One-loop corrections to the metastable vacuum ... - APS Link Manager

PHYSICAL REVIEW D 69, 025009 共2004兲

One-loop corrections to the metastable vacuum decay Ju¨rgen Baacke* Institut fu¨r Physik, Universita¨t Dortmund, D-44221 Dortmund, Germany

George Lavrelashvili† Department of Theoretical Physics, A.Razmadze Mathematical Institute, GE-0193 Tbilisi, Georgia 共Received 30 July 2003; published 30 January 2004兲 We evaluate the one-loop prefactor in the false vacuum decay rate in a theory of a self-interacting scalar field in 3⫹1 dimensions. We use a numerical method, established some time ago, which is based on a well-known theorem on functional determinants. The proper handling of zero modes and of renormalization is discussed. The numerical results show in particular that the quantum corrections strongly increase when one approaches the thin-wall case. In the thin-wall limit the numerical results are found to join into those obtained by a gradient expansion. DOI: 10.1103/PhysRevD.69.025009

PACS number共s兲: 11.27.⫹d, 02.60.⫺x, 03.65.Sq, 03.70.⫹k

I. INTRODUCTION

First-order phase transitions play an important role in various phenomena from solid state physics to cosmology. The basic theoretical concepts of these transitions have been developed long ago 关1– 6兴. The phase transition proceeds via formation of stable phase 共or true vacuum兲 bubbles within a metastable 共or false vacuum兲 environment, and via the subsequent growth of these bubbles. Two mechanisms of the first order phase transitions are known: quantum tunneling and thermal activation. In both cases the decay rate of a metastable state is given by the formula

␥ ⫽Ae ⫺B.

共1.1兲

For tunneling in a (3⫹1)-dimensional theory the quantity B in the exponent is given by the classical 4D Euclidean action evaluated on a bounce, a finite action Euclidean solution of classical equations of motion which asymptotically approaches the false vacuum. For thermal activation at nonzero temperature T the exponent is given by B⫽E/T, where E is the energy of a critical bubble 共sphaleron兲, which is a static solution ‘‘sitting’’ on a top of a barrier separating two vacua. The bounce as well as the sphaleron are unstable solutions with exactly one unstable mode. Bubbles smaller than critical collapse, and the ones bigger than critical expand and lead to the transition to a new phase. These static solutions and Euclidean solutions are related, namely the sphaleron in d⫹1 dimensions can be viewed as a bounce in d dimensions. The leading order estimate for the transition rate is easy to obtain; it just requires solving—in general numerically—an ordinary, though nonlinear differential equation. Analytic estimates can be obtained in the so-called thin-wall approximation. The pre-exponential factor A in Eq. 共1.1兲 is calculated taking into account quadratic fluctuations about the classical solution and is given as a ratio of the functional determi-

*Electronic address: [email protected] †

Electronic address: [email protected]

0556-2821/2004/69共2兲/025009共13兲/$22.50

nants. In general it is a very difficult task to calculate analytically the determinants if the background solution itself is not known in a closed form. It took two decades until the first 共numerical兲 computations of the quantum corrections to leading order semiclassical transition rates appeared 关7–12兴. Of course nowadays the CPU time requirements for such computations are, even for more involved systems, of the order of seconds. On the other hand the requirements of a precise renormalization, which compares exactly to the one of perturbative quantum field theory, and of the inclusion and careful treatment of high partial waves, have of course remained the same. The method used here has been developed and tested for various systems and has become a standard procedure. It is well suited for computations of coupled channel problems as well 关13兴. While the special technique used here applies only to the computation of functional determinants, the general approach can be used as well for computing zero point energies 关14 –16兴 via Euclidean Green functions. Of course functional determinants can be computed likewise using Euclidean Green functions 关12,17兴. Various other techniques for computing the exact quantum corrections were developed in the past decade. In Refs. 关11,18兴 the heat kernel is computed using a discretization of spectra, in Ref. 关19兴 Minkowskian instead of Euclidean Green functions are used, and in Ref. 关20兴 the zero point energy is computed via the ␨ function. The effective action may be computed approximatively by using gradient expansions. There is an ample literature on this subject. We just quote Refs. 关21–24兴 for expansions using advanced heat kernel techniques, and Ref. 关25兴 for expansions based on Feynman graphs. The leading quantum corrections, being essentially a one loop effect, can be viewed as a ‘‘summary’’ of the particle creation during the phase transition 关26兴. The question about the quantum corrections is a very important one; there are cases where particle creation is so strong that it drastically modifies the original classical tunneling solution 关27–29兴. The aim of the present paper is to calculate the pre-factor A for tunneling transitions in the quantum field theory of a self-interacting scalar field in 3⫹1 dimensions.

69 025009-1

©2004 The American Physical Society


J. BAACKE AND G. LAVRELASHVILI

next-to-leading order approximation a pre-exponential factor to the decay rate. The rate per volume and time is known to take the form 关5兴

␥⫽

冉

S cl 关 ␸ 兴 2␲

冊

2

兩 D兩 ⫺1/2exp兵 ⫺S cl 关 ␸ 兴 ⫺S ct 关 ␸ 兴其

共2.2兲

to one-loop accuracy. The coefficient D here is defined as D关 ␸ 兴 ⬅

FIG. 1. Potential U(⌽) in dimensionless form Eq. 共3.4兲. The curves are labeled with the value of ␣ .

The rest of this paper is organized as follows: In the next section we will describe our strategy for calculation of one loop effective action. In Sec. III we formulate our model, specify the form of the potential, write the equation of motion for the bounce and present our numerical results for the classical action S 关 ␸ 兴 . In Sec. IV we describe the calculation of the fluctuation determinant, Eq. 共2.3兲. There we also discuss regularization and renormalization. Our numerical results are presented and discussed in Sec. V. We end with some general remarks and conclusions in Sec. VI. Formulas describing the thin-wall approximation and gradient expansion are collected in the Appendixes A and B respectively. II. GENERAL STRATEGY

We will consider phase transitions in the quantum field theory of a self-interacting scalar field ␸ in 3⫹1 dimensions. The Euclidean action is given by S关␸兴⫽

冕冉 d 4x

冊

1 共 ⳵ ␸ 兲 2 ⫹U 共 ␸ 兲 , 2 ␮

共2.1兲

where the field potential U( ␸ ) is assumed to have two nondegenerate minima ␸ ⫽ ␸ ⫺ and ␸ ⫽ ␸ ⫹ ⬎0 共compare Fig. 1兲. U( ␸ ) will be given explicitly in the next section. For convenience we have fixed the value of ␸ in the unstable vacuum as ␸ ⫺ ⫽0. Any state built on the local minimum ␸ ⫺ is metastable. It can tunnel locally towards the ␸ ⫹ phase. The tunneling rate per unit volume per unit time, ␥ ⫽⌫/VT, is supposed to be dominated by the classical action S cl of a field configuration, the bounce ␸ b (x), which looks like a bubble of the ␸ ⫹ -phase within the ␸ ⫺ phase. In particular it can be shown 关30兴 that the bounce configuration ␸ b (x) which minimizes the action is spherically symmetric in four-dimensional Euclidean space. In the tree level approximation the decay rate is determined essentially by the tunneling coefficient, ␥ ⬀exp兵⫺Scl关␸b(x)兴其关39兴. The tree level tunneling rate receives corrections in higher orders of the semiclassical approximation. In quantum field theory the fluctuations around the bounce contribute in the

det⬘ „⫺ 共 ⳵ / ⳵ ␶ 兲 2 ⫺⌬⫹U ⬙ 共 ␸ 兲 … det„⫺ 共 ⳵ / ⳵ ␶ 兲 ⫺⌬⫹U ⬙ 共 0 兲 … 2

⫽

det⬘ 共 M兲 det共 M (0) 兲

. 共2.3兲

The prime in the determinant implies omitting of the four translation zero modes. With the second equation we have introduced the fluctuation operator in the background of the bounce M⫽⫺ 共 ⳵ / ⳵ ␶ 兲 2 ⫺⌬⫹U ⬙ 共 ␸ 兲

共2.4兲

and its counterpart M (0) in the unstable vacuum. The counterterm action S ct is necessary in order to absorb the divergences of the one-loop effective action 1 ef f S 1-loop 关 ␸ 兴 ⫽ ln兩 D关 ␸ 兴兩 . 2

