Abstract
A method of exact all-order summation of leading infrared logarithms in two dimensional massless Φ4-type non-renormalizable effective field theories (EFTs) is developed. The method is applied to the O(N)-symmetric EFT, which is a two-dimensional sibling of the four dimensional O(N + 1)/O(N) sigma-model. For the first time the exact all-order summation of the (E2ln(1/E))n contributions (chiral logarithms) for the 2 → 2 scattering amplitudes is performed in closed analytical form. The cases when the resulting amplitudes turn to be meromorphic functions with an infinite number of poles (Landau poles) are identified. This provides the first explicit example of quasi-renormalizable field theories.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Kivel, M.V. Polyakov and A. Vladimirov, Chiral Logarithms in the Massless Limit Tamed, Phys. Rev. Lett. 101 (2008) 262001 [arXiv:0809.3236] [INSPIRE].
J. Koschinski, M.V. Polyakov and A.A. Vladimirov, Leading Infrared Logarithms from Unitarity, Analyticity and Crossing, Phys. Rev. D 82 (2010) 014014 [arXiv:1004.2197] [INSPIRE].
N. Kivel, M.V. Polyakov and A. Vladimirov, Large-N Summation of Chiral Logs for Generalized Parton Distributions, Phys. Rev. D 79 (2009) 014028 [arXiv:0809.2064] [INSPIRE].
N.A. Kivel, M.V. Polyakov and A.A. Vladimirov, Leading Chiral Logarithms for Pion Form Factors to Arbitrary Number of Loops, JETP Lett. 89 (2009) 529 [arXiv:0904.3008] [INSPIRE].
I.A. Perevalova, M.V. Polyakov, A.N. Vall and A.A. Vladimirov, Chiral Inflation of the Pion Radius, arXiv:1105.4990 [INSPIRE].
B. Ananthanarayan, S. Ghosh, A. Vladimirov and D. Wyler, Leading Logarithms of the Two Point Function in Massless O(N) and SU(N) Models to any Order from Analyticity and Unitarity, Eur. Phys. J. A 54 (2018) 123 [arXiv:1803.07013] [INSPIRE].
M.V. Polyakov and A.A. Vladimirov, Leading Infrared Logarithms for σ-model with Fields on Arbitrary Riemann Manifold, Theor. Math. Phys. 169 (2011) 1499 [arXiv:1012.4205] [INSPIRE].
A.N. Vasilev, The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics, Chapman and Hall/CRC, Boca Raton, U.S.A. (2004) [INSPIRE].
J. Koschinski, Leading Logarithms in Four Fermion Theories, Ph.D. Thesis, Ruhr University, Bochum, (2015), unpublished [https://d-nb.info/1109051174/34].
M.V. Polyakov, K.M. Semenov-Tian-Shansky, A.O. Smirnov and A.A. Vladimirov, Quasi-Renormalizable Quantum Field Theories, arXiv:1811.08449 [INSPIRE].
E. Brézin and J. Zinn-Justin, Spontaneous Breakdown of Continuous Symmetries Near Two-Dimensions, Phys. Rev. B 14 (1976) 3110 [INSPIRE].
E. Brézin, J. Zinn-Justin and J.C. Le Guillou, Renormalization of the Nonlinear σ-model in 2+ϵ Dimension, Phys. Rev. D 14 (1976) 2615 [INSPIRE].
F. David, Cancellations of Infrared Divergences in the Two-dimensional Nonlinear σ-models, Commun. Math. Phys. 81 (1981) 149 [INSPIRE].
W.I. Smirnow, Lehrgang der höheren Mathematik, Teil 3/2, Deutscher Verlag der Wissenschaften, Berlin (1955).
R. Bacher and, P. Flajolet, Pseudo-factorials, Elliptic Functions, and Continued Fractions, Ramanujan J. 21 (2010) 71 [arXiv:0901.1379].
E. van Fossen Conrad and P. Flajolet, The Fermat Cubic, Elliptic Functions, Continued Fractions, and a Combinatorial Excursion, Sém. Lothar. Combin. 54 (2006) B54g [math/0507268] [https://www.mat.univie.ac.at/~slc/wpapers/s54conflaj.pdf].
V.Z. Enolski, E. Hackmann, V. Kagramanova, J. Kunz and C. Lammerzahl, Inversion of hyperelliptic integrals of arbitrary genus with application to particle motion in General Relativity, J. Geom. Phys. 61 (2011) 899 [arXiv:1011.6459] [INSPIRE].
D. Delphenich, J. Schechter and S. Vaidya, Pion pion scattering in two-dimensions, Phys. Rev. D 59 (1999) 056004 [hep-ph/9806349] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1811.12289
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Linzen, J., Polyakov, M.V., Semenov-Tian-Shansky, K.M. et al. Exact summation of leading infrared logarithms in 2D effective field theories. J. High Energ. Phys. 2019, 7 (2019). https://doi.org/10.1007/JHEP04(2019)007
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2019)007