The XX-model with boundaries. Part III:Magnetization profiles and boundary bound states

We calculate the magnetization profiles of the $\sigma_j^x$ and $\sigma_j^z$ operators for the XX-model with hermitian boundary terms. We study the profiles on the finite chain and in the continuum limit. The results are discussed in the context of conformal invariance. We also discuss boundary excitations and their effect on the magnetization profiles.Comment: 30 pages, 3 figure

Super congruences and Euler numbers

Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$ $$\sum_{k=0}^{(p-1)/2}\binom{2k}{k}^2/16^k=(-1)^{(p-1)/2}+p^2E_{p-3} (mod p^3)$$, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for $\pi^2$, $\pi^{-2}$ and the constant $K:=\sum_{k>0}(k/3)/k^2$ (with (-) the Jacobi symbol), two of which are $$\sum_{k=1}^\infty(10k-3)8^k/(k^3\binom{2k}{k}^2\binom{3k}{k})=\pi^2/2$$ and $$\sum_{k>0}(15k-4)(-27)^{k-1}/(k^3\binom{2k}{k}^2\binom{3k}k)=K.$

Zero modes, entropy bounds and partition functions

Some recent finite temperature calculations arising in the investigation of the Verlinde-Cardy relation are re-analysed. Some remarks are also made about temperature inversion symmetry.Comment: 12 pages, JyTe

Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms

A unified treatment is given of low-weight modular forms on \Gamma_0(N), N=2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations, which yields a single nonlinear third-order equation, called a generalized Chazy equation. As byproducts, a table of divisor function and theta identities is generated by means of q-expansions, and a transformation law under \Gamma_0(4) for the second complete elliptic integral is derived. More generally, it is shown how Picard-Fuchs equations of triangle subgroups of PSL(2,R) which are hypergeometric equations, yield systems of nonlinear equations for weight-1 forms, and generalized Chazy equations. Each triangle group commensurable with \Gamma(1) is treated.Comment: 40 pages, final version, accepted by Manuscripta Mathematic

Supersymmetric QCD corrections to e+e−→tbˉH−e^+e^-\to t\bar{b}H^- and the Bernstein-Tkachov method of loop integration

The discovery of charged Higgs bosons is of particular importance, since their existence is predicted by supersymmetry and they are absent in the Standard Model (SM). If the charged Higgs bosons are too heavy to be produced in pairs at future linear colliders, single production associated with a top and a bottom quark is enhanced in parts of the parameter space. We present the next-to-leading-order calculation in supersymmetric QCD within the minimal supersymmetric SM (MSSM), completing a previous calculation of the SM-QCD corrections. In addition to the usual approach to perform the loop integration analytically, we apply a numerical approach based on the Bernstein-Tkachov theorem. In this framework, we avoid some of the generic problems connected with the analytical method.Comment: 14 pages, 6 figures, accepted for publication in Phys. Rev.