Systematic uncertainties in the recent precise determination of the
strangeness magnetic moment of the nucleon are identified and quantified. In
summary, G_M^s = -0.046 \pm 0.019 \mu_N.Comment: Invited presentation at PAVI '04, International Workshop on Parity
  Violation and Hadronic Structure, Laboratoire de Physique Subatomique et de
  Cosmologie, Grenoble, France, June 8-11, 2004. 7 pages, 16 figure

A.G. Williams

A.W. Thomas

D.B. Leinweber

Donoghue

J.B. Zhang

J.M. Zanotti

Leinweber

Miller

R.D. Young

S. Boinepalli

Savage

Young

Zanotti

Nuclear Physics B - Proceedings Supplements

English

arXiv

© Societa Italiana di Fisica / Springer-Verlag 2005Systematic uncertainties in the recent precise determination of the strangeness magnetic moment of the nucleon are identified and quantified. In summary, GM8 = -0.046 ± 0.019 μN.D.B. Leinweber, S. Boinepalli, A.W. Thomas, A.G. Williams, R.D. Young, J.B. Zhang and J.M. Zanott

Leinweber, D.

Boinepalli, S.

Thomas, A.

Williams, A.

Young, R.

Zhang, J.

Zanotti, J.

Adelaide Research & Scholarship

The European Physical Journal A

Systematic uncertainties in the precise determination of the strangeness magnetic moment of the nucleon

Crossref

Systematic uncertainties in the recent precise determination of the strangeness magnetic moment of the nucleon are identified and quantified. In summary, 

$G_M^s = -0.046 \pm 0.019\ \mu_N$
.PACS: 13.40.Em  Electric and magnetic moments – 12.38.Gc  Lattice QCD calculations – 12.39.Fe  Chiral Lagrangian

EDP Sciences OAI-PMH repository (1.2.0)

Copyright © 2004 Elsevier B.V. All rights reserved. Printed in U.S.A. Submitted to Cornell University’s online archive www.arXiv.org in 2005 by Derek B. Leinweber. Post-print sourced from www.arxiv.org.Systematic uncertainties in the recent precise determination of the strangeness magnetic moment of the nucleon are identified and quantified. In summary, GMs = - 0.051 ± 0.021 μN. © 2004 Elsevier B.V. All rights reserved.http://www.elsevier.com/wps/find/journaldescription.cws_home/505717/description#descriptio

Zhang, J.B.

