Strange nucleon electromagnetic form factors from lattice QCD
C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, A. Vaquero Aviles-Casco
SStrange nucleon electromagnetic form factors from lattice QCD
C. Alexandrou , , M. Constantinou , K. Hadjiyiannakou , K. Jansen ,C. Kallidonis , , G. Koutsou , and A. Vaquero Avil´es-Casco Department of Physics, University of Cyprus,P.O. Box 20537, 1678 Nicosia, Cyprus Computation-based Science and Technology Research Center,The Cyprus Institute, 20 Kavafi Str., Nicosia 2121, 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,Stony Brook University, 100 Nicolls Road,Stony Brook, NY 11794, USA Department of Physics and Astronomy,University of Utah, Salt Lake City, UT 84112, USA
We evaluate the strange nucleon electromagnetic form factors using an ensemble of gauge con-figurations generated with two degenerate maximally twisted mass clover-improved fermions withmass tuned to approximately reproduce the physical pion mass. In addition, we present results forthe disconnected light quark contributions to the nucleon electromagnetic form factors. Improvedstochastic methods are employed leading to high-precision results. The momentum dependence ofthe disconnected contributions is fitted using the model-independent z-expansion. We extract themagnetic moment and the electric and magnetic radii of the proton and neutron by including bothconnected and disconnected contributions. We find that the disconnected light quark contributionsto both electric and magnetic form factors are non-zero and at the few percent level as comparedto the connected. The strange form factors are also at the percent level but more noisy yieldingstatistical errors that are typically within one standard deviation from a zero value.
PACS numbers: 11.15.Ha, 12.38.Gc, 24.85.+p, 12.38.Aw, 12.38.-tKeywords: Nucleon structure, Electromagnetic form factors, Disconnected, Strangeness, Lattice QCD
I. INTRODUCTION
The electromagnetic form factors of the nucleon are im-portant quantities encapsulating information about thedistribution of electric charge and magnetism inside theproton and neutron. Namely, at zero momentum trans-fer, electromagnetic form factors yield the electric chargeand magnetic moment, while from the slope of the formfactors at zero momentum transfer one extracts the nu-cleon radii. Obtaining the individual quark contribu-tions is a major theoretical and experimental challenge,which can reveal insights on the partonic structure ofthe nucleon. In particular, the strange quark contri-bution, which is subdominant compared to the up anddown quark contributions, is especially challenging tomeasure. The interference between the weak and electro-magnetic amplitudes leads to a parity-violating asym-metry in the elastic scattering cross section for right-and left-handed electrons, which gives information onthe strange form factors. Measuring the parity-violatingelectroweak asymmetry in elastic scattering of polarizedelectrons from protons, the HAPPEX collaboration [1]extracted the linear combination of strange form factors G sE +0 . G sM = 0 . ± . ± .
010 at Q = 0 .
48 GeV which was found to be compatible with zero, where G sE is strange electric and G sM the strange magnetic protonform factor. The A4 experiment at MAMI [2] finds acombination G sE + 0 . G sM = 0 . ± .
034 at Q =0 .
23 GeV , slightly non-zero within errorbars, while theSAMPLE experiment [3] determined the strange mag-netic form factor G sM ( Q = 0 .