共2.5兲

In order to evaluate the one loop effective action we decompose fluctuations around the bounce ␸ b into O(4) spherical harmonics, calculate the ratio of determinants J l of partial wave fluctuation operators and obtain ln D as 兺 l d l ln Jl , where d l is the O(4) degeneracy d l ⫽(l⫹1) 2 共see e.g. 关31兴兲. In calculating ln D we exclude the divergent perturbative contributions of first and second order in the external field of the bounce ␸ b . The regularized values of these contributions are then added analytically. All divergences of ln D appear in the standard tadpole and fish diagrams. We will not specify S ct explicitly, we will equivalently omit the divergent parts of ln D关 ␸ 兴 using the MS convention. III. THE TREE-LEVEL ACTION

In this section we specify our model, discuss the bounce solution and properties of corresponding classical action. We parametrize the ␸ 4 potential with two minima as 1 1 U共 ␸ 兲⫽ m 2␸ 2⫺ ␩ ␸ 3⫹ ␭ ␸ 4, 2 8

共3.1兲

and choose the same dimensionless variables as in Refs. 关10,32兴: x ␮ ⫽X ␮ /m for ␮ ⫽0,1,2,3, and ␸ ⫽(m 2 /2␩ )⌽. The classical action then takes the form S cl 共 ␸ 兲 ⫽ ␤˜S cl 共 ␸ 兲 ,

共3.2兲

where the rescaled classical action ˜S cl ( ␸ ) is

025009-2

˜S cl 共 ␸ 兲 ⫽

冕冉 d 4X

冊

1 共 ⵜ⌽ 兲 2 ⫹U 共 ⌽ 兲 , 2

共3.3兲


ONE-LOOP CORRECTIONS TO THE METASTABLE . . .

FIG. 2. Bounce profiles for different ␣ .

with

␣ 1 1 U共 ⌽ 兲⫽ ⌽ 2⫺ ⌽ 3⫹ ⌽ 4, 2 2 8

共3.4兲

and with the two dimensionless parameters 关40兴

␤⫽

m2 4␩2

,

␣ ⫽␭ ␤ .

共3.5兲

The parameter ␣ varies from 0 to 1 and controls the strength of self–interaction and the shape of the potential. For ␣ ⫽0 the second minimum disappears, whereas in the limit ␣ →1 the two minima become degenerate 共see Fig. 1兲. The parameter ␤ controls the size of the loop corrections. In order semiclassical approximation to be valid ␤ should not be too small 共see Sec. V for details兲. The bounce is a nontrivial, O(4)-symmetrical stationary point of S cl , Eq. 共3.3兲, obeying the Euler-Lagrange equation d 2⌽ dR

2

⫹

␣ 3 d⌽ 3 ⫺⌽⫹ ⌽ 2 ⫺ ⌽ 3 ⫽0, R dR 2 2

共3.6兲

tw ˜ cl FIG. 3. 共a兲 Classical action ˜S cl versus ␣ . 共b兲 The ratio ˜S cl /S for ␣ ⬎0.5.

˜S tw cl ⫽

冏

⫽0,

共3.8兲

3 共 1⫺ ␣ 兲 3

is displayed in Fig. 3共b兲. This ratio tends to unity for ␣ →1, as it should. Note that the radius of the bounce increases rapidly in this limit and numerical calculations become delicate. So, in the present article we restrict ourselves to the interval ␣ 苸关 0,0.95兴 . IV. CALCULATION OF THE FLUCTUATION DETERMINANT

and boundary conditions d⌽ dR

␲2

⌽ R→⬁ ⫽⌽ ⫺ .

共3.7兲

R⫽0

ជ 兩 2 …1/2. Equation 共3.6兲 can be easily Here R⫽„(X 0 ) 2 ⫹ 兩 X solved numerically, e.g., by the shooting method, as long as the value of ␣ is not too close to unity. We display some profiles ⌽(R) in Fig. 2 for various values of the parameter ␣. The classical action ˜S cl ( ␸ ) as a function of ␣ is plotted in Fig. 3共a兲. For small ␣ the classical action S goes to a constant and ˜S cl ( ␣ ⫽0)⫽90.857. In the limit ␣ →1 the thinwall case is realized 共see Appendix A兲 and the classical action diverges as (1⫺ ␣ ) ⫺3 . The ratio of the classical action computed numerically to the analytic thin-wall expression

In this section we discuss a method of computing the ratio of functional determinants 共2.3兲 which is based on earlier papers 关7,9,10兴. The explicit form of the operator in the nominator 共2.3兲 is M⫽⫺⌬ 4 ⫹m 2 ⫹V 共 r 兲 .

共4.1兲

Here ⌬ 4 is the 4-dimensional Laplace operator, and we have introduced the potential V as

025009-3

3 V 共 r 兲 ⫽U ⬙ 共 ␸ 兲 ⫺m 2 ⫽⫺6 ␩ ␸ 共 r 兲 ⫹ ␭ ␸ 2 共 r 兲 2

冋

册

3 ⫽m 2 ⫺3⌽ 共 R 兲 ⫹ ␣ ⌽ 2 共 R 兲 ⬅m 2 V 共 R 兲 . 2

共4.2兲



The ‘‘free’’ operator M (0) , corresponding to the metastable phase where ␸ ⫽0 and where m 2 ⫽U ⬙ ( ␸ ⫽0) takes the same form as Eq. 共4.1兲, but with V(r)⫽0. Due to the O(4) spherical symmetry of the bounce the operators M and M (0) can be separated with respect to O(4) angular momentum. We introduce the partial wave operators Ml 共 ␯ 兲 ⫽⫺

d2 dr

⫺ 2

l 共 l⫹2 兲 3 d ⫹ ⫹ ␯ 2 ⫹m 2 ⫹V 共 r 兲 , 2 r dr r

D关 ␸ 兴 ⬅

⬘ 兿 l,n

冋册兿冋 ⬁

⫽

l⫽0

det⬘ Ml 共 0 兲 det M(0) l 共0兲

册

共4.3兲

dl

,

共4.4兲

det Ml 共 ␯ 兲 det M(l ␯ ) 共 0 兲

⫽

兿n

冋

␻ 2ln ⫹ ␯ 2 2 ␻ ln(0) ⫹␯2

册

共4.5兲

A. Determinants of the radial operators

In order to find J l ( ␯ ) 共4.5兲 we make use of a known theorem 关6,33兴 whose statement is

det M(0) l 共␯兲

⫽ lim

␺ l 共 ␯ ,r 兲

(0) r→⬁ ␺ l 共 ␯ ,r 兲

.

共4.6兲

Here ␺ l ( ␯ ,r) and ␺ (0) l ( ␯ ,r) are solutions to equations Ml 共 ␯ 兲 ␺ ␯ ,l ⫽0,

(0) M(0) l 共 ␯ 兲 ␺ ␯ ,l ⫽0,

共4.8兲

I l⫹1 共 ␬ r 兲 . r

共4.9兲

Then, by the theorem 共4.6兲, the ratio of determinants 共4.5兲 can be expressed as J l 共 ␯ 兲 ⫽1⫹h l 共 ␯ ,⬁ 兲 .

共4.10兲

In terms of the h function the first equation 共4.7兲 reads

再冋 dr

2

⫹ 2␬

册冎

⬘ 共␬r兲 1 d I l⫹1 ⫹ h 共 ␯ ,r 兲 ⫽V 共 r 兲关 1⫹h l 共 ␯ ,r 兲兴 , I l⫹1 共 ␬ r 兲 r dr l 共4.11兲

⬘ ( ␬ r)⬅dI l⫹1 ( ␬ r)/d( ␬ r). where I l⫹1 In the following it will be convenient to consider the perturbation expansion ⬁

can be computed using the theorem on functional determi2 always nants as described in the next section. Note that ␻ ln denotes the eigenvalues of Ml (0), or more generally the ei2 . genvalues of M, the analogous definition holds for ␻ ln(0)

det Ml 共 ␯ 兲

I l⫹1 共 ␬ r 兲 , r

and we have

d2

where d l is the degeneracy of the O(4) angular momentum, d l ⫽(l⫹1) 2 . The prime denotes that for l⫽1 we have to remove the four translational zero modes. The ratio of determinants of the radial operators J l共 ␯ 兲 ⫽

␺ (0) l 共 ␯ ,r 兲 ⫽

␺ l 共 ␯ ,r 兲 ⫽ 关 1⫹h l 共 ␯ ,r 兲兴

with an additional variable ␯ that will be used later on. In terms of these operators we can write

