Kibble mechanism for electroweak magnetic monopoles and magnetic fields

The vacuum manifold of the standard electroweak model is a three-sphere when one considers homogeneous Higgs field configurations. For inhomogeneous configurations we argue that the vacuum manifold is the Hopf fibered three sphere and that this viewpoint leads to general criteria to detect electroweak monopoles and Z-strings. We extend the Kibble mechanism to study the formation of electroweak monopoles and strings during electroweak symmetry breaking. The distribution of magnetic monopoles produces magnetic fields that have a spectrum Bλ ∝ λ−2, where λ is a smearing length scale. Even as the magnetic monopoles annihilate due to the confining Z-strings, the magnetic field evolves with the turbulent plasma and may be relevant for cosmological observations.


Introduction
The distribution of topological defects formed after spontaneous symmetry breaking (SSB) is often analyzed by implementing the "Kibble mechanism" [1][2][3]. During SSB a field takes on a non-trivial vacuum expectation value (VEV) that lies on the "vacuum manifold". Distant spatial points are randomly selected on the vacuum manifold and if the vacuum manifold has non-trivial topology, the VEV of the field may end up in a non-trivial topological configuration, in which case a topological defect would be formed. Numerical simulations of the Kibble mechanism have been central to our understanding of topological defect formation during spontaneous symmetry breaking. Notably the cosmic string network was shown to be dominated by infinite strings that don't close on themselves, while the sub-dominant distribution of closed loops was found to be scale invariant [2] (for reviews see [4][5][6]).
Here we are interested in the implications of the Kibble mechanism when the electroweak Higgs field, denoted Φ, acquires a VEV. The electroweak vacuum manifold is a threesphere with trivial first and second homotopy groups and there are no topological magnetic monopoles or cosmic strings by these criteria. However, electroweak monopoles and Z-strings that connect the magnetic monopoles do exist in the model [7][8][9]. We will show that a suitably modified algorithm like that in the case of topological defects can still be used to obtain the distribution of electroweak monopoles and strings. The distribution can be used as an initial condition for further evolution. Since the monopoles and antimonopoles are confined by strings, they will quickly annihilate. Yet the annihilation will leave behind a distribution of magnetic fields [10,11] that can be of observational interest and may have important ramifications for cosmology [12][13][14][15].
In section 2 we describe our viewpoint that the electroweak vacuum manifold is better described as S 2 × S 1 , i.e. as a Hopf fibered S 3 , and thus contains electroweak monopoles and strings. We describe the prototype Nambu monopole in section 3 and implement the Kibble mechanism in section 4 to find a distribution of electroweak monopoles and Z-strings. With JHEP01(2022)059 evolution, the network of monopoles and strings will leave behind a distribution of magnetic fields that we characterize in section 5. We summarize our conclusions in section 6.

Electroweak vacuum manifold
The vacuum manifold of the electroweak manifold is the set of all spatially homogeneous and static Higgs fields for which the energy function vanishes. The Higgs VEV is an SU (2) and since the Higgs potential is, the vacuum manifold is an S 3 given by One issue is that the symmetry of the potential consists of rotations of the four dimensional vector (φ 1 , φ 2 , φ 3 , φ 4 ), hence it is O(4), whereas the electroweak symmetry is the smaller [SU(2) L × U(1) Y ]/Z 2 . The reduced symmetry is due to the derivative terms in the model and these are completely ignored in discussions that are based solely on the vacuum manifold. Derivative terms vanish for homogeneous Higgs configurations and so the S 3 vacuum manifold is appropriate for such configurations. On the other hand, the Kibble mechanism relies on VEVs that are different in different regions of space. Hence the Higgs configurations are necessarily inhomogeneous. We have learned from semilocal strings that the vacuum manifold does not give the complete picture when one considers inhomogeneous Higgs fields configuration for then the gradient energy terms can also be important.
