Doubly charged pseudo-goldstone bosons and dynamical SU(2) × U(1) breaking

Volume 182, number 2 PHYSICS LETTERS B 18 December 1986

DOUBLY CHARGED PSEUDO-GOLDSTONE BOSONS AND DYNAMICAL SU(2) x U( 1) BREAKING

R. Sekhar CHIVUKULA’ and Howard GEORGI

Department of Physics, Boston University, Boston, MA 02215, USA

and Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA

Received 10 October 1986

An alternative structure of electroweak symmetry breaking based on an SU(4)/SU(2) symmetry breaking pattern is

discussed. An effective chiral lagrangian describing the pseudoGoldstone bosons and gauge bosons is constructed and their

interactions are analyzed. It is shown that the model contains doubly charged pseudo-Goldstone bosons which couple to

two similarly charged W’s. The question of vacuum alignment is considered, and it is shown that if an additional discrete

symmetry is imposed, a custodial SU(2) preserving vacuum is a global minimum of the potential (for a range of coefficients

in the effective chiral lagrangian).

I. Introduction. No one knows what is responsible for breaking the SU(2) X U(1) gauge symmetry of the electroweak interactions. Until this is known it is important to explore different possibilities for the structure of SU(2) X U(1) breaking and to inves- tigate the corresponding range of possible experimen- tal signatures. A possible signature of interesting physics would be a doubly charged scalar particle coupling to pairs of similarly charged W’s [ 11.

In this paper we analyze a simple model of electroweak symmetry breaking in which such a coupling occurs. The model is based on the spontaneous breaking of an SU(4) symmetry to an SU(2) subgroup under which the fundamental 4 of SU(4) transforms as 3 d 1. Electroweak SU(2Xy X U(l)y is a gauged subgroup of SU(4) under which the fundamental transforms as 21j2 $2_,/, . Note that, since a subgroup of SU(4) is gauged, SU(4) is only an approximate global symmetry: instead of exact Goldstone bosons there are pseudoColdstone bosons (PGBs).

Using the methods of Coleman, Wess, Zumino, and Callan (CWZC) [2,3] an effective lagrangian describing the light particles (the PGBs and gauge bosons) can be constructed from this information alone. No assumption need be made about the dynamics

’ NSF Graduate Fellow.

that produces this symmetry breaking and none will be made here.

For this model to be phenomenologically accept- able the experimentally verified relation Mw = Mz cos Bw must be satisfied. This relation will hold if the unbroken SU(2) subgroup is “custodial” SU(2) - an approximate global symmetry (broken only by electromagnetism) under which the three gauge bosons of weak SU(2) transform as a triplet [4]. Whether the unbroken SU(2) is indeed custodial SU(2) is a question of vacuum alignment [S--7].

In the next section we assume that the unbroken SU(2) is custodial SU(2). We construct the PGB kinetic energy terms and consider the couplings of the PGBs and gauge bosons.

In the third section we analyze the stability of

custodial SU(2) preserving vacua by analyzing the potential: a set of nonderivative interactions constructed from the generators of the gauged symmetries. We show that, in models containing custodial SU(2) quintuplet PGBs whose doubly charged members couple to two similarly charged W’s, custodial SU(2) preserving vacua are not generally stable. In this model, however, we show that a discrete symmetry can be imposed which eliminates the instabil- ity .

In the fourth section we prove that a custodial

0370-2693/86/S 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

181


SU(2) preserving vacuum is a global minimum of the lowest order (in the gauge couplings) potential for a suitable range of coefficients in the effective lagrangian.

2. Kinetic energy terms. The generators of SU(4) can be represented as tensor products of Pauli matrices: ti X Z,Z X r6, ua X rb [under the commuting SU(2) subgroups generated by u X Z and Z X z the fundamental transforms as (2,2)]. It is convenient to choose the unbroken SU(2) to be generated by (a X Z) t (Z x r).

