Oscillons in dilaton-scalar theories
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Oscillons in dilaton-scalar theories
Gyula Fodor
MTA RMKI, H-1525 Budapest 114, PO..Box 49, Hungary, [email protected]
P´eter Forg´acs
MTA RMKI, H-1525 Budapest 114, P.O.Box 49, HungaryLMPT, CNRS-UMR 6083, Universit´e de Tours,Parc de Grandmont, 37200 Tours, France
Zal´an Horv´ath and M´ark Mezei
Institute for Theoretical Physics, E¨otv¨os University,H-1117 Budapest, P´azm´any P´eter s´et´any 1/A, Hungary
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 fieldtheory 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 field coupledto the dilaton. It is shown that in analogy to the previously considered cases with a singlenonlinear scalar field, 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:
Nonperturbative Effects, Solitons Monopoles and Instantons. ontents
1. Introduction 12. The scalar-dilaton system 33. The small amplitude expansion 4 ε powers in the expansion 83.3 Higher orders in the ε expansion 83.4 Free scalar field in 2 < D < ε order for D = 6 113.6 Total energy and dilaton charge of oscillons 12
4. Time evolution of oscillons 125. Determination of the energy loss rate 16 ε expansion 175.2 Fourier mode expansion 175.3 Fourier mode equations 195.4 ε → ε corrections near the pole 225.6 Extension to the real axis 24
6. Acknowledgments 26
1. Introduction
Long-living, spatially localized classical solutions in field theories containing scalar fieldsexhibiting nearly periodic oscillations in time – oscillons – [1]–[16] have attracted consid-erable 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 , ,
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 considerable– 1 –mportance [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 additionthey have a standing wave tail whose amplitude is minimized. At this point it is importantto emphasize that there are (infinitely) 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 [32], [30]. 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 fieldtheories coupled to a (massless) dilaton field (DS theory). Dilaton fields appear naturallyin low energy effective field theories derived from superstring models [33, 34, 35] and thestudy of their effects is of major interest. As the present study shows, the coupling of adilaton even to a free massive scalar field, referred to as the dilaton-Klein-Gordon (DKG)theory, 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 configurations is a major restriction since non-symmetricconfigurations 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, ε , 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 to ε . This is to be contrasted to self-interacting scalar theories whose oscillons are stable in D = 1 ,
2, unstable in D = 3, and their core amplitude is proportional to ε . The masterequations determining oscillons to leading order in the small amplitude expansion turn outto be the Schr¨odinger-Newton (SN) equations. The main analytical result of this paper isthe 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. [32], [36], [37], [30] and [31].The above results, namely the stability properties and the SN equations playing therˆole of master equation, show striking similarity to those obtained in the Einstein-Klein-Gordon (EKG) theory, i.e. for a free massive scalar field coupled to Einstein’s gravity,– 2 –here 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].
2. The scalar-dilaton system
The action of a scalar-dilaton system is A = Z dt d D x (cid:20)
12 ( ∂ µ ϕ ) + 12 ( ∂ µ Φ) − e − κϕ U (Φ) (cid:21) , (2.1)where ϕ is the dilaton field and Φ is a scalar field with self interaction potential U (Φ).The energy corresponding to the action (2.1) can be written as E = Z d D x E , E = 12 h ( ∂ t Φ) + ( ∂ i Φ) + ( ∂ t ϕ ) + ( ∂ i ϕ ) i + e − κϕ U (Φ) , (2.2)where E denotes the energy density. In the case of spherical symmetry E = Z ∞ dr π D/ r D − Γ( D/ h ( ∂ t Φ) + ( ∂ r Φ) + ( ∂ t ϕ ) + ( ∂ r ϕ ) + 2 e − κϕ U (Φ) i . (2.3)We assume that the potential can be expanded around its minimum at Φ = 0 as U (Φ) = ∞ X k =1 g k k + 1 Φ k +1 , U ′ (Φ) = ∞ X k =1 g k Φ k , (2.4)where g k are real constants. For a free massive scalar field with mass m the only nonzerocoefficient is g = m . If g k = 0 for integer k the potential is symmetric around itsminimum. In that case, as we will see, for periodic configurations the Fourier expansion ofΦ in t will contain only odd, while the expansion of ϕ only even Fourier components. Forspherically symmetric systems the field equations are − ∂ Φ ∂t + ∂ Φ ∂r + D − r ∂ Φ ∂r = e − κϕ U ′ (Φ) , (2.5) − ∂ ϕ∂t + ∂ ϕ∂r + D − r ∂ϕ∂r = − κ e − κϕ U (Φ) . (2.6)Since g = m is intended to be the mass of small excitations of Φ at large distances, welook for solutions satisfying ϕ → r → ∞ . Finiteness of energy also requires Φ → r → ∞ . Rescaling the coordinates as t → t/m and r → r/m we first set g = m = 1.Then redefining ϕ → ϕ/ (2 κ ) and Φ → Φ / (2 κ ) and appropriately changing the constants g k we arrange that 2 κ = 1. If for some reason we obtain a solution for which ϕ tends to