Semiclassical decay of topological defects
aa r X i v : . [ h e p - ph ] D ec Semiclassical decay of topological defects
Szabolcs Borsanyi ∗ and Mark Hindmarsh † Department of Physics & Astronomy, University of Sussex, Brighton BN1 9QH, UK
Perturbative estimates suggest that extended topological defects such as cosmic strings emitfew particles, but numerical simulations of the fields from which they are constructed suggest theopposite. In this paper we study the decay of the two-dimensional prototype of strings, domain wallsin a simple scalar theory, solving the underlying quantum field theory in the Hartree approximation.We conclude that including the quantum effects makes the picture clear: the defects do not directlytransform into particles, but there is a non-perturbative channel to microscopic classical structuresin the form of propagating waves and persistent localised oscillations, which operates over a hugeseparation of scales. When quantum effects are included, the microscopic classical structures candecay into particles.
PACS numbers: 03.65.Pm, 11.10.-z, 11.27.+dKeywords: Domain walls; Cosmic strings; Oscillons; Hartree approximation
I. INTRODUCTION
Our current fundamental theory of the universe sug-gests higher order symmetries which are successively bro-ken as the universe cools. This naturally leads to theformation of topological defect networks [1]. Such ex-tended objects could play a role in the early formation ofstructure in the universe [2, 3, 4]. Indeed, recent calcula-tions [5] show that models with cosmic strings and othertopological defects fit the Cosmic Microwave Backgrounddata better than the standard power-law ΛCDM model.Analytical estimates suggest that particle productionfrom decaying strings (or, in two dimensions: domainwalls) is suppressed by the separation of cosmological ( ℓ )and microscopic ( M ) scales. The former is the curvaturescale of the string, assumed to be of order the Hubbleradius, and the latter is determined by the underlyingparticle physics, typically on the GUT scale. Particleproduction would mean transforming the energy in thedeep infrared into ultraviolet excitations, over somethinglike 58 orders of magnitude in momentum scale. If thecosmic string network is prohibited from losing energy,its dynamics follows the Nambu-Goto action [3, 6]. Per-turbation theory suggest that the radiative energy loss is ∼ M/ℓ [7]. With more insight into the non-linear domainwall dynamics one discovers another channel throughcusp annihilation [8], at a rate of ∼ M / ℓ − / . Onceone takes into account the gravitational channel a simpleestimate gives ∼ M /M p , where M p is the Plack mass[9], which therefore seems to be dominant for sufficientlylarge ℓ .Numerical analysis of the Nambu-Goto action confirms[10, 11, 12] the analytical scaling assumption [13] thatimplies a string density M/t d − (in d = 2 , ∗ Electronic address: [email protected] † Electronic address: [email protected] network reveals that fragmentation to loops is the domi-nant decay process [14, 15], which proceeds down to thesmallest scale on the network, which may even be themicroscopic scale of the string width [16]. Simulationsin an expanding universe [11, 12] broadly confirm thispicture, although indicate that the relevant small scale isthe initial correlation length [17, 18, 19][52]However, for strings which are topological defects, wecan check this picture by solving the underlying field the-ory in the classical approximation. This means in prac-tice to integrate the non-linear wave equations on a spa-tial lattice and to average over an initial ensemble. Thisapproach offers a full insight into all non-perturbativephenomena, although it may be difficult to justify theomission of quantum effects at the microscopic scale.Nevertheless, this method has been successfully used else-where, e.g. to explore the dynamics of symmetry break-ing [20], or of non-thermal phase transitions [21] in thepost-inflationary Universe.The scaling behaviour suggested by Nambu-Goto sim-ulations is manifest in the classical field dynamics, eventhough only the microscopic scale appears in the equationof motion. It has been demonstrated in the context ofgauge strings in the Abelian Higgs model [22, 23] globalstrings [23, 24], non-Abelian global strings with junctions[25], and semilocal strings [26, 27], as well as domain walls[28, 29], including models with junctions [30]. The scal-ing is present in Minkowski as well as expanding spacetime, and in two as well as three dimensions. It seemsto be a universal feature of classical field theories withextended structures.For strings, a major difference between the numericalsolutions of the classical field dynamics and