Annihilation of electroweak dumbbells

We study the annihilation of electroweak dumbbells and the dependence of their dynamics on initial dumbbell length and twist. Untwisted dumbbells decay rapidly while maximally twisted dumbbells collapse to form a compact sphaleron-like object, before decaying into radiation. The decay products of a dumbbell include electromagnetic magnetic fields with energy that is a few percent of the initial energy. The magnetic field from the decay of twisted dumbbells carries magnetic helicity with magnitude that depends on the twist, and handedness that depends on the decay pathway.


Introduction
The "electroweak dumbbell" consists of a magnetic monopole and an antimonopole of the standard electroweak model connected by a string made of Z magnetic field [1,2].The existence of such non-perturbative field configurations in the electroweak model is of great interest as they can provide the first evidence for (confined) magnetic monopoles.In a cosmological context, dumbbells can source large-scale magnetic fields which can seed galactic magnetic fields and play an important role in the propagation of cosmic rays [3].
Electroweak dumbbells are often regarded as magnetic dipoles, with the magnetic field strength falling off as 1/r 3 with the distance r from the dipole.However the situation is richer: there is a one-parameter set of electroweak dumbbell configurations [4], all describing a confined monopole-antimonopole pair, but with additional structure called the "twist".In our previous work [5], we have shown that the magnetic field strength of the maximally twisted dumbbell falls off asymptotically as cos θ/r 2 (in spherical coordinates), a gross departure from the usual dipolar magnetic field.The twisted dumbbell configuration was proposed in [4] and is closely related to the electroweak sphaleron [6][7][8][9].
The dynamics of electroweak sphalerons and dumbbells are of particular interest in the efforts towards detecting them in experiments.In view of ongoing experiments like the Monopole and Exotics Detector (MoEDAL) at the Large Hadron Collider (LHC) [10], Ref. [11] recently studied the production of the electroweak sphaleron in the presence of strong magnetic fields that arise during heavy ion collisions.In the cosmological context, simulations of the dynamical decay of electroweak sphalerons have been conducted to study baryogenesis and magnetogenesis [12,13].Thus, there have been several efforts to numerically resolve the configuration and dynamics of electroweak sphalerons.The relevance of dynamics of electroweak dumbbells was first alluded to by Nambu [1] wherein he showed that electroweak dumbbells could be stabilized by rotation, potentially making them long lived enough to have significant implications in experimental searches.This, along with a lack of detailed study of the dynamics of electroweak dumbbells, has motivated our investigation into the dumbbell configurations and their dynamics.
In this article, we investigate the dynamical evolution of an initially stationary dumbbell configuration for a range of twists and initial dumbbell lengths.The initial condition is found by numerically relaxing a "guess" dumbbell configuration under the constraint that locations of the monopole and antimonopole remain fixed, according to the method outlined in our previous work [5].The main quantities of interest that we analyze are the dumbbell lifetimes and the magnetic field produced during the decay.The untwisted dumbbells are found to be unstable, with the monopoles immediately undergoing annihilation as expected.Twisted dumbbells, on the other hand, lead to the creation of an intermediate sphaleron configuration after the initial collapse, and subsequently decay into helical magnetic fields with relatively stronger field strength.
The article begins with a description of the model and our initial configuration in Section 2. The numerical simulation setup is described in Section 3. Our results for the lifetimes and the relic magnetic fields are given in Section 4, and we discuss our conclusions in Section 5.

Model
The Lagrangian for the bosonic sector of the electroweak theory is given by where Here, Φ is the Higgs doublet, W a µ are the SU(2)-valued gauge fields with a = 1, 2, 3 and, Y µ is the U(1) hypercharge gauge field.In addition, σ a are the Pauli spin matrices with Tr(σ a σ b ) = 2δ ab , and the experimentally measured values of the parameters that we adopt from [14] are g = 0.65, sin 2 θ w = 0.22, g ′ = g tan θ w = 0.35, λ = 0.129 and η = 174 GeV.
The classical Euler-Lagrange equations of motion for the model are given by where the gauge field strengths are given by (2.7) Electroweak symmetry breaking results in three massive gauge fields, namely the two charged W ± bosons and Z µ , and one massless gauge field, A µ , that is the electromagnetic gauge field.We define where are components of a unit three vector n.The weak mixing angle, θ w is given by tan θ w = g ′ /g, the Z coupling is defined as g Z ≡ g 2 + g ′2 , and the electric charge is given by e = g Z sin θ w cos θ w .The Higgs, Z and W boson masses are given by m H ≡ 2 GeV and m W ≡ gη/ √ 2 = 80 GeV, respectively.