1) = 0 . ± . ± . Z exchange provides more recent estimatesfor the electric and magnetic form factors (see Refs. [4–6]for some recent experimental results). These studies alsodeliver results consistent with zero for the strange quarkcontribution, and as such, provide limits on the contri-bution of strange quarks in the distribution of nucleoncharge and magnetization.Lattice QCD allows for a first principles calculationof the nucleon form factors. In lattice QCD, the calcu-lation of the individual quark contributions to nucleonmatrix elements requires the so called disconnected con-tributions, such as the one shown in Fig. 1. A number oflattice QCD calculations exist for the isovector form fac-tors, or equivalently the combination ( G pE,M − G nE,M ),in which the disconnected contributions cancel in theisospin limit, as well as the isoscalar ( G pE,M + G nE,M ) com- a r X i v : . [ h e p - l a t ] J a n bination neglecting disconnected contributions, of theelectric and magnetic Sachs form factors using simula-tions with near-physical [7–10] and higher than physi-cal [11–13] pion masses. Disconnected contributions haveonly recently been calculated, typically using larger thanphysical pion masses [14]. In this study, we evaluate boththe light and strange disconnected quark loops to high-statistical precision using an ensemble of two- degeneratetwisted mass fermions with a clover term with quark masstuned to yield a pion mass of about 130 MeV [15]. Thedisconnected quark loops are estimated using improvedstochastic techniques for several momenta and the nu-cleon two-point correlation functions are computed us-ing a number of final momenta, allowing us to obtainthe form factors from multiple nucleon moving frames.We extract the magnetic moment, electric and magneticradii by fitting the momentum dependence of the formfactors to the model-independent z-expansion [16]. Weuse the connected contributions as calculated in Ref. [10]to obtain results for the total quark contributions to thenucleon electromagnetic form factors, present new resultsfor the strange quark contributions, and update the dis-connected contributions for the light quarks.The remainder of this paper is organized as follows:In Section II, we explain how we compute the nucleonmatrix element within lattice QCD and in Section IIIwe provide the technical details of the calculation of thedisconnected contributions, the analysis and results. InSection IV, a comparison with other studies is performedand in Section V we summarize and tabulate our findings. II. LATTICE EXTRACTION
The electromagnetic nucleon matrix element is decom-posed in terms of two parity preserving form factors,the Dirac ( F ) and Pauli ( F ) form factors, given inMinkowski space by, (cid:104) N ( p (cid:48) , s (cid:48) ) | j µ | 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) ) (cid:20) γ µ F ( q ) + iσ µν q ν m N F ( q ) (cid:21) u N ( p, s ) . (1) N ( p, s ) is the nucleon state with initial (final) momentum p ( p (cid:48) ) and spin s ( s (cid:48) ), with energy E N ( (cid:126)p ) ( E N ( (cid:126)p (cid:48) )) andmass m N . q = q µ q µ is the momentum transfer squaredwhere q µ = ( p (cid:48) µ − p µ ) and u N is the nucleon spinor. Thevector current j µ is given by j µ ( x ) = j lµ ( x ) + j sµ ( x ) (2)with j lµ ( x ) = e u ¯ u ( x ) γ µ u ( x ) + e d ¯ d ( x ) γ µ d ( x ) , (3)and j sµ ( x ) = e s ¯ s ( x ) γ µ s ( x ) , (4) where ( e u , e d , e s ) = (2 / , − / , − /
3) are the electriccharges carried by the up, down and strange quarksrespectively. In this study, we use the local vectorcurrent, therefore renormalization is necessary and hasbeen computed non-perturbatively using the RI (cid:48)
MOM scheme [17, 18]. Lattice artifacts have been evaluated inperturbation theory to 1-loop level and all orders in thelattice spacing and have been subtracted before takingthe chiral and continuum limits [19].The nucleon matrix element on the lattice requires theevaluation of three- and two-point correlation functions.The three-point function in momentum space is given by C µ (Γ ν , (cid:126)q, (cid:126)p (cid:48) ; t s , t ins , t ) = (cid:88) (cid:126)x ins ,(cid:126)x s e i ( (cid:126)x ins − (cid:126)x ) · (cid:126)q e − i ( (cid:126)x s − (cid:126)x ) · (cid:126)p (cid:48) × Tr (cid:2) Γ ν (cid:104) J ( t s , (cid:126)x s ) j µ ( t ins , (cid:126)x ins ) ¯ J ( t , (cid:126)x ) (cid:105) (cid:3) , (5)and the two-point function is given by C (Γ , (cid:126)p ; t s , t ) = (cid:88) (cid:126)x s Tr (cid:2) Γ (cid:104) J ( t s , (cid:126)x s ) ¯ J ( t , (cid:126)x ) (cid:105) (cid:3) × e − i ( (cid:126)x s − (cid:126)x ) · (cid:126)p , (6)where J N is the standard nucleon interpolating field: J N ( (cid:126)x, t ) = (cid:15) abc u a ( x )[ u b (cid:124) ( x ) Cγ d c ( x )] , (7)with C = γ γ the charge conjugation matrix and u and d are the up- and the down-quark fields respectively. Γ ν is aprojector acting on spin space, with Γ = γ projectingto unpolarized nucleons and Γ k = iγ γ k Γ projecting tonucleons polarized in direction k .The three-point function receives contributions fromboth quark- connected and disconnected terms. As men-tioned, the connected contributions have been evaluatedand presented in Ref. [10] for the same ensemble as theone used here, as have preliminary results for the discon-nected light quark contributions. In this work, we presenta thorough analysis of the disconnected contributions, de-picted in Fig. 1, updating our results for light quarks andshowing results on the strange quark contributions notcalculated previously. We use Osterwalder-Seiler strangequarks [20] and tune the strange quark mass to reproducethe experimental Ω − mass. This yields aµ s = 0 . a = 0 . m Rs = 108 . .