Let us clarify this further by discussing the semilocal limit of the electroweak model. Then the SU(2) L gauge coupling is set to vanish: g = 0. In that case, one can consider Higgs configurations that lie entirely on the vacuum manifold but whose energy cannot vanish. This is because the gauged U(1) Y symmetry defines S 1 gauge orbits on the vacuum manifold. Only Higgs gradients along these orbits can be compensated by the gauge field so that the covariant gradient energy vanishes; if the Higgs VEV does not lie on a gauge orbit, the gradient energy cannot vanish. An alternative "semilocal" limit that has not previously been considered in this context is to take the U(1) Y coupling to vanish: g = 0. In that case the gauge orbits on the vacuum manifold are S 2 's. If we restrict attention to asymptotic Higgs fields configurations that have vanishing potential and gradient energy, the Higgs VEV would have to lie on an S 2 and this has the right topology for magnetic monopoles. The standard electroweak model has g = 0.65 and g = 0.34, so neither coupling vanishes, even though the SU(2) L coupling is larger. However the fibered structure still exists -the vacuum manifold S 3 has S 2 and S 1 gauge orbits. These gauge orbits are JHEP01(2022)059 precisely defined by the Hopf fibration of S 3 , as was originally pointed out in the g = 0 semilocal limit [16,17]. The Hopf fibration of S 3 provides a map from S 3 to S 2 with S 1 fibers. The electroweak monopole is due to winding around the S 2 base manifold and the Z-string is due to winding around the S 1 fiber. Because of the non-trivial global structure of the Hopf fibration, the Z-string is attached to the electroweak monopole.

Nambu monopole
It is instructive to first consider the explicit configuration for the Nambu monopole [7] for which the asymptotic Higgs VEV is, where θ, φ are spherical angles. Note that the configuration is singular at θ = π. To see the presence of the monopole in this configuration, construct whereΦ ≡ Φ/|Φ|, σ a (a = 1, 2, 3) are the Pauli spin matrices and the overall sign is chosen so thatn =ẑ when Φ T = v(0, 1)/ √ 2. Nown m is regular for all θ and φ and is in the (inner) radial direction. This is also called the "hedgehog" configuration and immediately implies the presence of a singularity ofn m at the origin that corresponds to a magnetic monopole [18,19]. Going back to Φ m , the singularity at θ = π signifies the Z-string attached to the monopole.
The above explicit example suggests that to apply the Kibble mechanism to the electroweak model we should start by considering a distribution of the vector fieldn. Sincê n lives on a two-sphere (S 2 ) that has non-trivial second homotopy, there will be hedgehog configurations ofn (e.g.n =r). As for 't Hooft-Polyakov monopoles [18,19], the topological winding ofn in a spherical volume of radius R is given by the surface integral The discrete winding number n M ∈ Z must remain constant as R → 0. For n M = 0 this implies thatn is singular within the spherical volume. Now we consider the Φ field that corresponds to the hedgehog configuration ofn. The relation between Φ andn for |Φ| = 0 is,n Therefore the singularity inn for non-trivial n M requires that Φ = 0 at the singular point where there is a magnetic monopole. The Z-string attached to the monopole appears when we try and invert (3.4) to obtain Φ. Forn with non-trivial winding (n M = 0), the reconstruction will necessarily give a singularity in Φ (as in (3.1)). This singularity is the location of the Z-string on the sphere surrounding the monopole. We will describe the explicit algorithm for finding the location of the Z-string in section 4.

JHEP01(2022)059 4 Kibble mechanism
The Higgs VEV of (2.1) can also be parametrized as, cos α e iβ sin α e iγ (4.1) where v = 246 GeV and α ∈ [0, π/2], β ∈ [0, 2π], γ ∈ [0, 2π] are Hopf angular coordinates on the vacuum manifold: Φ † Φ = v 2 /2. The volume measure on the vacuum manifold in terms of Hopf coordinates is (1/2)d(cos(2α))dβdγ. Hence in any given spatial region, the values of u ≡ cos(2α), β and γ are selected from uniform probability distributions in their respective ranges. In spatial regions that are separated by more than some correlation length, (u, β, γ) can be chosen independently. There is a lot of theoretical and experimental literature (for a review see [20]) on the determination of the correlation length and, more recently, a full quantum calculation for the growth of the correlation length [21,22]. However, the precise value of the correlation length is not a critical quantity for us since this only sets a length scale for the topological defects and does not affect the scaling laws for their distribution.
