Computing Nucleon Charges with Highly Improved Staggered Quarks
Yin Lin, Aaron S. Meyer, Steven Gottlieb, Ciaran Hughes, Andreas S. Kronfeld, James N. Simone, Alexei Strelchenko
FFERMILAB-PUB-20-551-T
Computing Nucleon Charges with Highly Improved Staggered Quarks
Yin Lin ( 林 胤 ),
1, 2, ∗ Aaron S. Meyer, † Steven Gottlieb, Ciaran Hughes, Andreas S. Kronfeld, ‡ James N. Simone, and Alexei Strelchenko (Fermilab Lattice Collaboration) Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA Fermi National Accelerator Laboratory, Batavia, IL 60510, USA Brookhaven National Laboratory, Upton, NY 11973, USA Department of Physics, Indiana University, Bloomington, IN 47405, USA (Dated: October 21, 2020)This work continues our program of lattice-QCD baryon physics using staggered fermions for boththe sea and valence quarks. We present a proof-of-concept study that demonstrates, for the firsttime, how to calculate baryon matrix elements using staggered quarks for the valence sector. Weshow how to relate the representations of the continuum staggered flavor-taste group SU(8) FT tothose of the discrete lattice symmetry group. The resulting calculations yield the normalizationfactors relating staggered baryon matrix elements to their physical counterparts. We verify thismethodology by calculating the isovector vector and axial-vector charges g V and g A . We use asingle ensemble from the MILC Collaboration with 2+1+1 flavors of sea quark, lattice spacing a ≈ .
12 fm, and a pion mass M π ≈
305 MeV. On this ensemble, we find results consistent withexpectations from current conservation and neutron beta decay. Thus, this work demonstrates howhighly-improved staggered quarks can be used for precision calculations of baryon properties, and,in particular, the isovector nucleon charges.
I. INTRODUCTION
Accurate first-principles calculations of nuclear crosssections are an important objective in the particle physicscommunity. In particular, heavy nuclei, such as C and Ar, are used as targets in neutrino-scattering and dark-matter detection experiments. In calculations of crosssections, a necessary component is the modeling of nucleias a collection of nucleons, opening up an opportunity forlattice QCD [1]. At the quasielastic peak, for example,the electromagnetic and axial-vector form factors of thenucleon, which characterize the electric charge and spindistribution within the nucleon, are key ingredients. Suchform factors can be obtained from the first-principleslattice-QCD framework. However, these hadronic inputsremain one of the largest sources of systematic error asthe experimental precision on these cross-sections contin-ues to improve [2–4].The electromagnetic form factors have been extractedprecisely from high statistics electron-nucleon scatteringexperiments [5, 6]. At zero momentum transfer, the pro-ton’s electric form factor becomes the total electric charge g V = 1, and the slope at the origin is related to the chargeradius. Recently, experiments that make use of the Lambshift of muonic hydrogen report significantly smaller pro-ton radii than those measured via scattering [7]. (For re-cent reviews of the proton radius puzzle, see Refs. [8, 9].)In addition, a recent reanalysis has demonstrated that ∗ [email protected] † [email protected]; present address: UC Berkeley andLawrence Berkeley National Laboratory, Berkeley, CA 94720,USA ‡ [email protected] the vector form factors at intermediate Q also exhibittensions outside of their quoted uncertainties [10]. Thesedisagreements could benefit from better knowledge of theStandard Model predictions, which necessitates using lat-tice QCD to calculate the form factor.In comparison, the nucleon axial-vector form factor ismuch less constrained from experimental data. A recentre-analysis [11] of the deuterium bubble-chamber datafound greater uncertainties than previously assumed.Again, lattice QCD can be illuminating here, computingthe axial-vector form factor from first principles as anindependent check on the form factor extracted from ex-perimental data. At zero momentum transfer, the axial-vector form factor gives the so-called nucleon axial charge g A = 1 . g A could shed light on the neutron lifetime puzzle [13].Lattice-QCD calculations of baryonic observables arehindered by the well-known exponential growth of thenoise relative to the signal, which sets in at largetimes [14, 15]. At early times, where the signal-to-noiseratio is favorable, the lattice-QCD correlator data con-tain significant contributions from several states in an in-finite tower. When using a fit to disentangle the higher-lying states from those of interest, some residual, un-wanted contamination remains in the parameters of in-terest. It is imperative, therefore, to demonstrate controlover both the noise and the excited-state contamination.In this work, we use an ensemble generated by theMILC Collaboration [16], which incorporates a sea withequal-mass up and down quarks, the strange quark,and the charm quark. MILC uses the highly improved a r X i v : . [ h e p - l a t ] O c t staggered-quark (HISQ) action [17] for the sea quarks;here we use the HISQ action for the valence quarks too.Because staggered fermions have only one component persite and retain a remnant chiral symmetry, they are com-putationally efficient. Nevertheless, staggered fermionsare complicated by the fermion doubling problem, lead-ing to four species, known as tastes, for each fermionfield. The four tastes become identical in the continuumlimit, leading to an SU(4 n f ) flavor-taste symmetry for n f flavors. Consequently, the spectrum of staggered latticebaryons is rich and intricate. For nucleons, the spectrumhas been classified [18–20], finding many states that havethe same properties as the physical nucleon.In a recent paper, we used staggered baryons to calcu-late the nucleon mass [20]. Computing nucleon chargesis the next step and a necessary one en route to the fullmomentum dependence of the form factors. As discussedin Ref. [20], it can be advantageous to use unphysicalnucleon-like states to carry out the calculation. Thesestates obtain the same properties as the physical nu-cleon in the continuum limit, where the full SU(8) F T flavor(isospin)-taste symmetry emerges. For matrix el-ements such as charges and form factors, however, onemust find the correct group-theoretic normalization fac-tors relating nucleon-like matrix elements to their phys-ical counterparts. This exercise is a straightforwardif complicated application of the generalization of theWigner-Eckart theorem to SU(8).To demonstrate this approach, we compute the nucleonvector and axial-vector charges on a single MILC HISQensemble with lattice spacing a ≈ .
12 fm and pion mass M π ≈
305 MeV. We employ local vector and axial-vectorcurrents. We also outline the steps needed to apply thismethod to matrix elements of other baryons, with an eyeto future studies including staggered baryons, such as N → ∆ transition form factors.This paper is organized as follows. In Sec. II, we dis-cuss staggered-baryon correlators, starting with a briefreview of the two-point correlator methodology [20]. Wethen present an overview of our three-point correlators.Here, we also present one of the key results of this pa-per: the correct normalization of the nucleon-like matrixelements. In Sec. III, we describe strategies for removingexcited-state contamination. Section IV provides the de-tails of our simulation, while Sec. V describes Bayesianfits to the correlator data. Our computational resultsare presented in Sec. VI, including the robustness ofour results under variations of our fitting procedure, therenormalization of the bare lattice charge to the physi-cal charges, and the final values for g V and g A on thesingle ensemble being used. Finally, we compare our re-sults to mixed-action results on the same ensemble andprovide our conclusions in Sec. VII. Appendices A and Bpresent the group theory relating the nucleon-like matrixelements to their physical counterparts, including a nu-merical demonstration that these derivations are correct. II. STAGGERED BARYON CORRELATORS
For simplicity, we focus here on two flavors, up anddown, with isospin symmetry. With staggered fermions,instead of the usual SU(2) F isospin symmetry, an en-larged SU(8) F T flavor-taste symmetry group emerges inthe continuum limit. It is important to note that the irre-ducible flavor-taste representations contain componentswith non-trivial taste and unexpected isospin. For ex-ample, Bailey has shown [19] that nucleon-like states ex-ist with unphysical isospin yet masses equal in the con-tinuum limit to the physical tasteless nucleon. In fact,all physics of such nucleon-like states can be related tothat of the physical nucleon. In particular, here we showhow to relate nucleon-like matrix elements to their phys-ical counterparts. As such, we are allowed to chooseany nucleon-like representation, for example, one thatreduces the computational complexity.We use the isospin- operators that transform in the16 irrep of the geometric timeslice group (GTS) [18, 21],as presented in Ref. [20]. They are less complicated toanalyze because only a single nucleon-like taste appearsin the spectrum. On the other hand, this irrep containscontributions from three ∆-like tastes. A. Two-point correlators
