High precision renormalization of the flavour non-singlet Noether currents in lattice QCD with Wilson quarks
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Abstract
We determine the non-perturbatively renormalized axial current for \(\hbox {O}(a)\) improved lattice QCD with Wilson quarks. Our strategy is based on the chirally rotated Schrödinger functional and can be generalized to other finite (ratios of) renormalization constants which are traditionally obtained by imposing continuum chiral Ward identities as normalization conditions. Compared to the latter we achieve an error reduction by up to one order of magnitude. Our results have already enabled the setting of the scale for the \({N_{\mathrm{f}}}=2+1\) CLS ensembles (Bruno et al. in Phys Rev D 95(7):074504. arXiv:1608.08900, 2017) and are thus an essential ingredient for the recent \(\alpha _s\) determination by the ALPHA collaboration (Phys Rev Lett 119(10):102001. arXiv:1706.03821, 2017). In this paper we shortly review the strategy and present our results for both \({N_{\mathrm{f}}}=2\) and \({N_{\mathrm{f}}}=3\) lattice QCD, where we match the \(\beta \)-values of the CLS gauge configurations. In addition to the axial current renormalization, we also present precise results for the renormalized local vector current.
1 Introduction
Lattice regularizations with Wilson type fermions [3] are widely used in current lattice QCD simulations [4, 5, 6, 7, 8, 9, 10]. The ultra-locality of the action enables numerical efficiency and thus access to a wide range of lattice spacings and spatial volumes. Furthermore, Wilson fermions maintain the full flavour symmetry of the continuum action, as well as the discrete symmetries such as parity, charge conjugation and time reversal. Unitarity is either realized exactly, or, in the case of Symanzik-improved actions, approximately up to cutoff effects which vanish in the continuum limit.
The price to pay for these advantages consists in the explicit breaking of all chiral symmetries by the Wilson term in the action. Well-known consequences include the additive renormalization of quark masses, the mixing under renormalization of composite operators in different chiral multiplets and discretization effects linear in a, the lattice spacing. Furthermore, the Noether currents of chiral symmetry are no longer protected against renormalization.
Over the last 30 years many efforts have been made to control the consequences of explicit chiral symmetry breaking with Wilson quarks. The main strategy consists in imposing continuum chiral symmetry relations as normalization conditions at finite lattice spacing [11, 12]. This is usually done using chiral Ward identities, which follow from an infinitesimal chiral change of variables in the QCD path integral. An example is the PCAC relation which determines the additive quark mass renormalization constant, as the “critical value” of the bare mass parameter, where the axial current is conserved. The fact that chiral symmetry is fully recovered only in the continuum limit implies that the choice of normalization condition matters at the cutoff level; at a fixed value of the lattice spacing the numerical results may occasionally differ substantially between any two such choices. Rather than interpreting this scatter as a systematic error, the modern approach consists in choosing a particular normalization condition and in fixing all dimensionful parameters (such as momenta or distances or background fields) in terms of a physical scale. This defines a so-called “line of constant physics” (LCP), along which the continuum limit is taken. As the lattice spacing a (or, equivalently, the bare coupling, \(g_0^2=6/\beta \)), is varied, this defines a function \(Z_{\mathrm{A}} = Z_{\mathrm{A}}(\beta )\). Obviously, another choice for the LCP will result in a different function \(Z'_{\mathrm{A}}(\beta )\). However, their difference will be, within errors, a smooth function of \(\beta \) which vanishes asymptotically \(\propto a\) or \(\propto a^2\) if \(\hbox {O}(a)\) improvement is implemented. Hence, following a LCP ensures that cutoff effects are smooth functions of \(\beta \) and the choice of LCP becomes irrelevant in the continuum limit. Adopting this viewpoint, the relevant systematic error is therefore determined by the precision to which a chosen LCP can be followed.
In this paper we apply a recently developed method to lattice QCD with \({N_{\mathrm{f}}}=2\) and \({N_{\mathrm{f}}}=3\) flavours, matching the lattice actions chosen by the CLS initiative [8, 10]. Our method is based on the chirally rotated Schödinger functional (\(\chi \mathrm{SF}\)) [13, 14]. The theoretical foundation of this framework has been explained in [14] and it has passed a number of perturbative and non-perturbative tests [15, 16, 17, 18, 19, 20]. In contrast to the Ward identity method the axial current renormalization conditions follow from a finite chiral rotation in the massless QCD path integral with Schrödinger functional (SF) boundary conditions. The renormalization constants are then obtained from ratios of simple 2-point functions. For the axial current, this represents a significant advantage over the Ward identity method [12, 21, 22] which involves 3- and 4-point functions. Hence, we observe a dramatic improvement in the attainable statistical precision for \(Z_{\mathrm{A}}\) and some care is required to ensure that systematic errors are under control at a similar level of precision. We also discuss the normalization procedure for the local vector current. While flavour symmetry remains unbroken on the lattice with (mass-degenerate) Wilson quarks, the corresponding Noether current lives on neighbouring lattice points connected by a gauge link, so that the use of the local vector current is often more practical.
