Gyula Fodor et al- Oscillons in dilaton-scalar theories

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Oscillons in dilaton-scalar theories

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Published by IOP Publishing for SISSA

Received : June 23, 2009 Accepted : August 7, 2009

Published : August 27, 2009

Oscillons in dilaton-scalar theories

Gyula Fodor,a

Peter Forg acs,a,b

Zalan Horvathc

and Mark Mezeic

a MTA RMKI,H-1525 Budapest 114, P.O. Box 49, Hungary

b LMPT, CNRS-UMR 6083, Universite de Tours,Parc de Grandmont, 37200 Tours, France

c Institute for Theoretical Physics, E otv os University,H-1117 Budapest, P azm any Peter set any 1/A, Hungary E-mail: [email protected] , [email protected] ,[email protected] , [email protected]

Abstract: It is shown by both analytical methods and numerical simulations that ex-tremely long living spherically symmetric oscillons appear in virtually any real scalar eldtheory coupled to a massless dilaton (DS theories). In fact such dilatonic oscillons arealready present in the simplest non-trivial DS theory a free massive scalar eld coupledto the dilaton. It is shown that in analogy to the previously considered cases with a singlenonlinear scalar eld, in DS theories there are also time periodic quasibreathers (QB) as-sociated to small amplitude oscillons. Exploiting the QB picture the radiation law of thesmall amplitude dilatonic oscillons is determined analytically.

Keywords: Solitons Monopoles and Instantons, Nonperturbative Effects

ArXiv ePrint: 0906.4160

c SISSA 2009 doi:10.1088/1126-6708/2009/08/106
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://arxiv.org/abs/0906.4160http://dx.doi.org/10.1088/1126-6708/2009/08/106http://dx.doi.org/10.1088/1126-6708/2009/08/106http://dx.doi.org/10.1088/1126-6708/2009/08/106http://arxiv.org/abs/0906.4160mailto:[email protected]:[email protected]:[email protected]:[email protected]


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Contents

1 Introduction 1

2 The scalar-dilaton system 3

3 The small amplitude expansion 43.1 The Schr odinger-Newton equations 53.2 Absence of odd powers in the expansion 83.3 Higher orders in the expansion 83.4 Free scalar eld in 2 < D < 6 dimensions 93.5 4 order for D = 6 113.6 Total energy and dilaton charge of oscillons 12

4 Time evolution of oscillons 13

5 Determination of the energy loss rate 165.1 Singularity of the small expansion 175.2 Fourier mode expansion 185.3 Fourier mode equations 195.4 0 limit near the pole 205.5 Order corrections near the pole 225.6 Extension to the real axis 24

1 Introduction

Long-living, spatially localized classical solutions in eld theories containing scalar eldsexhibiting nearly periodic oscillations in time oscillons [1][16] have attracted con-siderable interest in the last few years. Oscillons closely resemble true breathers of theone-dimensional ( D = 1) sine-Gordon (SG) theory, which are time periodic and are expo-nentially localized in space, but unlike true breathers they are continuously losing energy byradiating slowly. On the other hand oscillons exist for different scalar potentials in variousspatial dimensions, in particular for D = 1 , 2, 3. Just like a breather, an oscillon possessesa spatially well localized core, but it also has a radiative region outside of the core.Oscillons appear from rather generic initial data in the course of time evolution in an im-pressive number of physically relevant theories including the bosonic sector of the standardmodel [17][20]. Moreover they form in physical processes making them of considerableimportance [ 21][28]. In a series of papers, [12, 29][31], it has been shown that oscillonscan be well described by a special class of exactly time-periodic quasibreathers (QB).QBs also possess a well localized core in space (just like true breathers) but in addition

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they have a standing wave tail whose amplitude is minimized. At this point it is importantto emphasize that there are (innitely) many time periodic solutions characterized by anasymptotically standing wave part. In order to select one solution, we impose the conditionthat the standing wave amplitude be minimal. This is a physically motivated condition,which heuristically should single out the solution approximating a true breather as wellas possible, for which this amplitude is identically zero. The amplitude of the standingwave tail of a QB is closely related to that of the oscillon radiation, therefore its computa-tion is of prime interest. It is a rather non-trivial problem to compute this amplitude evenin one spatial dimensional scalar theories [30, 32]. In the limit when the core amplitudeis small, we have developed a method to compute the leading part of the exponentiallysuppressed tail amplitude for a general class of theories in various dimensions [ 31].

In this paper we show that oscillons also appear in rather general (real) scalar eldtheories coupled to a (massless) dilaton eld (DS theory). Dilaton elds appear naturallyin low energy effective eld theories derived from superstring models [3335] and the studyof their effects is of major interest. As the present study shows, the coupling of a dilatoneven to a free massive scalar eld, referred to as the dilaton-Klein-Gordon (DKG) the-ory, which is the conceivably simplest non-trivial DS theory, has some rather remarkableconsequences. This simple DKG theory already admits QBs and as our numerical investi-gations show from generic initial data small amplitude oscillons evolve. We concentrate onsolutions with the simplest spatial geometry spherical symmetry. We do not think thatconsidering spherically symmetric congurations is a major restriction since non-symmetriccongurations are expected to contain more energy and to evolve into symmetric ones [25].The dilatonic oscillons are very robust and once formed from the initial data they do noteven seem to radiate their energy, hence their lifetime is extremely long (not even detectableby our numerical methods).

Our means for constructing dilatonic oscillons will be the small amplitude expansion,in which the small parameter, , determines the difference of oscillation frequency fromthe mass threshold. The small amplitude oscillons of the DKG theory appear to be stablein dimensions D = 3 , 4, unstable in D = 5 , 6, and their core amplitude is proportional to2. This is to be contrasted to self-interacting scalar theories whose oscillons are stable inD = 1 , 2, unstable in D = 3, and their core amplitude is proportional to . The masterequations determining oscillons to leading order in the small amplitude expansion turn outto be the Schrodinger-Newton (SN) equations. The main analytical result of this paper is

the analytic computation of the amplitude of the standing wave tail of the dilatonic QBsfor any dimension D , and thereby the determination of the radiation law and the lifetimeof small amplitude oscillons in DS theories. The used methods have been developed inrefs. [30, 32, 36, 37] and [31].

The above results, namely the stability properties and the SN equations playing therole of master equation, show striking similarity to those obtained in the Einstein-Klein-Gordon (EKG) theory, i.e. for a free massive scalar eld coupled to Einsteins gravity,where also stable, long living oscillons (known under the name of oscillating soliton stars,or more recently oscillatons) have been found and investigated in many papers [38][44].