Following CWZC we construct the effective chiral lagrangian for the PGB fields from g = exp(in”X”), where the rr4 are the PGB fields and the Xa are a set of broken generators. Under the unbroken SU(2) the broken generators transform as 5 $3 @ 3 @ 1. In particular, the quintuplet generators are

Xl’ = a+ x T+ ) xg- = (x;+)t ,

x; = ; [(a’ x 73) + (a3 x T+)] )

x,- = (Xl)? )

x,0 = (l/G) [(u+ x r-) + (u- x r+) - (03 x r3)] ,

[a’ = (uI f iu2)/2, similarly for r+] ,

the triplet generators are

Eijkui x $ x3 =

axz-ZXI

2Jz ) (X3 3)’ = F<F ’

and the singlet generator is

ua x 3-a x, =- M-

(summed on a) .

The generators have been normalized such that TrXf(X!)t =6ii6ab.

The a/ction of an SU(4) transformation on .$ is given by

g ‘g&t g E SU(4) , h E unbroken SU(2) ,

(1)

(2)

(3)

(4)

where h is chosen such that g.$ht is of the same form as g: an exponential of broken generators only. For g E SU(4),g E unbroken SU(2), h depends on the Goldstone boson fields while for g E unbroken SU(2), h =g.

Consider first the construction of the kinetic energy terms for the Goldstone bosons in the ab- sence of the electroweak gauge symmetry. Under the transformation (4), the combination Et apt; transforms as

@aq+h(@aq)ht thaw2t . (5)

The inhomogeneous term in (5) is an element of the Lie algebra of the unbroken SU(2) symmetry. Tr ,$ t a p&J? transforms homogeneously for any broken generator Xa. The most general set of two derivative interactions invariant under (5) is

+ ;f;J$.J3p ‘Zf3f3’Js(‘.J3p

where fi , fs , f3, f3 1, and Z are arbitrary real coefficients *r and where

Jfp = -i Tr tl’ap&Yf . (7)

The terms quadratic in the Goldstone boson fields in &E are the kinetic energy terms for the Goldstone bosons.

By replacing ap in (7) by a covariant derivative Dp it is possible to gauge any subgroup of SU(4). The electroweak gauge symmetries may be generated by

Qa = (19 X Z)/2 and Q4 = (Z X r3)/2. In this case

Dr = ap t jgQaW: t ig’Q4&‘, (8)

where g and g’ are the coupling constants and IV,” and BF are the gauge fields of SU(2)w and U( l)r.

In general the true vacuum of the theory is < = lo for some C;u E SU(4), i.e., some of the Goldstone boson fields may have VEVs. To eliminate the VEVs it is necessary to perform an SU(4) transformation by ,$ . Eqs. (4) and

\ 8) imply that the gauge genera-

tors are then go Qa to . The orientation of the gauged subgroup with respect to the unbroken subgroup is given by the alignment of the vacuum - it is given by go. If .$,, = I, the unbroken SU(2) is custodial SU(2). For the remainder of this section it will be

*’ In order tainsure positivity of the PGB kinetic energy

terms, -1 <Z < 1.

182

Volume 182. number 2 PHYSICS LETTERS B I8 December 1986

assumed that &, = I is the true vacuum. Given the form of _-f& above it is convenient to

express .$ as

E = expMnLXL/fL + ns""gblfs + “3 *X,/f3

+“3’-X3’/f3’)1 . (9)

The fields rrL and rr5 ab then have properly normalized kinetic energy terms. The terms in &E can be expanded to give the interactions of the Goldstone bosons among themselves and with the gauge bosons.

The contributions from (Jj$)2 with no Goldstone bosons give rise to a gauge boson mass term

: f; (gw; - gW3B/J2 . (10)

Hence the usual formulae

M& = e2v2/4 sin28 w, Mw=Mzcos8,, (11)

hold if 2fi = v2 = (250 GeV)2. There is a one pion, one gauge boson mixing term

coming from the (J$)2 and J$ *J3 r,, terms:

(l/~)(gW~ -g’6i3B”)lf3 a,~: +Zf3 a,~;‘] . (12)

As usual, this term can be eliminated by choosing an appropriate (“unitary”) gauge [5]. Only one custodial SU(2) triplet Goldstone boson field is physical.

