We present recent Monte Carlo data for the axial charge of the nucleon
obtained by the QCDSF-UKQCD collaboration for N_f=2 dynamical quarks. We
compare them with formulae from chiral perturbation theory in finite and
infinite volume and find a remarkably consistent picture.Comment: 6 pages, 3 figures, talk presented at Lattice2005 (weak matrix
  elements), needs PoS.cl

Göckeler, M

Hemmert, TR

Horsley, R

Hägler, P

Irving, AC

Khan, A Ali

Pleiter, D

QCDSF-UKQCD Collaboration

Rakow, PEL

Schierholz, G

Schäfer, A

Stüben, H

Wollenweber, T

Zanotti, JM

English

arXiv

We present recent Monte Carlo data for the axial charge of the nucleon
obtained by the QCDSF-UKQCD collaboration for N_f=2 dynamical quarks. We
compare them with formulae from chiral perturbation theory in finite and
infinite volume and find a remarkably consistent picture

University of Liverpool Repository

PoS(LAT2005)349The axial charge of the nucleon on the lattice and inchiral perturbation theoryArifa Ali Khana, Meinulf Göckeler∗a, Philipp Häglerb, Thomas R. Hemmertb, RogerHorsleyc, Alan C. Irvingd, Dirk Pleitere, Paul E. L. Rakowd, Andreas Schäfera, GerritSchierholze f , Hinnerk Stübeng, Tim Wollenweberb and James M. Zanottiea Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germanyb Physik-Department, Theoretische Physik, Technische Universität München, 85747 Garching,Germanyc School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, UKd Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool,Liverpool L69 3BX, UKe John von Neumann-Institut für Computing NIC, Deutsches Elektronen-Synchrotron DESY,15738 Zeuthen, Germanyf Deutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germanyg Konrad-Zuse-Zentrum für Informationstechnik Berlin, 14195 Berlin, GermanyE-mail: meinulf.goeckeler@physik.uni-regensburg.deQCDSF-UKQCD CollaborationWe present recent Monte Carlo data for the axial charge of the nucleon obtained by the QCDSF-UKQCD collaboration for N f = 2 dynamical quarks. We compare them with formulae from chiralperturbation theory in finite and infinite volume and find a remarkably consistent picture.XXIIIrd International Symposium on Lattice Field Theory25-30 July 2005Trinity College, Dublin, Ireland∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(LAT2005)349The axial charge of the nucleon on the lattice and in chiral perturbation theory Meinulf GöckelerThe QCDSF and UKQCD collaborations have generated ensembles of gauge field configura-tions using N f = 2 non-perturbatively O(a) improved Wilson quarks and Wilson’s plaquette actionfor the gauge fields. Here we want to discuss the results obtained for the axial charge of the nu-cleon, gA. Their interpretation is not straightforward because the quark masses in the simulationsare larger than in nature, the volumes are somewhat smaller than infinity, the lattice spacings arelarger than 0 etc. So we need some guidance for the extrapolations towards the physical quarkmasses, the thermodynamic and continuum limits. Such guidance is provided by chiral effectivefield theory (ChEFT), which for selected quantities, e.g. for gA, yields parameterisations of thedependence on the quark mass and the volume which take into account the constraints imposed by(spontaneously broken) chiral symmetry. The dependence on the lattice spacing a can be included,but we shall not consider this possibility here. So we do not yet attempt to cope with the latticeartefacts remaining even after O(a) improvement.If ChEFT can be successfully applied, we gain control over the chiral extrapolation and theapproach to the thermodynamic