Euclidean action for vacuum decay in a de Sitter universe

0.24 MB·

by  V. Balek

Eu
lidean a
tion
for va
uum de
ay in a de Sitter universe
∗ †
V. Balek and M.Demetrian
5 Department of Theoreti
al Physi
s, Comenius University, Mlynská dolina
0
0 842 48 Bratislava, Slovakia
2

n
February 7, 2008
a
J

3
1

3
v
1
0 Abstra
t
0
9
0
4 Thebehaviorofthea
tionoftheinstantonsdes
ribingva
uumde
ayinadeSitterisinvesti-
0
/ gated. Foranear-to-limitinstanton(aColeman-deLu

iainstanton
losetosomeHawking-Moss
c
q
instanton) we (cid:28)nd approximate formulas for the Eu
lidean a
tion by expanding the s
alar (cid:28)eld
-
r
g and the metri
of the instanton in the powers of the s
alar (cid:28)eld amplitude. The order of the
:
v magnitude of the
orre
tion to the Hawking-Moss a
tion depends on the order of the instanton
i
X (thenumber of
rossingsofthebarrierbythes
alar(cid:28)eld): forinstantonsofoddandeven orders
r
a the
orre
tion is of the fourth and third order in the s
alar (cid:28)eld amplitude, respe
tively. If a
near-to-limit instanton of the (cid:28)rst order exists in a potential with the
urvature at the top of
2
the barrier greater than 4 × (Hubble
onstant) , whi
h is the
ase if the fourth derivative of
the potential at the top of the barrier is greater than some negative limit value, the a
tion of
the instanton is less than the Hawking-Mossa
tion and,
onsequently, the instanton determines
the out
ome of the va
uum de
ay if no other Coleman-de Lu

ia instanton is admitted by the
potential. A numeri
al study shows that for the quarti
potential the physi
al mode of the va
-
uum de
ay is given by the Coleman-de Lu

ia instanton of the (cid:28)rst order also in the region of
parameters in whi
h the potential admits two instantons of the se
ond order.
∗
e-mailaddress: balekfmph.uniba.sk
†
e-mailaddress: demetriansophia.dtp.fmph.uniba.sk
1
1 Introdu
tion
The va
uum de
ay in a de Sitter universe has been
onsidered as a me
hanism of transition to
a Friedman universe in the s
enario of old in(cid:29)ation [1℄, and as a triggering me
hanism in the
s
enario of open in(cid:29)ation [2℄. In both s
enarios, the instability of the false va
uum results in
the formation of rapidly expanding bubbles
ontaining the s
alar (cid:28)eld on the true-va
uum side
of the potential barrier. The pro
ess is des
ribed by the Coleman-de Lu

ia (CdL) instanton
[3℄. Another kind of va
uum de
ay is mediated by the Hawking-Moss (HM) instanton [4℄ and
pro
eedsinsu
hawaythatthe(cid:28)eldjumpsonthetopofthebarrierinsideahorizon-sizedomain.
(For this interpretation, see [5℄.) Re
ently this pro
ess was proposed as a starting point for the
in(cid:29)ation driven by the tra
e anomaly [6℄.
An important quantity
hara
terizingthe instanton is its a
tion. It determines the probabil-
B
ity of an instanton-indu
ed transition per unit spa
etime volume: if we denote the a
tion by ,
( B)
the probability is proportional to exp − . Thus, if more instantons exist for a given potential,
the one with the least a
tion prevails. In addition, the value of the a
tion may be relevant to
the question whether the instanton des
ribes a well de(cid:28)ned quantum me
hani
al transition at
all. The transition takes pla
e only if the instanton admits no negative modes of the perturba-
tion superimposed on it; and a

ording to [7℄, su
h modes are
ertainly not present only for the
instantonwith the least a
tion. If negativemodes exist, the preexponentialfa
torin the expres-
sion for the probability diverges,whi
hindi
ates that the wavefun
tion isnot
on
entratedin a
narrowtubearoundtheinstantonsolutionasit shouldbe. (Forananalogyin ordinaryquantum
me
hani
s, see [8℄.) Thus, the instanton with the least a
tion, in addition to being the most
probable, may be the only one with a physi
al meaning.
Thevaluesofthea
tionofCdLinstantonsweredis
ussedinthethinwallapproximationin[3℄
and [9℄, and
omputed fora one-parametri

