Dokument_1.

NONVACUUM PSEUDOPARTICLES, QUANTUM TUNNELING
AND METASTABILITY
JIU-QING LIANG
arXiv:hep-th/9505185 31 May 95
Department of Physics, University of Kaiserslautern, P.O.Box 3049
67653 Kaiserslautern, Germany
E-mail: jqliang@gypsy.physik.uni-kl.de
and
H.J.W. MU LLER-KIRSTEN
Department of Physics, University of Kaiserslautern, P.O.Box 3049
67653 Kaiserslautern, Germany
E-mail: mueller1@gypsy.physik.uni-kl.de
ABSTRACT
It is shown that nonvacuum pseudoparticles can account for quantum tunneling
and metastability. In particular the saddle-point nature of the pseudoparticles
is demonstrated, and the evaluation of path-integrals in their neighbourhood.
Finally the relation between instantons and bounces is used to derive a result
conjectured by Bogomolny and Fateyev.
1. Introduction
One of the most interesting applications of the Euclidean path-integral approach
is the study of semi-classical instabilities or tunneling processes as Hawking and Ross
emphasised recently. Instanton transitions related to the possibility of baryon{ and
lepton{number violation in electroweak theory have attracted widespread attention
. It has gradually been realised that vacuum instantons and vacuum bounces which
require vacuum boundary conditions may not be appropriate for the description of
tunneling at nite, nonzero energy . The investigation of quantum tunneling with a
new type of instanton{like congurations which are characterised by nonzero energy
and satisfy manifestly nonvacuum boundary conditions is therefore of great interest. In the following we consider the new type of instanton{like and bounce{like
congurations called periodic instantons or periodic bounces or sphalerons ; ; ; and
investigate their stability for various potentials in (1+1) dimensions. We then calculate their tunneling eects in (1+0) dimensions for various solvable models since their
appropriate transitions reduce to quantum mechanical tunneling problems ; ; ; . It
is well-known that the latter also play an important role in the investigation of the
large order behaviour of perturbation expansions. We close therefore with a discussion of the Bogomolny{Fateyev relation which relates the level splitting of one case
to the energy discontinuity of a related case, the main idea behind this being the fact
that the instanton is exactly half of the bounce. The latter point has been exploited
recently in the discussion of duality in gravity theory .
1
2
3
3456
7 8 9 10
11
1
2. Solitons, bounces and sphalerons on S and their stability
1
We recall rst the behaviour of vacuum pseudoparticles in (1+0) dimensional
quantum mechanics. For the double{well potential given by
V () = 2a ( ; a )
(1)
the instanton solution is the well-known expression
c = a tanh[( + )]
(2)
where = it is the Euclidean time. The conguration c which corresponds to a
transition between the degenerate vacua has nonzero topological charge and is stable.
The eect of tunneling appears in the level splitting for the potential with two
degenerate minima and in the band structure for the sine{Gordon potential with
an innite number of degenerate minima. In the case of the inverted double{well
potential
(3)
V () = ; 2a ( ; a ) + 2a a
the corresponding classical conguration is the bounce
p
p
c = a 2[cosh( 2 )];
(4)
This conguration has zero topological charge and is unstable. The tunneling eect
here is the decay of the (sometimes called \false") vacuum state (metastability) .
Calculating the imaginary part of the energy one obtains
p
p
4
2
4
=mE = g [ ] exp[; 3g2 ]
(5)
where g = a22 . One should note that the vacuum instanton is an odd function of
its argument whereas the bounce is even, so that their derivatives (which classically
represent velocities) are respectively even and odd. Since these derivatives are also
the translational zero modes, i.e. wave functions with eigenvalue zero, the instanton
is classically stable, wheras the bounce is not.
We now consider a eld (x; t) in (1+1) dimensions and static conguration (x)
with Lagrangian density
L = 21 @@ ; U ()
(6)
where U is respectively the double{well potential, the inverted double{well potential
or the sine{Gordon potential, i.e.
