Abstract
We discuss general one and two-loop banana diagrams and one-loop diagrams with external lines with arbitrary masses on the anti de Sitter spacetime by using methods of AdS quantum field theory in the dimensional regularization approach. The banana diagrams explicitly computed in this paper are indeed the necessary ingredients for the evaluation of the two-loop effective potential of the Standard Model and can be used to extend the flat space results in presence of a negative cosmological constant. In the one-loop case we also compute the effective potential for an O(N) model in d = 4 dimension as an explicit function of the cosmological constant Λ, both exactly and perturbatively up to order Λ. In the two-loop case we show the explicit calculation is possible thanks to a remarkable discrete Källén-Lehmann formula which we found and proved sometimes ago and whose domain of applicability we extend in the present paper.
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References
J.M. Maldacena, The AdS/CFT Correspondence, in Handbook of Quantum Gravity, Springer, Singapore (2024) [https://doi.org/10.1007/978-981-19-3079-9_65-1] [INSPIRE].
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].
O. DeWolfe, TASI Lectures on Applications of Gauge/Gravity Duality, PoS TASI2017 (2018) 014 [arXiv:1802.08267] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
J. Babington, D.E. Crooks and N.J. Evans, A Nonsupersymmetric deformation of the AdS/CFT correspondence, JHEP 02 (2003) 024 [hep-th/0207076] [INSPIRE].
O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large N field theories, string theory and gravity, Phys. Rep. 323 (2000) 183 [hep-th/9905111] [INSPIRE].
D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, Rev. Mod. Phys. 91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].
C. Copetti, L. Cordova and S. Komatsu, Non-Invertible Symmetries, Anomalies and Scattering Amplitudes, arXiv:2403.04835 [INSPIRE].
J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].
D. Carmi, Loops in AdS: From the Spectral Representation to Position Space. Part III, arXiv:2402.02481 [INSPIRE].
D. Carmi, Loops in AdS: from the spectral representation to position space. Part II, JHEP 07 (2021) 186 [arXiv:2104.10500] [INSPIRE].
D. Carmi, Loops in AdS: From the Spectral Representation to Position Space, JHEP 06 (2020) 049 [arXiv:1910.14340] [INSPIRE].
J.M. Drummond and H. Paul, Two-loop supergravity on AdS5 × S5 from CFT, JHEP 08 (2022) 275 [arXiv:2204.01829] [INSPIRE].
D. Carmi, L. Di Pietro and S. Komatsu, A Study of Quantum Field Theories in AdS at Finite Coupling, JHEP 01 (2019) 200 [arXiv:1810.04185] [INSPIRE].
E.T. Akhmedov, U. Moschella and F.K. Popov, Ultraviolet phenomena in AdS self-interacting quantum field theory, JHEP 03 (2018) 183 [arXiv:1802.02955] [INSPIRE].
I. Bertan and I. Sachs, Loops in Anti-de Sitter Space, Phys. Rev. Lett. 121 (2018) 101601 [arXiv:1804.01880] [INSPIRE].
S.L. Cacciatori, H. Epstein and U. Moschella, Banana integrals in configuration space, Nucl. Phys. B 995 (2023) 116343 [arXiv:2304.00624] [INSPIRE].
S.L. Cacciatori, H. Epstein and U. Moschella, Loops in de Sitter space, JHEP 07 (2024) 182 [arXiv:2403.13145] [INSPIRE].
R. Roiban, A. Tirziu and A.A. Tseytlin, Two-loop world-sheet corrections in AdS5 × S5 superstring, JHEP 07 (2007) 056 [arXiv:0704.3638] [INSPIRE].
P. Sundin and L. Wulff, One- and two-loop checks for the AdS3 × S3 × T4 superstring with mixed flux, J. Phys. A 48 (2015) 105402 [arXiv:1411.4662] [INSPIRE].
L.F. Alday, A. Bissi and X. Zhou, One-loop gluon amplitudes in AdS, JHEP 02 (2022) 105 [arXiv:2110.09861] [INSPIRE].
A. Herderschee, A new framework for higher loop Witten diagrams, JHEP 06 (2024) 008 [arXiv:2112.08226] [INSPIRE].
J. Bros, U. Moschella and J.P. Gazeau, Quantum field theory in the de Sitter universe, Phys. Rev. Lett. 73 (1994) 1746 [INSPIRE].
J. Bros and U. Moschella, Two point functions and quantum fields in de Sitter universe, Rev. Math. Phys. 8 (1996) 327 [gr-qc/9511019] [INSPIRE].
J. Bros, H. Epstein and U. Moschella, Towards a general theory of quantized fields on the anti-de Sitter space-time, Commun. Math. Phys. 231 (2002) 481 [hep-th/0111255] [INSPIRE].
