Ultra-relativistic oscillon collisions
Mustafa A. Amin, Indranil Banik, Carina Nagreanu, I-Sheng Yang
PPreprint typeset in JHEP style - HYPER VERSION
Ultra-relativistic oscillon collisions
Mustafa A. Amin ∗ , Indranil Banik , Carina Negreanu and I-Sheng Yang † , , Kavli Institute for Cosmology and Institute of Astronomy, Madingley Rd,Cambridge CB3 0HA, United Kingdom IOP and GRAPPA, Universiteit van Amsterdam, Science Park 904, 1090 GLAmsterdam, Netherlands
Abstract:
In this short note we investigate the ultra-relativistic collisions of smallamplitude oscillons in 1+1 dimensions. Using the amplitude of the oscillons and the in-verse relativistic boost factor γ − as the perturbation variables, we analytically calculatethe leading order spatial and temporal phase shifts, and the change in the amplitudeof the oscillons after the collisions. At leading order, we find that only the tempo-ral phase shift receives a nonzero contribution, and that the collision is elastic. Thiswork is also the first application of the general kinematic framework for understandingultra-relativistic collisions [1] to intrinsically time-dependent solitons. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] O c t ontents
1. Introduction 12. Small amplitude oscillons 3
3. Oscillon Collisions 4
4. Results 9
1. Introduction
The dynamics of soliton collisions can be complex because of the necessary nonlin-earity of the equations governing them. Some of us recently demonstrated that incertain nonlinear scalar field theories, an analytical formalism is possible for under-standing ultra-relativistic soliton collisions [1]. The accuracy of this formalism wasdemonstrated for colliding (anti)kinks in periodic potentials [2]. The examples in thosepapers emphasized the generality of the formalism with respect to general periodicpotentials (arbitrarily far away from known integrable ones). Although these werelimited to (1 + 1) dimensional stationary solitons (ie. intrinsically time-independentsolutions), neither the dimensionality nor the stationarity is a fundamental limitationof the formalism. Here, we demonstrate the applicability of this formalism to intrinsi-cally time-dependent solitons, which in addition do not rely on the periodicity of thepotential. As a concrete example, we study the ultra-relativistic collision of oscillons:spatially localized, oscillatory in time and unusually long-lived solutions of the nonlinearKlein-Gordon equation (for example, see [3–5]). Generalization to higher dimensionsis also an interesting direction , but we will leave that for future work. This includes, for example, the right-angle vortex scattering that is relevant for string intercom-mutation [6–9]. – 1 –he importance of this new example is twofold. First of all, due to Derrick’stheorem [10], many simple field theories cannot support stationary solitons in threedimensions or higher. Localized, time-dependent solitons are the most general objectsthat the kinematic scattering framework in [1, 2] applies to. Secondly, these particularsolitons (oscillons) appear in a wide range of physical scenarios. They can be producedcopiously at the end of inflation, in bubble collisions and in phase transitions in the earlyuniverse [11–15]. They can delay thermalization, play a role in baryogenesis [16], mightappear in dark-matter/dark energy models [17] and are also found in condensed mattersystems (for example [18]). Given their wide-ranging applications, it is worthwhile tounderstand their interactions.Oscillon collisions have been studied before (for example, see [19–21]). Controlledanalytic calculations, however, have not been provided (to the best of our knowledge).Here we take a step towards an analytic understanding of their interactions in thesmall amplitude and ultra-relativistic limit in 1+1 dimensions. We will use the smallamplitudes as well as the inverse relativistic boost factor γ − as small perturbationvariables to aid our calculations. Although ultra-relativistic, small-amplitude, 1+1dimensional oscillon collisions are not typical in the physical scenarios discussed earlier,we hope that our formalism and results will lead to a better understanding of the generalinteraction dynamics.We emphasize that our scenario has one important physical difference compared toearlier applications [2]. When the background object is a stationary soliton, perturba-tions around the soliton (generated by the collision) admit a well-defined expansion interms of separable eigenmodes. Here the oscillon background depends on both spaceand time, so the equation of motion for small perturbations is generically non-separable.That means there is no natural eigenbasis of perturbation modes. However, even inour intrinsically time-dependent situation, there are three modes which have a clearphysical interpretation. The first two are the zero modes corresponding to the spaceand time translational symmetries of the oscillon. The third one is a small change inits amplitude, which is always possible since oscillons exist for a continuous range ofsmall amplitudes. We will calculate the following leading order results for a stationaryoscillon with an amplitude (cid:15) (cid:28)
1, temporal oscillation frequency ω = √ − (cid:15) and spa-tial width ∼ (cid:15) − that undergoes a collision with an incoming, ultra-relativistic oscillon(with γ − = √ − v (cid:28)
1) and amplitude (cid:15) i (cid:28) • the change of internal oscillation phase, ω ∆ t = 4 (cid:15) i /γ , • the shift in position compared to the oscillon width, (cid:15) ∆ x = 0, • the relative change in amplitude, ∆ (cid:15)/(cid:15) = 0.– 2 –ote that the second point implies no velocity change (no time dependence in the posi-tion shift), and the last point means no change in the internal energy of the oscillation.Together, these two indicate that such collisions are elastic at leading order.
