Gauge symmetry breaking in supersymmetric gauge theories: Necessary and sufficient condition

Volume 118B, number 1, 2, 3

PHYSICS LETTERS

2 December 1982

GAUGE SYMMETRY BREAKING IN SUPERSYMMETRIC GAUGE THEORIES: NECESSARY AND SUFFICIENT CONDITION R. GATTO

D~partment de Physique Th~orique, Universit~ de Gen~ve, 1211 Geneva 4, Switzerland and G. SARTORI

Istituto di Fisica, Universit~ di Padova, Padua, Italy and INFN, Sezione di Padova, Padua, Italy Received 18 June 1982

A geometrical description is presented for gauge-symmetry breaking in supersymmetric gauge theories. We derive a neeessary and sufficient condition for spontaneous breaking of the gauge group when supersymmetry remains unbroken.

1. Introduction. Supersymmetric gauge theories have recently been the object of renewed interest, mainly based on the hope that the absence of renormalization for some o f the parameters entering the lagrangian could help in understanding the gauge hierarchy problem [1] ,1 Obviously supersymmetry has to be broken, but in a large class o f models it is assumed to break at a scale which is lower than the scale at which a first step of breaking of the gauge symmetry occurs [2]. This paper will describe a geometrical approach to the analysis o f the mechanism o f gauge symmetry breaking in supersymmetric gauge theories with compact semi-simple gauge group G. The gauge group acts linearly and unitarily on the complex scalar fields z - (Zl, ..., Zn) of the chiral supermultiplets o f the model, through a generally reducible representation R. We shall denote by z* the complex conjugate of z and use the notations g • z and T . z to denote the actions o f g E G and T E ~ ( ~ = Lie algebra o f G) on the n-dimensional complex vectors z E Cn. g and T are respectively unitary and anti, l The second paper also reports on results due to V. Glaser, R. Stora and B. Zumino. 0 031-9163[82/0000-0000/$02.75 © 1982 North-Holland

hermitean n X n complex matrices. We shall choose in q an orthonormal basis {Ta}a=l ..... r, r = dim ~ : tr TarT b = - t r TaT b = 6ab •

(1)

In the class of supersymmetric gauge theory models we are considering, spontaneous symmetry breaking is controlled b y a real non-negative G-invariant scalar potential V(z, z*), which can be written in the following way [ 1 ] :

V(z,z*)- D +F 1

-'--- ~ IDa(z, z*)] 2 + ~ ( F / ( z ) ) * F / ( z ) , 2 a /

(2a)

Da(z, z*) = i(z, Taz) = i j~k Z;TlakZk ,

(2b)

F/(z) - a z / ( Z ) ,

(2c)

where f(z) is a third degree G-invariant polynomial in z and the bracket denotes the standard hermitean scalar product in Cn. If the minimum o f the F-term in eq. (2a) is different from zero, V(z, z*) is positive at its minimum and 79

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supersymmetry broken. On the contrary, as shown by Wess and Zumino ,2, if a minimum of the F-term, say at z0, is zero, the absolute minimum of V(z, z*) will also be zero. In this case, in fact, F j ( z ) , which is covariant with respect to all the transformations of the complexification, Gc, of the group G, will be identically zero over the whole Gc-orbit through z0, ~2c(Z0). So, by means of the transformations of a one-parameter subgroup of Gc, or by their limit when the value of the parameter reaches infinity, the point z0 can be transformed into a point z~ ,3, where also the D-term annihilates ,2 All these points, which include the origin of Cn, will determine acceptable vacuum states of the theory in the tree approximation. Higher order corrections will not break supersymmetry and will not remove any possible degeneracy present at the tree level [1]. If the vacuum chosen by the real world is in correspondence with a null minimum of the potential at a point zo which is not stable under G, the gauge symmetry will be broken in the tree approximation. The possible presence of distinct absolute minima of V(z, z*) corresponds to generally different theoretically admissible patterns of spontaneous symmetry breaking. In order to determine all these possibilities in a given model, the mere analysis of the F-term is not sufficient in general and a determination of the zeros of the D-term is also required [3,4]. In this connection some particular solutions have been obtained in a SU(5) supersymmetric model in refs. [ 5 - 8 ] . In particular the results of ref. [8] have been obtained by the authors through a clever exploitation of the following condition A, proved by them to be sufficient in order that a point z E Cn be a zero of the Dterm: Condition A : there is a G invariant polynomial in z, l ( z ) , such that Ol(z)/az/=z;,

/ = 1 ..... n .

