Dynamical differential equations compatible with rational qKZ equations

For the Lie algebra $gl_N$ we introduce a system of differential operators called the dynamical operators. We prove that the dynamical differential operators commute with the $gl_N$ rational quantized Knizhnik-Zamolodchikov difference operators. We describe the transformations of the dynamical operators under the natural action of the $gl_N$ Weyl group.Comment: 7 pages, AmsLaTe

Difference Equations Compatible with Trigonometric KZ Differential Equations

The trigonometric KZ equations associated with a Lie algebra \g depend on a parameter \lambda\in\h where \h\subset\g is the Cartan subalgebra. We suggest a system of dynamical difference equations with respect to $\lambda$ compatible with the KZ equations. The dynamical equations are constructed in terms of intertwining operators of \g-modules.Comment: 23 pages, AmsTeX, third version, some misprints were correcte

Selberg Type Integrals Associated with sl3sl_3

We present several formulae for the Selberg type integrals associated with the Lie algebra $sl_3$.Comment: Errata added; 13 pages, amstex.tex 2.2 and amssym.tex require

Quark loop contribution to \pi^0 \to 4\gamma

We find the contribution of constituent quark loop mechanism to the branching ratio B_{4\gamma} = \Gamma_{4\gamma}/\Gamma_{4\gamma} \sim 5.45 \cdot 10^{-16} for the reasonable choice of constituent quark mass m \sim 280 MeV. This result is in agreement with vector-dominance approach result obtained years ago. Thus the main contribution arises from QED mechanism \pi^0 \to \gamma (\gamma^*) \to \gamma (3\gamma) including light-light scattering block with electron loop. This contribution was investigated in paper of one of us and gave B_{4\gamma} \sim 2.6 \cdot 10^{-11}.Comment: 7 pages, 2 figure

Counting master integrals: Integration by parts vs. functional equations

We illustrate the usefulness of functional equations in establishing relationships between master integrals under the integration-by-parts reduction procedure by considering a certain two-loop propagator-type diagram as an example.Comment: 8 pages, 1 figur