Instability of electroweak homogeneous vacua in strong magnetic fields

We consider the classical vacua of the Weinberg-Salam (WS) model of electroweak forces. These are no-particle, static solutions to the WS equations minimizing the WS energy locally. We study the WS vacuum solutions exhibiting a non-vanishing average magnetic field of strength $b$, and prove that (i) there is a magnetic field threshold $b_*$ such that for $bb_*$ they are not, (ii) for $b>b_*$, there are non-translationally invariant solutions with lower energy per unit volume and with the discrete translational symmetry of a 2D lattice in the plan transversal to $b$, and (iii) the lattice minimizing the energy per unit volume approaches the hexagonal one as the magnetic field strength approaches the threshold $b_*$. In the absence of particles, the Weinberg-Salam model reduces to the Yang-Mills-Higgs (YMH) equations for the gauge group $U(2)$. Thus our results can be rephrased as the corresponding statements about the $U(2)$-YMH equations.


Introduction
The Weinberg-Salam (WS) model of electroweak interactions [22] [35] [47] was the first triumph of the program to unify the four fundamental forces of nature.It is a key part of the standard model of elementary particles.It unifies electromagnetic and weak interactions, two of the three forces dealt with in the standard model.It involves particle, gauge and the Higgs fields.
While the gauge fields describe the electroweak interactions, the role of the Higgs field is to convert the original massless fields (zero masses are required by the relativistic invariance) to massive ones.This phenomenon is called the Higgs mechanism.This mechanism, together with the Goldstone theorem, leads to all gauge particles but one acquiring mass, resulting in two massive bosons -denoted W and Z -and a massless one -the photon.The W and Z particles where discovered experimentally 16 years after their theoretical prediction.
In this paper, we consider the vacuum solutions of the classical WS model with a non-vanishing average magnetic field ⟨ ⃗ b⟩.These are static, no-particle solutions minimizing the WS energy locally for a fixed ⃗ b.They are also no-particle solutions of the entire standard model. 1e prove that (i) there is a magnetic field threshold b * such that for | ⃗ b| < b * , the vacua are translationally invariant, while, for | ⃗ b| > b * , they are not, (ii) for | ⃗ b| > b * , there are non-translationally invariant solutions with lower energy per unit volume and with the discrete translational symmetry of a 2D lattice in the plane transversal to the magnetic field, and (iii) the lattice minimizing the energy of the latter solutions per unit volume approaches the hexagonal one as the magnetic field strength approaches the threshold b * .We expect that these solutions are stable under field fluctuations and, in fact, minimize the energy locally.
The phenomenon above was investigated extensively in the physics literature (see e.g.[12,19,26,28] and the references therein).It is similar to the one occurring in superconductivity and the solutions whose existence we establish are analogous to the superconducting Abrikosov vortex lattices ( [1], see e.g.[37], for a review).It is estimated in [26] that the spontaneous symmetry breaking takes place at the critical average magnetic field of approximately 10 24 Gauss = 10 20 Tesla.By comparison, the strongest magnetic field produced on Earth is 10 14 Tesla.
Note that, in the absence of particles, the WS system reduces to the Yang-Mills-Higgs (YMH) one with the gauge group U (2).So ultimately, these are the equations we deal with.
The only rigorous result ( [43,44]) on the classical WS model deals with the vortices in the self-dual regime, where the WS (or corresponding YMH) equations are equivalent to the first order equations, and it uses this equivalence in an essential way.(The self-dual regime in this context was discovered in [6,7,8], see also [41,42].) Open problems and further directions: For the stability and existence problems, (a) and (b), see e.g.[38,40] and [39], respectively.The last problem brings up the regime of 'sparse' vortex lattices as opposite to the case of | ⃗ b| close to (and >) b * resulting in densely packed vortices: the lattice step → 0 as | ⃗ b| → b * and → ∞ as | ⃗ b| → ∞.Hence the existence of vortex lattices at | ⃗ b| ≫ b * is closely related to the problem of existence of vortices (elementary excitations).
For the quantum corrections, problem (c), it would be natural to start with a BCS-type, or quasi-free, version of the WS model and a Bogoliubov-type expansion of a regularized (say, lattice) WS model around it, see e.g.[14,15].The paper is organized as follows.In Section 2, we formulate the problem and describe results.In Sections 3 -4, we fix the gauge and pass from the original Yang-Mills fields to the W and Z (massive boson) and A (photon) fields and rescale the resulting equations.The proofs of the main results are given in Section 5 (Theorem 2.1), Sections 6 -10 (Theorem 2.2) and Section 11 (Theorem 2.3).In Appendix A, we discuss various covariant derivatives used in the main text, and in Appendix B, we review the time-dependent YMH equations and derive the expression for the conserved energy as well as the YMH equations used in the main text.Furthermore, there we write the YMH equations in coordinate form and derive a convenient expression for the energy functional.In Appendices D.1 -D.2, we derive the WS equations in 3D and 2D, respectively, in terms of the fields W , Z, A and φ.In the remaining appendices, we carry out technical computations.
Throughout the paper, we use the Einstein convention of summing over repeated indices.
Acknowledgements The second author is grateful to Nicholas Ercolani, Jürg Fröhlich, Gian Michele Graf and Stephan Teufel for many instructive and stimulating discussions of the YMH equations.Both authors thank the anonymous referee for many constructive remarks.
2 No-particle and vacuum sectors of the Weinberg-Salam model The no-particle sector of the Weinberg-Salam (WS) model involves the interacting Higgs and SU (2) and U (1) gauge fields, Φ and V and X, while the particle fields are set to zero.The field Φ is a vector-function defined on the Minkowski space-time R 3+1 with values in C 2 , and the fields V and X are one-forms on R 3+1 with values in the algebras su(2) and u(1), respectively.We write where g and g ′ are coupling constants, which is a one-form with values in u (2).We consider SU (2) as a matrix group and U (1) as multiples of the identity matrix 1 acting on C2 .These fields satisfy the WS equations, which are the Euler-Lagrange equations for the action functional where M is a bounded domain in spacetime R 3+1 equipped with the Minkowski metric η of signature (−, +, +, +), λ and φ 0 are positive parameters, and the remaining symbols are defined as follows: ∇ Q is the covariant derivative mapping C 2 -valued functions (sections) into C 2 -valued one-forms defined as with d, the exterior derivative; F Q is the curvature 2-form of the connection one-form Q, given by where [A, B] is defined in local coordinates {x i } as with A = A i dx i and B = B i dx i ; Ω p U ≡ U ⊗ Ω p denotes the space of U -valued p-forms with the Minkowski, indefinite inner product, ⟨A, B⟩ η where A = A α (x)dx α and B = B α (x)dx α are U -valued p-forms, α is a p-form index and ⟨•, •⟩ U is the standard, positive definite inner product on U with the indices raised and lowered with help of the Minkowski metric η on M .For instance, for U = su(2), the inner product is given by := 2 Tr(A α (x) * B α (x)) = −2 Tr(A α (x)B α (x)). (2.6) Solutions of the no-particle WS equations solve also the full WS system as well as that for the standard model of the particle physics.
