We apply the methods of Chiral Perturbation Theory to the analysis of the
first moments of the Generalized Parton Distributions in a Nucleon, usually
known as generalized form factors. These quantities are currently also under
investigation in Lattice QCD analyses of baryon structure, providing simulation
results at large quark masses to be extrapolated to the "real world" via Chiral
Effective Field Theory. We have performed a leading-one-loop calculation in the
covariant framework of Baryon Chiral Perturbation Theory (BChPT), predicting
both the momentum and the quark-mass dependence for all the vector and axial
(generalized) form factors. In particular we discuss the results for the limit
of vanishing four-momentum transfer where the GPD-moments reduce to the well
known moments of Parton Distribution Functions (PDFs). We fit our results to
available lattice QCD data, extrapolating down to the physical point. We
conclude by presenting outstanding results from a combined fit to different
GPDs-moments.Comment: 7 pages, 4 figures, Proceedings of Lattice 2007 (July 30 - 4 August
  2007, Regensburg, Germany

Dorati, Marina

Gail, Tobias A.

Hemmert, Thomas R.

English

arXiv

We apply the methods of Chiral Perturbation Theory to the analysis of the first moments of the Generalized Parton Distributions in a Nucleon, usually known as generalized form factors. These quantities are currently also under investigation in Lattice QCD analyses of baryon structure, providing simulation results at large quark masses to be extrapolated to the "real world" via Chiral Effective Field Theory. We have performed a leading-one-loop calculation in the covariant framework of Baryon Chiral Perturbation Theory (BChPT), predicting both the momentum and the quark-mass dependence for all the vector and axial (generalized) form factors. In particular we discuss the results for the limit of vanishing four-momentum transfer where the GPD-moments reduce to the well known moments of Parton Distribution Functions (PDFs). We fit our results to available lattice QCD data, extrapolating down to the physical point. We conclude by presenting outstanding results from a combined fit to different GPDs-moments