2. Small amplitude oscillons
Oscillons are time-dependent, localized, (pseudo-)solitonic configurations that are foundin many scalar field theories with nonlinear couplings. In this paper we will focus on asimple and well-studied model in (1 + 1) dimensions: L = −
12 ( ∂ t φ ) + 12 ( ∂ x φ ) + V ( φ ) , (2.1) V ( φ ) = 12 φ − φ + ... (2.2)where we have assumed a symmetric potential . The minus sign in front of the quarticterm (the opening up of the potential) is necessary for spatially localized solutions toexist. The equation of motion is( ∂ x − ∂ t ) φ = V (cid:48) ( φ ) . (2.3)For the above equation, a long-lived, localized and oscillatory solution (an oscillon) isgiven by (see for example: [22]) φ ( x, t ; (cid:15) ) = (cid:15) (cid:114)
83 sech( (cid:15)x ) cos ( ωt ) + O ( (cid:15) ) , (2.4) ω = √ − (cid:15) , (2.5)where a single, small parameter (cid:15) (cid:28) ω ) andthe spatial width of this oscillon ( ∼ (cid:15) − ). We now consider small perturbations h ( x, t ) around this oscillon solution. We assumethat h is small compared to the leading order term in the oscillon profile, but largecompared to the higher order terms: (cid:15) (cid:28) h (cid:28) (cid:15) . The perturbation h satisfies( ∂ x − ∂ t ) h = [ V (cid:48) ( φ + h ) − V (cid:48) ( φ )] ≈ V (cid:48)(cid:48) ( φ ) h ≈ [1 + O ( (cid:15) )] h . (2.6) We started with the potential L = −
12 ( ∂ t φ ) + 12 ( ∂ x φ ) + m φ − λ φ . We then expressed spatial lengths and time intervals in units units of the mass m , and rescaled thefields by m/ √ λ . – 3 –ithin this function h ( x, t ), two parameters (∆ x and ∆ t ) quantify the spatial andtemporal shifts of the oscillon solution: φ ( x − ∆ x, t − ∆ t ; (cid:15) ) − φ ( x, t ; (cid:15) ) ≈ (cid:15) (cid:114)
83 sech( (cid:15)x ) (cid:20) ( ω ∆ t ) sin ( ωt ) + ( (cid:15) ∆ x ) sinh( (cid:15)x )cosh( (cid:15)x ) cos ( ωt ) (cid:21) . At the leading order of (cid:15) , these changes can be represented by two separable modefunctions: h ( x, t ) = g ( x ) f ( t ) + g ( x ) f ( t ) + ... , (2.7)where g ( x ) = (cid:114) (cid:15) (cid:15)x ) , (2.8) g ( x ) = (cid:114) (cid:15) (cid:15)x )cosh ( (cid:15)x ) , (2.9)and f ( t ) = ( ω ∆ t ) 4 (cid:114) (cid:15) ωt ) , (2.10) f ( t ) = ( (cid:15) ∆ x ) 43 √ (cid:15) cos ( ωt ) . (2.11)Note that the particular normalizations ensure (cid:82) g n ( x ) dx = 1 for n = 1 ,
2. We canproject any small perturbations onto these two modes and evaluate ∆ x and ∆ t . Modefunctions representing additional changes, which by definition are orthogonal to thespatial and temporal translations, are not needed for calculating the position and timeshifts due to the collision.For how a collision leads to the changes in the field profile, we do not need to includea change in amplitude ∆ (cid:15) explicitly in the leading order calculation. This is due to theenergy-conservation/optical-theorem of the formalism in [1]. At the leading order of γ − , the time dependence in the position shift, ∆ ˙ x , and the amplitude change ∆ (cid:15) , arethe only two contributions to the energy change. Other energy changes are “leaks”,which can only appear at the second order or higher. Therefore, if we explicitly calculate∆ x , we can then infer ∆ (cid:15) from it. In our particular case, as we will see, ∆ x = 0, whichdirectly means that ∆ (cid:15) = 0.
