On a classical solution to the Abelian Higgs model
OOn a classical solution to the Abelian Higgsmodel
N. Mohammedi ∗ Institut Denis Poisson (CNRS - UMR 7013),Universit´e de Tours,Facult´e des Sciences et Techniques,Parc de Grandmont, F-37200 Tours, France.
Abstract
A particular solution to the equations of motion of the Abelian Higgs model is given.The solution involves the Jacobi elliptic functions as well as the Heun functions. ∗ e-mail: [email protected] a r X i v : . [ h e p - t h ] J a n Introduction
The Abelian Higgs model is described by the Lagrangian L = − F µν F µν + ( ∂ µ φ (cid:63) − ieA µ φ (cid:63) ) ( ∂ µ φ + ieA µ φ ) − λ (cid:0) ϕ (cid:63) ϕ − v (cid:1) . (1.1)Here F µν = ∂ µ A ν − ∂ ν A µ is the field strength of the gauge field A µ and φ is the complexscalar field. We take v > λ > φ as φ = ρ e iθ . (1.2)The Lagrangian (1.1) becomes then L = − F µν F µν + e ρ (cid:18) A µ + 1 e ∂ µ θ (cid:19) (cid:18) A µ + 1 e ∂ µ θ (cid:19) + ∂ µ ρ ∂ µ ρ − λ (cid:0) ρ − v (cid:1) (1.3)and the local gauge symmetry is A µ −→ A µ + 1 e ∂ µ Λ , θ −→ θ − Λ , ρ −→ ρ . (1.4)One could use this gauge freedom to set the field θ to zero. Nevertheless, we will keep θ through out.The equations of motion for the Abelian Higgs model are ∂ µ (cid:101) F µν + 2 e ρ (cid:101) A ν = 0 , (1.5) ∂ µ ∂ µ ρ + λ ρ (cid:0) ρ − v (cid:1) − e ρ (cid:101) A µ (cid:101) A µ = 0 , (1.6) ∂ µ (cid:16) ρ (cid:101) A µ (cid:17) = 0 . (1.7)We have defined the gauge invariant variable (cid:101) A µ = A µ + 1 e ∂ µ θ . (1.8)and (cid:101) F µν = F µν = ∂ µ (cid:101) A ν − ∂ ν (cid:101) A µ . The last equation (1.7) corresponds to the field θ and is alsoa consequence of (1.5). Notice that we have expressed the equations of motion in terms ofthe two gauge invariant variables (cid:101) A µ and ρ . The space-time coordinates are x µ = (cid:0) x , x , x , x (cid:1) = (cid:0) x , (cid:126)x (cid:1) and the indices are raised and loweredwith the metric g µν = diag (1 , − , − , − V µ = (cid:0) V , V , V , V (cid:1) = (cid:16) V , (cid:126)V (cid:17) . The complex scalar field is multi-valued as φ = ρ e iθ = ρ e i ( θ +2 πN ) with N ∈ Z . In this note we assumethat ∂ µ ∂ ν θ − ∂ ν ∂ µ θ = 0. ρ is frozenat one of the two minima of the potential energy. That is, ρ = v . In this case the equationsof motion become ∂ µ (cid:101) F µν + 2 e v (cid:101) A ν = 0 , (1.9) (cid:101) A µ (cid:101) A µ = 0 , (1.10) ∂ µ (cid:101) A µ = 0 . (1.11)These equations have, for instance, the plane wave solution (cid:101) A µ = ε µ cos ( p ν x ν + w ) , ε µ ε µ = ε µ p µ = 0 , p µ p µ = 2 e v . (1.12)The polarisation vector ε µ has two independent components for a given wave vector p µ . Themass-shell relation p µ p µ = 2 e v is that of a massive particle with a positive mass squared.The other known solution is the ”kink” solution for which (cid:101) A µ = 0. The equations ofmotion come then to the single equation ∂ µ ∂ µ ρ + λ ρ (cid:0) ρ − v (cid:1) = 0 . (1.13)This is solved by ρ = ± v tanh ( p µ x µ + w ) , p µ p µ = − λv . (1.14)Here we have a mass-shell condition of a relativistic particle of negative mass squared.In this note, we will look for a