On dynamical supersymmetry breaking by chiral condensates

Volume 125, number 1

PHYSICS LETTERS

19 May 1983

ON DYNAMICAL SUPERSYMMETRY BREAKING BY CHIRAL CONDENSATES Ulrich ELLWANGER Department of Theoretical Physics, 1 Keble Road, Oxford, 0)(1 3NP, UK

Received 25 November 1982 Revised manuscript received l 1 February 1983

The renormalization of Green functions of composite chiral superfields is investigated. We show that the solution of the renormalization group equation for the corresponding generating functional does not allow for supersymmetry breaking vacuum expectation values.

Dynamical breakdown of supersynnnetry (DBS) has some very attractive features [1]. If it takes place in a supersymmetric grand unified theory, it could solve both kinds of "hierarchy problems": the huge difference of mass scales between the grand unified scale and the weak interaction scale, as well as its stability under quantum corrections. Models with assumed DBS by fermion condensates have akeady been proposed [2]. However, up to now nobody was able to construct a theory in four dimensions, in which DBS actually takes place [3]. (A model, for which DBS has been conjectured [4], was shown to suffer from negative norm states [5] .) Witten [6] has proposed a necessary criterion for DBS, which is very restrictive; unfortunately it could not yet be applied to the most interesting case of left-right unsymmetric super-GUTs (see also ref. [7] ). In this note we will deal with the question of whether the F-component of a local composite chiral superfield can have a non-vanishing VEV, which would break supersymmetry (SS) [2]. Our result can be appried to any renormallzable SS theory, including chiral super-GUTs. Since the vacuum energy in SS theories is positive semi-deFinite, spontaneous breakdown of SS has some special features [ 1 ] : It only takes place, if the SS vacuum state does not exist, since it would be automatically stable. In the case of SS breakdown at tree level, this can be achieved, if the equations of motion for the auxiliary fields do not possess a SS preserving solution [8]. In the case of DBS by condensates the 54

situation is similar: The corresponding SchwingerDyson equations must have no solutions, which preserve SS [9]. In the following we will study composite superfields, which are local in ordinary space as well as superspace. Since the renormalization of proper Green functions of local composite fields (and hence the corresponding effective action) is problematic [10], we will consider the generating functional of connected Green functions. Here renormalization is straightforward even for local composite fields [ 11 ], furthermore it also contains information about the corresponding VEV [12,13]. After all the VEV is a connected one-point function. In general a non-vanishing VEV of composite fields will be non4ocal in x-space. In restricting ourselves to local VEVs we have tacitly assumed, that a nonvanishing VEV has a non-vanishing local component, i.e. is t'mite at the origin. Two additional assumptions will be used for obtaining our result, the absence of DBS by chiral condensates: First we will require, that the parameters of the renormalization group are determined in perturbation theory, or, in other words, that the theory is perturbatively renormallzable. According to this assumption SS non-renormalization theorems [14] can be applied to the parameters of the renormalization group. Second, we will assume, that the Green functions of any SS theory can be obtained from the Green

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functions of a similar theory, in which SS has been strongly broken (e.g. by violating a SS relation between some dimensionless coupling constants), in the limit where SS is restored. We are aware of the fact, that phase transitions could occur, if we change the coupling constants of a theory. However, we do not require, that the Green functions of the theory with violated SS, which give the Green functions in the theory with restored SS, are the ones in the true vacuum of the theory - they could be the Green functions determined in any (e.g. unstable) phase of the theory. We take it for granted, that the SS theory exists, so that the corresponding limit exists, although it does not necessarily have to be analytical. (One could also put the theories in a finite volume with appropriate boundary conditions, where it is easy to satisfy conditions, under which analiticity in the coupling constants holds [6]. After obtaining our result, the absence of DBS, for any finite volume, the result - the vanishing of the vacuum energy - will also hold in the infinite volume limit, since the infinite volume limit of zero is zero [6] .) Let us start by introducing some notations and definitions. A composite superfield 4i], which is the product (local in superspace) of two left-handed chiral superfields with internal symmetry indices i and ], is again chiral and has the following scalar, fermionic and auxiliary components:

