Journal of High Energy Physics

, 2018:75 | Cite as

Unitarity constraint on the Kähler curvature

  • Yohei Ema
  • Ryuichiro Kitano
  • Takahiro TeradaEmail author
Open Access
Regular Article - Theoretical Physics


In supersymmetric theories, the signs of quartic terms in the Kähler potential control the stability of non-supersymmetric field configurations. In particular, in supersymmetric inflation models, the signs are important for the stability of an inflationary trajectory as well as for the prediction of the spectral index. In this paper, we clarify what properties of a UV theory determine the sign from unitarity arguments of scattering amplitudes. As non-trivial examples, we discuss the sign of a four-meson term in large N supersymmetric gauge theories and also those of the quartic terms obtained in the intersecting D-brane models in superstring theory. The UV origins of inflationary models and supersymmetry breaking models are constrained by this discussion.


Scattering Amplitudes Supersymmetric Effective Theories 1/N Expansion Compactification and String Models 


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]
    A. Martin, Unitarity and high-energy behavior of scattering amplitudes, Phys. Rev. 129 (1963) 1432 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    T.N. Pham and T.N. Truong, Evaluation of the Derivative Quartic Terms of the Meson Chiral Lagrangian From Forward Dispersion Relation, Phys. Rev. D 31 (1985) 3027 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    B. Ananthanarayan, D. Toublan and G. Wanders, Consistency of the chiral pion pion scattering amplitudes with axiomatic constraints, Phys. Rev. D 51 (1995) 1093 [hep-ph/9410302] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    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].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    A. Jenkins and D. O’Connell, The Story of O: Positivity constraints in effective field theories, hep-th/0609159 [INSPIRE].
  6. [6]
    A. Adams, A. Jenkins and D. O’Connell, Signs of analyticity in fermion scattering, arXiv:0802.4081 [INSPIRE].
  7. [7]
    G.R. Dvali, G. Gabadadze and M. Porrati, 4-D gravity on a brane in 5-D Minkowski space, Phys. Lett. B 485 (2000) 208 [hep-th/0005016] [INSPIRE].
  8. [8]
    M. Dine, G. Festuccia and Z. Komargodski, A Bound on the Superpotential, JHEP 03 (2010) 011 [arXiv:0910.2527] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    L. O’Raifeartaigh, Spontaneous Symmetry Breaking for Chiral Scalar Superfields, Nucl. Phys. B 96 (1975) 331 [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    S. Ray, Some properties of meta-stable supersymmetry-breaking vacua in Wess-Zumino models, Phys. Lett. B 642 (2006) 137 [hep-th/0607172] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    R. Kitano and Y. Ookouchi, Supersymmetry breaking and gauge mediation in models with a generic superpotential, Phys. Lett. B 675 (2009) 80 [arXiv:0812.0543] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    Z. Komargodski and D. Shih, Notes on SUSY and R-Symmetry Breaking in Wess-Zumino Models, JHEP 04 (2009) 093 [arXiv:0902.0030] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    I. Low, R. Rattazzi and A. Vichi, Theoretical Constraints on the Higgs Effective Couplings, JHEP 04 (2010) 126 [arXiv:0907.5413] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    A. Falkowski, S. Rychkov and A. Urbano, What if the Higgs couplings to W and Z bosons are larger than in the Standard Model?, JHEP 04 (2012) 073 [arXiv:1202.1532] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    B. Bellazzini, L. Martucci and R. Torre, Symmetries, Sum Rules and Constraints on Effective Field Theories, JHEP 09 (2014) 100 [arXiv:1405.2960] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    A.E. Nelson and N. Seiberg, R symmetry breaking versus supersymmetry breaking, Nucl. Phys. B 416 (1994) 46 [hep-ph/9309299] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    K.A. Intriligator, N. Seiberg and D. Shih, Dynamical SUSY breaking in meta-stable