In the numerical implementation we calculate the (discretized) topological winding for monopoles given by the surface integral in eq. (3.3) as was done for 't Hooft-Polyakov monopoles [23][24][25]. The implementation also assumes the "geodesic rule": a triangular plaquette of the spatial lattice gets mapped to a spherical triangle on the vacuum manifold, but three points on a two-sphere define two complementary spherical triangles and we choose the one with the smaller area [2,26].
We now turn to the Z-strings that connect the monopoles.
The VEV of Φ is invariant under the electromagnetic U(1) Q since QΦ = 0. Thus, for a fixedn, there is an entire circles worth of Φ's given by rotations by U(1) Z . As we go around a spatial plaquette, rotations of then vectors define "parallel transport" of the Φ fields, which may differ from the actual Φ by an element of U(1) Z , as explained in figure 1. Non-trivial winding of the U(1) Z phase factor implies the existence of a Z-string passing through the plaquette.
Consider one leg of a triangular plaquette as shown in figure 1. The vectorn 1 is rotated inton 2 , i.e.n 2 = R 21n1 , by an SO(3) rotation about the axisâ 21 and by angle θ 21 , < l a t e x i t s h a 1 _ b a s e 6 4 = " i 0 X I 8 p 2 K C t 2 k W H T s j 9 b 7 e W q i j c E = " > A A A C B 3 i c b V D L S g M x F M 3 U V 6 2 v U Z e C B I v g q s x U U T d C Q R c u q 9 g H t G X I p G k b m s k M y R 2 h D L N z 4 6 + 4 c a G I W 3 / B n X 9 j 2 g 6 o r Q c u n J x z L 7 n 3 + J H g G h z n y 8 o t L C 4 t r + R X C 2 v r G 5 t b 9 v Z O X Y e x o q x G Q x G q p k 8 0 E 1 y y G n A Q r B k p R g J f s I Y / v B z 7 j X u m N A / l H Y w i 1 g l I X / I e p w S M 5 N n 7 t 1 5 S d l O c t A c E s E w 9 F 1 / 8 P M q e X X R K z g R 4 n r g Z K a I M V c U Y S C i a 5 g Q n B n T 5 4 n 9 X L J P S 0 d 3 5 w U K 1 d Z H H m 0 h w 7 Q E X L R G a q g a 1 R F N U T R A 3 p C L + j V e r S e r T f r f d q a s 7 K Z X f Q H 1 s c 3 p S + X 6 Q = = < / l a t e x i t >  4). We find the rotation R 21 that takesn 1 ton 2 . This rotation in SO(3) also defines a rotation, D[R 21 ] in SU(2) L that acts on Φ 1 to give Φ 2 which in general differs from Φ 2 by rotation by an element P 2 ∈ U(1) Z , at vertex 2. Similarly we can obtain the rotations that take Φ 2 to Φ 3 , and Φ 3 to Φ 1 . The total rotation in going from vertex 1 around the triangle and back to vertex 1 is: , and this rotation acts on Φ 1 to give back Φ 1 .
If the net Z-phase rotation in going around the plaquette is ±2π, there is a Z-string (or anti-string) passing through the plaquette. and we take 0 ≤ θ 12 ≤ π. A corresponding SU(2) L rotation is 2 and rotates Φ 1 to, In general, Φ 2 = Φ 2 and an additional U(1) Z rotation, P 2 , may be necessary to rotate Φ 1 to Φ 2 , where P 2 = e iT Z2 δ 2 . T Z2 is as defined in (4.2) withn =n 2 , and δ 2 is a phase angle. To determine δ 2 we use, which can be derived using (4.4). We will choose δ 2 with the smallest value of |δ 2 | in accordance with the geodesic rule [2,26]. Note that P 2 = e iT Z2 δ 2 acts on Φ 2 to simply give a phase factor exp(iδ 2 ), In this way we can go around all the sides of the triangular plaquette and obtain with a, b, c = 1, . . . , 8. Here the T a are the generators of SU (3), normalized by tr(T a T b ) = 2δ ab , the f abc are structure constants defined by [T a , T b ] = 2if abc T c , and the integration is over the two sphere at infinity. Also note that the vector n a satisfies n a n a = 4/3. In Appendix A we show that the two forms for the topolgical charge are equivalent. It is simple to check that Q = 1 for the monopole configuration in Eq. (10) and Eq. (11). The formula in Eq. (14) will be useful to locate monopoles in our numerical work described in Sec. II.
