Baryon chiral perturbation theory with Wilson fermions up to $\mathcal{O}(a^2)$ and discretization effects of latest $n_f=2+1$ LQCD octet baryon masses

We construct the chiral Lagrangians relevant in studies of the ground-state octet baryon masses up to $\mathcal{O}(a^2)$ by taking into account discretization effects and calculate the masses up to $\mathcal{O}(p^4)$ in the extended-on-mass-shell scheme. As an application, we study the latest $n_f=2+1$ LQCD data on the ground-state octet baryon masses from the PACS-CS, QCDSF-UKQCD, HSC, and NPLQCD Collaborations. It is shown that the discretization effects for the studied LQCD simulations are at the order of one to two percent for lattice spacings up to $0.15$ fm and the pion mass up to 500 MeV.


Introduction
Over the past decade, lattice quantum chromodynamics (LQCD) has become an indispensable tool in studies of the non-perturbative regime of QCD from first principles [1,2]. As a numerical solution of QCD in the discrete Euclidean space-time in a finite hypercube, its main input parameters are the quark masses mq, the lattice box size L, and the lattice spacing a. Because computing time increases dramatically with decreasing quark masses, most past simulations have been performed with larger-than-physical light-quark masses. As a result, LQCD simulations require multiple extrapolations to the continuum (a → 0), to infinite space-time (L → ∞), and to the physical point with physical quark masses (mq → m phys. q ).
For many observables, these extrapolations have led to uncertainties comparable to or even larger than the inherent statistical uncertainties. Recently, simulations with physical lightquark masses have become available (see, e.g., Refs. [3,4]), which (will) largely reduce the systematic uncertainties related to chiral extrapolations to the physical light-quark masses.
It has long been employed to perform chiral extrapolations of and to study finite-volume corrections to LQCD simulations. Both of them are important for LQCD simulations. On the other hand, LQCD simulations with varying light-quark masses and lattice volume are extremely useful to help to fix the (sometimes many) unknown low-energy constants (LECs) of ChPT, which otherwise are difficult if not impossible to be determined. To apply ChPT to the study of LQCD simulations, in principle, one should first take the continuum limit of LQCD data, since ChPT describes the continuum QCD and is not valid for nonzero lattice spacing. However, nowadays it is a common practice to assume that lattice spacing arti- 3 facts for current LQCD setups of a ≈ 0.1 fm are small and can be treated as systematic uncertainties.
In order to study discretization effects on LQCD simulations, one can first write down Symanzik's effective field theory [17][18][19][20], a continuum effective field theory (EFT) which describes the lattice field theory close to the continuum limit, and then one can extend ChPT to be consistent with this EFT with additional symmetry breaking parameters. In this way, the chiral expansion results can naturally encode lattice spacing effects (see, e.g. Ref. [21]). Sharpe and Singleton [22] and Lee and Sharpe [23] first extended ChPT to include finite lattice spacing effects up to O(a) for Wilson fermions [1] (WChPT) and staggered fermions [24,25] (SChPT), respectively. Later, Munster and Schmidt [26] applied WChPT to the study of discretization artifacts of twisted mass fermions (tmChPT) [27,28].
In the past decade, discretization effects on the ground-state meson/baryon properties, such as masses, decay constants, electromagnetic form factors, etc., have been extensively studied in WChPT. 1 In the mesonic sector, the masses and decay constants of the Nambu-Glodstone mesons were first studied up to O(m 2 q ) and O(a) for the Wilson action [44] and for the mixed action [45], where Wilson sea quarks and Ginsparg-Wilson valence quarks are employed. These studies were subsequently extended to next-to-leading order (up to O(a 2 )) [46,47]. In the one-baryon sector, a systematic study of the nucleon properties up to O(a) was first performed by Beane and Savage for both the mixed and the unmixed action [48]. The electromagnetic properties of the octet mesons as well as of the octet and decuplet baryons were also studied up to O(a) for both the mixed and the unmixed action [49]. Discretization effects on the nucleon and ∆ masses [50] as well as on the vector meson masses [51] were also studied up to O(a 2 ). The EFT for the anisotropic Wilson lat- 1 We focus in this work on WChPT, but it should be noted that similar studies have been performed in  and tmChPT [37][38][39][40][41][42][43]. 4 tice action has been formulated up to O(a 2 ) [52] as well. In this context, it is interesting to note that recently several attempts have been made to determine the unknown LECs of WChPT [53][54][55][56][57].
In the past few years, fully dynamical n f = 2 + 1 simulations in the one-baryon sector have become available. The ground-state octet baryon masses might be one of the simplest observables to simulate in such a setting and serve as a benchmark for more sophisticated studies [58][59][60][61][62][63][64][65][66]. Many theoretical studies have been performed not only to understand the chiral extrapolations of and the finite-volume corrections to these simulations, but also to determine the many unknown LECs appearing in ChPT up to next-to-next-to-next-to-leading order (N 3 LO) [60,[67][68][69][70][71][72][73][74][75][76]. In Ref. [75], it is shown that the covariant baryon chiral perturbation theory (BChPT) together with the extended-on-mass-shell (EOMS) scheme [77,78] can describe reasonably well all the n f = 2 + 1 LQCD data. Nevertheless, discretization effects are ignored in all these studies, with the argument that they should be small. 2 In this work, we aim to study the discretization effects of the LQCD simulations of the ground-state octet baryon masses up to O(a 2 ) in covariant BChPT with the EOMS renormalization scheme. Although most of the LQCD simulations are performed at a single lattice spacing, a combination of the results from different collaborations enables one to examine finite lattice spacing effects by performing a global study. We limit ourselves to the unmixed action and, therefore, we will study those simulations based on the O(a)-improved Wilson action [19], i.e., those of the PACS-CS [59], QCDSF-UKQCD [65], HSC [61], and NPLQCD [66] Collaborations. 2 In Ref. [79], Alvarez-Ruso et al. performed a phenomenological study of the continuum extrapolation of the LQCD simulations of the nucleon mass by considering only O(a 2 ) terms, and they showed that finite-volume corrections and finite lattice spacing effects are of similar size. In our present work we will see that they are indeed of similar size, but the O(amq ) contributions are larger than the O(a 2 ) ones.

