# Form factors of descendant operators in the Bullough-Dodd model

## Abstract

We propose a free field representation for the form factors of descendant operators in the Bullough-Dodd model. This construction is a particular modification of Lukyanov’s technique for solving the form factors axioms. We prove that the number of proposed solutions in each level subspace of the chiral sectors coincide with the number of the corresponding descendant operators in the Lagrangian formalism. We check that these form factors possess the cluster factorization property. Besides, we propose an alternative free field representation which allows us to study analytic properties of the form factors effectively. In particular, we prove that the form factors satisfy non trivial identities known as the “reflection relations”. We show the existence of the reflection invariant basis in the level subspaces for a generic values of the parameters.

## Keywords

Field Theories in Lower Dimensions Integrable Field Theories## References

- [1]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].MathSciNetADSCrossRefGoogle Scholar - [2]F. Smirnov,
*The quantum Gelfand-Levitan-Marchenko equations and form-factors in the sine-Gordon model*,*J. Phys.***A 17**(1984) L873 [INSPIRE].ADSGoogle Scholar - [3]F.A. Smirnov,
*Form factors in completely integrable models of quantum field theory*, World Scientific, Singapore (1992).zbMATHCrossRefGoogle Scholar - [4]G. Delfino and G. Mussardo,
*The Spin spin correlation function in the two-dimensional Ising model in a magnetic field at T*=*T*_{c},*Nucl. Phys.***B 455**(1995) 724 [hep-th/9507010] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar - [5]G. Delfino, P. Simonetti and J.L. Cardy,
*Asymptotic factorization of form-factors in two-dimensional quantum field theory*,*Phys. Lett.***B 387**(1996) 327 [hep-th/9607046] [INSPIRE].MathSciNetADSGoogle Scholar - [6]G. Delfino and G. Niccoli,
*Matrix elements of the operator*\( T\overline{T} \)*in integrable quantum field theory*,*Nucl. Phys.***B 707**(2005) 381 [hep-th/0407142] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar - [7]G. Delfino and G. Niccoli,
*The Composite operator*\( T\overline{T} \)*in sinh-Gordon and a series of massive minimal models*,*JHEP***05**(2006) 035 [hep-th/0602223] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar - [8]R.K. Dodd and R.K. Bullough,
*Polynomial conserved densities for the sine-Gordon equations*,*Proc. Roy. Soc. Lond.***A 352**(1977) 481.MathSciNetADSGoogle Scholar - [9]A.V. Zhiber and A.B. Shabat,
*Klein-Gordon equations with a nontrivial group*,*Sov. Phys. Dokl.***24**(1979) 607.ADSGoogle Scholar - [10]C. Acerbi,
*Form-factors of exponential operators and exact wave function renormalization constant in the Bullough-Dodd model*,*Nucl. Phys.***B 497**(1997) 589 [hep-th/9701062] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar - [11]S.L. Lukyanov,
*Free field representation for massive integrable models*,*Commun. Math. Phys.***167**(1995) 183 [hep-th/9307196] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar - [12]V. Brazhnikov and S.L. Lukyanov,
*Angular quantization and form-factors in massive integrable models*,*Nucl. Phys.***B 512**(1998) 616 [hep-th/9707091] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar - [13]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].MathSciNetGoogle Scholar - [14]V. Fateev, S.L. Lukyanov, A.B. Zamolodchikov and A.B. Zamolodchikov,
*Expectation values of local fields in Bullough-Dodd model and integrable perturbed conformal field theories*,*Nucl. Phys.***B 516**(1998) 652 [hep-th/9709034] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar - [15]P. Baseilhac and M. Stanishkov,
*On the third level descendent fields in the Bullough-Dodd model and its reductions*,*Phys. Lett.***B 554**(2003) 217 [hep-th/0212342] [INSPIRE].MathSciNetADSGoogle Scholar - [16]P. Baseilhac and M. Stanishkov,
*Expectation values of descendent fields in the Bullough-Dodd model and related perturbed conformal field theories*,*Nucl. Phys.***B 612**(2001) 373 [hep-th/0104220] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar - [17]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].MathSciNetADSCrossRefGoogle Scholar - [18]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].MathSciNetADSCrossRefGoogle Scholar - [19]A. Arinshtein, V. Fateev and A. Zamolodchikov,
*Quantum s matrix of the*(1 + 1)*-dimensional Toda chain*,*Phys. Lett.***B 87**(1979) 389 [INSPIRE].ADSGoogle Scholar - [20]A. Fring, G. Mussardo and P. Simonetti,
*Form-factors of the elementary field in the Bullough-Dodd model*,*Phys. Lett.***B 307**(1993) 83 [hep-th/9303108] [INSPIRE].MathSciNetADSGoogle Scholar - [21]G. Delfino and G. Niccoli,
*Form-factors of descendant operators in the massive Lee-Yang model*,*J. Stat. Mech.***0504**(2005) P04004 [hep-th/0501173] [INSPIRE].MathSciNetCrossRefGoogle Scholar - [22]G. Delfino and G. Niccoli,
*Isomorphism of critical and off-critical operator spaces in two-dimensional quantum field theory*,*Nucl. Phys.***B 799**(2008) 364 [arXiv:0712.2165] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar - [23]O. Alekseev,
*Form factors in the Bullough-Dodd related models: the Ising model in a magnetic field*,*JETP Lett.***95**(2012) 201 [arXiv:1106.4758] [INSPIRE].ADSCrossRefGoogle Scholar - [24]V. Fateev, V. Postnikov and Y. Pugai,
*On scaling fields in Z*(*N*)*Ising models*,*JETP Lett.***83**(2006) 172 [hep-th/0601073] [INSPIRE].ADSCrossRefGoogle Scholar - [25]V. Fateev and Y. Pugai,
*Correlation functions of disorder fields and parafermionic currents in Z*(*N*)*Ising models*, arXiv:0909.3347 [INSPIRE]. - [26]M. Lashkevich,
*Resonances in sinh- and sine-Gordon models and higher equations of motion in Liouville theory*,*J. Phys.***A 45**(2012) 455403 [arXiv:1111.2547] [INSPIRE].MathSciNetADSGoogle Scholar