Abstract
We propose a proper definition of the vacuum expectation value of the stress energy tensor 〈0| Tμν| 0〉 for integrable quantum field theories in two spacetime dimensions, which is the analog of the cosmological constant in 4d. For a wide variety of models, massive or massless, we show \( {\rho}_{\textrm{vac}}=-{m}^2/2\mathfrak{g} \) exactly, where \( \mathfrak{g} \) is a generalized coupling which we compute and m is a basic mass scale. The kinds of models we consider are the massive sinh-Gordon and sine-Gordon theories and perturbations of the Yang-Lee and 3-state Potts models, pure \( T\overline{T} \) perturbations of infra-red QFT’s, and UV completions of the latter which are massless flows between UV and IR fixed points. In the massive case m is the physical mass of the lightest particle and \( \mathfrak{g} \) is related to parameters in the 2-body S-matrix. In some examples ρvac = 0 due to a fractional supersymmetry. For massless cases, m can be a scale of spontaneous symmetry breaking. The “cosmological constant problem” generically arises in the free field limit \( \mathfrak{g} \) → 0, thus interactions can potentially resolve the problem at least for most cases considered in this paper. We speculate on extensions of these results to 4 spacetime dimensions and propose \( {\rho}_{\textrm{vac}}={m}^4/2\mathfrak{g} \), however without integrability we cannot yet propose a precise manner in which to calculate \( \mathfrak{g} \). Nevertheless, based on cosmological data on ρvac, if \( \mathfrak{g} \) ~ 1 then it is worth pointing out that the lightest mass particle is on the order of experimental values of proposed neutrino masses.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Weinberg, The Cosmological Constant Problem, Rev. Mod. Phys. 61 (1989) 1 [INSPIRE].
A.B. Zamolodchikov, Thermodynamic Bethe Ansatz in Relativistic Models. Scaling Three State Potts and Lee-yang Models, Nucl. Phys. B 342 (1990) 695 [INSPIRE].
C.-N. Yang and C.P. Yang, Thermodynamics of one-dimensional system of bosons with repulsive delta function interaction, J. Math. Phys. 10 (1969) 1115 [INSPIRE].
T.R. Klassen and E. Melzer, The Thermodynamics of purely elastic scattering theories and conformal perturbation theory, Nucl. Phys. B 350 (1991) 635 [INSPIRE].
J. Martin, Everything You Always Wanted To Know About The Cosmological Constant Problem (But Were Afraid To Ask), Comptes Rendus Physique 13 (2012) 566 [arXiv:1205.3365] [INSPIRE].
G. Mussardo, Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics Oxford University Press (2010) [ISBN: 9780199547586].
A. Cavaglià, S. Negro, I.M. Szécsényi and R. Tateo, \( T\overline{T} \)-deformed 2D Quantum Field Theories, JHEP 10 (2016) 112 [arXiv:1608.05534] [INSPIRE].
F.A. Smirnov and A.B. Zamolodchikov, On space of integrable quantum field theories, Nucl. Phys. B 915 (2017) 363 [arXiv:1608.05499] [INSPIRE].
S. Dubovsky, V. Gorbenko and G. Hernández-Chifflet, \( T\overline{T} \) partition function from topological gravity, JHEP 09 (2018) 158 [arXiv:1805.07386] [INSPIRE].
L. McGough, M. Mezei and H. Verlinde, Moving the CFT into the bulk with \( T\overline{T} \), JHEP 04 (2018) 010 [arXiv:1611.03470] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models, Annals Phys. 120 (1979) 273.
A.B. Zamolodchikov, Thermodynamic Bethe ansatz for RSOS scattering theories, Nucl. Phys. B 358 (1991) 497 [INSPIRE].
C. Destri and H.J. de Vega, New exact results in affine Toda field theories: Free energy and wave function renormalizations, Nucl. Phys. B 358 (1991) 251 [INSPIRE].