␻ 2ln 2 ␻ ln(0)

constant value h l ( ␯ ,⬁) as r→⬁. The solutions ␺ (0) l ( ␯ ,r) are given in terms of modified Bessel functions as

共4.7兲

and have the same regular behavior at r⫽0. More exactly, the boundary conditions at r⫽0 must be chosen in such a way that the right-hand side of Eq. 共4.6兲 tends to 1 at ␯ →⬁. It is convenient to factorize the radial mode functions into the solution ␺ (0) l ( ␯ ,r) for V(r)⫽0 and a factor 1⫹h l ( ␯ ,r) which takes into account the modification introduced by the potential. If V(r) is of finite range the functions ␺ (0) l ( ␯ ,r) and ␺ l ( ␯ ,r) have the same behavior near r⫽0 and as r →⬁. Near r⫽0 they behave as r l and as r→⬁ they behave as exp(⫺␬r) where ␬ ⫽ 冑␯ 2 ⫹m 2 . Furthermore the requirement of an analogous behavior near r⫽0 introduces the initial conditions h(0)⫽h ⬘ (0)⫽0. The function h(r) then simply starts from zero at r⫽0 and goes smoothly to a finite

h l 共 ␯ ,r 兲 ⫽

兺

k⫽1

h (k) l 共 ␯ ,r 兲

共4.12兲

in powers of the potential V(r). This entails an analogous expansion for the ratios J l ( ␯ ) in the sense that J (k) l (␯) (k) ⫽h (k) ( ␯ ,⬁). The k-order contribution h obeys an equation l l

再冋 d2

⬘ 共␬r兲 I l⫹1

dr

I l⫹1 共 ␬ r 兲

⫹ 2␬ 2

⫹

册冎

1 d h (k) 共 ␯ ,r 兲 ⫽V 共 r 兲 h (k⫺1) 共 ␯ ,r 兲 , l r dr l 共4.13兲

where we defined h (0) l ⬅1. Since Eq. 共4.13兲 is a linear differential equation it holds also for linear combinations of (k) h (k) l . It is convenient to introduce the notation h l ⬁ (q) (1) ⫽ 兺 q⫽k h l . In this notation h l ⫽h l . A Green function that gives the solution to Eq. 共4.13兲 in the form h (k) l 共 r 兲 ⫽⫺

冕

⬁

0

˜˜r G l 共 r,r ˜ 兲 V 共˜r 兲 h (k⫺1) dr 共˜r 兲 l

共4.14兲

with the correct boundary condition at r⫽0 reads ˜ 兲⫽ G l 共 r,r

I l⫹1 共 ␬˜r 兲关 I 共 ␬ r ⬍ 兲 K l⫹1 共 ␬ r ⬎ 兲 I l⫹1 共 ␬ r 兲 l⫹1 ⫺I l⫹1 共 ␬ r 兲 K l⫹1 共 ␬˜r 兲兴 ,

共4.15兲

˜ 其, r ⬎ ⫽max兵r,r ˜ 其. where r ⬍ ⫽min兵r,r The first term on the right-hand side of Eq. 共4.15兲 does not contribute to h (k) l (⬁). The Green function 共4.15兲 gives rise to connected graphs as well as to disconnected ones. The

025009-4



latter are canceled in ln„1⫹h l (⬁)… whose expansion in k-order connected graphs J l(k)con ( ␯ ) reads ⬁

ln J l 共 ␯ 兲 ⫽ln„1⫹h l 共 ␯ ,⬁ 兲 …⫽

兺 k⫽1

共 ⫺1 兲 k⫹1 (k) J l con 共 ␯ 兲 . k 共4.16兲

This formula is analogous to the expansion of the full functional determinant in terms of Feynman diagrams ⬁

ln D⫽

兺 k⫽1

共 ⫺1 兲 k⫹1 (k) A , k

共4.17兲

where A (k) is the one-loop Feynman graph of order k in the external potential V(r). and, thereIndeed, it is obvious from Eq. 共4.14兲 that h (k) l fore, J l(k)con are of the order V k . Since the expansion of ln D in powers of V is unique, we conclude that ⬁

A (k) ⫽

兺共 l⫹1 兲 2 J l(k)con .

共4.18兲

l⫽0

B. Calculation of D „3…

Making use of a uniform asymptotic expansion of the modified Bessel functions in Eq. 共4.15兲 one can check that J l(k)con ⬃1/l 2k⫺1 as l→⬁. This results in the expected quadratic and logarithmic ultraviolet divergences in ln D due to the contribution of J l(1)con and J l(2)con . Our strategy is to compute analytically the first two terms in the sum Eq. 共4.17兲 and to add numerically computed ln D (3) , which is the sum without first and second order diagrams A (1) and A (2) . It reads explicitly

Each of the three terms on the right-hand side 共RHS兲 is now manifestly of order V 3 . The subtraction done in the square bracket is exact enough when the logarithm is calculated with double precision. We have determined h l (r) as solu(2) (3) tions of Eq. 共4.11兲 and h (1) l (r), h l (r) and h l (r) as those of Eq. 共4.13兲 using the Runge-Kutta-Nystro¨m integration method 关34兴. Of course we cannot integrate the differential equations until r⫽⬁. In fact we have integrated it up to the maximal value for which we know the profile ␾ (r), and therefore V(r). This value is such, that the classical field has well reached its vacuum expectation value, and therefore V(r) has become zero. This is the condition under which we can impose the asymptotic boundary condition for the classical profile. For such values the functions h (k) l (r) have already become constant; indeed for V(r)⫽0 they have the exact form a⫹bK l⫹1 ( ␬ r)/I l⫹1 ( ␬ r) and the second part decreases exponentially for rⰇ1/␬ . In praxi we used values of R up to R max⫽mrmax⯝20⫺30. Up to now we have neglected the existence of the negative mode ␻ 20 ⬍0 for l⫽0 and four zero modes ␻ 21 ⫽0 with l⫽1. The former results in a negative value of J 0 ( ␯ )⫽1 ⫹h 0 ( ␯ ,⬁) at ␯ ⫽0. According to Eq. 共2.2兲 one has to replace ␻ 20 by 兩 ␻ 20 兩 . This implies taking the absolute value of J 0 (0) in Eq. 共4.19兲; indeed J 0 (0) is found to be negative. The translational zero modes manifest themselves by the 2 ⫽0, the lowest radial excitation in the l vanishing of ␻ 10 ⫽1 channel with degeneracy (l⫹1) 2 ⫽4, and thereby by the vanishing of J 1 ( ␯ ) at ␯ ⫽0; see Eq. 共4.5兲. This represents a good check for both the classical solution and the integration of the partial waves. The factor ␯ 2 has to be removed according to the definition of det⬘ . So in the l⫽1 contribution we have to replace J 1 (0) by

⬁

ln D (3) ⫽

兺共 l⫹1 兲 2 „ln J l共 ␯ 兲 …(3) , l⫽0

共4.19兲

where „ln J l 共 ␯ 兲 …(3) ⫽ln„1⫹h l 共 ⬁ 兲 …⫺h (1) l 共⬁兲

冋

册

1 (1) 2 ⫺ h (2) l 共 ⬁ 兲 ⫺ „h l 共 ⬁ 兲 … . 2

共4.20兲

Here the terms in square brackets correspond to the fish diagram J l(2)con . Since all contributions to ln D (3) are ultraviolet finite, we need no regularization in computing them. The divergent contributions of the first and second order in V will be considered in Sec. IV C. In order to avoid a numerical subtraction that might be delicate we re-write the term 共4.20兲 to be summed up on the right-hand side 共4.19兲 in the form

冋

1

„ln J l 共 ␯ 兲 … ⫽ ln„1⫹h l 共 ⬁ 兲 …⫺h l 共 ⬁ 兲 ⫹ h l 共 ⬁ 兲 2 (3)

⫺

1 2

(1) h (2) l 共 ⬁ 兲 „h l 共 ⬁ 兲 ⫹h l 共 ⬁ 兲 ….

lim

2

册

␯ →0

␯

2

⫽

dJ 1 共 ␯ 兲 d␯

2

⫽

冏

d

h 1 共 ␯ ,⬁ 兲

d共 ␯2兲

.