2) MeV.To isolate the electromagnetic matrix element in thethree-point function, an optimized combination of two-point functions is constructed to form the ratio, R µ (Γ ν , (cid:126)p (cid:48) , (cid:126)p ; t s , t ins ) = C µ (Γ ν , (cid:126)p (cid:48) , (cid:126)p ; t s , t ins ) C (Γ , (cid:126)p (cid:48) ; t s ) × (cid:115) C (Γ , (cid:126)p ; t s − t ins ) C (Γ , (cid:126)p (cid:48) ; t ins ) C (Γ , (cid:126)p (cid:48) ; t s ) C (Γ , (cid:126)p (cid:48) ; t s − t ins ) C (Γ , (cid:126)p ; t ins ) C (Γ , (cid:126)p ; t s ) . (8)In the large time limit, R µ (Γ ν ; (cid:126)p (cid:48) , (cid:126)p ; t s ; t ins ) t s − t ins →∞ −−−−−−−→ t ins →∞ Π µ (Γ ν ; (cid:126)p (cid:48) , (cid:126)p ) yielding a time independent plateau. Note j FIG. 1. Disconnected three-point nucleon correlation functionwith source at x and sink at x s with vector insertion j µ at x ins . that in Eq. (8), t s and t ins are relative to the source, t , which is omitted, and we will adopt this conven-tion for the remainder of this paper. When taking largetime separations to obtain Π µ (Γ ν ; (cid:126)p (cid:48) , (cid:126)p ), one cannot setthe source-sink time separation to arbitrarily large val-ues since the noise-to-signal ratio grows exponentially.Therefore, one seeks a window within which the source-sink separation is large enough for the excited states tobe suppressed while small enough to yield a good sig-nal. We employ Gaussian smearing [22, 23] to increasethe overlap with the ground state and apply APE smear-ing [24] to the gauge links, with the same parametersused in Ref. [10].The Dirac and Pauli form factors, F and F , are re-lated to the electric Sachs ( G E ( Q )) and magnetic Sachs( G M ( Q )) form factors via: G E ( Q ) = F ( Q ) − Q (2 m N ) F ( Q ) , (9) G M ( Q ) = F ( Q ) + F ( Q ) (10)where Q = − q is the Euclidean momentum trans-fer squared. The combination of the projector Γ ν , thecurrent insertion and the initial and final momenta (cid:126)p , (cid:126)p (cid:48) leads to an overconstrained set of equations relatingΠ µ (Γ ν ; (cid:126)p (cid:48) , (cid:126)p ) to G E and G M . We solve by using the Sin-gular Value Decomposition of the minimization problemthat arises. The expressions used are given in AppendixA. The same procedure has been followed for extract-ing the axial and induced pseudo-scalar form factors inRef. [25], where more details can be found. For the re-sults that follow, the analysis combines two values of thefinal momentum, namely (cid:126)p (cid:48) = (cid:126) (cid:126)p (cid:48) = πL (cid:126) ˆ n .In what follows we use two analysis methods to assessexcited states contamination and extract the matrix ele-ment of the nucleon. Plateau method:
For specific t s one identifies a range of t ins where the value of the ratio remains unchanged andperforms a constant fit. This procedure is repeated forseveral t s seeking for convergence in the matrix elementof the ground state. Summation method:
