Scaling and nonscaling finite-size effects in the Gaussian and the mean spherical model with free boundary conditions

We calculate finite-size effects of the Gaussian model in a L\times \tilde L^{d-1} box geometry with free boundary conditions in one direction and periodic boundary conditions in d-1 directions for 2<d<4. We also consider film geometry (\tilde L \to \infty). Finite-size scaling is found to be valid for d3 but logarithmic deviations from finite-size scaling are found for the free energy and energy density at the Gaussian upper borderline dimension d* =3. The logarithms are related to the vanishing critical exponent 1-\alpha-\nu=(d-3)/2 of the Gaussian surface energy density. The latter has a cusp-like singularity in d>3 dimensions. We show that these properties are the origin of nonscaling finite-size effects in the mean spherical model with free boundary conditions in d>=3 dimensions. At bulk T_c in d=3 dimensions we find an unexpected non-logarithmic violation of finite-size scaling for the susceptibility \chi \sim L^3 of the mean spherical model in film geometry whereas only a logarithmic deviation \chi\sim L^2 \ln L exists for box geometry. The result for film geometry is explained by the existence of the lower borderline dimension d_l = 3, as implied by the Mermin-Wagner theorem, that coincides with the Gaussian upper borderline dimension d*=3. For 3<d<4 we find a power-law violation of scaling \chi \sim L^{d-1} at bulk T_c for box geometry and a nonscaling temperature dependence \chi_{surface} \sim \xi^d of the surface susceptibility above T_c. For 2<d<3 dimensions we show the validity of universal finite-size scaling for the susceptibility of the mean spherical model with free boundary conditions for both box and film geometry and calculate the corresponding universal scaling functions for T>=T_c.Comment: Submitted to Physical Review

Charmonium Production in Deep Inelastic Scattering at HERA

The electroproduction of J/psi and psi(2S) mesons is studied in elastic, quasi-elastic and inclusive reactions for four momentum transfers 2 Q^2 80 GeV^2 and photon-proton centre of mass energies 25 W 180 GeV. The data were taken with the H1 detector at the electron proton collider HERA in the years 1995 to 1997. The total virtual photon-proton cross section for elastic J/psi production is measured as a function of Q^2 and W. The dependence of the production rates on the square of the momentum transfer from the proton (t) is extracted. Decay angular distributions are analysed and the ratio of the longitudinal and transverse cross sections is derived. The ratio of the cross sections for quasi-elastic psi(2S) and J/psi meson production is measured as a function of Q^2. The results are discussed in terms of theoretical models based upon perturbative QCD. Differential cross sections for inclusive and inelastic production of J/psi mesons are determined and predictions within two theoretical frameworks are compared with the data, the non-relativistic QCD factorization approach including colour octet and colour singlet contributions, and the model of Soft Colour Interactions