Nucleon electromagnetic form factors using lattice simulations at the physical point
Constantia Alexandrou, Martha Constantinou, Kyriakos Hadjiyiannakou, Karl Jansen, Christos Kallidonis, Giannis Koutsou, Alejandro Vaquero Aviles-Casco
DDESY 17-085
Nucleon electromagnetic form factors using lattice simulations at the physical point
C. Alexandrou , , M. Constantinou , K. Hadjiyiannakou , K. Jansen ,Ch. Kallidonis , G. Koutsou , and A. Vaquero Aviles-Casco Computation-based Science and Technology Research Center,The Cyprus Institute, 20 Kavafi Str., Nicosia 2121, Cyprus Department of Physics, University of Cyprus,P.O. Box 20537, 1678 Nicosia, Cyprus Department of Physics,Temple University, 1925 N. 12th Street,Philadelphia, PA 19122-1801, USA NIC, DESY, Platanenallee 6,D-15738 Zeuthen, Germany Department of Physics and Astronomy,University of Utah, Salt Lake City, UT 84112, USA
We present results for the nucleon electromagnetic form factors using an ensemble of maximallytwisted mass clover-improved fermions with pion mass of about 130 MeV. We use multiple sink-source separations and three analysis methods to probe ground-state dominance. We evaluate boththe connected and disconnected contributions to the nucleon matrix elements. We find that thedisconnected quark loop contributions to the isoscalar matrix elements are small giving an upperbound of up to 2% of the connected and smaller than its statistical error. We present results for theisovector and isoscalar electric and magnetic Sachs form factors and the corresponding proton andneutron form factors. By fitting the momentum dependence of the form factors to a dipole formor to the z-expansion we extract the nucleon electric and magnetic radii, as well as, the magneticmoment. We compare our results to experiment as well as to other recent lattice QCD calculations.
PACS numbers: 11.15.Ha, 12.38.Gc, 24.85.+p, 12.38.Aw, 12.38.-tKeywords: Nucleon structure, Nucleon electromagnetic form factors, Lattice QCD
I. INTRODUCTION
Electromagnetic form factors probe the internal struc-ture of hadrons mapping their charge and magnetic dis-tributions. The slope of the electric and magnetic formfactors at zero momentum yields the electric and mag-netic root mean square radius, while the value of theform factors at zero momentum give its electric chargeand magnetic moment. Extensive electron scattering ex-periments have been carried out since the fifties for theprecise determination of the nucleon form factors, includ-ing recent experiments at Jefferson Lab, MIT-Bates andMainz. For a recent review on electron elastic scatteringexperiments, see Ref. [1]. The proton radius can also beobtained spectroscopically, namely via the Lamb shiftsof the hydrogen atom and of muonic hydrogen [2] andvia transition frequencies of electronic and muonic deu-terium. In these measurements, including a recent exper-iment using muonic deuterium [3], discrepancies are ob-served in the resulting proton radius between hydrogenand deuterium and their corresponding muonic equiva-lents. Whether new physics is responsible for this dis-crepancy, or errors in the theoretical or experimentalanalyses, a first principles calculation of the electromag-netic form factors of the nucleon can provide valuableinsight. Although nucleon electromagnetic form factorshave been extensively studied in lattice QCD, most ofthese studies have been carried out at higher than physi- cal pion masses, requiring extrapolations to the physicalpoint, which for the case of baryons carry a large system-atic uncertainty.In this paper, we calculate the electromagnetic formfactors of the nucleon using an ensemble of two degen-erate light quarks (N f = 2) tuned to reproduce a pionmass of about 130 MeV, in a volume with m π L (cid:39) O (10 ) measurements to re-duce the statistical errors, and multiple sink-source sepa-rations to study excited state effects using three differentanalyses. We extract the momentum dependence of theelectric and magnetic Sachs form factors for both isovec-tor and isoscalar combinations, i.e. for both the differ-ence ( p − n ) and sum ( p + n ) of proton and neutron formfactors. For the latter we compute the computationallydemanding disconnected contributions and find them tobe smaller than the statistical errors of the connectedcontributions. To fit the momentum dependence we useboth a dipole form as well as the z-expansion [7]. Fromthese fits we extract the electric and magnetic radii, aswell as, the magnetic moments of the proton, the neu-tron and the isovector and isoscalar combinations. Forthe electric root mean squared (rms) radius of the pro-ton we find (cid:112) (cid:104) r E (cid:105) p = 0 . a r X i v : . [ h e p - l a t ] S e p ple ensembles is required to assess accurately all latticeartifacts.The remainder of this paper is organized as follows:in Section II we provide details of the lattice set-up forthis calculation and in Section III we present our results.In Section IV we compare our results with other latticecalculations and in Section V we summarize our findingsand conclude. II. SETUP AND LATTICE PARAMETERSA. Electromagnetic form factors
The electromagnetic form factors are extracted fromthe electromagnetic nucleon matrix element given by (cid:104) N ( p (cid:48) , s (cid:48) ) |O Vµ | N ( p, s ) (cid:105) = (cid:115) m N E N ( (cid:126)p (cid:48) ) E N ( (cid:126)p ) ¯ u N ( p (cid:48) , s (cid:48) )Λ Vµ ( q ) u N ( p, s ) (1)with N ( p, s ) the nucleon state of momentum p and spin s , E N ( (cid:126)p ) = p its energy and m N its mass, (cid:126)q = (cid:126)p (cid:48) − (cid:126)p , thespatial momentum transfer from initial ( (cid:126)p ) to final ( (cid:126)p (cid:48) )momentum, u N the nucleon spinor and O V the vectorcurrent. In the isospin limit, where an exchange betweenup and down quarks ( u ↔ d ) and between proton andneutron ( p ↔ n ) is a symmetry, the isovector matrixelement can be related to the difference between protonand neutron form factors as follows: (cid:104) p |
23 ¯ uγ µ u −
13 ¯ dγ µ d | p (cid:105) − (cid:104) n |
23 ¯ uγ µ u −
13 ¯ dγ µ d | n (cid:105) u ↔ d −−−→ p ↔ n (cid:104) p | ¯ uγ µ u − ¯ dγ µ d | p (cid:105) . (2) Similarly, for the isoscalar combination we have (cid:104) p |
23 ¯ uγ µ u −
13 ¯ dγ µ d | p (cid:105) + (cid:104) n |
23 ¯ uγ µ u −
13 ¯ dγ µ d | n (cid:105) u ↔ d −−−→ p ↔ n (cid:104) p | ¯ uγ µ u + ¯ dγ µ d | p (cid:105) . (3)We will use these relations to compare our lattice results,obtained for the isovector and isoscalar combinations,with the experimental data for the proton and neutronmatrix elements.We use the symmetrized lattice conserved vector cur-rent, O Vµ = [ j µ ( x ) + j µ ( x − ˆ µ )], with j µ ( x ) = 12 [ ¯ ψ ( x + ˆ µ ) U † µ ( x )(1 + γ µ ) τ a ψ ( x ) − ¯ ψ ( x ) U µ ( x )(1 − γ µ ) τ a ψ ( x + ˆ µ )] , (4)where ¯ ψ = (¯ u, ¯ d ) and τ a acts in flavor space. We consider τ a = τ , the third Pauli matrix, for the isovector case,and τ a = / for the isoscalar case. ˆ µ is the unit vectorin direction µ and U µ ( x ) is the gauge link connectingsite x with x + ˆ µ . Using the conserved lattice currentmeans that no renormalization of the vector operator isrequired.The matrix element of the vector current can be de-composed in terms of the Dirac F and Pauli F formfactors as Λ Vµ ( q ) = γ µ F ( q ) + iσ µν q ν m N F ( q ) . (5) F and F can also be expressed in terms of the nucleonelectric G E and magnetic G M Sachs form factors via therelations G E ( q ) = F ( q ) + q (2 m N ) F ( q ) , and G M ( q ) = F ( q ) + F ( q ) . (6) B. Lattice extraction of form factors
On the lattice, after Wick rotation to Euclidean time,extraction of matrix elements requires the calculation ofa three-point correlation function shown schematically inFig. 1. For simplicity we will take x = ( (cid:126) ,
0) from hereon. We use sequential inversions through the sink, fixingthe sink momentum (cid:126)p (cid:48) to zero, which constrains (cid:126)p = − (cid:126)q : G µ (Γ; (cid:126)q ; t s , t ins ) = (cid:88) (cid:126)x s (cid:126)x ins e − i(cid:126)q.(cid:126)x ins Γ αβ (cid:104) ¯ χ βN ( (cid:126)x s ; t s ) |O µ ( (cid:126)x ins ; t ins ) | χ αN ( (cid:126)
0; 0) (cid:105) t s − t ins →∞ −−−−−−−→ t ins →∞ (cid:88) ss (cid:48) Γ αβ (cid:104) χ βN | N (0 , s (cid:48) ) (cid:105)(cid:104) N ( p, s ) | ¯ χ αN (cid:105)(cid:104) N (0 , s (cid:48) ) |O µ ( q ) | N ( p, s ) (cid:105) e − E N ( (cid:126)p ) t ins e − m N ( t s − t ins ) , (7)where Γ is a matrix acting on Dirac indices α and β and χ N is the standard nucleon interpolating operator given by χ αN ( (cid:126)x, t ) = (cid:15) abc u aα ( x )[ u b (cid:124) ( x ) Cγ d c ( x )] . (8) FIG. 1. Three-point nucleon correlation function with sourceat x , sink at x s and current insertion O µ at x ins . The con-nected contribution is shown in the upper panel and the dis-connected contribution in the lower panel. with C = γ γ the charge conjugation matrix. In thesecond line of Eq. (7) we have inserted twice a completeset of states with the quantum numbers of the nucleon,of which, after assuming large time separations, only thenucleon survives with higher energy states being expo-nentially suppressed. We use Gaussian smeared point-sources [8, 9] to increase the overlap with the nucleonstate with APE smearing applied to the gauge links, withthe same parameters as in Ref. [10], tuned so as to yielda rms radius of about 0.5 fm. These are the same param-eters as in Ref. [11], namely ( N G , α G ) = (50 ,
4) for theGaussian smearing and ( N APE , α
APE ) = (50 , .