Summing over t ins in the ratio of Eq. (8) between the source and the sink gives, t s − a (cid:88) t ins = a R µ (Γ ν , (cid:126)p (cid:48) , (cid:126)p ; t s , t ins ) = C + t s M + O ( e − ∆ Et s ) (11)where ∆ E is the energy gap between the ground state andthe first excited state. The nucleon matrix element, M , isextracted from the slope by fitting to a linear form. Thesummation method will be used to provide an estimateof the systematic error due to potential contaminationfrom excited states.In Table I we summarize the parameters of the simu-lation. Details on the determination of the nucleon andpion mass and the lattice spacing are given in Ref. [21]. TABLE I. Simulation parameters. First row gives the β -value,the value of the clover parameter c SW , the lattice spacing andthe Sommer parameter r . β =2.1, c SW =1.57751, a =0.0938(3) fm, r /a =5.32(5)48 × L =4.5 fm aµ l =0.0009 m π =0.1304(4) GeV m π L =2.98(1) m N =0.932(4) GeV m N /m π =7.15(4) In Table II we tabulate the statistics used in thiswork. The disconnected quark loop entering the dia-gram of Fig. 1 cannot be computed exactly, except forvery small lattices. In this work, we employ stochastictechniques combined with the so-called one-end trick [26]and specifically its generalized version explained in de-tail in Refs. [25, 27, 28] to estimate the disconnectedquark loops. The light quark loops are produced usinghigh-precision inversions employing deflation of the lowmodes to overcome critical slowdown. For the compu-tation of strange quark loops we employ the truncatedsolver method (TSM) [29] to increase the statistics atlow cost. Details for the tuning procedure followed canbe found in Ref. [25]. Note that we do not use any kindof dilution, therefore we invert each noise vector once.
TABLE II. The statistics of our calculation. N conf is the num-ber of gauge configurations analyzed and N src is the numberof source positions per configuration. N HP r is the number ofhigh-precision stochastic vectors used, and N LP r is the numberof low-precision vectors used when employing the truncatedsolver method. Flavor N conf N HP r N LP r N src light 2120 2250 - 100strange 2057 63 1024 100 III. ANALYSIS AND RESULTS
We demonstrate the quality of our plateaus in Figs. 2and 3. The disconnected part of the three-point func-tion can be computed for all source-sink time separations.However, very large time separations are not useful dueto the increased statistical error. Thus, we restrict toanalyzing separations up to t s = 1 .
31 fm for which thesignal-to-noise ratio is acceptable. In Fig. 2 the ratioyielding G lE ( Q ) is shown. Note that the upper index“ l ” is used to denote the light quarks combination intro-duced in Eq. (3). For demonstration purposes we choosea representative momentum, namely Q = 0 . ,having (cid:126)p (cid:48) = (cid:126)
0. In Fig. 3 the ratio yielding G lM ( Q ) is ins t s /2) [fm]0.0050.0000.0050.0100.0150.020 R ( t i n s , t s ) → G l E ( Q ) t s = 0. 75 fmt s = 0. 94 fmt s = 1. 13 fm FIG. 2. Results for the ratio from which G lE ( Q ) is extracted.This is a representative example for Q = 0 . . Thesource-sink time separations are for t s = 0 .
75 fm (open redcircles), t s = 0 .
94 fm (open blue squares) and t s = 1 .
13 fm(open black stars). Results for the two larger separations areshifted slightly to the right for clarity. The gray band is theextracted value using the plateau method for t s = 1 .
13 fm,using t ins -values indicated by the length of the error band. presented. Fitting the form factors within the plateauregion for several separations allows us to check conver-gence to the ground state. The extracted results areshown in Fig. 4 including also the result from the summa-tion method obtained using the fit range [0.56-1.31] fm.For the case of G lE , results using the plateau method upto t s = 1 .
