We calculate finite size effects on nucleon masses in chiral perturbation
theory. We confront the theoretical predictions with N_f=2 lattice results and
discuss chiral extrapolation formulae.Comment: talk at Lattice 03 (spectrum), 3 pages latex, 3 figures. Assignment
  of 2 data points to incorrect data sets in plot 1 and of 1 data point in plot
  2 corrected. 1 fm lattice result updated. Conclusions unchange

A. Ali Khan

A.C. Irving

Ali Khan

Aoki

Becher

Bernard

D. Pleiter

G. Schierholz

H. Stüben

Hasenfratz

M. Göckeler

P.E.L. Rakow

Procura

R. Horsley

T. Bakeyev

T.R. Hemmert

English

arXiv

Crossref

We calculate finite size effects on nucleon masses in chiral perturbation theory. We confront the theoretical predictions with N_f=2 lattice results and discuss chiral extrapolation formulae

Ali Khan, A

Bakeyev, T

Göckeler, M

Hemmert, TR

Horsley, R

Irving, AC

Pleiter, D

Rakow, PEL

Schierholz, G

Stüben, H

QCDSF Collaboration

UKQCD Collaboration

University of Liverpool Repository

arXiv:hep-lat/0309133v3  14 Jan 2004HU-EP-03/50DESY 03-143LU-ITP 2003/016LTH 590Edinburgh 2003/17Chiral Perturbation Theory and Finite Size Effects on the Nucleon Massin unquenched QCD ∗A. Ali Khana, T. Bakeyevb, M. Go¨ckelerc,d, T.R. Hemmerte, R. Horsleyf , A.C. Irvingg, D. Pleiterh,P.E.L. Rakowg, G. Schierholzh,i, and H. Stu¨benj (QCDSF and UKQCD Collaborations)aInstitut fu¨r Physik, Humboldt-Universita¨t zu Berlin, 12489 Berlin, GermanybJoint Institute for Nuclear Research, 141980 Dubna, RussiacInstitut fu¨r Theoretische Physik, Universita¨t Leipzig, 04109 Leipzig, GermanydInstitut fu¨r Theoretische Physik, Universita¨t Regensburg, 93040 Regensburg, GermanyePhysik-Department, Theoretische Physik T39, TU Mu¨nchen, 85747 Garching, GermanyfSchool of Physics, The University of Edinburgh, Edinburgh EH9 3JZ, UKgTheoretical Physics Division, Department of Mathematical Sciences, University of Liverpool,Liverpool L69 3BX, UKhJohn von Neumann-Institut fu¨r Computing NIC, 15738 Zeuthen, GermanyiDeutsches Elektronen-Synchrotron DESY, 22603 Hamburg, GermanyjKonrad-Zuse-Zentrum fu¨r Informationstechnik Berlin, 14195 Berlin, GermanyWe calculate finite size effects on nucleon masses in chiral perturbation theory. We confront the theoreticalpredictions with Nf = 2 lattice results and discuss chiral extrapolation formulae.1. INTRODUCTIONFinite size effects, in particular on dynamicalconfigurations, can be a serious impediment toprecision lattice calculations of hadron massesand matrix elements. It is of interest to find atheoretical description of finite size effects, to beable to determine the finite volume error at eachlattice size and, possibly, to find an extrapolationformula to infinite volume.On the lattice sizes used in production runs,the nucleons and pions are the relevant degreesof freedom for understanding the finite size ef-fects on the nucleon mass. To calculate them, weuse two-flavor relativistic baryon chiral perturba-tion theory (χPT ) at O(q3) in the chiral counting(one-loop order).∗presented by A. Ali Khan2. LATTICE PARAMETERSWe base our finite volume study on QCDSFand UKQCD configurations, using a plaque-tte gauge action with two flavors of non-perturbatively O(a)-improved Wilson fermions (adenotes the lattice spacing). The pion masses arein the interval 0.4−1 GeV. Valence and sea quarkmasses are equal. Lattices volumes are 1 − 2.2fm. The scale is set with r0, using the value atthe physical point r0 = 0.5 fm ≃ 1/(395MeV).We compare our results to JLQCD data with thesame lattice actions and range of simulation pa-rameters on varying lattice sizes [1]. In Fig. 1we plot nucleon masses of both groups. It isfound that the masses increase when the latticesize is decreased. The data from lattices of anextent ≥ 1.8 fm lie approximately on the same120 0.2 0.4 0.6 0.8 1mPS2[GeV2]0.811.21.41.61.82mN[GeV]L = 1-1.3 fmL = 1.4-1.7 fmL > 1.8 fmexperimentFigure 1. Nucleon masses from UKQCDSF(white symbols) and JLQCD (gray symbols).curve. In principle there are two major sourcesof systematic error: finite volume and discretiza-tion effects. Among the data points, the latticespacing varies between 0.4 and 0.6 GeV−1. Thea → 0 limit was not performed, and we have toascertain that there are no sizable discretizationerrors in the results. For example, between twodata points on approximately the same volumeand at the same pion mass, but with a varyingby ∼ 30%, the nucleon mass is found to remainunchanged within the statistical errors. We alsocompare our large lattice results (L ≥ 1.8 fm)with two CP-PACS data sets [2] using renormal-ization group improved gauge fields and tree-leveltadpole-improved clover quarks at a−1 ≈ 1.5 and2 GeV respectively. Their box sizes are ≥ 2.5fm. A compilation of the large L data is given inFig. 