aa r X i v : . [ h e p - t h ] O c t Numerical Simulation of an Electroweak Oscillon
N. Graham ∗ Department of Physics, Middlebury College, Middlebury, VT 05753
Numerical simulations of the bosonic sector of the SU (2) × U (1) electroweak Standard Model in3+1 dimensions have demonstrated the existence of an oscillon — an extremely long-lived, localized,oscillatory solution to the equations of motion — when the Higgs mass is equal to twice the W ± boson mass. It contains total energy roughly 30 TeV localized in a region of radius 0.05 fm. Adetailed description of these numerical results is presented. PACS numbers: 11.27.+d 11.15.Ha 12.15.-y
INTRODUCTION
While static, localized soliton solutions to the equations of motion of nonlinear field theories have been well studied,and are of interest in many applications [1, 2], no known examples exist in the electroweak Standard Model (althoughthere do exist extended electroweak string solutions [3, 4]). However, much less is known about the existence oflocalized solutions that oscillate in time, known as breathers or oscillons. (The latter term was originally introducedto describe similar phenomena in plasma physics [5].) In some models, such as the sine-Gordon breather [6] and Q -ball[7], one can use conserved charges to prove the existence of exact, periodic solutions. But oscillons have also beenfound in many nonlinear field theories that do not contain either static solitons or conserved charges. These solutionseither live indefinitely or for extremely long times compared to the natural timescales of the system.For scalar theories in one space dimension, oscillons have been found to remain periodic to all orders in a perturbativeexpansion [6] and are never seen to decay in numerical simulations [8], but can decay after extremely long times vianonperturbative effects [9] or by coupling to an expanding background [10]. In both φ theory in two dimensions[11, 12] and the abelian Higgs model in one dimension [13] and in two dimensions [14], oscillons have been found that arenot observed to decay. In φ theory in three dimensions, however, one finds long-lived quasi-periodic solutions whoselifetime depends sensitively on the initial conditions [15, 16, 17, 18, 19]. Similar behavior is present in other scalartheories in three dimensions [20] and in higher dimensions [21]. Phenomenologically, small Q -balls were considered asdark matter candidates in [22, 23, 24, 25], axion oscillons were considered in [26], and the effects of oscillons and otheraspects of nonequilibrium dynamics in and after inflation were studied in [27, 28, 29]. Oscillons and related solutionshave also been studied in connection with phase transitions [30], monopole systems [31], QCD [32], and gravitationalsystems [33].Recent work [34] demonstrated numerically the existence of an oscillon in the bosonic sector of the electroweakStandard Model, when the mass of the fundamental Higgs is exactly twice that of the W ± gauge bosons. (A similarmass relation also arises in the study of embedded defects [35].) This result was based on previous work [36], whichfound oscillons in spontaneously broken pure SU (2) Higgs-gauge theory with the same 2 : 1 mass ratio. In thatmodel, one can consider field configurations restricted to the spherical ansatz [37], meaning they are assumed to beinvariant under combined rotations in space and isospin, also known as grand spin rotations. Within this ansatz,the system can be described by an effective theory of fields depending only on r and t , which greatly simplifies thenumerical analysis. In [34] this numerical simulation was extended to a fully three-dimensional spatial lattice with noassumptions of rotational symmetry, making it possible to also include the U (1) hypercharge field (which breaks thegrand spin invariance of the spherical ansatz). The resulting simulation comprises the full electroweak sector of theStandard Model without fermions. Here we extend that analysis and describe its results in more detail. We use thesame SU (2) gauge coupling g and Higgs self-coupling λ as in the pure SU (2) theory, meaning that the Higgs mass istwice the mass of the W ± bosons, and set the U (1) coupling g ′ so that the mass of the Z boson matches its observedvalue.Ongoing analytic work [38] has shed some light on the 2 : 1 mass ratio by using a small amplitude approximation[6, 10, 32, 39] to construct oscillons in a simplified version of the spherical ansatz theory. In this analysis, one beginsby assuming that each field in the oscillon profile has large width, so that at large distances it falls like exp( − ǫmr ),where m is its mass. There, the amplitude is small and the oscillations obey a linear dispersion relation, which implies ω = m √ − ǫ . The linear, dispersive gradient terms in the equation of motion are then of order ǫ . They mustbe balanced by nonlinear terms to obtain a stable solution. Since the leading nonlinearity is typically given by aquadratic term in the equations of motion, this requirement implies that the field amplitudes must be proportionalto ǫ . In a multiple-field model, one must also ensure that the terms giving interactions between different fields areresonant with the dispersive linear terms, so that their