Running coupling

After performing the continuum limit of the SSF at the chosen values ofu, we use a fifth order polynomial to obtain the interpolating estimate ofσ(^u)at the whole range of couplings 0 ≤ u ≤ 2.7. As already mentioned, we use for this purpose

6.5 New result of the running coupling in Nf=4 theory

0 0.5 1 1.5 2 2.5 3

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5

u

σ(u)/u

Nf= 2 Nf= 4

2-loopβ-funct. forNf= 4 3-loopβ-funct. forNf= 4

Figure 6.7: Continuum step scaling function for N_f = 4 theory. The result for N_f = 2 is given for comparison. Dashed lines are the perturbative results from integrations with the 2-loop and 3-loop beta function.

the results coming from the constant fit of L/a=8 data (cf. Table 6.7) and obtain σ(^u) = ^u+^s0u²+^s1u³−0.012u⁴+0.012u⁵, 0≤^u≤2.7. (6.44) The fit of the non-perturbativeN_f =4 data is plotted in Figure 6.7, together with the non-perturbative result forN_f =2 theory [19] and the 2- and 3-loop perturba-tive result for N_f =4 theory. Coefficientss₀ands₁are universal (cf. eq. 5.25 and 5.26) and obtained from perturbation theory. We notice the agreement we obtain with the perturbative estimates of the SSF over the whole interval of couplingu.

Namely, our interpolation agrees with the PT result within one sigma, except at the largest coupling where the difference is slightly larger, but still smaller than two sigma. Finally, we complete step 2. of the strategy to extract the Lambda parameter described in section 6.2. Namely, we use the fit function obtained in 6.44 to give the estimate of ln(ΛLmax)inN_f =4 theory. Starting from the highest coupling u_max = g²(L_max), chosen such that the associated scale L_max is in the hadronic range, we recursively solve the following equationn-times

σ(^g2(L/2)) = ^g2(^L). (6.45)

i u_i ln(ΛLmax)

Table 6.8: Values estimated for ln(ΛLmax) for the value of L_max set with g²(L_max) =3.45. We take our final result from the stepi=9.

In this way, we obtain values forg²(L_max/2ⁱ),i=1 . . .n, where for a sufficiently large number of steps, we arrive to the regime where this coupling is perturbative and it is safe to expressΛin terms ofL_maxusing the perturbative expansion from eq. 2.54

The values for ln(ΛLmax)are given in Table 6.8. We take our final result from the stepi =9 from Table 6.8

ln(ΛLmax) = −2.027(80). (6.47) The choice of the step for quoting the final result (i = 9) is taken to be the same as in Ref. [21]. Note that, taking into account the data obtained in this work, the choicei=9 could be considered overconservative since the plateau in ln(ΛLmax) with our data arrises earlier than in [21]. Therefore, already reading off a final value from the stepi =5 is a viable choice.

As it was discussed in section 6.2, its still a significant challenge for future work to perform step 1. of the described strategy which would enable us to quote the value of theΛparameter in physical units.

Finally, in Figure 6.8 we show the running coupling in the SF scheme, expressed in units ofΛ. Even at the strongest coupling included in the computation, the ob-tained non-perturbative result shows the agreement with the perturbation theory

6.5 New result of the running coupling in Nf=4 theory estimates within one standard deviation. With the results of [21], a 2-sigma effect at the largest considered value of the SF coupling has been reported, but due to the error in the error analysis code, the final error estimate in this work is un-derestimated. Our results agree with the corrected result from [21] within one sigma. Nevertheless, it is expected that at energies higher than the ones consid-ered in this study the deviation of the non-perturbatively obtained SF coupling from the perturbative estimate will appear. For that reason it is very important to extend this study to lower energies and some interesting proposals for how this can be done will be discussed in Chapter 8.

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 6.8: The running coupling in N_f =4 obtained from the simulations sum-marized in section 6.5.1, employing a procedure described in sec-tion 6.5.3. The perturbative determinasec-tions of the coupling in the Schrödinger functional renormalization scheme to 2- and 3-loops are also given.