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2 The scalar-dilaton system

The action of a scalar-dilaton system is

A = dt dDx

12( )2 +

12( )2 e2 U() , (2.1)

where is the dilaton eld and is a scalar eld with self interaction potential U().The energy corresponding to the action ( 2.1) can be written as

E = dD x E , E = 12 ( t )2 + ( i )2 + ( t)2 + ( i)2 + e2U() , (2.2)where E denotes the energy density. In the case of spherical symmetry

E =

0dr

D/ 2r D 1

(D/ 2)( t )2 + ( r )2 + ( t)2 + ( r)2 + 2 e2U() . (2.3)

We assume that the potential can be expanded around its minimum at = 0 as

U() =

k=1

gkk + 1

k+1 , U () =

k=1

gkk , (2.4)

where gk are real constants. For a free massive scalar eld with mass m the only nonzerocoefficient is g1 = m2. If g2k = 0 for integer k the potential is symmetric around itsminimum. In that case, as we will see, for periodic congurations the Fourier expansion of in t will contain only odd, while the expansion of only even Fourier components. For

spherically symmetric systems the eld equations are

2t 2

+ 2r 2

+D 1

r r

= e2 U () , (2.5)

2t 2

+ 2r 2

+D 1

rr

= 2 e2 U() . (2.6)Since g1 = m is intended to be the mass of small excitations of at large distances, welook for solutions satisfying 0 for r . Finiteness of energy also requires 0 asr . Rescaling the coordinates as t t/m and r r/m we rst set g1 = m2 = 1. Thenredening / (2) and / (2) and appropriately changing the constants gk wearrange that 2 = 1. If for some reason we obtain a solution for which tends to a nonzeroconstant at innity then the dilatation symmetry of the system allows us to shift andrescale the coordinates so that it is transformed to a solution satisfying 0 for r .

An important feature of a localized dilatonic conguration is its dilaton charge, Q.It can be dened for almost time-periodic spherically symmetric congurations like oscil-lons as:

Q r 2D for r in D = 2 (2.7)Q ln r for r in D = 2 . (2.8)

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3 The small amplitude expansion

In this section we will construct a nite-energy family of localized small amplitude solutionsof the spherically symmetric eld equations ( 2.5) and ( 2.6) which oscillate below the mass

threshold [36]. It will be shown that such solutions exist for 2 < D < 6. The subtletiesof the case D = 6 will be dealt with in subsection 3.5. The result of the small amplitudeexpansion is an asymptotic series representation of the core region of a quasibreather oroscillon, but misses a standing or outgoing wave tail whose amplitude is exponentially smallwith respect to the core. The amplitude of the tail will be determined in section 5.

We are looking for small amplitude solutions, therefore we expand the scalar elds, and , in terms of a parameter as

=

k=1

kk , =

k=1

kk , (3.1)

and search for functions k and k tending to zero at r . The size of smoothcongurations is expected to increase for decreasing values of , therefore it is natural tointroduce a new radial coordinate by the following rescaling

= r . (3.2)

In order to allow for the dependence of the time-scale of the congurations a new timecoordinate is introduced as

= ()t . (3.3)

Numerical experience shows that the smaller the oscillon amplitude is the closer its fre-quency becomes to the threshold = 1. The function () is assumed to be analytic near = 1, and it is expanded as

2() = 1 +

k=1

kk . (3.4)

We note that there is a considerable freedom in choosing different parametrisations of the small amplitude states, changing the actual form of the function (). The physicalparameter is not but the frequency of the periodic states that will be given by . Afterthe rescalings eqs. ( 2.5) and ( 2.6) take the following form

2 2 2

+ 2 22

+ 2D 1

= e + k=2

gkk , (3.5)

2 2 2

+ 2 22

+ 2D 1

= e12

2 +

k=2

gkk + 1

k+1 . (3.6)

Substituting the small amplitude expansion ( 3.1) into ( 3.5) and ( 3.6), to leading order we obtain

21 2

+ 1 = 0 , 21 2

= 0 . (3.7)

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Since we are looking for solutions which remain bounded in time and since we are free toshift the origin = 0 of the time coordinate, the solution of ( 3.7) can be written as

1(, ) = P 1()cos , 1(, ) = p1() , (3.8)

where P 1() and p1() are some functions of the rescaled radial coordinate .The 2 terms in the expansion of ( 3.6) yield

22 2

=14

P 21 [1 + cos(2 )] . (3.9)

This equation can have a solution for 2 which remains bounded in time only if the timeindependent term in the right hand side vanishes, implying P 1 = 0 and consequently1 = 0. Then the solution of ( 3.9) is 2(, ) = p2(). The 2 terms in ( 3.5) yield

22

2 + 2 = 0 . (3.10)

Since 1 = 0 we are again free to shift the time coordinate, and the solution is 2(, ) =P 2()cos .

The 3 order terms in the expansion of ( 3.6) give

23 2

=d2 p1d2

+D 1

dp1d

. (3.11)

In order to have a solution for 3(, ) that remains bounded in time, the right hand sidemust be zero, yielding p1() = p11 + p122D when D = 2 and p1() = p11 + p12 ln forD = 2, with some constants p11 and p12 . Since we look for bounded regular solutions

tending to zero at , we must have p11 = p12 = 0. As we have already seen that1 = 0, this means that the small amplitude expansion ( 3.1) starts with 2 terms. Thesolution of ( 3.11) is then 3(, ) = p3(). The 3 order terms in the expansion of ( 3.5) give

23 2

+ 3 1P 2 cos = 0 . (3.12)This equation can have a solution for 3 which remains bounded in time only if the res-onance term proportional to cos vanishes, implying 1 = 0. After applying an 3 ordersmall shift in the time coordinate, the solution of ( 3.12) is 3(, ) = P 3()cos . Continu-ing to higher orders, the basic frequency sin term can always be absorbed by a small shift

in . It is important to note that after transforming out the sin terms no sin( k ) termswill appear in the expansion, implying the time reection symmetry of and at = 0.

3.1 The Schr odinger-Newton equations

The 4 terms in the expansion of ( 3.5) and ( 3.6) yield the differential equations

24 2

+ 4 =d2P 2d2

+D 1

dP 2d

+ ( p2 + 2)P 2 cos 12

g2P 22 [1 + cos(2 )] , (3.13)

24 2

=d2 p2d2

+D 1

dp2d

+14

P 22 [1 + cos(2 )] . (3.14)

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The function 4(, ) and 4(, ) can remain bounded only if the cos resonance termsin (3.13) and the time independent terms in ( 3.14) vanish,

d2P 2

d2 +

D 1

dP 2

d+ ( p2 + 2)P 2 = 0 , (3.15)

d2 p2d2

+D 1

dp2d

+14

P 22 = 0 . (3.16)

Then the time dependence of 4(, ) and 4(, ) is determined by ( 3.13) and ( 3.14) as

4(, )= P 4()cos +16

g2P 22 [cos(2 )3] , 4(, )= p4()116

P 2()2 cos(2 ) . (3.17)

Here we see the rst contribution of a nontrivial U() potential, the term proportional tog2 in 4. If (and only if) the potential is non-symmetric around its minimum, even Fouriercomponents appear in the expansion of .

