Nuclear Physics A 820 (2009) 231c–234c
www.elsevier.com/locate/nuclphysa
Semiclassical calculation of decay rates
A. Bessa a, C.A.A. de Carvalho b, and E. S. Fraga b
aInstituto de F́ısica, Universidade de São Paulo
C.P. 66318, São Paulo, SP, 05315-970, Brazil
bInstituto de F́ısica, Universidade Federal do Rio de Janeiro
C.P. 68528, Rio de Janeiro, RJ 21941-972, Brazil
Abstract
Several relevant aspects of quantum-field processes can be well described by semiclassical meth-
ods. In particular, the knowledge of non-trivial classical solutions of the field equations, and the
thermal and quantum fluctuations around them, provide non-perturbative information about
the theory. In this work, we discuss the calculation of the one-loop effective action from the
semiclasssical viewpoint. We intend to use this formalism to obtain an accurate expression for
the decay rate of non-static metastable states.
Key words: decay rates, finite-temperature field theory
PACS: 03.65.Sq, 64.60.My, 03.70.+k, 11.10.-z
1. Introduction
A theory for the description of metastable states was formulated by J. Langer long
ago[1,2]. The formalism and its quantum extensions[3,4] revealed the connection between
the decay rate and the free energy of a saddle-point configuration φs of the Euclidean
action with a single negative eigenvalue. In general, φs interpolates the local minimum
(the false, metastable vaccum) and the global minimum (the true vacuum) of the action.
The decay rate is given by
Γ = Ω exp −SE(φs) , (1)
where the pre-factor Ω is formally written in terms of the determinant of (quantum and
thermal) fluctuations around φs. In the end, we are led to the problem of calculating
the one-loop effective action around φs. In this paper, we present a finite-temperature
semiclassical procedure to obtain the pre-factor of the decay rate as an alternative to
the traditional approach which uses Matsubara sums[5–7]. Our approach appears more
appropriate to be generalized to the case of non-static saddle-point configurations.
0375-9474/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.nuclphysa.2009.01.057
232c A. Bessa et al. / Nuclear Physics A 820 (2009) 231c–234c
g E0(semiclassical) E0(exact) Error(%)
0.4 0.559258 0.559146 0.02
1.2 0.639765 0.637992 0.28
2.0 0.701429 0.696176 0.75
4.0 0.823078 0.803771 2.40
8.0 1.011928 0.951568 6.34
Table 1
Ground state energy of the quartic oscillator in quantum mechanics for different values of the coupling
g (~ = m = ω = 1) [from Ref. [8]].
2. The semiclassical method at finite T
Semiclassical methods have been successfully applied to quantum statistical mechanics.
In this approach, the path-integral expression for the partition function is calculated us-
ing the steepest descent method. Saddle-points of the action are solutions of the euclidean
equations of motion, and configurations in the vicinity of these classical solutions dom-
inate the path integral. The contribution of such configurations can be systematically
incorporated, defining a semiclassical series. In the particular case of one-dimensional
quantum-mechanical systems, it is possible to generate all the terms of the series using
the semiclassical propagator which, in turn, is determined by the classical solution[8].
Surprisingly, the first term of the semiclassical series can already produce accurate re-
sults. As an example, let us consider the single-well quartic potential:
1 1
V (x) = mω2x2 + λx4. (2)
2 4
Table 1 exhibts the ground-state energy for different values of the coupling g = λ 2 3~/m ω
[8]. We see that the semiclassical quadratic approximation is in good agreement with
numerical techniques that used optimized perturbation theory, even for large values of the
coupling. This serves as a motivation for the application of the semiclassical aproximation
to finite-temperature quantum field theory.