This paper is organized as follows: after a short reminder of the \(\chi \mathrm{SF}\) correlation functions in the continuum and the normalization conditions derived from them in Sect. 2, we define in Sect. 3 a couple of different LCPs which we have followed. We then present the \(Z_{\mathrm{A}}\) and \(Z_{\mathrm{V}}\) determinations for lattice QCD with \({N_{\mathrm{f}}}=2\) and \({N_{\mathrm{f}}}=3\) quark flavours in Sects. 4 and 5, respectively, together with various tests we have carried out. Section 6 contains a summary of the main results of this work and some concluding remarks. Finally, the paper ends with three technical appendices: Appendix A collects the parameters and results of the simulations, Appendix B provides a detailed discussion on the systematic error estimates for our determinations, and Appendix C gathers our set of chosen fit functions which smoothly interpolate our \(Z_{\mathrm{A,V}}\) results in \(\beta \).
The main results for \({N_{\mathrm{f}}}=2\) are collected in Table 4, while those for \({N_{\mathrm{f}}}=3\) are given in Tables 6 and 7. These results can be directly applied to data obtained from the CLS 2- and 3-flavour configurations, respectively [8, 10]. The \({N_{\mathrm{f}}}=3\) results have, in fact, already been used, and enabled the precision CLS scale setting in Ref. [1] and the accurate quark-mass renormalization of Ref. [23].
2 Renormalization conditions from universality relations
2.1 The Schrödinger functional and chiral field rotations
Regarding the case of QCD with \({N_{\mathrm{f}}}=3\) quark flavours we note that the very same steps can be taken provided the massless third quark does not take part in the chiral rotation and thus remains with standard SF boundary conditions [14]. Correlation functions are then considered for the doublet fields only, i.e. the third quark never appears as a valence quark.
2.2 Renormalization conditions
We also comment on the appearance of a second up-type flavour \(u'\) in (2.15). When applying the chiral rotation (2.8) to the diagonal components of \(f_{\mathrm{A}}\), the disconnected diagrams are mapped to disconnected diagrams on the \(\chi \hbox {SF}\) side which can be shown to add up to a pure cutoff effect. Their omission is thus perfectly legitimate, even if the formulation of the renormalization conditions then has an element of partial quenching to it. The situation is comparable with the Ward identity method in two-flavour QCD [12, 21], where a fictitious s-quark can be introduced to eliminate the disconnected diagrams.
Finally, we emphasize that similar renormalization conditions can be devised for other finite renormalization constants. An interesting example is the ratio \(Z_{\mathrm{P}}/Z_{\mathrm{S}}\), where \(Z_{\mathrm{P}}\) and \(Z_{\mathrm{S}}\) are the pseudo-scalar and scalar renormalization constant, respectively. We refer the reader to Ref. [20] for more details.
3 Lines of constant physics and choice of renormalization conditions
3.1 General considerations
Having set the scale one needs to ensure the correlation functions are calculated in the desired situation of massless QCD and for the chosen chirally rotated boundary conditions at \(\alpha =\pi /2\). This means one needs to tune the bare quark mass \(am_0\) and \(z_f\) as functions of \(\beta \). We will discuss this in more detail below. Finally, the correlation functions depend on kinematic parameters, such as \(x_0\) or background field parameters such as \(\theta \). We have already set \(x_0=L/2\) in Eqs. (2.15), (2.17) and we choose \(\theta =0\) and work with vanishing SU(3) background field.
3.2 Perturbative subtraction of cutoff effects
L / a | \( R_{\mathrm{A}}^{g(1)}(a/L)\) | \( R_{\mathrm{A}}^{l(1)}(a/L)\) | \( R_{\mathrm{V}}^{g(1)}(a/L)\) | \( R_{\mathrm{V}}^{l(1)}(a/L)\) |
---|---|---|---|---|
Wilson gauge action | ||||
6 | \( -\,0.104309 \) | \( -\,0.116808 \) | \( -\,0.118728 \) | \( -\,0.130549 \) |
8 | \( -\,0.109076 \) | \( -\,0.116640 \) | \( -\,0.122586 \) | \( -\,0.129838 \) |
10 | \( -\,0.111857 \) | \( -\,0.116595 \) | \( -\,0.125088 \) | \( -\,0.129662 \) |
12 | \( -\,0.113308 \) | \( -\,0.116564 \) | \( -\,0.126426 \) | \( -\,0.129588 \) |
16 | \( -\,0.114714 \) | \( -\,0.116526 \) | \( -\,0.127747 \) | \( -\,0.129519 \) |
\(\infty \) | \(Z_{\mathrm{A}}^{(1)}= -0.116458(2)\) | \(Z_{\mathrm{V}}^{(1)}= -0.129430(2)\) | ||
Lüscher–Weisz gauge action | ||||
6 | \( -\,0.078368 \) | \( -\,0.091011 \) | \( -\,0.089760 \) | \( -\,0.101750 \) |
8 | \( -\,0.083286 \) | \( -\,0.090737 \) | \( -\,0.093819 \) | \( -\,0.101006 \) |
10 | \( -\,0.085958 \) | \( -\,0.090650 \) | \( -\,0.096256 \) | \( -\,0.100811 \) |
12 | \( -\,0.087374 \) | \( -\,0.090604 \) | \( -\,0.097578 \) | \( -\,0.100730 \) |
16 | \( -\,0.088756 \) | \( -\,0.090557 \) | \( -\,0.098889 \) | \( -\,0.100657 \) |
\(\infty \) | \(Z_{\mathrm{A}}^{(1)}= -0.090488(5)\) | \(Z_{\mathrm{V}}^{(1)}= -0.100567(2)\) |
3.3 Choices of LCP for \({N_{\mathrm{f}}}=2\) and \({N_{\mathrm{f}}}=3\)
3.4 Topology freezing
On the lattice the topological charge is not unambiguously defined. We follow Refs. [40, 41] and define the trivial topological sector as the set of gauge field configurations for which \(|Q|<0.5\), where Q is discretized in terms of the Wilson flow and the clover definition of the field strength tensor [36]. The flow time t is then kept fixed in physical units by requiring \(\sqrt{8t}=0.6\times L\).
