Chiral extrapolations for nucleon electric charge radii
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Chiral extrapolations for nucleon electric charge radii
J. M. M. Hall, D. B. Leinweber, and R. D. Young Special Research Centre for the Subatomic Structure of Matter (CSSM),School of Chemistry and Physics, University of Adelaide, Adelaide, South Australia 5005, Australia ARC Centre of Excellence for Particle Physics at the Terascale, School of Chemistry and Physics,University of Adelaide, Adelaide, South Australia 5005, Australia
Lattice simulations for the electromagnetic form factors of the nucleon yield insights into the internal structureof hadrons. The logarithmic divergence of the charge radius in the chiral limit poses an interesting challengein achieving reliable predictions from finite-volume lattice simulations. Recent results near the physical pionmass ( m π ∼
180 MeV ) are examined in order to confront the issue of how the chiral regime is approached.The electric charge radius of the nucleon isovector presents a forum for achieving consistent finite-volumecorrections. Newly developed techniques within the framework of chiral effective field theory ( χ EFT) areused to achieve a robust extrapolation of the electric charge radius to the physical pion mass, and to infinitevolume. The chiral extrapolations exhibit considerable finite-volume dependence; lattice box sizes of L & are required in order to achieve a direct lattice simulation result within of the infinite-volume value at thephysical point. Predictions of the volume dependence are provided to guide the interpretation of future latticeresults. PACS numbers: 12.38.Gc 12.38.Aw 12.39.Fe 13.40.Em
I. INTRODUCTION
Much experimental progress has been made [1–5] in exam-ining the internal structure of hadrons, particularly with regardto the internal distribution of electric and magnetic charge dueto quarks. Current understanding of the internal charge distri-bution, characterized by the elastic form factors, is also for-tified by developments in supercomputing power and latticeQCD techniques. Lattice QCD has seen significant advancesin simulating electromagnetic form factors, and is now able toprobe the chiral regime [6–9].Recent results from the QCDSF Collaboration, using pionmasses of order ∼
180 MeV [10], provide a new opportu-nity for exploring the utility of chiral effective field theory( χ EFT)-based techniques in performing an extrapolation tothe physical point. Additional care must be taken in handlingfinite-volume effects relating to the electric charge radius [11–14]. In order to address this issue, a variety of
Ans¨atze for the Q behaviour of the form factor are examined in order to con-struct a finite-volume analogue. The finite-volume correctionsare applied directly to the electric form factors, and the electriccharge radii are then calculated at infinite volume. By com-bining these methods with new techniques within the frame-work of χ EFT, a robust extrapolation to the physical regimeis performed herein.In performing a chiral extrapolation, one should ideallyuse lattice simulation results that lie within the chiral power-counting regime (PCR) of chiral perturbation theory in orderto avoid a regularization scheme-dependent result. The PCRis defined by the range of quark (or pion) masses at which a χ EFT calculation is independent of the regularization scheme,and typically lies in a pion-mass range of . MeV [15–17]. Within the PCR, the chiral expansion of an observableis a controlled expansion, and the result is insensitive to treat-ments of higher-order terms, such as the resummation of thechiral series. Since lattice QCD results usually extend out-side the PCR, one is restricted by the available data when per- forming an extrapolation. An important application of finite-range regularization (FRR) is the ability to extrapolate usinglattice QCD results that extend beyond the PCR. One methodfor achieving this involves identifying a preferred regulariza-tion scale and an upper bound of the pion mass directly fromthe lattice QCD results, as demonstrated in Refs. [18, 19].In a previous investigation, a successful extrapolation