U [] = 2a ( ; a )
U [] = ; 2a ( ; a ) + 2a a
U [] = 1 + cos (7)
The static solution with nite, nonzero energy is given by
1 ( d ) ; U [] = ; 1 c
(8)
2 dx
2
2
2
2 2
2
0
2
2
2
2 2
2
4
2
1
7
3
3
0
2
2
2
1
2
2 2
2
2
2
2
2
2 2
2
2
3
2
2
4
where with i = 1; 2; 3
; U [0] ; 21 c ;U [a] = 0
;U [0] ; 12 c ;U [0] = 0
;U [0] ; 12 c ;U [] = 0
1
2
2
3
3
In these cases
2
1
1
2
2
2
(9)
3
= 0; = a; = 0; 2; :::
(10)
are trivial (constant) solutions and
= a; = 0; = (11)
are the corresponding vacuum solutions. The modulus k of the elliptic functions with
0 k 1 in the three cases is respectively given by
k)
c = a ( 11 ;
+k
;k )
c = a ( 11 +
k
c = 4(1 ; k )
(12)
Here and in the following k0 is the complementary elliptic modulus dened by k0 =
1 ; k . The expressions ci can be regarded as energies of the appropriate pseudoparticles. We now take xS , and we demand periodicity of the solutions, i.e.
(x) = (x + L). The nontrivial solutions are in the three cases respectively
(x) = akb(k) sn[b(k)x; k]
(x) = s (k)dn[ (k)x; ]
(x) = 2 arcsin[ksn(x); k]
(13)
where
b(k) = ( 1 +2 k ) 12 ; (k) = a s (k);
(14)
and
s (k) = a p1 + k ; = (1 +4kk)
(15)
1+k
In the limit k ! 1, i.e. c ! 0, we regain the vacuum solutions, i.e.
p(x) = a tanh(
p x;)
(x) = a 2[cosh( 2x)]
(x) = 2 arcsin[tanh x]
(16)
2
1
2
2
2
2
2
2
2
2
2
2
2
2
2
3
2
2
1 2
2
2
1
5
+
+
2
+
2
2
2
2
2
1
whereas in the limit k ! 0 the periodic solutions become the trivial solutions given
above. The periodicity requirement implies certain critical values of L, i.e. respectively
L = 4nK (k); L = 2nK (k); L = 4nK (k)
(17)
where K (k) is the complete elliptic integral of the rst kind dened by
Z K (k) = 2 q d
(18)
1 ; k sin and n = 1; 2; 3; ::: (i.e. note that n = 0 is excluded). Setting
2
0
2
2
(x; t) = c(x) +
X
i! t
m (x)e m
m
(19)
we obtain the stability or small uctuation equation
d + U 00[ (x)]) (x) = !
(; dx
(20)
c
m
m m(x)
with
(21)
m (x) = m (x + L)
For socalled \classical stability" we must have !m 0. For the vacuum solutions
m / sin or cos( L mx); m = 0; 1; 2; :::; these conditions are respectively in the three
cases
!m = 4 + 4L m > 0
!m = 2 + 4L m > 0
(22)
!m = 1 + 4L m > 0
It is seen that these conditions are satised. In the case of the trivial solutions
2
(23)
m / sin or cos( mx)
L
at the critical values of L the stability conditions are respectively
!m = ;2 + 4L m = 2 ( m
n ; 1)
!m = ;4 + 4L m = 4 ( m
n ; 1)
(24)
!m = ;1 + 4L m = ;1 + m
n
where m = 0; 1; 2; ::: and n = 1; 2; 3; :::. We see that for n > m: ! < 0, i.e. in that
case c is unstable (a sphaleron).
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
In the case of the nontrivial solutions the uctuation equation turns out to be in
each case a Lame equation , i.e.
d + [ ; N (N + 1) sn (z; )] = 0
(25)
dz
The discrete eigenvalues and eigenfunctions in each of the cases of the potentials Ui,
with i = 1; 2; 3; are:
U :Here N = 2, z = b(k)x, = k and = !b22 k2 , and the solutions are
5
2
2
2
2
2
1
+2
( )
2
= sn(z; k)cn(z; k)
= sn(z; k)dn(z; k)
= cn(z; k)dn(z; k)
q
[1 + k 1 ; k (1 ; k )]
(26)
; = sn (z; k ) ;
3k
with respectively the following eigenvalues
! = (1 6+k )
! = (16+ kk )
! = 0
q
2 1 ; k (1 ; k )
! ; = 2 (1 )
(27)
1+k
where the second last eigenvalue is seen to be negative and the third is that of the
zero mode.