effects are not washed out over many cycles. As shown in [38],the 2 : 1 mass ratio arises naturally in this analysis: Since the fields’ oscillation frequencies are tied to their masses,imposing a resonance condition on their frequencies is equivalent to fixing a particular mass ratio. Although thisanalysis has so far only been carried out in simplified models, we will see below that the oscillon observed numericallyin the full electroweak theory is of small amplitude and large width, so similar techniques are potentially applicablein this case as well.In all known oscillons, each field oscillates with a frequency below its mass, so that it couples to dispersive linearwaves (which have ω = √ k + m > m ) only through nonlinear interactions. The fields then converge to a configu-ration in which this decay channel is also suppressed. Because the electroweak theory includes the massless photonfield, which can radiate in arbitrarily low frequencies, one might expect the oscillon to decay rapidly by emittingelectromagnetic radiation, but it does not. Instead, after initially shedding some energy in this way, the system settlesinto a localized solution that no longer radiates and remains stable for as long as we can follow it in numerical sim-ulations. In preliminary work that provided motivation for the current investigation, similar behavior was observedboth when an additional massless scalar field was coupled to oscillons in one-dimensional φ theory and when anadditional spherically symmetric massless scalar field was coupled to oscillons in the spherical ansatz model. In eachcase, after shedding some energy into the massless field, the oscillon arranges itself in a neutral configuration thatno longer couples to the massless field. This mechanism may be similar to the suppression of nonlinear coupling todispersive waves that is common to all oscillons. CONTINUUM THEORY
We begin from SU (2) × U (1) electroweak theory in the continuum, ignoring fermions, and follow the conventionsof [40]. The Lagrangian density is L = − F µν F µν − F µν · F µν + ( D µ Φ) † D µ Φ − λ ( | Φ | − v ) , (1)where the boldface vector notation refers to isovectors. Here Φ is the Higgs field, a Lorentz scalar carrying U (1)hypercharge 1 / SU (2). The metric signature is + − −− .The SU (2) and U (1) field strengths are F µν = ∂ µ W ν − ∂ ν W µ − g W µ × W ν , F µν = ∂ µ B ν − ∂ ν B µ , (2)and the covariant derivatives are given by D µ Φ = (cid:18) ∂ µ + i g ′ B µ + i g τ · W µ (cid:19) Φ , D µ F µν = ∂ µ F µν − g W µ × F µν , (3)where τ represents the weak isospin Pauli matrices. We obtain the equations of motion ∂ µ F µν = J ν , D µ F µν = J ν , D µ D µ Φ = 2 λ ( v − | Φ | )Φ , (4)where the gauge currents are J ν = g ′ Im ( D ν Φ) † Φ , J ν = g Im ( D ν Φ) † τ Φ . (5)We work in the gauge B = 0, W = . With this choice, the covariant time derivatives become ordinary derivativesand we can apply a Hamiltonian formalism. The energy density is u = 12 X j = x,y,z ˙ B j + ˙ W j · ˙ W j + X k>j (cid:0) F kj + F kj · F kj (cid:1) + | ˙Φ | + X j = x,y,z ( D j Φ) † ( D j Φ) + λ (cid:0) | Φ | − v (cid:1) , (6)where dot indicates time derivative. The integral over space of this quantity is conserved by the time evolution. Fromthe equations for B and W , we obtain the Gauss’s Law constraints, X j = x,y,z ∂ j ˙ B j − J = 0 , X j = x,y,z D j ˙ W j − J = 0 , (7)where the charge densities are J = g ′ Im ˙Φ † Φ , J = g Im ˙Φ † τ Φ . (8)These constraints remain true at all times, at all points in space, assuming they are obeyed by the initial value data.Although the numerical calculation will be done using the underlying gauge fields W µ and B µ , because of spon-taneous symmetry breaking the physical content of the theory is better described by the fields of definite mass andelectric charge W ± µ = 1 √ W µ · ˆ x ) ± i ( W µ · ˆ y )] ,Z µ = ( W µ · ˆ z ) cos θ W − B µ sin θ W ,A µ = B µ cos θ W + ( W µ · ˆ z ) sin θ W , (9)where ˆ x , ˆ y , and ˆ z denote unit vectors in isospin space and θ W = arctan( g ′ /g ) is the weak mixing angle. The W ± µ fields have mass m W = gv/ √ ± e = ± g ′ cos θ W , the Z µ field has mass m Z = m W / cos θ W andzero electric charge, and the photon field A µ has zero mass and zero electric charge. The only other physical degreeof freedom in the theory is the magnitude of the Higgs field, with mass m H = 2 v √ λ and zero electric charge. LATTICE THEORY
To analyze the classical equations of motion numerically, we use the standard Wilsonian approach [41] for latticegauge fields (for a review see [42]), adapted to Minkowski space evolution as in [43, 44, 45]. The U (1) and SU (2)gauge fields live on the links of the lattice and the Higgs field lives at the lattice sites. We use a regular lattice withspacing ∆ x and determine the values of the fields at time t + = t + ∆ t based on their values at times t and t − = t − ∆ t .Throughout, we will use the same notation and conventions as [34].We associate the Wilson line U pj = e ig ′ B pj ∆ x/ e ig W pj · τ ∆ x/ (10)with the link emanating from lattice site p in the positive j th direction. We define the Wilson line for the link emanatingfrom lattice site p in the negative j th direction to be the adjoint of the corresponding Wilson line emanating