Abstract
The calculation of loop corrections to the correlation functions of quantum fields during inflation or in the de Sitter background presents greater challenges than in flat space due to the more complicated form of the mode functions. While in flat space highly sophisticated approaches to Feynman integrals exist, similar tools still remain to be developed for cosmological correlators. However, usually only their late-time limit is of interest. We introduce the method-of-region expansion for cosmological correlators as a tool to extract the late-time limit, and illustrate it with several examples for the interacting, massless, minimally coupled scalar field in de Sitter space. In particular, we consider the in-in correlator 〈ϕ2(η, q)ϕ(η, k1)ϕ(η, k2)〉, whose region structure is relevant to anomalous dimensions and matching coefficients in Soft de Sitter effective theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Phys. Lett. B 91 (1980) 99 [INSPIRE].
A.H. Guth, The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems, Phys. Rev. D 23 (1981) 347 [INSPIRE].
A.D. Linde, A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems, Phys. Lett. B 108 (1982) 389 [INSPIRE].
V.F. Mukhanov and G.V. Chibisov, Quantum Fluctuations and a Nonsingular Universe, JETP Lett. 33 (1981) 532 [INSPIRE].
S. Weinberg, Quantum contributions to cosmological correlations, Phys. Rev. D 72 (2005) 043514 [hep-th/0506236] [INSPIRE].
S. Weinberg, Quantum contributions to cosmological correlations. II. Can these corrections become large?, Phys. Rev. D 74 (2006) 023508 [hep-th/0605244] [INSPIRE].
L. Senatore and M. Zaldarriaga, On Loops in Inflation, JHEP 12 (2010) 008 [arXiv:0912.2734] [INSPIRE].
D. Seery, Infrared effects in inflationary correlation functions, Class. Quant. Grav. 27 (2010) 124005 [arXiv:1005.1649] [INSPIRE].
B. Allen, Vacuum States in de Sitter Space, Phys. Rev. D 32 (1985) 3136 [INSPIRE].
A. Rajaraman, On the proper treatment of massless fields in Euclidean de Sitter space, Phys. Rev. D 82 (2010) 123522 [arXiv:1008.1271] [INSPIRE].
M. Beneke and P. Moch, On “dynamical mass” generation in Euclidean de Sitter space, Phys. Rev. D 87 (2013) 064018 [arXiv:1212.3058] [INSPIRE].
A. Higuchi, D. Marolf and I.A. Morrison, On the Equivalence between Euclidean and In-In Formalisms in de Sitter QFT, Phys. Rev. D 83 (2011) 084029 [arXiv:1012.3415] [INSPIRE].
S. Hollands, Massless interacting quantum fields in deSitter spacetime, Annales Henri Poincare 13 (2012) 1039 [arXiv:1105.1996] [INSPIRE].
A.A. Starobinsky, Dynamics of Phase Transition in the New Inflationary Universe Scenario and Generation of Perturbations, Phys. Lett. B 117 (1982) 175 [INSPIRE].
A.A. Starobinsky, Stochastic de Sitter (inflationary) stage in the early Universe, Lect. Notes Phys. 246 (1986) 107 [INSPIRE].
A.A. Starobinsky and J. Yokoyama, Equilibrium state of a selfinteracting scalar field in the De Sitter background, Phys. Rev. D 50 (1994) 6357 [astro-ph/9407016] [INSPIRE].
B. Garbrecht, F. Gautier, G. Rigopoulos and Y. Zhu, Feynman Diagrams for Stochastic Inflation and Quantum Field Theory in de Sitter Space, Phys. Rev. D 91 (2015) 063520 [arXiv:1412.4893] [INSPIRE].
T. Prokopec, N.C. Tsamis and R.P. Woodard, Stochastic Inflationary Scalar Electrodynamics, Annals Phys. 323 (2008) 1324 [arXiv:0707.0847] [INSPIRE].
M. Baumgart and R. Sundrum, De Sitter Diagrammar and the Resummation of Time, JHEP 07 (2020) 119 [arXiv:1912.09502] [INSPIRE].
M. Mirbabayi, Infrared dynamics of a light scalar field in de Sitter, JCAP 12 (2020) 006 [arXiv:1911.00564] [INSPIRE].
H. Collins, R. Holman and T. Vardanyan, The quantum Fokker-Planck equation of stochastic inflation, JHEP 11 (2017) 065 [arXiv:1706.07805] [INSPIRE].
V. Gorbenko and L. Senatore, λϕ4 in dS, arXiv:1911.00022 [INSPIRE].
M. Mirbabayi, Markovian dynamics in de Sitter, JCAP 09 (2021) 038 [arXiv:2010.06604] [INSPIRE].
T. Cohen, D. Green, A. Premkumar and A. Ridgway, Stochastic Inflation at NNLO, JHEP 09 (2021) 159 [arXiv:2106.09728] [INSPIRE].
T. Cohen and D. Green, Soft de Sitter Effective Theory, JHEP 12 (2020) 041 [arXiv:2007.03693] [INSPIRE].
M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE].
N. Arkani-Hamed and J. Maldacena, Cosmological Collider Physics, arXiv:1503.08043 [INSPIRE].
N. Arkani-Hamed, D. Baumann, H. Lee and G.L. Pimentel, The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities, JHEP 04 (2020) 105 [arXiv:1811.00024] [INSPIRE].
