Embedding and breaking of gauge symmetries

Volume 222, number 1

EMBEDDING

PHYSICS LETTERS B

AND BREAKING OF GAUGE SYMMETRIES

11 May 1989

¢~

Varouzhan B A L U N I

Stanford Linear Accelerator Center. Stanford University, Stanford, CA 94309, USA Received 25 October 1988; revised manuscript received 14 February 1989

The effective potential of "weak" interactions of a gauge group R is studied in the framework of models whose global symmetry group G is dynamically broken down to its subgroup H. The properties of its critical points on the double coset R\G/H and their dependence on the embeddings R c G, H c G are investigated• All critical points with a nontrivial residual gauge symmetry H~=Rc~ H are identified in the series of models SU(n)\SU(N)/SO(N), n
This p a p e r concerns the e m b e d d i n g s o f gauge symmetries into d y n a m i c a l l y b r o k e n global symmetries in gauge field theory models without elementary scalars. It suggests a general prescription to identify and presents models to d e m o n s t r a t e the e m b e d d i n g s which are necessary for the realization o f specific breaking patterns of gauge symmetries. l will begin with a b r i e f review of the general framework underlying the results which will be established subsequently; for more details the reader is referred to the original literature cited in ref. [ 1 ]. The local gauge interactions are introduced into the field theory by means o f an e m b e d d i n g o f the corresponding gauge group R into its global s y m m e t r y group G. The group G is assumed to be d y n a m i c a l l y broken down to its subgroup H c G. The e m b e d d i n g R ( H ) c G is d e t e r m i n e d by two sets o f c o m p l i m e n tary conditions which m a y be characterized as kinematical or d y n a m i c a l in nature reflecting respectively mathematical or physical aspect o f the problem. The first set specifies the choice of groups and representations in question. In particular the R ( H )-representation content o f the G-representation, i.e. the d e c o m p o s i t i o n no,, = Z~ ® n ~ h ~ is an i m p o r t a n t condition as it d e t e r m i n e s the e m b e d d e d group R ( H ) up

to an inner a u t o m o r p h i s m o f G : R ~ g r R g ~ - ~ ( H - ~

g h H g ~ l ) , g r ~ h ~ G [2]. The second set is o f intrinsically d y n a m i c a l origin being often referred to as alignment conditions [ 3,4]. It is based on a m i n i m u m energy principle which removes the m a t h e m a t i c a l freedom in the choice o f the element g=grg~ ~. The potential energy ( d e n s i t y ) V(g) defined as a function o f the group element g is induced by gauge interactions to the leading order o f the fine structure constant C~R.It is given by an expectation o f the R-invariant product o f two N o e t h e r currents o f the gauge group gRg-~ in a H - i n v a r i a n t vacuum state. Obviously such expression is a function o f the coset element 1 ~ G / H only i.e. V(g) = V( 1 ) for g = lh, h ~ H and its 1-dependence m a y be always reduced by s t a n d a r d W i g n e r - E c k h a r t arguments. Here the discussion will be restricted to the subclass o f coset spaces i.e. compact symmetric manifolds G / H whose generators K={Km} along with those o f its c o m p l e m e n t H = {H+} obey the conditions [K, K] = H in a d d i t i o n to [H, K] = K ; furthermore it will be assumed that the group H is simple and its coset representation is irreducible. Then only two r e p r e s e n t a n•o n s -H - ~ , a = h , k a r e involved and therefore within a irrelevant additive constant the potential function m a y be further reduced to the form (c.f. refs. [5,6] )

Work supported by the Department of Energy, contract DEAC03-76SF00515. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )

V ( 1 ) = C Y ' R " ( 1 ) R ~ ' , ( I ) g , ....

C>O,

(1) 103

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PHYSICS LETTERSB

which is given in terms of an unknown constant C, R generators and Killing forms of R and G groups R"(1)=Tr(1R~

1-~Km),

(2)

r~b= - T r ( R ~ R b ) ,

(3)

gm, = -- Tr(KmK~).

(4)

11 May ! 989

Ra(xc) = 1 (Xc) R~ l t ( x c ) ~ R a the critical point x = xc ~ 0 can be translated to the origin. Therefore, it will be henceforth assumed x~=0. Thus the conditions (7), (8) can be used to determine critical configurations {R~} directly. The task is facilitated by the following straightforward deductions.

I will proceed to the mathematical formulation of the embedding conditions. To this end the potential function ( 1 ) will be expressed in a standard geometric language. This is immediately achieved by recognizing that the matrix elements R ~ ( 1 ) are directly related to the Killing vectors ~ ( 1 ) of G generators XM and vielbeins e~( 1 ) of the coset G / H . Parametrizing the coset element 1 = 1 (x) by local coordinates x = {x ~'} e.g. via 1 (x) =exp(xUKu) one finds

Corollary 1. The unbroken (Ha) and broken (Ka) components of critical gauge generators Ra = H a + Ka, Ha ell, KaeK obey the constraints

V(x)=r~b~(x)~g(x)gu~(x)--r~b(~,~b),

Corollary 2. Let the symmetry of the critical point x = 0 be Hc =Hw@Hg with the unbroken (residual) gauge subgroup Hw = R c~H and the global symmetry Hg c H defined by [ Hg, R ] = 0. Then the coset generators Km which are not neutral under Hc i.e. [ K,,, Hc ] ~ 0 satisfy the condition (9) identically.

(5)

~Ua(X)=e,~(x)Rm(X) ,

(6a)

q~(x) =e'~(x)e~(x)gm,.

(6b)

Note that the constant factor C of eq. ( 1 ) has been suppressed. Now the variational problem of the potential function can be addressed. In light of the Ansatz (5), it is convenient to consider variations induced by infinitesimal group motions (diffeomorphisms). Indeed the first variation is simply described by the Lie derivative LM as Lggu~ = 0, Lg~v = (f'MN], etc. The second variation may be realized by means of the Laplace-Beltrami operator AR----rabL~Lb. Hence one easily arrives at the following theorem.

Theorem. If the potential function (5) has a critical point (extremum) at the origin x = 0 , i.e., Vm(O) - OV(x)/OXm I~=0 = L ~ V(x) I~=o

=2r~b(~, ~[mb])lx=o,

(7)

then its hessian (mass matrix) at the critical point is given by

v,..(O) - 02V(x)lOx,. Ox~ I~=o,

(8a)

Vmn(O) =LmL, V(x)I~=o = {AR (~m, ~ ) -- (AR~m, ~,)+ (m~-~n) }~=o.

(8b) Clearly by redefinition 104

of R-generators

i.e.,

Vm = -- 2rab Tr{ [Ha, Rb]Km} = 0 ,

(9a)

or, alternatively,

rab[Ha, Kb ] =O .

(9b)

Corollary 3. The decomposition ~,, (x) =fmm~A(X) of the H-multiplet {~m(X)) in terms of R-multiplets {(A(X) IAR~A = --CA~A} reduces the hessian (8) to the form Vm. = f , ~ L c C A { ( 6 ~ + 6~c )

-2(BCIA)}(~,, ~c)x=o,

(10)

where the coefficients CA and (BCIA) stand for the second Casimir and Clebsh-Gordan coefficients of the group R.