5) for theAPE smearing.We construct an optimized ratio dividing G µ by a com-bination of two-point functions. The optimized ratio R µ is given by R µ (Γ; (cid:126)q ; t s ; t ins ) = G µ (Γ; (cid:126)q ; t s ; t ins ) G ( (cid:126) t s ) × (cid:34) G ( (cid:126) t s ) G ( (cid:126)q ; t s − t ins ) G ( (cid:126) t ins ) G ( (cid:126)q ; t s ) G ( (cid:126) t s − t ins ) G ( (cid:126)q ; t ins ) (cid:35) (9)with the two-point function given by G ( (cid:126)p ; t ) = (cid:88) (cid:126)x e − i(cid:126)p(cid:126)x Γ αβ (cid:104) ¯ χ βN ( (cid:126)x ; t ) | χ αN ( (cid:126)
0; 0) (cid:105) . (10)Γ is the unpolarized projector, Γ = γ . After tak-ing the large time limit, unknown overlaps and energy exponentials cancel in the ratio, leading to the time-independent quantity Π µ (Γ; (cid:126)q ), defined via: R µ (Γ; (cid:126)q ; t s ; t ins ) t s − t ins →∞ −−−−−−−→ t ins →∞ Π µ (Γ; (cid:126)q ) . (11)Having Π µ (Γ; (cid:126)q ), different combinations of current in-sertion directions ( µ ) and nucleon polarizations deter-mined by Γ yield different expressions for the form fac-tors [12, 13]. Namely, we haveΠ (Γ ; (cid:126)q ) = C E N + m N m N G E ( Q ) , Π i (Γ ; (cid:126)q ) = C q i m N G E ( Q ) , Π i (Γ k ; (cid:126)q ) = C (cid:15) ijk q j m N G M ( Q ) , (12)where Q = − q , is the Euclidean momentum transfersquared, C = (cid:113) m N E N ( E N + m N ) , and the polarized projectoris given by Γ k = iγ γ k Γ , and i, k = 1 , , µ from lattice data:i) Plateau method.
We seek to identify a range of valuesof t ins where the ratio R µ is time-independent (plateauregion). We fit, within this window, R µ to a constantand use multiple t s values. Excited states are consideredsuppressed when our result does not change with t s .ii) Two-state fit method.
We fit the time dependence ofthe three- and two-point functions keeping contributionsup to the first excited state. Namely, we truncate thetwo-point function of Eq. (10) keeping only the groundand first excited states to obtain G ( (cid:126)p ; t ) = c ( (cid:126)p ) e − E ( (cid:126)p ) t [1+ c ( (cid:126)p ) e − ∆ E ( (cid:126)p ) t + O ( e − ∆ E ( (cid:126)p ) t )] . (13)Similarly, the three-point function of Eq. (7) becomes G µ (Γ; (cid:126)q ; t s , t ins ) = a µ (Γ; (cid:126)q ) e − m ( t s − t ins ) e − E ( (cid:126)q ) t ins × (cid:20) a µ (Γ; (cid:126)q ) e − ∆ E ( (cid:126)q ) t ins + a µ (Γ; (cid:126)q ) e − ∆ m ( t s − t ins ) + a µ (Γ; (cid:126)q ) e − ∆ m ( t s − t ins ) e − ∆ E ( (cid:126)q ) t ins + O [min( e − ∆ m ( t s − t ins ) , e − ∆ E ( (cid:126)q ) t ins )] (cid:21) , (14)where ∆ E k ( (cid:126)p ) = E k ( (cid:126)p ) − E ( (cid:126)p ) is the energy difference be-tween the k th nucleon excited state and the ground stateat momentum (cid:126)p and m = E ( (cid:126)
0) and ∆ m k = ∆ E k ( (cid:126) µ (Γ; (cid:126)q ) = a µ (Γ; (cid:126)q ) (cid:113) c ( (cid:126) c ( (cid:126)q ) . (15)In practice, we fit simultaneously the three-point functionand the finite and zero momentum two-point functions TABLE I. Simulation parameters of the ensemble used inthis calculation, first presented in Ref. [4]. The nucleon andpion mass and the lattice spacing have been determined inRef. [14]. β =2.1, c SW =1.57751, a =0.0938(3) fm, r /a =5.32(5)48 × L =4.5 fm αµ =0.0009 m π =0.1304(4) GeV m π L =2.98(1) m N =0.932(4) GeV in a twelve parameter fit to determine m , E ( (cid:126)q ), ∆ m ,∆ E ( (cid:126)q ), c ( (cid:126)q ), c ( (cid:126) c ( (cid:126)q ), c ( (cid:126) a µ (Γ; (cid:126)q ), a µ (Γ; (cid:126)q ), a µ (Γ; (cid:126)q ) and a µ (Γ; (cid:126)q ). The two-point function is evalu-ated using the maximum statistics available at time sep-aration t s /a = 18.iii) Summation method . We sum the ratio of Eq. (9) overthe insertion time-slices. From the expansion up to firstexcited state of Eq. (14) one sees that a geometric sumarises, which yields: t s − a (cid:88) t ins = a R µ (Γ; (cid:126)q ; t s ; t ins ) t s →∞ −−−−→ c +Π µ (Γ; (cid:126)q ) t s + O ( t s e − ∆ m t s ) . (16)The summed ratio is then fitted to a linear form andthe slope is taken as the desired matrix element. Wenote that, in quoting final results, we do not use the val-ues extracted from summation method However, it doesprovide an additional consistency check for the plateauvalues. C. Lattice setup
The simulation parameters of the ensemble we use aretabulated in Table I. We use an N f = 2 ensemble oftwisted mass fermion configurations with clover improve-ment with quarks tuned to maximal twist, yielding a pionmass of about 130 MeV. The lattice volume is 48 × a =0.0938(3) fmyielding a physical box length of about 4.5 fm. The valueof the lattice spacing is determined using the nucleonmass, as explained in Ref. [14]. Details of the simulationand first results using this ensemble were presented inRefs. [4, 10].The parameters used for the calculation of the corre-lation functions are given in Table II. We use increasingstatistics with increasing sink-source separation so thatstatistical errors are kept approximately constant. Fur-thermore, as will be discussed in Section III, G E ( Q )is found to be more susceptible to excited states com-pared to G M ( Q ), requiring larger separations for ensur-ing their suppression. Therefore, we carry out sequentialinversions for five sink-source separations using the un-polarized projector Γ , which yields G E ( Q ) accordingEq. 12. To obtain G M ( Q ), we carry out three additional TABLE II. Parameters of the calculation of the form factors.The first column shows the sink-source separations used, thesecond column the sink projectors and the last column thetotal statistics ( N st ) obtained using N cnf configurations times N src source-positions per configuration. t s [ a ] Proj. N cnf · N src = N st , Γ k ·
16 = 924816 Γ ·
88 = 4664018 Γ ·
88 = 63800 sequential inversions, one for each polarized projector Γ k , k = 1 , ,
3, for each of the three smallest separations.
III. RESULTSA. Analysis
1. Isovector contributions
We use the three methods, described in the previoussection, to analyze the contribution due to the excitedstates and extract the desired nucleon matrix element.We demonstrate the quality of our data and two-statefits in Figs. 2 and 3 for the isovector contributions to G E ( Q ) and G M ( Q ) respectively, for three momentumtransfers, namely the first, second and fourth non-zero Q values of our setup. In these figures we show theratio after the appropriate combinations of Eq. (12) aretaken to yield either G u − dE ( Q ) or G u − dM ( Q ). We indeedobserve larger excited state contamination in the caseof G u − dE ( Q ), which is the reason for considering largervalues of t s for this case. We note that for fitting theplateau and summation methods, the ratios of Eq. (9) areconstructed with two- and three-point functions with thesame source positions and gauge configurations. For thetwo-state fit, as already mentioned, we use the two-pointcorrelation function at the maximum statistics available,namely 725 configurations times 88 source positions, asindicated in Table II. These are the ratios shown in Figs. 2and 3, which differ from those used for the plateau fits.The investigation of excited states is facilitated furtherby Figs. 4 and 5. These plots indicate that excited statecontributions are present in G u − dE ( Q ) for the first threesink-source separations of t s /a = 10, 12 and 14 in par-ticular for larger momentum transfer. For the two largersink-source separations we see convergence of the resultsextracted from the plateau method, which are in agree-ment with those from the summation method and thetwo-state fits when the lower fit range is t low s = 12 a =1 . G u − dM ( Q ), all results from the three sink-source separations are in agreement and consistent withthe summation and two-state fit methods within their er-rors. The values obtained at t s = 18 a = 1 . G u − dE ( Q ) and t s = 14 a = 1 .
10 5 0 5 10(t ins t s /2)/a0.40.50.60.70.80.9 X ~ q ∈ Q R u d ( ; ~ q ; t s , t i n s ) → G u d E ( Q ) t s = 12a = 1.1 fmt s = 14a = 1.3 fm t s = 16a = 1.5 fm t s = 18a = 1.7 fm FIG. 2. Ratio yielding the isovector electric Sachs form factor.We show results for three representative Q values, namelythe first, second and fourth non-zero Q values from top tobottom, for t s = 12 a (open circles), t s = 14 a (filled squares), t s = 16 a (filled circles) and t s = 18 a (filled triangles). Thecurves are the results from the two-state fits, with the fainterpoints excluded from the fit. The band is the form factorvalue extracted using the two-state fit.
