Download - Tunneling and the validity of the low-momentum expansion of the effective action

Nuclear Physics B256 (1985) 653-669 ~) North-Holland Publishing Company

T U N N E L I N G AND THE VALIDITY OF THE L O W - M O M E N T U M EXPANSION OF THE EFFECTIVE ACTION

Carl M. BENDER

Department of Physics, Washington University, St. Louis, Missouri 63130, USA

Fred COOPER

Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Barry FREEDMAN

Department of Physics, University of Illinois, Champagne-Urbana, Illinois 61801, USA

Richard W. HAYMAKER ~

Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA

Received 28 May 1984 (Revised 12 February 1985)

We investigate the validity of the low-momentum expansion of the effective action in a quantum H=~m~b +~mA +~A[~b - a ] +~e ~b A . mechanical version of scalar QED with hamiltonian l ' 2 ' 2 1 2 2 2 1 2 2 2

In a "Coleman-Weinberg" or Born-Oppenheimer approximation where the "photon" field A is treated exactly, we find that the validity of the low-momentum expansion of the effective action in determining instanton solutions is related to the validity of the WKB approximation for the "photon" propagator. We find that there is a regime in ~b, where the naive WKB approximation breaks down. A more careful WKB calculation including turning points leads to an improved local potential. We calculate in this model the improved local potential and determine the correction to the instanton action.

1. Introduction

Recently there has been renewed interest in calculating tunneling rates in quantum field theory using path integral techniques in order to explore phase transitions in the early universe. The inflationary scenario proposes that as the temperature of the universe dropped, the GUT Higgs field gets trapped in a false SU(5) vacuum, eventually tunneling through a barrier forming a bubble with the lower symmetry SU(3) xSU(2)xU(1). In the original work of Guth [1] the tunneling proceeds through a barrier in a Higgs classical potential and in the later work of Linde [2] and Albrecht and Steinhart [3] through a Coleman-Weinberg [4] effective potential.

The approximation of using the effective potential in a bounce calculation is based on the validity of the low-momentum expansion of effective action [6]. True bounces are solutions of 8S/8~b = 0 where S is the action correct to one loop. The

Work supported in part by the Department of Energy under contract DE-A50577ER05490.

653

654 C.M. Bender et aL / Tunneling

actual equations obtained from 6S/605 give coupled equations for the 05-field and the gauge field propagator. In this paper we show in a quantum mechanical example that the validity of using the first two terms in the low-momentum expansion of the effective action is related to the validity of the "no turning point" WKB approxima-

tion to the "pho ton" propagator equation. We find that when 05 ~ 0, there are turning points which cause a breakdown in the low-momentum approximation. This " infrared" problem is most conspicuous in quantum mechanics where the trace of the "pho ton" propagator is singular at 1/]051 as 05 approaches zero.

We organize our paper as follows. In sect. 2 we review the path integral approach to tunneling in a Coleman-Weinberg potential at zero temperature. In sect. 3 we consider a quantum mechanics problem with the two fields 05, A with hamiltonian

H 1 "2 1 " 2 + ~ e2052A 2 =2m05 +~_mA 4! (052- a2)2+ ~ . (1.1)

In this problem we need to put in the double well structure by hand. We then choose the A-field loop correction to be large and study the validity of the low- momentum expansion. We discuss in sect. 3 under what conditions the low- momentum expansion is valid and when it breaks down. We find for small 05/a,

that the photon propagator in naive WKB is incorrect. An improved WKB method is presented which cures this problem. We determine the correction to the potential and the instanton action, and find it is a few per cent.