3.5 On the tuning of \(am_0\) and \(z_f\)
Results for \(am_{\mathrm{cr}}(g_0,L/a)\) for four different values of \(z_f\), for \(L/a=8\) and \(\beta =5.3\). The weighted average of the results is also given in the last row of the table
\(z_f\) | \(am_{\mathrm{cr}}\) | \(\kappa _{\mathrm{cr}}\) |
---|---|---|
1.280 | \(-\,0.32808(13)\) | 0.1361685(47) |
1.283 | \(-\,0.32828(14)\) | 0.1361761(51) |
1.288 | \(-\,0.32808(12)\) | 0.1361687(45) |
1.293 | \(-\,0.32831(14)\) | 0.1361772(51) |
Average | \(-\,0.328179(65)\) | 0.1361722(24) |
Once the critical bare mass is fixed, a smooth interpolation of \(g_{\mathrm{A}}^{ud}(L/2)\) in \(m_0\) gives the results shown in Fig. 2. Over the chosen range, \(g_{\mathrm{A}}^{ud}(L/2)\) so interpolated is perfectly linear in \(z_f\), and it is thus straightforward to determine the point \(z_f^*\) where \(g_{\mathrm{A}}^{ud}(L/2)\) vanishes i.e. \(z^*_f= 1.2877(5)\) in this example.
The estimated values of \(am_{\mathrm{cr}}\) and \(z_f^*\) determined in this way turn out to be quite accurate in practice, cf. Table 8.^{3} We remark that results for \(m_{\mathrm{cr}}\) could also be taken from a different source, for instance from standard SF simulations. In this case only \(z_f\) needs to be tuned. The differences to the above procedure would be \(\hbox {O}(a)\) both in \(m_{\mathrm{cr}}\) and in \(z_f^*\) which, by the mechanism of automatic \(\hbox {O}(a)\) improvement, induce \(\hbox {O}(a^2)\) differences in observables such as the current normalization constants [14, 20]. One also expects that a precise tuning of \(m_0\) is less crucial in the \(\chi \mathrm{SF}\) than in the SF; the quark mass dependence of physical observables around the chiral limit is quadratic rather than linear [42].
3.6 Sources of uncertainties
- 1.
The LCP together with the set of values \(\beta _i\) translates to target values \((L/a)(\beta _i)\). At each \(\beta _i\) we choose lattices with even L / a straddling the target values. We here anticipate that with our choices of LCPs the required lattice sizes are in the range \(L/a=8\) to \(L/a=16\). Note that all target values \((L/a)(\beta _i)\) come with statistical errors except for \(\beta =\beta _{\mathrm{ref}}\), where, by definition, L / a is given as an (even) integer.
- 2.For given \(\beta \) and L / a we determine the solutions \(am_{0} = am_{\mathrm{cr}}\) and \(z_f=z_f^*\) of Eq. (2.13). In order to find their statistical errors which follow from the statistical uncertainties on m and \(g_{\mathrm{A}}^{ud}(L/2)\), we use estimates for the relevant derivatives,$$\begin{aligned} {\partial mL\over \partial m_0L},\quad {\partial mL\over \partial z_f},\quad {\partial g_{\mathrm{A}}^{ud}\over \partial m_0L},\quad {\partial g_{\mathrm{A}}^{ud}\over \partial z_f}. \end{aligned}$$(3.9)
- 3.We then determine the induced error on the Z-factors by estimating their derivatives with respect to the bare parameters,It turns out that the derivatives (3.9) and (3.10) scale quite well with lattice size and lattice spacing, so that it is unnecessary to evaluate them for all parameter choices. Some cross checks are sufficient. The errors coming from the uncertainties in \(m_0\) and \(z_f\) are then combined in quadrature and added, again in quadrature, to the statistical error.$$\begin{aligned} {\partial Z_{\mathrm{A,V}}\over \partial z_f},\quad {\partial Z_{\mathrm{A,V}}\over \partial m_0L}. \end{aligned}$$(3.10)
- 4.Where necessary, the results for \(Z_{\mathrm{A,V}}\) at the different L / a-values and fixed \(\beta _i\) are interpolated to the target \((L/a)(\beta _i)\); and the statistical error on \((L/a)(\beta _i)\) is propagated at this point. In the case where only one value of L / a has been simulated, an estimate for the derivativeis used to assign a systematic error due to the difference \(\Delta (L/a)\equiv L/a-(L/a)(\beta _i)\), also taking into account the statistical uncertainty on \((L/a)(\beta _i)\). The resulting systematic error is again added in quadrature.$$\begin{aligned} {\partial Z_{\mathrm{A,V}}\over \partial (L/a)} \end{aligned}$$(3.11)
4 Numerical results for \({N_{\mathrm{f}}}=2\) flavours
4.1 Lattice set-up and parameter choices
The CLS large volume simulations of 2-flavour QCD [8] were performed using non-perturbatively \(\hbox {O}(a)\) improved Wilson quarks and the Wilson gauge action. The matching to CLS data via the bare coupling requires that we use the same action in the \(\chi \hbox {SF}\). As for the details of the action near the time boundaries we refer to Ref. [20]. In particular the counterterm coefficients \(c_{\mathrm{t}}(g_0)\) and \(d_s(g_0)\) were set to their perturbative one-loop values using the results of that reference. In general, the incomplete cancellation of boundary \(\hbox {O}(a)\) artefacts implies some remnant \(\hbox {O}(a)\) effects in observables. However, for the estimators of the current normalization constants, Eqs. (2.15), (2.17), it can be shown that such \(\hbox {O}(a)\) effects only cause \(\hbox {O}(a^2)\) differences [20].