5
The paper is organized as follows. In Sect. 2, the Symanzik action up to O(a 2 ) is briefly introduced and the a-dependent chiral Lagrangians relevant to the study of the ground-state octet baryon masses are constructed. In Sect. 3, the discretization effects on the ground-state octet baryon masses are formulated up to O(a 2 ) for Wilson fermions. As an application, we then perform a simultaneous fit of the LQCD octet baryon masses and study the discretization effects. A short summary is given in Sect. 4.

BChPT at finite lattice spacing
In this section, we briefly review the continuum effective action up to and including O(a 2 ).
We will follow closely the procedure and notations of Ref. [50] and construct for the first time the chiral Lagrangians incorporating a finite lattice spacing for the Wilson action in the u, d, and s three-flavor one-baryon sector.

Continuum effective action
Close to the continuum limit, LQCD can be described by an effective action, the 'Symanzik action' [17,18], which is expanded in powers of the lattice spacing a as where L (4) is the normal (continuum) QCD Lagrangian and the two new terms L (5) and L (6) are introduced to include the discretization effects of LQCD. The Lagrangian L (5) contains chiral breaking terms only, while L (6) contains both chiral invariant and breaking terms. In the u, d, and s three-flavor sector, the QCD Lagrangian is 6 where the quark masses are encoded in a diagonal matrix M = diag(m l , m l , ms) in the isospin limit (mu = m d ≡ m l ), and / D = Dµγ µ with Dµ the covariant derivative.
At O(a), there is only the Pauli term left by using the equations of motion to redefine the effective fields [20] aL (5) where Gµν = [Dµ, Dν ] and c SW is the Sheikholeslami-Wohlert (SW) [19] coefficient that must be determined numerically. The ωq (q = u, d, s) is a constant which is determined by the kind of lattice fermions employed in LQCD simulations: ωq = 1 for Wilson fermions [1] and ωq = 0 for Ginsparg-Wilson (GW) fermions [80]. Similar to the quark masses, the ωq's are usually collected in the Wilson matrix W = diag(ω l , ω l , ωs) with conserved isospin symmetry (ωu = ω d ≡ ω l ). This term breaks chiral symmetry in precisely the same way as the quark mass term. It should be noted that the Pauli term can be canceled by adding the clover term to the lattice action [47], resulting in the O(a)-improved Wilson fermion action [19,20,81,82].

