Less Singular Terms and Small x Evolution in a Soluble Model

We calculate the effect of the less singular terms at small x on the evolution of the coefficient function in \phi^3 theory in six dimensions, which result from a complete solution of the ladder equation. Scale-invariant next-to-leading order contributions are also studied. We show that the small x approximation does not deliver the dominant contributions.Comment: 7 pages LATEX including 2 eps-file

Two-loop corrections to Higgs boson production

In this paper we present the complete two-loop vertex corrections to scalar and pseudo-scalar Higgs boson production for general colour factors for the gauge group ${\rm SU(N)}$ in the limit where the top quark mass gets infinite. We derive a general formula for the vertex correction which holds for conserved and non conserved operators. For the conserved operator we take the electromagnetic vertex correction as an example whereas for the non conserved operators we take the two vertex corrections above. Our observations for the structure of the pole terms $1/\epsilon^4$, $1/\epsilon^3$ and $1/\epsilon^2$ in two loop order are the same as made earlier in the literature for electromagnetism. However we also elucidate the origin of the second order single pole term which is equal to the second order singular part of the anomalous dimension plus a universal function which is the same for the quark and the gluon. [3mm]Comment: 27 pages LaTeX.We correct some misprints. Moreover we have discovered that the second order single pole term is equal to the singular part of the second order anomalous dimension plus a universal function which is the same for the quark and the gluon. This holds for vertex corrections as well as for scattering amplitude

Norm discontinuity and spectral properties of Ornstein-Uhlenbeck semigroups

Let $E$ be a real Banach space. We study the Ornstein-Uhlenbeck semigroup $P(t)$ associated with the Ornstein-Uhlenbeck operator $$Lf(x) = \frac12 {\rm Tr} Q D^2 f(x) + .$$ Here $Q$ is a positive symmetric operator from $E^*$ to $E$ and $A$ is the generator of a $C_0$-semigroup $S(t)$ on $E$. Under the assumption that $P$ admits an invariant measure $\mu$ we prove that if $S$ is eventually compact and the spectrum of its generator is nonempty, then \n P(t)-P(s)\n_{L^1(E,\mu)} = 2 for all $t,s\ge 0$ with $t\not=s$. This result is new even when $E = \R^n$. We also study the behaviour of $P$ in the space $BUC(E)$. We show that if $A\not=0$ there exists $t_0>0$ such that \n P(t)-P(s)\n_{BUC(E)} = 2 for all $0\le t,s\le t_0$ with $t\not=s$. Moreover, under a nondegeneracy assumption or a strong Feller assumption, the following dichotomy holds: either \n P(t)- P(s)\n_{BUC(E)} = 2 for all $t,s\ge 0$, \ $t\not=s$, or $S$ is the direct sum of a nilpotent semigroup and a finite-dimensional periodic semigroup. Finally we investigate the spectrum of $L$ in the spaces $L^1(E,\mu)$ and $BUC(E)$.Comment: 14 pages; to appear in J. Evolution Equation

Higgs Production at NNLO

We describe the calculation of inclusive Higgs boson production at hadronic colliders at next-to-next-to-leading order (NNLO) in perturbative quantum chromodynamics. We have used the technique developed in reference [4]. Our results agree with those published earlier in the literature.Comment: Talk given at PASCOS'03, TIFR, Mumbai, LaTeX, 5 page

Approximating the coefficients in semilinear stochastic partial differential equations

We investigate, in the setting of UMD Banach spaces E, the continuous dependence on the data A, F, G and X_0 of mild solutions of semilinear stochastic evolution equations with multiplicative noise of the form dX(t) = [AX(t) + F(t,X(t))]dt + G(t,X(t))dW_H(t), X(0)=X_0, where W_H is a cylindrical Brownian motion on a Hilbert space H. We prove continuous dependence of the compensated solutions X(t)-e^{tA}X_0 in the norms L^p(\Omega;C^\lambda([0,T];E)) assuming that the approximating operators A_n are uniformly sectorial and converge to A in the strong resolvent sense, and that the approximating nonlinearities F_n and G_n are uniformly Lipschitz continuous in suitable norms and converge to F and G pointwise. Our results are applied to a class of semilinear parabolic SPDEs with finite-dimensional multiplicative noise.Comment: Referee's comments have been incorporate

Second quantisation for skew convolution products of infinitely divisible measures

Suppose $\lambda_1$ and $\lambda_2$ are infinitely divisible Radon measures on real Banach spaces $E_1$ and $E_2$, respectively and let $T:E_{1} \rightarrow E_{2}$ be a Borel measurable mapping so that $T(\lambda_1) * \rho = \lambda_2$ for some Radon probability measure $\rho$ on $E_{2}$. Extending previous results for the Gaussian and the Poissonian case, we study the problem of representing the `transition operator' $P_{T}:L^{p}(E_{2}, \lambda_{2}) \rightarrow L^{p}(E_{1}, \lambda_{1})$ given by $$P_{T}f(x) = \int_{E_{2}}f(T(x) + y)d\rho(y) unify notations$$ as the second quantisation of a contraction operator acting between suitably chosen `reproducing kernel Hilbert spaces' associated with $\lambda_1$ and $\lambda_2$.Comment: Some typos have been corrected. To appear in IDAQ