lass of quarti
potentials admitting an instanton of
the(cid:28)rstorder(aninstantonwiththes
alar(cid:28)eld
rossingthetopofthebarrierjuston
e)in [10℄.
Furthermore,the behaviorofthea
tionofCdL instantons wasanalysedin the approximationof
negligibleba
krea
tionin[11℄;the propertiesofanear-to-limitinstanton(aCdLinstanton
lose
to some HM instanton) were studied with the ba
k rea
tion in
luded in [12℄; and the instanton
with the least a
tion was identi(cid:28)ed among the instantons of odd orders in [7℄. Here we link up
to this resear
h and develop it further. In se
tions 2 and 3 we (cid:28)nd a formula for the a
tion
of a near-to-limit instanton with the ba
k rea
tion in
luded, and in se
tion 4 we
ompute the
instanton a
tion for a
lass of quarti
potentials with a sub
lass admitting instantons of the
se
ond order. Thus, in the (cid:28)rst two se
tions we generalize the result of [11℄ and
omplete the
results of [12℄, while in the last se
tion we extend the results of [7℄ and [10℄.
2
2 The se
ond order
ontribution to the a
tion
φ
Consider a
ompa
t O(4) symmetri
4D spa
e and an O(4) symmetri
s
alar (cid:28)eld living in it.
φ(ρ) a(ρ) ρ
The system is
ompletelydes
ribed by the fun
tions and , where is the distan
efrom
a
the "north pole" (the radius of 3-spheres of homogeneity measured from the
enter) and is
thes
aleparameter(theradiusof3-spheresofhomogeneitydeterminedfromthe
ir
umferen
e).
ρ ρ a=0 ρ=0 ρ
f f
The
ompa
tnessimplies that runsfrom zero to some and at both and . The
~=c=1
Eu
lidean a
tion is ( )
ρf 1 1
B =2π2 φ˙2+V a2 (a˙2+1) a dρ,
Z (cid:20)(cid:18)2 (cid:19) − κ (cid:21) (1)
0
ρ V
where the overdot denotes di(cid:27)erentiation with respe
t to , is the e(cid:27)e
tive potential of the
κ=8πG/3
s
alar (cid:28)eld and . The extremization of the a
tion yields
a˙
φ¨+3 φ˙ =∂ V, a¨= κ(φ˙2+V)a.
φ
a − (2)
a
The equation for is obtained by di(cid:27)erentiating the equation
1
a˙2 =κ φ˙2 V a2+1,
(cid:18)2 − (cid:19)
φ¨ φ B
andinsertingfor fromtheequationfor . (Infa
t,whenputtingthevariationof equaltozero
a Ca−2 C
weobtainthe (cid:28)rst orderequationfor with anadditionalterm , where isan integration

onstant. To suppress this term one has to introdu
e the lapse fun
tion into the expression for
B φ a
and perform variation with respe
t to it.) Consider fun
tions and extremizing the a
tion
and su
h that the a
tion is (cid:28)nite for them. Fun
tions with these properties may be found, in
prin
iple, by solving equations (2) with the boundary
onditions
φ˙(0)=φ˙(ρ )=0, a(0)=0, a˙(0)=1.
f
(3)
φ φ˙ ρ = 0 ρ a˙ a¨
f
Sin
e and are (cid:28)nite at the points and , and are (cid:28)nite at these points, too.
Consequently, we may simplify the integral in (1) by repla
ing
1 2
aa˙2 aa˙2 a2a¨
→−3 − 3
a
and using both equations for to obtain
4π2 ρf
B = a dρ.
− 3κ Z (4)
0
Thisexpressionisappropriatefornumeri
al
al
ulationsbut whenexpandingthea
tionintothe
powers of the s
alar (cid:28)eld amplitude it is advisable to use expression (1) instead. Then one does
a B
not need to
ompute up to the same order of magnitude as ; as we shall see, one gets along
3
a
with the zeroth and se
ond order
ontributions to when
omputing the third and fourth order
B

