Validity of the kink approximation to the tunneling action

0.47 MB·English

by  Koushik Dutta

DESY-12-025
On the Existence of Tunneling Bounce Solutions in Piecewise Linear Potentials
Koushik Dutta, Cecelie Hector, Thomas Konstandin, Pascal M. Vaudrevange, and Alexander Westphal
DESY, Notkestrasse 85, 22607 Hamburg, Germany
(Dated: February 14, 2012)
Coleman tunneling in a general scalar potential with two non-degenerate minima is known to
haveanapproximationintermsofapiecewiselineartriangular-shapedpotentialwithsharp’kinks’
at the place of the local minima. This approximate potential has a regime where the existence of
the bounce solution needs the scalar field to ’wait’ for some amount of Euclidean time at one of
the ’kinks’. We discuss under which circumstances the correct bounce action can be consistently
obtainedasthelimitingcaseofaregularscalarpotentialwhere’kinks’areresolvedaslocallysmooth
’cap’ regions.
I. INTRODUCTION false vacuum. Having the field ’wait’ in such a manner
2 is only an approximation to the physical situation of re-
1
placing the tip of the potential by a smooth cap.
0 A semi-classical approach to quantum tunneling pro-
2 cesses in field theory has been presented in a series of However, within the context of the 4D effective field
pioneering papers [1], [2]. The role of gravity in the pro- theory approximation to a theory of quantum gravity
b
e cessoftunnelingwassubsequentlyconsideredin[3]. The such as string theory, the situation can become some-
F authorspresentedaschemeforcalculatingtunnelingam- what more involved. If the cap smoothing the potential
plitudesfortransitionsfromfalsetotruevacua. Thecal- hasacurvature∂2V/∂φ2thatislargerthanM2,effective
3 P
1 culation involves the evaluation of the Euclidean action fieldtheorycannotbetrusted. Inparticular,itcannotbe
ofthebouncesolutiontotheimaginary-timeequationsof guaranteedthatthecapisstrictlyconcave,i.e.largecor-

] motion. Theexistenceofthebouncesolutionwasproven rectionsmayinducelocalsaddlepointsorminimawithin
h
in generality in [1]. For almost degenerate vacuum en- thehighlycurvedregionofthecap. Thusthewholesemi-
t
- ergy, the thin-wall approximation can be used to calcu- classicalapproximationforthecalculationsofthebubble
p
late the tunneling amplitude without having to compute nucleation also breaks down and we can no longer talk
e
h the bounce solution. abouttunnelingintheusualpictureof[1]and[3]. Inthis
[ Given the fact that a bounce exists, a necessary and case, full quantum gravity effects must be incorporated

sufficient condition in this scheme for the false vacuum to calculate the tunneling amplitude.
1
v to be unstable and tunneling to occur is the existence IfthecurvatureofthesmoothcapissmallerthanM2,
P
1 of a single negative eigenvalue of the operator δ2S, the approximatingthissetupbykeepingthefieldfixedshould
2 secondvariationalderivativeoftheEuclideanactioneval- notintroducetoomuchofanerror. Themainaimofthe
7 uated at the bounce, [4]. Various authors examined sys- current work is to analyze under which circumstances
2
tems where the tunneling rate may become zero, either this expectation holds true.
.
2 through the appearance of singularities in multi-field se-
In the next Section we briefly review the original ar-
0 tupsincludinggravity[5],[6],[7],orthroughtheappear-
2 guments for the existence of bounce solutions. In Sec-
ance of intermediate vacua in single field setups [8], [9].
