Possible classical solutions in the Weinberg-Salam model

Volume 93B, number 1,2 PHYSICS LETTERS 2 June 1980

POSSIBLE CLASSICAL SOLUTIONS IN THE WEINBERG-SALAM MODEL

Vikram SONI Department of Theoretical Physics, University of Madras, Madras 600 025, India

Received 30 April 1979 Revised manuscript received 28 January 1980

Classical solutions to a spontaneously broken SU2 × U1 theory are considered in the absence of strings, using the for- malism introduced by Nambu. A different space dependent Higgs vacuum is found with interesting properties under space inversion. An intuitive method is used to calculate the characteristic length and energy (finite) of the extended objects that arise from this theory. Though these objects do not have topological stability they could yet be realised as local minima of the energy without an absolute stability against decay. Electric charge is introduced and some interesting features of these solutions are discussed.

This work is an attempt to find static solutions in the SU 2 X U 1 theory of Weinberg and Salam that have finite energy and like the Po lyakov- ' t Hooft monopole are solutions that explicitly avoid the introduction of strings. Nambu [ 1 ] has looked at monopole-like configurations in this theory which necessarily give rise to strings. Nambu goes on to construct a rotating dumbell object in which an SU 2 monopole and anti-monopole are tethered by a string. This object owes its stability to a fixed angular momentum constraint. It is clear that spherically symmetric classical solutions in the absence of strings are ruled out.

We thus look for static solutions that are not spherically symmetric and also do not have strings. We begin with a Higgs vacuum asymptotically given by the isodoublet

cos 0 ) ¢ = F

\sin 0 e i¢

The projection on SO 3 internal space is given by

= q~+~¢ = (sin 20 cos ¢, sin 20 sin ¢, cos 20}.

It is well known that we must satisfy [ ¢12 = F 2 where the constant F is determined on spontaneously breaking the symmetry by giving ¢ a vacuum expectation value to minimize the potential, and Due = 0 where Du stands for the covariant derivative. We go on to look for the asymptotic solutions for the gauge fields using the elegant prescription introduced by Nambu [ 1 ].

Following Nambu, we must satisfy for the Higgs isodoublet

D : - - + -ig iAi. , . , 0 - Au) , (1)

where A/u are the nonabelian (SU2) gauge fields and A0.the abelian (U1) gauge field that couple minimally to with the coupling strengths g and g ' , respectively, and r * are the familiar isospin Pauli matrices.

The following Fierz identities hold

= (:+:)z, =

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Volume 93B, number 1,2 PHYSICS LETTERS

(++rio)(++a~u+) - ++~++ri~u+ ) =i(++rJ+)au(++rk+)eil'k .

We can then solve eq. (1) for the gauge fields

gAiu+++ + g'(++ri+)A 0 =-i++rtauO'++,

and from eqs. (2)

= -i(++ri+/I +12)(++~'u+) - (++ri+/I+12) au(++rk+)ei/k.

This enables us to write down the expressions for the fields (as in ref. [1]):

1+14gau = - O × at*O- i(1 - r/)0(++a'-'u+), 1+12g'A ° =-iT(++*'au+),

2 June 1980

(2)

(3)

(4)

where r/is a (free) parameter that can vary from 0 to 1 and 0 = (++x+). We also have the usual definitions

Fiuv_ i _ ~ v A i + e - A j a k 0 = a u A 0 o - OuAv OK **+*v , F~v - OvA" . (5)

Asymptotically, we then have

, 0 g F~v = , f . v ,

gFL=- t (++++i (6) u\ 1+12 ] \ i + l z l e 6 k + ( 1 - ~ ) f ~ = ~ l + l z ] u=\ 1+12 1 ,

where

ft*v = -2 i (au++ av+ - av+ + at,+)/I+ 12 ,

which leads to

gFiu v + g'FOv(+ + ri+/ I +12) = 0 . (7)

But

gFiuv(++ri+/l+12) +g,FOv = (g2 + g 1,2 1/2F~v,Z

where FZv is identified with the neutral massive boson Z0, defined in a covariant way. The electromagnetic field is then identified with

t-(em)= [g,Fiuv(++ri+/l+12)_ 0 2 = g'2)l/2/g]FOuv (8) Jt*v gFuv]/(g +g,2)1/2 [r/(g2 +