of themagnetic moment of the nucleon to the physical point wasachieved using these techniques [20]. This analysis similarlyprovides a prediction of the pion-mass dependence of the elec-tric charge radius of the nucleon for a range of lattice volumes.The lattice QCD results from the QCDSF Collaboration[10] used in this analysis are displayed in Fig. 1. The simula-tion used a two-flavor O ( a ) -improved Wilson quark action,and the isovector nucleon ( p − n ) was calculated to avoidthe computational cost of disconnected loops that occur infull QCD. Only the simulation results that satisfy the crite-ria: L > . fm and m π L > , are shown. Of the nine pointsthat satisfy these criteria, the lattice size varies from . to . fm. The QCDSF results are displayed using a Sommer scaleparameter of r = 0 . fm, based on results from Ref. [21].Without consideration of chiral loop contributions, it is clearthat there would be a factor of two discrepancy between thelattice QCD simulations and the experimental value [22, 23]as shown by a linear trend line. II. CHIRAL EFFECTIVE FIELD THEORYA. Electromagnetic form factors
It is common to define the Sachs electromagnetic form fac-tors G E,M , which parametrize the matrix element for thequark current J µ . In the heavy-baryon limit, this can be writ- FIG. 1: (color online). Lattice QCD results for h r i E from QCDSF [10],using the Ansatz from Eq. (31), and the experimental value as marked [22,23]. The lattice results satisfy: L > . fm and m π L > . A na¨ıve lineartrend line is also included, which does not reach the experimental value. Thephysical point is shown with a vertical dotted line. ten as h B ( p ′ ) | J µ | B ( p ) i = ¯ u s ′ ( p ′ ) n v µ G E ( Q )+ iǫ µνρσ v ρ S σ v q ν m B G M ( Q ) o u s ( p ) , (1)where Q is defined as positive momentum transfer Q = − q = − ( p ′ − p ) . Lattice QCD results are often constructedfrom an alternative representation, using the form factors F and F , the Dirac and Pauli form factors, respectively. TheSachs form factors are simply linear combinations of F and F , G E ( Q ) = F ( Q ) − Q m B F ( Q ) , (2) G M ( Q ) = F ( Q ) + F ( Q ) . (3)In the heavy-baryon formulation of the quark current ma-trix element shown in Eq. (1), the spin operator S µ v = − γ [ γ µ , γ ν ]v ν is required. It has the useful properties in thatits commutation and anticommutation rules depend only onthe four-velocity of the baryon v µ [24, 25]. The momentum-dependent electric form factor G E ( Q ) allows a charge radiusto be defined in the usual manner, h r i E = lim Q → − ∂G E ( Q ) ∂Q . (4)For the leading-order contributions to the electric formfactor, the following first-order interaction Lagrangian fromheavy-baryon chiral perturbation theory ( χ PT) is used [24–29], L (1) χP T = 2 D Tr [ ¯ B v S µ v { A µ , B v } ] + 2 F Tr [ ¯ B v S µ v [ A µ , B v ] ]+ C ( ¯ T µ v A µ B v + ¯ B v A µ T µ v ) . (5)The pseudo-Goldstone fields ξ ( x ) are encoded in the adjointrepresentation of SU(3) L ⊗ SU(3) R , forming an axial vector combination, denoted A µ , ξ ≡ exp (cid:26) if π τ a π a (cid:27) , (6) A µ = 12 ( ξ ∂ µ ξ † − ξ † ∂ µ ξ ) . (7)The values for the D , F and C couplings in the interactionLagrangian are related through SU(6) flavor-symmetry [25,30], F = D and C = − D . Phenomenological values of theconstants D = 0 . and f π = 92 . MeV are used.
B. Finite-range regularization
In FRR χ EFT, a regulator function u ( k ; Λ) , with charac-teristic momentum scale Λ , is introduced in the numerators ofthe loop integrals. The regulators should be chosen such thatthey satisfy u | k =0 = 1 and u | k →∞ = 0 . The result of an FRRcalculation is independent of the choice of u ( k ; Λ) if the lat-tice simulation points are constrained entirely within the PCR.In this investigation, a dipole form is chosen, which takes thefollowing form, u ( k ; Λ) = (cid:18) k Λ (cid:19) − . (8)While conventional χ PT fails outside the PCR, FRR χ EFTremains effective, as the regulator takes on an additional rolein modelling the effect of higher-order terms in the expansion.Analyses have been undertaken previously for a range of pos-sible forms of regulator function [17, 18].
