DDistillation at High-Momentum
Colin Egerer, Robert G. Edwards, Kostas Orginos,
1, 2 and David G. Richards (On behalf of the HadStruc Collaboration ) Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA Physics Department, William and Mary, Williamsburg, Virginia 23187, USA
Extraction of hadronic observables at finite-momenta from Lattice QCD (LQCD) is constrainedby the well-known signal-to-noise problems afflicting all such LQCD calculations. Traditional quarksmearing algorithms are commonly used tools to improve the statistical quality of hadronic n -pointfunctions, provided operator momenta are small. The momentum smearing algorithm of Bali etal. extends the range of momenta that are cleanly accessible, and has facilitated countless novellattice calculations. Momentum smearing has, however, not been explicitly demonstrated withinthe framework of distillation. In this work we extend the momentum-smearing idea, by exploringa few modifications to the distillation framework. Together with enhanced time slice sampling andexpanded operator bases engendered by distillation, we find ground-state nucleon energies can beextracted reliably for | (cid:126)p | (cid:46) I. INTRODUCTION
Lattice field theory is now a thoroughly well-established scheme to quantitatively study strongly-interacting theories, such as Quantum Chromodynam-ics (QCD), from first-principles. With the exception ofthe lightest pseudoscalar mesons at rest, lattice QCD(LQCD) calculations of the spectrum and properties ofhadrons are afflicted by exponentially worsening signal-to-noise ratios as the Euclidean time extent between op-erators grows. It is thus a key demand of lattice cal-culations that the hadron of interest saturate correla-tion functions at as short a Euclidean time separationas possible. Key to satisfying this demand is identify-ing an operator whose overlap with the hadron of in-terest is maximized relative to those with other states: (cid:104) | ˆ O ( (cid:126)p ) | h ( (cid:126)p ) (cid:105) (cid:29) (cid:104) | ˆ O ( (cid:126)p ) | h (cid:48) ( (cid:126)p ) (cid:105) .The most widely used means of accomplishing this isthrough quark spatial smearing schemes, such as Wup-pertal [1] or Jacobi [2] smearing, which act as low-energyfilters of hadronic correlation functions, leading to a morerapid relaxation to low-energy eigenmodes. It is thusstandard practice to compute hadronic observables whereat least one interpolating operator of an N-point func-tion possesses a non-trivial spatial extent. However, aspointed out in ref. [3], spatial smearing of hadronic oper-ators is less than optimal and even detrimental for all butinterpolators projected to zero momentum. The authorsproposed a remedy, now known as momentum smear-ing, that involves the introduction of appropriately tunedphase factors onto the underlying gauge links, prior to thesubsequent spatial smearing of the quark fields. In effect,a tunable momentum space distribution is constructedby creating an oscillatory spatial profile. The remark-able effectiveness of this procedure was established in [3],wherein the pion and nucleon energies were reliably ex-tracted up to ∼ ∼ a r X i v : . [ h e p - l a t ] S e p is of operators implemented through distillation, hasproven essential in mapping the low-lying baryon spec-trum of QCD [27, 28] and exotic hadrons [29–32], as wellas exploring the glueball content in the isoscalar sector ofQCD [33]. Recently, the power of this approach has beendemonstrated in the calculation of the various nucleonisovector charges[34]. Calculational programs employingdistillation have generically limited the spatial momentato within the shell | a s (cid:126)p | (cid:46) π/L s ) , where a s is thespatial lattice spacing, and L s is the number of time slicesin the spatial directions. Here the resultant correlationfunctions have sufficient momentum-space overlap thatthe distillation framework does not necessitate modifica-tions. The goal of this work is to supplement distillationwith a realization of momentum smearing, thereby in-creasing the range of hadron momenta accessible, and inso doing demonstrate the efficacy of this approach bothfor the nucleon energies at higher spatial momenta andfor the nucleon charges derived at these high momenta.The remainder of this paper is organized as follows.We proceed in Section II with a brief summary of thedistillation framework, and the modifications needed toincorporate momentum smearing within that framework.In Section III, we describe its computational implemen-tation, and then proceed to a comparison of the nucleonenergies with and without momentum smearing on a lat-tice at the larger of our two pion masses, and identify anoptimal procedure for its implementation. In Section IV,we extend the investigation to a lighter pion mass, and inparticular highlight the efficacy of this approach by de-termining the renormalized isovector charges of the nu-cleon in both stationary and boosted-frames, with andwithout the momentum-smearing modifications. In Sec-tion V we discuss our results for the resultant matrix el-ements, and their interpretation in terms of both the ex-pected discretization effects, and the possible excited-to-ground-state transitions. Concluding remarks are givenin Section VI. II. DISTILLATION
Distillation [26] is a low-rank approximation toa gauge-covariant smearing kernel, conventionallytaken to be the Jacobi-smearing kernel J σ,n σ ( t ) = (cid:16) σ ∇ ( t ) n σ (cid:17) n σ [2]. The tunable parameters { σ, n σ } al-low for variable source “widths” and applications, respec-tively, such that in the large iteration limit, the kernel ap-proaches that of a spherically-symmetric Gaussian. Thelow-rank approximation is formed by isolating eigenvec-tors of the discretized three-dimensional gauge-covariantLaplacian −∇ ( t ) ξ ( k ) ( t ) = λ ( k ) ( t ) ξ ( k ) ( t )and ordering solutions according to the eigenvalue mag-nitudes λ k ( t ). The outer product of equal-time eigenvec- tors defines the distillation smearing kernel (cid:3) ( (cid:126)x, (cid:126)y ; t ) ab = R D (cid:88) k =1 ξ ( k ) a ( (cid:126)x, t ) ξ ( k ) † b ( (cid:126)y, t ) , (1)where R D is the chosen rank of the distillation space andcolor indices a, b are made explicit. Correlation functionsformed by Wick-contracting quark fields smeared via (1)can be factorized into distinct reusable components, the elementals and the perambulators . The elementalsΦ ( i,j,k ) αβγ ( t ) = (cid:15) abc (cid:16) D ξ ( i ) (cid:17) a (cid:16) D ξ ( j ) (cid:17) b (cid:16) D ξ ( k ) (cid:17) c ( t ) S αβγ , (2)shown here for the case of baryons, encode the oper-ator construction, where D i are covariant derivatives,and S αβγ are subduction coefficients encoding how aninterpolator with Dirac indices { α, β, γ } constructed inthe continuum will mix across irreducible representations(irreps) of a hypercubic lattice and its associated littlegroups. The perambulators τ ( l,k ) αβ ( t (cid:48) , t ) = ξ ( l ) † ( t (cid:48) ) M − αβ ( t (cid:48) , t ) ξ ( k ) ( t ) (3)encode the propagation of the quarks between elementsof the distillation space, where M is the Dirac operator.It is this factorization of the quark propagation from theconstruction of the interpolating operators that enablesthe computationally efficient implementation of the vari-ational method with an extended basis of operators. A. Momentum Smeared Distillation