ontributions to , respe
tively.
φ=0
Consider a potential with the global minimum, or the true va
uum, at , a maximum at
φ=φ >0 φ=φ >φ
M m M
andalo
alminimum, orthefalseva
uum, at . Suppose,furthermore,
that the potential equals zero at the true va
uum. There are two trivial (cid:28)nite-a
tion solutions
to equations (2) with a
ompa
t 4-geometry,
φ=φ , a=H−1sin(H ρ), 0<ρ<πH−1,
M M M M
and
φ=φ , a=H−1sin(H ρ), 0<ρ<πH−1,
m m m m
H H H = √κV
M m
where and are the values of (the Hubble
onstant
orresponding to a given
φ φ=φ φ
M m
value of ) at and respe
tively. Both solutions
onsist of a 4-sphere and a
onstant
s
alar(cid:28)eld. The(cid:28)rstsolutionistheHMinstanton;these
ondsolutiondes
ribesa"zeroquantum
transition"hen
eitmayberegardedasareferen
esolution. Inaddition,thepotentialmayadmit
nontrivial(cid:28)nite-a
tionsolutionstoequations(2)
alledtheCdLinstantons. Thesesolutionsmay
be
lassi(cid:28)ed a

ording to how many times the s
alar (cid:28)eld
rosses the barrier: if the number
l l
of
rossings is , the instanton may be
alled "the CdL instanton of the th order". Denote
ξ =∂2V /H2 ξ
φ M M. For − assuming one of the
riti
al values
λ =l(l+3)=4,10,... l=1,2,...,
l
for (5)
H−1
thepotentialadmitsanapproximateCdLinstanton
onsistingofa4-spherewiththeradius M
l
and a s
alar (cid:28)eld proportional to the th harmoni
on the 4-sphere [13℄. This solution may be
l ξ λ
l

