Topological solitons in the Weinberg-Salam theory

PHYSICA ELS I~'VI ER

Physica D 101 (1997)55-94

Topological solitons in the Weinberg-Salam theory * Yisong Yang 1 Department ~[Applied Mathematics and Physics, Polytechnic University, Broaklyn, NY 11201, USA

Received 22 April 1996; revised 4 September 1996; accepted 5 September 1996 Communicated by C.K.R.T. Jones

Abstract We establish the existence of multivortices arising in the self-dual phase of the standard model of Weinberg-Salam, and its two-Higgs-doublet extension, in the unified theory of electromagnetic and weak interactions. For the standard model, we prove the existence of solutions in a periodic lattice domain and study the effect of the gravitational coupling. We find an important connection of these self-dual vortices and the cosmological constant problem: the cosmological constant A may be written explicitly in terms of several fundamental parameters in electroweak theory and the two-dimensional surface on which the vortices reside becomes noncomplete. We then prove that such a gravitational background leads to the existence of finite-energy vortices on the full plane with a non-Abelian nature. For the extended electroweak model with two Higgs doublets, we solve the self-dual equations completely. For the periodic case, we prove an existence and uniqueness theorem under a necessary and sufficient condition. This result reveals some exact restrictions to the vortex charges. For the problem on the entire plane, we obtain existence, uniqueness, sharp decay estimates, and quantized fluxes. 1991 MSC: 35Q, 81E

1. Introduction The purpose of this paper is to present a fairly complete existence theory for the vortex-like solutions, in the critical B o g o m o l ' n y i phase, of the classical W e i n b e r g - S a l a m model (see the collected works edited by Lai[17]) and its two-Higgs-doublet extension unifying the electromagnetic and weak interactions. Solutions of such a nature give rise to multi-centered physical force concentrations which may be viewed as a many-particle system realized in two dimensions known as vortices [11,12,22,25-27]. These vortices are characterized by their local winding numbers or vortex charges and are topologically or energetically stable. In the ideal situations, vortices localize energy distributions and behave like classical solitons in interactions. In the context of cosmology, these topological solitons, known as cosmic strings [5,7,9,10,15,16,19,33,34,36,37], can also generate curvature fluctuations, apart * Supported in part by the National Science Foundation under grant DMS-9400243. I E-mail: [email protected]. 0167-2789/97/517.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII SO I 67-2789(96)002 12-6

56

Y. Yang/Physica D 101 (1997) 55-94

from the fluctuations of other forces, along a planar cross-section of space-time so that the vortices may serve as centers for matter accretion, around which, galaxy formation in the early universe might take place. This paper contains the following results. (I) Existence of doubly periodic multivortices realizing quantized Z-fluxlines in addition to quantized electromagnetic fluxes found earlier. These vortices arise from a Bogomol'nyi type "dual" system of elliptic equations in the Weinberg-Salam electroweak model, in which both self-dual and anti-self-dual structures are present. (II) The derivation of an expression that relates the cosmological constant to several fundamental parameters in the electroweak theory and the Newton constant of gravitation. This result shows in particular, in the context of cosmic strings, that the 2-surface that accommodates gravity is noncomplete. Therefore, the gravitational metric must decay sufficiently rapidly at infinity. Such a property enables us to prove the existence of nonAbelian multivortices with finite energies and nontrivial topology. (III) A complete resolution of the Bogomol'nyi multivortex problem in the extended electroweak model with two Higgs particles. There are two subcases: doubly periodic solutions and finite-energy solutions in the full plane. For the former, we establish an existence and uniqueness theorem under a necessary and sufficient condition. For the latter, we establish existence, uniqueness, sharp decay estimates, and flux quantizations. The presence of static doubly periodic vortex-like solutions in the Weinberg-Salam model was originally obtained by Ambjorn and Olesen [ l ] in their study of the production of electroweak strings through W-condensation. (See also [20,24] and references therein.) In the seminal work of Ambjorn and Olesen [1 ], the residual U(l)em-unitary gauge leads to a quantized magnetic flux but the weak Z-flux becomes trivial. Recently, Bimonte and Lozano [3] extended the Ambjorn-Olesen framework to accommodate quantized Z-fluxline lattices. However, their setting does not contain the Ambjorn-Olesen equations in the classical unitary gauge limit when the complex Higgs doublet becomes a real scalar particle. Motivated by the work of Bimonte-Lozano, we generalize the Ambjorn-Olesen self-dual equations to a system that allows quantized Z-fluxes as well as the original Ambjorn-Olesen quantized electromagnetic fluxes induced from their W-condensates, yet, contains the classical unitary gauge equations as a limit. The novelty of our system is that both self duality (characterized by W being a holomorphic section) a n d anti-self-duality (characterized by a complex component of the Higgs field being an anti-holomorphic section) are present at the same time. In Section 2, we first derive this new Bogomol'nyi type equations. As in the AmbjornOlesen case, these equations are fulfilled by solutions that saturate a topological energy lower bound. Under some conditions, we are able to prove by using the multiconstrained variational principle in [29] the existence of doubly periodic solutions realizing quantized magnetic flux and Z-flux at the same time. Our conditions for existence are necessary and sufficient when the vortex number N ---- 1,2. These solutions may be viewed as electroweak strings generated by both W- and Higgs condensations. The second topic of this paper, the study of electroweak model under the influence of gravity, is motivated mainly by the desire to produce finite-energy non-Abelian static vortices. The absence of such solutions in non-Abelian gauge theory is a typical situation. The work of Vachaspati [31,32] showed that one may embed the Nielsen-Olesen vortices of the Abelian Higgs model [22] into the electroweak model. However, the instability of the embedded vortices in physically desirable parameter regimes [ 13,14] has led people to attempt revisions of the standard model. Here we take the point of view that the presence of gravity will crucially alter the picture as was already witnessed earlier in the works of Bartnik and McKinnon [2], Smoller et al. [28], Chen et al. [6], and Yang [39-41]. Our goal now is to prove the existence of finite-energy electroweak vortices arising in the Ambjorn-Olesen framework (with the generality here containing the Z-fluxlines or the Higgs condensation) in a gravitational background in the Bogomol'nyi phase for the full range of physical parameters. In Section 3, we first briefly remark that there is no finite energy for our generalized W-condensate system over the full plane when gravity is absent. We then study the problem of how much difference, at least, we should expect for the electroweak model when gravity is present from the situation when there is no gravity. This goal will be remarkably achieved by a calculation of the

1~ Yang/Physica D IOl (1997)55-94

57

energy-momentum tensor of the electroweak model. In fact, this calculation allows us to find an explicit dependence of the cosmological constant on the Newton constant, the Weinberg mixing angle, the coupling constant, and the energy scale of broken symmetry for the Higgs field. It is interesting to compare this result with the result derived without the Z-fluxes in [38] in the classical unitary gauge along the line of the work of Ambjorn-Olesen. This result indicates that the cosmological constant could be significantly positive in an electroweak epoch. In such a situation the gravitational metric must decay sufficiently rapidly at infinity. In Section 4, we obtain decay estimates for the metric which will be used later to find multiwmices. We find that the governing system arising from the coupled Einstein equations and the dual equations in the electroweak theory may be reduced into a 2 x 2 system with superimposed exponential type nonlinearities. In Section 5, we prove the existence of multivortices in the Weinberg-Salam electroweak model under the influence of a gravitational metric. These solutions are of a nonAbelian nature and all carry finite energies. We show that when the conditions for existence are satisfied, we can prescribe the decay rates and the magnetic and Z-fluxes of the solutions. Thus there is also nonuniqueness. Various generalized electroweak models have been the focus of many latest studies. The common feature of these extensions is that more than one Higgs doublets are introduced so that it may be possible to obtain truly non-Abelian vortices in flat space-time. It has been argued by physicists that electroweak models with at least two-Higgs multiplets may arise from supersymmetric or supergravity grand unified theories and stability may be reached in physically relevant parameter regimes. Such a scenario has already tbund experimental support in particle physics laboratories 18,18]. Here we present a complete solution for the recently discovered self-dual equations by Bimonte and Lozano [4] governing two electroweak Higgs doublets, which may be viewed as a minimal extension of the standard model that allows a Bogomol'nyi phase. In Section 7, we first study the equations over a periodic lattice domain. We obtain necessary and sufficient conditions for the existence of multivortex solutions. The method is a constrained variational principle. In Section 8, we obtain existence, uniqueness, asymptotic decay, and quantized values of fluxes of multivortices in the full plane. The method is through the study of a convex variational functional. We mention that Perivolaropoulos [231 has conducted numerical and asymptotic analysis for some two-Higgs systems under radial symmetry ansS.tze. The comparison of the results obtained here for the W - and Higgs-condensate equations in the classical WeinbergSalam model and its two-Higgs-doublet extension of Bimonte-Lozano is interesting. In the doubly periodic setting, the conditions for existence for the former and latter are rather different. For the former, they imply that the area of the domain to confine multivortices with a given vortex number N must not be either too small or too large. While for the latter, there is no restriction to the upper bound of the area of a cell domain. Larger domains always accommodate more vortices, A common feature is that in both situations the solutions are minimizers of the energy functional and fluxes are quantized. Another is that the conditions for existence are independent of the locations of vortices but only depending on the total topological charge N. In the full plane setting, the tk)rmer does not allow the existence of finite-energy Bogomol'nyi type vortices. In order to get finite-energy solutions, it is important to put into consideration the effect of gravity through the coupling of the Einstein equations. Now the metric noncompleteness leads to a sufficiently rapid decay of the metric at infinity and the existence of finite-energy multivortices may be ensured. Our conditions for existence involve the vortex number N again as in the doubly periodic case and nonuniqueness often occurs. Besides, the fluxes can assume continuous values from some explicitly given intervals. For the latter, there exists a unique finite-energy solution for any prescribed vortex distribution and total topological charge N. There is no freedom to designate the values of fluxes: They are all quantized and represented by the vortex number N. It is also remarkable that these values are of the same form as those from an N-vortex solution in a periodic lattice cell. It is hoped that this part of the work would also lend hints to more generalized or realistic electroweak models.

58

Y.

Yang/Physica D 101 (1997) 55-94

2. Dualequationsandperiodicsolutions We begin our study of the standard electroweak model of Weinberg and Salam in the two-dimensional critical Bogomol'nyi phase. In Section 2.1, we introduce the model in its general form. In Section 2.2, we set up the model in the unitary gauge. In Section 2.3, we formulate the topological periodic boundary condition which is crucial for the quantization of various fluxlines. In Section 2.4, we derive in a natural way, by minimizing the energy, a new system of first-order equations which governs a periodic family of static vortices known as condensates. These equations contain the self-dual equations of Ambjorn and Olesen [ 1] as a special limiting case and are suggested by the recent work of Bimonte and Lozano [3]. The most interesting feature is that, beyond the range of real-valuedness for the second component of the Higgs doublet in the classical unitary gauge, the reduced Higgs scalar is now allowed to assume complex values and nontrivial winding around the boundary of a fundamental lattice cell. As a consequence, the Z-field may have a quantized flux as the magnetic field. Another interesting feature is that both self-duality and anti-self-duality are present as an underlying structural part of the equations. In Section 2.5, we reduce the system into a coupled pair of nonlinear elliptic equations. In Section 2.6, we calculate integral constraints associated with the vortex solutions. In Sections 2.7 and 2.8, we obtain our existence theorem. These two-dimensional periodic solutions are naturally characterized by the zeros of the W- and Higgs-fields. For clarity, we will use P in this section to denote the electromagnetic gauge potential as well as the field strength (after the Weinberg mixing) since it reminds us that the resulting interactions are mediated by photons. 2.1. General f r a m e w o r k

As usual, the Higgs scalar 4~ is a complex-doublet for which the isospin group SU(2) assumes its fundamental representation and the hypercharge group U ( I ) acts as diagonal 2 × 2 matrices. We u s e {Z'a}a=l.2,3 tO denote the Pauli matrices 2"1 =

(0 10) 1

,

r2

=

(0-i) i

0

'

l"3 =

(10 01) --

"

It is easily checked that the generators t, = it,,, a = 1,2, 3, of SU(2) satisfy the commutation rules [ta, tb ] = i~abctc.

We define the transformation rules of SU(2) x U ( I ) on q~ by 4~ w-~ exp(-iwata)qL

coa 6 1~,

a --- 1,2, 3,

¢ w-~ exp(-i~t0)qL

se 6 I~,

where, as mentioned already, we choose

l(,0

t0=~

as a generator of U ( 1) acting on ¢. Eventually we will concentrate on two-dimensional systems which are descendants of the model in the physical (3 ÷ 1)-dimensional Minkowski space-time. Assume that the signature of the Minkowski space-time is ( - + + ÷ ) . It is convenient to use the notation in [29] to denote the SU(2) and U(1) gauge fields by A u = Aauta and B~, respectively. Of course, both A~ and B u are real 4-vectors. The gauge-invariant field strength tensors and the mixed S U (2) × U (1) gauge-covariant derivative are given by Fu~ = OuAv - OvA u + ig[A u, A~,],

G~,, = OuBv - OrB u,

D~d? = Ou4) + igAa~ta49 + ig'Butodp,

Y. Yang/Physica D IOl (1997) 55-94

59

where g, g' > 0 are fixed coupling parameters. Using " t " to denote the Hermitian conjugate and neglecting the fermionic sector, we write down the Lagrangian density of the electroweak theory of Weinberg-Salam: ] llv £ = -~(F

.

