Time Dependence of Nucleon Correlation Functions in Chiral Perturbation Theory
aa r X i v : . [ h e p - l a t ] M a y UMD-40762-441
Time Dependence of Nucleon Correlation Functions in ChiralPerturbation Theory
Brian C. Tiburzi ∗ Maryland Center for Fundamental Physics,Department of Physics, University of Maryland,College Park, MD 20742-4111, USA (Dated: November 6, 2018)
Abstract
We consider corrections to nucleon correlation functions arising from times that are far from theasymptotic limit. For such times, the single nucleon state is contaminated by the pion-nucleonand pion-delta continuum. We use heavy baryon chiral perturbation theory to derive the spectralrepresentation of the nucleon two-point function. Finite time corrections to the axial currentcorrelation function are also derived. Pion-nucleon excited state contributions drive the axialcorrelator upward, while contributions from the interference of pion-delta and pion-nucleon statesdrive the axial correlator downward. Our results can be compared qualitatively to optimizednucleon correlators calculated in lattice QCD, because the chiral corrections characterize only low-energy excitations above the ground state. We show that improved nucleon operators can lead toan underestimation of the nucleon axial charge.
PACS numbers: 12.39.Fe, 12.38.Gc ∗ [email protected] . INTRODUCTION Lattice gauge theory simulations continue to make impressive progress towards addressingquantitatively the non-perturabative regime of QCD [1]. Two decades ago, it was thoughtimpossible that lattice QCD simulations would confront experimental data without ordersof magnitude of increased computing power, and considerable algorithmic advances. Today,however, state-of-the-art simulations have pushed forward on all fronts by: including fullydynamical quarks, shrinking the lattice spacing, enlarging the lattice volume, and decreasingthe size of the light quark masses (including a sprint to the physical point [2]). The currentstatus of these simulations is reviewed in [3]. Lattice QCD is approaching a point of notonly complementing existing experimental programs, but potentially guiding future ones. Athorough overview of hadron structure from recent and forthcoming lattice calculations ispresented in [4].Performing lattice QCD calculations with light pions is an exciting recent development,but one that is accompanied by new problems. At fixed lattice sizes, for example, finitesize effects from virtual pion fluctuations will become increasingly important. Anothernotorious issue in dealing with hadrons on the lattice is the signal-to-noise problem. Inthe chiral regime, this problem will become rather acute. Consider the nucleon two-pointcorrelation function, G ( τ ). Over long times, τ , the two-point function has an exponentialfalloff governed by the nucleon mass, M N . Over such long times, however, the statisticalnoise in the correlator, Σ( τ ), is dominated by its coupling to three-pions [5]. This leads tothe signal-to-noise ratio having the behavior G ( τ )Σ( τ ) τ ≫ ∼ √ N exp (cid:20) − (cid:18) M N − m π (cid:19) τ (cid:21) , with N as the number of independent measurements on gauge configurations. As pion massesenter the chiral regime, the long-time behavior of the correlation function is dominated bystatistical noise. Generally this limits lattice QCD measurements of hadronic correlationfunctions to times that are not ideally long. In this work, we consider corrections to hadronic correlation functions arising from thepion continuum. The Lehmann-Symanzik-Zimmerman (LSZ) reduction formula provides thefield theoretic recipe for producing hadronic states in the limit of asymptotically long times.Over shorter times, however, there are transient fluctuations to multiparticle states withina hadron. As the signal-to-noise problem restricts lattice QCD simulations to times farfrom the asymptotic regime where LSZ applies, we are motivated to investigate correctionsto the reduction formula. To focus our discussion, we consider the chiral dynamics ofthe nucleon. We derive