The vacuum sector of the Weinberg-Salam (WS) model consists of static, no-particle solutions.The static Higgs and SU (2) and U (1) gauge fields Φ, V and X are now defined on the physical space R 3 with the same respective values as in the time-dependent case.Geometrically, V, X and Q can be thought of as connection one-forms on the trivial bundles R 3 × SU (2), R 3 × U (1) and R 3 × U (2).
The fields Φ, V and X satisfy the static no-particle WS equations, which are the Euler-Lagrange equations for the static WS energy functional originating in (2.1) where N is a bounded domain in R 3 with appropriate boundary conditions (specified in (2.17) below) and ∥•∥ Ω p U is the standard norm on the space Ω p U := U ⊗ Ω p of U -valued p-forms at x ∈ N (e.g. for B = B i (x)dx i ∈ Ω 1 U , we have ∥B∥ Ω 1 U := ( i ∥B i (x)∥ 2 U ) 1/2 with the usual Euclidean metric and with the indices running through 1, 2, 3), while now, (2.5) (and (2.6)) become the usual inner products.The symbols ∇ Q and F Q are as defined above but without the time component.
Since Q = gV +g ′ X and X has the values in the centre, u(1), of the algebra u(2), we have F Q = gF V +g ′ F X , where are the curvatures of the connections V and X3 and .
We introduce the covariant derivative d Q mapping u(2)-valued k-forms into u(2)-valued (k + 1)-forms, k ≥ 1, as (2.9) This formula originates in the equation (δ , where δ Q is the Gâteaux derivative with respect to Q.For 0-forms, we set The Euler-Lagrange equations for energy functional (2.7) are given by (see Appendix B 4 ) where ∇ * Q is the adjoint of ∇ Q and maps C 2 -valued one-forms into C 2 -valued functions, d * Q is the adjoint of d Q and maps u(2)-valued two-forms into u(2)-valued one-forms, and J(Q, Φ) is the electroweak current, which is the u(2)-valued one-form given by where summing over repeated indices is understood, τ 0 := 1 and τ a , a = 1, 2, 3, are the Pauli matrices, where , and The physical quantities here are (a) the Higgs field density ∥Φ∥, (b) the magnetic field Tr F Q and (c) the YM current J(Q, Φ).It is easy to check that these quantities are gauge invariant.We say that a solution (Q, Φ) to (2.10)-(2.11) is homogeneous if ∥Φ∥, Tr F Q and J(Q, Φ) are independent of x. (We say that Tr F Q is independent of x, if it is a multiple of a constant 2-form, see (2.16).)Otherwise, we say that (Q, Φ) is inhomogeneous.
Furthermore, we say that a solution (Q, Φ) is gauge-translation invariant if it is invariant under translations up to gauge transformations.
Clearly, a solution (Q, Φ) which is gauge-translation invariant is also homogeneous.The converse in general might not be true.
We are interested in the vacuum solutions of the WS equations with a non-vanishing average magnetic field, lim i.e. solutions minimizing the WS energy locally under the constraint above.In physical field theories, one expects the vacua to have the maximal available symmetry.Consequently, we first consider gauge-translation invariant solutions with a fixed (constant) magnetic field.
where Φ ⃗ b is a constant field and Q ⃗ b is a connection with a constant magnetic field with the sum taken over all cyclic permutations of (1, 2, 3), and e := gg ′ √ g 2 +g ′2 .(e turns out to be the electron charge.)We specify this solution at the end of this section in equations ( 2 (iii) The lattice shape minimizing the energy per unit area approaches the hexagonal lattice as b approaches b * .
To formulate these results precisely, we introduce some definitions.Since we consider solutions which do not depend on the coordinate along ⃗ b, we can restrict our analysis to the plane ⊥ ⃗ b.We choose the x 3 -axis along ⃗ b and identify the plane ⊥ ⃗ b with R 2 .
We fix a lattice L in R 2 and say a triple (Φ(x), V (x), X(x)) is L-gauge-periodic, or, L-equivariant, if and only if it satisfies the equation (T gauge γs for some ).Here T gauge γ is given by (2.14) and T trans s is the group of translations, T trans s f (x) = f (x + s).(When L is clear, we omit it from the definition above.)We denote by H s L , s ∈ N, the Sobolev space of L-equivariant triples U ≡ (V, X, Φ) on R 2 , with the norm where Ω is an arbitrary fundamental domain of L, 5) (and (2.6)), and with corresponding the inner product.Note that L 2 L = H 0 L .The resulting Sobolev spaces H s L are independent (up to isomorphism) of the choice of the fundamental domain, Ω.All Sobolev embedding theorems are valid for H s L .They can be proven by passing to a vector bundle over the torus R 2 /L and then to the local charts and then using standard Sobolev embedding theorems.By the Sobolev embedding H 1 L ⊂ L p L , p < ∞, and the definitions 2.7 and 2.18, and is independent of a choice of Ω.We say a solution U * := (V * , X * , Φ * ) of the WS system (2.10) -(2.11) is energetically stable if and only if it is a local minimum of the WS energy E N , in the sense that the spectrum of the L 2 -Hessian of E N at U * on L 2 L (which is real) is non-negative.U * is said to be unstable if it is a saddle point of E N (so that the spectrum of its Hessian has a negative part).
For an L-equivariant triple U and a fundamental domain Ω of L, we define the energy per fundamental cell by where |Ω| denotes the area of Ω.This energy is independent of the choice of Ω.
In what follows, Ω denotes an arbitrary (but fixed throughout) fundamental domain of L, and |L|, the area of a fundamental cell of L, which is independent of the choice of the cell Ω and is called the covolume of L. Let The solutions described in this theorem can be reinterpreted geometrically as representing sections (Φ(x)) and connections ((V (x), X(x))) on a U (2) vector bundle over a torus (cf.[20]).However, a vector bundles over a torus is topologically equivalent to a direct sum of line bundles.In our case, this equivalence follows from equations (3.5) -(3.7) below.
For the next result, we use the topology on the space of (normalized) lattices induced by the standard parameterization of lattices defined as follows.Identifying R 2 with C via (x 1 , x 2 ) ↔ x 1 + ix 2 and viewing a lattice L ⊂ R 2 as a subset of C and using a translation and a rotation, any lattice L ⊂ C can be reduced to the form L = rL τ , where r > 0, L τ := Z + τ Z and τ ∈ H := {τ ∈ C : Im τ > 0}.Furthermore, any two τ 's produce the same lattice iff they are related by an element the modular group SL(2, Z) acting on the Poincaré half-plane H (see e.g.[4]).Hence, it suffices to restrict τ to the fundamental domain of SL(2, Z), Theorem 2.3.For M Z < M H , the lattice L * minimizing the average energy, E L (U L ), approaches the hexagonal lattice L hex as b → b * in the sense that the shape parameter τ * of the lattice L * approaches τ hex = e iπ/3 in C. Now, we construct explicitly the solution (2.15).We define where A ⃗ b (x) be a (U (1)-) magnetic potential of the constant magnetic field dA ⃗ b = ω ⃗ b and θ is Weinberg's angle, given by tan θ = g ′ /g.We have This gives (2.10) and reduces (2.11) Our approach is based on a careful examination of the linearization of the WS equations on the homogeneous vacuum.The spectrum of the linearized problem determines the domains of the linear, or energetic, stability and the transition threshold.In the instability domain, we apply an equivariant bifurcation theory.This gives Theorem 2.2(a) and (b).For Theorems 2.2(c) and 2.3, we carefully study the asymptotic behaviour of the energy functions for small values of the bifurcation parameter.