7 The strange quark mass and the Lambda parameter for

N _f = ^{2 theory}

The determination of the Lambda parameter in units of the hadronic scale Lmax

from chapter 6 is an important step towards the final comparison of this param-eter from the perturbation theory and from the lattice. Nevertheless, the impact of this determination on the phenomenological applications will be possible only after the scale of the simulations is set and the quantities in physical units are attained. Performing the set of large scale simulations at N_f =4 is a tedious and lengthy project, predicted to be finished in the next couple of years. Until this has been achieved, we go one step back and use the running od the coupling and running of the mass data from ALPHA collaboration [19, 20], in order to perform a determination of the Lambda parameter of and the strange quark mass for two flavour QCD. This represents the finalization of a long-term program of the AL-PHA collaboration of computing these parameters for N_f = 2 theory, using the Schrödinger functional strategy to overcome the multi-scale problem and keep the full control over the systematic errors. The results which will be presented here are obtained with the help of the precise scale determination from an ex-tensive set of large scale simulations produced with the DD-HMC code and the MP-HMC code discussed in chapter 4. In the scale setting a physical quantity used is the kaon decay constant f_K. With this scale we achieve a total error of approx. 2%, employing two different strategies for the chiral extrapolation which agree within errorbars. First part of this chapter will be dedicated to the im-proved scale setting and afterwards we move towards giving the physical values for the RGI values of the strange quark mass andΛ-parameter, in the setup with two dynamical flavors of light quarks. The results which will be presented here have already been published in Refs. [24, 109]. Text of sections 7.1 and 7.2 (with minor modifications to match the notation of Ref. [24]) is fully taken over from Ref. [109]. Text of the sections 7.3.1 and 7.4.1 as well as all the plots and tables in this chapter (7) are taken over from Ref. [24].

id L/a β κ κs R0 m_π[MeV] m_πL A2 32 5.2 0.13565 0.135438(20) 5.485(21) 630 7.7 A3 0.13580 0.135346(20) 5.674(32) 490 6.0 A4 0.13590 0.135285(20) 5.808(34) 380 4.7 A5 0.13594 0.135257(20) 5.900(24) 330 4.0

E4 32 5.3 0.13610 0.135836(17) — 580 6.2

E5 0.13625 0.135777(17) 6.747(59) 440 4.7 F6 48 0.13635 0.135741(17) 6.984(51) 310 5.0 F7 0.13638 0.135730(17) 7.051(43) 270 4.3 N4 48 5.5 0.13650 0.136278(08) 9.32(30) 550 6.5

N5 0.13660 0.136262(08) 9.31(26) 440 5.2

N6 0.13667 0.136250(08) 9.55(11) 340 4.0

O7 64 0.13671 0.136243(08) 9.68(10) 270 4.2 Table 7.1: Overview of the ensembles used in this study. We give the label, the

spatial extent of the lattice, β = 6/g²₀, the hopping parameterκ of the sea quarks, the hopping parameter κs of the strange quark, the scale R₀ =^r0/a, the mass of the sea pion m_π and the productm_πL, which is always larger or equal than 4. All lattices have dimensionT×L³ with T =2L.

7.1 Lattice parameters

The following study is based on ensembles generated within the CLS effort, with the Wilson plaquette gauge action together with N_f = 2 mass-degenerate fla-vors of O(a) improved Wilson fermions. The simulations are using either M.

Lüscher’s implementation of the DD-HMC algorithm (cf. section 4.1 and [68]), or our implementation of the MP-HMC algorithm (cf. chapter 4 and [74]). The list of ensembles used in the analysis is shown in Table 7.1. Lattice spacings are ranging from 0.05fm to 0.08fm and their precise determination will be presented in the following section. The ensembles cover a wide range of pion masses going down to 270MeV, whereas all lattice volumes satisfy the requirementm_πL ≥4 to keep finite volume effects under control.