alled "the limit instanton of the th order". If is
lose to − , at least from one side, the
potential should admit a CdL instanton
lose to the limit instanton with some small s
alar (cid:28)eld
amplitude. Inaone-parametri
familyofpotentials,thisnear-to-limitinstantonshouldapproa
h
ξ λ
l
the HM instanton as approa
hes − .
Introdu
e dimensionless variables
1
χ=H ρ, u= ∆φ, v =H a,
M M
A
∆φ=φ φ A=∆φ(0)
M
where − and , and write the potential as
1 1
V =H2 +A2 ξu2+ω .
M(cid:20)κ (cid:18)2 (cid:19)(cid:21)
The expression for the a
tion, when rewritten into the dimensionless variables, is
B = 2π2 χf A2 1u′2+ 1ξu2+ω v2 1(v′2 v2+1) v dχ,
H2 Z (cid:20) (cid:18)2 2 (cid:19) − κ − (cid:21) (6)
M 0
4
and the instanton equations are
′
v 1
u′′+3 u′ ξu=∂ ω, v′′+v = A2 u′2+ ξu2+ω v,
u
v − − (cid:18) 2 (cid:19) (7)
χ = H ρ χ s = sinχ
f M f
where and the prime denotes di(cid:27)erentiation with respe
t to . Denote
c=cosχ u=0 v =s χ =π
f
and . For the HM instanton , and , whi
h yields the a
tion
8π2
B = .
M −3κH2 (8)
M
u v =s ξ = λ
l
For the limit instanton, obeys the (cid:28)rst equation (7) with , − and suppressed right
hand side, d2 d
u=0, = 3 χ +λ .
D D d2χ −
otan dχ l
ThesolutionsareO(4)symmetri
spheri
alharmoni
sin5Dor,upto normalization,theGegen-
bauer polynomials with the parameter 3/2,
1
P =c, (5c2 1), ... l =1,2,...
l
4 − for (9)
ξ =ξ +Aξ +A2ξ +...
0 1 2
When
onstru
tinganear-to-limitinstantononeintrodu
esexpansions ,
u=Au +A2u +A3u +... v =v +A2v +A3v +... ξ = λ u =P v =s
1 2 3 0 2 3 0 l 1 l 0
and with − , and ,
and uses the expansion
1 1
ω = ηAu3+ ζA2u4+...,
6 24
η = ∂3V /H2 ζ = ∂4V /H2 u v
where φ M M and φ M M. >From the expanded equations for and one
u u ... v v ...
1 2 0 2
obtains the fun
tions , , and , , , and after inserting them into the expression for
B B B ... B =B +A2B +...
0 2 0 2
one(cid:28)ndsthe
oe(cid:30)
ients , , in theexpansion Inparti
ular,from
v =s B =B
0 0 M
one obtains .
B B B =B
2 M
Letusprovethatthetermproportionalto intheexpansionof vanishessothat
A2 B
2
up to the order . For we have
2π2 π 1 1
B = (P′2 λ P2)s3 [2scv′ +(c2 3s2+1)v ] dχ.
2 H2 Z (cid:26)2 l − l l − κ 2 − 2 (cid:27) (10)
M 0
(We have not in
luded here a term arising from the shift of the upper limit of integration sin
e
A4 P
l
su
h terms do not appear before the order .) The (cid:28)rst part, with the terms quadrati
in ,
P P
l l
mayberewrittenusingintegrationbypartstoanintegralof D ,hen
eitvanishes. These
ond
v
2
part, with the terms linear in , may be rewritten using integration by parts to an integral of
v
2
A with the
oe(cid:30)
ient of proportionality
= (2sc)′+c2 3s2+1=0,
A − −
hen
e it vanishes, too. This
an be seen alsowithout an expli
it
al
ulation. The se
ond part of
B B v δv =v
2 0 0 0 2
is just what weobtain if we perform the variation of with respe
t to and put ,
=0 v
0
thus A is in fa
t the equation determining . As a result we obtain
B =0.
2
(11)
5
3 Higher order
ontributions to the a
tion
B B
3
The (cid:28)rst higher order term in the expansion of whi
h may be nonzero is proportional to .
B u v P
3 2 3 l
Thetermsin linearin and
anberewrittentobeproportionaltoD andA, thus they
vanish and we are left with the expression
π2 1 π
B = ξ I + ηI , I = Pks3dχ.
3 H2 (cid:18) 1 2 3 3(cid:19) k Z l (12)
M 0
ξ u
1 2
To determine ,
onsider the equation for
1
u =ξ P + ηP2.
D 2 1 l 2 l (13)
The task simpli(cid:28)es
onsiderably by two observations: (cid:28)rst, the left hand side, when expanded
P = 1 P P ... P
0 1 2 l
into the system of harmoni
s , , , , does not
ontain a term proportional to
u
2
(this is easily seen if one expands into the harmoni
s and makes use of the fa
t that the
′
Pl′ l =l
operatorD, when applied on , reprodu
es it with a fa
tor that is zero for ); and se
ond,
s3 P
l
the harmoni
s are orthogonal with the weight . Thus, if we multiply equation (13) by and
s3
integrate it with the weight we obtain zero on the left hand side. As a result we have
1
ξ I + ηI =0,
1 2 3
2
hen
e π2
B = ηI .
3 −6H2 3
M
l cl cl−2 ... P3s3
The th harmoni
is a linear
ombination of , , , thus l is an odd fun
tion on the
(0,π) I l
3
interval and vanishes if is odd. On the other hand, from the general formula for the
I l
3
integral of the produ
t of three Gegenbauer polynomials [14℄ it follows that is nonzero if is
l ξ B ∆ξ =ξ+λ
1 3 l
even. Thus, for even values of both and are nonzero and the quantities and
∆B =B B A A3
M
− are of the order and respe
tively. Expli
itly,
. I3
∆ξ = ηA,
−2I (14)
2
and . π2I3 ηA3
∆B = .
− 6 H2 (15)
M
A ∆ξ
The a
tual small parameter of the theory is not but , therefore we have to interpret the
A ∆ξ l = 2 I = 2/21
2
(cid:28)rst equation as an approximate expression for in terms of . For , and
I = 2/63 ∆ξ ∆B
3
, thus the dimensionless
oe(cid:30)
ients in the expressions for and are 1/6 and
π2/189 I B
3 3
respe
tively. Note that we may avoid the
omputation of if we express in terms
ξ ξ P
1 1 l
of and determine from the requirement that the term proportional to on the right hand
side of (13) vanishes.
6
A3
To summarize, the
orre
tion to the HM a
tion of order vanishes for instantons of odd
η
orders, but it is nonzero for instantons of even orders provided is nonzero. If we
hange the
ξ η
parameters of the potential so that
hanges while stays (cid:28)xed, the
hara
ter of the instanton
ξ λ η >0 ξ < λ φ
l l