Having catalogued the various relevant fields we write down first the asymptotic fields corresponding to a static solution for

( c°s0 / + = F

\sin 0 ei¢'/ '

wi thA~ = A 0 = 0, for the time being,

gAiu = 2(1 - r/)ee r - I sin 0 (er cos 0 + e0 sin O) i - 2e O r - I cos 0 (er sin 0 - e0 cos O) i - 2e o r - l ( e ~ ) i , (9)

where the unhatted unit vectors refer to space indices and the hatted vectors, in parentheses, are reserved for is+spin indices. Similarly,

g'A '0 = 27 e~ r -1 sin 0 . (10)

It is clear that no strings appear in any of the fields. Also, it has been verified that these solutions satisfy the field equations asymptotically as r -+ oo. As we shall see, for r7 = 0 these solutions, without modification, lead to a new

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vacuum. These are the asymptotic expressions for the fields. When looking for finite energy solutions that are localised in space we introduce appropriate multiplicative functions to generalise the asymptotic solutions. It must be pointed out that, although the solution is not spherically symmetric, the simplest choice for these functions is to have only a radial dependence. We choose ~b' = f ( r ) G such that f ( r -~ oo) = 1. It has been ensured that, although the gauge fields involve derivatives of ~, the relations (4), (9) and (10) go over unchanged from the asymptotic case [f(r) = 1 ] to the case with any general f(r). We, therefore, have to use an independent multiplicative function g(r) for the gauge fields which must also satisfy g(r -~ oo) = 1 and g(0) = 0. The r -* oo condition is obvious, since the solution must have a finite energy, the r ~ 0 condition is to regulate the gauge fields at the origin to avoid any divergence in the energy. Also, for simplicity, we have used the same multiplicative function for A i and A 0. Thus, A 'i = g(r)A i and A '0 = g(r)A O. Such a choice of f ( r ) and g(r) will preserve all the asymptotic features, so without further ado we write down the hamiltonian in terms of these fields:

u=f ~FiuvFiuv+~FOvFO v +~Duc~'+Ducb ' + ~X2(q~'+$ ' - 1) 2 d3x . (11)

We have chosen (~)= 1 (as in ref. [1 ]) which simply sets the energy scale at 246 GeV. Also, we have the relations M W = g[2 and M H = ~, where M W and M H are the masses of the intermediate vector boson and the Higgs particle, respectively. Now we compute the constituent parts of H.

Let B i = ~elrnnFin and B 0 = XelrnnFO n . It follows that

gB i = 4r-2er{- (g(r )~ l + g(r)[g(r) - 1]) cos 0 (er cos 0 + e0 sin O) i

-- (1 - ~)g(r) [g(r) - 11 sin 0 (Co cos 0 - er sin O) i }

+ 2(dg(r) /dr)r -1 e o [ - (1 - 77) sin 0 (er cos 0 + eo sin O) i + cos 0 (er sin 0 - eo cos O) i]

+ 2 (dg(r)/dr) r - l e ~ ( - ~ ) i , (12)

and similarly

g'B 0 = 4 f i r - l e t cos 0 g(r) + e o 2~lr- l ( dg(r)/ dr) sin 0 . (13)

Then

¼(FiuvFiuv+ FOuFOu)= ~(B i" B i + e O . B ° ) : G 2 , (14)

a2 = g2(O cos20 + sin20 + 16 g2(r) - g ( O l 2 sinE0 g2r2

2[g(r) - 1]{cos20 + [g(r) 1] sin20}g2(r) + ~ r ) } sin20 ~ g 2 ~ 4

+ 16.__~1 g 2 ( r ) [ g ( r ) _ l ] 2 + 4 [.dg(r)] 2 2g2r4 g -~r 2 [ _ ~ J '

and

DutY+Dr --ure5'= ½{[df(r)/dr] 2 + 2 f2 ( r ) r -2 [1 - g(r)] 2} , ~;~2(q~'+0' - 1) = ~k2 [f2(r) - 1] 2 . (15)