Introducing the new variables

S =12

P 2 , s = p2 + 2 , (3.18)

(3.15) and ( 3.16) can be written into the form which is called the time-independentSchrodinger-Newton (or Newton-Schr odinger) equations in the literature:

d2S d2

+D 1

dS d

+ sS = 0 , (3.19)

d2sd2

+D 1

dsd

+ S 2 = 0 . (3.20)

We look for localized solutions of these equations, in order to determine the core part of small amplitude oscillons to a leading order approximation in . The main features of thesolutions depend on the number of spatial dimensions D . For D 6 positive monotonicallydecreasing solutions necessarily satisfy s = S , they tend to zero, furthermore, the Lane-Emden equation holds [45]

d2sd2

+D 1

dsd

+ s2 = 0 . (3.21)

For D > 6 solutions are decreasing as 1 / 2 for large , consequently they have inniteenergy. It can also be shown that solutions of the original Schr odinger-Newton system

with s = S , and a necessarily oscillating scalar eld, have innite energy, hence there isno nite energy solution for D > 6. For D = 6 the explicit form of the asymptoticallydecaying solutions of ( 3.21) are known

s = S =242

(1 + 22)2, (3.22)

where is any constant. Since the replacement of with and a simultaneous reectionof the potential around its minimum is a symmetry of the system, we choose the positivesign for S in (3.22). For D = 6 the total energy remains nite.

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If D < 6, then localized solutions have the property that for large values of thefunction S tends to zero exponentially, while s behaves as s s0 + s12D for D = 2 andas s s0 + s1 ln for D = 2, where s0 and s1 are some constants. Since we are interestedin localized solutions we assume 2 < D < 6. From ( 3.19) it is apparent that exponentiallylocalized solutions for S can only exist if s tends to a negative constant, i.e. s0 < 0. In thiscase the localized solutions of the Schr odinger-Newton (SN) equations ( 3.19) and ( 3.20)can be parametrized by the number of nodes of S . The physically important ones arethe nodeless solutions satisfying S > 0, since the others correspond to higher energy andless stable oscillons.

Motivated by the asymptotic behaviour of s, if D = 2 it is useful to introduce thevariables

=D 12 D

dsd

, = s 2D . (3.23)In 2 < D < 6 dimensions these variables tend exponentially to the earlier introducedconstants

lim

= s1 , lim = s0 . (3.24)

Then the SN equations can be written into the equivalent form

dd

+D12 D

S 2 = 0 , (3.25)

d d

+

D 2S 2 = 0 , (3.26)

d2S

d2 +

D 1

dS

d+ + 2D S = 0 . (3.27)

The SN equations ( 3.19) and ( 3.20) have the scaling invariance

(S (), s ()) (2S (), 2s()) . (3.28)If 2 < D < 6 we use this freedom to make the nodeless solution unique by setting s0 = 1.At the same time we change the parametrization by requiring

2 = 1 fo r 2 < D < 6 , (3.29)

ensuring that the limiting value of vanishes to 2

order. Going to higher orders, it canbe shown that one can always make the choice i = 0 for i 3, thereby xing the parametrization, and setting

= 1 2 for 2 < D < 6 . (3.30)If D = 6, since both s and S tend to zero at innity, we have no method yet to x the valueof in (3.22). Moreover, in order to ensure that tends to zero at innity we have to set

2 = 0 for D = 6 . (3.31)

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3.2 Absence of odd powers in the expansion

Calculating the 5 order equations from ( 3.5) and ( 3.6) and requiring the boundedness of 5 and 5 we obtain a pair of equations for P 3 and p3:

d2P 3d2

+ D 1 dP 3d + ( p2 + 2)P 3 + P 2( p3 + 3) = 0 , (3.32)d2 p3d2

+D 1

dp3d

+12

P 2P 3 = 0 . (3.33)

These equations are solved by constant multiples of

P 3 = 2 P 2 + dP 2d

, p3 + 3 = 2( p2 + 2) + dp2d

, (3.34)

corresponding to the scaling invariance ( 3.28) of the SN equations. In D > 2 dimensions p3 given by ( 3.34) tends to 3 2 for large . Since we are looking for solutions for which tends to zero asymptotically, after choosing 3 = 0 we can only use the trivial solutionP 3 = p3 = 0. The important consequence is that 3 = 3 = 0. Going to higher ordersin the expansion, at odd orders we get the same form of equations as ( 3.32) and ( 3.33),consequently, all odd coefficients of k and k can be made to vanish. Instead of the moregeneral form ( 3.1) we can write the small amplitude expansion as

=

k=1

2k2k , =

k=1

2k2k . (3.35)

3.3 Higher orders in the expansion

The 6

order equations, after requiring the boundedness of 6 and 6 , yield a pair of equations for P 4 and p4:

d2P 4d2

+D 1

dP 4d

+ ( p2 + 2)P 4 + P 2( p4 + 4)

12 p22P 2

132

P 32 +56

g22 34

g3 P 32 = 0 , (3.36)

d2 p4d2

+D 1

dp4d

+12

P 2P 4 14 p2P 22 = 0 . (3.37)

This is an inhomogeneous linear system of differential equations with nonlinear, asymp-totically decaying source terms given by the solutions of the SN equations. Since thehomogeneous terms have the same structure as in ( 3.32) and ( 3.33), one can always addmultiples of

P (h)4 = 2 P 2 + dP 2d

, p(h )4 + 4 = 2( p2 + 2) + dp2d

, (3.38)

to a particular solution of ( 3.36) and ( 3.37). If 2 < D < 6 we are interested in solutions forwhich at large radii P 4 decays exponentially, and p4 q0 + q12D with some constants q0and q1. We use the homogeneous solution ( 3.38) to make q0 = 0. Since similar choice canbe made at higher orders, this will ensure that the limit of will remain zero at .We note that, in general, it is not possible to make q1 also vanish, implying a nontrivial dependence of the dilaton charge Q.

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The resulting expressions for the original and functions are

= 2P 2 cos + 4 P 4 cos +16

g2P 22 [cos(2 ) 3] + 6 P 6 cos

+P 32256

1 +163

g22 + 8 g3 cos(3 ) g2 P 2P 4 ( p2 + 2)P 22 +dP 2d

2(3.39)

+g29

3P 2P 4 ( p2 + 2)P 22 dP 2d

2cos(2 ) + O(8) ,

= 2 p2 + 4 p4 P 2216

cos(2 ) + 6 p6 132

4P 2P 4 ( p2 + 2)P 22 dP 2d

2cos(2 )

+154

g2P 32 [9cos cos(3 )] + O(8) , (3.40)

where the functions P 2 and p2 are determined by the SN equations ( 3.15) and ( 3.16), P 4and p4 can be obtained from ( 3.36) and ( 3.37), furthermore, the equations for P 6 and p6can be calculated from the 8 order terms as

d2P 6d2

+D 1

dP 6d

+ ( p2 + 2)P 6 + ( p6 + 6)P 2 + p4 + 4 p222

P 4

3

32 52

g22 +94

g3 P 22 P 4 p2P 2 p4 +364

4918

g22 +34

g3 p2P 32 (3.41)

+1

64 179

g22 2P 32 +

16 p32P 2 + P 2

164

+199

g22dp2d

2

= 0 ,

d2 p6d2

+D 1

dp6d

+12

P 2P 6 +14

P 24 12 p2P 2P 4

14

P 22 p4 +18 p22P

22 +

P 4216

18

119

g22 +32

g3 = 0 . (3.42)

We remind the reader that the only non-vanishing k for 2 < D < 6 is 2 = 1, and wewill show in subsection 3.5, that in general, for D = 6 the only nonzero component is 4.The above expressions, especially those for and , simplify considerably for symmetricU() potentials, in which case g2 = 0.