共4.22兲

␯ ⫽0

Notice that replacement Eq. 共4.22兲 introduces a dimension into the functional determinant. Thereby the units used for ␯ become the units of the transition rate. Here we have used the scale m throughout; see Eqs. 共5.1兲 and 共5.2兲. Our next step is performing summation over l in Eq. 共4.19兲. For small bounces ( ␣ ⱗ0.8) we have found good agreement with the expected behavior, namely „ln J l 共 ␯ 兲 …(3) ⬀

1 共 l⫹1 兲 5

共4.23兲

.

So, the summation has been done by cutting the sum at some value l max and adding the rest sum from l max ⫹1 to ⬁ of terms fitted with ln J (3) l ⬇

⫹h (3) l 共⬁兲共4.21兲

J 1共 ␯ 兲

a 共 l⫹1 兲

5

⫹

b 共 l⫹1 兲

6

⫹

c 共 l⫹1 兲 7

.

共4.24兲

The summation was stopped when increasing the value of l max by unity did not change the result within some given accuracy ␦ . The required accuracy was decreased for higher

025009-5



␣ . The problem is that the convergence becomes worse as we get closer to ␣ ⫽1. This is related to the fact that the asymptotic behavior 共4.23兲 sets in only when lⰇmr eff , where r eff is the characteristic size of the bounce. It is of order 1/m at small values of ␣ and can be estimated as 1/(1⫺ ␣ )m near the thin-wall limit, ␣ →1. As the maximal value of the angular momentum we have used is l⫽25, our computations cease to be reliable beyond ␣ ⯝0.95. The value of ␦ was about 10⫺5 for small bounces, and of order of 10⫺3 for ␣ ⬎0.85. As we will see below, for larger values of ␣ the effective action is well approximated by the leading terms of a gradient expansion. C. Perturbative contribution and renormalization

We obtain A (2) ⫽

ln D (3) ⫽

共 ⫺1 兲 k⫹1 (k) A . k

兺 k⫽3

A (1) ⫽

冕

d 4⫺ ⑀ k

˜V 共 0 兲

where we have introduced the Fourier transform of the potential ˜V 共 k 兲 ⫽

冕

m2 16␲

2

冋

d xV 共 x 兲 e 4

⫺ikx

A (2) f in ⫽

⫻

冕

128

冋

4

⬁

0

Q 3 dQ 兩 ˜V 共 Q 兲兩 2

冑Q 2 ⫹4⫹Q 冑Q 2 ⫹4⫺Q

册

,

共4.32兲

with Q⫽q/m being the dimensionless momenta. For the numerical evaluation of A (2) we have to compute the Fourier transform of the external potential which is known numerically, the remaining computation is straightforward. V. NUMERICAL RESULTS

To summarize, we represented the false vacuum decay rate per unit time and per unit volume as

册冕

冉

S cl 关 ␸ 兴 2␲

冊

2

ef f

e ⫺S cl [ ␸ ]⫺S 1-loo p [ ␸ ] ,

d 4 xV 共 x 兲 , 共4.28兲

共5.1兲

冕

⬁

0

ef f ⫽ S 1-loop,p

冉

1 (1) 1 (2) A f in ⫺ A f in 2 2

冊

共5.3兲

and nonperturbative ef f ⫽ S 1-loop,n.p.

共4.29兲

R dRV 共 R 兲 . 3

共5.2兲

with perturbative

1

⬁

兺 2 k⫽3

共 ⫺1 兲 k⫹1

k

1 A (k) ⫽ ln兩 D (3) 兩 2

共5.4兲

contributions. It is useful to introduce the quantity G, ef f G 共 ␣ , ␤ 兲 ⫽S 1-loop 关 ␸ 兴 /S cl 关 ␸ b 兴 ,

兩˜ V共 q 兲兩 2 4⫺ ⑀

d 4⫺ ⑀ k

冕 ␲

⫻ 2⫺ 冑Q 2 ⫹4 ln

d 4⫺ ⑀ q 共2␲兲

共4.31兲

.

1 ef f ef f ef f ⫹S 1-loop,n.p. , S 1-loop 关 ␸ 兴 ⫽ ln兩 m 8 D关 ␸ 兴兩 ⫽S 1-loop,p 2

The second order terms takes the form

冕

册

where

where ␮ is the usual dimensional regularization parameter. We choose it to be equal to m. Then using the MS scheme we just retain the last contribution in the bracket 共see e.g. 关35兴, p. 377兲. Thus, the finite part of A (1) is

A (2) ⫽

q 3 dq 兩 ˜V 共 q 兲兩 2

1

共4.27兲

.

冕

d 4 x„V 共 x 兲 …2

冑q 2 ⫹4m 2 冑q 2 ⫹4m 2 ⫹q ln 2 q 冑q ⫹4m 2 ⫺q

␥ ⫽m 4

2 ␮2 ⫺ ␥ E ⫹ln 4 ␲ ⫹ln 2 ⫹1 ⑀ m

1 A (1) f in ⫽⫺ 8

128␲ 4

册冕

Again the MS scheme corresponds to omitting the first term on the right-hand side and for the finite part of A (2) we find

We obtain A (1) ⫽⫺

2 ␮2 ⫺ ␥ E ⫹ln 4 ␲ ⫹ln 2 ⑀ m

1

冋

共4.26兲

共 2 ␲ 兲 4⫺ ⑀ k 2 ⫹m 2

冋

⫻ 2⫺

共4.25兲

We now have to discuss the leading divergent contributions A (1) and A (2) . These are computed as ordinary Feynman graphs. Using dimensional regularization we have

16␲ 2 ⫹

We have described in the previous subsection the computation of the finite part ln D (3) which is the sum of all oneloop diagrams of the third order and higher, ⬁

1

1

共 2 ␲ 兲 4⫺ ⑀ 共 k 2 ⫹m 2 兲关共 k⫹q 兲 2 ⫹m 2 兴

. 共4.30兲

共5.5兲

which indicates how big the quantum corrections are. Since the classical action, Eq. 共3.2兲, depends linearly on the parameter ␤ we have G( ␣ , ␤ )⫽G( ␣ ,1)/ ␤ . The numerical calculation shows that G( ␣ ,1) varies from 0.0367 to 0.0448 as we vary ␣ from 0 to 0.95, with a shallow

025009-6



TABLE I. Numerical results for classical action and one loop effective action.

ef f FIG. 4. The ratio G( ␣ , ␤ )⫽S 1-loo p /S cl for ␤ ⫽1.

minimum G min ⬇0.033 at ␣ about 0.6 共see Fig. 4兲. Figure 4 suggests that G(1,1)⬇0.05, which means that for sufficiently big values of ␤ , namely ␤ ⬎0.1, the quantum corrections to the classical action are small 共less then 50%兲 for all values of ␣ . The corrections to the transition rate are given directly by ef f ), so even if the classical transition rate a factor exp(⫺S1-loop is sizable, as it happens for small ␤ , the quantum corrections suppress the decay of the false vacuum by factors exp (⫺3.3) at ␣ ⫽0 and exp(⫺291) at ␣ ⫽.9. Note that the main contribution to the effective action for all ␣ is coming from the A (1) f in 共cf. Tables I and II兲. For small ␣ the perturbative contribution is almost 100% of the total one-loop effective action 共see Fig. 5兲. In the limit ␣ →1 the leading terms of the gradient expansion 共Appendix B兲 give the dominant contribution to the oneloop effective action. Already for ␣ ⫽0.8 the sum of leading gradient terms ef f ef f ef f ⫽S grad,0 ⫹S grad,2 S grad,0⫹2

共5.6兲

␣

ef f S 1-loop,p

ef f S 1-loop,n.p.

ef f S 1-loop

˜S cl

0.00 0.02 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.83 0.85 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95

3.253⫻100 3.337⫻100 3.478⫻100 3.752⫻100 4.089⫻100 4.504⫻100 5.021⫻100 5.672⫻100 6.499⫻100 7.571⫻100 8.984⫻100 1.089⫻101 1.356⫻101 1.741⫻101 2.326⫻101 3.277⫻101 4.966⫻101 8.382⫻101 1.240⫻102 1.686⫻102 2.409⫻102 2.950⫻102 3.684⫻102 4.711⫻102 6.199⫻102 8.455⫻102 1.207⫻103 1.829⫻103 3.008⫻103