Using the same notation as in Ref. [20], the two-pointcorrelators read C ( r ,r )2pt = 116 (cid:88) s, (cid:126)D (cid:88) (cid:126)x (cid:68) B ( r ) s (cid:126)D ( (cid:126)x, t ) B ( r ) s (cid:126)D (0) (cid:69) , (2.1)using sink and source operators B ( r ) s (cid:126)D ( (cid:126)x, t ) and B ( r ) s (cid:126)D (0)defined in Ref. [20]. To increase the statistical precision,we average over the eigenvalues s = ± of the staggeredrotation in the x - y plane, and also the eight corners of thecube (cid:126)D ; together, s and (cid:126)D label the components of the16 irrep. Here, r , r = 2 , , , B. Staggered-baryon matrix elements
In this work, we are specifically interested in the isovec-tor nucleon vector and axial-vector charges, namely g V and g A , respectively. These are defined through the nu-cleon matrix elements (cid:104) N | (cid:0) ¯ u Γ J u − ¯ d Γ J d (cid:1) | N (cid:105) = g J ¯ u N Γ J u N , (2.2)where Γ A = γ z γ or Γ V = γ , u and d are continuum-QCD up- and down-quark fields, and u N is the nucleonspinor at zero momentum.We calculate these nucleon matrix elements using(highly improved) staggered quarks. To achieve this, wemust extend the mass relations of Bailey [19] to matrixelements. The baryon-like matrix elements and the phys-ical matrix elements are related through symmetry trans-formations in the continuum. In the appendices, we find the appropriate Clebsch-Gordan coefficients that relatethe single-taste baryon matrix elements to the physicaltasteless QCD matrix elements by applying the general-ized Wigner-Eckart theorem of SU(8) F T .The correctly normalized three-point correlators for our baryon-like operators are then C ( r ,r ) V ( t, τ ) = − (cid:88) (cid:126)D (cid:88) (cid:126)x,(cid:126)y S V ( (cid:126)D ) (cid:16)(cid:68) B ( r ) − (cid:126)D ( (cid:126)x, t ) V ( (cid:126)y, τ ) B ( r ) − (cid:126)D (0) (cid:69) + (cid:68) B ( r )+ (cid:126)D ( (cid:126)x, t ) V ( (cid:126)y, τ ) B ( r )+ (cid:126)D (0) (cid:69)(cid:17) , (2.3) C ( r ,r ) A ( t, τ ) = 116 (cid:88) (cid:126)D (cid:88) (cid:126)x,(cid:126)y S A ( (cid:126)D ) (cid:16)(cid:68) B ( r ) − (cid:126)D ( (cid:126)x, t ) A ( (cid:126)y, τ ) B ( r ) − (cid:126)D (0) (cid:69) − (cid:68) B ( r )+ (cid:126)D ( (cid:126)x, t ) A ( (cid:126)y, τ ) B ( r )+ (cid:126)D (0) (cid:69)(cid:17) , (2.4)where t is the source-sink separation time and τ is the current insertion time. The factor − C V andthe factor − C A come from the group theory just described. Without these factors,these correlators would not yield the desired nucleon charges. For baryon operators and currents in other GTS irreps,different prefactors arise. In Eqs. (2.3) and (2.4), we sum over unit-cube sites (cid:126)D with weights S J ( (cid:126)D ) ( J = V, A ) andhave a separate term for each value of s = ± V ( (cid:126)y, τ ) = S V ( (cid:126)y ) ( ¯ χ u ( (cid:126)y, τ ) χ u ( (cid:126)y, τ ) − ¯ χ d ( (cid:126)y, τ ) χ d ( (cid:126)y, τ )) , S V ( (cid:126)A ) = ( − ( A x + A y + A z ) /a , (2.5) A ( (cid:126)y, τ ) = S A ( (cid:126)y ) ( ¯ χ u ( (cid:126)y, τ ) χ u ( (cid:126)y, τ ) − ¯ χ d ( (cid:126)y, τ ) χ d ( (cid:126)y, τ )) , S A ( (cid:126)A ) = ( − A z /a , (2.6)where χ f is the field in the HISQ action of flavor f . The local vector and axial-vector currents, V and A , havespin-taste γ ⊗ ξ and γ z γ ⊗ ξ z ξ [21], respectively. The opposite-parity partners then arise from spin-taste γ ⊗ ξ and γ z γ ⊗ ξ z ξ , respectively. Because these local currents are not derived from Noether’s theorem, they require afinite renormalization, that is, Z V V and Z A A have the same matrix elements as the continuum isovector currents inEq. (2.2).In the limit τ → ∞ and t − τ → ∞ , the ratio of thethree-point to the two-point correlators approaches thedesired nucleon charge C ( r ,r ) g J ( t, τ ) C ( r ,r )2pt ( t ) τ →∞ −−−−−→ t − τ →∞ (cid:101) g J , (2.7)where (cid:101) g J is now the bare lattice charge, that is g J = Z J (cid:101) g J . In practice, of course, we compute the correlatorsfor several values of t and τ and fit the t and τ dependenceto extract the charges.The finite renormalization factors Z J are determinedfirst by noting that the remnant chiral symmetry re-quires Z A = Z V + O( m q a ) . At zero momentum trans-fer, the vector current simply counts the number of upquarks minus the number of down quarks, for the proton2 − Z V by demanding Z V (cid:101) g V = g V = 1. Here, however, we prefer to define Z V via a similar relation obtained from a pseudoscalar-mesonmatrix element [22], then use the result to renormalizeour nucleon matrix elements. With this choice our resultfor g V is a genuine test of our methodology. III. EXCITED-STATE CONTAMINATION
Excited-state contamination is one of the most diffi-cult challenges when accurately estimating nucleon ma- trix elements from lattice QCD. The problem is evenmore complicated with staggered nucleons because of thepresence of negative-parity and low-lying ∆-like states inthe spectrum, which non-staggered formulations do notcontain. These are both significant sources of excited-state contamination in the present calculation. We have,however, demonstrated control of excited-state contam-ination when extracting nucleon physics from two-pointstaggered-baryon correlators [20]. Here we describe ex-tensions of those techniques to the three-point correlatorsof the present work. In particular, we show how to sup-press contributions from the lowest-lying negative-paritystates and the lowest three ∆-like tastes.
A. Negative parity states
Let C ( t ) and C ( t, τ ) be any staggered-baryon cor-relators. The source-sink separation is denoted t and thecurrent insertion time is denoted τ . Any staggered op-erator that is local in time will create negative paritystates, which in turn causes the characteristic oscillationsin time. This is obvious from the correlators spectral de-composition C ( t ) = z + ¯ z + e − M + t + ( − t/a z − ¯ z − e − M − t + · · · , (3.1) C ( t, τ ) = z + A ++ ¯ z + e − M + t + ( − t/a z − A −− ¯ z − e − M − t + ( − ( t − τ ) /a z + A + − ¯ z − e − M + τ e − M − ( t − τ ) + ( − τ/a z − A − + ¯ z + e − M − τ e − M + ( t − τ ) + · · · , (3.2)where M ± are the lowest-lying ± parity masses, ¯ z ± and z ± are, respectively, the source and sink overlap factorsfor states of parity ± , and A ±± and A ±∓ are the tran-sition matrix elements. For simplicity, we have ignoredbackward propagating terms proportional to e − M ± ( T − t ) ,which are assumed to contribute negligibly in the follow-ing.Equation (3.2) shows that the terms involving negativeparity states change sign when either t/a or τ /a changeby one unit. With this in mind, a time-averaging pro-cedure can be applied to suppress the negative paritycontributions to the correlator. A similar scheme wasdeployed in Ref. [23]. The first ingredient is C (cid:48) ( t ) = e − aM snk C ( t ) + C ( t + a ) , (3.3) C (cid:48) ( t, τ ) = e − aM snk C ( t, τ ) + C ( t + a, τ ) , (3.4)where we call aM snk the time-averaging parameter. Sub-stituting this expression into the spectral decompositionin Eq. (3.2), one sees that the functional forms of primedcorrelators are unchanged except that the sink overlapfactors becomes z + → z + (cid:0) e − aM snk + e − aM + (cid:1) , (3.5) z − → z − (cid:0) e − aM snk − e − aM − (cid:1) . (3.6)If one chooses aM snk = aM − , then terms with the M − state at the sink will vanish, while the overlap factorsfor the positive parity states become slightly larger. Inpractice, the time-averaging parameter does not need tobe exact to suppress the negative-parity states.A similar time-averaging parameter, aM src , can be in-troduced to reduce the negative parity contributions atthe source via C (cid:48)(cid:48) ( t ) = e − aM src C ( t ) + C ( t + a ) , (3.7) C (cid:48)(cid:48) ( t, τ ) = e − aM src C ( t, τ ) + C ( t + a, τ + a ) . (3.8)Again, this step does not alter the functional forms of thetwo- and three-point correlators but replaces the sourceoverlap factors by¯ z + → ¯ z + (cid:0) e − aM src + e − aM + (cid:1) , (3.9)¯ z − → ¯ z − (cid:0) e − aM src − e − aM − (cid:1) . (3.10)If several negative parity states contribute signifi-cantly to the data, successive applications of this pro-cedure, with suitable parameters [ aM (1)src , aM (2)src , . . . ] and[ aM (1)snk , aM (2)snk , . . . ], can appreciably suppress them. On the other hand, because the relative error in the correla-tors becomes larger with time, too much time-averagingrenders the data statistically less precise. Moreover,time-averaging reduces the available τ range in the mod-ified correlators, thereby producing fewer data for the fit.For each data set, some study is necessary to strike anoptimal balance. B. ∆ -like states Another source of excited-state contamination arisesfrom the presence of the three ∆-like states in the 16-irrepcorrelators. With four different classes of interpolators atboth the source and the sink, we adopt the strategy fromRef. [24] and solve the generalized eigenvalue problem(GEVP) [25]. In Ref. [20], we applied the GEVP to ourtwo-point correlators and successfully disentangled thenucleon-like state from the ∆-like states. We extend thatstrategy to the three-point functions here.Given a matrix two-point correlator, C ( t ), the leftand right nucleon eigenvectors, u ( t , t ) and v ( t , t ), arethe solutions of C ( t ) u ( t , t ) = λ ( t , t ) C ( t ) u ( t , t ) , (3.11) v ( t , t ) C ( t ) = λ ( t , t ) v ( t , t ) C ( t ) . (3.12)Here, we focus on the eigenvectors for the nucleon-likestate, the ones with the lowest eigenvalues, and put theothers aside. These eigenvectors optimize the projectiononto the nucleon-like state in both the two- and three-point correlators via C ( t ) = v ( t , t ) C ( t ) u ( t , t ) , (3.13) C ( t, τ ) = v ( t , t ) C ( t, τ ) u ( t , t ) . (3.14)One has to decide which t and t to use in Eqs. (3.11)and (3.12). The stability of our results under such varia-tions will be discussed in Sec. IV. Below we call the cor-relators in Eqs. (3.13) and (3.14) the nucleon-optimizedtwo- and three-point correlators.To summarize our strategy, we start with the correla-tors in Eqs. (2.1), (2.4), and (2.3), and apply two itera-tions of time-averaging at both the source and sink, andthen project the time-averaged correlation matrix as inEq. (3.14). The time-averaging suppresses the negativeparity states contributions, and the projection suppressesthe ∆-like baryons contributions. IV. SIMULATION DETAILS