Values of \(af_K\) used to determine \((L/a)(\beta )\) such as to satisfy the condition (4.2) for the given \(\beta \). The \(\chi \mathrm{SF}\) simulations were performed at the neighbouring even integer L / a-values given in the last column
\(\beta \) | \(af_K\) | \((L/a)(\beta )\) | L / a |
---|---|---|---|
5.2 | 0.0593(7)(6) | 8 | 8 |
5.3 | 0.0517(6)(6) | 9.18(21) | 8, 10, 12 |
5.5 | 0.0382(4)(3) | 12.42(25) | 12 |
5.7 | 0.0290(11)\({}^{\dagger }\) | 16.35(67) | 16 |
Results for \(Z_{\mathrm{A,V}}\), both g and l definitions, for \({N_{\mathrm{f}}}=2\) non-perturbatively \(\hbox {O}(a)\) improved Wilson fermions and Wilson gauge action. The lower part of the table contains the same results after subtraction of the one-loop cutoff effects, cf. Eq. (3.6)
\(\beta \) | \(Z_{\mathrm{A}}^g\) | \(Z_{\mathrm{A}}^l\) | \(Z_{\mathrm{V}}^g\) | \(Z_{\mathrm{V}}^l\) |
---|---|---|---|---|
5.2 | 0.78022(55) | 0.76944(94) | 0.74673(47) | 0.73849(97) |
5.3 | 0.78411(61) | 0.77576(66) | 0.75220(70) | 0.74607(69) |
5.5 | 0.7945(13) | 0.7895(13) | 0.7663(14) | 0.7625(14) |
5.7 | 0.80526(97) | 0.80277(93) | 0.7800(11) | 0.77801(98) |
\(\beta \) | \(Z_{\mathrm{A,\,sub}}^g\) | \(Z_{\mathrm{A,\,sub}}^l\) | \(Z_{\mathrm{V,\,sub}}^g\) | \(Z_{\mathrm{V,\,sub}}^l\) |
---|---|---|---|---|
5.2 | 0.77262(54) | 0.76963(94) | 0.73986(47) | 0.73890(97) |
5.3 | 0.77847(42) | 0.77591(66) | 0.74706(49) | 0.74636(70) |
5.5 | 0.79138(89) | 0.7897(13) | 0.7634(11) | 0.7627(14) |
5.7 | 0.80358(63) | 0.80283(93) | 0.77836(79) | 0.7781(10) |
While the first 3 results for \(af_K\) in Table 3 have been directly measured [8] we have estimated \(af_K\) at the fourth value, \(\beta =5.7\), as follows: with \(af_K\) at \(\beta =5.5\) taken as starting point we used the three-loop \(\beta \)-function for the bare coupling [43], in order to determine the ratio of lattice spacings. The error is obtained by summing (in quadrature) the statistical error propagated from the result at \(\beta =5.5\), and a systematic error due to the use of perturbation theory. The latter is estimated as the difference between the non-perturbative result for \(af_K\) at \(\beta =5.5\), and the same perturbative procedure, applied between \(\beta =5.3\) and \(\beta =5.5\). This systematic error is about 2.7 times larger than the statistical one, and thus dominates the error on L / a at \(\beta =5.7\).
Except for \(\beta = 5.3\), the target values \((L/a)(\beta _i)\) resulting from condition (4.2), are very close to even integer values of L / a, so that interpolations between simulations at different L / a can be avoided. At \(\beta =5.3\) we simulated at the three L / a-values given in the last column of Table 3 and interpolated to the target value (see Appendix B.4 for more details). For each choice of \(\beta \) and L / a, along the lines of the discussion in Sect. 3.5, we have carried out various tuning runs covering a range of \(am_0\) and \(z_f\), so as to determine the parameters satisfying the conditions (2.13). The values of the tuned parameters and the results for m and \(g_{\mathrm{A}}^{ud}(L/2)\) at these parameters are given in Table 8.
4.2 Results and error budget
As discussed in Sect. 3.6, systematic errors result from uncertainties or deviations in following a chosen LCP, which correspond with statistical errors and deviations from zero in m and \(g_{\mathrm{A}}^{ud}(L/a)\), as well as uncertainties in the target lattice extent L / a and systematic errors arising from inter- or extrapolations from the simulated lattices sizes, if applicable. Tables 3 and 8 contain the relevant information for the case \({N_{\mathrm{f}}}=2\). The propagation of these uncertainties to the Z-factors is then performed following the steps outlined in Sect. 3.6. We have carried out some additional simulations to estimate the derivatives in Eqs. (3.9), (3.10), and some perturbative calculation to check the expected scaling of the derivatives with the lattice size. We delegate a detailed discussion to Appendix B. Here we just note that with our statistics and our rather conservative approach, the propagated uncertainties are typically larger than the statistical errors for the R-estimators Eqs. (2.15), (2.17) (cf. Tables 10, 11).