corrections to nucleon two- and three-point functions using chiralperturbation theory. In heavy baryon chiral perturbation theory, one expands about smallenergies above the nucleon mass. The nucleon operator should thus be thought of as somebaryon interpolating field that has been highly optimized through the removal of high-lying modes, i.e. modes k that are k & Λ χ above the nucleon mass, where Λ χ ∼ GeV is the chiral symmetry breaking scale. In the language of lattice QCD, the renormalizednucleon operator in chiral perturabtion theory behaves similarly to a quark-smeared baryon Excluding pion zero modes by using parity-orbifold boundary conditions has been suggested as a meansto overcome the signal-to-noise problem [6, 7]. G A .On the other hand, we describe a scenario in which the axial charge can be underestimatedby improving the nucleon overlap in two-point functions. Concluding remarks are given inSection IV. II. SPECTRAL REPRESENTATION
Consider a free fermion field, N ( x ), of mass M . In the limit of large mass, the velocitybecomes a good quantum number in coordinate space; and, to leading order in 1 /M , wecan write down a heavy fermion effective Lagrangian [8, 9, 10], L = N † D N , where N is atwo-component Pauli spinor. The coordinate space propagator, D ( x, D ( x,
0) = θ ( x ) δ ( x ) e − Mx . When interaction terms are includedin the Lagrangian, the full two-point function, G ( x, G ( x,
0) = θ ( x ) δ ( x ) e − Mx Z ∞ dE ρ ( E ) e − Ex , (1)which follows from taking the large mass limit of the K¨all´en-Lehmann spectral representa-tion. The spectral function, ρ ( E ), is a positive function in the distribution sense. Assumingthere is an isolated single-particle state corresponding to the N , we write ρ ( E ) = δ ( E )+ ρ ( E ).The residual spectral function, ρ ( E ), is then assumed to vanish below the energy E th , with E th >
0. With the single-particle state isolated, the spectral representation has the form G ( x,
0) = θ ( x ) δ ( x ) e − Mx (cid:20) Z ∞ E th dE ρ ( E ) e − Ex (cid:21) . (2)In the absence of bound states, E th is the threshold energy to create a multiparticle state.Suppose there is an excited state, N ∗ , of mass M ∗ contributing to the spectral function.The spectral weight around the value E = M ∗ − M then contains a delta-function. Forasymptotically large time separation between source and sink, x ≫
1, this excited-statecontribution is suppressed relative to the ground state by an exponentially small factor:exp[ − ( M ∗ − M ) x ]. The contributions from multiparticle states too are suppressed in thelimit of asymptotic time separation. If we assume the spectral weight near threshold is ofthe form ρ ( E ) ∝ ( E − E th ) n − , then the contribution from the multiparticle branch-cut issuppressed by a factor of ( x ) − n exp( − E th x ), for large time separation. The power n is3 IG. 1: One-loop diagrams contributing to the nucleon two-point function. Single (double) linesdenote nucleons (deltas), while the dashed lines denote pions. The filled circles denote axialcouplings from the interaction Lagrangian, Eq. (3). determined by the available phase space at threshold, and the required angular momentumfor multiparticle production. If correlation functions are not deduced at large enough timeseparations, the light pions will likely present difficulty for bound states in QCD, because E th = m π . A. Chiral Computation
Treated as heavy fermions, free nucleons can be described the heavy fermion Lagrangian,with N now upgraded to a two-component isospinor, N = ( p, n ) T . Interactions betweenpions and nucleons; as well as pions, nucleons and deltas can be included in a way consistentwith chiral symmetry [11, 12]. These interactions, moreover, can be systematically organizedin terms of a small expansion parameter ε , where the quantities: k/M with k as residualbaryon momentum, p/ Λ χ with p as pion momentum and Λ χ = 4 πf as the chiral symmetrybreaking scale, m π / Λ χ , and ∆ / Λ χ with ∆ as the nucleon-delta mass splitting, are all treatedto be of the size ε . The leading interactions among the baryons and pions are at O ( ε ), andcontained in the Lagrangian L = 2 g A N † S · A N + g ∆ N (cid:2) T † · A N + N † A · T (cid:3) . (3)Here S = σ / T . The axial-vector field of pions is at leading order A = f ∇ φ + . . . , with φ = τ a π a .Using Eq. (3), we can determine the interacting nucleon two-point function at one-looporder in the chiral expansion and thereby determine the spectral function. The contributingdiagrams are shown in Figure 1. Their computation is complicated by divergences which weregulate using dimensional regularization. To renormalize the two-point function, we chooseto work in coordinate space. This is particularly advantageous for our consideration of the It is instructive to carry out the computation additionally in momentum space. After the mass andwavefunction renormalization have been taken into account, the renormalized self-energy Σ ren ( E ) has theform Σ ren ( E ) = Σ( E ) − Σ(0) − E ddE
Σ(0) , but still contains divergences. These divergences only contribute to the extreme short-time behavior ofthe two-point function, i.e. contributions to G ( τ ) in Eq. (4) of the form δ ′ ( τ ), and δ ′′ ( τ ). The spectralfunction, however, can be deduced from the regulated self-energy from analyticity. The relation ρ ( E ) = − π ℑ m [ G ( E )] , produces the same spectral function as deduced in coordinate space. G ( τ ) ≡ Z d x G ( x , τ ; , , (4)where we have used spacetime translational invariance to locate the nucleon source at theorigin of our coordinate system. To renormalize the zero-momentum two-point function, werequire that there be a single nucleon contribution at asymptotically large τ . This consistsof two parts, the mass renormalizationlim τ →∞ (cid:20) − ddτ log G ( τ ) (cid:21) ≡ M phys , (5)that fixes the nucleon mass to its physical value (which thus absorbs its pion mass depen-dence), and the wavefunction renormalizationlim τ →∞ (cid:2) e M phys τ G ( τ ) (cid:3) ≡ , (6)that fixes the single nucleon probability to unity. Carrying out the one-loop chiral compu-tation of the nucleon two-point function, we find G ( τ ) = θ ( τ ) e − M phys τ (cid:20) Z ∞ m π dE ρ πN ( E ) e − Eτ + Z ∞ m π +∆ dE ρ π ∆ ( E ) e − Eτ (cid:21) , (7)where the pion-nucleon and pion-delta fluctuations are described the by spectral weights ρ πN ( E ) = 6 g A (4 πf ) [ E − m π ] / E , (8) ρ π ∆ ( E ) = 16 g N πf ) [( E − ∆) − m π ] / E . (9)For asymptotically large times, the decaying exponential from the single nucleon state isthe dominant contribution to the two-point function. Modification from pion interactions canbe deduced in this limit by considering the spectral weights near threshold. The pion-nucleonand pion-delta spectral functions both vanish as the 3 / ∝ ( E − E th ) / . This power-law behavior is due to the two-body phase space, ∝ It is straightforward to derive analogous results in a finite spatial volume. With periodic boundaryconditions, the lattice momentum modes are quantized in the form k = 2 π n /L , where n ∈ Z , and L issize of the lattice, which is assumed to be the same in each of the three spatial directions. Integrals overthe energy are replaced by sums over the momentum modes Z dE E f ( E ) → X n πnL f (cid:0)p k + m π (cid:1) , where n = √ n . We are assuming a large enough volume, m π L ≫
1, so that the sum can be replaced bythe integral for which the spectrum is continuous. Finite volume corrections to this approximation canbe calculated as the difference between the sum and integral. o D Full Τ (cid:144) a M e ff (cid:144) M N m Π =
140 MeV Τ (cid:144) a M e ff (cid:144) M N m Π =
300 MeV
FIG. 2: Interacting nucleon effective mass plots. The curve denoted by