To demonstrate the feasibility of nucleon matrix el-ements with staggered quarks, we use a single gaugeensemble, which was generated by the MILC collabo-ration [16]. MILC implemented the one-loop, tadpole-improved L¨uscher-Weisz gauge action [26], as well as theHISQ action [17] for the sea, which contains equal-massup and down quarks, the strange quark, and the charmquark. In this work, we also employ the HISQ action forthe valence quarks, with the same mass as the up-downsea quarks.The ensemble has dimension L × T = 24 × a = 0 . F p s mass-independent scheme [27]), a pion mass M π ≈
305 MeV, and a light-to-strange-quark mass ratioof 1 /
5. Other parameters of this ensemble are listed inRef. [27]. Note that the CalLat [28] and the PNDME [29]collaborations have both used this same ensemble to cal-culate g A , albeit with either the M¨obius domain wall orWilson-clover valence fermion actions, respectively.We generate the two- and three-point correlators ac-cording to Eqs. (2.1), (2.3), and (2.4). We measure eachcorrelator on 872 configurations, and further increase thestatistics by randomly placing the corner-wall sources oneight maximally separated timeslices to give a total of6976 measurements per correlator. (The τ /a = 7 corre-lators have only four time sources per configuration.)We block all measurements in a single gauge configu-ration and every four consecutive gauge trajectories toavoid autocorrelations. The covariance matrix betweendifferent correlator components are estimated with thenon-linear shrinkage method [30] to avoid ill-conditioningfrom finite sample sizes.As described in Ref. [20], we use corner-wall sourcesto optimize the signal-to-noise ratio and point sinks. Inthe present work, we remove the Coulomb-gauge fixedlinks, as we have empirically observed that leaving outthe links has little effect on correlators but with the addedadvantage of a simpler code. Here we also incorporate theWuppertal smearing [31, 32] at the sink by applying χ ( n ) = (cid:18) σ a N ∆ (cid:19) χ ( n − , (4.1)∆ χ ( (cid:126)x ) = − χ ( (cid:126)x ) + (cid:88) i =1 [ χ ( x i + 2 a ) + χ ( x i − a )] (4.2)in order to reduce excited state contamination. InEq. (4.1), n is the n th iteration of N total iterations; allshifts are stride 2 to preserve the staggered symmetries.We include the appropriate gauge transporters to makethe smearing gauge covariant [20], but for succinctnessthey are omitted from Eq. (4.2). We generate data withtwo different root-mean-squared (rms) smearing radii, σ ,which are about 0 . . τ /a = [3 , , , , aM (1)src , aM (2)src ] = [ aM (1)snk , aM (2)snk ] = [0 . , . . (4.3) t / a a M e ff Gr2.0N30 t / a = 4 t / a = 6 t / a = 8 t / a = 10 t / a a M e ff Gr6.0N70 t / a = 4 t / a = 6 t / a = 8 t / a = 10 FIG. 1. (Color online) Effective masses of the nucleon-optimized correlators as a function of the source-sink separa-tion time t . The top plot has Wuppertal sink smearing radius σ rms = 0 . σ rms = 0 . t when solving the GEVP equation in Eq. (3.12), as shownin the legends, and are offset slightly for clarity. These two numbers are based on an observation inRef. [20] that the lowest-lying negative parity state seemsto have energy around the S-wave
N π state, which inthis ensemble is about 0 . aM π ∼ . τ /a = [3 , , τ /a .After time-averaging, we solve for the left and righteigenvectors using Eq. (3.12) in order to optimize ourcorrelators as in Eq. (3.14). To ensure the robustnessof our fitting methodology, we test the stability of ourresults under variations of the choice of t . To do so, wecompute the effective mass of the optimized two-pointcorrelators, which we define as aM eff ( t ) ≡
12 ln (cid:18) C ( t ) C ( t + 2 a ) (cid:19) . (4.4)The t stability plots are shown in Fig. 1. All choicesproduce similar results, and so we choose t = 6 a for thesubsequent analyses. Similarly, we vary ( t − t ) /a from2 to 6 and find, again, that the differences are negligible,so we fix t − t = 2 a .Since we have normalized the nucleon-like three-pointcorrelators correctly, and each of the correlator transfor-mations that we perform preserve the functional formof the spectral decomposition, the optimized three-to-two-point correlator ratios converge to the desired nu-cleon charges in the large-time limits. In Figs. 2 and 3,we plot the ratio of the nucleon-optimized three-to-two-point correlators, with and without time-averaging atthe source and sink. The left column shows the opti-mized correlators without time-averaging, and the rightcolumn shows them time-averaged with the parametersgiven in Eq. (4.3). The two smearing are shown in thetop (Gr2.0N30) and bottom (Gr6.0N70) rows. Significantoscillations are clearly present in the unaveraged correla-tors, particularly for the vector current in Fig. 2. This isexpected, because the parity partner of the vector currentis the pseudoscalar current P , and the (cid:104) N π | P | N (cid:105) matrixelement gives a large contribution to the vector-currentdata and, thus, causes large oscillations. For the g A data,on the other hand, the parity partner of the axial currentis the tensor current T z , and when the nucleon is at rest (cid:104) N | T z | N (cid:105) = 0. Consequently, the first non-zero con-tribution in the axial-vector parity partner channel willlikely be from (cid:104) N π | T z | N (cid:105) , leaving small oscillations. V. CORRELATOR FITTING
We apply the Bayesian fitting methodology imple-mented in corrfitter [33] to extract the nucleon massand matrix elements. We observe in Fig. 2 that the vec-tor correlators have noticeable oscillatory contributions,whereas the axial-vector correlators shown in Fig. 3 donot. Further, the vector correlators seem relatively insen-sitive to our choice of Wuppertal smearing. We performseparate fits to the vector and axial-vector correlators,but include their correlations through bootstrapping.It should be stressed that, after applying the excited-state suppression techniques from Sec. III, the interpreta-tion of the higher exponentials in the correlators is am-biguous. For the positive-parity channel, the first “ex-cited state” could be a mixture of any leftover ∆-likestates, the P-wave
N π states, or other finite volume en-ergy levels higher up in the spectrum that are related toresonances. For the negative-parity channel, we found inRef. [20] that the ground state is likely to contain S-wave
N π states. The time-averaging procedure to cancel out
TABLE I. Summary of the prior choices for the fit Ans¨atzegiven in Eqs. (5.1), (5.2), and (5.3). All prior distributionsare Gaussian, except for M +1 − M +0 , which is log-normal.Quantity Prior value ± width M +0 = nucleon mass 1100 ±
200 MeV M +1 − M +0 ±
200 MeV M − ±
300 MeV A +0 , +0 = (cid:101) g A ± V +0 , +0 = (cid:101) g V ± A i,j ; i (cid:54) = +0 , j (cid:54) = +0 0.0 ± V i,j ; i (cid:54) = +0 , j (cid:54) = +0 0.0 ± the negative-parity states makes identification of thesestates even more ambiguous. Regardless of the origin, wecan treat the excited states as nuisance parameters andfit them away with an exponential fit function. In thiscase, each excited exponential mass parameter describesa conglomeration of several eigenstates of the Hamilto-nian. Still, we will refer to each exponential in the fitfunction as a state without necessarily identifying it withany single eigenstate. As discussed in Ref. [20], the sta-bility of the extracted fit parameters as a function t min indicates lack of excited-state contamination, as long asthey are modeled accurately. This t min stability plot isshown in Fig. 5, and is discussed in Sec. V B. A. Functional forms for fitting
For the g A analysis, we perform simultaneous fits tothe optimized two- and three-point correlators, and in-clude both the Gr2.0N30 and Gr6.0N70 sink smearings.Observation of the strong suppression of excited statesin Fig. 2 leads us to use a fit ansatz that contains twopositive-parity states and one negative-parity state: C σ, fit2pt ( t ) = z σ +0 ¯ z +0 e − M +0 t + z σ +1 ¯ z +1 e − M +1 t + ( − t/a z σ − ¯ z − e − M − t , (5.1) C σ, fit A ( t, τ ) = (cid:88) i,j =0 z σ + i A + i, + j ¯ z + j e − M + i τ e − M + j ( t − τ ) + z σ − A − , − ¯ z − i ( − t/a e − M − t (5.2)+ (cid:88) i =0 z σ − A − , + i ¯ z + i ( − ( t − τ ) /a e − M + i τ e − M − ( t − τ ) + (cid:88) i =0 z σ + i A + i, − ¯ z − ( − τ/a e − M − τ e − M + i ( t − τ ) . Here, M +0 = M N is the nucleon mass, M +1 is the massof the first residual positive-parity excited state, ¯ z + i and z σ + i are their source and sink overlap factors (with sinksmearing σ = 0 .