hanges as
rosses the value − . If, say, , for − the fun
tion starts and ends to
φ ξ > λ φ
M l
the right of (the instanton is "right handed"), while for − the fun
tion starts and
φ
M
ends to the left of (the instanton is "left handed"). No matter what the handedness of the
ξ < λ
l
instanton, thea
tionofthe instantonislessthanthe HM a
tionif − and greaterthan the
ξ > λ
l
HM a
tion if − .
B l
4
To
omplete the analysis we have to
ompute for odd values of . First, just as when we
B
3
were
omputing , we get rid of a large portion of the integrand by rewriting it in terms of
P u v
l 3 4
D and A; in this way we remove the terms proportional to and . Before we present the
B
remainingtermsletusmentionanewpointwhi
harisesinthisorderofexpansionof . Onemay
v χ=π l=1 δ v (π)= πκ/8
2 2
expe
tthatthevalueof at isnonzeroandindeed,for one(cid:28)nds ≡ −
χ π A2
f
(see the appendix). Consequently, di(cid:27)ers from by a quantity of order . As we shall see,
B B
4 4
this leads to additional terms in so that not the whole is stemming from the expansion
B u v
3 4
of the integrand of . After suppressing the terms proportional to and and integrating by
parts we arriveat
π2 1 π 2
B = ξ I + ζI + u ( u +P)s3+v v +Q dχ+b ,
4 H2 (cid:26) 2 2 12 4 Z (cid:20) 2 −D 2 2(cid:18)−κB 2 (cid:19)(cid:21) 4(cid:27) (16)
M 0
where d2 d
P =ηP2, Q=3(P′2 λ P2)s2, = s c 2s,
l l − l l B − d2χ − dχ −
b χ
4 f
and isaboundaryterm
onsistingoftwoparts,one
omingfromtheshift in and theother