Full corresponds to theeffective mass derived from the fully interacting two-point function in Eq. (7), while that denoted by No ∆ is the same two-point function evaluated without delta resonance contributions, i.e. g ∆ N = 0.The vertical bar symbolizes the region where we imagine noise might dominate the correlationfunction. ( E − E th ) / , and the requirement that the pion and baryon be in a relative p -wave. Becausethe pion-delta threshold is larger than the pion-nucleon threshold by the mass splitting ∆,the pion-delta contributions are suppressed relative to pion-nucleon contributions by theexponential factor exp( − ∆ τ ). Hence we shall neglect the delta contribution in asymptopia.Expanding the pion-nucleon spectral weight about threshold, we arrive at the asymptoticexpansion of the nucleon two-point function G ( τ ) τ ≫ −→ e − M phys τ " √ π (cid:18) g A m π πf (cid:19) e − m π τ ( m π τ ) / (cid:18) −
258 1 m π τ + 1785128 1( m π τ ) + . . . (cid:19) . (10)The . . . denotes power-law suppressed terms proportional to 1 /m π τ . The expansion for largetimes is only an asymptotic one, which is evidenced by the particularly large numericalcoefficients of higher-order terms. The times over which the expansion in 1 /m π τ resultsin a controlled approximation to Eq. (7) are quite large, τ & a , for a = 0 . fm . Forsmaller times, the first term in the expansion yields the best agreement with Eq. (7), as ischaracteristic of asymptotic expansions used outside their range of validity [13]. To analyzenucleon two-point functions, however, we shall not use the approximation in Eq. (10), butreturn to the full form in Eq. (7). B. Mass Extraction
Now we consider the effect of the pion continuum on the nucleon two-point function. Atypical quantity used for guiding the eye in fits of lattice two-point functions is the effectivemass, and we shall employ this tool to investigate the interacting nucleon two-point functionderived from chiral perturbation theory. The effective mass function is defined as M eff ( τ /a ) = − log G ( τ + a ) G ( τ ) . (11)6e employ lattice units using a temporal lattice spacing of a = 0 . fm , which is the spacingin some current-day lattices. For large τ /a , the effective mass should become flat with avalue corresponding to the nucleon mass. The onset of a plateau in the effective mass givesone an indication of when the ground state dominates the two-point correlation function.Of course, in actual lattice calculations the signal-to-noise problem for the nucleon limitsone to times that are not ideally long.In Figure 2, we make two effective mass plots using the interacting nucleon two-pointfunction in Eq. (7). The low-energy constants are fixed to their known values, specifically weuse g A = 1 . g ∆ N = 1 .
5, ∆ = 290
MeV , and f = 130 MeV . We plot the ratio of the effectivemass to the nucleon mass for two values of the pion mass, m π = 140, and 300 MeV . Theonset of a plateau in the effective mass hence corresponds the ratio M eff /M N approachingunity which is exhibited in both panels of the figure. We see, moreover, that the pion-delta continuum gives negligible contributions due to the exponential suppression factor,exp( − ∆ τ ). In a recent lattice study [14], the noise begins to dominate the signal around τ /a = 12 at m π ∼ MeV . We show a vertical line at this time to denote an imaginednoise barrier for our chiral computation. Fitting the nucleon two-point function with asingle exponential form at m π = 300 MeV for 6 ≤ τ /a ≤
12 will overestimate the nucleonmass by ∼ τ /a = 12. Fitting the two-point function on a smaller time interval, say8 ≤ τ /a ≤
12 results in the same quantitative overestimation of the nucleon mass. We cansee, moreover, that improving the signal-to-noise at light pion masses will expose effectivemasses which continue to fall with time. These qualitative features can be deduced directlyfrom the positivity of the spectral representation.One final observation about the nucleon two-point function is as follows. Improving theoverlap with the nucleon in two-point functions largely removes contamination from thepion-nucleon continuum, as evidenced by Figure 2. Due to the exponential suppression,pion-delta contributions are largely unaffected by the improvement. Such contributionsare not necessarily suppressed, however, in other nucleon correlation functions. The axialcorrelator provides such an example.
