Volume 206, number 4 PHYSICS LETTERS B 2 June 1988 UNITARITY LIMIT ON THE GAUGINO-GRAVITINO MASS RATIO Tanmoy BHATTACHARYA and Probir ROY Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India Received 22 March 1988 The imposition of tree-level unitarity at all energies upto the Planck scale on the amplitude for the scattering of two gauge bosons into two gravitinos is shown to put an upper bound on the ratio between gaugino and gravitino masses. The absolute masses of the lightest gauge singlet scalar and pseudoscalar bosons are also bounded from above. The nonrenormalizability of gravity generally makes gravitationally mediated tree amplitudes violate unitar- ity at the Planck scale Mp~. The amplitude for producing two Z-bosons in photon-photon collision can serve as a paradigm. At the tree level it proceeds by exchange of a single graviton in the incident channel. The growth of a typical jth partial wave amplitude aj with the centre-of-mass energy E is proportional in the leading term to l¢2E 2 where x- Mbq i = ( 8 XGNewton) 1 /2. The unitarity bound I as l < 1 then gets overshot beyond E = O (Mpl) where perturbation theory fails. There are many such examples. The archetype is the amplitude for the elastic scatter- ing of two gravitons [ 1 ]. This fact is not expected to change for the worse on account of the presence of local supersymmetry broken through the super-Higgs mechanism [2]. A spontaneously broken N= 1 supergravity theory is known to be nonrenormalizable. But the consequent violation of tree-level unitarity should once again take place beyond Mp~; it should not occur at lower energies. Yet, we show below that this is precisely what can happen with large hierarchies introduced between supersymmetry-breaking masses by adjustment of tree-level parameters, as done in certain no-scale theories [ 3 ]. We consider the amplitudes for the inelastic scattering of two identical gauge bosons of all possible helicities into a pair of gravitinos - each with mass m3/2 and helicity ~1 + ½. We find the critical energy for the breakdown of tree-level unitarity to be Ec = 12x~ x- lm3/2m~ 1 where m~ is the mass of the corresponding gaugino. Hence a large ratio between m~ and m3/2 (like 1015 as proposed in ref. [ 3 ] ) would imply that spontaneously broken supergravity becomes a strongly coupled system at a low (TeV) scale. On the other hand, the preservation of the weak coupling nature of such a theory all the way up to Mp~ will require m~m;/~2 not to be much in excess of 12x/~. Furthermore, this requirement will be shown to imply that the lightest scalar and pseudoscalar members of gauge singlet chiral supermultiplets must weigh less than (487rMplm2/2m~ t )1/2. Let us start with a locally supersymmetric nonabelian system of gauge bosons with structure tensors F~, and and the gravitino field ~u u. For definiteness, we take the gauge gaugino fields 2 a coupled to the vierbein field e u group to be simple though a generalization to nonsimple gauge groups is trivial. In order to generate gravitino and gaugino masses through the super-Higgs mechanism, we introduce chiral superfields Z i with their complex scalar component fields z i (and their conjugate fields g'-). These can be looked upon as taking values on a K~ihler manifold with a K~ihler potential fq(z,g). Their couplings to the previous fields are controlled, in addition, by fab (z) - the kinetic energy coefficient tensor (symmetric in its indices) of the unnormalized gauge superfields. The vacuum expectation values of its components are restricted by the requirements of CP-conservation in the ~ The scattering amplitudes involving any of the gravitinos with helicity + ~ do not have the requisite m ~-/~ factor in the leading terms and hence are uninteresting for our purpose. 655