Consider the transformation properties of the kinetic energy terms under a discrete G-parity symmetry, under which .$ transforms as

g + (02 x 72) (*(a2 x 72) . (13)

Under G-parity rrI , n31, and rr5 are odd while rr3 is even. All the kinetic energy terms in eq. (6) except

for the J$’ *J3 r,, cross term respect G-parity. The gauge bosons are even under G-parity.

The quintuplet Goldstone bosons include a com- plex conjugate pair of doubly charged scalars: rrl+, 7r-. A coupling between these and two similarly charged W’s is G-parity violating. Hence any such coupling must come from the J$ *J3 gcl cross term. Expanding this term to first order in the Goldstone boson fields and second order in the gauge fields yields

/&+(W-)p(W-), + h.c. (14)

where g = g2Zf3 f3 l/2f5 . As advertised this model contains a doubly charged scalar which couples directly to two similarly charged W’s.

The J$’ *J3 ecl term also yields a coupling of rrL to w+w- :

P’qw+>vup 9 (14’)

where ,u’ = 2g2Zf3 f3 l/fifl. While this coupling re- sembles that of a Higgs particle in a fundamental doublet Higgs model, the rrL field is not a Higgs particle and its couplings are not completely determined by the W and Z masses and the electroweak gauge

couplings. The dimensional analysis arguments in ref. [3]

imply that p and cc’ in eqs. (14) and (14’) are of order

g&v.

3. Stability of the vacuum. The question of vacuum alignment may be addressed by constructing a

potential: the vacuum is determined by minimizing the potential. The potential is a set of nonderivative terms constructed from .$ and the gauge generators. Eqs. (4) and (8) imply that the potential should be invariant under simultaneously performing an SU(4) transformation g on ,$ and replacing the generators Qa with gQag+ .

The most general potential at order g2 (g’2) is

I’(.$) = af4 (Tr AaX$ [Tr Aa (X$t ]

+ Pf4(Tr AaXI)2

(15)

(sums on the indices i and j ofX5 and a sum on a = 1,2,3,4 of Aa are implied) where 0, /3, and rii are

arbitrary coefficients, f is a dimensional constant of

orderfi ,f3 ,f3’, orfs and

Aa =g$Qal, A4 =g’.@Q4t,

r3 =axJ+Jxr/2fi. (16)

Note that since the potential has no derivatives there is no analogue of the inhomogeneous term in eq. (6) under an SU(4) transformation and the unbroken generators r3 may be used.

Consider ,$ = I as a possible vacuum. If, when g is expanded about I, there exist tadpoles [terms in V(l) linear in the Goldstone boson field] , then f = I is un-

183


stable and is not the true vacuum of the theory. The terms in V(t) built from the weak SU(2) gen-

erators do not violate custodial W(2) while terms built from the hypercharge generator do: Z X r3 transforms as a member of a custodial W(2) triplet. Since Z X r3 appears symmetrically in V(t), the terms in I’(.$) transform either as singlets or as members of quintuplets. Since C; = Z is a custodial SU(2) preserving vacuum, this implies that the only fields that can have tadpoles are the singlet and quintuplet fields.

Tadpoles for nI or rr5 violate G-parity. Hence there occur only the G-parity violating y12 and ~32 terms in eq. (15). Terms of the form Tr AO(u X Z)*Tr AaX,! (a = 1,2,3) contain rrI tadpoles and terms of the form Tr A4 (Z X s )*Tr A4X3 1 contain both n’l and rri tadpoles. This implies that the vacuum is unstable in the I$ direction. A VEV for the rrt field would be a phenomenological disaster: it would give large contributions to p - 1 (p = M$/Mi cos20w).

This is a general feature of models containing quintuplet Goldstone bosons which lack a G-parity symmetry: terms in the potential involving the hypercharge generator tend to destabilize the vacuum in the neutral quintuplet direction. In general, therefore, these models are phenomenologically unacceptable.