Wilson chiral Lagrangians
In order to construct the chiral Lagrangians of WChPT, one has to write down the most general Lagrangians that are invariant under the symmetries of the continuum EFT. This can be done by following the standard procedure of spurion analysis [46,47]. In practice, in order to obtain the corresponding a-dependent chiral Lagrangians, one only needs to know which symmetries are broken and how [50]. Before writing down the chiral Lagrangians up to O(a 2 ), one has to first specify a chiral power-counting scheme, which should be enlarged to include the lattice spacing a. In LQCD simulations, the following hierarchy of energy scales is satisfied: If one assumes that the size of the chiral symmetry breaking due to the light-quark masses and the discretization effects are of comparable size, as done in Refs. [47,48,50], one has the following expansion parameters: where p denotes a generic small quantity and Λ QCD ≈ 300 MeV denotes the typical low energy scale of QCD. Up to O(a 2 ), the a-dependent chiral Lagrangians contain terms of O (a, amq, a 2 ) and can be written as where The chiral Lagrangian at O(a) can be written as 9 whereb 0 ,b D , andb F are the unknown LECs of dimension mass −1 , X stands for the which introduces explicit chiral symmetry breaking because of the finite lattice spacing a. The The O(amq) Lagrangian has the following form: whereb 1,...,12 are unknown LECs of dimension mass −3 and χ + = u † χu † + uχ † u, where χ = 2B 0 M accounts for explicit chiral symmetry breaking with B 0 = − 0|qq|0 /F 2 φ . One can eliminate theb 3 term by use of the following identity valid for any 3 × 3 matrix A derived from the Cayley-Hamilton identity [83]: 3 The operator ρ + transforms under chiral rotation (R), parity transformation (P), charge conjugation transformation (C) and hermitic conjugation transformation in the following way: ρ + where 'perm' stands for permutation number. In the end, there are 11 independent terms left.
At O(a 2 ), the previous five operators in the Symanzik action can be mapped into the EFT with five classes of chiral Lagrangians L O(a 2 ) i (i = 1, . . . , 5). Following the notation of Ref. [50], the first class of chiral Lagrangians can be written as where the operator O + is defined as Because the second type of operators have an insertion of the quark mass mq, the chiral order of the corresponding chiral Lagrangians is at least O(p 6 ), which is beyond the present work and will not be shown.
There are seven independent terms in the third class of chiral Lagrangians where theē i are the unknown LECs of dimension mass −3 . Furthermore, we can eliminate theē 6 term by use of the Cayley-Hamilton identity [83]: Four-quark operators that break chiral symmetry can be mapped into the following chiral Lagrangian: with the seven unknown LECsd i of dimension mass −3 . Because the chiral transformation properties of ρ + and χ + are the same, the chiral Lagrangian has the same form as the corresponding fourth-order chiral Lagrangian of ChPT.
For the O(4) breaking operators, the mapped chiral Lagrangian can be written as where thef i are the unknown LECs of dimension mass −3 . Their contributions to the octet baryon masses can be absorbed by the terms of class one, i.e., Eq. (18).

Discretization effects on the octet baryon masses
In this section, we calculate the discretization effects on the octet baryon masses up to O(a 2 ) It should be stressed that we are not aiming at a precise determination of discretization effects on the octet baryon masses, given the fact that most LQCD simulations are performed at a single lattice spacing. On the contrary, we would like to get a rough estimate of discretization effects and to check whether the results of previous studies [69,70,75,76] are robust, which have neglected these effects.