Thus

H = f r 2 dr dg2 {G 2 + } { [df(r)/dr] 2 + 2 f2(r ) r -2 [1 - g(r)] 2} + ~_X2 [ f2(r ) _ 112}. (16)

First, we point out that there exists a zero energy solution in this theory. That is the asymptotic solution (9), (10)

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which follows trivially on taking g(r) = 1 and f(r) = 1 over all space, and then minimizing H with respect to r/. We find r7 = 0 and thus G 2 = 0. The contribution from the rest of H also vanishes. We have, in other words, a different vacuum. In this vacuum the asymptotic boundary conditions are space dependent and the gauge fields singular at the origin - nevertheless it is a vacuum. We defer elaboration on this point till later.

At this point one can write down the Euler-Lagrange equations for g(r) and f(r) and solve for a local minimum of H. Such a solution would be an extended object in space. However, the solution of such a complicated nonlinear equation is rather forbidding. So, here, we take leave of a formal solution and propose an intuitive way of looking for the existence of and estimating the mass of such an extended object. Such an object must have a characteristic length - let this length be D. We then propose that we divide the space into the regions 0 < r < D and D < r < o~. For the latter we may roughly take the asymptotic expression for g(r) and f(r). For the former it is clear that g(r ~ 0) goes to the origin at least as r to have a finite energy (also, there is no reason to expect 77 = 0 again; this will be determined by minimizing the energy with respect to r~). For the region 0 < r < D we propose the simple parameterization that g<(r) is a polynomial in r/D, and similarly for f<(r) . This polynomial must meet the condition that g<(D) = f<(D) = 1, where g<(r) and f< (r) are the functions in the region 0 < r < D and g>(r) and f> (r) are the functions in the region D < r < oo.

The simplest choices for g<(r) are (i) g<(r) = riD and (ii) g<(r) = r2/D2; with each of these choices for g<(r) we have used both f(r) = 1 and f(r) = riD. We have been totally cavalier, in that we have ignored the formality of even letting our solution come in smoothly to the origin, and also connect Smoothly to the asymptotic solution - but these contributions should not be significant. Of course, this can be rectified by using forms like 1 - e-r /D or tanh(r/D) for the functions.

The calculation is done in three steps. Firstly, we integrate out for the energy in terms of the given parameteriza- tions ofg(r) and f(r). This gives us a functional, dependent on 77 and D. We then minimize this functional with respect to ~ and substitute back to obtain an expression for-the energy depending simply on D. This is then mini- mized with respect to D and the value of D thus obtained gives us the characteristic length. Finally, the value of the energy follows on substituting this value o l D into the energy expression.

(i) g<(r) = r/D, f<(r ) = r/D, g>(r) = 1 and f>(r ) = 1. On substituting these forms for the functions above and

integrating we obtain

H = [(M2wD)-l(21.8 ~ 2 - 3.1 r? + 6.67) + 0 . 2 D + 0.01X2D3] 4rr.

On minimising with respect to r/we get ~7 = 0.07 which yields

n = 47r[(M20) -1 1.84 + 0 . 2 0 + 0.01 a2M2w D] ,

where we have substituted M w = g/2 and a = MH/M w. Minimising H with respect to D we finally get MwD ~ 2.25 where a has been set equal to 1 and H = E(4n/Mw) X 1.46 (with (~b) = 1) or alternatively E ~ 4#(Mw/e2) X 4 × 1.46.