anonzero constant at infinity then the dilatation symmetry of the system allows us to shift ϕ and rescale the coordinates so that it is transformed to a solution satisfying ϕ → r → ∞ .An important feature of a localized dilatonic configuration is its dilaton charge, Q . Itcan be defined for almost time-periodic spherically symmetric configurations like oscillonsas: ϕ ≈ Q r − D for r → ∞ in D = 2 (2.7) ϕ ≈ Q ln r for r → ∞ in D = 2 . (2.8)– 3 – . The small amplitude expansion In this section we will construct a finite-energy family of localized small amplitude solutionsof the spherically symmetric field equations (2.5) and (2.6) which oscillate below the massthreshold [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 fields, ϕ and Φ, in terms of a parameter ε as ϕ = ∞ X k =1 ε k ϕ k , Φ = ∞ X k =1 ε k Φ k , (3.1)and search for functions φ k and Φ k tending to zero at r → ∞ . The size of smoothconfigurations is expected to increase for decreasing values of ε , therefore it is natural tointroduce a new radial coordinate by the following rescaling ρ = εr . (3.2)In order to allow for the ε dependence of the time-scale of the configurations a new timecoordinate is introduced as τ = ω ( ε ) t . (3.3)Numerical experience shows that the smaller the oscillon amplitude is the closer its fre-quency becomes to the threshold ω = 1. The function ω ( ε ) is assumed to be analytic near ω = 1, and it is expanded as ω ( ε ) = 1 + ∞ X k =1 ε k ω k . (3.4)We note that there is a considerable freedom in choosing different parametrisations ofthe small amplitude states, changing the actual form of the function ω ( ε ). The physicalparameter is not ε but the frequency of the periodic states that will be given by ω . Afterthe rescalings Eqs. (2.5) and (2.6) take the following form − ω ∂ Φ ∂τ + ε ∂ Φ ∂ρ + ε D − ρ ∂ Φ ∂ρ = e − ϕ Φ + ∞ X k =2 g k Φ k ! , (3.5) − ω ∂ ϕ∂τ + ε ∂ ϕ∂ρ + ε D − ρ ∂ϕ∂ρ = − e − ϕ
12 Φ + ∞ X k =2 g k k + 1 Φ k +1 ! . (3.6)Substituting the small amplitude expansion (3.1) into (3.5) and (3.6), to leading ε order we obtain ∂ Φ ∂τ + Φ = 0 , ∂ ϕ ∂τ = 0 . (3.7)– 4 –ince we are looking for solutions which remain bounded in time and since we are free toshift the origin τ = 0 of the time coordinate, the solution of (3.7) can be written asΦ ( τ, ρ ) = P ( ρ ) cos τ , ϕ ( τ, ρ ) = p ( ρ ) , (3.8)where P ( ρ ) and p ( ρ ) are some functions of the rescaled radial coordinate ρ .The ε terms in the expansion of (3.6) yield ∂ ϕ ∂τ = 14 P [1 + cos(2 τ )] . (3.9)This equation can have a solution for ϕ which remains bounded in time only if the timeindependent term in the right hand side vanishes, implying P = 0 and consequentlyΦ = 0. Then the solution of (3.9) is ϕ ( τ, ρ ) = p ( ρ ). The ε terms in (3.5) yield ∂ Φ ∂τ + Φ = 0 . (3.10)Since Φ = 0 we are again free to shift the time coordinate, and the solution is Φ ( τ, ρ ) = P ( ρ ) cos τ .The ε order terms in the expansion of (3.6) give ∂ ϕ ∂τ = d p dρ + D − ρ dp dρ . (3.11)In order to have a solution for ϕ ( τ, ρ ) that remains bounded in time, the right hand sidemust be zero, yielding p ( ρ ) = p + p ρ − D when D = 2 and p ( ρ ) = p + p ln ρ for D = 2, with some constants p and p . Since we look for bounded regular solutionstending to zero at ρ → ∞ , we must have p = p = 0. As we have already seen thatΦ = 0, this means that the small amplitude expansion (3.1) starts with ε terms. Thesolution of (3.11) is then ϕ ( τ, ρ ) = p ( ρ ). The ε order terms in the expansion of (3.5)give ∂ Φ ∂τ + Φ − ω P cos τ = 0 . (3.12)This equation can have a solution for Φ which remains bounded in time only if the res-onance term proportional to cos τ vanishes, implying ω = 0. After applying an ε ordersmall shift in the time coordinate, the solution of (3.12) is Φ ( τ, ρ ) = P ( ρ ) cos τ . Contin-uing to higher orders, the basic frequency sin τ term can always be absorbed by a smallshift in τ . It is important to note that after transforming out the sin τ terms no sin( kτ )terms will appear in the expansion, implying the time reflection symmetry of Φ and ϕ at τ = 0. The ε terms in the expansion of (3.5) and (3.6) yield the differential equations ∂ Φ ∂τ + Φ = (cid:20) d P dρ + D − ρ dP dρ + ( p + ω ) P (cid:21) cos τ − g P [1 + cos(2 τ )] , (3.13) ∂ ϕ ∂τ = d p dρ + D − ρ dp dρ + 14 P [1 + cos(2 τ )] . (3.14)– 5 –he function Φ ( τ, ρ ) and ϕ ( τ, ρ ) can remain bounded only if the cos τ resonance termsin (3.13) and the time independent terms in (3.14) vanish, d P dρ + D − ρ dP dρ + ( p + ω ) P = 0 , (3.15) d p dρ + D − ρ dp dρ + 14 P = 0 . (3.16)Then the time dependence of Φ ( τ, ρ ) and ϕ ( τ, ρ ) is determined by (3.13) and (3.14) asΦ ( τ, ρ ) = P ( ρ ) cos τ + 16 g P [cos(2 τ ) − , ϕ ( τ, ρ ) = p ( ρ ) − P ( ρ ) cos(2 τ ) . (3.17)Here we see the first contribution of a nontrivial U (Φ) potential, the term proportional to g in Φ . If (and only if) the potential is non-symmetric around its minimum, even Fouriercomponents appear in the expansion of Φ.Introducing the new variables S = 12 P , s = p + ω , (3.18)(3.15) and (3.16) can be written into the form which is called the