Tobias A. Gail

Marina Dorati

Thomas R. Hemmert

Open Access Repository

Scientifica Acta 2, No. 2, 156 – 161 (2008)PhysicsChiral analysis of the Generalized Form Factors of the nucleonMarina DoratiDipartimento di Fisica Nucleare e Teorica, Università di Pavia, Via Bassi 6, 27100 Pavia, ItalyIstituto Nazionale di Fisica Nucleare, Sezione di Pavia, Via A. Bassi 6, Pavia, Italymarina.dorati@pv.infn.itWe apply the methods of Chiral Perturbation Theory to the analysis of the first moments of the GeneralizedParton Distributions in a Nucleon, usually known as generalized form factors. These quantities are currentlyalso under investigation in Lattice QCD analyses of baryon structure, providing simulation results at largequark masses to be extrapolated to the "real world" via Chiral Effective Field Theory. We have performeda leading-one-loop calculation in the covariant framework of Baryon Chiral Perturbation Theory (BChPT),predicting both the momentum and the quark-mass dependence for all the vector and axial (generalized)form factors. In particular we discuss the results for the limit of vanishing four-momentum transfer wherethe GPD-moments reduce to the well known moments of Parton Distribution Functions (PDFs). We fit ourresults to available lattice QCD data, extrapolating down to the physical point. We conclude by presentingoutstanding results from a combined fit to different GPDs-moments.1 IntroductionIn this paper we discuss the findings of references [1] and [2] where the generalized form factors of thenucleon were analyzed in the framework of covariant Baryon Chiral Perturbation Theory (BChPT). In stan-dard SU(2) BChPT the results for form factors typically depend on two variables, the momentum transfersquared and the quark-mass and on a number of low energy constants (LECs). In this work we studythe quark-mass dependence of the isoscalar- and isovector-vector as well as the isovector-axial general-ized form factor A2,0(t) at zero momentum transfer and predict the physical value of those quantities bydetermining previously unknown LECs by a fit of the BChPT results to lattice QCD data.Working in twist-2 approximation, the parity-even part of the structure of the nucleon is encoded viatwo Generalized Parton Distribution functions (GPDs) Hq(x, ξ, t) and Eq(x, ξ, t) [3].Moments of GPDs can be interpreted much easier and are connected to well-established hadron structureobservables. E.g. the zero-th order (Mellin-) moments in the variable x correspond to the contribution ofquark q to the well known Dirac and Pauli form factors F1(t), F2(t) of the nucleon:∫ 1−1dx x0 Hq(x, ξ, t) = F q1 (t) ,∫ 1−1dxx0 Eq(x, ξ, t) = F q2 (t). (1)Our aim is the application of the methods of ChPT to the analysis of the first moments in x of these nucleonGPDs∫ 1−1dxx Hq(x, ξ, t) = Aq2,0(t) + (−2ξ)2Cq2,0(t) ,∫ 1−1dxx Eq(x, ξ, t) = Bq2,0(t)− (−2ξ)2Cq2,0(t), (2)where one encounters three generalized form factors Aq2,0(t), Bq2,0(t), Cq2,0(t) of the nucleon for eachquark flavor q. For the case of 2 light flavours the generalized isoscalar (u+d) and isovector (u-d) formfactors have been already studied in a series of papers at leading-one-loop order in the non-relativisticframework of Heavy Baryon ChPT (HBChPT) [4]. We will provide the first analysis of these generalizedform factors utilizing the methods of covariant BChPT for 2 light flavors [5]. Our BChPT formalism [6]makes use of a variant of Infrared Regularization [7] for the loop diagrams and is constructed in such a waythat we exactly reproduce the corresponding HBChPT result of the same chiral order in the limit of small© 2008 Università degli Studi di PaviaScientifica Acta 2, No. 2 (2008) 157a) b)c) d)e)Fig. 1: Loop diagrams contributing to the first moments of the GPDs of a nucleon at leading-one-loop order in BChPT.The solid and dashed lines represent nucleon and pion propagators respectively. The solid dot denotes a coupling to anexternal tensor field.pion masses. For the complete analytical expressions of the discussed results and a detailed description ofthe formalism used for the calculation we refer to [1] and [2].2 The Generalized Form Factors of the Nucleon in ChPTIn ChPT one can direcly access the isoscalar (s) and isovector (v) contribution to the generalized formfactors of the nucleon by evaluating the following matrix elements [3]:i〈p′|qγ{µ←→D ν} q|p〉u+d = (3)u(p′)[As2,0(∆2)γ{µpν} −Bs2,0(∆2)2MN∆αiσα{µpν} +Cs2,0(∆2)MN∆{µ∆ν}]12u(p),(4)i〈p′|qγ{µ←→D ν} q|p〉u−d = (5)u(p′)[Av2,0(∆2)γ{µpν} −Bv2,0(∆2)2MN∆αiσα{µpν} +Cv2,0(∆2)MN∆{µ∆ν}]τa2u(p).