Abstract
In this paper we extend the study initiated in [1] to the computation of one-loop elastic amplitudes. We consider 1+1 dimensional massive bosonic Lagrangians with polynomial-like potentials and absence of inelastic processes at the tree level; starting from these assumptions we show how to write sums of one-loop diagrams as products and integrals of tree-level amplitudes. We derive in this way a universal formula for the one-loop two-to-two S-matrices in terms of tree S-matrices. We test our results on different integrable theories, such as sinh-Gordon, Bullough-Dodd and the full class of simply-laced affine Toda theories, finding perfect agreement with the bootstrapped S-matrices known in the literature. We show how Landau singularities in amplitudes are naturally captured by our universal formula while they are lost in results based on unitarity-cut methods implemented in the past [2, 3].
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References
D. Polvara, One-loop inelastic amplitudes from tree-level elasticity in 2d, JHEP 04 (2023) 020 [arXiv:2302.04709] [INSPIRE].
L. Bianchi, V. Forini and B. Hoare, Two-dimensional S-matrices from unitarity cuts, JHEP 07 (2013) 088 [arXiv:1304.1798] [INSPIRE].
L. Bianchi and B. Hoare, AdS3 × S3 × M4 string S-matrices from unitarity cuts, JHEP 08 (2014) 097 [arXiv:1405.7947] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Factorized s Matrices in Two-Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Models, Annals Phys. 120 (1979) 253 [INSPIRE].
R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [INSPIRE].
R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].
O.T. Engelund, R.W. McKeown and R. Roiban, Generalized unitarity and the worldsheet S matrix in AdSn × Sn × M10−2n, JHEP 08 (2013) 023 [arXiv:1304.4281] [INSPIRE].
S. Demulder et al., Exact approaches on the string worldsheet, arXiv:2312.12930 [INSPIRE].
P. Dorey and D. Polvara, Tree level integrability in 2d quantum field theories and affine Toda models, JHEP 02 (2022) 199 [arXiv:2111.02210] [INSPIRE].
B. Gabai et al., No Particle Production in Two Dimensions: Recursion Relations and Multi-Regge Limit, JHEP 02 (2019) 094 [arXiv:1803.03578] [INSPIRE].
S.R. Coleman and H.J. Thun, On the Prosaic Origin of the Double Poles in the Sine-Gordon S Matrix, Commun. Math. Phys. 61 (1978) 31 [INSPIRE].
P. Dorey, Root systems and purely elastic S matrices, Nucl. Phys. B 358 (1991) 654 [INSPIRE].
P. Dorey, Root systems and purely elastic S matrices. 2, Nucl. Phys. B 374 (1992) 741 [hep-th/9110058] [INSPIRE].
P. Dorey, Exact S matrices, in the proceedings of the Eotvos Summer School in Physics: Conformal Field Theories and Integrable Models, Budapest, Hungary, 13–18 August 1996 [hep-th/9810026] [INSPIRE].
S.P. Khastgir, Affine Toda field theory from tree unitarity, Eur. Phys. J. C 33 (2004) 137 [hep-th/0308032] [INSPIRE].
C. Bercini and D. Trancanelli, Supersymmetric integrable theories without particle production, Phys. Rev. D 97 (2018) 105013 [arXiv:1803.03612] [INSPIRE].
C.J. Goebel, On the Sine-Gordon S Matrix, Prog. Theor. Phys. Suppl. 86 (1986) 261 [INSPIRE].
I. Arefeva and V. Korepin, Scattering in two-dimensional model with Lagrangian L = 1/γ(1/2(∂μu)2 + m2(cos u − 1)), Pisma Zh. Eksp. Teor. Fiz. 20 (1974) 680 [INSPIRE].
H.W. Braden and R. Sasaki, Affine Toda perturbation theory, Nucl. Phys. B 379 (1992) 377 [INSPIRE].
H.W. Braden, E. Corrigan, P.E. Dorey and R. Sasaki, Affine Toda Field Theory and Exact S Matrices, Nucl. Phys. B 338 (1990) 689 [INSPIRE].
P. Christe and G. Mussardo, Elastic s Matrices in (1+1)-Dimensions and Toda Field Theories, Int. J. Mod. Phys. A 5 (1990) 4581 [INSPIRE].
M. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2014) [https://doi.org/10.1017/9781139540940].
B. Hoare and A.A. Tseytlin, On the perturbative S-matrix of generalized sine-Gordon models, JHEP 11 (2010) 111 [arXiv:1008.4914] [INSPIRE].
H.W. Braden and R. Sasaki, The S matrix coupling dependence for a, d and e affine toda field theory, Phys. Lett. B 255 (1991) 343 [INSPIRE].
A.E. Arinshtein, V.A. Fateev and A.B. Zamolodchikov, Quantum s Matrix of the (1+1)-Dimensional Todd Chain, Phys. Lett. B 87 (1979) 389 [INSPIRE].
E. Corrigan, P.E. Dorey and R. Sasaki, On a generalized bootstrap principle, Nucl. Phys. B 408 (1993) 579 [hep-th/9304065] [INSPIRE].
P. Dorey and D. Polvara, From tree- to loop-simplicity in affine Toda theories II: higher-order poles and cut decompositions, JHEP 10 (2023) 177 [arXiv:2307.15498] [INSPIRE].
H.W. Braden, E. Corrigan, P.E. Dorey and R. Sasaki, Multiple poles and other features of affine Toda field theory, Nucl. Phys. B 356 (1991) 469 [INSPIRE].
P. Dorey and D. Polvara, From tree- to loop-simplicity in affine Toda theories I: Landau singularities and their subleading coefficients, JHEP 09 (2022) 220 [arXiv:2206.09368] [INSPIRE].
A.V. Mikhailov, M.A. Olshanetsky and A.M. Perelomov, Two-Dimensional Generalized Toda Lattice, Commun. Math. Phys. 79 (1981) 473 [INSPIRE].
D.I. Olive and N. Turok, Local Conserved Densities and Zero Curvature Conditions for Toda Lattice Field Theories, Nucl. Phys. B 257 (1985) 277 [INSPIRE].
P.G.O. Freund, T.R. Klassen and E. Melzer, S Matrices for Perturbations of Certain Conformal Field Theories, Phys. Lett. B 229 (1989) 243 [INSPIRE].
C. Destri and H.J. de Vega, The Exact S Matrix of the Affine E8 Toda Field Theory, Phys. Lett. B 233 (1989) 336 [INSPIRE].
P. Christe and G. Mussardo, Integrable Systems Away from Criticality: The Toda Field Theory and S Matrix of the Tricritical Ising Model, Nucl. Phys. B 330 (1990) 465 [INSPIRE].
T.R. Klassen and E. Melzer, Purely Elastic Scattering Theories and their Ultraviolet Limits, Nucl. Phys. B 338 (1990) 485 [INSPIRE].
H.W. Braden, E. Corrigan, P.E. Dorey and R. Sasaki, Extended Toda Field Theory and Exact S Matrices, Phys. Lett. B 227 (1989) 411 [INSPIRE].
A. Fring and D.I. Olive, The Fusing rule and the scattering matrix of affine Toda theory, Nucl. Phys. B 379 (1992) 429 [INSPIRE].
G.W. Delius, M.T. Grisaru and D. Zanon, Exact S matrices for the nonsimply laced affine Toda theories \( {a}_{\left(2n-1\right)}^{(2)} \), Phys. Lett. B 277 (1992) 414 [hep-th/9112007] [INSPIRE].
G.W. Delius, M.T. Grisaru and D. Zanon, Exact S matrices for nonsimply laced affine Toda theories, Nucl. Phys. B 382 (1992) 365 [hep-th/9201067] [INSPIRE].
P. Dorey, A remark on the coupling dependence in affine Toda field theories, Phys. Lett. B 312 (1993) 291 [hep-th/9304149] [INSPIRE].
T. Oota, q deformed Coxeter element in nonsimply laced affine Toda field theories, Nucl. Phys. B 504 (1997) 738 [hep-th/9706054] [INSPIRE].
G.W. Delius, M.T. Grisaru, S. Penati and D. Zanon, The exact S matrices of affine Toda theories based on Lie superalgebras, Phys. Lett. B 256 (1991) 164 [INSPIRE].
