Geometrical hierarchy and spontaneous symmetry breaking

Volume 191, number 1,2 PHYSICS LETTERS B 4 June 1987

GEOMETRICAL HIERARCHY AND SPONTANEOUS SYMMETRY BREAKING

K. FARAKOS a, G. KOUTSOUMBAS b,l, M. S U R R I D G E c and G. ZOUPANOS Physics Department, National Technical University, Athens, Greece

b CERN, CH-1211 Geneva 23, Switzerland " Physics Department, Amsterdam University, Amsterdam, The Netherlands

Received 12 Februari 1987

A four-dimensional gauge theory, where Higgs fields and the corresponding potential appear naturally, is obtained by dimensionally reducing a pure gauge theory over a compact coset space S/R. We show, using an explicit example, that a hierarchy of the scales in the coset space can change the spontaneous symmetry breaking of the four-dimensional gauge theory.

The presently popular version of the old idea of unification o f all interactions is that such a unification is realized in higher dimensions. The most ambitious program in this direction is based on generalizations of the old Kaluza-Klein idea [ 1 ], while many new ideas have been introduced with the development of the superstring theory [ 2 ]. An important step, from the particle physics point o f view, is to obtain a realistic theory in four dimensions and eventually to obtain the standard model at low energies. In fact, there exists an interesting program, the Coset-Space Dimensional Reduction (CSDR), which permits a detailed examination of the var- ious features o f the four-dimensional theory such as its spontaneous breaking, fermionic spectrum etc. The CSDR scheme is based on the observation that by imposing symmetry conditions on a Yang-Mills theory one can obtain a Yang-Mills-Higgs theory in lower dimensions [ 3 ]. There exists already a considerable amount o f work [4-16 ] on the CSDR scheme that starts f rom the fundamental work of Forgacs and Manton [ 4 ] and includes attempts to embed the CSDR scheme in the superstring program [ 15 ].

Here we shall consider the Higgs potential o f the four-dimensional effective gauge theory, which is obtained from a pure Yang-Mills theory through CSDR. In particular, we shall demonstrate a novel feature, namely that a change of the scales in the coset space, when possible, can change the pattern o f the spontaneous symmetry breaking of the four-dimensi0nal gauge theory.

Let us briefly recall the dimensional reduction procedure using symmetric gauge fields [ 4]. One starts with a pure field theory with gauge group G defined on an extended space-t ime manifold M which is assumed to be a direct product:

M = M 4 X S / R , (1)

where M4 is a four-dimensional Minkowski space-time and S/R is a compact coset space. In general S is assumed to be a compact senti-simple Lie group and R is a Lie subgroup of S. The gauge group is also assumed to be compact and simple. One assumes that the metric o f M is block diagonal.

One defines symmetric gauge fields by demanding that the action o f S on the gauge field A u is compensated by a gauge transformation. Given a coset space S/R, it is possible to determine explicitly both the compensating gauge transformation and the dependence of A u on the coset space coordinates. When A u is inserted into the Yang-Mills lagrangian on M, one can integrate trivially over the coset space coordinates and obtain a Yang-Mills-Higgs theory in four dimensions. The four-dimensional theory has the following charcteristic fea-

On leave from Physics Department National Technical University, Athens, Greece.

0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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tures: (i) The gauge group, which we denote by H, is the centralizer in G of a subgroup Ro (isomorphic to R) of G. (ii) The Higgs fields can be determined, as well as the Higgs potential.

Formally, the Higgs potential takes the form:

V(~))oClf.rr'Kss' Tr{[fr~,~,- ( ~ , , ~r)] [fr, s,,, ¢~,,- (~ , , , ~ r ' ) ]} , (2)

where qbs(x) is a scalar field taking Values in the Lie algebra of the initial gauge group G, and x is a matrix related to the metric tensor of the coset space. In component form, ~ s ( X ) = ~ a (x)t a, where t ~ are generators of G. The indices s, s', r, r' correspond to generators of S and the f ' s are the structure constants of S.