time-independentSchr¨odinger-Newton (or Newton-Schr¨odinger) equations in the literature: d Sdρ + D − ρ dSdρ + sS = 0 , (3.19) d sdρ + D − ρ dsdρ + S = 0 . (3.20)We look for localized solutions of these equations, in order to determine the core part ofsmall amplitude oscillons to a leading order approximation in ε . The main features of thesolutions depend on the number of spatial dimensions D . For D ≥ s = S , they tend to zero, furthermore, the Lane-Emden equation holds [45] d sdρ + D − ρ dsdρ + s = 0 . (3.21)For D > /ρ for large ρ , consequently they have infiniteenergy. It can also be shown that solutions of the original Schr¨odinger-Newton systemwith s = S , and a necessarily oscillating scalar field, have infinite energy, hence there isno finite energy solution for D >
6. For D = 6 the explicit form of the asymptoticallydecaying solutions of (3.21) are known s = ± S = 24 α (1 + α ρ ) , (3.22)where α is any constant. Since the replacement of Φ with − Φ and a simultaneous reflectionof 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 finite.– 6 –f D <
6, then localized solutions have the property that for large values of ρ thefunction S tends to zero exponentially, while s behaves as s ≈ s + s ρ − D for D = 2 andas s ≈ s + s ln ρ for D = 2, where s and s 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. s <
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 are thenodeless solutions satisfying S >
0, since the others correspond to higher energy and lessstable oscillons.Motivated by the asymptotic behaviour of s , if D = 2 it is useful to introduce thevariables µ = ρ D − − D dsdρ , ν = s − ρ − D µ . (3.23)In 2 < D < ρ →∞ µ = s , lim ρ →∞ ν = s . (3.24)Then the SN equations can be written into the equivalent form dµdρ + ρ D − − D S = 0 , (3.25) dνdρ + ρD − S = 0 , (3.26) d Sdρ + D − ρ dSdρ + (cid:0) ν + ρ − D µ (cid:1) S = 0 . (3.27)The SN equations (3.19) and (3.20) have the scaling invariance( S ( ρ ) , s ( ρ )) → ( λ S ( λρ ) , λ s ( λρ )) . (3.28)If 2 < D < s = − ε parametrization by requiring ω = − < D < , (3.29)ensuring that the limiting value of ϕ vanishes to ε order. Going to higher orders, it canbe shown that one can always make the choice ω i = 0 for i ≥
3, thereby fixing the ε parametrization, and setting ω = p − ε for 2 < D < . (3.30)If D = 6, since both s and S tend to zero at infinity, we have no method yet to fix thevalue of α in (3.22). Moreover, in order to ensure that ϕ tends to zero at infinity we haveto set ω = 0 for D = 6 . (3.31)– 7 – .2 Absence of odd ε powers in the expansion Calculating the ε order equations from (3.5) and (3.6) and requiring the boundedness ofΦ and ϕ we obtain a pair of equations for P and p : d P dρ + D − ρ dP dρ + ( p + ω ) P + P ( p + ω ) = 0 , (3.32) d p dρ + D − ρ dp dρ + 12 P P = 0 . (3.33)These equations are solved by constant multiples of P = 2 P + ρ dP dρ , p + ω = 2( p + ω ) + ρ dp dρ , (3.34)corresponding to the scaling invariance (3.28) of the SN equations. In D > p given by (3.34) tends to ω − ρ . Since we are looking for solutions for which ϕ tends to zero asymptotically, after choosing ω = 0 we can only use the trivial solution P = p = 0. The important consequence is that Φ = ϕ = 0. Going to higher ordersin the ε expansion, at odd orders we get the same form of equations as (3.32) and (3.33),consequently, all odd coefficients of Φ k and ϕ k can be made to vanish. Instead of the moregeneral form (3.1) we can write the small amplitude expansion as ϕ = ∞ X k =1 ε k ϕ k , Φ = ∞ X k =1 ε k Φ k . (3.35) ε expansion The ε order equations, after requiring the boundedness of Φ and ϕ , yield a pair ofequations for P and p : d P dρ + D − ρ dP dρ + ( p + ω ) P + P ( p + ω ) − p P − P + (cid:18) g − g (cid:19) P = 0 , (3.36) d p dρ + D − ρ dp dρ + 12 P P − p P = 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 + ρ dP dρ , p ( h )4 + ω = 2( p + ω ) + ρ dp dρ , (3.38)to a particular solution of (3.36) and (3.37). If 2 < D < P decays exponentially, and p ≈ q + q ρ − D with some constants q and q . We use the homogeneous solution (3.38) to make q = 0. Since similar choice canbe made at higher ε orders, this will ensure that the limit of ϕ will remain zero at ρ → ∞ .– 8 –e note that, in general, it is not possible to make q also vanish, implying a nontrivial ε dependence of the dilaton charge Q .The resulting expressions for the original Φ and ϕ functions areΦ = ε P cos τ + ε (cid:26) P cos τ + 16 g P [cos(2 τ ) − (cid:27) + ε ( P cos τ + P (cid:18) g + 8 g (cid:19) cos(3 τ ) − g " P P − ( p + ω ) P + (cid:18) dP dρ (cid:19) (3.39)+ g " P P − ( p + ω ) P − (cid:18) dP dρ (cid:19) cos(2 τ ) ) + O ( ε ) ,ϕ = ε p + ε (cid:20) p − P
16 cos(2 τ ) (cid:21) + ε ( p − " P P − ( p + ω ) P − (cid:18) dP dρ (cid:19) cos(2 τ )+ 154 g P [9 cos τ − cos(3 τ )] ) + O ( ε ) , (3.40)where the functions P and p are determined by the SN equations (3.15) and (3.16), P and p can be obtained from (3.36) and (3.37), furthermore, the equations for P and p can be calculated from the ε order terms as d P dρ + D − ρ dP dρ + ( p + ω ) P + ( p + ω ) P + (cid:18) p + ω − p (cid:19) P − (cid:18) − g + 94 g (cid:19) P P − p P p + (cid:18) − g + 34 g (cid:19) p P (3.41)+ (cid:18) − g (cid:19) ω P + 16 p P + P (cid:18)
