# Non-perturbative determination of improvement coefficients \(b_\mathrm{m}\) and \(b_\mathrm{A}-b_\mathrm{P}\) and normalisation factor \(Z_\mathrm{m}Z_\mathrm{P}/Z_\mathrm{A}\) with \(N_\mathrm{f}= 3\) Wilson fermions

## Abstract

We determine non-perturbatively the normalisation parameter \(Z_\mathrm{m}Z_\mathrm{P}/Z_\mathrm{A}\) as well as the Symanzik coefficients \(b_\mathrm{m}\) and \(b_\mathrm{A}-b_\mathrm{P}\), required in \(\mathrm{O}(a)\) improved quark mass renormalisation with Wilson fermions. The strategy underlying their computation involves simulations in \(N_\mathrm{f}=3\) QCD with \(\mathrm{O}(a)\) improved massless sea and non-degenerate valence quarks in the finite-volume Schrödinger functional scheme. Our results, which cover the typical gauge coupling range of large-volume \(N_\mathrm{f}=2+1\) QCD simulations with Wilson fermions at lattice spacings below \(0.1\,\mathrm{fm}\), are of particular use for the non-perturbative calculation of \(\mathrm{O}(a)\) improved renormalised quark masses.

## 1 Introduction

Quark masses are amongst the fundamental parameters of the theory of strong interactions. Their high-precision determination is one of the main goals of lattice QCD (see Ref. [1] and references therein). These computations suffer from statistical and systematic uncertainties, which can be reduced in a controlled way. An important source of uncertainty are cutoff effects, which are removed by computing a given quantity at several lattice spacings, followed by continuum extrapolation. For several variants of lattice fermions (staggered, domain wall, overlap, twisted-mass) these uncertainties are \({\mathrm{O}}(a^2)\), while for Wilson fermions they are \({\mathrm{O}}(a)\). A related problem in the latter formulation is the loss of chiral symmetry, because it complicates the renormalisation properties of most quantities. A frequently cited example of these complications is the power divergence \(m_{\mathrm{crit}} \sim 1/a\) that must be subtracted from bare quark masses before they are renormalised multiplicatively. Another example is the fact that the normalisation factor \(Z_{\mathrm{A}}\) of the axial current and the ratio \(Z_{\mathrm{S}}/Z_{\mathrm{P}}\) of the scalar and pseudoscalar density renormalisation parameters are finite functions of the gauge coupling, which are equal to unity only in the continuum limit where chiral symmetry is fully recovered.

In spite of these shortcomings, Wilson fermions have advantages compared to other popular regularisations, namely strict locality (leading to relatively reduced computational costs) and preservation of flavour symmetry. It is the regularisation of choice of our collaboration, which is part of the effort by the CLS (Coordinated Lattice Simulations) cooperation to simulate QCD with \(N_\mathrm{f}=2+1\) flavours of non-perturbatively improved Wilson fermions [2, 3, 4, 5].

Wilson fermion \({\mathrm{O}}(a)\) discretisation effects are systematically removed by introducing so-called Symanzik counter-terms in the lattice action and composite operators. These counter-terms are higher dimensional operators with coefficients which are functions of the gauge coupling. The coefficients must be appropriately tuned so that \({\mathrm{O}}(a)\) improvement is achieved. Some of them (\(c_\mathrm{sw}, c_\mathrm{A}\), etc.) remove discretisation effects which are present also in the chiral limit, whereas others (\(b_\mathrm{m}, b_\mathrm{A}, b_\mathrm{P}\), etc.) are proportional to the quark masses and improve quantities off the chiral limit. The requirement for improvement in the fermionic sector has been noted early on [6], and only a few strategies to determine them non-perturbatively have been developed so far.

In the present work we compute non-perturbatively the coefficients \(b_\mathrm{m}\), \(b_\mathrm{A}- b_\mathrm{P}\) and the renormalisation parameter \(Z\equiv Z_\mathrm{m} Z_{\mathrm{P}}/Z_{\mathrm{A}}\) in a theory of three sea quark flavours. The methods we use can be traced back to Ref. [7]. In that work, renormalised quark masses were defined both through the PCAC bare quark masses and the subtracted bare Wilson masses. In both definitions \({\mathrm{O}}(a)\) improvement is introduced through the inclusion of all necessary *c*- and *b*-type counter-terms. Combining these results at constant bare gauge coupling provides estimates of \(b_\mathrm{m}\), \(b_\mathrm{A}- b_\mathrm{P}\) and *Z*. This work has been extended in Refs. [8, 9, 10], where results for other improvement coefficients were also reported. These computations were performed in large volumes with (anti)periodic boundary conditions. In parallel, in Ref. [11] (and subsequently in [12]) the method was extended and applied to small physical volumes with Schrödinger functional boundary conditions. These early analyses were carried out in the quenched approximation. More recently, in Ref. [13], \(b_\mathrm{m}\), \({b_\mathrm{A}-b_\mathrm{P}}\) and *Z* were measured in a theory with \(N_\mathrm{f}=2\) sea quarks, employing the Schrödinger functional scheme and working at a constant value of the renormalised coupling, so as to keep the physical extent of the lattice fixed. By thus imposing improvement and renormalisation conditions along a line in lattice parameter space, where all physical scales stay constant, it is ensured that any intrinsic higher-order lattice spacing ambiguities of *b*-coefficients and *Z*-factors vanish uniformly as the continuum limit is approached.