solution which satisfies the gauge invariant condition (cid:101) A µ (cid:101) A µ = 0 . (1.15)The equations of motion reduce then to ∂ µ (cid:101) F µν + 2 e ρ (cid:101) A ν = 0 , (1.16) ∂ µ ∂ µ ρ + λ ρ (cid:0) ρ − v (cid:1) = 0 , (1.17) ∂ µ (cid:16) ρ (cid:101) A µ (cid:17) = 0 . (1.18)We notice that the equation of motion for the scalar field ρ decouples from the rest. We willreport here on a non-trivial solution obeying the condition (1.15). The second equation (1.17) admits the ”kink” solution as written in (1.14). We would likehere to find the gauge field corresponding to it. We assume the following form for the gaugefield (cid:101) A µ : (cid:101) A µ ( w ) = ε µ h ( w ) . (2.1)Here (and in the rest of the paper) we will use the notation w = p µ x µ + w , p = p µ p µ (2.2)3ith w a constant. The four-vector p µ obeyes the mass-shell relation p = − λv . Thepolarisation vector ε µ is required to satisfy ε µ ε µ = p µ ε µ = 0 . (2.3)Equations (1.15) and (1.18) are then automatically obeyed.Substituting (2.1) into (1.16) results in the differential equation d hdw ( w ) − e λ tanh ( w ) h ( w ) = 0 . (2.4)By the change of variables z = tanh ( w ) (2.5)one transforms the differential equation (2.4) into (cid:0) − z (cid:1) d hdz ( z ) − z dhdz ( z ) + (cid:20) l ( l + 1) − m (1 − z ) (cid:21) h ( z ) = 0 . (2.6)This differential equation is known as associated Legendre equations [1, 2, 3]. It is, in general,singular at the points z = ± , ±∞ . In the case at hand, the variable z is real and lies inthe domain [ − , +1] and the real constants l and m are defined as l = − ± (cid:114)
14 + 4 e λ , m = 4 e λ . (2.7)The general solution to (2.6) is h ( z ) = a P ml ( z ) + b Q ml ( z ) , (2.8)where P ml ( z ), Q ml ( z ) are associated Legendre functions [1, 2, 3] of the first and second kind,respectively. The two constants a and b are arbitrary.When z is real and belonging to the interval [ − , +1], the associated Legendre function P ml ( z ) is real and is expressed as [1, 2, 3] P ml ( z ) = 1Γ (1 − m ) (cid:18) z − z (cid:19) m F (cid:18) − l, l + 1; 1 − m ; 1 − z (cid:19) , (2.9)where F ( a, b ; c ; x ) is the hypergeometric function and Γ ( ν ) is the gamma function. Similarly,the associated Legendre function Q ml ( z ) is given by Q ml ( z ) = π mπ ) (cid:20) cos ( mπ ) P ml ( z ) − Γ ( l + m + 1)Γ ( l − m + 1) P − ml ( z ) (cid:21) . (2.10)To summarise, a particular solution to the equations of motion of the Abelian Higgsmodel (1.1) is given by φ = ± v tanh ( w ) e iθ ,A µ = − e ∂ µ θ + ε µ [ a P ml (tanh ( w )) + b Q ml (tanh ( w ))] ,p = − λv , ε µ ε µ = p µ ε µ = 0 . (2.11) Greek indices ν and µ are usually used instead of l and m . θ ( x ) is any arbitrary smooth function (obeying ∂ µ ∂ ν θ − ∂ ν ∂ µ θ = 0). The gaugefield A µ has two independent polarisations for a given vector p µ . Furthermore, the condition ε µ ε µ = ε − (cid:126)ε = 0 implies that the solution carries both electric and magnetic fields (since ε = 0 implies (cid:126)ε = (cid:126) φ in terms of the Jacobi elliptic func-tions The equation of motion for a phi-to-the-four scalar field theory is given by ∂ µ ∂ µ ρ + λ ρ (cid:0) ρ − v (cid:1) = 0 . (3.1)We will look for solution which depend on the single variable w = p µ x µ + w . That is, ρ = ρ ( w ). The equation of motion becomes thend ρ d w + λp ρ (cid:0) ρ − v (cid:1) = 0 . (3.2)Notice that any solution to this equation is obviously subject to the following remark:if ρ (cid:0) w , p (cid:1) is a solution then ρ (cid:18) s w , p s (cid:19) is also a solution , (3.3)where s is a constant. Therefore, the mass-shell relation, p , will be determined up to aconstant.It is well-known that equation (3.2) is solved by the twelve Jacobi elliptic functions[1, 2, 3]. Indeed, The scalar field ρ ( w ) satisfies the first order differential equation (cid:18) d ρ d w (cid:19) = − λ p ρ (cid:0) ρ − v (cid:1) + av (3.4)with av a constant of integration. By writing ρ ( w ) = v ε y ( w ) , (3.5)where ε is a constant, one obtains the first order differential equation (cid:18) d y d w (cid:19) = − λv ε p y + λv p y + aε . (3.6)The twelve Jacobi elliptic functions are obtained as solutions to this last equation [1, 2, 3]for different choices of the three constants p , ε and a (and boundary conditions). The tablein Appendix B gather these values. For instance, the solution given in terms of the “sine”Jacobi elliptic function sn( x , m ) corresponds to the choice a = ε = 2 m m ,p = − λv m (3.7)5nd the differential equation (3.6) takes the form (cid:18) d y d w (cid:19) = (cid:0) − y (cid:1) (cid:0) − m y (cid:1) . (3.8)The general solution [1, 2, 3] to this last equation is y ( w ) = sn( w + d , m ), where d is anarbitrary constant. Hence our scalar field ρ is given by ρ ( w ) = ± v (cid:114) m m sn( w + d , m ) , p = − λv m . (3.9)The parameter m must be different from zero here. The mass-shell relation p = − λv m is that of a particle with negative mass squared. It is worth mentioning that the “kink”solution (1.14) corresponds to m = 1 assn( w + d ,
1) = tanh ( w + d ) . (3.10) The gauge field, A µ = ε µ h ( w ) with ε µ ε µ = p µ ε µ = 0, is determined by the differentialequation d h d w ( w ) + 2 e p ρ ( w ) h ( w ) = 0 . (4.1)For the moment, the expression of p is not fixed. This allows one to treat all possiblemass-shell conditions at the same time.Let us assume that h ( w ) = h ( Z ) , Z = 1 τ ρ ( w ) , (4.2)where τ is a constant to be properly chosen later. Then by using the fact that the field ρ ( w )satisfies the two equations d ρ d w = − λp ρ (cid:0) ρ − v (cid:1) , (cid:18) d ρ d w (cid:19) = − λ p ρ (cid:0) ρ − v (cid:1) + av (4.3)we arrive at the differential equationd h d Z + (cid:18) γZ − δ − Z − (cid:15)k − k Z (cid:19) d h d Z + ( s + αβk Z ) Z (1 − Z ) (1 − k Z ) h = 0 . (4.4)6he different constants are given by s = 0 ,δ = γ = (cid:15) = 12 ,α = α ± = 14 (cid:32) ± (cid:114) e λ (cid:33) ,β = β ∓ = 14 (cid:32) ∓ (cid:114) e λ (cid:33) ,τ = 2 k (1 + k ) v . (4.5)They satisfy the relation γ + δ + (cid:15) = α ± + β ∓ + 1 . (4.6)We also have the mass-shell relation p = − λv k a (1 + k ) . (4.7)The constant a will be fixed later.The equation (4.4) is a Fuchsian differential equation and its general solution is given by h ( Z ) = C Hn (cid:18) k , α + , β − , ,