Nambu-Gotosimulations is that defects decay into classical radiation[22], at a much faster rate than anticipated from per-turbation theory and cusp annihilation. One typicallyobserves a length of string ℓ in a volume ℓ d , and hencethat the string length density is L ∼ ℓ − d . Given a massper unit length of µ ∼ M , the energy density in string is M ℓ − d . Since scaling implies ℓ ∼ t , the energy loss rateper unit length M /ℓ . Hence, for a loop of size ℓ , theaverage energy loss rate is M , which is, in fact, greaterthan the gravitational estimate.We discover then that the classical scaling implies astrong radiative decay. This is very puzzling in view ofthe scale separation between the ℓ and M which grows asthe simulation proceeds. However, apart from confirmingthe scaling over more than three orders of magnitude, itis not our purpose to address this important question.Instead we note that up to now it has been unclear ifthe classical approach is valid here, where dynamics isdriven by an interplay between macroscopic and micro-scopic scales. A check for quantum corrections is crucial.An alternative to the classical approach is thetwo-particle-irreducible (2PI) effective action technique,which is based on a selective resummation of pertur-bative diagrams [31]. Preheating dynamics with non-perturbative particle production [32] and particle ther-malisation by scattering [33] are within the range of itsapplicability. The so far used homogeneous version of thiselaborate technique is, however, incapable of addressingthe question of defect formation [34].We can combine these techniques, using the classicalapproach to form defects and then studying their evolu-tion in the 2PI framework. If we keep the next-to-leadingorder diagrams in the 2PI effective action, we will gaininsight into the scattering and thermalisation of the pro-duced particles. The inhomogeneous variant of the 2PIequations, however, is technically hardly feasible. Keep-ing the lowest order 2PI diagram yields an approximationscheme, that is equivalent to the well-known Hartree ap-proximation [35, 36]. While scattering between the pro-duced particles is not included here, even the homoge-neous version of this scheme could account for the non-perturbatively rapid particle production in the early Uni-verse [37, 38]. The extension of the equations to inhomo-geneous backgrounds was historically motivated by thehope that the background field could mediate interactionbetween the freely streaming particles. Although numer-ics have shown that the opposite was true [39, 40, 41],this method can be still used for finding the leading quan-tum corrections to the evolution of classical structures,as has been suggested by a one-dimensional analysis ofmoving kings [42].In this paper we analyse the classical solution of the λ Φ theory in two space dimensions corrected by theHartree approximation. In the broken phase this toymodel features domain walls, which resemble strings inthis low dimensional setting. We check if there is a sig-nificant alteration to the kink dynamics by the inclusionof this type of quantum correction.First we recall the results from classical simulationsand demonstrate the scaling behaviour also found inRef. [28]. Then in section III we introduce the Hartreeapproximation of the considered model. Next, in sectionIV we numerically compute the domain wall evolutionboth in the classical and in the Hartree approximatedframework. We discuss possible interpretations of theresults in section V, and finally conclude in section VI. II. CLASSICAL DECAY OF DOMAIN WALLSA. Model details
The Lagrangian density of our scalar theory is as sim-ple as L = 12 [ ∂φ ] − m φ − λ φ (1)The theory has a Z (2) symmetry, this breaks sponta-neously if the thermal mass turns negative. By the choiceof the bare mass parameter m we make sure that the sys-tem is deeply in the broken phase at zero temperature.The used quartic potential is motivated by the simpleform of the classical kink solution:Φ( x, y ) = v tanh ( M x ) (2)with M = − m / v = p − m /λ . (3)The tension of the wall is inversely proportional to thecoupling: σ = 4 | m | / λ . In the classical limit the actualmagnitude of the coupling is irrelevant as it can be scaledout. In the numerics we used λ = 6 M . The only otherparameter, the mass sets the scale for the evolution, weuse the inverse wall width M to render all variables di-mensionless, this numerically means M = 1.We discretise the model on a spatial lattice. We solvea cut-off theory with a pre-set lattice spacing a . Sincemuch of the physics of our interest is in the infrared, a plays little role. Based on earlier numerical experience wecan use lattices as coarse as aM = 0 .
