1 Introduction

Behavior of nucleon electromagnetic form factors \(G_{Ep,n}\), \(G_{Mp,n}(Q^2)\) at high momentum transfer \(Q^2\approx 5\ldots 10\,\textrm{GeV}^2\) have implications for understanding and improving models of nucleon structure. Models involving vector meson dominance, chiral solitons, a pion cloud, and relativistic constituent quarks have been employed to predict form factor behavior at large \(Q^2\). Generally, while some models may describe data for the four nucleon form factors, their predictions differ in the region where data are unavailable (see, e.g., Ref. [1] for a review). Studies of nucleon form factors using Dyson-Schwinger and Faddeev equations have demonstrated the significance of diquark correlations for the nucleon electromagnetic structure at high momentum transfer [2]. In particular, the zero crossing in the electric Sachs form factors depend on quark correlations in Faddeev’s amplitude of the nucleon, thus data from experiment or nonperturbative lattice QCD calculations can be used to determine their magnitude. The experimental program to determine nucleon form factors up to \(Q^2\approx 18\,\textrm{GeV}^2\) is well underway [3,4,5,6,7], and the first results have been published for the proton magnetic form factor \(G_{Mp}(Q^2)\) for \(Q^2\) up to \(\approx 16\text { GeV}^2\) [8]. This calls for ab-initio theoretical calculations of nucleon form factors with rigorous control of systematic effects, which is possible using modern lattice QCD methods.

Until recently, studies of nucleon form factors on a lattice have been limited by \(Q^2\lesssim 1\ldots 2\,\textrm{GeV}^2\). One notable exception is the calculation of the \(G_{Ep}/G_{Mp}\) ratio using Feynman-Hellman method [9]. Lattice calculations involving hadrons with large momentum \(|\vec p|\gtrsim m_N\) are challenging for several reasons. First, Monte Carlo fluctuations of lattice hadron correlators are governed by the energy of the state [10]. The signal-to-noise ratio for the nucleon is expected to decrease \(\propto \exp \big [-(E_N(\vec p) - \frac{3}{2} m_\pi )\tau \big ]\) with Euclidean time \(\tau \), making high-momentum calculations especially “noisy”. At the same time, excited states of the nucleon, which are expected to introduce large systematic uncertainties, are less suppressed by Euclidean time evolution \(\propto \exp \big [-\Delta E_N(\vec p)\tau \big ]\) as the energy gap \(\Delta E(\vec p) = E_{N,\text {exc}}(\vec p) - E_N(\vec p)\) shrinks with increasing relativistic nucleon momentum \(|\vec p|\). Both these challenges are best adressed by choosing the Breit frame on a lattice, so that the initial and final momenta of the nucleon are equal to \(|\vec p^{(\prime )}|=\frac{1}{2}\sqrt{Q^2}\). For example, momentum transfer \(Q_1^2\approx 10\,\textrm{GeV}^2\) requires nucleon momentum \(p_1\gtrsim 1.6\,\textrm{GeV}\), which reduces the energy gap \(\Delta E_N(0)\approx 0.5\,\textrm{GeV}\) to \(E_N(p_1)\approx 0.3\,\textrm{GeV}\). Therefore, very large Monte Carlo statistics combined with rigorous analysis of excited states contaminations become absolutely necessary to obtain credible results.

Such large-statistics calculations have been pursued for a few years, with results previously reported in Refs. [11, 12]. These calculations have been performed with \(N_f=2+1\) (light and strange) dynamical quarks with the clover-improved Wilson fermion action with lattice spacing \(a\approx 0.09\,\textrm{fm}\). Two values of the pion masses \(m_\pi \approx 280\) and \(170\,\textrm{MeV}\) used in the calculations allowed to check for light quark mass dependence of the results. Recently, we have extended our work to a finer lattice spacing \(a\approx 0.073\text { fm}\) (“E5” ensemble), which is absolutely essential to understand discretization effects, a likely source of systematic errors in calculations involving large momenta. In this paper, we report results obtained on these finer lattices, as well as those from previous coarser lattices but with substantially increased statistics. Our current results rely on multi-state fits to to assess systematic effects from excited states.

2 Methods

We have performed large-statistics calculations on four ensembles of lattice gauge fields. The summary of our accumulated statistics is shown in Fig. 1 and Table 1.