2, 0 . M − is the mass of the resid-ual negative-parity state, and ¯ z − and z σ − the source and t / a C g V ( t , ) / C p t ( t ) Gr2.0N30 / a = 3/ a = 4/ a = 5/ a = 6/ a = 7 t / a Gr2.0N30 / a = 3/ a = 4/ a = 5 t / a C g V ( t , ) / C p t ( t ) Gr6.0N70 / a = 3/ a = 4/ a = 5/ a = 6/ a = 7 t / a Gr6.0N70 / a = 3/ a = 4/ a = 5 FIG. 2. (Color online) The g V three-point to two-point nucleon-optimized correlator ratio as a function of the source-sinkseparation time t , and current insertion time τ . Correlators are labeled by the rms Wuppertal smearing radii, σ rms = 0 . σ rms = 0 . τ, t − τ → ∞ , this ratio converges to the bare g V nucleon charge. Thecorrelators in the left column are not time-averaged with the oscillation suppressing procedure described in Sec. III. The rightcolumn shows data that are time-averaged with the parameters given in Eq. (4.3), yielding much smoother curves reminiscentof non-staggered fermion correlators. sink overlap factors. The terms A ± i, ± j are the unrenor-malized axial-vector matrix elements, with A +0 , +0 = (cid:101) g A the desired bare axial charge. Note that the two-pointcorrelator terms involving finite temporal T extent arenot included here since we average our data symmetri-cally around the T / ±
200 MeV for the nucleon mass on ourensemble with M π = 305 MeV. We put a wide priorof 300 ±
200 MeV centered at the ∆-like mass for themass splitting M +1 − M +0 to accommodate for potentialmixing of many physical states. For the same reason,we also impose a wide mass prior of 1600 ±
300 MeV for the negative-parity mass M − centered at the S-wave N π state. All prior choices are summarized in Table I.
A priori , we have no knowledge of the sign or mag-nitude of the overlap factors. Consequently, all overlapfactors are effectively unconstrained. Very wide priorsof 0 ± (cid:101) g A = A +0 , +0 , for which we choose a wide prior of 1 . ± . g A . As discussedbelow in Sec. VI, we know from other work with pseu-doscalar mesons that Z A is close enough to unity not toinfluence the choice of prior.For the g V analysis, we use the same two-point func-tional form as Eq. (5.1). However, for the three-pointcorrelators we use C σ, fit g V ( t, τ ) = (cid:88) i =0 z σ + i V + i, + i ¯ z + i e − M + i t + z σ − V − , − ¯ z − i ( − t/a e − M − t (5.3) t / a C g A ( t , ) / C p t ( t ) Gr2.0N30 / a = 3/ a = 4/ a = 5/ a = 6/ a = 7 t / a Gr2.0N30 / a = 3/ a = 4/ a = 5 t / a C g A ( t , ) / C p t ( t ) Gr6.0N70 / a = 3/ a = 4/ a = 5/ a = 6/ a = 7 t / a Gr6.0N70 / a = 3/ a = 4/ a = 5 FIG. 3. (Color online) Identical to Fig. 2 but with g A instead of g V . See the caption of Fig. 2 for further details. + (cid:88) i =0 z σ − V − , + i ¯ z + i ( − ( t − τ ) /a e − M + i τ e − M − ( t − τ ) + (cid:88) i =0 z σ + i V + i, − ¯ z − ( − τ/a e − M − τ e − M + i ( t − τ ) , where the notation is identical to that of Eq. (5.2). The V i,j are the unrenormalized vector matrix elements. The V + i, + j with i (cid:54) = j are omitted on the first line of Eq. (5.3),because they are forbidden by vector charge conservation,up to small discretization effects. The priors are alsoidentical to the g A fits except for the bare vector charge, (cid:101) g V . Given that the renormalization constant is close tounity, we choose the (cid:101) g V prior to be 1 . ± . B. Fit Stability
The most important part of the nucleon matrix ele-ment fitting procedure is separating the nucleon observ-ables of interest from the excited-state contributions. Todemonstrate the lack of excited-state contamination, weexamine the stability of the observables as choices in the
TABLE II. Summary of the nominal fit range parameters. t is the source-sink separation time, and τ is the current in-sertion time. t min and t max are the minimum and maximumsource-sink separation included in the nominal fits; ∆ τ min isthe minimum time after the current insertion time that weinclude in the three-point fits.Correlator Fit Parameter Nominal valueTwo-point t min /a t max /a τ min /a t max /a fit are varied. Specifically, we vary t min , ∆ τ min , and t max where t min is the minimum source-sink separation timethat we include in our two-point correlator fits, ∆ τ min isthe minimum source-sink separation time after the cur-rent insertion time, τ , that we include in our three-pointcorrelator fits, and t max is the maximum source-sink sep-aration time. The nominal parameters for the nominalfits are given in Table. II.We plot the stability of the extracted M N ( (cid:101) g V and (cid:101) g A ) t min / a a M N g V fits min / a =2 min / a =3 min / a =4 min / a =5 t min / a a M N g A fits min / a =2 min / a =3 min / a =4 min / a =5 FIG. 4. (Color online) The stability plot for the extracted nu-cleon mass, aM N , as a function of t min and ∆ τ min , obtainedfrom either the g V (top) or g A (bottom) fits. The definitionsof t min and ∆ τ min are described in the text. The maximumsource-sink separation time is fixed at t max = 13 a for all cor-relators. The solid squares are the nominal fit results, and alluncertainties are estimated with 1000 bootstrap samples. as a function of t min and ∆ τ min in Fig. 4 (Fig. 5). The x -axes are different choices of t min , and the y -axes are thecorresponding observables. The four different choices of∆ τ min are also shown slightly displaced for each t min . Thesolid squares are the nominal fits with parameters givenin Table II. As can be seen in Fig. 4, the extracted nu-cleon mass is stable as a function of t min /a and ∆ τ min /a ,which illustrates the lack of excited-state contaminationin these posteriors. Similar behavior is seen for g V inFig. 5. The only noticeable structure in the stability plotsis for g A , where the observable is stable for t min /a ≥ t min /a or ∆ τ min /a , fewer dataare available to fit, and, consequently, the results becomeless precise. Thus, as in Ref. [20], we have demonstratedcontrol over excited-state contamination when extractingmatrix elements from staggered-baryon correlators. t min / a g V min / a =2 min / a =3 min / a =4 min / a =5 t min / a g A min / a =2 min / a =3 min / a =4 min / a =5 FIG. 5. (Color online) The stability plot for the bare vectorcharge, (cid:101) g V (top), and bare axial charge, (cid:101) g A (bottom), as afunction of t min and ∆ τ min . See the caption of Fig. 4 forfurther details. VI. RESULTS
In this section, we present our Bayesian fitting resultsand our final renormalized values for the nucleon charges g V and g A . All fitting errors are estimated from 1000bootstrap samples. We take correlations into account byusing the same bootstrap samples for both (cid:101) g V and (cid:101) g A . A. Nucleon Mass
In Fig. 6, we plot the extracted posterior fitted valuefor the nucleon mass from simultaneous fits of bothsmearings of the optimized two-point correlator andthree-point correlator of a given current. We also plotthe nucleon-optimized effective masses. This effective-mass data is identical to the t /a = 6 data shown inFig. 1. The green-shaded bands are the posterior es-timates with the g A three-point correlators, while theyellow-shaded bands are with the g V three-point corre-lators. We obtain aM N = 0 . g A fit, and0 t / a a M e ff Gr2.0N30 g A fit g V fit t / a a M e ff Gr6.0N70 g A fit g V fit FIG. 6. (Color online) Nominal fit results for the effectivemasses of the optimized correlators as a function of source-sink separation time t . The open circles are excluded fromthe fits. Correlators are labeled by their Wuppertal smearingparameters, with RMS radii of 0 . . g A or the g V three-point correlators.Both sets of Wuppertal smeared correlators are included ineach fit. The green and yellow shading shows the 1 σ bandsfrom fits with either g A or g V , respectively. aM N = 0 . g V fit.There are some notable features in our fits. First, the g V fit has larger posterior uncertainties than the g A fit.Both fits include the same information from the two-point correlators, so the difference must arise from thethree-point correlators. As one can see in Fig. 2, the g V three-point correlators are less sensitive to the Wupper-tal smearing than the g A correlators. On the other hand,the g V three-to-two-point correlator ratios show remark-ably little curvature, even at the early times. This behav-ior implies that the vector three-point correlators becomequickly saturated by the ground state, and therefore pro-vide limited additional information about the overlap fac-tors and masses than what is contained in the two-pointcorrelators. The g A data does not share these features, and thus contains additional information about the two-point posteriors. This explains why the g V fit has a lessprecise nucleon mass than the g A fit.For these reasons, we quote the posterior nucleon massfrom the g A fits as the nominal result, which has value aM N = 0 . , M N = 1141(10) MeV , (6.1)where the error shown is statistical only. It is crucial tobear in mind that this result is for a lattice spacing of a =0 . M π = 305 MeV [27]. Forcomparison, a fit including only the two-point correlatorsyields aM N = 0 . g V .In Ref. [20], we computed the nucleon mass at the samelattice spacing but with a physical pion mass, obtain-ing M N = 960(9) MeV. The difference between thesetwo masses is ∆ M N = 181(13) MeV, assuming uncor-related statistical errors. Given that the pion mass dif-ference between these two ensembles is about 170 MeV,∆ M N agrees within 1 σ with the empirical observationthat M N = 800 MeV + M π within a few per cent [34]. B. Nucleon g V and g A charges In Figs. 7 and 8, we plot the optimized g V and g A three-to-two-point correlator ratios as a function of source-sinkseparation t . The raw data are identical to the right-hand plots of Figs. 2 and 3. The posterior fit results aresuperimposed as gray bands. In the limits τ, t − τ → ∞ ,the data points are seen to converge to these posteriors.It should be emphasized, however, that the ratio datapoints are shown only for illustration: we perform directfits to the optimized correlators, as discussed in Sec. V,in order to obtain results, namely (cid:101) g V = 1 . , (6.2) (cid:101) g A = 1 . . (6.3)It should be mentioned that the g V and g A fits havesome different features. First, the residual oscillationsfrom the parity partner matrix element are noticeable inthe g V fit. Second, the g V data turn out to be relativelyinsensitive to the Wuppertal smearing radius. Both ofthese features can be observed in Fig. 7. This highlightsthat there is less uncorrelated data available with whichto extract g V as compared to g A . In contrast, we observethat the vector correlators in Fig. 7 contain less positiveparity excited state contamination at early times thanthe axial-vector correlators in Fig. 8. As such, since theoscillations turn out to be easier to constrain and thereis less contribution from the same parity excited states,we obtain a more precise estimate for (cid:101) g V than for (cid:101) g A .As discussed in Sec. II B, the remnant chiral symmetryenforces Z A = Z V + O( am q ) . Therefore, the ratio ofbare charges is renormalized, and we obtain a value of g A g V = (cid:101) g A (cid:101) g V = 1 . . (6.4)1 t / a C g V ( t , ) / C p t ( t ) Gr2.0N30 / a = 3/ a = 4/ a = 5 t / a C g V ( t , ) / C p t ( t ) Gr6.0N70 / a = 3/ a = 4/ a = 5 FIG. 7. (Color online) Nominal fit results for the optimizedthree-to-two-point correlator ratio as a function of source-sinkseparation time t and current insertion time τ . In the limits τ, t − τ → ∞ , the optimized three-to-two-point correlator ra-tios converge to the bare axial charge (cid:101) g V . Data points fromdifferent current insertion times, τ , are slightly displaced forclarity. The filled data points are included in the nominal fit.Correlators are labeled by their Wuppertal smearing parame-ters with rms radii of 0 . . σ error bands for the different τ ’s are shown in blue, or-ange, and green, and the 1 σ error band for the (cid:101) g V posterioris shown in gray. where the correlation between (cid:101) g A and (cid:101) g V is taken intoaccount via bootstrapping. We can also obtain Z V byimposing current conversation on a pseudoscalar mesonvector-current matrix element [22]. Then the renormal-ized charges are g V = Z V (cid:101) g V = 1 . , (6.5) g A = Z A (cid:101) g A = 1 . , (6.6)based on Z V = Z A = 0 . t / a C g A ( t , ) / C p t ( t ) Gr2.0N30 / a = 3/ a = 4/ a = 5 t / a C g A ( t , ) / C p t ( t ) Gr6.0N70 / a = 3/ a = 4/ a = 5 FIG. 8. (Color online) Similar to Fig. 7, but for the axial-vector three-point correlators (cid:101) g A . VII. DISCUSSION AND CONCLUSIONS