C. Sleight, A Mellin Space Approach to Cosmological Correlators, JHEP 01 (2020) 090 [arXiv:1906.12302] [INSPIRE].
S. Melville and E. Pajer, Cosmological Cutting Rules, JHEP 05 (2021) 249 [arXiv:2103.09832] [INSPIRE].
D. Baumann et al., Linking the singularities of cosmological correlators, JHEP 09 (2022) 010 [arXiv:2106.05294] [INSPIRE].
L. Di Pietro, V. Gorbenko and S. Komatsu, Analyticity and unitarity for cosmological correlators, JHEP 03 (2022) 023 [arXiv:2108.01695] [INSPIRE].
T. Heckelbacher and I. Sachs, Loops in dS/CFT, JHEP 02 (2021) 151 [arXiv:2009.06511] [INSPIRE].
T. Heckelbacher, I. Sachs, E. Skvortsov and P. Vanhove, Analytical evaluation of cosmological correlation functions, JHEP 08 (2022) 139 [arXiv:2204.07217] [INSPIRE].
P. Benincasa, Amplitudes meet Cosmology: A (Scalar) Primer, arXiv:2203.15330 [https://doi.org/10.1142/S0217751X22300101] [INSPIRE].
S. Céspedes, A.-C. Davis and D.-G. Wang, On the IR Divergences in de Sitter Space: loops, resummation and the semi-classical wavefunction, arXiv:2311.17990 [INSPIRE].
T.S. Bunch and P.C.W. Davies, Quantum Field Theory in de Sitter Space: Renormalization by Point Splitting, Proc. Roy. Soc. Lond. A 360 (1978) 117 [INSPIRE].
J.S. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys. 2 (1961) 407 [INSPIRE].
L.V. Keldysh, Diagram technique for nonequilibrium processes, Zh. Eksp. Teor. Fiz. 47 (1964) 1515 [INSPIRE].
V.A. Smirnov and E.R. Rakhmetov, The strategy of regions for asymptotic expansion of two loop vertex Feynman diagrams, Theor. Math. Phys. 120 (1999) 870 [hep-ph/9812529] [INSPIRE].
A. Pak and A. Smirnov, Geometric approach to asymptotic expansion of Feynman integrals, Eur. Phys. J. C 71 (2011) 1626 [arXiv:1011.4863] [INSPIRE].
T.Y. Semenova, A.V. Smirnov and V.A. Smirnov, On the status of expansion by regions, Eur. Phys. J. C 79 (2019) 136 [arXiv:1809.04325] [INSPIRE].
E. Gardi et al., The on-shell expansion: from Landau equations to the Newton polytope, JHEP 07 (2023) 197 [arXiv:2211.14845] [INSPIRE].
J.C. Collins, D.E. Soper and G.F. Sterman, Factorization of Hard Processes in QCD, Adv. Ser. Direct. High Energy Phys. 5 (1989) 1 [hep-ph/0409313] [INSPIRE].
B. Jantzen, Foundation and generalization of the expansion by regions, JHEP 12 (2011) 076 [arXiv:1111.2589] [INSPIRE].
M. Beneke and T. Feldmann, Factorization of heavy to light form-factors in soft collinear effective theory, Nucl. Phys. B 685 (2004) 249 [hep-ph/0311335] [INSPIRE].
C.W. Bauer, D. Pirjol and I.W. Stewart, Factorization and endpoint singularities in heavy to light decays, Phys. Rev. D 67 (2003) 071502 [hep-ph/0211069] [INSPIRE].
J.-Y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory, JHEP 05 (2012) 084 [arXiv:1202.0814] [INSPIRE].
X. Chen, Y. Wang and Z.-Z. Xianyu, Schwinger-Keldysh Diagrammatics for Primordial Perturbations, JCAP 12 (2017) 006 [arXiv:1703.10166] [INSPIRE].
D. Green and A. Premkumar, Dynamical RG and Critical Phenomena in de Sitter Space, JHEP 04 (2020) 064 [arXiv:2001.05974] [INSPIRE].
T. Huber and D. Maitre, HypExp 2, Expanding Hypergeometric Functions about Half-Integer Parameters, Comput. Phys. Commun. 178 (2008) 755 [arXiv:0708.2443] [INSPIRE].
Acknowledgments
We thank Tim Cohen and Ivo Sachs for important discussions, and Dan Green for valuable comments on SdSET. This work has been supported in part by the Excellence Cluster ORIGINS funded by the Deutsche Forschungsgemeinschaft under Grant No. EXC - 2094 - 390783311 and by the Cluster of Excellence Precision Physics, Fundamental Interactions, and Structure of Matter (PRISMA+ EXC 2118/1) funded by the German Research Foundation (DFG) within the German Excellence Strategy (Project ID 390831469).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2312.06766
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.
About this article
Cite this article
Beneke, M., Hager, P. & Sanfilippo, A.F. Cosmological correlators in massless ϕ4-theory and the method of regions. J. High Energ. Phys. 2024, 6 (2024). https://doi.org/10.1007/JHEP04(2024)006
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2024)006