4.2.1 Effect of perturbative one-loop improvement
As discussed in Sect. 3.2, we have also computed the relevant \(\chi \hbox {SF}\) correlation functions in perturbation theory to order \(g_0^2=6/\beta \). Besides consistency checks and qualitative insight the main application consists in the perturbative subtraction of cutoff effects from the data. Note that this requires to emulate the non-perturbative procedure in all details, in particular the determination of \(am_{\mathrm{cr}}\) and \(z_f^*\) according to Eq. (2.13). The lower part of Table 4 contains the results for \(Z_{\mathrm{A,V}}\) after perturbative improvement. Comparing with the unimproved results in the upper part of Table 4, one can see that the g-definitions are more affected, and are brought closer to the corresponding l-definitions by the perturbative improvement (cf. also Fig. 5). In any case, the perturbative corrections are at the level of 1 per cent at most.
CLS \(\beta \)-values and corresponding results for \(t_0/a^2\) in the SU(3) flavour symmetric limit [1, 2]. The latter are used to determine the lattice sizes \((L_{1,2}/a)(\beta _i)\) which satisfy the conditions (5.1). The \(\chi \mathrm{SF}\) simulations are performed at the neighbouring L / a’s given in the last column of the table
\(\beta \) | \(t_0/a^2\) | \((L_1/a)(\beta )\) | \((L_2/a)(\beta )\) | L / a |
---|---|---|---|---|
3.40 | 2.8619(55) | 8 | 7.225(16) | 6, 8, 10, 12 |
3.46 | 3.662(13) | 9.049(18) | 8.172(22) | 6, 8, 10, 12 |
3.55 | 5.166(17) | 10.748(21) | 9.706(25) | 8, 10, 12, 16 |
3.70 | 8.596(31) | 13.864(29) | 12.521(34) | 8, 10, 12, 16 |
3.85 | 14.036(57) | – | 16 | 16 |
4.3 Universality and automatic \(\hbox {O}(a)\) improvement
The \(\chi \mathrm{SF}\) determinations (2.15) and (2.17) are expected to be automatically \(\hbox {O}(a)\) improved once the bare parameters \(m_0\) and \(z_f\) are properly tuned (cf. Sect. 2.2). This means that neither bulk nor boundary \(\hbox {O}(a)\) counterterms are necessary to cancel \(\hbox {O}(a)\) discretization errors in these quantities. This was confirmed to one-loop order in perturbation theory [20] and should hold generally. To this end we now look at the ratios between Z-factors coming from the g- and l-definitions. The expectation that these ratios converge to 1 with \(\hbox {O}(a^2)\) corrections is indeed very well borne out by the data, cf. Fig. 5, where we also include fits to this expected behaviour. We emphasize that this is a non-trivial result: even though the bulk action is improved to match the CLS set-up, we did not \(\hbox {O}(a)\) improve the currents entering the definitions (2.15), (2.17) and (2.13). This result thus confirms automatic \(\hbox {O}(a)\) improvement at the non-perturbative level, and, indirectly, the universality relations between the \(\chi \mathrm{SF}\) and SF formulations. A direct way to test universality between the \(\chi \mathrm{SF}\) and SF formulations would be simply to study the continuum scaling of ratios of Z-factors as obtained from one and the other formulation. Provided the SF determinations are properly improved, these should also approach 1 in the continuum limit with \(\hbox {O}(a^2)\) corrections. The large errors on the SF determinations do not allow us for a precise test of this expectation. However, the results in Figs. 3 and 4 clearly show that our determinations are in fact compatible with the SF ones within errors.
5 Numerical results for \({N_{\mathrm{f}}}=3\) flavours
5.1 Lattice set-up and parameter choices
In Table 5 we collect the relevant \(\beta \) values of the CLS simulations and the corresponding results for \(t_0/a^2\) [2]. The latter are evaluated for equal up-, down-, and strange-quark masses, which are close to the physical average quark mass (see Refs. [1, 2]). Table 5 also gives the lattice sizes \((L_{1,2}/a)(\beta )\) which satisfy the conditions (5.1). Compared to the \({N_{\mathrm{f}}}=2\) case (cf. Table 3), it is obvious that these \({N_{\mathrm{f}}}=3\) LCPs are much more accurately determined. In order to exploit this higher precision, we performed simulations for several L / a-values at each \(\beta \) (cf. Table 5). This allowed us to accurately interpolate the Z-factors to the target values (see Appendix B.5 for more details). Table 9 contains a summary of all simulations performed with the corresponding parameters. Due to both technical and historical reasons, we do not use the finest lattice spacing for the LCP defined in terms of \(L_1\). Following this LCP up to \(\beta =3.85\) would have required simulating lattices with \(L/a=18,20\), which are particularly inconvenient to parellelize with our current simulation program. Note also that CLS simulations at \(\beta =3.85\) are ongoing and currently limited to a single ensemble, so that the LCP with \(L_1\) may remain useful for a while. More importantly, however, the comparison between both LCPs allows us to perform additional tests on our results (cf. Sect. 5.3).