164 + 199 g (cid:19) (cid:18) dp dρ (cid:19) = 0 ,d p dρ + D − ρ dp dρ + 12 P P + 14 P − p P P − P p + 18 p P + P (cid:18) − g + 32 g (cid:19) = 0 . (3.42)We remind the reader that the only non-vanishing ω k for 2 < D < ω = −
1, and wewill show in Subsection 3.5, that in general, for D = 6 the only nonzero component is ω .The above expressions, especially those for Φ and ϕ , simplify considerably for symmetric U (Φ) potentials, in which case g = 0. < D < dimensions If Φ is a free massive field with potential U (Φ) = m Φ /
2, after scaling out m and κ noparameters remain in the equations determining P i and p i . 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 Figs. 1 and 2. The obtained central values of P i and p i for i = 2 , , D = 3 , , P k ρ P P P Figure 1:
The first three P k functions for the free scalar field case in D = 3 spatial dimensions. p k ρ p p p Figure 2:
The p k functions for the free scalar field case in D = 3 dimensions. The chosen central values make all functions D = 3 D = 4 D = 5 P c p c P c p c P c p c Table 1:
Central values of the first threefunctions P i and p i for the free scalar fieldin 3, 4 and 5 spatial dimensions. P i and p i , and consequently Φ i and ϕ i , tend tozero for ρ → ∞ . Although for i ≥ P i and p i are not monotonically decreasing functions, theircentral values represent well the magnitude ofthese functions. Generally, the validity domainof an asymptotic series ends where a higher or-der term starts giving larger contributions thanprevious order terms. For D = 3 the sixth order ε expansion can be expected to be valid even foras large parameter values as ε = 1. For D = 4this domain is ε < .
7, while for D = 5 it de-creases to ε < .
22. – 10 – .5 ε order for D = 6As we have already stated in Subsection 3.1, if D = 6 then ω = 0, s = S and the explicitform of the solution of the SN equations is given by (3.22). Introducing the new variables z and Z by P = 13 (2 Z + z ) , p + ω = 13 ( Z − z ) , (3.43)equations (3.36) and (3.37) decouple, d zdρ + 5 ρ dzdρ − sz + 34 β s = 0 , (3.44) d Zdρ + 5 ρ dZdρ + 2 sZ − β s = 0 , (3.45)where the constants β and β are defined by the coefficients in the potential as β = 1 + 809 g − g , (3.46) β = 1 − g + 83 g . (3.47)For a free scalar field with U (Φ) = Φ / β = β = 1. The general regular solutionof (3.44) can be written in terms of the (complex indexed) associated Legendre function P as z = 144 β α (14 + 6 α ρ + α ρ )13(1 + α ρ ) + C α ρ P i √ − / (cid:18) − α ρ α ρ (cid:19) , (3.48)where C is some constant. The limiting value at ρ → ∞ is z ∞ = − C cosh( π √ / /π ≈− . C . The regular solution of (3.45) is Z = 3888 β α (1 − α ρ )7(1 + α ρ ) ln(1 + α ρ ) (3.49)+ 324 β α ρ (220 + 100 α ρ − α ρ − α ρ )35(1 + α ρ ) + C α ρ − α ρ + 1) , The limiting value at ρ → ∞ is Z ∞ = − β α /
35, independently of C . Since P musttend to zero, according to (3.43), z ∞ = 648 β α /
35, fixing the constant C . Since the massof the field Φ is intended to remain m = 1, the limit of p also has to vanish, giving ω = − β α . (3.50)This expression is not enough to fix ω yet, since α is a free parameter. If β > ω = −
1, thereby fixing the free parameter α in the ε ordercomponent of Φ and ϕ . The change of the so far undetermined constant C correspondsto a small rescaling of the parameter α in the expression (3.22). Its concrete value willfix the coefficient ω in the expansion of the frequency. The homogeneous parts of thedifferential equations at higher ε order will have the same structure as those for P and p .Choosing the appropriate homogeneous solutions all higher ω k components can be set tozero, yielding ω = p − ε for D = 6 if β > . (3.51)– 11 –his expression is valid for the free scalar field case with potential U (Φ) = Φ / D = 6,since then β = 1. For certain potentials β <
0, and one can use (3.50) to set ω = 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 β = 0 the frequency differsfrom the fundamental frequency only in ε or possibly higher order terms. Substituting (3.39) and (3.40) into the expression (2.3) of the total energy, we get E = ε − D E + ε − D E + O ( ε − D ) , (3.52)where E = π D/ Γ( D/ Z ∞ dρ ρ D − P , E = π D/ Γ( D/ Z ∞ dρ ρ D − P (2 P − P ) . (3.53)Since P = 2 S , for 2 < D < E = 4 π D/ Γ( D/
2) ( D − s . (3.54)The numerical values of s , E and E for D = 3 , , ε − D , for D = 3 D = 4 D = 5 s E E Table 2:
The numerical values of s , E and E in 3, 4 and 5 spatial dimen-sions. D = 3 and D = 4 the energy is a monotonicallyincreasing function of ε , while for D = 5 there isan energy minimum at ε = 0 . D = 6 the leading order term in the totalenergy is E = 192 π α ε . (3.55)As we have already noted, for D > ε dependence of the dilaton charge for 2 < D < Q = s ε − D , (3.56)where we used the definition (2.7), (3.24) and the relation ρ = εr . The dilaton charge forthe D = 6 oscillon is infinite. In higher orders in ε the proportionality between the dilatoncharge and energy is violated.