2. Review of the path integral formalism and Coleman-Weinberg potential

Recently there has been much interest in applying euclidean path integral techniques to calculate energy splittings [7-8] and lifetimes of unstable states [9-12] due to barrier penetration. One uses the persistence probability for remaining in a state 05 which is given by the path integral in euclidean space:

z=<05.1e "T/"105_>=zl<05.1,,>l~e L.,-:,

= f D05 e -s ' /* , J

where F

SE = J (3~,05)2+ V[05] dx. (2.1)

If one can calculate Z and let T ~ o o one can obtain Eo, and E~ (or zlE = E~ - Eo)

in the case of an almost degenerate vacuum or in the case of a "false" vacuum one can determine the lifetime i V from

F = 2 Im/5o. (2.2)

In the semiclassical approximation one tries to saturate the path integral with

C.M. Bender et al. / Tunneling 655

solutions that minimize the action, i.e. S which satisfy

6S - - - - 0 . (2.3) 80

These solutions with relevant boundary conditions are called "instantons" in the case of a degenerate vacuum and "bounces" in the case of a transition from an unstable state (decay of a "false vacuum"). In a dilute gas approximation one can show that the barrier penetration factor is

e -s°/h , (2.4)

where So is the action of a single instanton or bounce. Namely, in quantum mechanical tunneling

A E = h K e -s°/~ , (2.5)

and in determining the lifetime of a state

Im Eo = ½I'= hlKI e • ( 2 . 6 )

The factor K comes from the determinant of the gaussian fluctuations about the stationary solution.

If we have a single field S with classical potential Vd[S] then the stationary points are solutions of the classical equations of motion

6S 0 V¢,[S] 0, (2.7) 8--~ =E22S 0S

with appropriate boundary conditions. Suppose, however, that there is no possibility of tunneling at the classical level

but only at the quantum level as in the Coleman-Weinberg picture of scalar QED [4]. There

SE(S, A) = f [D~,S D~'S + Ve,[S]+~F,,~F "~] d x ,

(2.8) V, _ A o, - ~ (S0)2 .

It is only after including one-loop quantum corrections in S and A that we can have broken symmetry and one obtains for the effective potential [4]

A ( 5A 2 + 3e ' '~ ( S~ 2,~ Vcf~=4-.tS4+h\l1521r2 647r2] S~_ ln M- i -6 - ] "

If one wants the radiative corrections to be important we need A ~ he 4 so that the one-loop 0-effects proportional to hA2 are of higher order and can be neglected

656 C.M. Bender el al. / Tunneling

c o m p a r e d to he 4. If one also wants a m i n i m u m at ~b = a then one obtains

33 he4, (2.9) A = 8"n" 2

V~tt = 3e4 [ , 4,~ 2 ,'~ 64~r2 ~b~ k m - ~ - ~} . (2.10)

Because the one- loop ~b-correction can be ignored c o m p a r e d to the A- loop correction, this a p p r o x i m a t i o n is equivalent to t reat ing the A-field exactly and the ~b-field semiclassical ly. Thus if one wants to s tudy tunnel ing in a C o l e m a n - W e i n b e r g scenario in the one- loop app rox ima t ion one has

z = ( ~ - I e - Sl~14~ .) = f d~b dA~, e-S[~, AJ/h

= f d(h e - s ' ~ l . (2.11)

To saturate the above equa t ion with bounce solut ions one needs to solve (A~, = 0)

aV¢l 2 ,, 8S'"-[-12ck,- -he rk,D~, (xx; q~)=0. (2.12) 8a~i a~bi

D~'~=[g~,~[~-O~,O~(1-1)-e2ch2g~,~]8"(x-x')=O. (2.13)

Ins tead of solving these two coupled equat ions one finds in the l i terature instead the equa t ion [4]

D2~b ' _ aVe~ = 0 , (2.14) c~4~i

where V ~ = Vd+he2j ck,D~,,(xx; cb~)dc~,. This is equivalent to solving (2.13) for constant ~b and plugging it into (2.12).