8.216⫻10⫺2 8.498⫻10⫺2 8.422⫻10⫺2 6.737⫻10⫺2 2.654⫻10⫺2 ⫺4.501⫻10⫺2 ⫺1.564⫻10⫺1 ⫺3.201⫻10⫺1 ⫺5.539⫻10⫺1 ⫺8.836⫻10⫺1 ⫺1.348⫻100 ⫺2.006⫻100 ⫺2.958⫻100 ⫺4.371⫻100 ⫺6.560⫻100 ⫺1.015⫻101 ⫺1.659⫻101 ⫺2.969⫻101 ⫺4.512⫻101 ⫺6.233⫻101 ⫺9.038⫻101 ⫺1.114⫻102 ⫺1.401⫻102 ⫺1.803⫻102 ⫺2.390⫻102 ⫺3.284⫻102 ⫺4.724⫻102 ⫺7.209⫻102 ⫺1.188⫻103

3.335⫻100 3.422⫻100 3.562⫻100 3.819⫻100 4.115⫻100 4.459⫻100 4.865⫻100 5.351⫻100 5.946⫻100 6.687⫻100 7.637⫻100 8.889⫻100 1.060⫻101 1.303⫻101 1.670⫻101 2.261⫻101 3.306⫻101 5.413⫻101 7.887⫻101 1.062⫻102 1.506⫻102 1.836⫻102 2.283⫻102 2.907⫻102 3.809⫻102 5.171⫻102 7.347⫻102 1.109⫻103 1.820⫻103

9.086⫻101 9.355⫻101 9.787⫻101 1.059⫻102 1.153⫻102 1.263⫻102 1.394⫻102 1.552⫻102 1.744⫻102 1.983⫻102 2.286⫻102 2.681⫻102 3.211⫻102 3.951⫻102 5.033⫻102 6.720⫻102 9.589⫻102 1.512⫻103 2.136⫻103 2.809⫻103 3.874⫻103 4.655⫻103 5.699⫻103 7.140⫻103 9.198⫻103 1.227⫻104 1.711⫻104 2.531⫻104 4.061⫻104

where a (2) is the following integral

ef f S 1-loop

within approximates the one-loop effective action 20%. So the gradient expansion reproduces well the behavior of the one-loop effective action when ␣ →1; see Fig. 5. As the numerical procedure described in the main part of this paper becomes precarious for ␣ ⲏ0.9 this expansion complements the computation of the transition rate in this region. As it is well known there is exactly one negative mode in the spectrum of fluctuations about the bounce. Its energy is plotted vs ␣ in Fig. 6. In the present paper we used dimensional regularization and we have chosen the parameter ␮ 2 , which can be understood as parametrizing a sequence of possible renormalization conditions, to be equal to m 2 . Choosing ␮ 2 differently (2) would result in the following corrections to A (1) f in and A f in

冉

A (1) f in → 1⫹ln

␮2 m2

冊

A (1) f in ,

(2) (2) ln A (2) f in →A f in ⫹a

␮2 m2

,

共5.7兲

a (2) ⫽

1 8

冕

⬁

0

R 3 dR„V 共 R 兲 …2 ,

共5.8兲

evaluated at the bounce solution. Numerical values for (2) (2) for different values of ␣ are collected in A (1) f in ,A f in and a Table II. With the present choice of ␮ 2 the perturbative terms represent the most important contributions to the effective action 共see above兲, this means at the same time that a modification of the regularization and renormalization procedures can result in large changes in the one-loop effective action. VI. DISCUSSION AND CONCLUSION

In the present paper we applied a previously developed technique for evaluations of functional determinants and calculated quantum corrections to the tunneling transitions in a model of a self-interacting scalar field in 3⫹1 dimensions. In the present toy model the decay rate is vanishingly small. The sign of quantum corrections is such that it decreases the false vacuum decay rate. The corrections can be thought as originating from particle creation during the phase

025009-7


J. BAACKE AND G. LAVRELASHVILI TABLE II. Numerical results for the first and second order contribution coefficients.

␣

A (1) f in

A (2) f in

a (2)

0.00 0.02 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.83 0.85 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95

5.178⫻100 5.379⫻100 5.706⫻100 6.325⫻100 7.060⫻100 7.942⫻100 9.013⫻100 1.033⫻101 1.199⫻101 1.410⫻101 1.686⫻101 2.056⫻101 2.570⫻101 3.311⫻101 4.438⫻101 6.269⫻101 9.526⫻101 1.613⫻102 2.391⫻102 3.256⫻102 4.660⫻102 5.711⫻102 7.137⫻102 9.134⫻102 1.203⫻103 1.643⫻103 2.347⫻103 3.561⫻103 5.861⫻103

⫺2.654⫻100 ⫺2.591⫻100 ⫺2.499⫻100 ⫺2.358⫻100 ⫺2.235⫻100 ⫺2.133⫻100 ⫺2.059⫻100 ⫺2.020⫻100 ⫺2.024⫻100 ⫺2.085⫻100 ⫺2.219⫻100 ⫺2.453⫻100 ⫺2.825⫻100 ⫺3.399⫻100 ⫺4.286⫻100 ⫺5.701⫻100 ⫺8.109⫻100 ⫺1.269⫻101 ⫺1.778⫻101 ⫺2.321⫻101 ⫺3.170⫻101 ⫺3.788⫻101 ⫺4.609⫻101 ⫺5.735⫻101 ⫺7.333⫻101 ⫺9.699⫻101 ⫺1.341⫻102 ⫺1.963⫻102 ⫺3.118⫻102

1.036⫻101 1.055⫻101 1.085⫻101 1.144⫻101 1.215⫻101 1.300⫻101 1.405⫻101 1.536⫻101 1.702⫻101 1.916⫻101 2.199⫻101 2.585⫻101 3.127⫻101 3.921⫻101 5.147⫻101 7.175⫻101 1.085⫻102 1.848⫻102 2.761⫻102 3.790⫻102 5.479⫻102 6.753⫻102 8.493⫻102 1.094⫻103 1.453⫻103 1.999⫻103 2.883⫻103 4.417⫻103 7.359⫻103

FIG. 6. The negative mode energy as a function of ␣ .

transition. The created particles take energy from the tunneling field and therefore decrease the tunneling probability. Analytical estimations show that particle creation is typically weak in the thin-wall approximation 关26兴. In the present paper it was found that the quantum corrections are even smaller away from the thin-wall case 共compare Fig. 4兲, which assumes that particle creation for ␤ ⬎0.1 is weak for all values of the coupling constant ␣ . On the other hand for ␤ ⬍0.1 the quantum corrections dominate, which means that in this regime one should look for a bounce solution taking into account the full effective action in the one-loop approximation 关27–29兴. Corrections to the false vacuum decay in a similar model in the 共3⫹1兲-dimensional theory in the thin wall approximation with the heat kernel expansion technique were calculated in 关36兴, but it is not straightforward to compare our results since we use a different renormalization scheme and a different parametrization of the potential. Powerful techniques for analytic calculations of the prefactor using different approximations were developed in 关24,37,38兴, but we cannot compare our results directly, since these calculations are within 3D theory. The technique described here can be applied to tunneling transitions in more realistic theories in 4 dimensions. ACKNOWLEDGMENTS

G.L. is thankful to the Theory Group of the University of Dortmund for their kind hospitality during his visit to Dortmund, where this work started and to the theory groups of the Max-Planck-Institute for Physics and Max-PlanckInstitute for Gravitational Physics for a stimulating and fruitful atmosphere during his visits to Munich and Golm, where part of the work was done. The work of G.L. was partly supported by a Grant of the Georgian Academy of Sciences. APPENDIX A: THE THIN-WALL APPROXIMATION FIG. 5. Our results for the effective action 共squares兲 toef f 共dotted line兲 and the leadgether with the perturbative part S 1-loop,p ef f ing parts of the gradient expansion S grad,0⫹2 共dashed line, ␣ ⫽0.45–0.95). All quantities shown are multiplied by the factor (1⫺ ␣ ) 3 . ef f S 1-loop

In the limit ␣ →1 the so called thin-wall case is realized. This is the case when the difference in energy density between the two vacua

025009-8

⑀ ⫽U 共 ⌽ ⫺ 兲 ⫺U 共 ⌽ ⫹ 兲 ,

共A1兲



is small compared to the height of the barrier. In this case the potential Eq. 共3.4兲 can be represented as U 共 ⌽ 兲 ⫽U 0 共 ⌽ 兲 ⫹O 共 ⑀ 兲 ,

共A2兲

where in our case the symmetric part U 0 is given by 1 U 0 共 ⌽ 兲 ⫽ ⌽ 2 共 2⫺⌽ 兲 2 , 8

U ⬙ „␾ 共 x 兲 …⫽m 2 ⫹V 共 x 兲 ,

冋

关 ln D兴 ⫽ln

and where

3S 1 , ⑀

˜S tw cl ⫽

2⑀3

⬁

⫽ ,

共A5兲

S 1 ⫽2

冕

dRU 0 共 ⌽ k 兲 ,

2 ¯

1⫹e (R⫺R )

S eff⫽

共A6兲

is the action of the one-dimensional kink solution corresponding to degenerate potential U 0 with the equal minima. For our choice of the potential, Eq. 共A3兲, the kink solutions is ⌽ k⫽

兺 N⫽1 ⬁

⫺⬁

共A7兲

.