Fur + Gl~VG,v) + (DUg)) * • (Du4)) + X(~0(~- ~b~q~) 2,

where Z and ¢P0 are positive constants.

2.2. Generalized unitao" gauge symmet~ We need to specify the unitary gauge to isolate the physics. For this purpose, introduce the real vector fields P, and Z u by the rotation relation P, = B, c o s 0 + A u3 sin 0,

Z u = - B , sin 0 + A 3u cos 0.

Consequently the gauge-covariant derivative takes the new form

D, = O, + ig(a~q + a~#2) + iPu(g sin Or3 + g' cos0to) + iZ~(g cos0t3 - g' sin Oto). Identify the field Pu with the electromagnetic potential (or the photon). We must require that the coefficient of Pu be the charge operator eQ = e(t3 + to) with - e being the electron charge. Thus we are led to the definition of the Weinberg angle 0: e = gsin0=

g ' COS O,

cos0-

g (g2 + g , 2 ) 1 / 2 '

In the generalized unitary gauge, the complex Higgs doublet 4' takes the form 0 where 0 is a complex scalar field. In the standard unitary gauge, ¢ is real. Such a choice prohibits any phase jump of ~ around the boundary of a lattice cell, which leads to a vanishing Z-fluxline in the cell domain. The crucial difference of the study here from that in [29] is to allow ¢ to become complex so that its phase change around the boundary of a cell region may give rise to quantized Z-fluxlines. It is straightforward to find that, in terms of ¢. we have the representation

Define the operators d . , D . (bundle connections), and the new fields W., P~,~, Z m, by the following expressions: g d, = 3 , - i 2 c o s 0 Z , , 1

W, = - ~ ( A .

I

•

D u = 3~ - igA~ = 3tz - ig(P, sin0 + Z u cos0), "~

+ IA~),

Puv = 3t~,Pv - 3vP.,

Zuv = ate,Z,, - 3,,Z..

Thus the original Lagrangian density becomes

£=½(D"W"-D~WU)t(D,W.-D~W.)+4=

l_71~v 7~ , , ' + ¼ P ' v P m '

I 2 t 2 - [WUWu][W"Wv] *) + ig(Z uv cos0 + p,V sinO)W~W,, + ~g ([W ~ W,]

I 2 1¢~12 W ,u w ,+ + (d'Ugr)+(du¢) + X((p2 -- 1¢12) 2. + ~g

(I)

Y. Yang/Physica D lO1 (1997) 55-94

60

It is seen in (1) that, at the classical level, the bosonic interactions of the electromagnetic and weak forces are described in terms of the fundamental Pu, Wu, and Z u fields. The unitary Higgs field ~0, whose physical existence is yet to be confirmed experimentally, serves as a link between these interactions. To obtain multivortices, we go to the ansatz that the temporal components of the vector fields are zero, i.e., Po = Wo = Zo = 0, the "magnetic" excitations are in the third direction, i.e., P3 = W3 = Z3 = 0 and Pj, Wj, Zj (j = 1, 2), ~p depend only on the spatial coordinates x I x 2, and Wj (j = 1,2) are represented by a complex scalar field W, i.e., Wi = W, W2 = i W. Therefore, the energy density of the vortex-lines calculated from (1) is written £ = [DI W + iD2W[ 2 + / P ? 2 + 1222 - 2g(g12 cos0 + PI2 sin 0)[W[ 2 + 2g2lW[ 4 + I dj~p[ 2 q- g2[~[2[W12 + ~.([~r[ 2 - ~o02)2.

(2)

The energy function (2) has an apparent residual U ( I ) gauge symmetry ~p w-~ e x p ( - i ~ ) ~ , 1

Pj w-~ Pj + - ( 1 + 2cos20)Oj~, e

W ~ exp(i~)W,

Zj ~ Zj

2 cos 0

g

(3)

aj~,

where se depends only on the coordinates x l, x 2 which may of course be shown to come from a suitable gauge symmetry in the original S U(2) × U(1) model.

2.3. The periodic bounda~, condition We need to get ready to derive the self-dual equations and quantized fluxlines in a periodic lattice cell. To this end, consider a doubly periodic region in ~2, say ~ , where ,.(2 ~---{x --- ( x l , x 2) G [~2]x = s i a l

-+- sZa2,0 < s l , s 2 < 11.

Here al and a2 are linearly independent vectors in 0~2. The boundary of ,f2 is given by 0~(2 = FI U F2 t.-J{al + F2} t-){a2 +-F'l}(-){O, al,a2, al + a 2 } , where

Fk={xE~2lx=skak,

O < s k < 1},

k=l,2.

In view of the gauge symmetry (3), the periodic boundary condition on 0S2 is stated as: (exp(i~k)W)(x + ak) = (exp(i~k)W)(x),

(exp(-i~k)~0)(x + ak) = (exp(-i~k)~)(x),

( Pj + - e(' 1 Zj

+ 2cos20)Oj,~k g

) (x +ak) = ( Pj + - e(' 1

Oj~k ( x + a k ) =

xcFIUF2-Fk,

k=

Zj

g

+ 2cos20)Oj~;k

Oj~k (x),

1,2,

where ~l, ~2 are real-valued functions defined in a neighborhood of F'2U{al -k-F2} respectively.

and

FI tA{a2+/-'l},

)

(x),

(4)

Y. Yang/PhysicaD I01 (1997)55-94

61

As in [29], denote the dependence of ~k on x = sial + s2a2 by ~k = ~k(Sl,S2). Since both ~p and W are single-valued, the total phase jump resulted from ~l, ~,_ as one travels around 3S2 must be an integral multiple of 27r:

~l(1,1 )-~t(l,0+)+~(0,0 +)-~(0,1-)

(5)

+ ~e2(0+, 1) - ~2( 1-, 1) + ~2(1 - , 0) - ~2(0 +, 0) + 27r N = 0. Hence

we have the flux conditions

• ,, = f Pi2dx = f Pj dxJ----27rN (le f2

+ 2 c°s2 0)'

(6)

0s2

• z=fZlzdx=fZ/dxJ-$2

4rrN cos0. g

812

(7)

2.4. Dual equation,; First, it is straightforward to verify the useful identities Idjgr ± id2~l 2 = I d t ~ l 2 + [ d2~Pl 2 ~ i(dl~k[d2~] + --[ dial+ d2~),

~{OJ(~Jk~+dk~P)}=

i ( d l ~ [ d 2 ~ ] t - [dl~P]t d2~P)

2 c og s 0 1~12Zi2.

Here and in the sequel, we use 3(.) or :if(.) to denote the imaginary or real component of a complex quantity. Therefore, after some manipulation, we obtain under the following Bogomol'nyi type criticality condition g2 ;~ -

-

-

(8)

8 cos 2 0

on the mass of the Higgs particle the energy function

g 2sin

g=IDIW+iD2W[2+~ 1 ( PI2

'(.,.

0 ~p°-2gsinOlW]2)2

g o(si~---~PI2 ' - co'o',2)

'

+ 2

2cos0

+ ~ g2 8 sin2 0qo4.

--idz~IZ--3{Oj(~jk~¢dk~r)}

+]dl~

(9)

Integrating (9) over f2 and using the gauge-periodic boundary condition (4) and (5), we find the lower bound

f

..o(,

Edx > ~

si-~-~-~p -

-

~2

= -

, )

cos 0

@z

gqOo[27rN_ 2 ~

~ se~ n 0 (1 q- 2cos2 0) +

sin 2 0

3~N-

~o21nl

,

..

8 sin 2 0

47)

~p~IE2i

g2 ~p4]n I 8sin20 (10)

Y. Yang/PhysicaD lOt (1997) 55-94

62

which is larger than that in the standard unitary gauge. Such a lower bound is attained if and only if the following first-order dual equations subject to the periodic boundary condition (4) are fulfilled: D I W + i D 2 W = 0,

dl~-id2~

PI2 -- 2 g~ s i °2 n ~ + 2g sin0 IWP2,

=0, Z]2 - cos--~ 2 g (1~12 _ ~p2) + 2g COS0 IW] 2.

(ll)

The third equation in (1 1) implies that the magnetic fluxlines are everywhere along the positive direction of the x3-axis: P12 > 0. Hence N ~ 0. In other words, the system prohibits the existence of trivial solutions (N = 0). Thus the P- and Z-fluxes can never be zero. The first (or second) equation in ( 11 ) says that the complex function W (or ~p) is holomorphic (or anti-holomorphic) in $2 modulo a nonvanishing factor. In two-dimensional gauge field theory, the reduction to a system of first-order equation in which the complex scalar field (frequently viewed as an order parameter) becomes a holomorphic (or anti-holomorphic) section with respect to the connection induced from a gauge potential is commonly referred as self-dual (or anti-self-dual). Thus, in this context, both self duality and anti-self duality are present in (11) by the properties of W and ~. Consequently, it is proper to call (11) simply a dual system. The analyticity of W and ~" indicates that both of them have finitely many zeros in S2 with integral multiplicities. For simplicity, assume that these functions do not vanish on aS2. Due to the boundary condition on the phase change, the condition (5), we see that both tb and W have exactly N zeros, say p], .. -, PN and q], • • -, qN, with possible algebraic multiplicities. We will combine the first-order system (11) into an equivalent second-order nonlinear elliptic system with two unknowns. We will see that some integral constraints then appear naturally.

2.5. Elliptic system reduction From the first two equations in (1 1), we have, away from the zeros of W and ~ , the relations - 2 i 0 ' In W = g sin 0(PI + iP2) + gcosO(Zi + iZ2), g -2iO In ~ -- - (Zj - iZ2), 2 cos 0

(12) (13)

where we have adopted the notation

o* --- ½(or + io2).

0 -- ½(O, - i a 2 ) ,

Note also that, with these operators, we have the convenient representation for the Laplacian A = 02 + 02 = 400*. Thus, differentiating the Eqs. (12) and (13), we obtain

1 gsinOPlz+gcosOZl2 = - = Z l l n l W ] 2

2,

g

--ZI2 cos 0

= Aln[~

[2.

Substituting these results into the remaining equations in (1 1), we arrive, away from the zeros of W and ~p, at a new system g2 zlln[~'[ 2 -

2cos20(l~12-q92)+2g21W[ 2,

AlnlWf 2=-g2]~f2-4g21WP2.

It is convenient to set u = In I~pl2, w = In JW] 2. Then the system becomes

Y. Yang/Physica D 101 (1997) 55-94

I, 2 COS2 0

-t- 2eW

63

2 cos 2 0 ~02 + 4zr Z 61'J" ./=1

N Au, ------g2e" - 4g2e u' + 47r ~ ~qj, j=l XEf2.

(14)

Note that u, w are doubly periodic in ~2 modulo the fundamental cell I2. In such a situation, it is also useful to view u, w as defined on the 2-torus R2/I2 - S2. Let uo and w0 be functions over the 2-torus S2 satisfying 47rN IS21

zauo-

U

+4rr Z3t~i'j=l zawo--

4rrN

IS?l

U

q-4YrZ3q/'.j=l

Setting u = uo + vt and w = wo + v2, the Eqs. (14) become At~l = g 2 ( ~ Ul ] e AV 2 = -g2UleVl

v] + 2U2e v2)

g2

2cos2 0

~ + 4- -r r N ~o~

IS?l

4rrN

_ 4g2U2eV2 + - -

Is?l

(15) '

.rEG. Here Ui = e "° and U2 = e v° are smooth functions. The problem now is reduced to a solution of (15). It will be instructive to derive some of its constraints first.

2.6. Constraints Integrating the two equations in (15) over the 2-torus ~2, we easily find the integral constraints g2 tan2 0

f

Ute v~ dx -

g20~o21~1-

cos 2

12rrN,

(16)

£2 4g 2 sin20 [ U2e v2 dx = 4zrN(1 + 2 cos2 0) - g2~ogLI2I.

(17)

£2 Above results also indicate that the physical parameters g, O, tS21, ~oo,and N must satisfy the necessary condition gZ~o21nl 4rr(l + 2 cos 2 0)


gZ~o21n I

(18)

12rr cos 2 0"

Suppose that vl, v2 solve (15). Then u = u0 + vl, w = w0 + v2 solve (14) over the 2-torus I2. As in [29], we would like to recover a solution of the original system self-dual equations (11) subject to the boundary conditions (4) and (5) by setting N ~p(x) = exP(½U(X) - iOp(X)),

Op(X) = ' ~ arg(x -

pj),

j=l U

W(x) = exP(½W(X) + iOq(X)),

(gq(X) = y ' ~ arg(x - q j ) , .j= 1

(19)

64

Y. Yang/Physica D I01 (1997) 55-94

and letting Pj and Zj be determined through solving (12) and (13) in terms of the definition (19). It is easily seen that the consistency is ensured when pj = qj, j = 1, • • •, N. This last condition is a basic constraint that we will observe in this section. 2. 7. Existence results

We will find a solution of the system (11), observing the constraints studied in the Section 2.6. Thus, we will look for a solution quartet 0P, W, Pj, Z j ) under the boundary condition (11) so that ~p and W vanish exactly at a given set of N points p], • •., PN. A solution of such a feature is called an N-vortex solution. The integer N is the winding number of ~p or W on ~I2 and, hence, is of a topological nature. We have the following theorem. Theorem 2.1. Let Y2 be a cell domain and pl, " " , PN be any given N points (with possible multiplicities) in 12. There is a doubly periodic N-vortex solution (Tt, W, Pj, Z j ) to (11) satisfying the boundary conditions (4) and (5) so that both ~p and W vanish precisely at Pl, • • •, PN, only if the inequalities (18) are fulfilled. If, in addition to (18), the total vortex number N observes the condition

47rN(1 + 3cot20) < 8zr +

g2go21K21 sin 2 0

(20)

then there exists an N-vortex solution verifying all aforementioned properties. Corollary 2.2. Condition (18) is necessary and sufficient for the existence of an N-vortex solution of the system (11) w h e n N = 1,2. 2.8. Proofs

To prove Corollary 2.2, we rewrite (20) in the form 4rr(N-2)

< c o t 2 0 \ { g2tpzly21 ~0

127rN)

From the right-hand side of (18), it is seen that (20) is automatically fulfilled when N -----1, 2. To prove Theorem 2.1, note that we can choose u0 = w0. Hence U] = U2 --- U. Let w~ = 2v~ + v2 and w2 = v2. Then v] = ½(wl - w2) and v2 = w2. Eqs. (15) become Awl = g2 tan z 0 Ue (~'t-w2)/2 _ H,

Aw2 = -gZUe(Wl-w2)/2

_

4zrN 4gZUeW2 + [~---T'

(21)

where H -- g2~°~ cos 2 0

127rN > 0 IS21

in view of (18). This system can be identified with (5.4) in [29] with minor differences. Consequently, following [29], Eqs. (21) have a solution if the function x(q, E) defined there satisfies x ( 1 , 0 ) > 0. In other words, 4zrN - H l ~ l c o t 2 0

< 8re,

which is the desired sufficiency condition (20).