Our strategy follows closely that of Ref. [13]. However, the extraction of the final estimates from our data has been improved by the introduction of several novelties in the data analysis, which enable us to obtain very reliable estimates in the chiral limit. The lattice action we employ consists of the tree-level Symanzik-improved gauge action [14] and the non-perturbatively improved Wilson-clover fermion action [15]. Our simulations are performed in the range of bare couplings, where gauge configurations on lattices with large physical volumes with \(N_\mathrm{f}=2+1\) sea quarks have been generated by CLS [2, 4]. These configurations are suitable for the computation of bare correlation functions, on the basis of which low-energy hadronic quantities can be evaluated. In Ref. [3] bare PCAC quark masses have been computed from these ensembles. To obtain renormalised up, down, and strange quark masses from these bare masses, one also needs the following: (i) The multiplicative mass renormalisation factor \(1/Z_{\mathrm{P}}\) at low energies and its non-perturbative running up to high energy scales; these are known in the Schrödinger functional scheme from Ref. [16]. (ii) The axial current improvement coefficient \(c_{\mathrm{A}}\) and its normalisation constant \(Z_{\mathrm{A}}\), which are known from Refs. [17, 18, 19], respectively. (iii) The improvement coefficient \(b_\mathrm{A}-b_\mathrm{P}\), which is one of the results of this work. (iv) The improvement coefficient \({\bar{b}}_{\mathrm{A}}-\bar{b}_{\mathrm{P}}\), which is particularly difficult to estimate but may be ignored, as it is sub-leading in perturbation theory.

Independent estimates of Symanzik *b*-coefficients, directly computed on CLS ensembles and obtained with a variant of the coordinate space method of Ref. [20], have been reported in [21]. A comparison of these results to ours may be found in Sect. 5. Preliminary results of the present work have been reported in Ref. [22]. For recent determinations of *b*-coefficients in the vector channel of three-flavour QCD with the same lattice action, see also Refs. [23, 24].

## 2 Quark mass renormalisation and improvement with Wilson fermions

*i*(\(i=1, \ldots , N_\mathrm{f}\)),

*a*is the lattice spacing. In terms of the subtracted masses \(m_{\mathrm{q},i}\), the \({\mathrm{O}}(a)\) improved, renormalised quark mass is given by

^{1}In a non-perturbative determination at non-zero quark mass, they are affected by \({\mathrm{O}}(am_{\mathrm{q},i})\) and \({\mathrm{O}}(a\mathrm {Tr}{[}M_\mathrm{q}{]})\) systematic effects, which are part of their operational definition. They have the following properties:

^{2}

- (i)The \((r_{\mathrm{m}}-1)\)-term multiplies \(\mathrm {Tr}{[}M_\mathrm{q}{]}\), so it arises from a mass insertion on a quark loop; i.e., from diagrams like where the filled square indicates a mass insertion. It is a two-loop effect, contributing at \({\mathrm{O}}(g_0^4)\). Its determination is beyond the scope of this paper.
- (ii)The \(b_\mathrm{m}\)-term multiplies \(m_{\mathrm{q},i}^2\). Thus it arises from: (a) the mass dependence of valence quark propagators and (b) the mass-independent contributions of the fermion loops. The former dependence begins at tree-level, so that \(b_\mathrm {m}=-1/2+{\mathrm{O}}(g_0^2)\), corresponding to Feynman diagrams like The latter dependence begins at two loops, contributing at \({\mathrm{O}}( g_0^4)\); cf. for example diagram The quenched \(b_\mathrm{m}\)-value differs from the one of the full \(N_\mathrm{f}\) theory by the mass-independent contributions of the fermion loops; consequently the difference arises at two loops and is \({\mathrm{O}}( g_0^4)\).
- (iii)The \({\bar{b}}_{\mathrm{m}}\)-term multiplies \(m_{\mathrm{q},i}\mathrm {Tr}{[}M_\mathrm{q}{]}\). The factor of \(m_{\mathrm{q},i}\) comes from the valence line, while the \(\mathrm {Tr}{[}M_\mathrm{q}{]}\) from a quark loop. It thus begins at two loops in perturbation theory (\({\bar{b}}_{\mathrm{m}} \sim {\mathrm{O}}( g_0^4)\)) and vanishes in the quenched approximation; e.g., diagram The determination of the \({\bar{b}}_{\mathrm{m}}\)-term is beyond the scope of this paper.
- (iv)The \((r_{\mathrm{m}} d_{\mathrm{m}} - b_\mathrm{m})\)-term multiplies \(\mathrm {Tr}{[}M_\mathrm{q}^2{]}\); so it must arise from two insertions of \(m_{\mathrm{q},i}\) on a single sea-quark loop. This combination begins at two loops, so it is \({\mathrm{O}}( g_0^4)\); cf. diagram But since it arises from sea quark propagators, while \(b_\mathrm{m}\) has tree-level and \({\mathrm{O}}( g_0^2)\) valence-quark contributions, it follows that also \(d_{\mathrm{m}}\) must get tree-level contributions and \({\mathrm{O}}( g_0^2)\) corrections from valence lines. The determination of \((r_{\mathrm{m}} d_{\mathrm{m}} - b_\mathrm{m})\) is beyond the scope of this paper.
- (v)The \((r_{\mathrm{m}} {\bar{d}}_{\mathrm{m}} - {\bar{b}}_{\mathrm{m}})\)-term multiplies \(\mathrm {Tr}{[}M_\mathrm{q}{]}^2\), so it can only arise from mass insertions on two separate quark loops. Thus this term begins at three loops and is \({\mathrm{O}}( g_0^6)\); cf. diagram But since \({\bar{b}}_{\mathrm{m}}\) itself begins at two loops, so must \(\bar{d}_{\mathrm{m}}\). The determination of \((r_{\mathrm{m}} {\bar{d}}_{\mathrm{m}} - \bar{b}_{\mathrm{m}})\) is beyond the scope of this paper.

*defined*through the following relation:

*i*,

*j*denoting two distinct flavours. The pseudoscalar density \(P^{ij}\) and the current \((A_{\mathrm{I}})^{ij}_\mu \equiv A^{ij}_\mu +ac_\mathrm{A}\partial _\mu P^{ij}\) are Symanzik-improved in the chiral limit, with the improvement coefficient \(c_\mathrm{A}(g_0^2)\) being in principle only a function of the gauge coupling.

^{3}

The operator \({\mathcal {O}}^{ji}\) is defined in a region of space-time that does not include the point *x*, thus avoiding contact terms. Our specific choice of correlation functions for Eq. (2.3) is discussed in Appendix B.

