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Register a new userSix Gluon Open Superstring Disk Amplitude, Multiple Hypergeometric Series and Euler-Zagier Sums
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The six gluon disk amplitude is calculated in superstring theory. This amplitude probes the gauge interactions with six external legs on Dp-branes, in particular including e.g. F^6-terms. The full string S-matrix can be expressed by six generalized multiple hypergeometric functions (triple hypergeometric functions), which in the effective action play an important role in arranging the higher order alpha gauge interaction terms with six external legs (like F^6, D^4 F^4, D^2 F^5, D^6 F^4, D^2 F^6, ...). A systematic and efficient method is found to calculate tree-level string amplitudes by equating seemingly different expressions for one and the same string S-matrix: Comparable to Riemann identities appearing in string-loop calculations, we find an intriguing way of using world-sheet supersymmetry to generate a system of non-trivial equations for string tree-level amplitudes. These equations result in algebraic identities between different multiple hypergeometric functions. Their (six-dimensional) solution gives the ingredients of the string S-matrix. We derive material relevant for any open string six-point scattering process: relations between triple hypergeometric functions, their integral representations and their alpha-(momentum)-expansions given by (generalized) Euler-Zagier sums or (related) Witten zeta-functions.
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We consider scattering processes involving N gluonic massless states of open superstrings with certain Regge slope alpha. At the semi-classical level, the string world-sheet sweeps a disk and N gluons are created or annihilated at the boundary. We present exact expressions for the corresponding amplitudes, valid to all orders in alpha, for the so-called maximally helicity violating configurations, with N=4, 5 and N=6. We also obtain the leading O(alpha^2) string corrections to the zero-slope N-gluon Yang-Mills amplitudes.
We discuss the amplitudes describing N-gluon scattering in type I superstring theory, on a disk world-sheet. After reviewing the general structure of amplitudes and the complications created by the presence of a large number of vertices at the boundary, we focus on the most promising case of maximally helicity violating (MHV) configurations because in this case, the zero Regge slope limit (alpha -> 0) is particularly simple. We obtain the full-fledged MHV disk amplitudes for N=4,5 and N=6 gluons, expressed in terms of one, two and six functions of kinematic invariants, respectively. These functions represent certain boundary integrals - generalized Euler integrals - which for N>= 6 correspond to multiple hypergeometric series (generalized Kampe de Feriet functions). Their alpha-expansions lead to Euler-Zagier sums. For arbitrary N, we show that the leading string corrections to the Yang-Mills amplitude, of order O(alpha^2), originate from the well-known alpha^2 Tr F^4 effective interactions of four gauge field strength tensors. By using iteration based on the soft gluon limit, we derive a simple formula valid to that order for arbitrary N. We argue that such a procedure can be extended to all orders in alpha. If nature gracefully picked a sufficiently low string mass scale, our results would be important for studying string effects in multi-jet production at the Large Hadron Collider (LHC).
The relations in the tautological ring of the moduli space M_g of nonsingular curves conjectured by Faber-Zagier in 2000 and extended to the moduli space of stable pointed curves by Pixton in 2012 are based upon two hypergeometric series A and B. The question of the geometric origins of these series has been solved in at least two ways (via the Frobenius structures associated to 3-spin curves and to CP1). The series A and B also appear in the study of descendent integration on the moduli spaces of open and closed curves. We survey here the various occurrences of A and B starting from their appearance in the asymptotic expansion of the Airy function (calculated by Stokes in the 19th century). Several open questions are proposed.
We propose a class of Pade interpolation problems whose solutions are expressible in terms of determinants of hypergeometric series.
We calculate the chiral string amplitude in pure spinor formalism and take four point amplitude as an example. The method could be easily generalized to $N$ point amplitude by complicated calculations. By doing the usual calculations of string theory first and using a special singular gauge limit, we produce the amplitude with the integral over Dirac $delta$-functions. The Bosonic part of the amplitude matches the CHY amplitude and the Fermionic part gives us the supersymmetric generalization of CHY amplitude. Finally, we also check the dependence on boundary condition for heterotic chiral string amplitudes.
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