10 5 0 5 10(t ins t s /2)/a2.02.22.42.62.83.03.23.4 X ~ q ∈ Q i j k q k R u d i ( k ; ~ q ; t s , t i n s ) → G u d M ( Q ) t s = 10a = 0.9 fm t s = 12a = 1.1 fm t s = 14a = 1.3 fm FIG. 3. Ratio yielding the isovector magnetic Sachs formfactor. We show results for three representative Q values,namely the first, second and fourth non-zero Q values fromtop to bottom, for t s = 10 a (open squares), t s = 12 a (opencircles), and t s = 14 a (filled squares). The curves are theresults from the two-state fits, with the fainter points excludedfrom the fit. The band is the form factor value extracted usingthe two-state fit. s [fm]0.20.30.40.50.60.70.80.91.0 G u d E ( Q ) Q = 0.074 GeV Q = 0.214 GeV Q = 0.345 GeV Q = 0.527 GeV PlateauSummation Two-state lows [fm]
FIG. 4. Isovector electric form factor, for four non-zero Q values, extracted from the plateau method (squares), the sum-mation method (circles) and the two-state fit method (trian-gles). The plateau method results are plotted as a function ofthe sink-source separation while the summation and two-statefit results are plotted as a function of t low s , i.e. of the smallestsink-source separation included in the fit, with t high s kept fixedat t s = 18 a = 1 . G u − dM ( Q ) are shown in Figs. 4 and 5 with the open sym-bols and associated error band that demonstrates con-sistency with the values extracted using the summationand two-state fit methods.Our results for the isovector electric Sachs form factorextracted using all available t s values and from the sum-mation and two-state fit methods are shown in Fig. 6.On the same plot we show the curve obtained from aparameterization of experimental data for G pE ( Q ) and G nE ( Q ) according to Ref. [15], using the parameters ob-tained in Ref. [16], and taking the isovector combination G pE ( Q ) − G nE ( Q ). We see that as the sink-source sepa-ration is increased, our results tend towards the experi-mental curve. The results from the two-state fit methodusing t low s =1 . t s = 1 . Q values. Resultsextracted using the summation method are consistentwithin their large errors to those obtained from fittingthe plateau for t s = 1 . G u − dE ( Q ), corroborating theconclusion drawn by observing Fig. 5. We also see agree-ment with the experimental curve for Q values largerthan ∼ . However, our lattice results underesti-mate the experimental ones at the two lowest Q values.Excited state effects are seen to be small for this quan-tity, and thus they are unlikely to be the cause of this s [fm]1.01.52.02.53.03.54.0 G u d M ( Q ) Q = 0.074 GeV Q = 0.214 GeV Q = 0.344 GeV Q = 0.526 GeV PlateauSummation Two-state lows [fm]
FIG. 5. Isovector magnetic form factor. The notation is thesame as that in Fig. 4. For the summation and two-state fitmethods, the largest sink-source separation included in the fitis kept fixed at t high s = 14 a = 1 . [GeV ]0.20.40.60.81.0 G u d E ( Q ) t s = 0.9 fmt s = 1.1 fmt s = 1.3 fmt s = 1.5 fmt s = 1.7 fmSummation,t s ∈ [0.9, 1.7] fmTwo-state,t s ∈ [1.1, 1.7] fmKelly parameterization FIG. 6. Isovector electric Sachs form factor as a functionof the momentum transfer squared ( Q ). Symbols for theplateau method follow the notation of Figs. 2 and 3. Resultsfrom the summation method are shown with open diamondsand for the two-state fit method with the crosses. The solidline shows G pE ( Q ) − G nE ( Q ) using Kelly’s parameterizationof the experimental data [15] with parameters taken from Al-berico et al. [16]. discrepancy given the consistency of our results at threeseparations, as well as with those extracted using thesummation and the two-state fit method. This small dis-crepancy could be due to suppressed pion cloud effects,due to the finite volume, that could be more significantat low momentum transfer. For example, a study of the magnetic dipole form factor G M in the N → ∆ transi-tion using the Sato-Lee model predicts larger pion cloudcontributions at low momentum transfer [17]. LatticeQCD computations also observe a discrepancy at lower Q for G M when compared to experiment [18]. Analy-sis on a larger volume is ongoing to investigate volumeeffects not only in G M ( Q ) but also for other nucleonmatrix elements and the results will be reported in sub-sequent publications. Our results for the form factorsat all sink-source separations and using the summationand two-state fit methods are included in Appendix A inTables VIII to XI. Preliminary results for the isovectorelectromagnetic form factors have been presented for thisensemble in Refs. [19, 20]. [GeV ]12345 G u d M ( Q ) t s = 0.9 fmt s = 1.1 fmt s = 1.3 fmSummation,t s ∈ [0.9, 1.3] fmTwo-state,t s ∈ [0.9, 1.3] fmKelly parameterization FIG. 7. Isovector magnetic Sachs form factor as a function ofthe momentum transfer squared. The notation is the same asthat of Fig. 6.
2. Isoscalar contributions
We perform a similar analysis for the isoscalar contri-butions, denoted by G u + dE ( Q ) and G u + dM ( Q ). As men-tioned, we use the combination ( u + d ) / in the matrixelement for the isoscalar such that it yields G u + dE,M ( Q ) = G pE,M ( Q ) + G nE,M ( Q ). Having also the isovector com-bination G u − dE,M ( Q ) = G pE,M ( Q ) − G nE,M ( Q ) the indi-vidual proton and neutron form factors can be extracted.While isovector matrix elements receive no disconnectedcontributions since they cancel in the isospin limit, theisoscalar form factors do include disconnected fermionloops, shown schematically in Fig. 1. These disconnectedcontributions are included for the first time here at thephysical point to obtain the isoscalar form factors.The connected isoscalar three-point function is com-puted using the same procedure as in the isovectorcase. We show results for the connected contribution s [fm]0.20.30.40.50.60.70.80.91.0 G u + d , c o nn . E ( Q ) Q = 0.074 GeV Q = 0.214 GeV Q = 0.345 GeV Q = 0.527 GeV PlateauSummation Two-state lows [fm]
FIG. 8. Connected contribution to the G u + dE ( Q ) form fac-tor, for four non-zero Q values, extracted from the plateaumethod (squares), the summation method (filled circles) andthe two-state fit method (filled triangles). The notation is thesame as in Fig. 4. to G u + dE ( Q ) and G u + dM ( Q ) in Figs. 8 and 9 respectively.These results are for the same momentum transfer val-ues as used in Figs. 4 and 5. In the case of the isoscalarelectric form factor, we observe contributions due to ex-cited states that are similar to those observed for theisovector case. Namely, we find that a separation ofabout t s =1.7 fm is required for their suppression. Forthe isoscalar magnetic form factor, we observe that thevalues extracted from fitting the plateau at time sepa-rations t s = 1 . t s = 1 . G u + dE ( Q ) and G u + dM ( Q ) in Fig. 10 for the firstnon-zero momentum transfer. The results are obtainedusing the same ensemble used for the connected contribu-tions, detailed in Table I, using 2120 configurations, withtwo-point functions computed on 100 randomly chosensource positions per configuration. 2250 stochastic noisevectors are used for estimating the fermion loop. Av-eraging the proton and neutron two-point functions andthe forward and backwards propagating nucleons yieldsa total of 8 · statistics. More details of this calculationare presented in Ref. [23], where results for the axial formfactors are shown.In the case of the electric form factor, we obtain G u + d, disc. E ( Q = 0 .
074 GeV ) = − . s [fm]0.30.40.50.60.70.80.9 G u + d , c o nn . M ( Q ) Q = 0.074 GeV Q = 0.214 GeV Q = 0.344 GeV Q = 0.526 GeV PlateauSummation Two-state lows [fm]
FIG. 9. Connected contribution to the G u + dM ( Q ) form factor.The notation is the same as in Fig. 5. tistical error. For the magnetic form factor, fitting tothe plateau we obtain G u + d, disc. M ( Q = 0 .
074 GeV ) = − . Q and half the value of the statisti-cal error. These values are consistent with a dedicatedstudy of the disconnected contributions using an ensem-ble of clover fermions with pion mass of 317 MeV [24]and a recent result at the physical point presented inRef. [25]. There it was shown that G u + d, disc. M ( Q ) isnegative and largest in magnitude at Q = 0 while G u + d, disc. E ( Q ) is largest at around Q = 0 . . Inour case, at our largest momentum transfer, we find G u + d, disc. E ( Q = 0 .
280 GeV ) = − . t s pointing to lesssevere excited state effects. ins t s /2)/a0.030.020.010.000.010.020.03 X ~ q ∈ Q R ( ; ~ q ; t s , t i n s ) → G u + d , d i s c E ( Q ) t s = 8a = 0. 8 fmt s = 10a = 0. 9 fm ins t s /2)/a0.100.080.060.040.020.00 X ~ q ∈ Q i j k q k R i ( k ; ~ q ; t s , t i n s ) → G u + d , d i s c M ( Q ) t s = 8a = 0. 8 fmt s = 10a = 0. 9 fm FIG. 10. Disconnected contribution to the electric (upperpanel) and magnetic (lower panel) isoscalar Sachs form fac-tors for sink-source separation t s = 8 a = 0 .
75 fm (invertedtriangles) and t s = 10 a = 0 .
94 fm (squares) for the first non-zero momentum transfer of Q = 0 .
074 GeV . The horizontalbands show the values obtained after fitting with the plateaumethod to the results at t s = 10 a = 0 .