Gauge fixing and W and Z bosons
In this section, we choose a particular gauge and pass from the fields (one-forms) V and X to more suitable gauge fields.We eliminate a part of the gauge freedom by assuming that the Higgs field Φ is of the form with φ real (this can be done using only the SU (2) part of the gauge group).Then where, recall, τ a , a = 1, 2, 3, are the Pauli matrices generating the Lie algebra su(2), and τ 0 = 1.However, there is one linear combination of τ a 's (unique up to a scalar multiple) which annihilates Φ: Thus, for the gauge Φ = (0, φ) the symmetries generated by τ 1 , τ 2 , τ 3 − τ 0 are broken and the U (1) symmetry generated by τ 3 + τ 0 remains unbroken.The unbroken gauge symmetry is given by transformations (2.14) with where γ ∈ C 1 (R 3 , R).
Continuing in the gauge Φ = (0, φ) and writing V = − i 2 τ a V a6 and X = − i 2 τ 0 X 0 , where X 0 and V a , a = 1, 2, 3, are real fields (since V takes values in su(2) and therefore V * = −V ), we pass to the new fields corresponding to the broken and unbroken generators, τ 3 − τ 0 and τ 3 + τ 0 , respectively: where, recall, θ is Weinberg's angle, defined by tan θ = g ′ /g.Note that Z and A are real fields Moreover, it is convenient to pass from the remaining two components, V 1 , V 2 , of V to a single complex field The gauge invariance of the original field equations with the unbroken gauge symmetry given by transformations (2.14) with (3.4) leads to the invariance under following gauge transformations: ) , where e iγ W = e iγ W i dx i for W = W i dx i , e is the electron charge.Here, we replaced Φ := (0, φ) by φ.
The WS energy in terms of W, Z, A and φ fields in 3D is given in (D.1), Appendix D.1.The WS equations in terms of W, Z, A and φ in 3D can be found by taking variational derivatives of this energy w.r.to different fields.
In terms of W, A, Z and φ fields, the vacua (2.15) of the Weinberg-Salam model become (up to a gauge symmetry): where, recall, A b (x) is a magnetic potential for the constant magnetic field of strength b in the x 3 -direction, dA b (x) = bdx 1 ∧ dx 2 , and φ 0 is a positive constant from (2.7).We choose the gauge so that A b (x) is of the form We will show that for a large magnetic field b, these homogeneous vacua become unstable and new, inhomogeneous vacua emerge from them.This is a bifurcation problem from the branch of gauge-translationally invariant (homogeneous) solutions, (3.8).
Since we consider the WS system with the fields independent of the third dimension x 3 , i.e. in R 2 , we can choose the gauge with V 3 = X 3 = 0 (and hence Also, we will work in a fixed coordinate system, {x i } 2 i=1 and write the fields as W = W i dx i , Z = Z i dx i and A = A i dx i .For ease of comparing our arguments with earlier results, and given that we use the standard Euclidean metric in R 2 , we identify (complex) one-forms W, Z and A with the (complex) vector fields (W 1 , W 2 ), (Z 1 , Z 2 ) and (A 1 , A 2 ).With this, we show in Appendix D.2 that in this case, WS energy functional (2.7) can be written as where It follows from (3.5) that V 3 = Z cos θ + A sin θ.Expanding (3.10) in φ around φ 0 , we see that the W , Z and ϕ (Higgs) fields have the masses 2 cos θ gφ 0 and M H = √ 2λφ 0 , respectively.Using the relation ξ ×η = Jξ •η, where • denotes the Euclidean scalar product in R 2 and J is the symplectic matrix, we find the Euler-Lagrange equations for (3.10), which give the WS system (2.10) - (2.11) in 2D in terms of the fields W , A, Z and φ where, recall, κ = g 2 2 cos 2 θ , V 3 = Z cos θ + A sin θ and ∆ is the standard Laplacian.(For a derivation of (3.12) -(3.15) from (3.10), see Appendix D.2 and also [26,43].)Of course, (3.12) -(3.15) can also be derived directly from WS system (2.10) - (2.11).
In terms of the (W, A, Z, φ) fields, the lattice gauge -periodicity (2.17) is expressed as ( T gauge γs for all s ∈ L, where γ s ∈ C 1 (R 2 , R) for all s ∈ L, T gauge γ given in (3.7) and T trans s is the group of translations, T trans s f (x) = f (x + s).We say that (W, A, Z, φ) satisfying (3.16) is an L-equivariant state.By evaluating the effect of translation by s + t in two different ways, we see that the family of functions γ s has the co-cycle property 7γ s+t (x) − γ s (x + t) − γ t (x) ∈ 2πZ, ∀s, t ∈ L.
(3.17) Since T trans s is an Abelian group, the co-cycle condition (3.17) implies that, for any basis {j 1 , j 2 } in L, the quantity is independent of x and of the choice of the basis {j 1 , j 2 }, and is an integer.This topological invariant is equal to the degree of the corresponding line bundle.Using Stokes' Theorem, one can show, for any A satisfying (3.16) - (3.18), that the magnetic flux through any fundamental domain Ω of the lattice L is quantized: where e is defined after (3.7) and n = c(γ s ) ∈ Z defined in (3.18).The left-hand side of ( Finally, we use the invariance of (3.12) -(3.15) under the gauge transformation (3.7) to choose a convenient gauge for the fields W (x) and A(x).We say that the fields (W, A, Z, φ) and (W Clearly, if (W, A, Z, φ) and (W ′ , A ′ , Z ′ , φ ′ ) are gauge-equivalent, then (W, A, Z, φ) solves (3.12) -(3.15) if and only if (W ′ , A ′ , Z ′ , φ ′ ) solves (3.12) - (3.15).The following proposition was first used in [31] and proven in [45] (an alternate proof is given in Appendix A of [46]): Proposition 3.2.Let (W ′ , A ′ , Z ′ , φ ′ ) be an L-equivariant state and let b be given by (3.20).Then there is a L-equivariant state (W, A, Z, φ), gauge-equivalent to (W ′ , A ′ , Z ′ , φ ′ ), which satisfies (3.16), with χ s (x) = eb 2 s ∧ x + k s , i.e. such that, ∀s ∈ L, div A = 0, ( Here k s satisfies the condition k s+t − k s − k t − eb 2 s ∧ t ∈ 2πZ, for all s, t ∈ L, the matrix J is given in (3.11).