1 tion III, we discuss the tunneling in a piecewise linear
Therefore, the decay process of false vacua via tunneling
: potential and explain how smoothing of the potential is
v in the semi-classical picture firstly depends on the exis-
important for certain parameter ranges of the potential.
i tence of a bounce under consistent approximations, and
X This subsequently leads us to a condition on the smooth
secondly on having only one negative eigenvalue of δ2S.
r potential for a meaningful approximation in terms of a
a In this short note, we would like to address the issue
piecewiselinearpotentialinSectionIV.InSectionV,we
of the existence of a bounce solution in piecewise linear
comment on the applicability of the shooting algorithm
potentials that act as approximations to locally smooth
in the smooth potential and we conclude in Section VI.
potentials. Itisclearthatviolatingtheconditionssetout
(implicitly)intheproofof[1]isdifficultinanyphysically
realisticsetup. Stillthisdoesnotguaranteethatthetwo
pictures lead to quantitatively similar actions.
Consider an effective potential that has sharp minima II. EXISTENCE OF BOUNCE SOLUTIONS
and maxima (’kinks’), see Figure 1. Tunneling in this
setupwasfirstdiscussedanalyticallyin[10]. Inthiscase,
In his pioneering paper [1], Coleman offered an exis-
for certain ranges of parameters, a consistent bounce so-
tence proof for the bounce solution. We briefly sketch
lutionexistsifweallowthefieldtorestforsomeamount
this proof before we specialize to piecewise linear poten-
of Euclidean time at the true minimum. With this field
tials in the next Section.
profile, it is possible for the relevant friction term to die
off sufficiently so that the field can roll back up to the In the inverted potential (see Figure 2), the bounce
2
VT fieldcansitalmostfixedatthatpositionforalongerand
longertime–untilfrictiondiesoff. Then, energyconser-
vationmakesthefieldrollpastφ aslongas∆V ≥∆V
+ − +
(φ being the location of the false and true vacuum re-
±
spectively,and∆V =V −V ). Toshowthatovershoot-
± T ±
V V(cid:43) ing past φ+ occurs for starting values φ0 close enough to
φ , we need to be somewhat quantitative: For analytic
−
potentials it is possible to linearize the equation of mo-
tion close to the true vacuum φ , giving
−
(cid:18) (cid:19)
3
V(cid:45) Φ Φ Φ ∂r2+ r∂r−µ2 (φ−φ−) = 0, (2)
(cid:43) T (cid:45)
Φ
with µ2 =V(cid:48)(cid:48)(φ ). It is solved by
−
FIG. 1: The piecewise linear potential first analyzed by [10].
I (µr)
Falowra|yφs+ex−istφ.T| < |φ− −φT|, the bounce solution does not φ(r)−φ− = 2(φ0−φ−) 1µr , (3)
where I is the Bessel function of the first kind, see [1].
1
Hence for φ ever closer to φ , the field can sit near φ
0 − −
for larger and larger r. Making the initial displacement
V
T from φ sufficiently small, r becomes large enough such
−
that the friction term effectively disappears. Thus by
energy conservation, φ can rush past φ .
+
V
On the other hand, starting far away from the top of
V
(cid:43) the inverted potential, the field does not have enough
energy to climb up to φ . Thus, by continuity, there
+
must be an initial value φ being φ < φ < φ such
0 T 0 −
that the field reaches φ with zero velocity.
+
V
(cid:45)
Φ Φ Φ
(cid:43) T (cid:45)
Φ III. PIECEWISE LINEAR POTENTIALS
FIG.2: Schematicviewofapotentialwithanexistingbounce
The existence of the bounce crucially depends on the
solution. The crucial part is that the true minimum is a
possibility of the field to spend arbitrarily long times ar-
smooth function of φ, i.e. ∂ V is continuous.
φ bitrarily close to the true vacuum. In other words, if
the equation of motion (2) takes on a different form, it
solutionφ 1 isthesolutiontotheequationofmotion(in is not guaranteed that the field can spend enough time
B
near the true minimum for the friction term to die out.