3.4 Free scalar eld in 2 < D < 6 dimensionsIf is a free massive eld with potential U() = m22/ 2, after scaling out m and noparameters remain in the equations determining P i and pi . The spatially localized nodelesspositive solution of the ordinary differential equations ( 3.15), (3.16), and the correspondingsolution of ( 3.36), (3.37), (3.41) and ( 3.42) can be calculated numerically. For D = 3 theobtained curves are shown on gures 1 and 2. The obtained central values of P i and pifor i = 2 , 4, 6 in D = 3 , 4, 5 are collected in table 1. The chosen central values make allfunctions P i and pi , and consequently i and i , tend to zero for . Although for i 4P i and pi are not monotonically decreasing functions, their central values represent well themagnitude of these functions. Generally, the validity domain of an asymptotic series ends

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0

0.5

1

1.5

2

0 2 4 6 8 10 12

P k

P 2P 4P 6

Figure 1 . The rst three P k functions for the free scalar eld case in D = 3 spatial dimensions.

0

0.20.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 5 10 15 20

p k

p2p4p6

Figure 2 . The pk functions for the free scalar eld case in D = 3 dimensions.

D = 3 D = 4 D = 5P 2c 2.04299 7.08429 28.0399 p2c 1.93832 4.42976 14.90729

P 4c 0.658158 -5.93174 -348.868 p4c 0.686532 -4.08270 -200.353P 6c 0.557141 27.3950 9532.72 p6c 0.541339 17.9090 5500.18

Table 1 . Central values of the rst three functions P i and pi for the free scalar eld in 3, 4 and 5spatial dimensions.

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where a higher order term starts giving larger contributions than previous order terms. ForD = 3 the sixth order expansion can be expected to be valid even for as large parametervalues as = 1. For D = 4 this domain is < 0.7, while for D = 5 it decreases to < 0.22.

3.5 4

order for D = 6As we have already stated in subsection 3.1, if D = 6 then 2 = 0, s = S and the explicitform of the solution of the SN equations is given by ( 3.22). Introducing the new variablesz and Z by

P 4 =13

(2Z + z) , p4 + 4 =13

(Z z) , (3.43)equations ( 3.36) and ( 3.37) decouple,

d2zd2

+5

dzd sz +

34

1s3 = 0 , (3.44)

d2Z d2 +

5

dZ d + 2 sZ

94 2s3 = 0 , (3.45)

where the constants 1 and 2 are dened by the coefficients in the potential as

1 = 1 +809

g22 8g3 , (3.46) 2 = 1

8027

g22 +83

g3 . (3.47)

For a free scalar eld with U() = 2/ 2 we have 1 = 2 = 1. The general regular solutionof (3.44) can be written in terms of the (complex indexed) associated Legendre function P as

z = 144 1 4(14 + 6 22 + 44)13(1 + 22)4

+ C 1 22

P 2(i231)/ 21 221 + 22 , (3.48)

where C 1 is some constant. The limiting value at is z = C 1 cosh(23/ 2)/ 297.495 C 1 . The regular solution of ( 3.45) is

Z =3888 24(1 22)

7(1 + 22)3ln(1 + 22) (3.49)

+324 2 62(220 + 100 22 16 44 66)

35(1 + 22)4+ C 2

22 1( 22 + 1) 3

,

The limiting value at is Z = 324 24

/ 35, independently of C 2. Since P 4 musttend to zero, according to ( 3.43), z = 648 24/ 35, xing the constant C 1. Since the mass

of the eld is intended to remain m = 1, the limit of p4 also has to vanish, giving

4 = 32435

2 4 . (3.50)

This expression is not enough to x 4 yet, since is a free parameter. If 2 > 0 then it isreasonable to use ( 3.50) to set 4 = 1, thereby xing the free parameter in the 2 ordercomponent of and . The change of the so far undetermined constant C 2 corresponds toa small rescaling of the parameter in the expression ( 3.22). Its concrete value will x thecoefficient 6 in the expansion of the frequency. The homogeneous parts of the differential

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D = 3 D = 4 D = 5s1 3.50533 7.69489 10.4038E 0 88.0985 607.565 1642.91E 1 123.576 2522.10 31374.2

Table 2 . The numerical values of s1 , E 0 and E 1 in 3, 4 and 5 spatial dimensions.

equations at higher order will have the same structure as those for P 4 and p4. Choosingthe appropriate homogeneous solutions all higher k components can be set to zero, yielding

= 1 4 for D = 6 if 2 > 0 . (3.51)This expression is valid for the free scalar eld case with potential U() = 2/ 2 in D = 6,since then 2 = 1. For certain potentials 2 < 0, and one can use ( 3.50) to set 4 = 1.This case is quite unusual in the sense that the frequency of the oscillon state is above thefundamental frequency m = 1. In the very special case, when 2 = 0 the frequency differsfrom the fundamental frequency only in 6 or possibly higher order terms.

3.6 Total energy and dilaton charge of oscillons

Substituting ( 3.39) and ( 3.40) into the expression ( 2.3) of the total energy, we get

E = 4D E 0 + 6D E 1 + O(8D ) , (3.52)where

E 0 =D/ 2

(D/ 2)

0d D1P 22 , E 1 =

D/ 2

(D/ 2)

0d D 1P 2(2P 4 P 2) . (3.53)

Since P 2 = 2 S , for 2 < D < 6 we can use (3.24) and ( 3.25) to get

E 0 =4D/ 2

(D/ 2)(D 2)s1 . (3.54)

The numerical values of s1, E 0 and E 1 for D = 3 , 4, 5 are listed in table 2. To thecalculated order, i.e. up to 6D , for D = 3 and D = 4 the energy is a monotonicallyincreasing function of , while for D = 5 there is an energy minimum at = 0 .2288. Thisresult can only be taken as an estimate, as the validity domain of an asymptotic series endswhen two subsequent terms are approximately equal.

For D = 6 the leading order term in the total energy is

E =1923

22. (3.55)

As we have already noted, for D > 6 there are no nite energy solutions.The leading order dependence of the dilaton charge for 2 < D < 6 is given by

Q = s14D , (3.56)where we used the denition ( 2.7), (3.24) and the relation = r . The dilaton charge forthe D = 6 oscillon is innite. In higher orders in the proportionality between the dilatoncharge and energy is violated.

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0

10

20

30

40

50

60

70

80

0 0.1 0.2 0.3 0.4 0.5

E

D=3

numerical evolutionE0

E0 +E 13

Figure 3 . Total energy of three-dimensional oscillons as a function of the parameter .

4 Time evolution of oscillons

In this section we employ a numerical time evolution code in order to simulate the actualbehaviour of oscillons in the scalar-dilaton theory. We use a fourth order method of line codewith spatial compactication in order to investigate spherically symmetric elds [ 46]. Ouraim is to nd congurations which are as closely periodic as possible. To achieve this, weuse initial data obtained from the leading 2 terms of the small amplitude expansion ( 3.39)and ( 3.40). The smaller the chosen is, the more closely periodic the resulting oscillatingstate becomes. However, for moderate values of , it is possible to improve the initial databy simply multiplying it by some overall factor very close to 1.