Distillation is quite costly initially both in computa-tional storage and the construction of its components.Moreover, the rank R D is expected to scale with the lat-tice spatial volume in order to maintain the same resolu-tion in correlation functions on different ensembles [26].This is particularly significant for the construction ofthe correlation functions, where the needed Wick con-tractions for meson and baryon two-point functions scaleas R D and R D , respectively. Thus an implementationof momentum smearing within distillation must seek tominimize the number of additional distillation vectors in-cluded in the basis, and in particular avoid the use of adistinct eigenvector basis for each momentum of the cor-relation functions.With such a scenario in mind, one might consider mod-ifying a set of eigenvectors according to:1. Single Phase ˜ ξ ( k ) a ( (cid:126)z, t ) = e i(cid:126)ζ · (cid:126)z ξ ( k ) a ( (cid:126)z, t )2. Opposing Phases ˜ ξ ( k ) a ( (cid:126)z, t ) = 2 cos (cid:16) (cid:126)ζ · (cid:126)z (cid:17) ξ ( k ) a ( (cid:126)z, t ) 2. Identity and Opposing Phases ˜ ξ ( k ) a ( (cid:126)z, t ) = (cid:104) (cid:16) (cid:126)ζ · (cid:126)z (cid:17)(cid:105) ξ ( k ) a ( (cid:126)z, t )4. Multiple Unidirectional Phases ˜ ξ ( k ) a ( (cid:126)z, t ) = (cid:104) e i (cid:126)ζ · (cid:126)z + e i (cid:126)ζ · (cid:126)z (cid:105) ζ (cid:54) = ζ ξ ( k ) a ( (cid:126)z, t ) , such that overlaps for several, potentially opposing,hadron momenta could be simultaneously improved. Aschematic qualitative picture of these candidate imple-mentations is depicted in Fig. 1.An important requirement of any modification of distil-lation is the preservation of translational invariance, sincethat is essential for the projection to states to definite mo-mentum. It is straightforward to show that the perambu-lators with the type-1 modification are indeed invariantunder the translation of the phase through (cid:126)x → (cid:126)x + (cid:126)d :˜ τ ijµν ( t (cid:48) , t ) = ξ ( i ) † ( (cid:126)x, t (cid:48) ) e − i(cid:126)ζ · ( (cid:126)x + (cid:126)d ) M − µν ( (cid:126)x, t (cid:48) ; (cid:126)y, t ) × e i(cid:126)ζ · ( (cid:126)y + (cid:126)d ) ξ ( j ) ( (cid:126)y, t )= ξ ( i ) † ( (cid:126)x, t (cid:48) ) e − i(cid:126)ζ · (cid:126)x M − µν ( (cid:126)x, t (cid:48) ; (cid:126)y, t ) e i(cid:126)ζ · (cid:126)y ξ ( j ) ( (cid:126)y, t ) . Such translation invariance fails for the other implemen-tations of momentum smearing, as we show below forphasing of Type 4:˜ τ ijµν ( t (cid:48) , t ) = ξ ( i ) † ( (cid:126)x, t (cid:48) ) { e − i (cid:126)ζ · ( (cid:126)x + (cid:126)d ) + e − i (cid:126)ζ · ( (cid:126)x + (cid:126)d ) }× M − µν ( (cid:126)x, t (cid:48) ; (cid:126)y, t ) { e i (cid:126)ζ · ( (cid:126)y + (cid:126)d ) + e i (cid:126)ζ · ( (cid:126)y + (cid:126)d ) } ξ ( j ) ( (cid:126)y, t )= ξ ( i ) † ( (cid:126)x, t (cid:48) ) e − i(cid:126)ζ · (cid:126)x e i ( (cid:126)ζ − (cid:126)ζ ) · (cid:126)d M − µν ( (cid:126)x, t (cid:48) ; (cid:126)y, t ) × e i(cid:126)ζ · (cid:126)y ξ ( j ) ( (cid:126)y, t ) + { (cid:126)ζ ↔ (cid:126)ζ } + T . I . where we find a combination of translationally invariant( T . I . ) and variant pieces for (cid:126)ζ (cid:54) = (cid:126)ζ . Thus in the remain- [ ] [ ][ ] [ ] FIG. 1. Qualitative momentum space overlaps following mod-ification of a computed eigenvector basis. Panels 2-4 expresslyviolate translation invariance, but would dramatically reducecomputational cost were translational symmetry preserved. ID a (fm) m π (MeV) L × N t N cfg N srcs R D a m
358 0 . ×
64 100 4 64 a m
278 0 . ×
64 259 4 64TABLE I. Lattice ensembles utilized throughout this work.The number of distillation eigenvectors R D and distinctsource positions N srcs per configuration are also indicated. der of this paper, we consider only phasing of type 1, andrefer to the modified eigenvector basis as “phased”.The momentum smearing scheme of ref. [3] reweightsgauge fields U µ [ x ] in a boost direction z µ with weight ζ = πL r according to˜ U µ [ x ] = e i πL r z µ U µ [ x ] (4)prior to quark source creation, where r ∈ R . As phasesare applied to the underlying gauge configurations priorto determination of the eigenvectors, the configurationscan safely be smeared with unallowed lattice momentaas highlighted in [3]. Thus it is sufficient to modify thepreviously computed eigenvectors, limiting phases to al-lowed lattice momenta. In particular we consider thephase factors (cid:126)ζ = 2 πL ˆ z, (5) (cid:126)ζ = 2 · πL ˆ z, (6)corresponding to one and two units of the allowed latticemomenta. We remark that phases applied in the − ˆ z -direction improve momentum space overlaps for ap z < III. DEMONSTRATION OF EFFICACY
We employ two isotropic clover ensembles, with 2 ⊕ ×
64, an inverse coupling β = 6 . a (cid:39) .
094 fm, and withpion masses of 358 and 278 MeV, respectively. Theseare cataloged in Table I; further details of the ensem-bles are contained in ref. [35, 36]. To first establish thefeasibility of our candidate implementation, we employthe ensemble at the heavier pion mass, herein denotedby a m a m
358 ensemble, with each configuration separatedby 10 HMC trajectories; this small number of configura-tions was found sufficient to quantitatively demonstratethe effectiveness of distillation for the nucleon energiesand dispersion relation. We employed R D = 64 eigen-vectors, where the gauge fields in the Laplacian were3moothed via 10 iterations of stout smearing [37] withsmearing parameter ρ ij = 0 .