Our goal is to prove the instability of the vacuum state (3.8) and the existence of L−equivariant (in the sense of (3.16)) solutions to transformed WS system (3.12)-(3.15) having the properties described in Theorems 2.2 and 2.3.

Rescaling
In this section, we rescale transformed WS system (3.12)-(3.15) to keep the lattice size fixed.Specifically, we define the rescaled fields (w, a, z, ϕ) to be where in the second equality (4.2), we used (3.20).Clearly, (W (x), where Now, the rescaled lattice L ′ is independent of b and the size of a fundamental domain, Ω ′ , of L ′ is fixed as Plugging the rescaled fields into (3.12) - (3.15) gives the rescaled Weinberg-Salem equations: where ξ := rφ 0 (with r given in (4.2)), ν := g(a sin θ+z cos θ) and, recall, curl 1)−valued vector-field iq) and, recall, w × w := w 1 w 2 − w 2 w 1 .We define the rescaled energy by with (W, A, Z, φ) related to (w, a, z, ϕ) by (4.1) and E Ω (W, A, Z, φ) given in (3.10).Explicitly, we have We note that after rescaling, the average magnetic flux per fundamental domain becomes n/e and the vacuum solution (3.8), where a n (x) ≡ A n (x) = n 2 Jx, .Furthermore, (3.16) and Proposition 3.2 imply that (w, a, z, ϕ) satisfy s×x+cs) w(x) for all s ∈ L ′ , (4.10) div a = 0, (4.12) where c s satisfies the condition c s+t − c s − c t − n 2 s × t ∈ 2πZ, for all s, t ∈ L ′ .Finally, the Sobolev spaces here, denoted again by H s L ′ , can be obtained by rescaling the Sobolev spaces defined above or defined directly, again as above, see (2.18) and the text around it.Similarly to (2.19), by a Sobolev embedding theorem, the rescaled energy is finite, and is independent of a choice of Ω ′ .

The linearized problem
In this section we prove Theorem 2.1, describing the stability properties of the vacuum (3.8).Equivalently, we will investigate the energetic stability of the rescaled vacuum solution (4.9) of the rescaled WS equations (4.3) -(4.6).
Let m := (w, a, z, ϕ) and denote by G(b, m) ≡ G(m) the map G : H 2 L ′ → C 7 given by the left-hand side of (4.3) -(4.6), written explicitly as where, recall, J is the symplectic matrix given in (3.11), ξ := rφ 0 (with r given in (4.2)), ν := g(a sin θ+z cos θ), ∆ is the standard Laplacian and the parameter b enters through periodicity conditions (4.10) -(4.13).Now, the WS system can be written as Recall the definition of stability given above Eq.(2.20).To apply it to the rescaled WS equations (4.3) -(4.6), we observe that the map G is the of E Ω ′ at m, defined on the space of variations Y tangent to the space of L 2 loc functions of the form (w, a, z, ϕ) satisfying the gauge -periodicity conditions (4.10) -(4.13): (5.8) Here L 2 n , L 2 0 and L 2 are given by , the L 2 -Hessian for E Ω ′ and m is the formally symmetric operator Denote the L 2 -Hessian at the vacuum solution m n,r (see (4.9)) by As seen from its explicit form given below, the operator L n,µ , acting on the space Y, is self-adjoint and therefore its spectrum is real.Thus, applied to the rescaled WS equations (4.3) -(4.6), the definition of stability can be rephrased as: the vacuum solution m n,r is energetically stable (respectively, unstable) if and only if inf spec(L n,µ ) ≥ 0 (respectively, inf spec(L n,µ ) < 0).
We consider the operator L n,µ on the space Y, with the domain where H s n , H s 0 and H s are the respective Sobolev spaces for the L 2 -spaces (5.9)-(5.11),with inner products given (for s ∈ Z ≥0 ) by where , Ω ′ is an arbitrary fundamental domain of the lattice L ′ and γ is a multi-index.The L ′ -equivariance of the above functions implies that these inner products do not depend on the choice of fundamental domain Ω ′ .
We compute the linear operator L n,µ explicitly.In what follows we use the notation ⊕ j A j for diagonal operator-matrices with the operators A j on the diagonal.Passing from the parameter ξ = rφ 0 , or r, to the parameter µ := g 2 ξ 2 /2 and using that ν a=a n /e,z=0 = 1 e a n g sin θ = a n , we find ) where, recall, curl The gauge invariance of Eq. (5.6) and the partial symmetry breaking of vacuum solution (4.9) imply that L n,µ=n has the gauge zero mode: L n,µ=n Γ f = 0, Γ f := (0, ∇f, 0, 0). (5.21) For a null vector Γ f defined in (5.21) to be in X , f must satisfy div(∇f ) = −∆f = 0.This implies that f is a linear function, f (x) = c • x + d for some c ∈ R 2 and d ∈ R, and so Γ f ∈ X =⇒ Γ f = (0, c, 0, 0).(5.22)In this section we shall prove the following result implying Theorem 2.1: Theorem 5.1.The operator L n,µ on the space X has purely discrete spectrum.For µ ̸ = n, L µ,n has the multiplicity 2 eigenvalue 0 with the eigenfuctions (0, e i , 0, 0), i = 1, 2, e 1 = (1, 0), e 2 = (0, 1) (see (5.22)).
Furthermore, the smallest non-zero eigenvalue given by µ − n, having multiplicity n.For µ = n, the eigenvalue 0 has the multiplicity n + 2.
Proof.The strict positivity of H 3 (µ) and H 4 (µ) and the non-negativity of H 2 (µ) are obvious.The discreteness of the spectra and the form of the null space of H 2 (µ) follow from the discreteness of the spectrum of the Laplacian on compact domains and the identity curl * curl v = −∆v when div(v) = 0. To compute the null space of H 2 (µ), we observe that the solutions of the equations ∆v = 0 and div(v) = 0 are constant vectors in R 2 .
(ii) The eigenspace of the eigenvalue −n + µ is n-dimensional and is spanned by functions of the form and the eigenspace of the eigenvalue µ is of the form In the proof of this proposition, we use the following standard result whose proof, for reader's convenience, is given in Appendix G: Proposition 5.4.The operator −∆ a n is self-adjoint on its natural domain and its spectrum is given by with each eigenvalue is of the multiplicity n.Moreover, In more detail, with z = (x 1 + ix 2 )/ 2π Im τ and τ coming from with div We write any w ∈ H 2 n as w = w 0 + ∇ a n f , where f solves the equation ∆ a n f = div a n w and w 0 is defined by this relation.By Proposition 5.4, 0 is not in the spectrum of ∆ a n and therefore the equation ∆ a n f = div a n w has the unique solution f ∈ H 3 n .Then, since ∆ a n := div a n ∇ a n , we have div a n w 0 = 0.This proves H 2 n = Y ⊕ Z. Now, recall that the operator H 1 (µ) acts on complex vectors w = (w 1 , w 2 ).The definition H 1 (µ) := curl * a n curl a n −niJ + µ and the relations curl * a n = −J∇ a n and which proves that the µ-eigenspace of H 1 (µ) is of the form (5.25) giving the second part of (ii).