The second stage of symmetry breaking is more involved. The fields Ψ j now also acquire VEVs, which are required to lie in the unbroken SU (2) subgroup, and hence commute with Φ. Their magnitudes tr(Ψ 2 j ) are fixed by the potential, and they are also required to be mutually orthogonal in the sense that tr(Ψ 1 Ψ 2 ) = 0. Given a value of Φ at some spatial point P , we need to identify this unbroken subgroup. The standard procedure is to work out commutators of Φ with SU (3) generators and to find linear combinations of the generators that commute. In practice, it is easier to first rotate Φ, say by an SU (3) rotation R, to the reference direction, Φ (0) . We discuss how to choose R below. Then the generators of the unbroken SU (2) sit in the 2 × 2 upper left corner while the generator T 8 of the unbroken U (1) is in the direction of Φ (0) itself. With respect to Φ (0) , the VEVs of Ψ 1 and Ψ 2 can be written in terms of two orthonormal 3-vectors, a and b, as Ψ and σ i are the Pauli spin matrices. Once Ψ  −1, 1). A string passes through a spatial contour if Ψ 1 and Ψ 2 are such that, on going around the contour, these fields are transformed by the element −1 2 and not by the identity element. The strings are of the Z 2 variety and there is no distinction between a string and an anti-string. Also, there is no known integral formula that can be used to evaluate the winding around the contour.

II. NUMERICAL IMPLEMENTATION
To simulate the formation of the monopole-string network, a 3-dimensional cubic lattice is chosen. Each cubic cell is further divided into 24 tetrahedral sub-cells, obtained by connecting the center of the cube to the 8 corners and the centers of the 6 faces (see Fig. 2).
or, in terms of the parameter choice of (7), Hence the SU (3)-invariant measure on CP 2 is √ g dθ dφ dα dβ = sin 3θ cosθ sinφ cosφ dθ dφ dα dβ. (20) Thus the assignment is done by drawing 0 ≤ sin 4θ ≤ 1, 0 ≤ sin 2φ ≤ 1, 0 ≤ α ≤ 2π and 0 ≤ β ≤ 2π from uniform distributions, and then constructing Z as in Eq. (7). The four vertices of a spatial tetrahedron then get mapped on to a tetrahedron in CP 2 which we will denote by (Z 1 , Z 2 , Z 3 , Z 4 ). To find out if this tetrahedron in CP 2 is topologically non-trivial (i.e. incontractable) we use a discrete version of the charge formula in Eq. (14) where the sum is over the four triangular faces of the tetrahedron (with positive orientation), and for each face, where we require α {ijk} to lie within the range [−π, π].
We can explicitly check that small changes in the Z i do Figure 2. The cubic lattice is divided into tetrahedral cells in our simulations.
The right-most rotation, D[R 21 ]Φ 1 , yields Φ 2 and, as in (4.8), the action of P 2 acting on Φ 2 simply gives a phase factor that commutes with all other rotations in (4.9). Hence the action of P 2 is to give an overall factor of e iδ 2 . Similar arguments apply to the action of P 1 and P 3 . Then the action of R on Φ 1 is equivalent to multiplication by, where h 123 denotes the phase angle due to the rotation D[ ]. This rotation implements the parallel transport of Φ 1 all the way around the triangular plaquette and gives the holonomy angle, h 123 , in this process. To determine h 123 we use From (4.9) we must have and a value of ±2π signals that a Z-string/anti-string passes through the plaquette. We have numerically implemented this algorithm to study the distribution of monopoles and strings on a discrete tetrahedral lattice. Each cell of a cubic lattice is divided into 24 tetrahedra [3] as shown in figure 2. At every lattice point, we assign random values of α, β and γ, from which we construct Φ andn. We find the monopoles on the lattice by evaluating the monopole winding in (3.3) for every tetrahedral cell, and the strings are found by evaluating the winding in (4.12) for every triangular plaquette. A sample of the monopole distribution with strings is shown in figure 3.