C. Loop integrals and definitions
The leading-order loop integral contributions to the electricform factor correspond to the diagrams in Figs. 2 through 4.The electric charge radius itself is also renormalized by con-tributions from loop integrals, obtained from χ EFT. The loopintegrals can be simplified to a convenient form by taking theheavy-baryon limit, and performing the pole integrations for k , T EN ( Q ) = − χ EN π Z d k ( k − ~k · ~q ) u ( ~k ; Λ) u ( ~k − ~q ; Λ) ω ~k ω ~k − ~q ( ω ~k + ω ~k − ~q ) , (9) T E ∆ ( Q ) = − χ E ∆ π Z d k ( k − ~k · ~q ) u ( ~k ; Λ) u ( ~k − ~q ; Λ)( ω ~k + ∆)( ω ~k − ~q + ∆)( ω ~k + ω ~k − ~q ) , (10) T E tad ( Q ) = − χ Et π Z d k u ( ~k ; Λ) ω ~k + ω ~k − ~q , (11)where ω ~k = q ~k + m π and ∆ is the baryon mass splitting.The coefficients χ EN , χ E ∆ and χ Et , for both proton ( p ) andneutron ( n ), are related to the constants D , F , C and f π from FIG. 2:
The pion loop contributions to the electric charge radius of a nucleon.All charge conserving pion-nucleon transitions are implicit.
FIG. 3:
The pion loop contribution to the electric charge radius of a nucleon,allowing transitions to the nearby and strongly-coupled ∆ baryons. FIG. 4:
The tadpole contribution at O ( m q ) to the electric charge radius of anucleon. the chiral Lagrangian in Eq. (5), χ E,pN = − π f π ( D + F ) = − χ E,nN , (12) χ E,p ∆ = + 516 π f π C − χ E,n ∆ , (13) χ E,pt = − π f π = − χ E,nt . (14) D. Finite-volume corrections
Finite-volume corrections cannot be applied directly to thecharge radius itself [14]. Instead, the electric form factors G E ( Q ) are corrected to infinite-volume. To obtain the in-tegrals T E that contribute to the electric charge radius, onetakes the derivative of T E with respect to momentum transfer ~q , as ~q → , T E = lim ~q → − ∂ T E ( ~q ) ∂~q , (15)which is equivalent to the derivative in Eq. (4) in the Breitframe, defined by q = (0 , ~q ) .The finite-volume corrections to the electric form factors are achieved by subtracting the electric charge symmetry-preserving finite-volume correction, defined as ∆ L ( Q ,
0) = δ L (cid:2) T E ( Q ) (cid:3) − δ L (cid:2) T E (0) (cid:3) . (16)The functional δ L is defined through the convention [31]: δ L [ T E ( Q )] = χ (2 π ) L x L y L z X k x ,k y ,k z − Z d k I E ( Q ) , (17)for an integrand I E . The second term of Eq. (16) ensures thatboth infinite- and finite-volume electric form factors are cor-rectly normalized, i.e. G E (0) = 1 . This normalization pro-cedure exploits the lattice Ward Identity that ensures chargeconservation is satisfied in a finite volume. It has been shownpreviously that this is realised in practice; numerically andthrough χ EFT analyses [14]. Thus, the infinite-volume elec-tric form factor can be calculated using the equation: G ∞ E ( Q ) = G LE ( Q ) − ∆ L ( Q , . (18)The infinite-volume charge radius h r i ∞ E can be recoveredfrom the form factor by choosing an Ansatz for the extrap-olation in Q , analogous to the procedure typically performedat finite volume.In applying FRR to the finite-volume corrections, the valueof ∆ L ( Q , stabilises as Λ becomes large. Applying thesame technique as in Ref [18], the asymptotic result of ∆ L ( Q , is achieved numerically by evaluating it with adipole regulator, using a relatively large value of Λ ′ = 2 . GeV. This method is similar to the algebraic approach out-lined in Ref. [32], and has been successfully demonstrated inprevious studies [33].
