In the first two chapters, we explained some aspects of the continuum QCD briefly and gave then an overview of the lattice formulation. After discussing numerical and algorithmic issues we gave a brief summary about the theoretical foundations on which this thesis is based. Then we discussed our idea of extending the ALPHA code with two flavors ofO(a) improved Wilson quarks to four flavors ofO(a) improved Wilson quarks.
We presented then the results of our studies. The first part of our results concerned the determination of the improvement coefficientcsw with four flavors (with plaquette gauge action). As one can see in equation (2.50) and Table 2.2, this coefficient was only known from perturbation theory up to 1-loop and non-perturbatively for Nf = 0,2,3 [54–56].
Hence, it was essential to determine this coefficient for four flavors to O(a)-improve the action. So, we have performed simulations for the calculation of the improvement coefficientcswfor four flavors of Wilson fermions in the range ofβ ≥5.0. A determination of csw beyondβ = 5.0 was not possible because we could not show unambiguously that our improvement condition ∆M = 0 (7.13) was fulfilled atβ= 4.8. The errorbars of the raw data became too large and we were not able to determine the value for csw where
∆M vanishes. The raw data can be found in the appendix B. Then we gave a detailed description of how we determinedcsw from our raw data in section 7.1.5 and quoted
csw(g02) = 1−0.1372g20−0.1641g40+ 0.1679g06
1−0.4031g20 , 0≤g02 ≤1.2.
as a suitable parametrization of our data (eq. (7.18)). We compared three simple-minded different Padé-approximation formulae for our data and could show that a small extrapolation of csw beyond β = 5.0 to β = 4.5 with the above formula may still be acceptable.
In the second part we gave an estimation of critical κ (abbreviated with κc) where the quark mass vanishes. This estimation was extracted from the raw data of the csw
determination (Table 7.2). We quoted
κc= 1/8 +κ(1)c g02+ 0.000129g04+ 0.007470g60−0.007716g08+ 0.002748g010, 0≤g02≤1.2
where κ(1)c is the 1-loop value κ(1)c = 0.008439857 [120] as our interpolation for κc. We emphasize that the above formula for κc was used as a first estimate in the simula-tions for the step scaling function and we tuned κ explicitly such that the PCAC mass vanishes. We compared then the non-perturbative estimations for κc with Nf = 2,3,4 [55, 56] and the 1-loop perturbation theory [52, 120]. As one can see in Figure 7.5, the
8 Summary and Outlook
non-perturbative results are never far from the 1-loop perturbative result but a small dependence on the number of flavors can be seen.
In the third part we used our determination of csw with our estimation of κc to determine the non-perturbative running of the Schrödinger functional coupling forNf= 4 massless flavors. For this purpose we computed first the step scaling function of the QCD coupling in the Schrödinger functional scheme with four massless flavors. As can be seen in Figure 7.7 and Table 7.3 the resulting cutoff effects were very small allowing for a continuum extrapolation. While the data are compatible with a constant forL/a≥6, we assumed this form only forL/a≥8; the smaller lattices thus only entered the analysis by demonstrating that cutoff effects were small. We emphasize that this statement refers to the present level of statistical errors. If in the future statistical errors are further reduced, larger L/a will be necessary at the same time. It will be very interesting to see also the efficiency of computations with different regularizations of the Schrödinger functional as well as the corresponding test of the universality of the continuum limit.
Most notably there are chirally rotated boundary conditions for the quarks [151–153]
and staggered quarks [154, 155] for which results are expected soon.
Our interpolation of the step scaling function based on our data resulted in σ(u) =u+s0u2+s1u3+ 0.0036u4−0.0005u5, 0≤u≤2.7.
which is close to the perturbative results as shown in Figure 7.8. Using this parametriza-tion we calculated the Λ parameter in units ofLmax with≈8% precision and quoted
ln(ΛLmax) =−2.294(83) atumax= 3.45
as our final result. Due to the undetermined technical scale Lmax in physical units in our studies, we showed the running of the coupling in Schrödinger functional scheme in units of Λ (Figure 7.9). As one can see in Figure 7.9, we observed a small but significant deviation from 3-loop perturbation theory at the largest reached coupling. It is about 10% (three standard deviations) and the Schrödinger functional coupling has a value of αSF ≈0.28. ForNf= 2 a similar effect was visible only for larger coupling [113]a. These findings underline the necessity of going to weak coupling before applying renormalized perturbation theory in the continuum.
The present work has brought us a good step closer to the computation of the Λ-parameter in 4-flavor QCD, which may then be perturbatively connected to e.g. the 5-flavor MS coupling at the Z-pole. However, the technically introduced scaleLmaxremains to be expressed in physical units through large volume 4-flavor simulations. Apart from the challenge of tuning more parameters, one needs to treat a massive charm quark at small enough lattice spacing. Presently this appears to be a considerable challenge due to a severe slowing down of lattice simulation algorithms at small lattice spacings [156, 157].
aForαSF≈0.45 a similar deviation is visible but there are no non-perturbative data points in between αSF≈0.28 andαSF≈0.45 to see better where this sets in.
9 Publications
• Fatih Tekin, Rainer Sommer, and Ulli Wolff. Symanzik improvement of lattice QCD with four flavors of Wilson quarks. Phys. Lett., B683:75, 2010.
• Fatih Tekin, Rainer Sommer, and Ulli Wolff. The running coupling of QCD with four flavors. Nucl. Phys., B840:114, 2010
• Rainer Sommer, Fatih Tekin, Ulli Wolff. Running of the SF-coupling with four massless flavours. PoS, LAT2010, 2010.