Y. Yang/PhysicaD I01 (1997) 55-94

65

3. Cosmic strings and the cosmological constant We will consider the effect of gravitation to the electroweak vortices. The original motivation of such a study is to produce vortices in full [~2. As remarked in Section 1, there have been some earlier works to embed the Nielsen-Olesen vortices in the electroweak model. The embedding makes some parameters lie outside physically relevant regimes and the solutions obtained are of an Abelian nature. It is natural to ask whether we can use (1 1) to get full space solutions. Unfortunately this does not work because the finite-energy condition is violated in R 2. Such a phenomenon makes us look for what is missing. We find that, if we consider the gravitational sector, the existence of finite-energy multivortices will be automatically ensured. The coupled electroweak and gravitational interactions to be studied in this section will, of course, yield strong influence to each other. The electroweak sector affects the gravitation through its energy-momentum tensor while gravity affects the electroweak sector through its distortion of the underlying space-time metric. In our two-dimensional setting, we are in the context of cosmic strings. In this section, we study the influence of the electroweak vortices on the gravitational sector. We show that a positive value of the cosmological constant may be derived in the model. Later we will see that it is this result that leads to the existence of multivortices in the full [R2 (equipped with the induced gravitational metric). We now state the main result of this section.

Theorem 3.1. We assume that the space-time is uniform along the time axis x ° and a vertical direction, say x 3, and the non-trivial geometry arising from gravity is coded on a conformally Euclidean 2-surface M = ( ~ 2 e,13#.). More precisely, the space-time = [~1,1 × M and the metric is given by gu" dxlz dxv ------( dx0)2 "+- (dx3)2 + e°[( dxl )2 Jr- (dx2)2], where the conformal exponent ~/ depends only on (xl, x2) 6 [~2 (the string ansatz). Then coupling the Einstein equations with the Weinberg-Salam electroweak model in the Bogomol'nyi phase (8) via the W- and Higgscondensate equations similar to (1 1) leads to a positive cosmological constant which is explicitly determined by several fundamental parameters in the electroweak model. In Section 3.1, we show through a simple calculation that any Bogomol'nyi dual solutions over [~2 necessarily carry infinite energy. In Section 3.2, we show that the coupling of the gravitational sector via the Einstein equations leads to a positive cosmological constant (the proof of Theorem 3.1) which inevitably implies a finite geodesic distance to infinity from any local region (the noncompleteness of metric). As mentioned earlier, an important consequence of this property is that, in such a background, there are W- and Higgs-condensate electroweak vortices with finite energies. Naturally, these solutions are governed by the curved-space version of the dual system ( 1 1).

3. I. Infinite vortex energy in the full plane We shall consider the energy density of a dual vortex solution (~p, W, Pj, Wj) of (11). Thus, neglecting the kinetic and potential energy terms of the scalar field ~p and W, we have the lower estimates for (2): i 2 i 2 C ~>_5P12 + ~ZI2 -- 2g(Zl2cOsO + PI2 sin0)lWl 2 + 2glWl 4 + g21~121WI 2 =

g~ o 0~ ,~ + 2g sin 0 [W12)2 + 1 2l (2- - sin

-

- 2glWI 2 (sin0 [22 -sig ~ p0o

(2c~sO(l~.12~p~))+2gcos(91Wl2)2

-t- 2g sin 01WI 2] + cos0 [ ~ ( l ~ P l

2 - ~Po) + 2gc°sOIWI2])

66

Y. Yang/Physica D 101 (1997) 55-94

÷ 2g2[W] 4 ÷ g2[~[2[W[2 > -

g2q94

> 0.

(22)

8 sin 2 0

Consequently, the total energy per unit length of the vortex-lines E =

f E=

e~.

Note. The inequality (22) only says that a solution of (11) over the full plane will carry infinite energy. It is already known that there are finite-energy vortex solutions in the electroweak model such as those embedded NielsenOlesen vortices [31,32]. Whether there exist finite-energy non-Abelian vortices is an interesting open question. The complication with the electroweak model is that the topological winding of the Higgs field at spatial infinity is no longer sufficient in distinguishing solutions. Other fields, such as the W-field treated as an order parameter in our study here, may give rise to vortices as well.

3.2. Presence of the gravitational sector." Cosmic strings In this section we will show that the presence of gravity realized by a cosmic string metric gives us a unique opportunity to obtain finite-energy electroweak vortices with nontrivial topology in the full space. This special feature arises essentially from non-Abelian gauge theory because the Abelian Higgs model in the Bogomol'nyi phase necessarily leads to the vanishing of the cosmological constant and the metric becomes complete when the string number is not too large [39,40]. In this situation the metric may not decay rapidly enough to ensure finite energy for any electroweak vortices. Our result is another indication that the effect of gravity may often lead to something that is not expected when gravity is absent. Similar situations are already known in other studies relating gravity to gauge fields. We may recall the work of Bartnik and McKinnon [2] and Smoller et al. [28] on the existence of static finite-energy regular solutions of the coupled Einstein and S U (2) Yang-Miils equations in three dimensions. It has long been known that there is no such regular solution in the pure Yang-Mills sector when gravity is absent. Another recent work is Ref. [40], where it is shown that the presence of gravity confines the range of the vortex charges for the Abelian Higgs model. This phenomenon is more distinct in the compact case when the underlying 2-manifold is a Riemann surface M. The gravity sector relates the total gravitational curvature of M to the vortex number N which implies that the topology of M can only be that of the 2-sphere S 2. Furthermore, the energy scale ~P0 > 0 of the broken symmetry must assume a quantized spectrum of values in order to ensure the existence of a solution. Thus these results show that no matter how weak the gravity is, its presence often leads to significantly different phenomena globally. Suppose that g~,~ is the Riemannian metric of a four-dimensional space-time with signature ( - + + ÷ ) . Although we are actually interested in cosmic string solutions for which the gravitational metric guy takes the diagonal form stated in Theorem 3.1, we need to assume a general form of guy in order to calculate the energy-momentum tensor. Therefore, in such a metric background, the Lagrangian action density of the electroweak gauge-matter sector now takes the modified scalar form

£ = gg =

g

¼g~,. g t

ruvru, v, + ag put

1

puvpu,v, + ~g

g ~/£t

g

t~uvt~#v, + gUV(Du~b)t - ( D ~ ) + X(~p2 - ~bt~b)2 Vpt ~

~

P

t

z.vLu,v, + ½gUU gVV (DuW v _ DvWu)(Du, Wv, _ Dv, Wu,)t

1 2 [(g tar W/zWv) t 2 -(g + g U V ( d u ~ ) ( d v ~ ) t +½g2gUVWuWtv]~]2+~g

+ iggUU'gVV(Zu,v, cos0 + Pu'v' sinO)Wtu Wv + X(l~p]2 _ ~o2)2,

Iz#'

WuWu,)(g vv' W~W~,) t ] (23)

Y. Yang/Physica D IOl (1997)55-94

67

where we have inserted the fields ~r, Pu, Zu, and Wu in place of q~, A~, and B~,. To obtain the energy-momentum tensor T~,, of the gauge-matter sector from/2, we vary the metric gt, v in (23). Then

Tl,,' = g " ¢ Pl,~,' P,,v + gl,'v' Zt,, , Z~, + 2!)t {g~'v' (Du Wl~, - Dl~, Wt, )(D,, Wv, - D,,, W,, )t} + 2:}t{( dz, ~ ) ( d y e ) ~} + g2 (Iqzl2 + 2g z''v' wj,, WvT,):~{w~, W,+,} + ig cosOgU'V'[Z,,,,(Wtv, Wv - WtvWv,) + Z,,,v(W+.,W,, - W,,, W~)I + ig sin OgU'~"[Pl,,~(W~,W~ - WfW~,,) + Py~(W~, W,, - W,,, Wz~)]

- 2g2!)t{(g ~'¢ Wu, W~,)W~ W~} - gu~£.

(24)

The gauge-matter sector determines the gravitational metric through the Einstein equations

G l~, - Agll,, = -87rGT~v,

(25)

where GI~, = RI~ v - ½guvR is the Einstein tensor with R and Ru. the scalar curvature and the Ricci tensor of the metric guv. On the other hand, varying the gauge and matter fields Pu, Zl~, WI~. ~ in (23), we obtain the Yang-Mills equations (26)

8 f £ = 0 !

defined in a space-time background with the metric gu~. The coupled equations (25) and (26) describe a unified model of the gravitational, electromagnetic, and weak interactions at the classical level, and, in general, are difficult to solve. Fortunately, we will show that there is a dual structure as in Section 2, which allows us to make a remarkable reduction from (25) and (26). In particular, the detailed structure of (26) is of little concern here. The dual reduction of (25) and (26) occurs also at the critical coupling (8) of the Higgs potential energy density. To see this, suppose now the space-time metric has the property stated in Theorem 3.1. In this situation, the fields PI~, Zu, Wj,, ~r are as prescribed in Section 2 for vortices. Therefore 8" =

T0{} = / 2

I ,~-2r/D2 I ,=-2r/72 = ~,~ - 1 2 q- ~,., ~12+e-Oldj~r12+e-2OlDiW+iD2WI

2

- 2ge-2"(Zi2 cos0 + PI2 sin0)lW[ 2 + 2gZe-2'l[W] 4 + g2e-'llCrl2lW]2 + X(1612 - {P0)2

-l°-'l

= 2~

g~°2 e° - 2gsinOlWI 2 2sin0

PI2

+

'

-

+ e - 2 O I D I W + i D 2 W ] 2 + e - ' l l d l ~ - id2~l 2 - e O~{Oj(6jk~fdk~)} + g~o2 e - o

T-

(

1

s~noel2

1 Z~2 cos 0

i

g-~oo . 8 sin -) 0

(27)

Thus we obtain the curved-space version of the dual equations D1W + i D 2 W = 0,

dlO - idz~r = 0. (28)

g 0 {pZeo +2gsinOIWI 2, PI2 - 2 sin

g O_(l~l 2 - {p2)eO + 2gcosOlWI 2. Zl2 - 2cos

68

Y. Yang/Physica D I01 (1997) 55-94

This system is a reduction of the Yang-Mills equations (26) because any solution of (28) also satisfies (26). We next examine the reduced form of the Einstein equations (25). The string metric or the structure of the space-time = [~2 x M implies that the Einstein tensor has the following simplification: -Goo = G33 = ~ R,

Gu~ = 0

for other values of/~, v,

(29)

where, from now on, R is the scalar curvature of the conformally Euclidean surface M = (~2, e,J&jk). It is known that R = --e-"Arl

forx c M ~ ~2.

(30)

The constraints expressed in (29) render some stringent restrictions to the energy-momentum tensor Tu~ of the gauge-matter sector due to the consistency requirement in the Einstein equations (25). In other words, we must have Ae ~ = 8zrG TII = 8rrG T22,

(31)

l

R=A+8zrGE, whereC=T00=-T33, (32) 2 and other components of Tu~ vanish exactly. Indeed, from the assumed form of the gauge and Higgs fields and the dual equations (28), it is not hard to verify that Tu~ = 0 whenever # #: v. Moreover, in view of (28) again, we have 1 ,:.07 2 TI1 = 7"22= ~e-°P22 + 2" ~12 -{- ] dllpl2 - - ] d2~12 + 2 g e e - " l W I 4 g2 - 2ge -~ sin0 &21WI z - 2ge -'J cosOZl21W] 2 - 8cos 2 0e°(l~l 2 - ~p02)2

= 2l e - ' j

+2

(

PI2 - 2 g s i n O J W I 2 - 2g~°2sinOe~

)(

g~°02 e,j~ P12 - 2 g s i n O I W ] 2 + 2sinO /

g eO(f#t[2_~o02)) 1 e - ~ (Zl2 -- 2gcos0lWI 2 _ 2cos0

x ( Z I 2 - 2gcosOlW] 2 +

g

2 cos 0

e~(l~l 2-q92)) +

g2~°4 e" 8 sin 2 0

_-- g2~p4 e ~. 8sin20

(33)

Combining (31) and (33), we see that it is necessary to impose the condition A = rrG g29904 sin 2 ~9

(34)

Thus (32) is the only remaining equation in the Einstein equations. Namely, the Einstein equations (25) become a single equation relating the scalar curvature of the unknown gravitational 2-surface M to the energy density of the gauge and matter fields and the cosmological constant is expressed in terms of G, g, 0, and ~P0.This completes the proof of Theorem 3.1. The expression (34) may stimulate some cosmologists and lead to various theoretical speculations. Here we would only restrain ourselves to the following comments: Remarks.