08 and ρ µ = ρ µ = 0. A. Interpolator Construction
The regularization of QCD through lattice discretiza-tion explicitly breaks continuum rotational symmetry,and consequently baryons at rest are now cataloged ac-cording to the double-cover irreps of the octahedral group O Dh . Thus mass eigenstates once cataloged by J P mustnow be isolated according to their patterns of subductionacross the finite number of irreps Λ of O Dh . The con-struction of the nucleon operators follows the procedureintroduced in refs. [27, 28], which we summarize now,and are expressed in terms of the baryon elementals in-troduced in Eq. 2. These operators are projections ontothe lattice irreps of discretized continuum-like operators,which we classify according to the spectroscopic nota-tion N (2 S +1) L P J P , where S represents the Dirac spin, L the angular momentum introduced via derivatives, P the permutational symmetry of such derivatives, and J P the total angular momentum and parity of the nucleoninterpolator N .To best capture the ground-state J P =
12 + nucleon atrest, which trivially subduces into the G g irrep of O Dh ,we use a basis of non-relativistic interpolators [27, 28]: B (cid:126)p = (cid:126) = { N S S
12 + , N S M
12 + , N S (cid:48) S
12 + , N P A
12 + ,N P M
12 + , N P M
12 + , N D M
12 + } (7)that admit a flexible description of the radial/orbital nu-cleon structure - we note N P M
12 + and N P M
12 + are ofhybrid construction.Projection of the lattice interpolating fields to non-zerospatial momenta ( (cid:126)p (cid:54) = (cid:126)
0) further breaks the O Dh symme-try group to little groups dependent on the * ( (cid:126)p )[38], andfurthermore mixes states of different parities. Here weconsider only boosts along a spatial axis, which are espe-cially important for PDF calculations in the LaMET andpseudo-PDF frameworks. In this case, the little group isthe order-16 dicyclic group or Dic . The framework forthe construction of the operators, specialized to the caseof mesons, is given in ref. [39]. The genesis is the classi-fication of operators of definite helicity, and therefore weextend our basis both to include those of higher spins,and of negative parity, which are then subduced to thelittle group. In particular, our basis is extended as fol-lows, based on the study of the nucleon spectrum and the dominant operators in ref. [28]: B (cid:126)p (cid:54) = (cid:126) = { N S S
12 + , N S M
12 + , N P A
12 + , N P M
12 + ,N P M
12 + , N D M
12 + , N S M
32 + , N D S
52 + ,N P M − , N P M − , N P M − , N P M − ,N P M − , N D S
32 + , N D M
32 + , N D M
32 + } . (8)We emphasize that the density of the (discrete) energyspectrum for the nucleon is expected to be considerablygreater for states in motion compared with those at restfor the following reasons. Firstly, as the spatial momen-tum is increased the separation between the energies ofa given state is compressed. Secondly, through the re-duced symmetries, even in the continuum, that enablesmore states to contribute within a given symmetry chan-nel. B. Variational Analysis
The factorization of a correlation function intrinsic todistillation facilitates the use of an extended basis of in-terpolators at source and sink, without re-computationof quark propagators as in standard smearing schemes.We are then able to perform a variational analysis in thenucleon G g channel at rest (Eq. 7), and for all boostedframes in the Dic little group (Eq. 8). We start with amatrix of correlation functions C ij ( T, (cid:126)p ) = (cid:104) | O i ( T, − (cid:126)p ) O † j (0 , (cid:126)p ) | (cid:105) , (9)where (cid:126)p is the momentum projection, and O † selectedfrom some interpolator basis B ; we reiterate that distil-lation enables momentum projections at both source andsink time slices, respectively. The variational methodcorresponds to solution of a generalized eigenvalue prob-lem (GEVP) of the form C ( T, (cid:126)p ) v n ( T, T ) = λ n ( T, T ) C ( T , (cid:126)p ) v n ( T, T ) . (10)Optimal operators, in the variational sense, for the en-ergy eigenstates | n (cid:105) are defined by (cid:80) i v i n O † i . Associatedwith each eigenvector is a principal correlator λ n ( T, T ).We will obtain the energy associated with each state | n (cid:105) by fitting its principal correlator according to λ n ( T, T ) = (1 − A n ) e − E n ( T − T ) + A n e − E (cid:48) n ( T − T ) . (11)The inclusion of a second exponential serves to quantifythe extent to which a principal correlator is dominatedby a single state, for which any deviation is encapsulatedby the amplitude A n and “excited” energy E (cid:48) n . Furtherdetails, and in particular regarding the selection of t andthe conditions used to enforce orthogonality of eigenvec-tors v n ( T, T ), are contained in refs. [30, 34]. Note N S (cid:48) S
12 + is removed from our interpolator basis a) (b) FIG. 2. The left-hand (a) and right-hand (b) plots show the effective energies for the nucleon, obtained on the a m N S S
12 + , subduced to the relevant little group, constructed with unphased(a) and phased (b) distillation eigenvectors, respectively. Data are shown for points where the signal-noise ratios are ≥ . ≥ (cid:126)p = 0 correlator. C. Efficacy of Phased Distillation & NucleonDispersions
We benchmark the standard distillation implementa-tion, without phasing, by first computing ground-statenucleon energies using the single, local interpolating op-erator N S S
12 + , the analog to standard nucleon interpo-lators, for ap z ≤ π/L ). We fit the two-point functionsto the two-exponential form C ( T, (cid:126)p ) = e − E ( (cid:126)p ) T (cid:0) a + be − ∆ ET (cid:1) , (12)where ∆ E is the gap between the ground and excited-state energies, and priors are introduced to ensure thepositivity of the overlap parameters { a, b } . To avoid pos-sible contact terms arising from the use of the Wilson-clover action, only temporal separations greater thanone are included in the fit. The data and the result-ing fits are shown in Figure 2a. For the lowest momenta ap z ≤ π/L ), the data exhibit a clear signal over thelarge range of T /a , and are well described by a two-statefit. Furthermore, the resulting ground-state energies arein excellent agreement with the expectations from thecontinuum dispersion relation E = m + p . However,for momenta ap z = { , } × (2 π/L ), not only does thesignal-to-noise ratio degrade rapidly, but a two-state fitbecomes insufficient to capture the contributions of ex-cited states to the correlator signal. The latter is seen bythe tension between the fit and correlator for Euclidean separations T /a ≤
5. Inclusion of additional states inthe functional of (12) would undoubtedly better describeearly times in the ap z = { , } × (2 π/L ) signals, but thelack of statistically meaningful signal beyond T /a (cid:39) N S S
