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. References [1] J. S. Langer, Annals Phys. 41 (1967) 108 [Annals Phys. 281 (2000) 941]. [2] J. S. Langer, Annals Phys. 54 (1969) 258. [3] C. G. Callan and S. R. Coleman, Phys. Rev. D 16 (1977) 1762. [4] I. Affleck, Phys. Rev. Lett. 46 (1981) 388. [5] A. D. Linde, Nucl. Phys. B 216 (1983) 421 [Erratum-ibid. B 223 (1983) 544]. [6] A. D. Linde, Phys. Lett. B 100 (1981) 37. [7] M. Gleiser, G. C. Marques and R. O. Ramos, Phys. Rev. D 48 (1993) 1571. [8] C. A. A. de Carvalho, R. M. Cavalcanti, E. S. Fraga and S. E. Joras, Annals Phys. 273 (1999) 146. [9] A. Bessa, C. A. A. de Carvalho, E. S. Fraga and F. Gelis, JHEP 0708 (2007) 007. [10] C. A. A. de Carvalho, Phys. Rev. D 65 (2002) 065021 [Erratum-ibid. D 66 (2002) 049901]. [11] A. Bessa, C. A. A. de Carvalho and E. S. Fraga, to appear.