(i) The cosmological constant problem is one of the most important ones in contemporary physics. The observed value of A today shows that it is almost zero. However, A is believed to take a significantly positive value in

Y Yang/Physica D 101 (1997) 55 94

69

the early universe when nuclear interactions were dominant and the temperature was extremely high because it is related to the vacuum energy level of the universe. The result (34) complies, at least qualitatively, with such a picture. In particular, it is interesting to notice the dependence of A on the vacuum energy-breaking scale ~00 which, in regard with the Newton constant G, makes a contribution to gravitation. (ii) One may argue that the Bogomol'nyi phase considered here restricts the applicability of (34) to realistic situations. In superconductivity, the Bogomol'nyi condition divides two types of superconductivity so that the intermediate phase is considered unrealistic. In particle physics, however, one cannot simply rule out such a phase since the Bogomol'nyi criticality condition mainly imposes a mass value for the Higgs bosons which are yet to be determined experimentally in the future. (iii) One may raise the question whether (34) gives the correct value of the cosmological constant A in a universe dominated by electroweak forces. At this moment, we are unable to speculate on this issue in either way. We only emphasize that (34) comes as an exact mathematical result from coupling the Einstein equations with the bosonic Weinberg-Salam model under the static string ansatz for the gravitational metric. In fact the string ansatz imposes stringent restrictions to the form of the energy-momentum tensor of the gauge and Higgs sector (see the discussion surrounding (31) and (32)) and the dual system (28) is so far the only known consistent structure that meets these restrictive requirements on the energy-momentum tensor and thus makes the whole (static string) framework valid. With (28), one is led immediately to the relation (34) and other paths are forbidden, In this sense, (34) is significant. 4. Decay estimates of the conformal factor In this section, we study the asymptotic behavior of the gravitational metric, which is important for the existence of finite-energy electroweak vortices in the induced background. In Section 4.1, we reduce the coupled Einstein and dual equations, (32) and (28), into a second-order nonlinear elliptic system. In Section 4.2, we derive suitable decay rates for the conformal factor e" of the gravitational string metric.

4. I. Nonlinear elliptic equations Since we are looking for cosmic string solutions in the dual system (28) and (32) of the unified theory coupling the Einstein gravity with the Weinberg-Salam model on the full (0~2, e'T&jk), there is no restriction to the phase jumps of the complex scalar fields ~p and W around infinity. Thus the numbers of zeros of ~p and W are arbitrary. Let p j . . . . , PN. and ql, - " , qN2 be the zeros of ¢r and W, respectively. Following Section 2.5, we see that the system (28) becomes, after setting u = In I~plz and w = In IWI 2,

Au = g2 ( e,7+,

)

COS2-----~ 2 + 2e"'

2 7 N. 2gcos29960e~ + 4rr Z &Pi" j=l

N~

zSu, = -g2e'l+u - 4g2e w + 4rr ~

(35)

&qj.

j=[

In order to understand the Einstein equation (32) where the cosmological constant A has been determined by (34), we first calculate the energy density C of the gauge-matter sector using (28) in (27):

E :-

g 2__~( -e

°~{Oj(fjklP t dklP)} -+- ~qg0c

= e_,l([dllp[2 + ]d2/p]2, +

1 ~PI2

l cos 0

ZI2

)

g2~o4 8 sin 2 0

g.~ e_~Pl2 + g___~,e_O([~p] 2 _q)2)Zl2 z sm t~ 2cos~

A . 8rr G

Y Yang/Physica D 101 (1997) 55-94

70

Thus, (32) takes the form Ari + 16rra(] dl ~]2 + ] d2~pl2) + 8JrGgg°2pt2 + 8Jra g (l~l 2 - ~02)Z12 = 0. sin 0 cos 0 Expressing the above equation in terms of u and w given in (35), we obtain

g2 ( ~o4 A k 8 z r0__~ G l +eUlVul2+_~eO ~gl_T_~

[eU ~o2]2~ + 2g2eU+ w -c~s~O J

~

(36)

0.

Eqs. (35) and (36) are the reduced Einstein-Weinberg-Salam equations governing static cosmic strings. We next show that (36) can be completely eliminated. In fact, it is straightforward to see that (35) and (36) leads to the equation A

- (2u + w) 2 sin 2 0

+

--

28pj + E aqj

sin 2 0

j=l

which implies immediately the existence of an entire harmonic function h so that

~ ~°° 2u + w) - D r G e " - 4 r c G ~°! 21nix - pj 12 + } - - ~ l n [ x - q j l 2 ri = 4Jro----~--( sin~0 sin k 0 \ j = l j=l

+h.

(37)

The function h clearly defines a gravitatibnal "background" which should be independent of the Newton constant G. In order to recover the flat Minkowski space-time corresponding to/7 = 0 in the event that gravity is turned off by setting G = 0, we must choose h --- 0. Therefore, the background freedom is fixed. For convenience, let

riO = -Cl

~

- pj

j-----l

+m

qJ

4:r Gq92 cr----, sin 2 0

'

j=l

c2=8zrG

(38)

in (37) with h = 0. Then ri is represented by ri = Cl (2u + w) - c2e u + rio.

(39)

Inserting (39) in (35), we see that the unified Einstein-Weinberg-Salam equations (35) and (36) are reduced to the following 2 x 2 system with superimposed nonlinearities of exponential type: Nt

Au -- ~ g (e u -- ~o02)e~0+cj(2u+w)-c2e" + 2g2eW + 4rr E 2 cos 2 0

8p~,

j=l

N2

A w = --g2eO°+(2cj +l)u+cj w-c2e" _ 4g2e u, + 4zr E

(40)

6qj.

j=l

It is seen that the difficult structure comes from eliminating the Einstein scalar curvature type equation (36).

4.2. Decay of the conformal factor and its implication Physically, the amplitude functions I1/i[2 = e u and I WI 2 = e w cannot exceed their vacuum expectation values. Therefore these quantities should stay bounded. In other words, u and w may be assumed to have upper bounds. As a

Y. Yang/Physica D 101 (1997) 55-94

71

consequence, the asymptotic behavior of the conformal factor e '7 of the gravitational metric is seen to be dominated by e o°. Hence, using (38), we have K = e'

=

O(r-O(212Nn+N2l+2au+a2)),

r = Ixl,

x ¢ R2

(41)

near infinity. Here al _> 0, c~2 > 0 are suitable decay rate constants so that e" = O(r -'~1)

and e '' = O(r -'~2)

for r = Ix I large.

Thus e'~ vanishes asymptotically like a power function. The decay estimate (41) has an interesting implication. To see it, recall that the Eq. (32) and condition (34) indicate that the scalar curvature R of (~2, e0ajk) is bounded away from a positive constant, which is at least 2A. Thus the metric eO3jk suffers noncompleteness. Combining this property with the decay estimate (41), we obtain the interesting restriction cl (212N1 + N21 + 2eel + or2) > 2. In other words, these decay rates of I~l 2, IWI 2, and the string numbers should satisfy the following lower bound estimate 2Ni + N2 + al

+

1 ~0/2 >

sin 2 0 4rr Gcp2

(42)

•

Since the universal gravitational constant G is a very small number, inequality (42) says that the number of cosmic strings arising in the unified Einstein-Weinberg-Salam dual equations, or the decay rate of the fields 7* or W, must have been sufficiently large if such strings ever existed. The importance of (42) motivates the following rigorous analysis which may be viewed as a partial proof of (41 ) or (42). To overcome the difficulties of the nonlinear equations just derived, we now restrict our attention to radially symmetric solutions. We assume that the strings are all superimposed at the origin. Then the most general form of the harmonic function h is that h is a constant. Since we are only interested in obtaining the asymptotic behavior of e 'l, the value of h is not crucial. Without loss of generality, we may assume h = 0. Therefore (39) becomes r / = cl (2u + w) - c2e u - 2Cl (2N1 + N2) lnr,

(43)

where u and w satisfy the radial version of the system (35): Ur,. +

1

g2

-- - (e u - ~02)e~ + 2g2e u', r u" 2cos2 0

lim rut = 2NI, r--+0

1

Wrr

+ - w r = -g2e'l+u - 4g2e u', r

r > 0,

(44)

lim rwr = 2N2. r-+O

We consider the full range of radially symmetric solutions including the trivial ones, N1 >_ 0, N2 > 0. The following result is crucial: Lemma 4.1. Let (0, u, w) be a solution of (43) and (44). Then it is necessary that u(r) < 2In ~o0 for all r > 0 in order that the solution leads to a finite gauge-matter energy per unit length of strings, namely E = f £ e " d x < oo. d ~2

Y Y a n g / P h y s i c a D I01 (1997) 5 5 - 9 4

72

Proof Suppose otherwise that there is some ro > 0 to make u(ro) > 2 In tpo. Notice that the boundary condition in (44) gives us the property that u(r) < 2 In ~oo for r small if Nr > 0 or if N~ : 0 but u(0) < 2 In qgo. Hence, in either of these two cases, we can assume ro to be the smallest such number at which u(ro) > 2 In tpo. Of course, u(ro) = 2 In ~oo. Using the first equation in (44), it is seen that ro is an isolated point for which u(ro) = 2 In ~oo and u(ro) is not a local maximum. Consequently, there is some number 3 > 0 so that u(r) > 2 In ~oo for r ~ (r0, r0 + 3). The maximum principle clearly prohibits the existence of an rr > r0 to make u(rl) --- 2 in ~oo. So u(r) > 2 In ~Po for all r > ro. If Nj ----0 and u (0) > 2 In ~oo,we have two possible situations: either u (r) > 2 In ~Pofor all r > 0 or u (rt) < 2 In ~P0 for some r! > 0. In the first situation, we arrive at the same conclusion as above for ro = 0. In the second situation, we have two subcases.

Subcase 1: u (0) = 2 In ~Po. By the first equation in (44), u cannot have a local maximum greater than or equal to 2 In ~oo. Hence u(r) < 2 In ~oo for all 0 < r < rl. If u(r) < 2 In ~o0 for all r > 0, the statement in the lemma is obtained. Assume there is some r2 > rl to make u(r2) = 2 In ~oo. Suppose such an r2 is the smallest. Then u(r2) is not a local maximum and there is a 3 > 0 so that u(r) > 2 In ~P0 for r ~ (r2, r2 + 6). Thus, again, u(r) > 2 In ~o0 for all r > r2.

Subcase 2: u(0) > 2 In ~oo. The first equation in (44) implies that (rUr)r > 0 for r > 0 as long as u(r) > 2 In ~oo. Thus Ur > 0 there as well which leads to u(r) > u(0), in particular. Hence u(r) > 2 In qgo for all r > 0. In summary, we have obtained a suitable r0 >_ 0 so that u(r) > 21n ~oo for all r > ro. We now claim ur(r) >_ 0 (r > ro). In fact, ifro > 0, it is easily seen that Ur(ro) >_0 because otherwise we would have ur(ro) < 0, violating the definition of ro. If ro -----0, we only have the subcase 2. There we already observed that Ur > 0 everywhere. From the first equation in (44), we have

(rUr)r

>

0 whenever r > r0. Thus

rur(r)

is strictly increasing for

r >__r0. The property Ur(ro) > 0 implies that

rur(r) > (ro + 1)ur(ro + 1) ~ cr > 0,

V r > ro + 1.

Thus, an integration of the above gives us the lower bound

u(r)>u(ro+l)+cr(lnr-ln[ro+l]),

r >ro+l.

As a consequence, we have the estimate [ d l ~ 1 2 + [ d 2 ~ r [ 2 = .~e 1 u U r2 > -~eU(ro+l)(ro+l)-Cr~2rc~-e=C(ro)rO-2 '

r>ro+l

which immediately leads to the energy blow-up O~

E>_

f t /

Ixl>ro+l

([dl~[ 2+lde~kle) dx_>C(r0) [

r~-ldr

t /

ro+l

contradicting the finite-energy assumption made. This completes the proof of the lemma.

[]

Theorem 4.2. For a radially symmetric finite-energy cosmic string solution of the coupled Einstein and WeinbergSalam dual system, namely Eqs. (32) and (28) with Nl and Ne being the numbers of Z and P fluxlines or strings, there holds the power decay law (41). Proof We have seen in Lemma 4.1 that u remains bounded from above. On the other hand, the second equation in (44) says that ( r t O r ) r < 0. Hence w is also bounded from above. Inserting these results in (39), we find (41) and (42) as before and the proof is concluded. []

73

Y Yang/Physica D 101 (1997)55-94

The decay exponent ce2 for e u' actually satisfies or2 > 2.

(45)

In fact, integrating the second equation in (44) gives us ?/1

r w r ( r ) : 2N2 - g 2 ](eO+U + 4eU')p dp. ,s

0 From this equation, it is easily shown that there is a number ~2 > 2 so that r w r ( r ) --+ - a 2 as r --+ cx~ as expected. Hence, we may rewrite (41 ) as K-----er ; = O ( r

~),

r=lxl,

xc[~2

/~ > 2 c I ( 2 N I + N 2 + l )

(46)

in view of (45). For convenience, we often implement the weaker statement/~ > 2 in (46) by virtue of condition (42).

