Abstract
We derive the leading exponential finite volume corrections in two dimensional integrable models for non-diagonal form factors in diagonally scattering theories. These formulas are expressed in terms of the infinite volume form factors and scattering matrices. If the particles are bound states then the leading exponential finite-size corrections (μ-terms) are related to virtual processes in which the particles disintegrate into their constituents. For non-bound state particles the leading exponential finite-size corrections (F-terms) come from virtual particles traveling around the finite world. In these F-terms a specifically regulated infinite volume form factor is integrated for the momenta of the virtual particles. The F-term is also present for bound states and the μ-term can be obtained by taking an appropriate residue of the F-term integral. We check our results numerically in the Lee-Yang and sinh-Gordon models based on newly developed Hamiltonian truncations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
H. Babujian and M. Karowski, Towards the construction of Wightman functions of integrable quantum field theories, Int. J. Mod. Phys.A 19S2 (2004) 34 [hep-th/0301088] [INSPIRE].
P. Dorey, Exact S matrices, in Conformal field theories and integrable models. Proceedings, Eotvos Graduate Course, Budapest, Hungary, 13–18 August 1996, pg. 85 [hep-th/9810026] [INSPIRE].
F. Smirnov, Form-factors in completely integrable models of quantum field theory, Adv. Ser. Math. Phys.14 (1992) 1 [INSPIRE].
M. Lüscher, Volume dependence of the energy spectrum in massive quantum field theories. 2. Scattering states, Commun. Math. Phys.105 (1986) 153 [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].
M. Lüscher, Volume dependence of the energy spectrum in massive quantum field theories. 1. Stable particle states, Commun. Math. Phys.104 (1986) 177 [INSPIRE].
T.R. Klassen and E. Melzer, On the relation between scattering amplitudes and finite size mass corrections in QFT, Nucl. Phys.B 362 (1991) 329 [INSPIRE].
R.A. Janik and T. Lukowski, Wrapping interactions at strong coupling: the giant magnon, Phys. Rev.D 76 (2007) 126008 [arXiv:0708.2208] [INSPIRE].
Z. Bajnok and R.A. Janik, Four-loop perturbative Konishi from strings and finite size effects for multiparticle states, Nucl. Phys.B 807 (2009) 625 [arXiv:0807.0399] [INSPIRE].
C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie, TBA, NLO Lüscher correction and double wrapping in twisted AdS/CFT, JHEP12 (2011) 059 [arXiv:1108.4914] [INSPIRE].
D. Bombardelli, A next-to-leading Luescher formula, JHEP01 (2014) 037 [arXiv:1309.4083] [INSPIRE].
P. Dorey and R. Tateo, Excited states by analytic continuation of TBA equations, Nucl. Phys.B 482 (1996) 639 [hep-th/9607167] [INSPIRE].
Z. Bajnok, Review of AdS/CFT integrability, chapter III.6: thermodynamic Bethe ansatz, Lett. Math. Phys.99 (2012) 299 [arXiv:1012.3995] [INSPIRE].
B. Pozsgay and G. Takács, Form-factors in finite volume I: form-factor bootstrap and truncated conformal space, Nucl. Phys.B 788 (2008) 167 [arXiv:0706.1445] [INSPIRE].
B. Pozsgay and G. Takács, Form factors in finite volume II: disconnected terms and finite temperature correlators, Nucl. Phys.B 788 (2008) 209 [arXiv:0706.3605] [INSPIRE].
Z. Bajnok and C. Wu, Diagonal form factors from non-diagonal ones, arXiv:1707.08027 [INSPIRE].
A. Leclair and G. Mussardo, Finite temperature correlation functions in integrable QFT, Nucl. Phys.B 552 (1999) 624 [hep-th/9902075] [INSPIRE].
H. Saleur, A comment on finite temperature correlations in integrable QFT, Nucl. Phys.B 567 (2000) 602 [hep-th/9909019] [INSPIRE].
B. Pozsgay, Mean values of local operators in highly excited Bethe states, J. Stat. Mech.1101 (2011) P01011 [arXiv:1009.4662] [INSPIRE].
B. Pozsgay, Form factor approach to diagonal finite volume matrix elements in integrable QFT, JHEP07 (2013) 157 [arXiv:1305.3373] [INSPIRE].
B. Pozsgay, I.M. Szecsenyi and G. Takács, Exact finite volume expectation values of local operators in excited states, JHEP04 (2015) 023 [arXiv:1412.8436] [INSPIRE].