*b*-type Symanzik counter-terms. The renormalised and \({\mathrm{O}}(a)\) improved axial current and pseudoscalar density are given by [25, 26]

- (vi)The \(b_\mathrm{A}\)- and \(b_\mathrm{P}\)-terms multiply \(m_{\mathrm{q},ij}\). Thus they arise from: (a) the mass dependence of valence quark propagators and (b) the mass-independent contributions of the fermion loops. The former dependence begins at tree-level, so that \(b_\mathrm{A}, b_\mathrm{P}\, = \, 1/2 + {\mathrm{O}}( g_0^2)\); cf. diagrams The latter dependence begins at two loops, contributing at \({\mathrm{O}}( g_0^4)\); cf. diagram The difference \(b_\mathrm{A}- b_\mathrm{P}\), appearing in Eq. (2.8) below, is therefore \({\mathrm{O}}( g_0^2)\). The quenched values differ from those of the full \(N_\mathrm{f}\) theory by the mass-independent contributions of the fermion loops; consequently the difference arises at two loops and is \({\mathrm{O}}( g_0^4)\).
- (vii)The \({\bar{b}}_{\mathrm{A}}\)- and \({\bar{b}}_{\mathrm{P}}\)-terms multiply \(\mathrm {Tr}{[}M_\mathrm{q}{]}\). They arise from the mass dependence of quark fermion loops. They begin at two loops in perturbation theory (\({\bar{b}}_\mathrm{A}, {\bar{b}}_{\mathrm{P}} \sim {\mathrm{O}}( g_0^4)\)) and vanish in the quenched approximation; cf. diagram The determination of these coefficients is beyond the scope of this paper.

*b*-coefficients listed above Eqs. (2.2), (2.5), (2.6) and (2.8) also apply to the non-unitary theory, where valence and sea quarks of the same flavour have different masses. We saw that all terms containing traces of the fermion matrix refer to sea quarks, while the others refer to valence quarks. This is shown somewhat more explicitly in Appendix A. The present and previous works, such as Ref. [13], rely on this property in order to obtain reliable non-perturbative estimates of the Symanzik coefficients \(b_\mathrm{m}\) and \(b_\mathrm{A}-b_\mathrm{P}\), as well as the combination

### 2.1 Non-perturbative determination of \(b_\mathrm{m}\), \(b_\mathrm{A}-b_\mathrm{P}\) and *Z*

*Z*from the ratios

Note that the leading improvement coefficients obtained from \(R_\mathrm{AP}\) and \(R_\mathrm{m}\) suffer from mass-dependent \(\mathrm{O}(a)\) effects, which introduce only \({\mathrm{O}}(a^2)\) uncertainties in the quark masses; cf. Eqs. (2.2) and (2.8). The \(\mathrm{O}(a \mathrm {Tr}{[}M_\mathrm{q}{]})\) effects appearing on the r.h.s. of Eqs. (2.11a), (2.11b) and (2.11c) arise from the presence of a residual non-zero sea quark mass in realistic simulations. This uncertainty is removed once simulations are performed for several sea quark masses and the chiral limit is reached by extrapolation. In our setup we are also able to simulate negative current quark masses, so in practice interpolation to the chiral limit is also possible. In the chiral limit, the leading normalisation factor *Z* of the estimator \(R_Z\) suffers from \(\mathrm{O}\big (a^2\big )\) effects; this is easily derived from Eqs. (2.10) and (2.11c).

The above ratios are not the only possible estimators of the quantities of interest. For example, we can modify \(R_\mathrm{AP}\) and \(R_\mathrm{m}\) by replacing the denominator \(\left( m_{11}-m_{22}\right) \) by any of the following PCAC mass differences: \(2(m_{22}-m_{33})\), \(2(m_{33}-m_{11})\), \(2(m_{22}-m_{12})\), or \(2 (m_{12}-m_{11})\). As shown in Ref. [22], these new ratios also provide estimates of \((b_\mathrm{A}- b_\mathrm{P})\), \(b_\mathrm{m}\) and *Z*, with different finite cutoff effects. In practice these differences were found to be orders of magnitude smaller than other systematic effects. So we have retained the original estimators \(R_\mathrm{AP}\), \(R_\mathrm{m}\) and \(R_{Z}\) in the present work.

As previously stated, improvement coefficients are short distance quantities, which can be determined in small physical volumes, using the Schrödinger functional setup, with \(L^3 \times T\) lattices having periodic boundary conditions in space and Dirichlet boundary conditions in time. Definitions of boundary operators and related correlation functions are given in Appendix B. For reasons explained above, sea quark masses are tuned closely to the chiral limit, in line with the usual ALPHA Collaboration choice of mass-independent renormalisation schemes.

### 2.2 The strategy revisited

In Refs. [11, 13] the computation of the Symanzik *b*-coefficients proceeds as follows: the PCAC quark masses \(m_{11}\), \(m_{22}\), \(m_{12}\) and \(m_{33}\) are first determined in standard fashion from Eq. (B.3), and are subsequently fed into the ratios \(R_X\) (with \(X=\mathrm{AP}, \mathrm{m}, Z\)) defined in Eqs. (2.11a), (2.11b) and (2.11c). In some cases, this procedure turns out to suffer from numerical instabilities: the numerators of \(R_X\) are current quark mass differences, i.e., constructed so that the leading contributions in powers of the lattice spacing *a* cancel, isolating the *b*-counter-terms. If these delicate cancellations in the mass differences occur with inadequate precision, the signal may be lost to the noise. Moreover, as we decrease the heavier quark mass \(m_{\mathrm{q},2}\) towards \(m_{\mathrm{q},1}\), striving to reduce discretisation effects, both numerator and denominator of the three estimators \(R_X\) will contrive to give a noisy signal. We will show in Appendix C (cf. Fig. 10) examples of this instability. In order to overcome this problem, we introduce here a more elaborate method of analysis of quark masses evaluated from correlator measurements, which should ameliorate the stability of the results.