The just if icat ion of this app rox ima t ion is the so-cal led l o w - m o m e n t u m expans ion o f the effective act ion [6]

s~,[~d = f v~,~[q,]+z[~d(a,,,/,¢)~+ . . . , (2.15)

where

0 D - t[4)¢, p2] Z-'[¢o] = 7p~ ; = o

where D -~ is the inverse ~b-field propagator in a constant background field ~ . There have been several attempts [13] to calculate Se. [4~] using this low-

m o m e n t u m expans ion in Monte Car lo s imulat ions but the expans ion is hard to control because one does not know a priori the size of the (a,,~b) 4 term. To s tudy

C.M. Bender et aL / Tunneling 657

the validity o f this expansion as it relates to the Co leman-Weinbe rg scenario it is

sufficient to s tudy the one- loop equations (2.12), (2.13) and see when they can be

replaced by eq. (2.14). To do this we restrict ourselves in this paper for simplicity

to a quan tum mechanics model. A numerical study of eqs. (2.12) and (2.13) will be

presented elsewhere.

We want to strongly point out that in the tunnel ing problem it is Se~[~b] with tunnel ing boundary condi t ion and not F[~b] the Legendre t ransform of In Z[j], the

true vacuum persistence functional, which we are considering, so that the Ved~b]

is not the usual effective potential. A properly defined effective potential is convex [6], it cannot have a barrier and thus cannot be used to describe tunneling! The

Co leman-Weinbe rg potential has a barrier but that is only because it is calculated in a low-order o f approximat ion [14-16]. In a loop expansion if V(~b) has a domain

with a negative second derivative, that domain will be complex in the next order.

The usual effective potential is defined in terms of the Legendre t ransform o f the

energy functional o f the g round state, E ( j ) w h e r e j is an external source. Nonconvex

domains o f 4' cor respond to choosing the wrong branch of E( j ) , i.e. not the one o f lowest energy [11].

The quest ion then arises: wherein lies the difference between a path integral

calculation o f the effective potential and a path integral calculation o f tunnel ing? The difference lies in boundary condit ions; the path integral in each case is domi-

nated by solutions with different boundary conditions. For example, consider fig.

1. There are constant solutions to the euclidean equat ion of mot ion ~b = a and ~b = b. It is shown in refs. [14-16] that both solutions must be included in the calculation

o f the effective potential and that this gives a convex result. However in a tunneling

problem there is no sum over two solutions; the relevant solution is the instanton

v(~)

Constant solution

a Instonton -~ b

Fig. 1. A potential with degenerate minima at q~ = a and Ob = b. Euclidean solutions going from a to b dominate the euclidean path integral for the tunneling process. Static solutions ~b = a, ¢b = b, dominate

the path integral for the effective potential.


for which 4) = a at r = -co and 4) = b at r = +oo(r = - i t ) . The potential that arises

is not the effective potential but rather we call it the static part of the effective action for fields having tunneling boundary conditions.

3. Quantum mechanical model

The model we will use to study the validity of the low-momentum expansion is

governed by the hamihonian

H = ½m(ck 2+ A 2 )+2~A(4 )2 - a2)2+½e24)2A 2. (1.1)

Near 4)= +a one has

n ~ naoprox = '~m(d~ 2 + A 2) + ~ ~a24) ~ + ) e 2 a 2 A ~ •

We see that when

e2~,~A ,

the oscillations of the photon field A are rapid compared to the oscillations of the

4)-field (tOA ~" t%) and one is led to consider a Born-Oppenheimer approximation in which one integrates out the fast degrees of freedom (the A-field) and is left with

an effective theory for the slow (4)) coordinate. We will see that when toAd-to6

4)-quantum-loop corrections to the effective action are much smaller than the A-loop

corrections. When e = 0, eq. (1.1) reduces to the standard double well anharmonic oscillator.

