Stringy KLT relations, global symmetries, and E_7(7) violation

TL;DR

This work analyzes stringy KLT relations in four dimensions to expose how α'^3 corrections induce SU(8)–violating amplitudes and how the SU(8)–invariant completion of R^4 must satisfy E7(7) constraints. By isolating SU(8)–violating pieces with both averaging over SU(8) and superamplitude methods, the authors demonstrate that the proposed R^4 counterterm cannot be compatible with E7(7) due to nonvanishing single-soft scalar limits, while confirming the quadratic moduli dependence of its SU(8)–invariant completion satisfies the predicted Laplace eigenvalue equation. The paper also develops a comprehensive SU(4)×SU(4)–restricted superamplitude framework, identifies independent local operators in various MHV sectors, and derives linear relations among F^4 matrix elements from KLT. - Main approach: use KLT to relate open- and closed-string amplitudes, decompose SU(8) breaking, construct SU(4)×SU(4)–symmetric superamplitudes, and compare with automorphic constraints. - Key findings: (i) e^{-6φ}R^4 contains SU(8)–violating contributions, (ii) the SU(8)–invariant completion of R^4 has nonzero single-soft scalar limits, incompatible with E7(7), (iii) the SU(8)–invariant coupling satisfies the Laplace equation to quadratic order, and (iv) linear relations among F^4 matrix elements follow from closed-string vanishing at α'^2. - Significance: solidifies the role of E7(7) in constraining counterterms in N=8 supergravity, clarifies the interplay between string moduli, duality, and ultraviolet behavior, and provides a detailed toolkit for SU(4)×SU(4)–restricted amplitude analysis.

Abstract

We study consequences of the Kawai-Lewellen-Tye (KLT) relations applied to tree amplitudes in toroidal compactifications of string theory to four dimensions. The closed string tree amplitudes with massless external states respect a global SU(4)xSU(4) symmetry, which is enhanced to the SU(8) R-symmetry of N=8 supergravity in the field theory limit. Our analysis focuses on two aspects: (i) We provide a detailed account of the simplest SU(8)-violating amplitudes. We classify these processes and derive explicit superamplitudes for all local 5- and 6-point operators with SU(4)xSU(4) symmetry at order alpha'^3. Their origin is the dilatonic operator exp(-6 phi) R^4 in the closed-string effective action. (ii) We expand the 6-point closed string tree amplitudes to order alpha'^3 and use two different methods to isolate the SU(8)-singlet contribution from exp(-6 phi) R^4. This allows us to extract the matrix elements of the unique SU(8)-invariant supersymmetrization of R^4. Their single-soft scalar limits are non-vanishing. This demonstrates that the N=8 supergravity candidate counterterm R^4 is incompatible with continuous E_7(7) symmetry. From the soft scalar limits, we reconstruct to quadratic order the SU(8)-invariant function of scalars that multiplies R^4, and show that it satisfies the Laplace eigenvalue equation derived recently from supersymmetry and duality constraints.