By the above the subspace Y is invariant under H 1 (µ).To compute the spectrum of the operator H 1 (µ) on the subspace Y, we use the definitions of curl a n and curl * a n and recall the relation By above, we have ) Thus, we conclude that ) Identifying one-forms with vector-fields, we compute which gives By Proposition 5.4, we know that and so the spectrum of H 1 (µ) on Y is given by the first set on the r.h.s. of (5.23).Hence, by H 2 n = Y ⊕ Z, (5.23) follows, giving (i).
Furthermore, by (5.35) and (5.36), any eigenvector χ of h 1 corresponding to the eigenvalue −n must be of the form where β satisfies This relation, together with the equation Null(−∆ Furthermore, by Proposition 5.4, the space of such functions is n-dimensional.Thus (after rescaling ω by a factor of √ 2) χ is of the form (5.24).This gives also the first part of (ii) completing the proof of the proposition.
We see that the operator H 1 (µ) is non-negative for the magnetic fields satisfying b < b * := g
In conclusion of this section, we investigate properties of the map F (µ, u).For f = (f 1 , f 2 , f 3 , f 4 ) and δ ∈ R, define the global transformation (6.12) Proposition 6.2.The map F (µ, u) defined in (6.7) has the following properties: (i) F : R >0 × X → Y is continuously Gâteau differentiable of all orders; (ii) F (µ, 0) = 0 for all µ ∈ R >0 ; (iii) Proof.(i) follows because F is a polynomial in the components of u and their first-and second-order (covariant) derivatives.(ii), (iii) and (iv) follow from an easy calculation (in fact, u and L n,µ were defined so that (ii) and (iii) hold).For (v), it suffices to show that ⟨w, To simplify notation we return to the coordinates (w, a, z, ϕ) = (w, 1 e a n + α, z, The first, second and fourth terms are clearly real, while the third term is real because ν is real and w × w is imaginary.
Recall that L n,µ is defined in (5.16).Let P be the orthogonal projection onto K := Null(L n,µ=n ), which can be written explicitly as where H 1 (n) is defined in (5.17), γ n is any simple closed curve in C containing the eigenvalue 0 and no other eigenvalues of H 1 (n) (see Proposition 5.3), and ⟨α⟩ is the mean value of α in Ω ′ , ⟨α⟩ := 1 |Ω ′ | Ω ′ α.P 1 is a projection onto Null(H 1 (n)) (spanned by vectors of the form (5.24)).Since H 1 (n) is self-adjoint, P 1 is an orthogonal projection (relative to the inner product of L 2 n ).By Theorem 5.1, K := Null(L n,µ=n ) is (n + 1)-dimensional.
Let P ⊥ = 1 − P be the projection onto the orthogonal complement of K. Applying P and P ⊥ to the equation F (µ, u) = 0 (see (6.7)), we split it into two equations for two unknowns as where v := P u, u ′ := P ⊥ u.
Our next goal is to solve (7.5) for u ′ in terms of µ and v.For n = 1, K is two-dimensional and we write . Furthermore, this solution has the following properties: ) ) ) ) ) (7.12)By Proposition 6.2 (i) and (ii), F ⊥ is continuously differentiable of all orders as a map between Banach spaces and F ⊥ (µ, 0, 0) = 0 for all µ ∈ R >0 .Furthermore, which is invertible for µ = n because P ⊥ is the projection onto the orthogonal complement of K = Null(L n,µ=n ).By the Implicit Function Theorem (see e.g.[16]), there exists a function u ′ (µ, v) with continuous derivatives of all orders such that for (µ, v) in a sufficiently small neighbourhood U ⊂ R >0 × K of (n, 0), (µ, v, u ′ ) solves (7.5) if and only if u ′ = u ′ (µ, v).This proves the first statement and property (7.6).
We define the operator Then by (6.7) and (7.13), we can write equation (7.5) as L ⊥ n,µ u ′ = −P ⊥ P ′ J(µ, u).By Theorem 5.1 and the relation ) ) is a polynomial in the components of u and their first-order (covariant) derivatives consisting of terms of degree at least 2, so the left-hand side of (7.16) can be bounded above by a sum of products of one L 2 -norm and at least one L ∞ -norm of these terms.H 1 is trivially continously embedded in L 2 , and by the Sobolev Embedding Theorem, H 1 is continuously embedded in L ∞ .Therefore, (7.17) Recalling that u = v + u ′ , this proves (7.7) and (7.8) when m = 0.The other case is proven similarly.For v = (v 1 , . . ., v 4 ), we let v i ≡ v| v i =0 , i = 1, . . ., 4. By the Taylor theorem for Banach spaces (see e.g.[16]), we have Let (µ, v) ∈ U with ∥v i ∥ = ∥v i ∥ = 1, and let ϵ > 0. Then with the norm taken in the appropriate space for v i .Taking the supremum over all v i with ||v i || = 1 gives proving (7.9) -(7.10) for m = 0.The other cases are proven in exactly the same way.
Since F : R >0 × X → Y and u ′ : R >0 × K → Y ⊥ have been shown to be continuously differentiable of all orders, we conclude: Furthermore, γ(µ, v) inherits the following symmetry of F (µ, u), which we will use to find a solution of (7.27): Lemma 7.4.Let T δ be given by (6.12).For every δ ∈ R and (µ, v) in a neighbourhood of (n, 0), we have Proof.For equation (7.28), we note that by Proposition 6.2 (iv) (Here we used P ⊥ T δ = T δ P ⊥ , which follows because T δ = e iδ ⊕ 1 ⊕ 1 ⊕ 1 and P ⊥ = 1 − P where P is defined in (7.1).)Since For equation (7.29), we note that by (7.28) and Proposition 6.2 (iv), (where again we used P T δ = T δ P ).
Proof of Theorem 8.1.For the proof below, recall that we denote the partial (real) Gâteaux derivatives with respect to # by δ # , and let By Proposition 6.1, solving equation (6.1) is equivalent to solving (6.7).By Corollary 7.2, solving (6.7) is equivalent to solving the bifurcation equation (7.27).Hence, we address the latter equation.
Using Equation (8.25), we calculate the second term on the right-hand side of (8.24) at (n, s, 0): The inner product term is real.Integrating it by parts and using that, by Equation (5.38), β satisfies curl a n curl * a n β = −∆ a n β = nβ and using that ∥β∥ 2 The last two equations and the relation Im This, together with (8.24), gives Therefore, (8.36) and (8.23) (with l = 1) imply proving that ∂ c γ 2 (n, 0, 0) is invertible, as required.