Simply imposing the G-parity symmetry would eliminate the interesting rr~‘(W-)p(W-), coupling. In this model, however, it is possible to impose a discrete symmetry other than G-parity which restricts the form of the potential and eliminates the offending tadpoles. The discrete symmetry is the symmetry under which the two SU(2) subgroups of SU(4) are interchanged: e X Z *I X s , and analogously for all

the SU(4) generators. The gauged SU(2) X U( 1) electroweak symmetry breaks this u+* r symmetry - however, the potential must break the symmetry in exactly the same way. This implies that y13 (= yjl) and ~23 = (~32) are zero. Note that the kinetic energy terms of the previous section respect this discrete symmetry: Jr and (Jy)” are even while both J$ and J$ are odd.

The only dangerous term remaining is the y12 (yzl) term. Consider the following change of variable:

t = t exp(ioX1) . (17)

XI is a singlet under the unbroken SU(2) and is even under e+tt . Hence { and E’ transform the same way under all the symmetries and (in particular) .$’ = Z is

184

a custodial SU(2) preserving vacuum for any (Y. XI commutes with X5 and itself and therefore the OL and /3 terms in the potential are the same for .$” as for .$. Under conjugation by exp(ioX1), X3 and X3 I rotate into one and other as a two vector under SO(2). Such a change of variable can therefore be used to eliminate the y12 (7~~) cross term in V(.$).

Performing this change of variable and relabeling .$’ by ,$ yields a potential of the form in eq. (15) with ~12 = y13 = ~23 = 0. This potential respects G-parity! Since the potential respects G-parity, there can be no tadpoles for rl and “50. In this basis .$u = Z [a custodial SU(2) preserving vacuum] is an extremum of the potential.

Under the change of variable in eq. (17) J.$’ and J.$‘e also rotate into one and other as a two-vector under SO(2). There is no reason, however, to assume that the rotation which eliminates the r12 cross term will eliminate the corresponding Jf *J3 I~ term in

%E. To summarize: the b*r symmetry was imposed

so as to restrict the form of the potential. The lowest order (g2 or g’2) potential respecting this symmetry possesses an accidental G-parity symmetry, and this G-parity symmetry insures that t = Z is an extremum of the potential. G-parity is not a symmetry of the kinetic energy terms nor is it, in general, a symmetry of higher order terms in the potential *’ .

4. Minimizing the potential. Incorporating the field redefinition in eq. (17) the lowest order potential can be written as

V(g) =(a + 2161)f4(Tr AaX!) [Tr AQ(XY)t]

+ (p’ + 216 l)f4 (Tr AQX,)2

t (ri2 + 2 I6 l)f4 (Tr AaX 9)2

t6(TrAQaXZ)*(TrAQZXr). (18)

Here terms of the form [Tr An (u X Z)] 2 t [Tr Aa (Z X r)] 2 (from the yll and y33 terms) have been elimmated in favor of the terms of the form (Tr AQXy) [Tr Aa (X$t ] + (Tr AaX ,)2 t (Tr AQX1)2 and constants independent oft. This

*’ This will give rise to corrections to p-l of order

aem/dn.

Volume 182. number 2 PHYSICS LETTERS B 18 December 1986

was done using the SU(4) Fierz identity 1Tr $‘Qa~(eXr)*Tr $Q”[(lXz)l

C(TrPTa)2 = TrP2 -i (TrP)2 , a

(19) < 2(Tr .$’ QQ gX!$ [Tr Et Q[(X$t ]

where the To are a complete set of SU(4) generators ((OX 1)/2,(1X t)/2,XI ,X,.,Xy) normalized such that Tr TaTb = 6Qb and P is an arbitrary 4 X 4 matrix (Au in this case).

When [ is equal to I, V(t) vanishes. Furthermore, we shall show that V(l) is positive definite if (Y’, p’, and y;2 are positive - this implies that .$ = I [a custodial SU(2) conserving vacuum] is aglobal minimum of the (lowest order) potential.

If a’, p’, and yi2 are all positive the first three terms are manifestly positive definite (they can be written as squares of traces of hermitian matrices). The last term, though not positive definite, is bounded.