5. Multivortices in the presence of gravity

In this section, we study the existence of finite-energy W- and Higgs-condensate multivortices in the WeinbergSalam theory in the presence of a gravitational background. The results in Section 4, say that the existence of a solution with vortex charges Ni, N2 implies that the conformal metric factor e ~ should decay like (46). We now establish an existence theorem under such a condition which says that there are finite-energy solutions for any vortex charges Nl, N2 provided that the physical parameters in the model lie in specific regions. Our result gives an interesting relation between the vortex charges Ni, N2, the symmetry-breaking scale qgo, the coupling parameter g, and the Weinberg angle 0. The solutions we are to obtain are actually a two-parameter continuous family with designated asymptotics. In Section 5.1, we calculate the full-space version of integral constraints which determine the regimes of several physical parameters. In Section 5.2, we set up our variational principle for the problem. In Section 5.3, we obtain our main existence theorem concerning the existence of multivortices in the standard model and comment on some implications of the results. 5.1. Constraints

Let uo, w0 and h I, h2 be defined by the expressions NI U0

N2

- ~ . ] ln(l + Ix - p/I-2), .j= I

NI hl=4~(l+lx-pjl2) j=l

-2.

w0 = - --~_~ln(1 + Ix j= I

-

qjl-2),

N2 h2 =4~-"~(1 + Ix - q j l 2 ) -2. j=l

Thus, following (35), vl = u - uo, v2 = w - wo satisfy 9

A v l --

g"

2 cos 2 0

e~(e,O+~] _ (p2) + 2g2e~,O+~,2 + h i ,

Choose vo 6 C ~ ( ~ 2) so that v0(x) = - I n Ixl,

Ixl ~ 1,

x E R 2.

AV2 =

--g2e°+U°+VL -- 4g2e u'°+v3 +

h2.

(47)

Y Yang/Physica D 101 (1997) 55-94

74

Then an integration by parts gives us

-- f Aoodx = -- f

Avodx =

Ix[_
R2

Define now for ~], u2 > 0 the transformed unknown functions fl = 2(Vl -- ul v0) + (l)2 -- O'2V0),

f2 = I)2 -- 0t2V0.

It is seen that the system (47) becomes

Afl = tanZOUle(ft-f2)/2 - cos2-------~o2e" g2 + HI,

Af2 =- -Ule (f'-fz)/2 - U2e f2 + H2,

(48)

where, for r = ]xl large,

UI = gZe~+U°+Cqv°= O ( r - ( 3 + m ) ) ,

U2 =

4g2ew°+u2v°= O(r-C~2),

(49)

and Hj = 2hi + h2 - (2Otl + ~2)A1)0,/-/2 = h2 - otzA1)0 satisfy

f H! dx = 47r(2Nl + N2) + 27r (2otl +

(50)

f H2 dx = 27r (2N2 + or2).

0t2),

R2

R2

We would need in (49) the restrictions fl+cq

> 2,

or2 > 2

(51)

in order to have a suitable weighted Sobolev space setting. However, it is clear that the first condition in (51) is already contained in (42) for any oq > 0 when one uses the noncompleteness of the metric e'6jk. Integrating (48), formally inserting f Afj = 0, f Aye -- 0, and using (50), we have tan 2 0 f UI e(fl-fz)/2

g2 _2 ]f e , dx - 4~r(2Nl + dx -- e~2~q)O

N 2 ) -- 2zr(2al + 0t2),

t /

R2

R2

f Ule(f'-f2)/2 dx + f U2ef2 dx = 27r(2N2 +et2), R2

(52)

R2

or

f

U2 ef2 d x = 27r

(

212NI + o q ] c o t 2 0 +

2N2+c~2~ g2g fe"dx siG 2--'~

R2

2'

(53)

sin20

R2

As a consequence, we arrive at the basic constraints from (52) and (53): 2:a'(212Nl + N2]+[2Otl +ot2])cot2O < -g2~°2 sin 2 0

f e q d x < 2rr (212NI +~l]COt2O + 2N2-k-_~2~sin2O /

"

N2

It is clear that the left-hand and right-hand sides of the above inequality are consistent.

(54)

g Yang/Physica D 101 (1997) 55-94

75

5.2. A variational formulation From (51 ), there is an e > 0 such that 2 + e = min{fl + ~1, ot2}.

(55)

Let h0 be a positive-valued C~-function on ~2 satisfying

ho(x)

= Ixl -(2+*)

for Ix[ > 1.

(56)

Denote by L P ( d # ) the space of LP-functions under the weighted measure d/, = h0 dx and 7-/the Hilbert space of ") Li-oc-fUnctlons for which I l f l l ~ -- IlVfllLe(dx) 2 2 -F IlfllLZ(du) < e~.

Since (56) holds, so 7-/contains the constants. Thus

A2

is a closed subspace of 7-(. The following Trudinger-Moser type inequality is due to McOwen [21].

Lemma 5.1.

Let y > 0 be such that g < min{4zr, 2Jre} where e > 0 is as given in (56) for the weight function of the measure d#. Then there is a constant C = C(V) > 0 depending only on V so that e "lfl

drt <_C(y)

exp

IlVfl122
¥f •

(57)

~2

Thus (52) and (53) are well defined for f l , f2 • 7-g in view of (49), (55) and (56). Introduce now the functional

l(fl../'2)

=

dx

~1l V f l 12 +

tan 2 OlVf2] 2 -- [\ C O g-~°6 S 2 0 e n -- Hi

fl +tanZOHzf2

and the optimization problem min{l(ft,

f2)lfl, f2 • 7-gand

f l , f2 satisfy (52) and (53)}.

(58)

It can be shown that a solution of (58) verifies the system (48). In fact, let ( f l , f2) solve (58). Since the constraints (52) and (53) are independent, there are numbers )q and/1. 2 in • so that for any ~l, ~2 • ~ there hold

f dx

V fl . V ~ I -

~1

~2

=2-1 ~2

tan2OflVf2.V~a+Ha~2ldx=--~.,fUlef'-f2)/i~2dx+~.ifu2ef2~idx. ~2

R2

~2

Set ~l = 1 and ~2 = I in the above two equations. We obtain by using (52) and (53) the solution for the Lagrange multipliers ;vl and ~-2 as follows: ~-

I

=

--

tan 2 O,

)v2 = tan 2 0.

Y. Yang/PhysicaD 101 (1997)55-94

76

These values imply that the pair ( f l , f2) is actually a week solution of (48). Using the standard elliptic regularity theory we see that ( f l , f2) is a smooth solution of the system as well. Hence we need only solve the optimization problem (58).

5.3. Existence results The constrained minimization problem (58) is similar to the problem (3.14) in [30]. The difference is that we now have e > 0 here instead of e > 2 in [30]. This difference only leads to a more restricted range for F: y < min{4zr, 2zre} in Lemma 5.1, instead o f f < 4zr in [30]. As in [30], we know that the quantities 0-1 and 0-2 defined by

2:rrcr I =

f

{ g-~06 e ° |cos 2 -- HI \ ~2

)

g2q92

dx --

COS2 0

/

e ~ dx - 2zr(212N~ + N2] + [2Otl + or2]),

~2

23"/'02 = f H2 dx = 2rr(2N2 + or2), R2 where we have used (50), are important to the existence of a minimizer of (58). By virtue of (54), we have 0-2

crl tan 2 0

- - 2(2N! + ~l)COt20 +

2N2 + ore

g2q92

sin 2 0

2Jr sin 2 0

f

J

e ' d x > 0.

~2

Consequently the method of [30] shows that (58) has a solution if 0-1 Y < --. tan 2 0 zr

0-2 -- - -

In other words, if in addition to (54), there holds 2 N 2 + ~t2

2(2N1 + oq) cot 2 0 + -

sin 2 0

g2~02

f

27r sin 2 0

J R2

e ° dx < min{4, 2e},

(59)

where e > 0 is as defined in (55), then (58) has a solution pair ( f l , f2), which necessarily is a solution of (48). Moreover, it can be shown that fr, f2 go to some finite limits as [xl ~ ~ . To this end, we need to cite a few results from [21 ]. Let a c ~ and s be a natural number. Define the weighted Sobolev space Ws.a as the closure of the set of C ~ functions with compact supports in the norm

Ilfl12,.a = Z

I1(1 + [xDa+l'*lOC'fl122.

I~l<_s Lemma5.2. For s > 1 and 6 > - 1 there holds W~,a C Co(R 2) (the set of continuous functions that vanish at infinity).

Lemma 5.3. For - 1 < a < 0, the Laplace operator A sends W2.a to Wo,a+2. This map is 1-1 and the range of A is given by

A(W2,a) = {h E Wo,a+2 f hdx =O}. ~2

Y Yang/Physica D 101 (1997) 55-94

77

Lemma 5.4. If .f • H and A.f = 0, then f is a constant. For the solution pair .fq ¢ 7-g, q = 1,2 of (48), we see that the right-hand sides of the two equations all lie in W0,a+2 when we choose fi=-l+½min{s,

1},

where s is defined in (55). Obviously 8 satisfies 0 > a > - 1 required by Lemma 5.3. From (52) and (53), it is seen that the right-hand sides of the two equations in (48), for which we use gl and g2 to denote, both have vanishing integrals. Using Lemma 5.3, let FI, F2 ¢ W2.a satisfy AFI = gt, AF2 = g2, respectively. Then FI and F2 vanishes at infinity by Lemma 5.2. In particular, FI, F2 ¢ H. Therefore A(fq - Fq) = 0 and .lq - Fq e H, q = 1,2. In view of Lemma 5.4 it is seen that ,fq - Fq -----constant for q = 1,2. In other words, f l , .re tend to some finite limits at infinity. We now calculate the P- and Z-fluxes, ¢,e and g'z. From (28), we have ,)

v.g~-)Of e?~dx-~-ggsinOf ]w]gdx"

qDp = 2sin-----# ~2

(60)

~2

C ~ z - g ~ fcos ( 1 001 2 - ~ ° Z ) e U d x + 2 g c ° s O / I W l E d x 2 A2

(61)

t~2

Using the relations 1012

e"

e u ° + u l v ° + ( f ' - f2)/2

IWI 2

e"'

e ''°+'~'~'°+ I;

(621

definition (49), and Eqs. (52) and (53), we obtain f 1~12e '1 dx D~2 f

IWI2dx

UI e ~ .tl -.re J dx --

=

g-2

jR2

lfu2el2dx

4g 2

N2

sin 2 0

2yr e '1 dx - -- 5- c°t2 0 (212N1 + N21 + [2o~1 + ~21) g[~2

Jr (212NI + oil ]cot2 0 + 2N2 + o~2) 2g 2 sm" 0

-~0 -o 4 sin- 0

f e '1 dx. N2

Inserting these results into (61) and (62), we have Jr @p = --(212NI + o t j ] c o s 2 0 + [2N2 + ot2]), e

2jr q~z = - - - cos0(2Nl + oq ). g

(63)

It will be interesting to compare (63) with (6) and (7). We then study the finiteness of the energy. Recall that, according to Section 4.1, the energy density C for the gauge-matter sector of a solution of (28) is C=e

'(]d10]2+]d2012)+

g~°2 - , i n g ]2 ~ 2sin~----~e rl2 + ~ c o s 0 e - ' J ( t 0 - ~P(5)ZI2

A 8jrG"

(64)

From (28) and the decay properties of e n, I~P12, and IWI 2, we see that there hold the asymptotic estimates PI2 --= O(r-(2+s)),

ZI2 = O(r -(2+e))

for r = Ixl large,

where s > 0 is as given in (55). Besides, since u = In 10l 2 satisfies IVul 5 IVuol + ~ l l V v o l + ~lV./ll + ½1V f21

(65)

Y. Yang/Physica D 101 (1997) 55-94

78

with IVu0f = O ( r -3) and IVy0] = O ( r - I ) for ]dl~rl2 + id2~[2

1

12eU <

= ~]gu

--

r

=

Ix[ large. As a consequence,

{O(r-6)+C(lVfll2+lVfzl2), O(r -(2+cq)) + C ( I V f l 12 -+- IV f212),

Otl = 0 , o'1

>

0.

Therefore, E = f ~'e ~ dx is finite as desired. Similar to the derivation of (22), it can be shown that the density function (64) has the same constant lower bound, > -

g2~04

A

8sin 20 = 8JrG"

The finiteness of the energy E is due to the decay of the gravitational metric. Using (32), we see that the total curvature f Re 0 dx is also finite. In condition (59), the number e > 0 can be chosen to verify a relaxed inequality, i.e., 2 + e _< min{/3 + ~1, c¢2}. All the conclusions above may be arrived at without any difficulties. Thus, in order to simplify the discussion, we may require de >_/3 and set e = / 3 - 2. We summarize the results obtained in this section as follows.

Theorem 5.5. Suppose that the vortex numbers Nl, N2 for the vortex locations Pl, "" ", the parameters oq >_ 0, c¢2 _>/3 satisfy inequalities (54) and (59), namely

PN~ and

ql,

" • ",

qN2 and

g2cp2 2(2N1 + N2) + (2Otl + or2) < 2rr cos 2 0 f e ° dx d ~2 < 2(2Nt + N2) + (2oq + or2) + tan 2 0 (2N2 + or2) g2~o2 f e ° dx + tan 2 0 min{4, 2(3 - 2)}, < 2rr cos 2 0 t , ,

(66)

~2

where/3 is the decay exponent of the conformal factor e ° of the gravitational metric, e ~ ---O(r-t~) for r = Ix I large, and, the condition/3 > 2 is imposed to ensure the geodesic noncompleteness due to the presence of a positive cosmological constant A determined by (34). Then the dual equations (28) have finite-energy multivortex solutions (Pj, Zj, lp, W) to realize these prescribed vortices so that ~p and W vanish precisely at the points Pl, " ", PN~ and ql, " " , qN2, respectively. The total P- and Z-fluxlines are determined by (63), the P and Z field strengths vanish at infinity according to the rate (65), and ~p, W satisfy the sharp decay estimates I~pl2 = O(r-Cq),

IWl 2

=

O(r -a2)

as r = fxf --+ oo.