12 + correlators where theunderlying eigenvectors are phased with one unit of mo-mentum, as in Eq. 5. While there is only a modest im-provement in the statistical precision of large-
T /a signalfor ap z = { , } × (2 π/L ), a dramatic improvement isseen for the ap z = { , } × (2 π/L ) signals. The improvedstatistical precision with phasing also serves to exposedeviations of the energies from the expectations of thecontinuum dispersion relation. These discrepancies couldarise from discretization effects, or from incomplete de-termination of the ground state correlation function. It isthis latter possibility that we now try to control throughthe use of the variational method.We performed the variational analysis on the matrix ofcorrelation functions formed by interpolators in the B (cid:126)p = (cid:126) (Eq. 7) and B (cid:126)p (cid:54) = (cid:126) bases (Eq. 8). We first applied thevariational method to the unphased basis to determinethe improvement this provides with respect to the singleoperator used above. We then performed the same anal-yses with distillation spaces modified according to (5)(one unit of momentum) and (6) (two units of momen-tum), over the momentum ranges 1 ≤ (2 π/L ) − ap z ≤ ≤ (2 π/L ) − ap z ≤
8, respectively. These mo-5 a) (b)
FIG. 3. The ground-state nucleon principal correlators for the a m
358 ensemble using a projected interpolator within eachmomentum channel obtained from the B (cid:126)p (cid:54) = (cid:126) interpolator basis subduced into the relevant little group. The left-hand andright-hand panels are obtained from the unphased and phased eigenvectors, with one unit of momentum, respectively. Theground-state principal correlator for the unphased B (cid:126)p = (cid:126) basis is shown for reference (blue). In each case, data are shown forsignal-to-noise ratios ≥
2. The bands show the two-exponential fits of Eq. 11, with data excluded from the fits in grey. Boththe data and fits are shown as λ e E ( T − T ) , where E is the lowest-lying energy obtained from the fit. mentum ranges were chosen to emphasize that, althoughone would naively expect eigenvectors modified accord-ing to (5) to have optimal overlap with momenta ap z =3 (2 π/L ) and (6) with ap z = 6 (2 π/L ), a broad cover-age in momentum is possible within each modified space,thereby obviating the need to use many distillation baseseach with its own computational cost.The principle correlators, together with the two-statefits of Eq. 11, are shown in the left and right-hand plotsof Figure 3 for the cases of unphased eigenvectors, andphased eigenvectors with one unit momentum, respec-tively. Compared to the use of phasing with the single N S S
12 + interpolator, the gains afforded by a variationalanalysis of the phased operator basis appear less dra-matic than use of an unmodified basis. The principal correlators in each case demonstrate a rather uniformplateau very close to unity, indicative of single eigen-state dominance. However, the phased principal corre-lators are much better determined and lead to more pre-cise determinations of the ground-state nucleon energies.For example in the ap z = 4 (2 π/L ) case, the extractednucleon energy from the phased principal correlator is ∼
35% more precise than the unphased equivalent.For the highest momenta 4 ≤ (2 π/L ) − ap z ≤ ≤ (2 π/L ) − ap z ≤
8, where now the eigenvectors are phasedwith two units of allowed lattice momenta (6). Thoughthe principle correlators for the higher excited states6ould not be resolved in such highly boosted frames,the resolution of the ground-state nucleon to at least ap z = 6 (2 π/L ) marks a considerable improvement in thedistillation/GEVP infrastructure for the study of hadronstructure. FIG. 4. The ground-state nucleon principal correlators forthe a m
358 ensemble using a projected interpolator withineach momentum channel obtained from the B (cid:126)p (cid:54) = (cid:126) interpolatorbasis subduced into the relevant little group. The eigenvec-tors are phased with two units of momentum (6). Principalcorrelator fits (11) are shown with colored bands, while ex-cluded data are in grey. Data is shown for signal-to-noiseratios ≥ The results for our variational analyses of the unphasedand phased bases for different momenta are summarizedin Fig. 5, where we plot the extracted nucleon energies,together with expectations from both the continuum dis-persion relation, and the lattice dispersion relation fora free scalar particle. It is evident, even with the useof an extended operator basis and the correspondinglyimproved isolation of the ground state, distillation with-out phasing is unable to cleanly resolve the ground-statenucleon energy for ap z = 4 (2 π/L ), where the signal isdominated by noise whether the single or variationally ~ ζ = π L ˆ z ~ ζ = · π L ˆ z FIG. 5. The ground-state nucleon dispersion relation for the a m
358 ensemble, together with expectations from the con-tinuum dispersion relation (blue), and free lattice scalar dis-persion relation (purple). Energies without the use of phasingare shown in magenta for a single, N S S
12 + operator, and or-ange for the variational analysis using the bases B (cid:126)p = (cid:126) , B (cid:126)p (cid:54) = (cid:126) .The energies obtained by applying the variational method onthe phased B (cid:126)p = (cid:126) , B (cid:126)p (cid:54) = (cid:126) bases are shown in green, and for a ba-sis of purely local operators in red. The squares and trianglesdenote the (cid:126)ζ = πL ˆ z and (cid:126)ζ = 2 · πL ˆ z phasing, respectively. Theground-state nucleon energies for momentum ap z = 4 (2 π/L )are shown in the inset plot, shifted for legibility. optimized operator is used.Energies from the low-momentum phasing (5) werefound to be consistent with those determined from theunphased GEVP, but are of substantially higher statis-tical quality. Most encouraging is that we are now ableto map the ground-state nucleon dispersion relation upto p z (cid:39) (cid:126)ζ = 2 · πL ˆ z phased distillationspace, even within the limited statistics. Moreover, sig-nificant uncertainty in the nucleon energies accrues onlyfor the highest momenta ap z = { , } × (2 π/L ), wherefor discretization effects are considerable.Confidence in our extracted nucleon energies is bol-stered by a separate variational analysis of an extendedoperator basis containing only the spatially-local inter-polators, in particular the N S S
12 + and seven explicitlyrelativistic interpolators. These results are shown in redof Fig. 5, and are again consistent with the (un)phaseddeterminations when using the B (cid:126)p (cid:54) = (cid:126) operator basis. Theslightly higher values for the nucleon energies at largemomenta are not surprising, as the purely local opera-tor basis did not include negative-parity operators northose of continuum spin J > , certainly contaminat-ing the true ground-state nucleon signal. Nonetheless,7 consistent determination of the nucleon dispersion re-lation when using two distinct operator bases validatesthe union of distillation with momentum smearing, andin particular confirms that the addition of phase factorsdoes not spoil the group theory required to construct ourinterpolating operators. IV. MATRIX ELEMENTS AT HIGHMOMENTUM