III. AXIAL THREE-POINT FUNCTION
We now treat the nucleon axial-current correlation function at finite times using thecoordinate-space approach. Formally there are no spectral representations for three-pointfunctions. Consequently contamination from excited states can drive the correlator up ordown depending on the underlying dynamics.
A. Chiral Computation
Nucleon three-point functions are constructed using the LSZ reduction formula to iso-late single-nucleon contributions from the external legs. We are interested in a three-pointfunction formed from inserting the axial current J +5 µ ( y , t ) between two nucleon states. Fol-lowing the typical lattice procedure (for recent lattice calculations of the axial charge,see [15, 16, 17, 18, 19]), and projecting both source and sink onto vanishing three mo-7 IG. 3: One-loop diagrams contributing to the axial-vector current matrix element of the nucleon.The open squares denote axial-vector current insertions, while the remaining diagram elements areas in Figure 1. mentum, we form the ratio R µ ( τ, t ) = R d x R d y h | N ( x , τ ) J +5 µ ( y , t ) N † ( , | i R d x h | N ( x , τ ) N † ( , | i . (12)Using the axial current at tree-level in the effective theory, namely J +5 µ = 2 g A N † S µ τ + N , wearrive at R µ ( τ, t ) = θ ( τ − t ) θ ( t ) 2 g A u † S µ u, (13)where u is a Pauli spinor. Working beyond tree level, there are divergences. After regular-ization, we renormalize to the physical axial charge, G A , namely R µ ( τ, t ) τ ≫ t ≫ −→ G A u † S µ u, (14)which additionally absorbs the pion-mass dependence of the axial charge into the physicalcoupling G A .The leading one-loop diagrams contributing to the three-point function of the axial cur-rent are shown in Figure 3. The required terms of the baryon axial current appear as [9, 10] J +5 µ = 2 g A N † S µ τ + N + g ∆ N (cid:0) T † µ τ + N + N † τ + T µ (cid:1) − g ∆∆ T † S µ τ + · T , (15)while the pion-nucleon and pion-nucleon-delta interaction terms are contained in the La-grangian Eq. (3). Additionally to calculate the ratio R µ ( τ, t ), we must divide by thetwo-point function calculated to one-loop order. The result of the chiral computation canbe cast in the form R µ ( τ, t ) = 2 u † S µ u h G A + F A ( τ, t ) i θ ( τ − t ) θ ( t ) , (16)where the function F A ( τ, t ) encodes the deviation from the limit of infinite time separation.The form of this function is F A ( τ, t ) = 8 g A Z ∞ m π dE ρ Nπ ( E ) (cid:2) e − E ( τ − t ) + e − Et − e − Eτ (cid:3) − g ∆ N √ Z ∞ m π dE p ρ Nπ ( E ) ρ ∆ π ( E + ∆) h e − E ( τ − t ) (cid:0) e − ∆( τ − t ) (cid:1) + e − Et (cid:0) e − ∆ t (cid:1) − e − Eτ (cid:0) e − ∆ t + e − ∆( τ − t ) (cid:1) i + (cid:18) g A + 25 g ∆∆ (cid:19) Z ∞ m π +∆ dE ρ ∆ π ( E ) (cid:2) e − E ( τ − t ) + e − Et − e − Eτ (cid:3) . (17)The nucleon-pion, ρ Nπ ( E ), and delta-pion, ρ ∆ π ( E ), spectral weights have been given abovein Eqs. (8) and (9). In the limit { τ, t } → ∞ with τ > t , accordingly we have F A ( τ, t ) → . Axial Charge Extraction To investigate the effect of finite times on the extraction of the nucleon axial charge, weplot the ratio of axial three-point to two-point functions, R µ ( τ, t ), given simply by R µ ( τ, t ) = G A + F A ( τ, t ) . (18)Scaling R µ ( τ, t ) by the axial charge, G A , we thus expect a plateau at unity for asymptot-ically large time separations, τ ≫ t ≫
1. We fix the sink time at τ /a = 12, which is atypical source-sink separation in lattice QCD computations of three-point functions. Withthe sink time τ fixed, the current insertion time dependence of R µ is shown in Figure 4.We compare the behavior with and without delta resonances. Including the resonances re-quires knowledge of the delta axial charge, g ∆∆ . This low-energy constant is poorly known,see [20], and we assume the value g ∆∆ = − .
