One-loop Case Research Articles

Operator approach to bosonic string: Multiloop calculation and b-ghost insertion.

We construct the /ital g/-loop /ital N/-string vertex using the Feynman-likerules in the operator formalism with bosonized /ital bc/ ghosts. The measurefactor is analyzed in detail, which consists of the ghost /ital g/ vacuum andthe contour integration associated with the /ital b/-ghost insertion. Thegeneral form of the /ital g/ vacuum is derived using the handle operators. Thehandle operator is formulated by the geometrical quantities: period matrix,first Abelian integral, prime form, 1/2 differential, andRiemann constant. Explicit results are given up to two-loop level. The contourintegrations associated with the ghost insertions can be performed explicitlyin the one-loop case, in which we reproduce the well-known formula of theone-loop /ital N/-tachyon amplitude.

FUNCTIONAL INTEGRAL METHODS FOR FERMIONIC STRING AND SUPER RIEMANN SURFACES

We investigate the light-cone gauge formulation of fermionic string of Mandelstam in a superspace. To formulate the fermionic string in a superspace, we use the theory of super Riemann surfaces (SRS). We define the Neumann functions and the Mandelstam mappings in a superspace by means of the so-called Abelian differentials of the first and the third kinds on SRS. In the super Schottky parametrization of SRS these superdifferentials are constructed at the multiloop level. The functional integral measure of a super light-cone diagram, which consists of 6g−6+2N even moduli parameters and 4g−4+2N odd ones, are specified in our formulation. In one-loop case with even spin structures we explicitly evaluate the superdeterminants on the super light-cone diagrams and calculate the one-loop amplitudes with N massless vector states. It is shown that the result is written in the form of a correlation function of the vertex operators. Furthermore, we evaluate the 3- and 4-particle amplitudes explicitly, which agree exactly with those calculated in the Green-Schwarz formulation.

Operator construction of ghost g-vacuum, handle operator and BRST-invariant multiloop amplitude of the bosonic string

Applying the Feynman-like rules in the operator formalism with bosonized bc-ghost, we construct the g-loop, N-string vertex out of three-string vertices and propagators. The measure factor is analyzed in detail. It is shown that a part of the ghost contribution can be represented by using g 2 -differential cand Riemann constants. The resulting measure factor is represented by the ghost g-vacuum with the b-ghost insertion accompanied by a contour integration. This contour integration can be performed explicitly in the one-loop case, in which we reproduce the well-known formula of the one-loop N-tachyon amplitude.