U :Here N = 2 with z = (k)x and = = kk 2 and = 6 + !2;2 k2 . In this
case the solutions have respectively the same form as in the rst case but with elliptic
modulus instead of k and the corresponding eigenvalues are
1
2
3
2
2
2
45
2
2
2
2
1
2
2
2
2
2
2
2
3
2
45
2
2
2
2
2
2
! = 0
! = ; 3 1(1+;k k)
! = ; 31(1++k k)
2
( )
4
(1+ )
2
2
1
2
2
2
2
2
2
2
3
2
2
p
k +k
! ; = ;2 2 1 +1 14
+k
where ! ; ! ; ! are seen to be negative.
U :Here N = 1; = k; z = x; = ! + 1, and the eigenfunctions are
= cn(x; k); = dn(x; k); = sn(x; k)
2
45
2
2
2
3
2
4
2
2
2
2
4
(28)
2
3
1
2
3
(29)
with respectively the following eigenvalues
! = 0; ! = k ; 1; ! = k
(30)
We see that in this case the second eigenvalue is negative. We also observe that
in each case the number of negative eigenvalues is odd and some eigenvalues merge
with others in the limit k ! 1. A negative eigenvalue implies,of course, that the
corresponding conguration is a saddle point. We mention nally that the supersymmetrised versions of the three models discussed here have also been investigated .
2
1
2
2
2
2
3
2
2
12
3. Nonvacuum instantons and tunneling
We now consider the calculation of the level-splitting for the double-well potential
by summing contributions originating from nonvacuum instantons and corresponding
nonvacuum instanton{ anti-instanton pairs . An analogous calculation can be performed for the sine-Gordon potential . Finally we consider the limiting cases of high
and low energies, high meaning here energies approaching the top of the tunneling
barrier.
We consider a scalar eld in (1+0)-dimensions with mass = 1 and Lagrangian
L = 21 ( d
(31)
dt ) ; V ()
with potential
V () = 2a ( ; a )
(32)
Integrating the classical equation we obtain with = it
1 ( dc ) ; V ( ) = ;E
(33)
c
c
2 d
Integrating we obtain
c = akb(k) sn[b(k)( + )]
(34)
8
10
2
2
2
2 2
2
2
0
where we have suppressed the elliptic modulus k. The solution c has been dubbed
\periodic instanton" , \sphaleron" and \bounce" . It is convenient to introduce
a new parameter u dened by
u
k = 11 ;
(35)
+u
3
4
13
2
p2
with u = aEc and b(k) = ( k2 ) 12 . The Jacobian elliptic function sn has period
T = 4nK (k) for n = 1; 2; 3; :::. Setting b(k)T = K (k) we can dene as the analogue
of the topological charge the quantity
2
1+
s
Q = 21a [c(T ) ; c (;T )] = k 1 +2 k
(36)
We consider the half period part of the solution from = ;T to = +T (and so
with = 0) as the trajectory of the nonvacuum instanton (as we prefer to call it).We
2
0
are interested in the transition amplitude A ;; for the transition from one side of
the central barrier of the double{well potential to the other. We let jE > be the
eigenstates of the same energy E in the two wells (with minima ) if the presence
of the other well is ignored. The nite height of the potential barrier in between splits
the degeneracy so that the eigenstates become the odd and even states
jE >o;e= p12 [jE > jE >;]
(37)
with eigenvalues Eo 4E . The desired amplitude then becomes
+
0
+
1
2
A ;; = < E j exp(;2HT )j E >; = ; exp(;2E T ) sinh(T 4E )
(38)
The problem is to calculate the shift 4E .We do this with the help of the path-integral
method. In this case the amplitude A ;; can be written
+
+
0
+
A
+
;;
=
Z
E +(f ) E ; (i )K (f ; f ; i ; i )df di
(39)
where f ; i = 2T . The kernel K is given by the Feynman path-integral
Z
K (f ; f ; i; i) < f ; f ; i; i >= D[]exp[;S ]
(40)
where the Euclidean action S is given by
Z f
S = [ 21 ( d
(41)
d ) + V ()]d
i
We write the amplitude as a sum over nonvacuum instanton contributions, i.e.
