Mellin amplitudes for fermionic conformal correlators

  • Josua Faller
  • Sourav SarkarEmail author
  • Mritunjay Verma
Open Access
Regular Article - Theoretical Physics


We define Mellin amplitudes for the fermion-scalar four point function and the fermion four point function. The Mellin amplitude thus defined has multiple components each associated with a tensor structure. In the case of three spacetime dimensions, we explicitly show that each component factorizes on dynamical poles onto components of the Mellin amplitudes for the corresponding three point functions. The novelty here is that for a given exchanged primary, each component of the Mellin amplitude may in general have more than one series of poles. We present a few examples of Mellin amplitudes for tree-level Witten diagrams and tree-level conformal Feynman integrals with fermionic legs, which illustrate the general properties.


Conformal Field Theory AdS-CFT Correspondence 


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. Mack, D-independent representation of conformal field theories in D dimensions via transformation to auxiliary dual resonance models. Scalar amplitudes, arXiv:0907.2407 [INSPIRE].
  2. [2]
    G. Mack, D-dimensional conformal field theories with anomalous dimensions as dual resonance models, Bulg. J. Phys. 36 (2009) 214 [arXiv:0909.1024] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  3. [3]
    J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  4. [4]
    M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    A.L. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju and B.C. van Rees, A natural language for AdS/CFT correlators, JHEP 11 (2011) 095 [arXiv:1107.1499] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    L. Rastelli and X. Zhou, The Mellin formalism for boundary CFT d, JHEP 10 (2017) 146 [arXiv:1705.05362] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  7. [7]
    A.L. Fitzpatrick and J. Kaplan, Analyticity and the holographic S-matrix, JHEP 10 (2012) 127 [arXiv:1111.6972] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  8. [8]
    M.F. Paulos, Towards Feynman rules for Mellin amplitudes, JHEP 10 (2011) 074 [arXiv:1107.1504] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  9. [9]
    D. Nandan, A. Volovich and C. Wen, On Feynman rules for Mellin amplitudes in AdS/CFT, JHEP 05 (2012) 129 [arXiv:1112.0305] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    O. Aharony, L.F. Alday, A. Bissi and E. Perlmutter, Loops in AdS from conformal field theory, JHEP 07 (2017) 036 [arXiv:1612.03891] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    E.Y. Yuan, Loops in the bulk, arXiv:1710.01361 [INSPIRE].
  12. [12]
    C. Cardona, Mellin-(Schwinger) representation of one-loop Witten diagrams in AdS, arXiv:1708.06339 [INSPIRE].
  13. [13]
    E.Y. Yuan, Simplicity in AdS perturbative dynamics, arXiv:1801.07283 [INSPIRE].
  14. [14]
    L. Rastelli and X. Zhou, Mellin amplitudes for AdS 5 × S 5, Phys. Rev. Lett. 118 (2017) 091602 [arXiv:1608.06624] [INSPIRE].
  15. [15]
    L. Rastelli and X. Zhou, How to succeed at holographic correlators without really trying, arXiv:1710.05923 [INSPIRE].
  16. [16]
    X. Zhou, On superconformal four-point Mellin amplitudes in dimension d > 2, arXiv:1712.02800 [INSPIRE].
  17. [17]
    D. Ponomarev, A note on (non)-locality in holographic higher spin theories, Universe 4 (2018) 2 [arXiv:1710.00403] [INSPIRE].
  18. [18]
    X. Bekaert, J. Erdmenger, D. Ponomarev and C. Sleight, Bulk quartic vertices from boundary four-point correlators, in Proceedings, International Workshop on Higher Spin Gauge Theories, World Scientific, Singapore, (2017), pg. 291 [arXiv:1602.08570] [INSPIRE].
  19. [19]
    M. Taronna, Pseudo-local theories: a functional class proposal, in Proceedings, International Workshop on Higher Spin Gauge Theories, World Scientific, Singapore, (2017), pg. 59 [arXiv:1602.08566] [INSPIRE].
  20. [20]
    M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap. Part I: QFT in AdS, JHEP 11 (2017) 133 [arXiv:1607.06109] [INSPIRE].
  21. [21]
    R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, Conformal bootstrap in Mellin space, Phys. Rev. Lett. 118 (2017) 081601 [arXiv:1609.00572] [INSPIRE].
  22. [22]
    R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, A Mellin space approach to the conformal bootstrap, JHEP 05 (2017) 027 [arXiv:1611.08407] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    V. Gonçalves, J. Penedones and E. Trevisani, Factorization of Mellin amplitudes, JHEP 10 (2015) 040 [arXiv:1410.4185] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    H.-Y. Chen, E.-J. Kuo and H. Kyono, Towards spinning Mellin amplitudes, arXiv:1712.07991 [INSPIRE].