J. Bros, Complexified de Sitter space: Analytic causal kernels and Kallen-Lehmann type representation, Nucl. Phys. B Proc. Suppl. 18 (1991) 22 [INSPIRE].
J. Bros, H. Epstein, M. Gaudin, U. Moschella and V. Pasquier, Triangular invariants, three-point functions and particle stability on the de Sitter universe, Commun. Math. Phys. 295 (2010) 261 [arXiv:0901.4223] [INSPIRE].
M.S. Costa, V. Gonçalves and J. Penedones, Spinning AdS Propagators, JHEP 09 (2014) 064 [arXiv:1404.5625] [INSPIRE].
J. Bros, H. Epstein, M. Gaudin, U. Moschella and V. Pasquier, Anti de Sitter quantum field theory and a new class of hypergeometric identities, Commun. Math. Phys. 309 (2012) 255 [arXiv:1107.5161] [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, Analyticity and the Holographic S-Matrix, JHEP 10 (2012) 127 [arXiv:1111.6972] [INSPIRE].
C. Ford, I. Jack and D.R.T. Jones, The Standard model effective potential at two loops, Nucl. Phys. B 387 (1992) 373 [hep-ph/0111190] [INSPIRE].
M. Bertola, J. Bros, U. Moschella and R. Schaeffer, A general construction of conformal field theories from scalar anti-de Sitter quantum field theories, Nucl. Phys. B 587 (2000) 619 [INSPIRE].
R.F. Streater and A, S. Wightman, PCT, Spin and Statistics, and All That, Princeton University Press (2000).
S.J. Avis, C.J. Isham and D. Storey, Quantum Field Theory in anti-de Sitter Space-Time, Phys. Rev. D 18 (1978) 3565 [INSPIRE].
A. Erdélyi ed., The Bateman project: Higher Transcendental Functions. Volume I, McGraw-Hill, New York, NY, U.S.A. (1953).
G. Szegö, Orthogonal Polynomials, in Colloquium Publications, 4th edition, American Mathematical Society, Providence, RI, U.S.A. (1975).
C. Fronsdal, Elementary particles in a curved space. Ii, Phys. Rev. D 10 (1974) 589 [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Stability in Gauged Extended Supergravity, Ann. Phys. 144 (1982) 249 [INSPIRE].
S. Cacciatori, V. Gorini, A. Kamenshchik and U. Moschella, Conservation laws and scattering for de Sitter classical particles, Class. Quant. Grav. 25 (2008) 075008 [arXiv:0710.0315] [INSPIRE].
N.F. Svaiter and C.A.D. Zarro, A comment on Schwinger functions in Euclidean Rindler space, Class. Quant. Grav. 25 (2008) 095008 [INSPIRE].
M.E.X. Guimaraes and B. Linet, Selfinteraction and quantum effects near a point mass in three-dimensional gravitation, Class. Quant. Grav. 10 (1993) 1665 [INSPIRE].
M.E.X. Guimaraes and B. Linet, Scalar Green’s functions in an Euclidean space with a conical-type line singularity, Commun. Math. Phys. 165 (1994) 297 [INSPIRE].
B. Linet, Euclidean scalar and spinor Green’s functions in Rindler space, gr-qc/9505033 [INSPIRE].
C. García-Recio and L.L. Salcedo, The perturbative scalar massless propagator in Schwarzschild spacetime, Class. Quant. Grav. 30 (2013) 097001 [arXiv:1303.6620] [INSPIRE].
I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press (1996).
P.L. Butzer, P.J.S.G. Ferreira, G. Schmeisser and R.L. Stens, The Summation Formulae of Euler-Maclaurin, Abel-Plana, Poisson, and their Interconnections with the Approximate Sampling Formula of Signal Analysis, Results Math. 59 (2011) 359.
D.W. Düsedau and D.Z. Freedman, Lehmann Spectral Representation for Anti-de Sitter Quantum Field Theory, Phys. Rev. D 33 (1986) 389 [INSPIRE].
National Institute of Standards and Technology, NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov/.
F.G. Tricomi and A. Erdélyi, The asymptotic expansion of a ratio of Gamma functions, Pacific J. Math. 1 (1951) 133.
F.W.J. Olver, Asymptotics and Special Functions, A K Peters, Wellesley, MA, U.S.A. (1997).
Acknowledgments
We thank the anonymous referee for his careful reading and his suggestions to improve the paper. U.M. gratefully thanks the Department of Theoretical Physics of CERN for hospitality and support while writing this paper.
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ArXiv ePrint: 2403.13142
To Jacques Bros and Michel Gaudin.
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Cacciatori, S.L., Epstein, H. & Moschella, U. Loops in anti de Sitter space. J. High Energ. Phys. 2024, 109 (2024). https://doi.org/10.1007/JHEP08(2024)109
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DOI: https://doi.org/10.1007/JHEP08(2024)109