MHV graviton scattering amplitudes and current algebra on the celestial sphere

Abstract

The Cachazo-Strominger subleading soft graviton theorem for a positive helicity soft graviton is equivalent to the Ward identities for \( \overline{\mathrm{SL}\left(2,\mathrm{\mathbb{C}}\right)} \) currents. This naturally gives rise to a \( \overline{\mathrm{SL}\left(2,\mathrm{\mathbb{C}}\right)} \) current algebra living on the celestial sphere. The generators of the \( \overline{\mathrm{SL}\left(2,\mathrm{\mathbb{C}}\right)} \) current algebra and the supertranslations, coming from a positive helicity leading soft graviton, form a closed algebra. We find that the OPE of two graviton primaries in the Celestial CFT, extracted from MHV amplitudes, is completely determined in terms of this algebra. To be more precise, 1) The subleading terms in the OPE are determined in terms of the leading OPE coefficient if we demand that both sides of the OPE transform in the same way under this local symmetry algebra. 2) Positive helicity gravitons have null states under this local algebra whose decoupling leads to differential equations for MHV amplitudes. An n point MHV amplitude satisfies two systems of (n − 2) linear first order PDEs corresponding to (n − 2) positive helicity gravitons. We have checked, using Hodges’ formula, that one system of differential equations is satisfied by any MHV amplitude, whereas the other system has been checked up to six graviton MHV amplitude. 3) One can determine the leading OPE coefficients from these differential equations.

This points to the existence of an autonomous sector of the Celestial CFT which holographically computes the MHV graviton scattering amplitudes and is completely defined by this local symmetry algebra. The MHV-sector of the Celestial CFT is like a minimal model of 2-D CFT.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    F. Cachazo and A. Strominger, Evidence for a new soft graviton theorem, arXiv:1404.4091 [INSPIRE].

  2. [2]

    S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  3. [3]

    M. Pate, A.-M. Raclariu, A. Strominger and E. Y. Yuan, Celestial operator products of gluons and gravitons, arXiv:1910.07424 [INSPIRE].

  4. [4]

    Y. T. A. Law and M. Zlotnikov, Poincaré constraints on celestial amplitudes, JHEP 03 (2020) 085 [Erratum ibid. 04 (2020) 202] [arXiv:1910.04356] [INSPIRE].

  5. [5]

    S. Banerjee, S. Ghosh and R. Gonzo, BMS symmetry of celestial OPE, JHEP 04 (2020) 130 [arXiv:2002.00975] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  6. [6]

    A. Fotopoulos, S. Stieberger, T. R. Taylor and B. Zhu, Extended BMS algebra of celestial CFT, JHEP 03 (2020) 130 [arXiv:1912.10973] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  7. [7]

    A. Strominger, On BMS invariance of gravitational scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  8. [8]

    A. Strominger, Asymptotic symmetries of Yang-Mills theory, JHEP 07 (2014) 151 [arXiv:1308.0589] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  9. [9]

    A. Strominger, Lectures on the infrared structure of gravity and gauge theory, arXiv:1703.05448 [INSPIRE].

  10. [10]

    D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Semiclassical Virasoro symmetry of the quantum gravity S -matrix, JHEP 08 (2014) 058 [arXiv:1406.3312] [INSPIRE].

    ADS  Article  Google Scholar 

  11. [11]

    D. Kapec, P. Mitra, A.-M. Raclariu and A. Strominger, 2D stress tensor for 4D gravity, Phys. Rev. Lett. 119 (2017) 121601 [arXiv:1609.00282] [INSPIRE].

    ADS  Article  Google Scholar 

  12. [12]

    T. He, D. Kapec, A.-M. Raclariu and A. Strominger, Loop-corrected Virasoro symmetry of 4D quantum gravity, JHEP 08 (2017) 050 [arXiv:1701.00496] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  13. [13]

    A. Ball, E. Himwich, S. A. Narayanan, S. Pasterski and A. Strominger, Uplifting AdS3/CFT2 to flat space holography, JHEP 08 (2019) 168 [arXiv:1905.09809] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  14. [14]

    H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  15. [15]

    R. K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103 [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  16. [16]

    R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  17. [17]

    G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett. 105 (2010) 111103 [arXiv:0909.2617] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  18. [18]

    G. Barnich and C. Troessaert, Supertranslations call for superrotations, PoS(CNCFG2010)010 (2010) [arXiv:1102.4632] [INSPIRE].