  25. [25]
    L. Iliesiu, F. Kos, D. Poland, S.S. Pufu, D. Simmons-Duffin and R. Yacoby, Bootstrapping 3D fermions, JHEP 03 (2016) 120 [arXiv:1508.00012] [INSPIRE].
  26. [26]
    P. Kravchuk and D. Simmons-Duffin, Counting conformal correlators, JHEP 02 (2018) 096 [arXiv:1612.08987] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    J. Penedones, TASI lectures on AdS/CFT, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: new frontiers in fields and strings (TASI 2015), World Scientific, (2017), pg. 75 [arXiv:1608.04948] [INSPIRE].
  28. [28]
    S. Weinberg, Six-dimensional methods for four-dimensional conformal field theories, Phys. Rev. D 82 (2010) 045031 [arXiv:1006.3480] [INSPIRE].
  29. [29]
    L. Iliesiu, F. Kos, D. Poland, S.S. Pufu, D. Simmons-Duffin and R. Yacoby, Fermion-scalar conformal blocks, JHEP 04 (2016) 074 [arXiv:1511.01497] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  30. [30]
    D. Simmons-Duffin, Projectors, shadows and conformal blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    E. Elkhidir, D. Karateev and M. Serone, General three-point functions in 4D CFT, JHEP 01 (2015) 133 [arXiv:1412.1796] [INSPIRE].
  32. [32]
    G.F. Cuomo, D. Karateev and P. Kravchuk, General bootstrap equations in 4D CFTs, JHEP 01 (2018) 130 [arXiv:1705.05401] [INSPIRE].
  33. [33]
    A. Castedo Echeverri, E. Elkhidir, D. Karateev and M. Serone, Seed conformal blocks in 4D CFT, JHEP 02 (2016) 183 [arXiv:1601.05325] [INSPIRE].
  34. [34]
    P. Kravchuk, Casimir recursion relations for general conformal blocks, JHEP 02 (2018) 011 [arXiv:1709.05347] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    D. Karateev, P. Kravchuk and D. Simmons-Duffin, Weight shifting operators and conformal blocks, JHEP 02 (2018) 081 [arXiv:1706.07813] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B 599 (2001) 459 [hep-th/0011040] [INSPIRE].
  37. [37]
    S. Kharel and G. Siopsis, Tree-level correlators of scalar and vector fields in AdS/CFT, JHEP 11 (2013) 159 [arXiv:1308.2515] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    T. Kawano and K. Okuyama, Spinor exchange in AdS d+1, Nucl. Phys. B 565 (2000) 427 [hep-th/9905130] [INSPIRE].
  39. [39]
    A.M. Ghezelbash, K. Kaviani, S. Parvizi and A.H. Fatollahi, Interacting spinors-scalars and AdS/CFT correspondence, Phys. Lett. B 435 (1998) 291 [hep-th/9805162] [INSPIRE].
  40. [40]
    M.F. Paulos, M. Spradlin and A. Volovich, Mellin amplitudes for dual conformal integrals, JHEP 08 (2012) 072 [arXiv:1203.6362] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    D. Nandan, M.F. Paulos, M. Spradlin and A. Volovich, Star integrals, convolutions and simplices, JHEP 05 (2013) 105 [arXiv:1301.2500] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    A.A. Nizami, A. Rudra, S. Sarkar and M. Verma, Exploring perturbative conformal field theory in Mellin space, JHEP 01 (2017) 102 [arXiv:1607.07334] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    K. Symanzik, On calculations in conformal invariant field theories, Lett. Nuovo Cim. 3 (1972) 734 [INSPIRE].CrossRefGoogle Scholar
  44. [44]
    D. Pappadopulo, S. Rychkov, J. Espin and R. Rattazzi, OPE convergence in conformal field theory, Phys. Rev. D 86 (2012) 105043 [arXiv:1208.6449] [INSPIRE].
  45. [45]
    M. Henningson and K. Sfetsos, Spinors and the AdS/CFT correspondence, Phys. Lett. B 431 (1998) 63 [hep-th/9803251] [INSPIRE].
  46. [46]
    W. Mueck and K.S. Viswanathan, Conformal field theory correlators from classical scalar field theory on AdS d+1, Phys. Rev. D 58 (1998) 041901 [hep-th/9804035] [INSPIRE].
  47. [47]
    M. Henneaux, Boundary terms in the AdS/CFT correspondence for spinor fields, in Mathematical methods in modern theoretical physics. Proceedings, International Meeting, School and Workshop, ISPM’98, Tbilisi Georgia, (1998), pg. 161 [hep-th/9902137] [INSPIRE].
  48. [48]
    D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Correlation functions in the CFT d /AdS d+1 correspondence, Nucl. Phys. B 546 (1999) 96 [hep-th/9804058] [INSPIRE].
  49. [49]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Institut für Physik, Humboldt-Universität zu BerlinBerlinGermany
  2. 2.Institut für Mathematik, Humboldt-Universität zu BerlinBerlinGermany
  3. 3.Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-InstitutPotsdamGermany
  4. 4.Harish-Chandra Research Institute, HBNIAllahabadIndia
  5. 5.International Centre for Theoretical Sciences, TIFRBengaluruIndia

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