E. Renormalization
The procedure for the renormalization of the low-energycoefficients of the chiral expansion in FRR χ EFT will now beoutlined. A thorough discussion can be found in Ref. [18].Each loop integral contributing to the electric charge radiusmay be expanded out as an analytic polynomial plus a nonan-alytic term, T EN ( m π ; Λ) = b Λ ,N + χ EN log m π µ + b Λ ,N m π + O ( m π ) , (19) T E ∆ ( m π ; Λ) = b Λ , ∆0 + b Λ , ∆2 m π + χ E ∆ m π log m π µ + O ( m π ) , (20) T E tad ( m π ; Λ) = b Λ ,t + χ Et log m π µ + b Λ ,t m π + O ( m π ) , (21)where µ is a mass scale associated with the chiral logarithm.Once the lattice results have been converted into infinite-volume charge radii, the chiral behaviour of the electric chargeradius can be written in terms of an ordered expansion in pion-mass squared, through use of the Gell-Mann − Oakes − RennerRelation, m q ∝ m π [34], h r i ∞ E = { a Λ0 + a Λ2 m π } + T EN ( m π ; Λ) + T E ∆ ( m π ; Λ)+ T E tad ( m π ; Λ) + O ( m π ) . (22)This expansion contains an analytic polynomial in m π plusthe leading-order chiral loop integrals, from which nonana-lytic behaviour arises. The scale-dependent coefficients a Λ i are the residual series coefficients, which correspond to directquark-mass insertions in the full Lagrangian. Upon renormal-ization of the divergent loop integrals, these will correspondwith low-energy coefficients of χ EFT [35].In order to obtain the renormalized chiral coefficients, c i ,one must add the b Λ i terms from each of the loop integrals tothe residual series coefficients a Λ i , c = a Λ0 + b Λ ,N + b Λ , ∆0 + b Λ ,t , (23) c = a Λ2 + b Λ ,N + b Λ , ∆2 + b Λ ,t . (24)The resultant coefficients, c and c , are the renormalized low-energy coefficients of the chiral expansion at the scale, µ . Byevaluating the loop integrals, the renormalized chiral expan-sion can also be written in terms of a polynomial in m π andthe nonanalytic terms, h r i ∞ E = c ( µ )0 + ( χ EN + χ Et ) log m π µ + c m π + χ E ∆ m π log m π µ + O ( m π ) , (25)reproducing χ PT in the PCR. Since the chiral expansion ofEq. (25) contains a logarithm, the value of c can only be ex-tracted relative to the mass scale, µ , which is chosen to be GeV in this case.To achieve a chiral extrapolation, it is convenient to subtractthe b Λ0 coefficients from the respective loop integrals, thus au-tomating the renormalization procedure to chiral order O (1) , ˜ T EN = T EN − b Λ ,N , (26) ˜ T E ∆ = T E ∆ − b Λ , ∆0 , (27) ˜ T E tad = T E tad − b Λ ,t . (28)This removes the dependence on the regularization scale Λ inthe leading low-energy coefficient. Thus, the chiral formulaused for fitting lattice QCD results takes the form: h r i ∞ E = { c ( µ )0 + a Λ2 m π } + ˜ T EN + ˜ T E ∆ + ˜ T E tad + O ( m π ) . (29)To ascertain the presence of an optimal regularization scale Λ scale , the renormalization flow of the leading low-energy co-efficient c ( µ )0 will be considered in Sec. III B, using the pre-scription detailed in Refs. [18–20]. III. RESULTSA. Q extrapolation In extracting an electric charge radius from typical latticeQCD results on periodic volumes, one must choose an
Ansatz to model the finite-volume corrected Q behaviour of the elec-tric form factor. A common choice is the dipole form, definedby G E ( Q ) = G E (0)(1 + Q / Λ D ) , (30)where the dipole mass Λ D is a free parameter, related to theelectric charge radius by Λ D = 12 / h r i E . This Ansatz tightlyconstrains the Q dependence and leads to small errors inthe radius h r i E compared with other Ans¨atze . These dipole-constrained radii are shown in Fig. 5.A modification may be made to account for higher orderterms in Q . An inverse quadratic with two fit parameters, asinspired by Kelly [36], may be chosen. This form is used inthe analysis by the QCDSF Collaboration [10], G E ( Q ) = G E (0)1 + αQ + βQ . (31)The charge radius is obtained through h r i E = 6 α . This Ansatz was originally chosen for modelling the large Q be-haviour of F [10, 36]. However, it is of greater interest hereto examine and compare the small Q behaviour of this Ansatz with that of the dipole. Furthermore, this will provide a guideto the expected variation in h r i E due to the choice of Ansatz .A demonstration of an infinite-volume chiral extrapolationusing each
Ansatz is shown in Figs. 5 and 6. In each case,the smallest three Q values available are considered in fit-ting the Ansatz parameters. For illustrative purposes, FRR isperformed with a dipole regulator with
Λ = 1 . GeV. A di-rect comparison of the finite-volume-corrected lattice valuesof h r i E , using the dipole Ansatz from Eq. (30), and the vari-ant
Ansatz from Eq. (31), is shown in Fig. 7.In Fig. 5, the estimate of the uncertainty in h r i E is muchsmaller than for the other Ansatz , raising concerns of an unac-counted for systematic uncertainty. The electric charge radiiobtained using the variant
Ansatz from Eq. (31), as shown inFig. 6, appear to be the more cautious, in that the error barencompasses a range of variation from the choice of
Ansatz .This can be seen most clearly in Fig. 7.In order to assess the low Q behaviour of the variant Ansatz in Eq. (31), a comparison of the Q extrapolation us-ing this Ansatz is shown in Fig. 8 at the point: m π = 0 . GeV . Both Q extrapolations are plotted on the same axes.The lightest three values of Q are used in constraining theparameters. The merit of the extra fit parameter in Eq. (31) isevident. B. Renormalization flow analysis
The QCDSF results for the electric charge radius, displayedin Fig. 1, include a linear extrapolation, which does not take
FIG. 5: (color online). Infinite-volume chiral extrapolation of h r i E , us-ing the dipole Q extrapolation Ansatz from Eq. (30). The infinite-volumecorrected lattice points are also shown.
