The groundbreaking discovery of atmospheric and solar neutrino oscillations more than two decades ago by the Super-Kamiokande [1] and SNO [2] experiments, respectively, required an adjustment of the Standard Model to accommodate massive neutrinos. The present generation of terrestrial long baseline neutrino oscillation experiments, aimed at a more precise determination of the neutrino masses and mixing parameters, NOvA [3] and T2K [4] as well as the future DUNE [5] experiment and the upgrade of T2K to the Hyper-Kamiokande detector [6] determine the fluxes of muon and antimuon neutrinos via their interaction with nuclear targets in a near and a far detector. The neutrino energies of the scattering events are reconstructed via Monte Carlo event generators [7,8], which require knowledge of the differential neutrino-nucleon cross section. For neutrino energies below 1 GeV this is dominated by (quasi-)elastic scattering, while from about 400 MeV onward also resonance production with nucleon-pion (Nπ) final states sets in [9].

Focusing on low energies, the cross section is proportional to the square of a combination of nonperturbative nucleon vector and axial matrix elements, which can be parametrized in terms of form factors. The two vector form factors as functions of the squared four-momentum transfer (Q2) are sufficiently well known from experiment. However, the two isovector axial form factors GA(Q2) and G˜P(Q2) are much less well constrained experimentally, apart from GA in the forward limit (Q2=0, i.e. the axial charge gA [10]) and G˜P(0.88mμ2)=gP* at the muon capture point of muonic hydrogen [11]. Fortunately, these form factors can be computed directly from QCD via lattice simulation. However, there is a tension [12,13] between recent lattice results [14–18] and analyses of neutrino-deuteron scattering experiments [19]. Therefore, it is important to establish the reliability of the lattice determinations. This requires the investigation of all systematics and, in particular, the one associated with extracting the nucleon matrix elements from correlation functions at finite Euclidean times. The latter receive contributions also from single- and multiparticle states with the same quantum numbers as the nucleon (normally referred to as excited states). At zero momentum, the lowest excitations with positive parity include Nπ P-wave and Nππ S-wave scattering states, whereas, at nonvanishing momentum, parity is not a good quantum number and also Nπ in an S-wave can contribute. Toward small pion masses, the mass gap between the ground state and the first excitation decreases, and the spectrum becomes more dense. Bearing in mind that the signal-to-noise ratio of correlation functions decreases exponentially with the Euclidean time separations, it can be very challenging to reliably extract nucleon matrix elements. In order to control the leading excited state contributions to these nucleon to nucleon form factors, we will, for the first time, explicitly calculate matrix elements that are also related to nucleon to Nπ transition form factors, which are required for a firm understanding of the resonance production regime.

Reliable continuum limit results for the axial form factors should reproduce the experimentally known values of gA and gP* and also be consistent with the partially conserved axial current (PCAC) relation [also referred to as the axial Ward identity (AWI)], which relates the axial form factors to the pseudoscalar form factor. In many previous simulations gA was reproduced; however, gP* (defined at Q2>0) was found to be smaller than the experimental value, and also the AWI between form factors was significantly violated [16,20–23]. Since the AWI was found to be satisfied on the level of the correlation functions in the continuum limit [24], the inconsistency had to be related to the difficulty of isolating the ground state contribution when extracting the form factors [14,15,24]. While the interpolator that is used to create the nucleon was found to have little overlap with excited states, as evidenced by analyses of two-point functions, transition matrix elements between different states, contributing to the spectral decomposition of the three-point function, appeared to be enhanced. Indeed, in chiral perturbation theory (ChPT) the axial and pseudoscalar currents directly couple to the pion. Regarding the pseudoscalar current or the time component of the axial current, N to Nπ transitions can contribute substantially to the three-point functions [25–27] (see also Refs. [28,29]). At a small but nonvanishing momentum, the leading such contribution increases in proportion to the ratio of the nucleon mass over the pion energy, mN/Eπ [25,26]. These terms were taken into account in recent analyses of the Euclidean time dependence of lattice correlation functions, where form factors were obtained, that are consistent with the AWI [14,15,17]. However, the size of the excited state contamination, found in these analyses, is quite large in some channels for the Euclidean times that are accessible at present. A more reliable approach would be to construct optimized interpolators to minimize the dominant (Nπ) excited state contributions.