in thepositive direction from the neighboring site, U p − j = ( U p − jj ) † , where the notation p ± j indicates the adjacent latticesite to p , displaced from p in direction ± j . At the edges of the lattice we use periodic boundary conditions.The equation of motion for the Higgs field at site p isΦ p ( t + ) = 2Φ p ( t ) − Φ p ( t − ) + ∆ t ¨Φ p ( t ) , (11)where ¨Φ p ( t ) = X j = ± x, ± y, ± z U pj ( t )Φ p + j ( t ) − Φ p ( t )∆ x + 2 λ (cid:0) v − | Φ p ( t ) | (cid:1) Φ p ( t ) . (12)For the gauge fields, we have U pj ( t + ) = exp log U pj ( t ) U pj ( t − ) † − X j ′ = j log U p (cid:3) ( j,j ′ ) ( t ) + log U p (cid:3) ( j, − j ′ ) ( t )∆ x + i ∆ x g ′ J pj + g J pj · τ ) ∆ t U pj ( t ) , (13)where U p (cid:3) ( j,j ′ ) ( t ) = U pj ( t ) U p + jj ′ ( t ) U p + j + j ′ − j ( t ) U p + j ′ − j ′ ( t ) and J pj = g ′ Im Φ p ( t ) † U pj ( t )Φ p + j ( t )∆ x , J pj = g Im Φ p ( t ) † τ U pj ( t )Φ p + j ( t )∆ x , (14)are the gauge currents. Here we have defined the logarithm of a 2 × U pj = i ∆ x g ′ B pj + g W pj · τ ) , (15)which gives the more familiar gauge fields in terms of the link variables. We note that log XY = log X + log Y whenthe matrices do not commute.The U (1) and SU (2) matrices in Eq. (10) are stored separately in the numerical code. To represent the U (1) matrix U = e iθ , just the real quantity θ = g ′ B pj ∆ x/ SU (2) matrix can be written as U = (cid:18) x x − x ∗ x ∗ (cid:19) , (16)so only the two complex elements of the top row need to be stored. (This representation is redundant, since | x | + | x | = 1, but more efficient computationally than storing three real quantities and reconstructing the fourth.) Thelogarithms and exponentials needed to convert between the group and the algebra can be computed efficiently using U = e iθ ˆ n · ~τ = cos θ + i ˆ n · ~τ sin θ = (cid:18) cos θ + i ˆ n z sin θ i ˆ n x sin θ + ˆ n y sin θi ˆ n x sin θ − ˆ n y sin θ cos θ − i ˆ n z sin θ (cid:19) , (17)where ˆ n is a unit vector and the link matrices have ˆ n θ = W pj g ∆ x/ θ → θ and cos θ → √ − θ when computing both the logarithm andthe corresponding exponential. While the approach we are using corresponds a little more directly to the continuumequations, any differences are of higher order in the lattice spacing. Numerical experiments show that their approachyields completely equivalent results, and is somewhat more efficient computationally, since it avoids the need tocompute trigonometric functions in this conversion.The energy density at p is then u p ( t ) = 12 X j = x,y,z (cid:13)(cid:13) exp (cid:0) log U pj ( t + ) − log U pj ( t − ) (cid:1)(cid:13)(cid:13) (2∆ t ) + X j ′ >j (cid:13)(cid:13)(cid:13) U p (cid:3) ( j,j ′ ) ( t ) (cid:13)(cid:13)(cid:13) ∆ x + | Φ p ( t + ) − Φ p ( t − ) | (2∆ t ) + X j = x,y,z (cid:12)(cid:12) U pj ( t )Φ p + j ( t ) − Φ p ( t ) (cid:12)(cid:12) ∆ x + λ (cid:0) | Φ p | − v (cid:1) , (18)whose integral over the whole lattice is conserved. Here we have defined (cid:13)(cid:13) U pj (cid:13)(cid:13) = (cid:12)(cid:12) Tr log U pj (cid:12)(cid:12) g ′ ∆ x + (cid:0) Tr τ log U pj (cid:1) † · (cid:0) Tr τ log U pj (cid:1) g ∆ x = | B pj | + W pj · W pj (19)for any U (2) link matrix.At every lattice point, Gauss’s Law, X j = x,y,z log U pj ( t + ) U pj ( t ) † + log U p − j ( t + ) U p − j ( t ) † i ∆ x ∆ t − ( g ′ J p + g J p · τ ) = 0 , (20)is also maintained throughout the evolution, where the charge densities are given by J = g ′ Im (cid:18) Φ p ( t + ) − Φ p ( t )∆ t (cid:19) † Φ p ( t ) , J = g Im (cid:18) Φ p ( t + ) − Φ p ( t )∆ t (cid:19) † τ Φ p ( t ) . (21)This requirement will provide a stringent check on the correctness of the numerical simulation. Here we have computedGauss’s Law at time t + ∆ t/
2, which is obeyed exactly by the discrete equations of motion for any time step andlattice spacing. In [34], Gauss’s Law at time t was used; it is only obeyed to order ∆ t , but as a result it also providesa rough estimate of whether the time step is small enough. SPHERICAL ANSATZ
With the U (1) field included, the grand spin symmetry of the spherical ansatz used in [36] is broken and fieldconfigurations will not maintain this symmetry under time evolution. The continuum theory does still preserveinvariance under grand spin rotations around the z -axis, but the Cartesian lattice provides a small breaking of allrotational symmetries. As a result, field configurations that start within the spherical ansatz are not constrained to liein any reduced ansatz at later times. (We will also demonstrate the oscillon’s stability under explicitly nonsphericaldeformations below.) Nonetheless, because we will use the spherical ansatz as a starting point to obtain our initialconditions, it will be helpful to analyze it in more detail. We will see that the electroweak oscillon retains much ofthe structure it inherits from these initial conditions.For our choice of gauge, the spherical ansatz takes the form [37] τ · W j = 1 g (cid:20) a ( r, t ) τ · ˆ r ˆ r j + α ( r, t ) r ( τ j − τ · ˆ r ˆ r j ) − γ ( r, t ) r ( ˆ r × τ ) j (cid:21) , Φ = 1 g [ µ ( r, t ) − iν ( r, t ) τ · ˆ r ] (cid:18) (cid:19) , (22)where r is the position vector, r = | r | is the distance from the origin, and ˆ r = r /r is the unit radial vector.Configurations