(6)The brackets {. . .} denote the completely symmetrized and traceless combination of all indices in anoperator. u (u) is a Dirac spinor of the incoming (outgoing) nucleon of mass MN , for which the quarkmatrix-element is evaluated.From a powercounting analysis we find that the Feynman diagrams contributing to the first moments ofGPDs of a nucleon at leading-one-loop order [O(p2)] in covariant ChPT are the ones depicted in Fig.1.© 2008 Università degli Studi di Pavia158 Scientifica Acta 2, No. 2 (2008)3 Analysis of the results for A(v,s)2,0 (t = 0)In this section we present an analysis of the generalized isovector- and isoscalar-vector form factors Av,s2,0(t)in the forward limit t → 0. The details of the ChPT calculation as well as a complete analysis of the formfactors Bv,s2,0(t), Cv,s2,0 (t) and their connection to the spin physics sector can be found in ref[1, 2].In the forward limit t → 0 the generalized form factors As,v2,0(t = 0) can be understood as moments ofthe ordinary Parton Distribution Functions (PDFs) q(x), q̄(x) [3]:〈x〉u±d = As,v2,0(t = 0) =∫ 10dxx (q(x) + q̄(x))u±d . (7)Experimental results are available for 〈x〉 in proton- and “neutron-” targets, from which one can estimatethe isoscalar and isovector quark contributions at the physical point [8] at a regularization scale µ. Wechoose µ = 2 GeV for our comparisons with phenomenology.For the PDF-moment Av2,0(t = 0) we obtain to O(p2) in BChPTAv2,0(0).= 〈x〉u−d= av2,0 +av2,0m2π(4πFπ)2{− (3g2A + 1) logm2πλ2− 2g2A + g2Am2πM20(1 + 3 logm2πM20)−12g2Am4πM40logm2πM20+ g2Amπ√4M20 −m2π(14− 8 m2πM20+m4πM40)arccos(mπ2M0)}+∆av2,0gAm2π3(4πFπ)2{2m2πM20(1 + 3 logm2πM20)− m4πM40logm2πM20+2mπ(4M20 −m2π)32M40× arccos(mπ2M0)}+ 4m2πc(r)8 (λ)M20+O(p3).Most LECs in this expression are well known from analyses of chiral extrapolation functions [9]. However,the sizes of av2,0, ∆av2,0 and c(r)8 (λ) are only poorly known at this point. The coupling ∆av2,0 is related tothe spin-dependent analogue of the mean momentum fraction, namely 〈∆x〉u−d (see Section 4 and [2]). Ina first fit (Fit I) to lattice data we constrain ∆av2,0 from the phenomenological value of 〈∆x〉phen.u−d ≈ 0.21and perform a 2-parameter fit with the couplings av2,0, c(r)8 (1GeV) at the regularization scale λ = 1 GeV.We fit to the LHPC lattice data for this quantity as given in ref.[10], including lattice data up to effectivepion masses of mπ ≈ 600 MeV. The resulting values for the fit parameters together with their statisticalerrors are given in table 1 and the resulting chiral extrapolation function is shown as the solid line in theleft hand side of Fig.2. The extrapolation curve tends towards smaller values for small quark-masses, butdoes not quite reach the phenomenological value at the physical point, which is not included in the fit.Since BChPT is based on a systematic perturbative expansion, it does not only provide us with the resultat a certain order, but also allows for an estimate of possible higher order effects. From dimensionalanalysis we know that the leading chiral contribution to 〈x〉u−d beyond our calculation takes the formO(p3) ∼ δA m3πΛ2χM0+ .... Constraining δA between values−1, . . . , +1 (the natural scale of all couplings inthe observables considered here is below 1) and repeating the fit with this uncertainty term included leadsto the grey band indicated in Fig.2. As one can see the phenomenological value for 〈x〉u−d lies well withinthat band of possible next-order corrections, giving us no indication that something may be inconsistentwith the large values for 〈x〉u−d typically found in lattice QCD simulations for large quark-masses.As stated in the Introduction, the covariant BChPT scheme used in this analysis is able to reproduceexactly the corresponding non-relativistic HBChPT result at the same order by the appropriate truncation in© 2008 Università degli Studi di PaviaScientifica Acta 2, No. 2 (2008) 1590 0.1 0.2 0.3 0.4 0.5 0.6mπ [GeV]00.10.20.3<x>u-d0 0.2 0.4 0.6 0.8mπ [GeV]00.10.20.3<x>u-dFig. 2: Left panel. Fit I of the O(p2) result of Eq.