Abstract
We reexamine genus one super-Green functions with general boundary conditions twisted by $(\alpha, \beta)$ for $(\sigma, \tau)$ directions in the eigenmode expansion and derive expressions as infinite series of hypergeometric functions. Using these, we compute one-loop superstring amplitudes with non-maximal supersymmetry, taking an example of massless vector emissions of open string type I ${\cal Z}_2$ orbifold.
Highlights
The study of one-loop superstring amplitudes [1, 2] having their bosonic predecessors [3, 4, 5] and that of the attendant genus one Green functions have a long history
These Green functions are needed in order to study scattering properties of particles in superstring compactifications [9, 10, 11] which carry non-maximal supersymmetry and which are soluble by free fields
Our final expression is given by an infinite series consisting of a hypergeometric function, which is relevant to the genus zero Green function1
Summary
The study of one-loop superstring amplitudes [1, 2] having their bosonic predecessors [3, 4, 5] and that of the attendant genus one Green functions have a long history. The number of articles devoted to computations and (phenomenological) applications of superstring amplitudes in recent years are, relatively small and the generalized Green functions that do not satisfy ordinary periodicity or anti-periodicity on the genus one Riemann surfaces in σ or τ directions do not seem to have been systematically studied, according to our search [8], despite that they are after all two point functions of QFT free fields. The terms in the bracket [...] vanish when acting on ∆ = 4∂z∂z We divide this sum into n1 = 0 part and n1 = 0 part to use the partial fraction decomposition in eq (2.6). Where z, z′, θand θdenote respectively the conjugate points of z, z′, θ and θ′
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