12 ; Z (cid:19) + C Hn (cid:18) k , α − , β + , ,
12 ; Z (cid:19) ,Z = 1 τ ρ ( w ) = (1 + k )2 k v ρ ( w ) , (4.8)where Hn ( k , s ; α, β, γ, δ ; z ) is the Heun function and C C ρ ( w ) is any solution to theequation of motion (3.2), that is, any of the twelve Jacobi elliptic functions.When, for instance, the scalar field ρ ( w ) is expressed in terms of the Jacobi ellipticfunction sn ( w + d, m ) in (3.9) then the expression of p there has to match that written in(4.7). This leads to the identifications a = 2 k (1 + m )(1 + k ) = 2 m (1 + m ) = ⇒ k = m or k = 1 m . (4.9)If we choose k = m , for example, then, using (3.9) and (4.8), the two fields of the Abelian Sometimes the notation a = 1 /k and q = − s/k is used. A useful note on Heun’s functions and furtherreferences can be found in [4]. φ = ± v (cid:114) m m sn( w + d , m ) e iθ ,A µ = − e ∂ µ θ + ε µ (cid:20) C Hn (cid:18) m , α + , β − , ,
12 ; sn ( w + d , m ) (cid:19) + C Hn (cid:18) m , α − , β + , ,
12 ; sn ( w + d , m ) (cid:19) (cid:21) ,p = − λv m , ε µ ε µ = p µ ε µ = 0 . (4.10)The two constants α ± and β ± depend on the Abelian Higgs parameters and are listed in(4.5). The field θ ( x ) is arbitrary.Finally, when m = 1 one has sn( w + d ,
1) = tanh ( w + d ), and the particular solutiongiven in (2.11) is recovered by converting the Heun functions into hypergeometric functionswith the help of the relation [4] Hn (1 , s ; α, β, γ, δ ; z ) = (1 − z ) r F ( r + α, r + β ; γ ; z ) ,r = ξ − (cid:112) ξ − αβ − s , ξ = ( γ − α − β ) . (4.11)We have represented in Figure 1 the Jacobi elliptic function sn( x,
2) and in Figure 2 theHeun function Hn (cid:0) , , − , , ; sn ( x, (cid:1) . xK4 K2 0 2 4K0.4K0.20.20.4 Figure 1: A sketch of the Jacobi elliptic function sn( x ,
K4 K2 0 2 41.011.021.031.041.051.061.071.08
Figure 2: A sketch of the Heun function Hn (cid:0) , , − , , ; sn ( x, (cid:1) for (cid:113) e λ = 2. We have presented classical solutions to the equations of motion of the Abelian Higgs model(1.1). The complex scalar field φ and the gauge field A µ are given by φ = ± v ε pq( w + d , m ) e iθ ,A µ = − e ∂ µ θ + ε µ (cid:20) C Hn (cid:18) k , α + , β − , ,
12 ; Z (cid:19) + C Hn (cid:18) k , α − , β + , ,
12 ; Z (cid:19) (cid:21) ,Z = ε (1 + k )2 k pq ( w + d , m ) ,w = p µ x µ + w , ε µ ε µ = p µ ε µ = 0 . (5.1)Here pq( w + d , m ), with pq any pair of the letters (c,d,n,s), is one of the twelve Jacobielliptic functions and Hn ( k , s ; α, β, γ, δ ; z ) is the Heun function. We have listed in table 1the values p (the mass-shell relation) and ε for each function pq( w + d , m ). Table 1 givesalso the expression of k in terms of the parameter m entering the Jacobi elliptic functionpq( w + d , m ). Finally, α ± = (cid:18) ± (cid:113) e λ (cid:19) and β ∓ = (cid:18) ∓ (cid:113) e λ (cid:19) .Notice that the differential equation (4.1) is linear in h ( w ). Therefore, the gauge field A µ is in fact a linear combination of all the polarisation vectors ε µ satisfying ε µ ε µ = p µ ε µ = 0.Finally, the solution (5.1) could have been expressed in terms of the Weierstrass ellipticfunction as shown in Appendix A.What remains to be