Two-loop corrections to false vacuum decay in scalar field theory

We consider radiative corrections to false vacuum decay in a four-dimensional scalar field theory with cubic and quartic potential. Using planar thin wall approximation we were able to get analytical expression for the decay rate up to two loop order. The results obtained employ dimensional regularization and $\overline{MS}$ renormalization scheme.


Introduction
Recently, first-order phase transitions driven by scalar fields attracted a lot of attention in high-energy, astro-particle physics and cosmology communities. In particular, a lot of recent research was connected with the possible metastability 1 of Standard Model vacuum at a scale around 10 11 GeV [15][16][17][18][19][20][21][22].
The purpose of this paper is to extend the methods of [50][51][52][53][54][55] and [38][39][40][41][42][43] for the computation of higher order radiative corrections to false vacuum decay in four-dimensional scalar field theory. Here we will concentrate on two-loop corrections within dimensional regularization and M S scheme. The generalization to Coleman-Weinberg (CW) scheme and cut-off regularization will be presented in our subsequent paper. The paper is organized as follows. The section 2 and its subsections contain all the material of the present paper. First, following [40] we describe our model with single scalar field experiencing cubic and quartic self-interactions and false vacuum decay in it at one-loop order. Next, in subsection 2.1 we present required Green functions in the bounce background and false vacuum in a planar thin wall approximation. Most of these results are already known from [40], expect the expression of bounce Green function in a special kinematical limit which is required in a subsequent Feynman diagrams calculation. Then, in subsections 2.2 and 2.3 we present the details of one and two loop calculations in dimensional regularization. Finally in section 3 we come with our conclusion. The A contains the details of calculation of the most complicated two-loop sunset diagram.

False vacuum decay in scalar field theory
Let us consider false vacuum decay in a four-dimensional field theory with a single real scalar field Φ ≡ Φ(x) and a lagrangian 3 1 For tunneling rates calculations in Standard Model and its extensions see [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and references therein. 2 See [30] and references therein. 3 We refer the reader to [40] for more details. where The potential U has two minima ϕ When g → 0 the minima at v ± = ±v become degenerate as ∼ gv 2 6 → 0. It is precisely this limit which allows analytical treatment and which was considered in [40]. In what follows it is convenient to chose U 0 = µ 2 v 2 4 − gv 3 6 , so that the potential vanishes in the false vacuum ϕ = +v.
The false vacuum decay in this theory accounting for semi-classical tunneling between false (ϕ = v) and true (ϕ = −v) vacua together with first quantum corrections was first considered in [23,24]. In a path integral formulation the false vacuum decay probability is given by the ratio of the path integrals evaluated around bounce and false vacuum solutions. In the present model the bounce corresponds to a four dimensional bubble of some radius R separating true vacuum inside bubble from a false vacuum outside. The latter is given by a O(4) symmetric solution of the classical equations of motion where denotes the derivative with respect to the field ϕ. Rewriting the above equation in hyperspherical coordinates and using thin-wall approximation, that is neglecting both cubic self-interaction gφ 3 and damping term in (4), it is easy to see that the bounce is given by the well-known kink solution [61]: The radius of the bubble is then obtained by extremizing the bounce action [40]: The classical action itself evaluated at bounce solution is given by Next, introducing partition function the false vacuum decay rate could be written as where T is the Euclidean time of the bounce and partition function being evaluated around bounce solution. Note, that here the partition function should be evaluated in one bounce approximation, see [3,24] for more details. The latter goes through saddle point approximation and at one-loop order is given by [40]: where λ 0 = − 3 R 2 is the negative eigenvalue of G −1 (ϕ) operator and det (5) denotes the determinant calculated only over the continuum of positive-definite eigenvalues, that is omitting zero and negative eigenmodes. Here and below, the parameter is introduced simply in order to count loops, the actual perturbation theory is constructed as series in λ.
The inverse of Green functions G −1 at the bounce ϕ and in false vacuum v are defined as ( (4) is the four-dimensional Laplacian): The spectrum of operator G −1 (ϕ) at bounce solution is given by [40]: and contains one negative mode at λ 0 = λ 20 and four zero modes at λ 21 . The "continuum" of positive-definite modes starts at λ 10 ≈ λ 11 = 2γ 2 . Knowing the expression for the partition function evaluated at bounce solution (10) the expression for false vacuum decay (tunneling probability per unit volume) at one-loop order takes the form [40]: where and tr (5) denotes the trace only over positive-definite eigenmodes.