5. We repeated thepresented numerical analysis on a coarser lattice ( aM =0 .
7) and found no significant difference. The lattice size L , however, matters. In order to avoid the interaction ofa pair of signals originating from the same site we stop thesimulation at t = L/
2. This assumes that at t = 0 thereis no correlation between any of the sites. Our initialcondition will approximately satisfy this condition.The initial condition can introduce other scales. Westart the dynamics from a low energy density randomconfiguration with a rich domain wall structure. Ourmain interest is how these walls evaporate under the re-alistic assumption that the scale in the initial conditionseparates from the microscopic scale M .We designed the following numerical experiment. Westart from a white noise configuration at t = 0, deep inthe symmetric phase. We also checked the invariance ofour results under replacing the initial noise by a (classi-cal) thermal equilibrium of the same energy density. Wethen apply a cooling by adding a friction term to the ki-netic term in the equation of motion: ∂ φ → ∂ φ +2 γ∂ φ .This evolution is non-physical, and we switch off at a con-veniently chosen time when the particle content is neg-ligible and the domain wall density reached a desirablevalue [53]. We starting the numerical observations onlyafter a short period of relaxation after the non-physicaldynamics has been switched off. This corresponds to thepre-thermalisation time scale [43].The field configuration at this instant is the initial con-dition of the dynamics of interest. We could set the originof time to this instant, but we choose not to. The t = 0point marks the onset of cooling, because at that pointthe correlation length is known to be of the microscopicscale. B. Scaling solutions
The solution of classical dynamics is a straightforwardcomputational task and being restricted to two spatialdimensions our resources allows for larger lattices ( L ≥ ℓ ) of the domain wall network decouplesfrom the microscopic scale, one may expect from dimen-sional reasons: ℓ − ∼ L ∼ t − . The domain wall density L we define as the total length of domain walls on thelattice divided by the volume. A link on a lattice is con-sidered as a part of a domain wall if the sites at its bothends have field amplitudes of opposite sign. i n v e r s e do m a i n w a ll den s i t y timeSide Length | runs averaged4000 | 242000 | 1281000 | 1024250 | 16000 FIG. 1: Inverse domain wall density as a function of time.There is a perfect linear correspondence over at more thantwo orders of magnitude ( t = 10 . . . t = 10 to the end of thesimulation, when the walls’ mean curvature radius is over 10 times larger than their width. As always in classical field theory we always display anaverage over an ensemble of runs. (This is the thermal orwhite noise ensemble at t = 0. At positive times there isno randomness in the dynamics.) The averaged domainwall densities start deviating near t ≈ L , slightly laterthan expected.This scaling is a manifestation of a more generic featurein classical field theories, as it has been found in flat or curved space-time, and in two or three dimensions [28].So that we gain more insight into the observed scalingwe show a pair of lattice configurations in Fig. 2. Thesimilarly looking snapshots were taken at times 50 and100, respectively, but the earlier configuration we halvedin linear size and scaled up accordingly. The time evolu-tion appears to be equivalent to zooming. FIG. 2: Two snapshots of the lattice field configurations( L = 2000). Blue and orange regions show the domains ofthe degenerate vacua. To the left, we show the configurationat t = 50, we cropped the region (0 , L/ × (0 , L/
2) and scaledup by a factor of two. To the right we show the uncroppedlattice at t = 100. There is no qualitative difference betweenthe snapshots. This feature can be made more formal in terms of thecorrelation function. We define our C ( r, t ) correlationfunction as C ( r, t ) = 1 L Z dxdydz h φ ( x, z, t ) φ ( y, z + r, t ) i , (4)where h·i denotes an ensemble averaging. If the moreconventionally defined correlation function [54] G ( | ~x − ~y | , t ) = h φ ( ~x, t ) φ ( ~y, t ) i scales as G ( r, t ) = t α G ( r/t ), it iseasy to see that C ( r, t ) = t α +1 C ( r/t ). Indeed, Fig. 3shows a numerical evidence for the scaling of C , with α ≈
0. (If we require t = 0 to be the origin, the scalinglaw is only accurate to 10 %. If we fit the location of theorigin of time and drop the initial evolution ( t < −
26 ).