(3.2) to the LHPC lattice data of ref.[10]. Right panel. Fit II of theO(p2) BChPT result of Eq.(3.2) to the LHPC lattice data of ref.[10] and to the physical point (solid line). The dashedcurve shown corresponds to O(p2) result in the HBChPT truncation.1/(16π2F 2πM0). In order to also compare the O(p2) HBChPT result of refs.[11] with the O(p2) covariantBChPT result of Eq.(8) we perform a second fit (Fit II): We fit the covariant expression for 〈x〉u−d ofEq.(8) again to the LHPC lattice data and we constrain the coupling ∆av2,0 in such a way, that the resultingchiral extrapolation curve reproduces the phenomenological value of 〈x〉phen.u−d = 0.160±0.006 [8] exactlyfor physical quark masses. The parameter values for this Fit II are again given in table 1, whereas theresulting chiral extrapolation is shown as the solid line in the right hand side of Fig.2. We would like toemphasize that the curve looks very reasonable, connecting the physical point with the lattice data of theLHPC collaboration in a smooth fashion. For the comparison with HBChPT we now utilize the very samevalues for av2,0 and c(r)8 of Fit II. The resulting curve based on the O(p2) HBChPT truncation is shown asthe dashed curve in Fig.2. One observes that this leading-one-loop HBChPT expression agrees with thecovariant result between the chiral limit and the physical point, but is not able to extrapolate on towardsthe lattice data.We therefore conclude that the smooth extrapolation behaviour of the covariant O(p2) BChPT expres-sion for 〈x〉u−d of Eq.(8) between the chiral limit and the region of present lattice QCD data is due to aninfinite tower of(mπM0)nterms.The same analysis can also be performed for the isoscalar generalized form factor As2,0(t) which in theforward limit reduces to As2,0(0) = 〈x〉u+d. Fig.3 shows a 2-parameter fit of the O(p2) covariant BChPTresults for this observable [1] to LHPC and QCDSF data of reference [10] and [12]. We want to stress thatthe experimental value of the quantity 〈x〉u+d is not included in the fit. As the plot shows, the obtainedchiral extrapolation curve is very satisfying, consistently linking the lattice data at large quark-masses withthe phenomenological value. In contrast, the correspondent result in the heavy baryon limit representedTable 1: Values of the couplings resulting from the two fits to the LHPC lattice data for 〈x〉u−d [1]. The errors shownare only statistical and do neither include uncertainties from possible higher order corrections in ChEFT nor fromsystematic uncertainties connected with the lattice simulation.av2,0 ∆av2,0 cr8(1GeV)Fit I (4 points - 2 parameter) 0.157 ± 0.006 0.210 (fixed) -0.283 ± 0.011Fit II (6+1 points - 3 parameter) 0.141 ± 0.0057 0.144 ± 0.034 -0.213 ± 0.03© 2008 Università degli Studi di Pavia160 Scientifica Acta 2, No. 2 (2008)0 0.2 0.4 0.6mπ [GeV]00.20.40.60.81<x>u+dFig. 3: Two-parameters Fit of theO(p2) BChPT result of ref.[1] to the LHPC lattice data of ref.[10] and to the QCDSFdata of ref.[12]. We obtain as20 = 0.513±0.006 and c9 = −0.064±0.005 as values for the free parameters. The bandshown indicate estimate of higher order possible corrections. The dashed line correspond to the respective HBChPTresults at this order.by the dashed line in Fig.3 not even allows for an interpolation between the presently available lattice dataand results from experiments.4 Combined fitWe have extended the analysis to the first moments of the axial GPDs H̃q(x, ξ, t) and Ẽq(x, ξ, t)∫ 1−1dx x H̃q(x, ξ, t) = Ãq2,0(t) ,∫ 1−1dxx Ẽq(x, ξ, t) = B̃q2,0(t). (8)Again, in the limit of vanishing four-momentum transfer the isovector form factor Ãv2,0(t → 0) is directlyconnected to the spin dependent analogue of the mean momentum fraction 〈∆x〉u−dÃv2,0(0) = 〈∆x〉u−d =∫ 10dx x (q↓(x)− q↑(x))∣∣u−d (9)= ∆av2,0 +O(p2)where ∆av2,0 corresponds to the chiral limit value of 〈∆x〉u−d.Looking at the O(p2) BChPT expression for 〈∆x〉u−d [2] one can easily observe that each isovectormoment (〈x〉u−d and 〈∆x〉u−d) depends on 3 unknown parameters: 2 couplings (av2,0, ∆av2,0) and onecounterterm. As the same couplings contribute in both moments, it is hoped that a simultaneous fit ofour BChPT results to the lattice data of ref.