Ordinary perturbation theory in A about the minima at 4) = +a leads to an incorrect

calculation of the energy spectra since it ignores the tunneling between the wells, and just gives two separate degenerate harmonic oscillators. As is well known the

energy splitting between the symmetric and anti-symmetric combination of the "harmonic" oscillator solutions is due to tunneling and can be obtained from the

euclidean path integral formalism using a semiclassical approximation. The

euclidean version of Feynman's sum over histories is

z,~ - - <4)rle- '"/"l 4),) (3.1)

= z<4)A.><.I4),) e- ~,,T/. (3.2)

= N f dd~ d A e -s /~ (3.3) 3

where

½m d4) +~a A (4)2 _ (3.4)

r = - i t . (3.5)

If we can evaluate the path integral, then at large T we can pick off from Z~ r the


quant i ty E , - E o , the energy splitting. For e = 0 , Polyakov and Gi ldener and Patrascioiu [8] showed that one could app rox ima te the path integral by the contr ibu- t ion of ins tanton and ant i - instanton pairs and obta ined in that case

E , - Eo = AE = (hK) e -so/~ , (3.6)

where So is the act ion of a single instanton solut ion to

8S - 0 (e = 0 ) , a,/,

i.e.

, o,an ( ) ar , (3.7)

and K is related to the de te rminant resulting f rom the gaussian f luctuations about the ins tanton solution.

In this work we would like to do a similar calculat ion with e # 0 in the Born- O p p e n h e i m e r approx ima t ion , which is the approx ima t ion used by Co leman and Weinberg [4] in scalar QED. Of course in our case there is tunnel ing even before quan tum fluctuations. Integrat ing out the A-field we obtain

Z = N ' f [d~b] e - S " / h , (3.8) J

where

Se,= I~_ ½mqb2+ Vcl[d~]+½h Trln D -l , (3.9)

Vc, = ~Z (~b 2 - a2) 2 , (3.10)

D-l(r , r'; 4)) = m-~r2-e24~2 8 ( r - r ' ) . (3.11)

The effective potent ial is obta ined from

Ve,[4,] = Vc,[4,] + ½h Tr In D -l , (3.12)

where we solve for D -l assuming 4, is a constant in eq. (3.11). This gives

- e x p [ -1~4' (~- ~')l/,/m] 1 D(r , r ' ) - 2Umlee, I • D(rr; 4,)--- (3.13)

Therefore , the effective potent ial is

ve.[ 4,7-- ~ a ( ~ 2 - a2)2+ ~ - -~ - lee,{ • (3.14) 24 -1

2~le~l"

660 C.M. Bender et al. / Tunneling

We must impose a n u m b e r of restrictions on the pa ramete r s in eq. (1.1) in order to model the C o l e m a n - W e i n b e r g approx imat ion :

(i) I f e is too large, the double well s t ructure of V, eq. (3.14), d isappears . The condi t ion that there are two min ima is

2 A a ~ / m e < ec,, ec 'm 9x/3h (3.15)

If e,~ ecr, then the loop contr ibut ion is un impor tan t , hence we choose

e ~< eer. (3.16)

(ii) The A- loop contr ibut ion should domina te over the ~b-loop contr ibut ion. The steps leading to eq. (3.14) can be repeated to include the ~b-loop giving*

A ~ / 2 A ) '/2 = - - - ~b - ~ u ) . ( 3 . 1 7 ) V,~[d~] 4!(~b 2 a 2 ) 2 + le~,l+_~htm (2 ,_2,,/2

w e require that the second term domina te the third term at ~b = a, giving

e 2 ~ , ~ A . (3.18)

This is an equivalent to the validity condi t ion for a B o r n - O p p e n h e i m e r app rox ima- tion, i.e. the f requency for small oscil lat ions of the fast coord ina te ~oA, be much larger than for the co r respond ing slow coordinate , w~:

e2a 2 Aa 2 2 2 ¢ o A = - - •" = w ~ . (3.19)

m 3m

(iii) Finally there should be many osci l lator states in the well since we are taking the lowest one to be essentially at the bo t tom

A h~o~, ,~ - - a 4 . (3.20)

4 t

We can e l iminate m in favor of ecr in the three condi t ions giving in s u m m a r y

(i) e2<~ e2~ (3.21a)

( i i ) e 2 ' ~ A , (3.21b)

(iii) 2 ( 16)2h2 ec,~" , (3.21c)

2 , ~ a 3 , / m

e c ~ 9~/3h (3.15)

There is no inconsis tency in satisfying these condit ions.

