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Supergravity Unification 1 Supergravity Unification a b cd Ali H. Chamseddine ,R.Arnowitt and Pran Nath ∗ aCenter for Advanced Mathematical Sciences (CAMS) and Physics Department, American University of Beirut, Lebanon bCenter for Theoretical Physics, Department of Physics, Texas A& M University, College Station TX 77843,-4242, USA cTheoretical Physics Division, CERN CH-1211, Geneve 23, Switzerland d Department of Physics, Northeastern University, Boston, MA 02115-5000, USA A review is given of the historical developments of 1982 that lead to the supergravity unified model (SUGRA) with gravity mediated breaking of supersymmetry. Further developments and applications of the model in the period 1982-85 are also discussed. The supergravity unified model and its minimal version (mSUGRA) are currently among the leading candidates for physics beyond the Standard Model. A brief note on the developments from the present vantage point is included. 1. Introduction efforts to break global supersymmetry sponta- neously lead to some unpleasant features as dis- cussed below and consequently such efforts were The main advantage of supersymmetry [1] in abondoned. An alternative suggestion, advanced the context of building models of particle inter- however, was to introduce in an ad hoc fashion actions is that it offers a technical solution to the soft breaking terms consisting of dimension 2 and hierarchy problem of masses[2]. Thus, for exam- dimension 3 operators in the theory which break ple, the Higgs field is not protected from acquir- supersymmetry but nonetheless respect the ul- ing large masses though loop corrections and in traviolet behavior of the theory and respect the a grand unified theory the radiative corrections gauge hierarchy since such terms are super renor- 14 would give it a GUT size mass of O(10 )GeV. malizable[4,5]. The list of such soft terms con- However, if the scalar fields appear in a super- sists of scalar mass terms such as for squarks and multiplet then supersymmetry would require a sleptons and for the Higgs, bilinear scalar terms, cancellation of the loop corrections in the super- Majorana mass terms for the gauginos, as well as symmetric limit. When supersymmetry is bro- cubic bosonic terms involving products of scalar ken the loop corrections to the Higgs mass and fields. However, such a procedure introduces gen- to other masses will then have a dependence on erally an enormous number of free parameters in the SUSY breaking scale. So one finds that the the theory. For the so called Minimal Supersym- breaking of supersymmetry is central to getting metric Standard Model this number is around a meaningful result out of supersymmetric theo- 104. A theory such as this is not predictive. ries. Up to 1982 models in particle physics were in To understand the origin of soft breaking terms the framework of global supersymmetry[3]. How- and to derive them from the basic model one ever, this effort faced an important hurdle, i.e., needs to break supersymmetry spontaneously. it is difficult to break global supersymmetry in However, the spontaneous breaking of supersym- a phenomenologically acceptable fashion. Thus metry leads to a Goldstone fermion, or a Gold- stino and it is important to absorb this Goldstino ∗Permanent address of P.N 2 to achieve a meaningful theory which is not in couplings and on the soft parameters. The evo- contradiction with experiment. To absorb the lution of these parameters from the GUT scale Goldstino one needs a vector-spinor, i.e., a mass- down to the electro-weak scale turns the deter- less spin 3/2 field or a gravitino. The necessity minant of the Higgs mass square matrix negative to get rid of the undesirable Goldstino requires which triggers the breaking of the electro-weak that we consider an extension of supersymmetry symmetry[13]. In the framework of MSSM there to include gravity. In this extension the gravitino are two Higgs doublets in the theory, one which is the superparter of the graviton and thus one gives mass to the up quark and the other which is led to consider supergravity[6] as the starting gives mass to the down quark. These two dou- point for a phenomenologically viable breaking of blets provide a bilinear term with a mixing pa- supersymmetry. However, to build models within rameter µ in the superpotential. Although this supergravity one needs