3+1 dimensional Euclidean space time)
In particular, it is intuitively clear that this is the case
3 for piecewise linear potentials, see Figure 1. In the fol-
φ(cid:48)(cid:48)+ φ(cid:48)−∂ V = 0, (1)
r φ lowing, we discuss the tunneling solutions in detail for a
piecewise linear potential, pointing out several subtleties
where φ(cid:48) ≡ ∂rφ with the following properties. At the before we analyze the transition to the smooth and reg-
center of the bubble, at r = 0, the field sits with zero ular potential where the kinks are resolved by caps.
velocity at position φ0 somewhere between φT, the lo- Tunneling in a piecewise linear potential
cation of the top of the potential barrier, and φ , the
−
(cid:26)
location of the true vacuum. For r > 0, the field moves V +λ (φ−φ ), φ<φ
V = T + T T , (4)
towardsthefalsevacuumφ ,reachingitwithzerospeed V −λ (φ−φ ), φ≥φ
+ T − T T
for r → ∞. Owing to the friction term in Eq. (1), it is
not immediately clear that the field can ever reach φ . hasbeenanalyzedby[10],seeFigure1. Wepresenttheir
+
Inside of the bubble, starting ever closer to φ , the analysis in a slightly different form.
−
First of all, we set φ = V = 0 as shifts in the field
T T
and in the zero point energy do not change the physics
– ignoring the effects of gravity. Solving the equation of
1 The term bounce stems from the description of this process in motion inside of the bubble
QM,wherethefieldsitsinthefalsevacuumfort→−∞,reaches
the true vacuum at some −∞<t∗ <∞, and rolls back to the φ(cid:48)(cid:48)+ 3φ(cid:48) +λ = 0, (5)
falsevacuumfort→∞. i r i −
3
subject to φ(0)=φ , φ(cid:48)(0)=0, we find
0
φ = φ − λ−r2. (6) VT
i 0 8
Enforcing the matching condition φi(RT)=0 gives V(cid:43)
V
(cid:115)
2φ
R = 2 0 . (7)
T λ
−
Solvingtheequationofmotionontheoutsideofthebub-
V
ble (cid:45)
Φ Φ Φ
(cid:43) (cid:42) (cid:45)
3 Φ
φ(cid:48)(cid:48)+ φ(cid:48) −λ = 0, (8)
o r o +
FIG.3: Shapeofatrueminimumforwhichnobounceexists.
subject to φ (R )=0, φ(cid:48)(R )=φ(cid:48)(R ), we find
o T o T i T If the wings of the trough at φ− are polynomials of at least
order 4, the bounce solution in the inverted potential cannot
(R2 −r2)(R2(λ +λ )−r2λ )
φ = T T − + + . (9) roll past the point φ∗ on the left.
o 8r2
On the outside, the field settles in the false vacuum at
almost degenerate minima, the field would need to wait
radius R >R , for which φ(cid:48)(R )=0
+ T o + near the true minimum for the friction term to fall off.
But the linear potential makes it impossible for the field
(cid:18) λ (cid:19)1/4
R = 1+ − R . (10) to stay longer at that initial position. Therefore, with
+ λ+ T a small energy difference between the true and the false
vacuum, the field can not roll up the hill due to friction,
At this position, the field has the value
and no solution exists that reaches φ .
+
√ For ∆<1/4 and α>1, although the field has a large
1+c−1
φ (R ) ≡ φ =−φ √ , (11) potentialenergytostartwith, thenon-vanishingfriction
o + + 0
1+c+1
term still prevents it from climbing up the long shallow
part to reach φ .
where c = λ /λ . As φ < φ , this gives a restriction +
− + 0 − Another example in which a large difference in po-
on the shape of the potential
tential energy does not guarantee a bounce solution are
√
−φ −φ 1+c−1 scalarpotentialswithlocalφ4 (orhigherpower)behavior
α≡ + ≤ + = √ <1. (12)
φ− φ0 1+c+1 V = −c4(φ−φ )4+higher order, c >0. (14)
4 ∗ 4
This is equivalent to
For solutions to the equation of motion
∆V 1 1
∆V−+ ≤ 2+c+2√1+c ≤ 4. (13) φ(cid:48)(cid:48)+ 3rφ(cid:48)+c4(φ−φ∗)3 = 0, (15)
Inotherwords,thebouncesolutionsgivenbyEq.(6)and with initial conditions φ(0)=φ and φ(cid:48)(0)=0, the field
0
Eq. (9) with initial condition φ(0) = φ , φ(cid:48)(0) = 0 are reaches φ = φ with zero speed [11] independent from
0 ∗
only valid for the parameter range α<1 and ∆< 1. the release point φ . If the φ4 behavior ends with a kink
4 0
Before proceeding, let us take a look at the physical (as depicted in Figure 3) no bounce solution exists.