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– 12 –ode with spatial compactification in order to investigate spherically symmetric fields [46].Our aim is to find configurations which are as closely periodic as possible. To achieve this,we use initial data obtained from the leading ε terms of the small amplitude expansion(3.39) and (3.40). The smaller the chosen ε is, the more closely periodic the resultingoscillating state becomes. However, for moderate values of ε , it is possible to improve theinitial data by 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 configuration, but its amplitude and frequency will oscillate with a lowfrequency modulation. We employ a fine-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 thefine-tuning method to suppress this decay mode, and make long living oscillon states withwell defined 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 flux of energy, the other isthrough collapsing to a central region first.Having calculated several closely periodic oscillon configurations, 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 ε values the small amplitude expansion remains valid. Theparameter ε for the evolving oscillon is calculated from the numerically measured frequencyby the expression ε = √ − ω . The results for D = 3 are presented on Fig. 3. In contrast E ε D=3numerical evolutionE ε E ε +E ε Figure 3:
Total energy of three-dimensional oscillons as a function of the parameter ε . 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 ε dependence of the energy for D = 5 is presented on Fig. 4. There is an energyminimum of the numerically obtained states, approximately at ε = 0 .
21, above which– 13 – E ε D=5numerical evolutionE / ε E / ε +E ε Figure 4:
Total energy of five-dimensional oscillons as a function of the parameter ε . The verticalline at ε = 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. oscillons are stable. The place of the minimum agrees quite well with the value ε = 0 . E ε D=5numerical evolutionE / ε +E ε Figure 5:
The region of Fig. 4 near the energy minimum.
We have also constructed oscillon states for D = 6 dimensions. These oscillons havequite large energy, due to the slow spatial decay of the functions Φ and ϕ . For free massivescalar fields, oscillons have frequency given by (3.51), i.e. an initial data with a given ε value will evolve to an oscillon state with frequency approximately following ω = √ − ε .However, there are potentials, for which the oscillation frequency is above the threshold ω = 1. For example, this happens for the potential U (Φ) = Φ (Φ − / κ = 1 / ε decreases withdecreasing energy, oscillons are stable, while if ε increases with decreasing energy, oscillonsare unstable. In other words if the time evolution (i.e. energy loss) of an oscillon leadsto spreading 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 configurations, is well described by the first 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 ε = 0 .
05. In order to make the unstable state long living, a fine tuning procedure isapplied, multiplying the amplitude of the initial data by a factor with value close to 1 . . × − . Oneof the two initial data develops into a configuration decaying with a uniform outwardcurrent of energy, the other through collapsing to a high density state first. On Fig. 6the time evolution of the difference of the central value of the dilaton fields in the twostates ∆ ϕ = ϕ − ϕ is shown. The curve follows extremely well the exponential increase ∆ ϕ t Figure 6:
Increase in the difference of the dilaton field ϕ for two similar configurations. described by ∆ ϕ = 6 . × − exp(0 . t ) , (4.1)showing that there is a single decay mode growing exponentially. The difference of thescalar fields, ∆Φ = Φ − Φ , grows with the same exponent. The spatial dependence ofthe decaying mode is illustrated on Fig. 7, where ∆Φ is plotted at several moments of timecorresponding to the maximum of Φ at the center.– 15 – ∆ Φ r t=6669.8t=6870.1t=7071.4t=7272.7 Figure 7:
Radial behaviour of the difference of the scalar fields of two very similar configurations.At the chosen moments of time the scalar is maximal at the center, and the subsequent momentsare separated by 32 oscillations.
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 fieldsystem the instability arises for
D > U ( φ ) = φ (1 − ln φ ) admits exactly time-periodic breathers in any dimensions. Thestability 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 field.
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 configurations. To simplify the expressions in this section we consider amassive free scalar field, i.e. U (Φ) = m Φ /
2. We assume that 2 < D <
6, since then thescalar field tends to zero exponentially for large ρ .The outgoing radiation will dominantly be in the dilaton field and the radiation am-plitude will have the ε dependence: ε exp ( − Q D /ε ). In Refs. [30] and [31] we have usedtwo different methods for determining the ε independent part of the radiation amplitude:Borel summation and solution of the complexified 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-ric around their minima can be treated analogously. If the potential is asymmetric only– 16 –he numerical method could be used. This phenomenon is in complete analogy with theproblem arising with a single scalar field considered in Ref. [30]. ε expansion We first investigate the complex extension of the functions obtained by the small amplitudeexpansion in Sec. 3. Extending the solutions s and S of the Schr¨odinger-Newton equations(3.19) and (3.20) to complex ρ coordinates they both have pole singularities on the imag-inary axis of the complex plane. We consider the closest pair of singularities to the realaxis, since these will give the dominant contribution to the energy loss. They are locatedat ρ = ± iQ D . The numerically calculated values of Q D are listed in Table 3 for spatialdimensions D = 3 , , D Q D Table 3:
The distance Q D between the real axisand the pole of the funda-mental solution of the SNequation for various spa-tial dimensions D . coordinate R defined as ρ = iQ D + R . (5.1)Close to the pole we can expand the SN equations, and obtainthat s and S have essentially the same behaviour, s = ± S = − R − i ( D − Q D R − ( D − D − Q D + O ( R ) , (5.2)even though they clearly differ on the real axis. Since for sym-metric potentials we can always substitute Φ by − Φ, we choose the positive sign for S in (5.2). This choice is compatible with the sign of S used on the real axis at the smallamplitude expansion section. We note that for D > s and S , starting with terms proportional to R ln R . According to (3.18),the functions determining the leading ε parts of ϕ and Φ in this case are p = s + 1 , P = 2 S . (5.3)Substituting these into the equations (3.36) and (3.37), the ε order contributions p and P can also be expanded around the pole p = − R + 324 i ( D −