Corollary 4. A necessary condition for the embedding R c G to be physical is that the potential function V(x) has a relative minimum at the critical point x = 0, i.e., it possesses a positive hessian Vmn> 0 along with Vm(O)=0. At this point it is appropriate to dwell upon some special features of the hessians in question. In general, hessians can be degenerate with a spectrum consisting of positive, negative as well as zero eigenvalues. However, mathematical discussions (e.g. in Morse theory) are usually restricted to the functions with non-degenerate hessians as they are known to be

Volume 222, number 1

PHYSICS LETTERS B

dense in the class of smooth functions. Nevertheless in what follows I will not be bound by such restrictions; moreover, the search of potential functions with degenerate hessians will be a major thrust of the discussion. A physical significance of the degeneracy may be appreciated by delineating prominent characteristics of the generators { X , , = K , , ( m o d H ) } representing zero modes. Obviously R-singlet generators {X,,I [Am, R,] =0} have a trivial nature and it is sufficient to focus upon R-nonsinglets {X,,I [Am, R~] # 0}. They can be classified according to the properties {K(g)}={Ka}, [K(mn) , Hw] = 0 a n d t'~(~)~,., Hw] # 0 d e scribing respectively gauge, neutral and charged zero modes. An insight into the intricate origin of these modes may be gleaned by identifying positive (H,,~) and negative ( - K , , , , ) components of the hessian [5 ]. One easily infers from the definition (8a) that

V,,,,(O)=H,,,~-Km~,

(11)

Zm~ = 2r~b Tr{ [Z~, K,,] [Zb, Kn]},

Z=H,K.

(12)

Evidently all zero modes {K,,} arise as a result of a subtle conspiracy of broken (K~) and unbroken (H~) components of R-generators leading to an exact cancellation of their contributions i.e. K,,, = H=,. However, the origin of the conspiracy underlying gauge zero modes is fundamentally distinct from that of charged/neutral zero modes. In the former case it is a manifestation of the universal property i.e. R-gauge invariance; the potential function V(x) is altogether independent of the gauge coset parameters {x~g~} (as well as R-singlet ones) or more precisely in the vicinity of the critical point it is defined on the left-right coset R \ G / H . On the other hand for charged/neutral zero modes {x,.(c) , x,,~n~} such conspiracy is an expression of flatness of the potential in certain directions enforced by particular features of the double coset in question. Furthermore, it is important to realize that the potential V(z) restricted to its flat directions 0 = ~ -..(c) , , , x£"~ } is, in general, a non-trivial function V(0) of 0:

V(x) = V( O) + ½V,,,~x,,x, + O ( x 2) ,

(13a)

V(0) = V ( 0 ) + O ( 0 2 ) , 0 = (x(~), x(C)) .

(13b)

The general features described above will be demon-

11 May 1989

strated on a series of models of the class introduced in refs. [7,8] R\G/Hs =SU(n)\SU(M)/SO(M) -=K~(nIM),

n
(14a)

R \ G / H a =SU(n) \ S U ( M ) / S p ( N ) ---Ka(n[N) ,

n
(14b)

The embeddings, both H c G and R ~ G, will be closely examined with a special attention to the resulting intersection Hw= R c~H, i.e., residual gauge symmetry. For this purpose the choice of the Weyl basis for U ( M ) generators {X~4} is most suitable: X:

i m (x~). = &i 6 jm ,

M=2N(2N+I).

i,j=(0), +I,...,+N,

(15)

Indeed the H~-generators can be now simply identifled by means of an involutive automorphism e:e G:

H:ZH'j=X~-e*,.g}e-,

eH=-~qe,

(16a)

K::K~=X~ +e*:X}e-,

eK=Ke.

(16b)

Here the tilde indicates transposed matrices and (e-) ..... =~,,,&,,+~, e~,,=l,

z=s,a,

ema--sign(m).

(17)

An important property of the matrix [ (e-),,,,] is that it defines the : H invariant (anti)symmetric bilinear form for the fundamental representation ;nn, z = s (a). The following lemma is a direct consequence of this fact.

Lemma 1. A necessary and sufficient condition for the embedding R ~ - - H with the content -'nn = ~ ® n ~ ~ to exist is to have a R invariant (anti)symmetric bilinear form on the space -'nil, z=s (a).

Hence one is immediately led to a simple and yet very effective prescription to identify breaking patterns.