meaning of this condition. It corresponds to a potential It is important to note that the arguments illustrated
profile where the energy difference between the false and in Section II do not immediately hold here: there is no
the true vacuum is large (i.e. not thin-wall) and |φ |< quadraticapproximationofthepotentialaroundthetrue
+
|φ |. In this case, it is always possible to find a field minimumtogetEq.(2). Inalinearpotential,puttingthe
−
position φ <φ with zero initial velocity such that the initial position ever closer to the true minimum φ does
0 − −
field can roll up the hill to φ . not force the field to spend an ever longer time there.
+
On the other hand, if either ∆ > 1/4 for any value of As [10] pointed out, we thus need to modify the initial
α, or ∆ < 1/4 and α > 1, it is not immediately clear conditions. Inside the bubble, the field should be arti-
that a bounce solution exists for φ ≤φ . The physical ficially fixed at the true minimum φ until radius R .
0 − − 0
picture is as follows: This should be understood as an approximation. In par-
For ∆ > 1/4 (and any value of α), there is a smaller ticular,thiscanbeinterpretedasmimickingtheeffectof
energy difference between the true and the false mini- removing the kink and replacing it with a smooth cap.
mum–intheextremecase,makingtheenergydifference In this suitably capped potential, the field can sit arbi-
infinitesimally small for ∆ → 1, the thin-wall limit. For trarily close to the true minimum and spend ever longer
4
time there. It is clear that the original argument of the
existenceofbouncesolutionasoutlinedinSectionIIim-
mediately holds.
This waiting period can be realized by a change in the
boundary conditions as first done by [10] for a piecewise
linear potential. The modified initial conditions become
V
φ (r)=φ , φ(cid:48)(r)=0 for 0≤r ≤R , giving
i − i 0
(r2−R2)2
φ (r) = φ − 0 λ (16)
i − 8r2 −
for R ≤ r < R . Outside of the bubble at R > R ,
0 T + T
the field comes to rest in the false vacuum φ(cid:48)(R ) = 0,
o +
φ(cid:48)o(R+)=0, giving Φ
(r2−R2)2
φ (r) = φ + + λ . (17) FIG. 4: The piecewise linear potential and different regu-
o + 8r2 +
lar potentials in the cap region. The three potentials on the
Now matching the two solutions φ (R ) = 0 = φ (R ), bottompassthecriterion(24)whilethepotentialontopcan
i T o T
φ(cid:48)(R )=φ(cid:48)(R ) gives lead to large deviations in the bounce action.
i T o T
φ (cid:115) 2 (cid:114) (cid:104) √ (cid:105)
R = −√ (1+α) α−1+2 ∆ ,
0 1− ∆ V− IV. CAPS VERSUS KINKS
(cid:115)
R = φ−√ 2 (1+α), (18) InthisSection,wediscussinwhichcasesreplacingthe
T 1− ∆ V− tip of a piecewise linear potential with a smooth cap can
bewellapproximatedbykeepingthefieldartificiallyfixed
(cid:115) (cid:115)
(cid:20) (cid:18) (cid:19)(cid:21)
R = φ−√ 2 (1+α) 1+α √2 −1 , at the minimum as outlined in the previous Section.
