Abstract
Four-particle tree-level scattering amplitudes in string theory are magically consistent with unitarity, reflected in the non-trivial fact that beneath the critical dimension, the residues of the amplitudes on massive poles can be expanded in partial waves with all positive coefficients. While this follows (rather indirectly) from the no-ghost theorem, the simplicity of the statement and its fundamental importance for the physical consistency of string theory begs for a more direct and elementary understanding. In this note we take a step in this direction by presenting a new expression for the partial wave coefficients of string amplitudes, given by surprisingly simple double/triple contour integrals for open/closed strings. This representation allows us to directly prove unitarity of all superstring theories in D ≤ 6 spacetime dimensions, and can also be used to determine various asymptotics of the partial waves at large mass levels.
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References
G. Veneziano, Construction of a crossing-symmetric, Regge behaved amplitude for linearly rising trajectories, Nuovo Cim. A 57 (1968) 190 [INSPIRE].
M.A. Virasoro, Alternative constructions of crossing-symmetric amplitudes with Regge behavior, Phys. Rev. 177 (1969) 2309 [INSPIRE].
J.A. Shapiro, Electrostatic analog for the virasoro model, Phys. Lett. B 33 (1970) 361 [INSPIRE].
D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm, Heterotic String Theory. 2. The Interacting Heterotic String, Nucl. Phys. B 267 (1986) 75 [INSPIRE].
P. Goddard and C.B. Thorn, Compatibility of the Dual Pomeron with Unitarity and the Absence of Ghosts in the Dual Resonance Model, Phys. Lett. B 40 (1972) 235 [INSPIRE].
K. Aoki, E. D’Hoker and D.H. Phong, Unitarity of Closed Superstring Perturbation Theory, Nucl. Phys. B 342 (1990) 149 [INSPIRE].
M. Soldate, Partial Wave Unitarity and Closed String Amplitudes, Phys. Lett. B 186 (1987) 321 [INSPIRE].
M.B. Green and C. Wen, Superstring amplitudes, unitarily, and Hankel determinants of multiple zeta values, JHEP 11 (2019) 079 [arXiv:1908.08426] [INSPIRE].
S. Caron-Huot, Z. Komargodski, A. Sever and A. Zhiboedov, Strings from Massive Higher Spins: The Asymptotic Uniqueness of the Veneziano Amplitude, JHEP 10 (2017) 026 [arXiv:1607.04253] [INSPIRE].
T.L. Curtright and C.B. Thorn, Symmetry Patterns in the Mass Spectra of Dual String Models, Nucl. Phys. B 274 (1986) 520 [INSPIRE].
T.L. Curtright, C.B. Thorn and J. Goldstone, Spin Content of the Bosonic String, Phys. Lett. B 175 (1986) 47 [INSPIRE].
J.G. Russo and L. Susskind, Asymptotic level density in heterotic string theory and rotating black holes, Nucl. Phys. B 437 (1995) 611 [hep-th/9405117] [INSPIRE].
H. Kawai, D.C. Lewellen and S.H.H. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].
G.N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press (1995).
P. Flajolet and R. Sedgewick, Analytic combinatorics, Cambridge University Press (2009) [DOI].
P. Maity, Positivity of the Veneziano Amplitude in D = 4, arXiv:2110.01578 [INSPIRE].
H.L. Krall and O. Frink, A new class of orthogonal polynomials: the Bessel polynomials, Trans. Am. Math. Soc. 65 (1949) 100.
S. Mizera, Aspects of Scattering Amplitudes and Moduli Space Localization, Ph.D. Thesis, Princeton, Inst. Advanced Study (2020) [DOI] [arXiv:1906.02099] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer (1997) [DOI].
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Arkani-Hamed, N., Eberhardt, L., Huang, Yt. et al. On unitarity of tree-level string amplitudes. J. High Energ. Phys. 2022, 197 (2022). https://doi.org/10.1007/JHEP02(2022)197
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DOI: https://doi.org/10.1007/JHEP02(2022)197