vacua, JHEP 04 (2006) 021 [hep-th/0602239] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    A. Katz, Y. Shadmi and T. Volansky, Comments on the meta-stable vacuum in N(f) = N(c) SQCD and direct mediation, JHEP 07 (2007) 020 [arXiv:0705.1074] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    A. Hetz and G.A. Palma, Sound Speed of Primordial Fluctuations in Supergravity Inflation, Phys. Rev. Lett. 117 (2016) 101301 [arXiv:1601.05457] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    G.R. Dvali, Q. Shafi and R.K. Schaefer, Large scale structure and supersymmetric inflation without fine tuning, Phys. Rev. Lett. 73 (1994) 1886 [hep-ph/9406319] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    Planck collaboration, Y. Akrami et al., Planck 2018 results. X. Constraints on inflation, arXiv:1807.06211 [INSPIRE].
  23. [23]
    M. Bastero-Gil, S.F. King and Q. Shafi, Supersymmetric Hybrid Inflation with Non-Minimal Kähler potential, Phys. Lett. B 651 (2007) 345 [hep-ph/0604198] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    K. Kumekawa, T. Moroi and T. Yanagida, Flat potential for inflaton with a discrete R invariance in supergravity, Prog. Theor. Phys. 92 (1994) 437 [hep-ph/9405337] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    K.I. Izawa and T. Yanagida, Natural new inflation in broken supergravity, Phys. Lett. B 393 (1997) 331 [hep-ph/9608359] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    K.I. Izawa, Supergravity minimal inflation and its spectral index revisited, Phys. Lett. B 576 (2003) 1 [hep-ph/0305286] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    M. Kawasaki, M. Yamaguchi and T. Yanagida, Natural chaotic inflation in supergravity, Phys. Rev. Lett. 85 (2000) 3572 [hep-ph/0004243] [INSPIRE].
  28. [28]
    L. Álvarez-Gaumé, C. Gomez and R. Jimenez, Minimal Inflation, Phys. Lett. B 690 (2010) 68 [arXiv:1001.0010] [INSPIRE].
  29. [29]
    L. Álvarez-Gaumé, C. Gomez and R. Jimenez, A Minimal Inflation Scenario, JCAP 03 (2011) 027 [arXiv:1101.4948] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    A. Achucarro, S. Mooij, P. Ortiz and M. Postma, Sgoldstino inflation, JCAP 08 (2012) 013 [arXiv:1203.1907] [INSPIRE].
  31. [31]
    K.I. Izawa and Y. Shinbara, Supersymmetric tuned inflation, arXiv:0710.1141 [INSPIRE].
  32. [32]
    S.V. Ketov and T. Terada, Inflation in supergravity with a single chiral superfield, Phys. Lett. B 736 (2014) 272 [arXiv:1406.0252] [INSPIRE].
  33. [33]
    S.V. Ketov and T. Terada, Generic Scalar Potentials for Inflation in Supergravity with a Single Chiral Superfield, JHEP 12 (2014) 062 [arXiv:1408.6524] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    S.V. Ketov and T. Terada, On SUSY Restoration in Single-Superfield Inflationary Models of Supergravity, Eur. Phys. J. C 76 (2016) 438 [arXiv:1606.02817] [INSPIRE].
  35. [35]
    R. Kallosh, A. Linde and D. Roest, Superconformal Inflationary α-Attractors, JHEP 11 (2013) 198 [arXiv:1311.0472] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    D. Roest and M. Scalisi, Cosmological attractors from α-scale supergravity, Phys. Rev. D 92 (2015) 043525 [arXiv:1503.07909] [INSPIRE].
  37. [37]
    A. Linde, Single-field α-attractors, JCAP 05 (2015) 003 [arXiv:1504.00663] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    K. Nakayama, K. Saikawa, T. Terada and M. Yamaguchi, Structure of Kähler potential for D-term inflationary attractor models, JHEP 05 (2016) 067 [arXiv:1603.02557] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    L. Covi, M. Gomez-Reino, C. Gross, J. Louis, G.A. Palma and C.A. Scrucca, Constraints on modular inflation in supergravity and string theory, JHEP 08 (2008) 055 [arXiv:0805.3290] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    N. Seiberg, Exact results on the space of vacua of four-dimensional SUSY gauge theories, Phys. Rev. D 49 (1994) 6857 [hep-th/9402044] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    G. Veneziano, Some Aspects of a Unified Approach to Gauge, Dual and Gribov Theories, Nucl. Phys. B 117 (1976) 519 [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    C. Csáki, M. Schmaltz and W. Skiba, Confinement in N = 1 SUSY gauge theories and model building tools, Phys. Rev. D 55 (1997) 7840 [hep-th/9612207] [INSPIRE].