5.2 Results and error budget
\({N_{\mathrm{f}}}=3\) results for \(Z_{\mathrm{A,V}}\) using the \(L_1\)-LCP, both for g and l definitions. The lower part of the table contains the results after subtraction of the one-loop cutoff effects, cf. Eq. (3.6)
\(\beta \) | \(Z_{\mathrm{A}}^g\) | \(Z_{\mathrm{A}}^l\) | \(Z_{\mathrm{V}}^g\) | \(Z_{\mathrm{V}}^l\) |
---|---|---|---|---|
3.40 | 0.76847(35) | 0.75446(68) | 0.72923(27) | 0.71940(70) |
3.46 | 0.77128(44) | 0.76018(80) | 0.73392(36) | 0.72637(77) |
3.55 | 0.77703(30) | 0.76879(42) | 0.74261(23) | 0.73758(44) |
3.70 | 0.78831(30) | 0.78327(43) | 0.75833(31) | 0.75521(44) |
\(\beta \) | \(Z_{\mathrm{A,\,sub}}^g\) | \(Z_{\mathrm{A,\,sub}}^l\) | \(Z_{\mathrm{V,\,sub}}^g\) | \(Z_{\mathrm{V,\,sub}}^l\) |
---|---|---|---|---|
3.40 | 0.75702(35) | 0.75485(68) | 0.71882(27) | 0.72008(70) |
3.46 | 0.76245(44) | 0.76048(80) | 0.72578(36) | 0.72683(77) |
3.55 | 0.77103(29) | 0.76900(42) | 0.73701(23) | 0.73789(44) |
3.70 | 0.78485(30) | 0.78340(43) | 0.75506(31) | 0.75538(44) |
Same as Table 6 but for the \(L_2\)-LCP
\(\beta \) | \(Z_{\mathrm{A}}^g\) | \(Z_{\mathrm{A}}^l\) | \(Z_{\mathrm{V}}^g\) | \(Z_{\mathrm{V}}^l\) |
---|---|---|---|---|
3.40 | 0.77129(39) | 0.75592(72) | 0.73368(30) | 0.72164(74) |
3.46 | 0.77371(51) | 0.76132(93) | 0.73721(42) | 0.72782(89) |
3.55 | 0.77856(31) | 0.76953(43) | 0.74468(25) | 0.73846(46) |
3.70 | 0.78925(31) | 0.78362(47) | 0.75936(33) | 0.75552(48) |
3.85 | 0.79985(31) | 0.79657(47) | 0.77304(33) | 0.77061(49) |
\(\beta \) | \(Z_{\mathrm{A,\,sub}}^g\) | \(Z_{\mathrm{A,\,sub}}^l\) | \(Z_{\mathrm{V,\,sub}}^g\) | \(Z_{\mathrm{V,\,sub}}^l\) |
---|---|---|---|---|
3.40 | 0.75741(38) | 0.75642(72) | 0.72120(29) | 0.72259(74) |
3.46 | 0.76288(51) | 0.76169(93) | 0.72732(41) | 0.72845(89) |
3.55 | 0.77115(31) | 0.76979(43) | 0.73780(24) | 0.73886(46) |
3.70 | 0.78499(31) | 0.78378(47) | 0.75534(33) | 0.75574(48) |
3.85 | 0.79734(31) | 0.79667(47) | 0.77065(33) | 0.77074(49) |
Like in the \({N_{\mathrm{f}}}=2\) case, the high statistical precision requires a careful assessment of the systematic errors in order to arrive at reliable error estimates. Tables 5 and 9 contain information on the accuracy with which the chosen LCPs are realized for our simulation parameters. Our estimates for the systematic uncertainties due to deviations from the chosen LCP were then obtained analogously to the case of \({N_{\mathrm{f}}}=2\); we refer the reader to Appendix B for the details. Here it is worth noting that, similarly to this case, the propagated uncertainties are typically larger than the statistical errors for the R-estimators, Eqs. (2.15), (2.17), cf. Table 12.
5.2.1 Effect of perturbative one-loop improvement
In the lower halves of Tables 6 and 7 we give the results for \(Z_{\mathrm{A,V}}\) after perturbatively subtracting the lattice artefacts to one-loop order. The results have been obtained by first improving the \(Z_{\mathrm{A,V}}\) determinations for each L / a and \(g_0\) value, and then interpolating to the proper \((L_{1,2}/a)(\beta )\) (see Appendix B.5).
Comparing the results for \(Z_{\mathrm{A,V}}\) before and after perturbative improvement, one sees that the g-definitions are the most affected, and are brought closer to the corresponding l-definitions. All in all, the effect of the perturbative improvement is at most at the level of a couple of percent (cf. Fig. 7). Hence, not too surprisingly perhaps, the situation is very much the same as for the \({N_{\mathrm{f}}}=2\) case.
In conclusion, our final results for \(Z_{\mathrm{A,V}}\) are very precise for both LCPs. Similarly to the \({N_{\mathrm{f}}}=2\) case, the results for \(Z_{\mathrm{A}}\) are significantly more accurate than the standard SF determination based on Ward identities [22]. This can be appreciated in Fig. 6, where the results from Table 7 are displayed together with the 2 alternative definitions \(Z_{\mathrm{A,0}}\) and \(Z_{\mathrm{A,0}}^\mathrm{con}\) of Ref. [22].
5.3 Universality and automatic \(\hbox {O}(a)\) improvement
It is also interesting to consider the continuum limit of the ratio between the definitions belonging to different LCPs i.e. the \(L_1\)- and \(L_2\)-LCP. An example of such a ratio is shown in Fig. 8. Also in this case, the continuum scaling of this ratio is the one expected, and the initial difference is at the 2 per cent level. Apart from providing an important check of universality and automatic \(\hbox {O}(a)\) improvement, these results show that considering one definition or the other for the renormalization of matrix elements of the axial and vector currents, will only introduce small \(\hbox {O}(a^2)\) differences over the whole range of lattice spacings covered.