2
A
;;
+
=
1
X
n=0
A
n
;;
(2 +1)
+
(42)
where A ;n; denotes the contribution of the amplitude for one nonvacuum instanton
and n nonvacuum instanton-anti-instanton pairs.
We consider rst the one-nonvacuum{instanton contribution. We set
( ) = c( ) + X ( )
(43)
where X ( ) is the deviation of ( ) from the classical trajectory c with xed endpoints X (i) = X (f ) = 0. We also write S = Sc + S where
Sc = W ((f ); (i); E ) + 2Ec T
(44)
Here
1
2 [E (k ) ; uK (k )]
W ;! 4a
(45)
(1
+
u
)
3
in the limits (i) = i ! ;a~ and (f ) = f ! a~, where ;a~ and a~ are the two
middle turning points (where i = ;T; f = T ) (;a~0 and a~0 are the two other outside
(2 +1)
+
2
turning points) and E (k) is the complete elliptic integral of the second kind.In the
one-loop approximation we have
Z f
; )] Z f (X MX
^ )d
)
+
X
(3
(46)
S = d [ 21 ( dX
d
a c
i
i
where M^ is the small uctuation operator evaluated at the classical conguration,i.e.
2
2
2
2
2
2
M^ = ; 21 dd + ( 3ac ; 1)
2
2
2
2
(47)
2
The kernel K dened above is then given by
K exp[;Sc]:I
where
Z X f
I=
DfX gexp(;S )
(
(48)
)=0
(49)
X (i)=0
We now have to evaluate the integral I . The usual analysis starts as follows. One
expands the uctuation X in terms of the complete set of eigenfunctions
n of the
small uctuation operator M^ with M^ n = !n n . Then X = Pn Cn n and
Z
@X ) exp(; X C ! )
I = DfCn g det( @C
n n
n
n
@X ) Y[ ]
= det( @C
n n !n
@X ) = det( @C
^
n det M
(50)
Here one expects a problem with the negative eigenvalue of the small uctuation
operator obtained above. However, the boundary conditions X (i ) = X (f ) = 0
remove this negative eigenvalue ! . This can be seen as follows. The two boundary
conditions imply the equations
(51)
; C k0 + C (1 ; 4 3+k 4 ) + C (1 ; 4 3;k 4 ) = 0
and
C k0 + C (1 ; 4 3+k 4 ) + C (1 ; 4 3;k 4 ) = 0
(52)
2
2
2
2
2
4
2
1
4
2
2
1
4
2
5
2
q
1
5
2
2
2
1
2
2
where 4 = 1 + k and 4 = 1 ; k (1 ; k ). These equations imply C = 0 and
1
2
2
2
2
2
C (1 ; 4 3+k 4 ) = ;C (1 ; 4 3;k 4 )
4
1
2
2
5
1
2
2
(53)
From the denition of the coecients Cn we obtain
Z
Z
Z
C = X ?d = sn [b(k) ]Xd ; Xd 4 3+k 4
4
and
Z
C = X
5
R
1
2
4
2
2
?d
5
= sn [b(k) ]Xd ; Xd 4 3;k 4
Z
Z
2
1
2
2
(54)
(55)
Here if ;TT XR( )d = 0 if we require the uctuation to be orthogonal to the zero
mode ddc , i.e. ;TT ddc X ( )d = 0, so that X has to satisfy X ( ) = ;X (; ). The
above equations therefore imply that C = C . The previous equation therefore
implies that C = C = 0. Using the shift{method{transformation we can set
Z _ 0
X ( ) = Y ( ) + N ( ) NN (( 0)) Y ( 0)d 0
(56)
i
with
kb (k)a cn[b(k) ]dn[b(k) ]
c
N ( ) d
=
(57)
d
=
=
( )
4
4
5
14
5
2
2
and
1R
I = p1 [
] 12
(58)
d
f
2 N (i)N (f ) i N 2 This expression is singular at the turning point values of f and i, since the \velocities"expressed by the zero modes vanish at the turning points;this is dierent from the
case of vacuum instantons or vacuum bounces in which case the turning points can be
reached only asymptotically. Our procedure here is to use the end-point integrations