Fig. 1
figure 1

Lattice ensembles and statistics accumulated for each value of a and \(m_\pi \). The circle areas are proportional to the number of samples. Lighter pion-mass calculations (D5) require significantly more statistics

Full size image
Table 1 Summary of ensembles, kinematics, and statistics
Full size table

In order to obtain nucleon form factors, we calculate nucleon matrix elements of the quark vector current with large-momentum nucleons in the in- and out-states,

$$\begin{aligned} C_{N V_q^\mu \bar{N}}(\vec p', \vec q ; t_\text {sep},t_\text {ins}) = \sum _{\vec y,\vec z} e^{-i\vec p'\vec y + i\vec q\vec z }\, \langle N(\vec y,t_\text {sep})\, [\bar{q} \gamma ^\mu q]_{\vec z,t_\text {ins}} \, N(0)\rangle \,, \end{aligned}$$
(1)

where \(N=\epsilon ^{abc}[\tilde{u}^{aT} C\gamma _5 \tilde{d}^b] \tilde{u}^c\) is the nucleon interpolating field on a lattice constructed with “momentum-smeared” quark fields \(\tilde{q}\) to improve their overlap with the ground state of the boosted nucleon [13]. Nucleon matrix elements are extracted from nucleon-current three-point correlation functions using well-established methods of lattice QCD (see, e.g. Ref. [14]). Wick contractions of lattice quark fields generate two types of diagrams: quark-connected and quark-disconnected. The latter have lattice quark “loops” that are connected to the valence quark lines only by the gluons and are more difficult to compute. Their contributions to nucleon form factors at \(Q^2\lesssim 1.2\,\mathrm {GeV^2}\) were found small (\(\lesssim 1\%\)) [15], but remain to be explored at higher momenta; these contributions are omitted in the current work.

Fig. 2
figure 2

Nucleon effective energies and ground-state fits computed on ensembles E5 (left) and D6 (right)

Full size image
Fig. 3
figure 3

Dependence of nucleon ground-state energies on momentum computed on ensembles E5 (left) and D6 (right)

Full size image
Fig. 4
figure 4

Fits to the nucleon three-point functions (3) for the D5 ensemble. The colored bands show fits to Eq. (3), the dark-gray bands are the ground-state values \({\mathcal A}_{0'0}\) from these fits, and the light-gray bands are overdetermined fits of these matrix elements to the form factors \(F_{1,2}(Q^2)\)

Full size image

The nucleon correlators become dominated with the ground state \(C(t) = \langle N(t)\ldots \bar{N}(0)\rangle \propto e^{-E_N t}\) as the Euclidean time \(\tau \) is inscreased. As expected, there are substantial contributions from nucleon excited states. Although more than one excited state is expected to contribute, the data are not precise enough to constrain more than one, especially at large momenta. Therefore, we impose a simple two-state model on lattice data

$$\begin{aligned} \langle N(\vec p, t) \bar{N}(0)\rangle&\sim C_0^2 e^{-E_{N0} t} + C_1^2 e^{-E_{N1} t}\,, \end{aligned}$$
(2)
$$\begin{aligned} \langle N(\vec p^\prime , t) J(\vec q, \tau ) \bar{N}(0)&\sim {\mathcal A}_{0^\prime 0} C_{0^\prime } C_0 e^{-E_{N0}^\prime (t-\tau ) - E_{N0}\tau } + {\mathcal A}_{1^\prime 0} C_{1^\prime } C_0 e^{-E_{N1}^\prime (t-\tau ) - E_{N0}\tau } \nonumber \\&\quad + {\mathcal A}_{0^\prime 1} C_{0^\prime } C_1 e^{-E_{N0}^\prime (t-\tau ) - E_{N1}\tau } + {\mathcal A}_{1^\prime 1} C_{1^\prime } C_1 e^{-E_{N1}^\prime (t-\tau ) - E_{N1}\tau } \end{aligned}$$
(3)

to extract ground-state nucleon energies \(E_{N0}^{(\prime )}\) and momentum-dependent matrix elements of nucleon operators \(C_{0^{(\prime )}}=\langle \textrm{vac}|N|N(\vec p^{(\prime )})\rangle \) and vector current density \({\mathcal A}_{0^\prime 0}=\langle N(\vec p^\prime )|J|N(\vec p)\rangle \). The fits and the ground-state energies from the former are shown in Fig. 2, together with effective-energy estimators \(E_N^\text {eff}(t) = \frac{1}{a} \log \big [C(t)/C(t+a)\big ]\). The dispersion relation on a lattice \(E^2(p^2)\) shown in Fig. 3 indicates that discretization effects in the spectrum of moving nucleons are under control. A representative set of fits of Eq. 3 to three-point proton nucleon-current correlator data from the D5 ensemble is shown in Fig. 4. The ground-state matrix elements \({\mathcal A}_{0'0}\) from fits (3) are decomposed into form factors \(F^q_{1,2}\) separately for each flavor q. The data points in Fig. 4 show correlator ratios estimating nucleon matrix elements for \(t\rightarrow \infty \), and the bands of the respective color bands show fits to Eqs. (3). The dark-gray bands show ground-state matrix elements \({\mathcal A}_{0'0}\), and the light-gray bands show the overdetermined fits of these matrix elements to the form factor values \(F_{1,2}(Q^2)\).

