Finite volume dependence of hadron properties and lattice QCD

Abstract

Abstract. Because the time needed for a simulation in lattice QCD varies at a rate exceeding the fourth power of the lattice size, it is important to understand how small one can make a lattice without altering the physics beyond recognition. It is common to use a rule of thumb that the pion mass times the lattice size should be greater than (ideally much greater than) four (i.e., mπL ≫ 4). By considering a relatively simple chiral quark model we are led to suggest that a more realistic constraint would be mπ(L − 2R) ≫ 4, where R is the radius of the confinement region, which for these purposes could be taken to be around 0.8-1.0 fm. Within the model we demonstrate that violating the second condition can lead to unphysical behaviour of hadronic properties as a function of pion mass. In particular, the axial charge of the nucleon is found to decrease quite rapidly as the chiral limit is approached. IntroductionOur current capacity to compute hadron properties in lattice QCD is constrained by the need to take many limits. For example, we need to take the spacing a → 0, the size of the lattice L → ∞ and the quark mass to around 5 MeV. Of course, these limits are not unconnected because the size of a region of space big enough to contain (say) a proton will need to grow with the Compton wavelength of the pion. At present, with lattice spacings that represent a reasonable approximation to the continuum limit and for full QCD with reasonable chiral symmetry, the time for a lattice simulation scales roughly like m −9 π and we are limited to pion masses larger than 0.4-0.5 GeV. This has led to much effort to explore the application of chiral perturbation theory as a tool for extrapolating hadron properties to the physical pion mass [1][2][3].In such an environment there is great interest in seeing whether one can lower the pion mass without increasing the size of the lattice, thereby saving a factor of m 4 π . Indeed, there have been many calculations for which even the rather optimistic rule of thumb that m π L > 4 has not been satisfied. Considerable attention has been devoted to studying such systems within the framework of effective field theory in order to understand how the relevant path integral might change as we go from one regime to another [4,5].The question we ask is somewhat different. We consider a simple chiral quark model upon which we impose boundary conditions which roughly approximate those on a lattice. Simple inspection of the solutions naturally leads one to conclude that the condition noted above is incorrect. Indeed, the pion cloud of the nucleon does not even begin until one is outside the region of space in which the valence quarks are confined. Within the bag model this is characterised by the bag radius, R, and within the cloudy bag model as well as the model

Nucleon form factors with2+1flavor dynamical domain-wall fermions

“…We suspect that this pion mass dependence driving g A away from the experimental value is caused by the finite volume of our calculation. Similar behavior was observed in quenched DWF studies [1,20] and was predicted in a model calculation [50]. However, for pion masses close to our lightest point such a sizable shift is not observed when V is larger than about (2.4 fm) 3 , not only in the quenched case, but also the 2+1 flavor, mixed action calculation in [26] and their updated results [29].…”

Axial Form Factor of the Nucleon in a Finite Spherical Cavity

“…Careful studies of the axial charge of the proton by the RBC-UKQCD collaboration [5] have shown a very strong dependence on the size of the lattice used, with the axial charge decreasing rather rapidly at small pion mass (below 0.3-0.4 GeV) as the lattice size goes below 3 fm. Hints of such behaviour had been seen in earlier lattice simulations [6,7,8] and a study of the axial charge for a proton in a finite volume using a chiral bag model did indeed predict results very similar [9] to those found in Ref. [5].…”

“…By combining the linear and non-linear boundary conditions, we obtain two eigenvalue equations to be solved for Ω and R The solutions to these equations were shown in Ref. [9], with the pion field, which is of particular interest, growing from the origin to the bag radius and only then starting to decay. For this reason the condition that the lattice volume be large enough to contain the hadron including its pion field is not d ≡ 2L 4m −1 π but rather d ≡ 2L 2R + 4m −1 π .…”

Nucleon Axial Charge in (2+1)-Flavor Dynamical-Lattice QCD with Domain-Wall Fermions

“…However, it is difficult to control finite-volume effects (FVE's) in g A , as suggested by quenched calculations [8]. The FVE's have been investigated in effective models [9,10], and also in heavy baryon chiral perturbation theory (HBChPT) [11][12][13]. The FVE's in HBChPT are inconsistent with lattice calculations unless contributions of the baryon resonance are included.…”