The main characteristics of the evolution of small amplitude initial data depend onthe number of spatial dimensions D . For D = 3 and D = 4 oscillons appear to be stable.If there is some moderate error in the initial data, it will still evolve into an extremely longliving oscillating conguration, but its amplitude and frequency will oscillate with a lowfrequency modulation. We employ a ne-tuning procedure to minimize this modulationby multiplying the initial data with some empirical factor. For D = 5 and D = 6 smallamplitude oscillons are not stable, having a single decay mode. In this case we can use thene-tuning method to suppress this decay mode, and make long living oscillon states withwell dened amplitude and frequency. Without tuning in D = 5 and D = 6, in general,an initial data evolves into a decaying state. The tuning becomes possible because thereare two possible ways of decay. One with a steady outwards ux of energy, the other isthrough collapsing to a central region rst.

Having calculated several closely periodic oscillon congurations, it is instructive tosee how closely their total energy follow the expressions ( 3.52)(3.55). Apart from checkingthe consistency of the small amplitude and the time-evolution approaches, this also givesinformation on how large values the small amplitude expansion remains valid. Theparameter for the evolving oscillon is calculated from the numerically measured frequencyby the expression = 1 2 . The results for D = 3 are presented on gure 3. In contrast

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0

10000

20000

30000

40000

50000

60000

70000

0 0.05 0.1 0.15 0.2 0.25

E

D=5

numerical evolutionE0 /

E0 / +E 1

Figure 4 . Total energy of ve-dimensional oscillons as a function of the parameter . The vertical

line at = 0 .21 shows the place of the energy minimum. States to the right of it are stable, whilethose to the left have a single decay mode.

14200

14400

14600

14800

15000

15200

15400

15600

15800

0.16 0.18 0.2 0.22 0.24

E

D=5

numerical evolutionE0 / +E 1

Figure 5 . The region of gure 4 near the energy minimum.

to general relativistic oscillatons, there is no maximum on the energy curve. This indicatesthat all three dimensional oscillons in the dilaton theory are stable.

The dependence of the energy for D = 5 is presented on gure 4. There is anenergy minimum of the numerically obtained states, approximately at = 0 .21, abovewhich oscillons are stable. The place of the minimum agrees quite well with the value = 0 .2288 calculated in subsection 3.6 using the rst two terms of the small amplitudeexpansion. The behaviour of the energy close to the minimum is shown on gure 5. Wehave also constructed oscillon states for D = 6 dimensions. These oscillons have quite largeenergy, due to the slow spatial decay of the functions and . For free massive scalarelds, oscillons have frequency given by ( 3.51), i.e. an initial data with a given value willevolve to an oscillon state with frequency approximately following = 1 4. However,

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1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

0 1000 2000 3000 4000 5000 6000 7000 8000 900010000

t

Figure 6 . Increase in the difference of the dilaton eld for two similar congurations.

there are potentials, for which the oscillation frequency is above the threshold = 1. Forexample, this happens for the potential U() = 2( 2)2/ 8 with the choice = 1 / 2.

In conclusion, in the dilaton-scalar theory oscillons follow the stability pattern observedin the case of self-interacting scalar and Einstein-Klein-Gordon theory; if decreases withdecreasing energy, oscillons are stable, while if increases with decreasing energy, oscillonsare unstable. In other words if the time evolution (i.e. energy loss) of an oscillon leads tospreading of the core, the oscillon is stable, while oscillons are unstable, if they have tocontract with time evolution. The decreasing or increasing nature of the energy, and henceempirically the stability of the oscillating congurations, is well described by the rst twoterms of the small amplitude expansion ( 3.52). The result following from eq. ( 3.52) showsthe existence of an energy minimum for D > 4. This provides an analytical argument forthe existence of at least one unstable mode. In particular, for D = 5 spatial dimensions thefrequency separating the stable and unstable domains is determined by the small amplitudeexpansion to satisfactory precision.

In order to study the instability in more detail numerically, we compared the evolutionof two almost identical initial data obtained from the small amplitude expansion with = 0 .05. In order to make the unstable state long living, a ne tuning procedure is applied,multiplying the amplitude of the initial data by a factor with value close to 1 .0178. Themultiplicative factors used in the two chosen initial values differ by 2 .2

10

16 . One of

the two initial data develops into a conguration decaying with a uniform outward currentof energy, the other through collapsing to a high density state rst. On gure 6 the timeevolution of the difference of the central value of the dilaton elds in the two states =1 2 is shown. The curve follows extremely well the exponential increase described by

= 6 .583 1016 exp(0 .003157 t) , (4.1)showing that there is a single decay mode growing exponentially. The difference of thescalar elds, = 1 2, grows with the same exponent. The spatial dependence of the

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-1.2e-05

-1e-05

-8e-06

-6e-06

-4e-06

-2e-06

0

2e-06

0 10 20 30 40 50 60

r

t=6669.8t=6870.1t=7071.4t=7272.7

Figure 7 . Radial behaviour of the difference of the scalar elds of two very similar congurations.

At the chosen moments of time the scalar is maximal at the center, and the subsequent momentsare separated by 32 oscillations.

decaying mode is illustrated on gure 7, where is plotted at several moments of timecorresponding to the maximum of 1 at the center.

Our numerical results strongly indicate that oscillons in the scalar-dilaton theory areunstable for D > 4, and they admit a single decay mode. For the single scalar eldsystem the instability arises for D > 2 (see ref. [31]), but the decay modes have beencalculated analytically only in some very special cases. The scalar theory with potentialU() = 2(1

ln 2) admits exactly time-periodic breathers in any dimensions. The

stability of these breathers in three dimensions has been investigated in detail in [ 47, 48].It has been found that these breathers always admit a single unstable mode. It needsfurther studies whether an analysis along the lines of ref. [ 47] can also be applied to moregeneral potentials in the small amplitude limit, and whether it can be generalized to thecase when the scalar is coupled to a dilaton eld.

5 Determination of the energy loss rate

Although oscillons are extremely long living, generally they are not exactly periodic. Inthis section we calculate how the energy loss rate depends on the oscillon frequency forsmall amplitude congurations. To simplify the expressions in this section we consider amassive free scalar eld, i.e. U() = m22/ 2. We assume that 2 < D < 6, since then thescalar eld tends to zero exponentially for large .

The outgoing radiation will dominantly be in the dilaton eld and the radiation am-plitude will have the dependence: exp(2QD / ). In refs. [30] and [31] we have usedtwo different methods for determining the independent part of the radiation amplitude:Borel summation and solution of the complexied mode equations numerically. In thispaper we will use an analytic method based on Borel-summing the asymptotic series in theneighborhood of its singularity in the complex plane. Other potentials which are symmet-

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D Q D3 3.977364 2.304685 1.23595

Table 3 . The distance QD between the real axis and the pole of the fundamental solution of theSN equation for various spatial dimensions D.

ric around their minima can be treated analogously. If the potential is asymmetric onlythe numerical method could be used. This phenomenon is in complete analogy with theproblem arising with a single scalar eld considered in ref. [ 30].