As in earlier simulations of monopole formation [23][24][25],n is uniformly distributed on an S 2 and the magnetic charge within a volume, ∼ L 3 , is given by a surface integral due to Gauss' law, with N ∼ (L/ξ) 2 independent domains of size ξ on the surface. Hence the root-mean-square magnetic charge within the volume goes as √ N ∼ L/ξ. We have confirmed this scaling in our simulations.
We   where the length is measured in units of the step length in going from one tetrahedral cell to its neighboring cell. The number density of closed loops also follows an exponential with, A c = 0.66 ± 0.07, l c = 7.79 ± 0.08. (4.14)

Magnetic field
As in the case of topological defects, the Kibble mechanism only provides initial conditions for the evolution of the system. In the case of cosmic strings, small loops formed during the JHEP01(2022)059 symmetry breaking will quickly collapse and dissipate, while longer loops and infinite strings will persist and eventually reach a scaling solution. In the electroweak case, monopoles and anti-monopoles will be brought together by the confining strings and rapidly annihilate [27]. However their annihilation will leave behind a magnetic field. Since Maxwell equations hold after electroweak symmetry breaking, the magnetic field can then be evolved with the usual Maxwellian magneto-hydrodynamical (MHD) equations [28]. We now turn to a characterization of the initial magnetic field. The electromagnetic field strength is defined as where A µ ≡ sin θ wn a W a µ + cos θ w Y µ and the last term in (5.1) is required for a suitable gauge invariant definition of A µν [10,18]. The definition breaks down at points where |Φ| = 0, i.e. in the symmetry restored phase, becausen andΦ are not well-defined.
The magnetic field of the monopole is With Φ = Φ m of eq. (3.1) and A = 0 we find the monopole magnetic field outside the core of the monopole, B m = sin θ wr /(gr 2 ) where r is the radial coordinate. Around the Z-string at θ = π we findΦ m → e iφ (0, 1) T . Using this form in (5.2) we see that there is no electromagnetic field associated with the Z-string at locations where Φ = 0. We can extend the formula (5.2) to the point where Φ = 0 in the Z-string by using continuity, and then the magnetic field vanishes everywhere for the Z-string. The usual characterization of stochastic isotropic magnetic fields is in terms of the two point correlators, In Maxwell theory, the correlation functions M N and M L are related by the condition that the magnetic field is divergence free, In our case, however, the magnetic field is not divergence-free and M N and M L are independent functions. The helical correlator, M H , vanishes for us since we have not included any source of parity violation in the system. We have evaluated the magnetic field correlator numerically and find with f (r) exhibiting anti-correlations at small scales. This makes physical sense since it is known that defects are preferentially surrounded by anti-defects [24]. studied this evolution. Instead we use a "smearing procedure" to estimate the volume averaged magnetic field due to monopoles, where the last expression for the surface integral follows from using (5.2) together with an integration by parts. Note that (5.2) assumes |Φ| = 0 and hence is not valid in the interior of the integration volume V in the presence of monopoles. The volume integral in (5.6) is ambiguous because of the divergent magnetic field at the locations of the monopoles. However the surface integral given in (5.6) still applies as the surface of integration does not intersect any monopole cores. The surface may intersect Z-strings but the formula in (5.2) holds by continuity as discussed below (5.2). For the integration in (5.6) we will consider cubical volumes with side λ. If ξ denotes the size of domains in which the random variableΦ † ∇Φ is tightly correlated, the discretized surface integral in (5.6) consists of a sum of (λ/ξ) 2 independent random terms and the sum itself will go like the square root of this number. Therefore we expect the magnitude B λ ≡ | B V | to grow as B λ ∝ λ/V ∝ 1/λ 2 . We have numerically evaluated B λ and the result is plotted in figure 5. The fit shows indeed shows that B λ ∝ 1/λ 2 .
As a final comment, note that the numerical calculation of the magnetic field does not directly use the network of monopoles and strings discussed in the previous sections. All that is needed is to evaluate the final term of (5.2) from the random distribution of the Higgs VEV.