*exactly*to their critical value, we would have \(C_{00} = 0\) and \(C_{01} = Z\). Moreover, all \(C_{nk}\) would be functions of only \(g_0^2\) in perturbation theory. A non-perturbative determination of the \(C_{nk}\), however, depends on the imposed improvement condition.

*x*-axis of Fig. 3. The heavier mass is \(m_{\mathrm{q},2}\) and the average of the two is \(m_{\mathrm{q},3}\), i.e., we have \(m_{\mathrm{q},2}> m_{\mathrm{q},3} > m_{\mathrm{q},1}\). Starting from Eq. (2.12), we now write the current quark masses \(m_{11}, m_{22}\) and \(m_{12}\) as power series, with \(m_{22}\) and \(m_{12}\) re-expressed in terms of \(m_{\mathrm{q},1}\) and the (partially-quenched) mass-splitting \(\Delta \equiv \tfrac{1}{2}(m_{\mathrm{q},2} - m_{\mathrm{q},1}) = m_{\mathrm{q},3} - m_{\mathrm{q},1}\):

The original estimator \(R_Z\) was constructed in order to cancel an \(\mathrm{O}(a\Delta )\) effect [11], i.e., to reduce the largest bias in the determination of the leading order factor *Z*. In the unquenched theory, uncancelled \(\mathrm{O}(a\mathrm {Tr}{[}M_\mathrm{q}{]})\) effects remain in all quantities. They are typically supressed by 1–2 orders in magnitude due to the sea quark mass tuning (\(m_{11}\approx 0\)) required in a mass-independent renormalisation scheme. In the determination of the estimators \(R_X\), a renormalised trajectory, or line of constant physics (LCP), has to be employed such that they adopt the proper scaling behaviour when the bare gauge coupling \(g_0^2\) is varied. This means that, besides \(m_{11}=0\), a value for the mass-splitting \(\Delta \), which fixes the LCP in the valence sector, has to be specified. In principle, any sensible choice \(\Delta \ne 0\) is sufficient to define a valid set \(\{R_\mathrm{AP},R_{\mathrm{m}},R_{Z}\}_\Delta \) that achieves \(\mathrm{O}(a)\) improvement in physical quantities. Different choices lead to somewhat different approaches to the continuum limit and are equivalent in the framework of Symanzik’s effective theory. Their relative difference is a higher-order cutoff effect that vanishes for \(a\rightarrow 0\) as can be easily seen in Eq. (2.21). A non-perturbative determination of these estimators inherits a non-trivial all-order dependence on \(a\Delta \) if the limit \(\Delta \rightarrow 0\) is not taken. In that sense, an explicit choice of \(\Delta \) constitutes an ambiguity in their definition. In Ref. [13], for instance, \(\Delta \) has been held constant by requiring \(Lm_{22}\approx 0.5\) at constant physical *L* with \(L/a\in [12,24]\).

In the present paper we aim at eliminating this \(\Delta \)-ambiguity in the definition of \(R_{\mathrm{AP}}\), \(R_{\mathrm{m}}\) and \(R_{Z}\), because it can potentially lead to larger cutoff effects in the physics of light quarks. By noting that Eqs. (2.18) and (2.19) for \(am_{ij}\) can literally be used as joint ansatz for interpolating fit functions (polynomials in \(a\Delta \)), we are able to build the standard estimators according to Eq. (2.21) by dropping the \(\mathrm{O}(a\Delta )\) terms explicitly. In this case only the first few parameters of the polynomials are relevant, which can be well controlled by sufficiently scanning the diagonal and non-diagonal current quark masses as functions of \(a\Delta \). The sub-leading effects in the sea quark mass of order \(a\mathrm {Tr}{[}M_\mathrm{q}{]}\) can be removed by extrapolation or interpolation. The presented proposal has the additional advantage that no iterative tuning of the second (and subsequently third) mass is required in advance.

In Sect. 4 we will elaborate on both variants of the data analysis and the achieved control over the systematic effects.

## 3 Gauge configuration ensembles

Overview of the simulation parameters of the \(N_\mathrm{f}=3\) ensembles (labeled by ID) that represent our data. Subsequent columns refer to the lattice dimensions \(L^3T/a^4\), the inverse gauge coupling \(\beta =6/g_0^2\), the light (sea) quark hopping parameter \(\kappa _1\), the number of replica \(N_{\mathrm{r}}\), the number of configurations per ensemble, both in total (\(N_\mathrm{cfg}\)) and in the subset of configurations with zero topological charge (\(N_\mathrm{cfg}^{(0)}\)), and the corresponding PCAC sea quark masses. All ensembles have configurations separated by 8 molecular dynamic units (MDU), except for A1k3 and D1k4 that have 4 and 16 MDU, respectively. Compared to the data base of [17, 18], we have generated and used the nearly chiral ensembles A1k3, A1k4, B1k4 and D1k4, and significantly increased statistics for E1k1 and E1k2

ID | \(\textstyle {\frac{L}{a}}\) | \(\textstyle {\frac{T}{a}}\) | \(\beta \) | \(\kappa _1\) | \(N_{\mathrm{r}}\) | \(N_\mathrm{cfg}\) | \(N_\mathrm{cfg}^{(0)}\) | \(am_{11}\) | \(am_{11}^{(0)}\) |
---|---|---|---|---|---|---|---|---|---|

A1k1 | 12 | 17 | 3.3 | 0.13652 | 20 | 2560 | 935 | \(-\,0.00166(61)\) | \(-\,0.00278(80)\) |

A1k3 | 12 | 17 | 3.3 | 0.13648 | 5 | 1719 | 614 | \(0.00262(130)\) | \(0.00079(118)\) |

A1k4 | 12 | 17 | 3.3 | 0.13650 | 20 | 12080 | 4424 | \(0.00030(29)\) | \(-\,0.00110(36)\) |

E1k1 | 14 | 21 | 3.414 | 0.13690 | 32 | 4800 | 1694 | \(0.00308(22)\) | \(0.00262(26)\) |