25. Rather fortuitously the time-dependenceof the axial correlator is largely insensitive to the value of the delta axial charge. To agood approximation, we can drop all terms in Eq. (17) that arise from diagrams with onlyintermediate state deltas, that is F A ( τ, t ) ≈ g A Z ∞ m π dE ρ Nπ ( E ) (cid:2) e − E ( τ − t ) + e − Et − e − Eτ (cid:3) − g ∆ N √ Z ∞ m π dE p ρ Nπ ( E ) ρ ∆ π ( E + ∆) h e − E ( τ − t ) (cid:0) e − ∆( τ − t ) (cid:1) + e − Et (cid:0) e − ∆ t (cid:1) − e − Eτ (cid:0) e − ∆ t + e − ∆( τ − t ) (cid:1) i , (19)which is independent of g ∆∆ . The figure confirms that Eq. (19) well approximates the time-dependence of the axial correlation function. Excluding the delta completely (by setting g ∆ N = 0), however, has a dramatic effect on the time-dependence of the axial correlationfunction. The figure shows that the contributions from pion-nucleon excited states drive theaxial correlator upward. Those from pion-delta states are negligible, while contributions fromthe interference of pion-delta and pion-nucleon states drive the axial correlator downward.We see the effect of insufficient time can lead to spurious plateaus for three-point func-tions. At both pion masses depicted, the derived correlation function flattens out, butabove the asymptotic value of unity. Our expectation from chiral perturbation theory isthat the nucleon axial charge would be overestimated, with the overestimation worsening asthe pion mass is lowered. Rather large couplings of the pion to the nucleon, and nucleon-delta transition complicate extraction of quantities from lattice three-point functions. Assuch systematic errors, moreover, depend on the pion mass, the resulting chiral behavior of G A , for example, will be specious. Lattice QCD calculations often extract the ratio of theaxial-to-vector charges, G A /G V . The vector charge, G V , is determined from the three-pointfunction of the vector current which is also susceptible to an effect from insufficient time.The one-loop calculation of the vector three-point function in heavy baryon chiral perturba-tion theory, however, does not produce any finite time corrections due to exact cancellationborne in by the vector Ward identity. This cancellation is only guaranteed at zero momen-tum transfer. Thus away from this limit, the vector three-point function too is subject tofinite-time corrections.As a final comment, there has been debate about underestimation of the axial charge inlattice computations [24, 25]. The most recent study of the axial charge in fully dynamical9 o Exp D No D Full t (cid:144) a R Μ (cid:144) G A m Π =
140 MeV t (cid:144) a R Μ (cid:144) G A m Π =
300 MeV
FIG. 4: Time-dependence of the axial current correlation function. For sink time τ /a = 12, weplot the ratio of the axial three-point to two-point correlators as a function of the current insertiontime t/a . This ratio we scale by the axial charge G A . The curve denoted by Full corresponds tothe axial correlation function determined from Eq. (17), while No ∆ corresponds to the correlatorwithout delta-resonance contributions, i.e. g ∆ N = 0. The curve No Exp ∆ excludes from the axialcorrelator contributions that are suppressed by an exponential factor involving the mass splitting∆, see Eq. (19). three-flavor simulations [18] finds G A falling as the pion mass is decreased. This behaviorwas attributed to a finite volume effect, but not that described by effective field theory [21,22, 23]. We undertook the present study with the hope that the falling value of G A couldbe attributed to excited state contamination: i.e. although the axial charge is calculatedin [18] on two volumes, these volumes have differing temporal extents, and hence differingeffects from insufficient source-sink separation. Given our results, there is a plausible wayin which such underestimation is possible. Underestimation of the axial correlation functioncan arise from improvement of the nucleon overlap in two-point functions. Designing baryonoperators with the best ground-state overlap in a two-point function does not necessarilyoptimally improve three-point functions. In a two-point function, improvement of the thenucleon operator decreases contamination from the pion-nucleon continuum, but not fromthe the pion-delta continuum, because the latter makes only negligible contributions. In theaxial three-point function, however, the dominate excited state contributions arise from boththe square of the overlap with pion-nucleon states, and the interference term between pion-nucleon and pion-delta states. Improving the nucleon operator diminishes the former fasterthan the latter. As the latter is negative, the axial correlator can thus be underestimatedfor highly optimized nucleon interpolating fields. IV. SUMMARY
