Field sparsening for the construction of the correlation functions in lattice QCD
FField sparsening for the construction of the correlation functions inlattice QCD
Yuan Li, Shi-Cheng Xia, Xu Feng,
1, 2, 3, 4
Lu-Chang Jin,
5, 6 and Chuan Liu
1, 2, 31
School of Physics, Peking University, Beijing 100871, China Collaborative Innovation Center of Quantum Matter, Beijing 100871, China Center for High Energy Physics, Peking University, Beijing 100871, China State Key Laboratory of Nuclear Physics andTechnology, Peking University, Beijing 100871, China Department of Physics, University of Connecticut, Storrs, CT 06269, USA RIKEN-BNL Research Center, Brookhaven NationalLaboratory, Building 510, Upton, NY 11973 (Dated: September 3, 2020)
Abstract
Two field-sparsening methods, namely the sparse-grid method and the random field selectionmethod, are used in this paper for the construction of the 2-point and 3-point correlation functionsin lattice QCD. We argue that, due to the high correlation among the lattice correlators at differentfield points associated with source, current, and sink locations, one can save a lot of computationaltime by performing the summation over a subset of the lattice sites. Furthermore, with this strategy,one only needs to store a small fraction of the full quark propagators. It is found that the numberof field points can be reduced by a factor of ∼
100 for the point-source operator and a factor of ∼ ∼ a r X i v : . [ h e p - l a t ] S e p . INTRODUCTION Lattice QCD provides a non-perturbative approach to the numerical solution of QuantumChromodynamics (QCD), which is believed to be the basic theory of strong interactionsamong quarks and gluons. With the development of the cutting-edge supercomputers, thenew algorithms and the advanced methodologies, lattice QCD is now playing an increasinglyimportant role in the understanding of low-energy QCD.A typical way to gain a better efficiency in a numerical lattice QCD calculation is toreduce the redundant costs. Here are some examples. • To guarantee an accurate generation of a quark propagator, the residual criterion inthe conjugate gradient (CG) inversion for the quark propagator is usually set to − oreven smaller. When realizing that most of the CG iterations can be saved, the all modeaveraging (AMA) technique [1] is proposed, where the residual criterion is raised to alevel of ∼ − . In this way the computational time is significantly reduced while thephysical quantities can still be obtained with no bias by performing a correction, whichcompensates the systematic effects from the approximated propagators by adding thedifference between some samples of the precise correlators and the approximated ones. • In the calculation of hadronic light-by-light contributions to the muon anomalousmagnetic moment, a four-point hadronic function with vector-currents is required. Toconstruct such a four-current correlator, one expects a spacetime summation over thelocations of at least three of the four currents which is very challenging numerically.Realizing that in the connected diagram, when the locations of two vector currentsare separated with a spacetime distance r , the hadronic function falls exponentiallywith the increase of r , an importance sampling is introduced to evaluate the stochasticsum over r efficiently [2]. Therefore, in the important regions where r (cid:46) fm, thesummation is run in a complete way while in the other regions r > fm, the con-tributions are calculated with a probability of p ( r ) ∝ /r . . In this way, much lesscomputational resources are spent in the very long distance region, where the latticecorrelation functions mainly contribute noise rather than the signal. • In many cases, it is appealing to utilize the translational invariance and construct thecorrelation function using the all-to-all propagators [3]. As a result, the information2ver the whole spacetime volume is summed and one expects to gain a good precisionfor the correlation function. On the other hand, generating all-to-all propagators isquite time consuming. Since the correlation functions from the same configuration ishighly correlated, one can achieve nearly the same precision by averaging over part ofthe source and sink locations. Such kind of techniques, called field-sparsening methods here, are the main focus of this paper.The field-sparsening techniques have been studied by Detmold and Murphy in Ref. [4],where a sparse-grid technique is introduced. An earlier application of the sparse