1) ln R Q D R + c − R + O (cid:18) ln RR (cid:19) (5.4) P − p = 8113 R + 18 i ( D − Q D R + O (cid:18) R (cid:19) , (5.5)where the constant c − can only be determined from the specific behaviour of the functionson the real axis, namely from the requirement of the exponential decay of P for large real ρ . 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– 17 –egion 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 configuration 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 ω by Fourier expanding the scalar and dilaton field asΦ = N F X k =0 Ψ k cos( kωt ) , ϕ = N F X k =0 ψ k cos( kωt ) . (5.6)Although, in principle, the Fourier truncation order N F should tend to infinity, one canexpect 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 ε expansioncomponents in (3.1). Since in this section we only deal with an self-interaction free scalarfield with a trivially symmetric potential,Ψ k = 0 , ψ k +1 = 0 , for integer k . (5.7)We note that the absence of sine terms in (5.6) is equivalent to the assumption of timereflexion symmetry at t = 0. This assumption appears reasonable physically, and we haveseen in Sec. 3 that it holds in the small amplitude expansion framework.For small amplitude configurations we can establish the connection between the ex-pansions (3.1) and (5.6) by comparing to (3.39) and (3.40), obtainingΨ = ε P + ε P + ε P + O ( ε ) , (5.8)Ψ = ε P
256 + O ( ε ) , (5.9) ψ = ε p + ε p + ε p + O ( ε ) , (5.10) ψ = − ε P − ε " P P − ( p − P − (cid:18) dP dR (cid:19) + O ( ε ) . (5.11)Let us define a coordinate y for an “inner region” by R = εy . This coordinate willhave the same scale as the original radial coordinate r , since they are related as r = iQ D ε + y . (5.12)The “inner region” | R | ≪ y coordinate; if ε → ε | y | = | R | ≪ | 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 ofthe Fourier modes for | y | → ∞ can be written asΨ = − y − y + ε ln ε i ( D − Q D y + ε (cid:20) i ( D − Q D y (cid:18) y + 108 ln y y − (cid:19) + 2 c − y (cid:21) + . . . , (5.13)Ψ = − y − ε i ( D − Q D y + . . . , (5.14)– 18 – = − y − y + ε ln ε i ( D − Q D y + ε (cid:20) i ( D − Q D y (cid:18)
54 ln y y − (cid:19) + c − y (cid:21) + . . . , (5.15) ψ = − y − ε i ( D − Q D y + . . . . (5.16)These expressions are simultaneous series in 1 /y and in ε . In order to obtain finite number of Fourier mode equations with finite number of terms,when substituting (5.6) into the field equations (2.5) and (2.6) we Taylor expand andtruncate the exponential e − ϕ = N e X k =0 k ! ( − ϕ ) k . (5.17)We need to carefully check how large N e should be chosen to have only a negligible influenceto the calculated results. For n ≤ N F the Fourier mode equations have the form (cid:18) d d r + D − r dd r + n ω − (cid:19) Ψ n = F n , (5.18) (cid:18) d d r + D − r dd r + n ω (cid:19) ψ n = f n , (5.19)where we have collected the nonlinear terms to the right hand sides, and denoted them with F n and f n . These are polynomial expressions involving various Ψ k and ψ k , with quicklyincreasing complexity when increasing the truncation orders N F and N e . The solution of(5.18) and (5.19) yields the intended quasibreathers, with a localized core and a very smallamplitude oscillating tail. For small amplitude configurations the functions Ψ k and ψ k willhave poles at the complex r plane, just as we have seen in the small amplitude expansionformalism. In order to calculate the tail amplitude it is necessary to investigate the Fouriermode equations instead of the equations obtained in Sec. 3. Although in the Fourierdecomposition method we have not defined a small amplitude parameter yet, motivatedby (3.30), we can, in general, define ε as ε = p − ω . (5.20)Dropping O ( ε ) terms, in the neighborhood of the singularity the mode equations take theform (cid:18) d d y + ε D − iQ D dd y + n − (cid:19) Ψ n = F n , (5.21) (cid:18) d d y + ε D − iQ D dd y + n (cid:19) ψ n = f n . (5.22)We look for solutions of these equations that satisfy (5.13)-(5.16) as boundary conditionsfor | y | → ∞ for − π/ < arg y <
0. This corresponds to the requirement that the functions– 19 –ecay 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. ε → limit near the pole For very small ε values one can neglect the terms proportional ε 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 first, and consider finite but small ε corrections later as perturbations to it. We expand the solution of (5.21) and (5.22) (with ε = 0) in even powers of 1 /y ,Ψ k +1 = ∞ X j = k +1 A ( j )2 k +1 y j , ψ k = ∞ X j = k +1 a ( j )2 k y 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 toprint: d ψ dy = 14 ( ψ − + Ψ ) + 18 ψ Ψ (Ψ + 2Ψ ) , (5.24) d Ψ dy = − ψ Ψ − ψ (Ψ + Ψ ) , (5.25) d ψ dy + 4 ψ = 14 Ψ ( ψ − + 2Ψ ) + 14 ψ (Ψ + Ψ Ψ + Ψ ) , (5.26) d Ψ dy + 8Ψ = − ψ Ψ − ψ Ψ . (5.27)When looking for solution of these equations in the form of the 1 /y expansion (5.23), onlyone ambiguity arises, the sign of A (1)0 . Choosing it to be negative, the first few terms ofthe expansion turn out to be ψ = − y − y + O (cid:18) y (cid:19) , (5.28)Ψ = − y − y + O (cid:18) y (cid:19) , (5.29) ψ = − y − y + O (cid:18) y (cid:19) , (5.30)Ψ = − y − y + O (cid:18) y (cid:19) . (5.31)The first terms agree with those of (5.13)-(5.16) obtained by the small amplitude expansion.The difference in the 1 /y terms of ψ and Ψ are caused by the too low truncation for theTaylor expansion of the exponential. For N e ≥ 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. These– 20 –omplicated mode equations can be calculated and 1 /y 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 system(5.24)-(5.27). The large n behaviour of these coefficients will be essential for the calculationof 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 (large n ) behavior of the coefficients, a ( n )2 ∼ k ( − n (2 n − n (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 ofthe Fourier expansion is given in Table 4.The results strongly indicate that in the N f , N e → ∞ N F N ( min ) e k . × − . × − . × − . × − Table 4:
Dependence of the con-stant k on the considered Fouriercomponents N F . The second col-umn lists the minimal exponentialexpansion order N e which is nec-essary to get the k value with thegiven precision. limit k = 0. We do not yet understand what is the deeperreason or symmetry behind this. Hence, instead of (5.32)and (5.33), the correct asymptotic behavior is A ( n )3 ∼ K ( − n (2 n − n (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 valueof the constant turns out to be K = − . ± .
