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RNS derivation of N-point disk amplitudes from the revisited S-matrix approach

Luiz Antonio Barreiro, Ricardo Medina

TL;DR

The authors show that the tree-level open-string N-point disk amplitudes (N=3–7) and the bosonic part of the OSLEEL can be derived in the Ramond-Neveu-Schwarz formalism via a revisited S-matrix approach, reproducing the MSS closed form $A(1,\ldots,N)=\sum_{\sigma_N\in S_{N-3}} F^{\{\sigma_N\}}(α')\, A_{YM}(1,\{2_{\sigma},\ldots,(N-2)_{\sigma}\},N-1,N)$. The method exploits a kinematic constraint that forbids $(\zeta\cdot k)^N$ terms, yielding a (N-3)!-dimensional basis ${\cal B}_N$ of YM subamplitudes and enabling a linear-algebra projection to obtain MSS, including BCJ relations, within the RNS framework for N up to 7. α' expansions of the momentum factors $F^{\{σ_N\}}(α')$ are derived using tree-level unitarity and analyticity, letting higher-point corrections be determined from the well-known 4-point (and 5-point) data without explicit polylogarithm calculations, with explicit results through ${α'}^6$ for N=5–7. The findings highlight a purely kinematic route to MSS and BCJ structures in string amplitudes and point toward extending the approach to arbitrary N.

Abstract

In the past year, in arXiv:1208.6066 we proposed a revisited S-matrix approach to efficiently find the bosonic terms of the open superstring low energy effective lagrangian (OSLEEL). This approach allows to compute the ${α'}^N$ terms of the OSLEEL using open superstring $n$-point amplitudes in which $n$ is very much lower than $(N+2)$ (which is the order of the required amplitude to obtain those ${α'}^N$ terms by means of the conventional S-matrix approach). In this work we use our revisited S-matrix approach to examine the structure of the scattering amplitudes, arriving at a closed form for them. This is a RNS derivation of the formula first found by Mafra, Schlotterer and Stieberger in arXiv:1106.2645, using the Pure Spinor formalism. We have succeeded doing this for the 5, 6 and 7-point amplitudes. In order to achieve these results we have done a careful analysis of the kinematical structure of the amplitudes, finding as a by-product a purely kinematical derivation of the BCJ relations (for N=4, 5, 6 and 7). Also, following the spirit of the revisited S-matrix approach, we have found the $α'$ expansions for these amplitudes up to ${α'}^6$ order in some cases, by only using the well known open superstring 4-point amplitude, cyclic symmetry and tree level unitarity: we have not needed to compute any numerical series or any integral involving polylogarithms, at any moment.

RNS derivation of N-point disk amplitudes from the revisited S-matrix approach

TL;DR

The authors show that the tree-level open-string N-point disk amplitudes (N=3–7) and the bosonic part of the OSLEEL can be derived in the Ramond-Neveu-Schwarz formalism via a revisited S-matrix approach, reproducing the MSS closed form . The method exploits a kinematic constraint that forbids terms, yielding a (N-3)!-dimensional basis of YM subamplitudes and enabling a linear-algebra projection to obtain MSS, including BCJ relations, within the RNS framework for N up to 7. α' expansions of the momentum factors are derived using tree-level unitarity and analyticity, letting higher-point corrections be determined from the well-known 4-point (and 5-point) data without explicit polylogarithm calculations, with explicit results through for N=5–7. The findings highlight a purely kinematic route to MSS and BCJ structures in string amplitudes and point toward extending the approach to arbitrary N.

Abstract

In the past year, in arXiv:1208.6066 we proposed a revisited S-matrix approach to efficiently find the bosonic terms of the open superstring low energy effective lagrangian (OSLEEL). This approach allows to compute the terms of the OSLEEL using open superstring -point amplitudes in which is very much lower than (which is the order of the required amplitude to obtain those terms by means of the conventional S-matrix approach). In this work we use our revisited S-matrix approach to examine the structure of the scattering amplitudes, arriving at a closed form for them. This is a RNS derivation of the formula first found by Mafra, Schlotterer and Stieberger in arXiv:1106.2645, using the Pure Spinor formalism. We have succeeded doing this for the 5, 6 and 7-point amplitudes. In order to achieve these results we have done a careful analysis of the kinematical structure of the amplitudes, finding as a by-product a purely kinematical derivation of the BCJ relations (for N=4, 5, 6 and 7). Also, following the spirit of the revisited S-matrix approach, we have found the expansions for these amplitudes up to order in some cases, by only using the well known open superstring 4-point amplitude, cyclic symmetry and tree level unitarity: we have not needed to compute any numerical series or any integral involving polylogarithms, at any moment.

Paper Structure

This paper contains 58 sections, 274 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review of the revisited S-matrix method
  3. Finding terms of the OSLEEL in an efficient way
  4. Using the $4$ and the $5$-point $\alpha'$ information to obtain $N$-point information
  5. Basis for open superstring and Yang-Mills subamplitudes
  6. Some general facts about the structure of scattering amplitudes of gauge bosons
  7. Finding a basis for ${\cal V}_N$, when $3 \leq N \leq 7$
  8. Case of $N=3$
  9. Case of $N=4$
  10. Case of $N=5, 6, 7$
  11. Finding the components of an element of ${\cal V}_N$ with respect to the basis ${\cal B}_N$
  12. Closed expression for the $N$-point disk amplitude using the RNS formalism
  13. For gauge bosons only
  14. Case of $N=4$
  15. Case of $N=5$
  16. ...and 43 more sections

Figures (3)

  • Figure 1: Two type of Feynman diagrams contribute to the YM 5-point amplitude. Permutations of the legs of these diagrams should also be considered in order to account for the full amplitude.
  • Figure 2: Three type of Feynman diagrams contribute to the YM 6-point amplitude. Permutations of the legs of these diagrams should also be considered in order to account for the full amplitude.
  • Figure 3: Tree kind of Feynman diagrams used to calculate the YM subamplitude $A_{YM}(1,2,3,4,5,6,7)$.