grid can betraced back to a study from χ QCD collaboration in 2010 [5]. In this work, in addition to thesparse-grid approach, we developed another field-sparsening approach, which we will referto as the random field selection method. We will study and compare these two methods insome detail. The paper is organized as follows. We start with Sect. II by introducing thetwo field-sparsening techniques mentioned above. In Sect. III we discuss the results of the2-point functions for pion and proton with both point-source and Gaussian-smeared-sourceoperators. The advantages and disadvantages for each field-sparsening method are alsodiscussed. In Sect. IV we employ a simple model to analyze the sources of the uncertaintiesfor the random field selection method. In Sect. V we extend the study to the 3-point function,where the proton axial charge g A is used as an example to demonstrate the efficiency of thefield sparsening methods. II. FIELD SPARSENING METHODS
In the lattice QCD calculation of a generic n -point function, (cid:88) (cid:126)x ,(cid:126)x , ··· ,(cid:126)x n − ∈ Λ full (cid:104) O ( x ) O ( x ) · · · O n ( x n ) (cid:105) , (1)with O i ( x i ) being the interpolating operators at temporal-spatial point x i = ( x i , (cid:126)x i ) and Λ full the full set of spatial lattice points, one needs to perform the volume summation ( n − times. This results in a computational cost of ( L ) n − in the quark contraction, with L being the spatial lattice size. If one wants to gain a better precision by making anotherspatial-volume average over the locations of x n , then the cost becomes ( L ) n . The typicalsize of L is about - for practical lattice QCD simulations. Given a relatively largelattice, the complexity of O ( L ) usually exceeds the capability of the current lattice QCD3alculations. In many cases, the techniques of all-to-all propagators [3] or sequential-sourcepropagators [6] are used to reduce the computational costs. On the other hand, there aresome limitation for the usage of these propagators. For example, the all-to-all propagatorswork more efficiently in the mesonic sector than in the baryonic sector. For the sequential-source propagators, although it allows to reduce a complexity of O ( L ) to a level of O ( L ) ,the computational cost can increase dramatically if one wants to build the sequential-sourcepropagators with multiple time slices, momentum insertions and gamma matrix structures.Given a gauge configuration from Monte Carlo simulation, the correlation functions atdifferent source and sink locations are usually highly correlated statistically. Therefore onecan save computational time by only summing over a small subset of all the possible thesource or sink locations. In fact, as will be shown below, utilizing less source locations onecan efficiently reduce the costs to generate the quark propagators. For each propagator,one can use less sink locations and reduce the computational cost for the Wick contractionsin the construction of the correlation functions. Since the numbers of both source andsink locations are reduced, it also saves the disk space to store these quark propagatorsand also reduces the pressure for the input and output (I/O) of the large-size data on thesupercomputers. It is the task of this paper to show how much fewer location points one canuse to maintain a comparable precision of the correlation functions that use the full locationpoints.For simplicity, we start with the 2-point correlation function as an example to introducethe field-sparsening techniques. The standard 2-point correlation function with zero spatialmomentum insertion is written as C full ( t ) = (cid:88) (cid:126)x ∈ Λ full (cid:104) O ( (cid:126)x, t + t ) O † ( (cid:126)x , t ) (cid:105) , (2)where the subscript “full” indicates that the summation of (cid:126)x runs over all the sink locationpoints. By using field sparsening, one can replace the summation (cid:80) (cid:126)x ∈ Λ full by L N Λ (cid:80) (cid:126)x ∈ Λ , where Λ is a subset of Λ full , which contains only N Λ location points. Due to the high correlationin the lattice data, we expect that the replacement does not increase the noise much butreduces the propagator storage and contraction time for modest size N Λ . In Eq. (2) onlyone source location ( (cid:126)x , t ) is used. In principle one can use multiple source locations and4rite the correlation functions as C sparse ( t ) = L N Λ N Λ N Λ t (cid:88) (cid:126)x ∈ Λ (cid:88) (cid:126)x ∈ Λ (cid:88) t ∈ Λ t (cid:104) O ( (cid:126)x, t + t ) O † ( (cid:126)x , t ) (cid:105) , (3)where the source spatial location takes the value from the set Λ and source time slice takesthe value from Λ t . The size of the set Λ and Λ t is given by N Λ and N Λ t , respectively.In this work we will compare two different sets for Λ ( Λ ), namely the sparse-grid methodand the random field selection method, and determine the optimal values for N Λ and N Λ in each case. A. Sparse-grid method
