False vacuum decay rates, more precisely

We develop a method for accurately calculating vacuum decay rates beyond the thin-wall regime in a pure scalar field theory at the one-loop level of the effective action. It accounts for radiative effects resulting from quantum corrections to the classical bounce, including gradient effects stemming from the inhomogeneity of the bounce background. To achieve this, it is necessary to compute not only the functional determinant of the fluctuation operator in the background of the classical bounce but also its functional derivative evaluated at the classical bounce. The former is efficiently calculated using the Gel'fand-Yaglom method. We illustrate how the latter can also be calculated with the same method, combined with a computation of various Green's functions.


Introduction
One of the most striking features of the Standard Model (SM) is that the electroweak vacuum is metastable [1][2][3][4][5][6][7][8][9][10][11].The measured 125 GeV Higgs boson [12,13], together with the 173 GeV top quark [14], suggests that the Higgs potential in the SM develops a lower minimum due to the renormalisation running of the Higgs self-interaction coupling.The electroweak vacuum could then decay to the true vacuum via tunnelling, which is a first-order phase transition in quantum field theory.State-of-the-art calculations suggest that the electroweak vacuum lies at the edge of the stable region, having a lifetime longer than the present age of the Universe [11,[15][16][17][18][19].
In contrast to the running couplings, the radiative corrections to tunneling are computed less accurately.The transition rate is sensitive to a solitonic field configuration in Euclidean space, called the "bounce" [20,21].This bounce configuration also encodes the information of the nucleated critical bubble.Typically, the bounce is calculated from the tree-level scalar potential with the running coupling constants evaluated at the scale corresponding to the bounce radius.However, the bounce is an inhomogeneous configuration, and the beta functions for the couplings do not account for the effects of the background inhomogeneity.The inhomogeneity of the bounce requires us to go beyond the effective potential.Such effects from background inhomogeneity, as well as the corresponding higher-loop radiative corrections, have not yet been included in the full SM computations but are only studied in simpler scalar [22][23][24], Yukawa [25], and gauge [26] theories in the planar thin-wall approximation.These studies are based on a Green's function method developed earlier in Ref. [27].See also Refs.[28][29][30] for the case of two-dimensional spacetime.
We extend previous work by going beyond the thin-wall regime.In order to calculate the quantum-corrected bounce, we need both the functional determinants of the fluctuation operators and their functional derivatives evaluated at the classical bounce (see Eq. (20) or Eq. ( 30) below).The functional determinants in the O(4)-symmetric backgrounds can be efficiently computed by the Gel'fand-Yaglom method [31,32].We show that the functional derivative of a functional determinant can be computed with the same method, combined with a computation of various Green's functions.This method is closely related to the Green's function method used in Refs.[22][23][24][25][26][27].Our method is applied numerically to compute the corresponding radiative effects in an archetypal scalar model.
The outline of the paper is as follows.In the next section, we briefly review the standard Callan-Coleman formalism for false vacuum decay in terms of the classical bounce.In Sec. 3, we reformulate the decay rate using the one-particle-irreducible (1PI) effective action [33] in a way that radiative corrections to the decay rate can be incorporated.In Sec. 4, we introduce the powerful Gel'fand-Yaglom method for computing functional determinants.In Sec. 5, we show how the Gel'fand-Yaglom method, in combination with a Green's function method, is used to compute the quantum corrections to the classical bounce and the decay rate.In Sec. 6 we give the results of numerical calculations using our method.In Sec. 7, we give our conclusions.For completeness, some technical details are presented in the appendices.In Appendix A, we give a general introduction to how to solve the Green's functions.In Appendix B, we discuss how to renormalise the effective one-loop action using the dimensional renormalisation scheme.In the paper, we use ℏ = c = 1.