III. HARTREE APPROXIMATION OF SCALARFIELDS
In this section we review the inhomogeneous Hartreeapproximation and its application to our model. Thereader can find a more detailed introduction in Ref. [39].Its other name, Gaussian approximation, reflects theessence of the truncation of the dynamics: we disregardany connected higher n -point functions. Note in the con-text of the N -component scalar field the leading order in1 /N expansion leads to very similar (also Gaussian), butinequivalent approximation [44].The operator equation in Heisenberg picture that we -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 f ( t ) C (r /t,t ) rescaled length (r/t) t= 30t= 60t=120t=240t=480 f ( t ) time FIG. 3: Scaling of the equal time correlation function. Noticethat the function has an approximate Gaussian shape, whichis due to the nature of spinodal instability that created thedomains. (There are deviations at small distance.) ( L =1000, 1024 runs averaged) have to solve reads:( ∂ + m ) ˆ φ ( ~x, t ) + λ φ ( ~x, t ) = 0 . (5)We split off the quantum expectation value: ˆ φ ( ~x, t ) =¯Φ( ~x, t ) + ˆ ϕ ( ~x, t ), with h ˆ ϕ ( ~x, t ) i ≡ ϕ degrees of freedomto Gaussian at all times. For such degrees of freedom ˆ ϕ i we have the following identities: (cid:10) ˆ ϕ i (cid:11) = 0 and (cid:10) ˆ ϕ i ˆ ϕ j (cid:11) =3 (cid:10) ˆ ϕ i (cid:11) h ˆ ϕ i ˆ ϕ j i .We can simply take the quantum average of Eq. (5),or multiply from the left by ˆ ϕ ( ~y, t y ) and get an equationfor the Wightman propagator G < ( ~x, t x ; ~y, t y ) ≡ G < ( x, y )by averaging that, too: (cid:20) ∂ x + m + λ G < ( x, x ) (cid:21) ¯Φ( x ) + λ ( x ) = 0 , (6) (cid:20) ∂ x + m + λ ( x ) + λ G < ( x, x ) (cid:21) G < ( x, y ) = 0 , (7)It is remarkable that this simple truncation of the hier-archy of n -point functions leads to a self-consistent set ofequations. Indeed, these are the Schwinger-Dyson equa-tions for the propagator in the leading order truncationof the two-particle irreducible (2PI) effective action [45].Gaussianity also implies that the Heisenberg operatorsat a finite time relate to the initial operators by a Bo-golyubov transformation:ˆ ϕ ( ~x, t ) = Z d d k (2 π ) d (cid:16) ˆ a ~k ψ ~k ( ~x, t ) + ˆ a + ~k ψ ~k ∗ ( ~x, t ) (cid:17) . (8)So that we have the text-book operator at t = 0 weset ψ ~k ( ~x,
0) = e − i~k~x / √ ω k and ˙ ψ ~k ( ~x,
0) = − iω k ψ ~k ( ~x, d is the number of space dimensions and ω k = ~k + m r , where m r is the renormalized mass, often amended with a contribution from the background field. The lad-der operators obey the usual [ˆ a ~k , ˆ a + ~k ] = (2 π ) d δ ( ~k − ~k )commutation relation. The initial particle spectrum aregiven by D ˆ a + ~k ˆ a ~k E = n ~k . These numbers appear in theequal time two-point function: G < ( ~x, t, ~x, t ) = Z d d k (2 π ) d | ψ ~k ( ~x, t ) | (2 n ~k + 1) (9)This obviously diverges even in two space dimensions.The initial infinite mass shift we compensate by a massrenormalisation and introduce the finite mass squared m r in the equation for ψ ~k : h ∂ x + m r + λ ~x, t )+ λ Z d d k (2 π ) d (cid:2) | ψ ~k ( ~x, t ) | − ψ ~k ( ~x, | (cid:3) (2 n ~k + 1) (10) i ψ ~p ( ~x, t ) = 0A coupling renormalisation is also necessary in threedimensions [46], but in this simple case we do not needto go beyond mass renormalisation.The traditional way of solving the dynamics in Gaus-sian approximation involves Eqs. (6), (9) and (10). Onenormally discretizes the equations on a space lattice. Aconsistent Bogolyubov transformation requires that the ~k index of the mode functions runs in the entire Fourierspace of the lattice. In addition to the trivial backgroundequation on an N lattice this means N complex equa-tions.But mode function expansion is just one of the possi-ble ways of solving Eqs. (6) and (7). Alternatively, weconsider an ensemble of N e classical trajectories ϕ i ( ~x, t ),solutions of the equation (cid:18) ∂ x + m + λ (cid:2) ¯Φ ( ~x, t ) + (cid:10) ϕ ( ~x, t ) (cid:11) E (cid:3)(cid:19) ϕ i ( ~x, t ) = 0 . (11)Here h·i E stands for the ensemble average. Indeed, mul-tiplying the equation with ϕ i ( ~y, t y ) and averaging over i (ensemble average), will bring us back to Eq. (7) with G < → G e = h ϕ ( ~x, t x ) ϕ ( ~y, t y ) i E . But there will be noexact equivalence between ensemble and quantum aver-ages: the quantum two-point function G < is complex, G e is real. Notice, however, that the imaginary part of G < entirely decouples in Eq. (7), since the equal timepropagator is always real.Of course, ϕ i ( ~x, t ) must be properly initialized to forma Gaussian ensemble of the correct standard deviation: h ϕ ( ~x, ϕ ( ~y, i = ¯ h Z d d k (2 π ) d e − i~k ( ~x − ~y ) ω k (cid:18) n ~k + 12 (cid:19) , h ˙ ϕ ( ~x,
