Abstract
Form factors in the sinh-Gordon model are studied semiclassically for small values of the parameter b ~ ħ1/2 in the background of a radial classical solution, which describes a heavy exponential operator placed at the origin. For this purpose we use a generalization of the radial quantization scheme, well known for a massless boson field. We introduce and study new special functions which generalize the Bessel functions and have a nice interpretation in the Tracy-Widom theory of the Fredholm determinant solutions of the classical sinh-Gordon model. Form factors of the exponential operators in the leading order are completely determined by the classical solutions, while form factors of the descendant operators contain quantum corrections even in this approximation. The construction of descendant operators in two chiralities requires renormalizations similar to those encountered in the conformal perturbation theory.
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References
M. Karowski and P. Weisz, Exact Form Factors in (1+1)-Dimensional Field Theoretic Models with Soliton Behavior, Nucl. Phys. B 139 (1978) 455 [INSPIRE].
B. Berg, M. Karowski and P. Weisz, Construction of Green Functions from an Exact S Matrix, Phys. Rev. D 19 (1979) 2477 [INSPIRE].
F.A. Smirnov, The quantum Gelfand-Levitan-Marchenko equations and form factors in the sine-Gordon model, J. Phys. A 17 (1984) L873 [INSPIRE].
F.A. Smirnov, Form factors in completely integrable models of quantum field theory, Adv. Ser. Math. Phys. 14 (1992) pp. 1–208 [INSPIRE].
A. Fring, G. Mussardo and P. Simonetti, Form factors for integrable Lagrangian field theories, the sinh-Gordon theory, Nucl. Phys. B 393 (1993) 413 [hep-th/9211053] [INSPIRE].
G. Mussardo, Correlation functions in two-dimensional integrable quantum field theories, in the proceedings of the Advanced Research Workshop on Integrable Quantum Field Theories: Conformal Field Theories and Current Algebra, Integrable Models, 2D Quantum Gravity, Matrix Models and String Theory, (1992) [hep-th/9212119] [INSPIRE].
A. Koubek and G. Mussardo, On the operator content of the sinh-Gordon model, Phys. Lett. B 311 (1993) 193 [hep-th/9306044] [INSPIRE].
S.L. Lukyanov, Form factors of exponential fields in the Sine-Gordon model, Mod. Phys. Lett. A 12 (1997) 2543 [hep-th/9703190] [INSPIRE].
H.M. Babujian and M. Karowski, The Exact quantum sine-Gordon field equation and other nonperturbative results, Phys. Lett. B 471 (1999) 53 [hep-th/9909153] [INSPIRE].
H. Babujian and M. Karowski, Sine-Gordon breather form factors and quantum field equations, J. Phys. A 35 (2002) 9081 [hep-th/0204097] [INSPIRE].
B. Feigin and M. Lashkevich, Form factors of descendant operators: Free field construction and reflection relations, J. Phys. A 42 (2009) 304014 [arXiv:0812.4776] [INSPIRE].
M. Lashkevich and Y. Pugai, On form factors and Macdonald polynomials, JHEP 09 (2013) 095 [arXiv:1305.1674] [INSPIRE].
M. Lashkevich and Y. Pugai, Form factors in sinh- and sine-Gordon models, deformed Virasoro algebra, Macdonald polynomials and resonance identities, Nucl. Phys. B 877 (2013) 538 [arXiv:1307.0243] [INSPIRE].
M. Jimbo, T. Miwa and F. Smirnov, Fermionic screening operators in the sine-Gordon model, Physica D 241 (2012) 2122 [arXiv:1103.1534] [INSPIRE].
M. Jimbo, T. Miwa and F. Smirnov, Fermionic structure in the sine-Gordon model: Form factors and null-vectors, Nucl. Phys. B 852 (2011) 390 [arXiv:1105.6209] [INSPIRE].
H. Boos, M. Jimbo, T. Miwa and F. Smirnov, Hidden Grassmann Structure in the XXZ Model IV: CFT limit, Commun. Math. Phys. 299 (2010) 825 [arXiv:0911.3731] [INSPIRE].