94 fm. B. Q -dependence of the form factors
1. Isovector and isoscalar form factors
We fit G E ( Q ) and G M ( Q ) to both a dipole Ansatzand the z-expansion form. The truncated z-expansion isexpected to model better the low- Q [7] dependence ofthe form factors, while the dipole form is motivated byvector-meson pole contributions to the form factors [27].For the case of the dipole fits, we use G i ( Q ) = G i (0)(1 + Q M i ) , (17)with i = E, M , allowing both G M (0) and M M to varyfor the case of magnetic form factor, while constraining G E (0) = 1 for the case of the electric form factor. For [GeV ]0.20.40.60.81.0 G u + d , c o nn . E ( Q ) t s = 0.9 fmt s = 1.1 fmt s = 1.3 fmt s = 1.5 fmt s = 1.7 fmSummation,t s ∈ [0.9, 1.7] fmTwo-state,t s ∈ [1.1, 1.7] fmKelly parameterization FIG. 11. Connected contribution to the isoscalar electricSachs form factor as a function of the momentum trans-fer, using the notation of Fig. 11. The solid line shows G pE ( Q ) + G nE ( Q ) using the Kelly parameterization of ex-perimental data from Ref. [15] with parameters taken fromAlberico et al. [16]. [GeV ]0.00.20.40.60.81.0 G u + d , c o nn . M ( Q ) t s = 0.9 fmt s = 1.1 fmt s = 1.3 fmSummation,t s ∈ [0.9, 1.3] fmTwo-state,t s ∈ [0.9, 1.3] fmKelly parameterization FIG. 12. Connected contribution to the isoscalar magneticSachs form factor as a function of the momentum transfer.The notation is the same as in Fig. 11. the z-expansion, we use the form [7] G i ( Q ) = k max (cid:88) k =0 a ik z k , where z = (cid:112) t cut + Q − √ t cut (cid:112) t cut + Q + √ t cut (18)and take t cut = 4 m π . For both isovector and isoscalar G E ( Q ) we fix a E = 1 while for G M ( Q ) we allow all max a E a M a M FIG. 13. Results from fitting using the z-expansion as a func-tion of k max for a E (lower panel), a M (center panel) and a M (top panel) of Eq. 18. [GeV ]0.00.20.40.60.81.0 G u d E ( Q ) Dipole
Lattice results(plateau method)Experiment [GeV ] z-expansion FIG. 14. Isovector electric Sachs form factor as a function ofthe momentum transfer extracted from the plateau method at t s = 18 a = 1 . G pE ( Q ) from Ref. [29]and for G nE ( Q ) from Refs. [30–44]. parameters to vary. We use Gaussian priors for a ik for k ≥ w = 5 max( | a i | , | a i | ) as proposed inRef. [28]. We observe larger errors when fitting with thez-expansion compared to the dipole form. In Fig. 13we show a M and a M from fits to the magnetic isovectorform factor and a E from fits to the electric as a functionof k max and observe no significant change in the fittedparameters beyond k max ≥
2. We also note that theresulting values for a ik for k ≥ | a ik | (cid:28) | a i | , | a i | ). Wetherefore quote results using k max = 2 from here on.Fits to the Q dependence of G u − dE ( Q ) are shown in [GeV ]1234 G u d M ( Q ) Dipole
Lattice results(plateau method)Experiment [GeV ] z-expansion FIG. 15. Isovector magnetic Sachs form factor as a functionof the momentum transfer extracted from the plateau methodat t s = 14 a = 1 . Q values, while the largerband is obtained after omitting the two smallest values. Theblack points are obtained using experimental data for G pM ( Q )from Ref. [29] and for G nM ( Q ) from Refs. [45–50]. [GeV ]0.200.150.100.050.00 G u + d , d i s c . M ( Q ) t s = 8a = 0. 8 fmt s = 10a = 0. 9 fm FIG. 16. Disconnected contribution to the isoscalar magneticSachs form factor as a function of the momentum transfer for t s = 8 a = 0 . t s = 10 a = 0 . k max = 1. Fig. 14 using the values extracted from the plateau at t s = 18 a = 1 . Q values. Both dipole and z-expansion formdescribe the lattice QCD results well. In this plot we alsoshow results from experiment, using data for G pE ( Q ) ob-tained from Ref. [29] and data for G nE ( Q ) from Refs. [30–0 [GeV ]0.00.20.40.60.81.0 G u + d E ( Q ) Dipole
Lattice results(plateau method)Experiment [GeV ] z-expansion FIG. 17. Isoscalar electric Sachs form factor with fits to thedipole form (left) and to the z-expansion (right). We showwith triangles the sum of connected and disconnected contri-butions, with the plateau result for t s = 18 a = 1 . t s = 10 a = 0 . G pE ( Q ) to the Q valuesfor which G nE ( Q ) is available.For both dipole and z-expansion fit, the resulting curvelies about one standard deviation above the experimentaldata. This small discrepancy may be due to small resid-ual excited state effects, which would require significantincrease of statistics at larger sink-source separations toidentify. Having only performed the calculation using oneensemble we cannot check directly for finite volume andcut-off effects. However, in a previous study employing N f = 2 twisted mass fermions at heavier than physicalpion masses and three values of the lattice spacing, wefound no detectable cut-off effects in these quantities for alattice spacing similar to the one used here [13]. We havealso performed a volume assessment using the aforemen-tioned heavier mass twisted mass ensembles with m π L values ranging from 3.27 to 5.28. Namely, we found novolume dependence within our statistical accuracy be-tween two ensembles with m π L = 3 .
27 and m π L =4.28respectively and similar pion mass of m π (cid:39)
300 MeV. Weplan to carry out a high accuracy analysis of the vol-ume dependence at the physical point on a lattice size of64 ×
128 keeping the other parameters fixed in a forth-coming publication.The same analysis carried out for G u − dE ( Q ) is alsoperformed for G u − dM ( Q ) in Fig. 15, where we use the re-sult from fitting to the plateau at the largest sink-sourceseparation available, namely t s = 14 a = 1 . G u − dE ( Q ), both the dipole Ansatz and z- [GeV ]0.00.20.40.60.81.01.2 G u + d M ( Q ) Dipole
Lattice results(plateau method)Experiment [GeV ] z-expansion FIG. 18. Isoscalar magnetic Sachs form factor with fits to thedipole form (left) and to the z-expansion (right). We showwith triangles the sum of connected and disconnected contri-butions, with the plateau result for t s = 14 a = 1 . t s = 10 a = 0 . expansion describe well the lattice QCD data. The plotsshow two bands, one when including all Q values, re-sulting in the smaller error band, and one in which thefirst two Q values are omitted, resulting in the largerband. The experimental data shown are obtained using G pM ( Q ) from the same experiment as for G pE ( Q ) shownin Fig. 14, namely Ref. [29], and G nM ( Q ) from Refs. [45–50].In both dipole and z-expansion fits of G u − dM ( Q ) wefind that the Q dependence is consistent with experi-ment after Q (cid:39) . . We suspect that the devia-tion at the two smallest Q values is due to finite volumeeffects. As already mentioned, we plan to further inves-tigate this using an ensemble of N f = 2 twisted massfermions on a larger volume of 64 × Q values results in alarger error for G u − dM (0), in particular in the case of thez-expansion.We show the momentum dependence of the discon-nected contribution to G u + dM ( Q ) in Fig. 16. The largeerrors do not permit as thorough analysis as for the con-nected contribution. Since the disconnected isoscalarcontributions do not follow a dipole form, and in theabsence of any theoretically motivated form for the dis-connected contributions, we use a z-expansion fit with k max = 2, fixing a = 0 for G u + d, disc. E ( Q ) and with k max = 1, allowing both a and a to vary. For thecase of G u + d, disc. E ( Q ) we find results consistent with zero.For the magnetic case, the disconnected contribution de-creases the form factor by at most 3% at Q = 0.We add connected and disconnected contributions to1 TABLE III. Results for the isovector electric charge radiusof the nucleon ( (cid:104) r E (cid:105) u − d ) from fits to G u − dE ( Q ). In the firstcolumn we show t s for the plateau method and the t s fit rangefor the summation and two-state fit methods. t s [fm] dipole z-expansion (cid:104) r E (cid:105) u − d [fm ] χ d . o . f (cid:104) r E (cid:105) u − d [fm ] χ d . o . f Plateau0.94 0.523(08) 2.0 0.562(19) 1.21.13 0.562(14) 1.9 0.677(37) 0.71.31 0.580(26) 1.2 0.718(75) 0.71.50 0.666(33) 0.9 0.61(10) 0.31.69 0.653(48) 0.6 0.52(14) 0.2Summation0.9-1.7 0.744(55) 0.3 0.79(14) 0.2Two-state1.1-1.7 0.623(33) 1.0 0.56(10) 0.8 obtain the isoscalar contributions shown in Figs. 17and 18. There are small discrepancies between our lat-tice data and experiment at larger Q values. Whetherthese are due to volume effects or other lattice artifactswill be investigated in a follow-up study.The slope of the form factors at Q =0 is related to theisovector electric and magnetic radius as follows ∂∂Q G i ( Q ) | Q =0 = − G i (0) (cid:104) r i (cid:105) , (19)with i = E, M for the electric and magnetic form factorsrespectively. For the z-expansion, this is given by (cid:104) r i (cid:105) = − t cut a i a i (20)and for the dipole fit (cid:104) r i (cid:105) = 12 M i . (21)Furthermore, the nucleon magnetic moment is defined as µ = G M (0) and is obtained directly from the fitted pa-rameter in both cases. As for the form factors, we willdenote the isovector radii and magnetic moment with the u − d superscript and for the isoscalar with u + d . We tab-ulate our results for the isovector radii and magnetic mo-ment from both dipole and z-expansion fits in Tables IIIand IV, and from fits to the isoscalar form factors inTables V and VI. For the isoscalar results shown in Ta-bles V and VI, we show two results for each case, namelythe result of fitting only the connected contribution inthe first column of each case and the total contribution,by combining connected and disconnected, in the secondcolumn.For our final result for the isovector electric charge ra-dius, we use the central value and statistical error of theresult from the plateau method at t s = 18 a = 1 . Q values. We also include a TABLE IV. Results for the isovector magnetic charge radiusof the nucleon ( (cid:104) r M (cid:105) u − d ) and the isovector magnetic moment G M (0) = µ u − d from fits to G u − dM ( Q ). In the first column weshow t s for the plateau method and the t s fit range for thesummation and two-state fit methods. The two smallest Q values are omitted from the fit. t s [fm] dipole z-expansion (cid:104) r M (cid:105) u − d [fm ] χ d . o . f (cid:104) r M (cid:105) u − d [fm ] χ d . o . f Plateau0.94 0.404(10) 0.3 0.59(13) 0.31.13 0.434(22) 0.3 0.82(23) 0.31.31 0.536(52) 0.3 0.79(40) 0.3Summation0.9-1.3 0.68(16) 0.1 1.83(49) 0.1Two-state0.9-1.3 0.470(31) 0.3 1.15(25) 0.3 t s [fm] dipole z-expansion G u − dM (0) χ d . o . f G u − dM (0) χ d . o . f Plateau0.94 3.548(52) 0.3 3.85(16) 0.31.13 3.595(90) 0.3 4.13(31) 0.31.31 4.02(21) 0.3 4.31(57) 0.3Summation0.9-1.3 4.32(57) 0.1 6.35(1.35) 0.1Two-state0.9-1.3 3.74(14) 0.3 4.71(42) 0.3TABLE V. Results for the isoscalar electric charge radius ofthe nucleon ( (cid:104) r E (cid:105) u + d ). In the first column we show t s for theplateau method and the t s fit range for the summation andtwo-state fit methods. For each t s and for each fit Ansatz,we give the result from fitting to the connected contributionin the first column and to the total contribution of connectedplus disconnected in the second column. t s [fm] dipole z-expansion (cid:104) r E (cid:105) u + d [fm ] χ d . o . f (cid:104) r E (cid:105) u + d [fm ] χ d . o . f Connected Total Connected TotalPlateau0.94 0.440(3) 0.449(49) 4.5 0.418(9) 0.427(49) 0.91.13 0.469(6) 0.478(49) 1.9 0.464(17) 0.474(52) 0.71.31 0.494(12) 0.503(50) 0.9 0.485(34) 0.495(59) 0.51.50 0.502(14) 0.512(50) 0.3 0.494(41) 0.503(63) 0.41.69 0.527(22) 0.537(53) 0.9 0.493(60) 0.503(77) 0.8Summation0.9-1.7 0.565(20) 0.576(53) 0.9 0.555(54) 0.564(72) 0.6Two-state1.1-1.7 0.490(16) 0.499(51) 0.5 0.453(77) 0.462(91) 0.7 TABLE VI. Results for the isoscalar magnetic charge radiusof the nucleon ( (cid:104) r M (cid:105) u + d ) and the isoscalar magnetic moment G u + dM (0). The notation is as in Table V. t s [fm] dipole z-expansion (cid:104) r M (cid:105) u + d [fm ] χ d . o . f (cid:104) r M (cid:105) u + d [fm ] χ d . o . f Connected Total Connected TotalPlateau0.94 0.392(13) 0.302(34) 0.2 0.41(19) 0.32(20) 0.21.13 0.419(29) 0.329(47) 0.1 0.84(28) 0.78(32) 0.11.31 0.476(59) 0.394(82) 0.4 0.4(1.0) 0.4(1.1) 0.5Summation0.9-1.3 0.50(18) 0.42(24) 0.2 1.94(92) 2.0(1.3) 0.2Two-state0.9-1.3 0.439(44) 0.353(65) 0.2 0.89(47) 0.83(52) 0.2 t s [fm] dipole z-expansion G u + dM (0) χ d . o . f G u + dM (0) χ d . o . f Connected Total Connected TotalPlateau0.94 0.838(16) 0.808(18) 0.2 0.867(50) 0.837(50) 0.21.13 0.841(29) 0.811(30) 0.1 0.981(90) 0.951(90) 0.11.31 0.900(59) 0.870(60) 0.4 0.90(19) 0.87(19) 0.5Summation0.9-1.3 0.88(16) 0.85(16) 0.2 1.51(45) 1.48(45) 0.2Two-state0.9-1.3 0.861(47) 0.831(48) 0.2 1.01(14) 0.98(14) 0.2 systematic error from the difference of the central valueswhen comparing with the two-state fit method to accountfor excited states effects. Similarly, for the magnetic ra-dius and moment, we take the result from the dipole fitsto our largest sink-source separation, which for this caseis t s = 14 a = 1 .