01. Theabove results indicate that the outgoing radiation is inthe Ψ scalar mode instead of being in the ψ dilatonmode. This conclusion is valid only in the framework ofthe approximation employed in the present subsection,i.e. when dropping the terms proportional to ε in (5.21) and (5.22). As we will see in thenext subsection, the situation will change to be just the opposite when taking into account ε 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 Ψ . The first stepis to define a Borel summed series by V ( z ) = ∞ X n =2 A ( n )3 (2 n )! z n ∼ ∞ X n =2 K ( − n n (cid:18) z √ (cid:19) n = − K (cid:18) z (cid:19) . (5.36)– 21 –his series has logarithmic singularities at z = ± i √
8. The Laplace transform of V ( z ) willgive us the Borel summed series of Ψ ( y ) which we denote by b Ψ ( y ) b Ψ ( y ) = Z ∞ d t e − t V (cid:18) ty (cid:19) . (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 partIm b Ψ ( y ) = Z ∞ i √ y d t e − t Kπ Kπ (cid:16) − i √ y (cid:17) . (5.38)A similar calculation for the ψ dilaton mode yieldsIm b ψ ( y ) = kπ − iy ) . (5.39)Since k = 0, this mode is vanishing now. However, as we will show in the next subsection,when taking into account order ε corrections a similar expression for ψ with exp ( − iy ) 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. ε 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 field equation on the real axis we deal withthe corrections arising when taking into account the terms proportional to ε in the modeequations (5.21) and (5.22). We denote the solutions obtained in the previous subsectionby ψ (0) n and Ψ (0) n , and linearize the mode equations around them by defining ψ n = ψ (0) n + e ψ n , Ψ n = Ψ (0) n + e Ψ n . (5.40)The mode equations take the form (cid:18) d d y + n (cid:19) e ψ n + ε D − iQ D d ψ (0) n d y = X m ∂f n ∂ψ m e ψ m + X m ∂f n ∂ Ψ m e Ψ m , (5.41) (cid:18) d d y + n − (cid:19) e Ψ n + ε D − iQ D dΨ (0) n d y = X m ∂F n ∂ψ m e ψ m + X m ∂F n ∂ Ψ m e Ψ m , (5.42)where the partial derivatives on the right hand sides are taken at Ψ n = Ψ (0) n and ψ n = ψ (0) n .The small dimensional corrections e ψ n and e Ψ n have parts of order both ε ln ε and ε .The linearized equations (5.41) and (5.42) are solved to ε ln ε order by the followingfunctions: e ψ n = ε ln ε C d ψ (0) n d y , (5.43) e Ψ n = ε ln ε C dΨ (0) n d y , (5.44)– 22 –here C is an arbitrary constant. The reason for this is quite simple: in ε ln ε order theterms proportional to ε on the left hand sides are negligible and we get the ε = 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 continuingback our functions to the real axis. This can be ensured by requiring agreement with thefirst few terms of the small amplitude expansion formulae (5.13)-(5.16), yielding C = 27 i ( D − Q D . (5.45)In the small amplitude expansion (5.13)-(5.16) to every term of order ε ln ε correspondsa term of order ε which we get by changing ln ε to ln y . Thus, we define the new variables ψ n and Ψ n to describe the ε order small perturbations by e ψ n = ε ln ε C d ψ (0) n d y + ε C ln y d ψ (0) n d y + ψ n ! , (5.46) e Ψ n = ε ln ε C dΨ (0) n d y + ε C ln y dΨ (0) n d y + Ψ n ! . (5.47)Substituting into the linearized equations (5.41) and (5.42) we see that all terms containingln y cancel out, (cid:18) d d y + n (cid:19) ψ n + Cy y d ψ (0) n d y − d ψ (0) n d y ! + D − iQ D d ψ (0) n d y == X m ∂f n ∂ψ m ψ m + X m ∂f n ∂ Ψ m Ψ m , (5.48) (cid:18) d d y + n − (cid:19) Ψ n + Cy y d Ψ (0) n d y − dΨ (0) n d y ! + D − iQ D dΨ (0) n d y == X m ∂F n ∂ψ m ψ m + X m ∂F n ∂ Ψ m Ψ m . (5.49)If C is given by (5.45), ψ n and Ψ n turn out to be algebraic asymptotic series which areanalytic in y . Let us write their expansion explicitly:Ψ k +1 = ∞ X n = k +1 B ( n )2 k +1 y n − , ψ k = ∞ X n = k +1 b ( n )2 k y n − . (5.50)Substituting these and the expansions (5.23) for ψ (0) n and Ψ (0) n into (5.48) and (5.49), it ispossible to solve for the coefficients b ( n ) k and B ( n ) k , up to one free parameter. Comparingto (5.13)-(5.16) it is natural to choose this free parameter to be b (2)0 = c − . 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 ψ k and Ψ k yields exactly the value of C given in (5.45). – 23 –q. (5.48) is consistent with the asymptotics b ( n )2 ∼ ik D ( − n (2 n − n − (cid:20) O (cid:18) n (cid:19)(cid:21) , (5.51)where k D is some constant. Since the leading order result for A ( n )3 is given by (5.34), if k D = 0, the coefficients follow the hierarchy b ( n )2 ≫ A ( n − . In order to be able to extractthe value of k D we have calculated b ( n )2 by solving the mode equations to high orders in1 /y , obtaining k D = 1 . D − Q D . (5.52)The displayed four digits precision for k D can be relatively easily obtained by setting N F ≥ N e ≥ b ( n )2 to orders n ≥