We have presented two key results in this work. First,we have shown how to analytically relate the staggerednucleon-like matrix elements with non-trivial tastes tothe physical nucleon matrix elements. This step is crucialfor our on-going program of extracting high-precision nu-cleon results from staggered fermions. The general proce-dure, which can be applied to any staggered baryon ma-trix element, is outlined in Appendices A and B. Specif-ically, for the nucleon charges g V and g A , we summa-rize our key results for the zero-momentum isovector (ax-ial) vector three-point correlators in Eqs. (2.3) and (2.4).These equations explicitly show the non-trivial normal-izations needed to relate the nucleon-like matrix elementsto their physical counterparts. Our successful computa-tion of g V and g A shows that continued use of the 16 irrepof the staggered symmetry group GTS is feasible, whichis convenient because the 16 contains a single nucleon-liketaste in the spectrum [20].This finding is encouraging, because the additionalcomplexity of staggered baryons, compared with stag-gered mesons is probably the reason staggered-baryon2matrix elements have not been explored until now. Thereare as many meson tastes (16) as bosonic irreps of GTS.As such, each staggered meson interpolating operator ex-cites only a single taste of meson. In contrast, there are64 = 4 different tastes of a staggered baryon, yet onlythree unique irreps of GTS, denoted 8, 8 (cid:48) , and 16 af-ter their dimension. Consequently, there are not enoughunique components of these irreps to accommodate all64 tastes of baryons, and more than one taste of thesame baryon can appear in each irrep’s tower of states.Choosing an irrep with only one nucleon taste simplifiesthe correlator analysis and, as we have shown in this pa-per, allows for accurate and precise results for nucleonmatrix elements.The second key result of this work is demonstratingthe practicality of staggered baryons by computing theisovector nucleon vector and axial-vector charges. Forthis purpose, we choose a single ensemble with a ≈ .
12 fm, 2+1+1 flavors in the sea, and, when using iden-tical sea and valence HISQ quarks, M π = 305 MeV.With approximately 7000 measurements and techniquesdesigned to handle staggered correlators, we find few-percent statistical uncertainty. Our final values for g V , g A , and g A /g V on this ensemble are g V = 1 . , (7.1) g A = 1 . , (7.2) g A g V = 1 . . (7.3)The conservation of the vector charge, g V = 1, is a non-trivial verification of our methodology.As discussed in Sec. V, we include two positive-paritystates and one negative-parity state in our fit function.The number of matrix elements included in the fit growsquadratically as a function of the states included. Withmore precise data, we could constrain more matrix ele-ments. Alternatively, we could also impose tighter priorson the transition matrix elements and overlap factors,for example with the empirical Bayes method [29]. Thisproof-of-concept study does not attempt a full calcula-tion with all errors included, so we leave exploration ofthose options for future work.The same ensemble has been used by both theCalLat [28] and PNDME [29] collaborations in their cal-culations of g V and g A . CalLat uses M¨obius domain-wall fermions for the valence quarks, while PNDME usesWilson fermions with the clover action. CalLat defines Z V by demanding g V = Z V (cid:101) g V = 1 and uses the rem-nant chiral symmetry to set Z A = Z V . They thenquote (cid:101) g V = 1 . g A = 1 . Z V and Z A independently via the regularization-independent symmetric momentum-subtraction scheme,commonly known as RI-sMOM, and quote g V = 0 . g A = 1 . g A /g V = 1 . a , M π , and physical volume closeenough to ours to allow a straightforward comparison.With an eye towards sub-percent determinations of theaxial charge, it is instructive to compare how the preci-sion on g A is influenced by each collaboration’s data andmethodology. Presumably influenced by the common en-semble, the three analyses share a few common aspects.First is the use of eight sources (with high-precision so-lutions of the Dirac equation) per gauge-field configura-tion, so the raw statistics are about the same. Second,the time range of the central fits for the two-point cor-relators turns out to be the same: t max + 1 − t min = 8.Third, all three collaborations simultaneously fit a corre-lator containing the matrix element with the two-pointcorrelators. Last, PNDME and we use time ranges in thecentral fits of the three-point correlators, such that thereare 21 data points in the fit.In addition, each collaboration employs techniques toimprove the signal. We have two smeared sinks and startwith 4 × × a -independent number of steps of agradient flow [37]. In the future, we could easily take ad-vantage of the truncated-solver method, while the gradi-ent flow would prevent us from using numerous technicalresults from the Fermilab Lattice and MILC collabora-tions, such as lattice-spacing and renormalization-factordeterminations.A more striking difference is CalLat’s introduction ofthe currents into a propagator in a way inspired by theFeynman-Hellmann theorem [38]. A key feature of thetechnique is that instead of a three-point function, thematrix element lies within another two-point function.Thus, the CalLat method requires a fit to a single timevariable instead of two; indeed the matrix element popsout of a fit to the ratio of the two two-point correlators.In the end, the relative precision on g A is quoted as 1%,3%, and 4% for CalLat [28], PNDME [29], and this work,respectively. One should bear in mind, however, the ef-fective number of components per site, which are fourfor Wilson fermions, eight for staggered fermions (corre-sponding to the corners of the unit cube), and 4 L fordomain-wall fermions (where L is the extent of the fifthdimension; L = 8 in Ref. [28]). Taking the number ofcomponents into account but ignoring algorithmic speed-ups from the code or specific features of each action, thecost for given precision is roughly the same. It would,therefore, be interesting to explore the truncated-solverand Feynman-Hellmann-inspired methods with staggeredfermions.This work sets the foundation needed to continue aprogram of precise nucleon form-factor calculations. Cal-culations of the vector and axial-vector form factors atnonzero momentum transfer are indeed underway on the3same ensemble as used here. Further we, have startedcomputing g V and g A on the same ensembles used inRef. [20]. These ensembles have physical pion masses,and a range of lattice spacings to enable a continuumextrapolation. Appendix A: Relating Staggered-QCD MatrixElements to QCD Matrix Elements
Lattice gauge theory with staggered fermions can bethought of as an extension of QCD with four degener-ate flavors, called tastes, for each quark. The associatedtaste symmetry allows for many more composite stateswhich can have non-trivial taste structures. We callstates that have non-trivial taste “baryon-like” states,to distinguish from the physical single-taste baryons. Inthis work, we focus on the nucleon and restrict ourselvesto that case going forward. The nucleon-like states canbe mapped onto the physical nucleon states through ap-propriate flavor-taste symmetry transformations. Thisallows the freedom to choose which nucleon-like state tostudy in order to extract observables. As highlighted inRef. [20], the two-point correlator data constructed fromnucleon-like states are easier to analyze than their physi-cal counterparts due to the smaller multiplicity of tastesin the spectrum. However, one needs the mapping fromthe specific nucleon-like state to the physical state.We use isospin- , GTS-16 nucleon-like interpolatingoperators to extract nucleon observables, since the spec-trum contains only a single nucleon-like state. The re-lationship between the nucleon-like matrix elements andthe physical nucleon matrix elements is, unfortunately,not at all transparent. In this and the following appen- dices, we will establish the relationship between the 16-irrep nucleon-like matrix elements and the single-tastephysical nucleon matrix elements.Bailey [19] inferred the spectrum of staggered baryonsby subducing nucleon-like representations of the fullSU(8) F T flavor-taste symmetry of the continuum limitinto GTS. We expand that work to matrix elements.Specifically, we will demonstrate how one can apply thegeneralized Wigner-Eckart theorem to SU(4) and relatethe lattice nucleon-like matrix elements to the physi-cal tasteless nucleon matrix elements through appropri-ate normalization factors, which are generalized Clebsch-Gordan coefficients. The procedure outlined here can beapplied to any staggered baryon matrix elements in anySU( n f ) × GTS flavor-taste irrep.Following the notation from Ref. [20], we first deter-mine the continuum quantum numbers of the nucleon-like states that subduce into the 16 irrep of GTS. Thisstep is needed for the generalized Wigner-Eckart the-orem. We focus on the continuum symmetry groupSU(2) S × SU(8)
F T , where SU(2) S is the spin symme-try and SU(8) F T is the flavor ( F ) and taste ( T ) sym-metry for two equal-mass flavors. This group breakson a discrete lattice to the unbroken flavor symmetrysubgroup SU(2) F and the “geometric timeslice group”(GTS) [18, 19]. GTS can be decomposed into [20, 39]GTS = (( Q (cid:111) SW ) × D ) / Z , (A1)where Q is generated by the discrete taste transforma-tions { Ξ , Ξ } , SW by the cubic rotations { R , R } ,and D by the discrete taste and spatial inversion trans-formations { Ξ , I S } . (These symbols are all defined inthe appendix of Ref. [20].)The subgroup chain we work with is SU(2) S × SU(8)
F T × P ⊃ SU(2) S × SU(2) F × SU(4) T × P ⊃ SU(2) S × SU(2) F × SU(2) Q × SU(2) D × U(1) D × P ⊃ SU(2) F × GTS × P, (A2)where P = I S Ξ becomes the usual parity operation in the continuum limit [18]. The factor SU(2) D on the secondline arises from decomposing the SU(4) T taste symmetry onto a discrete lattice, which leads to the factor D inEq. (A1), combined with the U(1) D phase factor. Note that in Ref. [20] we omitted the U(1) D factor, but here wemake it explicit. The other the groups are defined and explained in Ref. [20].
