Ali Khan, A

Bakeyev, T

Go¨ckeler, M

Horsley, R

Irving, AC

Pleiter, D

QCDSF Collaboration

Rakow, P

Schierholz, G

Stu¨ben, H

UKQCD Collaboration

University of Liverpool Repository

arXiv:hep-lat/0209111v1  12 Sep 20021HU-EP-02/39LU-ITP 2002/020Edinburgh 2002/10LTH 559Finite Size Effects in Nucleon Masses in Dynamical QCD ∗A. Ali Khana, T. Bakeyevb, M. Go¨ckelercd, R. Horsleye, A. C. Irvingf , D. Pleiterg, P. Rakowdf ,G. Schierholzgh, and H. Stu¨beni (QCDSF and UKQCD Collaborations)aInstitut fu¨r Physik, Humboldt-Universita¨t zu Berlin, 10115 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, GermanyeSchool of Physics, The University of Edinburgh, Edinburgh EH9 3JZ, UKfTheoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, LiverpoolL69 3BX, UKgJohn von Neumann-Institut fu¨r Computing NIC, 15738 Zeuthen, GermanyhDeutsches Elektronen-Synchrotron DESY, 22603 Hamburg, GermanyiKonrad-Zuse-Zentrum fu¨r Informationstechnik Berlin, 14195 Berlin, GermanyFor lattice calculations with light dynamical quarks, finite size effects have become an important aspect. Westudy finite size effects in nucleon masses on Nf = 2 dynamical lattices of 1− 2 fm. Predictions for the finite sizeeffects are obtained in one-loop chiral perturbation theory.1. INTRODUCTIONFor a lattice calculation of physical quantitiessuch as hadron masses it is desirable to have a the-oretical understanding of finite size effects and tobe able to extrapolate to infinite volume. On lat-tices used in typical quenched calculations, finitesize effects of hadron masses seem to be small,but they may be considerable in the dynamicalcase [1]. A source of finite size effects is a virtualpion going around the boundary of the lattice be-fore it is absorbed again. At sufficiently smallpion masses and large volumes this effect can bedescribed by chiral perturbation theory, the low-energy effective theory of nucleons and pions, in∗presented by A. Ali Khana finite box. Here, we discuss finite size effectson nucleon masses generated by the QCDSF andUKQCD collaborations.2. THE SIMULATIONThe simulations were done using the pla-quette gauge action and Nf = 2 dynamical non-perturbatively O(a)-improved Wilson fermions.We consider lattices of size ∼ 1 fm, ∼ 1.6 fm,and ∼ 2 fm. Pion masses are in the range ofabout 0.6− 1 GeV. The lattice spacing is a ≈ 0.1fm. We use r0 = 0.5 fm to set the scale. Valenceand sea quark masses are taken to be equal.The lattice data for the nucleon mass are plot-ted in Fig. 1. We see significant finite size effects.We expect O(a2) effects to be small. Indeed, our2two mass points at r0mPS ≃ 2 on the 1.6 fmlattice differ by less than a percent, whereas a2differs by O(20)%.0 4 8(mPSr0)222.42.83.23.644.44.85.2mNr 0Ls = 1 fmLs = 1.6 fmLs = 2 fmexperimentFigure 1. Nucleon mass as a function of the pionmass with extrapolation to the physical value ac-cording to Eq. (1). The solid line denotes a fit tothe 1.6 fm lattices. The dashed line correspondsto the 2 fm lattices, without error bar on the cor-responding chiral extrapolation, since there areonly three points. The heaviest point at 1.6 fm isnot included in the fit.Chiral perturbation theory in the infinite vol-ume predictsmNr0 = a+ b(mPSr0)2 + c(mPSr0)3, (1)where the cubic term reflects the leading non-analytic (LNA) behaviour of the nucleon self-energy. The chiral extrapolation is performedseparately for different lattice sizes using Eq. (1).At 1.6 fm, the chiral extrapolation gives a valuethat is 8% or 2σ higher than experiment, at 2fm no discrepancy is found. The coefficient c dis-agrees significantly with the infinite volume LNAprediction.3. THEORETICAL PREDICTIONSThe finite size effects are calculated from thedifference of the