in this ansatz are then described by reduced fields a , α , γ , µ , and ν , all of which depend only on r and t . The field definitions have been chosen so that the reduced fields match those used in [36], even though theconventions for the three-dimensional theory used here are slightly different.These configurations are in the grand spin zero channel, meaning they are symmetric under simultaneous rotationsin space and isospin. The gauge field W j has isospin i = 1 and internal angular momentum s = 1. These two spinscan be coupled together to yield total generalized angular momentum 0, 1, and 2. To obtain grand spin G = 0,these combinations must then be coupled with equal orbital angular momenta ℓ = 0, ℓ = 1, and ℓ = 2 respectively,corresponding to monopole, dipole and quadrupole spatial distributions. These three possibilities are reflected in Eq.(22) through the three terms α ( r, t ), γ ( r, t ), and a ( r, t ).We have written the Higgs field as a matrix times a fixed isospinor. This matrix transforms under both the gauged SU (2) L and global SU (2) R isospin transformations. (We are only considering global rotations in both cases, however.)Under both transformations it has isospin i = 1 /
2, giving total isospin i = 0 or i = 1. Since the Higgs is a Lorentzscalar, with zero internal angular momentum, to obtain G = 0 these two possibilities must be coupled to ℓ = 0 and ℓ = 1 respectively, corresponding to monopole and dipole spatial distributions. These possibilities appear in Eq. (22)as the terms µ ( r, t ) and ν ( r, t ).Although the spherical ansatz does not contain the U (1) field, to leading order in θ W we can find the electric chargedensity created by a spherical ansatz configuration for our choice of gauge [46], J = 2 ezr g ( γ ˙ α − α ˙ γ ) . (23)The charge shows a dipole structure centered on the z axis — as we would expected since the electromagneticinteractions break the grand spin symmetry by selecting the z direction in isospin. We note that this electric chargedensity is time independent (and thus does not radiate) if the α and γ fields vary sinusoidally in time with the samefrequency. NUMERICAL SIMULATION
The initial conditions for the simulation are obtained starting from an approximate functional fit to the solutionsthat were found in SU (2)-Higgs theory using the spherical ansatz [36]. These results, with slight modifications,provide the initial data for the W j and Φ fields, and the initial B j field is chosen to vanish. In order to guarantee thatthe initial configuration obeys Gauss’s Law in the full SU (2) × U (1) theory, we generate the spherical ansatz fit at apoint in the cycle where the time derivatives are smallest, and then set all time derivatives to zero. We note that inpure SU (2) Higgs-gauge theory, this restriction would not be necessary, because even though an approximate fit withnonvanishing time derivatives will not obey Gauss’s Law, we can restore Gauss’s Law by adjusting Φ( t + ) slightly viaan SU (2) transformation at each point, Φ new ( t + ) = (cid:12)(cid:12)(cid:12)(cid:12) Φ old ( t + )Φ( t ) (cid:12)(cid:12)(cid:12)(cid:12) U p Φ( t ) , (24)with U p = exp X j = x,y,z log U pj ( t + ) U pj ( t ) † + log U p − j ( t + ) U p − j ( t ) † g ∆ x | Φ old ( t + ) || Φ( t ) | / † . (25)This procedure has been used successfully to reproduce spherical ansatz solutions with nonvanishing time derivativesat t = 0 in a fully three-dimensional simulation of pure SU (2) Higgs-gauge theory, but it cannot be extended to the SU (2) × U (1) theory because Φ carries both charges, and thus cannot be adjusted to satisfy both constraints at once.Therefore we will consider only initial conditions in which all fields have zero time derivatives, so that Gauss’s Law istrivially satisfied.To construct the initial conditions, we begin from the spherical ansatz form of Eq. (22). We work in units where v = 1 / √
2. Since we are dealing with purely classical dynamics, we can rescale the fields to fix the SU (2) couplingconstant at g = √
2, so that the W ± mass is then m W = gv/ √ / √
2. With this rescaling, we must also introducean overall factor of g /g W multiplying the total energy, where g W = 0 .
634 is the true weak coupling constant. (Thisfactor was incorrectly omitted in the original version of [34].) We choose λ = 1, so that the Higgs mass is twice the W ± mass, m H = 2 v √ λ = √
2. Finally, we fix g ′ = 0 . g ′ /g matches its observed value and the Z boson has the correct mass. With these choices, one unit of energy is 114 GeV, one unit of time is 5 . × − sec,and one unit of length is 1 . × − m. In these units, we take the following initial configuration for the radial fields, a ( r ) = χ (0 . χ + 0 . χr ) ( sech 2 χr ) / ,µ ( r ) = 1 − . χ sech χr . ,ν ( r ) = 0 . χr sech χr ,α ( r ) = 0 . χ r sech χr ,γ ( r ) = 0 , (26)where the adjustable parameter χ allows us to include a combined rescaling of the fields’ amplitudes and r -dependence,as is commonly used in a small amplitude analysis [6, 32, 39]. While χ = 1 gives an approximation to the sphericalansatz solution of [36], a slightly larger value appears to be necessary for the configuration to settle into a stablesolution in the full SU (2) × U (1) model. Here we will use χ = 1 .