arXiv:hep-lat/0502004v1  8 Feb 2005U. Adelaide preprint: ADP-05-03/T613Jefferson Lab. preprint: JLAB-THY-05-296DESY preprint: DESY 05-014Systematic uncertainties in the precise determination of thestrangeness magnetic moment of the nucleonD. B. Leinweber12, S. Boinepalli1, A. W. Thomas12, A. G. Williams1, R. D. Young12, J. B. Zhang1, and J. M. Zanotti31 Special Research Center for the Subatomic Structure of Matter, and Department of Physics, University of Adelaide,Adelaide SA 5005, Australia2 Jefferson Lab, 12000 Jefferson Ave., Newport News, VA 23606, USA3 John von Neumann-Institut fu¨r Computing NIC, Deutsches Elektronen-Synchrotron DESY, D-15738 Zeuthen, GermanyReceived: 3 December 2004Abstract. Systematic uncertainties in the recent precise determination of the strangeness magnetic momentof the nucleon are identified and quantified. In summary, GsM = −0.046 ± 0.019 µN .PACS. 13.40.Em Electric and magnetic moments 12.38.Gc Lattice QCD calculations 12.39.Fe Chiral La-grangians1 IntroductionRecent low-mass lattice QCD simulation results combinedwith new chiral extrapolation techniques and the principleof charge symmetry have enabled a precise determinationof the strangeness magnetic moment of the nucleon GsM[1]. In this paper, the systematic errors of the approach areexplored and quantified. In particular, we examine the sen-sitivity of GsM and the magnetic moments of the baryonoctet to the regulator-mass scale of finite-range regular-ized chiral effective field theory, the lattice scale determi-nation, the finite-volume of the lattice, and the quenchedapproximation.2 Charge SymmetryThe approach [1] centres around two equations for thestrangeness magnetic moment of the nucleon, GsM , ob-tained from charge symmetryGsM =(ℓRsd1− ℓRsd)[2p+ n− upuΣ(Σ+ −Σ−)], (1)GsM =(ℓRsd1− ℓRsd)[p+ 2n− unuΞ(Ξ0 − Ξ−)]. (2)Here the baryon labels represent the experimentally mea-sured baryon magnetic moments and ℓRsd ≡ GsM/ℓGdM isthe ratio of s- and d-quark sea-quark loop contributions,depicted in the right-hand diagram of Fig. 1. ℓRsd lies inthe range (0,1). The ratios up/uΣ and un/uΞ are ratios ofvalence-quark contributions to baryon magnetic momentsin full QCD as depicted in the left-hand diagram of Fig. 1.The latter are determined by lattice QCD calculations [1,3,4,5,6].Fig. 1. Diagrams illustrating the two topologically differentinsertions of the current within the framework of lattice QCD.These skeleton diagrams for the connected (left) and discon-nected (right) current insertions are dressed by an arbitrarynumber of gluons and quark loops.Equating Eqs. (1) and (2) provides a linear relationshipbetween up/uΣ and un/uΞ which must be obeyed withinQCD under the assumption of charge symmetry – itselftypically satisfied at the level of 1% or better [2]. There areno other systematic uncertainties associated with this con-straint. Figure 2 displays this relationship by the dashedand solid line. Since this line does not pass through thepoint (1.0, 1.0), corresponding to the simple quark modelassumption of universality, there must be an environmenteffect exceeding 12% in both ratios or approaching 20% ormore in at least one of the ratios. To determine the sign ofGsM , it is sufficient to determine where on this constraintcurve, QCD resides.2 D. B. Leinweber et al.: Systematic uncertainties in the precise determination. . .Fig. 2. The straight line (dashed GsM (0) < 0, solid GsM (0) > 0)indicates the constraint on the ratios up/uΣ and un/uΞ impliedby charge symmetry and the experimentally measured mag-netic moments. The assumption of environment independentquark moments is indicated by the square. The dependence ofthe extrapolated ratios from lattice QCD simulations on theparameter Λ = 0.7, 0.8 and 0.9 GeV, (governing the size ofpion-cloud corrections) is illustrated by the cluster of pointswith Λ increasing from left to right.3 up/uΣ and un/uΞ DeterminationOur present precise analysis has been made possible by asignificant breakthrough in the regularization of the chiral-loop contributions to hadron observables [7,8,9]. Throughthe process of regulating loop integrals via a finite-rangeregulator (FRR) [8,10], the chiral expansion is effectivelyre-summed to produce an expansion with vastly improvedconvergence properties.The chiral expansion for the u-quark contribution tothe proton magnetic moment in quenched QCD (QQCD),has the formup = aΛ0 + up χη′ Iη′(mπ , Λ) + χπB IB(mπ , Λ)+χKB IB(mK , Λ) + aΛ2 m2π + aΛ4 m4π, (3)where the repeated index, B, sums over allowed baryonoctet and decuplet intermediate states. The dependenceof the unrenormalized coefficients, aΛi , and the associateddipole-vertex regulated loop integrals, I(mπ, Λ), on theregulator parameter, Λ, is emphasized by the explicit ap-pearance of Λ. The loop integrals are defined asIB(m,Λ) = (4)− 23pi∫dk(2√k2 +m2 +∆BN)k4 u2(k, Λ)(k2 +m2)3/2(√k2 +m2 +∆BN)2 ,Iη′(mπ, Λ) = −∫∞0dkk4(k2 +m2π)52u2(k, Λ) , (5)where ∆BN is the relevant baryon mass