Callan-Coleman formalism and the tree-level bounce
We consider the archetypal model, where η µν is the Minkowski metric with signature {+, −, −, −} and The couplings m2 , g, λ all take positive values.When g 2 > 8λm 2 /3, the potential has two local minima.For 8λm 2 /3 < g 2 < 3λm 2 , φ = 0 2 is the global minimum; for g 2 > 3λm 2 , the global minimum becomes When g 2 = 3λm 2 , the two local minima become degenerate.Below we consider the parameter region g 2 > 3λm 2 .The potential (see Fig. 1) has two minima at φ FV = 0 and φ TV , corresponding to the false and true vacua respectively.In the potential, we have chosen U (φ FV ) = 0 for convenience.The decay rate is obtained from the Euclidean transition amplitude where H is the Hamiltonian, T is the amount of Euclidean time taken by the transition and S is the Euclidean action.In terms of Z[0], the decay rate per volume is given by [21] The partition function Z[0] is evaluated by using the method of steepest descent for path integrals.The classical equation of motion in Euclidean spacetime reads with the boundary conditions φ| τ →±∞ = φ FV and φ| τ =0 = 0, where the dot denotes the derivative with respect to the Euclidean time τ .A trivial solution is φ(x) = φ FV .There is a nontrivial solution-the bounce-which is O(4) symmetric in Euclidean spacetime.The bounce solution is relevant for false vacuum decay and describes a field configuration that starts and ends in Euclidean time in the false vacuum.Working in four-dimensional hyperspherical coordinates, the equation of motion reads where r = √ τ 2 + x 2 .The boundary conditions become φ(r → ∞) = φ FV and dφ(r)/dr| r=0 = 0. We denote this classical bounce solution as φ (0) b .Aside from the single bounce solution, there are also multi-bounce solutions that go back and forth from the false vacuum N times, for arbitrary N .These multi-bounce configurations exponentiate [21,34,35] in analogous to disconnected vacuum Feynman diagrams.Integrating the fluctuations about the stationary points requires one to analyse the eigenfunctions of the fluctuation operators.In particular, the operator −∂ 2 + U ′′ (φ (0) b ) contains negative and zero modes (see Sec. 4 below).Carefully dealing with some subtleties from these particular modes, one obtains the decay rate per unit volume [21] where det ′ means that the zero eigenvalues are omitted from the determinant and a prefactor B (0) /2π is included for each of the four collective coordinates that are related to the zero modes corresponding to spacetime translations [36].Using S[φ FV ] = 0 we have 3 Quantum-corrected bounce and radiative corrections to the decay rate We reformulate Eq. ( 5) such that radiative effects can be systematically considered, using the 1PI effective action [33].To keep the discussion general, we work in d-dimensional spacetime.But we will return to four-dimensional spacetime later.First, we introduce the partition function in the presence of a source J, The one-point function in the presence of the source J is given by The effective action then is defined as the Legendre transform From the effective action, one obtains the quantum equation of motion for the one-point function, δΓ[φ J ]/δφ J (x) = J(x).Note that Γ has no explicit dependence on J. Denoting φ ≡ φ J=0 , one obtains the equation of motion for the vacuum expectation value in the absence of sources by setting J = 0, A nontrivial O(4)-symmetric solution to Eq. ( 13) would give the quantum-corrected bounce φ b .
In terms of the effective action, the decay rate per volume (5) can be written as This is the formula used in Refs.[22,34,37,38] where it is implicitly assumed that Γ[φ FV ] = 0. Since adding constant and linear terms in the Lagrangian does not affect the dynamics (when gravity is not considered), in the renormalisation procedure one can always implement φ FV = 0 and Γ[φ FV ] = 0, even beyond the tree level.On denoting the fluctuation operator evaluated at a general configuration φ as the one-loop effective action reads where The bounce at the one loop, φ b , satisfies Expanding this equation about φ where we ignored perturbatively small higher-order terms.Substituting φ b + ∆φ b into Eq.( 16), we obtain Γ (1) [φ Using Eq. ( 18), we have When taking the functional derivative of D[φ], we just focus on its φ dependence and note that D In function space, we consider M[φ] as a "diagonal matrix" with one continuous index, M[φ(x)]. 4 Then where the "inverse matrix" with G(x 1 , x 2 ; φ) being the Green's function to the operator M[φ].Therefore, we have where a prime on M denotes the derivative with respect to the field.With this formula, one needs to calculate the Green's function G(x, y; φ (0) b ) and take the coincident limit x = y.From Eq. (18), ∆φ b can be expressed as where we have used Eq.(22).Upon using M ′ [φ] = U ′′′ (φ) = (−g + λφ) (cf.Eq. ( 15)), the last term in Eq. ( 20) reads If one uses G(x, y; φ b ) as the propagator, the above expression can be pictorially represented as where a crossed wheel denotes the classical bounce φ b ) as the propagator, we say that the last term in Eq. ( 20) corresponds to a partial two-loop corrections, but not to full two-loop corrections.
Making use of O(d)-symmetry.Although D[φ] and also the effective action are functionals of general configurations φ(x), it is assumed that the corrected bounce remains O(d)-symmetric.For this reason, it is sufficient to obtain D[φ] for O(d)-symmetric configurations φ(r) and take the functional derivative of the former with respect to the latter.In this case, we can treat φ(r) as a one-dimensional function and derive its equation of motion by directly varying the action S[φ] and D[φ] with respect to the one-dimensional field φ(r).To illustrate this, we first consider the classical equation of motion for the bounce, ( 7) generalised to d-dimensional spacetime.As explained in Sec. 2, one can derive it from Eq. ( 6) by assuming O(d)-symmetry.However, one can also derive it by applying the O(d)-symmetry to the classical action where Γ(x) is the Gamma function, and varying it with respect to the one-dimensional field φ(r).
Similarly, we can do the same procedure for D[φ].Then the one-loop equation of motion reads where again we have expanded the functional derivative at the classical bounce and neglected perturbatively small higher-order terms.Note that in this procedure δD ; they even have different mass dimensions due to the different Dirac delta functions generated in the two procedures for taking the functional derivative.The reason for introducing this new procedure is as follows.Below we have explicit expressions of D[φ] only for O(4)-symmetric configurations φ(r) instead of general four-dimensional configurations φ(x).As a consequence, we can take the functional derivative of D[φ] only with respect to φ(r), rather than with respect to the general four-dimensional function φ(x).Even when one assumes O(d)-symmetry for the one-point function, one still integrates over all possible fluctuations about φ(r) in deriving the expression of D[φ], as indicated by the sum over l in Eqs. ( 45), (46) below.
From Eq. ( 27), where G(r, r ′ ) is the Green's function, satisfying This procedure finally gives where the last term is a one-dimensional integral.Note that the last term above is not obtained from the last term in Eq. ( 20) by simply taking φ(x) = φ(r).As we commented above, b and we should have From Eqs. ( 28) and ( 30), we see that, at the one-loop level of the effective action, a selfconsistent bounce solution and the corrected decay rate are derived from D[φ  18) and ( 20))).
In the following, we use the Gel'fand-Yaglom method to compute the functional determinant D[φ] as well as its functional derivative.
4 Functional determinants for an O(4) background and the Gel'fand-Yaglom method The powerful Gel'fand-Yaglom method [31] is widely employed in calculations for tunneling in theoretical as well as phenomenological studies, see e.g.Refs.[7,18,19,32,[40][41][42].It is used in BubbleDet [43], a recent Python package that aims to compute functional determinants.For a general O(d)-symmetric configuration φ that approaches zero (the false vacuum φ FV ) for r → ∞, we have the decomposition where5 and For each l, one can calculate the functional determinant ratio using the Gel'fand-Yaglom theorem [31].Define the functions ψ l (r) and ψl (r) as the solutions to the following equations with the same leading regular behavior at r = 0.The Gel'fand-Yaglom theorem states that First, we show an asymptotic analysis for ψl (r) and ψ l (r).For r → 0, one can ignore the potential term in the operators M l and M l [φ].Therefore, the solutions regular at r = 0 are proportional to r l .Without loss of generality, both ψl (r) and ψ l (r) are normalised to r l as r → 0. Then we obtain I l+d/2−1 (x) are the modified Bessel functions of the first kind, satisfying where The boundary conditions are We note that Eq. ( 39) can be written in a slightly different form due to the recursion relations of the modified Bessel functions.
In the following, we take d = 4.