Fig. 5
figure 5

Comparison of lattice results for Dirac \(F_1\) (left) and Pauli \(F_2\) (right) form factors of the proton (top) and the neutron (bottom) to phenomenological fits of experimental data [16] (dashed curves). Disconnected quark contractions are neglected

Full size image

3 Results

Individual proton and neutron form factors are shown in Fig. 5, similarly compared to phenomenological fits. Although the lattice results have qualitatively similar \(Q^2\) behavior, they overshoot the phenomenological fits by a factor of \((2\ldots 2.5)\). This substantial difference may be due to discretization effects. Without a calculation on a smaller lattice spacing, these effects are difficult to assess. A detailed study of O(a)-improved current operators and calculations at different lattice spacings are required to control this source of systematic effects.

Fig. 6
figure 6

Ratio of proton Pauli and Dirac form factors \(Q^2 F_{2p}(Q^2) / F_{1p}(Q^2)\), compared to phenomenological fits of experimental data [16] (dashed curves). Disconnected quark contractions are neglected

Full size image
Fig. 7
figure 7

Ratio of proton (left) and neutron (right) Sachs form factors \(\mu G_E / G_M\), compared to phenomenological fits of experimental data [16] and quark+diquark Faddeev equation calculations [2]. Disconnected quark contractions are neglected

Full size image

In Fig. 6, the ratio of proton Pauli and Dirac form factors is shown. In perturbative QCD calculations, this ratio is expected to scale as \(F_{2p} / F_{1p} \sim \frac{\log ^2(Q^2/\Lambda ^2)}{Q^2}\) [17]. The lattice data are compared with the phenomenological fits [16] based on proton experimental data available at \(Q^2\lesssim 8.5\,\textrm{GeV}^2\) (shown with black symbols). Although the general trend in the data is compatible with the logarithmic growth, the current precision is insufficient to validate it.

The ratios of Sachs electric and magnetic form factors for the proton and the neutron are shown in Figs. 7, and again compared to the phenomenological fits [16] and experimental data, as well as calculations using quark+diquark Faddeev equations [2]. The agreement between lattice data and experiment (phenomenology) for the ratios in the proton case is reassuring, although better precision is certainly required in light of upcoming new experiments at JLab. In the case of the neutron, the \(G_{En}/G_{Mn}\) ratio is below the experimental values, although it demonstrates qualitative agreement in its \(Q^2\) behavior. Since the neutron is neutral, its electric form factor may be much more sensitive to the systematic effects in this calculation, in particular the omission of disconnected quark contractions and unphysical heavy pion masses. We observe, however, that at high momenta where the results should depend less on the masses of the light quarks, the lattice data agrees with extrapolations from phenomenological fits. Better motivated comparisons will be possible with future neutron form factor data with extended \(Q^2\) range.

Fig. 8
figure 8

Contributions of u and d quarks to Dirac \(F_1\) (left) and Pauli \(F_2\) (right) nucleon form factors, scaled by \(Q^4\). The scales are adjusted for comparison to figures in Ref. [18]. Disconnected quark contractions are neglected. The phenomenological fits to experimental data (dashed curves) are limited to \(Q^2\le 3.4\,\textrm{GeV}^2\) in the neutron case [16]

Full size image

Finally, in Fig. 8 we show contributions to nucleon form factors from u and d quarks separately. For comparison, these contributions are shown rescaled in the fashion similar to Ref. [18]. In experiment, this can be studied by combining proton and neutron data and relying on \(SU(2)_f\) symmetry, which is exact in our lattice QCD calculations. Since both the neutron and the proton data are required, the fit can only be relied upon for \(Q^2\lesssim 3.4\,\textrm{GeV}^2\). Similarly to the nucleon form factors, lattice results for their flavor consituents overshoot experimental fits by a large factor. Still, it is reassuring that their \(Q^2\) behavior and the relative u and d quark contributions are in qualitative agreement.

4 Conclusions

To summarize, results of these initial lattice QCD calculations of nucleon form factors are overestimating the results of experiment by a large factor. However, the ratios of these form factors are in much better agreement with experiment and phenomenology. Calculations with smaller lattice spacings, which are underway, will lead to better understanding of this disagreement, validate lattice QCD methods for high-momentum nucleon states on a lattice, and shed light on nucleon structure in the important region of transition from nonperturbative to perturbative quark-gluon dynamics.