Initial dumbbell configuration
We construct a suitable initial configuration for dumbbell dynamics by first choosing a "guess" field configuration that contains a monopole and antimonopole separated by a distance 2d and with relative twist γ [2].In the asymptotic region, (2.11) The monopole and antimonpole are located along the z-axis at z = ±d, with θ m and θ m being the spherical polar angles as measured from the monopole and antimonpole, respectively; ϕ is the azimuthal angle.The coordinate system is illustrated in Figure 1.In the limit θ m → 0, (2.11) reduces to the monopole configuration, and in the limit θ m → 0 to an antimonopole.In between the monopole and antimonopole, θ m → π and θ m → 0, the configuration is that of a Z−string.
The asymptotic gauge field configurations are obtained by setting the covariant derivative of the Higgs field to vanish, To correctly account for the spatial dependence of the Higgs field around the monopoleantimonopole pair, we attach spatial profiles.Including this in the ansatz, the initial monpole-antimonopole scalar field configuration is given by where r m and r m are the radial coordinates centered on the monopole and antimonpole, respectively, given by The function k(⃗ r ) is the Z-string profile.Similar to the Higgs, we include spatial profiles for the gauge fields as ) We have previously used numerical relaxation to solve for the Higgs and gauge field profile of a static dumbbell in with the constraint that the topology of the monopole and antimonopole remain fixed during the relaxation process [5].We use the same procedure to find the initial field configurations that we will then evolve to study dumbbell dynamics.

Numerical simulation
We have used a numerical relativity technique adapted from [15], and previously used in [16] to study monopole-antimonopole scattering in the SO(3) model.Adopting the temporal gauge for convenience in numerical implementation, W a 0 = Y 0 = 0, the evolution equations (2.3)-(2.5)can be written as Straightforward discretization of the evolution equations leads to numerical instabilities.To control the instabilities, we introduce "Gauss constraint variables" Ξ = ∂ i Y i and Γ a = ∂ i W a i , with their respective evolution equations Here, we have introduced the numerical stability parameter g 2 p .These equations reduce to the Gauss constraints in the continuum, regardless of the choice of g 2 p .However, once the system of equations is discretized for numerical evolution, the term in the curly brackets do not always vanish, and a non-zero value of g 2 p ensures numerical stability as outlined in [15].The equations in (3.1)-(3.3)are now written with the Gauss constraint variables as Our simulations are conducted on a regular cubic lattice with Dirichlet boundary conditions and the fields are evolved in time using the explicit Crank-Nicholson method with two iterations [17].We adopted phenomenological values for the electroweak model as given in Sec. 2. We choose to work in units of η and set its numerical value to η = 1 in our simulations.Then the Higgs mass of 125 GeV in these units is given by 2η For most of our runs, we use a lattice of size 400 3 with lattice spacing δ = 0.25η −1 = 0.25, and time step dt = δ/8.The Compton radius of the W boson is m −1 W = 2.17.This is also approximately the radius of the monopole and string in the dumbbell, implying that their profiles have a resolution of roughly 10 grid points in our setup.
The Higgs field vanishes at the centers of the monopole, antimonopole and Z−string.Since the initial dumbbell profile involves delicate cancellation of zeros on the dumbbell, we offset the dumbbell location away from the z−axis by half a lattice spacing.That is, the monopole and antimonopole are at the coordinates x = y = δ/2, z = ±(d + δ/2), while the Z−string is located at x = y = δ/2 and parallel to the z−axis.
To ensure that the Dirichlet boundary conditions do not significantly affect the dynamics of the annihilating dumbbell, we only consider initial dumbbell lengths 2d that are sufficiently smaller than the lattice size.The maximum separation considered in our simulations was slightly less than half the lattice width and we ensure that the Higgs field is close to the vacuum expectation value near the boundaries |Φ| ≈ 1.We run our simulations for ∼ (5 − 10) × T c , where T c is the dumbbell lifetime, to study the relic energies in the different fields.We ensure that there are no significant effects on the dumbbell dynamics due to reflections from the Dirichlet boundary conditions in the time span considered here.We have tested this by varying the lattice spacing δ and the total simulation box size.Thus we ensure that our choice of numerical parameters and boundary conditions do not affect the dumbbell dynamics.