Green function in bounce background
As was mentioned in introduction here we will restrict ourselves with the so called planarwall approximation, which is good one in the case of large bubble radius R, see Fig. 1.
Here, the coordinates z are parallel to the bubble surface, while z ⊥ is the one orthogonal to it. To find a Green function in a bounce background we need to solve the following inhomogeneous Klein-Gordon equation To do that, it is convenient to perform Fourier transform with respect to coordinates parallel to the bubble Then the Green function G(ϕ; z ⊥ , z ⊥ , k) satisfies the equation (z, z = z ⊥ , z ⊥ ): Making change of variables and the above equation takes the form where m = 1 γ 4γ 2 + k 2 . Here and below to shorten notation we will write G(x, y, k) instead of G(ϕ; x, y, k). The equation (20) is solved differently depending on whether m > 2 or m = 2. When m > 2 the solution of homogeneous equation is given by where P m 2 (x) and Q m 2 (x) are the associated Legendre functions of the first and second kind, respectively. Applying the boundary conditions we get Finally, employing the representation of associated Legendre function of the first kind in terms of Jacobi polynomials and using the fact that for n = 2 the polynomial expansion of the latter terminates we get [40]: In the case m = 2 the differential operator acting on Green function in (20) has zero mode and its inversion is consistently defined only on the subspace of functions orthogonal to this zero mode (Fredholm alternative). The corresponding equation for m = 2 was already considered in [51,62] and is obtained by adding the product of properly normalized zero modes 4 (ϕ 0 = 3γ 4 1 cosh 2 (γz) ) to the right-hand side of the equation (18), so that and equation (20) becomes The solution of this equation at x = y is given by or in other form In this case G(x, y, k) must satisfy the following boundary conditions 1) G(x, y, k) → 0 when x, y → ±1 2) continuity at x = y 3) orthogonality of G(x, y, 0) to the zero mode which allow us to fix functions C 1,2 (y) and the result reads Similarly in the case of false vacuum we have 5 Transforming this expression back to z, z variables we get Now, if we take the Fourier transforms in variables z, z we recover ordinary momentum space propagator we used to Here, momentum k is four-dimensional already. This later property is of great importance to us as it allows us to use for computations in false vacuum ordinary four-dimensional propagators.

One-loop expression
To get one-loop expression for false vacuum decay rate we need to evaluate difference of two traces (15). In this work to regulate ultraviolet divergences we will use the dimensional regularization. Using the latter together with heat kernel method [34] we get 6 Here k s denotes the substitution k 2 → k 2 + s, d = 4 − 2ε and L −1 s is the inverse Laplace transform with respect to s. The inverse Laplace transform and all integrals except over τ variable could be easily taken and the expression for I (1) takes the form To perform this final integration it is convenient to use the series representation of the error function and after a term-by-term integration of the resulting series we get The only functions with ε singularities are Γ(ε − 1) and Γ(ε), so that the pole is generated only by first two terms in the series above. The remaining part of this series can be easily summed by putting ε = 0, which is allowed because this part is not multiplied by the functions containing singularities.
Finally, gathering everything together for the unrenormalized one-loop contribution we get To get finite expression we need to account for ultraviolet renormalization, which is usual procedure in renormalizable quantum field theories. The needed counterterms could be conveniently introduced with the following additional interaction terms in the lagrangian The mass and coupling counterterms could be determined by considering renormalization of effective Coleman-Weinberg (CW) potential, while wave function renormalization is determined through renormalization of two-point function. In M S scheme at one-loop order we get (γ E is the Euler constant) which is in agreement with previously known results [63][64][65]. Then adding action counterterm the final expression for I (1) takes the form where µ M S is M S renormalization scale. If we would use the Coleman-Weinberg renormalization scheme prescription, than at one-loop we would recover the result of [40]. Note, that in [40] the authors used regularization by cut-off. However, the final result in this particular scheme is independent of regularization prescription.