0) ˙ ϕ ( ~y, i = ¯ h Z d d k (2 π ) d e − i~k ( ~x − ~y ) ω k (cid:18) n ~k + 12 (cid:19) , h ϕ ( ~x,
0) ˙ ϕ ( ~y, i = 0 . (12)Technically, we initialize ϕ in momentum space by a ran-dom phase and amplitude at the t = 0 and t = δt timeslices.We intentionally introduced the factor ¯ h , as a controlparameter for the fluctuation ϕ . This way we can tunestrength of the back reaction of the quantum fluctuationsto the background. In the classical theory it was possibleto scale out λ , here rescaling the field with 1 / √ λ wouldalso require to rescale ¯ h with λ . If we stick to λ = 6 M in the numerics, it is the ¯ h in Eq. (12) that one can useto vary the coupling, effectively.Numerically, it is much simpler to solve N e N realequations than N complex ones, we found that even anensemble of 1 ≪ N e ≪ N was big enough. The simplestructure of Eq. (11) allows high speed implementations[55]. We note that the equation is not stable withoutmanually fixing h ϕ i ( ~x ) i = 0 after every leap-frog timestep.Before embarking into the analysis of numerical re-sults, let us pause to discuss in what sense the Hartreeequations represent a quantum correction to the classicaldynamics.Notice that we can arrive at Eqs. (6) and (7) also froma different concept. Let us start a number of classi-cal trajectories from an initial Gaussian ensemble (e.g.Eq. (12)). Instead of following the individual trajectorieswe can write down the equations for the n -point func-tions. Simply discarding the three or higher order cor-relators we get a closed set of equations, that coincidewith Eqs. (6) and (7). We would also arrive to the sameequations by truncating the 2PI effective action for theclassical (or quantum) field theory to leading order.Indeed, whether we start from a classical or quantumGaussian ensemble, the genuine quantum features startto appear if we keep the four-point equation at least. Aself-consistent set of equations follows from the next-to-leading order truncation of the 2PI effective action, whereone easily identifies the term, responsible for quantumeffects [47].This statement, however, means that to Hartree or-der it is only the initial condition that reflects quantumphysics. Do the mode function equations (10) or thepropagator equation (7) introduce quantum correctionsat all?The answer is yes. If we consider one single classicaltrajectory ¯Φ, switching on ¯ h in Eqs. (12) will definitelyenable many quantum phenomena, such as vacuum par-ticle production. Instead of doing Hartree, one can, ofcourse, consider an ensemble of ¯Φ fields, initialized (asusual) with the just-the-half rule (analogous to Eqs. 12)and evolve them classically. This classical ensemble willequally enable the same quantum phenomena, but it willbring in several classical artefacts, too, such as the decayof the quantum zero-point energy. These artefacts canmost simply eliminated by shutting down all higher loopdiagrams, down to the order where quantum and classicalapproximations agree: this is the Hartree approximation.Although it is possible to properly include higher or- der corrections [32, 33], they are not inevitable in thefollowing two extremes: If the particle numbers are low,the higher order quantum corrections, that account forscattering of the quantum fluctuations, are not very im-portant compared to the dynamics of other energetic ob-jects, such as defects. If the particle numbers are veryhigh, higher order quantum corrections are crucial, butthey can be well estimated by a classical ensemble, here n ~k dominates in n ~k + 1 / IV. CLASSICAL VERSUS HARTREEDYNAMICS