FIG. 6: (color online). Infinite-volume chiral extrapolation of h r i E , usingthe variant Q extrapolation Ansatz from Eq. (31). into account the nonanalytic behaviour of the chiral loop in-tegrals, nor the finite-volume corrections. Neglecting theseimportant effects [37], it is not surprising that the linear trendline does not approach the experimental value of the electriccharge radius at the physical pion mass. Since these latticeQCD results extend outside the PCR, the result of an extrap-olation will be regularization scale dependent. However, thescale dependence may be constrained using a procedure [18–20] that obtains an optimal regularization scale, and an esti-mate of its uncertainty, as constrained by the lattice results.In order to obtain an optimal regularization scale, the low-energy coefficient, c ( µ )0 from Eq. (29), will be calculatedacross a range of values of the regularization scale, Λ . Multi-ple renormalization flow curves may be obtained by constrain-ing the fit window by a maximum value, m π, max , and sequen-tially adding points to extend further outside the PCR. Therenormalization flow curves for a dipole regulator are plottedon the same set of axes in Fig. 9. Within the PCR, c will beinsensitive to the value of Λ , and appear as a horizontal line inFig. 9. In contrast, variation of c with respect to Λ becomes FIG. 7: (color online). A comparison of the infinite-volume chiral extrapo-lations of h r i E using the dipole Q extrapolation Ansatz from Eq. (30), andthe variant Q extrapolation Ansatz from Eq. (31).
FIG. 8: (color online). A comparison of the Q extrapolation of the electricform factor G E , using the normal dipole Ansatz from Eq. (30), and the variant
Ansatz , defined in Eq. (31). The smallest three values of Q are used (thesmallest two being almost coincident). The fits are shown for m π = 0 . GeV . Error bands are shown with dotted lines. larger as one moves further from the PCR. The correct valueof c , and thus the optimal value for Λ , is identified by theintersections of the curves, where their deviation is minimal[18].Unlike the results from the analysis of the nucleon mass[18] or magnetic moment [20], the regularization scale-dependence is relatively weak for Λ > . GeV. There is nodistinct intersection point in the renormalization flow curves.This lack of sensitivity to the regularization scale is a conse-quence of the logarithm in the chiral expansion of Eq. (25),which is slowly-varying with respect to Λ .An optimal regularization scale for the dipole regulator canbe obtained using a χ dof analysis, taking the degrees of free-dom to be the curves of c corresponding to different valuesof m π, max . For the six different values of m π, max consideredin Fig. 9, each curve is described by c i (Λ) , where i takes val-ues through . δc i (Λ) denotes the uncertainty in c i obtainedwhen fitting the lattice results. The χ dof for each value of Λ is expressed as χ dof (Λ) = 1 n − n X i =1 ( c i (Λ) − ¯ c (Λ)) ( δc i (Λ)) , (32) ¯ c (Λ) = P ni =1 c i (Λ) / ( δc i (Λ)) P nj =1 / ( δc j (Λ)) , (33)with the statistically weighted average ¯ c (Λ) given byEq. (33). The χ dof is illustrated in Fig. 10. The valueof the optimal scale, obtained using a dipole regulator, is Λ scaledip = 1 . +0 . − . GeV, which is consistent with the optimalregularization scale values obtained for the nucleon mass us-ing a dipole regulator [18]. The value is also consistent withthe result obtained for the nucleon magnetic moment, basedon these QCDSF simulations [20]. This provides evidencethat, for a given functional form of the regulator, the optimalregularization scale may be associated with an intrinsic scale,characterizing the finite size of the nucleon.