+ 1− ∆ V− ∆ Supposethatapiecewiselinearpotentialisobtainedas
the limit of a regular smooth potential. The scale δφ on
with R < R < R , α = −φ+, ∆V = V −V and which the kink is resolved in the regular potential serves
0 T + φ− ± T ±
∆= ∆V+. NotethattheconditionthatR isrealimplies asexpansionparameterδφ(cid:28)|φT−φ−|. Apparently,the
∆V− √ 0 bounceactionscanbeverydifferent,ifthesmoothpoten-
that α>1−2 ∆.
tial varies strongly in the cap region. For example, if the
The tunneling amplitude can then be computed as
potential has a large positive spike, the bounce solution
B = B +B +B , (19) can leave the cap region with a sizable velocity that can
a b c
alter the bounce action significantly. Hence we demand
with that the potential in the cap does not differ too much
B = 2π2(cid:90) R0drr3(−λ φ −λ φ ) , (20) from the corresponding value V− in the kink potential
a − − + + at least up to the first local minimum in the regularized
0
potential
(cid:90) RT (cid:18)1 (cid:19)
B = 2π2 drr3 φ(cid:48)2−λ φ −λ φ ,(21)
b R0 2 L − L + + |V−cap−V−kink|<∼λ−δφ. (24)
(cid:90) R+ (cid:18)1 (cid:19)
B = 2π2 drr3 φ(cid:48)2+λ φ −λ φ ,(22) Still, the potential can vary strongly in the cap region in
c RT 2 R + R + + the sense that its derivative does not need to be small.
giving Some examples are given in Figure 4.
(cid:16) √ (cid:17) Now,considerthebouncesolutionsforthekinkpoten-
2π2 φ4 (1+α)3 (α−3) ∆+1−3α tialandthecappotentialoutsidethecapregion. Evenfor
B = − √ .(23) large r these two solutions only coincide approximately.
3 ∆V− ( ∆−1)3 It might well be that one of the two solutions passes a
For certain choices of parameters of a piecewise linear given point in the potential a little bit later but com-
potential, we just saw that the bounce solutions exist pensates by a slightly smaller velocity. The former effect
only when we keep the field artificially fixed at the true leads to a reduced friction that is compensated by the
minimum. Holding it there for a sufficiently long time, latter effect.
the damping term becomes small enough such that the Even a small difference between the two bounces can
field can reach the false vacuum with zero velocity. Of have a large effect on the action. To see this, consider a
course, holding the field fixed at the tip of a piecewise piecewiselinearpotentialwithtwoslightlydifferentkink
linear potential is all but impossible. This should be positions φ but same slope λ . Suppose that we would
− −
thought of as an approximation to the physical situation arrange this shift in φ such that the bounce solution of
−
of smoothing the tip with a cap. thismodifiedkinkpotentialandthebounceoftheregular
5
potential coincide. The potential with the more remote V. EXOTIC CAPS
kink position φ has a smaller R . According to (23),
− 0
a shift ∆φ− leads generically to a change in the bounce IntheprecedingSections,wedemonstratedthatunder
action of order δB/B ∼ ∆φ−/φ− and hence be small. most circumstances, smoothing the kink in a piecewise
Only in the thin wall regime where ∆ (cid:39) 1 this change linear potential is equivalent to holding the field fixed at
can be large and of order someradiusR . Thisstatementdependscruciallyonthe
0
choiceofcapthatreplacesthekink. Theshapeofthecap
δB 3 1 ∆φ
(cid:39) √ −. (25) must be such that the field can spend an arbitrarily long
B 2 ∆−1 φ− timeRclosetothetruevacuumtoallowthefrictionterm
to get sufficiently small. This time R is approximately
Fortunately, ∆φ cannot be larger than δφ. We prove
− (assuming that friction dominates over acceleration in
thisbycontradiction. Thesolutiontothefieldequations
the equation of motion)
of motion are (see Eq. (16))
3dφ (cid:90) φ(R) dφ 1
φi(r) = φ−+∆φ−− (r2−(R80r+2 δR0)2)2λ− .(26) r dr ≈V(cid:48)(φ) ⇒ φ0 V(cid:48)(φ) = 6R2. (34)
Now assume δφ (cid:28) ∆φ (cid:28) φ : the field leaves the cap To spend an arbitrary and potentially infinite amount of
− −
at time near the cap, the integral must diverge in the limit
φ → φ . Certainly this is true for analytic potentials
0 −
r2−(R0+δR0)2 (cid:39)(cid:112)8∆φ−(R0+δR0)2/λ− (27) with a finite mass |V(cid:48)(cid:48)(φ−)| < ∞. For example, taking
V(φ) = 1m2φ2, it is clear that the integrand becomes
2
with velocity 1/V(cid:48)(φ)=1/m2φ which is logarithmically divergent.