Lemma 2. A necessary condition for the breaking R--,Hw=RC~H with the content M c = n n = Y~@n~w~) ( M ~ = n n = ~ @ n ~ ~)) to occur is the existence (absence) of a bilinear form which is both Hw (R) and H invariant. 105

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PHYSICS LETTERS B

The familiar example of the group R = SU (2) provides a simple demonstration of the first lemma. Any irreducible representation of a (half)-integer spin admits an (anti)symmetric bilinear form. Thus it can be embedded into SO (Sp)-groups but cannot be embedded into Sp ( SO )-groups. Evidently a reducible representation containing both integer and halfinteger spins cannot be accommodated by either SO or Sp groups. However one should bear in mind that conjugate representations of SU(2) are equivalent. Therefore a reducible representation with two identical spins admits both symmetric and anti-symmetric bilinear forms and its embedding into either SO or Sp is permitted. Now the results described in the table 1 should not be difficult to verify. ALl possible embeddings of R = SU (n), n = 2, 3 into G = SU (M), M~< 6 have been examined and critical points of the potential function of corresponding models in the series (14) have been identified. A crucial feature of all cases considered is the special choice of the R content which by lemma 2 obstructs the embedding R c H a priori. The content of table 1, in general, is not intended to be exhaustive. However, the listing of critical points with Uw ( 1 ) or SUw (2) gauge symmetries is complete. A comprehensive discussion of these results will be presented elsewhere. Here the discussion will be restricted to the potentials with flat directions. The models of interest are as follows:

11 May 1989

2V(x) =NR +rabTr[e*RaeU*(x)RbU(x) ] ,

(19)

where

NR = r~brab,

(20a)

U ( x ) = 1 (x) e* T (x) e.

(20b)

Thus the potential is defined as a bilinear function of matrix elements { Umn}. In particular the representations of R-generators given in table 1 reduce the potentials of the models ( 18 ) to the form

2Va(x)=3-(lUl~12+l U, _212+1U2,12), (21a) 3VB(x) = 6 - ( [ U ~ + U22 ] 2+ ] U23 - Ul -3 [2 (21b)

..]_ 1U32.31_ U_31 [2) ,

6 Vc(x) = 1 2 - ( I U~2 - U2~ 12+ I U~3 - U3~ 12

+ I U23 - U3212) ,

(21c)

2VD(X ) = 10-- (1UII -- U22 [2+ ] UI 3 _ U32 [2 31-] U 3 1 - U23 [ 2) .

Obviously it is parametrization

U(x)=12(x),

(21d)

sufficient

to

introduce

the

l(x)=exP(½i/xoK~),

(22)

to infer the hessians of the above potentials (c.f. eq. (8))

VA(X)=I"~-(IXl_3[Z"]-IXl312)-~O(x2) ,

(23a)

3VB(X) = 2 + (]Xll --X22 ]2"+'4]X1212+41X, -212 Ka(216),

6G=3R+3X1R,

He = SUg(2) ®Uw(1 ) ,

+2]Xl -1 ]2+2]X2-2 [2+2]X13 ]2"]-2 IX2-3 ]2) (18a)

K~(216) , 6 G = 2 X 2 R + 2 ~ , He = U g ( 1 ) ® U w ( 1 ) , IQ(216),

(18b)

(18c)

(18d)

The analysis should be extended beyond the calculation of their hessians and the following expression of the potential (5) is more suitable for this purpose

106

-{-[X23--X32] 2 ) "~ 0 (X 2) ,

(23c)

2 V~(x) = 1 0 - ( I x ~ - x 2 : 12+ Ix~3 -x32 12

IQ(316) , 6 G = 2 × 3 R , He=Ug(1)QSOw(3).