13 fm have a good agreement with the sum-mation method while larger separations become noisy.For G lM , the value increases in magnitude as t s increasesand becomes compatible with the summation method for t s = 1 .
13 fm. Therefore, we show final results extractedusing the plateau method at t s = 1 .
13 fm to which weperform our Q -fits in what follows. The same procedureis followed to extract the disconnected contributions tothe form factors at several Q values where the analysisis extended to allow for non-zero final nucleon momen-tum yielding a large number of closely spaced values for Q . To display the results we do a weighted average on results with close values of Q . In particular, we usebins with width of 0 .
02 GeV for the light disconnectedquark contributions and 0 .
04 GeV for the strange sincefor the latter we have results available up to higher Q compared to the light. A systematic error due to excitedstates contamination is given by the difference betweenthe plateau and the summation values. ins t s /2) [fm]0.070.060.050.040.030.020.010.000.01 R ( t i n s , t s ) → G l M ( Q ) t s = 0. 75 fmt s = 0. 94 fmt s = 1. 13 fm FIG. 3. Ratio leading to G lM ( Q ) for Q = 0 . . Thenotation is as in Fig. 2. G l E Summation Plateau s [fm]0.040.030.020.01 G l M FIG. 4. Extracted values for G lE and G lM at Q = 0 . using the plateau method (red points) and summationmethod (gray band). Open symbols show our chosen valuefrom the plateau method. The dipole form is widely used to fit the proton electricand magnetic form factors [30, 31] yielding the expectedbehavior in the large- Q region where the form factors areexpected to decrease like Q − [32]. The z-expansion [16,33] is a model independent Ansatz that has been appliedrecently to fit experimental results. Using a conformalmapping of Q to a variable z defined as, z = (cid:112) t cut + Q − √ t cut (cid:112) t cut + Q + √ t cut (12)one can expand the form factor into a polynomial G ( Q ) = k max (cid:88) k =0 a k z k , (13)where t cut is the cut in the time-like region of the formfactor. For light disconnected form factors t cut = (2 m π ) is used while for the strange t cut = (2 m K ) with m K the kaon mass. The z-expansion should converge as weincrease k max and the coefficients a k should be boundedin size for this to happen. The form factor at Q = 0 isobtained from the first coefficient, i.e. G ( Q = 0) = a .We define the radius as, r = − dG ( Q ) dQ (cid:12)(cid:12)(cid:12) Q =0 , (14)which is related to the second coefficient, via r = − a / t cut . In the case of the proton and neutron elec-tric form factors the mean square radius is the same asEq. (14), whereas for the magnetic, one has to dividewith the total value of the form factor at Q = 0.In our fitting procedure, the coefficients a , a are freeto vary, while for a k> we impose Gaussian priors forthe series to converge. The priors are imposed using anaugmented χ where the additional term is χ pr = k max (cid:88) k> ( a k − ˜ a k ) w a k (15)for parameter a k , which is centered at ˜ a k with width w a k .To compute ˜ a k we start by setting k max = 1 to obtain anestimate for a and a using jackknife ensemble averages.Then, for k max = 2, ˜ a is set to max( | a | , | a | ) and thewidth is chosen as w a k = 2 | ˜ a k | . This procedure is gener-alized for any k max and the priors are used to restrict a k inside the jackknife bins. In Fig. 5 we show two repre-sentative observables extracted from the electromagneticform factors using the z-expansion as a function of k max .We seek for convergence in both mean value and erroras we increase k max . In the case of the magnetic mo-ment µ l , increasing k max does not affect the result whilein the case of the radius ( r ) lE one needs up to k max = 3to converge. Therefore, we choose to use k max = 3 forall the extracted quantities where we have checked theconvergence of Eq. (13).In Fig. 6 we present the light quarks disconnected con-tribution to the nucleon electric form factor. The formfactor is shown up to Q ∼ . . The fits of theform factor yield a monotonically increasing dependenceon the Q that flattens out for Q > . . In thecase of G lE ( Q ) we impose a = 0. Fitting the results ex-tracted from the plateau method at t s = 1 .