Conclusions
Vacuum configurations of a field theory should include all configurations with minimum energy. Conventional considerations focus on homogeneous fields and then the vacuum manifold is given by the minima of the potential. However, in gauge theories, inhomogeneous JHEP01(2022)059 configurations can also have minimum energy provided they lie on gauge orbits on the vacuum manifold. Thus the vacuum manifold has additional structure. In particular, by minimizing the potential of the electroweak model the vacuum manifold is seen to be an S 3 . However the gauge orbits map the S 3 to S 2 with S 1 fibers, i.e. the vacuum manifold is a Hopf fibered S 3 . The topology of S 2 × S 1 leads to electroweak magnetic monopoles that are confined by Z-strings whose distribution we have determined by an extension of the Kibble mechanism. Since the electroweak monopoles are confined by Z-strings, they will annihilate rapidly even as they are formed, leaving behind a cosmological magnetic field whose spectrum falls off slowly with increasing wavelength: B k ∝ k 2 .
An alternative approach to deriving the properties of the magnetic field is to directly simulate the electroweak symmetry breaking, as has been done in several works [29][30][31][32][33]. These field theory simulations are much more computationally intensive than the present approach and are limited by computer resources. On the flip side, an advantage is that they more completely account for the dynamical evolution during the symmetry breaking, including magnetic fields that may be generated independently of the monopoles (the A µ terms in (5.1)).
The MHD evolution of magnetic fields depends significantly on the helicity of the field, described by the parity odd M H correlator in (5.3). There is, however, no source of parity violation in the formulation of the Kibble mechanism, and indeed in the bosonic sector of the electroweak model. Hence the magnetic field will be (globally) non-helical. (The process of monopole annihilation can induce local helicity because, in general, the monopole and antimonopole will be relatively twisted [34].) It is an interesting open question if parity violation from the fermionic sector or extensions of the standard model can be incorporated in the Kibble mechanism, that can then be used to study the generation of helical magnetic fields. Parity violating effects are also necessary for generating cosmic matter-antimatter asymmetry and the connection with magnetic helicity has already been noted [35][36][37][38][39][40][41].
The evolution of the magnetic field from the electroweak epoch to the present epoch is affected by several factors: turbulence, cosmic expansion, dissipation, and perhaps novel chiral effects. Magneto-hydrodynamical evolution does not apply initially because the magnetic field is not divergence-free. From general arguments that are supported by numerical simulations, a few percent of the electroweak false vacuum energy goes into magnetic fields during spontaneous symmetry breaking [14]. The coherence scale of the magnetic field at the electroweak epoch, ξ(t EW ), will depend on the dynamics during electroweak symmetry breaking. To obtain estimates we use an upper bound on the coherence and take it to be the horizon size at the electroweak scale: ξ(t EW ) ∼ t EW ∼ 1 cm. Then the magnetic field on length scale λ at the present epoch is given by where λ(t EW ) = λ(t 0 )T 0 /T EW and T denotes the cosmic temperature. With T 0 ∼ 10 −4 eV, T EW ∼ 10 11 eV, ρ γ (t 0 ) ∼ 10 −6 G, we get B 1 kpc (t 0 ) ∼ 10 −18 G, (6.2)

JHEP01(2022)059
and on Mpc scales the magnetic field is ∼ 10 −24 G. This estimate is much smaller than the blazar lower bounds in the literature: B 1 Mpc 10 −16 − 10 −19 G [42][43][44][45]. Hence the monopoles by themselves cannot provide magnetic fields of the observed strength. Additional ingredients are necessary if the magnetic fields generated during electroweak symmetry breaking are to explain observations. In particular, magnetic helicity can be this necessary ingredient as it can stretch the coherence scale of the magnetic field by a large factor ∼ 10 7 and increase the field strength estimate to ∼ 10 −11 G on 10 kpc scales [14].
In summary, we have extended the Kibble mechanism and applied it to the electroweak model. Then topological considerations lead to a distribution of magnetic monopoles and Z-strings that we can characterize. The distribution of magnetic monopoles immediately implies the presence of magnetic fields. We have derived the (smeared) magnetic field distribution as a function of the smearing length scale, λ, and find B λ ∝ λ −2 . The role of early universe magnetic fields for cosmological observations has been recently reviewed in refs. [12][13][14][15].