On the third level descendent fields in the Bullough-Dodd model and its reductions

Exact vacuum expectation values of the third level descendent fields $<(\partial\phi)^3({\bar\partial}\phi)^3e^{a\phi}>$ in the Bullough-Dodd model are proposed. By performing quantum group restrictions, we obtain $$ in perturbed minimal conformal field theories.


Introduction
In 2-D integrable quantum field theories which can be considered as conformal field theories (CFTs) perturbed by a relevant operator, two-point correlation functions are complicated objects to study. However, using operator product expansion (OPE) in the short-distance limit one can reduce down their expression in terms of vacuum expectation values (VEVs) of local fields. Since four years, important progress has been made in this direction, as exact VEVs either of primary fields [1,2,3,4] or their first descendent [5,6,4] have been obtained explicitly. However it remains an open important problem to find all higher level VEVs of descendent fields and study their properties. Although a general method is still lacking, a case by case study based on CFT data provides a useful tool in order to determine some of the simplest higher level VEVs. Beyond the technical aspects, the knowledge of any of such quantities improves the analytical prediction for short-distance expansion of two-point functions, which can be better compared with the results obtained from the numerical study of the model (see [7] for instance).
Recently [6], we considered the Bullough-Dodd (BD) model and its quantum group restrictions, following the approach of [5] concerning the sinh-Gordon or sine-Gordon models. In Euclidean space, the action associated with the BD model writes Here, the parameters µ and µ ′ are introduced, as the two operators do not renormalize in the same way, on the contrary to any simply-laced affine Toda field theory. The purpose of this letter is to provide an exact expression for the VEV of the third level descendent fields (next to leading order in the UV limit of the two-point function) in the BD model, in order to complete the short-distance expansion of the two-point function calculated in [6]. It should be stressed that differently from sine-Gordon (SG) model, this VEV is nonzero, due to the existence of a local conserved current of spin 3 in the BD model. Finally, it is well-known that c < 1 minimal CFT with action perturbed by the operator Φ pert ∈ {Φ 12 , Φ 21 , Φ 15 } can be obtained by a quantum group (QG) restriction of imaginary Bullough-Dodd model [8,9,10,11] with special values of the coupling. Here we denote respectively Φ 12 , Φ 21 and Φ 15 as specific primary operators of the unperturbed minimal model M p/p ′ and introduce the parameter λ which characterizes the strength of the perturbation. Using this correspondence and the previous VEVs in the BD model, we will deduce 0 s |L −3 L −3 Φ lk |0 s in the perturbed minimal model (2).

VEVs of the third level descendent fields
The BD model can be regarded as a relevant perturbation of a Gaussian CFT in which case the field is normalized such that ϕ ( z, z)ϕ(0, 0) Gauss = −2 log(zz). For imaginary coupling b = iβ, the perturbation is relevant for 0 < β 2 < 1. Although the model (1) for real coupling is very different from the one with imaginary coupling in its physical content (this latter model contains solitons and breathers), there are good reasons to believe that the expectation values obtained in the real coupling case provide also the expectation values for the imaginary coupling. Then, let us now consider the two-point function in the BD model with imaginary coupling G α 1 α 2 (r) = e iα 1 ϕ (x)e iα 2 ϕ (y) BD with r = |x − y|. It can be expanded in the short-distance limit (r → 0) which, as mentioned above, contains a term corresponding to the third descendent contribution. The result reads (see [6] for details) where we defined H(α) and K(α) by the ratios and G α = e iαϕ BD is the VEV of the exponential field in the BD model. A closed analytic expression for G α and H(α) has been proposed in ref. [2] and ref. [6], respectively. Their expression involves an integral representation which is well defined if and obtained by analytic continuation outside this domain. Here we used the notations of [6] for the Dotsenko-Fateev integrals j n (a, b, ρ) and F n,m (a, b, ρ). In particular, the integrals j n (a, b, ρ) have been evaluated explicitly in [12] with the result where the notation γ(x) = Γ(x)/Γ(1 − x) is used. Also, the integral F 1,1 (a, b, ρ) can be obtained from the result of [13]. Instead, the integral F 1,2 (a, b, ρ) is a quite complicated object, and its explicit calculation goes beyond the purposes of this letter.
In the (Gaussian) free field theory, the composite fields (∂ϕ) 3 (∂ϕ) 3 e iαϕ are spinless with scale dimension For 0 < β 2 < 1 the perturbation is relevant and a finite number of lower scale dimension counterterms are sufficient to cancel the divergences arising in the VEVs of third level descendent fields. However, this procedure is regularization scheme dependent, i.e. one can always add finite counterterms. For generic values of α this ambiguity in the definition of the renormalized expression for these fields can be eliminated by fixing their scale dimensions to be (7). It exists however a set of values of α for which the ambiguity still remains. In the BD model with imaginary coupling, this situation arises if two fields, say O α and O α ′ , satisfy the resonance condition associated with the ambiguity In this case one says that the renormalized field O α has an (n|n ′ )-th resonance [5,6] with the field O α ′ . Due to the condition (5) we find immediately that a resonance can appear between the third level descendent field (∂ϕ) 3 (∂ϕ) 3 e iαϕ and the following primary fields: If we now look at the expression (3), we notice that the contribution brought by the third level descendent field in (4), and that of any of the exponential fields in (i), (ii), (iii) and (iv), have the same power behavior in r (r 4α 1 α 2 +6 ) at short-distance for the corresponding values of α. The integrals which appear in these contributions are, respectively: As we will see, K(α) (and similarly for the real coupling case) exhibits the same poles in order that the divergent contributions compensate each other. This last requirement leads for instance to a set of relations for K(α). The third one reads which is used to fix the α-independent part (normalization) of K(α).
On the other hand, to determine the explicit form of the α-dependent part of K(α), we use the reflection relations method. Indeed, the BD model (1) can be regarded as two different perturbations of the Liouville field theory [2]. First, one can consider the Liouville action where the perturbation is identified with e − b 2 ϕ . The holomorphic stress-energy tensor which ensures the local conformal invariance of the Liouville field theory with coupling b can be written in terms of the standard Virasoro generators T (z) = n∈Z L n z −n−2 and T (z) = n∈Z L n z −n−2 . Then, using the OPE of the stress-energy tensor of the Liouville part with any primary field, we have the relation Furthermore, taking the expectation value of the combination above and using the (Gaussian) equations of motion ∂∂ϕ = 0 we obtain Alternatively, we can consider e bϕ as a perturbation. Using both pictures and CPT framework, we deduce reflection relations between operators with the same quantum numbers. We report the reader to [2,3,5,6] for details about this approach. Consequently, if we denote then we obtain the following two functional relations Notice that these equations are invariant with respect to the symmetry b → − 2 b with a → −a in agreement with the well-known self-duality of the BD-model. Assuming that K(a) is a meromorphic function in a, we find that the "minimal" solution which follows from (11), (16) is: where h = 6 + 3b 2 is the "deformed" Coxeter number [14,15]. Here we have used the exact relation between the parameters µ and µ ′ in the action (1) and the mass of the fundamental particle m [2] : Notice that K(a) is invariant under the duality transformation b → −2/b as expected, and contains all the expected poles. Accepting this conjecture and taking a = 0, we obtain for instance: where f BD is the bulk free energy of the Bullough-Dodd model, obtained in [2].