Following the sparse-grid method introduced in Ref. [4], the set Λ is chosen as,type I: { ( n , n , n ) (cid:12)(cid:12) ≤ n i < L ; n i = 0 (mod k ) } , (4)where k is an integer factor of L . By using this setup, a L -point spatial lattice is reducedto a ( L/k ) -point one. For all the time slices t and t + t used in Eq. (2), the same sparsegrid set is implemented. One can also extend the type of the sparse grid totype II : { ( n , n , n ) (cid:12)(cid:12) ≤ n i < L ; n i = 0(mod k ); n + n + n = 0(mod 2 k ) } , type III : { ( n , n , n ) (cid:12)(cid:12) ≤ n i < L ; n i = 0(mod k ); n i + n j = 0(mod 2 k ) , i (cid:54) = j } . (5)With the above definitions, we have N Λ = L k , L k , L k for type I, II and III, respectively. Inour numerical study, we use a lattice gauge ensemble with L = 24 and pick up 15 values for N Λ . For convenience, these 15 cases are labelled by an integer denoted as N th , running from to , and the corresponding values for N Λ are given by the following list, N Λ ( N th ) = (cid:26) , , , , , , , , , , , , , , (cid:27) . (6)It means that we have N Λ = 24 for N th = 0 and N Λ = 1 for N th = 14 .Note that the sets of type I, II, III always include the location point of ( n , n , n ) =(0 , , . We therefore can consider it as a reference point. To reduce the correlation in thelattice calculation, for each configuration, one can shift the reference point randomly with L /N Λ choices. 5 . Random field selection Another choice of Λ is called random field selection, with Λ chosen as Λ = { ( n , n , n ) (cid:12)(cid:12) ≤ n i < L ; n i are random numbers } . (7)The random numbers for n i vary when the time slice t and configuration trajectory change.In principle, we can use any value for N Λ . To favor a comparison with the sparse-gridmethod, we use the same choices of N Λ as that in Eq. (6). III. 2-POINT FUNCTION
In this section we will present the results for 2-point correlation functions. The calculationis performed using a gauge ensemble with clover-improved Wilson twisted massquarks, generated by the ETM Collaboration [7]. The lattice volume is × with a pionmass m π ≈ MeV and a lattice spacing a ≈ . fm. In total 91 configurations areutilized in this analysis.We use Gaussian-smeared-source propagators to construct the smeared-source smeared-sink 2-point functions C α ( t ) = L N Λ N Λ N Λ t (cid:88) (cid:126)x ∈ Λ (cid:88) (cid:126)x ∈ Λ (cid:88) t ∈ Λ t (cid:104) O α ( (cid:126)x, t + t ) O † α ( (cid:126)x , t ) (cid:105) (8)with α = π for pion and α = p for proton. We use 24 time slices for t . For each time slice,we use 8 random source locations for (cid:126)x . Thus we have N Λ t = 24 and N Λ = 8 . For the sinklocation (cid:126)x , it can be summed over the full set Λ full or field-sparsening set Λ . Fig. 1 showsthe effective masses for the pion (left panel) and the proton (right panel) with the point set Λ full and Λ at N Λ = 2 . The effective masses m α at the time slice t are obtained using C α ( t ) and C α ( t + 1) as inputs. For the two field-sparsening methods, the data points are slightlyshifted horizontally to favor a comparison with that from the full set.From Fig. 1 a clear enhancement of the excited-state contamination is found in the sparse-grid method at N Λ = 2 . This is due to the mixing of the hadron states with high momenta.Let us take the sparse-grid set type I as an example. The summation over Λ can be writtenas L N Λ (cid:88) (cid:126)x ∈ Λ = (cid:88) (cid:126)m ∈ Γ (cid:88) (cid:126)x ∈ Λ full e i πk (cid:126)m · (cid:126)x , (9)6 .5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 t / a a m /24 /24 grid2 /24 random t / a a m p /24 /24 grid2 /24 random Figure 1. Effective masses for the pion (left panel) and the proton (right panel) with full set Λ full (blue), sparse-grid (dark green) and random-field selection (cyan). For the two field-sparseningmethods, we use N Λ = 2 . The green and cyan data points are shifted horizontally for an easiercomparison. with Γ = { ( m , m , m ) (cid:12)(cid:12) m i = 0 , , · · · , k − } . Therefore the