Divergences and renormalisation
The sum is divergent.This is not surprising because we are calculating the one-loop effective action whose divergence can only be removed by counterterms.Formally, we can write the renormalised oneloop effective action as where S 0 [φ] is the classical action, ∆S[φ] = D[φ]/2, and S ct [φ] is the contribution from the counterterms.In the renormalised effective action, all the couplings (including mass parameters) are now renormalised ones.The details of the counterterms and renormalised couplings depend on the renormalisation scheme..In the MS scheme, we have where S MS 0 is the lowest-order action expressed in terms of the renormalised MS couplings, and (∆S[φ]) pole is the pole of the divergent part of ∆S defined according to the MS renormalisation prescription.
In the literature, two slightly different methods of obtaining a (∆S[φ]) reg are used.The first was used by Dunne and Min [40] (see also [44,45]), and followed by Refs.[41][42][43].The second one was used even earlier by Baacke and Kiselev [46], and followed in Refs.[7,19,47,48].Below, we shall show that when the functional derivative with respect to φ is taken, the first method gives a UV divergent result.This means that the first method is at most valid only "on shell", i.e., only for the classical bounce solution but not for variations about it.The second method still gives a UV-convergent functional derivative.However, we observe that the regularised one-loop functional determinant contains a UV-finite tadpole term (linear term in φ) for our model, which would shift the position of the false vacuum from zero.For self-consistency, we therefore propose removing such a non-vanishing tadpole by a linear counterterm.This would modify one of the finite terms (A (1) fin ) to be discussed below.
Before we write down the explicit expressions for (∆S[φ]) reg , we discuss why there are two different points of view on the divergence in Eq. (42).The first one is that the divergence is caused by the large l behaviour of ln T l (∞; φ) and one needs to derive the analytical expression of ln T l (∞; φ) as a series of l.This can be done by the WKB method for solving Eq. ( 39) for large l [40,43].Then one can use the dimensional regularisation to subtract the pole terms in the sum.This leads to the Dunne-Min formula given below.The second point of view is that the divergence is caused by W (φ) ̸ = 0, i.e., by a nonvanishing interaction term.If W (φ) = 0, then the sum automatically gives zero.In this case, one expands ln T l (∞; φ) in powers of W .The divergence is found to be contained in the terms of order O(W ), O(W 2 ) (each as a sum over l).These terms however correspond to the standard UV divergences of the effective action and thus can be expressed by loop integrals.(The inhomogeneity of the background field becomes irrelevant in the deep UV.) Once again, one can obtain pole terms in the loop integrals by dimensional regularisation which will be then subtracted from the sum.For more details see Appendix B. This procedure leads to the Baacke-Kiselev formula, Eq. ( 46).
The first method gives [40,44] ( where γ E is Euler's constant, and µ is a renormalisation scale.
The second method gives [47] ( where h (1) with boundary conditions h (r; φ)/dr| r=0 = 0 , and where In Appendix B, we derive the above equations.Since φ(x) depends on r = |x| only, we can write where J 1 is the Bessel function of the first kind, defined as We observe that A fin contains a linear term in φ for our model since W (φ) = −gφ + λφ 2 /2.As a consequence, is not zero even for φ = 0.This means that the position of the false vacuum would be shifted if there are no other linear terms in Eq. (46).We show below that such additional linear terms are absent.To be self-consistent, we have to add a linear counterterm to remove the observed tadpole term by requiring This is achieved by simply doing the following replacement, where an overline means the subtraction of tadpole terms.For false vacuum decay, it is natural to take µ = m (as we will do in the following) so that fin [φ] and A fin [φ] simplify further.One can numerically solve Eq. (47a) to obtain h   l (∞; φ) as well as its functional derivative with respect to φ can be computed via a Green's function method with known h (1) l (r; φ), see below.

