Multistring Solutions of the Self-Graviting Massive W Boson

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1 Multistring Solutions of the Self-Graviting Massive W Boson Dongho Chae Department of Mathematics Sungkyunkwan University Suwon , Korea chae@skku.edu Abstract We consider a semilinear elliptic system which include the model system of the W strings in the cosmology as a special case. We prove existence of multi-string solutions obtain precise asymptotic decay estimates near infinity for the solutions. As a special case of this result we solve an open problem in [4 Introduction Let λ,,, λ 4 > be given. We consider the following system for (u, η) in. u = λ e η e u + 4π N δ(z z j ), (.) η = λ 3 e η λ 4 e u (.) equipped with the boundary condition e u dx + e η dx <, (.3)

2 where we denoted z = x + ix C =. The system (.)-(.) reduces to the Bogomol nyi type of equation modelling the cosmic strings with matter field given by the massive W boson of the electroweak theory if we choose the coefficients as, λ = m W, = 4e, λ 3 = 6πGm4 W e, λ 4 = 3πGm W, (.4) where m W is the mass of the W boson, e is the charge of the electron, G is the gravitational constant([,[4). The points {z,, z N } corresponds to the location on the (x, x ) plane of parallel (along the x 3 axis) strings. See [4 for the derivation of this system from the corresponding Einsten- Weinberg-Salam theory as well as interesting physical backgrounds of the model. In [4 the construction of radially symmetric solution(in the case z = = z N ) is discussed by further reduction the system into a single equation, solving the ordinary differential equation. When the locations of strings are different to each other, however, we cannot assume the radial symmetry of the solutions, no existence theory is available. In particular, the author of [4 left the construction of solution in this case as an open problem. One of our main purpose in this paper is to solve this problem. Actually, we solve the existence problem for more general coefficient cases as in (.)-(.). The following is our main theorem. Theorem. Let N N {}, Z = {z j } N be given in allowing multiplicities. Then, there exists a constant ε > such that for any ε (, ε ) any c > there exists a family of solutions to (.)-(.3), (u, η). Moreover, the solutions we constructed have the following representations: u(z) = ln ρ I ε,a ε (z) + ε w (ε z ) + ε v,ε(εz), (.5) η(z) = ln ρ II ε,a ε (z) + ε w (ε z ) + ε v,ε(εz), (.6) where the functions ρ I ε,a(z), ρ II ε,a(z) are defined by with ρ I ε,a(z) = ρ II ε,a(z) = f(z) = (N + ) 8ε N+ f(z) ( + ε N+ F (z) + a ), (.7) ε N+ c ε 4 ( + ε N+ F (z) + a λ4 (.8) ) ε N+ N (z z j ), F (z) = z f(ξ)dξ (.9) for k =,, ε > a = a +ia C. The smooth radial functions, w, w in (.5) (.6) respectively satisfy the asymptotic formula, w ( z ) = C ln z + O(), w ( z ) = C ln z + O() (.)

3 as z, where C = C = c λ λ 4 (N + )( + λ 4 )( + λ 4 ), (.) [ C λ 4 c λ λ 4 = if λ λ 4 λ 3 =, (N + )( + λ 4 )( + λ 4 ) C λ 4 (λ ( λ 4 λ 3 )c B (N + ) N +, λ 4 ) if N + > N + λ 4 (.) with the beta function(euler s integral of the first kind) defined by B(x, y) = t x ( t) y dt. x, y > (See [3.) The function v,ε, v,ε in (.5) (.6) respectively satisfy sup z v,ε(εz) + v,ε(εz) ln(e + z ) o() as ε. (.3) emark.. In the physical model of the cosmic strings of W boson, (.4), we note that the first case of (.) holds(λ λ 4 λ 3 = ), we have C > as well as C >. Thus, we have extra(additional) contributions from the second terms of to the decays of u η in (.5) (.6) respectively. Proof of Theorem. We note that for any ε > a C, ln ρ I ε,a(z), is a solution of the Liouville equation. N ln ρ I ε,a(z) = ρ I ε,a(z) + 4π δ(z z,j ). (.) We consider the following equation for ρ II a,ε(z) From (.) we have [ ln ρ I a,ε(z) ln ρ II a,ε(z) = λ 4 ρ I a,ε(z). (.) N ln z z j = ρ I a,ε(z). (.3) Combining (.) with (.3), we obtain [ N {λ 4 ln ρ I a,ε(z) ln z z j 3 ln ρ II a,ε(z) } =,