nucleon self-energy Σ in finiteand infinite spatial volume,δmN = Σ(Ls)− Σ(∞). (2)In the perturbative calculation, the x4 directionis taken to be of infinite extent. We calculate theself-energy at one loop in Heavy Baryon χPT ac-cording to the lattice prescription given in Ref. [2]adapted to the case of 2-flavours. Additionally,we include ∆ intermediate states. For this we dis-cretise the N∆pi interaction Lagrangian given inRef. [3]. The numerical values for fpi, gA = D+F ,and the N∆pi coupling cA are also taken fromthis reference. The relevant Feynman diagramsare given in Fig. 2.To regularize the lattice integrals we vary thecut-off Λ between pi/a = 6 GeV and ∞. Thedifference between using Λ = 6 GeV and the con-tinuum limit is around 10% for mPS ≃ 600 MeV.The values presented here are for Λ ≈ 30 GeV,which practically amounts to infinite cut-off. Re-sults for the smallest pion mass used in the simu-lation at 1.6 fm are given in Fig. 3. For Ls ≥ 1.6fm, the Ls dependence can be approximated byan exponential decay,δmN = (c0/Ls) exp(−c1Ls), (3)where the fit with Eq. (3) describes the χPT re-sult with an accuracy of several percent.The finite size effect of the nucleon mass hasbeen estimated previously, relating the mass shiftto the piN scattering amplitude [4]. A comparisonwith our one-loop result is given in Fig. 3.In Fig. 4, we show mN − δmN . We see thatthe masses on the smaller volumes are broughtdown towards a universal curve. In particular,50% of the finite size corrections at Ls = 1 fm,and 60% at Ls = 1.6 fm, are accounted for bychiral perturbation theory. We consider this aremarkable result, considering that the pion massis relatively heavy.4. CONCLUSIONS AND DISCUSSIONWe find indications for finite size effects in thenucleon mass on lattices of 1.6 − 2 fm size. We3Ν Νpia)Ν Ν Νpib)Ν Νpi∆c)Figure 2. Self-energy diagrams relevant for thecalculation at one loop.calculated the finite volume corrections in lattice-regularised χPT at one loop. At Ls ≥ 1.6 fm,the correction is of the order of a few percentfor the pion masses considered. χPT describesabout 50 − 60% of the finite size effects of thedata. The mass dependence is exponential ratherthan power-like even for intermediate lattices andpion masses. Higher order corrections to the one-loop calculation may be important at present pionmasses.ACKNOWLEDGEMENTSWe acknowledge useful discussions withTh. Hemmert. The numerical calculations wereperformed on the Hitachi SR8000 at LRZ (Mu-nich), the APEmille at NIC (Zeuthen), the CrayT3E at EPCC (Edinburgh) and NIC (Ju¨lich).We wish to thank all institutions for their sup-port. This work has been supported in part bythe European Community’s Human PotentialProgram under contract HPRN-CT-2000-00145,Hadrons/Lattice QCD.REFERENCES1. A.C. Irving, these proceedings,T. Kaneko, Nucl. Phys. Proc. Suppl. 106, 133(2002).2. R. Lewis and P.A. Ouimet, Phys. Rev. D 64,0340051 (2001).3. Th. Hemmert and W. Weise, hep-lat/0204005.4. M. Lu¨scher, Carge`se Lectures 1983.0 1 2 3 4Ls[fm]1e-061e-050.00010.0010.010.1δmN[GeV]N only, 1 loop internal ∆, 1 loopRef. 4Figure 3. Ls dependence of δmN for mPS = 578MeV. Circles denote the sum of diagrams a) andb) of Fig. 2, triangles the contribution of diagramc) of Fig. 2, and plusses the result of Ref. [4]. Fitsto δmN = (c0/Ls) exp(−c1Ls) are drawn as solidlines.0 2 4 6 8(mPSr0)222.42.83.23.644.44.85.2(mN − δmN)r 0Ls = 1 fmLs = 1.6 fmLs = 2 fmexperimentFigure 4. Nucleon masses minus the finite sizeeffect calculated in one-loop χPT . As in Fig. 1,lines denote fits according to Eq. (1).

Finite size effects in nucleon masses in dynamical QCD

Finite size effects in nucleon masses in dynamical QCD

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