Oij = (AiA/,Ai~ / +Aj~i, AiF] +A/f t. - ~idd]), where Ai, d/i and F/are the corresponding compo-

(1)

nents of the individual superfields. In the following, internal symmetry indices will be suppressed for simplicity, although 4 must not necessarily transform as a singlet e.g. under gauge transformations. Next we define the generating functional G(J, J+) of connected Green functions of composite chiral superfields in Minkowski space:

G(J,J+) = lntTexp(ifdSz[J(-D2/4[])O

with z = (x, 0,0). The sources J (J+) are again lefthanded (right-handed) chiral superfields. In general they will carry internal symmetry indices corresponding to the ones of 4 (4+)- From now on we will restrict ourselves to sources, which do not depend on

19 M a y 1 9 8 3

the space-time variable x, and where only the scalar component is different from zero. This restriction preserves SS, since scalar constants transform as chiral superfields. These sources generate exactly the Green functions we are interested in, i.e. the Green functions of space-time independent auxiliary field components of 4. Using the commutation relations of the SS charges Q~ and Q~ with the components of a chiral superfield, it can easily be verified, that a space-time independent VEV of the auxiliary field component of 4 implies a non-supersymmetric vacuum [2]. For constant scalar sources G(J, J+) becomes a simple function o f J and J + with the corresponding n-point functions as coefficients of a Taylor expansion. (It is useful to rescale the Green functions by appropriate powers of the space-time volume V.) •k

•

G(J, J+) = ~ ~

1

G(k, l)jkj +l,

k,l=l

•

(3)

.

with

G(k, l) = (1/vk +l-1)(4(O1)...O(Ok) +

-

+

-

x ¢ (01)._4 (0l)>conn.10....o2k v21 ""~'l

(4)

Since the Green functions G(k, l) are not necessarily unique, G(J, J+) can be a multivalued function near J = J+ = 0 [12]. The interesting VEVs are given by i(4)j,j.[02 = d G(J, J+)/ dJ, -i(4+)JJ+, 1~2 = dG(J,

J+)/dJ +.

(5)

As mentioned at the beginning, DBS implies that the SS preserving solution (4) = (4 +) = 0 does not exist. At these points the vacuum energy would vanish and would necessarily have a minimum. The extrema of the vacuum energy are given by the extrema of the Legendre transform of G(J, J+), the generating functional of proper Green functions P(4, 4 ÷) [ 1 0 - 1 3 ] . The extrema of P(¢, ¢ +) correspond to the values J = J+ = 0 [ 1 0 - 1 3 ] . Hence the first derivatives of G(J, J+) cannot vanish for values of J, J+ different from zero in any SS theory; DBS would imply, that the frst derivatives of G(J, J+) do not vanish anywhere. Using the restrictions imposed on G(J, J+) by the renormalization group, we will show, that this cannot happen. 55

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Renormalization of composite fields will, in general, induce operator mixing [11 ] and the Green functions of all operators, which do mix under renormalization, have to be treated together. Furthermore, n-point functions with Q-fields only, will diverge even in free field theory for n ~< 4 for the composite fields under consideration [10,11,13]. Accordingly a polynomial P(J, J+) with degree four and divergent coefficients has to be added to G(J, J+) [10,11,13] : 4

P(J, J+) =

~ P(i,j)jij +i. i,]=O

(6)