(67)

In other words, the decay rates of the ~ and W scalars can be prescribed.

Remarks. (i) In (67), the case where Oil = 0 corresponds to the sector that the Higgs amplitude I~P12 tends to a nonzero limit at infinity. Thus there is a spontaneous symmetry-breaking phase in the solutions obtained. In this situation, the flux formulas (63) say that, one of the fluxes, the Z-flux q~z, is quantized. On the other hand, the case otr > 0 leads to the solutions in the symmetric phase ~ = 0, asymptotically. In either case, the P-flux q3~, is not quantized but is labeled by two continuous parameters oq, or2. Solutions of this latter type are actually nontopological.

Y Yang/Physica D 101 (1997) 55-94

79

(ii) Since (66) are strict inequalities for existence, we see immediately that whenever there are values of ul, or2 to verify (66), there must be some open intervals in which ott, ~2 still verify these inequalities. Therefore this shows that whenever there are suitable ott, ~2 satisfying (66) to ensure existence of finite-energy multivortex solutions, non-uniqueness will take place. (iii) It is easily checked that (66) is ensured when the condition 2(2Nt + N2) + (2ott + ~2) + tan2 0 (2N2 + ~2 -- ~'0) g2¢p2 < 2rr COS2 0

/ e ~ dx < 2(2N] + N2) + (2oq + u2) + tan 2 0 (2N2 + ~2),

(68)

i /

Re

E0 = min{2N2 + ot2, 4, 2(/~ - 2)}. holds. Thus our result says that for any vortex data N], N2, otl, ot2, there are suitable ranges of the symmetrybreaking scale ~00, the coupling strength parameter g, and the Weinberg angle 0 to ensure the existence of a family of finite-energy multivortex solutions with designated values of P- and Z-fluxlines given by (63).

6. Generalized electroweak model In the next three sections, we present a complete resolution of the dual equations governing electroweak vortices in a generalized model with two-Higgs complex doublets originally found in [4]. The goal of this section is to introduce the governing system of equations to be solved. Since in [4] the equations are derived for radially symmetric solutions, but here, we are interested mostly in solutions with arbitrarily prescribed vortex locations (without radial symmetry), we need to obtain these dual equations in our generality. Thus, in Section 6. l, we fix our notation. In Section 6.2, we record the dual equations due to Bimonte and Lozano.

6.1. Two-complex doublet model Let q~t and 4~2 be two-complex doublets in the fundamental representation of SU (2). Of course, we still let U (1) act on ~bq (q = l, 2) trivially as before. The gauge group is now SU(2) x U ( I ) r × U(I)y, with Y, Y' the two U(l)-hypercharge labels. The gauge fields associated to SU(2), U (1)Y, and U(I)y, are denoted by A u, Bt,, and /~l~, respectively, with the corresponding field strengths Fu~, Gu,~, and (3t,~- The Lagrangian density is

1 2 £ = -~(Fuv • Fuv + GuvG t~v + G~vG ~v) + ~-~(D(q)IZq~q) t . (o(q)llqbq) q- V(~bl, ~b2), q=l

where, with no summation convention assumed on the index q = 1,2, the gauge-covariant derivatives are defined by D(q)rh 1. au + -~lg 1- ; YqB u + ½i2gl YqBu)qOq, i ~ l~ "t'q = (Oft q- ~lgraA

q = 1,2,

which mixes the weak and electromagnetic interactions. Recall that r, are the Pauli matrices given in Section 2. The quantities g, g', gl and Yq, Yq are positive physical constants. The Higgs potential density V is determined by the expressions V(~bl,~b2) = ~R I a R a + I R 2 + ½/~2,

Ra .

~1 g ( + l +Taq}l .

+ ;.T a ~ 2 ) ,

.R

½ g t.( y , , ; , ,

.

Y2';*2 .

P),

/~

½gl(Y;+;q~l

Y2*2*2' l" _ / 3 ) .

Y Yang/Physica D 101 (1997) 55-94

80

6.2. Dual equations Assume that the field configurations are independent of the time variable x ° and the vertical direction x 3 with A 3, Bu,/}u ----0 (for # -----0, 3). Then the following Bogomol'nyi type dual equations (see [4]) are satisfied in order to saturate a topological energy lower bound: F~2:--R

a,

GI2=-R,

D0)~¢l = i D O l ) e l ,

GI2=-/~,

D(12)~O2:-iD~2)~b2.

(69)

It has been shown in [4] that the asymptotic behavior of ~bl, qb2 of a finite-energy solution requires the existence of a positive number v0 so that I 2 p : ~vo(Yi - V2),

fi = ½ v 2 ( V ; - r~).

(70)

To simplify (69), we impose the ansatz

Ajl = Aj2=O,

j=l,2;

4 ~ q = ( 0 )~/fq ,

q = 1, 2.

(71)

It is easily seen that the ansatz (71) is self-consistent and F32 = Oi A~ - O2A~ (the commutator vanishes). Consequently we arrive from (69) at the following dual equations: -2i0~kl = o t ( l ) * ~ l ,

-2i0t~p2 = 0t(2)~2,

F32 : ½g(l~112 _ 1~212), GI2 : - ~ 1g , (YII~I 12 - Y21~212 - P),

(72) G I 2 = -½gl(Y~llPll 2 - Y~l~2l z - / 5 ) ,

where the complex-valued vector fields Ot (q) (q = l, 2) are defined by tgj(q) = ~gAj 1 3 - ~g I ¢YqBj - ½gl YqBj, '-

j=l

, 2,

Ot(q) = Ot(iq) -t'- iff~q) •

(73)

The first two equations in (72) say that g,, (or ~2) is an anti-holomorphic (or holomorphic) section. Thus both antiself-duality and self-duality are present as in the single Higgs particle case containing Z-fluxlines studied earlier in this paper. The last three equations imply that F132, G 12, and G 12 are linearly dependent as observed already by Bimonte and Lozano [4]:

g' gl(Vl Y~ - Y2Y;)F32 + ggl(Y2 - Y~)G12 - g ' g ( Y 2 - Yl)(~12 = 0. Hence we can invoke another ansatz

g'g, (Y, Y(~ - Y2 Y~)A 3 + ggl (Y~ - Y~)Bj - g'g(Y2 - Y1)/)j = 0,

j = l, 2.

(74)

We will assume (74) throughout the rest of this paper. We are to obtain multivortex solutions of (72) on a periodic lattice cell as well as on the full plane.

7. Existence o f periodic multivortices In this section, we establish an existence theory for the Eqs. (72) subject to a periodic boundary condition. Our condition for existence is necessary and sufficient. In Section 7.1, we first derive from (72) the nonlinear elliptic system governing multivortex solutions. In Section 7.2, we prove the existence of solutions by a variational approach. In Section 7.3, we state our results and obtain the exact quantized values for various fluxlines.

Y. Yang/Physica D 101 (1997) 55-94

81

7. I. Nonlinear equations From (72) and the formula a'llq) = --i(Oa'(q) -- Otot{q)t),

q = 1,2,

we obtain 2/11n I~112 =- I~Ol]2(g2+ g ' 2 y ? + g l 2Y- It2~) - ] 0 2 ] 2 ( g 2 + g ' 2 y i Y 2 + g l Y2) Y,2 ) :- ( g ' 2 y i p + g ' f Y l p"~) , f ~ 2/1 In 1~212 = - I ~ 1 1 2 ( g 2 + g'ZYl Y2 q- g2y(y~) q_ 1~212(g2 _+_g,2Y2 + g2y~2) q_ (g,2y2 p q_ g~Y2P). We are to look for multivortex solutions so that ~pj and

~2 vanish

at the prescribed vortex locations Pt, " ,

(75)

PNI

and qt, • •, qN,, respectively. For convenience, we now use the notation M =

all

o12

\a21

a22

02

g2 _.~ g12y2 _{_ g~y;2 2.,,.,:,

_(g2+g'2yiY2+glfiY2)

= 7

~ 2-

g +g

12

2 I I

/2,,22,,,2 x2~_glx 2

(76)

and (2L~j12'~ ul = In ~ - - ~ - o 2 / ,

{21~212~ u2=ln\T/,

IMl=det(M)>0.

Then (75) becomes i

NI

A u l = a l l ( e ul -- 1) + a 2 1 ( e u2 -- 1) + 4 7 r ~_~@i, i

j=l

N~

Au2

=

a 2 1 ( e ul - -

1) +

a 2 2 ( e u2 - -

(77)

1) +4zr ~--'~aqj. j=l

We will find a necessary and sufficient condition to ensure the existence of a doubly periodic solution of these equations.

7.2. Proof of existence by a variational method We are to look for vortex solutions of the dual system (72) over a f u n d a m e n t a l periodic lattice cell, say ,"2, as in Section 2. Such a situation requires us to solve (77) on ~ so that ~2 is treated as a 2-torus. As before, the measure of S'2 is denoted by I ~ l . Let u o' and u o be source functions satisfying

, Au 0 --

4rr N i N~ j ~ + 47r J=IE@/'

,, Au° --

4zr N2 N2 la"2l + 4n J='~ aqj,

Then, on the 2-torus $2, the functions Vl = ul - u 0' and v2 = u~_ - u 0" verify a slightly modified version of (77) of the form ' u" v 47rN1 Avl = a l l ( e uo+v~ -- 1 ) + a l 2 ( e o+ 2 _ 1 ) + - 1S2[ ' , , 4rr N2 A v2 = a21(e uo+vl -- 1 ) + a22 (eUo+v2 -- 1 ) + -

lS?l

(78)

Y. Yang/Physica D 101 (1997) 55-94

82

Integrating (78) over ~ and remembering that there is no boundary term involved, we easily obtain by simple linear algebra the following two basic constraints: f e% +v' dx = I.Qf - i4zr ~(a22Ni

- a12N2),

(79)

= IS2] - T4:rr ~ (all N2 - a21NI ).

(80)

£2

f

e,,,,o+v2dx

~2

Therefore we see that a necessary condition for the existence of a solution is 4rr

IMI(a22NI -

47r /]M~--7(allN2- a21Nl) < I~1.

a12N2) < IS2l,

(81)

We shall show that (81) is also sufficient. To this end, we introduce the transform Wl

1 [,~

I)l,

Vl ~--~I V / ~ Wl,

(82)

1

1

1)2 = - - ( [ M I / 2 all

w2 = ] - - ~ ( a l l v 2 - - a 2 1 v l ) ,

+ a21~Wl).

In view of (82), the system (78) takes the form

AWl

=

all

e.o+ i . / ~ w , +

a12

eUo+(a2,r~lw,+lMiw2)/a,n _

Cl '

,/IMi

(83)

AW2 = e uo+(a21 I'/~lw~+lMIw2)/aIJ --C2, where

,(

CI -- x / ~

all + a 1 2 - - - -

4rrNt "~

C2 = 1

4re - -

lY2tlMI

1~1 J '

(all N2 - a21NI).

Thus constraints (79) and (81) become all

f

e"°+P'/V~lUldx=Cll~l-~ a12

C 2 .(.2, --- C3lI2l > 0,

(84)

~2

f e uo+(aztl'/T~]lwl+lMPw2)/ajl dx = C2IS21 > 0.

(85)

I2

Consider the functional l ( w l , W2) =

//1

~[Vwjl2 +

[Vw2[2-CIwI-C2w2

}

dx.

~2

Let W 1,2 ($2) be the space of L2-functions over the 2-torus ~ so that their distributional derivatives also lie in L 2. We show that the solution to (83) can be reduced to the following optimization problem: m i n { l ( w l , w2) [ w~, w2 a WJ'2(I2); wl, w2 satisfy (84) and (85) }. The following well-known Trudinger-Moser inequality will be crucial to our development in this section.

(86)

Y Y a n g / P h y s i c a D 101 (1997) 5 5 - 9 4

83

Lemma 7.1. There are constants F > a positive optimal lower bound and C (F) > 0 so that e f dx < C ( g ) e

-

,

V f ~ W1'2(52) satisfying

f2

f d x = 0. £2

Thus the two constraints (84) and (85) are well defined. Lemma 7.2. Let (wl,

tO2)be a solution to (86). Then (wl, w2) also solves (83).

Proof Let (wl, w2) be a critical point of I in W1'2($2) satisfying the constraints (84) and (85). Then the Lagrange multiplier rule says that there are numbers ~.l and ),2 so that AWl = )~le "i~+'/~u'~ + ~.2 a21 ~/IM~leU~;+(a21"/Mql~'+hMLu'2)/' " -- C I , all Aw2 = L2 IMleuii+(a21~wt+lMIw2)/all -- C2. all

(87)

Integrating the second equation in (87) and using (85), we have )~2 = a l l / I M I . Inserting this result into the first equation in (87) and taking integration, we obtain in view of the constraint (84) that ,kl = a jl/~/I M I. Therefore the original system (83) is recovered and the lemma is proven. [] To proceed, we write any pair Wl, 1192in the admissible class C = {(wl, w2) I wl, w2 c W1'2(~2): Wl, w2 satisfy (84) and (85)} in the form ! 1/)q ~---"~q -I- //)q,

Wqdx = 0,

tOq E ~, f I2

q=

1,2.