3. Oscillon Collisions
Consider a stationary oscillon φ with an amplitude (cid:15) centered around the origin. Atearly times, there is an incoming oscillon φ i with amplitude (cid:15) i moving towards the– 4 – -
100 0 100 200 - x j - -
100 0 100 200 - x j - -
100 0 100 200 - x j - -
100 0 100 200 - x j t = -175t = 150t = - 60t = 25 Figure 1:
The above figure shows an ultra-relativistic collision between two small amplitudeoscillons. The incoming oscillon shows a large number of spatial oscillations due the Lorentztransformation of the oscillatory (temporal) part of the oscillon. The relative narrowness ofthe incoming oscillon is due to the Lorentz contraction of the spatial profile. The dashedline indicates the unperturbed solution for the stationary oscillon. Note that there is adistinct shift in the temporal phase of the stationary oscillon after the collision. This shift intemporal phase can be seen as the difference between the orange and black-dashed profiles ofthe stationary oscillons after the collision. – 5 –tationary one with a speed v and boost factor γ = 1 / √ − v . Since field profiledecays exponentially at the length scale of (cid:15) − , we can simply add the profiles ofoscillons separated by a distance (cid:29) (cid:15) − . Hence, when the two oscillons are still farapart before the collision, we have φ + φ i = φ ( x, t ; (cid:15) ) + φ [ γ ( x + vt ) , γ ( t + vx ); (cid:15) i ] . (3.1)These solitons collide with each other at around t = 0. We capture four snapshots froma numerical simulation of this collision process in Fig.1. The changes in the stationarysoliton, in particular a temporal phase shift, due to the collision is also clearly visible.We turn to the calculation of these collision related changes next. In the presence of an incoming oscillon, Eq. (2.6) is modified as follows:( ∂ x − ∂ t − h = V (cid:48) ( φ + φ i ) − V (cid:48) ( φ ) − V (cid:48) ( φ i ) , (3.2)= − φ φ i + φ φ i ) , ≡ S ( x, t ) . This equation is accurate at the “leading order” in three small parameters, (cid:15) , (cid:15) i and1 /γ . In the full equation of motion for h , the incoming oscillon also modifies the LHS.However, as argued in [1], since h started to be zero, it must first be sourced beforeany self-coupling becomes important. So modifications to the LHS can only affect theresult at higher order in the small parameters.Solving Eq. (3.2) is not significantly different from solving the full problem, since itstill involves solving a PDE in two variables x and t . The useful insight from [1] is thatto calculate changes such as phase shifts, we can first “project” this equation of motionusing the relevant spatial mode functions and then solve the corresponding ODE forthe time-dependent amplitude of these spatial mode functions. The projection ontothe g mode yields (cid:90) dxg ( x )( ∂ x − ∂ t − h ( x, t ) = (cid:90) dxg ( x ) S ( x, t ) , = ⇒ − ( ∂ t + 1) f ( t ) + (cid:90) dx∂ x g ( x ) h ( x, t ) = S ( t ) , = ⇒ − ( ∂ t + 1) f ( t ) + O ( (cid:15) ) = S ( t ) , (3.3)where S ( t ) ≡ (cid:90) dxg ( x ) S ( x, t ) . (3.4)– 6 –ote that we have integrated by parts twice, and ∂ x acting on the slowly varying profile g suppresses it by a factor of (cid:15) . The remaining leading order equation becomes asimple ODE for f . Similarly, we can project onto g ( x ) to get − ( ∂ t + 1) f ( t ) + O ( (cid:15) ) = S ( t ) , (3.5)where S ( t ) ≡ (cid:90) dxg ( x ) S ( x, t ) . (3.6)Solving Eq. (3.3) and (3.5) will allow us to calculate ∆ x and ∆ t created by the collision. It is straightforward to write the solution of Eq. (3.3) and (3.5) as f n ( t ) = − (cid:90) t ∞ dτ sin( t − τ ) S n ( τ ) . (3.7)where n = 1 ,