In this work, we take into account directly the Nπ contribution by constructing nucleon-pion-like interpolators [(qqq)(q¯q) with the quarks q∈{u,d}], O5q, and computing the associated two-point and, for the first time, three-point correlation functions between the standard three-quark nucleon interpolator O3q and O5q. Using this basis, that has good overlap both with the nucleon ground state and the lowest lying Nπ state, nucleon to nucleon three-point functions can be constructed with minimized Nπ contributions, enhancing the reliability of the extraction of the nucleon matrix elements. As mentioned above, this is the first step toward determining nucleon to nucleon-pion matrix elements, associated with neutrino scattering in the resonance production regime [9,13,30]. In this pilot study we carry out the analysis for a single unphysical pion mass mπ=429  MeV. It turns out that even at this relatively large value the Nπ contribution is very significant and that this can effectively be removed with our approach. We expect this method to work even better at the physical pion mass: ChPT becomes more reliable as the pion mass is reduced, and the tree-level Nπ contribution is even more dominant, which is consistent with the observations made in Refs. [14,15,25–27].

We define local isovector pseudoscalar and axial currents, P=d¯γ5u and Aμ=d¯γμγ5u, respectively. The Lorentz decompositions into form factors of the respective matrix elements read as ⟨np′|P|pp⟩=u¯p′GP(Q2)γ5up,(1)⟨np′|Aμ|pp⟩=u¯p′[γμGA(Q2)+qμ2mNG˜P(Q2)]γ5up,(2)where we assume isospin symmetry (i.e. mN=mp=mn and mℓ=mu=md for the quark masses), up is the spinor of a nucleon with three-momentum p, qμ=pμ′-pμ is the four-momentum transfer and Q2=-qμqμ. Note that the above decomposition of the axial matrix element does not hold if the two states differ in their mass, e.g., if a nucleon is on the right-hand side and a Nπ on the left. The AWI ∂μAμ=2imℓP implies the relation between form factors, mNGA(Q2)=mℓGP(Q2)+(Q2/4mN)G˜P(Q2), which is exact in the continuum limit but will be affected by moderate discretization effects at our lattice spacing a≈0.098  fm [14,24]. In addition, the pion pole dominance (PPD) assumption gives the approximate relation, G˜P(Q2)≈4mN2GA(Q2)/(mπ2+Q2). While this only holds exactly for mπ=0, in Ref. [14] it was found to hold within uncertainties of 1%–2% at the physical point in the continuum limit, with violations of less than 3% up to mπ≈420  MeV. Deviations from these relations can be quantified in terms of the differences from unity of the combinations rPCAC=4mNmℓGP(Q2)+Q2G˜P(Q2)4mN2GA(Q2),(3)rPPD=(mπ2+Q2)G˜P(Q2)4mN2GA(Q2).(4)

We construct the matrices of two- and three-point correlation functions (see the Supplemental Material [31]), C2pt(p,t)ij=⟨Oi(p,t)O¯j(p,0)⟩,(5)C3ptJ(p′,t;q,τ)ij=⟨Oi(p′,t)J(q,τ)O¯j(p,0)⟩,(6)where we indicate the three-momentum transfer in the argument of the local current J∈{P,Aμ}. The interpolators Oi∈{O3q,O5q} are projected onto the G1 representation of the double cover of the cubic group O2h (or, for nonvanishing momentum, the relevant little group) [32–34], corresponding to spin and helicity 1/2 in the continuum, as well as to definite momentum and isospin. For instance, I3=-1/2 corresponds to O3q∼n and O5q∼1/3nπ0-2/3pπ-. The Wick contractions of the correlation functions are evaluated using the sequential method [35] for quark-line connected topologies, while the stochastic “one-end-trick” [36–38] is used for disconnected diagrams.

For the results shown here, about 200 propagators (see the Supplemental Material [31]) for each of the six source-sink separations have been computed on 800 gauge configurations. In view of the computational cost, we carry out the analysis on a single coordinated lattice simulations ([39]) ensemble (A653, see Ref. [40]) with the spatial volume L3=(24a)3, employing Nf=3 nonperturbatively improved Wilson fermions with a lattice spacing a≈0.098  fm and the pion mass mπ=429  MeV. The best results were obtained, using extended (smeared) quark fields in the nucleon and pion interpolators. (For details on the smearing, see Appendix C.1 and Table 15 of Ref. [40]).

