Nucleon Electromagnetic Form Factors at Large Momentum Transfer from Lattice QCD

Nucleon form factors at large momentum transfer are important for understanding the transition from nonperturbative to perturbative QCD and have been the focus of experiment and phenomenology. We calculate proton and neutron electromagnetic form factors GE,M(Q2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{E,M}(Q^2)$$\end{document} from first principles using nonperturbative methods of lattice QCD. We have accumulated large Monte Carlo statistics to study form factors up to momentum transfer Q2≲8GeV2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q^2\lesssim 8\text { GeV}^2$$\end{document} with a range of lattice spacings as well as quark masses that approach the physical point. In this paper, results of initial analyses are presented and compared to experiment, and potential sources of systematic uncertainty are discussed.


Introduction
Behavior of nucleon electromagnetic form factors G E p,n , G Mp,n (Q 2 ) at high momentum transfer Q 2 ≈ 5 . . . 10 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 ≈ 18 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 ≈ 16 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 1 . . . 2 GeV 2 .One notable exception is the calculation of the G E p /G Mp ratio using Feynman-Hellman method [9].Lattice calculations involving hadrons with large momentum | p| 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 signalto-noise ratio for the nucleon is expected to decrease ∝ exp − (E N ( p) − 3 2 m π )τ with Euclidean time τ , 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 ∝ exp − E N ( p)τ as the energy gap E( p) = E N ,exc ( p) − E N ( p) shrinks with increasing relativistic nucleon momentum | 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 | p ( ) | = 1 2 Q 2 .For example, momentum transfer Q 2 1 ≈ 10 GeV 2 requires nucleon momentum p 1 1.6 GeV, which reduces the energy gap E N (0) ≈ 0.5 GeV to E N ( p 1 ) ≈ 0.3 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 ≈ 0.09 fm.Two values of the pion masses m π ≈ 280 and 170 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 ≈ 0.073 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.

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.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, where N = abc [ ũaT Cγ 5 db ] ũc is the nucleon interpolating field on a lattice constructed with "momentumsmeared" quark fields 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 1.2 GeV 2 were found small ( 1%) [15], but remain to be explored at higher momenta; these contributions are omitted in the current work.
The nucleon correlators become dominated with the ground state C(t) = N (t) . . .N (0) ∝ e −E N t as the Euclidean time τ 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 to extract ground-state nucleon energies E ) 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 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 → ∞, and the bands of the respective color bands show fits to Eqs. (3).The dark-gray bands show ground-state matrix elements 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 ).

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 . . .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.
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 Q 2 [17].The lattice data are compared with the phenomenological fits [16] based on proton experimental data available at Q 2 8.5 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  [16] and quark+diquark Faddeev equation calculations [2].Disconnected quark contractions are neglected 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.
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  3.4 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.

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.

Fig. 1
Fig. 1 Lattice ensembles and statistics accumulated for each value of a and m π .The circle areas are proportional to the number of samples.Lighter pion-mass calculations (D5) require significantly more statistics

Fig. 4 Fig. 5
Fig. 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 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 )

1 a
momentum-dependent matrix elements of nucleon operators C 0 ( ) = vac|N |N ( p ( ) ) and vector current density A 0 0 = N ( p )|J |N ( p) .The fits and the groundstate energies from the former are shown in Fig. 2, together with effective-energy estimators E eff N (t) = log C(t)/C(t + a) .The dispersion relation on a lattice E 2 ( p 2

Fig. 8
Fig. 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 ≤ 3.4 GeV 2 in the neutron case[16]

Table 1
Summary of ensembles, kinematics, and statistics