studied, from the physical point of view, is of course the stabilityand the interpretation of the solutions presented here. The most sought configuration inthe Abelian Higgs model is certainly the vortex solution [5] as it has direct applications incondensed matter physics and cosmology [6, 7]. Since no analytic vortex solution is known,the present paper might give hints on how to look for these solutions. The other facet ofthe present article is also mathematical. It shows how some special functions might enterphysical contexts like the Abelian Higgs model.9 PPENDICES
A Solutions to φ in terms of the Weierstrass ellipticfunction The Weierstrass elliptic function is a general solution to the first order differential equation[1, 2, 3] (see also [8] for some lectures on the suject) (cid:18) d ℘ d z (cid:19) = 4 ℘ ( z ) − g ℘ ( z ) − g = 4 ( ℘ − e ) ( ℘ − e ) ( ℘ − e ) . (A.1)The constants g and g are known as the lattice invariants and e i , i = 1 , , z − g z − g = 0.In order to obtain a Weierstrass differential equation in the case of φ theory, we startfrom equation (3.6) (cid:18) d y d w (cid:19) = − λv ε p y + λv p y + aε (A.2)and multiply both sides with 4 y to get (cid:18) d f d w (cid:19) = − λv ε p f + 4 λv p f + 4 aε f ,f ( w ) ≡ y ( w ) . (A.3)The next step is to get rid of the term proportional to f . This is achieved by the change offunctions f ( w ) = g ( w ) + 23 1 ε . (A.4)The resulting differential equation is given by (cid:18) d g d w (cid:19) = − λv ε p g + 4 ε (cid:18) λv p + a (cid:19) g + 83 1 ε (cid:18) λv p + a (cid:19) . (A.5)In order to obtain a Weierstrass differential equation, we choose the constant ε such that p = − ε λv . (A.6)This leads to (cid:18) d g d w (cid:19) = 4 g + 4 ε (cid:18) ε a − (cid:19) g + 83 1 ε (cid:18) ε a − (cid:19) , = 4 (cid:20) g − ε (cid:18)
13 + √ − ε a (cid:19)(cid:21) (cid:20) g − ε (cid:18) − √ − ε a (cid:19)(cid:21) (cid:20) g + 23 1 ε (cid:21) . (A.7)10he solution to this differential equation is given by the Weierstrass elliptic function g ( w ) = ℘ (cid:18) w + d ; − ε (cid:18) ε a − (cid:19) , −
83 1 ε (cid:18) ε a − (cid:19)(cid:19) , (A.8)where d is a constant. Recalling that ρ ( w ) = vεy ( w ), the square of our scalar field ρ ( w ) isfinally given by ρ ( w ) = v (cid:20)
23 + ε ℘ (cid:18) w + d ; − ε (cid:18) ε a − (cid:19) , −
83 1 ε (cid:18) ε a − (cid:19)(cid:19)(cid:21) = v (cid:20)
23 + ℘ (cid:18) ε ( w + d ) ; − (cid:18) ε a − (cid:19) , − (cid:18) ε a − (cid:19)(cid:19)(cid:21) ,p = − ε λv . (A.9)In the last equality we have used the homogeneity relation [1, 2, 3, 8] ℘ ( z ; g , g ) = µ − ℘ (cid:0) µ − z ; µ g , µ g (cid:1) . (A.10)Therefore we could have used the Weierstrass elliptic function (instead of the Jacobielliptic functions) to express the solution to the Abelian Higgs model given in (5.1). In thiscase, the identification of the two expressions of p in (A.9) and (4.7) leads to ε a = 4 k (1 + k ) . (A.11)Finally, we should mention that the Weierstrass elliptic function could be converted intoJacobi elliptic functions [1, 2, 3, 8]. B The twelve Jacobi elliptic functions Solutions to φ (cid:18) d y d w (cid:19) = − λv ε p y + λv p y + aε (B.1)and gives the corresponding values of the parameters p (the mass-shell relation), ε and a (see also [9, 10] and [11] for some particular cases).It is important to notice that ε has to be strictly positive (for ρ ( w ) to be a real fieldand different from zero). This, consequently, puts restrictions on the allowed values of theparameter m for some of the solutions. 