G.W. Delius, M.T. Grisaru, S. Penati and D. Zanon, Exact S matrix and perturbative calculations in affine Toda theories based on Lie superalgebras, Nucl. Phys. B 359 (1991) 125 [INSPIRE].
H.W. Braden et al., Singularity analysis in An affine Toda theories, Prog. Theor. Phys. 88 (1992) 1205 [hep-th/9207025] [INSPIRE].
R. Sasaki and F.P. Zen, The affine Toda S matrices versus perturbation theory, Int. J. Mod. Phys. A 8 (1993) 115 [INSPIRE].
E. Corrigan, Recent developments in affine Toda quantum field theory, in the proceedings of the CRM-CAP Summer School on Particles and Fields ’94, Banff, Canada, 16–24 August 1994 [hep-th/9412213] [INSPIRE].
A. Fring, H.C. Liao and D.I. Olive, The mass spectrum and coupling in affine Toda theories, Phys. Lett. B 266 (1991) 82 [INSPIRE].
A. Fring, Couplings in affine Toda field theories, hep-th/9212107 [INSPIRE].
B. Kostant, The principal three-dimensional subgroup and the betti numbers of a complex simple lie group, Am. J. MAth. 81 (1959) 973.
R. Steinberg, Finite Reflection Groups, Trans. Am. Math. Soc. 91 (1959) 493.
M.D. Freeman, On the mass spectrum of affine Toda field theory, Phys. Lett. B 261 (1991) 57 [INSPIRE].
H.J. de Vega and J.M. Maillet, Renormalization Character and Quantum S Matrix for a Classically Integrable Theory, Phys. Lett. B 101 (1981) 302 [INSPIRE].
B. Hoare, Integrable deformations of sigma models, J. Phys. A 55 (2022) 093001 [arXiv:2109.14284] [INSPIRE].
B. Hoare, N. Levine and A.A. Tseytlin, On the massless tree-level S-matrix in 2d sigma models, J. Phys. A 52 (2019) 144005 [arXiv:1812.02549] [INSPIRE].
R. Roiban, P. Sundin, A. Tseytlin and L. Wulff, The one-loop worldsheet S-matrix for the AdSn × Sn × T10−2n superstring, JHEP 08 (2014) 160 [arXiv:1407.7883] [INSPIRE].
P. Sundin and L. Wulff, The complete one-loop BMN S-matrix in AdS3 × S3 × T4, JHEP 06 (2016) 062 [arXiv:1605.01632] [INSPIRE].
R. Borsato et al., Dressing phases of AdS3/CFT2, Phys. Rev. D 88 (2013) 066004 [arXiv:1306.2512] [INSPIRE].
S. Frolov and A. Sfondrini, New dressing factors for AdS3/CFT2, JHEP 04 (2022) 162 [arXiv:2112.08896] [INSPIRE].
R.W. Carter, Simple groups of Lie type, vol. 22, John Wiley & Sons (1989).
Acknowledgments
We thank Patrick Dorey, Ben Hoare, Anton Pribytok and Alessandro Sfondrini for related discussions. DP especially thanks Ben Hoare for useful discussions and the kind hospitality in Durham where this work was started. The authors also thank the participants of the workshop “Integrability in Low-Supersymmetry Theories” in Filicudi, Italy, for a stimulating environment where part of this work was carried out. The authors acknowledge support from the European Union — NextGenerationEU, from the program STARS@UNIPD, under the project “Exact-Holography — A new exact approach to holography: harnessing the power of string theory, conformal field theory and integrable models”, also from the PRIN Project n. 2022ABPBEY, “Understanding quantum field theory through its deformations”, and from the CARIPLO Foundation “Supporto ai giovani talenti italiani nelle competizioni dell’European Research Council” grant n. 2022-1886 “Nuove basi per la teoria delle stringhe”.
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Fabri, M., Polvara, D. One-loop elastic amplitudes from tree-level elasticity in 2d. J. High Energ. Phys. 2024, 104 (2024). https://doi.org/10.1007/JHEP06(2024)104
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DOI: https://doi.org/10.1007/JHEP06(2024)104