higher-momentum modes with (cid:126)m (cid:54) = (cid:126) mix with the zero-momentum mode. As a consequence, the excited-state contami-nations increase as N Λ decreases. The situation becomes even more problematic when onetargets on the calculation of the correlation functions with large momentum transfer. In thiscase it is possible that a state with an assigned momentum mixes with the states carryingsmaller momenta and consequently the correlation functions are distorted by the low-lyingstates.For the random-field selection method, there is no enhancement of the excited-statecontribution. On the other hand, the statistical errors become larger due to the randomnoise arising from the field sparsening. Note that the correlation among C α ( t ) at various t isweakened by the random-field selection. As a result, the uncertainty for the effective masscan be further reduced if one performs a fit over a temporal window.In Fig. 2, we show the effective mass from the correlated fit as a function of N th , orequivalently the different choices of N Λ . The fitting windows are determined using the datawith Λ full . To be specific, we use the fitting window ≤ t ≤ for the pion correlator andobtain m π = 0 . from a correlated fit with χ / dof = 0 . . We use the fitting window ≤ t ≤ for the proton and obtain m p = 0 . with χ / dof = 0 . . We denotethese effective masses as m full π and m full p . The reasonable values of χ / dof suggest that theexcited-state contributions are well under control. We thus fix these fitting windows for the7 N th a m gridrandom N th a m p gridrandom Figure 2. Effective masses for the pion (left panel) and proton (right panel) from the correlated fitas a function of N th . For each N th , the number of field points is given in Eq. (6). The data pointsfrom sparse-grid method and random field selection are printed in blue and green color, respectively. field-selection cases. For the sparse-grid method, the additional excited-state contributionsfrom higher momenta start to make an obvious impact on the effective mass when N th ≥ (or N Λ ≤ ). For the random field selection, although the effective masses at each time slicecarries larger errors, the uncertainties are very close to the ones from sparse-grid methodafter a correlated fit. We find that the effective masses at N th = 10 ( N Λ = 16 ) is givenby m π = 0 . and m p = 0 . . By using the random field selection method,one can reduce in the summation the number of the sink points by a factor of ( N Λ = L → ), while the lattice results are still very precise. For the pion effective mass, thestatistical uncertainty is consistent with that from Λ full , and for the proton, the uncertaintyonly increases by about %.The efficiency of the field sparsening likely depends on the type of interpolating oper-ators used in the lattice calculation. As the Gaussian-smeared-source propagator is morecorrelated than the point-source propagator, we expect that the field sparsening works moreefficiently for the former case. To confirm this conjecture we calculate the effective masses forthe pion and proton using both Gaussian-smeared-source and point-source propagators. Tosave the cost, we keep N Λ t at 24 while reducing N Λ to 1 for both smeared-source smeared-sink (s-s) and point-source point-sink (p-p) correlation functions. The effective masses fromthe correlated fit are shown in the left panels of Fig. 3, while the ratios between the statisticaluncertainty and the effective masses m full α for α = π and p are shown in the correspondingright panels. With the same statistics, the results of p-p correlators are noisier than that8 N th a m p-p randoms-s random N th a m / a m f u ll p-p randoms-s random N th a m p p-p randoms-s random N th a m p / a m f u ll p p-p randoms-s random Figure 3. Effective masses for the pion and proton from the smeared-source smeared-sink (s-s)and point-source point-sink (p-p) correlation functions. In the left panel, the effective masses as afunction of N th are shown. Here we use the random field selection. In the right panel, the ratiobetween the statistical uncertainty and the effective mass m full α for α = π and p is shown. of the s-s ones. The field-sparsening method works less efficiently in the p-p corelators aswe expect. Nevertheless, we find that at N th = 5 ( N Λ = 108 ) the uncertainty of the p-peffective mass increases only 15% for the pion and 18% for the proton. This implies that onecan reduce the field points by a factor of 128 in trade with a small increase of the statisticalerror. Two order of magnitude reduction in the propagator storage and I/O data transferwould make a typical lattice calculation much easier. IV. NOISE FROM RANDOM FIELD SELECTION METHOD