E1k2 | 14 | 21 | 3.414 | 0.13695 | 47 | 7050 | 2653 | \(0.00034(18)\) | \(-\,0.00022(22)\) |

B1k1 | 16 | 23 | 3.512 | 0.13700 | 3 | 3328 | 1336 | \(0.00562(14)\) | \(0.00549(21)\) |

B1k2 | 16 | 23 | 3.512 | 0.13703 | 2 | 1151 | 395 | \(0.00481(19)\) | \(0.00444(25)\) |

B1k3 | 16 | 23 | 3.512 | 0.13710 | 2 | 2048 | 938 | \(0.00164(16)\) | \(0.00107(20)\) |

B1k4 | 16 | 23 | 3.512 | 0.13714 | 1 | 3482 | 1401 | \(0.00002(14)\) | \(-\,0.00057(19)\) |

C1k2 | 20 | 29 | 3.676 | 0.13700 | 4 | 1904 | 857 | \(0.00619(7)\) | \(0.00600(11)\) |

C1k3 | 20 | 29 | 3.676 | 0.13719 | 4 | 1934 | 1249 | \(-\,0.00086(8)\) | \(-\,0.00109(11)\) |

D1k2 | 24 | 35 | 3.810 | 0.13701 | 2 | 803 | 357 | \(0.00084(8)\) | \(0.00079(10)\) |

D1k4 | 24 | 35 | 3.810 | 0.137033 | 8 | 5313 | 3469 | \(-\,0.00002(3)\) | \(-\,0.00007(3)\) |

Most of the ensembles in this study coincide with those of Refs. [17, 18], where the improvement coefficient \(c_\mathrm{A}\) and the normalisation constant \(Z_\mathrm{A}\) of the axial vector current are determined. In these works the constant physics condition is fixed by setting \({L \approx 1.2\,\mathrm{fm}}\). This is achieved by beginning with a particular pair of \(g^2_0\) and *L* / *a* (\(\beta = 6/g^2_0=3.3\) at \(L/a =12\) here). Next we chose the bare couplings for subsequent smaller lattice spacings according to the universal two-loop \(\beta \)-function. In this way, lattice spacings are covered in the range from \(a \approx 0.09\,\mathrm{fm}\) to \(a \approx 0.045\,\mathrm{fm}\). At each bare coupling, we generate ensembles for a few small values of the bare sea current quark mass \(am_{11}\), in order to obtain an estimate of the critical point \(\kappa _{\mathrm{crit}}\) and to be able to extrapolate to \(am_{11}=0\) at a subsequent stage of the analysis.

Table 1 gives an overview of the ensembles used in this work. The labelling of these ensembles, based on an alphanumeric four-symbol code such as A1k1, follows the conventions: the first letter (A–E) represents a specific lattice geometry \(L^3 T/a^4\), while different choices of \(\beta \) for a given geometry are distinguished by the subsequent number. In the present work, we have a single \(\beta \) for each geometry. Separated by a “k”, the final integer labels the sea quark hopping parameter \(\kappa _1 = \kappa _\text {sea}\). In addition to the ensembles available to us from previous ALPHA Collaboration simulations [17, 18], we have generated ensembles A1k3, A1k4, B1k4, D1k2 and D1k4, with \(\kappa _\text {sea}\) tuned so that the corresponding PCAC masses are closer to the chiral limit. Furthermore, the replica lengths of ensembles E1k1 and E1k2 were increased for larger statistics and a more reliable estimation of autocorrelations.

Note that the values of \(\beta \) are in the same range as those of the large-volume ensembles produced with the same lattice action by the CLS (Coordinated Lattice Simulations) effort [2, 3, 4]. Therefore, our results can be applied in, e.g., determinations of \(\mathrm{O}(a)\) improved phenomenological quantities such as quark masses and decay constants.

### 3.1 Topological charge

Chirally extrapolated LCP-0 results, both for the vanishing topological charge sector, \(\smash {R_X^{(0)}}\), and without zero-charge projection, \(\smash {R_X^{(\text {all})}}\). Only the former are plotted in Fig. 7

\(\beta \) | \(R_\text {AP}^{(0)}\) | \(R_\text {AP}^{(\text {all})}\) | \(R_\text {m}^{(0)}\) | \(R_\text {m}^{(\text {all})}\) | \(R_Z^{(0)}\) | \(R_Z^{(\text {all})}\) |
---|---|---|---|---|---|---|

3.300 | \(-\,0.769(101)\) | \(-\,0.656(55)\) | \(1.303(90)\) | \(1.244(43)\) | 0.7462(56) | 0.7468(28) |

3.414 | \(-\,0.812(53)\) | \(-\,0.770(53)\) | \(0.291(53)\) | \(0.364(44)\) | 0.8762(40) | 0.8719(37) |

3.512 | \(-\,0.515(49)\) | \(-\,0.536(36)\) | \(-\,0.291(39)\) | \(-\,0.177(32)\) | 0.9764(33) | 0.9672(26) |

3.676 | \(-\,0.291(46)\) | \(-\,0.279(37)\) | \(-\,0.671(43)\) | \(-\,0.583(35)\) | 1.0588(31) | 1.0536(23) |

3.810 | \(-\,0.156(20)\) | \(-\,0.144(17)\) | \(-\,0.738(19)\) | \(-\,0.700(18)\) | 1.0882(11) | 1.0866(10) |

Chirally extrapolated LCP-1 results, both for the vanishing topological charge sector, \(\smash {R_X^{(0)}}\), and without zero-charge projection, \(\smash {R_X^{(\text {all})}}\). Only the former are plotted in Fig. 7

\(\beta \) | \(R_\text {AP}^{(0)}\) | \(R_\text {AP}^{(\text {all})}\) | \(R_\text {m}^{(0)}\) | \(R_\text {m}^{(\text {all})}\) | \(R_Z^{(0)}\) | \(R_Z^{(\text {all})}\) |
---|---|---|---|---|---|---|

3.300 | \(-\,0.356(24)\) | \(-\,0.376(10)\) | \(-\,0.025(19)\) | \(-\,0.002(7)\) | 0.7896(36) | 0.7846(16) |