C. Chiral extrapolations
The identification of an optimal regularization scale allowsan accurate chiral extrapolation to be performed. Further-more, a range of box sizes may be considered, thus providingan estimate of the finite-volume effects. In order to determinethe most suitable number of points to be used for fitting thelattice results, the method described in Refs. [19, 20] is used.In extrapolating the electric charge radius, the statistical un-certainty comprises contributions from the fit coefficients. Inthe case of the systematic uncertainty, the axial coupling andthe pion decay constant are assumed to be sufficiently well-determined experimentally. Thus, the dominant contributionto the systematic uncertainty in the extrapolation is associatedwith the optimal regularization scale.A second source of systematic uncertainty is due to thechoice of the regulator functional form, which is combinedin quadrature, (cid:16) δ h r i sys E (cid:17) = (cid:16) δ h r i Λ E (cid:17) + (cid:16) δ h r i reg E (cid:17) . (34) δ h r i reg E is obtained by comparing the result of a dipole reg-ulator to that of using a sharp cutoff, which has an intrinsicscale of Λ scalesc = 0 . +0 . − . GeV, determined using the samemethods described for the dipole regulator. The systematicuncertainty is taken as half the difference between the cen-tral values of each case. Though the sharp cutoff regulatordoes not provide higher-order nonanalytic contributions in thechiral expansion [38] and is less physical than the dipole reg-ulator, a comparison between the two regulators provides themost cautious evaluation of the dependence of the result onthe functional form of the regulator.The value of the extrapolation of h r i E to the physical pointis shown in Fig. 11 for different values of m π, max . Statisticaland systematic errors have been added in quadrature. Fig. 12shows the magnitude of the statistical and systematic error FIG. 9: (color online). The renormalization flow of c ( µ )0 , obtained using adipole regulator, and based on QCDSF simulation results. c ( µ )0 is calculatedrelative to the mass scale, µ = 1 GeV. For each curve, two arbitrary values of Λ are chosen to indicate the general size of the error bars. FIG. 10: (color online). A χ dof analysis for the renormalization flow of c ( µ )0 , obtained using a dipole regulator, and based on QCDSF simulation re-sults. The dotted line illustrates the upper limit: ( χ + 1) /dof . bars separately, in addition to the total uncertainty. Theseplots allow the identification of the optimal number of latticeresults to be used for an extrapolation, which is signified bythe best compromise between the statistical and the systematicuncertainties. Figure 12 indicates that, in this case, the lightestseven lattice points should be used, corresponding to a valueof m π, max ≃ . GeV . Table I summarizes the breakdownof each error bar into its source components.Note that there is a discrepancy between the experimentalvalue and the extrapolation results. This could be a conse-quence of excited state contamination in the lattice calcula-tion of the three-point function; the use of only two flavours,and/or neglecting O ( a ) effects. D. Finite-volume effects in future lattice simulations
To predict the finite-volume dependence of future latticesimulations, consider again the electric charge radii from thelattice, corrected to infinite volume obtained using the variant
TABLE I:
Results for the isovector nucleon electric charge radius, extrapolated to the physical point using different values of m π, max , as illustrated in Fig. 11.The uncertainty in h r i E ( m π, phys ) is provided in the following order: the statistical uncertainty, the uncertainty due to Λ scale , the uncertainty due to thechange in regulator functional form, and the total uncertainty, respectively. The value of Λ scale is calculated for each choice of regulator functional form. m π, max (GeV ) h r i E ( m π, phys ) (fm ) δ h r i stat E δ h r i Λ E δ h r i reg E δ h r i tot E .
205 0 .
705 0 .
055 0 .
017 0 .
006 0 . .
255 0 .
731 0 .
042 0 .
019 0 .
007 0 . .
266 0 .
755 0 .
040 0 .
020 0 .
007 0 . .
483 0 .
753 0 .
028 0 .
031 0 .
008 0 . .
497 0 .
751 0 .
028 0 .
032 0 .