2. We find a good scaling and conclude thatin this data O(a) uncertainties are small.3. CHIRAL EXTRAPOLATIONThe one-loop contribution is generated by theO(q1) Lagrangian L(1)N :L(1)N = Ψ¯ (iγµDµ −m0)Ψ +12gAΨ¯γµγ5uµΨ, (1)with Dµ = ∂µ +12 [u†, ∂µu], uµ = iu†∂µUu† andu2 = U .We use the infrared regularization schemewhich is discussed in detail in [3]. To computethe renormalized nucleon mass mN , we add attree-level the L(2)N term −4c1m2PS and an addi-tional term of the form e1m4PS , which is, strictlyspeaking, derived from L(4)N . The renormalizationprocedure is detailed in [4]. In this calculation weuse gA = 1.2 [5], and F = 92.4 MeV. We deter-mine the nucleon mass in the chiral limit, m0, thevalue of c1 and the renormalized e1 (er1) by a fit tosix lattice data points at the smallest masses, andfind m0 = 0.85(14)GeV, c1 = −0.80(18)GeV−1and er1(1GeV) = 2.8(1.1)GeV−3. For the defini-tion of er1 see [4]. In Fig. 2, the result is comparedwith the lattice data on volumes > 1.8 fm. TheχPT result shown here differs numerically slightlyfrom the one quoted in [4] since they used thevalue gA = 1.267 and a fit to a larger set of latticepoints. Expanding the χPT result up to O(m3PS),one obtains the non-relativistic (NR) approxima-tion. The NR theory is valid only in the limitmPS ≪ mN . This is reflected in the breakdownof the curves already at small pion masses, whichis also shown in Fig. 2. A method to push thevalidity of the NR approximation to higher mo-mentum scales within a cutoff scheme is describedin [6].4. FINITE SIZE EFFECTSWe calculate the finite size effect from theone-loop O(q3) contribution to the self energy.Putting the external nucleon line on-shell, thisis given by [3]Σ(q/ = m0) = −i3g2Am0m2PS2F 2∫ ∞0dx∫d4p(2π)4[p2 −m20x2 −m2PS(1− x) + iǫ]−2(2)in Minkowski space. We defineδ =1m0(Σ(q/ = m0, L)− Σ(q/ = m0,∞)) (3)where the temporal extent of the lattice is as-sumed to be infinite. Using [7], one can express δas an integral over Bessel functions:δ =3g2Am2PS16π2F 2∫ ∞0dx×30 0.2 0.4 0.6 0.8 1mPS2[GeV2]0.811.21.41.61.82mN[GeV]JLQCDCP-PACSUKQCDSFexperimentFigure 2. Nucleon mass compared to χPT . Thesolid curve denotes the fit with relativistic χPT ,and the dashed curve the non-relativistic limit us-ing the same values ofm0 and c1. The dot-dashedcurve shows a non-relativistic result with esti-mated parameters,m0 = 0.81 GeV and c1 = −1.1GeV−1.×∑~n6=0K0(L|~n|√m20x2 +m2PS(1− x)). (4)To calculate the nucleon mass in a finite box,mN(L) = (1 + δ)mN (∞), (5)we first extrapolate to mN (∞) by using the lat-tice result on the largest available lattice as inputto Eq.(5). We are then able to calculate mN (L)at finite lattice extent. Lattice results for the nu-cleon mass at fixed values of β and κ, but dif-ferent lattice sizes are compared with chiral per-turbation theory in Fig. 3. It is found that thefinite size effect can be well described by rela-tivistic χPT . In contrast, in NR χPT at O(q3)we obtain only roughly ∼ 30% of the finite vol-ume effect in the lattice data [8]. At pion masses≥ 500 MeV, large loop momenta give a substan-tial contribution also to the finite size effect. Itis of interest to study whether this dependence isreduced in the relativistic formalism.AcknowledgementsThis work is supported in part by the Deutsche0 0.5 1 1.5 2 2.5 3L[fm]1.21.41.61.822.2mN[GeV]mPS=0.717 GeV mPS=0.545 GeVmPS=0.732 GeVFigure 3. Comparison of the finite size depen-dence of lattice nucleon mass data sets at a fixed(β, κ) value and χPT . The data sets are labeledby the pion mass on the largest volume.Forschungsgemeinschaft. The simulations weredone on the APEmille at NIC (Zeuthen), theHitachi SR8000 at LRZ (Munich) and the CrayT3E at EPCC (Edinburgh) and NIC (Ju¨lich). Wethank all institutions for their support.REFERENCES1. JLQCD Collaboration, S. Aoki et al.,Phys.Rev. D68 (2003) 054502.2. CP-PACS Collaboration, A. Ali Khan et al.,Phys.Rev. D65 (2002) 054505; Erratum ibid.D67 (2003) 059901.3. T. Becher and H. Leutwyler, Eur.Phys.J. C9(1999) 643.4. M. Procura, T.R. Hemmert and W. Weise,hep-lat/0309020.5. M. Procura, T.R. Hemmert and W. Weise,hep-lat/0303002.6. V. Bernard, T.R. Hemmert and U.-G.Meißner, hep-ph/0307115.7. P. Hasenfratz and H. Leutwyler, Nucl. Phys.B343 (1990) 241.8. A. Ali Khan et al., Nucl. Phys. B (Proc.Suppl.) 119 (2003) 419.

Chiral perturbation theory and finite size effects on the nucleon mass in unquenched QCD

Chiral Perturbation Theory and Finite Size Effects on the Nucleon Mass
  in unquenched QCD

Chiral Perturbation Theory and Finite Size Effects on the Nucleon Mass in unquenched QCD

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