Zero and negative modes
When taking φ = φ (0) b , there are four zero modes in the sector l = 1.The integral over the fluctuations in the zero-mode directions can be traded for an integral over the collective coordinates of the bounce, giving [43,44] where A is defined by the normalisation of the asymptotic large r behavior of the classical bounce: Here K 1 (r) is the modified Bessel function of the second kind.Using the equation of motion for φ (0) b , Eq. ( 7), one has where we have used ∂ r φ b (r) as r → 0. So, formally, we have the replacement Note that the factor of V T in Eq. ( 55) will be cancelled by the same factor in γ/V (cf.Eq. ( 14)), leading to a finite decay rate per volume.The lowest eigenvalue mode in the l = 0 sector is a negative mode and is responsible for the instability of the false vacuum.Naively, the negative mode leads to divergence in the Gaussian functional integral about the bounce.However, this divergence is simply due to an improper application of the method of steepest descent [21,32,49].With a suitable deformation of the contour into the complex plane whose real axis is generated by the negative mode, one actually obtains the following replacement where the additional factor is due to the contour deformation and gives rise to an imaginary part of the effective action with a particular factor of 1/2.The sign depends on the direction in which one deforms the contour into the complex plane.Usually, one chooses the way that leads to a positive decay rate.But in our formula ( 14), we have taken the absolute value, and the sign in Eq. ( 58) does not matter.
In the following, we show how to take the functional derivative in Eqs. ( 45) and ( 46).