Octet baryon masses up to O(p 4 )
The octet baryon masses up to N 3 LO and with finite lattice spacing a contributions up to O(a 2 ) can be expressed as where m 0 is the chiral limit octet baryon mass and m Here, we need to mention that virtual decuplet contributions are not explicitly included, since their effects on the chiral extrapolation and the finite-volume corrections are relatively small [76].
In the case of the unmixed Wilson action, where the u, d, and s quarks are all Wilson fermions, the Wilson matrix can be written as W = diag(1, 1, 1). One can easily compute the O(a) contributions of the diagram Fig. 1a to the octet baryon masses, 13 where B = N, Λ, Σ, and Ξ.
The O(amq) contributions can be written as and the coefficients ξ l and ξs are tabulated in Table 1. We have introduced the following combinations of LECs:b 1 +b 2 + 3b 7 + 3b 8 =B 1 ,b 4 − 3b 7 + 3b 8 =B 2 , and 2b 10 + 3b 11 +b 12 =B 3 . Hence, there are 3 independent combinations. In obtaining the above results, the light-quark masses have been replaced by the leading-order pseudoscalar meson The O(a 2 ) contributions are not only from the fourth-order tree-level diagram Fig. 1-( but also from the one-loop diagrams of Fig. 1c whereC =c 1 + 4(3c 2 + 2c 3 ),D = 4d 3 + 9d 7 + 3d 8 , andĒ = 4ē 3 + 9ē 7 + 3ē 8 . We introduceC + 16D + 16Ē = 16X as one free LEC in the fitting process. The second line of Eq. (28) is for the contributions from the tadpole diagram of Fig. 1c, and the corresponding coefficients ξ (c) B,φ are listed in Table 2. The last term is for the contributions from the oneloop diagram of Fig. 1d, and the coefficients ξ (d) BB ′ ,φ can be found in Table 5 of Ref. [75].
is for the leading-order discretization effects of Eq. (26). Table 2 Coefficients of the tadpole diagram contributions to the octet baryon masses (Eq. (28)). It should be noted that both the HSC [61] and the NPLQCD [66] simulations employed the anisotropic clover fermion action [84]. In this action, the temporal lattice spacing is chosen to be much smaller than the spatial lattice spacing. The EFT for such a LQCD setup has been worked out in Ref. [52], which in principle is more appropriate to be employed to study the HSC and NPLQCD simulations. On the other hand, this EFT has to introduce more LECs to discriminate the temporal and spatial lattice spacing effects. As we will see, present limited LQCD data do not allow us to perform such a study. Therefore, in our study we assume that these simulations are performed with a single lattice spacing, as, and we treat the difference between as and a t as higher-order effects.
As in Refs. [76,85], we focus on the LQCD data from the above four collaborations with In the O(a)-improved Wilson action the Pauli term aL (5) is eliminated. As a result, discretization effects originate only from the O(amq) and O(a 2 ) terms. Therefore, only the fourth-order tree-level diagrams contribute, while the leading order tree-level diagram and Table 3 Values of the LECs from the best fit to the LQCD data and the experimental data at O(p 4 ) with and without discretization effects.

BChPT WBChPT
BChPT WBChPT   are in total 23 free LECs that need to be fixed. 4 As in Ref. [75], the meson decay constant is fixed at its chiral limit value F φ = 0.0871 GeV. For the baryon axial coupling constants, we use D = 0.8 and F = 0.46 [86]. The renormalization scale is set at µ = 1 GeV.
In order to study the discretization effects on the octet baryon masses, we perform two fits. First, we use the continuum octet baryon mass formulas to fit the LQCD and experimental data. Second, the mass formulas of Eq. (24) with discretization effects taken into account are employed to fit the same data. In both fits, the finite-volume corrections to the LQCD simulations are always taken into account self-consistently [75]. The LECs, together 4 In our fits, we set W 0 at 1 GeV 3 . Later a more proper value will be used to check the naturalness of the particularly,X are rather large. This shows clearly the need to perform LQCD simulations at multiple lattice spacings in order to pin down more precisely discretization effects, which has long been recognized [87].  Table 3 and Fig. 2. For a lattice spacing up to a = 0.15 fm, the finite lattice spacing effects on the baryon masses are less than 2%, consistent with the LQCD study of Ref. [89].
The above results can be naively understood in the following way. Recall that  Table 3 appear to be small because we have set the dimensional quantity W 0 to be 1 GeV 3 . Its more 'proper' value can be estimated by noting the following relations W 0 a ∼ B 0 mq and M 2 π ∝ 2B 0 mq (in the leading-order ChPT), which yields W 0 ≈ 0.02 GeV 3 . With this value, the LECs turn out to beB 1 = −0.0605 GeV −3 ,B 2 = −0.234 GeV −3 ,B 3 = 0.172 GeV −3 , and X = 0.152 GeV −3 , which are of natural size as expected.

Conclusions
We have studied discretization effects on the octet baryon masses. With the LECs of Wilson ChPT fixed from the best fit, we have also studied the evolution of discretization effects with the lattice spacing and the pion mass. It was shown that the 21 discretization effects on the octet baryon masses are less than 2% for lattice spacings up to 0.15 fm, in agreement with other LQCD studies.
Nevertheless, future lattice simulations performed at multiple lattice spacings will be extremely valuable to pin down more precisely discretization effects (on the octet baryon masses) and to check the validity of Wilson ChPT in the one-baryon sector.