31 fm and as in the case of the electriccharge radius, we take the difference with the two-statefit method as an additional systematic error. In this case,the values at the two lowest momenta are not includedin the fit. Our final values for the isovector radii andisovector nucleon magnetic moment are: (cid:104) r E (cid:105) u − d = 0 . , (cid:104) r M (cid:105) u − d = 0 . , and µ u − d = 4 . , (22)where the first error is statistical and the second error is asystematic obtained when comparing the plateau methodto the two-state fit method as a measure of excited stateeffects. For the isoscalar radii and moment we follow asimilar analysis after adding the disconnected contribu-tion from the plateau method for t s = 10 a = 0 . (cid:104) r E (cid:105) u + d = 0 . , (cid:104) r M (cid:105) u + d = 0 . , and µ u + d = 0 . . (23) [GeV ]0.00.20.40.60.81.0 G p E ( Q ) Lattice results(plateau method)Experiment
FIG. 19. Proton electric Sachs form factor as a function ofthe momentum transfer. We show with triangles the sum ofconnected and disconnected contributions, with the plateauresult for t s = 18 a = 1 . t s =10 a = 0 . C. Proton and neutron form factors
Having the isovector and isoscalar contributions to theform factors, we can obtain the proton ( G p ( Q )) andneutron ( G n ( Q )) form factors via linear combinationstaken from Eqs. (2) and (3) assuming isospin symmetrybetween up and down quarks and proton and neutron.Namely, we have: G p ( Q ) = 12 [ G u + d ( Q ) + G u − d ( Q )] G n ( Q ) = 12 [ G u + d ( Q ) − G u − d ( Q )] (24)where G p ( Q ) ( G n ( Q )) is either the electric or magneticproton (neutron) form factor. In Figs. 19 and 20 we showresults for the proton electric and magnetic Sachs formfactors respectively. As for the isoscalar case, the dis-connected contributions have been included. The bandsare from fits to the dipole form of Eq. (17). In theseplots we compare to experimental results from the A1collaboration [29]. We observe a similar behavior whencomparing to experiment as for the case of the isovectorform factors. Namely, the dipole fit to the lattice datahas a smaller slope for small values of Q as compared toexperiment, while G pM ( Q ) reproduces the experimentalmomentum dependence for Q > . .In Figs. 21 and 22 we show the same for the neutronform factors. For the neutron electric form factor we fit3 [GeV ]0.00.51.01.52.02.53.0 G p M ( Q ) Lattice results(plateau method)Experiment
FIG. 20. Proton magnetic Sachs form factor as a function ofthe momentum transfer. We show with squares the sum ofconnected and disconnected contributions, with the plateauresult for t s = 14 a = 1 . t s =10 a = 0 . to the form [15]: G nE ( Q ) = τ A τ B Q Λ ) (25)with τ = Q / (2 m N ) and Λ = 0 .
71 GeV and allow A and B to vary. This Ansatz reproduces our data well.We compare to a collection of experimental data fromRefs. [30–44]. For G nM ( Q ), we agree with the experimen-tal data for Q > . , however we underestimatethe magnetic moment by about 20%. Experimental datafor G nM ( Q ) shown in Fig. 22 are taken from Refs. [45–50].We use Eq. (19) to obtain the radii using the dipolefits. For the case of G nE ( Q ), the neutron electric radius isobtained via: (cid:104) r E (cid:105) n = − A m N , where A is the parameterof Eq. (25). In all cases we have combined connected anddisconnected. We obtain: (cid:104) r E (cid:105) p = 0 . , (cid:104) r M (cid:105) p = 0 . , and µ p = 2 . , (26)for the proton, and: (cid:104) r E (cid:105) n = − . , (cid:104) r M (cid:105) n = 0 . , and µ n = − . , (27)for the neutron, where as in the case of the isoscalar andisovector, the first error is statistical and the second is a [GeV ]0.000.020.040.060.080.100.120.14 G n E ( Q ) Lattice results(plateau method)Experiment
FIG. 21. Neutron electric Sachs form factor as a functionof the momentum transfer. Triangles are from the sum ofconnected and disconnected contributions, with the plateauresult for t s = 18 a = 1 . t s =10 a = 0 . systematic obtained when comparing the plateau methodto the two-state fit method as a measure of excited stateeffects. IV. COMPARISON WITH OTHER RESULTSA. Comparison of isovector and isoscalar formfactors
Recent lattice calculations for the electromagneticform factors of the nucleon include an analysis from theMainz group [51] using N f = 2 clover fermions down to apion mass of 193 MeV, results from the PNDME collabo-ration [52] using clover valence fermions on N f = 2+1+1HISQ sea quarks down to pion mass of ∼
220 MeV and N f = 2 + 1 + 1 results from the ETM collaboration downto 213 MeV pion mass [53]. Simulations directly at thephysical point have only been possible recently. TheLHPC has published results in Ref. [54] using N f = 2 + 1HEX smeared clover fermions, which include an ensemblewith m π =149 MeV. Preliminary results for electromag-netic nucleon form factors at physical or near physicalpion masses have also been reported by the PNDMEcollaboration in Ref. [55] using clover valence quarkson HISQ sea quarks at a pion mass of 130 MeV andby the RBC/UKQCD collaboration using Domain Wallfermions at m π = 172 MeV in Ref. [56].In Fig. 23 we compare our results for G u − dE ( Q ) fromthe plateau method using t s = 18 a = 1 . [GeV ]2.01.51.00.50.0 G n M ( Q ) Lattice results(plateau method)Experiment
FIG. 22. Neutron magnetic Sachs form factor as a functionof the momentum transfer. We show with squares the sum ofconnected and disconnected contributions, with the plateauresult for t s = 14 a = 1 . t s =10 a = 0 . [GeV ]0.00.20.40.60.81.0 G u d E ( Q ) Kelly parametrizationETMC (this work)LHPC (2014)
FIG. 23. Comparison of G u − dE ( Q ) between results from thiswork (circles) denoted by ETMC and from the LHPC takenfrom Ref. [54] (squares). The dashed line shows the parame-terization of the experimental data. from the summation method using three sink-source sep-arations from 0.93 to 1.39 fm for their ensemble at thenear-physical pion mass of m π =149 MeV. We note thattheir statistics of 7752 are about six times less than oursat the sink-source separation we use in this plot (see Ta-ble II).In Fig. 24 we plot our results for G u − dM ( Q ) from theplateau method using t s = 14 a = 1 . [GeV ]12345 G u d M ( Q ) Kelly parametrizationETMC (this work)LHPC (2014)
FIG. 24. Comparison of G u − dM ( Q ) between results from thiswork (circles) and Ref. [54] (squares). The dashed line showsthe parameterization of the experimental data. [GeV ]0.20.40.60.81.0 F u d ( Q ) Kelly parametrizationETMC (this work)LHPC (2014)
FIG. 25. Comparison of F u − d ( Q ) between results from thiswork (circles) and Ref. [54] (squares). The dashed line showsthe parameterization of the experimental data. are larger, possibly due to the fact that the summationmethod is used for their final quoted results. Withinerrors, we see consistent results at all Q values.In Figs. 25 and 26 we compare our results for theisovector Dirac and Pauli form factors F u − d ( Q ) and F u − d ( Q ) with those from Ref. [54]. We use Eq. (6)to obtain F u − d ( Q ) and F u − d ( Q ) from G u − dE ( Q ) and G u − dM ( Q ) extracted from the plateau method at thesame sink-source separations used in Figs. 23 and 24.As in the case of G u − dE ( Q ) and G u − dM ( Q ) we see agree-ment between these two calculations. We also note thatthe discrepancy with experiment of G u − dM ( Q ) at low Q values carries over to F u − d ( Q ).For the isoscalar case, we compare the connected con-tributions to the Sachs form factors with Ref. [54] inFigs. 27 and 28. The agreement between the two latticeformulations is remarkable given that the results have notbeen corrected for finite volume or cut-off effects. The5 [GeV ]12345 F u d ( Q ) Kelly parametrizationETMC (this work)LHPC (2014)
FIG. 26. Comparison of F u − d ( Q ) between results from thiswork (circles) and Ref. [54] (squares). The dashed line showsthe parameterization of the experimental data. [GeV ]0.20.40.60.81.0 G u + d E ( Q ) Kelly parametrizationETMC (this work)LHPC (2014)
FIG. 27. Comparison of G u + dE ( Q ) between results from thiswork (circles) and Ref. [54] (squares). The dashed line showsthe parameterization of the experimental data. gauge configurations used by LHPC were carried out us-ing the same spatial lattice size as ours but with a coarserlattice spacing yielding m π L = 4 . m π L = 3. Although the LHPC results for the isovectormagnetic form factor at low Q are in agreement with ex-periment, they carry large statistical errors that do notallow us to draw any conclusion as to whether the originof the discrepancy in our much more accurate data is dueto the smaller m π L value.For the radii and magnetic moment, we compare ourresult to recent published results, which are available forthe isovector case, from Refs. [51–54]. We quote their val-ues obtained before extrapolation to the physical point,using the smallest pion mass available. In Fig. 29 wesee that the two results at physical or near-physical pionmass, namely the result of this work and from LHPC,are within one standard deviation from the spectroscopicdetermination of the charged radius using muonic hydro-gen [2]. [GeV ]0.20.40.60.81.01.21.4 G u + d M ( Q ) Kelly parametrizationETMC (this work)LHPC (2014)
FIG. 28. Comparison of G u + dM ( Q ) between results from thiswork (circles) and Ref. [54] (squares). The dashed line showsthe parameterization of the experimental data.