Application to perturbed conformal field theories
For imaginary value of the coupling b = iβ, with the substitutions µ → −µ and µ ′ → −µ ′ in (1) the BD model possesses quantum group symmetry U q (A 2 ) with deformation parameter q = e iπ/β 2 [8,9]. At roots of unity, it is used to describe Φ 12 , Φ 21 or Φ 15 perturbed CFTs (2). Let us consider the first case i.e. the Φ pert ≡ Φ 12 perturbation, obtained for β 2 = p/p ′ with 1 < p < p ′ relative prime integers. In the following, Φ lk will denote a primary field of the minimal model M p/p ′ . The exact relation between the parameters λ in (2) and the mass of the fundamental kink M can be found in [2]. Here we denote For unitary minimal models ξ > 1 which, for Im(λ) = 0, corresponds to a massive phase [2]. Using the particle-breather identification [2] m = 2M sin ( πξ 3ξ+6 ) and parameter a = i( l−1 2β − k−1 2 β) in K(a) it is then straightforward to get the VEV: Here |0 s is one of the degenerate ground states of the QFT (2) (see [2] for a detailed discussion of the vacuum structure of the model).
For the second restriction β 2 = p ′ /p which leads to the action (2) with Φ pert ≡ Φ 21 , the exact relation between the parameter λ and the mass of the fundamental kink M has been obtained in [2]. The VEV of the third order descendent field immediately follows from (20) with the replacement ξ → −1 − ξ.
Another subalgebra of U q (A 2 ) is the subalgebra U q 4 (sl 2 ). One can again restrict the phase space of the complex BD with respect to this subalgebra for a special value of the coupling β 2 = 4p/p ′ with 2p < p ′ relative prime integers in order to describe the third case, i.e. Φ pert = Φ 15 . The calculations are straightforward so we will not report them here.

Conclusion
In conclusion, we have conjectured an exact expression for the VEV of the third level descendent in the BD model. The highly nontrivial check of the residue conditions corresponding to the poles (10), (11) strongly supports our conjecture.
As explained above, the computation of the (UV behavior) of the two-point function involves an infinite tower of VEVs of descendent fields. It is not clear how to solve this problem in general. Even in the simplest case as the SG theory, a system of functional equations appear for the 4-th level descendent. A solution of this problem is still an open question.
Yet, for practical reasons, the computation of any higher order descendent VEV gives new information. Our results for example can be used in improving the comparison with the numerical computations in some interesting statistical models around their critical point: the critical Ising model in a magnetic field [7], the tricritical Ising model perturbed by its energy operator [13] or by its subleading magnetic operator,...

Aknowledgements
We are grateful to Al. B. Zamolodchikov for discussions. MS's work is supported by KISTEP exchange program 12-69-002. PB's work is supported by JSPS fellowship.