Hadron structure calculations within lattice QCD pro-ceed through calculation of matrix elements betweenhadrons of interest, implemented through the calcula-tion of three-point, or higher, correlation functions. Asemphasized in the introduction, many of the key mea-sures of hadron structure, such as the parton distribu-tion functions computed in the LaMET, pseudo-PDF orlattice-cross-section frameworks, require that the result-ing three-point functions be computed for hadrons at aslarge a momentum, or over as large a range of momen-tum, as possible in order to have the best control over sys-tematic uncertainties in their approaches. Thus the re-mainder of this paper is devoted the addressing this issuethrough the calculation of the nucleon isovector charges,in the forward direction, both for the nucleon at rest andfor the nucleon in a moving frame of increasing boosts.For our study of the nucleon charges, we use an ensem-ble at a somewhat lighter pion mass, which we denote by a m ap z = { , } × (2 π/L )), we use the vanillaform of distillation, without phasing. As we demonstratebelow, at high momentum, where phasing is essential, weuse two units of phasing, as implemented in Eq. 6. For ap z = 4 (2 π/L ), we compare our results both with andwithout phasing as a consistency check of the method. A. Nucleon Effective Energies
We begin by presenting in Fig. 6 the nucleon effec-tive energies computed on the a m
278 ensemble usingground-state interpolating operators obtained from thevariational method with the B (cid:126)p = (cid:126) and B (cid:126)p (cid:54) = (cid:126) bases, follow-ing the procedure described for the a m
358 ensemble.At all values of the momenta shown (i.e. ap z ≤ π/L ))we show the results without phasing; for ap z = 4 (2 π/L ),we also show the results using the phased eigenvectors,as described above. The need for phasing at this valueof the momenta (green) and above is striking, where theplateau in the effective energy is clear at far greater tem-poral separations, and the resulting energy far more pre-cisely determined. We observe that at such a lighterpion mass, the variational method without phasing isinsufficient to extract the ground-state nucleon energyfor ap z ≥ π/L ) (red), but arguably ap z ≥ π/L ) (brown). We do not expound further on nucleon ener-gies for this ensemble, however this demonstration under-scores the need for variational improvement of a phaseddistillation space in order to study physical observablesat high-momenta. B. Charges
We isolate forward isovector matrix elements by con-structing nucleon three-point functions C ( T, τ, (cid:126)p ) = (cid:88) (cid:126)x,(cid:126)y,(cid:126)z e i(cid:126)p · ( (cid:126)y − (cid:126)x ) P βα ×(cid:104)N α ( (cid:126)y, T ) O u − d Γ ( (cid:126)z, τ ) N β ( x, (cid:105) , (13)with O u − d Γ an isovector insertion introduced at time τ be-tween nucleon interpolators with temporal separation T ,and P βα = P (1 + iγ γ ) a z -polarized positive-parityprojector. To study the asymptotic 0 (cid:28) τ (cid:28) T behavior,we parameterize our two-point and three-point (13) cor- FIG. 6. Nucleon effective energies for the a m
278 ensem-ble using a projected interpolator obtained from the B (cid:126)p = (cid:126) and B (cid:126)p (cid:54) = (cid:126) bases subduced into the relevant little group, to-gether with continuum expectations (dashed), and 2-state fits(bands), where in each case the darker region denotes the timeseries included in the fit. No phasing was used to extract theground-state nucleon energy for lattice momenta ap z ∈ Z ,while ap z = 4 (2 π/L ) was also determined with two units ofphasing (6). In the case of ap z = 4 (2 π/L ), the results withand without phased eigenvectors are shown as the green andred points respectively, clearly demonstrating the need forphasing. Data shifted for legibility, and shown for signal-to-noise ratios greater than 1.35. C ( T ) = e − E T (cid:0) a + b e − ∆ ET (cid:1) (14) C ( T, τ ) = e − E T (cid:0) A + B e − ∆ ET + C e − ∆ E T cosh (cid:20) ∆ E (cid:18) τ − T (cid:19)(cid:21)(cid:19) , (15)where ∆ E is the energy gap between the ground-state( E ) and an effective first-excited ( E ) state, B and C respectively contain excited and transition matrix ele-ments, and A contains the desired forward matrix ele-ment. Priors are again introduced to enforce the posi-tivity of { a, b } . With these parametrizations, the desiredground-state matrix element is then g Γ00 = A /a in thelarge- T limit, as shown in [34]. We perform simultaneouscorrelated fits to the computed two-point and three-pointcorrelators according to (14) and (15) to extract these pa-rameters. Contact terms arising from the fermion actionare excluded from the simultaneous fits by fitting in thewindows τ fit /a ∈ [2 , T −
2] and T fit /a ∈ [2 , T maxfit ], where T maxfit is set by the maximal temporal range for whichthe associated principal correlators have signal-to-noiseratios exceeding unity: • (2 π/L ) − ap z = 0: T maxfit = 16 • (2 π/L ) − ap z = 1: T maxfit = 16 • (2 π/L ) − ap z = 4 - no phase: T maxfit = 7 • (2 π/L ) − ap z = 4 - phased: T maxfit = 12.When computing hadronic charges, the degree ofexcited-state contamination present in the three-pointcorrelators for a given interpolator separation T is of-ten quantified (c.f. [34, 36]) via definition of an effectivecharge g Γeff ( T, τ ) = C ( T, τ ) /C ( T ) , where the numerator is a three-point correlation functionwith inserted Dirac structure Γ computed for intermedi-ate times τ /a = [0 , T − C ( T ) is the two-pointfunction fit evaluated at the source-sink interpolator sep-aration T . This ratio has the advantage of plateauing to g Γ00 as τ and T − τ become large, but is only useful inso far as C is well-determined and sufficiently cap-tures the ground-state. We find this ratio, particularlyin the high-momentum frames considered, to be mislead-ing when juxtaposed with the ratio of the simultaneous C ( T, τ ) and C ( T ) fit. We instead illustrate thequality of our data by forming a direct ratio of the com-puted correlation functions R Γ ( T, τ ) = C ( T, τ ) /C ( T ) . (16)All following figures depict these ratios (16) together withratios of the fitted three-point and two-point functions foreach T /a , as well as the extracted renormalized isovectorcharge indicated with a black line and grey errorband.Data excluded from fits are in grey. All errors are de-termined via a simultaneous jackknife resampling of thedata.