When using the random field selection method, the correlation function receives two typesof the noise. The first is the gauge noise, δ gauge , and the second is the noise from the selectionof the random field points, δ rand . The increase of the gauge noise is an inevitable price when9he number of the field points is reduced in the summation. Since the lattice data at thedifferent points are highly correlated, we expect the gauge noise only increases mildly. Wetherefore focus on the estimation of δ rand .We write the time dependence of the 2-point correlation function as C ( (cid:126)x, t ) = (cid:104) O ( (cid:126)x, t ) O † ( (cid:126) , (cid:105) . = 1 L (cid:88) (cid:126)p e i(cid:126)p · (cid:126)x ˜ A ( (cid:126)p )2 E (cid:0) e − Et + e − E ( T − t ) (cid:1) , (10)where the sign of . = reminds us that the excited-state contributions are neglected at suffi-ciently large t . The energy E satisfies the dispersion relation E = (cid:112) m + (cid:126)p with m thehadron’s mass.To fully understand the impact on the uncertainty of the correlation function from therandom field selection, one needs to determine the weight function ˜ A ( (cid:126)p ) . Considering thefact that we have used the Gaussian-smeared source and sink in our calculation, we assumethat the weight function in the coordinate space is given by a Gaussian distribution A ( (cid:126)x ) = A e − (cid:126)x /σ . (11)Then the weight function in the momentum space is given by the Fourier transformation of A ( (cid:126)x ) ˜ A ( (cid:126)p ) = (cid:90) d (cid:126)x e − i(cid:126)p · (cid:126)x A ( (cid:126)x ) = A (cid:16) πσ (cid:17) e − σ (cid:126)p . (12) r / a C ( r , t = )
1e 9 modellattice
Figure 4. Lattice results of C ( r, t ) at t = 10 together with a fit to Eq. (10). In the following context we use the pion correlation function as an example. The analysisfor the case of the proton is similar. Plugging Eq. (12) into Eq. (10) we can use the resulting10 .5 0.0 0.5 1.0 1.5 2.0 2.50.00.51.01.52.02.53.0 C ( t )
1e 7 t=10 C ( t )
1e 7 t=15 C ( t )
1e 7 t=20
Figure 5. Uncertainties for the three types of 2-point functions at t = 10 , and . The blue barindicates the uncertainty of the correlation function with random field selection for N Λ = 1 . Theorange bar shows the uncertainty for the data using Λ full . The green bar shows the δ rand only. formula to fit the pion lattice correlator with three free parameters, namely A , σ and m .We obtain m = 0 . which is consistent with the effective mass m full π calculated before.Furthermore, we can obtain the result for C ( r, t ) by averaging the lattice data of C ( (cid:126)x, t ) ina range of r ≤ | (cid:126)x | < r + 1 . In Fig. 4 we show the C ( r, t ) at t = 10 together with the fittingcurve. The good agreement suggests that the functional form given in Eq. (10) describesthe lattice data well at large distances.As a next step, we use the parameters A , σ and m and Eqs. (10) and (12) to constructthe correlation function ˆ C ( (cid:126)x, t ) . As the gauge noise is eliminated in ˆ C ( (cid:126)x, t ) , we can isolatethe noise δ rand by replacing the lattice correlator C ( (cid:126)x, t ) with ˆ C ( (cid:126)x, t ) in the random fieldselection method. In Fig. 5 we compare the uncertainties for the three different types of2-point functions at t = 10 , and : The blue bar indicates the uncertainty of thecorrelation function with random field selection method with N Λ = 1 . The orange barshows the uncertainty for the data with N Λ = Λ full while the green bar indicates the sizeof δ rand only. It is noticed that the size of δ rand is much smaller than the gauge noise andthus can be safely neglected. We have thus reached a conclusion that the precision of thepion 2-point functions using the random field selection method is equally good as that usingthe sparse-grid method. The random field selection is theoretically cleaner as there is noenhancement of the excited-state contamination.11 N th C p t / C p t N th [ C p t / C p t ] Figure 6. The left panel shows the ratio between 3-point and 2-point functions, C /C , asa function of N Λ s (the x -axis is labeled by N th ). The 3-point functions are constructed using thesequential-source propagators. The time separation t i and t s are fixed as t i = 5 and t s = 10 . In theright panel, the uncertainty of the ratio as a function of N th is shown. V. 3-POINT FUNCTION