" The ,b-loop contribution to V is complex in the nonconvex domain of a~ as advertised in sect. 2.


The minima of the effective potential are solutions of

,~, = . a' 2 h e OVefo~b = 0 = ~ A [ q ~ 3 - a 4 , ] + ~ m m * = = . ' ( 3 . 2 2 )

We have approximately

a ' = ± a ( 1 3he 2Aa3-~m / . (3.23)

Now that we have determined the boundary conditions on the instanton, i.e.

&(t)[, ~o~ = ± a ' , (3.24)

we are in a position to solve the Born-Oppenheimer approximation to the instanton equation, and can check the validity of the low-momentum expansion. From (3.8)- (3.11) we have that the instanton in the Born-Oppenheimer approximation is the

solution to

6Sen = 0 , (3.25)

64,

with boundary condition (3.24). We thus obtain

- m ~ + ~Acb(~b 2 - a 2) - h e 2 c k D ( r r ; ok) = O, (3.26a)

m-~r2-e2d~2 D(r, r ' ; 40 = 6(% r ' ) . (3.26b)

To obtain the low-momentum approximation, one takes 4, to be a constant in eq. (3.26b) as described above, eq. (3.13). Allowing 4~ now to be a dynamical variable we get in the low-momentum approximation the equation of motion

6ge~ = ~ _ _ _ g~, = (½42+ Ve,(~)) dr . (3.27) 0 = 6~ a,/, '

In order to get a handle on the validity of the low-momentum expansion of the action it is useful to look at the WKB expansion. Of course the semiclassical tunneling approximation is itself a WKB approximation to the full quantum mechanical problem. However we mean here the WKB approximation applied to a solution of the A-propagator equation (eq. (3.26b)). To apply WKB we introduce an expansion parameter e 2 multiplying the r-derivative:

e2mdr--7-e24)(r)2 D(r, r ' ; 4~) = 6 ( z - r ' ) , (3.28)

where e is assumed small but set to one in the end. This then is a "Schr6dinger equation" in the coordinate r except that we do not know the r-dependence of the

662 C.M. Bender et al. / Tunneling

"po ten t i a l " because we do not know a priori the funct ion ~b(r). Nevertheless there exists a W K B series solution. Let us write O(r)= D(r, r'; 49) where

~b( r) = ~7( r', e) e ( I / ' , l s ' ' s ' l T j , (3.29)

with b o u n d a r y condi t ions

g , ( r ' ) - ~ ( r ' ) = 0 , (3.30)

~b'(r'+) - 6'(r' ) = l / e2m, (3.31)

t / , ( r ) + 0 as Irl--,oo. (3.32)

(r/ is a normal iza t ion factor that can depend on e and r ' .) Keeping So and $1 we obtain for the two linearly independent solut ions

exp{+(1/e~/m) I f le4)(s) lds }

to. ( , ) = F . ( , ) -= le,/,(r)l (3.33)

Using the b o u n d a r y condi t ions at ~ and r ' we obtain

exp [ - ( 1 / e , / m ) I ; ,e~b(s). d s l ]

D(r, r': 49) = 2ele249(r)49(r,)[,/2 / ~ , (3.34)

which differs f rom eq. (3.27) unless 49(r) is a constant . Hence in the naive W K B a pp rox ima t ion we obtain, setting r = r ' , e = 1,

D(r, r; 49)= -[2e,/mrb(r)[- ' . (3.35)

This is what goes into the dynamica l equat ion (3.26a) for 49(r) and is identical to the l o w - m o m e n t u m app rox ima t ion to D(rr, 49). Hence for the specific p rob lem of finding the instanton, this WKB result is identical to the l o w - m o m e n t u m approx ima- tion even though we have not taken O to be cons tant in r. in the earlier derivat ion we as sumed it to be a constant and later p romo ted it to be a dynamica l variable. Therefore , whenever the naive WKB approximation to D(rr&) is valid, the low- momentum expansion to S ~ rb ] is also valid.