5.1 Singularity of the small expansion

We rst investigate the complex extension of the functions obtained by the small ampli-tude expansion in section 3. Extending the solutions s and S of the Schr odinger-Newtonequations ( 3.19) and ( 3.20) to complex coordinates they both have pole singularities onthe imaginary axis of the complex plane. We consider the closest pair of singularities tothe real axis, since these will give the dominant contribution to the energy loss. They arelocated at = iQ D . The numerically calculated values of QD are listed in table 3 forspatial dimensions D = 3 , 4, 5. Let us measure distances from the upper singularity by acoordinate R dened as

= iQ D + R . (5.1)

Close to the pole we can expand the SN equations, and obtain that s and S have essentially

the same behaviour,s = S =

6R2

6i(D 1)5QD R

(D 1)(D 51)50Q2D

+ O(R) , (5.2)even though they clearly differ on the real axis. Since for symmetric potentials we canalways substitute by , we choose the positive sign for S in (5.2). This choice iscompatible with the sign of S used on the real axis at the small amplitude expansionsection. We note that for D > 1 there are logarithmic terms in the expansion of s and S ,starting with terms proportional to R4 ln R. According to ( 3.18), the functions determiningthe leading 2 parts of and in this case are

p2 = s + 1 , P 2 = 2 S . (5.3)

Substituting these into the equations ( 3.36) and ( 3.37), the 4 order contributions p4 andP 4 can also be expanded around the pole

p4 = 116152R4

+324i(D 1)ln R

35QD R3+

c3R3

+ Oln RR2

(5.4)

P 4 2 p4 =81

13R4+

18i(D 1)5QD R3

+ O1

R2, (5.5)

where the constant c3 can only be determined from the specic behaviour of the functionson the real axis, namely from the requirement of the exponential decay of P 4 for large real .

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5.2 Fourier mode expansion

Since all terms of the small amplitude expansion ( 3.1) are asymptotically decaying, i.e.localized functions, the small amplitude expansion can be successfully applied to the core

region of oscillons. However it cannot describe the exponentially small radiative tail re-sponsible for the energy loss. Instead of studying a slowly varying frequency radiatingoscillon conguration it is simpler to consider exactly periodic solutions having a largecore and a very small amplitude standing wave tail. We look for periodic solutions withfrequency by Fourier expanding the scalar and dilaton eld as

=N F

k=0

k cos(kt ) , =N F

k=0

k cos(kt ) . (5.6)

Although, in principle, the Fourier truncation order N F should tend to innity, one can

expect very good approximation for moderate values of N F . In (5.6) we denoted theFourier components by psi instead of phi to distinguish them from the small expansioncomponents in ( 3.1). Since in this section we only deal with an self-interaction free scalareld with a trivially symmetric potential,

2k = 0 , 2k+1 = 0 , for integer k . (5.7)

We note that the absence of sine terms in ( 5.6) is equivalent to the assumption of timereexion symmetry at t = 0. This assumption appears reasonable physically, and we haveseen in section 3 that it holds in the small amplitude expansion framework.

For small amplitude congurations we can establish the connection between the ex-pansions ( 3.1) and ( 5.6) by comparing to ( 3.39) and ( 3.40), obtaining

1 = 2P 2 + 4P 4 + 6P 6 + O(8) , (5.8)3 = 6

P 32256

+ O(8) , (5.9)0 = 2 p2 + 4 p4 + 6 p6 + O(8) , (5.10)2 = 4

P 2216

6 132

4P 2P 4 ( p2 1)P 22 dP 2dR

2+ O(8) . (5.11)

Let us dene a coordinate y for an inner region by R = y. This coordinate willhave the same scale as the original radial coordinate r , since they are related as

r =iQ D

+ y . (5.12)

The inner region |R |1 is not small in the y coordinate; if 0 then |y| = |R|1but |y| . Using the coordinate y and substituting ( 5.2)(5.5) into the small amplitudeFourier mode expressions ( 5.8)(5.11), we obtain that the leading asymptotic behaviour of

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the Fourier modes for |y| can be written as

1 = 12y2

99926y4

+ ln 648i(D 1)

35QD y3

+ 6i(D 1)5QD y 3y2 + 108ln y7y2 2 + 2c3y3 + . . . , (5.13)3 =

274y6

81i(D 1)20QD y5

+ . . . , (5.14)

0 = 6y2

116152y4

+ ln 324i(D 1)

35QD y3

+ 6i(D 1)

5QD y54ln y

7y2 1 +c3y3

+ . . . , (5.15)

2 = 9y4

18i(D 1)5QD y3

+ . . . . (5.16)

These expressions are simultaneous series in 1 /y and in .

5.3 Fourier mode equations

In order to obtain nite number of Fourier mode equations with nite number of terms,when substituting ( 5.6) into the eld equations ( 2.5) and ( 2.6) we Taylor expand andtruncate the exponential

e =N e

k=0

1k!

()k . (5.17)

We need to carefully check how large N e should be chosen to have only a negligible inuenceto the calculated results. For n N F the Fourier mode equations have the form

d2

dr 2+

D 1r

ddr

+ n22 1 n = F n , (5.18)d2

dr 2+

D 1r

ddr

+ n22 n = f n , (5.19)

where we have collected the nonlinear terms to the right hand sides, and denoted them withF n and f n . These are polynomial expressions involving various k and k , with quicklyincreasing complexity when increasing the truncation orders N F and N e . The solutionof (5.18) and ( 5.19) yields the intended quasibreathers, with a localized core and a verysmall amplitude oscillating tail. For small amplitude congurations the functions k andk will have poles at the complex r plane, just as we have seen in the small amplitudeexpansion formalism. In order to calculate the tail amplitude it is necessary to investigatethe Fourier mode equations instead of the equations obtained in section 3. Although inthe Fourier decomposition method we have not dened a small amplitude parameter yet,motivated by ( 3.30), we can, in general, dene as

=

1 2 . (5.20)

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Dropping O(2) terms, in the neighborhood of the singularity the mode equations takethe form

d2

dy2+

D 1iQ D

ddy

+ n2 1 n = F n , (5.21)d2

dy2+

D 1iQ D

ddy

+ n2 n = f n . (5.22)

We look for solutions of these equations that satisfy ( 5.13)(5.16) as boundary conditionsfor |y| for / 2 < arg y < 0. This corresponds to the requirement that the functionsdecay to zero without any oscillating tails for large r on the real axis. The small correctioncorresponding to the nonperturbative tail of the quasibreather will arise in the imaginarypart of the functions on the Re y = 0 axis.

5.4 0 limit near the poleFor very small values one can neglect the terms proportional on the left hand sidesof (5.21) and ( 5.22). In this limit the there is no dependence on the number of spatialdimensions D . We investigate this simpler system rst, and consider nite but small corrections later as perturbations to it. We expand the solution of ( 5.21) and ( 5.22) (with = 0) in even powers of 1 /y ,

2k+1 =

j = k+1

A( j )2k+11

y2 j, 2k =

j = k+1

a ( j )2k1

y2 j. (5.23)

We illustrate our method by a minimal system where radiation loss can be studied, namelythe case with N F = 3 and N e = 1. Then the mode equations are still short enough to print:

d20

dy2= 1

4(0 1)( 21 + 23) + 1821(1 + 2 3) , (5.24)

d21dy2

= 01 12

2(1 + 3) , (5.25)

d22dy2

+ 4 2 =14

1(0 1)( 1 + 2 3) +14

2(21 + 13 + 23) , (5.26)

d23dy2

+ 8 3 = 12

21 03 . (5.27)When looking for solution of these equations in the form of the 1 /y 2 expansion ( 5.23), onlyone ambiguity arises, the sign of A(1)0 . Choosing it to be negative, the rst few terms of the expansion turn out to be

0 = 6y2

83752y4

+ O1y6

, (5.28)

1 = 12y2

45926y4

+ O1y6

, (5.29)

2 = 9y4

184552y6

+ O1y8

, (5.30)

3 = 274y6

2565416y8

+ O1

y10. (5.31)

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N F N (min )e k

3 5 3.71 1034 7 3.12 1065 8 6.03

109

6 10 4.61 1013

Table 4 . Dependence of the constant k on the considered Fourier components N F . The secondcolumn lists the minimal exponential expansion order N e which is necessary to get the k value withthe given precision.