We will now use the branch of solutions to (8.7) -(8.8), provided by Lemmas 8.2 and 8.4, and Corollary 7.2 to obtain the corresponding unique branch, (µ s , u s ), of solutions to (6.7), with imply that (u ′ s ) 1 is an odd function of s and (u ′ s ) 2 , (u ′ s ) 3 and (u ′ s ) 4 are even functions of s.Arguing as in the case of Lemma 8.2 above shows that the functions:  Recall that M W , M Z , M H are the masses of the W , Z and Higgs bosons, respectively, and that τ is the shape parameter of the lattice L (see the paragraph before Theorem 2.3 of Section 2).We introduce the notation the average of f over fundamental domain Ω ′ = 2π |Ω| Ω.Furthermore, we introduce the function (cf.[26]) Here χ is defined in (5.24) and G m,m ′ is the operator-family on the space (5.11) given by Proposition 9.1.If M Z < M H , the parameter s of the branch (8.1) is related to the magnetic field strength by where R s (ω) is a real, smooth function of ω satisfying Consider the solution branch (µ s , w s , a s , z s ) given in equation (8.1) and described in Theorem 8.1.Using Taylor's theorem for Banach space-valued finctions (see e.g.[16]) and recalling the relation ξ = √ 2µ/g, we may expand this branch in s as follows: where w ′ , a ′ , z ′ , ψ ′ , ξ ′ and a ′′ are the coefficients of s 2 and s 4 , respectively, in the Taylor expansion of g j (s 2 ), j = 0, ..., 5, in (8.1), and χ is defined in (5.24).Here O(|s| p ) stand for various error terms which, together with their (covariant) derivatives, have norms of order O(|s| p ) when taken in the appropriate spaces.
10 Asymptotics of the Weinberg-Salam energy near b = M 2 W /e 2 gφ 0 , and η mz,m h (τ ) is defined in (9.2).The main result of this section is the following: Theorem 10.1.If M Z < M H , then the WS energy (3.10) of the branch of solutions (8.1) has the following expansion: where R E (ω) is a real function with continuous derivatives of all orders satisfying Before proving Theorem 10.1, we derive from it Theorem 2.2 (c).
Proof of Theorem 10.1.
, where E Ω ′ is the rescaled WS energy given in (4.8).In Appendix F, we derive the following expansion (to order s 4 ) of E ′ evaluated at family (9.7) of solutions: where R ε (s) = O(|s| 6 ) and has continuous derivatives of all orders, ν ′ := g(a ′ sin θ + z ′ cos θ) and, recall, ξ s = √ 2µ s /g.To simplify notation, in what follows, we shall suppress the argument (w s , a s , z s , ψ s + ξ s ; r) of E ′ .We claim the following relation: Proof of (10.4).We simplify the integral at order s 4 in (10.3) by applying equations (9.9) for a ′ , z ′ and ψ ′ to convenient groupings of terms.First, we address the z ′ terms in (10.3).Integrating by parts and factoring out z ′ gives Applying (9.9) for z ′ gives Integrating by parts again gives Next, we address the a ′ term in (10.3).Integrating by parts gives Inserting into this expression (9.9) for a ′ gives Integrating by parts again gives Next, we address the ψ ′ terms.Integrating by parts and factoring out ψ ′ gives Inserting into this expression (9.9) for ψ ′ gives For the ξ ′ term in (10.3), we have by (9.11) and (9.14), where, recall, ν ′ := g(a ′ sin θ + z ′ cos θ).Finally, there are two remaining terms of the integral at order s 4 in (10.3), which we will not presently simplify.Adding equations (10.7), (10.10), (10.12), (10.13) and (10.14) and remembering (10.3) gives Eq. ( 10.4), as required.
For the WS energy (3.10), evaluated at (W b , A b , Z b , φ b ), we recall that Eq. (10.1) follows by plugging (10.15) into (10.16).Since the remainder term Rε of (10.15) has continuous derivatives of all orders, so does the remainder term R E of (10.1).

Shape of lattice solutions
In this section we shall prove Theorem 2.3.Recall the shape parameter τ described in the paragraph preceding (2.22).We return briefly to working with the rescaled fields to prove that E Ω ′ (u; r), u = (w, α, z, ψ), given in (4.8), (and hence E Ω (U )) is continuously Gâteau differentiable of all orders in the shape parameter τ (restricted to domain (2.22)), which enters through Ω ′ (and Ω), as well as the spaces containing u (and U ). Below, we write to emphasize the dependence of the family of solutions (9.7), the corresponding energy (4.8) (respectively (3.10)) and the space (5.12) containing these solutions on the shape parameter τ , the magnetic field strength b and the position in space x ∈ R 2 .Also, recall the notation r := n/eb.
To get rid of the dependency of the space X τ containing u τ,b , on the shape parameter τ , we make the change of coordinates mapping Ω ′ into a square of area 2π.This allows us to define the functions where, recall, G(b, v) is the map given by the left-hand side of (4.3) -(4.6), given explicitly in (5.1), and ε W S (b, u) := ε(u; r) is the rescaled energy density given by the integrand in (4.8), see (11.2) (ε depends on the magnetic field strength b but does not directly depend on the shape parameter τ ).For the τ -derivatives, note that are continuously differentiable of all orders in Re(τ ) and Im(τ ).Since G(b, u) and ε W S (b, u) are polynomials in the components of u and their (covariant) derivatives, G ′ and Σ are simply G and ε W S with the coefficients of the derivative-containing terms multiplied by smooth functions of Re(τ ) and Im(τ ).Therefore G ′ (τ, b, v) and Σ(τ, b, v) have continuous Re(τ )-and Im(τ )-derivatives of all orders.
Proof.Let τ 0 be an arbitrary shape parameter, and recall that δ # denotes the partial (real) Gâteaux derivative with respect to #.Then invertible, and by Lemma 11.1, G ′ is continuously Gâteau differentiable of all orders in τ , b and v. Therefore, by the Implicit Function Theorem, the unique solution v τ,b to the equation G(τ, b, v) = 0 is continuously differentiable of all orders in Re(τ ) and Im(τ ) near (Re(τ ), Im(τ )) = (Re(τ 0 ), Im(τ 0 )).Since τ 0 was arbitrary, this proves the result.Proof.To get rid of the dependency of E(τ, b, u τ,b ) on the domain of integration Ω ′ , we again make the change of coordinates y = m −1 τ x.Then where, recall, For both expansions to hold, we must have ), as required.
Euler-Lagrange equations.The Euler-Lagrange equations (called Yang-Mills-Higgs equations) for the fields Ψ and A are where ∇ * η A and d * η A are the adjoints of ∇ A and d A in the appropriate inner products involving the metric η and J(Ψ, A) is the YMH current given by where γ a is an orthonormal basis of g and Proof of (B.4) -(B.5).For convenience, we assume periodic or Dirichlet boundary conditions and that Ψ and A are T -periodic in t and calculate the Gâteaux derivatives formally.
Recall that δ # denotes the partial (real) Gâteaux derivative with respect to #.First we calculate the (complex) Gâteaux derivative of (B.3) in the Ψ-direction.Define Integrating the first term by parts and factoring out Ψ ′ gives For this derivative to be zero for every variation Ψ ′ , (B.4) must hold.