The initial conditions given in section II A define a(highly non-Gaussian) ensemble of domain wall config-urations at t = 15. For each member ¯Φ i we define a(Gaussian) sub-ensemble of fluctuations. We follow thedynamics of this sub-ensemble in the Hartree approxima-tion. The final averaging over the domain wall configu-rations occurs at the very end of the calculation. At thetime we switch on the quantum equations (11) we renor-malize the mass and thereby allow a smooth transitionto quantum evolution.In Fig. 4 we show the evolution of the power spectrumof the background field. In classical field theory this is theonly degree of freedom, whereas in the Hartree approxi-mation energy may drift into the “modes” (the ensembleof quantum fluctuations).The correlation length in Fig. 5 is defined by a Gaus-sian fit to the correlation function shown in Fig. 3. In har-mony with Fig. 4 we see no impact of the quantum fluc-tuations on the evolution of the macroscopic degrees offreedom. If all the domain wall loops were macroscopic,this would suggest that inverse total length of domainwalls shown in Fig. 5 receives no significant quantumcorrection. Indeed, we again find a linear scaling, andwe could not find a significant correction to the slope pa-rameter for t > ϕ x ( ~x, t )functions in Eq. 11 reflect the created particles. Thisspectrum does not scale, and performs a “boring” evolu-tion: only the amplitude changes slightly and always re-sembles the vacuum power spectrum. This confirms theassumption that the particles are created on the massscale and not e.g. in the infrared. At and beyond the < | φ ( k ) | > t - tkhbar=0 t=18t=30t=60t=120t=2401e-071e-061e-051e-041e-031e-021e-011e+001e+01 0.1 1 10 100 1000 < | φ ( k ) | > t - tkhbar=1 t=15t=30t=60t=120t=240 FIG. 4: The scaling of the power spectrum. The macroscopicpart (domain walls) clearly follows the scaling law both in theclassical (top) and in the quantum (bottom) case. The scalingbreaks at the tail of the spectrum (small classical structures).( L = 500, average of 16000 and 192 runs for ¯ h = 0 and 1,respectively.) mass scale the quantum fluctuations dominate over theclassical structures in the power spectrum.By construction of the initial conditions the energydensity transferred from the domain walls to the fluc-tuations does not significantly raise the temperature andso the thermal mass. It was an important assumption inour analysis that the wall width M is constant in time.The domain wall density and the correlation length arethe key observables when we discuss scaling. Irrespec-tively to the coupling ( λ ) or the strength of the fluctua-tions (¯ h ) we fit ξ ≈ . · t . For the domain wall densitywe find L = l/L ≈ . /t for our initial condition,where l is the total counted length of domain walls onthe lattice at a given time. It is remarkable that thesedimensionless coefficients are robustly insensitive to thevariation of the coupling or the value of ¯ h . Also it doesnot depend on the lattice spacing nor on the initial noiseamplitude or the details of the cooling procedure.To gain more insight into the small discrepancy be-tween the quantum and classical domain wall density wecount the number of domains, and use this number toestimate the loop number density ( n ( t ) = N ( t ) /L ). Weapplied a cluster algorithm on the lattice and plotted c o rr e l a t i on l eng t h time / do m a i n w a ll den s i t y timeL= 500, hbar=0L= 500, hbar=1 FIG. 5: The scaling of the correlation length (top) and thedomain wall density (bottom) the resulting number density in Fig. 6. The effect of thequantum fluctuations is now striking.In the approximated quantum theory we get what wethink we should: if the correlation length ξ ∼ t scaleslinearly, any number density must scale as n q ∼ t − .The conclusion from Fig. 6 is that in the classical theorythe domain number is dominated by microscopic struc-tures. One hint for the smallness of these “mini-domains”is that the inverse wall width M must appear in the n cl ( t ) ∼ M/t scaling rule for dimensional reasons, and itscoefficient is not extreme. We can directly measure an av-erage defect loop size by counting the loops (or domains,practically) that are in excess in the classical solution.The total wall length is also bigger in the classical casethan with quantum correction. We used their quotient toestimate the size of these loops in Fig. 6. Unfortunatelyour numerics is not conclusive at later times, we expectthis ratio to settle at a positive value ∼ M − .To find out more about these small classical structureslet us compare the lattice snapshots taken from the samerun with and without quantum fluctuations. We pickedthe time t = 40 and cropped a larger lattice appropriatelyso that we can show the most phenomena in one image:Fig. 7In these images the black regions