2E u d [fm ] ETMC 2013, m = 213 MeVPNDME 2014, m = 220 MeVLHPC 2014, m = 149 MeVMainz 2015, m = 193 MeVETMC, m = 130 MeV (this work) C O D A T A H FIG. 29. Our result for (cid:104) r E (cid:105) u − d at m π =130 MeV (cir-cle) compared to recent lattice results from LHPC [54] at m π = 149 MeV (square), PNDME [52] at m π = 220 MeV(triangle), the Mainz group [51] at m π = 193 MeV (diamond)and ETMC [53] (pentagon). We show two error bars whensystematic errors are available, with the smaller denoting thestatistical error and the larger denoting the combination ofstatistical and systematic errors added in quadrature. Thevertical band denoted with µ H is the experimental result us-ing muonic hydrogen from Ref. [2] and the band denoted withCODATA is from Ref. [57].
2M u d [fm ] ETMC 2013, m = 213 MeVPNDME 2014, m = 220 MeVLHPC 2014, m = 149 MeVMainz 2015, m = 193 MeVETMC, m = 130 MeV (this work) P D G FIG. 30. Comparison of results for (cid:104) r M (cid:105) u − d with the notationof Fig. 29. The experimental band is from Ref. [57]. u d ETMC 2013, m = 213 MeVPNDME 2014, m = 220 MeVLHPC 2014, m = 149 MeVMainz 2015, m = 193 MeVETMC, m = 130 MeV (this work) P D G FIG. 31. Comparison of results for the isovector nucleon mag-netic moment µ u − d with the notation of Fig. 29. [GeV ]0.20.40.60.81.0 G p E ( Q ) Kelly parametrizationETMC (this work)LHPC (2014)
FIG. 32. Comparison of G pE ( Q ) between results from thiswork (circles) and Ref. [54] (squares). The dashed line showsthe parameterization of the experimental data. A similar comparison is shown in Fig. 30 for the mag-netic radius. We see that all lattice results underestimatethe experimental band by at most 2 σ , with the exceptionof the LHPC value that used the summation method.Similar conclusions are drawn for the isovector magneticmoment G M (0) = µ u − d = µ p − µ n , which we show inFig. 31. B. Comparison of proton form factors
Published lattice QCD results for the proton form fac-tors at physical or near-physical pion masses are availablefrom LHPC [54]. We compare our results in Figs. 32and 33 for the proton electric and magnetic Sachs formfactors respectively. We see agreement with their resultsand note that their relatively larger errors at small Q for the case of the magnetic form factor are consistentwith both the experimentally determined curve and ourresults. [GeV ]0.51.01.52.02.53.0 G p M ( Q ) Kelly parametrizationETMC (this work)LHPC (2014)
FIG. 33. Comparison of G pM ( Q ) between results from thiswork (circles) and Ref. [54] (squares). The dashed line showsthe parameterization of experimental data. V. SUMMARY AND CONCLUSIONS
A first calculation of the isovector and isoscalar elec-tromagnetic Sachs nucleon form factors including the dis-connected contributions is presented directly at the phys-ical point using an ensemble of N f = 2 twisted massfermions at maximal twist at a volume of m π L (cid:39) G E ( Q ) between0.94 fm and 1.69 fm, we confirm our previous find-ings that excited state contributions require a separationlarger than ∼ G M ( Q ) we use three sink-source separationsbetween 0.94 fm and 1.31 fm and observe that for theisovector no excited state effects are present within statis-tical errors, while for the connected isoscalar, the largestseparation of t s = 1 .
31 fm is sufficient for their suppres-sion. Our results for both the isovector and isoscalar G E ( Q ) lie higher than experiment by about a standarddeviation. This may be due to small residual excitedstate contamination since this difference is found to de-crease as the sink-source separation increases. Our re-sults for G u − dM ( Q ) at the two lowest Q values under-estimate the experimental ones but are in agreement for Q > . . Volume effects are being investigatedto determine whether these could be responsible for thisdiscrepancy.The isoscalar matrix element requires both connectedand disconnected contributions, the latter requiring anorder of magnitude more statistics. We have com-puted the disconnected contributions to G u + dE ( Q ) and G u + dM ( Q ) for the first four non-zero momentum trans-fers up to Q = 0 .
28 GeV and find that their magni-tude is smaller or comparable to the statistical error ofthe connected contribution. We include the disconnectedcontributions to combine isovector and isoscalar matrixelements and obtain the proton and neutron electromag-netic Sachs form factors at the physical point.We have used two methods to fit the Q -dependence7 TABLE VII. Our final results for the isovector ( p − n ),isoscalar ( p + n ), proton ( p ) and neutron ( n ) electric radius( (cid:104) r E (cid:105) ), magnetic radius ( (cid:104) r M (cid:105) ) and magnetic moment ( µ ).The first error is statistical and the second a systematic dueto excited state contamination. (cid:104) r E (cid:105) [fm ] (cid:104) r M (cid:105) [fm ] µp - n p + n p n -0.038(34)(6) 0.586(58)(75) -1.58(9)(12) of our data, both a dipole Ansatz and the z-expansion.These two methods yield consistent results, however thelatter method yields parameters with larger statisticalerrors. Using the dipole fits to determine the electric andmagnetic radii, as well as the magnetic moment, we findagreement with other recent lattice QCD results for theisovector case, and are within 2 σ with the experimentaldeterminations. Our result for the proton electric chargeradius (cid:104) r E (cid:105) p = 0 . , is two sigmas smallerthan the muonic hydrogen determination [58] of (cid:104) r p (cid:105) =0 . , which may be due to remaining excitedstate effects or volume effects, which will be investigatedfurther.Our final results are collected in Table VII. We planto analyze the electromagnetic form factors using both an ensemble of N f = 2 twisted mass clover-improvedfermions simulated at the same pion mass and latticespacing as the ensemble analyzed in this work but witha lattice size of 64 × m π L = 4 as well aswith an N f = 2 + 1 + 1 ensemble of finer lattice spacing.In addition, we are investigating improved techniques forthe computation of the disconnected quark loops at thephysical point. These future calculations will allow forfurther checks of lattice artifacts and resolve the remain-ing small tension between lattice QCD and experimentalresults for these important benchmark quantities. Acknowledgments:
We would like to thank the mem-bers of the ETM Collaboration for a most enjoyable col-laboration. We acknowledge funding from the EuropeanUnion’s Horizon 2020 research and innovation programunder the Marie Sklodowska-Curie Grant AgreementNo. 642069. Results were obtained using Jureca, via aJohn-von-Neumann-Institut fr Computing (NIC) alloca-tion ECY00, HazelHen at H¨ochstleistungsrechenzentrumStuttgart (HLRS) and SuperMUC at the Leibniz-Rechenzentrum (LRZ), via Gauss allocations with ids44066 and 10862, Piz Daint at Centro Svizzero di CalcoloScientifico (CSCS), via projects with ids s540, s625 ands702, and resources at Centre Informatique National delEnseignement Suprieur (CINES) and Institute for Devel-opment and Resources in Intensive Scientific Computing(IDRIS) under allocation 52271. We thank the staff ofthese centers for access to the computational resourcesand for their support. [1] V. Punjabi, C. F. Perdrisat, M. K. Jones, E. J. Brash,and C. E. Carlson. The Structure of the Nucleon: ElasticElectromagnetic Form Factors.
Eur. Phys. J. , A51:79,2015. doi:10.1140/epja/i2015-15079-x.[2] R. Pohl, A. Antognini, F. Nez, F. D. Amaro, F. Biraben,et al. The size of the proton.
Nature , 466:213–216, 2010.doi:10.1038/nature09250.[3] Randolf Pohl et al. Laser spectroscopy of muonicdeuterium.
