Starting from a simple atomic model giving the potential between electrons and atoms as V(r) = Ze 2 a s−1/sr s with the empirical value s=fraction six-fifths, we combine the diffusion effect due to multiple collisions and the energy retardation in accordance with a modified Thomson-Whiddington law, with the scattering cross section in the Lenard absorption law. On this basis, consistent expressions are obtained for the fraction of transmitted electrons in amorphous solid targets, the backscattering fraction with depth, the fraction of electrons absorbed per unit mass-thickness and the depth-dose function, which are in good agreement with experiments over the energy range 10-1000 keV. A diffusion model represented by a sphere whose centre is located at the maximum energy dissipation depth, related to the diffusion depth and the range, is found to agree well with experiments.

We present the first results of the PACS-CS project which aims to simulate 2 þ 1 flavor lattice QCD on the physical point with the nonperturbatively OðaÞ-improved Wilson quark action and the Iwasaki gauge action. Numerical simulations are carried out at ¼ 1:9, corresponding to the lattice spacing of a ¼ 0:0907ð13Þ fm, on a 32 3 Â 64 lattice with the use of the domain-decomposed HMC algorithm to reduce the up-down quark mass. Further algorithmic improvements make possible the simulation whose up-down quark mass is as light as the physical value. The resulting pseudoscalar meson masses range from 702 MeV down to 156 MeV, which clearly exhibit the presence of chiral logarithms. An analysis of the pseudoscalar meson sector with SU(3) chiral perturbation theory reveals that the next-to-leading order corrections are large at the physical strange quark mass. In order to estimate the physical up-down quark mass, we employ the SU(2) chiral analysis expanding the strange quark contributions analytically around the physical strange quark mass. The SU(2) low energy constants " l 3 and " l 4 are comparable with the recent estimates by other lattice QCD calculations. We determine the physical point together with the lattice spacing employing m , m K and m as input. The hadron spectrum extrapolated to the physical point shows an agreement with the experimental values at a few % level of statistical errors, albeit there remain possible cutoff effects. We also find that our results of f , f K and their ratio, where renormalization is carries out perturbatively at one loop, are compatible with the experimental values. For the physical quark masses we obtain m MS ud and m MS s extracted from the axial-vector Ward-Takahashi identity with the perturbative renormalization factors. We also briefly discuss the results for the static quark potential.

A report is presented on our continued effort to elucidate the continuum limit of BK using the quenched Kogut-Susskind quark action. By adding to our previous simulations one more point at β = 6.65 employing a 56 3 × 96 lattice, we now confirm the expected O(a 2 ) behavior of BK with the Kogut-Susskind action. A simple continuum extrapolation quadratic in a leads to BK (NDR, 2 GeV) = 0.598(5). As our final value of BK in the continuum we present BK (NDR, 2 GeV)=0.628(42), as obtained by a fit including an α M S (1/a) 2 term arising from the lattice-continuum matching with the one-loop renormalization.

We present results of a large-scale simulation for the flavor non-singlet light hadron spectrum in quenched lattice QCD with the Wilson quark action. Hadron masses are calculated at four values of lattice spacing in the range a ≈ 0.1 -0.05 fm on lattices with a physical extent of 3 fm at five quark masses corresponding to m π /m ρ ≈ 0.75 -0.4. The calculated spectrum in the continuum limit shows a systematic deviation from experiment, though the magnitude of deviation is contained within 11%. Results for decay constants and light quark masses are also reported.

The problem of whether there is a constraint on the number of flavors for quark confinement in QCD is numerically investigated on a lattice with Wilson fermions as quarks. It is shown that even in the strong coupling limit, when the number of flavors exceeds 7, quarks are not confined and chiral symmetry is not spontaneously broken for light quarks. PACS numbers: 12.38.Gc, ll.30.RdThe fundamental properties of QCD are quark confinement, asymptotic freedom, and spontaneous breakdown of chiral symmetry. It is well known that if the number of flavors exceeds 17, asymptotic freedom is lost. Then the question which naturally arises is whether there is a constraint on the number of flavors for quark confinement and/or the spontaneous breakdown of chiral symmetry. Here we would like to investigate numerically quark confinement and chiral symmetry versus the number of flavors, taking the Wilson formalism [1] of fermions on the lattice for quarks, because this is the only known formalism which describes any number of flavors in terms of a local action. We use the same method as in a previous paper [2] to discriminate the phases of QCD with various numbers of flavors: With the quark mass defined through the axial-vector-current Ward identity, the pion mass at zero quark mass determines whether chiral symmetry is spontaneously broken or not. It will turn out that confinement is closely related with the spontaneous breakdown of chiral symmetry.We generate gauge configurations using the hybridmolecular-dynamics R algorithm [3] with the molecular dynamics time step AT =0.01, unless otherwise stated. The inversion of the quark matrix (x =Z) ~xb) is made by a minimal residual method or a conjugate gradient (CG) method. The lattice sizes are 8 2 xl0xT (r=4, 6, or 8) and 18 2 x24xr (r = 18). When the hadron spectrum is calculated in the former case, the lattice is duplicated in the direction of the lattice size 10, which we call the z direction. We use an antiperiodic boundary condition for quarks in the / direction and periodic boundary conditions otherwise.We investigate confinement in the strong coupling limit 0=0.0 (g"* 00 , /3=6/g 2 ). Although quark confinement is rigorously proved at 0=0.0 in the pure gauge theory when the action is local as in the case of the standard one-plaquette action [4], there is no proof for confinement in full QCD.Let us begin with the case of A^ = 18, because asymptotic freedom is lost for N/>\1 and therefore we may expect quark nonconfinement here. We take 7=4. Shown in Fig. 1 (a) are the results of the Polyakov loop and the Wilson loop W(\ x l) for various hopping parameters. The data with large symbols are for long runs: They are obtained by averaging over the last r =500 [at each T = integer here and in the following for W(\x\) and Polyakov loop] after a thermalization of r=300-

By Monte Carlo simulation we study the critical exponents governing the transition of the three-dimensional classical O(4) Heisenberg model, which is considered to be in the same universality class as the finite-temperature QCD with massless two flavors. We use the single cluster algorithm and the histogram reweighting technique to obtain observables at the critical temperature. After estimating an accurate value of the inverse critical temperature K c = 0.9360(1), we make nonperturbative estimates for various critical exponents by finite-size scaling analysis.They are in excellent agreement with those obtained with the 4−ǫ expansion method with errors reduced to about halves of them.

We investigate the phase structure of lattice QCD for the general number of flavors in the parameter space of gauge coupling constant and quark mass, employing the one-plaquette gauge action and the standard Wilson quark action. Performing a series of simulations for the number of flavors N F ϭ6 -360 with degenerate-mass quarks, we find that when N F у7 there is a line of a bulk first order phase transition between the confined phase and a deconfined phase at a finite current quark mass in the strong coupling region and the intermediate coupling region. The massless quark line exists only in the deconfined phase. Based on these numerical results in the strong coupling limit and in the intermediate coupling region, we propose the following phase structure, depending on the number of flavors whose masses are less than ⌳ d which is the physical scale characterizing the phase transition in the weak coupling region: When N F у17, there is only a trivial IR fixed point and therefore the theory in the continuum limit is free. On the other hand, when 16уN F у7, there is a nontrivial IR fixed point and therefore the theory is nontrivial with anomalous dimensions, however, without quark confinement. Theories which satisfy both quark confinement and spontaneous chiral symmetry breaking in the continuum limit exist only for N F р6.

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