Two-loop corrections
The evaluation of higher order corrections for false vacuum decay is similar to quantum mechanical case considered in [55], see also [50][51][52][53][54] for calculation of other quantities. There are however technical complications related to the dimension of spacetime our quantum field theory lives in and the need for renormalization. The required Feynman rules needed for the computation of partition function around false vacuum and bounce solution are given by The vertex with a plus sign inside a circle is a tadpole vertex coming from integration measure 7 , while all other vertexes come from lagrangian. At two-loop order we need to calculate diagrams presented in Fig. 2   All diagrams except sunset diagram a11 could be calculated more or less straightforwardly. For example for diagram a12 we have Going from z ⊥ , z ⊥ to x = tanh(γz ⊥ ), y = tanh(γz ⊥ ) variables and taking all integrations except over x and y the expression for I a12 takes the form Finally, using the integral expressions and This example shows the use of the expression for Green function G(x, y, 0) in the special case with m = 2, see section 2.1 8 . The values of other diagrams could be found in a mathematica file accompanying the article and A contains details of sunset diagram evaluation.
The two-loop counterterms are again determined from the renormalization of the effective potential and two-point function. In M S scheme at two-loop order we get which are again in agreement with previously obtained results [63][64][65]. Summing contribution of all diagrams 9 and adding two-loop action counterterm we finally get 10 or where Here s 0 is the finite part of the sunset diagram a11, see A. We would like to note that part of two-loop corrections related to bounce renormalization considered in [40] is contained in the sum of our diagrams a12, b12 and b13. These diagrams are obtained by inserting tadpole contribution for scalar field self-energy found in [40] into corresponding one-loop diagram for false vacuum decay rate. Numerically, this contribution is of the order of 10% of full two-loop result. Vacuum decay rate is a measurable quantity and as such it should not depend from the scale choice. This means that our result need to satisfy the following equation: We have verified that it is indeed the case up to order by expressing S b and I (1) in terms of the Lagrangian parameters γ, g, λ and applying to them the renormalization group equations [4]: Note, that since we are using thing wall approximation (R → ∞, g → 0) we should leave only leading order terms in coupling g in renormalization group equations. Moreover, it turns out that within thin-wall approximation the knowledge of one-loop beta-functions is sufficient to check the coefficients in front of logarithms with µ M S dependence at two-loop order also. So, we see that there is no renormalization-scale uncertainty in this method 11 . Finally, in order to understand the qualitative significance of our result, let us consider the ratio of I (2) to λI (1) as a function of µ M S /γ. From Fig. 3 we see that at reasonable values of µ M S 2γ, where neither I (1) or I (2) are close to zero, this ratio is about 11 As usual there could be some residual scale dependence left at one perturbation order higher.
0.02 − 0.03. This means that the two loop correction have very little impact on the vacuum decay rate and can be safely neglected whenever we do not interested in accuracy greater than 2 − 3%.

Conclusion
In this work we computed for the first time two-loop quantum corrections to false vacuum decay in a four-dimensional scalar field theory with cubic and quartic potential. Using planar thin wall approximation we were able to get analytical expression for the latter.
The results obtained employ dimensional regularization and M S renormalization scheme. It is shown that the obtained decay rate is independent from renormalization scale variation. It turns out that two-loop corrections is approximately 2 − 3% of one-loop result. So, we may conclude that one-loop approximation accounting for prefactor to exponent of the bounce action is a well defined approximation to false vacuum decay in the model considered. In a subsequent paper we are planning to present the generalization of the obtained results to Coleman-Weinberg (CW) scheme and cut-off regularization.
and taking integrals over z we will for example obtain Finally taking integrals over x and y we get Here . . . denote higher order terms in the expansion. To further evaluate integrals over k 1 and k 2 we derived Mellin-Barnes representation for the integral where d = 3 − 2ε, m i = (k 2 i + 4) 1/2 , M = m 1 + m 2 + m 3 as before and a i are arbitrary indexes. The latter is given by The resulting Mellin-Barnes integrals where evaluated numerically with the help of [70]. Finally for the sunset diagram we got where s −1 = 0.39522 and s 0 ≈ 0.71. The value of s −1 = 0.39522 which we got from the first 20 terms of the series 12 (67) is actually several percent less then the exact value which could be found from the cancellation of 1/ε poles in the process of renormalization ≈ 0.410995 (71) 12 We have written explicitly only two of them in (67) which is explained by slow convergences of the mentioned series. It could be certainly further improved but this goes beyond the goal of the present paper.