in the expression for the transition amplitude in order to smooth out the singularities
in I . One can show that in approaching the limits f;i ! T the following expression
holds formally
@ Sc = [N ( )N ( ) Z f d ];
(59)
f
i
@ ( )
N ( )
( )
8
2
2
In the integral
A
;;
+
=
Z
1
i
f
E + (f ) E ; (i ) exp [;Sc (f ; i ; f
2
; i)]I (f ; i; f ; i)df di
(60)
we replace the wave functionals by their respective WKB approximations in the barrier (from (;T ) = ;a~ to (T ) = a~) , i.e.we set
_ ]
C exp [; Raf d
C
exp
[
;
(
)]
f
q
q
(
)
=
E
f
N (f )
N (f )
R i _
C
exp
[
;
(
)]
C
exp
[
;
;
i
;
;a d]
q
q
E ; (i ) =
N (i )
N (i )
(61)
+
+
+
~
~
where the constants C ; C; can be calculated from integrals over the two domains
(;a~0; ;a~) and (~a; ~a0) neighbouring the barrier and are given by
+
2
C
+
31
6
= C; = 64 R a~
a~
0
7
7
5
1
2
p dE;V
2(
2
)
p
"
#1
u 2 C
= 2K1(+
k0)
(62)
We are interested in the limits
i ! (;T ) ;a~; f ! (T ) a~0
(63)
We therefore use Taylor expansion in f around a nearby point ( ) so that
exp [;Sc(f ; i; f ; i)] = exp [;Sc(( ); i; ; i)] @ Sc )
exp [; 21 ( @
(64)
f 0 (f ; ( )) ]
f
and
(f ) )
exp [;
(f )] = exp [; 21 ( @ @
(65)
f 0 (f ; ( )) ]
0
0
0
2
= (
2
0
)
2
2
f
= (
)
0
2
2
in the Gaussian approximation. We also have with ( ) = ;(; )
N_ (( )) +
1R
( @@Sc )f 0 = ; N
(( )) N (( )) ;00 Nd2 f
_ (( ))
( @@
)f 0 = + N
N (( ))
f
2
2
= (
2
= (
0
)
2
0
2
0
( )
0
)
0
(66)
(here the contribution is seen to be one degree less divergent than that of Sc) and
the relations
Sc((T ); (;T ); 2T ) = W ((T ); (;T ); Ec) + 2Ec T
1
2 [E (k ) ; uK (k )]
W = 4a
(1
+
u
)
3
(67)
2
i
and write N d
i = d .The integration with respect to f becomes Gaussian and can
be carried out rst. Then the limit ! T is taken. Finally integrating with respect
to from ;T to T we obtain for the amplitude in the one-loop approximation
A ;; = 2TC exp [;W ]exp [;2EcT ] S ;; exp [;2EcT ]
(68)
(For comparison with the S-matrix calculation to be mentioned below we note here
that the factor exp[;2EcT ] represents the free eld evolution part (here in Euclidean
time) so that in the limit of vacuum boundary conditions the remaining part S ;; can
(
( ))
0
+
2
+
+
be looked at as the (here rather unconventional) S-matrix element in the one vacuum
instanton approximation between the low-lying nth excited states in the two wells,
i.e.S :; 2TC exp[;W ] for k ! 1).
Proceeding similarly in the case of amplitude contributions stemming from one
nonvacuum instanton and respectively one or n nonvacuum instanton pairs we obtain
2
+
A
2
(3)
+
;;
ZT
Z 1
d
d
dC exp [;3W ]exp [;2EcT ]
;T
;T
;T
= (23!T ) C exp [;3W ]exp [;2EcT ]
(2T ) n C n exp [;(2n + 1)W ] exp [;2E T ]
= (2
c
n + 1)!
=
1
3
A
n
;;
(2 +1)
+
Z 2
3
2
3
2 +1
2 +1
(69)
at the
Summing over n and comparing the expression with the expression for A ;;
beginning, we obtain the WKB level-splitting formula
p
4E = K1(k+0) u exp [;W ]
(70)
We dene as weak coupling those values of a which are such that g a2 << 1.