In this section, we extend the random field selection method to the 3-point function,where we calculate the proton axial charge g A as an example which is one of the mostfundamental quantities in nuclear physics. A global average of the lattice results of g A canbe found in the latest FLAG review [8]. In this study we mainly focus on the efficiency ofthe random field selection method rather than the full control of various systematic effectsfor g A . We start with the 3-point and 2-point functions as C ( t i , t s ) = L N Λ N Λ s N Λ N Λ t (cid:88) (cid:126)x ∈ Λ (cid:88) (cid:126)x s ∈ Λ s (cid:88) (cid:126)x ∈ Λ (cid:88) t ∈ Λ t (cid:104)P [ O p ( t + t s , (cid:126)x s ) A ( t + t i , (cid:126)x ) O † p ( t , (cid:126)x )] (cid:105) (13)and C ( t s ) = L N Λ s N Λ N Λ t (cid:88) (cid:126)x s ∈ Λ s (cid:88) (cid:126)x ∈ Λ (cid:88) t ∈ Λ t (cid:104)P (cid:2) O p ( t s + t , (cid:126)x s ) O † p ( t , (cid:126)x ) (cid:3) (cid:105) , (14)where O p is the Gaussian-smeared operator for the proton, A is the axial vector currentwith the polarization in the z direction and P is the spin projection operator.In order to compare the field-sparsening results with the full-size ones, we use thesequential-source propagators, which start from the source ( t , (cid:126)x ) , go through the currentinsertion point ( t + t i , (cid:126)x ) and end at the sink location ( t + t s , (cid:126)x s ) . We use 24 time slicesfor t . For each t , one random source is used and the time t i and t s are fixed as t i = 5 and12 t i t s /2 C p t / C p t N s = N [ C p t / C p t ] Figure 7. The left panel shows the results of C /C as a function of t i − t s / , where t s is chosento be 10. The right panel shows the uncertainty of the ratio (with t s = 10 and t i − t s / ) as afunction of N Λ = N Λ s . In these two plots, we set N Λ = L . t s = 10 . In total we generate 24 sequential-source propagators for each configuration. Thesepropagators allow us to obtain the 3-point functions with N Λ = 1 , N Λ t = 24 , N Λ = L andarbitrary values of N Λ s . By comparing the lattice results at different N Λ s , one can estimatehow large the correlation among the sink points in the 3-point function. In the currentstudy, we fixed t i and t s . The excited-state effects will be taken into consideration later. InFig. 6, it shows the ratio between 3-point and 2-point functions, C /C , as a function of N Λ s (the x -axis is labeled by N th ). We find that at N th = 10 ( N Λ s = 16 ), the uncertaintyof the ratio C /C only increases by 12% compared to the case of N th = 0 ( N Λ s = L ).This is very consistent with the observation in the effective masses of the 2-point functions.Thus we can conclude that the field-sparsening method seems to work equally well for both2-point and 3-point functions.As a next step, we want to determine the optimal values of N Λ and N Λ s for the sourceand sink location points. Here the 3-point functions are constructed using the Gaussian-smeared propagators only, which start from both source and sink locations and end at thecurrent insertion point ( t + t i , (cid:126)x ) . These Gaussian-smeared propagators are placed at 24time slices, which results in N Λ t = 24 . At each time slice we calculate 32 Gaussian-smearedpropagators. It allows us to build the correlators with the N Λ - N Λ s pair changing from - to - . We perform the summation of current insertion location (cid:126)x over the whole spatialvolume and have N Λ = L . In the left panel of Fig. 7, we show the results of C /C asa function of t i − t s / , where t s is chosen to be 10 and the values of t i − t s / vary in therange of [ − , . The data points with various N Λ - N Λ s pairs are plotted using the different13ymbols. In the right panel of Fig. 7, the statistical uncertainty of C /C as a functionof N Λ = N Λ s are shown. We find that with N Λ changing from to , the statistical errordrops relatively fast. From N Λ = 4 to , the error decreases slower. Due to the highcorrelation, the change of the uncertainty is very mild from N Λ = 8 to 32. It is unnecessaryto move on to larger N Λ as one can expect that the precision at N Λ = N Λ s = 8 is close tothe best precision using the all-to-all setup ( N Λ = N Λ s = L ). In practise, we can choose N Λ = 4 or and invest the additional computational resources in accumulating data frommore gauge configurations. t i t s /2 C p t / C p t t s = 08 t s = 10 t s =12 Figure 8. The ratio of C /C