We extract the generalized eigenvalue and eigenvector matrices Λ(p,t;t0)=diag(λ1(p,t;t0),λ2(p,t;t0)) and V(p,t,t0)=(v1(p,t;t0),v2(p,t;t0)), respectively, by solving the generalized eigenvalue problem (GEVP) [41–44] for the matrix of two-point functions, C2pt(p,t)V(p,t,t0)=C2pt(p,t0)V(p,t,t0)Λ(p,t,t0), for fixed reference times t0, where we employ the normalization vα⊺C2pt(t0)vα=1. For large times t the eigenvalues will decay exponentially with the energy of the state: λα(p,t;t0)→dα(p;t0)e-Eα(p)(t-t0), where dα≲1.

The effective energies Eeffα(t)=a-1ln[λα(t)/λα(t+a)] are shown in Fig. 1 for p=0 and t0=0.2  fm. The lowest energy coincides with the nucleon mass on this ensemble [40], while the second level is close to the sum of the nucleon and pion energies for the lowest P-wave momentum combination. Therefore, we will identify N with α=1 and Nπ with α=2. Note that the eigenvectors are very stable in t and that the contribution of O5q (subscript i=2) to the nucleon state is suppressed by more than 1 order of magnitude relative to O3q. Nevertheless, as we will see, the impact on three-point functions can be significant. We also solve the GEVP for moving frames, in particular for p=ez≕2πL(0,0,1) (|ez|≈530  MeV). Regarding O5q, we consider the combinations O3q(ez)Oq¯q(0) and O3q(0)Oq¯q(ez), with Oq¯q being a pion interpolator. Solving the GEVP, in both cases we find the effective energy of the second eigenvalue for t>0.5  fm to be consistent with the N(ez)π(0) and N(0)π(ez) noninteracting energies, respectively.

Considering these results, we employ the generalized eigenvectors for t0=0.2  fm and t=0.5  fm to construct the GEVP-optimized correlation functions, C2pt(p,t)α=viα(p)C2pt(p,t)ijvjα(p),(7)C3ptJ(p′,t;q,τ)αβ=viα(p′)C3ptJ(p′,t;q,τ)ijvjβ(p),(8)where α,β∈{N,Nπ}. Note that here we only present results for α=β=N, and we neglect the i=j=2 element (5q to 5q) of C3ptJ that, in this case, is suppressed by the second power of the small eigenvector component v5qN. In addition, from ChPT we would only expect nondiagonal elements of the matrices of correlators to be enhanced. The nucleon matrix elements of interest are then extracted by forming the GEVP ratios [45–47] RJ(p′,t;q,τ)=C3ptJ(p′,t;q,τ)NNC2pt(p′,t)N×C2pt(p′,τ)NC2pt(p′,t)NC2pt(p,t-τ)NC2pt(p,τ)NC2pt(p,t)NC2pt(p′,t-τ)N∝⟨Np′|J|Np⟩(t≫τ≫0),(9)where excited state contributions of the type N→Nπ and Nπ→N are explicitly removed. Forming the same ratio for the usual two- and three-point functions, C2pt,11 and C3pt,11J, will give the same result in the limit of large t and τ. Any time dependence observed for these ratios (GEVP-improved or not) is an indication of remaining excited state contamination.

For p′=p the combination under the square root in Eq. (9) cancels. We consider two kinematic combinations: p′=p=0 and p′=p=ez. Regarding the rest frame, the three-point functions vanish due to parity for J=A4 and J=P, while J=Ai (with the spin projected in the i direction) at large Euclidean time separations gives the axial charge gA. Contamination from the coupling to Nπ states exists, however, only as a loop effect in ChPT. Indeed, even when using the standard ratio, for t>0.8  fm the data near τ=t/2 show no time dependence within their errors. Fitting the (unimproved) ratio for 1.15  fm<t<1.4  fm, we find gA=1.156(7) at our unphysical pion mass.