3.414 | \(-\,0.362(13)\) | \(-\,0.363(12)\) | \(-\,0.264(12)\) | \(-\,0.237(10)\) | 0.8992(26) | 0.8950(24) |

3.512 | \(-\,0.227(12)\) | \(-\,0.244(9)\) | \(-\,0.469(11)\) | \(-\,0.429(9)\) | 0.9861(23) | 0.9785(18) |

3.676 | \(-\,0.125(14)\) | \(-\,0.133(12)\) | \(-\,0.643(14)\) | \(-\,0.607(12)\) | 1.0611(23) | 1.0564(17) |

3.810 | \(-\,0.070(7)\) | \(-\,0.071(6)\) | \(-\,0.684(7)\) | \(-\,0.669(6)\) | 1.0884(8) | 1.0871(8) |

Figure 1 shows Monte Carlo histories of the topological charge *Q* and its distribution on three exemplary ensembles. The effect of topology freezing is clearly visible: while the topological charge is appropriately sampled for the coarse lattice spacing in the A1 ensembles (top), the HMC algorithm is not able to properly tunnel between different topological charge sectors at finer lattice spacings (\(L\approx \mathrm {const}\)); this becomes most pronounced for the D-ensembles (bottom). Sectors with \(Q=-1,0,1\) are mostly sampled, and the charge remains for a longer Monte Carlo time in single sectors when the lattice spacing is decreased. Such a behaviour is in line with similar findings, e.g., in [17, 18, 19]. As in these references, we thus have confined the analysis to the sector with zero topological charge, in order to avoid potential bias from improper sampling and associated, unresolved large autocorrelation times that could affect a reliable statistical error estimation for our observables. Note that this procedure is also theoretically sound, since our strategy to extract \(b_\mathrm{m}\), \(b_\mathrm{A}-b_\mathrm{P}\) and *Z* relies on PCAC quark masses defined through Ward identities which, being operator relations, hold also within a single topological sector. Although this projection to zero topological charge typically comes at the expense of larger statistical uncertainties and a slightly modified cutoff dependence, it is not expected to induce a noticeable difference in the final results. This is indeed confirmed in Tables 2 and 3.

*O*onto the sector of trivial (\(Q=0\)) topology was introduced in [32, 33] via

### 3.2 Statistical error analysis

The statistical uncertainties are determined using the \(\Gamma \)-method [35, 36], so as to take the autocorrelations of all observables into account. An independent analysis, using jackknife error estimation, was done as follows. First the replica of an ensemble are concatenated and subsequently subdivided into bins of width ten. Then, the standard jackknife error is computed by eliminating a single bin average at a time. The bin width was specified by varying the bin size and choosing the minimal value at which the jackknife error stabilises. Error estimates from both methods are in very good agreement.

The chiral extrapolations discussed in the next section are based on independent datasets of ensembles belonging to the same group (e.g. of A1k1, A1k3 and A1k4 belonging to the A1-group). In the context of the jackknife error analysis, we exploit the embedding trick for combining statistically independent runs, described in Appendix A.3 of [37].

## 4 Data analysis

The definitions of the Schrödinger functional (SF) correlation functions \(f_\mathrm{A}\), \(f_\mathrm{P}\) and the PCAC quark masses \(m_{ij}\) are standard; therefore, we defer all details to Appendix B.

For each ensemble we compute valence quark propagators for \(m_{\mathrm{q},1}\) (which is fixed by the sea quark hopping parameter of the simulation) and for \(\mathrm{O}(15)\) values of \(m_{\mathrm{q},2}\) in the range \({0\le L\Delta _{22}\le 1}\), as well as for the corresponding \(m_{\mathrm{q},3}\equiv \tfrac{1}{2}(m_{\mathrm{q},1}+m_{\mathrm{q},2})\). Earlier approaches [11, 13] relied upon the three distinct masses, in order to evaluate the estimators \(R_X\) by direct use of Eq. (2.11). In the strategy adopted in the present work, \(m_{33}\) is simply another current quark mass diagonal in flavour, so it is on an equal footing with \(m_{22}\), thereby enriching the density of points in the low mass region. Results from the earlier method and the issues related to it are discussed in Appendix C.

The SF correlation functions for the appropriate mass combinations are obtained be utilizing the “\(\texttt {sfcf}\)” program [38].

We compute the PCAC masses \(m_{11}\), \(m_{22}\), \(m_{12}\) and \(m_{33}\) for each time-slice \(x_0\) using the improved lattice derivatives of Eq. (B.4). Then we average over the middle third of the time extent \(T=(3/2)L\), i.e., \(x_0/a\in [L/(2a),L/a]\) (keeping the physical plateau length constant); an example is shown in Fig. 9. The more standard choice of the mass definition at \(x_0=T/2\) with standard derivatives has been taken into account for comparison.

### 4.1 Interpolating functions for PCAC quark masses

We proceed by applying the method described in Sect. 2.2. With the measurements done as explained above, we have \(\mathrm{O}(15)\) estimates of flavour non-diagonal masses \(m_{12}\) and \(\mathrm{O}(30)\) estimates of diagonal ones \(m_{22}\) and \(m_{33}\). An example is shown in Fig. 2, where data points for diagonal and non-diagonal PCAC masses are plotted as functions of \({\Delta \equiv \tfrac{1}{2}(m_{\mathrm{q},2}-m_{\mathrm{q},1})}\) for the nearly chiral ensemble B1k4. To simplify the notation, all quantities connected with and derived from the diagonal masses \(m_{22}\) and \(m_{33}\) will henceforth be denoted by the subscript “22”.

Following Eqs. (2.18) and (2.19), we minimise for the parameters of two polynomials of a given degree (\(am_{11}, N_1, N_2, \ldots , D_2, \ldots \)), constrained to have the same intercept and related first derivatives. In order to avoid over-constraining our fits, we prefer treating \(am_{11}\) as a free parameter, rather then keeping it fixed to its measured mean value.