Results
We ran the simulation for a range of initial dumbbell lengths and twists.We show several snapshots of the energy density in the xz-plane for the untwisted case (γ = 0) in Figure 2, and for the twisted case (γ = π) in Figure 3.As can be inferred from these slices, the untwisted dumbbell promptly undergoes annihilation.However, the twisted dumbbell forms an intermediate object that appears as an over-density in the center, before eventually decaying away.We will discuss the relevance of this object (most likely the electroweak sphaleron, as discussed in Ref. [4]) in the context of the dumbbell lifetimes and the relic magnetic energies in the following sections.

Estimating lifetimes
The magnetic monopoles are zeros in the Higgs field and we leverage this to estimate the lifetime of the dumbbell by tracking the zeros and finding when they disappear in the simulation box.Since the dumbbell is offset by δ/2 in the positive x and y directions in our setup, the zeros of the Higgs field are never located at lattice points.We instead borrow the approach from [16] used in studying the creation of monopoles via classical scattering.We track the evolution of the minimum value of the Higgs field over the entire lattice, Min[|Φ|].Once Min[|Φ|] exceeds a threshold, we tag the timestep in our simulation as the dumbbell lifetime.The lifetime obtained by this criterion depends on the chosen value of the threshold.As we will see, the result is sensitive to the threshold in the untwisted case but is quite insensitive in the twisted case.Additionally, we have tested the dependence of lifetimes on the spatial and time resolution of the simulation and demonstrated that there were no significant dependencies on the choice of our simulation parameters.
In Figure 4, we show the evolution of Min[|Φ|] for various values of initial separation 2d.Comparing the untwisted (left panel) and twisted (right panel) cases, respectively, we see that the curves for the untwisted case rise slowly, implying greater sensitivity of the dumbbell lifetime to the chosen threshold.In the twisted case, the curves rise very sharply and the dumbbell lifetime is not sensitive to our choice of threshold.After tagging the timestep when the threshold condition on Min[|Φ|] is satisfied in our simulations, we interpolate the time evolution of Min[|Φ|] in a small range around the tagged timestep.The interpolation was performed via a fourth order polynomial curve fit, and we evaluate the interpolated function at the threshold value (Min[|Φ|] = 0.25) to find the lifetime.The interpolated Min[|Φ|] are shown in Figure 4 as solid lines.We plot the dumbbell lifetime against the initial dumbbell length for untwisted and maximally twisted dumbbells in Figure 5.We find that the lifetime grows with initial length but appears to saturate beyond a certain length.To check the saturation we have performed one run for very long dumbbells (2dm W = 55.2).These runs are computationally very expensive.The saturation is clearer in the case of twisted dumbbells.We interpret these plots recalling the instability analysis of Z-strings [18,19]: the untwisted dumbbell decays on a timescale ∼ m −1 W due to the instability, while the twisted dumbbell survives about 10 times longer, first collapsing to an electroweak sphaleron that eventually decays due to its instability on a longer timescale.To verify that the instability is dynamical and not a result of numerical artifacts, we followed [2,20] and ran test simulations with large values of sin 2 θ w and small values of m H /m Z for which Z-strings are known to be stable [18,19].The results of these simulations are given in Appendix A. In contrast to the simulations with physical parameters, the Z-string instability is absent and the dynamics of the monopoles is clearly visible in these "semilocal" simulations.