!

Hence, from (84) and (85), we have

(88)

(89)

Using (89) in the functional I (wl, w2), we arrive at

l(wl, W2)=

f ,VW,l[2 { +

dx - ]S2](C1NI +

C2~2)

I2 I

r

= ~(llVw11122 + IlVw~ll~2) - 63112[ Wl

(90)

Y. Yang/Physica D I01 (1997) 55 94

84

Using C3 > 0 in (90) and (88), along with the Jensen inequality to get In

( f e uo+ ' ' IvU~lwj ) dx

>_ ~ '

S 'u odx,

In

(feU°+("2'~u'~+lMlw'2dx) )l"">~liu°dx' 1

I2

!!

S2

we find the estimate l

/(u,~, w2) >

/

")

~(llVw~ I1~_~+ IlVw~llec2) - C,

(91)

where C > 0 is a constant independent of wl, w2. In particular I is bounded from below in the admissible space C.

Lemma 7.3. If condition (81) holds, then (86) has a solution. In other words, the system (78) has a solution if and only if (81) is fulfilled.

Proof When (81) is satisfied, the constants

C2

and C3 are positive in (84) and (85). Hence the admissible class

C is not empty. Let {(wllk), w~k))} be a minimizing sequence of (86). Inequality (91) says that {w(lk)'} and {w~k)'} are bounded sequences in W J'2(I2). From Lemma 7.1 and Eqs. (88) and (89), it is seen that {~(ikl} and {~k)} are bounded sequences in JR. Then a weak compactness argument shows that there is a subsequence of {(w~jk), w~k))} that goes to a minimizer of (86). The lemma is proven. []

Lemma 7.4. If (83) has a solution, then the solution must be unique. Proof Consider the following functional:

J(w~.w2) = ½11Vw~11~2+ ½11Vw2112c2- Is21(CtW,+ C2~2) f[alleu'o+r./~M]w,+alle.i;+,a2,I./F~w,+lMIw2'/dx. a,,] +

IMI

IMI

S2

It is straightforward to check by calculating the Hessian that J is strictly convex. Thus J has at most one critical point. However, any solution of (83) must be a critical point of J. This proves the lemma.

7.3. Existence of periodic vortices for the two-Higgs system In order to construct solutions for the dual system (72) from solutions of (77) obtained in Section 7.2, we need to examine the residual gauge symmetry of (72) because any periodic boundary condition modulo gauge transformations may impose additional restriction on the phase jumps of ~pq (q = 1, 2) along the boundary of the cell region I2 as observed in Section 2. We set formally the gauge symmetry 1/tq ~

ei~q~q,

q = 1,2,

3 Aj3 P--~Aj+OjXi,

Bjw-~ Bj+OjX2,

Bjw+ Bj+OjX3,

(92) j=l,2,

where ~q (q = 1,2) are real-valued functions and Xk (k = 1,2, 3) are to be determined accordingly. Clearly the only thing we need to achieve is to obtain the gauge symmetry in the first two equations of (72). This requirement can be fulfilled when Xk (k = l, 2, 3) satisfy by virtue of (73) and (74) the relations

E Yang/Physica D 101 (1997) 55-94 gO iXI - g'YIOjX2 - glr(Ojx3 = 2 a i ~ l ,

g'OjX1 - g'Y2OjX2 - glY2OjX3 = 2"0i~2,

gt,~,l~,.YiY2, -- Y2YI)OjXI + ggl (Y2 - Y1)OjX2'

- g'g(Y2 - YI)OjX3 = O.

85 (93)

It is easily examined that the coefficient matrix of (93) is nonsingular if and only if 9

t9

t

g-g "(YI - Y2)2 + g2g2(y( _ y~)2 + g,2g~(yi Y2

--

t

9

Y2Y i)- ~ O.

which is equivalent to the condition

YI ~ Y2

t

or

t

YI # Y 2 .

(94)

Whenever (94) is verified, the system (93) has a unique solution for any ~q (q = l, 2). On the other hand, if (94) is violated so that Yj = Y2 and Y( = Y~, then D (l) = D (2) and the two Higgs scalars are just a duplicate of one another. This is obviously a trivial case one should avoid. Thus (94) is a general condition we should observe for the two Higgs system under discussion. Consequently there are no restrictions to ~q (q = I, 2). Hence the locations and numbers of the zeros of Ol, ~2 confined in a periodic cell domain may be arbitrary when the condition (81) is satisfied. We now calculate the fluxes of the weak and magnetic fields. Using (79) and (80) and 2 ~-~]~ll 2,

e"'

e"2=

v/i

2

v6

we obtain the quantities

/l~,t2dx=VO

~-

(

47r ) [S'21- 7 ~ [ a a 2 N l + ]a12[N2] ,

s2 f 1~212 d x = v2 4:7l2 (1S2[ - ~-~[lal21N1 +altN2] ) . ~2 These results combined with the last three equations in (72) give us the quantized fluxes: grr ,-} g'F = f F 3 dx = v~([lal21- a22]Nl + [all --lal21lN2) IMI £2 gzr -- 4tMI v:(g'Z[Y2Nl + gl N2][YI - Y21 + g~[g~Ni + Y(NzllY~ - Y2]), g'Jr Pc = / Gi2dx = V2o([Yla22 - Y2Ial21]NI + [Yllal21 - Y2alj IN2) /MI J s2 =

g'zr v4(g2lNl + N2I[YI - Y21 + g2[Y2N| + Y~N2I[YI Y~ - YIY2]),

41MI

dp~; = 1 ~ 1 2 d x = gIMI 17r v°([Yla22 2 t : - Y2laI2I]N1 + [ Y / l a l 2 l - Y~al l lNz) d s2 _ glTr v:(gZ[Ul + N2][Y/ - Y~] + g'Z[Y2NI + YI N2IIY~Y2 - Yl ~ ] ) 4[MI In summary, under the general nontriviality condition (94), we have tlae following existence results.

(95)

Y Yang/PhysicaD 101 (1997) 55-94

86

Theorem 7.5. Let Nj, N2 be two positive integers and p j, . - . , PN~, ql, • "', qN2 be points in the periodic cell domain S2 (periodicity up to the gauge symmetry transform (92) and (93)). We are to obtain multivortex solutions of (72) of the extended electroweak model with two-Higgs scalars, 7q, ~2, so that ~l and 7t2 vanish precisely at prescribed vortex locations P l , • "", PNI a n d q l , " ' " , qN2, respectively. (i) Uniqueness: For any prescription of vortices there is at most one solution. (ii) Existence: Given any prescription of vortices, there is a solution if and only if the topological charges Ni, N2 satisfy (81), namely, g Z ( N l -Jr-N2) qt-

g'ZYq(Y2NI -Jr-Y I N 2 ) + gl2 Yq(Y~NI , , + Y~N2) <

[S2i IMI

zrv2

,

q=

1,2.

(96)

Besides, the solution carries the quantized fluxes of the weak and magnetic fields over S2 given by the formulas in (95).

Remark. Condition (96) says that small energy scales of the symmetry-breaking characterized by small values of v0 allow the existence of large vortex numbers N], N2. Thus one may expect to have arbitrary numbers of vortex charges Nl, N2 when symmetry is restored by setting v0 = 0. In fact, we can show, to the contrary, that there is no solution in such a situation.

8. Finite-energy solutions on the plane In Section 7, we obtained an existence theorem for doubly periodic vortices resembling the Abrikosov vortices in superconductivity. The conditions we found indicate that the numbers of B- and/~-fluxlines confined in a lattice cell a'-2are confined by the size of a'2 and larger cells can accommodate more vortices. Consequently it is natural to expect that when the problem is considered over the full plane, the obstructions to the vortex charges should disappear. The result in the present section confirms such an expectation. In the following we shall prove the existence and uniqueness of a multivortex solution of Eqs. (77) over full ~2 for an arbitrarily given vortex prescription. Since there are no constraints of the form (79) and (80), we will use a direct variational method. In Section 8.1, we formulate a functional so that it is strictly convex and its critical point solves (77) on R 2. In Section 8.2, we show that the functional indeed has a global minimizer in a natural admissible class. Of course, this minimizer is the unique solution of (77). In Section 8.3, we obtain sharp exponential type decay estimates at infinity for the solution and some implied flux quantization results. In Section 8.4, we state our existence theorem for the two-Higgs model and some concluding remarks.

8.1. Variational principle Similar to Jaffe and Taubes [ 12], we need to introduce some background functions depending on a real parameter /x>0: NI

u; = - Z I n ( 1 j=l

N2

+/*Ix -pjl-2),

u~ = - Z l n ( 1

+ # I x - qjl-2),

j=l

Ni

(97)

N2

"

j=l ( / z + Ix--- PJ 12)2'

"=4E

go

"

j=l (Iz+lx--qJ 12)2"

Y. Yang/Physica D 101 (1997) 55-94 Setvl = U l - U

87

" The equations (77) on I~2 take the form 0' and v2 = u2 - u o. P

:

k v l = all(e"o +v' -- I) + a 1 2 t e I

)

U 0 -{'- 13"~

,'

" -- I) + g o '

It

(98)

1I

/1102 = a21 (e u°+~l -- 1) + a22(e uo+v2 -- 1) + go. It is important to notice that the integrals of g~, g~' over R 2 are independent of the value of it. In fact, we have go dx = 4rr Ni,

4rr N2.

~2

(99)

~2

Using the transformation (82), we rewrite (98) as awl

4 all ~-~(e% +'/~u'

--

- 1) +

a12 (e uo+(a21 I'/~-~-ul+lMIw2)/alt -- l ) + h l , ( 1 oo)

AW2 = (e uii+(a211"/~4uq+lMIwz)/all -- 1) + h2, where 1

1

h, - iv/T_~g ~,

!

h2--- [~-~(allg~'-a21go).

(101)

It is clear that system (100) are the variational equations of the functional

l(tOl, w2) : f { ~

]VWl

_1 2+ 12+ ~lVwz[

a " e%(e ,M[ " b'/FMiw' - 1)

R2 + alleuo(e(azl iv~lw~+lmlw2)/all _ 1)] dx

IMI

"

+ / {(h, _ ~ [ a l l l

/

+ a12]) to, + ( h 2 - 1 ) w 2 } d x

~2

(102)

I 2 + /-~all-_ ( e U ° ' e ~ W ' - l - l v / - ~ l w l ) 2 = lllVWl[122 + ~IIVw2[IL2 + ~atl (,,eUo, e(a211v~lw'+lMIw2)/a"--I -- --alll[a21 ~ / ~ t t ' l + ( wl, hi + 7,ql:vtlall" , ~ l e u'o - l l + ~ a12 t e f u"( o - 1]

+ IMIw2]) 2

+ (//02, h2 -+- [e uo" - 1])2, 2

where (-, ")2 denotes the inner product of L2(•2). This functional is similar to the one for the self-dual vortex solutions in the Abelian Higgs model (see [12, III.3, Eq. (3. I)]). Thus we will borrow some techniques used there with suitable adaption.

8.2. Existence of a global minimizer First, we know that 1 is differentiable and strictly convex for wl, w2 6 W1'2(•2). Next, we prove the following result.

Lemma 8.1. There is/z > 0 in (97) so that --Ci + C2(1[wl [[w,.2 -}- Ilw211w~.2) ~ (M)¢,~,w2)(wl, w2)

Vwl, w2 E W1'2([~2),

Y Yang/Physica D 101 (1997) 55-94

88

where Ci, C2 > 0 are constants independent of wl, w2 and (M)(w),w2) denotes the Fr&het derivative of the functional I at the point (wl, w2).

Lemma 8.2. The functional I defined in (102) has a unique critical point (wi, w2) in WI.2(IR 2) x WI.2(~2). This point is a global minimizer and solves the nonlinear elliptic system (100) as a smooth solution.