correspond to the¯Φ > < do m a i n nu m be r den s t i y time hbar=10.7 t -2 hbar=00.045 Mt -1 a v e r age l eng t h o f l oop s i n e xc e ss timel cl -l qu N cl -N qu FIG. 6: Number of domains over lattice volume (top) inthe classical and quantum framework. The classical scalingis counter intuitive. Dividing the difference of the total looplength by the number of loops (estimated by the number ofdomains) we get an average loop size (bottom). merous dark spots where ¯Φ goes locally close or beyondzero. In the middle (quantum) plot we see fewer ripplesand some (but not all) of the dark spots of the classicalplot are missing here. To the right, we demonstrate theexcitation of the quantum modes by plotting (cid:10) ϕ ( ~x, t ) (cid:11) E .These fluctuations are the strongest on the domain walls,which we interpret as particles in the bound state.Looking at a sequence of such snapshots gives moredetail about how these ripples are produced. As the do-main walls shrink they emit classical waves, with a wavelength of few times the wall width. These waves areequally present in the quantum case as well, where theyare more damped. In the quantum fluctuation plot wealso find traces of the classical ripples, but the spatial dis-tribution of the produced particles appears to be smooth,and the ripples in (cid:10) ϕ ( ~x, t ) (cid:11) E are an order of magnitudesmaller than in ¯Φ .On the snapshots we marked the interesting places byletters. In both sides of the letter “A” the ripples are lo-cally so high in amplitude that these spots are counted asa domain by the cluster algorithm and they contributeto the total length of domain walls. But they are notcounted into the statistics of walls in the quantum case,since then their amplitude is within the threshold of zero field value. The amplitude of these spots actually oscil-lates, this is why we do not see the one on the right handside of the letter “A” in the quantum plot. “B” marks thecentre of ripples emitted earlier by the collapsing bubblemarked with “D”. These are mostly damped in the quan-tum run. The waves around the bubble “D” are higherin amplitude than in the quantum case. Finally, there isa spot with strongly oscillating amplitude, marked with“C”. The magnitude of the quantum fluctuations oscil-late coherently with the background field value. V. DISCUSSION
Let us summarize the numerical findings: The correla-tion length, which is fitted from the correlation functionin direct space, reflects the macroscopic evolution. Wefind that the known scaling behaviour is unperturbedby quantum effects. On the microscopic level, however,where the scaling is broken, we find stronger quantumeffects, as expected.We also find, that there are “mini-domains” in theclassical simulation, that (at least partly) disappear ifwe switch on the quantum degrees of freedom. Its sim-plest explanation is that there are classically stable smallstructures that decay in a quantum field theory. Now wecan speculate what these could be. Natural candidatesare oscillons, localised oscillating wave packets, whichare (quasi-)stable solutions of the classical field theory[48, 49].If these small structures are indeed oscillons, their sta-bility is enhanced by low dimensionality. If in three di-mensions they are subject to a more rapid decay [50, 51],making the quantum decay channel less significant andhence the quantum correction to the scaling even smaller.Indeed, a closer look on the lattice field revealed thatthere are small regions (with a diameter of O(5) do-main wall width) that oscillate with a frequency ∼ M .But oscillons are not the only structures that appear.The shrinking and collapsing domain walls emit classicalwaves with a wave length ∼ M − . We see these waveson the lattice snapshots as circular ripples. These ripplesfrom various sources interfere and at the points of con-structive interference the field value may locally exceedzero and will then be counted as a small domain.Classical waves are emitted in the quantum field the-ory, too. In quantum mechanics this classical excitationis known as coherent state, which transforms into an en-larged width of the wave function, or into particles in fieldtheory language. This is the point where quantum cor-rections enter: the classical waves are damped and theirinterference results in fewer and less stable localised os-cillating wave packets.In