2<~i+/~4 The extension of eq. (6), taking internal symmetry indices into account, is trivial. The coefficients P(i,]), like usual counterterms, will depend on the coupling constants of the theory, the renormalization point/1 and an appropriate regularization. They are determined with the help of the following renormalization prescriptions: In order to be consistent with zeroth order perturbation theory, the corresponding two-, three- and four-point functions have to vanish at the renormalization point [11 ]. In our formalism the renormalization point is defreed by J = J+ =/~ [13]. (The renormalization prescriptions for wavefunction renormalization constants may also require non-vanishing external momenta for the corresponding two point functions due to infrared divergences; in the following we will assume, that such conditions are satisfied at the renormalization point.) The vanishing of the second, third and fourth derivatives of G(J, J+) at J = J+ =/~ has to be used as boundary conditions for the integration of the renorrealization group equation. As it will become clear below, the renormalization group equation for the fourth derivative of G(J, J+) with respect to left-handed (or right-handed) chiral fields only is of particular importance. Therefore we will reintroduce internal symmetry indices and use the following definitions:

G4i ~ d4G/(dJi )4 ,

Gi, 3i = d4G/dJ/(d J/) 3"

For P 4i , PI, 3i correspondingly.

56

71k=ZliIJd(Z-1)ik/dIJ=-u[d(Z

19 May 1983

1)li/dla]Zik.

(7)

gi denote all coupling constants of theory, including masses. Z is the matrix of composite field renormalization constants [11 ]. The corresponding renormalization group equation becomes [13] :

Eq. (8) is a system of inhomogeneous partial differential equations for the Green functions G4i(J, J+) and GI, 3i(J, J+). Renormalizability implies the finiteness of the rhs. In the preceding discussions of renormalization we have never made use of any simplifications which would occur for SS theories due to non-renormalization theorems [14]. Hence the equations above will also hold for an auxiliary theory, in which SS has been strongly broken e.g. by violating a supersymmetfic relation between dimensionless coupling constants (preserving renormalizability). In the following we will assume, that a corresponding change in the parameters of the SS theory under investigation has been performed. However, we will not change the field content of the theory, so that the generating functional discussed above in any case generates Green functions of a combination of composite fields, which would correspond to any auxiliary field component of a composite chiral superfield in the SS theory. If we restore SS by changing the parameters of the theory, we will assume, in view of the discussion at the beginning, that the Green functions and parameters of the renormalization group change continuously. For the latter the following results hold: Using the power counting rules derived in ref. [ 14], it can be seen, that the Green functions with C-insertions and two external elementary chiral superfields are superficially convergent. Hence the renormalization constant Z of the composite field ~ is simply the product of the square roots of the wavefunction renormalization constants of the elementary fields contained in q~. No operator mixing takes place, and the non-diagonal anomalous dimensions 7i] (i 4=/) are zero. Furthermore the coefficients P(i,/) of the polynomial P vanish for i or/" equal to zero. According to our assumption discussed at the beginning, perturba-

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five non-renormalization theorems can be applied; in this case it follows from 0-counting that Green functions with external left-handed or right-handed chiral fields only are zero [14]. It is important, that these Green functions vanish exactly, so that not even finite coefficients P(O,j) or P(L 0) remain. Since the rhs of eq. (8) is proportional to P(i, 0), it will approach zero in the SS limit. Due to our assumption o f continuity of the Green functions the vanishing of d4G/dJ 4 at J = J+ =/~, which served as a boundary condition for the integration of eq. (8), will also persist in the SS limit. [Here this assumption is crucial; it would have been inconsistent, to require such a boundary condition ad hoc in a situation, where the rhs o f eq. (8) vanishes.] Now the renormalization group equation for d4G/dJ 4 reads after eliminating /~/O/J with the help of dimensional considerations (taking the absence of mixing into account): ([(/3i - digi)O/Ogi + (7 - 1)JO/OJ] + h.c. + 4"/} d4G/dJ 4 = O,

(9)

with d i = dim (gi). From the general solution o f eq. (9) together with d4G/dJ 4 = 0 at J = J+ =/~ it can be seen that d4G/dJ 4 (and, by hermiticity, d4G/dJ +4) must vanish everywhere. Hence G(J, J+) can only be a finite polynomial in J and J + : 3

G(J, J+) = ~ CklJkJ+l. k,l=O

(10)

