Abstract
The Coon amplitude is a q-deformed generalization of the Veneziano amplitude exhibiting a semi-infinite sequence of poles that converge on an accumulation point, from which a branch cut emerges. A number of recent papers have provided compelling evidence that the residues of this amplitude satisfy the positivity requirements imposed by unitarity. This paper investigates whether positivity is also satisfied along the branch cut. It is demonstrated for a wide range of q-values that positivity violations occur in a region of the branch cut exponentially close to the accumulation point according to a scale set by q. The closing section of the paper discusses possible interpretations of this fact and strategies for excising negativity from the partial wave coefficients.
An appendix presents derivations of instrumental identities relating the q-gamma and q-polygamma functions to the Weierstrass elliptic and quasiperiodic functions.
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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].
D.D. Coon, Uniqueness of the Veneziano representation, Phys. Lett. B 29 (1969) 669 [INSPIRE].
D.D. Coon, U.P. Sukhatme and J. Tran Thanh Van, Duality and proton proton scattering at all angles, Phys. Lett. B 45 (1973) 287 [INSPIRE].
D.D. Coon and S. Yu, Dual four point functions with no negative residues, Phys. Rev. D 10 (1974) 3780 [INSPIRE].
D.B. Fairlie and J. Nuyts, A fresh look at generalized Veneziano amplitudes, Nucl. Phys. B 433 (1995) 26 [hep-th/9406043] [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].
F. Figueroa and P. Tourkine, Unitarity and low energy expansion of the Coon amplitude, Phys. Rev. Lett. 129 (2022) 121602 [arXiv:2201.12331] [INSPIRE].
J. Chakravarty, P. Maity and A. Mishra, On the positivity of Coon amplitude in D = 4, JHEP 10 (2022) 043 [arXiv:2208.02735] [INSPIRE].
R. Bhardwaj, S. De, M. Spradlin and A. Volovich, On unitarity of the Coon amplitude, arXiv:2212.00764 [INSPIRE].
N. Arkani-Hamed, L. Eberhardt, Y.-T. Huang and S. Mizera, On unitarity of tree-level string amplitudes, JHEP 02 (2022) 197 [arXiv:2201.11575] [INSPIRE].
N. Geiser and L.W. Lindwasser, Properties of infinite product amplitudes: Veneziano, Virasoro, and Coon, JHEP 12 (2022) 112 [arXiv:2207.08855] [INSPIRE].
L.-Y. Chiang et al., (Non)-projective bounds on gravitational EFT, arXiv:2201.07177 [INSPIRE].
C. Lovelace, A novel application of Regge trajectories, Phys. Lett. B 28 (1968) 264 [INSPIRE].
J.A. Shapiro, Narrow-resonance model with Regge behavior for ππ scattering, Phys. Rev. 179 (1969) 1345 [INSPIRE].
C. Fernandez, A. Pomarol, F. Riva and F. Sciotti, Cornering large-Nc QCD with positivity bounds, arXiv:2211.12488 [INSPIRE].
J. Albert and L. Rastelli, Bootstrapping pions at large N, JHEP 08 (2022) 151 [arXiv:2203.11950] [INSPIRE].
H. Chen, A.L. Fitzpatrick and D. Karateev, Nonperturbative bounds on scattering of massive scalar particles in d ≥ 2, JHEP 12 (2022) 092 [arXiv:2207.12448] [INSPIRE].
C. Cheung and G.N. Remmen, Stringy dynamics from an amplitudes bootstrap, arXiv:2302.12263 [INSPIRE].
C. Cheung and G.N. Remmen, Veneziano variations: how unique are string amplitudes?, JHEP 01 (2023) 122 [arXiv:2210.12163] [INSPIRE].
N. Geiser and L.W. Lindwasser, Generalized Veneziano and Virasoro amplitudes, JHEP 04 (2023) 031 [arXiv:2210.14920] [INSPIRE].
J. Maldacena and G.N. Remmen, Accumulation-point amplitudes in string theory, JHEP 08 (2022) 152 [arXiv:2207.06426] [INSPIRE].
J. Thierry-Mieg and P. Jarvis, SU(2/1) superchiral self-duality: a new quantum, algebraic and geometric paradigm to describe the electroweak interactions, JHEP 21 (2020) 001 [arXiv:2012.12320] [INSPIRE].
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].
I.R. Klebanov, J.M. Maldacena and C.B. Thorn, Dynamics of flux tubes in large N gauge theories, JHEP 04 (2006) 024 [hep-th/0602255] [INSPIRE].
L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].
Y.-T. Huang and G.N. Remmen, UV-complete gravity amplitudes and the triple product, Phys. Rev. D 106 (2022) L021902 [arXiv:2203.00696] [INSPIRE].
I. Mező, A q-Raabe formula and an integral of the fourth Jacobi theta function, J. Number Theory 133 (2013) 692.
I. Cherednik, On q-analogues of Riemann’s zeta, Selecta Math. 7 (2001) 447 [math/9804099].
M. Kaneko, N. Kurokawa and M. Wakayama, A variation of Euler’s approach to values of the Riemann zeta function, Kyushu J. Math. 57 (2003) 175 [math/0206171].
A. Fitouhi, N. Bettaibi and K. Brahim, The Mellin transform in quantum calculus, Construct. Approx. 23 (2005) 305.
Acknowledgments
I am grateful to Zohar Komargodski and Nikita Nekrasov for illuminating discussions and to Zohar Komargodski for incisive comments on this work.
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Jepsen, C.B. Cutting the Coon amplitude. J. High Energ. Phys. 2023, 114 (2023). https://doi.org/10.1007/JHEP06(2023)114
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DOI: https://doi.org/10.1007/JHEP06(2023)114