Relic Magnetic Fields
The definition for the electromagnetic field strength tensor in the symmetry broken phase (|Φ| = η) that reduces to the usual Maxwell definition in unitary gauge is [21,22], This definition implies the presence of non-zero electromagnetic fields even for A µ = 0 due to the Higgs gradient term.In our study of the decay of dumbbells, ∂ µ Φ vanishes at late times and then the expression in (4.1) agrees with the Maxwell definition.The total magnetic field energy is given by We show the time evolution of magnetic energy E M (t) for twisted and untwisted dumbbells of various initial lengths in Figure 6.After an initial phase of annihilation, the total magnetic field energy reaches a steady value that depends on the initial length of the dumbbell.An important observation is that the relic magnetic energy depends on the twist, in addition to the initial dumbbell lengths.In Figure 7 we plot the fractional magnetic field energy at a late time after annihilation T c + ∆t, where we have chosen ∆t = 25η −1 .Since we use Dirichlet boundary conditions, the simulation time has a upper bound, after which reflections occurring at the boundaries would affect quantities of interest.The time at which we evaluate the relic magnetic field energy, T c + ∆t is large enough such that the relic magnetic energy has reached an asymptote but is still smaller than the time at which a significant fraction of the energy is reflected.From Figure 7, we infer the fractional magnetic energy, after annihilation, is about twice as large for the twisted case when compared to the untwisted case, for the same initial (large) dumbbell length.In addition to the magnetic energy density, we are interested in the helicity of the magnetic field, which has been shown to have significant implications for cosmic baryogenesis and magnetogenesis [3,4,23].The total magnetic helicity is defined as where A is given by the spatial components of the vector potential (2.9) and B is derived from the EM tensor (4.1).The evolution of total magnetic helicity for an initially twisted dumbbell is shown by the blue curve in the left panel of Figure 8.It is clear that the helicity does not approach a specific value within the span of our simulation.A possible explanation for the behavior is that the definition of H M is gauge independent only if the magnetic field is orthogonal to the areal vectors everywhere on the boundaries of the simulation domain.(Alternately, if the magnetic field vanishes on the boundaries.)This is certainly not true in our simulations.Hence we cannot assign physical meaning to our evaluation of H M at late times when the magnetic field is not small at the boundaries.Nonetheless, we can infer that the relic magnetic field has significant net helicity when compared to the untwisted case, which had a helicity H ∼ 10 −13 , consistent with numerical roundoff.An alternative measure of the parity violation in the magnetic field is provided by the "physical helicity" defined as, In contrast to the magnetic helicity, the physical helicity does not have issues with gauge invariance.It is also better behaved in a finite domain as the magnetic field B falls off faster than the gauge field A. The evolution of H B is shown in Figure 8 for γ = π, and it asymptotes to a constant value at late times.We also plot H B at late times after annihilation for various initial dumbbell lengths and twists in the right panel of Figure 8.
From this, we can infer that the H B for the untwisted case γ = 0 is consistent with zero.In the twisted case with γ = π, it takes on finite positive values, ∼ 3η 2 , as shown in Figure 8.One feature, that is immediately clear, is that the signs of the physical helicities, H B , for γ = π and 0 < γ < π are opposite to each other.The opposite signs are related to the untwisting dynamics of the dumbbell configuration as it annihilates.Dumbbells with twist γ = π are maximally twisted and carry maximal energy as a function of twist [5].Dumbbells with twist slightly less than π will tend to untwist by reducing the twist angle from π− → 0, while those with twist somewhat greater than π will untwist by increasing the twist angle from π+ → 2π.These decay modes lead to magnetic field helicity of opposite signs.We tested this argument by running the simulation for twists slightly larger and smaller than π, and confirmed that the signs of the physical helicity are indeed opposite to each other.The plots for the tests we conducted, for two different initial dumbbell lengths, are shown in Figure 9, where we see that the evolution of H B for twist π+ mirrors the one for twist π−.