splitting and thefunction u(k, Λ) is the dipole-vertex regulator. The coef-ficients, χ, denote the known model-independent coeffi-cients of the non-analytic terms for pi and K mesons inQQCD [11,12].Figure 3 illustrates a fit of FRR, quenched chiral per-turbation theory (χPT) to our lattice results (solid curve),where only the discrete momenta allowed in the finite vol-ume of the lattice are summed in evaluating the chiral loopintegrals. The long-dashed curve that also runs throughthe lattice results corresponds to replacing the discretemomentum sum by the infinite-volume, continuous mo-mentum integral. For all but the lightest quark mass, finitevolume effects are negligible.Having determined the analytic coefficients aΛ0,2,4 for aparticular choice of Λ, one can correct the chiral propertiesof the pion-cloud contribution from QQCD to full QCD [9,13] by changing the coefficients of the loop integrals, χη′ ,χπB, χKB of Eq. (3), to their full QCD counter parts [11,12]. Valence quark contributions in full QCD are indicatedby the long-dash-dot curve in Fig. 3 (i.e. sea-quark loopcharges are zero) and the full u-quark sector includingthe u-sea-quark loop contributions are indicated by theshort-dash-dot curve for Λ = 0.8 GeV. Figures 4, 5 and6 show similar results for the u quark in n, Σ+, and Ξ0respectively.The importance of correcting for both finite-volumeand quenching artifacts is illustrated in Figs. 7 and 8,where the one standard deviation agreement between thechirally corrected lattice QCD simulation results and theexperimentally measured baryonmagnetic moments is high-lighted.4 Systematic Errors4.1 Regulator DependenceIt is important to investigate systematic errors associatedwith the regulator-mass dependence of FRR χPT . Theextrapolated results of finite-volume quenched chiral ef-fective field theory should be insensitive to the choice ofregulator parameter. When working to sufficient order inthe chiral expansion, changes in the regulation of loop in-tegrals should be absorbed by changes in the unrenormal-ized coefficients, aΛ0,2,4, in a manner which preserves theinvariant renormalized coefficients. The latter are reflectedin Fig. 9 which illustrates the insensitivity of quenchedbaryon magnetic moments in a finite-volume to the regu-lator parameter Λ. This systematic error is small relativeto the statistical error.Since the finite-volume and quenching corrections areapplied only to the loop integral contributions, the finalresults are Λ dependent. In this case, the regulator ofthe loop integral has become a model for the axial-vectorform factor of the nucleon, describing the coupling of pi-ons to a core described by the analytic terms of the FRRexpansion. This approach describes the relation betweenquenched and full QCD N and ∆ mass simulation resultsas a function ofmπ very accurately [13]. Figures 10 and 11display results for positive and negative baryon magneticmoments in full QCD respectively. One-standard devia-tion agreement is achieved for 0.7 ≤ Λ ≤ 0.9.Because the strangeness magnetic moment of Eqs. (1)and (2) depends only on ratios of magnetic moments, mostD. B. Leinweber et al.: Systematic uncertainties in the precise determination. . . 3Fig. 3. The contribution of a single u quark (with unit charge)to the magnetic moment of the proton. Lattice simulation re-sults (square symbols for m2π > 0.05 GeV2) are extrapolatedto the physical point (vertical dashed line) in finite-volumeQQCD as well as infinite volume QQCD. Estimates of the va-lence u quark contribution in full QCD and the full u-quarksector contribution in full QCD are also illustrated. Extrapo-lated values at the physical pion mass (vertical dashed line),are offset for clarity.Fig. 4. The contribution of the u quark (with unit charge) tothe magnetic moment of the neutron. Curves and symbols areas described in Fig. 3.of this Λ dependence cancels in the final ratios, as illus-trated by the close clustering of points in Fig. 2.4.2 Role of the Decuplet in χPTIt is often argued that next-to-leading-order non-analytic(NLNA) contributions from the baryon decuplet are es-sential in describing the mass dependence of nucleon mag-netic moments. While the decuplet baryon contributionsare not necessarily small, we find the non-analytic curva-ture induced by these contributions is sufficiently subtleFig. 5. The contribution of a single u quark (with unit charge)to the magnetic moment of Σ+. Curves and symbols are as forFig. 