Journal of High Energy Physics

, 2019:79 | Cite as

Superstring amplitudes, unitarily, and Hankel determinants of multiple zeta values

  • Michael B. Green
  • Congkao WenEmail author
Open Access
Regular Article - Theoretical Physics


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.


Effective Field Theories Scattering Amplitudes Superstrings and Heterotic Strings 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited


  1. [1]
    G. Veneziano, Construction of a crossing-symmetric, Regge behaved amplitude for linearly rising trajectories, Nuovo Cim.A 57 (1968) 190 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    M.A. Virasoro, Alternative constructions of crossing-symmetric amplitudes with Regge behavior, Phys. Rev.177 (1969) 2309 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    G. ‘t Hooft, A planar diagram theory for strong interactions, Nucl. Phys.B 72 (1974) 461 [INSPIRE].
  4. [4]
    A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP10 (2006) 014 [hep-th/0602178] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    Z. Komargodski and A. Schwimmer, On renormalization group flows in four dimensions, JHEP12 (2011) 099 [arXiv:1107.3987] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The string landscape, black holes and gravity as the weakest force, JHEP06 (2007) 060 [hep-th/0601001] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    C. Cheung and G.N. Remmen, Infrared consistency and the weak gravity conjecture, JHEP12 (2014) 087 [arXiv:1407.7865] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    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].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    Y. Hamada, T. Noumi and G. Shiu, Weak gravity conjecture from unitarity and causality, Phys. Rev. Lett.123 (2019) 051601 [arXiv:1810.03637] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    B. Bellazzini, M. Lewandowski and J. Serra, Amplitudes' positivity, weak gravity conjecture and modified gravity, arXiv:1902.03250 [INSPIRE].
  11. [11]
    N. Arkani-Hamed, T.-Z. Huang and Y.-T. Huang, in preparation.Google Scholar
  12. [12]
    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.Google Scholar
  13. [13]
    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.Google Scholar
  14. [14]
    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].
  15. [15]
    C. de Rham, S. Melville and A.J. Tolley, Improved positivity bounds and massive gravity, JHEP04 (2018) 083 [arXiv:1710.09611] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  16. [16]
    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].
  17. [17]
    N. Arkani-Hamed, Y.-T. Huang and S.-H. Shao, On the positive geometry of conformal field theory, JHEP06 (2019) 124 [arXiv:1812.07739] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    K. Sen, A. Sinha and A. Zahed, Positive geometry in the diagonal limit of the conformal bootstrap, arXiv:1906.07202 [INSPIRE].
  19. [19]
    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].MathSciNetCrossRefGoogle Scholar
  20. [20]
    R.C. Brower, Spectrum generating algebra and no ghost theorem for the dual model, Phys. Rev.D 6 (1972) 1655 [INSPIRE].ADSGoogle Scholar
  21. [21]
    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].ADSCrossRefGoogle Scholar
  22. [22]
    C.B. Thorn, A proof of the no-ghost theorem using the Kac determinant, MSRI Publ.3 (1985) 411 [INSPIRE].MathSciNetzbMATHGoogle Scholar
  23. [23]
    H. Monien, Hankel determinants of Dirichlet series, arXiv:0901.1883.
  24. [24]
    A. Haynes and W. Zudilin, Hankel determinants of zeta values, SIGMA11 (2015) 101 [arXiv:1510.01901].ADSMathSciNetzbMATHGoogle Scholar
  25. [25]
    R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix, Cambridge University Press, Cambridge, U.K. 1966.zbMATHGoogle Scholar
  26. [26]
    D. Zagier and F. Zerbini, Genus-zero and genus-one string amplitudes and special multiple zeta values, arXiv:1906.12339 [INSPIRE].
  27. [27]
    M.E. Hoffman, The algebra of multiple harmonic series, J. Alg.194 (1997) 477.Google Scholar
  28. [28]
    J.A. Shapiro, Electrostatic analog for the Virasoro model, Phys. Lett.B 33 (1970) 361 [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    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].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    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].
  31. [31]
    F. BROWN, Single-valued motivic periods and multiple zeta values, Forum Math. Sigma2 (2014) e25.MathSciNetCrossRefGoogle Scholar
  32. [32]
    F. Brown and C. Dupont, Single-valued integration and double copy, arXiv:1810.07682 [INSPIRE].
  33. [33]
    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].
  34. [34]
    P. Vanhove and F. Zerbini, Closed string amplitudes from single-valued correlation functions, arXiv:1812.03018 [INSPIRE].
  35. [35]
    M.B. Green, J.G. Russo and P. Vanhove, Low energy expansion of the four-particle genus-one amplitude in type- II superstring theory, JHEP02 (2008) 020 [arXiv:0801.0322] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    O. Schlotterer and S. Stieberger, Motivic multiple zeta values and superstring amplitudes, J. Phys.A 46 (2013) 475401 [arXiv:1205.1516] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  37. [37]
    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].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    E. D’Hoker and M.B. Green, Exploring transcendentality in superstring amplitudes, JHEP07 (2019) 149 [arXiv:1906.01652] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Centre for Research in String Theory, School of Physics and AstronomyQueen Mary University of LondonLondonU.K.
  2. 2.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeU.K.

Personalised recommendations