The path-integral formula for the partition function admits a direct extension to quan-
tum field theories. Indeed, the partition function of a given (scalar) system can be cast
in the form: ∫ ∫
Z = [Dϕ(x)] [Dφ(τ,x)] e−S (φ)E , (3)
φ(0,x)=φ(β,x)=ϕ(x)
where S (φ) is the Euclidean action of the field:
E
∫ β [ ]1 1
S (φ) = dτd3x ∂ φ∂µφ+ m2φ2µ + U(φ) . (4)E
0 2 2
We assume that φs is a saddle-point of the action. It obeys the equation of motion
−φs(τ,x) + U ′ (φs(τ,x)) = 0
φs(0,x) = φs(β,x) = ϕs(x) , (5)
where we denote by ≡ (∂2 +∇2) the Euclidean d’Alembertian operator. In principle,
E τ
φs is a general non-static solution of (5). Now, we proceed to calculate the contribution to
A. Bessa et al. / Nuclear Physics A 820 (2009) 231c–234c 233c
Fig. 1. Snapshot at τ = 0 of profiles contributing to the partition function. The quadratic expansion of
the action around the kink (solid line) along the direction of a non-static fluctuation (dashed line) will
have non-zero boundary corrections.
the partition function coming from quadratic fluctuations around φs. We write φ(τ,x) =
φs(τ,x) + η(τ,x). The only condition on the fluctuation η is that η(0,x) = η(β,x). In
practice, one can restrict the calculation to those configurations with finite action. As an
example, one can think of φs as being a kink-like static profile. The finite action condition
imposes that η goes to zero at spatial infinity. Figure 1 ilustrates a typical configuration
with finite action in the vicinity of the kink at τ = 0.
A careful expansion of the euclidean action around φs up to quadratic order produces:
S (φ) = S (φs) + δ
(1)S + δ(2)S +O(η3) , (6)
E E E E
where
∫ [ ]τ=β
δ(1)S = d3x φ̇s(x)η(x) (7)E
τ=0
and ∫ ∫
1 [ ]
δ(2)S = d3
τ=β 1
x [η(τ,x)∂τη(τ,x)]
4
τ=0 + (d x) η(x) − + V
′′
 (φs(x)) η(x) .E 2 2 E E
(8)
The boundary terms do not vanish because the boundary value of the fluctuation η at
τ = 0, β is not zero. In other words, there are configurations close to φs whose boundary
value is not the same as ϕs(x) = φs(0,x). In order to integrate (3) over boundary values
in the neighborhood of ϕs(x), we can use the techniques of Ref. [9], which incorporate
fluctuations of boundary conditions. We write the boundary field as ϕ(x) = ϕs(x)+ξ(x),
and expand the action up to quadratic order in ξ. To be consistent with the quadratic ap-
proximation, we introduce a number of important simplifications which make the problem
tractable. It is possible to show that Z around φs is given by the following formula:
Z ≈ e−S (φs) (detG)−1/2E , (9)
where
[− ′′ + V (φs)]G(x;x′) = δ(4)(x − x′) (10a)E
G(τ,x; 0,x′) = G(τ,x;β,x′) = 0 . (10b)
234c A. Bessa et al. / Nuclear Physics A 820 (2009) 231c–234c
From (9), we obtain the pre-factor defined in (1). In special cases, the Green function
(10) can be analytically calculated. For instance, we consider a scalar theory with a
quartic potential, and a static kink solution which interpolates between the two equivalent
minima: [ ]
λ
ψ′′ +m2ψ − ψ3 = 0 → m mxψ(x) = √ tanh √ . (11)
4 2 λ 2
It is possible to show that the Green function we need has the form:
2 ∑∞
G(τ,x; τ ′,x′) = G̃(ω ,x,x′) sin(ω τ) sin(ω τ ′n n n ) , (12)
β
n=1
where ωn = πn/β. Following [10], we obtain:
1
G̃ = [ρ+(u)ρ−(u
′)Θ(u′ − u) + ρ+(u′)ρ−(u)Θ(u− u′)] , (13)
b
√ n √
with ξ = mx/ 2, b = 4 + ω2n n, u = (1− tanh ξ)/2, and
( )±bn/2u
ρ±(u) = − p±(u) , (14)1 u
where p± are quadratic polynomials. Therefore, we have all the ingredients to calculate
the determinant of G. Numerical results will be presented in a future publication[11].
3. Conclusions
We presented a systematic procedure to calculate decay rates of metastable states in
finite temperature quantum field theory using semiclassical methods. Decay rates are
directly related to the one-loop effective action around a saddle point of the Euclidean
action. We illustrated the method in the simple case of a static kink profile. We claim
that our approach is particularly useful to deal with non-static saddle-points.
Acknowledgment
The authors would like to thank the support of CNPq, FAPERJ, FUJB and FAPESP
for financial support.
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