The rst terms agree with those of ( 5.13)(5.16) obtained by the small amplitude expan-sion. The difference in the 1 /y 4 terms of 0 and 1 are caused by the too low truncationfor the Taylor expansion of the exponential. For N e 2 these terms agree as well.

When increasing N F and N e growing number of additional terms appear on the righthand sides of ( 5.24)(5.27), and the number of mode equations rise to N F + 1. Thesecomplicated mode equations can be calculated and 1 /y 2 expanded using an algebraic ma-nipulation program. However, apart from a factor, the leading order behaviour of thecoefficients a (n )2 and A

(n )3 for large n will remain the same as that of the minimal sys-

tem ( 5.24)(5.27). The large n behaviour of these coefficients will be essential for thecalculation of the nonperturbative effects resulting in radiation loss for oscillons.

Starting from the free system, consisting of the linear terms on the left hand sides, itis easy to see that the mode equations are consistent with the following asymptotic (largen) behavior of the coefficients,

a (n )2 k (1)n (2n 1)!4n (5.32)a (n )0 , A

(n )1 , A

(n )3 a

(n )2 , (5.33)

where k is some constant. The value of k can be obtained to a satisfactory precisionby substituting the 1 /y expansion into the mode equations and explicitly calculating thecoefficients to up to high orders in n. In practice, using an algebraic manipulation software,we have calculated coefficients up to order n = 50. The dependence of k on the order of theFourier expansion is given in table 4. The results strongly indicate that in the N f , N e limit k = 0. We do not yet understand what is the deeper reason or symmetry behind this.

Hence, instead of ( 5.32) and ( 5.33), the correct asymptotic behavior is

A(n )3 K (1)n(2n 1)!

8n(5.34)

a (n )0 , A(n )1 , a

(n )2 A

(n )3 . (5.35)

Taking at least N F = 6 and N e = 9, the numerical value of the constant turns out to beK = 0.57 0.01. The above results indicate that the outgoing radiation is in the 3scalar mode instead of being in the 2 dilaton mode. This conclusion is valid only in theframework of the approximation employed in the present subsection, i.e. when dropping

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the terms proportional to in (5.21) and ( 5.22). As we will see in the next subsection, thesituation will change to be just the opposite when taking into account corrections.

All terms of the expansion ( 5.23) are real on the imaginary axis Re y = 0. However,using the Borel-summation procedure it is possible to calculate there an exponentiallysmall correction to the imaginary part. We will only sketch how the summation is done,for details see [30] and [37]. We illustrate the method by applying it to 3. The rst stepis to dene a Borel summed series by

V (z) =

n =2

A(n )3(2n)!

z2n

n =2K

(1)n2n

z8

2n=

K 2

ln 1 +z2

8. (5.36)

This series has logarithmic singularities at z = i8. The Laplace transform of V (z) willgive us the Borel summed series of 3(y) which we denote by 3(y)

3(y) =

0

dt et V ty

. (5.37)

The choice of integration contour corresponds to the requirement of exponential decay onthe real axis. The logarithmic singularity of V (t/y ) does not contribute to the integraland integrating on the branch cut starting from it yields the imaginary part

Im 3(y) = i8 y dt et K2 = K2 exp i8 y . (5.38)A similar calculation for the 2 dilaton mode yields

Im 2(y) =k2

exp(2iy) . (5.39)Since k = 0, this mode is vanishing now. However, as we will show in the next subsection,when taking into account order corrections a similar expression for 2 with exp ( 2iy) be-haviour arise, which, due to its slower decay, will become dominant when Im y . Thecontinuation to the real axis of these imaginary corrections turns out to be closely relatedto the asymptotically oscillating mode responsible for the slow energy loss of oscillons.

5.5 Order corrections near the pole

Before discussing the issue of matching the imaginary correction calculated in the neigh-borhood of the singularity to the solution of the eld equation on the real axis we deal withthe corrections arising when taking into account the terms proportional to in the modeequations ( 5.21) and ( 5.22). We denote the solutions obtained in the previous subsectionby (0)n and (0)n , and linearize the mode equations around them by dening

n = (0)n + n , n = (0)n + n . (5.40)

The mode equations take the form

d2

dy2+ n2 n +

D 1iQ D

d(0)ndy

=m

f n m

m +m

f n m

m , (5.41)

d2

dy2+ n2 1 n +

D 1iQ D

d (0)ndy

=m

F n m

m +m

F n m

m , (5.42)

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where the partial derivatives on the right hand sides are taken at n = (0)n and n =

(0)n .

The small dimensional corrections n and n have parts of order both ln and .The linearized equations ( 5.41) and ( 5.42) are solved to ln order by the follow-

ing functions:

n = ln C d(0)n

dy, (5.43)

n = ln C d (0)n

dy, (5.44)

where C is an arbitrary constant. The reason for this is quite simple: in ln order theterms proportional to on the left hand sides are negligible and we get the = 0 equationlinearized about the original solution. Our formula simply gives the zero mode of thisequation. The constant C is determined by the appropriate behaviour when continuing

back our functions to the real axis. This can be ensured by requiring agreement with therst few terms of the small amplitude expansion formulae ( 5.13)(5.16), yielding

C =27i(D 1)

35QD. (5.45)

In the small amplitude expansion ( 5.13)(5.16) to every term of order ln correspondsa term of order which we get by changing ln to ln y. Thus, we dene the new variablesn and n to describe the order small perturbations by

n = ln C d(0)n

dy+ C ln y

d(0)n

dy+ n , (5.46)

n = ln C d (0)n

dy+ C ln y

d (0)ndy

+ n . (5.47)

Substituting into the linearized equations ( 5.41) and ( 5.42) we see that all terms containingln y cancel out,

d2

dy2+ n2 n +

C y2

2yd2(0)n

dy2 d(0)n

dy+

D 1iQ D

d(0)ndy

=m

f n m

m +m

f n m

m ,

(5.48)

d2

dy2+ n21 n +

C y2

2yd2(0)n

dy2 d (0)n

dy+

D 1iQ D

d (0)ndy

=m

F n m

m +m

F n m

m .