One finds that S 1 ⫽2/3, and correspondingly

␲ 1 , ˜S tw . cl ⫽ 1⫺ ␣ 3 共 1⫺ ␣ 兲 3

兺

N⫽1

共 ⫺1 兲 N⫹1 tr关共 ⫺⌬ 4 ⫹m 2 兲 ⫺1 V 共 x 兲兴 N . 2N

We introduce the Fourier transform ˜V 共 q 兲 ⫽

冋

⫺⌬ 4 ⫹U ⬙ 共 0 兲

册

冕

e ⫺iq•x V 共 x 兲 d 4 x.

共B5兲

The individual terms in the expansion of the effective action have the form of Feynman diagrams with external sources V(q j ) with j⫽1 . . . k. The momentum that has flown into the line l is Q l⫽

共A8兲

兺

j⫽1

共B6兲

qj ,

of course the total momentum must be zero, i.e., Q N ⫽0. With these notations we can write the Nth term in the effective action, omitting the factor (⫺1) N⫹1 /2N as

We want to derive an approximation to the effective action of a scalar field on the background of a bounce solution. The strategy is to expand first the effective action with respect to external vertices, and to expand in a second step the resulting Feynman amplitudes with respect to the external momenta. This approach is fairly standard, and has been used, e.g., in Ref. 关25兴. We note that we will retain all powers in the external vertices; such a summation was found to yield a very good approximation for the sphaleron determinant 关11,12兴; see Fig. 1 in the second entry of Ref. 关12兴. We have to compute the trace log or log det of a generalized Euclidean Klein-Gordon operator ⌬ 4 ⫹U ⬙ ( ␾ ) where ⌬ 4 is the four-dimensional Laplace operator. Formally 关 ln D兴 ⫽ln

共B4兲

l

APPENDIX B: THE LEADING TERMS OF THE GRADIENT EXPANSION

⫺⌬ 4 ⫹U ⬙ 共 ␾ 兲

共B3兲

共 ⫺1 兲 N⫹1 关共 ⫺⌬ 4 ⫹m 2 兲 ⫺1 V 共 x 兲兴 N , N

2

¯R ⫽

册

and the effective action is given by

where ⬁

⫺⌬ 4 ⫹m 2

⫽ln关 1⫹ 共 ⫺⌬ 4 ⫹m 2 兲 ⫺1 V 共 x 兲兴

In the thin-wall approximation the radius ¯R of the bounce and the Euclidean action S cl are given analytically 关4,6兴 as ¯R ⫽

⫺⌬ 4 ⫹m 2 ⫹V 共 x 兲

⫽ln关共 ⫺⌬ 4 ⫹m 2 兲 ⫺1 „⌬ 4 ⫹m 2 ⫹V 共 x 兲 …兴

共A4兲

27␲ 2 S 41

共B2兲

For the bounce the potential only depends on r⫽ 兩 x 兩 but we will not use this now. The logarithm can be expanded with respect to the potential V(x). We write

共A3兲

⑀ ⫽2 共 1⫺ ␣ 兲 .

U ⬙ 共 0 兲 ⫽m 2 .

A N⫽

冕

d4p 共2␲兲

N

兿

4 j⫽1

冋冕

d 4q j 共2␲兲

册兿冋 N

˜V 共 q j 兲 4

l⫽1

1 共 p⫹Q l 兲 2 ⫹m 2

⫻共 2 ␲ 兲 4 ␦ 共 Q N 兲 .

册

共B7兲

The four-momentum delta function arises from taking the trace. We obtain a gradient expansion by expanding the denominators (p⫹Q l ) 2 ⫹m 2 with respect to the momenta Q l . The leading term is of course A N,0⫽

冕

d4p

冋册兿冋冕 1

N N

共 2 ␲ 兲 4 p 2 ⫹m 2

j⫽1

d 4q j 共 2␲ 兲4

˜V 共 q j 兲

册

⫻共 2 ␲ 兲4␦共 Q N 兲 .

共B1兲 ⫽

We introduce a potential V(x) via 025009-9

冕

d4p

冋册冕 1

共 2 ␲ 兲 4 p 2 ⫹m 2

N

d 4 x 关 V 共 x 兲兴 N .

共B8兲



The zero-gradient contribution to the effective action is obtained by resuming this series; one finds eff ⫽ S grad,0

1 2

冕冕

再

d4p

p 2 ⫹U ⬙ 共 ␾ 兲

冎

1 ln ⬅ 2 p 2 ⫹U ⬙ 共 0 兲共 2␲ 兲4

4

d x

冕

4

d xK

(4)

冉冊

⫽ ⌫

冉冊

冕

dpp D⫺1 共 2␲ 兲D

2

D 共 4 ␲ 兲 D/2 2

⫺

2 D

冕

冋

ln

p 2 ⫹U ⬙ 共 ␾ 兲 p 2 ⫹U ⬙ 共 0 兲

共B9兲

1 p ⫹U ⬙ 共 ␾ 兲 2

⫺

册冏

p ⫹U ⬙ 共 0 兲

冉冊

0

册冎

.

N

⌸ N⬅

x

⫽

D⫹1

x 2 ⫹1

⫽⫺

⫺ 关 U ⬙ 共 0 兲兴

⌫ 共 ⫺D/2兲共 4 ␲ 兲 D/2

⫹

其

再

1 2 3 ⫺ ␥ E⫹ 2 ⑀ 2

冉冊

冎

to obtain K (4⫺ ⑀ ) ⫽

冋

⫺1 2 3 ⫺ ␥ E ⫹ln 4 ␲ ⫹ 2 ⑀ 2 32␲

⌸ N,2⫽

冋

⯝ „m 2 ⫹V 共 r 兲 …2

册冋

⑀ m ⫹V 共 r 兲 ⑀ m ⫻ 1⫺ ln ⫺m 4 1⫺ ln 2 2 2 2 ␮ ␮ 2

冕

⬁

0

冋

冋

R 3 dR „1⫹V 共 R 兲 …2 ln

␮2

冎

.

1⫹V 共 R 兲 ˜2 ␮

册

共 p⫹Q l 兲 2 ⫹m 2

册

1

1

1

⫺

共B11兲

共p

2

兺

⫹m 2 兲 N⫹1 j⫽1

2 p•Q j

Q 2j

N⫺1

N

兺兺

共 p 2 ⫹m 2 兲 N⫹2

j⫽1 k⫽ j⫹1

4 共 p•Q j 兲共 p•Q k 兲

N

1 共p

兺

j⫽1

N

1

2

N

共 p 2 ⫹m 2 兲 N⫹1

1

兺

⫹m 2 兲 N⫹2 j⫽1

1 共 p ⫹m 兲 2

2 N⫹2

⫹4 p ␮ p ␯

册再

⫹m ln

m2

4 共 p•Q j 兲 2 ⫹O 共 Q 3 兲 .

共B12兲

Under O(4)-symmetric integration 4p ␮ p ␯ ⯝ p 2 ␦ ␮ ␯ , and p ␮ ⯝0. So the one-gradient term vanishes and the complete two-gradient contribution becomes

兵关 U ⬙ 共 ␾ 兲兴 D/2⫺ 关 U ⬙ 共 0 兲兴 D/2其 .

