A quantitative prediction of Conformal Field Theory (CFT), which relates the
second moment of the energy-density correlator away from criticality to the
value of the central charge, is verified in the sine-Gordon model. By
exploiting the boson-fermion duality of two-dimensional field theories, this
result also allows to show the validity of the prediction in the strong
coupling regime of the Thirring model.Comment: 5 pages, no figures, late

Naón, Carlos

Salvay, Mariano

English

arXiv

A quantitative prediction of Conformal Field Theory (CFT), which relates the second moment of the energy-density correlator away from criticality to the value of the central charge, is verified in the sine-Gordon model. By exploiting the boson-fermion duality of two-dimensional field theories, this result also allows to show the validity of the prediction in the strong coupling regime of the Thirring model. Some time ago Cardy [1] derived a quantitative prediction of conformal invariance  where ε is the energy-density operator, ∆ ε is its scaling dimension and t ∝ (T −T c ) is the coupling constant of the interaction term that takes the system away from criticality. It is interesting to notice that a similar sum rule has been recently obtained by Jancovici  The main purpose of this note is to show that (1) also holds for a bosonic QFT with highly non-trivial interactions, the well-known sine-Gordon (SG) model with Euclidean Lagrangian density given by where α and λ are real constants. In this work we shall perform a perturbative computation up to second order in λ. We are then naturally led to consider the renormalization of this theory. Fortunately this issue has been already analyzed by many authors  Since we want to verify equation (1) for the model given by (2), we will take ε = (1 − cos √ λΦ)/λ and t = α. Taking into account that the model of free massless scalars has a conformal charge c = 1 and that the scaling dimension of ε for this case is equal to λ/4π, (1) read