  19. [19]

    T. He, V. Lysov, P. Mitra and A. Strominger, BMS supertranslations and Weinberg’s soft graviton theorem, JHEP 05 (2015) 151 [arXiv:1401.7026] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  20. [20]

    A. Strominger and A. Zhiboedov, Gravitational memory, BMS supertranslations and soft theorems, JHEP 01 (2016) 086 [arXiv:1411.5745] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  21. [21]

    M. Campiglia and A. Laddha, Asymptotic symmetries and subleading soft graviton theorem, Phys. Rev. D 90 (2014) 124028 [arXiv:1408.2228] [INSPIRE].

    ADS  Article  Google Scholar 

  22. [22]

    L. Donnay, S. Pasterski and A. Puhm, Asymptotic symmetries and celestial CFT, JHEP 09 (2020) 176 [arXiv:2005.08990] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  23. [23]

    M. Campiglia and J. Peraza, Generalized BMS charge algebra, Phys. Rev. D 101 (2020) 104039 [arXiv:2002.06691] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  24. [24]

    G. Compère, A. Fiorucci and R. Ruzziconi, Superboost transitions, refraction memory and super-Lorentz charge algebra, JHEP 11 (2018) 200 [Erratum ibid. 04 (2020) 172] [arXiv:1810.00377] [INSPIRE].

  25. [25]

    D. Kapec and P. Mitra, A d-dimensional stress tensor for Minkd+2 gravity, JHEP 05 (2018) 186 [arXiv:1711.04371] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  26. [26]

    S. Pasterski, S.-H. Shao and A. Strominger, Flat space amplitudes and conformal symmetry of the celestial sphere, Phys. Rev. D 96 (2017) 065026 [arXiv:1701.00049] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  27. [27]

    S. Pasterski and S.-H. Shao, Conformal basis for flat space amplitudes, Phys. Rev. D 96 (2017) 065022 [arXiv:1705.01027] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  28. [28]

    C. Cheung, A. de la Fuente and R. Sundrum, 4D scattering amplitudes and asymptotic symmetries from 2D CFT, JHEP 01 (2017) 112 [arXiv:1609.00732] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  29. [29]

    J. de Boer and S. N. Solodukhin, A holographic reduction of Minkowski space-time, Nucl. Phys. B 665 (2003) 545 [hep-th/0303006] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  30. [30]

    S. Banerjee, Null infinity and unitary representation of the Poincaré group, JHEP 01 (2019) 205 [arXiv:1801.10171] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  31. [31]

    S. Banerjee, Symmetries of free massless particles and soft theorems, Gen. Rel. Grav. 51 (2019) 128 [arXiv:1804.06646] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  32. [32]

    S. Banerjee, S. Ghosh, P. Pandey and A. P. Saha, Modified celestial amplitude in Einstein gravity, JHEP 03 (2020) 125 [arXiv:1909.03075] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  33. [33]

    S. Pasterski, S.-H. Shao and A. Strominger, Gluon amplitudes as 2d conformal correlators, Phys. Rev. D 96 (2017) 085006 [arXiv:1706.03917] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  34. [34]

    A. Schreiber, A. Volovich and M. Zlotnikov, Tree-level gluon amplitudes on the celestial sphere, Phys. Lett. B 781 (2018) 349 [arXiv:1711.08435] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  35. [35]

    C. Cardona and Y.-T. Huang, S-matrix singularities and CFT correlation functions, JHEP 08 (2017) 133 [arXiv:1702.03283] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  36. [36]

    H. T. Lam and S.-H. Shao, Conformal basis, optical theorem, and the bulk point singularity, Phys. Rev. D 98 (2018) 025020 [arXiv:1711.06138] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  37. [37]

    N. Banerjee, S. Banerjee, S. Atul Bhatkar and S. Jain, Conformal structure of massless scalar amplitudes beyond tree level, JHEP 04 (2018) 039 [arXiv:1711.06690] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  38. [38]

