Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality

Abstract

In an earlier paper, we constructed the genus-two amplitudes for five external massless states in Type II and Heterotic string theory, and showed that the alpha' expansion of the Type II amplitude reproduces the corresponding supergravity amplitude to leading order. In this paper, we analyze the effective interactions induced by Type IIB superstrings beyond supergravity, both for U(1)_R-preserving amplitudes such as for five gravitons, and for U(1)_R-violating amplitudes such as for one dilaton and four gravitons. At each order in alpha', the coefficients of the effective interactions are given by integrals over moduli space of genus-two modular graph functions, generalizing those already encountered for four external massless states. To leading and sub-leading orders, the coefficients of the effective interactions D^2 R^5 and D^4 R^5 are found to match those of D^4 R^4 and D^6 R^4, respectively, as required by non-linear supersymmetry. To the next order, a D^6 R^5 effective interaction arises, which is independent of the supersymmetric completion of D^8 R^4, and already arose at genus one. A novel identity on genus-two modular graph functions, which we prove, ensures that up to order D^6 R^5, the five-point amplitudes require only a single new modular graph function in addition to those needed for the four-point amplitude. We check that the supergravity limit of U(1)_R-violating amplitudes is free of UV divergences to this order, consistently with the known structure of divergences in Type IIB supergravity. Our results give strong consistency tests on the full five-point amplitude, and pave the way for understanding S-duality beyond the BPS-protected sector.

91 pages, 3 figures, dedicated to the life, science, and art of Professor Jean-Loup Gervais, v2: minor improvements throughout the paper, published version, v3: corrected normalization factors of parity-even and/or parity-odd terms in equations (2.30), (4.34), (4.44), (4.46) and footnote 20

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Cited by in corpus (21)

• Tree-level amplitudes from the pure spinor superstring
• Superstring loop amplitudes from the field theory limit
• The SAGEX Review on Scattering Amplitudes, Chapter 10: Selected topics on modular covariance of type IIB string amplitudes and their $\mathcal{N}=4$ supersymmetric Yang-Mills duals
• Perfecting one-loop BCJ numerators in SYM and supergravity
• Unitarity Cuts of the Worldsheet
• Poincaré series for modular graph forms at depth two. I. Seeds and Laplace systems
• Modular graph forms from equivariant iterated Eisenstein integrals
• Two-loop superstring five-point amplitudes III, construction via the RNS formulation: even spin structures
• Elliptic modular graph forms I: Identities and generating series
• Poisson equations for elliptic modular graph functions
• One-loop diagrams with quadratic propagators from the worldsheet
• Relations between elliptic modular graphs
• One-loop Bern-Carrasco-Johansson Numerators on Quadratic Propagators from the Worldsheet
• Integrating three-loop modular graph functions and transcendentality of string amplitudes
• Cyclic products of Szegö kernels and spin structure sums I: hyper-elliptic formulation
• Advanced tools for basis decompositions of genus-one string integrals
• Pure spinor formulation of the superstring and its applications
• Poisson equation for genus two string invariants: a conjecture
• Supermoduli and PCOs at Genus Two
• Integrating simple genus two string invariants over moduli space
• Singular Spin Structures and Superstrings