1. Using shift symmetries to relate staggeredcorrelators
The goal is to assign continuum quantum numbers ofSU(2) S × SU(2) F × SU(4) T to each nucleon-like state cre- Various Z N quotient factors are often omitted for clarity. Theyare only necessary to avoid overcounting group elements (for ex-ample, SU(4) T ⊃ (SU(2) Q × SU(2) D ) / Z ). ated by every component of the 16 irrep. We begin by in-vestigating the continuum quantum numbers of the sim-plest nucleon-like states created by the 16 irrep. After-wards, we can use the lattice symmetry transformationsto obtain the remaining components.We can form nonvanishing two-point correlation func-tions is by contracting any one of the 16 irrep componentswith the same component on a later timeslice. One can4then apply lattice rotations and shifts to show that these16 two-point correlators are identical in the ensemble av-erage.The 16 irrep components split into two sets of 8 differ-ent components that reside on the eight corners of a cube(see the appendix of Ref. [20] for explicit constructions).The construction of nonvanishing three-point correlatordata also depends on the current insertion. For the lo-cal vector and axial-vector currents we use in this work,the zero-momentum three-point correlators do not van-ish if and only if the source and sink interpolators areidentical. Correlators constructed from the same set of 8components can be related to each other with the latticeshift symmetries.To summarize, this means that the nonvanishing two-point correlators satisfy (cid:88) (cid:126)x (cid:68) B (cid:126)M ( (cid:126)x, t ) B (cid:126)M (0) (cid:69) = (cid:88) (cid:126)x (cid:68) B s (cid:126)N ( (cid:126)x, t ) B s (cid:126)N (0) (cid:69) (A3)where the superscript denotes the 16 irrep operators, (cid:126)M and (cid:126)N are equal to any one of the eight corners of thecube, and s = ± R for (cid:126)M = (cid:126)N = (cid:126)
0. The notation is defined indetail in Ref. [20].We are using local currents J = V, A in this work, sothe nonvanishing three-point correlators satisfy (cid:88) (cid:126)x,(cid:126)y (cid:68) B s (cid:126)M ( (cid:126)x, t ) J ( (cid:126)y, τ ) B s (cid:126)M (0) (cid:69) = (A4) S J ( (cid:126)N − (cid:126)M ) (cid:88) (cid:126)x,(cid:126)y (cid:68) B s (cid:126)N ( (cid:126)x, t ) J ( (cid:126)y, τ ) B s (cid:126)N (0) (cid:69) , where S J ( (cid:126)A ) = ± J and (cid:126)A . Its specific value can be determined by applyinga lattice shift symmetry transformation between (cid:126)M and (cid:126)N . For the currents used in this work, it is identical to thesign factor appearing in the construction of the staggeredcurrent J . For example, S V ( (cid:126)A ) = ( − ( A x + A y + A z ) /a forthe γ ⊗ ξ vector current and S A ( (cid:126)A ) = ( − A z /a for the γ z γ ⊗ ξ z ξ axial current. The currents and phase factorsare also defined in Eqs. (2.5) and (2.6). For a generalcurrent (other than the local currents used here), how-ever, it might be necessary to have different interpolatingoperators at the source and sink. In that case, the phasefactors in the general version of Eq. (A4) would still beobtained from the lattice shift symmetries.Going forward, it is sufficient to study the correlatorwith component (cid:126)N = 0 located at the origin of the stag-gered unit cube, (cid:80) (cid:126)x B ± (cid:126) ( (cid:126)x, t ). Then, owing to Eq. (A4),the other seven components follow immediately. We use the convention of staggered phases, η ( x ) = ( − x , η ( x ) = ( − x + x , η ( x ) = ( − x + x + x , and η ( x ) = 1 [40],which affects the phases appearing in the lattice rotations andshifts. The quantum numbers of the nucleon-likestates created by (cid:80) (cid:126)x B ± (cid:126) ( (cid:126)x, t ) will be denoted as (cid:12)(cid:12) [ , ] F [16 , ± (cid:126) GTS (cid:11) . The first bracket gives the un-broken SU(2) F flavor quantum numbers, which herehas total and z -component isospins , and the secondbracket denotes the 16 irrep with the eigenvalues of R .
2. Quantum numbers of nucleon-like states
Next, we must find a convenient basis for the contin-uum nucleon-like states and then subduce them down tothe (cid:12)(cid:12) [ , ] F [16 , ± (cid:126) GTS (cid:11) lattice states. From Eq. (A2), wewant to track the quantum numbers of SU(2) S × SU(2) F × SU(4) T and may ignore the passive phase U(1) D andparity P = +1 factors. From the group subduction pre-sented in Ref. [19, 20], the 16 irrep is subduced from thecontinuum spin-flavor-taste irrep viaSU(2) S × SU(2) F × SU(4) T ⊃ SU(2) S × SU(2) F × SU(2) Q × SU(2) D (A5) (cid:18) , , M (cid:19) → (cid:18) , , , (cid:19) ⊕ (cid:18) , , , (cid:19) ⊕ (cid:18) , , , (cid:19) . (A6)Here we have adopted a convention that labels non-SU(2) group irreps by their dimensions and subscript M (mixed), S (symmetric), or A (antisymmetric). Theirreps of SU(2) are denoted with standard spin notation.The task of classifying a general irrep of SU(4) amountsto finding the maximal set of commuting operators anduniquely labeling the states by their eigenvalues; fora general SU(4) irrep, there are 6 eigenvalues to clas-sify [41]. Because there are no degenerate irreps whendecomposing any of the irreps in this work from SU(4)into SU(2) × SU(2), we can use the eigenvalues of thepair of SU(2) factors to identify SU(4) states. Therefore,only 4 of those 6 eigenvalues are necessary to completelycharacterize the states. As such, the 4 eigenvalues ofeach state can be uniquely identified with two pairs ofthe | L , L z (cid:105) quantum numbers.Given Eq. (A6), we notice that 20 M → (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) . We seek to find the four quantum numbers forthe states after decomposition of the 20 M irrep of SU(4) T into the subgroup SU(2) Q × SU(2) D . We write the con-tinuum nucleon-like states kets (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , m S (cid:21) S (cid:20) , (cid:21) F (cid:20) j Q , m Q (cid:21) Q (cid:20) j D , m D (cid:21) D (cid:43) . (A7)Each bracket represents the standard spin quantum num-bers of one of the SU(2) group factors, distinguished bythe superscripts and subscripts: S (spin), F (flavor), Q (SU(2) Q ), and D (SU(2) D ). This ket serves asthe irrep basis for both SU(2) S × SU(2) F × SU(4) T andSU(2) S × SU(2) F × SU(2) Q × SU(2) D .5
3. Matching the continuum and lattice nucleon-likestates
Now that we have established an appropriate basis forthe nucleon-like states, both on the lattice and in the con-tinuum, we are ready match the two sets. In particular,we are interested in which linear combination of statesfrom Eq. (A7) combine to subduce into the lattice states (cid:12)(cid:12) [ , ] F [16 , ± (cid:126) GTS (cid:11) of interest. For the 16 irrep nucleon-like states, we have shown in Ref. [20] that j Q = and j D = . Consequently, we only need to determine m S , m Q , and m D .We start with determining m D of SU(2) D fromEq. (A7). To do so, it is illuminating to study the de-composition SU(2) D × U(1) D × P → { I S } , (A8)where { I S } is the group generated by the lattice spatialinversion. As Eq. (A8) shows, I S receives contributionsfrom three different factors: the taste factor SU(2) D ,a phase factor e − iπ/ = − i from U(1) D to match theeigenvalues of I S , and the continuum-limit parity P = I S Ξ . For the spin- irreps of SU(2) D , which includethe 16 irrep nucleons [20], the matrix representation of I S is the tensor product of those three factors e iσ π/ ⊗ e − iπ/ ⊗ +1 = (cid:34) − (cid:35) = σ = I S , (A9)where σ is the third Pauli matrix. The representationin Eq. (A9) can be mapped onto the groups in Eq. (A8).The first factor arises from the 180 degrees rotation inthe “ x - y plane” of the spin- representation of SU(2) D ,the second e − iπ/ phase is from U(1) D , and the +1 fromparity. As can be seen from Eq. (A9), for the spin- irrepof SU(2) D , the I S matrix admits ± m D = ± components of σ . Since thenucleon is a positive-parity state with I S = 1, we assign m D = to the (cid:12)(cid:12) [ , ] F [16 , ± (cid:126) GTS (cid:11) lattice states.We now consider the quantum numbers of m S and m Q . The 16 irrep components can be labeled by theirreps of W = SW × { , I S } , where SW is the cubicrotation group, as [18]16 → E + ⊕ E − ⊕ T +1 ⊕ T − ⊕ T +2 ⊕ T − , (A10) where E is the two-dimensional irrep of SW , T and T are the different three-dimensional irreps of SW , and thesuperscripts show the eigenvalues of I S . By applying lat-tice rotations to (cid:12)(cid:12) [ , ] F [16 , ± (cid:126) GTS (cid:11) , we can show theybelong to the two-dimensional E + irrep of W .Subducing SU(2) SW ⊂ SU(2) Q × SU(2) S to the lat-tice angular momentum of SW is a problem commonto all fermion formulations [42]. We can write the irrepcomponents of SU(2) SW that subduce into E as [42] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , (cid:21) SW (cid:20) , (cid:21) D (cid:43) → (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , + (cid:126) (cid:21) GTS (cid:29) → (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) E + , + (cid:21) W (cid:43) (A11)1 √ (cid:32)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , (cid:21) SW (cid:20) , (cid:21) D (cid:43) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , − (cid:21) SW (cid:20) , (cid:21) D (cid:43)(cid:33) → (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , − (cid:126) (cid:21) GTS (cid:29) → (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) E + , − (cid:21) W (cid:43) (A12)where the irreps of SU(2) SW are again labeled by thetotal and z -component of angular momentum, and thearrows indicate the subduction from continuum to lat-tice states. | [ E + , ± ] W (cid:105) is a state that transforms inthe E + irrep of W with a +1 eigenvalue under spa-tial inversion and ± R . Weidentify SU(2) SW as the diagonal subgroup of SU(2) S × SU(2) Q [18]. Then, by using the Clebsch-Gordan coeffi-cients, the components are related as (cid:12)(cid:12) [2 , SW (cid:11) =1 √ (cid:32)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , (cid:21) S (cid:20) , − (cid:21) Q (cid:43) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , − (cid:21) S (cid:20) , (cid:21) Q (cid:43)(cid:33) , (A13)and1 √ (cid:18) (cid:12)(cid:12) [2 , SW (cid:11) + (cid:12)(cid:12) [2 , − SW (cid:11) (cid:19) =1 √ (cid:32)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , (cid:21) S (cid:20) , (cid:21) Q (cid:43) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , − (cid:21) S (cid:20) , − (cid:21) Q (cid:43)(cid:33) . (A14)Taking all the results of this appendix together, wehave (cid:12)(cid:12)(cid:12) , + (cid:126) (cid:69) ≡ √ (cid:32)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , (cid:21) S (cid:20) , (cid:21) F (cid:20) , − (cid:21) Q (cid:20) , (cid:21) D (cid:43) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , − (cid:21) S (cid:20) , (cid:21) F (cid:20) , (cid:21) Q (cid:20) , (cid:21) D (cid:43)(cid:33) → (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , (cid:21) F (cid:20) , + (cid:126) (cid:21) GTS (cid:29) , (A15)6 (cid:12)(cid:12)(cid:12) , − (cid:126) (cid:69) ≡ √ (cid:32)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , (cid:21) S (cid:20) , (cid:21) F (cid:20) , (cid:21) Q (cid:20) , (cid:21) D (cid:43) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , − (cid:21) S (cid:20) , (cid:21) F (cid:20) , − (cid:21) Q (cid:20) , (cid:21) D (cid:43)(cid:33) → (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , (cid:21) F (cid:20) , − (cid:126) (cid:21) GTS (cid:29) . (A16)Here, (cid:12)(cid:12)(cid:12) , ± (cid:126) (cid:69) have been introduced as shorthand nota-tion for the continuum states for future reference.