Finally, we look at ratios between \(\chi \mathrm{SF}\) and standard SF determinations. Towards the continuum limit these should also scale like \(1+\mathrm{O}(a^2)\), if the SF determinations are \(\hbox {O}(a)\) improved. In Fig. 9 we show the continuum limit of the ratios between the standard SF determinations of Ref. [22] and the \(\chi \mathrm{SF}\) results of table 7. We here consider both definitions of this reference, and label them as \(Z_{\mathrm{A}}^\mathrm{SF} = Z_{\mathrm{A,0}}\) and \(Z_{\mathrm{A,\,con}}^\mathrm{SF} = Z_{\mathrm{A,0}}^\mathrm{con}\), respectively (cf. [22] for the exact definitions).
As one can see in Fig. 9, for their preferred definition, \(Z^\mathrm{SF}_{\mathrm{A}}\), the expected scaling is only setting in around \(a^2/t_0<0.2\), where the SF and \(\chi \mathrm{SF}\) determinations differ by a couple of per cent. At the coarsest lattice spacing, corresponding to \(\beta =3.4\), the deviation from the \(\hbox {O}(a^2)\) scaling is significant. The results for \(Z_{\mathrm{A}}^g\) show the largest deviation from the SF determination, which is about 6%. Considering the perturbatively improved \(\chi \mathrm{SF}\) results this difference is somewhat reduced to 4–5%, but \(\hbox {O}(a^2)\) scaling is not observed either. If we consider instead the alternative definition, \(Z^\mathrm{SF}_{\mathrm{A,\,con}}\), the deviation is reduced to about 2 per cent at the coarsest lattice spacing for \(Z_{\mathrm{A}}^l\), while, remarkably, the results for \(Z_{\mathrm{A}}^g\) and \(Z^\mathrm{con}_{\mathrm{A,0}}\) are compatible within errors. In particular, the difference between this SF and both our \(\chi \mathrm{SF}\) determinations is perfectly compatible with an \(\hbox {O}(a^2)\) effect over the whole range of lattice spacings considered. While discretization effects can only be defined with respect to some reference definition, we conclude that the alternative SF definition \(Z_{\mathrm{A,\,con}}^\mathrm{SF}\) is, within errors, perfectly scaling with \(a^2\) for \(\beta \ge 3.4\) relative to all \(\chi \mathrm{SF}\) definitions, whereas the preferred definition \(Z_{\mathrm{A}}^\mathrm{SF}\) of Ref. [22] requires much finer lattices before this expected asymptotic behaviour sets in. With hindsight, \(Z_{\mathrm{A,\,con}}^\mathrm{SF}\) seems to be a better choice within the SF framework and also has been the preferred SF definition within the \({N_{\mathrm{f}}}=2\) setup of Refs. [8, 44].
6 Summary and conclusions
We have used a new method [20] based on the chirally rotated Schrödinger functional [14] to obtain high precision results for the normalization constants of the Noether currents corresponding to non-singlet chiral and flavour symmetries. The matrix elements of these axial and vector currents play a crucial rôle in various contexts of hadronic physics. Our method differs from the traditional Ward identity method [11, 12] in that it compares correlation functions which are related by finite chiral or flavour rotations, rather than infinitesimal ones. The major advantage compared to the Ward identity method consists in the avoidance of 3- and 4-point functions in favour of simple 2-point functions. This very significantly improves on the precision achieved in previous determinations [8, 21, 22, 44, 47]. In particular, for the case of \(Z_{\mathrm{A}}\), we obtain a reduction of the error by up to an order of magnitude (cf. Figs. 3, 6). The relatively poor precision obtained for \(Z_{\mathrm{A}}\) with the traditional Ward identity methods [8, 21, 22, 44] (around the percent level at the coarsest lattice spacings of interest), has now become a limiting factor in several applications. For this reason, our results are in high demand and have already been used in several works [1, 23, 48]. In particular, the precise \({N_{\mathrm{f}}}=2+1\) scale setting from a linear combination of \(f_K\) and \(f_\pi \) in Ref. [1] crucially relies on our values of \(Z_{\mathrm{A}}^l\) in Table 6 and the associated uncertainty is negligible compared to the statistical error of the bare hadronic matrix elements. In turn, the precise scale setting result of [1] is entering almost all studies done with CLS gauge configurations: in particular it has enabled the precise result for the 3-flavour QCD \(\Lambda \)-parameter and thus \(\alpha _s(m_Z)\) by the ALPHA-collaboration [2, 41, 49, 50]. Further applications of our \(Z_{\mathrm{A}}\)-results include the non-perturbative quark mass renormalization factor in [23] and the related determination of the light and strange quark masses [48]. Regarding the \({N_{\mathrm{f}}}=2\) case, the potential improvement of the scale setting in Ref. [8] due to our \(Z_{\mathrm{A}}\)-results would be very significant, too. Tentative estimates anticipate a gain by a factor 3–6 in precision, when going from the finest to the coarsest lattice spacing [51].
