan to perform. This facilitates the setting up of new simulations, since the predefined values of κcr from [21] may be used. The new parameter sets that we add to the previous study have to be tuned to satisfy the derived bound from 6.35. Some more details of the additional tuning are given in Appendix 7. We also show in Tables 3 - 5 of Appendix 6 that, in the final set of our new simulations, we have reached the precision required by expression 6.35.

6.5 New result of the running coupling in