Also in the moving frame the Nπ contributions to Ai only appear as loop effects, and we find that the corresponding standard ratio is almost constant (black stars in Fig. 2). However, regarding A4 and P, N to Nπ transitions appear at tree level and are enhanced by one power of mN/Eπ, relative to the N to N matrix elements of interest. The ratio RA4 will be proportional to gA at large times too; however, using the O3q interpolators, we find substantial excited state contamination, which is indicated by its strong dependence on the source-sink separation; see the blue symbols in the upper panel of Fig. 2. A difference between the ratios for Ai and A4 at p′=p≠0 was also observed, e.g., in Ref. [48], using standard interpolators. In contrast, the GEVP-improved ratios (red symbols) already agree for t>0.6  fm with the value extracted from the standard ratios for Az obtained at p≠0 (light green band) and at p=0 (dark green band).

A similar, dramatic reduction of the excited state contamination is observed for the pseudoscalar current for the GEVP-optimized correlation functions. For this current, due to charge conjugation symmetry, diagonal matrix elements vanish. Therefore, the deviation from zero of the standard ratio in the lower panel of Fig. 2 is entirely an excited state effect, which is removed within the present errors for the GEVP-optimized ratio. This demonstrates that the dominant contribution is from N to Nπ transitions, which is consistent with the tree-level ChPT expectation.

When determining the axial and pseudoscalar form factors, the excited state contamination is prominent for correlation functions involving the currents J∈{A4,P} [14,25–27], that can transfer the momentum to a pion at tree level in ChPT. This contribution, which is proportional to mN/Eπ [25,26], is largest at small momentum transfer. Therefore, we consider these two currents and set q=ez to a single unit of lattice momentum (|q|≈530  MeV). With p′=0 and p=-q, this corresponds to Q2≈0.3  GeV2. Our results for the two ratios for the standard and optimized correlation functions are shown in Fig. 3. Clearly, the time dependence is much reduced for the GEVP-optimized results: at the source, excited states are effectively removed; however, at the sink (that is at rest) there are clearly residual effects from higher excitations.

The ratios at large times (green bands) are proportional to the respective matrix elements which, using the decompositions (1) and (2), are related to a linear combination of GA and G˜P for A4 and GP for P, respectively.

The axial form factor GA=0.91±0.01 is extracted from the standard correlation functions with Ai and ei⊥q. These show ground state dominance within our range of t and τ. Indeed, the large tree-level Nπ ChPT diagrams do not contribute to this channel, and only Nπ loop diagrams appear [25]. The ground state matrix element is proportional only to GA, and we use the value we extract as a prior in fits to the other channels. We fit the GEVP-optimized ratios for the pseudoscalar and the temporal axial currents simultaneously to constants plus exponentials ∝e-ΔE(t-τ). We find ΔE≈2mπ for the gap between the nucleon ground state and this first excitation. The resulting matrix elements then give the pseudoscalar and induced pseudoscalar form factors at Q2≈0.3  GeV2, the latter after subtracting the GA contribution. The results for GP and G˜P as well as for the PCAC and PPD ratios of Eqs. (3) and (4) are shown in Fig. 4. We also include results that are obtained using the ChPT guided methods of Ref. [14] (M1) and a simultaneous fit to the channels A4, P and Ai with q=ei, inspired by [15] (M2). In spite of the large excited state contributions, the results using modern multistate analysis techniques agree within errors with the GEVP results, at least at mπ=429  MeV. At this single lattice spacing, the ratio rPCAC somewhat differs from 1.

Given the current tension [12,13] between results for the axial form factor obtained from lattice QCD and from reanalyses of historical neutrino-deuteron scattering experiments [19], it is important to rigorously investigate the systematics associated with the lattice approach. The PCAC relation between form factors has only recently been verified in some studies [14,15,17], and this provides an important cross check. We have shown that the very large excited state contributions encountered can be removed by including Nπ-type interpolators. This confirms ChPT expectations, even at our relatively large pion mass, and supports assumptions made in recent determinations of the axial form factor [14–18]. In the near future, we will repeat the study at a smaller pion mass where excited state contributions are even larger, with the aim of also determining N to Nπ matrix elements that are relevant for the scattering of neutrinos with energies larger than 400 MeV.