15. The first term in parentheses in the definitionof a ( r ) is scaled with an additional χ so that it matches the coefficient of α , ensuring that α , a − α/r , γ/r , and ν all vanish as r →
0, as required for regularity of the fields at the origin. Within the spherical ansatz simulation,these initial conditions converge to a long-lived oscillon in the pure SU (2)-Higgs theory, which is never observed todecay. As a check of the numerical calculation, the full three-dimensional simulation agrees with the spherical ansatzsimulation when the U (1) interaction is turned off.Although initial conditions of this form do settle into stable oscillon configurations in the SU (2) × U (1) theory, itis helpful to make a minor modification to them that is outside the spherical ansatz: setting the τ z -component of W j to zero brings the initial conditions significantly closer to the localized solution that the fields ultimately converge to.While we obtain an equivalent oscillon solution in both cases, this modification reduces the energy shed as the oscillonforms. Doing so provides a significant technical benefit, because the radiation emitted as the configuration settlesinto the oscillon solution can wrap around the periodic boundary conditions, return to the region of the oscillon, andpotentially destabilize it. To avoid this problem, the energy density in this radiation, which spreads throughout thevolume of the simulation, must be small compared to the oscillon’s energy density. As long as the lattice volume islarge enough compared to the oscillon size, this radiation is sufficiently diffuse that it does not affect the oscillon’sevolution. We use a lattice of size L = 144 on a side in natural units, which is more than enough to satisfy thiscriterion. For L > ∼ L < SU (2) Higgs-gauge models, because in the electroweak model the radiatedenergy ends up almost entirely in the electromagnetic field, while the oscillon arranges itself to be electrically neutral.For this reason, it is not necessary to use absorptive techniques such as adiabatic damping [11] or an expandingbackground [10], although both have been applied successfully to this problem as well. However, clearly it is helpfulto adjust the initial conditions to be as close as possible to the true oscillon configuration, to minimize the amountof unwanted energy emitted as the configuration settles into the oscillon solution, and therefore limit the numericalcosts associated with a larger lattice.Starting from the modified spherical ansatz initial conditions, we let the system evolve for as long as is practicalnumerically, and see no sign of oscillon decay. We use lattice spacing ∆ x = 0 .
75, though ∆ x = 0 .
625 and ∆ x = 0 . t = 0 .
1. Timesteps of 0 .
05 and 0 .
025 also gave equivalent results, although in this case one must take into account the fact that ene r g y i n bo x o f r ad i u s λ =1 λ =0.95 FIG. 1: Energy in a spherical box of radius 28 as a function of time in natural units. The initial conditions are given by themodified spherical ansatz form given in the text, in which the τ z component of the gauge field is set to zero, with χ = 1 . λ are shown. For λ = 1, the masses of the Higgs and W fields are in the 2 : 1 ratio neededfor oscillon formation and the solution remains localized throughout the simulation. Here one unit of energy is 114 GeV, oneunit of time is 5 . × − sec, and one unit of length is 1 . × − m, giving a total energy of roughly 30 TeV within thebox radius of roughly 0.05 fm. A transient beat pattern is also visible. For λ = 0 .
95, the mass ratio is 1 .
95 : 1. In that case,there is no stable object and the energy quickly disperses. ene r g y i n bo x o f r ad i u s λ =0.95 λ =0.975 λ =0.9875 λ =0.99375 0 500 1,000 1,500 2,000 2,50050100150200250300 time ene r g y i n bo x o f r ad i u s λ =1.05 λ =1.025 λ =1.0125 λ =1.00625 FIG. 2: Energy in the spherical box for a variety of values of λ . For λ = 1, the Higgs mass is twice the W ± mass and no decayis observed. When the Higgs mass is just below this value, we see a region of meta-stability. For λ <
1, the fields decay by firstcollapsing inward before dispersing, while for λ > this change also slightly alters the initial conditions: To set the initial time derivatives to zero, the simulation setsthe first two time slices equal. Changing the time step thus changes the time at which the field configuration matchesits value at t = 0, representing a slight perturbation of the initial conditions. This change slightly alters the initialtransient behavior as the fields approach the oscillon, but these differences quickly disappear and the simulationsapproach equivalent oscillon configurations.Total energy is conserved to a few parts in 10 for ∆ t = 0 .