4. Quantum numbers of the current operators
The last ingredient needed for the Wigner-Eckart the-orem is the irreducible tensor current operator. In thiswork, we use the local isovector axial current, A and lo-cal isovector vector current, V , which have spin-tastes γ z γ ⊗ ξ z ξ and γ ⊗ ξ respectively. We will need theirSU(2) S × SU(2) F × SU(4) T quantum numbers, just as inthe above sections.The spin and flavor quantum numbers of the currentsare straightforward. By construction, both currents havea total isospin equal to one, with I z = 0 components. A isa spin-1 current with S z = 0, and V is a spin scalar. Thenontrivial part of the identification comes from mappingthe quantum numbers of SU(2) Q × SU(2) D to the fullSU(4) T group. The quark bilinears we use transform inthe 15 (adjoint) irrep of SU(4) T . The decomposition ofthe 15 irrep into SU(2) Q × SU(2) D irreps occurs via15 → (1 , ⊕ (1 , ⊕ (0 , . (A17)Just as above, the quantum numbers of SU(2) Q × SU(2) D can label the 15 irrep of SU(4) T as there areno degenerate irreps in Eq. (A17). It should be notedthat on the lattice, bosonic irreps can be classified ac-cording to a subgroup of the GTS group called the RFgroup.We will first examine the continuum quantum numbersof the local lattice vector current, V . At zero-momentum,it has spin-taste γ ⊗ ξ . Within the RF group, V trans-forms as the trivial irrep, 1 [21]. We can decompose RFinto the discrete rotational subgroup, SW , to get1 → A , (A18)where A is the trivial irrep of SW .We denote as V the continuum operator correspond-ing to V and apply the same subduction procedure asin the previous session by following the subgroup chainSU(2) S × SU(2) Q → SU(2) SW → SW . The spin-0 ir-rep of SU(2) SW subduces into the trivial irrep of SW [42]. Consequently, V needs to be in the trivial irrep We do not use the taste-scalar current as it is a multilink oper-ator, which has been empirically observed to have more noise. of SU(2) S × SU(2) Q , and matching Q factors, V canonly transform as (0 ,
1) irrep of SU(2) Q × SU(2) D fromEq. (A17).We have just found that V is a triplet of SU(2) D , andso we need to determine its z -component quantum num-ber. With positive parity, the three m D componentsof the (0 , ,
1) irrep from SU(2) S × SU(2) Q × SU(2) D subduce into the lattice currents γ ⊗ γ , γ ⊗ ξ ξ ,and γ ⊗ ξ . Each transforms trivially in RF. The firstlattice current is local and the other two are non-localwith multi-link connections between the quarks and an-tiquarks. The eigenvalues of I S are +1 for the local cur-rent, and − I S in the continuum can beconstructed from the tensor product of representationsof SU(2) D , U (1) D , and P to give e iπ × diag(1 , , − ⊗ ⊗ − − = I S , (A19)where the SU(2) D factor is in a spin triplet as discussed, U (1) D is a trivial factor to give the correct I S eigenval-ues, and the parity is also trivial by construction. Conse-quently, to get the correct I S = 1 eigenvalue on the lat-tice, the local γ ⊗ γ current must have zero z -componentin the triplet irrep of SU(2) D in the continuum limit.This completes the subduction of V into V .The procedure is similar subducing the continuumaxial-vector current A into the lattice version A . On thelattice, A transforms as a three-dimensional irrep, 3 (cid:48)(cid:48)(cid:48)(cid:48) ,of RF, which decomposes into the3 (cid:48)(cid:48)(cid:48)(cid:48) → A ⊕ E (A20)irreps of SW . The linear combination A ∝ ( γ x γ ⊗ ξ x ξ ) + ( γ y γ ⊗ ξ y ξ ) + ( γ z γ ⊗ ξ z ξ )(A21)transforms trivially under discrete rotations so it lives inthe A irrep. The remaining linear combinations are E + ∝ ( γ x γ ⊗ ξ x ξ ) + ( γ y γ ⊗ ξ y ξ ) − γ z γ ⊗ ξ z ξ )(A22) E − ∝ ( γ x γ ⊗ ξ x ξ ) − ( γ y γ ⊗ ξ y ξ ) , (A23)where the subscript on the left-hand side is the eigenvalue ± of R .In the continuum, A is a spin-1 operator of SU(2) S .The A irrep subduces from the spin-0 irrep of SU(2) SW E irrep subduces from the spin-2 irrep ofSU(2) SW . With the rules for the addition of angularmomentum, this requires A to be in the irrep (1,1) ofSU(2) S × SU(2) Q with zero z -component spins in bothSU(2) factors.Now, according to Eq. (A17), A can either be a spin-0or 1 operator of SU(2) D . Recall that on the lattice, D is generated by the transformations I S and Ξ [20]. A is an eigenvector of both these symmetries with respec-tive eigenvalues 1 and −
1. Because SU(2) D subducesinto the D factor of the GTS group, these non-trivialeigenvalues mean that A cannot transform trivially un-der SU(2) D . As such, A can only belong to spin-1 irrepof SU(2) D . Further, it has zero z -component followingthe same argument in Eq. (A19).In summary, we have determined the continuum quan-tum numbers of A and V , which subduce into the desiredlattice current operators, A and V respectively. Usingthe same notation as in Eq. (A7), the continuum cur-rents transform as −A (1 , S (1 , F (1 , Q (1 , D4 ≡ A / √ n t → A/ √ n t , (A24) V (0 , S (1 , F (0 , Q (1 , D4 ≡ V / √ n t → V / √ n t . (A25)The spin and flavor quantum numbers of the tensoroperators are denoted by the superscripts, whereas thetaste quantum numbers are given in the subscripts. n t =4 is the number of tastes and √ n t = 2 is required toproperly normalize tensor operators. The minus sign infront of the axial current is a convention that we followaccording to Table I of Ref. [41].As an aside, there is an easy way to obtain the con-tinuum taste quantum numbers of an arbitrary quarkbilinear without explicit group subduction. Table I ofRef. [41] outlines the SU(4) generators and their cor-responding tensor operators. Once we adopt the Eu-clidean Dirac representation for the taste gamma ma-trices ξ = σ ⊗ I, ξ j = σ ⊗ σ j (where σ j are the usualPauli matrices), those generators give the components ofthe continuum taste matrices. For example, the local ax-ial and vector currents we use have taste gamma matricesof ξ z ξ = − − (A26) ξ = − − . (A27)They are proportional to the generators ( A − A − A + A ) and ( A + A − A − A ). By identifying S as D in Table I of Ref. [41], and similarly T as Q , we can recognize the tensor product S ⊗ T = σ ⊗ σ and σ ⊗ I , indicating a spin-1 representation whenever a σ appears in the tensor product. This yields the continuumtaste quantum numbers of these states as (1 , Q (1 , D and (0 , Q (1 , D . Appendix B: Wigner-Eckart Theorem and thePhysical Matrix Elements