2. We can calculate the S n (and hence f n ) explicitly if we assume (cid:15) i γ (cid:29) (cid:15) .This condition is satisfied when the two oscillons are about the same size and thecollision is fast. The condition is slightly more general than that; we require that thelength contracted profile of the incoming oscillon is much narrower than the stationaryone. With these assumptions: S ( t ) = − (cid:90) dx φ φ i + φ φ i ) g ( x ) , ≈ − (cid:90) dx φ φ i g ( x ) , (3.8) ≈ − (cid:15) (cid:114)
83 sech( − (cid:15)vt ) cos (cid:16) √ − (cid:15) t (cid:17) g ( − vt ) (cid:90) dx (cid:15) i
83 sech [ (cid:15) i γ ( x + vt )] . = − (cid:15)(cid:15) i γ (cid:114) (cid:15) ( − (cid:15)vt ) cos (cid:16) √ − (cid:15) t (cid:17) . Note that a highly boosted oscillon, φ i , will have a rapidly oscillatory profile in x with a spatial oscillation frequency ∝ γ . As a result the term linear in φ i does notcontribute to the integral in the second line. φ i on the other hand, is positive definiteand will contribute. Nevertheless, we can treat φ i as an averaged envelope instead ofa rapidly oscillating profile, which is the approximation used in the third line. Due tothe highly Lorentz contracted extent of φ i centered around x = − vt , we treat otherfactors which are varying much more slowly in space as a time dependent height (with– 7 – → − vt ) of this averaged φ i envelope. In the fourth line we have carried out the x integral explicitly and use the normalized mode function g ( x ) (see Eq. (2.8)).A similar calculation for the source term S ( t ) in Eq. (3.5) yields S ( t ) = − (cid:90) dx φ φ i + φ φ i ) g ( x ) , ≈ − (cid:15)(cid:15) i √ (cid:15)γ cos (cid:16) √ − (cid:15) t (cid:17) sinh( − (cid:15)vt )cosh ( − (cid:15)vt ) . (3.9)We can now solve for the f n ’s explicitly: f ( t ) = − (cid:90) t −∞ sin( t − τ ) S ( τ ) dτ , = 16 (cid:15)(cid:15) i √ (cid:15)γ √ (cid:90) t −∞ sin( t − τ )sech ( − (cid:15)vτ ) cos (cid:16) √ − (cid:15) τ (cid:17) dτ , = 16 (cid:15)(cid:15) i √ (cid:15)γ √ t ) (cid:90) ∞−∞ cos( τ )sech ( − (cid:15)vτ ) cos (cid:16) √ − (cid:15) τ (cid:17) dτ , = 16 (cid:15)(cid:15) i √ (cid:15)γ √ t ) (cid:90) ∞−∞ sech ( − (cid:15)vτ )2 dτ , = 16 (cid:15) i √ (cid:15)γ √ t ) . (3.10)In the third line we are assuming that t (cid:29) (cid:15) − , i.e. there is no significant overlapbetween the solitons and the collision is complete. In the last line, we combined twocosines, since they have the same frequency (to the leading order in (cid:15) ) and oscillatemuch faster than the slowly varying ‘sech’ envelope. An almost identical calculationfor f ( t ) yields f ( t ) = − (cid:90) t −∞ sin( t − τ ) S ( τ ) dτ , = 16 (cid:15)(cid:15) i √ (cid:15)γ (cid:90) t −∞ sin( t − τ )sech ( − (cid:15)vτ ) sinh( − (cid:15)vτ ) cos (cid:16) √ − (cid:15) τ (cid:17) dτ , = − (cid:15)(cid:15) i √ (cid:15)γ √ λ cos( t ) (cid:90) ∞−∞ sin( τ )sech ( − (cid:15)vτ ) sinh( − (cid:15)vτ ) cos (cid:16) √ − (cid:15) τ (cid:17) dτ , ≈ − (cid:15)(cid:15) i √ (cid:15)γ √ λ cos( t ) (cid:90) ∞−∞ sin(2 τ )sech ( − (cid:15)vτ ) sinh( − (cid:15)vτ ) dτ , = 0 . (3.11)We have used the same approximations as in Eq. (3.10). For example, in the thirdand fourth lines we have set the frequency of the cosine to be 1, which is correct in the– 8 – .02 0.04 0.06 0.08 0.100.010.020.030.04 Ε i Ω (cid:68) t (cid:144) Γ Ω (cid:68) t Figure 2:
The dependence of the temporal phase shift on the amplitude of the incomingoscillon (cid:15) i and the inverse boost factor γ − . The black dots indicate the results from numericalsimulations, whereas the orange line is the expected leading order behavior in these smallparameters based on our analytic calculation. In the right panel, the deviation is visible at γ ∼
4, which shows that higher order terms in the expansion become important. leading order of (cid:15) . Note an important difference from Eq. (3.10). Instead of a cos ()which is positive definite, we have an oscillation (sin()) around zero. This leads to theintegral being zero at the leading order .
4. Results
With the f n ( t ) solutions at hand, we are now ready to calculate the explicit expressionsfor ∆ t and ∆ x . Comparing Eq. (3.10) with Eq. (2.10) and Eq. (3.11) with Eq. (2.11),we get the leading order change in the position and temporal phase of the stationaryoscillon after the collision: ω ∆ t = (cid:15) i γ [4 + O ( (cid:15) ) + O ( (cid:15) i )] , (4.1) (cid:15) ∆ x = (cid:15) i γ [0 + O ( (cid:15) ) + O ( (cid:15) i )] . (4.2)We have chosen to uphold the condition (cid:15) i γ (cid:29) (cid:15) and kept the leading order terms in (cid:15), (cid:15) i and γ − , a good approximation for ultra-relativistic, small amplitude collisions. Note that without setting √ − (cid:15) →
1, one could have gotten a nonzero answer from this integral.However, such answer is higher order in the (cid:15) expansion, which cannot be trusted in our approximation. – 9 –nly the temporal phase shift gets a nonzero contribution at the leading order. As weexplained in the end of Sec.2, the lack of time dependence in ∆ x (more explicitly avelocity term which would be linear in time) implies that the amplitude change ∆ (cid:15) isalso zero. Thus, this collision is elastic at the leading order, same as the collision ofkinks [2]. These are the main results of our short paper.To test our formalism and our analytical results, we carried out detailed 1+1 di-mensional lattice simulations of the oscillon collisions. For the temporal phase shift ω ∆ t (which is nonzero at leading order), we compared the results from numerical sim-ulations (black dots) with the result from our analytic calculation for several values of γ − and (cid:15) i . Excellent agreement with our analytic answer in Eq. (3.10) can be seen inFig. 2. The energy conservation in our simulations was better than 1 part in 10 .We have thus confirmed that the kinematic framework put forth in [1] can beapplied to the case of time-dependent solitons. The excellent agreement between thenumerical and analytical results is encouraging. We used small amplitude oscillons in1+1 dimensions as our specific example of time dependent solitons. Understandingoscillon interactions is interesting in its own right, given their ubiquitous appearancein many physical scenarios from the end of inflation to condensed matter systems. Itwould be interesting to see if the agreement between our analytic results and simulationscontinues to hold beyond the 1+1 dimensional example considered here. A similaranalysis should also be feasible for other time dependent solitons such as Q-balls [23,24]. Acknowledgemements
MA is supported by a Senior Kavli Fellowship at the University of Cambridge. ISY issupported by the research program of the Foundation for Fundamental Research onMatter (FOM), which is part of the Netherlands Organization for Scientific Research(NWO).
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