1, which improves with ∆ t as expected for our second-order algorithm. We check Gauss’s Law by monitoring the left-hand side of Eq. (20), which we verify vanishes tomachine precision throughout the simulation. It is necessary, however, to use double precision to avoid gradual One can instead evaluate Gauss’s Law at time t instead of t + ∆ t/ | Φ | a t o r i g i n R e ( W y + + i W y − ) a t o r i g i n | Φ | a t o r i g i n R e ( W y + + i W y − ) a t o r i g i n FIG. 3: Decay of the oscillon for λ = 1. One of the gauge fields and the magnitude of the Higgs field at the origin are shownas functions of time. In the left panel λ = 0 . λ = 1 . degradation in this result. For the parameters as given above, a run to time 10 ,
000 takes roughly 40 hours using 24parallel processes, each running on a 2 GHz Opteron processor core. ene r g y i n bo x r box =28r box =36 FIG. 4: Energy in the spherical box for two different box radii in the simulation of Fig. 1, with λ = 1. The transient beatpattern represents a “breathing” perturbation in which the oscillon stretches and compresses slightly. For the larger box size,less energy flows in and out of the box during this process, and so the observed beat amplitude is smaller. Fig. 1 shows the energy in a spherical box of radius 28 as the fields are evolved from these initial conditions. Whenthe Higgs mass is twice the W ± mass, a small amount of energy is initially emitted from the central region, withthe rest remaining localized for the length of the simulation. If the masses are not in this ratio, however, the initialconfiguration quickly disperses. Fig. 2 shows the growth in oscillon lifetime as λ approaches this critical value. Wesee a region of meta-stability when the Higgs mass is just below the 2 : 1 ratio. For λ <
1, the fields first collapsetoward the origin before dispersing, while for λ > m H = 2 m Z , did not form stable objects from these initial conditions. take its trace, and then take the square root of the result. For a typical run with ∆ t = 0 .
1, the integral of this quantity over the latticenever exceeds 0 .
025 and shows no upward trend over time. For smaller ∆ t , we see the expected O (∆ t ) improvement in this result. The parallel C++ code used for these simulations is available from http://community.middlebury.edu/~ngraham . The spherical box contains approximately 3% of the total volume available to the simulation. Its radius has beenchosen to be just large enough to enclose nearly all of energy density associated with the stable oscillon. As a resultof this choice, the λ = 1 graph also shows a transient beat pattern. It represents a “breathing” or “ringing” motion,in which the oscillon gradually expands and contracts slightly over many periods, accompanied by a correspondingmodulation of the field amplitudes. This process causes a small amount of the oscillon’s energy to move in and outof the box. As we would expect, when a larger box size is used, the “breathing” is more completely contained withinthe box and the graph of the energy in the box flattens out, as shown in Fig. 4. Similar beats appear in the SU (2)spherical ansatz oscillon [36], but in the electroweak oscillon their amplitude decays much more rapidly. Re (W x+ + iW x− )−50 0 50−50050 00.0050.010.0150.020.025 Im (W x+ + iW x− )−50 0 50−50050 00.0050.010.0150.020.025Re (W y+ + iW y− )−50 0 50−50050 −0.05−0.04−0.03−0.02−0.01 Im (W y+ + iW y− )−50 0 50−50050 00.020.04Re (W z+ + iW z− )−50 0 50−50050 −505x 10 −3 Im (W z+ + iW z− )−50 0 50−50050 −505x 10 −3 Z −50 0 50−50050 −3−2−1012x 10 −3 Z −50 0 50−50050 −505x 10 −3 Z −50 0 50−50050 01020x 10 −3 A x −50 0 50−60−40−200204060 −505x 10 −3 A y −50 0 50−50050 −0.0200.02A z −50 0 50−50050 −0.08−0.06−0.04−0.020 FIG. 5: A snapshot of the gauge fields in the x = 0 plane for the simulation of Fig. 1 at time t = 50 , To illustrate the field configurations that make up the oscillon, we graph the fields at time t = 50 ,
000 for thetwo-dimensional slice x = 0. Fig. 5 shows the gauge field components. It is most illustrative to consider a linearsuperposition of the W ± j fields, as shown in the figure. Fig. 6 shows the electric fields, which are given by the timederivatives of the gauge fields for our choice of gauge. Fig. 7 shows the components of the Higgs field and its first timederivative, and Fig. 8 shows the magnitude of the Higgs field and the first time derivative of this quantity, togetherwith the total energy density. The oscillon is constructed primarily out of the lower component of the Higgs field, theimaginary part of the upper component of the Higgs field, the x and y spatial components of the W ± j fields, and the z spatial component of the Z j field. We see the multipole structures we anticipated from the spherical ansatz analysis.The Higgs field contains monopole and dipole fluctuations. The photon field