Science , 353(6300):669–673, 2016. doi:10.1126/science.aaf2468.[4] A. Abdel-Rehim et al. Simulating QCD at the PhysicalPoint with N f = 2 Wilson Twisted Mass Fermions atMaximal Twist. 2015.[5] R. Frezzotti and G. C. Rossi. Chirally improving Wilsonfermions. 1. O(a) improvement. JHEP , 08:007, 2004. doi:10.1088/1126-6708/2004/08/007.[6] Roberto Frezzotti, Pietro Antonio Grassi, Stefan Sint,and Peter Weisz. Lattice QCD with a chirally twistedmass term.
JHEP , 08:058, 2001.[7] Richard J. Hill and Gil Paz. Model independentextraction of the proton charge radius from electronscattering.
Phys. Rev. , D82:113005, 2010. doi:10.1103/PhysRevD.82.113005.[8] C. Alexandrou, S. Gusken, F. Jegerlehner, K. Schilling,and R. Sommer. The Static approximation of heavy- light quark systems: A Systematic lattice study.
Nucl. Phys. , B414:815–855, 1994. doi:10.1016/0550-3213(94)90262-3. [9] S. Gusken, U. Low, K. H. Mutter, R. Sommer, A. Patel,and K. Schilling. Nonsinglet Axial Vector Couplings ofthe Baryon Octet in Lattice QCD.
Phys. Lett. , B227:266–269, 1989. doi:10.1016/S0370-2693(89)80034-6.[10] A. Abdel-Rehim et al. Nucleon and pion structure withlattice QCD simulations at physical value of the pionmass.
Phys. Rev. D (in press) , 2015.[11] A. Abdel-Rehim, C. Alexandrou, M. Constantinou,K. Hadjiyiannakou, K. Jansen, Ch. Kallidonis, G. Kout-sou, and A. Vaquero Aviles-Casco. Direct Evaluation ofthe Quark Content of Nucleons from Lattice QCD at thePhysical Point.
Phys. Rev. Lett. , 116(25):252001, 2016.doi:10.1103/PhysRevLett.116.252001.[12] C. Alexandrou, G. Koutsou, John W. Negele, andA. Tsapalis. The Nucleon electromagnetic form factorsfrom Lattice QCD.
Phys.Rev. , D74:034508, 2006. doi:10.1103/PhysRevD.74.034508.[13] C. Alexandrou, M. Brinet, J. Carbonell, M. Constanti-nou, P.A. Harraud, et al. Nucleon electromagnetic formfactors in twisted mass lattice QCD.
Phys.Rev. , D83:094502, 2011. doi:10.1103/PhysRevD.83.094502.[14] Constantia Alexandrou and Christos Kallidonis. Low-lying baryon masses using N f = 2 twisted mass clover-improved fermions directly at the physical point. 2017.[15] J. J. Kelly. Simple parametrization of nucleonform factors. Phys. Rev. , C70:068202, 2004. doi:10.1103/PhysRevC.70.068202.[16] W. M. Alberico, S. M. Bilenky, C. Giunti, and K. M. Graczyk. Electromagnetic form factors of the nucleon:New Fit and analysis of uncertainties.
Phys. Rev. , C79:065204, 2009. doi:10.1103/PhysRevC.79.065204.[17] Toru Sato, D. Uno, and T. S. H. Lee. Dynamical model ofweak pion production reactions.
Phys. Rev. , C67:065201,2003. doi:10.1103/PhysRevC.67.065201.[18] C. Alexandrou, G. Koutsou, J. W. Negele, Y. Proestos,and A. Tsapalis. Nucleon to Delta transition form factorswith N F = 2 + 1 domain wall fermions. Phys. Rev. , D83:014501, 2011. doi:10.1103/PhysRevD.83.014501.[19] Constantia Alexandrou, Martha Constantinou, KyriakosHadjiyiannakou, Karl Jansen, Christos Kallidonis, Gi-annis Koutsou, Konstantin Ottnad, and Alejandro Va-quero. Nucleon electromagnetic and axial form factorswith N f =2 twisted mass fermions at the physical point. PoS , LATTICE2016:154, 2016.[20] Abdou Abdel-Rehim, Constantia Alexandrou, MarthaConstantinou, Kyriakos Hadjiyiannakou, Karl Jansen,and Giannis Koutsou. Nucleon electromagnetic formfactors from twisted mass lattice QCD.
PoS , LAT-TICE2014:148, 2015.[21] A. Abdel-Rehim, C. Alexandrou, M. Constantinou,V. Drach, K. Hadjiyiannakou, et al. Disconnectedquark loop contributions to nucleon observables in lat-tice QCD.
Phys.Rev. , D89(3):034501, 2014. doi:10.1103/PhysRevD.89.034501.[22] C. Alexandrou, K. Hadjiyiannakou, G. Koutsou,A. O’Cais, and A. Strelchenko. Evaluation of fermionloops applied to the calculation of the η (cid:48) mass andthe nucleon scalar and electromagnetic form factors. Comput. Phys. Commun. , 183:1215–1224, 2012. doi:10.1016/j.cpc.2012.01.023.[23] Constantia Alexandrou, Martha Constantinou, KyriakosHadjiyiannakou, Karl Jansen, Christos Kallidonis, Gian-nis Koutsou, and Alejandro Vaquero Aviles-Casco. Thenucleon axial form factors using lattice QCD simulationswith a physical value of the pion mass. 2017.[24] Jeremy Green, Stefan Meinel, Michael Engelhardt, Ste-fan Krieg, Jesse Laeuchli, John Negele, Kostas Orginos,Andrew Pochinsky, and Sergey Syritsyn. High-precisioncalculation of the strange nucleon electromagnetic formfactors.
Phys. Rev. , D92(3):031501, 2015. doi:10.1103/PhysRevD.92.031501.[25] Raza Sabbir Sufian, Yi-Bo Yang, Jian Liang, TerrenceDraper, and Keh-Fei Liu. Sea Quarks Contribution tothe Nucleon Magnetic Moment and Charge Radius atthe Physical Point. 2017.[26] Abdou Abdel-Rehim, Constantia Alexandrou, MarthaConstantinou, Jacob Finkenrath, Kyriakos Hadjiyian-nakou, Karl Jansen, Christos Kallidonis, Giannis Kout-sou, Alejandro Vaquero Avils-Casco, and Julia Volmer.Disconnected diagrams with twisted-mass fermions.
PoS ,LATTICE2016:155, 2016.[27] C. F. Perdrisat, V. Punjabi, and M. Vanderhaeghen. Nu-cleon Electromagnetic Form Factors.
Prog. Part. Nucl.Phys. , 59:694–764, 2007. doi:10.1016/j.ppnp.2007.05.001.[28] Zachary Epstein, Gil Paz, and Joydeep Roy. Model in-dependent extraction of the proton magnetic radius fromelectron scattering.
Phys. Rev. , D90(7):074027, 2014. doi:10.1103/PhysRevD.90.074027.[29] J. C. Bernauer et al. Electric and magnetic form factorsof the proton.
Phys. Rev. , C90(1):015206, 2014. doi:10.1103/PhysRevC.90.015206.[30] J. Golak, G. Ziemer, H. Kamada, H. Witala, and Walter Gloeckle. Extraction of electromagnetic neutron form-factors through inclusive and exclusive polarized electronscattering on polarized He-3 target.
Phys. Rev. , C63:034006, 2001. doi:10.1103/PhysRevC.63.034006.[31] J. Becker et al. Determination of the neutron electricform-factor from the reaction He-3(e,e’ n) at mediummomentum transfer.
Eur. Phys. J. , A6:329–344, 1999.doi:10.1007/s100500050351.[32] T. Eden et al. Electric form factor of the neu-tron from the H ( −→ e , e −→ n ) H reaction at Q = 0.255(GeV/c) . Phys. Rev. , C50(4):R1749–R1753, 1994. doi:10.1103/PhysRevC.50.R1749.[33] M. Meyerhoff et al. First measurement of the electricform-factor of the neutron in the exclusive quasielas-tic scattering of polarized electrons from polarized He-3.
Phys. Lett. , B327:201–207, 1994. doi:10.1016/0370-2693(94)90718-8.[34] I. Passchier et al. The Charge form-factor of the neu-tron from the reaction polarized H-2(polarized e, e-prime n) p.
Phys. Rev. Lett. , 82:4988–4991, 1999. doi:10.1103/PhysRevLett.82.4988.[35] G. Warren et al. Measurement of the electricform-factor of the neutron at Q = 0.5 and 1.0 GeV /c . Phys. Rev. Lett. , 92:042301, 2004. doi:10.1103/PhysRevLett.92.042301.[36] H. Zhu et al. A Measurement of the electric form-factorof the neutron through polarized-d (polarized-e, e-primen)p at Q = 0.5-(GeV/c) . Phys. Rev. Lett. , 87:081801,2001. doi:10.1103/PhysRevLett.87.081801.[37] B. Plaster et al. Measurements of the neutron electricto magnetic form-factor ratio G(En) / G(Mn) via theH-2(polarized-e, e-prime,polarized-n)H-1 reaction to Q = 1.45-(GeV/c) . Phys. Rev. , C73:025205, 2006. doi:10.1103/PhysRevC.73.025205.[38] R. Madey et al. Measurements of G(E)n / G(M)n fromthe H-2(polarized-e,e-prime polarized-n) reaction to Q = 1.45 (GeV/c) . Phys. Rev. Lett. , 91:122002, 2003. doi:10.1103/PhysRevLett.91.122002.[39] D. Rohe et al. Measurement of the neutron electric form-factor G(en) at 0.67-(GeV/c) via He-3(pol.)(e(pol.),e’n). Phys. Rev. Lett. , 83:4257–4260, 1999. doi:10.1103/PhysRevLett.83.4257.[40] J. Bermuth et al. The Neutron charge form-factor andtarget analyzing powers from polarized-He-3 (polarized-e,e-prime n) scattering.
Phys. Lett. , B564:199–204, 2003.doi:10.1016/S0370-2693(03)00725-1.[41] D. I. Glazier et al. Measurement of the electric form-factor of the neutron at Q = 0.3-(GeV/c) to 0.8-(GeV/c) . Eur. Phys. J. , A24:101–109, 2005. doi:10.1140/epja/i2004-10115-8.[42] C. Herberg et al. Determination of the neutron electricform-factor in the D(e,e’ n)p reaction and the influenceof nuclear binding.