However, potentials such that the integral is finite in
(cid:112)
φ(cid:48) (cid:39) 2∆φ λ . (28) the limit φ → φ do also exist. One class of examples
i − − 0 −
arepotentialsoftheformV ∝|φ−φ |α,with1<α<2,
−
However, due to energy conservation and the condition whichweshallnowanalyzeinmoredetail. Thederivative
(24), the energy at the border of the cap cannot exceed ofthepotentialattheminimumisV(cid:48)(φ )=0inthelimit
∗
the potential energy. This implies φ →φ both from the left and from the right.
∗ −
To set up the full picture, let us assume a piecewise
φ(cid:48)i <∼(cid:112)2λ−δφ, (29) potential, where we examine the area around the cap.
Weassumethattheotherpiecewisepartsofthepotential
and hence ∆φ− <∼δφ such that contain at least one other local minimum. The complete
bounceisgivenbymatchingthesolutionsineachpartof
δB 3 1 δφ
<∼ √ . (30) the potential. We solve the equations of motion for the
B 2 ∆−1φ− bounce in the part of the potential describing the cap
where V(φ) = λ|φ−φ |α, 1 < α < 2. The equation of
−
Thus we demand
motion reads
√ δφ 3
∆−1(cid:29) , (31) φ(cid:48)(cid:48)+ φ(cid:48)−λαsgn(φ−φ )|φ−φ |α−1 = 0, (35)
φ− r − −
in order to obtain accurate results for the bounce action subjecttotheboundaryconditionsφ(0)=φ , φ(cid:48)(0)=0.
0
in the kink approximation. Even in the limit φ → φ , the solution spends only a
0 −
Even though our reasoning above seems very conser- finite amount of time in the cap region and is given by
vative, no better upper bound on the variation of the
action exists in the thin-wall regime. The boundary of (cid:18)4αλ(α−2)2(cid:19)2−1α
the relation (31) is equivalent to φ(r) = φ−+ 3−α r2−2α . (36)
|V+−V−|(cid:39)λ−δφ, (32) Hence,eventhoughthefieldstartsofffromtheextremum
where there is no force (V(cid:48)(φ ) = 0), it begins rolling
−
but in this case the potential difference between the true away in finite time. However, this does not imply that
and false vacua in the regular capped potential can be there is no bounce. The field can wait for some time at
very different from that in the kink potential. This can φ and then still leave the cap in finite time (friction is
−
lead to grossly different bounce actions. For this special even less important than in the previous case). This so-
situation, the constraint in (30) is saturated. lutioncanbeobtainedbyusingtheboundaryconditions
Combining the criteria (24) and (31), our two condi- φ(R ) = φ and φ(cid:48)(R ) = 0, and sending the release
0 0 0
tions on the kink approximation read point subsequently to φ . If the waiting period R is
− 0
chosen appropriately, the field reaches the false vacuum,
|V−cap−V−kink|<∼λ−δφ(cid:28)|V+kink−V−kink|. (33) thus constituting a bounce solution.
6
Numerically, the bounce solution is often determined kinetic energy in the cap; the second inequality results
using the shooting algorithm [1]. In this case, a release from the fact that the bounce action is very sensitive to
positionforthefieldischosenandthecorrespondingini- the potential difference between the true and the false
tialvalueproblemusingtheequationofmotionissolved. vacuum in the thin-wall regime.