11 ε aε y ( w ) ρ ( w ) k or k − λv m m m w, m ) ± v ε sn ( w + d, m ) m − λv − m − m − m − m cn ( w + d, m ) ± v ε cn ( w + d, m ) m − m λv − m − m m − w + d, m ) ± v ε dn ( w + d, m ) 1 − m − λv m m m w, m ) ± v ε cd ( w + d, m ) m λv m − m (1 − m )2 m − w, m ) ± v ε sd ( w + d, m ) m − m λv − m − m )2 − m − w, m ) ± v ε nd ( w + d, m ) 1 − m − λv m m m dc ( w, m ) ± v ε dc ( w + d, m ) m λv m − − − m )2 m − − m nc ( w, m ) ± v ε nc ( w + d, m ) m − m λv − m − − m )2 − m w, m ) ± v ε sc ( w + d, m ) 1 − m − λv m m m ns ( w, m ) ± v ε ns ( w + d, m ) m λv m − − m − − m (1 − m ) ds ( w, m ) ± v ε ds ( w + d, m ) m − m λv − m − − m − m cs ( w, m ) ± v ε cs ( w + d, m ) 1 − m Table 1: The twelve Jacobi elliptic functions solutions to (B.1) and (3.2) and their corre-sponding parameters p (the mass-shell relation), ε and aε . The table gives also the relationbetween k and m appearing in (5.1). C Solutions to φ in terms of trigonometric and hyper-bolic functions We have seen that the equation of motion for the field ρ ( w ) is given by the differentialequation d ρ d w + λp ρ (cid:0) ρ − v (cid:1) = 0 . (C.1)The general solution to this equation is expressed in terms of the twelve Jacobi ellipticfunctions depending on a parameter m . On the other hand, the Jacobi elliptic functionsreduce to ordinary trigonometric or hyperbolic functions for the two special values m = 0and m = 1 [1, 2, 3].There are twelve different trigonometric and hyperbolic functions corresponding to thevalues m = 0 and m = 1. However, only five of them are solutions the above differential The other seven functions, for m = 0 and m = 1, lead either to ρ ( w ) = 0 or to a complex ρ ( w ). Forexample, ds( w + d ,
1) = w + d ) but ε = − m = 1, as can be seen from table 1. This leads to acomplex scalar field ρ ( w ). ρ ( w ) = ± v sn [ τ ( w + α ) + β , ± v tanh [ τ ( w + α ) + β ] , p = − λ v τ , (C.2) ρ ( w ) = ± v ns [ τ ( w + α ) + β , ± v tanh [ τ ( w + α ) + β ] , p = − λ v τ , (C.3) ρ ( w ) = ±√ v cn [ τ ( w + α ) + β , ±√ v cosh [ τ ( w + α ) + β ] , p = λ v τ , (C.4) ρ ( w ) = ±√ v ds [ τ ( w + α ) + β , ±√ v sin [ τ ( w + α ) + β ] , p = − λ v τ , (C.5) ρ ( w ) = ±√ v dc [ τ ( w + α ) + β , ±√ v cos [ τ ( w + α ) + β ] , p = − λ v τ . (C.6)The solutions depend on a parameter τ and a constant τ α + β . However, sometimes thesesolutions are found under different writings [12, 13, 14, 15]. For instance, the first solution(C.2) can be expressed as ρ ( w ) = ± v tanh [ τ ( w + α ) + β ] = ± v µ + tanh [ τ ( w + α )]1 + µ tanh [ τ ( w + α )] , µ = tanh ( β ) . (C.7)Other expressions can be reached by simply expanding the arguments of the trigonometricand hyperbolic functions. References [1]
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