(ii) g<(r) = r/D, f<( r ) = 1, g>_(r) = 1 and f>( r ) = 1. Similarly we get MwD ~ 2.35 and E = (4rr/Mw) X 1.55 or

alternatively E = 47r(Mw/e 2) X ~- × 1.55. (iii) g<(r) = r2/D 2, f<( r ) = r/D, g>(r) = 1 and f>( r ) = 1. a = 1 gives MwD ~ 2 and E ~ (4rr/Mw) × 1.26 or,

alternatively, E ~ 4rr(Mw/e 2) × 4 X 1.26. (iv) g<(r) = r2/D 2, f<(r)4--- 1, f>( r ) = g>(r) = 1. This case gives MwD ~ 1.6 and E ~ (4rr/M w) × 1.68 or, alter-

natively, E ~ 4n(Mw/e 2) × ~ X 1.68. Cases (ii) and (iv) are for f<( r ) = 1 which also implies X = 0 and a = 0. For (i) and (iii) we have used a =MH/M w

= 1. As the results indicate, there is only a small variation from the dependence on this parameter. (As in ref. [1]

we take sin20w = ~ ,z . ) As a test case we have applied this method to the 't Hoof t -Polyakov monopole. Using g<(r) = riD and f<( r )

= r/D gives a value for the mass of the monopole ~(4/e2)Mw X 1.4 (very close to the exact one).

, i The existence of such solutions is borne out by Derrick's theorem [2] (see discussion).

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Firstly, we want to point out that the asymptot ic expressions for the Higgs and gauge fields reduce H to

H = f r 2 d r d ~ 2 G 2 = f r 2 d r d ~ 7 1 2 5 - ~ - ~ 2 c o s 2 0 .

It is clear that the value ~ = 0 will give the minimum energy which is, of course, E = O. This must be an absolute minimum and is thus a space dependent vacuum, with a Higgs configuration

" cos 0 ) qS= IO) s = F

\s in 0 e iO

Unlike the usual space independent Higgs vacuum ~ = [ 0) = F(1 ) , where the gauge fields are trivially zero, in this case we have nontrivial SU 2 gauge fields that are singular at the origin (which, however, do not give rise to physical fields). This could well be associated with a pure gauge, for the gauge transformation U = cos 0 + i a • ( i × t:) will

take ~b = 10) to ~b = [0)s. This gauge transformation is singular at the origin. The most interesting proper ty of the Higgs configuration of the new vacuum, 10)s, is with respect to the space inversion operation. Let 10) be even under space inversion, by convention. I 0) s, however, is space dependent and has the obvious proper ty under space inversion ,2

~--,0+n 10) s , _ 10) s .

0 ~ r r - 0

This leads to the interesting possibili ty of pari ty violation arising from the vacuum. This question needs more elaborate at tention, especially in the context of quantum states and operators, and will be addressed separately. Further, I 0) s does not carry any topological charge - otherwise, of course, it could never have zero energy. It seems that there are different classical vacua in most theories whose properties have yet to be explored. We also find that r? ~ 0 effectively removes the U 1 degrees of freedom and therefore this vacuum is in reality the complete- ly broken SU 2 vacuum. The difference is, that in broken SU 2 there is no unbroken generator and thus electromagnetic charge cannot be introduced into the vacuum as we shall briefly discuss.

It has been observed that there are two length scales in these extended object solutions, one set by the Higgs mass and the other by the W boson mass. The foregoing calculation shows, however, that the characteristic length, D, of the extended object is determined by both, A(MH) and M w together. In our calculation we have, for simplic- i ty, considered only fixed cz = MH/M w = 0 or 1.

Meanwhile, a digression. Putting 77 = 0, a priori, removes the U 1 degrees of freedom reducing a SU 2 ® U 1 theory to just SU 2. In a strong, broken SU 2 theory the mass of an extended object solution can be found by the method enunciated earlier with r /= 0 and the further modification g = g(strong) ~ 1. (Note: r7 = 0 by itself is not a suffi- cient condit ion for the vacuum solution (it just removes U1); for the vacuum solution we must have the asymptotic fields found before to be the fields over all space.) This results in a mass " rap where mp is the mass of the massive gauge boson in such a theory. I f such a particle may be speculatively associated with the p meson, then we fired the mass of such an extended object to be ~mp with a characteristic length 1~rap.