The above expression for V(4) is only formal because ~s must satisfy the constraints [4]

(11)m, ~ s ) - - f rs t fl) t = 0 , (3)

where s takes all values, but the index m is restricted to values corresponding to generators of R. These constraints imply that some components ¢~ are zero, some are constants, and the rest can be identified with components of the genuine Higgs fields. One can easily find the genuine Higgs content of the dimensionally reduced theory. The gauge fields, which become scalars in four dimensions, transform under R as a vector as specified by the embedding

ad S=ad R + v , (4)

and v= 2sk, where each sk is an irreducible representation of R. Then the multiplets under H of independent Higgs fields can be found by decomposing the adjoint representation of G under R e × H :

ad a= Z(rk, hk), (5)

where rk is some irreducible representation of Re and hk is some irreducible representation of H. For each pair of (rk, sj), where rk and sj are identical irreducible representations, there is a Higgs multiplet hk in the four- dimensional theory. The rule given above for finding which Higgs multiplets occur in four dimensions is a consequence of solving eq. (3). When V(~) is expressed in terms of the unconstrained independent Higgs fields, it remains a quartic polynomial which is invariant under gauge transformations of the final gauge group H, and its minimum determines the vacuum expectation value of the Higgs fields.

The minimization of this potential remains in general a difficultproblem even though the work which has been done and the methods that have been developed considerably facilitate the problem [ 16 ]. However, if S has an isomorphic image St within G which contains Re in a consistent way, then it is possible to allow the ~s to become generators of St. In that case the potential V(4~) is zero, which is clearly the absolute minimum value that it can take. These nonzero vacuum expectation values of the Higgs fields break the symmetry from H to K, where all elements of K commute with the generators (qOs) of St. Thus one deduces that the final unbroken symmetry group K is just the centralizer of S t in G [ 14]. Recently it was shown [ 12] that the fer- mions of such a theory become all very massive after spontaneous symmetry breaking. Therefore this attractive possibility is ruled out in searches to obtain the standard model from CSDR of a gauge group in higher dimensions. Thus we return to the original problem to derive the scalar potential of the four-dimensional theory in terms of the genuine Higgs fields and to minimize it.

Our observation is that in the general case (when S ¢ G), by changing the scales in the coset space, when it is possible, one can change the pattern of symmetry breaking of the gauge theory (including the case that the theory remains unbroken). On the other hand, when S ~ G one cannot change the pattern of symmetry breaking. In the following we shall demonstrate our points in two examples. In the first example let us consider a pure Yang-Mills with G = SU5 in ten dimensions, where the extradimensions belong to the nonsymmetric six- dimensional coset space Sp4/(SU2 X U1) . . . . . . . We are going to identify Re as the SU2 × U I subgroup appearing in the decomposition

SU5 = SU3 xSU2 ×U1, 24=(8 , 1 ) ( 0 ) + ( 1 , 3 ) ( 0 ) + ( 3 , 2 ) ( 1 ) + ( 3 , 2 ) ( - 1 ) + ( 1 , 1 ) (0 ) . (6)

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From eq. (6) it is obvious that the centralizer of SU2 × U~ is SU3 × U,, which according to the previously given rules is the gauge group of the effective theory in four dimensions. The decomposition of the adjoint of Sp4 under (SUzXU~) . . . . . . is:

SP4 ~ (SU2 X U I ) . . . . . . . 10=3o + 12, o, _2 +21. _1 . (7)

Therefore, from eqs. (6) and (7) and according to the rules we find that the genuine Higgs fields fl appearing in the four-dimensional theory transform as a complex 3_ ~ of H = SU3 × U~. Note that SP4 has no isomorphic image in SUs. The decomposition (6) suggests the introduction of the following generators of SUs:

Qsv5 = {Qp(0), Q~(0), Qac(1 ), QaC(_ 1), Q ( 0 ) ) , (8)

where p= 1, ..., 8: i= 1, 2, 3; a = 1, 2, 3; c= 1, 2. The generators Qsv5 are in full correspondence with the decomposition (6). Their commutation relations are given in appendix 1 of ref. [ 16]. for the construction of the Higgs potential we shall also need to introduce the following generators of Sp4 according to the decomposition (7)

Tsv4 -= { Ti(0), T+ (2), To(0), T_ ( - 2 ) , TL ( - 1 ), To+ (1)}, (9)

where i= 1, 2, 3; c= 1, 2. Their commutation relations can be found in appendix 6 of ref. [ 16]. In order to express the formal Higgs potential ofeq. (2) in terms of the genuine Higgs fields, we need explicit

expressions for the sclar fields q~s. Again the decomposition (7) suggests the following change in the notation of the scalar fields qL:

{qss: s= 1 ..... 10}-,{q~, q)+, ~o, q) , q~c, q)c+ }. (10)

Then the solution of the constraints (4) in terms of the genuine Higgs field is

qg'=Q', ~c+=f la (_ l )Qac(1 ) , ~ + = 0 , qbo=~Q, ~ C = f l a ( 1 ) Q a c ( - 1 ) , ~b_=0 . (11)

Finally let us consider the matrix x in eq. (2). Since the constraints (4) have to be satisfied, the indices r, r', s, s' in eq. (2) correspond to generators of S which are not generators of R. In the new notation introduced in eq. (10) the constraints (4) exclude four values [corresponding to Ti(0) and To(0)]. Therefore the matrix x is 6×6 and it is the metric tensor of the coset space Spa/SU2XUr. SU2XUI invariance requires that x in the notation of eq. (9) take the form

Tl+ T2+ T 1- T 2- T_ T+

f 0! 0 RF 2 0 0 0 O 0 0 Ri -2 0 0

~-2 0 0 0 0 0

~c= Ri -2 0 " 0 0 0 0 0 0 0 R~ -2

0 0 0 R2 -2 0

TI+ T2+ T 1_ T 2_

T T+

(12)

where R,, R2 are in principle arbitrary radius parameters. In order for the kinetic term of the genuine Higgs fields fl to have the ordinary form we have to make the

redefinition

fl' = ( x/~/gRl )fl , (13)

where g is the coupling constant of the resulting four-dimensional theory. Now we can rewrite the potential (2), taking into account the solution of the constraints (11 ), as

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Volume 191, number 1,2

Table 1

PHYSICS LETTERS B 4 June 1987

( Q , F a ) = - ~ F ~

( Q , G " ) = - ~ _ G ~

{R, F") = (1/..~)F ~ (R, G g = - ( l & / g ) a o (R,R+ )=., /2 R+ ( R, Q,,,+ ) =x/2 Q .... (R,,, ,F.)=_(R,,,)~I'F~' (R'", G") = - (R'")~I'G~'

(R'", Q"+ ) =f,..,Z, Qp+ (R,., Q"- ) =f""P Qp- (R'", Q")=f,,,,WQZ, (Q,,~,F~)=_(Q,n)abF b (Q,,~, G.)=(Q,,,),~,Gh {Q,, , ,Qn+)=_~f2~,, . ,R+ ( R , F " ) = - G ~ (R+, G ~) = - F "

(Qm_, F~) = _ ~_2(Q,~) ~bGb (Qm+, G ~) = _ ~,(?m)~bFb (R+, Qm) =x/~ Q {R+, Q ' - ) = -x /2Q 'n (R+,R)=.fir (Qm+, Q. )=.~,~,,. ,Rq_f,,, .pR p ( F a, Ft,) = - ~fi.b Q+ (1/x/~) cs~b R _ ( Rm)at, Rm( Q m),~b O,n

(6% 6~,) = - ~ci"~Q- (1/~)fi~bR- (Rm)"t'R"+ (Qm)"bQm fi'~bR 2 m ab m+ (F%Gb) - +-x/Z(Q ) Q

(1/g 2) V(~b) = (2 /R4)Tr( q b i ~ + ~bo2 ) + (2 /R4)Tr~8 + ( 4 / R 4 ) ( 1 / R ~ _ 1/R ~)Tr(qb ~_ ~c+ )

+ (2 /R4)Tr[(q)c+, q}a+ )( q}a_, q)L ) + ( q)c+, q}~- )(q}a+, q}L ) ] . (14)

Expressing the q}'s in eq. (14) in terms of the genuine Higgs fl according to eq. (11 ) and using the redefinition (13) we obtain:

(1/g 2) V( f l ' ) = ( 1 / g 2 ) ( 3 / R 4 + 2/R4) 6 - 4(2/R~ - 1 / R ~ ) ( f l ' + f l ') + 3g2( f l ' +fl , )2 . (15)

It is clear f rom the mass term in the potential that the theory will remain unbroken in the vacuum if R2 < R 1/x/2, and will break to SU2 × U1 if R2 > Ri/x/2. It is also clear that such choices of R 1 and R2, since they hold for a huge range of the parameters , are natural.