(23b)

6Vc.(X ) = 1 2 - ([x12-x21 ]2+ [Xl3-X3112

6~=3×2R,

H~ = SOg(3) ®Uw( 1 ) ,

+ O ( x 2) ,

+ 1X31--X2312) "~-O( x2 ) •

(23d)

Hence one readily identifies charged and neutral zero modes by their He-content: H, = {SUg(2)IH~/v/2, X3_3}®{Uw( 1 ) IHI }, [fiN/2 ~ XI2 "~-X2 --1 ~lg®l*w , ~---Xll --X22 ~ lg G10w,

(24a)

Volume 222, number 1

PHYSICS LETTERS B

11 May 1989

Table 1 Critical points of the models defined in eq. (14) are described by their R-content {n~.")} of R ~ G and nature as a relative minimum (min), maximum (max) or saddle (ext) point. Symmetry (H~) with global (Hg) and residual gauge (H~) components. R-generators with zero (Ro) and positive (R+) roots. The realization of the generators ( 15 ) is switched to the form Xj =zta/Oz ~. {n~") }

H~= H,®H,.

Ro

R+

Ks(213) Kd213) K~(214) K~(214) Ka(214) K~(215) Kd215) K.(215) K~(215) K.(216) K.(216)

(2+1) .... (2+ 1)max (2+2×1)m~. (4)~,. (3+1)e~¢ (2+3)min (2+3)max (2+3)~, (4+1),~, (3+3×1)~x, (3+2+l)min

O Uw(l) Ug(I)®U~(1) U~(I) Uw(l) Uw(l)

(g°--X-{)12 HI~2 H[/2 (3H~+H])/2

X °_ ,/,,//2 XLi/x/2

H]

H~/2+H~ H 2 / 2 + ( X I - X °)

X~-X2~ X22/.fi + H~ X2_21x/2+ (XL~ - X~-~)

(X]-X°)12+(Xz[-X~)

X~/x//2 + (X~_ I --X2 -2 )

(3H2+H])/2

K~(216) K~(216) K~(216) K~(216) Ks(216) K~(216)

D

x'_,l~/~ (.f3H] +2X~, )/x/2

o Uw(1) SUg(2)®Uw(1 )

HI

(.f3g~ + 2X~_,) / x ~ X,' - X2 ,

O

(x]-x-l)/2+(xI-x~)

X~,lx/2 + ( x~ -x~)

(3+2+1)~ (3+2+1)~x, (5+l)~x, (2×2+2")mm (3X2)m~x (3+2+l)min

U,(1) U~(I) U~(1) Us(1 )®U~(2) SOg(3)®U~(I ) 0

K~(2]6)

(3+2+ 1)~,~t

Uw(1)

(x-~-xzl)/2+(x2-xl) H312+HI 2H~+H[ (H~+HZ-HI)/2 (H~+H2+HI)/2 (X~-X~)/2+HI (.fZz°=z~+z ~) H3/2+HI

K,(216)

(4+2)rain

o

Kd216) K,(216) K~(316)

(4+2)~xt (6)m~. (3+ 3× 1)~,,,

U~(1) U~(I) SU~®SU~(2)

(X~-Xy_3+3H~+X~-X-I)/2 (3z"=z~+x/8z ~) (H3+3H2+Hl)/2 (5H3+3H~+HI)/2 H~/2

K.(316)

(2X3)max

Ug(1)®SOw(3)

(HI-H~)/2

(x/5~o=~ ~+~3)

X_~l~/i + ( x ~ - x~) x~_~l,/2 + (x~ - x~_, )

~H~+./5(x~-x~_,) (x~_~ +I4~)/./5 ( X l l + X2_ 2 + X3_ 3) l N / 2

x~/ ,/22+ ( x'~ - x~_ , ) (x/2zt'= - - z 2 + z -2 )

X~l,/5+(x'~-x",) (.~z.=z2+z 2) ~X ~2 - ~ X-_2)1x/2 ~ 2 (XL3 +2Xh-, +.,/3 (3z~=-x/8z~+z ~) (X33 + 2 X ' , +xf3H~)lxf2 (.v,/5H3+ 2.v/2H2- 3 X