:1:2j. Wess and B. Zumino, quoted in the second paper of ref. [1]. ~3 zb lies on ~c(z0) or outside it, but on its topological closure and its isotropy subgroup Gz~) is not in general conjugated in G to Gzo even if Gc,z~ is conjugated to Gc,zo when z~ ~ S2c(Zo). 80

2 December 1982

In this letter we shall give a geometrical characterization o f the zeros of the D-term. The geometrical approach will allow us to get some information on their little groups and to prove, with the aid of a theorem recently proved by one of us [9,10], that the following condition B is necessary and sufficient so that z be a zero of the D term. Condition B: there is a G-invariant real polynomial I ( z , z*) in z and z*, such that:

~7-(Z, Z*)/~Zj = i z l , ] = 1 ..... n . Condition B is easily seen to be implied by condition A if only I(z, z*) = i[l(z) - (I(z))*] .

(3)

We have not however been able to derive condition A from condition B in a general case. 2. A geometrical characterization o f the zeros o l D . In order to present our geometrical approach it will be advantageous to translate the problem in an entirely real formalism. This will be done through the following one-to-one correspondence:

Cn Dz = (z 1..... an) +~ ~"=- (~'1..... ~'2n) E Iq2n = (Re z 1.... , Re z n, Im z 1..... Im Zn) ,

(4a)

n X n complex matrix L *~ 2n X 2n real matrix X L ] -ImL Re L )"

(4b)

In particular one finds:

(i) ~

z '+ e~',

where the real antisymmetric matrix e, e

\lnl

0]

(5)

commutes with all the matrices )t as defined in eq.

(4b): [e, X] = 0 .

(6)

(ii) Denoting by ( , } the euclidean scalar product in FI2n: (z, z') = (~', ~") + i(e~', ~").

(7)

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(iii) The group G is represented in R2n by a group r of real orthogonal matrices and its Lie algebra ~ by a Lie algebra G of real antisymmetric matrices. The real images ra of the elements T a of the basis of ~ yield an orthonormal basis for G tr zar b = - 2 8 ab .

(8)

The complexification Gc of G and its Lie algebra q c will be represented respectively by a group of real matrices (that we shall call rc) and a Lie algebra of real matrices (that we shall call Gc) a basis in Gc being given by ( ~ , era)a=l .... r. We shall denote by ~2(~) [respectively ~2c(~)] the r-orbit (respectively re-orbit through ~', i.e. the set of all the points 3' "~, with 3' E r (respectively 3' E re). ~2(~') and ~2c(~") are smooth (C=_) submanifolds of R~" and the tangent planes to these manifolds at the point are respectively ~ + Tt [10,11] and ~"+ Tc, t, where T~-and Tc,~ are the vector spaces formed by all the vectors r • ~, r E G and respectively r ~ Gc: T~= ( r . ~ ' I r E G ) , Tc,~= {r • ~tr ~ Go}.

N~. = (~ ~ R2nl(~, r • ~') = 0, all r ~ G ) ,

(lOa)

Nc, t = (~ E R2nl(~, r • ~') = 0, all r ~ Ge}.