oming from the integration by parts whi
h has to be performed when deriving the term with
B u
4 2
the operator B. The expression for simpli(cid:28)es further if we exploit the equations for and
v
2
,
1 1
u = P, v = κQ.
2 2
D 2 B 4 (17)
B
4
Bothequationsaremosteasilyderivedbyperformingthevariationof andusingthefa
tthat
s3
the operators D and B aresymmetri
; however,they may be obtained by expandingthe exa
t
u v ξ =0
1
equations for and as well. The former equation is identi
al to (13) provided , whi
h is
what we presently assume, and the latter equation is a linear
ombination of the two equations
v P
2
for presentedin the appendix. If wemakeuseof equations(17) and of the de(cid:28)nition of , we
B
4
(cid:28)nd that the integral in redu
es to
1 1 π π
ηJ + Qv dχ, J = P2u s3 dχ.
2 κZ 2 Z l 2
0 0
B
4
Letusnowshowthattheboundarytermin iszero. We
an(cid:28)ndthe
orre
tiontotheupper
v χ = π 1
0
limit of integration by noti
ing that the derivative of at is − , so that to
ompensate
7
.
∆v =A2v χ=π χ ∆=A2δ
2 f
for in the neighborhood of we have to shift by . The
ontribution
b
4
to arising from this shift is
π+∆
bshift =A−4 2(F +A2F ) dχ,
4 × the leading term in Z 0 2
π
F F B B
0 2 0 2
where and are the integrands in and ,
1 1
F =... (c2+1)s, F =... (c2+1)v .
0 2 2
− κ − κ
(Theomittedtermsareirrelevantforthepresentdis
ussion.) Theleadingtermintheintegralof
F F′(π)∆2/2=∆2/κ F (π)=0 F
0 equals 0 , sin
e 0 , and the leading term in the integralof 2 equals
F (π)∆= 2δ∆/κ
2
− , hen
e 2δ2
bshift = .
4 − κ
B
Ontheotherhand, fromthe expressionoftherelevantpart of givenintheappendixit follows
b
4
that the
ontribution to
oming from the integration by parts is
2δ2
bint = .
4 κ
Putting this together we (cid:28)nd
b =0.
4
(18)
u ξ
3 2
Nextwehaveto writedowntheequationfor in ordertodetermine . When
onstru
ting
δ δ
this equation we en
ounter a subtlety that is again
onne
ted with the presen
e of . If is
′
v s χ=π a˙/a=H v /v
2 M
nonzero, is not small with respe
t to in the vi
inity of and the fa
tor
′
φ v /v
in the equationfor
annotbe treated perturbatively. In fa
t, if we formallyexpand up to
A2
the order , v′ =. χ+A2q, q = v2 ′,
v
otan (cid:16) s (cid:17)
χ=π
we
anseethatthe
orre
tiontermevendivergesrelativelytothezerothordertermin . (It
∆/ǫ2 ǫ=π χ 0 1/ǫ
behaveslike− for − → ,whilethezerothordertermbehaveslike .) To(cid:28)xthat,
δˆ v χ=π
2
notethatthe term in whi
his nonzeroat and thereforeis responsibleforthenonzero
δ δχv′/π δˆ v
value of equals − 0 (see the appendix), thus may be absorbed into 0 by rede(cid:28)ning
χ (1 ∆/π)χ
→ − . Su
h pro
edure is well known in the perturbation theory of an anharmoni
δˆ
os
illatorwherethetermsofthetype are
alled"resonan
eterms"[15℄. The
orre
tedequation
v q qred
3
for isobtainedinsu
hawaythatwerepla
e by ("red"standingfor"redu
ed"),de(cid:28)ned
q v vred =v δˆ ξ κ χ
as with 2 repla
edby 2 2− , andaddto 2 theterm arisingfromtherede(cid:28)nitionof
u v
1 3
in the equationfor . However, these
orre
tions, ne
essaryas they arewhen one
omputes ,
v s3
3
aresuper(cid:29)uous if one is interested only in the integral of the equation for with the weight .
q
Thepointisthatthe weightwashesoutthesingularityin , thereforethereversetransformation
χ (1+δ/π)χ
→ by whi
h one passes from the
orre
ted equation to the original one may be
8
treated on equal footing with the regular transformations (that is, transformations that do not
χ=π
movethepoint ). To
on(cid:28)rmthis,onemay
he
kthatthetwo
orre
tionsmentionedabove
χ s3