H. Boos, Fermionic basis in conformal field theory and thermodynamic Bethe ansatz for excited states, SIGMA 7 (2011) 007 [arXiv:1010.0858] [INSPIRE].
S. Negro and F. Smirnov, Reflection relations and fermionic basis, Lett. Math. Phys. 103 (2013) 1293 [arXiv:1304.1860] [INSPIRE].
H. Boos and F. Smirnov, New results on integrable structure of conformal field theory, J. Phys. A 51 (2018) 374003 [arXiv:1610.09537] [INSPIRE].
A.B. Zamolodchikov, Higher Order Integrals of Motion in Two-Dimensional Models of the Field Theory with a Broken Conformal Symmetry, JETP Lett. 46 (1987) 160 [INSPIRE].
A.B. Zamolodchikov, Integrable field theory from conformal field theory, Adv. Stud. Pure Math. 19 (1989) 641 [INSPIRE].
Al.B. Zamolodchikov, Two-point correlation function in scaling Lee-Yang model, Nucl. Phys. B 348 (1991) 619 [INSPIRE].
A.B. Zamolodchikov and Al.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE].
V. Fateev, S.L. Lukyanov, A.B. Zamolodchikov and Al.B. Zamolodchikov, Expectation values of boundary fields in the boundary Sine-Gordon model, Phys. Lett. B 406 (1997) 83 [hep-th/9702190] [INSPIRE].
C.A. Tracy and H. Widom, Fredholm determinants and the mKdV/Sinh-Gordon hierarchies, Commun. Math. Phys. 179 (1996) 1 [solv-int/9506006].
Al.B. Zamolodchikov, Mass scale in the sine-Gordon model and its reductions, Int. J. Mod. Phys. A 10 (1995) 1125 [INSPIRE].
B.M. McCoy, C.A. Tracy and T.T. Wu, Painlevé Functions of the Third Kind, J. Math. Phys. 18 (1977) 1058 [INSPIRE].
S. Cecotti, P. Fendley, K.A. Intriligator and C. Vafa, A New supersymmetric index, Nucl. Phys. B 386 (1992) 405 [hep-th/9204102] [INSPIRE].
Al.B. Zamolodchikov, Painlevé III and 2-d polymers, Nucl. Phys. B 432 (1994) 427 [hep-th/9409108] [INSPIRE].
E.L. Basor and C.A. Tracy, Asymptotics of a tau-function and Teplitz determinants with singular generating functions, Int. J. Mod. Phys. A 7S1A (1992) 83 [INSPIRE].
O. Gamayun, N. Iorgov and O. Lisovyy, How instanton combinatorics solves Painlevé VI, V and IIIs, J. Phys. A 46 (2013) 335203 [arXiv:1302.1832] [INSPIRE].
S.L. Lukyanov and A.B. Zamolodchikov, Exact expectation values of local fields in quantum sine-Gordon model, Nucl. Phys. B 493 (1997) 571 [hep-th/9611238] [INSPIRE].
V. Fateev et al., Expectation values of descendent fields in the sine-Gordon model, Nucl. Phys. B 540 (1999) 587 [hep-th/9807236] [INSPIRE].
M. Jimbo, T. Miwa and F. Smirnov, On one-point functions of descendants in sine-Gordon model, arXiv:0912.0934 [INSPIRE].
S. Negro and F. Smirnov, On one-point functions for sinh-Gordon model at finite temperature, Nucl. Phys. B 875 (2013) 166 [arXiv:1306.1476] [INSPIRE].
Acknowledgments
The authors are grateful to M. Bershtein, A. Litvinov and, especially, to Ya. Pugai for discussions. The work of M.L. was supported by the Russian Science Foundation under the grant 23-12-00333.
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Lashkevich, M., Lisovyy, O. & Ushakova, T. Semiclassical approach to form factors in the sinh-Gordon model. J. High Energ. Phys. 2023, 157 (2023). https://doi.org/10.1007/JHEP07(2023)157
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DOI: https://doi.org/10.1007/JHEP07(2023)157