B. Pozsgay, Lüscher’s μ-term and finite volume bootstrap principle for scattering states and form factors, Nucl. Phys.B 802 (2008) 435 [arXiv:0803.4445] [INSPIRE].
Z. Bajnok, J. Balog, M. Lájer and C. Wu, Field theoretical derivation of Lüscher’s formula and calculation of finite volume form factors, JHEP07 (2018) 174 [arXiv:1802.04021] [INSPIRE].
B. Pozsgay and I.M. Szécsényi, LeClair-Mussardo series for two-point functions in integrable QFT, JHEP05 (2018) 170 [arXiv:1802.05890] [INSPIRE].
A. Cortés Cubero and M. Panfil, Thermodynamic bootstrap program for integrable QFT’s: form factors and correlation functions at finite energy density, JHEP01 (2019) 104 [arXiv:1809.02044] [INSPIRE].
Z. Bajnok and R.A. Janik, String field theory vertex from integrability, JHEP04 (2015) 042 [arXiv:1501.04533] [INSPIRE].
B. Basso, S. Komatsu and P. Vieira, Structure constants and integrable bootstrap in planar N = 4 SYM theory,arXiv:1505.06745 [INSPIRE].
Z. Bajnok and R.A. Janik, From the octagon to the SFT vertex — gluing and multiple wrapping, JHEP06 (2017) 058 [arXiv:1704.03633] [INSPIRE].
B. Basso, V. Goncalves and S. Komatsu, Structure constants at wrapping order, JHEP05 (2017) 124 [arXiv:1702.02154] [INSPIRE].
J.L. Cardy and G. Mussardo, S matrix of the Yang-Lee edge singularity in two-dimensions, Phys. Lett.B 225 (1989) 275 [INSPIRE].
J. Teschner, On the spectrum of the sinh-Gordon model in finite volume, Nucl. Phys.B 799 (2008) 403 [hep-th/0702214] [INSPIRE].
Z. Bajnok, O. el Deeb and P.A. Pearce, Finite-volume spectra of the Lee-Yang model, JHEP04 (2015) 073 [arXiv:1412.8494] [INSPIRE].
V.P. Yurov and A.B. Zamolodchikov, Truncated conformal space approach to scaling Lee-Yang model, Int. J. Mod. Phys.A 5 (1990) 3221 [INSPIRE].
A. Coser, M. Beria, G.P. Brandino, R.M. Konik and G. Mussardo, Truncated conformal space approach for 2D Landau-Ginzburg theories, J. Stat. Mech.1412 (2014) P12010 [arXiv:1409.1494] [INSPIRE].
M. Hogervorst, S. Rychkov and B.C. van Rees, Truncated conformal space approach in d dimensions: a cheap alternative to lattice field theory?, Phys. Rev.D 91 (2015) 025005 [arXiv:1409.1581] [INSPIRE].
S. Rychkov and L.G. Vitale, Hamiltonian truncation study of the ϕ 4theory in two dimensions, Phys. Rev.D 91 (2015) 085011 [arXiv:1412.3460] [INSPIRE].
A.B. Zamolodchikov, Mass scale in the sine-Gordon model and its reductions, Int. J. Mod. Phys.A 10 (1995) 1125 [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys.B 477 (1996) 577 [hep-th/9506136] [INSPIRE].
A.B. Zamolodchikov, On the thermodynamic Bethe ansatz equation in sinh-Gordon model, J. Phys.A 39 (2006) 12863 [hep-th/0005181] [INSPIRE].
S. Rychkov and L.G. Vitale, Hamiltonian truncation study of the ϕ 4theory in two dimensions. II. The Z 2-broken phase and the Chang duality, Phys. Rev.D 93 (2016) 065014 [arXiv:1512.00493] [INSPIRE].
Z. Bajnok and M. Lajer, Truncated Hilbert space approach to the 2d ϕ 4theory, JHEP10 (2016) 050 [arXiv:1512.06901] [INSPIRE].
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].
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].
Z. Bajnok and F. Smirnov, Diagonal finite volume matrix elements in the sinh-Gordon model, Nucl. Phys.B 945 (2019) 114664 [arXiv:1903.06990] [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: 1904.00492
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
Bajnok, Z., Lájer, M., Szépfalvi, B. et al. Leading exponential finite size corrections for non-diagonal form factors. J. High Energ. Phys. 2019, 173 (2019). https://doi.org/10.1007/JHEP07(2019)173
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2019)173