JHEP11(2018)204
At the end of this section we note that the symmetry algebra of CFT considered above
can be extended by the introduction of additional field φ
m+1
, and the additional term to
the action (2.4)
A
0
CF T
= A
CF T
+
Z
1
8π
(∂
µ
φ
m+1
· ∂
µ
φ
m+1
)
d
2
x. (2.10)
In next section we will consider integrable perturbed QFT’s which can be viewed as per-
turbed CFT’s (2.4) and (2.10).
3 Integrable perturbed CFT
In this section we consider QFTs, which can be derived from CFTs (2.4), (2.10) by ne-
glecting the dilaton and adding the affine terms. Namely we denote
U
(m)
odd
(ϕ, φ) = U
(m)
(ϕ, φ) + µ exp(−bϕ
m
− iaφ
m
),
U
(m)
ev
(ϕ, φ) = U
(m)
(ϕ, φ) + µ
exp(−bϕ
m
+ iaφ
m+1
) + exp(−bϕ
m
− iaφ
m+1
)
.
(3.1)
The notations are related with the property that U
(m)
odd
(ϕ, φ) and U
(m)
ev
(ϕ, φ) correspond
to QFT’s dual to the deformed O(n + 2)−models with odd n = 2m −1 and even n = 2m.
As the non-local integrals of motion and S−matrices for these cases are slightly different
we consider them separately. We give examples of local Integrals of Motion in appendix B.
We note also, that the Lagrangians (3.1) inherit the non-unitarity of the corresponding
conformal field theories.
3.1 The case n = 2m − 1
In the case m = 1, the potential U
(1)
odd
(ϕ, φ) is special. It has the form
U
(1)
odd
(ϕ, φ) = 2µ exp(bϕ) cos(aφ) + 2µ exp(−bϕ) cos(aφ)
The QFT with the action
A =
Z
1
8π
(∂
µ
ϕ · ∂
µ
ϕ) + (∂
µ
φ · ∂
µ
φ)
+ U
(1)
odd
(ϕ, φ)
d
2
x. (3.2)
is known to be dual to the deformed O(3) sigma-model, or the sausage sigma-model. We
note that this action, contrary to the general case m > 1, is real and corresponds to unitary
QFT. The scattering theory, TBA analysis and the metric of the corresponding deformed
O(3)-sigma model can be found in [1, 6]. Besides the local Integrals of Motion this QFT
has the non-local ones generated by the non-local fields
J
±
= e
±i
ˆ
φ/a
(b∂ϕ + ia∂φ); I
±
= e
± ˆϕ/b
(b∂ϕ + ia∂φ)
where ˆϕ and
ˆ
φ are the chiral parts of fields ϕ and φ. To develop the perturbation theory
for small b with the action (3.2) it is convenient to use 2D Mandelstam-Coleman corre-
spondence [11, 12] between bosons and fermions
1
2
∂
µ
φ∂
µ
φ → i
¯
ψγ
µ
∂
µ
ψ; ∂
µ
φ → iε
µν
¯
ψγ
ν
ψ, e
±iφ
=
¯
ψ(1 ± γ
5
)ψ, (3.3)
– 5 –
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