limit. At the same time we can determine not only the physicalvalue of the quantity of interest, gA in our case, but also some effective coupling constants. Thesemay occur in the ChEFT expressions for other observables and be of phenomenological interestthere. Establishing the link between Monte Carlo results and ChEFT will thus enable us to extractconsiderably more information from our simulations than just the physical value of the quantityunder study.In its standard form, ChEFT describes low-energy QCD by means of an effective field theorybased on effective pion, nucleon, . . . fields. Since the effective Lagrangian does not depend on thevolume, besides the quark-mass dependence the very same Lagrangian governs also the volumedependence, and finite size effects can be calculated by evaluating the theory in a finite (spatial)volume. Thus the finite volume does not introduce any new parameters and the study of the finitesize effects yields an additional handle on the coupling constants of ChEFT. The effective descrip-tion will break down if the box length L becomes too small, just as it fails for pion masses that aretoo large.The simulation parameters are listed in Table 1. Note that we have two groups of three ensem-bles each which differ only in the volume.We compute gA from forward proton matrix elements of the flavour-nonsinglet axial vectorcurrent Au−dµ = u¯γµγ5u− ¯dγµγ5d:〈p,s|Au−dµ |p,s〉 = 2gAsµ . (1)The required bare matrix elements are extracted from ratios of 3-point functions over 2-point func-tions in the standard fashion. Compared to the computation of hadron masses, additional difficul-ties arise in the calculation of nucleon matrix elements such as gA: In general there are quark-linedisconnected contributions, which are hard to evaluate, the operators must be improved and renor-malised etc. Fortunately, in the limit of exact isospin invariance, which is taken in our simulations,all disconnected contributions cancel in gA, because it is a flavour-nonsinglet quantity. The im-proved axial vector current is given byAimpµ (x) = q¯(x)γµ γ5q(x)+acA∂µ q¯(x)γ5q(x) , (2)349 / 2PoS(LAT2005)349The axial charge of the nucleon on the lattice and in chiral perturbation theory Meinulf GöckelerColl. β κsea volumeQCDSF 5.20 0.1342 163×32UKQCD 5.20 0.1350 163×32UKQCD 5.20 0.1355 163×32QCDSF 5.25 0.1346 163×32UKQCD 5.25 0.1352 163×32QCDSF 5.25 0.13575 243×48QCDSF 5.40 0.1350 243×48QCDSF 5.40 0.1356 243×48QCDSF 5.40 0.1361 243×48Coll. β κsea volumeUKQCD 5.29 0.1340 163×32QCDSF 5.29 0.1350 163×32QCDSF 5.29 0.1355 123×32QCDSF 5.29 0.1355 163×32QCDSF 5.29 0.1355 243×48QCDSF 5.29 0.1359 123×32QCDSF 5.29 0.1359 163×32QCDSF 5.29 0.1359 243×48Table 1: Simulation parameters.and hence the improvement term, i.e. the term proportional to cA, does not contribute in forwardmatrix elements. The renormalised improved axial vector current can be written asAµ = ZA (1+bAam)Aimpµ (3)with the bare quark mass m = (1/κsea−1/κc)/(2a).While the coefficient bA will be computed in tadpole improved one-loop perturbation theory,we calculate the renormalisation factor ZA non-perturbatively by means of the Rome-Southamptonmethod [1, 2]. Thus ZA is first obtained in the so-called RI’-MOM scheme. Using continuumperturbation theory we switch to the MS scheme. For sufficiently large renormalisation scalesµ , ZA should then be independent of µ . However, unless µ  1/a lattice artefacts may spoilthis behaviour. Since our scales do not always satisfy this criterion, we try to correct for thismismatch by subtracting the lattice artefacts perturbatively with the help of boosted one-loop latticeperturbation theory. Some lattice artefacts still remain, but we can nevertheless estimate ZA. InTable 2 we compare our results with a recent determination of ZA by the ALPHA collaboration [3].