In this limit the two minima of the potential are widely separated and the central barrier becomes very high. In the following we shall consider high energies as
those associated with high quantum states. We therefore replace Ec by the oscillator approximation En = (n + )! where !q = 2 . In that approximation we have
R
V () ' 2 ( ; ) ; = a and aa 2(E ; V )d = (n + ). We consider
separately the cases of low and high
energies. q
p
Low energies. We have u = 2Ec =a = 2g n + ; k = ;uu = 1 ; k0 . The
appropriate expansions (for small u or k0 ) of the elliptic integrals are
E (k) = 1 + 12 k0 fln( k40 ) ; 21 g + :::
K (k) = ln( k40 ) + 41 k0 fln( k40 ) ; 1g + :::
(71)
+
2
1
2
2
2
2
~0
1
1
2
~
1
2
2
2
1
1+
2
2
2
With these expansions we obtain
W = 34g + 2(n + 12 ) ln( g4 ) + (n + 12 ) ln(n + 21 ) ; (n + 12 )
and hence
4
4En = 2 [ g (n2 +e ) ]n 12 e; 3g2
2
4
2
+
1
2
Using the Stirling relation
p
1
2
e
1
2
e
n
2
[ n + ] n!
(1 + n )n
1
2
+
1
2
p
1
+2
n2!
(72)
(73)
(74)
we see that
p
4
(75)
4En = 2n!p2 [ 2g ]n 12 e; 3g2
This expression agrees, as expected,with the WKB-equivalent result in the low
energy limit, i.e. for u = 0. We also observe that under the condition g (n + ) <<
1; k0 ! 0 the amplitude A as well as 4En grow with energy (due to the second term
in W above).
These low energy results agree with those of Bachas et al. who estimated the
S-matrix element Sn!n for the vacuum instanton transition from the nth asymptotic
oscillator state on one side of the barrier to the nth asymptotic oscillator state on the
other side using the LSZ procedure. Thus, dening by
:= a ; c(x) ! 0; x ! 1
(76)
4
+
2
19
2
1
2
15
we can construct eective boson creation and annihilation Heisenberg operators a^y; ^a
in the wells \+" and \;" given by
$
@ (x = it) ;!
1 4ap
^a = ; pi e it @t
(77)
These operators are such that in the asymptotic limits they correspond to the harmonic oscillator operators a^ in
(t) = p1 [^ae;i!t + a^yei!t]
(78)
2!
with ! = 2. The transition amplitude through the central barrier induced by a
vacuum instanton is then
< 1; outj1; in >; = < 0ja^ ^ay;j0 >
$
$
@
;
i
i
it
; it @ )G
e
)(
e
= t!;1lim
(
p
p
0
;t ! 1 @t @t
p
= (4a ) I
= S ;;
(79)
where G =< 0j (x0);(x)j0 > and I is the vacuum instanton tunneling propagator
High energies. High energies here means those approaching the top of the barrier,the latter generally being
called the sphaleron mass. In the present context this
implies k ! 0 and E ! a22 . In this limit
2
+
+
2
0
2
+
0
2
+
+
2
p
W g2 k ! 0
2
2
and
p
p
1 + u 2 ! 0
K (k0) ln( k )
4
(80)
(81)
Thus at these energies the amplitude and the splitting 4E are no longerp suppressed by
the typical vacuum instanton factor exp(; g2 ) but by the prefactor K k u ! 0.Thus
in the high energy limit
p
(82)
A K1(k+0)u exp [;W ] ! 0
Similar results can be expected for various other potentials, in particular for the
sine-Gordon potential which has been considered in the literature .
1+
( 0)
4
3
10
4. Nonvacuum bounces and tunneling
With methods similar to those described above one can consider an amplitude
in the neighbourhood of a bounce which
is the classical conguration in the case of
the inverted double{well potential ; ; ; and calculate the imaginary part of the
energy. In this case the small uctuation equation has three negative eigenmodes, of
which two do not contribute to the amplitude in view of boundary conditions and
the remaining one; is responsible for the imaginary part. For further details we refer
to the literature .