  43. [43]
    I.I. Pomeranchuk, Equality of the Nucleon and Antinucleon Total Interaction Cross Section at High Energies, Sov. Phys. JETP 7 (1958) 499.MathSciNetGoogle Scholar
  44. [44]
    S. Weinberg, Cross Sections at High Energies, Phys. Rev. 124 (1961) 2049 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    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].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    S.J. Brodsky and G.R. Farrar, Scaling Laws at Large Transverse Momentum, Phys. Rev. Lett. 31 (1973) 1153 [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    G.P. Lepage and S.J. Brodsky, Exclusive Processes in Perturbative Quantum Chromodynamics, Phys. Rev. D 22 (1980) 2157 [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    O. Andreev and W. Siegel, Quantized tension: Stringy amplitudes with Regge poles and parton behavior, Phys. Rev. D 71 (2005) 086001 [hep-th/0410131] [INSPIRE].
  49. [49]
    G. Veneziano, S. Yankielowicz and E. Onofri, A model for pion-pion scattering in large-N QCD, JHEP 04 (2017) 151 [arXiv:1701.06315] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    M. Froissart, Asymptotic behavior and subtractions in the Mandelstam representation, Phys. Rev. 123 (1961) 1053 [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    I.I. Pomeranchuk, The Conservation of Isotopic Spin and the Scattering of Antinucleons by Nucleons, Sov. Phys. JETP 3 (1956) 306.Google Scholar
  52. [52]
    B. Okun and I.I. Pomeranchuk, The Conservation of Isotopic Spin and the Cross Section of the Interaction of High-Energy π-Mesons and Nucleons with Nucleons, Sov. Phys. JETP 3 (1956) 307.Google Scholar
  53. [53]
    L.L. Foldy and R.F. Peierls, Isotopic Spin of Exchanged Systems, Phys. Rev. 130 (1963) 1585 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  54. [54]
    A.B. Kaidalov, High-energy hadronic interactions (20 years of the quark gluon strings model), Phys. Atom. Nucl. 66 (2003) 1994 [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    C. Ewerz, The Odderon in quantum chromodynamics, hep-ph/0306137 [INSPIRE].
  56. [56]
    S. Okubo, Phi meson and unitary symmetry model, Phys. Lett. 5 (1963) 165 [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    G. Zweig, An SU(3) model for strong interaction symmetry and its breaking. Version 2, in Developments in the quark theory of hadrons. Vol. 1. 1964-1978, D. Lichtenberg and S.P. Rosen eds., pp. 22-101 (1964) [INSPIRE].
  58. [58]
    J. Iizuka, Systematics and phenomenology of meson family, Prog. Theor. Phys. Suppl. 37 (1966) 21 [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    I.R. Klebanov and E. Witten, Proton decay in intersecting D-brane models, Nucl. Phys. B 664 (2003) 3 [hep-th/0304079] [INSPIRE].
  60. [60]
    I. Antoniadis, K. Benakli and A. Laugier, Contact interactions in D-brane models, JHEP 05 (2001) 044 [hep-th/0011281] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  61. [61]
    B. Bellazzini, Softness and amplitudes’ positivity for spinning particles, JHEP 02 (2017) 034 [arXiv:1605.06111] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  62. [62]
    K. Higashijima and E. Itou, Unitarity bound of the wave function renormalization constant, Prog. Theor. Phys. 110 (2003) 107 [hep-th/0304047] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    S. Weinberg, The Quantum theory of fields. Vol. 1: Foundations, Cambridge University Press (2005) [INSPIRE].
  64. [64]
    E. Witten and J. Bagger, Quantization of Newton’s Constant in Certain Supergravity Theories, Phys. Lett. B 115 (1982) 202 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Theory Center, KEKTsukubaJapan
  2. 2.Department of Particle and Nuclear PhysicsGraduate University for Advanced Studies (Sokendai)TsukubaJapan

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