Proof Since I is convex and differentiable in (wl, w2) c WI'2(R 2) x WI'2(~ 2) = X, it is weakly lower semicontinuous. L e m m a 8.1 says that for any 6 > 0 there is an R > 0 so that < inf{l(wi,w2)(wl, w2) I wl, w2 c WI'2(R2), Ilwl IIw, 2 + Ilw2PIw, 2 _> e l . Using [12, VI, Proposition 8.6], it is seen that I has a local minimum in the open ball of radius R. However, the strict convexity of I implies that 1 can have at most one critical point, thus we have shown that 1 has exactly one critical point. Of course, this critical point is a solution of (100) which must be smooth by the standard regularity theory of elliptic equations. []

Proof of Lemma 8.1. A simple calculation gives us the difference (al)(w~,we)(wj, w2) - (llVw11122 + IlVw21122)

, i'/Vfl]wl -- 1))2 + ~ a21 all :(wl, eUo(e twl, ,

_

,/IMI

e u"o[e (a211"f~lw~+lMIw2)/'ql -- 1])2

+ (W2, eUo[e(a21I~wl+lMlw2)/alt _ 1])2 +

=

( wl,

(

oil

hi + ~ [ e

,

u° - 1] +

a'2 " u" ~le0

l])

~ all t c r^U'+ u I~IWl --1]-t- ~q-M--~,a21 [eU°+(a2i~wi+lMIw2)/ai'

2

q'- (//)2, eU°+(a2il~lwi+lMlw2)/alt -- I + h2)2 all

= (

l~U'+~w I _ 1]+hi

/' a21 + ~ w l

+

a21 . "~

m2, e u'~+(a2'l'/~w'+lMIw2)/a''

-

1 + h2)

2

(using the proof of III. Lemma 3.8 in [ 12]

with the undetermined parameter # > 0 sufficiently large)

f ~bl

~2

w---~21 dx + b2 l+lWll

f

~2

(a2i [q/-~wi+[Mlw2)2 d x - b o ,

l+lazll.v/~Wl+lMlwzl

where b0, bl, b2 > 0 are constants. Using an elementary interpolation inequality, we obtain from (103) the lower bound

(I03)

89

Y. Yang/Physica D I01 (1997) 55-94 (a/)(,,~,,,u,2)(Wl,

//)2)

--

(llVw11122 + I1Vw21122) >_ b 'I f

~2

(I

+ W~lu++ ~,Ll u'2lw2l) 2 dx

- b 0, '

(104)

where, as before, bl), b'I > 0 are suitable constants. We need the embedding inequality

ff4dx<_2ffedxflvfledx, ~2 [~2 [~2

f E WI'2(~ 2 )

(105)

to extract useful information from the inequality (104). We will show that (104) and (1105) are just good enough to enable us to arrive at the conclusion of this lemma. In view of (105), we have

(11wlllZ2+l[w211L2)2< ( / --

_< C

[wtl-t-Iw21

~2 (1 + Iw~l+ Iwz[) 2 ~2

"Y+ [to2[) ' < C~2 (l -{-{1/01[-{2 d. <

(1 + Iu-q I + Iw21)(lWlL-t-lw21) d x ) -

I -{- Itoll -t- I//)2[

2 dx

_

- 2

Wq

(!)/(!2 1 Z-

q=l

)2 + C (IN2f

d.

,+

(1 + lu'll + Iw21) 2

dx

_ q=l

j4 q- [Rfq~ LV Wq 12dx

+,),

where C > 0 denotes a uniform constant which may vary its value at different places. Hence,

ZllWqllL2
q:=l

l-t-Zl[Vwql[224 " dx q=l ~2 (l q-lWllq-lw2t)2

(106)

"

Combing (104) and (106), we see that the proof of the lemma is complete.

[]

8.3. Asymptotic behavior

In the Section 8.2, we proved the existence of a classical solution of (100). The next step is to show that such a solution give rise to a finite-energy vortex solution of the two-Higgs dual system (72) which is not automatically guaranteed by the study made earlier. We now elaborate on this problem. Lemma 8.3. Let (wl, w2) be obtained as in Lemma 8.2. Then

1/3q~

0 as Ixl --~ ~ , x C II~2, q = 1,2.

Proof We recall first the standard embedding inequality for p > 2:

IlfllL, <

(

:r

Hfllwl.2,

f

C

WI'2(R2).

(107)

Y. Yang/Physica D 101 (1997) 55-94

90

We show that e f - 1 E L 2 for f e W 1'2. In fact, the Taylor expansion gives us (e f _ 1)2 = f 2 +

~-~2 k - 2 k.r f k . k=3

Using (107), we obtain formally the series [[ef

-

l[[2z -< ][f1122 + E k=3

k~

rr

-

-

[if IIkWe2"

(108)

Calculus methods may be used to show that (108) is a convergent power series in IIf IIw~.2. This verifies the claim we made. Return to the system (100). Since by the above observation and the property that wl, w2 ~ WJ'2(~ 2) the terms e"; + f ' / V ~ ' - 1 = eUo(e I ' / ~ w ' - 1) + (e"; - 1), (109) eUd+(a2j x/~wl+lMIwz)/atl _ 1 = eU~(e (a21Ix/~wl +lMlw2)/alt _ 1) + (eU~ -- 1) in (100) both lie in L2(R2), the definitions of hi, h2 (see (97) and (101)) imply that the right-hand sides of the two equations in (100) are all in L 2 (1~2). The well-known L2-estimates for elliptic equations show that w l, w2 are elements in W2'2(R2). Then an embedding theorem says that Wl, w2 both approach zero as Ixl ~ o~ because we are in two dimensions. The lemma is proven. []

Lemma 8.4. Let Wq (q = 1, 2) be as stated in L e m m a 8.3. We also have the property that IVWq(X)J ~ [xl ~ oo, q = 1,2.

0 as

Proof Since Wq(X) ~ 0 as Ixt ~ ~ , q = 1, 2, the conclusion that the terms in (109) all lie in L2(I~ 2) may be generalized to the conclusion that they all lie in LP(I~ 2) for any p > 2. Thus the right-hand sides of (100) all lie in L p (p > 2). The proof of L e m m a 8.3 already shows that Wq ~ W2'2(~2), q = 1, 2. Thus the Sobolev embedding indicates that Wq ~ WJ'P(I~ 2) for p > 2, q = 1,2. Hence the elliptic LP-estimates imply that Wq E w e ' p ( ~ 2) for any p > 2, q = 1, 2. Consequently ]Vwq(X)[ ~ 0 as Ixl ~ ~ , q = 1, 2, as expected. [] Let 3.1, ~.e > 0 be the eigenvalues of the positive definite matrix M defined in (76) and set )~0 -- 2rain{Z1, ~-2}. From Wl, w2 in L e m m a 8.3, we get the pair v j, v2 by (82). Then a solution (u r, u2) is obtained as a solution of (77) on the full R 2.

Lemma 8.5. For the pair u l, u2 stated above, there holds the exponential decay estimate u2(x) + u2(x) < C(s)e-O-s)vcZSIxl when Ixl is sufficiently large, where s ~ (0, 1) is arbitrary and C(s) > 0 is a constant.

Proof Let O be an orthogonal 2 x 2 matrix so that OrMO=()q 0

O) )~2 "

(l 1o)

Y. Yang/PhysicaD 101 (1997) 55-94

91

C o n s i d e r (77) over ~2 outside the disk DR0 = {x E ~2 I lxl _< go} where R > max{IPjl, Iqjl}. J Then (77) takes the truncated form

Aul = aliui + al2u2 + all(e ut -- 1 -- u l ) + a~2(e u2 -- 1 -- u2), Au2 = a21ul + a22u2 + a 2 1 ( e u~ -- 1 -- Ul) + a22(e u2 -- 1 -- u2),

(1 11)

Introduce a new set of variables Ui,/-/2 so that

(::) Then, using (1 10) and (11 1), we get

AUt = XIUI + b l l ( x ) U l +bl2(x)U2,

AU2 = )~2U2 +b2j(x)Ui +b22(x)U2,

(112)

where bjk d e p e n d on UI, U2 and bjk(x) --~ 0 (j, k = 1,2) as Ixl ~ ~ because UI, U2 enjoy this property by virtue of L e m m a 8.3. Therefore we arrive from (1 12) at the elliptic inequality +

>_ Z o ( U , +

where b(x) -+ 0 as Ixl ~

-

+

oo. Consequently, for any s > 0, there exits a suitable Ro > 0 so that

A(U 2 + U2) > (1 - Is)),o(U2 + U2),

Ixl > Ro.

(1 13)

Thus it follows from using a suitable c o m p a r i s o n function and applying the m a x i m u m principle in ( 1 13) that there is a constant C(e) > 0 to make

U((x) + U~(x) <_ C ( e ) e - ( l - e ) ' / ~ l x l

(114)

hold for Ix[ > R0. Since O is orthogonal, we have u~ + u~ = U 2 + U 2. Using (114), we see that the l e m m a is proven. []

Lemma 8.6. For u l, u2 in L e m m a 8.5, we have in addition IVul 12 -}- IVu212 _< C(e)e-¢l-e),/~lxl where e, C(e) are as described there.

Proof. Differentiate (1 1 1) outside DR. We have A(Ojul) = all(~jUl) + al2(Oju2) + aml(e u' -- l ) ( 0 j u l ) + a l 2 ( e u2 -- i)(Oju2), Z~(0jtt2) ~--- a21 (OjUl) -~- a22(Oju2) q- a21 (e "l - I)(OjU 1) q- a22( eu2 -- 1)(OjU2). C o m p a r i n g (1 15) with (i 1 1) and using L e m m a 8.4, we see that the stated decay estimate holds.

(!15)

[]

8.4. Multivortices and quantized fluxes on ~2 We can now construct the solutions of the dual system (72) over the full ~ 2 using u l , u2. In fact, let O)p(x) (N = N i ) and fOq(X) ( N = N 2 ) be as defined in (19) on ~2. Set ~01(x) = exP(½Ul(X) - i(Op(X)) and ~ 2 ( x ) =

Y. Yang/Physica D I01 (1997) 55-94

92

exP(½U2(X) + Oq(X)). Then ~Pl, lP2 a r e smooth functions which vanish precisely at Pl, " " , PNt and qI, " " , qN2, 3 Bj, Bj by solving the linear system respectively. With these functions, we obtain A j,

gA: - g'YIBj - glY;Bj = "(l)vtj ,

gA; - g'Y2Bj - glY~Bj =c~) 2)

g'g' (Yl Y'~ - Y2Y~)A 3 + ggl (Y~ - Y[)Bj - g'g(Y2 - Y, )Bj = O, j=

(116)

1,2,

where we define o~(J) I = rtt{2ia t In ~p~},

~ ' ) = -~{2i0 t In ¢~},

oil 2) = -.qt{2i0* In ~2},

o4 2) = -,~{2iO + In ~ 2 } .

Recall that condition (94) ensures that (116) has a unique solution. The quintuplet (¢Sl, ¢s2, A j3, Bj, /~j) solves (72) on entire ~2. Besides, using Lemmas 8.5 and 8.6, the system (72), and the equations

ID(q)cbql 2 = l e u q l v u q l 2 ,

q=

1,2,

we find the following exponential type decay estimates for the physical fields: I~bql 2 -

llJ2 ,

F?2,

612,

Gi2 = O(e-4'~lxl/2),

(117) [o(q)~ql2 = O ( e - ' / ~ l x l ) ,

q = 1, 2,

where the precise meaning of h(x) = O ( e - ~ l x l ) , for example, is that, for any e c (0, 1), there is a constant C(e) > 0 to make Ih(x)l < C(e)e - ( l - E ) ~ t x l valid. These estimates imply that the solution just constructed carries a finite energy. Next, we calculate the fluxes.

Lemma 8.7. Let v l, P2 be the solution of (98)just obtained. Then f Avi dx ~2

f I Al)2 dx = O. ,) ~2

Proof For vi, we have vl = ui - u 0. ' Hence, in view of L e m m a 8.5, we have IVvll =O(Ix1-3) as Ixl --+ because IVu&l decays like that at infinity. Using the divergence theorem, we easily show that Avj has zero integral on R2. The same is true for v2. [] Applying L e m m a 8.7 and (99) in (98), we find ]~0jl 2 -

dx =

(e u~ - 1)dx = 7 - ~ 7 - ( a l 2 N 2 - a22Ni), ~2

s(

R2

17x21' 2 -

-D -s dx = 002 2

~2

(e u2 - l ) d x =

[M[

(al2Ni - ailN2).

Y. Yang/Physica D lOl (19971 55-94

93

Integrating the last three equations in (72) over R 2 and using the above results, we see that the fluxes cl)r., ~ ; , ~ are given by the same formulas (95) as for the fluxes in a periodic cell domain. This is certainly an interesting property. We summarize the study of this section as follows. Theorem 8.8. Consider the dual system (72) in the entire ~2 arising in the extended Weinberg-Salam model with two-Higgs doublets. Given any prescribed points Pl, "'", PNI and ql, " " , qN:, the system (72) has a unique finiteenergy multivortex solution (~Pl, 42, A 3, Bj, Bj) so that ~Pl and ~P2 vanish precisely at the points Pt, " " , PN, and q l , ' " , qN: and their winding numbers (the topological vortex charges) around a circle near infinity are NI and N2, respectively. Furthermore the physical field strengths decay exponentially at infinity according to (1171 and the weak and the two electromagnetic fluxes over [~2 obey the same quantization property (95) as that for periodic vortices. We now conclude the paper with a few remarks. (i) For dual equations (1 l) in the standard model of Weinberg-Salam, although we obtained necessary and sufficient conditions tbr the existence of doubly periodic vortices when the vortex number N = I, 2, we only established sufficiency conditions when N >_ 3. The basic obstacle in achieving a better understanding of this existence problem arises from the optimal constant in the Trudinger-Moser inequality over a compact 2-surface. We may expect to improve our existence theorem by looking for saddle points of the corresponding constrained variational problem. (ii) It is seen that the dual equations (1 1) cannot produce finite-energy solutions over ~2 unless gravitational effect is considered. It may be important to investigate this problem when the color group SU(3) is also put into the coupling in the context of grand unified theory. (iii) The connection of the cosmological constant A with the non-Abelian gauge theory may turn out to be a general feature since when the universe is in a phase dominated by nuclear interactions the high energy density naturally leads to a positive A. Hence the metric becomes noncomplete and decays rapidly at infinity which makes the existence of finite-energy vortices possible. In view of this property, it would be interesting to study the existence of vortices in non-Bogomol'nyi phase, even tor radially symmetric solutions. (iv) The cosmological constant formula (34) takes the same tk~rm as the one derived in [38] without the topological winding of the Higgs scalar ~p. Whether such a phenomenon indicates that cosmology is indifferent between the classical unitary gauge and the generalized unitary gauge is an interesting question. (v) Finally, we point out that the new nonlinear elliptic equations (40) seem to present intriguing challenges to mathematical analysts at the moment.

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