13 fm, we find l max ( r ) l E [ f m ] FIG. 5. Extracted values for µ l and ( r ) lE as a function of k max , where results from the plateau method at t s = 1 .
13 fmhave been used. Q [GeV ] G l E ( Q ) t s = 0.75 fmt s = 0.94 fmt s = 1.13 fmSummation FIG. 6. Disconnected light quarks contribution to the nucleonelectric form factor denoted as G lE ( Q ). Results are extractedusing the plateau method for three source-sink time separa-tions with t s = 0 .
75 fm (red open circles), t s = 0 .
94 fm (blueopen squares) and t s = 1 .
13 fm (black open stars). Resultsusing the summation method in the fit range of [0.56-1.31] fmare depicted with the green open triangles. Results shownare obtained after a binning of neighboring Q values as ex-plained in the text. Results are shifted slightly to the rightfor clarity. The gray band is a fit to the results extracted fromthe plateau method using t s = 1 .
13 fm. a value for the radius ( r ) lE = − . , whereas us-ing the summation method we find ( r ) lE = − . . We assign a systematic error due to possible excitedstates from the difference between the values extractedusing the plateau and summation methods obtaining avalue for the electric squared charge radius( r ) lE = − . . (16)It is interesting to check how much the proton and neu-tron charge radii are affected by the disconnected contri-butions. Using results for the connected contributionsfrom Ref. [10], tabulated in Table III, we find that theconnected plus disconnected light quark contributions are( r ) pE (total) = 0 . , (17)( r ) nE (total) = − . . (18)Although the light disconnected contribution to the pro-ton charge radius is small, it is important to calculateaccurately enough when comparing to experiment, espe-cially in light of the discrepancy observed in the exper-imental value of proton charge radius between the con-ventional and the muonic hydrogen measurement. Forthe neutron, disconnected quark contributions are moreimportant making the value of the charge radius morenegative, albeit with large statistical errors.In Fig. 7 we show our results for G lM ( Q ), which asnoted above, shows a clear trend to decrease by increas-ing source-sink time separation, especially at small val-ues of Q . Fitting G lM ( Q ) using the z-expansion we find Q [GeV ] G l M ( Q ) t s = 0.75 fmt s = 0.94 fmt s = 1.13 fmSummation FIG. 7. Disconnected light quarks contribution to the nucleonmagnetic form factor G lM ( Q ). The notation is as in Fig. 6. that disconnected contributions to the nucleon magneticmoment and radius are µ l = − . , ( r ) lM = − . . In Fig. 8, we show results for thestrange nucleon electric form factor, which receives onlydisconnected contributions. We find that the strangecharge radius of the nucleon is( r ) sE = 0 . , (19)which is consistent with zero if one takes into account the systematic error due to the estimate of excited statecontributions.The strange magnetic form factor G sM ( Q ) is shown inFig. 9. We find a strange nucleon magnetic moment of µ s = 0 . . (20) Q [GeV ] G s E ( Q ) t s = 0.75 fmt s = 0.94 fmt s = 1.13 fmSummation FIG. 8. Strange nucleon electric form factor, G sE ( Q ). Thenotation is as in Fig. 6. The strange magnetic radius is ( r ) sM =0 . , consistent with zero, as expectedfrom the flat behavior of the form factor in Fig. 9. Ourresults for the proton and neutron magnetic momentsand radii are given in Table III. Q [GeV ] G s M ( Q ) t s = 0.75 fmt s = 0.94 fmt s = 1.13 fmSummation FIG. 9. Strange nucleon magnetic form factor, G sM ( Q ). Thenotation is as in Fig. 6. IV. COMPARISON WITH OTHER STUDIES