16 17 7 9
79
5. The Bogomolny-Fateyev relation and conclusions
In the case of systems with more than one classical ground state, the classical
vacuum state chosen as the perturbation theory vacuum in general does not coincide
with the true quantum mechanical ground state. Thus although the exact ground
state is stable, the corresponding perturbation theory vacuum is only metastable due
to the possibility of tunneling to the other vacuum states. For example, the doublewell potential
V () = 2 ( ; 1 )
(83)
leads to the level splitting calculated above. But the following distorted form of the
potential
V () = 21 ( ; 1 ) for 1
= ; 12 ( ; 1 ) for > 1
(84)
2
2
2
2
2
2
2
2
2
2
2
2
results in an imaginary part of the energy, =mE , for a \real" metastable ground
state. The shape of the potential is then similar to that of a cubic potential which is
the easiest example of a potential with a bounce . (The unphysical shape of V ()
for > is irrelevant here;equivalently one could assume a at behaviour or even a
rising one far away so that in any case the tunneling particle would behave as free over
some distance suciently far away from = ). Bogomolny and Fateyev observed
that (to leading order)
4E = 2i(E )
(85)
where 4E is the discontinuity of the ground state energy at the cut 0 with
4E = 2i=mE while E is the instanton contribution to the real part of the ground
18
1
1
11
2
2
state energy (i.e. for the double-well potential), namely the level shift due to quantum
tunneling. The Bogomolny-Fateyev relation has been veried and extended to excited
states by comparing the explicit expressions of the two quantities for both the doublewell potential and the periodic potential and their appropriately distorted versions in
the above sense . The formula serves as a crucial test of the validity of calculating
quantum tunneling eects with nonvacuum instantons and nonvacuum bounces.In
the considerations above the level splitting for the excited states of the double-well
potential was obtained with nonvacuum instantons . The classical solution which
extremises the Euclidean action is
c ( ) = kb(k) sn[b(k); k]
(86)
where
u ; u = p2E; b(k) = [ 2 ] 21
(87)
k = 11 ;
+u
1+k
The Jacobian elliptic function sn(z; k) has period T = 4nK (k). In the calculation of
the level splitting the solution for a half period is regarded as a nonvacuum instanton
conguration. The level shift (i.e. half of the level splitting ) is obtained as
E = B exp[;W 0]
(88)
where the prefactor B is given by
19
2
2
1
2
[1
+
u
]
B = 2K (k 0 )
(89)
and W 0 by
W 0 = 34 (1 + u) 12 [E (k) ; uK (k)]
(90)
If we regard the conguration over the full period (eectively a nonvacuum instanton{
anti-instanton pair) as a bounce conguration which returns to its original position
(such a consideration has also been discussed by Hawking and Ross )we can write it
~c( ) = kb(k) sn[b(k) + K (k); k]
(91)
so that this is zero at b(k) = ;K (k), i.e. at = ;T , and at b(k) = K (k), i.e.
at = +T (since snu vanishes for u = 0; 2K (k)). This motion is allowed for a
physical system with the distorted potential above. The motion of the bounce starts
at = ;2T and ends at = +2T . Since sn[u + K (k); k] = cn[u; k]=dn[u:k] we see
that this bounce is an even function of u. The zero mode , i.e. the derivative of the
bounce (which corresponds classically to its velocity), is therefore odd, i.e. a wave
function with eigenvalue zero which passes through zero at u = 0. Thus the ground
state eigenfunction of the corresponding uctuation equation must have a negative
eigenvalue. This negative eigenvalue is the one which is responsible for the instability
of the conguration. The imaginary part of the energy is now obtained the way we
obtained it elsewhere . We then have
=mE = B exp[;2W 0]
(92)
2
1
0
7
and so
=mE = B1 (E )
(93)
2
In the low energy limit B = , and the Bogomolny{Fateyev relation holds exactly.
We conclude, therefore, that the consideration of classical nite energy nonvacuum
congurations is applicable to numerous tunneling phenomena. Given E one can use
the Bogomolny{ Fateyev relation (by inserting =mE into the appropriate moment
integral) in order to derive the behaviour of a large order term of the perturbation
expansion (in Borel nonsummable cases) of the eigenvalue E in the case with splitting.
The Bogomolny{Fateyev relation has also been observed in other related contexts
and computationally . The contribution of sphalerons to the large-order behaviour
of perturbation expansions in quantum mechanical models derived from nonlinear
sigma models with symmetry{breaking potentials has been investigated recently by
Rubakov and Shvedov . We also mention that it is possible to develop a BRSTinvariant approach to quantum mechanical tunneling which avoids the degeneracy
problem of ill dened path integrals due to zero modes. So far we have applied
this method only to the sine{Gordon potential . Finally we remark that sphaleron
congurations analogous to those discussed here arise also in other theories such as
Skyrme-like models and Yang-Mills and sigma-model theories .
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