4 from which we derive ln ρ II a,ε(z) = λ 4 [ ln ρ I a,ε(z) N ln z z j + h(z), where h(z) is a harmonic function. Choosing h(z) as the constant, h(z) λ ( 4 ln ε 4 λ N ) λ 4 λ 4 [8(N + ) λ c 4, we get the form of ρ II a,ε(z) given in (.8). We set g I ε,a(z) = ε ρi ε,a define ρ (r) ρ (r) by ( z, g ε) ε,a(z) II = ( z ε 4 ρii ε,a, ε) ρ (r) = 8(N + ) r N ( + r N+ ) = lim ε gi ε,(z), ρ (r) = c ( + r N+ ) λ 4 = lim g II ε ε,(z) respectively. We transform (u, η) (v, v ) by the formula u(z) = ln ρ I ε,a(z) + ε w (ε z ) + ε v (εz), (.4) η(z) = ln ρ II ε,b(z) + ε w (ε z ) + ε v (εz), (.5) where w w are the radial functions to be determined below. Then, using (.), the system can be written as the functional equation, P (v, v, a, ε) = (, ), where P (v, v, a, ε) = v + λ g II a,ε(z)e ε (w +v ) + g I ε,a(z) ε (e ε (w +v ) ) + w, (.6) P (v, v, a, ε) = v + λ 3 gε,a(z)e II ε (w +v gε,a(z) I ) + λ 4 (e ε (w +v ) ) + w ε. (.7) Now we introduce the functions spaces introduced in [. For α > the Banach spaces X α Y α are defined as X α = {u L loc( ) ( + x +α ) u(x) dx < } 4

5 equipped with the norm u X α = ( + x +α ) u(x) dx, Y α = {u W, loc ( ) u u(x) X α + + x + < } α L ( ) equipped with the norm u Y α = u X α + u(x) + x + α following propositions proved in [. L (. We recall the ) Proposition. Let Y α be the function space introduced above. Then we have the followings. (i) If v Y α is a harmonic function, then v constant. (ii) There exists a constant C > such that for all v Y α v(x) C v Yα ln(e + x ), x. Proposition. Let α (, ), let us set where We have where we denoted L = + ρ : Y α X α. (.8) ρ(z) = ρ( z ) = 8(N + ) z N ( + z N+ ). KerL = Span {ϕ +, ϕ, ϕ }, (.9) ϕ + (r, θ) = rn+ cos(n + )θ + r N+, ϕ (r, θ) = rn+ sin(n + )θ + r N+, (.) Moreover, we have ϕ = rn+. (.) + rn+ ImL = {f X α fϕ ± = }. (.) Hereafter, we fix α =, set X 4 = X Y = Y. 4 4 Using Proposition. (ii), one can check easily that for ε > P is a well defined continuous mapping from B ε into X, where we set B ε = { v Y + v Y + a < ε }, for sufficiently small ε. In order to extend continuously P to ε = the radial functions w (r), w (r) should satisfy w + ρ w + λ ρ = (.3) w + λ 4 ρ w + λ 3 ρ = (.4) For the existence asymptotic properties of w w we have the following lemma, which is a part of Theorem.. 5