⑀ D 1 ⫽ ⌫ ⌫ ⫺ 2 共 ⫺2⫹ ⑀ /2兲共 ⫺1⫹ ⑀ /2兲 2 ⫽

1 32

共 p 2 ⫹m 2 兲 N

⫹

D/2

Now we set D⫽4⫺ ⑀ and use

冉冊

兿 l⫽1

⫺

冉冊

⫻ 兵关 U ⬙ 共 ␾ 兲兴

␮2

4

Integrating over 4D Euclidean space we finally obtain

⌫ 共 D/2⫹1 兲 ⌫ 共 ⫺D/2兲 ⫺2 ⫽ D 2⌫ 共 1 兲 D⌫ 共 4 ␲ 兲 D/2 2 D/2

m 2 ⫹V 共 r 兲

共B10兲

The first term in the parenthesis vanishes for 0⬍D⬍2 and is defined to vanish in general by analytic continuation. The second term can be rewritten as dx

2

˜ ⫽ ␮ /m. with ␮ Let us now consider the one- and two-gradient contributions. We expand the denominators up to second order in the gradients, i.e., in the momenta Q j . We obtain

p⫽0

⬁

3 关 2m 2 V 共 r 兲 ⫹V 2 共 r 兲兴 2

⫺„m ⫹V 共 r 兲 … ln

⬁

1

冕

再

3 ˜2 , ⫺ „2V 共 R 兲 ⫹V 2 共 R 兲 …⫹ln ␮ 2

2

⫺2 兵关 U ⬙ 共 ␾ 兲兴 D/2⫺ 关 U ⬙ 共 0 兲兴 D/2其 D D⌫ 共 4 ␲ 兲 D/2 2

32␲

2

2

册

1 D p 2 ⫹U ⬙ 共 ␾ 兲 p ln 2 D p ⫹U ⬙ 共 0 兲

冋

⫺1

ef f ⫽ S grad,0

再冋

d pp D⫹1

K (4) ⫽

.

Of course this integral has to be regularized, e.g., via dimensional regularization. The divergences arise from the terms with N⫽1 and N⫽2, which are standard divergent one-loop integrals. We find 2 ␲ D/2 K (D) ⫽ D ⌫ 2

Using MS subtraction we get

2

册冎

.

⫺ 共 p 2 ⫹m 2 兲

兺j Q 2j

兺 Q j ␮ Q k ␯ ⫹4 p ␮ p ␯ 兺j Q j ␮ Q j ␯

k⬎ j

1 共 p ⫹m 兲 2

冋

2 N⫹2

冋

p2

册

册

兺 Q j •Q k ⫺m 2 兺j Q 2j .

k⬎ j

共B13兲

We now have to rewrite this in terms of the momenta q j that represent the gradients on the functions V(q j ). After having used the fact that ⌸ 2 appears under the integral over d 4 p we will now use the fact that it appears under the product of 025009-10



integrals 兰 d 4 q j V(q j ) which implies permutation symmetry in the indices j. So if we expand the products Q j •Q k and Q 2j we will encounter just two kinds of terms: products q l •q m with l⫽m and squares q 2l , which may be replaced by q 1 •q 2 and by q 21 , respectively. We have to do some combinatorics in order to find

兺j Q 2j ⯝ 兺

k⬎ j

N 共 N⫹1 兲 2 共 N⫺1 兲 N 共 N⫹1 兲 q 1 •q 2 ⫹ q1 3 2 共B14兲

A N,2⫽⫺

共 N⫺1 兲 N 共 N⫹1 兲 2 q1 . 6

Now we may use momentum conservation to rewrite 共B16兲

so that

eff ⫽ S grad,2

兺j

共 N⫺1 兲 N 共 N⫹1 兲 q 1 •q 2 6

兺

冋册 m

N⫺2

2

N 12m 2

„ⵜV 共 x 兲 …2 . 共B23兲

冕 ␲

1 32

d 4x

2

1 „ⵜV 共 x 兲 …2 , 共B24兲 12 m ⫹V 共 x 兲 1

2

1 192

冕

⬁

R 3 dR

0

1 „V ⬘ 共 R 兲 …2 . 1⫹V 共 R 兲

共B25兲

An alternative derivation starts with a technical step that frees us from the denominator 1/N. We take the derivative of the effective action with respect to m 2 , a step that we can revert later on. We then obtain, using the cyclic property of the trace,

共B17兲

dS eff dm 2 ⬁

⫽ k⬎ j

d x

V共 x 兲

or finally in dimensionless variables

G⬅ Q 2j ⯝⫺

16

eff ⫽ S grad,2

共B15兲

q 21 ⫽⫺q 1 • 共 q 2 ⫹•••⫹q N 兲 ⯝⫺ 共 N⫺1 兲 q 1 •q 2

2

4

The term A 1,2 is zero. The sum over all terms yields

共 N⫺1 兲 N 共 N⫹1 兲共 3N⫺2 兲 q 1 •q 2 Q j •Q k ⯝ 24

⫹

冕 ␲

1

共 N⫺2 兲共 N⫺1 兲 N 共 N⫹1 兲 q 1 •q 2 Q j •Q k ⯝⫺ 24 共B18兲

兺

N⫽0

1 ⫽ 2

共 ⫺1 兲 N tr兵关共 ⫺⌬ 4 ⫹m 2 兲 ⫺1 V 共 x 兲兴 N 共 ⫺⌬ 4 ⫹m 2 兲 ⫺1 其 2 ⬁

兺

N⫽0

共B26兲

BN .

and ⌸ N,2⯝

1 共 p 2 ⫹m 2 兲 N⫹2

We note that we have included the N⫽0 term, which can be removed later on if necessary. So we have arrived at the trace of the exact Green function in the external field. The terms B N have the form

共 N⫺1 兲 N 共 N⫹1 兲 q 1 •q 2 24

⫻ 关 ⫺ 共 N⫺2 兲 p 2 ⫹4m 2 兴 .

共B19兲 B N ⫽ 共 ⫺1 兲

The momentum integrals are

冕冕

d4p

p2

共 2 ␲ 兲共 p ⫹m 兲 4

2

2 N⫹2

4

d p

m

2

⫽

1 16␲

2

m 2⫺2N

2 共 N⫺1 兲 N 共 N⫹1 兲共B20兲

1 ⫽ m 2⫺2N 4 2 2 N⫹2 2 N N⫹1 兲共 16␲ 共 2 ␲ 兲共 p ⫹m 兲

N

冕

˜ 共q1兲 ⫻V

共B21兲

兿共 2 ␲ 兲 4 j⫽1

冋冕册 d 4q j

共2␲兲

1 共 p⫹Q 1 兲 ⫹m

˜ 共qN兲 ⫻V

1

N

d4p

2

2

˜V 共 q 2 兲

1 共 p⫹Q N 兲 2 ⫹m 2

4

1 p ⫹m 2 2

1 共 p⫹Q 2 兲 2 ⫹m 2

共 2 ␲ 兲4␦共 Q N 兲.

˜V 共 q 3 兲 . . . 共B27兲

Assume we have expanded the fraction 1/关 (p⫹Q k ) 2 ⫹m 2 兴 to first order in 2p•Q k ⫹Q 2k , yielding a factor

and, therefore,

冕

1 4

d p 共 2␲ 兲4

⌸ N,2⫽q 1 •q 2

1 16␲ 2

m

2⫺2N

N . 12

p ⫹m 2

共B22兲

The momenta are converted into gradients; so we finally obtain as the expansion terms of the two-gradient part of the effective action

2

关 ⫺2 p•Q k ⫺Q 2k 兴

1 p ⫹m 2 2

共B28兲

at the k th place in the product of propagators and vertices, in other words we have obtained an insertion of ⫺2 p•Q k ⫺Q 2k . Consider the part of the product to the right of this insertion. We rewrite it as

025009-11


J. BAACKE AND G. LAVRELASHVILI N

冋冕册兿冋冕 d 4q j

兿 j⫽k⫹1

共 2␲ 兲4

N

⫻

d 4x jV共 x j 兲

j⫽k⫹1

e ⫺iq j •x j p 2 ⫹m 2

Obviously part ⫺i2p• ⳵ vanishes upon symmetric integration over p. It can also be written as a boundary term for the x integration. If we want to obtain the second order gradient term we have to take into account the Q 2k term of the first order expansion, i.e.

1

关 ⫺2 p•Q k ⫺Q 2k 兴

p 2 ⫹m 2

册

⫻ 共 2 ␲ 兲 4 ␦ 共 Q k ⫹q k⫹1 ⫹•••⫹q N 兲 .

冕

共B29兲

d 4x

We furthermore rewrite the delta function as

冕

d 4 xe i(Q k ⫹q k⫹1 ⫹•••⫹q N )•x .