as a function of t i − t s / together with a two-state fit. Theshadowed band indicates the result of g A obtained from the fit. We use N Λ s = N Λ = 32 , N Λ = L ,and N Λ t = 24 . In Fig. 7, the ratio of C /C at t s = 10 is shown. As a next step we add thelattice results at another two values of t s ( t s = 8 and 12) and vary t i − t s / in a range of [ − t s + 1 , t s − . The corresponding results are shown in Fig. 8. At large time separation,the time dependence of C /C can be approximated by a two-state form as C ( t i , t s ) C ( t s ) = g A + c e − ∆ t s + c ( e − ∆( t s − t i ) + e − ∆ t i )1 + c e − ∆ t s , (15)where ∆ is the energy difference between the excited state and the ground state. Thecoefficients of c , c and c arise from the excited-state contamination. We determine thevalues of ∆ and c from the 2-point function and then perform a two-state fit of C /C to Eq. (15) using three free parameters g A , c and c . The lattice data for three t s areused in the fit simultaneously with t i − t s / ranging from [ − t s / , t s / − . As a finalresult we obtain the axial charge g A = 1 . at the pion mass m π ≈ MeV using 9114onfigurations and N Λ s = N Λ = 32 , N Λ = L , and N Λ t = 24 for each configuration. N th g A Figure 9. The proton axial charge g A as a function of N th . Here N th is related to the number offield points at the location of current insertion. We use N Λ s = N Λ = 32 and N Λ t = 24 . As a last step, we calculate g A with different value of N Λ , which is the number of the fieldpoints at the location of current insertion ( t + t i , (cid:126)x i ) . In Fig. 9 we show g A as a function of N th . By increasing N th from 0 to 5 (or equavilently reducing N Λ from L to 108), we findthat the uncertainty of g A only increases by 10%. Therefore, by using the field sparsening,one can reduce the size of the Gaussian-smeared source point-sink propagators by a factorof 128 at the expense of 10% increase in the statistical error of the correlation function. VI. CONCLUSION
In this work we make an exploratory study on the field-sparsening methods. The ob-servables under investigation include the pion and proton 2-point correlation functions andthe proton axial charge g A involving the 3-point functions. For the sparse-grid method, theresults are not affected by the noise from field sparsening but receive additional excited-statecontamination from the higher-momenta states. For the random field selection method, thesituation is just the opposite. There is no enhancement of the excited-state contamination,but the correlation functions are affected by the noise from random selection. Fortunately,we confirm, in both numerical lattice results and a model analysis mimicking pion correlator,that the noise from the random selection can be safely neglected.In the calculation, we construct the correlation function using both Gaussian-smearedoperator and the point-source operator. We find that Gaussian-smeared correlators do have15igher correlation than the point-source ones. At the expense of a ∼
15% increase in thestatistical error, we can reduce the number of field points by a factor of ∼
100 for thepoint-source operator and a factor of ∼ ACKNOWLEDGMENTS
We thank ETM Collaboration for sharing the gauge configurations with us. X.F. andL.C.J. gratefully acknowledge many helpful discussions with our colleagues from the RBC-UKQCD Collaboration. X.F. and S.C.X. were supported in part by NSFC of China underGrant No. 11775002. L.C.J acknowledges support by DOE grant DE-SC0010339. Y.L.and C.L. are supported in part by CAS Interdisciplinary Innovation Team, NSFC of Chinaunder Grant No. 11935017, and the DFG and the NSFC through funds provided to the Sino-Germen CRC 110 “Symmetries and the Emergence of Structure in QCD”, DFG grant no.TRR 110 and NSFC grant No. 11621131001. The calculation was carried out on TianHe-3(prototype) at Chinese National Supercomputer Center in Tianjin. [1] T. Blum, T. Izubuchi, and E. Shintani, Phys. Rev. D , 094503 (2013), 1208.4349.[2] T. Blum et al. , Phys. Rev. D , 014503 (2016), 1510.07100.[3] J. Foley et al. , Comput. Phys. Commun. , 145 (2005), hep-lat/0505023.[4] W. Detmold et al. , (2019), 1908.07050.[5] xQCD, A. Li et al. , Phys. Rev. D , 114501 (2010), 1005.5424.[6] G. Martinelli and C. T. Sachrajda, Nucl. Phys. B , 355 (1989).[7] C. Alexandrou et al. , Phys. Rev. D , 054518 (2018), 1807.00495.[8] Flavour Lattice Averaging Group, S. Aoki et al. , Eur. Phys. J. C , 113 (2020), 1902.08191., 113 (2020), 1902.08191.