(lOb)

The subspace formed by the vectors of N¢ (respectively No, 0, which are invariant with respect to the isotropy P-subgroup of ~', rr, ,4, will be denoted by N (0), respectively N(0.). 2N~0) can be characterized as the vector subspace of R spanned by the gradients at ~"of the r-invariant real polynomials in ~', [9,101 or equivalently ,s by the following property: ~ E N~0' if and only if: q

a,~8Oa(~)/8~i,

] = I ..... 2n,

for some reals al ..... aq [9,10]. N(0) c,s can be easily shown to contain the gradients at ~"of every r-invariant real function of ~, which is the real (or imaginary) part of a r-invariant, analytic function ofz ,6, but it is not known to our knowledge, whether N (0) is completely spanned by such gradients, and consequently if it is generated by the gradients at ~"of the real and imaginary parts of the elements of an integrity basis for the ring of G-invariant polynomials in z. The following proposition gives a complete geometrical characterization of the zeros of the D-term. Proposition 1. The zeros ~ of the D-term are characterized by the validity of the following orthogonal decomposition of R2n: R 2n = T~- ~ T e t ~ (~'} ~ (e~') ~ N~ es ,

(12)

where (~) denotes the one-dimensional vector subspace of R2n generated by ~, and N~es is a subspace of Nr, which may also consist of only the null vector.

(9a,b)

The normal spaces to ~2(~') and ~2c(~) will be denoted by Nr and Nc,$ respectively. They are formed by all the vectors of Rzn which are orthogonal to all the vectors of T~ and Tc,~ respectively:

~/= ~

2 December 1982

(11)

~:4 The largest subgroup of r which leaves invariant ~" (little group of ~'). , s Using a theorem by Hilbert, every l~-invariant polynomial in ~ can be written as a polynomial in a set of fundamental real r-invariant homogeneous polynomials 0 x (-¢)..... Oq(~'), [an homogeneous integrity basis (abbreviated IB] for the ring of r-invariant polynomials in ~'].

Note that T~ • Te~ = Tc,~, by definition, since [e,r] = 0 for all r E G. Moreover, in eq. (12) ~ is always orthogonal to e~ and to T~-,and e~"is orthogonal to Te~-,since e and all r E G are antisymmetric matrices and e 2 = - 1 , so that, for all ~ E R2n and all r E G: (~, e~) = 0 = (~, r~) = (e~, er~).

(13)

Thus, what really characterizes the zeros of the Dterm is each of the equivalent relations: ~'lTe~-,

e~'lT~-,

T~-ITe~-.

(14a,b,c)

Proof: D(z, z*) = 0 if and only if(z, Tz) = 0 for all T E ~ , or, in the real formalism, if and only if 0 = (e~', r~) = -(~, er~') for all r E G, where we have exploited the antihermiticity of T E ~ and the antisymmetry of e. So eqs. (14a) and (14b) are proved. As for eq. (14c) one has, from eq. (6) and the antisymmetry of e and of all r, r' E G: (r~, er'~'> = -(~, e[r, r'] ~'>

(15)

, 6 If h (z) = h (1)(~.) + ih (2)(j.) is the decomposition of the analytic G-invariant function h (z) in its real and imaginary parts then the identity: ah(Z)(~) = eah(1)(J -) characterizes hO)(.¢) and h ( z ) ~ ) . Therefore b o t h ah (i)(~') and e ah(O(~), i = 1, 2 belong to Nc,~.(0).