In this appendix we need to relate, for each current,the s = ± M V ± ≡ (cid:68) , ± (cid:126) |V| , ± (cid:126) (cid:69) , (B1) M A ± ≡ (cid:68) , ± (cid:126) |A| , ± (cid:126) (cid:69) . (B2)Since we know the continuum quantum numbers ofeach state and current, we can apply the Wigner-Eckarttheorem to relate the different components. To furtherreduce the number of independent matrix elements fromfour to two, we apply the Wigner-Eckart theorem to theSU(2) Q part of the irreps in Eqs. (A15), (A16), (A24),and (A25) to find M A − = − M A + , (B3) M V − = M V + . (B4)This result is consistent with the discussion aroundAppendix (A 1). On the lattice, we have found exactsymmetries for the local vector currents (cid:28)(cid:20) , (cid:21) F (cid:20) , + (cid:126) (cid:21) GTS (cid:12)(cid:12)(cid:12)(cid:12) V (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , (cid:21) F (cid:20) , + (cid:126) (cid:21) GTS (cid:29) = (cid:28)(cid:20) , (cid:21) F (cid:20) , − (cid:126) (cid:21) GTS (cid:12)(cid:12)(cid:12)(cid:12) V (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , (cid:21) F (cid:20) , − (cid:126) (cid:21) GTS (cid:29) , (B5)which comes from Eq. (A3) and (cid:88) (cid:126)x,(cid:126)y (cid:68) B (cid:126) ( (cid:126)x, t ) V ( (cid:126)y, τ ) B (cid:126) (0) (cid:69) = (cid:88) (cid:126)x,(cid:126)y (cid:68) B − (cid:126) ( (cid:126)x, t ) V ( (cid:126)y, τ ) B − (cid:126) (0) (cid:69) , (B6)derived from applying lattice rotations and shifts. Forthe local axial-vector current, there are no symmetriesrelating the matrix elements on the lattice, but the rela-tionship in Eq. (B3) emerges in the continuum.To demonstrate this observation, we have plotted theratio of optimized g A three-point correlators created with There is a typo in Table I of Ref. [41]. The irreducible tensor com-ponents at line 3 should read − T [211](1 , , instead of − T [211](0 , , . / + Gr2.0N30 / a = 3/ a = 4/ a = 5 / + Gr6.0N70 / a = 3/ a = 4/ a = 5 FIG. 9. (Color online) The ratio of the three-point g A cor-relators, built with (cid:80) (cid:126)x B − (cid:126) ( (cid:126)x, t ) (”16-” with eigenvalues of − x − y plane rotation) and (cid:80) (cid:126)x B (cid:126) ( (cid:126)x, t ) (”16+”with eigenvalue of +1) interpolating operators, as a functionof source-sink separation time t and current insertion time τ .Both interpolators are used in Eq. (2.4) to compute g A . Eachplot represents a different Wuppertal smearing at the sink,with parameters 0 . . t, τ → ∞ limits that the ratio is equal to −
3, which is shownas a dashed lines. (cid:80) (cid:126)x B − (cid:126) ( (cid:126)x, t ) and (cid:80) (cid:126)x B (cid:126) ( (cid:126)x, t ) interpolators in Fig. 9.In the limits τ, t − τ → ∞ and a →
0, the ratio shouldconverge to the dashed lines at − | M V phy | = | (cid:104) B |V| B (cid:105) | , (B7) | M A phy | = | (cid:104) B |A| B (cid:105) | , (B8)where M V phy and M A phy are the physical vector and axialmatrix elements. Here | B (cid:105) ≡ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) , (cid:21) S (cid:20) , (cid:21) F (cid:20) , (cid:21) Q (cid:20) , (cid:21) D (cid:43) (B9)is the single-taste nucleon, e.g., | B (cid:105) has the correctisospin of and transforms as the symmetric 20 S ir-rep of SU(4) T . The 20 S irrep of SU(4) T contains stateswith a single-taste baryon . To relate single-taste baryonmatrix elements to the physical one, we also need thetaste-diagonal current operators which have tastes ξ z ξ , ξ , ξ ξ , or . These constructions must coincide withthe physical matrix elements, up to a sign, if the tasterestoration is valid in the continuum limit.Again, we can use the quantum numbers of SU(2) Q × SU(2) D to uniquely label components in 20 S becausethere are no degenerate irreps in the decomposition20 S → (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) . We apply the Wigner-Eckart the-orem to normalize the matrix elements, M V − and M A − ,to M V phy and M A phy . This boils down to finding the cor-rect Clebsch-Gordon coefficients to rotate | , − (cid:126) (cid:105) to thesingle-taste baryon | B (cid:105) while leaving the taste-diagonalcurrents unchanged. An SU(4) T rotation alone is insuf-ficient because these states belong to different SU(4) T irreps. However we can embed flavor and taste into alarger group and perform rotations in this larger groupto accomplish the task. If we consider the relevant groupfactors SU(4) F × D ⊃ SU(2) F × SU(2) D , both | , − (cid:126) (cid:105) and | B (cid:105) belong to the same 20 M irrep of SU(4) F × D ,and so we can apply the Wigner-Eckart theorem to thisgroup.The details of the generalized Wigner-Eckart theoremfor SU(4) are described in Ref. [41]. We will only need theWigner-Eckart theorem in Eq. (33) of that reference, andthe Clebsch-Gordon coefficients in Table A4.5 of Ref. [41],to conclude that | M V − | = |(cid:104) B |V| B (cid:105)| = | M V phy | (B10) | M A − | = |(cid:104) B |A| B (cid:105)| = | M A phy | . (B11)We can understand the trivial normalization factor byrealizing that in the continuum, SU(2) F , SU(2) Q , andSU(2) D are indistinguishable from one another becauseof the enlarged SU(8) F T symmetry. This means thatthe matrix elements are invariant under the exchange ofD and F labels in Eq. (B7) and Eq. (B8). This showsthat Eq. (B7) and Eq. (B8) are identical to Eq. (B1) andEq. (B2), and hence, the trivial normalization factors.Combining the shift symmetry relationship in the corre-lators from Eqs. (A4) with Eqs. (B11) gives a key resultfor this paper, which is presented in Eq. (2.3), (2.4). As an analogy, the single-taste of SU(2) T is similar to the ∆ ++ (consisting of three valence up-quarks) in SU(2) F ACKNOWLEDGMENTS
We are grateful to the MILC collaboration for the useof the source code adapted to generate the correlators inthis study and for permission to use their 2+1+1-flavorgauge-field ensemble. Computations for this work werecarried out on facilities of the USQCD Collaboration,which are funded by the Office of Science of the U. S. Department of Energy. This manuscript has been au-thored by Fermi Research Alliance, LLC under ContractNo. DE-AC02-07CH11359 with the U. S. Department ofEnergy, Office of Science, Office of High Energy Physics.Brookhaven National Laboratory is supported by theU. S. Department of Energy under Contract No. DE-SC0012704. Additional support was provided under U. S.DOE Contract No. DE-SC00190193. [1] A. S. Kronfeld, D. G. Richards, W. Detmold, R. Gupta,H.-W. Lin, K.-F. Liu, A. S. Meyer, R. Sufian, andS. Syritsin (USQCD), Eur. Phys. J.
A55 , 196 (2019),arXiv:1904.09931 [hep-lat].[2] R. Ruiz de Austri and C. P´erez de los Heros, JCAP ,049 (2013), arXiv:1307.6668 [hep-ph].[3] L. Alvarez-Ruso et al. (NuSTEC), Prog. Part. Nucl.Phys. , 1 (2018), arXiv:1706.03621 [hep-ph].[4] J. Ellis, N. Nagata, and K. A. Olive, Eur. Phys. J. C ,569 (2018), arXiv:1805.09795 [hep-ph].[5] J. Arrington, C. Roberts, and J. Zanotti, J. Phys. G ,S23 (2007), arXiv:nucl-th/0611050.[6] Z. Ye, J. Arrington, R. J. Hill, and G. Lee, Phys. Lett.B , 8 (2018), arXiv:1707.09063 [nucl-ex].[7] G. Lee, J. R. Arrington, and R. J. Hill, Phys. Rev. D , 013013 (2015), arXiv:1505.01489 [hep-ph].[8] R. J. Hill, EPJ Web Conf. , 01023 (2017),arXiv:1702.01189 [hep-ph].[9] H.-W. Hammer and U.-G. Meißner, Sci. Bull. , 257(2020), arXiv:1912.03881 [hep-ph].[10] K. Borah, R. J. Hill, G. Lee, and O. Tomalak, (2020),arXiv:2003.13640 [hep-ph].[11] A. S. Meyer, M. Betancourt, R. Gran, and R. J. Hill,Phys. Rev. D , 113015 (2016), arXiv:1603.03048 [hep-ph].[12] P. Zyla et al. (Particle Data Group), PTEP ,083C01 (2020).[13] R. J. Hill, P. Kammel, W. J. Marciano, and A. Sirlin,Rept. Prog. Phys. , 096301 (2018), arXiv:1708.08462[hep-ph].[14] G. Parisi, Phys. Rept. , 203 (1984).[15] G. P. Lepage, in Theoretical Advanced Study Institute inElementary Particle Physics (1989) pp. 97–120.[16] A. Bazavov et al. (MILC), Phys. Rev. D , 054505(2013), arXiv:1212.4768 [hep-lat].[17] E. Follana, Q. Mason, C. Davies, K. Hornbostel, G. P.Lepage, J. Shigemitsu, H. Trottier, and K. Wong(HPQCD), Phys. Rev. D75 , 054502 (2007), arXiv:hep-lat/0610092.[18] M. F. L. Golterman and J. Smit, Nucl. Phys.
B255 , 328(1985).[19] J. A. Bailey, Phys. Rev.
D75 , 114505 (2007), arXiv:hep-lat/0611023.[20] Y. Lin, A. S. Meyer, C. Hughes, A. S. Kronfeld, J. N.Simone, and A. Strelchenko, (2019), arXiv:1911.12256[hep-lat].[21] M. F. Golterman, Nucl. Phys. B , 663 (1986).[22] Z. Gelzer et al. (Fermilab Lattice, MILC), PoS