Carlos Naon

Mariano Salvay

CiteSeerX

Proceedings of Institute of Mathematics of NAS of Ukraine 2002, Vol. 43, Part 2, 641–644
On a CFT Prediction in the Sine-Gordon Model
Carlos NAON and Mariano SALVAY
Instituto de F́ısica La Plata, Departamento de F́ısica, Facultad de Ciencias Exactas,
Universidad Nacional de La Plata, C.C. 67, La Plata 1900, Argentina
E-mail: naon@fisica.unlp.edu.ar, salvay@fisica.unlp.edu.ar
A quantitative prediction of Conformal Field Theory (CFT), which relates the second mo-
ment of the energy-density correlator away from criticality to the value of the central
charge, is verified in the sine-Gordon model. By exploiting the boson-fermion duality of
two-dimensional field theories, this result also allows to show the validity of the prediction
in the strong coupling regime of the Thirring model.
Some time ago Cardy [1] derived a quantitative prediction of conformal invariance [2, 3] for
2D systems in the scaling regime, away from the critical point. Starting from the so called
‘c-theorem’ [4], he was able to relate the value of the conformal anomaly c, which characterizes
the model at the critical point, to the second moment of the energy-density correlator in the
non-critical theory:∫
d2x |x|2 〈ε(x)ε(0)〉 = c
3 π t2 (2 − ∆ε)2 , (1)
where ε is the energy-density operator, ∆ε is its scaling dimension and t ∝ (T−Tc) is the coupling
constant of the interaction term that takes the system away from criticality. It is interesting
to notice that a similar sum rule has been recently obtained by Jancovici [5] in the context of
Classical Statistical Mechanics. This author considered the correlations of the number-density
of particles in a 2D two-component plasma (Coulomb gas).
The validity of (1) has been explicitly verified for the Ising model [1], for 2D self-avoiding
rings [6], and for the Baxter model [7]. In this last case the formula could be checked only in
the weak-coupling limit, by describing the system in terms of a massive Thirring model [8] and
performing a first-order perturbative computation.
The main purpose of this note is to show that (1) also holds for a bosonic QFT with highly
non-trivial interactions, the well-known sine-Gordon (SG) model with Euclidean Lagrangian
density given by
L =
1
2
(∂µΦ)2 − α
λ
cos
(√
λΦ
)
+
α
λ
, (2)
where α and λ are real constants.
In this work we shall perform a perturbative computation up to second order in λ. We are
then naturally led to consider the renormalization of this theory. Fortunately this issue has
been already analyzed by many authors [9–14]. One of the main conclusions is that for λ < 8π
a normal order procedure that eliminates the contributions of the tadpoles is enough to have
a finite theory. The only effect of this prescription is to renormalize the constant α. We shall
then restrict our study to this case.
Since we want to verify equation (1) for the model given by (2), we will take ε = (1 −
cos
√
λΦ)/λ and t = α. Taking into account that the model of free massless scalars has a con-
formal charge c = 1 and that the scaling dimension of ε for this case is equal to λ/4π, (1) reads
F (α, λ) =
∫
d2x |x|2 〈ε(x)ε(0)〉 = 1
3 π α2
(
2 − λ4π
)2 . (3)
642 C. Naon and M. Salvay
Expanding the interaction term up to order λ2 one has ε = 12Φ
2 − λ4!Φ4 + λ
2
6! Φ
6. Replacing
this expression in (3) we obtain
F (α, λ) = A(α, λ) + B(α, λ) + C(α, λ) + D(α, λ),
where
A(α, λ) =
1
4
∫
d2x |x|2 〈Φ2(x)Φ2(0)〉α, B(α, λ) = − λ4!
∫
d2x |x|2 〈Φ2(x)Φ4(0)〉α,
C(α, λ) =
λ2
(4!)2
∫
d2x |x|2 〈Φ4(0)Φ4(x)〉α, D(α, λ) = λ
2
6!
∫
d2x |x|2 〈Φ2(x)Φ6(0)〉α.
At this point we notice that D(α, λ), up to this order, contains only tadpoles which, as
explained above, were already considered in the renormalization of α. Then we must disregard
this contribution in the present context. Now, in order to illustrate the main features of the
computation, we shall briefly describe the evaluation of A(α, λ), which involves both analytical
and numerical procedures. In the above equations 〈 〉α means v.e.v. with respect to the SG
Lagrangian expanded up to second order in λ. From now on we will decompose this Lagrangian
into free and interaction pieces as
L0 =
1
2
(∂µΦ)2 +
α
2
Φ2, Lint = −αλ4! Φ
4 +
αλ2
6!
Φ6.
Using Wick’s theorem and the well-known expression for the free bosonic propagator
〈Φ(x)Φ(0)〉0 = 1/(2π)K0(
√
α|x|) (K0 is a modified Bessel function of zeroth order), after a con-
venient rescaling of the form x → x√
α
, we obtain
A(α, λ) =
1
2α2(2π)2
∫
d2x |x|2 K20 (|x|) +
λ
4α2(2π)4
∫∫
d2x d2x1 |x|2 K20 (|x1 − x|)K20 (|x1|)
+
λ2
α2(2π)6
∫∫∫
d2x d2x1 d
2x2 |x|2
[
1
8
K20 (|x2 − x1|)K20 (|x1|)K20 (|x2 − x|)
+
1
6
K30 (|x2 − x1|)K0(|x2|)K0(|x1 − x|)K0(|x|)
+
1
4
K20 (|x2 − x1|)K0(|x1|)K0(|x2 − x|)K0(|x2|)K0(|x1 − x|)
]
.
The first two terms are related to tabulated integrals [15] yielding the first order analytical
expression F (α, λ) = (1+λ/(4π))/
(
12πα2
)
which can be shown to verify (3) in a straightforward
way. Concerning the remaining second order terms we have three contributions corresponding
to the prefactors 1/8, 1/6 and 1/4 in the last integrand. Due to its symmetry, the first one can
be analytically computed by repeatedly using∫
d2x |x|2 K20 (|x − y|) =
2π
3
+ π|y|2.
The result is 2π3. The computation of the other two contributions is more involved. In fact we
could not find analytical results in these cases. However we were able to considerably simplify
these multiple integrals in order to facilitate their numerical evaluation. Indeed, using the Fourier
transform of K0 and the integral representation of the first kind Bessel function J0:
J0(|x|) = 12π
∫ 2π
0
dθ exp (−i|x| cos θ) ,
On a CFT Prediction in the Sine-Gordon Model 643
we obtain∫∫∫
d2x d2x1 d
2x2 |x|2 K30 (|x2 − x1|)K0(|x1|)K0(|x2 − x|)K0(|x|)
= 4(2π)3
∫
dr
∫
dp
r p
(
1 − p2) K30 (r)J0(pr)
(p2 + 1)5
.
Using NIntegrate in the program Mathematica for the double integral in this expression one
obtains the value 0.04874. Similar manipulations with the last contribution to A(α, λ) led us to
the following numerical results:
∫
dr
∫
dp
∫
dk
r p k
(
1 − p2) K30 (r)J0(pr)J0(kr)
(p2 + 1)4 (k2 + 1)2
= 0.0169622,
and ∫
dr r5 K20 (r)K
2
1 (r) = 0.08783.
Putting all this together we have
A(α, λ) =
1
12πα2
(
1 +
λ
4π
)
+
λ2
α2(2π)6
[
1
8
2π3 +
1
6
4 (2π)3 0.04874
+
1
4
(2π)3
(
8 × 0.0169622 − 1
8
0.08783
)]
.
Working along the same lines with B(α, λ) and C(α, λ) we find
B(α, λ) =
−32λ2
α2(2π)34!
∫
dr
∫
dp
r p
(
1 − p2)K30 (r)J0(p r)
(p2 + 1)4
=
−32λ2
α2(2π)34!
× 0.0501125
and
C(α, λ) =
λ2
α2(2π)34!
∫
dr r3 K40 (r) =
λ2
α2(2π)34!
× 0.0754499.
All these numerical values were confirmed by using Fortran.
Finally, inserting these results in the left hand side of (3) we get
F (α, λ) =
1
12πα2
+
λ
48π2α2
+
λ2
4!(2π)3α2
(
3
4
+ 5.4 × 10−6
)
.
Comparing this expression with the expansion in λ of the right hand side of (3) one sees that
they are equal up to first order and differ in a small quantity
(
O
(
10−6
))
up to second order
in λ.
In summary, we have verified the validity of a quantitative prediction of CFT in the context
of the SG model. Taking into account the well-known bosonization identity between the SG
theory and the massive Thirring model (characterized by a coupling g2) [9] which takes place
for β2/(4π) =
(
1 + g2/π
)−1, it becomes apparent that our result implies that (1) also holds in
the strong coupling limit of the Thirring model. This, in turn, allows us to improve the proof
of the validity of (1) for the Baxter and Ashkin–Teller models which, up to now, was restricted
to the weak coupling limit [7].
644 C. Naon and M. Salvay
Acknowledgements
This work was partially supported by Universidad Nacional de La Plata (Argentina) and CONI-
CET (Argentina). MS thanks CICPBA (Argentina) for a Fellowship for Students. CN is in-
debted to Bernard Jancovici for useful correspondence and stimulating e-mail exchanges. The
authors are also grateful to Marta Reboiro for helping them with Fortran, and Vicky Fernández
and Ańıbal Iucci for helpful comments.
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On a CFT Prediction in the Sine-Gordon Model

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On a CFT prediction in the sine-Gordon model

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