A j contains delocalized backgroundradiation that was emitted as the oscillon formed from the initial conditions. As we would expect from Eq. (23), ithas a dipole structure. In the spherical ansatz, the W ± j and Z j fields can potentially contain monopole, dipole, andquadrupole components. Here we see significant monopole and quadrupole structures, but only a very small dipolecomponent, which appears in Z j . As a result, the electric charge we estimate from Eq. (23) is very small, as is the0 (d/dt) Re (W x+ + iW x− )−50 0 50−50050 −1−0.500.511.5x 10 −3 (d/dt) Im (W x+ + iW x− )−50 0 50−50050 −10123x 10 −3 (d/dt) Re (W y+ + iW y− )−50 0 50−50050 −0.05−0.04−0.03−0.02−0.01 (d/dt) Im (W y+ + iW y− )−50 0 50−50050 00.020.04(d/dt) Re (W z+ + iW z− )−50 0 50−50050 −4−2024x 10 −4 (d/dt) Im (W z+ + iW z− )−50 0 50−50050 −4−2024x 10 −4 (d/dt) Z −50 0 50−50050 −10−505x 10 −4 (d/dt) Z −50 0 50−50050 −505x 10 −4 (d/dt) Z −50 0 50−50050 −10−505x 10 −4 (d/dt) A x −50 0 50−50050 −3−2−1012x 10 −4 (d/dt) A y −50 0 50−50050 −1.5−1−0.500.51x 10 −4 (d/dt) A z −50 0 50−50050 −2−101x 10 −4 FIG. 6: A snapshot of the electric fields (time derivatives of the gauge potentials) in the x = 0 plane for the simulation of Fig.1 at time t = 50 , Re Φ −50 0 50−50050 −0.0500.05 Re Φ −50 0 50−50050 0.7080.710.7120.714Im Φ −50 0 50−50050 −6−4−2024x 10 −4 Im Φ −50 0 50−50050 −0.04−0.0200.02(d/dt) Re Φ −50 0 50−50050 −2−1012x 10 −3 (d/dt) Re Φ −50 0 50−50050 −0.08−0.06−0.04−0.02(d/dt) Im Φ −50 0 50−50050 −2−1012x 10 −4 (d/dt) Im Φ −50 0 50−50050 −4−202x 10 −3 FIG. 7: A snapshot of the Higgs field and its time derivatives in the x = 0 plane for the simulation of Fig. 1 at time t = 50 , true value from the numerical simulation; the oscillon is decoupled from the electromagnetic background. Each excited field oscillates at a frequency just below its mass. In our units, these oscillations have typical amplitudeof order 0 . While the multipole analysis is instructive as a description of the field configuration, is is important to note that because the oscillonhas large spatial extent compared to its period of oscillation, it is in exactly the domain where the standard multipole expansion for theelectromagnetic radiation emitted is invalid. | Φ |−50 0 50−60−40−200204060 0.7070.7080.7090.710.7110.7120.7130.714(d/dt)| Φ |−50 0 50−60−40−200204060 −0.09−0.08−0.07−0.06−0.05−0.04−0.03−0.02−0.01 energy density−60 −40 −20 0 20 40 60−60−40−200204060 0.010.020.030.040.050.06 FIG. 8: Left panel: A snapshot of the magnitude of φ and its first time derivative in the x = 0 plane for the simulation ofFig. 1 at time t = 50 , x = 0 plane for the simulation of Fig. 1 at time t = 50 , R e ( W y + + i W y − ) a t o r i g i n R e ( W y + + i W y − ) a t o r i g i n | Φ | a t o r i g i n | Φ | a t o r i g i n FIG. 9: Oscillon fields at the origin as functions of time. The left side shows one component of the SU (2) gauge field. Theupper graph shows the full extent of the simulation. On this scale, the individual oscillations are too small to be seen. Instead,we see the decaying beat pattern from the transient “breathing” motion. The lower graph shows the oscillation of the field fora short time at the end of the simulation (when the transient effects have decayed away). The right side shows the magnitudeof Φ in the same way. It oscillates with fundamental frequency twice that of the gauge field. ω H = 1 .
404 for the Higgs field components and ω W = 0 .
702 for the gauge field components. These properties are allvery similar to the spherical ansatz oscillon. They are also consistent with a small-amplitude analysis, as describedin the Introduction, with ǫ of order 0 .
1. In Fig. 9, oscillon fields at the origin are shown as functions of time. Thefundamental oscillation of each field is modulated by the decaying beat pattern.The oscillon we have seen is not significantly altered by small perturbations of the initial conditions. As anexample, in Fig. 10 we show the results of a run in which the rotational symmetry has been explicitly broken. Wetake initial conditions as before, except we introduce different rescalings of the x , y and z coordinates in the definitionof r . As an additional numerical check, this run also uses a smaller time step, ∆ t = 0 .
05. Although the beatpattern is slightly enhanced, likely indicating that we have started further away from the true oscillon because of the2 ene r g y i n bo x o f r ad i u s λ =1 FIG. 10: Energy in a box of radius 28 as in Fig. 1, but with initial conditions that have been deformed to break rotationalsymmetry. The spatial coordinate r = x ˆ x + y ˆ y + z ˆ z has been everywhere replaced by r ′ = 0 . x ˆ x + 1 . y ˆ y + 0 . z ˆ z andsimilarly r and ˆ r have been replaced by r ′ = | r ′ | and r ′ = x ′ /r ′ . As a further check of the numerics, this run also uses a smallertime step, ∆ t = 0 .