this picture there is a non-perturbative classicalmechanism that converts the energy stored in the string(or domain wall) to microscopic objects. In a field theory,these objects are neither loop fragments, nor particles,but coherent oscillations of the field expectation value. FIG. 7: Snapshot from a classical (left) and the corresponding quantum (middle) run. To the right the particle content ischaracterized by (cid:10) ϕ ( ~x, t ) (cid:11) E . The darker points mean higher value. (These images were taken at t = 40 on a L = 128 lattice.We cropped out a piece of 75 × Our numerics suggests that the scale of these classicalwaves are on the microscopic scale M . We observe thatthese waves are emitted from structures of size ℓ , present-ing us with the challenge of explaining energy transportover a huge scale separation, M ℓ ∼ at the end thesimulation.It is clear from the shown numerics that the domainwall decay was not enhanced by the quantum fluctuationsand this conclusion we checked to stay true with ¯ h = 2or λ = 12. There is no indication for a direct decay chan-nel into particles. A direct decay might also manifest inthe sensitivity to the choice of the lattice spacing as weswitched between aM = 0 . .
7, but we found nosignificant difference. However, the decay of the classi-cal waves and oscillons is no longer protected by scaleseparation.Finally, let us attempt to understand Fig. 6. Theenergy density associated to macroscopic D -dimensionaldefects in d dimensions is ∼ M D t D − d . Their decay re-leases energy at a rate of ∼ M D t D − d − . This energy isused to produce high amplitude classical structures (e.g.oscillons) that may have been counted as small domains.Since they emerge on the microscopic scale, their numberdensity has a source of C source M D t D − d − , where C source and the other constants we introduce here are dimension-less numbers of O (1). These small structures can decayin various ways: a) In the quantum calculation we includethe direct quantum mechanical decay into particles witha rate of Γ ∼ M ; b) The small objects can be hit by adomain wall or string, its rate is proportional to the de-fect density: C defect M D − d +1 t D − d ; c) These objects canalso hit each other and annihilate. The probability of agiven small object to meet an other one is proportionalto its density n , which gives a rate of C coll M − d n . These together give the following equation for the density n ˙ n +Γ n + C defect M D − d +1 t d − D n + C coll M − d n = C source M D t d − D +1 (13)If the quantum decay into particles dominates giving afinite life time to these small classical structures, the den-sity n simply follows the source. Indeed we see n ∼ t − in Fig. 6. In the absence of Γ, however, we find that n ∼ M/t solves Eq. (13) in consistence with our observa-tion. Since in this case n shows the same scaling as thedomain wall density, counting them as defects does notspoil the observation of scaling. The classical approxi-mation of Eq. (13) suggests that for d > n ∼ t − . In higher dimen-sions, however, oscillons and other analogous structuresare less stable, which introduces a decay term of classicalnature bluring difference between classical and quantumscaling. VI. CONCLUSION
In this paper we integrated the classical field equa-tions as well as the Hartree approximated quantum evo-lution of a scalar field in the broken phase, starting froma network of domain walls. The scaling of macroscopicobservables was manifest also in the quantum theory.Our numerical results suggest that the direct decay ofdomain walls into particles is insignificant, as the per-turbative estimates predict. We can instead attributethe decay to the emergence of classical waves and otherstructures, such as oscillons. Since these coherent exci-tations of the quantum field theory are produced at themicroscopic scale, their perturbative decay is no longersuppressed by the separation of scales. The productionof large-amplitude classical oscillations is a genuine non-perturbative phenomenon that deserves further investi-gation, as a similar effect is seen to drive the decay ofcosmic strings in three dimensional field theory simula-tions [22]. Understanding the dominant decay channelof strings is of crucial importance for computing theirobservational signals.
Acknowledgments
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