Abstract
The interplay of unitarity and analyticity has long been known to impose strong constraints on scattering amplitudes in quantum field theory and string theory. This has been highlighted in recent times in a number of papers and lecture notes. Here we examine such conditions in the context of superstring tree-level scattering amplitudes, leading to positivity constraints on determinants of Hankel matrices involving polynomials of multiple zeta values. These generalise certain constraints on polynomials of single zeta values in the mathematics literature.
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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].
G. ‘t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hep-th/0602178] [INSPIRE].
Z. Komargodski and A. Schwimmer, On renormalization group flows in four dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The string landscape, black holes and gravity as the weakest force, JHEP 06 (2007) 060 [hep-th/0601001] [INSPIRE].
C. Cheung and G.N. Remmen, Infrared consistency and the weak gravity conjecture, JHEP 12 (2014) 087 [arXiv:1407.7865] [INSPIRE].
S. Andriola, D. Junghans, T. Noumi and G. Shiu, A tower weak gravity conjecture from infrared consistency, Fortsch. Phys. 66 (2018) 1800020 [arXiv:1802.04287] [INSPIRE].
Y. Hamada, T. Noumi and G. Shiu, Weak gravity conjecture from unitarity and causality, Phys. Rev. Lett. 123 (2019) 051601 [arXiv:1810.03637] [INSPIRE].
B. Bellazzini, M. Lewandowski and J. Serra, Amplitudes' positivity, weak gravity conjecture and modified gravity, arXiv:1902.03250 [INSPIRE].
N. Arkani-Hamed, T.-Z. Huang and Y.-T. Huang, in preparation.
N. Arkani-Hamed, Positive geometry of effective field theory, lectures at CERN Winter School on Supergravity, Strings and Gauge Theory, CERN, Geneva, Switzerland, 4-8 February 2019.
Y.-T. Huang, The space of EFT and CFT: life behind the facets of cyclic polytopes, in Amplitudes 2018, SLAC, U.S.A., 19 June 2018.
C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, Positivity bounds for scalar field theories, Phys. Rev. D 96 (2017) 081702 [arXiv:1702.06134] [INSPIRE].
C. de Rham, S. Melville and A.J. Tolley, Improved positivity bounds and massive gravity, JHEP 04 (2018) 083 [arXiv:1710.09611] [INSPIRE].
W.-M. Chen, Y.-T. Huang, T. Noumi and C. Wen, Unitarity bounds on charged / neutral state mass ratios, Phys. Rev. D 100 (2019) 025016 [arXiv:1901.11480] [INSPIRE].
N. Arkani-Hamed, Y.-T. Huang and S.-H. Shao, On the positive geometry of conformal field theory, JHEP 06 (2019) 124 [arXiv:1812.07739] [INSPIRE].
K. Sen, A. Sinha and A. Zahed, Positive geometry in the diagonal limit of the conformal bootstrap, arXiv:1906.07202 [INSPIRE].
S. Fallat, C.R. Johnson and A.D. Sokal, Total positivity of sums, Hadamard products and Hadamard powers: results and counterexamples, Linear Alg. Appl. 520 (2017) 242 [arXiv:1612.02210].
R.C. Brower, Spectrum generating algebra and no ghost theorem for the dual model, Phys. Rev. D 6 (1972) 1655 [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].
C.B. Thorn, A proof of the no-ghost theorem using the Kac determinant, MSRI Publ. 3 (1985) 411 [INSPIRE].
H. Monien, Hankel determinants of Dirichlet series, arXiv:0901.1883.
A. Haynes and W. Zudilin, Hankel determinants of zeta values, SIGMA 11 (2015) 101 [arXiv:1510.01901].
R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix, Cambridge University Press, Cambridge, U.K. 1966.
D. Zagier and F. Zerbini, Genus-zero and genus-one string amplitudes and special multiple zeta values, arXiv:1906.12339 [INSPIRE].
M.E. Hoffman, The algebra of multiple harmonic series, J. Alg. 194 (1997) 477.
J.A. Shapiro, Electrostatic analog for the Virasoro model, Phys. Lett. B 33 (1970) 361 [INSPIRE].
H. Kawai, D.C. Lewellen and S.-H. Henry Tye, A relation between tree amplitudes of closed and open strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].
Z. Bern, J.J.M. Carrasco and H. Johansson, New relations for gauge-theory amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].
F. BROWN, Single-valued motivic periods and multiple zeta values, Forum Math. Sigma 2 (2014) e25.
F. Brown and C. Dupont, Single-valued integration and double copy, arXiv:1810.07682 [INSPIRE].
O. Schlotterer and O. Schnetz, Closed strings as single-valued open strings: a genus-zero derivation, J. Phys. A 52 (2019) 045401 [arXiv:1808.00713] [INSPIRE].
P. Vanhove and F. Zerbini, Closed string amplitudes from single-valued correlation functions, arXiv:1812.03018 [INSPIRE].
M.B. Green, J.G. Russo and P. Vanhove, Low energy expansion of the four-particle genus-one amplitude in type- II superstring theory, JHEP 02 (2008) 020 [arXiv:0801.0322] [INSPIRE].
O. Schlotterer and S. Stieberger, Motivic multiple zeta values and superstring amplitudes, J. Phys. A 46 (2013) 475401 [arXiv:1205.1516] [INSPIRE].
C.R. Mafra, O. Schlotterer and S. Stieberger, Complete N -point superstring disk amplitude I. Pure spinor computation, Nucl. Phys. B 873 (2013) 419 [arXiv:1106.2645] [INSPIRE].
E. D’Hoker and M.B. Green, Exploring transcendentality in superstring amplitudes, JHEP 07 (2019) 149 [arXiv:1906.01652] [INSPIRE].
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ArXiv ePrint: 1908.08426
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Green, M.B., Wen, C. Superstring amplitudes, unitarily, and Hankel determinants of multiple zeta values. J. High Energ. Phys. 2019, 79 (2019). https://doi.org/10.1007/JHEP11(2019)079
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DOI: https://doi.org/10.1007/JHEP11(2019)079