Functional derivative of a functional determinant
To compute the quantum corrections to the bounce as well as the corresponding corrections to the decay rate, we still need to compute δD[φ]/δφ(r)| φ (0) b (cf.Eqs. ( 28) and ( 30)).We will use Eq. ( 46), which leads At the end of this section, we shall comment on the problem if one uses Eq. ( 45).We need to compute δT l (∞; φ)/δφ, δh (∞; φ)/δφ.To simplify the notation, we define Then Taking the functional derivative of the first equation with respect to φ(r) gives Viewing r as a fixed number, the above equation in the coordinate r ′ can then be solved by the Green's function method and we obtain where G l (r, r ′ ; φ) satisfies with the boundary conditions6 One can apply a similar analysis for the second and third equations in (61).But it is more straightforward to take the functional derivative with explicit expressions of h (1,2) l (∞; φ).In this case, we need to solve the Green's function with the same boundary conditions as in Eq. ( 65).Then The same expressions are valid if one replaces ∞ with r above.Taking the functional derivative of these expressions, one obtains (2) l (r; φ) Ĝl (r, r ′ ) can be obtained analytically (see Appendix A) and reads where K l+1 (x) is the modified Bessel function of the second kind.The Green's functions G l (r, r ′ ; φ) depend on φ and can only be obtained numerically.
From Appendix A, we can see Therefore, one immediately has If one uses Eq. ( 45), one would obtain a divergent and cutoff-dependent result from the second and third terms in the sum over l at r = ∞.Therefore, Eq. ( 45) cannot give a meaningful result once the functional derivative with respect to φ(r) is taken.On the contrary, Eq. ( 46) gives a finite result because we also have due to the properties in Eq. ( 71).This also shows that there is no additional tadpole term in the sum over l in Eq. ( 46), otherwise one would obtain a term proportional to r 3 in the functional derivative, which is not vanishing for r → ∞.
At last, the functional derivatives of A fin [φ] and A fin [φ] with respect to φ(r) read where W (|k|) is defined in Eq. (50).

Numerical example
With the analysis given in the previous sections, here we discuss the procedure for numerical calculations and provide a numerical example.We leave a more delicate phenomenological study for future work.Denoting we can write the decay rate per unit volume as . Both here and below, D[φ] is regularised even though the superscript "reg" is suppressed.We use the Baacke-Kiselev formula for the regularised one-loop functional determinant (cf.Eq. ( 46)) with the modification (54).
In the numerical calculations, all the parameters that have mass dimension one take values in the unit of m so that m = 1.As a benchmark point, we choose g = 0.66 , λ = 0.1 .
The profile of the potential is shown in Fig. 1.
We summarise the procedure of computing the bounce and decay rate at the level of the Callan-Coleman formula.
• Step 1: Solve the classical EoM (7) and obtain the classical bounce φ The zero and negative modes in computing T l (∞; φ (0) b ) need to be properly dealt with as discussed in Sec. 4. With these three steps, one can obtain B (0) and B (1) and thus the Callan-Coleman decay rate.
The classical bounce is obtained via a shooting algorithm and is shown in Fig. 2. The classical bounce action is B (0) = 12942.7.
For B (1) , we first compute f l [φ b ] for l ≥ 2. We find that l max = 50 is already a good cutoff.We plot f l [φ b ]| drops as fast as or faster than 1/l 4 , ensuring a convergent result for the sum in Eq. (78).We obtain For the contribution from l = 0, 1, we find B (1) ⊃ − ln (πAU ′ (φ Finally, A fin [φ
Again, in calculating the functional derivatives with respect to φ, the zero and negative modes need special treatment.Formally, where 0 is a function that equals to zero identically.In the above equation, one therefore needs to look into the spectrum for both −∂ 2 +U ′′ (φ b +δφ).We expect that for an infinitesimal deformation of the classical bounce δφ → 0, the operator −∂ 2 + U ′′ (φ (0) b + δφ) still contains the same number of zero and negative modes.And we assume that Eq. ( 58) is still valid for φ infinitely close to φ For the zero mode, it is more subtle because Eq. ( 57) is expressed in terms of φ with φ and take the functional derivative of the second term in Eq. (57).We thus will neglect the zero-mode contribution to the functional derivative. 7However, we have checked that δh the decay rate is obtained as B (2) = −16.42(Step 6), which is 53.4% of B (1) .The correction ∆φ b (r) to the classical bounce is maximal around rm ≃ 3.5, which also corresponds to roughly the steepest part of the bubble wall profile.This indicates that these higher-order corrections receive large contributions from the gradients of the bubble wall.This observation is consistent with that made in Ref. [25] for thin-wall bubbles.

