PHYSICAL REVIEW D VOLUME 56, NUMBER 7 1 OCTOBER 1997 Rb in supergravity grand unification with nonuniversal soft supersymmetry breaking Tarakeshwar Dasgupta and Pran Nath Department of Physics, Northeastern University, Boston, Massachusetts 02115 ~Received 27 February 1997! An analysis of supersymmetric contributions to Rb in supergravity grand unification with nonuniversal boundary conditions on soft supersymmetry breaking in the scalar sector is given. Effects on Rb of Planck scale corrections on gaugino masses are also analyzed. It is found that there exist regions of the parameter space where positive corrections to Rb of size ;1s can be gotten. The region of the parameter space where enhancement of Rb occurs is identified. Predictions of sparticle masses for the maximal Rb case are given. The analysis has implications for the discovery of supersymmetric particles at colliders. @S0556-2821~97!00719-4# PACS number~s!: 12.60.Jv, 12.10.Dm, 13.38.Dg SM SUSY The standard model ~SM! predicts a value of Rb of Numerically Rb (0,0)50.2196 and ¹b (mt ,mb) is given by SM Rb 50.2159, mt5175 GeV ~1! SUSY a vLFL1vRFR ¹b ~mt ,mb!5 , ~4! with dRb /dmt520.0002. The experimental value of Rb has 2psin2u S ~v !21~v !2 D W L R been in a state of flux over the past couple of years. The experimental analyses in 1994–1995 indicated 1 1 2 expt where vL is defined by vL52 2 1 3 sin uW , and vR by Rb50.220860.0024. However, more recently Rb has 1 2 drifted downwards, and currently, assuming the value of Rc vR5 3 sin uW . In supersymmetric models the quantities F L,R at its SM value of Rc50.172, one finds @1# receive one-loop contributions from the charged Higgs, the chargino, the neutralinos, and the gluino. The most dominant Rexpt50.217860.0011, ~2! terms are those arising from the chargino exchange and we b exhibit these below @3#: which is about 1.8s higher than the SM value. ~We note that 4 W˜ ai 2 iai L,R L,R in the most recent analysis of data three of the four CERN F 5 B vL,R2 sin uWC L L* 1 2 L,R S 1 3 24 D ia ia e e collider LEP detectors observe a significant Rb anomaly and ALEPH alone does not show any deviation 1Cia jT* T LL,RL*L,R from the SM results @2#.! The possibility of a discrepancy 24 i1 j1 ia ja expt between Rb and the SM value has aroused much interest, aib L,R L,R L,R aib 1M ˜ M ˜ C O L L* 1 2C since if valid the result would signal the onset of new physics Wa Wb 0 ab ia ib S 24 beyond the SM. There have been several analyses recently to understand the possible origin of potentially large R correc- 1 b 2M 2~Caib1Caib!2 OR,LLL,RL*L,R , ~5! tions. Specifically supersymmetric contributions to this pro- Z 12 23 2D ab ia ib cess have been analyzed within minimal supersymmetric standard model ~MSSM!@3–7#. A variety of other sugges- where a,b (i, j) are the chargino ~stop! indices. tions have also been made, such as corrections from addi- B1 ,C0 ,C12 , etc., are given in terms of the Passarino- tional Z8 and from additional fermion generations. Veltman functions @15# and LL,R are given by In this work we give the first analysis of the maximal ia supersymmetry ~SUSY! corrections within supergravity uni- fication @8,9# with radiative breaking of the electroweak sym- L mt Lia5Ti1Va*12 Ti2Va*2 , ~6! metry with non-universal boundary conditions @10–13# in- A2M Wsinb cluding Planck scale corrections to the gauge kinetic energy function in supergravity @14#. For comparison with the pre- vious work we also give results for the maximum SUSY R mb Lia52 Ti1Ua*2 , ~7! corrections in MSSM, and in minimal supergravity. One de- A2MWcosb ¯ fines Rb5G(Z bb)/G(Z hadrons) and the supersymmet- → → SM SUSY ric corrections to Rb by Rb5Rb (mt ,mb)1DRb , where where tanb5^H2&/^H1& is the ratio of the Higgs vac- SUSY L,R DRb can be written in the form @3# uum expectation values ~VEV’s!, Oab are defined by L 2 1 R 2 Oab52cos uWdab1 2 Ua*2Ub2 and Oab52cos uWdab SUSY SM SM SUSY 1 * DRb 5Rb ~0,0!@12Rb ~0,0!#@¹b ~mt ,mb! 1 2 Va2Vb2, where U ab ,Vab are the matrices that diagonal- ize the chargino mass matrix and Tij is the matrix that di- SUSY 2 2¹b ~0,0!#. ~3! agonalizes the stop mass matrix: i.e., 0556-2821/97/56~7!/4194~4!/$10.0056 4194 © 1997 The American Physical Society 56 Rb IN SUPERGRAVITY GRAND UNIFICATION WITH . 