There are wel l -developed criteria for the validity of the WKB approx imat ion . There are two types of condit ions:

e l & . ,t "~ ls°l , (3.36a)

e°lS. , , I .~ 1. (3.36b)

Let us write D in the form of eq. (3.29), taking t o < r ' < r:

exp ( l / e , / ~ ) le491 ds exp - ( l / e x / ~ ) le491 ds

D(r, r'; 4 ' ) - " . . . . . __ , (3.37) 2 e l e , b ( r ' ) l ' / 2 , / m le,b(r)[ '/2


where ro is an arbitrary reference point. Hence

II ISol = edp(s) ds , (3.38) -r o

Is, I--IIn le ,(ol'/21. (3.99)

No matter what ~'o we pick it is clear that both of these inequalities are violated for n = 0 if z coincides with a zero of q~(r). Hence whenever ~ ( z ) is near zero one cannot justify the use of the naive WKB approximation (nor the low-momentum expansion). In the inflationary scenario one expects a breakdown in the low- momentum expansion at • = 0 since this is the place where the G U T gauge bosons have zero mass and long-range correlations can take place.

This failure of the WKB approximation is just due to the well-known fact that in potential theory WKB fails at the classical turning points of the potential. Our "potent ial" e2~2(r) has a turning point whenever ~ ( r ) = 0, in fact for this Vc~(~)

it is a quadratic turning point. There are well-developed methods of handling these special points [17]. What we can do is assume one or more turning points, solve the A-propagator equation with the appropriate WKB refinements and then check to see that the behavior is self-consistent when considering the coupled equations [eqs. (3.26a, b)]. The procedure for handling quadratic turning points is to isolate a neighborhood around each turning point and solve the harmonic oscillator equation exactly there. One then connects the solutions in various regions by the method of asymptotic matching across the boundaries.

Let us assume that 4 ( r ) has a zero at r = 0.

~b(r)-~flz for - A < r < A , (3.40)

where /3 will be determined later. This choice is suggested by the result in the classical approximation (A = 0), ~ d ( z ) = a tanh [az(A/12m) '/2] which has a linear zero at ~" = 0, • - a2r(A/12rn) '/2. A defines the region where the quadratic approximation to ~2( r ) is valid.

Consider first r ' > A, fig. 2a. In the four regions the solutions are respectively

I ~ b = A F . 0 " ) , (3.41a)

II 0 = BD_1/2(z)+ CD.,/2(-z), (3.41b)

III ~b = EF÷(7")+ GF_(r), (3.41c)

IV ~ = HF (z), (3.41d)

where F~ are given by eq. (3.33), and D_,/2(z) is the parabolic cylinder function:

( 2/3e ~ '/2 z =

To fix the constants we use eqs. (3.30)-(3.32) and asymptotic matching at z - - - A

664

(o)

Region

C.M. Bender et al. Tunneling

v,x-O (r, r'; @ )

,

i I i / i I

-A 0 A r '

(b) Region

i i I I rw I ~ I j u ~ , a t i

~ l l ~ - D ( r , r ' ; ~ ) I I I I ~ ~ r

-ZL 0 r' /L

Fig. 2. Regions needed to calculate D( r r ' ; ~) in the WKB approximation. (a) A < ~-', (b) [~"l < A.

and +A. This is done by noting that for e --, 0 the functional form of the solutions are identical on each side of the boundary and one merely matches coefficients. The asymptotic forms of D_,/2 for z ~ o o are

e - Z2/4

D ,/2(z)- z,/2 • (3.42a)

~/2 eZ2/4. D_,/2(-Z)=z,/'--- 5 (3.42b)

The matching between regions I and I! gives for example

A exp [e~r2/2e,/m] I (3.43) (e~r),/2

B(e4-m~'/" exp[-e~r2/2e4m] (e4m~'/',f2exp[e~r2/2e4-~] II \ - ~ ] ~ + C \ 2fie ] r,/2 , (3.44

which gives B = 0 and relates A to C. Determining the coefficients and comparing them with the previous WKB solution, eq. (3.34), one finds D(z, r ' ; @) different in regions I and I1 but identical in III and IV. But this means that for ~" away from the turning point D(r~-; qb) is again given by the same result eq. (3.35).