In contrast to the previous case, when S has an isomorphic image SG within G, as we have already mentioned, it is expected [ 14, 5 ] that the four-dimensional gauge group G breaks into K = CG(S~). This result should hold even when one introduces different radii. The reason is that the potential (2) being positive definite takes its m i n i m u m value zero when

( ~ s ) = ( q ) ~ ) Q a = Q , = Q G . (16)

The latter is due to the fact that each term vanishes independently of its proport ionali ty constant, which contains the radii.

Let us demonstra te the above result in an explicit example. Consider the same G = SU5 and the same coset space S/R = Sp4/(SU2 X U~ ) . . . . . . . However, Sp4 is now embedded in SU5 as follows:

SU5 D Sp4 • (17)

Then R = SU2 X U~ can be obtained f rom the following decomposit ion:

SU 5 D SU4 XUI D (SU2 ×SU2) ×U1 D (SU2 ×U1) × U I , (18)

and the decomposi t ion of the adjoint of SU5 under (SU2 × U l ) × U, is:

2 4 = 1 ( 0 , 0 ) + 2 ( 1 / 2 , - 1 ) + 2 ( - 1 / 2 , - 1 ) + 2 ( - 1 / 2 , 1 )+2(1 /2 , 1 ) + 1 ( 1 , 0 )

+ I ( 0 , 0 ) + 1 ( - 1 , 0 ) + 3 ( 0 , 0 ) + 3 ( 1 , 0 ) + 3 ( 0 , 0 ) + 3 ( - 1 , 0 ) . (19)

Therefore in this example H = Csu~ ( SU2 × U l ) = U i N U ~ and K = Csu5 (Sp4 = 0. Let us demonstrate that K = 0 by considering the potential and the breakdown of H.

We introduce the following generators of SU5 according to the decomposi t ion (19):

Osu~ = {Q, F% G% f~, a~, R+ , R, R _ , R' , Q~+, Q~, Q ' - }, (20)

where a = 1, 2 and i = 1, 2, 3. The commuta t ion relations of Qsu~ are given in table 1. From eqs. (7) and (19)

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we f ind dat the genuine Higgs fields appear ing in the four-d imensional theory t ransform as r = ( - 1, 0) , 7 1 = ( - 1 / 2 , - 1 ) , 7 2 - - ( - 1 / 2 , 1) under H = U I × U 1 .

The Solution of the constraints (4) in terms o f the genuine Higgs fields is now:

~i = R i, c19+ =fiR+, ~ c - = (1/x/~)( 7 ~- G c _y+ ~CdFd) , (21)

q)o =R, q) - = f l + R - , qbc+ = ( 1 / ~ ) ( 7 1 G c - 7 2 e ~ a F d) . (22)

At this poin t it is worth ment ioning that by choosing r , 7~, 72 equal to one, the embedding o f Sp4 in SU5 is determined.

The matr ix x in eq. (2) is given by eq. (12) as before. The potent ia l (2) , using the solut ion o f the constraints, is

V(~o) = (4 /R4) ( /3 - y t 72) + (1~--7172) -}- (2/R4)( 2 - 7 + 71 --7~- 72) 2 + (4/R4) (7 7 71 - -7J- 72) 2.

+ ( 4 / R { ) ( f l + f l - 1 )2 _]_ (2/R~R2)(7 , _f ly+ ) + (7~ - f17~ ) + (2/R2R2)(72 - f lY+) + (72 f lY?) •

We observe that for fl = 7 ~ = 72 = 1 this potent ia l takes its m i n i m u m value zero independent ly of the values o f the radi i R~, R2.

In conclusion, in the f ramework o f the CSDR scheme, we have po in ted out that the in t roduct ion of a suitable but natural hierarchy o f scales in the coseI space, when possible, can change the symmetry breaking pa t te rn of the resulting four-d imensional gauge theory. In add i t ion we have shown that, in the case that S has an isomorphic image in G, the in t roduct ion of radi i does not change the breaking pattern. In the light o f this new result many of the negative results that have been obta ined previously in the a t tempts to obta in a realist ic four- d imens iona l theory from CSDR of a gauge theory in higher d imensions should be reexamined.

We would like to thank P. Forgacs for interest ing discussions and S. Fer ra ra for interest ing discussions and reading the manuscr ipt .

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Geometrical hierarchy and spontaneous symmetry breaking