In an interacting quantum field theory, single particle states emerge only in the limit oflong times. The existence of particle interactions necessarily implies that at shorter times a While the volume dependence employed in [18] is inspired by the p -regime of chiral perturbation theory,the size of the corrections is not consistent with expectations [21]. τ ∼ /m π , over which pion branch-cuts can be important.Using the nucleon in chiral perturbation theory as an example, we investigated multiparticlecontributions to two- and three-point functions. Specifically we found the nucleon mass issubject to overestimation, which is directly related to the positivity of the spectral function.For the nucleon axial charge, G A , we found that three-point functions calculated away fromasymptotic time separations are affected by the competition between pion-nucleon and pion-delta contributions. Pion-nulceon excited states drive the axial correlation function upward,while the interference between pion-delta and pion nucleon excited states drives the axialcorrelation function downward. The sign of these corrections depends on the underlyingchiral dynamics, and cannot be deduced from general principles alone.The formulae derived in this work apply to the renormalized nucleon operator in chiralperturbation theory. Because high-lying excitations are absent, this operator behaves like aquark-smeared lattice QCD interpolating field. While not quantitative, the analytic insightoffered by our computation, however, is useful for comparing qualitatively with lattice QCDcomputations, nearly all of which use optimized baryon operators with smeared quark fields.Using a basis of interpolating operators, a variational method can be employed to isolateground states from contamination by excited states [26, 27]. Work along these lines hasbeen pursued, for example, the baryon spectrum has been investigated in [28, 29, 30, 31].Removing pion-nucleon contamination from the nucleon two-point function should improvethe mass determination, although the study in [30] seems to overestimate the nucleon mass.Despite eliminating a large part of the excited-state contamination, the nucleon effectivemass continues to drop (their Figure 6 has the same qualitative behavior as our Figure 2),signaling the need for a two-exponential fit. Only a single exponential fit, however, wascarried out.Our calculation of the nucleon axial current makes plausible that the use of improvednucleon operators can lead to an underestimation of the axial charge. The extraction ofnucleon observables from three-point functions is potentially fallible due to the pion contin-uum, which is important owing to the lightness of the pion, as well as the comparativelylarge axial coupling, g A . The same observation can be made for three-point functions in-volving deltas due to the size of the axial transition coupling, g ∆ N . As pion masses approachthe physical point, the effect of insufficient time in lattice QCD calculations of nucleon anddelta properties will become pronounced. Optimizing the overlap with the ground state intwo-point functions will not necessarily lead to better determination of observables fromthree-point functions because of interference terms. We hope our work can qualitativelyguide the extraction of nucleon and delta observables from lattice QCD.11 cknowledgments We acknowledge the Institute for Nuclear Theory for their hospitality, and support fromthe U.S. Department of Energy, under Grant No. DE-FG02-93ER-40762. [1] T. DeGrand and C. DeTar,
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