Discussion and Conclusion
We have studied the collapse, annihilation, and decay products of electroweak dumbbells as a function of their length and twist.
The untwisted case has the expected dynamics, where the dumbbell collapses in a time that is very short, ∼ m −1 W , and comparable to the instability time scale of the Zstring [18,19].The energy of the dumbbell that is converted into magnetic field energy depends on the length of the dumbbell.For short untwisted dumbbells, the magnetic field energy can be ∼ 0.1%; for longer dumbbells it is ∼ 1%.The magnetic fields produced in the untwisted case are not helical.
Twisted dumbbells collapse on a time-scale that saturates to ∼ 10m −1 W for long dumbbells.They collapse to form a long-lived object that presumably is an electroweak sphaleron, see Figure 3.Eventually, the sphaleron decays as well.The decay products include magnetic  field energy that is large compared to the untwisted case.The fractional energy converted to magnetic field energy is roughly independent of the dumbbell length and is ∼ 4%.The produced magnetic field is helical though our calculation of the magnetic helicity is not reliable, especially at late times, due to the finiteness of the simulation box.As an alternative we have calculated the integrated physical helicity and this asymptotes to a constant (∼ 3η 2 ), at late times for the maximally twisted dumbbell.
In future work, we propose to study the dynamics of rotating dumbbells as this pertains to their production and detection in a laboratory setting as first discussed by Nambu [1].We expect the study to be technically challenging as it would require significant improvements in implementing boundary conditions, especially if rotating dumbbells survive for a long time.

Figure 1 :
Figure 1: The location of the monopole (blue dot), antimonopole (red dot) and the Zstring (purple line) in a cartesian grid.

Figure 2 :
Figure 2: The snapshots of energy density in xz-plane for γ = 0 are shown for times 0dt, 1000dt and 5000dt in the left, middle and right panels, respectively.The colors represent the energy density and the corresponding values are given by the scale, in units of η 4 .Here, the simulation parameters are dt = δ/100, with δ = 0.25η −1 , and the initial dumbbell length is 2d = 160δ.The horizontal (z) and vertical (x) axis are in lattice units.

Figure 3 :
Figure 3: The energy density snapshots, similar to Figure 2, for the twisted case (γ = π) and times 0dt, 5000dt and 10000dt in the left, middle and right panels, respectively.

Figure 4 :
Figure 4: The evolution of Min[|Φ|] in the simulation box for twists γ = 0 (left panel) and π (right panel).The line colors correspond to different values of dumbbell lengths expressed as a multiple of the approximate monopole width m −1 W .The hollow circles correspond to the data points from the simulation and the solid curves are 4 th order polynomial fits around the threshold Min[|Φ|] = 0.25.We show our threshold choice, Min[|Φ|] = 0.25, as a dotted horizontal line.

Figure 5 :
Figure 5: The lifetimes T c of the dumbbells as a function of their initial separation are shown for twists γ = 0 and π in the left and right panels, respectively.

Figure 6 :
Figure 6: The total magnetic energy as a function of time for twists γ = 0 and π in the left and right panels, respectively.The time along the horizontal axis is shifted by the lifetime T c and t = T c is shown by a vertical dotted line.Different colored curves correspond to different initial dumbbell lengths, as stated in the legends.

Figure 7 :Figure 8 :
Figure 7: The magnetic energy at a late time, E M (T c +25), as a function of initial dumbbell length.The different colors correspond to different values of twists γ as stated in the legend.

Figure 9 :
Figure 9: The evolution of physical helicity H B for twists γ = 5.5π/6 (red curve) and 6.5π/6 (blue curve) are shown for initial dumbbell lengths 2d = 80δ and 160δ in the left and right panels, respectively.The annihilation time T c is marked with a black dotted vertical line.The horizontal red and blue curves correspond to the physical helicities at time t = 20m −1 W . Here, the time step size in the simulation is dt = δ/8, with δ = 0.25η −1 .

Figure 11 :
Figure 11: The energy density snapshots, similar to Figure 2, for the twisted case (γ = π) and times 0dt, 450dt and 1000dt in the left, middle and right panels, respectively.

Figure 12 :
Figure 12: The evolution of Min[|Φ|] in the simulation box for twists γ = 0 (red curve) and π (blue curve).The hollow circles correspond to the data points from the simulation and the solid lines are 4 th order polynomial fits around the threshold Min[|Φ|] = 0.25.We show our threshold choice, Min[|Φ|] = 0.25, as a dotted horizontal line.