1 , )~

(KI-2X2 )/2x/3

K~/ ~2, ( X~ + XT_~) I.v/2

(KI +K22- 2K3)/2x/3

(x~ + x:~)/,/~

Ka(316)

(2×3)ex,

Ug(1 )®SUw(2)

(H]-H~)/2

H~/,fi, ( Xi + X : ~) /,(~

(KI +K22- 2K33)/2x/3 Ks(316)

(3+3× 1)max

SOg(3)®SOw(3)

HI~2, (K]-2Xg)/2x/3

(x~ - x - ~ ) l . f 5 x~,/,/i, x~/,~, x,, ,/,/i

Kd316)

(2 ×3),..~x

Ug(l )®SOw(3)

(HI-H2)~2

Kd316)

(2×3)m,n

Ug( 1)®SU~(2)

H~+H~)/2

(./~z°=~+~ -~)

(K', +Ki- 2K~)/2,,/~ (K', + K ~ - 2K~)/2~/'3

(X I, Xz) -= (x23, Xz-3)E2g ®1 ° ,

(24a c o n t ' d )

q=X2361g*(~l ° ,

H~ = {SOg(3)1%mH~, m = 1, 2, 3} ® { U w ( 1 ) l (HI + H ~ +H33)/2},

H e = { U g ( 1 ) I H I +H22} ® { U w ( 1 ) l ( n 3 + H 22- H I ) / 2 }

(X 3 +X_3)/~/2 H' I./2, (X~ + X:I)I.,,~ (x~ -x_~)/,/2

,

Z=--X3_I~Ig*®I° ,

q(I ~- Xt--j -- I (~uX . . . .

(24b)

i,j=l,2, 3,

~Sg®l~, (24c)

107

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PHYSICS LETTERS B

H c = { U g ( 1 ) l (HI +H2 + H 3 ) / 2 } ®{SOw(3) IHI -H2,2 H31-H3} 1

l~®Sw,

i,j= 1, 2, 3.

(24d)

Now the evaluation of the potentials restricted to the flat directions ( 2 4 a ) - ( 2 4 d ) becomes straightforward. It is easily accomplished by means of the following parametrization of the coset element 1 (x) ( see eq.(20b)). 1A(X) ~ l ( O ) l ( ~ ) l ( z / 2 ) ,

(25a)

l n ( x ) ~ l(q/2, Z / 2 ) ,

(25b)

lc(x) ~ l(~J2),

(25c)

1D(X)~ 1(Z¢/2).

(25d)

Note that every factor 1 (¢) on the right-hand side of eq. (25) may be represented as an exponential of an "off diagonal" matrix. Therefore the 1 (~) is amenable to some standard manipulations, i.e., 1 (~) = e x p [ i ( ~ K + h . c . ) ] _ exp [ _ O

=

[ (1-ZZ?)I/2 -z*

z

(l_ztz)l/2

1,

B]

(26)

where

z=Bsin(B*B)I/Z/ (B*B)I/2 .

(27)

Returning to eq. (21 ) with the parametrization of the matrix U given by eqs. (20), ( 2 5 ) - ( 2 7 ) and performing some simple algebra one derives (c.f. eq. (13b)): 2VA(~) =2+sin2l~vl sin2 IZ[ = 2 + I~ul 2 Izl 2 " ~ 0 ( ~ 4) ,

(28a)

12 VB(O) = 8 + 16 sin 4 ½( It/12+ IZ12) 1/2 -----8+ ([r/12+ 1ZI2)2+O(04) ,

(28b)

12 Vc(~) = 2 4 - 41Tr t cos (r/tr/) 1/212 = 2 4 - [ Tr(r/tr/*)12+O(~ 4) , 4 VD(~) = 20--4 ITr tcos(z*Z)

1/312

= 2 0 - ITr(ztZ*)12+O(~4) . 108

(28c)