4195 ˜ ˜ t 2 cosu˜t sinu˜t t L 5 . ~8! S˜t D S 2sinu˜t cosu˜t DS˜t D 1 R To set the stage for the analysis in supergravity unifica- tion we discuss first the general features that lead to a large DRb contribution in SUSY models. The maximum contribu- SUSY tion to DRb comes from the terms involving light masses SUSY ˜6 ˜ in Eq. ~5!. So a large DRb will require light x 1 , t 1 and specific mixings requiring the stop which is mostly right handed and charginos which have significant Higgsino com- ponents. An effect of these constraints is to optimize the contributions of the top Yukawa couplings in the vertices FIG. 1. Maximum Rb for various models as a function of the light m˜6. involving the stop loops @3–7#. For low tanb we find x 1 M˜6'M˜6 @6# which is possible for M 2'2m, where M 2 is x 2 x 1 the SU~2! gaugino mass and m is the Higgs mixing param- salities in the Higgs sector and in the third-generation sector. For this reason we shall focus in the present analysis on the eter ~for an overview of large tanb case see Ref. @6#!. Fur- L L,R nonuniversalities in these sectors and assume universality in ther, Lij’s and Oij ’s that give large weights to the domi- SUSY the remaining sectors. It has recently been shown that the nant terms in Eq. ~5! lead to a large DRb . Large weights L L nonuniversalities in the Higgs sector and in the third- for the dominant terms require a large L11 (L12) and a large L,R L,R generation sector are strongly coupled because of the large negative O11 (O22 ) implying a large T12 ,V12 (V22) and a top Yukawa coupling @13#. This phenomenon will play an small U12 (U22) for tanb,1 (tanb.1). We find that for important role in our analysis. It is convenient to parametrize SUSY 6 tanb.1 DR is maximum for u˜.29° and a ˜x which the nonuniversalities in the Higgs sector by d ,d where b t 2 H1 H2 is mixture of a large up Higgsino and a gaugino state m2 (0)5m2(11d ), and m2 (0)5m2(11d ). Similarly H1 0 H1 H2 0 H2 (uV22u.0.9, uU22u,0.1). Our results are in accord with the opt we parametrize the nonuniversalities in the third-generation 2 analysis of Ref. @6# except for u˜t where our value supports 2 sector by d˜t and d˜t where m˜ (0)5m0(11d˜t ), and L R t L L the result of Ref. @3#. Our best value of DRb in MSSM then 2 SUSY 2 m˜ (0)5m0(11d˜t ). We also include in the analysis is DRb <0.0028 for tanb>1.16, comparable with previ- t R R ous determinations @4–7#. Planck scale corrections which arise via corrections to the Although, as discussed above, one can generate a signifi- 1 a bmn SUSY gauge kinetic energy, i.e., 2 4 f abFmnF , where f ab con- cant DRb correction in MSSM, it is not a priori clear tains the corrections from the Planck scale. Planck correc- what part of the parameter space, if any, which gives large tions in fab contribute to gauge coupling unification in super- corrections is compatible with the constraints of grand unifi- gravity grand unified theory ~GUT!@14# and also generate cation and radiative breaking of the electroweak symmetry. corrections to the gaugino masses which can be parametrized This is the issue we address in this work. The analysis we by M i5(ai /aG)@11c8(M/M P)ni#m1/2 , where M is the carry out includes radiative breaking of the electroweak sym- GUT mass, M is the Planck mass, c8 parametrizes the metry, constraints to avoid color and charge breaking, ex- P Planck scale correction, and ni5(21,23,22) @14#. Thus perimental constraints on the superparticle spectrum, and the for the nonminimal model we have the set of parameters c8, b s1g experimental constraint as given by the CLEO Col- d , d , d˜ , and d˜ , in addition to the parameters of the laboration→ @16#. We also include the constraint arising from H1 H2 t L t R ˜ ˜0 minimal model. the decay t t 1x i and assume that the branching ratio of In Fig. 1 we display R in the standard model and the → ˜ ˜0 b the top decay into stops satisfies B(t t 1x i ),0.4. We dis- maximal R that can be achieved in MSSM and supergravity → b cuss first the minimal supergravity case which is param- models with universal and nonuniversal boundary condi- etrized by m0, m1/2 , A0, and tanb, where m0 is the universal tions.