05. Except for these modifications, the simulation is the same as in Fig. 1. nonspherical deformation, we see that the system nonetheless converges to a very similar configuration to the casewithout the rescaling. Equivalent behavior is seen when we make these two changes individually and when we makeother perturbations, such as variations of χ .Finally, we consider the topological properties of the electroweak oscillon. Unfortunately, as shown in [47], there isno unambiguous definition of the topological charge for solutions to the equations of motion. (Topological propertiesare typically studied using vacuum-to-vacuum paths [48], which are clearly not solutions to the equations of motionsince they do not conserve energy.) However, for any localized spatial configuration in which the Higgs field nevervanishes, the Higgs winding number is unambiguously defined as n = 124 π Z ǫ ijk Tr (cid:2) U † ( ∂ i U ) U † ( ∂ j U ) U † ( ∂ k U ) (cid:3) d x , (27)where U is the unique SU (2) matrix associated with a nonvanishing Higgs field Φ, so thatΦ = | Φ | U (cid:18) (cid:19) . (28)The Higgs winding number is a topological invariant, which can only change with time if the Higgs field passes throughzero at some point in space. The change in the Higgs winding is physically meaningful and measures whether the fieldshave crossed the sphaleron barrier. Because the electroweak oscillon contains only small-amplitude field fluctuations,its Higgs winding is always zero and it does not approach the sphaleron barrier. Correspondingly, its topologicaldensity q = g π ǫ µνλσ F µν · F λσ (29)is small as well. But the restriction to small amplitude does not apply to its decays (induced, for example, bycollision with another oscillon), when the fields frequently exhibit an implosion to small radii and large amplitudesbefore ultimately dispersing. This behavior is seen in Fig. 3 for the oscillon’s decay when λ is slightly less than one.However, both this particular decay and limited experiments with oscillon collisions have not led to winding in thefinal Higgs field. Current work continues to investigate this possibility. CONCLUSIONS
We have seen in detail the results of a numerical simulation describing a long-lived, localized, oscillatory solution tothe equations of motion in the bosonic sector of the electroweak Standard Model, for a Higgs mass that is twice the3 W ± mass. Compared to the natural scales of the system, this solution has small field amplitudes, large spatial extent,and large total energy. In the quantized theory, it would represent a coherent superposition of many elementaryparticles, and thus is well described by the classical analysis undertaken here. Quantization of the small oscillationsaround the classical solution would nonetheless be of interest, as has been done for Q -ball oscillons in [49]. It wouldalso be desirable to incorporate fermion couplings, which have been ignored here. Such an analysis would requireintroducing chiral fermions on the lattice, which is well known to be a difficult problem, but one on which significantprogress has been made in recent years. While one might expect the oscillon to be destabilized by decay to lightfermions, in the case of the photon coupling we have seen that the analogous decay mechanism is highly suppressed.Because it would require bringing many Higgs and gauge particles together at once, forming such an oscillonwould likely require large energies available only in the early universe. If extremely long-lived, such an oscillon couldbe a dark matter or ultra-high energy cosmic ray candidate. A slow fermion decay mode would be of interest forbaryogenesis, since it could provide a mechanism for fermions to be produced out of equilibrium, as is necessary toavoid washout of particle/antiparticle asymmetry. The oscillon has small amplitude everywhere and thus remainsfar from the sphaleron configuration, even though it has energy above the height of the sphaleron barrier. However,when induced to decay, for example by a collision with another oscillon, the fields typically collapse to a configurationwith small radius and large energy density and field amplitudes before dispersing. Such decays could potentially crossthe sphaleron barrier and produce fermion number violation. For baryogenesis applications, one would also need toincorporate interactions containing C and CP violation in the classical effective action.The spherical ansatz provided a crucial tool for obtaining the electroweak oscillon solution. However, any searchfor oscillons using a particular ansatz cannot guarantee that all solutions have been found. “Emergent” techniques,in which oscillons form from generic initial conditions, offer the opportunity for more comprehensive searches foroscillons, albeit at a higher computational cost. In simpler models, oscillons have been shown to emerge from phasetransitions [30] and from thermal initial conditions in an expanding universe [50]. Clearly, it would be desirable toextend these techniques to the electroweak model.The electroweak oscillon remains stable even when one would expect it to decay, suggesting that there might existother stable, oscillatory solutions in the electroweak theory or its extensions, either for generic or specific mass ratios.While results for generic mass ratios are clearly of broader applicability, a compelling result for a specific mass ratiomight suggest a preferred value of the Higgs mass. ACKNOWLEDGMENTS
It is a pleasure to thank E. Farhi, F. Ferrer, M. Gleiser, A. Guth, R. R. Rosales, R. Stowell, J. Thorarinson, and T.Vachaspati for helpful discussions, suggestions and comments; P. Lubans, C. Rycroft, S. Sontum, and P. Weakleimfor Beowulf cluster technical assistance; and the Massachusetts Institute of Technology (MIT) Center for TheoreticalPhysics for hospitality and support while this work was being carried out. N. G. was supported by National ScienceFoundation (NSF) grant PHY-0555338, by a Cottrell College Science Award from Research Corporation, and byMiddlebury College.Computational work was carried out on the Hewlett-Packard (HP) Opteron cluster at the California NanoSys-tems Institute (CNSI) High Performance Computing Facility at the University of California, Santa Barbara (UCSB),supported by CNSI Computer Facilities and HP; the Hoodoos cluster at Middlebury College; and the Applied Math-ematics Computational Lab cluster at MIT. Access to the CNSI system was made possible through the UCSB KavliInstitute for Theoretical Physics Scholars Program, which is supported by NSF grant PHY99-07949. ∗ Electronic address: [email protected][1] S. Coleman,
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