V. CHARGE BEHAVIORA. g u − dS The isovector scalar S = q τ q current within nucleonstates decomposes trivially as (cid:104) N | S | N (cid:105) = 12 M N u N ( p f ) G u − dS (cid:0) q (cid:1) u N ( p i ) , (17)where G u − dS is the isovector scalar form factor. The am-plitude G u − dS is Lorentz-invariant and should thus be in-dependent of the nucleon boost, absent excited-state, dis-cretization and finite-volume effects. In particular, in theforward limit one should, in principle, be able to access G u − dS (0) = g u − dS regardless of frame. Figure 7 illustratesthe R S ( T, τ ) ratios needed to access the scalar chargeand associated fits within our considered nucleon frames,demonstrating the degree to which this supposition is re-alized. In the rest frame a clear plateau is observed in theratio by
T /a = 10, while determinations at larger valuesof
T /a deviate from this trend and exhibit increased un-certainty; the latter being consistent with the observedvariability of the nucleon effective energies at these sametimes. Most notable is a reduction in value and uncer-tainty of g u − dS when compared with standard, high statis-tics, smearing schemes on the same a m
278 ensemble.Namely in [36], it was found g u − dS = 0 . ∼ ap z = (2 π/L ) frame, we observe sta-tistical consistency with the ap z = 0 determination,with a plateau emerging for T /a ∼ −
12. The ex-pected increase of excited-state contamination is evidentin Fig. 7b, where there exists greater curvature of theratio data for a given T and the difference between each R S ( T, τ ) plateau and the asymptotic charge is seen to in-crease relative to the rest case. This amounts to markedincreases in B and C of Eq. 15 which capture excited-state (cid:104) N (cid:48) | S | N (cid:48) (cid:105) and transition (cid:104) N (cid:48) | S | N (cid:105) matrix elements, re-spectively.Without introduction of appropriate momentumphases into the distillation space, attempts to access thescalar charge in a highly-boosted frame are utterly mean-ingless (Fig. 7c). Isolation of the scalar charge in the ap z = 4 × (2 π/L ) frame is however dramatically improvedwhen a phased distillation space is used. The statis-tical precision of the R S ( T, τ ) data improves consider-ably, provided the two-point function is well-determined.However, the extracted charge is dubious - the phaseddetermination differs by 25% from the average of the ap z = { , } × (2 π/L ) cases. The close proximity ofthe R S ( T, τ ) plateaus for each
T /a and the asymptoticcharge suggest that at the level of the 2-state fits con-sidered herein, the first excited-state matrix element issmall. However, without performing R S ( T, τ ) compu-tations for additional
T /a and performing higher statefits, this cannot be rigorously confirmed. We do point9 a) (b)(c) (d)
FIG. 7. Extracted renormalized R S ( T, τ ) and isovector scalar charges for momenta (a) ap z = 0, (b) ap z = (2 π/L ), (c) ap z = 4 × (2 π/L ) without phasing, and (d) ap z = 4 × (2 π/L ) with two units of allowed lattice momentum applied toeigenvectors. Variationally improved operators were used within each momentum channel. out the statistical noise evident in the T /a = 10 datais not surprising, as the phased two-point function losessignal at
T /a ∼
10 (c.f. Fig. 6). Furthermore, deter-minations of Z S found in [36] vary below the 2% leveland thus also cannot explain the observed discrepancy.One may be tempted to attribute this dramatic differ-ence to a mixing of the scalar current with the deriva-tive of the vector current D µ { ψγ µ ψ ( x ) e − iq · x } . Giventhe explicit zero 3-momentum transfer with the prob-ing current, it is evident this derivative mixing is onlypossible for q (cid:54) = 0 or when unwanted excited-to-groundstate transitions are present. This possibility is capturedby C of (15), and is reflected in the overall curvatureof R S ( T, τ ) rather than vertical shifts of the computedmatrix element. We are left to attribute this puzzlingdiscrepancy to statistical fluctuations and the lack of ad- g Γ ap z = 0 ap z = 2 π/L ap z = 8 π/L ap phase z = 8 π/Lg u − dS χ r ditional T /a data. As will be shown, the other chargeswe explore exhibit much greater consistency in the stud-ied momentum frames, and observed deviations can beattributed to known systematic effects. Table. II cata-logs the isolated scalar charges and the correlated figureof merit for the simultaneous fits of each frame. 10 . g u − dV Among the currents considered, the vector current V µ = qγ µ τ q is unique given that it is a conserved quan-tity in the continuum. Our decision to adopt purely localcurrents in this work necessarily violates this conserva-tion. However the derived vector current renormalizationconstant [36] reestablishes the desired conservation upto quadratic corrections in the lattice spacing - namely, Z V g u − dV, bare = 1 + O (cid:0) a (cid:1) . Considering the vector currentLorentz structure between the ground-state nucleon andan arbitrary state N (cid:48) with nucleon quantum numbers (cid:104) N (cid:48) | V µ | N (cid:105) = u N (cid:48) ( p f ) (cid:20) F u − d (cid:0) q (cid:1) (cid:18) γ µ − q µ q /q (cid:19) + σ µν q ν M N (cid:48) + M N F u − d (cid:0) q (cid:1)(cid:21) u N ( p i ) , it is clear for (cid:126)q = 0 the temporal component of the vectorcurrent simply yields the baryon number of the nucleonand all its excitations. A useful sanity check then for thephasing considered herein, is to ensure the renormalized g u − dV is unity in the V = qγ q channel for each forwardframe considered. As illustrated in Figures 8,9b & 9f,we indeed find Z V g u − dV , bare to be unity and temporally in- FIG. 8. Extracted renormalized R V ( T, τ ) and isovector vec-tor charges determined for momenta ap z = 0. A variationallyimproved operator was used in these determinations. variant, most notably even as the nucleon momentumis increased and phasing is employed. A highly-boostednucleon interpolator without phasing exhibits poor over-lap with the ground-state nucleon (Fig. 9d) and is suffi-ciently noisy such that Z V g u − dV , bare (cid:54) = 1. The extracted g u − dV are presented in Tab III, with consistent deter-minations observed in the ap z = { , } × (2 π/L ) and ap phase z = 4 (2 π/L ) momentum channels.Non-zero nucleon momenta while still with (cid:126)q = 0,opens the V z = qγ τ q channel as an additional means to quantify the ground-state Dirac form factor F u − d (0).However, any such attempt to isolate the ground-stateDirac form factor F u − d signal will be contaminated withthe transition form factor F u − d (cid:0) q (cid:1) signal in propor-tion to q / ( M N (cid:48) + M N ). In the ideal scenario that ex-cited states are completely removed, the energy transfer q will vanish and F u − d (0) can be directly accessed with V z . Figure 9a illustrates R V z ( T, τ ), which features a cleardependence on { T, τ } and whose asymptotic limit differsfrom g u − dV by ∼ F u − d contamination, the bestwe can extract here is F u − d (cid:0) q (cid:1) − q γ M N (cid:48) + M N F u − d (cid:0) q (cid:1) - which we will denote as g u − dV z for brevity. To the ex-tent this pollution is unchanging in other forward framesis bore out in Figs. 9c & 9e. As for the scalar charge,the unphased ap z = 4 (2 π/L ) determination is meaning-less and is dominated by uncertainty in the unalteredtwo-point function. The phased ap z = 4 (2 π/L ) deter-mination, although statistically consistent with g u − dV , isconstrained by only two values of T and is character-ized by a curious flip in concavity of R V z ( T, τ ). As thisdependence is captured by C of Eq. 15, it is clear the ef-fect of phasing has apparently identified the conjugate ofthe ground-to-first-excited state transition. This behav-ior warrants repeated calculations for additional values of T /a with increased statistics to elucidate whether this be-havior is merely fluctuations or a clear trend. That said,the R V z ( T, τ ) appears to be trending below unity withinthe well-determined values of