We have opted for third-order polynomials. From Eqs. (2.21a)–(2.21c) we see that for the determination of the estimators at the unitary point (i.e., \(a\Delta =0\)) we only need the polynomial coefficients up to second order. By fitting with third-order polynomials we take into account possible higher-order effects, without contaminating the lower order coefficients. The dependence of the results on the polynomial order is investigated in Sect. 4.3.2.

In the two upper panels of Fig. 2 we display the difference between the interpolating fit and the data points for the diagonal and non-diagonal masses, respectively. Comparing the deviation of the mean values from zero, we see that the combined fit is an excellent representation of the data down to \(10^{-5}\).

### 4.2 Estimators from the interpolation method

For the unitary case LCP-0 with \(a\Delta =0\) (or, equivalently, \(L\Delta _{22}=0\)), these estimators are built from the parameters \(N_1\), \(N_2\) and \(D_2\) according to Eqs. (2.21). We gather these results for all ensembles in Tables 5 and 6 of Appendix D.

The next step towards a fully massless calculation is to extrapolate the results to the chiral limit \(m_{11}=0\); note that the knowledge of the exact value of the critical hopping parameter is not required here. Since an ensemble with a small negative \(m_{11}\) exists for all gauge couplings in the trivial topological sector with \(Q=0\) (cf. Table 1), this leads in practice to chiral *inter*polations rather than extrapolations. The coefficients \(N_i\) and \(D_i\) of the expansions (2.18) and (2.19) have an implicit dependence on \(am_{11}\), which only affects \({\mathrm{O}}(a^2)\) terms of these expansions. Thus a linear fit appears to be sufficient for our purposes. An example is reproduced for ensemble group B1 (\(\beta =3.512\)) in Fig. 3, where chiral interpolations of \(R_Z\) are presented.^{4} Results are listed in Table 2, for all topological sectors and after projecting to \(Q=0\).

We repeat the full analysis for LCP-1, i.e., \(L\Delta _{22}=1\). The errors of the respective estimators are typically smaller than in the unitary case LCP-0, cf. Tables 7 and 8 of Appendix D. The interpolation to vanishing sea quark mass \(m_{11}=0\) is also illustrated in Fig. 3 for \(R_Z\) of the B1 ensembles, where one can infer that the difference of the two chiral (red) points is statistically significant and represents an \(a\Delta \)-ambiguity. The results for all chiral estimators of LCP-1 are given in Table 3.

### 4.3 Ambiguity checks

#### 4.3.1 Standard vs. improved derivatives

#### 4.3.2 Degree of the polynomial fits

#### 4.3.3 Determination with non-unitary valence quark masses

*a*/

*L*roughly scales with an integer power of the ratio of \(L\Delta _{22}\), i.e., 1 / 4 for \(X=\mathrm{AP}\) and \((1/4)^2\) for \(X=Z\).

We note in passing that there is also an implicit dependence of the results on our choice of mass range \(0\le L\Delta _{22} \le 1\).

## 5 Results

Based on our non-perturbative calculation of the estimators \(R_X^{(0)}\) listed in Tables 2 and 3, we now provide interpolating formulae to make them accessible also at other values of the gauge coupling.

Interpolated values of our estimators for couplings employed in CLS simulations along the two renormalised trajectories LCP-0 and LCP-1 considered in this work. Statistical uncertainties are as described in the text and match the confidence band in Fig. 7

LCP-0 | LCP-1 | |||||
---|---|---|---|---|---|---|

\(\beta \) | \(R_\mathrm{AP}\) | \(R_\mathrm{m}\) | \(R_Z\) | \(R_\mathrm{AP}\) | \(R_\mathrm{m}\) | \(R_Z\) |

3.85 | \(-\,0.155(36)\) | \(-\,0.781(38)\) | 1.0975(25) | \(-\,0.073(12)\) | \(-\,0.708(15)\) | 1.0971(18) |

3.70 | \(-\,0.258(42)\) | \(-\,0.640(31)\) | 1.0591(23) | \(-\,0.119(14)\) | \(-\,0.630(11)\) | 1.0612(17) |

3.55 | \(-\,0.432(46)\) | \(-\,0.358(47\)) | 0.9937(42) | \(-\,0.196(14)\) | \(-\,0.498(15)\) | 1.0015(30) |

3.46 | \(-\,0.590(53)\) | \(-\,0.044(65)\) | 0.9320(50) | \(-\,0.265(14)\) | \(-\,0.376(17)\) | 0.9468(35) |

3.40 | \(-\,0.726(67)\) | \(+\,0.290(76)\) | 0.8758(52) | \(-\,0.324(17)\) | \(-\,0.266(17)\) | 0.8981(35) |

We have also produced a gauge configuration ensemble at \(\beta =8.0\), in order to evaluate the estimators \(R_X\) in the deeply perturbative region. Even though the physical volume of this ensemble is smaller than the LCP one, these \(\beta =8.0\) results qualitatively agree with our fit ansätze and their shape as \(g_0^2\rightarrow 0\). This corroborates our interpolating fit functions, which asymptotically approach the one-loop perturbative predictions at small gauge couplings.

We cover the \(g_0^2\)-range typical of large-volume computations of bare quark masses, matrix elements and other phenomenological applications. In particular, the \(N_\mathrm{f}= 2+1\) couplings of the CLS effort in Refs. [2, 3, 4, 5, 41] are \({\beta \in \{3.85,3.70,3.55,3.46,3.4\}}\). In Fig. 7 we indicate the CLS \(g_0^2\)-values as vertical dashed lines, and in Table 4 we provide our interpolated results at the corresponding \(\beta \)-values. Note that the smallest coupling employed in the CLS simulations, being slightly outside our range \({\beta \in [3.3,3.81]}\), can be reached by extrapolation of our interpolating functions. In these cases, near the edges of our \(\beta \)-range, the results are more sensitive to the choice of the specific fit ansatz. The covariance matrices of Eqs. (5.3) and (5.5), as well as the statistical errors in Table 4, are large enough to cover the results obtained from different fit ansätze we have tried out. Moreover, it can be seen from Fig. 7 that the error band at the CLS couplings is consistent with the errors of the neighbouring data points. We have verified explicitly that any effect on the error of a typical combination of our estimators coming from the correlations among the \(R_X\)’s (which in principle would have to be taken into account when, for instance, calculating renormalised quark masses) is negligible compared to the inflated statistical uncertainties of the fits. Nevertheless, for the sake of completeness, we quote in Appendix E the correlation matrices among the \(R_X\)’s.