As mentioned below eq. (5), a zero of the first derivatives of G(J, J+) for J, J+ 4 : 0 is impossible in SS theory. It follows, that the coefficients ckt must vanish except for Coo, col and clo. However, non-vanishing coefficients col and clo would correspond to the following pathological situation: In that case the VEV o f ~ and 4 + would be constant (equal to - i c l o , icol respectively) independent of the sources coupled to ~ and ~+. The sources simply correspond to mass terms of the elementary fields contained in 4- (If q5transforms as a non-singlet under internal symmetry transformations, the sources are symmetry breaking mass terms.) Hence, for non-vanishing col and c 10, the VEVs of q~and ~+ would be non-zero and independent o f the mass of the component fields, even if the mass

19 May 1983

goes off to infmity - a situation we disregard as unphysical. If she (or he) wishes, the reader can keep these cases in mind as exceptions to our result, but we are lead to conclude, that col and Cl0 must be zero. Hence the VEVs o f 0 and 4 + must vanish, the effective action o f ~ and 4 + is even only defined at ~ = 4 + = 0, and no DBS takes place - the result announced above. As mentioned frequently, the composite field ¢ is allowed to transform non-trivially under gauge transformations; accordingly this result can be applied to chiral GUTs in contrast to the index-theorem of Witten [6]. Ultraviolet divergences and renormalization, which are ignored in ref. [6], play a central role in this approach. In contrast to ref. [9] we have never assumed the existence of a supersymmetric solution of the Schwinger-Dyson equation, in which case we consider the absence o f DBS to be trivial. The reason, why we had to restrict ourselves to chiral condensates, becomes clear from the use of non-renormalization theorems, which would not hold for vector-like composite superfields. Clearly, our result implies the absence of DBS by fermion condensates, which are formed by components of matter multiplets [2,3]. All in all we think, that evidence for the absence o f DBS in globally SS theories is increasing alarmingly, and that it might well be necessary to go beyond global SS (e.g. supergravity [15] ), if one wishes to solve the hierarchy problems.

R eferen ces [1] E. Witten, Nucl. Phys. B188 (1981) 513. [2] M. Dine, W. Fischler and M. Srednicki, Nucl. Phys. B189 (1981) 575; S. Dimopoulos and S. Raby, Nucl. Phys. B192 (1981) 353. [3] H.P. Nilles, Phys. Lett. l12B (1982) 455; W. Buchm/iller and S. Love, Nucl. Phys. B204 (1982) 213; G. Domokos and S. K6vesi-Domokos, Johns Hopkins University preprint (1982) JHU-HET 8210. [4] D. Zanon, Phys. Lett. 104B (1981) 127. [5] P. Salomonsen, Nucl. Phys. B207 (1982) 350; D. Amati and K. Chou, Phys. Lett. l14B (1982) 129; A. Higuchi and Y. Kazama, Nuel. Phys. B206 (1982) 152; U. Ellwanger, Oxford preprint 67/82 (1982). [6] E. Witten, Nucl. Phys. B202 (1982) 253.

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[7] S. Cecotti and L. Girardello, Phys. Lett. l l 0 B (1982) 39. [8] L. O'Raifeartaigh, Nucl. Phys. B96 (1974) 331; P. Fayet and J. Iliopoulos, Phys. Lett. 51B (1974) 461. [9] G. Domokos and S. K6vesi-Domokos, Johns Hopkins University preprint (1982) JHU-HET 8211. [ 10] J. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D10 (1974) 2428; T. Banks and S. Raby, Phys. Rev. D14 (1976) 2182.

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[ 111 C. Itzykson and J.B. Zuber, Quantum field theory (McGraw Hill, New York, 1980). [12] G. Jona-Lasinio, Nuovo Cimento 34 (1964) 1760. [13] U. Ellwanger, Nucl. Phys. B207 (1982) 447. [14] M. Grisaru, W. Siegel and M. Ro~ek, Nucl. Phys. B159 (1979) 429. [15] H.P. Nilles, Phys. Lett. l15B (1982) 193.