[10] can considerably reduce the statistical errors. As one cansee from Fig.4, the results of this procedure are pretty outstanding, given that the values at the physicalpion mass were not included in the fit! The chiral curvature in both observables naturally bends down tothe phenomenological value for lighter quark masses, leading to a very satisfactory extrapolation curve.The extrapolated values at the physical pion mass together with the ones known from phenomenology arereported in table 2. As one can easily see, the obtained BChPT values from our O(p2) leading-one-loopanalysis are clearly consistent with the experimental ones!We would like to stress that this efficient cross-talk between the ChPT results for 〈x〉u−d and 〈∆x〉u−doccurs only in the covariant framework, while in the non-relativistic approach both observables are com-pletely independent at this order.We conclude that combined fits of several observables characterized by a common subset of ChEFTcouplings are the winning strategy towards the most reliable chiral extrapolations of lattice QCD results.© 2008 Università degli Studi di PaviaScientifica Acta 2, No. 2 (2008) 161Table 2: The phenomenological values for the observables 〈x〉u−d and 〈∆x〉u−d, together with the extrapolated valuesobtained from the combined fit shown in fig.4.〈x〉u−d 〈∆x〉u−dPHENOMENOLOGY 0.16± 0.006[8] 0.21± 0.025 [13]EXTRAPOLATED VALUES AT mphysπ 0.16± 0.01 0.19± 0.010 0.2 0.4 0.6 0.8mΠ @GeVD00.050.10.150.20.250.3<x>u-d<x>u-d0 0.2 0.4 0.6 0.8mΠ @GeVD00.050.10.150.20.250.3<Dx>u-d<Dx>u-dFig. 4: Combined FIT of the O(p2) results of ref. [2] to the lattice data of ref. [10]. Note that the phenomenologicalvalues at physical pion mass were not included in the fit. The bands shown indicate estimate of higher order possiblecorrections.References[1] M. Dorati, T. A. Gail and T. R. Hemmert, Nuclear Physics A798, 96 (2008).[2] M. Dorati, The structure of the nucleon by electromagnetic probes in Chiral Effective Field Theory (PhD Thesis,University of Pavia, 2008).[3] X. Ji, Journal of Physics G24 1181 (1998) and Annual Review of Nuclear and Particle Science 54, 413 (2003);M. Diehl, Physics Reports 388, 41 (2003); A.V. Belitsky and A. V. Radyushkin, Physics Reports 418, 1 (2005).[4] J.-W. Chen and X. Ji, Physical Review Letters 88, 052003 (2002); A. V. Belitsky and X. Ji, Physics Letters B538,289 2002; S. Ando, J.-W. Chen and C.-W. Kao, Physical Review D74, 094013 (2006); M. Diehl, A. Manashovand A. Schäfer, European Physical Journal A29, 315 (2006) and European Physical Journal, A31, 335 (2007).[5] J. Gasser, M. Sainio, and A. Svarc, Nuclear Physics B307, 779 (1988).[6] T. A. Gail and T. R. Hemmert, in Proceedings of ECT* Workshop “lattice QCD, ChPT and Hadron Phenomenol-ogy”, Trento, Italy, 2-6 Oct 2006, hep-ph/0611072 and T. A. Gail (PhD Thesis, Technical University ofMunich, 2007).[7] T. Becher and H. Leutwyler, European Physical Journal C9, 643 (1999).[8] see http://www-spires.dur.ac.uk/hepdata/pdf3.html[9] M. Procura, T. R. Hemmert and W. Weise, Physical Review D69, 034505, (2004); QCDSF-UKQCD Collabora-tion (A. Ali Khan et al.), Nuclear Physics, B689, 175 (2004); V. Bernard, T. R. Hemmert and U.-G. Meißner,Physics Letters B622, 141 (2005); M. Procura, B. U. Musch, T. Wollenweber, T. R. Hemmert and W. Weise,Physical Review D73, 114510 (2006).[10] LHPC Collaboration (P. Hägler et al.), Physical Review D77, 094502 (2008).[11] D. Arndt and M. J. Savage, Nuclear Physics A697, 429 (2002); J.-W. Chen and X. Ji, Physics Letters B523, 107(2001).[12] QCDSF Collaboration (M. 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Chiral Analysis of the Generalized Form Factors of the Nucleon

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