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On the field theory expansion of superstring five point amplitudes

Rutger H. Boels

TL;DR

This paper develops a recursive, symbol-level method to obtain the α' (field theory) expansion of tree-level five-point open-string amplitudes and extends the framework to closed strings via KLT, with a focus on MRV amplitudes. It leverages hypergeometric function recursions, Mellin-Barnes representations, and Molien’s theorem to organize high-weight MZV structures and invariant polynomials, yielding explicit results up to weight 21 for open and MRV closed-string cases. A nine-element τ-basis for completely symmetric five-point polynomials enables compact representation of symmetric kinematics, and the work provides concrete MRV expressions up to α'^{10} alongside insights into transcendental structures and potential inconsistencies. The methods and data open pathways for higher-point generalizations, effective-action reconstruction, and deeper connections between string amplitudes, permutation-invariant polynomials, and modular forms.

Abstract

A simple recursive expansion algorithm for the integrals of tree level superstring five point amplitudes in a flat background is given which reduces the expansion to simple symbol(ic) manipulations. This approach can be used for instance to prove the expansion is maximally transcendental to all orders and to verify several conjectures made in recent literature to high order. Closed string amplitudes follow from these open string results by the KLT relations. To obtain insight into these results in particular the maximal R-symmetry violating amplitudes (MRV) in type IIB superstring theory are studied. The obtained expansion of the open string amplitudes reduces the analysis for MRV amplitudes to the classification of completely symmetric polynomials of the external legs, up to momentum conservation. Using Molien's theorem as a counting tool this problem is solved by constructing an explicit nine element basis for this class. This theorem may be of wider interest: as is illustrated at higher points it can be used to calculate dimensions of polynomials of external momenta invariant under any finite group for in principle any number of legs, up to momentum conservation.

On the field theory expansion of superstring five point amplitudes

TL;DR

This paper develops a recursive, symbol-level method to obtain the α' (field theory) expansion of tree-level five-point open-string amplitudes and extends the framework to closed strings via KLT, with a focus on MRV amplitudes. It leverages hypergeometric function recursions, Mellin-Barnes representations, and Molien’s theorem to organize high-weight MZV structures and invariant polynomials, yielding explicit results up to weight 21 for open and MRV closed-string cases. A nine-element τ-basis for completely symmetric five-point polynomials enables compact representation of symmetric kinematics, and the work provides concrete MRV expressions up to α'^{10} alongside insights into transcendental structures and potential inconsistencies. The methods and data open pathways for higher-point generalizations, effective-action reconstruction, and deeper connections between string amplitudes, permutation-invariant polynomials, and modular forms.

Abstract

A simple recursive expansion algorithm for the integrals of tree level superstring five point amplitudes in a flat background is given which reduces the expansion to simple symbol(ic) manipulations. This approach can be used for instance to prove the expansion is maximally transcendental to all orders and to verify several conjectures made in recent literature to high order. Closed string amplitudes follow from these open string results by the KLT relations. To obtain insight into these results in particular the maximal R-symmetry violating amplitudes (MRV) in type IIB superstring theory are studied. The obtained expansion of the open string amplitudes reduces the analysis for MRV amplitudes to the classification of completely symmetric polynomials of the external legs, up to momentum conservation. Using Molien's theorem as a counting tool this problem is solved by constructing an explicit nine element basis for this class. This theorem may be of wider interest: as is illustrated at higher points it can be used to calculate dimensions of polynomials of external momenta invariant under any finite group for in principle any number of legs, up to momentum conservation.

Paper Structure

This paper contains 13 sections, 1 theorem, 70 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Brief review of five point superstring amplitudes
  3. Open superstring
  4. Closed superstring
  5. Field theory expansion of open string five point amplitudes
  6. Warmup: expanding the ${}_2F_1$ hypergeometric function
  7. Expanding ${}_3F_2$ hypergeometric functions
  8. Verifying conjectures up to and including weight $21$.
  9. Application to closed string five point amplitudes in the MRV sector
  10. Classifying completely symmetric polynomials
  11. Obtaining explicit results for MRV amplitudes at five points
  12. Discussion and conclusion
  13. Mathematica implantation of the algorithm

Key Result

Theorem 4.1

(Molien) Let $V$ be a vector space and $M$ a matrix representation of a group P, which is a subgroup of the permutation group on this space. Let $f_k$ be a function on the space $V$ invariant under the action of $M$, with fixed homogeneity e.g. Let $d_k$ be the number of independent polynomials of degree $k$. Let $g(t)$ be the generating function for these numbers, Then this generating function

Theorems & Definitions (1)

  • Theorem 4.1