an
el when the equation is integrated over with the weight . The un
orre
ted equation for
v
3
reads
1
v =ξ P +η P u + ζP3 3qP′,
D 3 2 l 1 l 2 6 l − l (19)
B
3
and by the same argument as we have used when
al
ulating we
an dedu
e from it the
identity 1 π
ξ I +ηJ + ζI + 3qPP′s3 dχ=0.
2 2 6 4 Z l l
0
P
l
Finally,byintegratingbypartsandusingtheequationfor itmaybeshownthattheintegrand
Qv
2
inthelastterm
anberepla
edby− . This
ompletestheproofthatinourinitialexpression
B ξ I ξ I /2
4 2 2 2 2
for the sum of all terms following equals − , and
π2
B = I ξ .
4 2H2 2 2
M
B ∆ξ
3
Similarlyaswhenwewere
omputing we
annowobtainapproximateexpressionsfor and
∆B A ξ v
2 3
in terms of . Again, we may determine also by analysing the equation for ; however,
q qred ξ ξ +κ
2 2
forthat purposethe
orre
tedequationhasto be used, with repla
edby and by .
l=1
Let us present the results for the most important
ase . In this
ase,
. 1 1
∆ξ = η2+ζ+32κ A2
−14(cid:18)12 (cid:19) (20)
and
. 2π2 ∆ξA2
∆B = .
15 H2 (21)
M
∆ξ ξ
The expression for implies that a near-to-limit instanton exists only if − approa
hes the
λ =4 η ζ
1
value from a given side. Whi
h side, that depends on the parameters and . De(cid:28)ne
1
ζ = η2 32κ.
crit
−12 − (22)
ζ > ζ ξ < 4 ζ < ζ ξ > 4
crit crit
If the instanton exists for − , while if the instanton exists for − .
The a
tion of the instanton is less than the HM a
tion in the former
ase and greater than the
HM a
tion in the latter
ase.
∆ξ κ
The expression for is identi
al to that obtained in [12℄, and if we suppress in the
∆ξ H η ζ
M
expression for and insert for , and the values for the quarti
potential, the resulting
∆B ∆B
expressionfor isidenti
altothatderivedin [11℄. Ourformulafor is
onsistentalsowith
the behavior of the (cid:28)rst perturbation mode of the near-to-limit instanton established in [12℄,
sin
e this mode is positive (and,
onsequently,
ontributes to the a
tion by a positive quantity)
B B
M
in the same range of parameters in whi
h we found that − is positive.
9
4 The instanton with the least a
tion
The question whi
h instanton has the least a
tion is addressed in [7℄ where it is shown that
amongtheinstantonsofoddorders,theonewiththeleasta
tionisne
essarilyofthe(cid:28)rstorder.
The proof is based on the analysis of a graph that may be regarded as a phase diagram for
φ−(ρ) φ+(ρ)
the solutions of the Eu
lidean theory. One introdu
es noninstanton solutions and
φ ∆φ π =2π2a3φ˙
M φ
startingtotheleftandtotherightof ,andassignstothemthevaluesof and
ρ a(ρ)
at su
h at whi
h the fun
tion rea
hes maximum. In this way one obtains two oriented
(∆φ,πφ) C− ( φM,0)

urvesinthe plane: the
urve startingatthepoint − (thetrueva
uum)and
C
+
ending at the origin (the top of the barrier), and the
urve starting at the origin and ending
(φ φ ,0)
m M
at the point − (the false va
uum). An instanton of an odd order is represented by
C+ C− ∆B
an interse
tion of the
urves and , and its "net a
tion" equals the area inside a loop
C
+
passingfromthe originto the interse
tionandba
k,(cid:28)rst alongthe
urve and then alongthe
C− z r dr r

urve . (The area is regarded here as the -
omponent of the ve
tor × , where is the
H
(x,y)
radius ve
tor in the plane. A

ording to this de(cid:28)nition, the area inside a given
urve is
positiveif the
urveis oriented
ounter
lo
kwiseand negativeif the
urveisoriented
lo
kwise.)
C−
As to the instantons of even orders,they arerepresented by the points where the
urves (for
C ∆φ
+
lefthandedinstantons)and (forrighthandedinstantons)interse
ttheaxis , andtheirnet
a
tion equals twi
e the area inside a loop passing from the origin to the interse
tion and ba
k,
C− ∆φ
(cid:28)rstalongthe
urve andthenalongtheaxis iftheinstantonislefthanded,and(cid:28)rstalong
∆φ C
+
the axis and then along the
urve if the instanton is right handed. Three typi
al phase
1
diagramsaredepi
tedin(cid:28)g. 1. Alldiagrams
ontainaninstantonofthe(cid:28)rstorderdenotedby ;
C
C
C 1
p 1 1
C
+ C C
f + +
2' 2'' 2 2
+
Df
Figure 1: Phase diagrams for the solutions of the Eu
lidean theory
besides, the middle diagram
ontains two left handed instantons of the se
ond orderdenoted by
′ ′′
2 2
− −
and ,andtherightdiagram
ontainsonelefthandedandonerighthandedinstantonofthe
10