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and the rhs is zero if eq. (14a) is satisfied, since It, r'] E G. Thus eq. (14a) implies eq. (14c). Viceversa ifeq. (14c) is satisfied, then (r~', er'~') = 0 for all r, r ' E G, and from eq. (15) one gets (~', e[r, r'] ~')= 0 for all r, r' E G. Since G is by assumption semi-simple, its derivative algebra G' = [G, G] coincide with G [12], so that every element of G can be written as a commutator and we have therefore proved that eq. (14c) implies eq. (14a). Three more or less immediate implications of proposition 1, we would like to stress, are the following: (a) at the zeros ~" of the D-term, the vectors ~"and e~" are orthogonal to the P-orbit through ~', so they belong to N (0) and the Pc-orbit through ~', being tangent to the sphere of radius ~"centered in the origin, behaves locally like an orbit of a compact group, in spite of the fact that I"c is never compact; :moreover: (b) N~0) has at least dimension two, so ~ cannot lie on a critical orbit [10,11]. (c) If a zero ~ of the D-term is a principal point ,7 then [10,11] N~0) = N~. so that T d- C N~-and therefore T~ = eTch., must be formed only by singlets of P~.. This implies, in particular, that the isotropy subalgebra of G at ~', G~., must be an ideal in G. (d) The isotropy sub-algebra of Gc at ~', Gc,~- ,8 is the complexification of the isotropy subalgebra of G at ~', G~. This property, which extends obviously to the connected component of the unity of the little groups P~. and Pc,~', is assured in at least one point of every closed (in the Zarisky topology) Pc-orbit [13]. Let us now prove that condition B, stated above is necessary and sufficient in order that z ~ ~"be a zero of the D-term. From proposition 1, we get e~"E Nc e, or better e~"E N(0~ C N~0) since e~" is P~-invarian~[recall eq. (6)!]. Therefore from eq. (11) [9,10]

2 December 1982

formalism to q

iz* +

a:l

a OzO-, (z, *) = o,

where O-a(z, z*) -- Oa(f). So we can restate our result more rigorously as: Proposition 2. z is a zero of the D-term if and only if eqs. (17), in the unknown real variables as, admit solutions. It is evident from the proof of condition B and the consequence (i) of proposition 1 that condition A could be proved to be also necessary if only one could prove that N(0,~ is generated by the gradients of the real P-invariant polynomials, which are the real parts of G-invariant (analytic) polynomials in z. 3. Invariant formulation o f condition B. The equation D = 0 is G-invariant and homogeneous, thus it must be possible to express the conditions under which eqs. (16) admit solutions in a G-invariant homogeneous form. This is what we shall do below. Neglecting the trivial solution G = 0, and defining, for ~"4= 0: ~

~-/(~-2)1/2,

b0 = 1 ,

b~ = (~2)(dJ2)-la~,

q c~=l

(16)

which is easily seen to be equivalent, in the complex

,7 A principal point is a "generic" point of R 2n, and has minimal symmetry with respect to G-transformations

[10,11]. ,8 God. (respectively G~-) is the subalgebra formed by the elements of Gc(G) which annihilate ~'.

82

a = 1 ..... q ,

(18)

where &, = degree 0a(g"), eq. (15) can be rewritten in the following homogeneous way: q

0 = (c-Olb0 + ~

a]Ool(-0 boe

q

= ~

Qja,(}-) b~,

43¢t= 0

e~ = ~_j a,~OO~(D,

(17)

]= 1

2n.

(19)

~ -.,,

Eq. (19) can be rephrased in a P-invariant form by requiring that the squared norm of the 2n-vector in the rhs annihilates. In this way, after defining the q-vector H(~') = (II 1 (D, ..-, Hq (~')): 1I~(~') - -(e~', 00=(~')),

oe = 1 ..... q ,

(20a)

and the real symmetric semipositive definite matrices P(~') [10] andP'(~'):

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Paa(~) --- ,

(20b)

1 IIT(~") II(~') P(~') '

p,([)=QT([)Q(~)=

(20c)

so that the q-vector N (0) is, by det'mition, orthogonal to all the eigenvectors o(/)(0). So after putting in eqs. (22): n+

where the symbol T denotes transposition, one obtains from eq. (16):

n0

ba = ~ x(k)(O) u~k)(o) + ~ yO)(O) og)(O), k=l

/=1

n+

q

(21)

ba'P~'o,([) b O' = O. a~, 13'---0

In eq. (21), the coefficients P~,¢,(~'), defined in eqs. (20) as euclidean scalar products, turn out to be Finvariant polynomials in -(, since 1-' is a group of orthogonal transformations. So, by Hilbert's theorem the P~,~,(-()'s can be identified to polynomials Pa,¢,(O(-()) in the elements 0 a of the integrity basis. Taking also into account the semipositivity property ofP'(-(), eq. (21) can be equivalently written: q #'=0