Disconnected quark loop contributions to the nucleonelectromagnetic form factors are available from two re- cent works beyond the current one. In Ref. [14], LHPC
TABLE III. Our final results for ( r ) E (first row), µ (middle row) and ( r ) M (last row). In the first and second columns wegive the light and strange disconnected contributions, in the third and fourth, the proton connected and total values and inthe fifth and sixth the corresponding ones for neutron. The radius is defined in Eq. (14). In the case of the magnetic radiusone has to divide with the total value of magnetic moment to extract the mean square radius. Results for the connected aretaken from Ref. [10].Quantity Disc. light Strange p (conn.) p (total) n (conn.) n (total)( r ) E [fm ] -0.022(9)(13) 0.0012(6)(7) 0.584(30)(28) 0.563(31)(31) -0.042(23)(6) -0.063(25)(14) µ -0.040(9)(3) 0.006(4)(1) 2.455(127)(155) 2.421(127)(155) -1.564(94)(123) -1.598(95)(123)( r ) M [fm ] -0.071(24)(4) 0.0014(27)(2) 1.284(183)(218) 1.214(185)(218) -0.875(139)(180) -0.945(141)(180) has analyzed an ensemble of N f = 2 + 1 Wilson clover-improved fermions simulated for heavier than physicalpion mass, namely m π = 317 MeV. The other study, from χ QCD, used valence overlap fermions on four N f = 2 + 1domain-wall fermion ensembles with pion masses in therange m π ∈ (135 , l (r ) l E [fm ] LHPC (2015), m =317 MeVQCD (2017), m ∈ (135, 403) MeV ETMC, m =130 MeV (this work) FIG. 10. Comparison of our results (blue star) for µ l withresults from LHPC (red circle) and χ QCD (green square) andfor ( r ) lE with χ QCD. We multiply by a factor of 1/3 theresults from LHPC to match our convention. The inner errorband is the statistical error, while the outer band is the totalerror.
In Fig. 10, we compare our result for µ l to the onefrom χ QCD, while for ( r ) lE to those from both χ QCDand LHPC. The dark, inner band indicates the statisti-cal error, while the outer band is the statistical and sys-tematic error added in quadrature. The good agreementwith χ QCD, for which a continuum and infinite volumeextrapolation has been performed, indicates that latticeartifacts due to finite lattice spacing and volume on thesequantities are small for our ensemble. On the other hand,the result for µ l from LHPC at higher than physical pionmass is smaller, as expected from chiral perturbation the-ory arguments. In Fig. 11 we compare the strange µ s and( r ) sE with the corresponding results from the two otherstudies. For ( r ) sE , results from the three studies are ingood agreement, whereas for µ s , the result from χ QCDagrees within one standard deviation. Given the largestatistical errors on the strange quark contributions suchan agreement among lattice QCD results is welcoming. s (r ) sE [fm ] LHPC (2015), m =317 MeVQCD (2017), m ∈ (135, 403) MeV ETMC, m =130 MeV (this work) FIG. 11. Comparison of our results (blue star) for ( r ) sE and µ s with results from LHPC (red circles) and χ QCD (greensquare). The convention is as in Fig. 10.
V. CONCLUSIONS
In this study, we compute the disconnected quarkloop contributions from up, down and strange quarks tothe nucleon electromagnetic form factors using N f = 2maximally twisted mass fermions at the physical point.While all source-sink time separations accessible, we optto use up to t s = 1 .