Next we calculate the Gâteaux derivative of (B.3) in the A-direction.Using the definition δ A f (A)B = ∂ s f (A s )| s=0 , where A s = A + sA ′ , s ∈ R, we find which gives I = M ⟨B, J(Ψ, A)⟩ Ω 1 g .For the second term on the r.h.s. of (B.10), integrating by parts yields Collecting the last two equations gives For this derivative to be zero for every variation B, (B.5) must hold.
Conserved energy.Again, the Gâteaux derivative calculations in the following subsection are formal.
Recall that M := Ω × [0, T ] ⊂ R d+1 .To find the expression for the energy, we use, as in classical mechanics, the (partial, i.e. without passing to the momentum fields) Legendre transform of (B.2) is given by where the norms are taken using the Euclidean metric on R d+1 (rather than the Minkowski metric).
Proof.Let ∂ # denote the partial derivative with respect to the symbol #, and recall that δ # denotes the partial (real) Gâteaux derivative with respect to #.We calculate  ).This can be done by using the (partial) Legendre transform (B.15) as in classical mechanics, or by a direct computation.We proceed in the second way.Applying the chain rule gives where, recall, ∂ i ≡ ∂ x i .We now calculate the first term using (B.4).
Integrating the second term by parts gives By (B.4), we have , where the second equality follows because the representation of g is unitary.Therefore, Similarly, and so where J 0 (Ψ, A) is the time component of the YMH current (B.6).One may show using (B.5) that For the gauge invariance, recall that U (Ψ) is g-invariant, and that the representations g → ρ g (on V ) and the adjoint representation g → ad g (on g) are unitary.Therefore, to prove invariance under the gauge transformation (B.29), it suffices to show that ∇ ρgA gΨ = g∇ A Ψ, (B.30) We shall use the equation For (B.31), computing in coordinates {x i } and writting F ρgA := (F ρgA ) ij dx i ∧dx j and F A := (F A ) ij dx i ∧dx j , we find where, recall, ∂ i ≡ ∂ x i .Expanding the partial derivative and commutators gives Expanding the product on the second line gives Cancelling terms symmetrical in i and j and simplifying gives as required.
The YMH equations in coordinate form.In coordinate form, the differential form (gauge field) entering the YMH Lagrangian (B.2) is written as A = A i dx i .The local coordinate expression for the curvature is Furthermore, for the covariant derivatives ∇ A and d A , we have We write F ij = F a ij γ a for an orthonormal basis γ a of g and the lower case roman indices run over the spatial components 1, 2, . . ., d.Note that Let Ω be either a bounded domain in R d or R d+1 .In the former case, we assume either periodic or Dirichlet boundary conditions.
Proposition B.3.The Lagrangian and energy for the YMH model are given in coordinates by (with different ranges of indices as mentioned above).The YMH equations are given in coordinates by  on u(2), for which − i 2 τ a , a = 0, 1, 2, 3, (where τ a , a = 1, 2, 3, are the the Pauli matrices together with τ 0 := 1) form an orthonormal basis.It is customary to factor out the coefficient of − i 2 .In coordinates, we write

C The WS equations in coordinate form
We specify equation (B.41) -(B.44) for to the Weinberg-Salam (WS) model, which has the gauge group G = U (2) = SU (2) × U (1).As was mentioned in Appendix A, in this case, there is a slight discrepancy in the definition of the covariant derivative due to the fact that U (2) is not simple, but a (semi-)direct product of the simple group SU (2) and U (1), with each component having a coupling constant, see (C.2)-(C.6).
Using Eqs (C.2)-(C.6),we express the Lagrangian and the energy in coordinates as (with indices ranging from 0 to d and 1 to d, respectively, as mentioned above), and the Euler-Lagrange equations are written in coordinates as We work in a fixed coordinate system, {x i } 3 i=1 and write the fields as W = W i dx i , Z = − i 2 Z i dx i and A = − i 2 A i dx i .We show Proposition D.1.Energy (2.7), written in terms of the fields W, A, Z and φ and coordinates {x i } 3 i=1 , is given by (see also [43]): where where V 3 := Z cos θ + A sin θ and V 3 ij := ∂ i V j − ∂ j V i , with the important property that T (W, A, Z) is invariant under the gauge transformation (3.7).
Proof of (D.1).We proceed by rewriting the terms in the coordinate expression of the WS energy (C.8), in terms of the fields We simplify the matrix representing the connection's action on Φ: In terms of the fields Z, A and W (see equations (3.5) -(3.6) for the definitions of these fields), (D.3) becomes Hence, for Φ = (0, φ), Therefore, the first term of (C.8), written in terms of the fields W, A, Z and φ, becomes The second term of (C.8) becomes For the third term of (C.8), we will use the fact that Tr , where V ij and X ij are defined in (C.5) and (C.6).Furthermore, we have and, with Adding (D.9) and (D.10), using that Since V ij and X ij are Hermitian, Tr V ij V ij and Tr X ij X ij are the sum of the squared absolute values of the matrix coefficients of V ij and X ij , respectively.Thus ) and expanding the first term gives Writing the first line of (D.13) in terms of these fields gives Expanding the first term of the second line, and using Recalling the definition (D.2) of T (W, A, Z) gives Proof of (3.10).Now, we consider the Weinberg-Salam (WS) model in R 2 with fields independent of the third dimension x 3 , and correspondingly choose the gauge with V 3 = X 3 = 0 (and hence W 3 = A 3 = Z 3 = 0).In this case the summation in (D.1) contains only two terms, (ij) = ( 12) and (ij) = ( 21), and we use this to simplify (D.1).We proceed by simplifying the terms of (D.2) and the first line of (D.1); the remaining terms are unchanged.
Replacing corresponding terms in (D.1) -(D.2) with (D.17 Proof of (3.12) -(3.15).We proceed by calculating the (complex) Gâteaux derivatives of (3.10).Let δ # denote the partial (real) Gâteaux derivative with respect to #.Let W z = W + zW ′ , z ∈ C, and define Integrating the first term by parts and factoring out W and W ′ gives For the derivative to be zero for every variation W ′ , (3.12) must hold.
Using A ′ × W = −JW • A ′ in the first two terms, and integrating the last two terms by parts, gives which simplifies to For the derivative to be zero for every variation A ′ , (3.13) must hold.The proof of (3.14) is essentially the same as the proof of (3.13), so we omit it.Let φ s = φ + sφ ′ , s ∈ R. Then Integrating the third term by parts and factoring out 2φ For the derivative to be zero for every variation φ ′ , (3.15) must hold.
E Proof of (9.11) In the proof below, we will use the following result: Lemma E. ) for some k, l ∈ Z. Then for i, j = 1, 2 and p, q = 0, 1, Furthermore, if f s and g s have continuous derivatives of all orders in s, then so does the above integral.
Proof.Equation (E.1) follows from the following chain of inequalities: If f s and g s have continuous derivatives of all orders in s, then their s-derivatives of all orders are in H 2 per .In particular, this means that ∂ k s (f s g s ), k ∈ Z ≥0 , remains integrable, so the s−derivatives of the above integral (obtained by differentiation under the integral sign) are well-defined.