25. We note that there is also a termproportional to the unknown b (2)0 = c − in each b ( n )2 , giving a c − dependent k D . Luckily,the influence of this term to k D quickly becomes negligible as N F and N e grow, makingthe concrete value of c − irrelevant for our purpose.The Borel summation procedure can be done similarly as in Eqs. (5.36)-(5.38). On theimaginary axis ψ is real to every order in 1 /y , however it gets a small imaginary correctionfrom the summation procedure given byIm b ψ ( y ) = ε k D π − iy ) . (5.53) Solutions of the Fourier mode equations (5.18) and (5.19) can be considered to be the sumof two parts. The first part corresponds to the result of the small amplitude expansion,the second to an exponentially small correction to it. The small amplitude expansionis an asymptotic expansion, it gives better and better approximation until reaching anoptimal order, but higher terms give increasingly divergent results. The smaller ε is, thehigher the optimal truncation order becomes, and the precision also improves. The smallamplitude expansion procedure gives time-periodic localized regular functions to all orders,characterizing the core part of the quasibreather. Their extension to the complex plane isreal on the imaginary axis. Furthermore, the functions obtained by the ε expansion aresmooth on large scales, missing an oscillating tail and short wavelength oscillations in thecore region. On the imaginary axis, to a very good approximation, the small second partof the solution of the mode equations (5.18) and (5.19) is pure imaginary, and satisfiesthe homogeneous linear equations obtained by keeping only the left hand sides of theseequations, because the quasibreather is a small-amplitude one. In the ”inner region” it isof order 1 /y , while on the real axis its amplitude is of order ε , hence to leading order thequasibreather core background does not contribute. In the previous subsection we havedetermined the behaviour of this small correction close to the poles. Now we extend it tothe real axis.In the “inner region”, close to the pole, the function Im b ψ 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 small– 24 –mplitude result mentioned in the previous paragraph. We intend to find the solution b ψ of the left hand side of (5.19), which reduces to the value given by (5.53) close to the upperpole, where r = iQ D /ε + y , and behaves asIm b ψ ( y ) = − ε k D π iy ) . (5.54)near the lower pole, where r = − iQ D /ε + y . We follow the procedure detailed in [31]. Theresulting function for large r is b ψ = ε ik D π (cid:18) Q D εr (cid:19) ( D − / exp (cid:18) − Q D ε (cid:19) h i ( D − / exp( − ir ) − ( − i ) ( D − / exp(2 ir ) i . (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 ν ( x ) → r πx cos (cid:16) x − νπ − π (cid:17) , (5.56) Y ν ( x ) → r πx sin (cid:16) x − ν π − π (cid:17) , (5.57)for x → + ∞ . The solution satisfying the asymptotics given by (5.55) is b ψ = √ π α D r D/ − Y D/ − (2 r ) , (5.58)where the amplitude at large r is given by α D = επk D (cid:18) Q D ε (cid:19) ( D − / exp (cid:18) − Q D ε (cid:19) . (5.59)For D > ε , and its size to 1 /ε , for small ε it is possible to extend the function b ψ in its form (5.58)to the real axis into a region which is outside the domain where b ψ 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 ofthe quasibreather core, the nonlinear source terms on the right hand side of (5.19) are notnegligible anymore, and the expression (5.58) for b ψ cannot be used. What actually happensis that b ψ tends to zero exponentially as r → ∞ . This follows from the special choice ofthe “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 expansiongives 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 solution– 25 –as 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 δψ 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 φ QB = −√ π α D r D/ − Y D/ − (2 r ) cos(2 t ) (5.60) ≈ − α D r ( D − / sin h r − ( D − π i cos(2 t ) . (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, φ osc = −√ π α D r D/ − (cid:2) Y D/ − (2 r ) cos(2 t ) − J D/ − (2 r ) sin(2 t ) (cid:3) (5.62) ≈ − α D r ( D − / sin h r − ( D − π − t i . (5.63)The radiation law of the oscillon is easily obtained now,d E d t = − k D π π D/ Γ (cid:0) D (cid:1) ε (cid:18) Q D ε (cid:19) D − exp (cid:18) − Q D ε (cid:19) , (5.64)where the constant k D is given by (5.52). If we assume adiabatic time evolution of the ε parameter determining the oscillon state, using Eqs. (3.52) and (3.54) giving E as afunction of ε , we get a closed evolution equation for small amplitude oscillons, determiningtheir energy as the function of time.For the physically most interesting case, D = 3 we write the evolution equation for ε and its leading order late time behavior explicitly:d ε d t = − .
29 exp (cid:18) − . ε (cid:19) (5.65) ε ≈ . t , E ≈ . t . (5.66)
6. Acknowledgments
This research has been supported by OTKA Grants No. K61636, NI68228.
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