The release point is then varied until the correct bound-
For example, this includes potentials that fluctuate
aryconditionsofthebouncesolutioninthefalsevacuum
strongly or that do not have a finite second derivative
are fulfilled. Obviously, a bounce of the kind described
in the true vacuum, as for potentials that behave close
above cannot be found with the conventional shooting
to the true vacuum as V (cid:39) (φ−φ )α with 1 < α < 2.
algorithm. −
In particular, for the latter class of potentials, the kink
This situation might be rather surprising, since the
approximation yields accurate results even though the
potential is smooth and even differentiable everywhere.
bounce solutions cannot be obtained from the regular
This special situation arises because the equation of mo-
potential using the conventional shooting algorithm.
tion for the bounce in the potential (35) with α < 2
doesnotfulfilltheLipschitzcondition. HencethePicard- Violations of the condition Eq. (37) can appear for in-
Lindel¨oftheoremdoesnotholdandasolutiontotheini- stance within the context of the 4D effective field theory
tialvalueproblemisnotnecessarilyunique. Thiskindof approximation to a theory of quantum gravity such as
situation is well known in the philosophy of science com- string theory. It is not clear how to self-consistently de-
munity [12] in the context of classical mechanics. For- scribe caps with curvature larger than Mp2 within this
tunately, here we need not concern ourselves with the framework. In particular, in such a high curvature cap,
initial value problem. We are interested in computing steep local maxima acting as large positive spikes as dis-
the tunneling amplitude, and the bounce solution with cussed before cannot be excluded. In a situation where
the usual boundary conditions is indeed unique. So for the cap is confined to such a region of strong quantum
this kind of potential, one can be in the situation that gravity effects, we can no longer guarantee that condi-
the kink approximation describes the tunneling process tion (37) is satisfied from a calculation of the cap from
quite well, while the common shooting algorithm in the effective field theory. Thus, a description of the tunnel-
smooth and regular potential fails. ing using the description of [1] and [3] entirely within
the realm of effective field theory is not possible for a
high curvature cap. In this case, full quantum gravity
VI. CONCLUSIONS effects must be incorporated to calculate the tunneling
amplitude.
We have addressed the issue of the existence of a Possibleexamplesinstringtheoryaree.g.warpedcom-
bounce solution in an effective potential that has sharp pactificationswithD3-branes. Insuchcompactifications,
minima and maxima (’kinks’), see Figure 1. Tunneling thewarpfactorcontributestothe4Deffectivescalarpo-
in this setup was first discussed analytically in [10]. For tential. To leading order, the warp factor, and in turn
certain ranges of parameters, a consistent bounce solu- its contribution to the 4D scalar potential, may develop
tion exists only if the field can rest for some amount of ’kinks’ at the position of a D3-brane, similar to the case
Euclidean time at the true vacuum. With this field pro- of 5D warped Randall-Sundrum compactifications [13].
file, it is possible for the relevant friction term to die off String theory effects can smooth such ’kinks’, but the
sufficiently so that the field can roll back up to the false curvature of the smoothing cap may then be too large,
vacuum. Having the field ’wait’ in such a manner is only asmentionedabove,foratreatmentwithineffectivefield
an approximation to the full bounce solution where the theory alone.
tip represents a smooth cap of a regular potential.
We found that replacing a regular smooth potential
by its piecewise linear approximation is a very robust
procedure. A sufficient criterion for the bounce action of
the kink potential to yield accurate results is given by Acknowledgments
|V−cap−V−kink|<∼λ−δφ(cid:28)|V+kink−V−kink|. (37)
TheauthorswishtothankL.Coviforstimulatingdis-
Here, δφ is the scale on which the kink is resolved in cussions and S. Bobrovskyi for bringing the work [12]
the smooth and regular potential, λ denotes the slope to our attention. This work was supported by the Im-
−
in the kink potential close to the true minimum and V puls und Vernetzungsfond of the Helmholtz Association
∓
denotevaluesofthepotentialinthetrue(false)vacuum, of German Research Centers under grant HZ-NG-603,
respectively. and German Science Foundation (DFG) within the Col-
The first inequality reflects the fact that the bounce laborative Research Center 676 “Particles, Strings and
actionvariesstronglyifthefieldcanaccumulateasizable the Early Universe”.
7
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