6 Lemma. There exist radial solutions w ( z ), w ( z ) of (.3)-(.4) belonging to Y, which satisfy the asymptotic formula in (.),(.),(.). Proof: Let us set f(r) = ρ (r). Then, it is found in [ that the ordinary differential equation(with respect to r), w + C ρ w = f(r) has a solution w (r) Y given by { r φ f (s) φ f () w (r) = ϕ (r) ds + φ } f()r (.5) ( s) r with ( ) + r N+ ( r) φ f (r) := r N+ r r ϕ (t)tf(t)dt, where φ f () w () are defined as limits of φ f (r) w (r) as r. From the formula (.5) we find that r w (r) = ϕ (r) as r, where ( ) + s N+ I(s) ds + (bounded function of r) s N+ s s I(s) = λ ϕ (t)tρ (t)dt. Since ϕ (r) as r, the first part of (.) follows if we show I = I( ) = λ ϕ (r)rρ (r)dr = C. Changing variable r N+ = t, we evaluate I = λ ϕ (r)ρ (r)rdr [ r N = c λ = = c λ (N + ) c λ (N + ) ( + r N + ) 3+ λ 4 [ + λ 4 ( + t) 3+ λ 4 dt r 4N+ ( + r N + ) 3+ λ 4 t ( + t) 3+ λ 4 ( ) ( ) + λ 4 + λ 4 rdr dt = c λ λ 4 (N + )( + λ 4 )( + λ 4 ) = C. (.6) In order to obtain C we find from (.3) (.4) that (λ 4 w w ) = ( λ λ 4 + λ 3 )ρ, 6

7 from which we have w (z) = λ 4 w (z) + λ λ 4 λ 3 π = λ 4C ln z + λ λ 4 λ 3 π ln( z y )ρ ( y )dy [ ρ ( y )dy ln z + O() (.7) as z. In the case λ λ 4 λ 3 =, we have C = λ 4C. In the case λ 4 >, we compute the integral as follows. N+ r ρ ( y )dy = πc dr ( + r N+ ) λ 4 = πc t N N+ N + = πc N + B dt (r N+ = t) ( + t) λ 4 ( N +, λ 4 N + where we used the formula(see pp. 3[3) for the beta function x µ dx = B(µ, ν µ), where ν > µ. ( + x) ν ), (.8) Substituting (.8) into (.7), we have w (z) = C ln z +O() as z, where C is given by (.). This completes the proof of Lemma. Now we compute the linearized operator of P. By direct computation we have ga,ε(z) I g I a,ε(z) lim = 4ρ ϕ +, lim = 4ρ ϕ, ε a ε a a= a= ga,ε(z) II g II a,ε(z) lim = 4ρ ε a ϕ +, lim = 4ρ ε a ϕ. a= a= Let us set P u,η,a(,,, ) = A. Then, using the above preliminary computations, we obtain A [ν, ν, α = ν + ρ ν 4( w ρ + λ ρ )(ϕ + α + ϕ α ), A [ν, ν, α = ν + λ 4 ρ ν 4(λ 4 w ρ + λ 3 ρ )(ϕ + α + ϕ α ). We establish the following lemma for the operator A. 7

8 Lemma. The operator A : Y C + defined above is onto. Moreover, kernel of A is given by KerA = Span{(, ); (ϕ ±, λ 4 ϕ ± ), (ϕ, λ 4 ϕ )} {(, )}. Thus, if we decompose Y C = U KerA, where we set U = (KerA), then A is an isomorphism from U onto X. In order to prove the above lemma we need to establish the following. Proposition.3 I ± := (λ w ρ + λ ρ )ϕ ± dx. (.9) Proof: In order to transform the integrals we use the formula [ L = (N + ) r 4N+ 6( + r N+ ) ( + r N+ ), 4 N Z + which can be verified by an elementary computation. Using this, we have the following π { (λ w ρ + λ ρ )ϕ r N+ cos ±dx = ( w ρ + λ ρ ) (N + )θ ( + r N+ ) sin (N + )θ [ 8(N + ) r N r N+ = π ( + r N+ ) w + λ ρ ( + r N+ ) rdr [ { } = π L w ( + r N+ ) + λ ρ r N+ rdr ( + r N+ ) [ = π Lw ( + r N+ ) + λ ρ r N+ rdr ( + r N+ ) [ ρ = πλ c ( + r N+ ) + ρ r N+ rdr ( + r N+ ) = πλ c r N+ rdr = πλ c t N+ dt (r = t) ( + r N+ ) + λ 4 4 ( + t N+ ) + λ 4 [ = πλ c t N+ t N+ dt + dt 4 ( + t N+ ) + λ 4 ( + t N+ ) + λ 4 (Changing variable t /t in the second integral,) [ = πλ c t N+ ( t N+ )t λ 4 dt + dt 4 ( + t N+ ) + λ 4 ( + t N+ ) + λ 4 = πλ c 4 (t N+ )( t λ 4 ) dt <. ( + t N+ ) + λ 4 8 } rdrdθ