The interesting case is when r ' is near the turning point at 0 as in fig. 2b, r ' < A. Using the same general procedure we calculate D(r , r ; 4 ) and obtain

1 1 D.I/2(-z)D_I/2(z) D ( T T ; ( ~ ) - /~N/- ~ e 3 / 2 m 3/4 N/2 ' (3.45)


where [ 2/3e ~ '/2

z = , .

Using the asympto t ic expans ions of the pa rabo l ic cyl inder funct ions we find that the quant i ty in brackets in (3.45) is accurately app rox ima ted as follows:

- 1 . 0 4 + 0 . 1 7 1 z - 4 D - 1 / z ( z ) D - I / 2 ( - z ) = , z<~ 1.5

t - 1 / z , z >t 1.5. (3.46)

However (3.45) was der ived in the region where • =/3r, and if we replace r by 4 / /3 in (3.45) we get a un i form approx ima t ion to D( r~ '~ ) valid for all ~ ! Thus we set

z = 4 . (3.47)

As long as z > 1.5 we have f rom (3.46)

D(~: 4)= 12e,f-mme40")l"

We are now in a posi t ion to calculate the improved WKB potent ial " V ' ~ ] . The only unknown pa rame te r is/3, which is the s lope o f the instanton solut ion at • = 0, i.e.

/3 = ~l~=o. (3.48)

However for the ins tanton

Thus

½m6 2 = " V ' I 4 ] • (3.49)

/3 = . (3.50)

Once 13 is de te rmined we can de termine where the l o w - m o m e n t u m expans ion breaks down f rom (3.46) and (3.47). This turns out to be all 4 for which

4 * " V57- ) '

setting e = 1 we have

t ' 2e,/- 141'

D(~', r 4 ) = 0.74 - - + 0.242,~e 4~,) m~/4,,'~-~ ~/~'m~;" , ~ ~ ~ * .


The improved WKB potential is

"v"[4,]=~,~(4, -~ a2)2 ' 2 I - -2he ~bD[~] d~b + C,

where the constant is chosen consistency one must have

so that

(3.52)

so that at the minimum, ~ = a ' , "V"[a']=O. For

~ * < a ' ,

a"V'Ia'] avo.[a'] - - = 0 . (3.53)

a,/, a4, This condition insures that the instanton connects vacuua determined by Ve~[~b].

At this point it is useful to rescale the potential in terms of dimensionless parameters in order to study the different criteria to be satisfied in parameter space. We introduce a via

x / A a /3 = a V l - ~ m " (3.54)

So a is unity in the classical case. We define R, via

(3.55)

R, \ Xo<-/

Rt controls the validity of the Born-Oppenheimer approximation which requires

e 2 -~.~. (3.18) A

The other dimensionless variable is e/ecp where

h e~, , / ~ A a 3 - ~ . (3.15)

This ratio controls the size of the radiative correction. In terms of these parameters we obtain

@2 .] Aa 4{ ~ ~ [ O.523 -~ -O.O855 R~ ( ~)

+ C,} --- V,, ~ < ~b* (3.56) v~,[~]

( e ) ] L~/7--a - ~ +C2 -=I,'2, ,b>,~*. (3.57)

At ~b = ~b*< a ' , "V"[~b] and a"V"/a,/, are continuous. For ~b > ~*, "V"[~b] = Ve,[~b].