T /a . Results of these si-multaneous fits are cataloged in Tab. IV. C. g u − dA The axial charge of the nucleon is perhaps the mostenigmatic of the isovector charges given its long historyas a benchmark in LQCD, and only recent efforts fallingto within 1% of experiment [40–42]. At zero-momentumthe nucleon expectation of the axial current is vanishingexcept for components along the direction of polarization.Thus for our z -polarized nucleons, we must use γ γ atrest to access g u − dA - which we denote as g u − dA z . Togetherwith a Lorentz decomposition of the axial current (cid:104) N | A µ | N (cid:105) = u N ( p f ) (cid:2) γ µ γ G u − dA (cid:0) q (cid:1) − i q µ M N γ (cid:101) G u − dP (cid:0) q (cid:1)(cid:21) u N ( p i ) (18) g Γ ap z = 0 ap z = 2 π/L ap z = 8 π/L ap phase z = 8 π/Lg u − dV χ r γ at rest and in boosted frames. a) (b)(c) (d)(e) (f) FIG. 9. Extracted renormalized R V µ ( T, τ ) and isovector vector charges determined from γ (left panel) and γ (right panel)insertions. External nucleon momentum according to (a),(b) ap z = (2 π/L ), (c),(d) ap z = 4 (2 π/L ) without phasing, (e),(f) ap z = 4 (2 π/L ) with two units of allowed lattice momenta applied to eigenvectors. Variationally improved operators were usedwithin each momentum channel. Γ ap z = 0 ap z = 2 π/L ap z = 8 π/L ap phase z = 8 π/Lg u − dV z – 0.915(15) 0.63(8) 0.995(23) χ r – 1.216 12.544 2.150TABLE IV. Renormalized g u − dV z determined via γ in boostedframes. By definition, g u − dV z = 0 at rest. and (cid:126)q = 0, it is evident the axial matrix element at restreceives contributions only from the axial form factor andnot the induced pseudoscalar form factor. We plot inFig. 10 the renormalized R γ γ ( T, τ ) and g u − dA z isolatedat rest from our simultaneous fits. We observe notice-able contamination from excited-states for T /a = { , } ,but broad consistency for the remaining T /a values. Theobserved ∼
7% deviation from the experimental valueof 1 . γ γ channel is particularly sensitive to closely-spacedexcited states, which when incorrectly identified leads tonot only a discrepancy of g u − dA z with experiment but alsoa violation of the operator derived PCAC relation. Ourdeviation of g u − dA z from experiment is, however, consistentwith [36], where with standard smearing schemes on thesame a m
278 ensemble it was found g u − dA z = 1 . g u − dA z have long been explored, such as use of O ( a )-improved currents [45]. In that work, however, it wasfound use of an O ( a )-improved axial current only mildlyimproved the experiment-lattice discrepancy, bolsteringthe presumed preponderance of excited-state and finite-volume effects. These same authors explored the degreeto which g u − dA z = g u − dA could be satisfied, just as we nowexplore based on Eq. 18.Figure 11 illustrates the R γ γ ( T, τ ) and R γ γ ( T, τ )ratios isolated in the boosted frames we have considered.As with the scalar and vector charges, the lack of phas-ing at high-momentum degrades the two-point correlatorsuch that the resulting matrix element signals containessentially no information (Figs. 11c & 11d). Comparedto the rest frame, we observe ∼
3% difference in g u − dA z when computed in the ap z = (2 π/L ) frame. This differ-ence is indicative of q (cid:54) = 0, despite (cid:126)q = 0, and hencemild radiative transitions with excited-states affecting g Γ ap z = 0 ap z = 2 π/L ap z = 8 π/L ap phase z = 8 π/Lg u − dA z χ r γ γ at rest and in boosted frames. FIG. 10. Extracted renormalized R γ γ ( T, τ ) and isovectoraxial charge determined via γ γ at rest. A variationally op-timized operator was used in these determinations. this determination. Furthermore, we do observe a dra-matic difference of ∼
15% between the determination of g u − dA z and g u − dA in the ap z = (2 π/L ) frame. This is un-surprising given the observation of q (cid:54) = 0 in the moving γ γ channel, all but ensuring the outsized influence of (cid:101) G u − dP [46, 47]. The increased separation between each R γ γ ( T, τ ) in Fig. 11b again points to this increasedexcited-state contamination. We do remark that despitethe different vertical scales chosen in Figs. 11a & 11b,the fitted energy gap ∆ E is consistent within error. Themomentum smeared ap z = 4 (2 π/L ) charges (Figs. 11e& 11f) again exhibit improved statistical quality, yet su-perficially appear to agree with each other and oddlywith experiment. Each determination does not howeverseem to indicate a plateau in R Γ ( T, τ ) has been found,especially in light of the noisy R Γ ( T = 10 , τ ) determina-tions. Moreover, the R γ γ ( T, τ ) and R γ γ ( T, τ ) ratiosare clearly trending away from each other within the il-lustrated data, and suggests the extracted charges in thisphased frame would indeed be distinct were calculationsperformed with improved statistics and, especially, finer
T /a . The results for our simultaneous fits for the g u − dA z and g u − dA axial charges are presented in Tab. V & VI,respectively. g Γ ap z = 0 ap z = 2 π/L ap z = 8 π/L ap phase z = 8 π/Lg u − dA – 0.970(14) 0.71(9) 1.302(24) χ r – 1.148 12.353 1.990TABLE VI. Renormalized isovector axial charges determinedvia γ γ in boosted frames. By definition, g u − dA = 0 at rest. a) (b)(c) (d)(e) (f) FIG. 11. Extracted renormalized R A µ ( T, τ ) and isovector axial charges using γ γ (left) and γ γ (right). External nucleonmomentum according to (a),(b) ap z = (2 π/L ), (c),(d) ap z = 4 (2 π/L ) without phasing, (e),(f) ap z = 4 (2 π/L ) with two unitsof allowed lattice momentum applied to eigenvectors. Variationally improved operators were used within each momentumchannel. Γ ap z = 0 ap z = 2 π/L ap z = 8 π/L ap phase z = 8 π/Lg u − dT xy χ r T at rest and in boosted frames. D. g u − dT The isovector tensor current within nucleon states in-duces the following form factor decomposition: (cid:104) N | T µν | N (cid:105) = u N ( p f ) (cid:2) iσ µν A u − d (cid:0) q (cid:1) +[ γ µ , q ν ]2 M N B u − d (cid:0) q (cid:1) +[ P µ , q ν ]2 M N (cid:101) A u − d (cid:0) q (cid:1)(cid:21) u N ( p i ) , where T µν = qiσ µν τ q and P = p f + p i . At restonly the T = qσ τ q matrix element is non-vanishing;apart from kinematic factors, this particular tensor cur-rent continues to be non-vanishing within the nucleonin motion. This has the fortunate consequence thatall form factors outside the desired A u − d (cid:0) q (cid:1) , where g u − dT xy ≡ A u − d (0), do not contribute to the matrix elementsignal. We indeed find g u − dT xy determined in each momen-tum frame to be statistically consistent across the boostsconsidered, and in the case of ap z = { , } × (2 π/L )the charge is especially well determined and in fantasticmutual agreement (see Figs. 12a & 12b). The lack of aclean signal for g u − dT xy for ap z = 4 (2 π/L ) (Fig. 12c) is bynow expected, and underscores the need for phasing athigh-momentum (Fig. 12d). We note the slightly larger,though no less consistent, value found for g u − dT xy in thephased ap z = 4 (2 π/L ) frame appears to be a result ofthe noisy T /a = 10 data. We anticipate future calcula-tions with improved statistics will help to bring down thisvalue. The ratios R T xy ( T, τ ) and simultaneous fit resultsare compared in Fig. 12 and the extracted tensor chargesare gathered in Tab. VII. In the interest of completeness,we note our best determined g u − dT xy is ∼
8% larger than g u − dT xy = 0 . a m m π L (cid:39) .