Finally we note that, since \(Z_\mathrm {S}/Z_\mathrm {P}=(Z_\mathrm {m}Z_\mathrm {P})^{-1}=(Z_\mathrm {A}Z)^{-1}\), the (scale independent) ratio of renormalisation constants \(Z_\mathrm {S}/Z_\mathrm {P}\) may be obtained by combining \(R_Z\) from this work with the interpolation formula for \(Z_\mathrm {A}\) from [18, 19]. A direct determination of \(Z_\mathrm {S}/Z_\mathrm {P}\) based on the Ward identity approach is in progress [42].

## 6 Conclusions

The present paper is part of a series of publications dedicated to the non-perturbatively \({\mathrm{O}}(a)\) improved quark mass renormalisation in three-flavour lattice QCD with Wilson fermions. It complements previous determinations of the axial current improvement and normalisation [17, 18, 19] and the renormalisation factor of the pseudoscalar density [16] by a non-perturbative calculation of the improvement coefficients \(b_\mathrm{A}-b_\mathrm{P}\) and \(b_\mathrm{m}\) – multiplying associated additive, quark mass dependent Symanzik counter-terms – as well as of the normalisation factor \(Z\equiv Z_mZ_\mathrm{P}/Z_\mathrm{A}\). We work in the framework of lattice QCD with \(N_\mathrm{f}=3\) flavours of mass-degenerate, non-perturbatively \({\mathrm{O}}(a)\) improved Wilson-clover sea quarks and tree-level Symanzik-improved gluons.

Our computational setup to determine \(b_\mathrm{A}-b_\mathrm{P}\), \(b_\mathrm{m}\) and \(Z_mZ_\mathrm{P}/Z_\mathrm{A}\) consists in small physical volume simulations, with Schrödinger functional boundary conditions, and exploiting the PCAC relation with mass non-degenerate valence quarks. Valence quark masses and lattice volumes have been varied ensuring that we approach the chiral and continuum limits while staying on a line of constant physics. Although we have based our work on an earlier \(N_\mathrm{f}=2\) publication [13], we have extended that analysis by introducing a series of novelties as explained in the main part of the paper. The final results obtained refer to massless sea quarks. These can be inferred from Tables 2 and 3, together with the formulae (5.1), (5.2) and (5.4), which provide smooth parameterisations of \(b_\mathrm{A}-b_\mathrm{P}\), \(b_\mathrm{m}\) and \(Z_mZ_\mathrm{P}/Z_\mathrm{A}\) in terms of the bare gauge coupling squared. Several checks have been performed to address the various systematics involved and to guarantee the stability of the analysis strategy as well as the reliability of the quoted error estimates.

Since our range of couplings matches that of the large-volume CLS simulations [2, 3, 4, 5], our results are currently being applied in a \((2+1)\)-flavour computation of light, strange and charm quark masses (see [43, 44] for a preliminary account). They are also useful in the computation of other physical quantities, such as certain combinations of QCD matrix elements involving the axial current and the pseudoscalar density. Our results for \(b_\mathrm{A}-b_\mathrm{P}\), \(b_\mathrm{m}\), and *Z* for the specific bare couplings of the CLS ensembles are collected in Table 4.

## Footnotes

- 1.
In an improved mass-independent scheme, the coupling is to be defined as \({\tilde{g}}_0^2 \equiv g_0^2 (1+b_\mathrm {g}a\mathrm {Tr}{[}M_\mathrm{q}{]})\), see Ref. [25]. As this coupling redefinition affects

*b*-counter-terms by \({\mathrm{O}}(a^2)\) terms, we need not take it into consideration in the present work. - 2.
The first three properties enumerated below are discussed in Ref. [26], while those of the last two are due to S.R. Sharpe (private communication).

- 3.
To be precise, for the divergence of the improved axial current we use \({\partial _\mu (A_{\mathrm{I}})^{ij}_\mu \equiv {\widetilde{\partial }^{}_{\mu }}A^{ij}_\mu +ac_\mathrm{A}\partial _\mu ^*\partial _\mu P^{ij}}\), where \({\widetilde{\partial }^{}_{\mu }}\) denotes the average of the usual forward and backward derivatives defined as \({a\partial _\mu f(x) \equiv f(x+a{\hat{\mu }}) - f(x)}\) and \({a\partial _\mu ^*f(x) \equiv f(x) - f(x-a{\hat{\mu }})}\).

- 4.
Alternatively, such an interpolation can first be performed on the individual fit parameters, before combining them into \(R_X\). This leads to exactly the same results in the chiral limit.

## Notes

### Acknowledgements

We wish to thank Maurizio Firrotta for participating in the early stages of this project. A. V. wishes to thank Steve Sharpe for illuminating discussions and the University of Münster for its hospitality. We also like to thank Christian Wittemeier, Fabian Joswig and Carlos Pena for many useful discussions and particularly Fabian for his valuable contributions in extending the set of ensembles used in our computations. Computer resources were provided by the ZIV of the University of Münster (PALMA, PALMA-NG and PALMA-II HPC clusters), the INFN (GALILEO cluster at CINECA) and the HTCondor Cluster of the Institut für Theoretische Physik of the University of Münster. This work is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Research Training Group 11*GRK 2149: Strong and Weak Interactions - from Hadrons to Dark Matter”* (J. H. and S. K.). C. C. K., scholar of the German Academic Scholarship Foundation (Studienstiftung des deutschen Volkes), gratefully acknowledges their financial and academic support.

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