A.B. Zamolodchikov, Mass scale in the sine-Gordon model and its reductions, Int. J. Mod. Phys. A 10 (1995) 1125 [INSPIRE].
G.M. Gandenberger and N.J. MacKay, Remarks on excited states of affine Toda solitons, Phys. Lett. B 390 (1997) 185 [hep-th/9608055] [INSPIRE].
G. Takács, Quantum affine symmetry and scattering amplitudes of the imaginary coupled \( {d}_4^{(3)} \) affine Toda field theory, Nucl. Phys. B 502 (1997) 629 [hep-th/9701118] [INSPIRE].
D. Bernard and A. Leclair, Quantum group symmetries and non-local currents in 2D QFT, Commun. Math. Phys. 142 (1991) 99.
A. LeClair and C. Vafa, Quantum affine symmetry as generalized supersymmetry, Nucl. Phys. B 401 (1993) 413 [hep-th/9210009] [INSPIRE].
D. Bernard and A. Leclair, Residual quantum symmetries of the restricted sine-gordon theories, Nucl. Phys. B 340 (1990) 721.
R.J. Baxter, Exactly solved models in statistical mechanics, Dover (1989).
G. Camilo et al., On factorizable S-matrices, generalized TTbar, and the Hagedorn transition, JHEP 10 (2021) 062 [arXiv:2106.11999] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Massless factorized scattering and sigma models with topological terms, Nucl. Phys. B 379 (1992) 602 [INSPIRE].
A. LeClair, deformation of the Ising model and its ultraviolet completion, J. Stat. Mech. 2111 (2021) 113104 [arXiv:2107.02230] [INSPIRE].
A.B. Zamolodchikov, From tricritical Ising to critical Ising by thermodynamic Bethe ansatz, Nucl. Phys. B 358 (1991) 524 [INSPIRE].
C. Ahn, C. Kim, C. Rim and A.B. Zamolodchikov, RG flows from superLiouville theory to critical Ising model, Phys. Lett. B 541 (2002) 194 [hep-th/0206210] [INSPIRE].
C. Ahn and A. LeClair, On the classification of UV completions of integrable \( T\overline{T} \) deformations of CFT, JHEP 08 (2022) 179 [arXiv:2205.10905] [INSPIRE].
A. Leclair and G. Mussardo, Finite temperature correlation functions in integrable QFT, Nucl. Phys. B 552 (1999) 624 [hep-th/9902075] [INSPIRE].
A. LeClair, Comment on the cosmological constant for λϕ4 theory in d spacetime dimensions, to appear.
Astronomical data is taken from WMAP results, http://pdg.lbl.gov/2012/reviews/rpp2012-rev-astrophysical-constants.pdf.
M.C. Gonzalez-Garcia and Y. Nir, Neutrino Masses and Mixing: Evidence and Implications, Rev. Mod. Phys. 75 (2003) 345 [hep-ph/0202058] [INSPIRE].
M. Montero, T. Van Riet and G. Venken, Festina Lente: EFT Constraints from Charged Black Hole Evaporation in de Sitter, JHEP 01 (2020) 039 [arXiv:1910.01648] [INSPIRE].
M. Montero, C. Vafa, T. Van Riet and G. Venken, The FL bound and its phenomenological implications, JHEP 10 (2021) 009 [arXiv:2106.07650] [INSPIRE].
Acknowledgments
We wish to thank Changrim Ahn for discussions, as our work together [26] indirectly led in part to the considerations of this paper. We also wish to thank Miguel Montero and Gerben Venken for pointing out their potentially related work [31, 32] after this work was originally completed. Finally we thank the referee at JHEP for several expert and insightful remarks that substantially improved the original version of this article.
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: 2301.09019
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
LeClair, A. Mingling of the infrared and ultraviolet and the “cosmological constant” for interacting QFT in 2d. J. High Energ. Phys. 2023, 222 (2023). https://doi.org/10.1007/JHEP05(2023)222
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2023)222