24. Nonetheless, calculations at different physical volumes required to con-firm this expectation are planned.
VI. CONCLUSIONS
We have expounded upon the seminal Gaussian mo-mentum smearing scheme developed by Bali et al.,demonstrating momentum space overlaps of distilled in-terpolators can likewise be improved by introducing ap-propriate spatial phase factors onto eigenvectors of thegauge-covariant Laplacian. We elected to introducephases onto a pre-computed eigenvector basis, ratherthan rotating the underlying gauge transporters. Con-sequently, the introduced phase factors were limited toallowed lattice momenta in the numerical investigationsherein. Regardless of when the phase factors are in-troduced, all components forming the scaffolding of adistillation-smeared correlation function (e.g. elementalsand perambulators) must be recomputed. This moti-vated our choice to smear pre-computed eigenvectors.We established the efficacy of this approach by isolat-ing the ground-state nucleon dispersion relation using astandard eigenvector basis, and two modified bases; mod-ified with one and two units of allowed lattice momenta,respectively. Despite variational optimization of unmod-ified interpolators within the J λ = λ = ± / channel, thenucleon dispersion relation was only meaningfully satis-fied up to (cid:39) .
75 GeV. Variational analyses within thephase modified distillation spaces yielded agreement withthe nucleon dispersion relation in excess of 3 GeV.The determination of several renormalized isovectorcharges of the nucleon was used as further evidence forthe utility of merging distillation with momentum smear-ing. Matrix elements at rest and for ap z = (2 π/L ) werecomputed without phasing. These were then comparedto identical matrix elements computed in a boosted frame( ap z = 4 (2 π/L )) with and without momentum phases.Our aim was to demonstrate consistency between chargescomputed in different (forward) frames. This is an espe-cially nuanced venture, as numerous form factors beginto compete as the momentum frame is varied. Further-more, the momentum smearing procedure certainly im-proves overlap onto unwanted single- and multi-particleexcited states. A proper treatment of this consistency re-quires dedicated calculations of nucleon form factors atseveral lattice spacings/volumes, and pion masses. Theseencouraging results nevertheless establish the feasibilityof future calculational paradigms requiring distillation athigh-momenta. Our attention is now turned to such stud-ies. Acknowledgments
We thank members of the
HadStruc
Collaboration forinvaluable discussions and scrutiny. We acknowledge the15 a) (b)(c) (d)
FIG. 12. Extracted renormalized R T xy ( T, τ ) and isovector tensor charge determined for (a) ap z = 0, (b) ap z = (2 π/L ), (c) ap z = 4 (2 π/L ) without phasing, and (d) ap z = 4 (2 π/L ) with two units of allowed lattice momentum applied to eigenvectors.Variationally improved operators were used within each momentum channel. facilities of the USQCD Collaboration used for this re-search in part, which are funded by the Office of Sci-ence of the U.S. Department of Energy. This work usedthe Extreme Science and Engineering Discovery Environ-ment (XSEDE), which is supported by the National Sci-ence Foundation under grant number ACI-1548562 [48].We further acknowledge the Texas Advanced Comput-ing Center (TACC) at the University of Texas at Austinfor HPC resources on Frontera that have contributedgreatly to the results in this work. We gratefully acknowl-edge computing cycles provided by facilities at Williamand Mary, which were provided by contributions from theNational Science Foundation (MRI grant PHY-1626177),and the Commonwealth of Virginia Equipment TrustFund. The authors acknowledge William and Mary Re-search Computing for providing computational resources and/or technical support that have contributed to the re-sults reported within this paper. Calculations were per-formed using the
Chroma [49], QUDA [50, 51], QDP-JIT [52] and
QPhiX [53, 54] software libraries which weredeveloped with support from the U.S. Department of En-ergy, Office of Science, Office of Advanced Scientific Com-puting Research and Office of Nuclear Physics, ScientificDiscovery through Advanced Computing (SciDAC) pro-gram. This research was also supported by the Exas-cale Computing Project (17-SC-20-SC), a collaborativeeffort of the U.S. Department of Energy Office of Scienceand the National Nuclear Security Administration. Thismaterial is based upon work supported by the U.S. De-partment of Energy, Office of Science, Office of NuclearPhysics under contract DE-AC05-06OR23177. C.E. issupported in part by the U.S. Department of Energy un-16er Contract No. DEFG02-04ER41302, a Department ofEnergy Office of Science Graduate Student Research fel-lowships, through the U.S. Department of Energy, Officeof Science, Office of Workforce Development for Teach-ers and Scientists, Office of Science Graduate StudentResearch (SCGSR) program, and a Jefferson Science As-sociates graduate fellowship. The SCGSR program isadministered by the Oak Ridge Institute for Science andEducation (ORISE) for the DOE. ORISE is managed byORAU under contract number DE-SC0014664. KO wassupported in part by U.S. DOE grant No. DE-FG02-04ER41302 and in part by the Center for Nuclear Fem-tography grants C2-2020-FEMT-006, C2019-FEMT-002-05. 17
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