β 5.20 5.25 5.29 5.40this work 0.765(5) 0.769(4) 0.772(4) 0.783(4)ALPHA 0.719 0.734 0.745 0.767Table 2: Values of ZA from this work and from the ALPHA collaboration.Our results for gA are plotted in Fig. 1. Here mpi has been taken from the largest available latticeat each (β , κsea) combination. The scale has been set by means of the force parameter r0 with r0 =0.467fm, and r0/a has been taken at the given quark mass. Obviously there are considerable finitesize effects. A similar volume dependence has already been observed in quenched simulations [4].In order to describe or fit these data we use ChEFT. More specifically, we employ the so-called small scale expansion (SSE) [5], which is one possibility to include explicit ∆(1232) degrees349 / 3PoS(LAT2005)349The axial charge of the nucleon on the lattice and in chiral perturbation theory Meinulf Göckeler0.0 0.2 0.4 0.6 0.8 1.0 1.2m2 [GeV2]0.60.81.01.21.4g A=5.20=5.25=5.29=5.40Figure 1: Results for gA. The red triangles correspond to a spatial box length L≈ 1.0fm, the blue trianglesbelong to L≈ 1.3fm, and for the black symbols the volumes are larger. The red star represents the physicalpoint.of freedom in ChEFT. The small expansion parameter in the SSE is called ε , and in O(ε3) the massdependence of gA is given for infinite volume by [6]gSSEA (m2pi) = g0A +[4CHB(λ )− (g0A)316pi2F2pi−25c2Ag1324pi2F2pi+19c2Ag0A108pi2F2pi]m2pi−m2pi4pi2F2pi[(g0A)3+12g0A]ln mpiλ +4c2Ag0A27pi∆0F2pim3pi+[25c2Ag1∆20−57c2Ag0A∆20−24c2Ag0Am2pi]√m2pi −∆2081pi2F2pi ∆0arccos∆0mpi+25c2Ag1(2∆20−m2pi)162pi2F2piln 2∆0mpi+c2Ag0A(3m2pi −38∆20)54pi2F2piln 2∆0mpi+O(ε4) ,(4)where g0A denotes the chiral limit value of gA. This expression depends on several coupling con-stants, all referring to the chiral limit: Fpi is the pion decay constant with the physical value of about92.4 MeV, ∆0 denotes the real part of the N∆ mass splitting, cA and g1 are N∆ and ∆∆ axial couplingconstants, respectively. Finally, CHB(λ ) is a counterterm at the renormalisation scale λ , which canbe expressed in terms of the more conventional heavy-baryon couplings Br9(λ ) and Br20(λ ):CHB(λ ) = Br9(λ )−2g0ABr20(λ ) . (5)Evaluating the underlying ChEFT in a finite spatial volume yields an expression for the L depen-dence of gA [7, 8].Phenomenology provides some information on the parameters appearing above. The analysisof (inelastic) pi N scattering, in particular the process piN → pipiN, suggests that choosing thephysical pion mass as the scale λ one has [6]Br9(λ = mphyspi ) = (−1.4±1.2)GeV−2 , Br20(λ = mphyspi )≡ 0 . (6)349 / 4PoS(LAT2005)349The axial charge of the nucleon on the lattice and in chiral perturbation theory Meinulf Göckeler0.0 0.2 0.4 0.6 0.8 1.0m2 [GeV2]0.81.01.21.4g A=5.20=5.25=5.29=5.40Figure 2: Results for gA with the finite size correction subtracted. The curve represents the first fit describedin the text. The data points shown as open symbols have not been fitted.Therefore we set λ = 0.14GeV in the following and identify CHB(λ = mphyspi ) = Br9(λ = mphyspi ). Inthe real world one has ∆0 = 0.2711GeV, and from an O(ε3) SSE analysis of the ∆ width one findscA = 1.5. At the physical pion mass we have gA = 1.267, while g0A ≈ 1.2 [6]. Little is known aboutg1. In the SU(6) quark model one finds g1 = 95g0A ≈ 951.2 = 2.16. For Fpi one expects in the chirallimit Fpi ≈ 86.2MeV.Unfortunately, we cannot fit all parameters. So we fix ∆0 = 0.2711GeV, cA = 1.5, Fpi =86.2MeV and fit g0A, Br9, g1 taking into account only pion masses below approximately 600 MeV. Incontrast to Ref. [6] the physical point is not fitted. We find g0A = 1.15(12), Br9 =−0.71(18)GeV−2,g1 = 2.6(8) with χ2/dof= 4.23/3. Remarkably enough, these values are very well compatible withour phenomenological prejudices above. In Fig. 2 we plot the data with the finite size correctionsubtracted together with the fit curve. So, if the fit would be perfect the data points which differonly in the volume would collapse onto a single point.The uncertainty in g0A is large enough to cover the experimental point. To exemplify thiscircumstance we fix g0A = 1.225, a value well within the range favoured by the above fit, and useonly Br9 and g1 as fit parameters. We find Br9 = −0.66(17)GeV−2 and g1 = 3.0(3) with χ2/dof =4.61/4. In Fig. 3 we plot the data without subtracting the finite size corrections. Using the resultsof the last fit we show curves not only for L = ∞, but also for the values taken in the simulationsfor β = 5.29, κsea = 0.1359. Of course, many more variations of the fit procedure are possible, butthe overall pattern remains remarkably stable yielding g0A ∼ 1.2, Br9 ∼−0.5 . . .−0.7GeV−2, g1 ∼ 3and Fpi ∼ 90MeV, in accordance with phenomenological expectations.AcknowledgementsThe numerical calculations have been performed on the Hitachi SR8000 at LRZ (Munich), onthe Cray T3E at EPCC (Edinburgh) [9], and on the APEmille at NIC/DESY (Zeuthen). This work349 / 5PoS(LAT2005)349The axial charge of the nucleon on the lattice and in chiral perturbation theory Meinulf Göckeler0.0 0.2 0.4 0.6 0.8 1.0m2 [GeV2]0.60.81.01.21.4g AL=L= 1.91 fmL= 1.27 fmL= 0.95 fmFigure 3: Results for gA with curves for several values of L.is supported in part by the DFG (Forschergruppe Gitter-Hadronen-Phänomenologie) and by the EUIntegrated Infrastructure Initiative Hadron Physics under contract number RII3-CT-2004-506078.References[1] G. Martinelli, C. Pittori, C. T. Sachrajda, M. Testa and A. Vladikas, A General method fornonperturbative renormalization of lattice operators, Nucl. Phys. B445 (1995) 81[hep-lat/9411010].[2] M. Göckeler et al., Nonperturbative renormalisation of composite operators in lattice QCD, Nucl.Phys. B544 (1999) 699 [hep-lat/9807044].[3] M. Della Morte, R. Hoffmann, F. Knechtli, R. Sommer and U. Wolff, Non-perturbative renormalizationof the axial current with dynamical Wilson fermions, JHEP 0507 (2005) 007 [hep-lat/0505026].[4] S. Sasaki, K. Orginos, S. Ohta and T. Blum, Nucleon axial charge from quenched lattice QCD withdomain wall fermions, Phys. Rev. D68 (2003) 054509 [hep-lat/0306007].[5] T. R. Hemmert, B. R. Holstein and J. Kambor, Heavy baryon chiral perturbation theory with lightdeltas, J. Phys. G24 (1998) 1831 [hep-ph/9712496].[6] T. R. Hemmert, M. Procura and W. Weise, Quark mass dependence of the nucleon axial-vectorcoupling constant, Phys. Rev. D68 (2003) 075009 [hep-lat/0303002].[7] S. R. Beane and M. J. Savage, Baryon axial charge in a finite volume, Phys. Rev. D70 (2004) 074029[hep-ph/0404131].[8] T. Wollenweber, Diploma Thesis, Technische Universität München (2005).[9] C. R. Allton et al. [UKQCD Collaboration], Effects of non-perturbatively improved dynamicalfermions in QCD at fixed lattice spacing, Phys. Rev. D65 (2002) 054502 [hep-lat/0107021].349 / 6

The axial charge of the nucleon on the lattice and in chiral
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The axial charge of the nucleon on the lattice and in chiral perturbation theory

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