If we demand a static solution, charge can be introduced into this theory by choosingA~ and A 0 to be nonzero and space dependent. From the definition of 5 rem uv we can include a charge Q = 137eX' by solving for the A 0 fields which satisfy Gauss' law for Q = 137e~ ' and also satisfy DOS = 0. This leads to

A 0 = X'(g')-lr-mgO(r), Aio = -X'g-lr-lgO(r)(~r cos 0 + e0 sin 0) i ,

giving ~ em uv or E = (h'/e)r-2er which is E = 137eA'r-2er in conventional units. The energy can be calculated in a way similar to the chargeless case. Using a parameterization gO(r) = riD we get almost the same values for the masses of these charged extended objects as we did before. However, though charges can be thus introduced into

,2 Note also that U(x) has under space inversion the property: U(-x) = -U(k).

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a space independent vacuum, with the Higgs field ~ = 10) = F(01), we find that it cannot be localised - there is no solution for gO(r). This is because a given charge would like to distribute itself as thinly as possible giving a "D"

- characteristic length - that diverges. These effects shall be dwelt on at length in a later work. Finally, we come to the r/D, r2/D 2 parameterization. The reason for looking at g<(r) ~-- r2/D 2 is that the differ-

ential equation at the origin for g(r) cannot be satisfied by an riD dependence. As can be seen from the results, this gives a lower energy, though for a crude estimate g<(r) ~ riD is adequate. The rationale behind such a naive method has already been spelt out; also, it is found to work admirably for the Po lyakov- ' t Hooft monopole. One cannot rigorously prove that the existence of a D that minimizes E will yield a solution. But since no D could be found (except D -+ oo) in the case of the electrically charged "vacuum" ~ = 10), this method seems to be corrobo- rated by the correct physics.

We cannot argue for the absolute stability against decay of this state as it carries no conventional topological charge, but nevertheless the expectation of a local minimum means that such a state should be realizable. This is vindicated by the following considerations. Consider our hamiltonian H = - L , H = - L = G + {2) + {3), where

G= f(1F~vF~v+~FO~FOv) d3x, { 2 } = } f(D~,~D~,~+)d3x, (3}=~ 2 f(~+~-l)Zd3x;

G, {2} and {3) are all individually positive definite. For time independent solutions we consider a scale variation with one scale parameter for all fields since we are

interested in investigating the stability of the composite object. As the gauge fields transform like gradients we consider the following one-parameter family of field configurations (with A~ = A 0 = 0):

~ x ) = ( ~ ( ~ , x ) = ~ ( ~ x ) a a _ a , AO(x)=~AO(X,x)=XAO(Xx) , A i ( x ) = ~ A i ( ~ , x ) - X A i ( X x )

From Hamilton's principle L (or/4) must be stationary under such variations on the fields, or (aH/ah)x= 1 = 0. We get

H(X) = )~4-3G + ,'k2-3 {2) + X-3 { 3 ) , d H / d ~ = G - ) ~ - 2 { 2 ) - - 3 2 ~ - 4 { 3 ) , d2H/d)~2 = 2~-3 {2) + 12)~-5 {3)

Thus at X = 1 we find

dn/dXIx=l = G - {2) - 3 ( 3 ) , d2H/dX21 x= 1 = 2{2) + 12{3 ) .

Thus the condition (dH/dX)x= 1 = 0 can be satisfied by the appropriate values of G, {2) and {3), and furthermore, since (d2H/dX2)x= 1 is positive definite we have a minimum in E.

Lastly, in this context we are led to the following expectation. Just a charge (topological) by itself is only the first moment of a charge distribution. Even in the absence of charge, other moments can exist, e.g. dipole, quadru- pole etc. and to such will correspond solutions with different asymptotic boundary conditions in the same topological sector. There should thus be a hierarchy of such solutions which may give rise to different classical vacua, all of which may have to be taken into account to construct the actual vacuum. Even without topological charge they can, as we have shown, have other very interesting attributes.

The author thanks especially G. Rajasekaran and V. Srinivasan for discussions. N.D. Hari Dass and Romesh Kaul and members of the Theoretical Physics group at Madras University are acknowledged for useful exchanges.

References

[1] Y. Nambu, Nucl. Phys. B130 (1977) 505. [2] See S. Coleman, Classical lumps and their quantum descendants, Ettore Majorana Lectures (1975).

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Possible classical solutions in the Weinberg-Salam model