9 This completes the proof of the proposition. We are now ready to prove Lemma.. Proof of Lemma.: Given (f, f ) X, we want first to show that there exists (ν, ν ) Y, α, α such that which can be rewritten as Let us set A(ν, ν, α, α ) = (f, f ), ν + ρ ν 4( w ρ + λ ρ )(ϕ + α + ϕ α ) = f, (.) ν + λ 4 ρ ν 4(λ 4 w ρ + λ 3 ρ )(ϕ + α + ϕ α ) = f. (.) α = f 4I + ϕ + dx, α = f 4I ϕ dx, (.) where I ± is defined in (.9). We introduce f by Using the fact f = f α ϕ + α ϕ. (.3) π ϕ + ϕ dθ =, (.4) we find easily f ϕ ± dx =. (.5) Hence, by (.) there exists ν Y such that ν + ρ ν = f. Thus we have found (ν, α, α ) Y satisfying (.). Given such (ν, α, α ), the function ν (z) = ln( z y )g(y)dy + c π, (.6) where g = f λ 4 ρ ν + 4(λ 4 w ρ + λ 3 ρ )(ϕ + α + ϕ α ), c is any constant, satisfies (.), belongs to Y. We have just finished the proof that A : Y X is onto. We now show that the restricted operator(denoted by the same symbol), 9

10 A : (KerL Span{}) X is one to one. Given (ν, ν, α, α ) (KerL Span{}), let us consider the equation, A(ν, ν, α, α ) = (, ), which corresponds to ν + ρ ν 4( w ρ + λ ρ )(ϕ + α + ϕ α ) =, (.7) ν + λ 4 ρ ν 4(λ 4 w ρ + λ 3 ρ )(ϕ + α + ϕ α ) =. (.8) Taking L ( ) inner product of (.7) with ϕ ±, using (.9), we find α = α =. Thus, (.7) implies ν KerL. This, combined with the hypothesis ν (KerL) leads to ν =. Now, (.8) is reduced to ν =. Since ν Y, Proposition. implies ν = constant. Since ν (Span{}) by hypothesis, we have ν =. This completes the proof of the lemma. We are now ready to prove our main theorem. Proof of Theorem.: Let us set U = (KerL Span{}). Then, Lemma. shows that P (v,v,α) (,,, ) : U X is an isomorphism. Then, the stard implicit function theorem(see e.g. [5), applied to the functional P : U ( ε, ε ) X, implies that there exists a constant ε (, ε ) a continuous function ε ψ ε := (v,ε, v,ε, a ε) from (, ε ) into a neighborhood of in U such that P (v,ε, v,ε, a ε) = (, ), for all ε (, ε ). This completes the proof of Theorem.. The representation of solutions u, u, the explicit form of ρ I ε,a (z), ρii ε ε,a (z),, together with the asymptotic behaviors of w, w described in Lemma., the fact that v,ε, v,ε ε Y, combined with Proposition., implies that the solutions satisfy the boundary condition in (.3). Now, from Proposition. we obtain that for each j =,, v j,ε(x) C v j,ε Y (ln + x + ) C ψ ε U (ln + x + ). (.9) This implies then v j,ε(εx) C ψ ε U (ln + εx + ) C ψ ε U (ln + x + ).cxxc From the continuity of the function ε ψ ε from (, ε ) into U the fact ψ = we have ψ ε U as ε. (.3) The proof of (.3) follows from (.9) combined with (.3). This completes the proof of Theorem.

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