F Proof of (10.3) Proof of (10.3).We shall calculate each term in the integral (4.8) up to order s 6 using Lemma E.1 and the Taylor expansions (9.7).
Plugging the Taylor expansions (9.7) into the first term of (4.8) gives ) remainder terms.R ε has continuous derivatives of all orders because it is a sum of integrals of the form (E.1) with f s and g s coming from the continuously differentiable remainder terms O(|s| p ) of (9.7).
Proof of Proposition 5.4.The self-adjointness of the operator −∆ a n is well-known.To find its spectrum, we introduce the complexified covariant derivatives (harmonic oscillator annihilation and creation operators), ∂a n and ∂ * a n = −∂ a n , with One can redily verify that these operators satisfy the following relations: As for the harmonic oscillator (see e.g.[23]), this gives explicit information about the spectrum of −∆ a n , namely (5.26), with each eigenvalue is of the same multiplicity.Furthermore, the above properties imply (5.27).We find Null ∂a n .A simple calculation gives the following operator equation ∂a n e n 2 (ix 1 x 2 −(x 2 ) 2 ) = ∂ x 1 + i∂ x 2 .
(The transformation on the left-hand side is highly non-unique.)This immediately proves that ∂a n ψ = 0, (G.4) if and only if θ = e − n 2 (ix 1 x 2 −(x 2 ) 2 ) ψ satisfies (∂ x 1 + i∂ x 2 )θ = 0. We now identify x ∈ R 2 with z = x 1 + ix 2 ∈ C and see that this means that θ is analytic and We observe that θ(−z, τ ) = θ(z, τ ) and therefore ψ 0 (−x) = ψ 0 (x).Indeed, using the expression (G.7), we find, after changing m to −m ′ , we find (a) Stability of the emerging solutions.(b) Existence of vortex lattices at | ⃗ b| ≫ b * .(c) Quantum corrections to the values of the classical critical magnetic field b * and the optimal lattice shape parameter τ * .

For ⃗ b = (b 1 , b 2 , b 3 )
̸ = 0, Eqs.(2.10) -(2.11) have the gauge-translation invariant solution given (up to a gauge symmetry) by .23) and (2.24).(For it, Q ⃗ b solves the YM equation d * Q F Q = 0.) Fixing the average magnetic field breaks the full special Euclidean symmetry (i.e.translations and rotations but not reflections) but maintains the special Euclidean symmetry in the plane orthogonal to ⃗ b and the translational symmetry along ⃗ b.Looking for the simplest non-trivial solutions, we consider solutions which do not depend on the coordinate along ⃗ b and look for solutions spontaneously breaking the transversal translational symmetry.With the notation b = | ⃗ b|, we show that for appropriate perturbations: (i) (2.15) is linearly stable for b < b * and unstable for b > b * , where b * := g 2 φ 2 0 /2e; (ii) At b = b * , a new inhomogeneous solution (breaking the gauge-translational invariance) bifurcates, and this solution has the discrete translational symmetry of a lattice in the plane orthogonal to ⃗ b and has lower energy per unit area;

Theorem 2 . 1 . 2 W
where θ is the Weinberg angle defined by cos θ = g √ g 2 +g ′2 .These are the masses of the W, Z and Higgs bosons, respectively (this nomenclature will be explained in the discussion following Eq.(3.10)).Finally, let b = g sin θ.(2.21)With the above definitions, we will prove the following: The gauge-translational invariant solution (2.15) is energetically stable for b < b * and unstable for b > b * .Theorem 2.2.Let L be a lattice satisfying 0 < 1 − M 2π |L| ≪ 1 and assume that M Z < M H .5 Then there exist δ > 0 such that the following holds: (a) Equations (2.10) -(2.11) have an inhomogeneous solution U L ∈ H 2 L in the δ-ball B H 2 L (U ⃗ b * ; δ) in H 2 L around the homogeneous solution (2.15);(b) U L is the unique, up to gauge symmetry transformation, inhomogeneous solution in the δ-ball B H 2 L (U ⃗ b * ; δ); (c) U L has energy per unit area less than vacuum solution (2.15): E

9
Proof of Theorem 2.2(a), (b) ) with, recall, m w := √ n, m z := √ n cos θ and m h := √ 4λn g the masses of the rescaled W, Z and Higgs boson fields, w, z and ϕ, respectively, and

. 6 ) 2 . 2 W 2 W 2 We
Before proving Proposition 9.1, we shall see how it implies statements (a) and (b) of Theorem 2.Proof of Theorem 2.2(a), (b).Since the operator G mz,m h is positivity preserving, the function G mz,m h (|χ| 2 ) is positive for M Z < M H , and hence α mz,m h (τ ) and η mz,m h (τ ) are positive.Furthermore, when the right-hand side of (9.5) is positive, we solve (9.5) for s as a function of b, s = s(b), having continuous derivatives of all orders.When |1 − M eb | ≪ 1, the right-hand side of (9.5) is positive if and only if 1 − M eb > 0. 9 Plugging s = s(b) into (8.1)(i.e.passing from the bifurcation parameter s to the physical parameter b), undoing the rescaling (4.1), and recalling that b * = M , we arrive at the branch, U L ≡ (W b , A b , Z b , φ b ), of solutions of (3.12) -(3.15), which has the properties listed in statements (a) and (b) of Theorem 2.2.The following statement follows from the proof above: Lemma 9.2.U L is continuously differentiable of all orders in b for b in an open right half-interval of b * Proof of Proposition 9.1.

Lemma 11 . 1 .
G ′ (τ, b, v) and ε ′ (τ, b, v) are continuously Gâteau differentiable of all orders in Re(τ ), Im(τ ), b and v. Proof.Since G(b, u) and ε(b, u) have continuous b and u derivatives of all orders, and M τ is a linear map independent of b and v, it follows that G ′ (τ, b, v) and ε ′ (τ, b, v) have continuous b-and v-derivatives of all orders.

27 )
Hence, by (B.19) we have d dt E(Ψ, A) = 0, as required.Gauge symmetries.We define the local action, ρ g A13 , of the group G on A, by the equationd ρgA = gd A g −1 , for all g ∈ C 1 (N, G),where N is either M or Ω.We computeρ g A = gAg −1 + gdg −1 .(B.28)Proposition B.2.The Lagrangian (B.2) is invariant under the Poincaré group and the gauge transformationsT gauge g : (Ψ, A) → (gΨ, ρ g A), ∀g ∈ C 1 (M, G). (B.29)Proof.The invariance under the Poincaré group follows from the definition of this group and the choice of the Minkowski metric on M ⊂ R d+1 .

44 )
41oof.Equations (B.41) and (B.42) follow from the coordinate expressions d A Ψ = ∇ k Ψdx k and F A = F a ij γ a ⊗ dx i ∧dx j , together with the fact that dx k and γ a ⊗dx i ∧dx j form orthonormal bases for Ω 1 and Ω 2 g , respectively.Equations (B.43) -(B.44) follow from equations (B.4) -(B.6) and the coordinate expressions for d A and d * A above.