冕

共B30兲

Inserting this in Eq. 共B29兲 we can carry out the integrations over the q j and the x j to obtain

冕

d 4 xe iQ k •x 关 ⫺2p•Q k ⫺Q 2k 兴

冋

N

1

兿

p 2 ⫹m 2 j⫽k⫹1

V共 x 兲

1 p 2 ⫹m 2

册

Now the Q k, ␮ in 2 p•Q k ⫹Q 2k can be written as ⫺i ⳵ / ⳵ x ␮ ⫽ ⫺i ⳵ ␮ on the exponential. Integrating by parts they can be written as i ⳵ ␮ acting on the product to their right. So the whole string to the right of the insertion can be written as

冕

N

1

兿

p 2 ⫹m 2 j⫽k⫹1

冋

V共 x 兲

1 p 2 ⫹m 2

册

.

We now consider the sum over N; we split N⫽k⫹l and (⫺1) N ⫽(⫺1) k (⫺1) l . The sum over l is independent of k and runs from 0 to ⬁ and, putting in the factor (⫺1) l we obtain d 4 xe iQ k •x 关 ⫺2ip• ⳵ ⫹ ⳵ 2 兴 ⬁

l

⫻

兺兿 l⫽0 j⫽1

⫽

冕

冋

⫺V 共 x 兲

p ⫹m 2

d 4x

1 p ⫹m ⫹V 共 x 兲 2

2

p ⫹m ⫹V 共 x 兲 2

共 ⫺2ip• ⳵ 兲


1 p ⫹m ⫹V 共 x 兲 2

1 p ⫹m 2

2

关 ⫺2 p•Q k ⫺Q 2k 兴

2

共B35兲

,

1 p ⫹m ⫹V 共 x 兲 2

2

共B36兲

.

1 p ⫹m 2

2

关 ⫺2 p•Q k ⫺Q 2k 兴

1

, p ⫹m 2 共B37兲 2

a term that is needed for obtaining the complete propagator 1/„p 2 ⫹m 2 ⫹V(x)… between the two insertions. We now have the two-gradient term G (2) ⫽

1 2

冕

⫹

d4p 共2␲兲

4

冕

d 4x

1 p ⫹m ⫹V 共 x 兲 2

2

⫻ 共 ⫺2ip• ⳵ 兲

册

d 4 xe iQ k •x 关 ⫺2ip• ⳵ ⫹ ⳵ 2 兴

1 2

Here is included the term arising from expanding one propagator to second order. Indeed this yields

p 2 ⫹m 2

再

1 p ⫹m ⫹V 共 x 兲 2

2

共 ⫺2ip• ⳵ 兲

1 p ⫹m ⫹V 共 x 兲 2

2

⳵2

1 p ⫹m ⫹V 共 x 兲 2

2

1 p ⫹m ⫹V 共 x 兲

冎

2

2

.

共B38兲

The first term can be written, after one integration by parts, as 1 p ⫹m ⫹V 共 x 兲 2

2

.

共B33兲

Note that the sum starts with l⫽0, which corresponds to the case k⫽N; in this case the product over j reduces to 1. Now we do the analogous operations on the part to the left of the insertion, using in the exponent Q k ⫽q 1 ⫹•••⫹q k ; we now can carry out the summation over k and we finally find for the case that we have taken into account the first order expansion of one of the denominators (p⫹Q k ) 2 ⫹m 2

冕

1 2

1

1 2

d 4x

⫻ 共 ⫺2ip• ⳵ 兲

共B32兲

冕

⳵2

.

共B31兲

d 4 xe iQ k •x 关 ⫺2ip• ⳵ ⫹ ⳵ 2 兴


the terms ⫺2ip• ⳵ arising if two denominators are expanded to first order, yielding

共 2 ␲ 兲 4 ␦ 共 Q k ⫹q k⫹1 ⫹•••⫹q N 兲

⫽

1 2

关 ⫺2ip• ⳵ ⫹ ⳵ 2 兴

1 2

. p ⫹m ⫹V 共 x 兲共B34兲 2

d4p 共2␲兲

4

冕

d 4x

⫺1 „p ⫹m 2 ⫹V 共 x 兲 …4 2

关 ⳵ V 共 x 兲兴 2 . 共B39兲

In the second term we remark that the derivatives in the first insertion act on the complete part to the right of it. Therefore an integration by parts lets it act onto the part to the left of it. Using symmetric integration over p the second part yields 1 2

1

2

冕

冕

d4p 共 2␲ 兲4

冕

d 4x

p2 „p 2 ⫹m 2 ⫹V 共 x 兲 …5

关 ⳵ V 共 x 兲兴 2 . 共B40兲

Now we integrate with respect to m 2 to obtain the twogradient contribution to the one-loop effective action

025009-12



冕

冕

冋

which coincides with the previous result Eq. 共B24兲.

The terms of the gradient expansion can be evaluated in a straightforward way. We note, however, that the term m 2 ⫹V(x) vanishes, depending on the value of ␣ , at one or two points, and that therefore the expressions are ill-defined a priori. This is a reflection of the fact that the effective action has an imaginary part, due to the negative mode. An expansion of the effective action has to reflect this feature. With an m 2 ⫺i ⑀ prescription this becomes apparent. When computing these terms we have used the principal value prescription for eff and taken the absolute value in the logarithm appearS grad,2 ef f ing in S grad,0 .

关1兴 J.S. Langer, Ann. Phys. 共N.Y.兲 41, 108 共1967兲. 关2兴 J.S. Langer, Ann. Phys. 共N.Y.兲 54, 258 共1969兲. 关3兴 M.B. Voloshin, I.Y. Kobzarev, and L.B. Okun, Sov. J. Nucl. Phys. 20, 644 共1975兲. 关4兴 S. Coleman, Phys. Rev. D 15, 2929 共1977兲. 关5兴 C.G. Callan and S.R. Coleman, Phys. Rev. D 16, 1762 共1977兲. 关6兴 S. Coleman, in The Whys of Subnuclear Physics, edited by A. Zichichi 共Plenum, New York, 1979兲. 关7兴 V.G. Kiselev and K.G. Selivanov, JETP Lett. 39, 85 共1984兲. 关8兴 V.G. Kiselev and K.G. Selivanov, Sov. J. Nucl. Phys. 43, 153 共1986兲. 关9兴 K.G. Selivanov, Sov. Phys. JETP 67, 1548 共1988兲. 关10兴 J. Baacke and V.G. Kiselev, Phys. Rev. D 48, 5648 共1993兲. 关11兴 L. Carson, X. Li, L.D. McLerran, and R.T. Wang, Phys. Rev. D 42, 2127 共1990兲. 关12兴 J. Baacke and S. Junker, Phys. Rev. D 49, 2055 共1994兲; 50, 4227 共1994兲. 关13兴 J. Baacke, Phys. Rev. D 52, 6760 共1995兲. 关14兴 J. Baacke, Acta Phys. Pol. B 22, 127 共1991兲 and references therein. 关15兴 J. Baacke, Z. Phys. C 53, 402 共1992兲. 关16兴 J. Baacke and H. Sprenger, Phys. Rev. D 60, 054017 共1999兲. 关17兴 J. Baacke and T. Daiber, Phys. Rev. D 51, 795 共1995兲. 关18兴 D. Diakonov, M.V. Polyakov, P. Sieber, J. Schaldach, and K. Goeke, Phys. Rev. D 53, 3366 共1996兲. 关19兴 N. Graham, R.L. Jaffe, V. Khemani, M. Quandt, M. Scandurra, and H. Weigel, Nucl. Phys. B645, 49 共2002兲. 关20兴 M. Bordag, Phys. Rev. D 67, 065001 共2003兲. 关21兴 M. Hellmund, J. Kripfganz, and M.G. Schmidt, Phys. Rev. D 50, 7650 共1994兲. 关22兴 D. Fliegner, M.G. Schmidt, and C. Schubert, Z. Phys. C 64, 111 共1994兲.

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eff S grad,2 ⫽

1 2

⫺

⫽

d4p 共 2␲ 兲4

d 4x

1 1 2 2 3 „p ⫹m ⫹V 共 x 兲 …3

册

1 p2 关 ⳵ V 共 x 兲兴 2 4 „p 2 ⫹m 2 ⫹V 共 x 兲 …3

冕 ␲

1 32

2

d 4x

1 关 ⳵ V 共 x 兲兴 2 , 12 m ⫹V 共 x 兲 1

2

共B41兲

025009-13