31 fm for which statistical errorsare not prohibitively large. Both the plateau and thesummation methods are employed to estimate contam-ination due to the excited states. Three-point func-tions produced with final nucleon momenta of (cid:126)p (cid:48) = (cid:126) (cid:126)p (cid:48) = πL (cid:126) ˆ n and analyzed to increase statistics. Theform factors, G lE ( Q ) and G lM ( Q ), are computed up to Q (cid:39) . while G sE ( Q ) and G sM ( Q ) are com-puted up to Q (cid:39) . . The model independentz-expansion is used to fit the Q dependence of the formfactors and extract the electric and magnetic radii as wellas the magnetic moment. The size of the individual con-tributions as well as the total values for the extractedquantities are tabulated in Table III. While the contri-bution of the light quark-disconnected diagram is clearlynon-zero, strange quark contributions are almost consis-tent with zero within the current errors.We plan to analyze an N f = 2 + 1 + 1 twisted massensemble with a clover term at the physical point tocheck possible quenching effects of the strange and charmquarks in the sea. Further improvements for the compu-tation of disconnected quark loops are under investiga-tion to improve the accuracy of the disconnected loopdetermination. Acknowledgments:
We would like to thank the mem-bers of the ETM Collaboration for a productive collab-oration. We acknowledge funding from the EuropeanUnion’s Horizon 2020 research and innovation programunder the Marie Sklodowska-Curie Grant Agreement No.642069.We gratefully acknowledge the Gauss Centre forSupercomputing e.V. ( ) for fund-ing this project by providing computing time on the GCSSupercomputer SuperMUC at Leibniz SupercomputingCentre ( ). Results were obtained using PizDaint at Centro Svizzero di Calcolo Scientifico (CSCS),via projects with ids s540, s625 and s702. We thank thestaff of CSCS for access to the computational resourcesand for their constant support as well as the J¨ulich Su-percomputing Centre (JSC) for the tape storage. MCacknowledged financial support by the US National Sci-ence Foundation under Grant No. PHY-1714407. [1] K. A. Aniol et al. Parity violating electroweak asymmetryin polarized-e p scattering.
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Appendix A: Extraction of form factors from lattice QCD ratios
In this Appendix we generalize the equations from which the form factors are extracted for a nucleon with non-zerofinal momentum (cid:126)p (cid:48) . All expressions are given in Euclidean space.Π µ (Γ , (cid:126)p (cid:48) , (cid:126)p )) = − iG E ( Q ) C m (4 m + Q ) (cid:16) ( p (cid:48) µ + p µ ) (cid:104) m (cid:16) E ( (cid:126)p (cid:48) ) + E ( (cid:126)p ) + m (cid:17) − p (cid:48) ρ p ρ (cid:105)(cid:17) + G M ( Q ) C m (4 m + Q ) (cid:16) δ µ (cid:0) m + m Q + 4 m p (cid:48) ρ p ρ + Q p (cid:48) ρ p ρ (cid:1) +2 im p (cid:48) µ (cid:0) E ( (cid:126)p (cid:48) ) − E ( (cid:126)p ) (cid:1) − im ( p (cid:48) µ + p µ ) − E ( (cid:126)p ) iQ p (cid:48) µ − E ( (cid:126)p (cid:48) ) iQ p µ − imQ ( p (cid:48) µ + p µ ) − im p µ (cid:0) E ( (cid:126)p (cid:48) ) − E ( (cid:126)p ) (cid:1) − imp (cid:48) ρ p ρ ( p (cid:48) µ + p µ ) (cid:17) ; (A1)Π µ (Γ k , (cid:126)p (cid:48) , (cid:126)p )) = − G E ( Q ) C m (4 m + Q ) (cid:16) m ε µk ρ ( p (cid:48) ρ − p ρ ) − iε µkρσ p (cid:48) ρ p σ (cid:0) E ( (cid:126)p (cid:48) ) + E ( (cid:126)p ) (cid:1) + ε µ ρσ p (cid:48) ρ p σ ( p (cid:48) k + p k ) − ε µk ρ p (cid:48) σ p σ ( p (cid:48) ρ − p ρ ) (cid:17) − G M ( Q ) C m (4 m + Q ) (cid:16) mε µk ρ ( p (cid:48) ρ − p ρ )(2 m + Q )+2 imε µkρσ p (cid:48) ρ p σ (cid:0) m + E ( (cid:126)p (cid:48) ) + E ( (cid:126)p ) + Q m (cid:1) − mε µ ρσ p (cid:48) ρ p σ ( p (cid:48) k + p k ) + 2 mε µk ρ p (cid:48) σ p σ ( p (cid:48) ρ − p ρ ) (cid:17) , (A2)where C = 2 mE ( (cid:126)p )( E ( (cid:126)p (cid:48) ) + m ) (cid:115) E ( (cid:126)p )( E ( (cid:126)p (cid:48) ) + m ) E ( (cid:126)p (cid:48) )( E ( (cid:126)p ) + m ) ..