In order to maximize the usefulness of our results we have chosen the same actions and the same \(\beta \)-values for \({N_{\mathrm{f}}}=2\) and \({N_{\mathrm{f}}}=3\) lattice QCD as used by the CLS initiative [8, 10]. Hence, anyone working with CLS gauge configurations will be able to directly use our results: for \({N_{\mathrm{f}}}=2\) we recommend to use \(Z_{\mathrm{A,V,\,sub}}^l\) from Table 4, and for \({N_{\mathrm{f}}}=3\) we recommend using \(Z_{\mathrm{A,V,\,sub}}^l\) either of Tables 6 or 7. Although the results for \(Z_{\mathrm{A,V,\,sub}}^l\) are slightly less precise than those for \(Z_{\mathrm{A,V,\,sub}}^g\), their L / a-interpolations turn out to be more robust. Furthermore, the effect of the perturbative subtraction of cutoff effects is rather small and only marginally significant with current errors. While the precise choice of the \(\chi \mathrm{SF}\) results for the Z-factors is not crucial, it is however very important to be consistent and to not switch definitions when changing \(\beta \). Only then cutoff effects are guaranteed to vanish smoothly at a rate \(\propto a^2\).
Our determination of \(Z_{\mathrm{A,V}}(\beta )\) was carried out for each \(\beta \)-value independently, in order to avoid adding statistical correlation between physics results at different lattice spacings. However, it is straightforward to fit our Z-factors to a smooth function of \(\beta \) (or \(g_0^2\)), which interpolates to any intermediate \(\beta \)-value. We have included a few such fits in Appendix C to our preferred definitions \(Z_{\mathrm{A,V,\,sub}}^l\). We also include fits which incorporate the expected perturbative behaviour to 1-loop order. However, the high precision obtained in the \(\beta \)-range covered by the data cannot be guaranteed outside this range. If a similar precision is required at higher \(\beta \), an extension of our non-perturbative determination will be required. If \(t_0/a^2\) was known for higher \(\beta \)-values one could extend our chosen line of constant physics covering another factor of 2 or so in the lattice spacing. The required simulations of the \(\chi \hbox {SF}\) for lattice sizes up to \(L/a=32\) would be feasible with current resources. Going beyond this range it may be advisable to choose a different line of constant physics from a finite volume observable, or at least estimate the errors incurred by deviating from the original choice.
In applications to hadronic physics one would also like to control the \(\hbox {O}(a)\) effects cancelled by the counterterms to the currents. Close to the chiral limit, one essentially requires the counterterm coefficients \(c_{\mathrm{A,V}}\) [26, 27]. We emphasize that our method of determining the Z-factors does not rely on any assumptions about these counterterms and can therefore be combined with results for \(c_{\mathrm{A,V}}\) from other studies, e.g. [47, 52]. The same remark applies to the b-coefficients multiplying O(am) counterterms, which have recently been determined for the vector current in Ref. [53].
Looking beyond direct applications of our results in the CLS context, it is quite obvious that the precision gains of this method are generic and could be implemented with any other choice of Wilson type fermions. One would need to implement the \(\chi \hbox {SF}\) boundary conditions following Ref. [14], as well as the \(\chi \hbox {SF}\) correlation functions [20]. We also note that the computer resources required are rather modest: in fact our largest lattice size was \(16^4\); indeed, the main work for the present results went into painstakingly following lines of constant physics and the determination of the corresponding uncertainties and their propagation to the Z-factors. We have reported many technical details in the hope that any further applications of the method will be able to benefit from our experience. One possible improvement we did not explore was to measure the derivatives (3.9), (3.10) by computing the corresponding operator insertions into the correlation functions directly on the tuned ensembles; this was done e.g. in Refs. [1, 2] for the PCAC mass, \(t_0\), and other observables, and this would certainly allow one to further improve on the precision, as no assumptions on the derivatives need to be made.
Possible future applications of the \(\chi \hbox {SF}\) include the determination of the ratio between pseudo-scalar and scalar renormalization constants, \(Z_{\mathrm{P}}/Z_{\mathrm{S}}\). Advantages of the \(\chi \hbox {SF}\) are also expected for scale-dependent problems, such as the renormalization of 4-quark operators, where the contamination by \(\hbox {O}(a)\) effects could be significantly reduced by the mechanism of automatic \(\hbox {O}(a)\) improvement [19]. Finally the \(\chi \hbox {SF}\) offers new methods for the determination of \(\hbox {O}(a)\) improvement coefficients, which we hope to explore in the future.
Footnotes
- 1.
The choice of the scale from \(f_K\) seems somewhat circular, as its measurement requires the correctly normalized axial current. We use the results from Ref. [8] which were obtained using \(Z_{\mathrm{A}}\) from a standard SF Ward identity determination.
- 2.
For ease of notation, in the following the subscript Q is implicitly understood, and we assume that all relevant correlation functions are restricted to the \(Q=0\) sector.
- 3.
Note that the \(L/a=8\), \(\beta =5.3\), simulations listed in Table 8, use slightly different values for \(am_{\mathrm{cr}}\) and \(z_f^*\) from a previous, less precise determination.
- 4.
Even though the fermionic contributions were calculated with the Wilson gauge action, to this order the calculation only depends on the gauge background field, which is not modified when using the LW action with option B of [32].
- 5.
Note that perturbative results without explicit group factors assume gauge group SU(3) and fermions in the fundamental representation.
- 6.
We recall that at leading order in PT the x-derivatives of the Z-factors are of O(\(g^2_0\)) (cf. Eq. (B.21)). They are thus expected to diminish, and eventually vanish, as \(g_0\rightarrow 0\).
Notes
Acknowledgements
We would like to thank Rainer Sommer and Stefano Lottini for useful discussions, and Patrick Fritzsch, Tim Harris, and Alberto Ramos for helpful comments on the analysis of the data. We are moreover thankful to the computer centers at ICHEC, LRZ (project id pr84mi), and DESY-Zeuthen for the allocated computer resources and support. The code we used for the simulations is based on the openQCD package developed at CERN [38].
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