3.Fig. 6. The contribution of the u quark (with unit charge) tothe magnetic moment of the Ξ0 hyperon. Curves and symbolsare as described in Fig. 3.that it may be accurately absorbed by the analytic termsof the chiral expansion. Figure 12 illustrates the insensitiv-ity of finite-volume quenched chiral effective field theoryto NLNA decuplet-baryon contributions.Figure 13 confirms that the NLNA decuplet-baryoncontributions are indeed large in some cases and as suchare important in correcting the artifacts of the quenchedapproximation. However, other more highly-excited baryonresonances have small couplings to the ground-state baryonoctet relative to that for the decuplet and provide negli-gible corrections.4 D. B. Leinweber et al.: Systematic uncertainties in the precise determination. . .Fig. 7. The one standard deviation agreement between thechirally corrected lattice QCD simulation results (square sym-bols) and the experimentally measured baryon magnetic mo-ments (circular symbols) having positive values. Finite-volumequenched results (crossed boxes) and infinite-volume quenchedresults (diamonds) are also illustrated to highlight the impor-tance of correcting for both finite-volume and quenching arti-facts.Fig. 8. The one standard deviation agreement between thechirally corrected lattice QCD simulation results and the ex-perimentally measured baryon magnetic moments having neg-ative values. Symbols are as in Fig. 7.4.3 Scale DependenceSetting the scale in quenched QCD simulations is some-what problematic. Different observables lead to differentlattice spacings, a. If one is explicitly correcting the one-loop pion-cloud contributions to hadronic observables, aswe are here, then clearly one must set the scale using anobservable insensitive to chiral physics. This excludes ob-servables such as the rho-meson mass, nucleon mass, orthe pion decay constant commonly used in the literatureto hide the artifacts of the quenched approximation. Onthe other hand, the heavy-quark phenomenology of thestatic-quark potential provides an optimal case. In par-Fig. 9. The FRR χPT regulator mass dependence of finite-volume quenched chiral effective field theory. Experimentalmeasurements are illustrated at left for each baryon for ref-erence. Results for Λ = 0.6, 0.7, 0.8, 0.9 and 1.0 GeV are illus-trated from left to right for each baryon. The small systematicdependence on Λ relative to the statistical error bars illustratedindicate the order of the chiral expansion is adequate for thisanalysis.ticular, the Sommer parameter, r0, is ideal as it sets thescale by equating the force between two static quarks inQQCD and full QCD at a precise separation of r0 = 0.49fm.The string tension might also be used, but screeningof the potential in full QCD makes this measure poorlydefined. However, we consider it here as a measure of thesystematic error encountered in setting the scale of thelattice QCD results. Figure 14 illustrates this scale depen-dence on baryon magnetic moments and Fig. 15 illustratesthe rather minor impact this systematic uncertainty hason the valence-quark moment ratios vital to determiningthe sign of GsM .Collecting the variation of the valence-quark momentratios from variations in Λ (which maintain one-standard-deviation agreement with experiment), variations in set-ting the lattice scale and statistical errors determined by athird-order single-elimination jackknife analysis, one findsupuΣ= 1.092± 0.030 andunuΞ= 1.254± 0.124 . (6)Using the experimental magnetic moments, one ob-serves that Eq. (2)GsM =(ℓRsd1− ℓRsd)[−1.033 − unuΞ(−0.599)], (7)is least sensitive to variation in the valence-quark momentratio, and hence provides the most precise determinationfor GsM . Figure 16 plots GsM as a function ofℓRsd withstandard error limits provided by Eq. (6).D. B. Leinweber et al.: Systematic uncertainties in the precise determination. . . 5Fig. 10. The dependence of positive octet-baryon magneticmoments on the parameter Λ = 0.6, 0.7, 0.8, 0.9 and 1.0 GeV,governing the size of pion-cloud corrections associated withthe finite-volume of the lattice and artifacts of the quenchedapproximation. Experimental measurements, illustrated at leftby the filled circle for each baryon, indicate that optimal cor-rections are obtained for 0.6 ≤ Λ ≤ 0.9 GeV.Fig. 11. The dependence of negative octet-baryon magneticmoments on the parameter Λ = 0.6, 0.7, 0.8, 0.9 and 1.0 GeV,governing the size of pion-cloud corrections associated withthe finite-volume of the lattice and artifacts of the quenchedapproximation. Experimental measurements, illustrated at leftby the filled circle for each baryon, indicate that optimal cor-rections are obtained for 0.7 ≤ Λ ≤ 0.9 GeV.5 Estimating ℓRsd ≡ GsM/ℓGdMThe symmetry of the three-point correlation function [14]describing the sea-quark loop contributions to the nucleon,depicted in the right-hand side of Fig. 1, ensure that thechiral expansion for this quantity is identical for all threequark flavours, up to simple charge factors. For the d- ors-quark loop contributions, the only difference that canarise is whether one evaluates the chiral expansion at thepion or kaon mass. The leading non-analytic contributionFig. 