Conclusions
FVD is a very important phenomenon in quantum field theory and there continue to be many developments both conceptual and methodological [32,[50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67].In this paper, we propose a systematic procedure to compute the self-consistent bounce solution and its resulting radiative correction to the decay rate at the one-loop level of the effective action.This is achieved by evaluating the one-loop effective action at the one-loop saddle point configuration for the bounce (the self-consistent bounce), instead of the classical one.The method requires the computation of both conventional functional determinants and their functional derivative, both evaluated at the classical bounce.Earlier studies on this are restricted to the thin-wall regime [22,25,26] and are usually based on the "resolvent method" of calculating functional determinants by treating Green's function as a spectral sum [32,[68][69][70][71].In this work, we have demonstrated that the powerful Gel'fand-Yaglom method can be used to compute both the functional determinants and their functional derivatives, making the computation extremely efficient.In particular, for O(4)symmetric configurations that asymptotically approach the false vacuum, an explicit expression for the renormalised one-loop effective action in pure scalar field theory is known.We have shown how it is directly used in the computation.
Although the theoretical developments here are general, we provide an explicit numerical study of our archetypal model.Our chosen benchmark point for the mass and coupling parameters is such that the bounce is far away from the thin-wall regime.In this example, the correction B (1) (the decay rate due to the functional determinant evaluated at the classical bounce) is small compared with the classical bounce action B (0) .However, the additional correction due to the consideration of the self-consistent bounce, B (2) , is of the same order of magnitude (about 50%) as the purely one-loop correction B (1) .This indicates that in some precision phenomenological studies when the conventional functional determinants are important, one may be required to consider the self-consistent bounce as well.
Another motivation for considering the self-consistent bounce is to obtain a gauge-independent tunneling rate when gauge fields are involved.It is known that the effective action is gaugedependent.By the Nielsen identity [34,72,73], where ξ is the gauge parameter, the effective action is only gauge independent at its extrema.Then Eq. ( 14) gives a gauge-independent decay rate in principle only at the self-consistent bounce [34].Since the effective action must be truncated, this issue is usually more complicated.For more discussions related to this topic, see Refs.[74][75][76][77][78][79][80].
Our method may find applications in physics of the early Universe (see, e.g., Refs.[81][82][83]), where bubble nucleation can occur at finite temperature [84][85][86][87][88]. 8 At high temperature, the phase transition is dominated by a static, i.e., Euclidean time independent, O(3)-symmetric bounce and the transition rate can be written as [86] where Z[β] is the thermal partition function with the temperature T = 1/β, and λ − is the negative eigenvalue for the three-dimensional fluctuation operator at the classical bounce.The factor |λ − |/(2π) above may be generalised to include the Langevin damping coefficient [85,97,98].The calculation presented in this paper can be easily generalised to lower-dimensional bounce configurations, e.g., O(3)-symmetric static bounces that are relevant for thermal first-order phase transitions, and therefore can be applied to study radiative corrections in thermal first-order phase transitions [99].However, at finite temperatures, there are many more theoretical uncertainties [100,101].The quantum corrections due to the self-consistent bounce might be detectable in experiments involving analogue systems in condensed matter [102][103][104][105].
Our goal is to find two different expressions of ∆S (2) such that the finite quantity [∆S − ∆S (2) ] can be computed numerically while the other finite quantity [∆S (2) − (∆S) pole ] can be computed analytically.For this purpose, we assign the perturbation W with a bookkeeping factor ε and expand ln det(−∂ 2 + m 2 + εW ) in ε.
Numerically, we can write where T l (∞) is viewed as a function of ε and expanded in powers of ε, T l (∞; ε) = 1 + εh (1) This gives l (∞) + h (2) To obtain h l (∞) and h l (∞), we substitute T l (r; ε) = 1 + εh where the perturbation term is associated with a factor of ε now.Tracing the equations at the order of O(ε) and O(ε 2 ), we then obtain Eqs.(47) for h (1) l (r) and h (2) l (r).To obtain the analytical part [∆S (2) − (∆S) pole ], we note that Eq. (B19) can be written as The first term is To do the MS regularization, we perform the momentum integral in d-dimensional spacetime, using the useful formula (B28) Again, the momentum integral inside the square brackets can be performed in d-dimensional spacetime by using the Feynman-parameter method, which in our case is The final result of the momentum integral inside the square brackets is Finally, subtracting the pole term (1/ϵ together with −γ E + ln 4π) and doing the integral over s gives A fin .

b
that this two-loop diagram is obtained from the expansion φ + ∆φ b of the one-loop diagram on using G(x, y; φ (1) b ) as the propagator.When using G(x, y; φ (0)

Figure 3 :
Figure 3: The behavior of f l as a function of l.
one can obtain δh

b
at r = 0 and is not a standard functional of φ (0) b (r).One even cannot formally replace φ (0) b