The cons tan t C2 is de t e rmined from the cond i t ion

"v '~4 , = a ' ] = 0 ,

where a ' is the largest so lu t ion of

(3.58)

One then de te rmines C, by cont inui ty o f

where

" v ' I 4 , ] at - - = - - , (3.60) a a

&* 1.5 a ~/2R1 (3.61)

F i n a l l y we have

~/2" v"~ = o] /3---

which leads to

ot = x/1 + 24C2(a, e2 /h )

o r

12e 2 ,~R~ = 1 + 24Cd R , ] . (3.62)

The value 4'* at which "V'~@] switches from Vc,[4,]+ V,[4,] to Veal&] occurs at

1.5a &* - . (3.63)

42R, TO s tudy the results for different e2/h we set e 2 = 3ec, to get a large radia t ive

correct ion. Fo r that value o f e/ecr, we find

a ~

- - = 0.8315,

C2 = - 0 . 0 3 9 5 3 .

The effective po ten t ia l a lso has relat ive m a x i m a at

4, - - = +0.27811. a

a, 3 _ a, + 324_~ e = 0 . (3.59) ecr


TABLE 1

Parameters for the improved WKB potential for e~ ec, =

e2/A R l a ~ * / a

io 4.055 0.3845 0.2615 10 5.6103 0.3480 0.1890

100 10.821 0.2958 0.0980

In table 1 we show how Rt, a and 4)* vary as a funct ion o f e2/A for e/ecr =2.

From table 1 we see that the regime where the effective potential breaks down varies slowly with e2/h and gets smaller with increasing e2/A. We also see that the

entire calculat ion is self-consistent since 4)*< a ' and thus the improved potential

agrees with the effective potential long before we are at the min imum o f Ve,[4)]. Thus the instanton always has the correct bounda ry condit ions.

In order to display the size o f the correct ion we have chosen e2/h = 10 which

clearly satisfies the criteria that the B o r n - O p p e n h e i m e r approximat ion is valid. For

that choice we plot both Ve,[4)]/Aa 4, and "V"wKB[4)]/AO 4 in fig. 3. We note for

4 ) / a < 4 ) * / a = O . 1 8 9 0 , ';V"wKB is quite different from Ve,[4)] especially in that it does not have a discontinuity in V'[4)] at 4 )=0 . In fact, V, , /Aa4[4)=0]=0.0021 w h e r e a s "V " / ,~a4 1 4 ) = 0] = 0 . 0 0 5 0 .

An impor tant quant i ty to determine is the change in the instanton action as a result o f the infrared problem.

We have

So a3~/Arn f"'/" 42I/[4)/ h h d_,,./,, A4~a 4 a] d ( 4 ) / a ) . (3.64)

0.008 I i ~ I - ~ I I I I l _

0.006 ~ / / ~

i -

0.000 I I 1 I I I I " - - J 0 .I .2 .3 .4 .5 .6 .7 .8 .9 1.0

4,/° Fig. 3. Plot of the various potentials V($)/Aa 4 versus ¢b/a for e/ecr = ~ and e2/A = 10. We notice that

"V"[ch] differs significantly from V,~$] for ~b/a < 0.1.


For "V'~] the integral So/(a3,CC~m) is 0.1537 whereas for V~[~b] the integral is 0.1506. Thus the action is only changed by.2% even though the change in shape of the potential for 4)< ~b* is dramatic. This is because the integral depends only on the square root of the potential.

4. Conclusions

We have shown that in the Born-Oppenheimer approximation, the validity of the low-momentum expansion of the effective action is related to the validity of the naive WKB expansion for the "photon" propagator. That is unless the "photon" mass is small the low-momentum expansion is valid. However for small ~b ~< 4)* the naive WKB breaks down and must be replaced by a uniform WKB expansion which takes into account turning points. The improved WKB potential "V'~qb] differed in this simple quantum mechanical model significantly from Ve~[~b] near ~b=0. However the effect on the instanton action which determines the tunneling rate was found to be only a few per cent in this example.

We would like to thank our colleagues for interesting discussions. We would also like to thank Sidney Coleman for some stimulating remarks on the probable size of the errors of using the naive WKB approximation. We would also like to thank Su-Wen Wang for helpful discussions, and the referee for making us think about the relevant parameter space. R. Haymaker, C. Bender and B. Freedman would like to thank Los Alamos National Laboratory for its hospitality during the summer of 1983. We also thank the Department of Energy for support.

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