    S. Stieberger and T. R. Taylor, Strings on celestial sphere, Nucl. Phys. B 935 (2018) 388 [arXiv:1806.05688] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  39. [39]

    S. Stieberger and T. R. Taylor, Symmetries of celestial amplitudes, Phys. Lett. B 793 (2019) 141 [arXiv:1812.01080] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  40. [40]

    S. Banerjee, P. Pandey and P. Paul, Conformal properties of soft operators. Part I. Use of null states, Phys. Rev. D 101 (2020) 106014 [arXiv:1902.02309] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  41. [41]

    S. Banerjee and P. Pandey, Conformal properties of soft-operators. Part II. Use of null-states, JHEP 02 (2020) 067 [arXiv:1906.01650] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  42. [42]

    A. Fotopoulos and T. R. Taylor, Primary fields in celestial CFT, JHEP 10 (2019) 167 [arXiv:1906.10149] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  43. [43]

    L. Donnay, A. Puhm and A. Strominger, Conformally soft photons and gravitons, JHEP 01 (2019) 184 [arXiv:1810.05219] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  44. [44]

    M. Pate, A.-M. Raclariu and A. Strominger, Conformally soft theorem in gauge theory, Phys. Rev. D 100 (2019) 085017 [arXiv:1904.10831] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  45. [45]

    W. Fan, A. Fotopoulos and T. R. Taylor, Soft limits of Yang-Mills amplitudes and conformal correlators, JHEP 05 (2019) 121 [arXiv:1903.01676] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  46. [46]

    D. Nandan, A. Schreiber, A. Volovich and M. Zlotnikov, Celestial amplitudes: conformal partial waves and soft limits, JHEP 10 (2019) 018 [arXiv:1904.10940] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  47. [47]

    T. Adamo, L. Mason and A. Sharma, Celestial amplitudes and conformal soft theorems, Class. Quant. Grav. 36 (2019) 205018 [arXiv:1905.09224] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  48. [48]

    A. Puhm, Conformally soft theorem in gravity, JHEP 09 (2020) 130 [arXiv:1905.09799] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  49. [49]

    A. Guevara, Notes on conformal soft theorems and recursion relations in gravity, arXiv:1906.07810 [INSPIRE].

  50. [50]

    E. Himwich, S. A. Narayanan, M. Pate, N. Paul and A. Strominger, The soft \( \mathcal{S} \)-matrix in gravity, JHEP 09 (2020) 129 [arXiv:2005.13433] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  51. [51]

    A. Bagchi, R. Basu, A. Kakkar and A. Mehra, Flat holography: aspects of the dual field theory, JHEP 12 (2016) 147 [arXiv:1609.06203] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  52. [52]

    A. Hodges, New expressions for gravitational scattering amplitudes, JHEP 07 (2013) 075 [arXiv:1108.2227] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  53. [53]

    A. Hodges, A simple formula for gravitational MHV amplitudes, arXiv:1204.1930 [INSPIRE].

  54. [54]

    P. Paul, unpublished work.

  55. [55]

    Z. Bern, L. J. Dixon, M. Perelstein and J. S. Rozowsky, Multileg one loop gravity amplitudes from gauge theory, Nucl. Phys. B 546 (1999) 423 [hep-th/9811140] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  56. [56]

    D. Nandan, J. Plefka and W. Wormsbecher, Collinear limits beyond the leading order from the scattering equations, JHEP 02 (2017) 038 [arXiv:1608.04730] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  57. [57]

    L. Rodina, Scattering amplitudes from soft theorems and infrared behavior, Phys. Rev. Lett. 122 (2019) 071601 [arXiv:1807.09738] [INSPIRE].

    ADS  Article  Google Scholar 

  58. [58]

    S. Banerjee, S. Ghosh, P. Pandey, P. Paul and A. P. Saha, unpublished work.

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Shamik Banerjee.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2008.04330

Rights and permissions

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Banerjee, S., Ghosh, S. & Paul, P. MHV graviton scattering amplitudes and current algebra on the celestial sphere. J. High Energ. Phys. 2021, 176 (2021). https://doi.org/10.1007/JHEP02(2021)176

Download citation

Keywords

  • Gauge-gravity correspondence
  • Models of Quantum Gravity
  • Scattering Amplitudes
  • Space-Time Symmetries