Consider 1Tr @‘QQE(o’ X I t1 X ri)l for a given a. For fixed .$ the matrix .$ Qa[ has a projection onto particular directions it, *(a X I f I X 5) (n, are unit vectors) and no projection onto the orthogonal directions. Then

t 2(Tr $Qa.$X1)2 + 2(Tr @‘QQ{X3t)2 . (24)

Hence V(t) is positive definite if (Y’$, and y;z are all positive, and therefore t = I is a global minimum of the potential if these conditions are satisfied.

The terms in the potential give rise to mass terms for the pseudoGoldstone bososn. The dimensional analysis arguments in ref. [3] imply that LY’, S’, y&, and 6 are all of order one. This implies that the pseudoGoldstone boson masses are all of order Mw .

q [Tr .$Qat(ui XI *IX r’)12

= ITr $Qa!$+*(uX1?lX t)12 .

Now consider the following inequality:

(20)

5. Conclusions. An alternative structure of electroweak symmetry breaking based on an SU(4)/SU(2) symmetry breaking pattern was discussed. An effective lagrangian describing the pseudoGoldstone bosons and gauge bosom was constructed. It was shown that the relation Mw = M, cos 0, holds for a custodial SU(2) preserving vacuum and that the model contains doubly charged pseudoColdstone bosons which couple directly to two similarly charged W’s.

ITrMNI G c l$ll~l, i

(21)

where the (9) and (4) are the eigenvalues of the hermitian matricesM and N respectively ordered such

that IX, I 2 IA, I > . . . . lpI I> Ip2 I > . . . . For any unit vectors n,, it - (U X I + I X t ) have the same eigenval- uesas(u3 X1*1X r3): 2,0,0,-2.Thematrix @QQ.$ has eigenvalues l/2, l/2, -l/2, -l/2. Eqs. (20) and (21) then imply

The question of vacuum alignment was addressed by constructing a potential. It was shown that (for a range of coefficients in the effective lagrangian) a custodial SU(2) conserving vacuum is a global minimum of the lowest order U++T invariant potential, and that the pseudoGoldstone bosons have a mass

of order M,.

TITr$QQt(oiXI-+IX ri)12

~(~‘2+:.2+:.0+~.0)2~4. (22)

Expanding the square in eq. (22) yields

+2(Tr$QQ~eXZ)*(Tr @QQglXs)

<4-(Tr~~QQ~aXZ)2-(Tr@‘Qa~1Xt)2. (23)

Using the Fierz identity (19) this implies that

Several questions remain to be explored: Can a technicolor theory based on SU(4)/SU(2) be constructed, i.e., is there a strongly interacting theory with a SU(4)/SU(2) chiral symmetry breaking pattern (respecting the e+ e symmetry but not G-parity) *’ ? Can an extended technicolor theory be constructed utilizing an SU(4)/SU(2) technicolor sector? If so, are there interesting fermion-pseudoGoldstone boson interactions?

We gratefully acknowledge discussions with Michael Chanowitz, Junegone Chay, Jonathan Flynn, Mitchell Golden, David Kosower, and Lisa Randall. This research is supported in part by the National _ Science Foundation under Grant No. PHY-82-15249.

*3 Progress has been made on this question [ 81.

185


References

[l] H. Georgi and M. Machacek, Nucl. Phys. B 262 (1985)

463.

[2] S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177

(1968) 2239;

C.G. Callan, S. Coleman, J. Wess and-B. Zumino, Phys.

Rev. 177 (1968) 2247.

[ 31 H. Georgi, lectures at Les Houches Summer School

(1985).

[4] P. Sikivie, L. Susskind, M. Voloshin and V. Zakharov,

Nucl. Phys. B 173 (1980) 189.

[5] S. Weinberg, Phys. Rev. D 13 (1976) 974. [6] M.E. Peskin, Nucl. Phys. B 175 (1980) 197. [7] J. Preskill, Nucl. Phys. B 177 (1981) 21.

[ 81 D. Kosower, private communication.

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Doubly charged pseudo-goldstone bosons and dynamical SU(2) × U(1) breaking