2 December 1982

P~,#,(O) b#, = O,

c~' = O, ..., q ,

(22)

and our problem reduces to finding the values of the variables 0a, for which eqs. (22) admit solutions with b0 = 1, the value of 0 a being in the range of the function 0a(-() ,9 (in the following the last condition will always be understood). To this end let us call (u(k)(o),vq)(O))k=l

.....

n + ; l = l ..... no;

n+>O, no~>O,

a complete orthonormal set of eigenvectors of P(O): P(O)u(k)(O)=p(k)u(k)(o), p(k)>o, k = l ..... n+, (23a) P(O) vq)(o) = O,

l = 1, ..., n o .

(23b)

From eq. (23b) and the definition of P(O) we get: q vq)(o(-()) boa(-() = O,

l : 1, ..., n o ,

Ha(0 ) = ~ II(k)(0)u(ak)(o), ot = 1 .... , q , k=l

(24)

one immediately realizes that, for 0 = 0-, there are solutions with bo = 1 if and only if n+

[II(k)(o-)12/p(k)(o) = 1.

(25)

k=l

A simple example to illustrate eq. (25): G = SO3, fl = 3 complex. Then C 3 9 z = x + i y , x , y E R 3 and an integrity basis is 01 = (z, z) = x 2 +y2, 02 = Re(z*, z) = (x, x ) - ( y , y ) , 03 = Im(z*, z) = 20c, y). For 01 = 1 :

(1023)

P(0)= 02 1

,

03 0

1i(0) = (o, 03, -02), so that P(O) N (0) = i'1 (0) and eq. (25) gives 02 + 02 = 1, which is satisfied by all z 4: 0, with x 11y. 4. Conclusions. We have shown that the existence of a gauge invariant real polynomial I ( z , z*) satisfying iz 1 = aT(z, z*)/bz i, for each scalar field zi, is a necessary and sufficient condition for z to be a zero of the D-term. This condition has been translated into an equivalent algebraic one involving only the elements of a complete set of fundamental invariants (integrity basis).

~=1

which multiplied internally by e~-, implies also, with eq. (20a): q ~_l vq)(o) Ha(0) = 0 ,

~=1

l = 1 ..... n o ,

One of us (R.G.) would like to thank Raymond Stora for introducing him to the subject of symmetry breaking in supersymmetric theories. This research was partly supported by the Swiss National Science Foundation.

,9 A general way of determining this range has been proposed in ref. [10]. 83

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References [ l] See for instance: B. Zumino, Intern. Conf. on High energy physics (Lisbon, July 1981), CERN preprint TH 3167; LBL preprint 13691, and references therein. [2] For a recent review see: S. Weinberg, invited paper Intern. Conf. on Unified theories and their experimental tests (Venice, March 1982). [3] F. Buccella, J.P. Derendinger, S. Ferrara and C.A. Savoy, CERN preprint TH 3212 (1981). [4] P.H. Frampton and T. Kephart, Phys. Rev. Lett. 48 (1982) 1234. [5] H. Georgi and S. Dimopoulos, Nucl. Phys. B193 (1981) 150.

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[6] N. Sakai, Z. Phys. Cl1 (1981) 153. [7] N. Dragon, Heidelberg preprint THEP-82-3 (1982). [8] F. Buccella, J.P. Derendinger, S. Ferrara and C.A. Savoy, Phys. Lett. 115B (1982) 375. [9] G. Sartori, Padova preprint IFPD 65/81 (1982). [lo] M. Abud and G. Sartori, Phys. Lett. 104B (1981) 147; and in preparation. (111 See for instance: L. Michel, CERN preprint TH 2716 (1979). 1121 N. Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, VoL No. 10 (Wiley, New York, 1962). [ 131 G.W. Schwarz, Lifting smooth homotopies of orbit spaces, Inst. Hautes Etudes Sci. Publ. Math. (1980).