(28d)

11 May 1989

An actual difference in representations of the SO(3) generators {t"lTr(tat b) =2~ ab} in eqs. (28c) and (28d) has been ignored as it can be compensated by an appropriate redefinition of the zero modes q or Z. Note that the evaluation of the potential (23a), (28a) is consistent with the previous calculations of the model (18a) carried out in somewhat different parametrization [9 ]. The results (23), (28) lead to the main conclusion of the present article: There exist potentials with critical points (minima or maxima) described by a residual gauge symmetry Hw = R c~H ~ o and flat directions (degenerate hessians) either neutral (28b) or charged (28a), (28c), (28d) under Hw; These potentials restricted to the flat directions are nontrivial functions of the zero modes. For purposes of comparison one may recall flat potentials entertained previously. For example in field theory models with elementary scalars the flatness has been either postulated or enforced via supersymmetry [ 10 ]. On the other hand, in the framework of dynamical models with a semisimple gauge group R it has been achieved by a fine tuning of gauge couplings [ 11 ]. Therefore the approach to fiat potentials pursued in this work may be viewed as a natural development of the latter attempts. Apparently, a physical significance of the above results predicated on the existence of a program which incorporates the flat potential as a necessary first step toward a dynamical realization of the Higgs potential. Such program has been advanced recently i.e. it has been proposed to construct models which possess a potential function ( 1 ) with the following features [9] ~1. (1) Its minimum is described by a residual gauge symmetry and charged flat direction. (2) It mimics a Higgs potential without mass terms when restricted to flat directions. (3) Its higher order corrections O (as) are responsible for Higgs mass terms. Evidently the potential (28a) represents the only model which satisfies the first condition. Unfortunately none of the potentials (28) meets the second condition. The potential (28b) fails only because its flat directions are neutral under the residual gauge symmetry. A consideration of the third condition is ~t A special realization of this program has been discussed in ref. [ 12]. I thank H. Georgi for bringing this work to my attention.

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b e y o n d the scope o f this discussion. A f u r t h e r e x a m i n a t i o n has r e v e a l e d n u m e r o u s exa m p l e s o f flat p o t e n t i a l s h o w e v e r the r e a l i z a t i o n o f a b o n a - f i d e Higgs p o t e n t i a l has p r o v e n to be a difficult task. A n e x t e n s i v e d i s c u s s i o n o f these a n d related p r o b l e m s will be p r e s e n t e d in f o r t h c o m i n g reports. I w o u l d like to a c k n o w l e d g e the h o s p i t a l i t y o f m y colleagues at t h e Physics D e p a r t m e n t o f U C L A and the T h e o r y G r o u p o f S L A C .

References

11 May 1989

[2] E.B. Dynkin, Arner. Math. Soc. Trans. Ser. 2, 6 ( 1975 ) 111, 245. [3] R. Dashen, Phys. Rev. D 3 ( 1971 ) 1879. [4] S. Weinberg, Phys. Rev. D 13 (1976) 974. [5] M.E. Peskin, Nucl. Phys. B 175 (1980) 197. [ 6 ] J.D. Preskill, Nucl. Phys. B ! 77 (1980) 21. [7 ] S. Dimopoulos and L. Susskind, Nucl. Phys. B 155 (1979) 237. [8] S. Dimopoulos, Nucl. Phys. B 168 (1980) 69. [9] V. Baluni, Dynamical gauge hierarchies, FNL Report 86/ 169-T (1986), unpublished. [ 10 ] S. Coleman and E. Weinberg, Phys. Rev. D 7 ( 1973 ) 1883. [ l 1 ] M.J. Dugan, H. Georgi and D.B. Kaplan, Nucl. Phys. B 254 (1985) 299. [12] D.A. Kosower, Nucl. Phys. B 294 (1987) 255.

[ 1 ] v. Baluni, Ann. Phys. 165 ( 1985 ) 148.

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