12. The insensitivity of finite-volume quenched chiral ef-fective field theory to NLNA decuplet-baryon contributions.Results including decuplet intermediate states in chiral effec-tive field theory (squares) are compared with results exclud-ing the decuplet (diamonds). Experimental measurements (cir-cles), are illustrated at left for each baryon for reference.Fig. 13. The dependence of octet-baryon magnetic momentson the inclusion of NLNA decuplet-baryon contributions in theprocess of correcting the pion cloud of quenched chiral effectivefield theory. Preferred results including decuplet intermediatestates in chiral effective field theory (squares) are comparedwith results excluding the decuplet (diamonds). Experimentalmeasurements (circles), are illustrated at left for each baryonfor reference.to the chiral expansion involves two pseudoscalar mesonpropagators, and therefore one expects contributions toℓRsd in the ratio m4π/m4K ∼ 0.1.To be more precise, one can use the same successful(single-parameter) model, previously used to correct thequenched simulation results to full QCD, as highlightedin Figs. 7 and 8, to provide an estimate for ℓRsd. Evalu-ating the loop integrals with Λ = 0.8 ± 0.2 GeV yields6 D. B. Leinweber et al.: Systematic uncertainties in the precise determination. . .Table 1. Sources of uncertainty and their contribution to the strangeness magnetic moment of the nucleon, GsM , in units ofnuclear magnetons, µN . Uncertainties are documented for GsM obtained from the valence-quark ratio up/uΣ in Eq. (1), fromthe valence-quark ratio un/uΞ in Eq. (2) and from a statistically weighted (SW) average of these two determinations.up/uΣ , Eq. (1) un/uΞ , Eq. (2) SW AverageUncertainty Source Parameter Range GsM = −0.045 GsM = −0.046 GsM = −0.046Statistical Errors 0.016 0.009 0.008Chiral corrections 0.7 ≤ Λ ≤ 0.9 GeV 0.001 0.002 0.002Scale Determination 0.122 ≤ a ≤ 0.134 fm 0.001 0.002 0.002ℓRsd Determination 0.096 ≤ ℓRsd ≤ 0.181 0.016 0.017 0.017Total Uncertainty 0.023 0.019 0.019Fig. 14. The dependence of octet-baryon magnetic momentson the matching criteria for determining the lattice spacing, a.While ar0 = 0.128 fm set by r0 (square symbols) is the pre-ferred method for determining the scale, results for aσ = 0.134fm set by the string tension (triangles) and a third spacing of0.122 fm (diamonds) are also illustrated. Experimental mea-surements (filled circles), are illustrated at left for each baryon.ℓRsd = 0.139 with 0.096 ≤ℓRsd ≤ 0.181. This uncertaintydominates the final uncertainty in GsM .Table 1 summarizes the sources of uncertainty andtheir contributions to the final determinationGsM = −0.046± 0.019 µN , (8)for the strange quark contribution to the magnetic mo-ment of the nucleon.AcknowledgmentsWe thank the Australian Partnership for Advanced Com-puting (APAC) for generous grants of supercomputer timewhich have enabled this project. Support from the SouthAustralian Partnership for Advanced Computing (SAPAC)and the National Facility for Lattice Gauge Theory isalso gratefully acknowledged. DBL thanks Jefferson LabFig. 15. The charge symmetry constraint line (dashedGsM (0) < 0, solid GsM (0) > 0) on the ratios up/uΣ and un/uΞ .The dependence of the ratios from chirally-corrected quenchedlattice QCD on the scale parameter, a = 0.134, 0.128, and0.122 fm, is illustrated by the cluster of points with a decreas-ing from left to right.Fig. 16. The dependence of GsM on the strange to light sea-quark loop ratio ℓRsd. Standard error limits have there originin the systematic error summary of Eq. 6.D. B. Leinweber et al.: Systematic uncertainties in the precise determination. . . 7for their kind hospitality where the majority of this re-search was performed. This work is supported by the Aus-tralian Research Council and by DOE contract DE-AC05-84ER40150 under which SURA operates Jefferson Lab.References1. D. B. Leinweber et al., arXiv:hep-lat/0406002.2. G.A. Miller, B.M. Nefkens and I. Slaus, Phys. Rept. 194(1990) 1.3. J. M. Zanotti et al., Phys. Rev. D 65, 074507 (2002)[hep-lat/0110216].4. D. B. Leinweber et al., Eur. Phys. J. A 18, 247 (2003)[nucl-th/0211014].5. J. M. Zanotti, et al., hep-lat/0405015.6. J. M. Zanotti, et al., hep-lat/0405026.7. R. D. Young et al., Prog. Part. Nucl. Phys. 50, 399 (2003)[hep-lat/0212031].8. D. B. Leinweber, A. W. Thomas and R. D. Young, Phys.Rev. Lett. 92, 242002 (2004) [hep-lat/0302020].9. R. D. Young, D. B. Leinweber and A. W. Thomas,arXiv:hep-lat/0406001.10. J. F. Donoghue, B. R. Holstein and B. Borasoy, Phys. Rev.D59, 036002 (1999).11. D. B. Leinweber, Phys. Rev. D 69, 014005 (2004)[hep-lat/0211017].12. M. J. Savage, Nucl. Phys. A 700, 359 (2002)[nucl-th/0107038].13. R. D. Young, D. B. Leinweber, A. W. Thomasand S. V. Wright, Phys. Rev. D 66 (2002) 094507[arXiv:hep-lat/0205017].14. D.B. Leinweber, Phys. Rev. D53, 5115 (1996)hep-ph/9512319.

G.A. Miller

J.F. Donoghue

M.J. Savage

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Systematic uncertainties in the precise determination of the strangeness
  magnetic moment of the nucleon

Systematic uncertainties in the precise determination of the strangeness magnetic moment of the nucleon

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