008 0 . .
898 0 .
746 0 .
019 0 .
051 0 .
007 0 . FIG. 11: (color online). Behaviour of the extrapolation of h r i E to thephysical point, vs. m π, max . The value of Λ scale is used, as obtained fromthe χ dof analysis. The error bars include the statistical and systematic uncer-tainties added in quadrature. FIG. 12: (color online). Magnitude of the statistical, systematic and totalerror bars in the extrapolation of h r i E to the physical point, vs. m π, max .In each case of regulator, the value of Λ scale is used, as obtained from thecorresponding χ dof analysis. At a maximum pion mass of ˆ m π, max = 0 . GeV , the best compromise between statistical and systematic uncertainty isachieved. Ansatz defined in Eq. (31). As shown in Fig. 13, this time theexperimental value is included in the fit, and the chiral extrap-olation at infinite volume is shown.With the fit parameters determined, extrapolations at a vari-ety of finite volumes are shown in Fig. 14. The extrapolationsuse the lightest seven data points, and are only calculated forvalues m π L > , as in the initial selection of the lattice simu-lation results. These finite-volume results allow comparisonswith current lattice simulations, and also allow estimates offinite-volume effects at arbitrary box sizes to be made. Forexample, using a box size of L ∼ fm, a significant deviationfrom the infinite-volume limit is observed. In this case, thefinite-volume radius is h r i nuc − isov E = 0 . fm; significantlybelow the physical value of . fm used to constrain the fit.In addition, the finite-volume extrapolations can provide abenchmark for lattice QCD simulations at large and currentlyuntested box sizes. The extrapolation curves indicate that abox length of L & is required to achieve an extrapolationwithin of the infinite-volume result.This extrapolation method may be used to provide specificpredictions for the charge radius based on lattice configura-tions from the PACS-CS Collaboration [39], freely availablevia the International Lattice Data Grid (ILDG). By choosingthe lattice volume and the m π values to match the PACS-CSdata, an estimate of the expected charge radii to be observedin future lattice simulations are shown in Fig. 15. It is note-worthy that the predicted values of the charge radius near thephysical point do not approach the experimental point at thePACS-CS lattice volume of L = 2 . fm. This emphasizesthe importance of using χ EFT to correct for finite-volume ef-fects, until very large lattice volumes can be used to resolvethe correct chiral nonanalytic behaviour of hadrons.
IV. CONCLUSION
Newly developed techniques within the framework of chiraleffective field theory were applied to recent precision latticeQCD results from the QCDSF Collaboration for the chargeradius of the isovector nucleon. The inclusion of chiral loopcontributions is vital for reconciling lattice simulations withthe experimental result. It was discovered that the logarithmicdivergence in the chiral expansion of the charge radius drivesthe large finite-volume corrections encountered near the phys-ical point. Lattice box sizes of L & are required in orderto achieve a direct lattice simulation result within of thevalue at the physical point.A discrepancy was found between the experimental valueand the extrapolation results, which may be a consequenceof excited state contamination; the use of only two flavours,and/or neglecting O ( a ) contributions.Finite-volume chiral extrapolations provide a benchmarkfor future lattice simulations. Specific predictions can bemade by choosing lattice volumes and pion masses to match those of a lattice calculation. By using this method, estimatesof the electric charge radii simulations were obtained basedon the PACS-CS configurations, which provide a guide forthe interpretation of future lattice results. Acknowledgments
We would like the thank James Zanotti for many helpfuldiscussions. This research is supported by the Australian Re-search Council through Grants No. DP110101265 and No.FT120100821 (R.D.Y.). [1] H.-y. Gao, Int.J.Mod.Phys.
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FIG. 13: (color online). Extrapolation of h r i E at infinite volume. The experimental value has been included in the fit, in preparation for making futurefinite-volume corrections. FIG. 14: (color online). Extrapolations of h r i E at different finite volumes, and at infinite volume. The curves are based on lattice QCD results from QCDSF,lattice sizes: . − . fm, and the experimental value. The provisional constraint m π L > is used. The experimental value [22, 23] is marked as a square. FIG. 15: (color online). Predictions of h r i E based on the volume ( L = 2 . fm) and pion masses from the PACS-CS lattice QCD configurations [39]. Theerror bars represent the total uncertainties. The points are estimated only within the constraint m π L >3