(5.49)

If C is given by (5.45), n and n turn out to be algebraic asymptotic series which areanalytic in y. Let us write their expansion explicitly:

2k+1 =

n = k+1

B (n )2k+11

y2n1, 2k =

n = k+1

b(n )2k1

y2n1. (5.50)

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Substituting these and the expansions ( 5.23) for (0)n and (0)n into ( 5.48) and ( 5.49), it is

possible to solve for the coefficients b(n )k and B(n )k , up to one free parameter. Comparing

to (5.13)(5.16) it is natural to choose this free parameter to be b(2)0 = c3. Similarly tothat case, b(2)0 will only be determined by the requirement that the extension to the realaxis represent a localized solution. Furthermore, leaving C a free constant and requiringthe absence of logarithmic terms in the expansion of k and k yields exactly the value of C given in (5.45).

Eq. (5.48) is consistent with the asymptotics

b(n )2 ikD (1)n(2n 2)!

22n11 + O

1n3

, (5.51)

where kD is some constant. Since the leading order result for A(n )3 is given by (5.34), if

kD = 0, the coefficients follow the hierarchy b(n )2 A

(n1)3 . In order to be able to extractthe value of kD we have calculated b

(n )

2by solving the mode equations to high orders in

1/y , obtaining

kD = 1 .640D 1

QD. (5.52)

The displayed four digits precision for kD can be relatively easily obtained by settingN F 4, N e 5 and calculating b

(n )2 to orders n 25. We note that there is also a term

proportional to the unknown b(2)0 = c3 in each b(n )2 , giving a c3 dependent kD . Luckily,

the inuence of this term to kD quickly becomes negligible as N F and N e grow, makingthe concrete value of c3 irrelevant for our purpose.

The Borel summation procedure can be done similarly as in eqs. ( 5.36)(5.38). On theimaginary axis 2 is real to every order in 1 /y , however it gets a small imaginary correctionfrom the summation procedure given by

Im 2(y) = kD

2exp(2iy) . (5.53)

5.6 Extension to the real axis

Solutions of the Fourier mode equations ( 5.18) and ( 5.19) can be considered to be the sumof two parts. The rst part corresponds to the result of the small amplitude expansion,the second to an exponentially small correction to it. The small amplitude expansion is anasymptotic expansion, it gives better and better approximation until reaching an optimalorder, but higher terms give increasingly divergent results. The smaller is, the higher theoptimal truncation order becomes, and the precision also improves. The small amplitudeexpansion procedure gives time-periodic localized regular functions to all orders, charac-terizing the core part of the quasibreather. Their extension to the complex plane is real onthe imaginary axis. Furthermore, the functions obtained by the expansion are smooth onlarge scales, missing an oscillating tail and short wavelength oscillations in the core region.On the imaginary axis, to a very good approximation, the small second part of the solutionof the mode equations ( 5.18) and ( 5.19) is pure imaginary, and satises the homogeneouslinear equations obtained by keeping only the left hand sides of these equations, becausethe quasibreather is a small-amplitude one. In the inner region it is of order 1 /y 2, while

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on the real axis its amplitude is of order 2, hence to leading order the quasibreathercore background does not contribute. In the previous subsection we have determined thebehaviour of this small correction close to the poles. Now we extend it to the real axis.

In the inner region, close to the pole, the function Im 2 given by ( 5.53) solvesthe homogeneous linear differential equations given by the left hand side of ( 5.22). Theextension of this function to the real axis will provide the small correction to the smallamplitude result mentioned in the previous paragraph. We intend to nd the solution 2of the left hand side of ( 5.19), which reduces to the value given by ( 5.53) close to the upperpole, where r = iQ D / + y, and behaves as

Im 2(y) = kD

2exp (2iy) . (5.54)

near the lower pole, where r = iQ D / + y. We follow the procedure detailed in [ 31]. Theresulting function for large r is

2 = ikD

2QDr

(D1)/ 2exp

2QD

i(D1)/ 2 exp(2ir )(i)(D1)/ 2 exp(2 ir ) . (5.55)

The general solution of the left hand side of ( 5.19) can be written as a sum involving Besselfunctions J n and Y n , which have the asymptotic behaviour

J (x) 2x cos x 2 4 , (5.56)Y (x) 2x sin x 2 4 , (5.57)

for x + . The solution satisfying the asymptotics given by ( 5.55) is

2 = D

r D/ 21Y D/ 21(2r ) , (5.58)

where the amplitude at large r is given by

D = k DQD

(D 1)/ 2exp

2QD

. (5.59)

For D > 2 the function given by ( 5.58) is singular at the center, due to the usual centralsingularity of spherical waves. Since the amplitude of the quasibreather core is proportionalto 2, and its size to 1 / , for small it is possible to extend the function 2 in its form ( 5.58)to the real axis into a region which is outside the domain where 2 gets large, but whichis still close to the center when considering the enlarged size of the quasibreather core.When extending this function further out along the real r axis, because of the large size of the quasibreather core, the nonlinear source terms on the right hand side of ( 5.19) are notnegligible anymore, and the expression ( 5.58) for 2 cannot be used. What actually happensis that 2 tends to zero exponentially as r . This follows from the special choice of the inner solution close to the singularity; namely, we were looking for a solution whichagreed with the small amplitude expansion for Re y . The small amplitude expansion

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gives exponentially localized functions to each order and we also required decay beyond allorders when choosing the contour of integration in the Borel summation procedure.

By the above procedure we have constructed a solution of the mode equations which issingular at r = 0. The singularity is the consequence of the initial assumption of exponentialdecay for large r . The asymptotic decay induces an oscillation given by ( 5.58) in theintermediate core, and a singularity at the center. In contrast, the quasibreather solutionhas a regular center, but contains a minimal amplitude standing wave tail asymptotically.Considering the left hand side of ( 5.19) as an equation describing perturbation aroundthe asymptotically decaying solution, we just have to add a solution 2 determined bythe amplitude ( 5.59) with the opposite sign of ( 5.58) to cancel the oscillation and thesingularity in the core. This way one obtains the regular quasibreather solution, whoseminimal amplitude standing wave tail is given as

QB = D

r D/ 21Y D/ 21(2r )cos(2t) (5.60)

Dr (D1)/ 2 sin 2r (D 1) 4 cos(2t). (5.61)Adding the regular solution, where Y is replaced by J , would necessarily increase theasymptotic amplitude.

If we subtract the incoming radiation from a QB and cut the remaining tail at largedistances, we obtain an oscillon state to a good approximation. Subtracting the regularsolution with a phase shift in time, we cancel the incoming radiating component, and obtainthe radiative tail of the oscillon,

osc = D

r D/ 21Y D/ 21(2r )cos(2t) J D/ 21(2r )sin(2 t) (5.62)

Dr (D1)/ 2 sin 2r (D 1) 4 2t . (5.63)The radiation law of the oscillon is easily obtained now,

dE dt

= k2D 24D/ 2

D22

QD

D 1exp

4QD

, (5.64)

where the constant kD is given by (5.52). If we assume adiabatic time evolution of the parameter determining the oscillon state, using eqs. ( 3.52) and ( 3.54) giving E as a functionof , we get a closed evolution equation for small amplitude oscillons, determining theirenergy as the function of time.

For the physically most interesting case, D = 3 we write the evolution equation for and its leading order late time behavior explicitly:

ddt

= 30.29 exp 15.909

(5.65)

15.909

ln t, E

1401.6ln t

. (5.66)

Acknowledgments

This research has been supported by OTKA Grants No. K61636, NI68228.

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Gyula Fodor et al- Oscillons in dilaton-scalar theories

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