Eur. Phys. J. , A5:131–135, 1999. doi:10.1007/s100500050268.[43] R. Schiavilla and I. Sick. Neutron charge form-factorat large q . Phys. Rev. , C64:041002, 2001. doi:10.1103/PhysRevC.64.041002.[44] M. Ostrick et al. Measurement of the neutron elec-tric form-factor G(E,n) in the quasifree H-2(e(pol.),e’n(pol.))p reaction.
Phys. Rev. Lett. , 83:276–279, 1999.doi:10.1103/PhysRevLett.83.276.[45] B. Anderson et al. Extraction of the neutron magneticform-factor from quasi-elastic polarized-He-3(polarized-e, e-prime) at Q = 0.1 - 0.6 (GeV/c) . Phys. Rev. , C75: Phys. Rev. , C50:R546–R549, 1994. doi:10.1103/PhysRevC.50.R546.[47] H. Anklin et al. Precision measurement of the neutronmagnetic form-factor.
Phys. Lett. , B336:313–318, 1994.doi:10.1016/0370-2693(94)90538-X.[48] H. Anklin et al. Precise measurements of the neutronmagnetic form-factor.
Phys. Lett. , B428:248–253, 1998.doi:10.1016/S0370-2693(98)00442-0.[49] G. Kubon et al. Precise neutron magnetic form-factors.
Phys. Lett. , B524:26–32, 2002. doi:10.1016/S0370-2693(01)01386-7.[50] R. Alarcon. Nucleon form factors and the BLAST ex-periment.
Eur. Phys. J. , A32(4):477–481, 2007. doi:10.1140/epja/i2006-10395-x, 10.1140/epja/i2006-10268-4. [Eur. Phys. J.A32,477(2007)].[51] S. Capitani, M. Della Morte, D. Djukanovic, G. von Hip-pel, J. Hua, B. Jger, B. Knippschild, H. B. Meyer, T. D.Rae, and H. Wittig. Nucleon electromagnetic form fac-tors in two-flavor QCD.
Phys. Rev. , D92(5):054511, 2015.doi:10.1103/PhysRevD.92.054511.[52] Tanmoy Bhattacharya, Saul D. Cohen, Rajan Gupta,Anosh Joseph, Huey-Wen Lin, and Boram Yoon. Nu-cleon Charges and Electromagnetic Form Factors from2+1+1-Flavor Lattice QCD.
Phys. Rev. , D89(9):094502,2014. doi:10.1103/PhysRevD.89.094502.[53] C. Alexandrou, M. Constantinou, S. Dinter, V. Drach,K. Jansen, C. Kallidonis, and G. Koutsou. Nucleon formfactors and moments of generalized parton distributionsusing N f = 2 + 1 + 1 twisted mass fermions. Phys. Rev. ,D88(1):014509, 2013. doi:10.1103/PhysRevD.88.014509.[54] J.R. Green, J.W. Negele, A.V. Pochinsky, S.N. Syrit-syn, M. Engelhardt, et al. Nucleon electromagneticform factors from lattice QCD using a nearly physi-cal pion mass.
Phys.Rev. , D90(7):074507, 2014. doi:10.1103/PhysRevD.90.074507.[55] Yong-Chull Jang, Tanmoy Bhattacharya, Rajan Gupta,Boram Yoon, Huey-Wen Lin, and Pndme Collaboration.Nucleon Vector and Axial-Vector Form Factors.
PoS ,LATTICE2016:178, 2016.[56] Michael Abramczyk, Meifeng Lin, Andrew Lytle, andShigemi Ohta. Nucleon structure from 2+1-flavor dy-namical DWF ensembles.
PoS , LATTICE2016:150, 2016.[57] Peter J. Mohr, David B. Newell, and Barry N. Tay-lor. CODATA Recommended Values of the Fundamen-tal Physical Constants: 2014.
Rev. Mod. Phys. , 88(3):035009, 2016. doi:10.1103/RevModPhys.88.035009.[58] Aldo Antognini et al. Proton Structure from theMeasurement of 2 S − P Transition Frequencies ofMuonic Hydrogen.
Science , 339:417–420, 2013. doi:10.1126/science.1230016. Appendix A: Tables of Results
TABLE VIII. Results for the isovector G E ( Q ) using the plateau method for five sink-source separations and the summationand two-state fit methods fitted to all separations. Results where the operand of the square root in Eq. 9 becomes negative aredenoted with “NA”. Q [GeV ] Plateau, t s [fm] Summation Two-state0.94 1.13 1.31 1.50 1.69 [0.94, 1.69] [1.13, 1.69]0.000 0.9982(08) 0.998(2) 0.996(41) 1.003(4) 1.006(8) 1.000(6) -0.074 0.8460(31) 0.832(6) 0.826(11) 0.819(15) 0.841(23) 0.798(20) 0.849(18)0.145 0.7337(34) 0.713(6) 0.703(12) 0.701(16) 0.711(24) 0.664(21) 0.717(17)0.214 0.6423(45) 0.618(8) 0.598(15) 0.615(21) 0.608(29) 0.556(26) 0.617(19)0.280 0.5753(54) 0.553(10) 0.549(19) 0.514(24) 0.521(36) 0.483(35) 0.535(22)0.345 0.5222(43) 0.503(7) 0.497(15) 0.461(19) 0.478(26) 0.435(26) 0.474(16)0.407 0.4761(49) 0.456(8) 0.450(17) 0.391(20) 0.378(30) 0.357(30) 0.407(19)0.527 0.4000(62) 0.380(12) 0.379(22) 0.326(31) 0.291(39) 0.283(49) 0.334(24)0.584 0.3676(57) 0.353(10) 0.365(22) 0.265(27) 0.287(37) 0.269(42) 0.296(22)0.640 0.3500(67) 0.338(13) 0.339(25) 0.256(35) 0.260(53) 0.229(52) 0.292(25)0.695 0.3273(72) 0.320(13) 0.303(26) 0.236(31) 0.219(43) 0.208(54) 0.279(26)0.749 0.284(11) 0.282(20) 0.343(47) 0.138(68) NA NA 0.181(46)0.802 0.2847(85) 0.262(16) 0.215(28) 0.196(49) 0.12(21) 0.058(79) 0.203(35)0.853 0.2707(81) 0.273(15) 0.257(34) 0.144(33) 0.156(52) 0.160(74) 0.186(35)TABLE IX. Results for the isovector G M ( Q ) using the plateau method for three sink-source separations and the summationand two-state fit method fitted to all separations. Q [GeV ] Plateau, t s [fm] Summation Two-state0.94 1.13 1.31 [0.94, 1.31] [0.94, 1.31]0.074 3.225(36) 3.220(54) 3.230(99) 3.18(18) 3.292(82)0.145 2.841(28) 2.807(38) 2.832(70) 2.73(13) 2.847(54)0.214 2.538(26) 2.505(38) 2.596(67) 2.53(13) 2.546(54)0.280 2.288(26) 2.281(39) 2.262(75) 2.23(13) 2.294(59)0.344 2.098(21) 2.042(31) 2.037(55) 1.85(11) 2.033(46)0.407 1.941(19) 1.873(32) 1.899(56) 1.67(12) 1.863(45)0.526 1.665(20) 1.611(35) 1.583(70) 1.37(15) 1.593(50)0.583 1.565(17) 1.515(31) 1.469(64) 1.29(14) 1.483(40)0.640 1.481(22) 1.420(39) 1.304(77) 1.14(17) 1.354(53)0.694 1.387(20) 1.339(37) 1.219(68) 1.12(17) 1.299(52)0.748 1.330(25) 1.275(54) 1.23(14) 0.99(29) 1.247(58)0.800 1.218(21) 1.128(44) 0.999(83) 0.86(22) 1.063(77)0.852 1.173(20) 1.140(46) 1.054(98) 0.98(23) 1.116(46) TABLE X. Results for the connected contribution to the isoscalar G E ( Q ) using the plateau method for five sink-sourceseparations and the summation and two-state fit methods fitted to all separations. Results where the operand of the squareroot in Eq. 9 becomes negative are denoted with “NA”. Q [GeV ] Plateau, t s [fm] Summation Two-state0.94 1.13 1.31 1.50 1.69 [0.94, 1.69] [1.13, 1.69]0.000 0.999(0) 1.000(1) 0.999(1) 1.000(2) 1.004(3) 1.000(2) -0.074 0.870(1) 0.863(2) 0.855(4) 0.852(5) 0.839(9) 0.834(7) 0.874(16)0.145 0.768(2) 0.755(3) 0.746(5) 0.746(6) 0.738(10) 0.721(9) 0.756(16)0.214 0.688(2) 0.671(4) 0.657(7) 0.665(9) 0.670(15) 0.638(11) 0.672(12)0.280 0.624(2) 0.609(4) 0.600(9) 0.588(12) 0.585(18) 0.570(15) 0.603(12)0.345 0.567(2) 0.552(3) 0.544(7) 0.535(10) 0.532(16) 0.508(12) 0.537(12)0.407 0.524(2) 0.506(4) 0.499(9) 0.476(10) 0.467(17) 0.446(14) 0.487(11)0.527 0.450(3) 0.429(6) 0.413(13) 0.430(21) 0.373(29) 0.362(24) 0.412(11)0.584 0.421(3) 0.408(6) 0.398(14) 0.387(19) 0.361(26) 0.356(23) 0.381(10)0.640 0.399(4) 0.385(7) 0.357(16) 0.369(26) 0.318(40) 0.308(29) 0.357(10)0.695 0.375(4) 0.358(8) 0.322(15) 0.321(20) 0.270(29) 0.247(26) 0.338(10)0.749 0.352(5) 0.339(12) 0.328(31) 0.336(93) NA NA 0.314(16)0.802 0.331(5) 0.302(10) 0.261(20) 0.306(41) 0.235(176) 0.189(40) 0.286(12)0.853 0.315(5) 0.304(11) 0.275(24) 0.265(31) 0.227(55) 0.239(42) 0.278(11)TABLE XI. Results for the connected contribution to the isoscalar G M ( Q ) using the plateau method for three sink-sourceseparations and the summation and two-state fit method fitted to all separations. Q [GeV ] Plateau, t ss