to couple N=1 supergrav- term is supersymmetric the µ parameter is of the ity with chiral multiplets and a vector multiplet size of the soft breaking terms. An important leading to the so called applied supergravity[7,8]. aspect of the radiative breaking of the electro- Remarkably the framework of applied supergrav- weak symmetry is that the spontaneous symme- ity allows one to break supersymmetry in a spon- try breaking conditions allow one to determine taneous fashion and also to build realistic mod- the µ parameter and this eliminates one of the els. One of the basic ingredients in the analysis is arbitrary parameters in the theory. that the scalar potential of the theory is not pos- The radiative breaking of the electro-weak sym- itive definite which was one of the main hurdles metry which arises in SUGRA model relates the in finding a global vacuum with broken supersym- weak scale MW with the soft SUSY breaking scale metry in the framework of global supersymmetry. which in turn depends on the Planck constant. For the case of supergravity grand unification pro- The SUGRA model in this fashion connects two posed in 1982[9] it is possible to break supersym- fundamental scales of physics, i.e., the Fermi scale metry spontaneously in a so called hidden sector and the Planck scale. Further, the fact that and arrange the vacuum energy to vanish. The MW is determined in terms of soft parameters information of supersymmetry breaking from the up to gauge and Yukawa coupling constant fac- hidden sector to the physical sector is communi- tors means that the soft parameters must lie in cated by gravity resulting in the growth of soft the TeV range, or else, one is faced with a new breaking terms in the physical sector[9,10]. For kind of a fine tuning problem. This result leads to definiteness the GUT group chosen in the 1982 the remarkable prediction in SUGRA GUTS that work was SU(5) although the main results of the one must have sparticles lying in the TeV mass effective theory are not tied to any specific GUT range. group[9,11,12]. Supergravity grand unification model proposed 2. SUGRA Unification in 1982 also resolves another problem, i.e., the puzzle of the breaking of the electro-weak We discuss now the main features of the work of SU(2)L U(1)Y symmetry. There is no natural Ref.[9]. To build models containing gravity one mechanism× for the breaking of the electro-weak needs to couple supergravity with an arbitrary symmetry in the Standard Model, and one ar- number of chiral multiples χi(x),zi(x) (where { } ranges for the breaking by the introduction of a χi(x) are left (L) Weyl spinors and zi(x) are scalar tachyonic mass term. However, the introduction fields) and a vector multiplet which belongs to of such a tachyonic mass term is rather ad hoc. the adjoint representation of the gauge group. Supergravity grand unification provides a natural Such a coupling scheme depends on three arbi- mechanisms for the generation of the tachyonic trary functions: : the superpotential g(zi), the mass term. In SUGRA unification one starts at gauge kinetic function fαβ(zi) (which enters in α µνβ the GUT scale with prescribed boundary condi- the Lagrangian as fαβFµν F with α, β = gauge tions on the gauge coupling constants, Yukawa indices) and the Kahler potential d(zi,zi†). The 3 bosonic part of the Lagrangian is given by and one has in this case that the gravitino mass is e e a α b 2 2 LB = R + G, DαzaD z m 3 = κ <g2 >= κm (7) −2κ2 κ2 b 2 e α µνβ In addition to the bosonic interactions one also fαβFµν F eV (1) −4 − has interaction structures in the theory which in- 18 volve gauge couplings of the gauginos to the chiral Here κ 1/MP` where MP lanck =2.4 10 GeV, G≡ depends on the superpotential g(z),× and multiplets and these take the form on the Kahler potential d(z,z†)sothat α α T j Lλ = igαλ¯ zi( )ij χ + h.c. (8) 2 6 − 2 κ κ 2 G = d(z,z†) ln( g )(2) − 2 − 4 | | Further, the superpotential contributes to both and V is the effective potential for the scalar com- the fermi and the bose interactions. Thus the i ponents of the chiral multiplets in the theory and Fermi interactions for the Weyl spinors χ are is given by given by[14] 1 1 G 1a b 1 α a 2 iC j V = e− (3+G− G;aG, )+ (G, aT z ) (3) Lg = (¯χ gij χ + h.c.)(9) −κ4 b 8κ4 −2 As noted in the introduction the potential is no We now elaborate on some of the further details longer positive definite because of the Planck in the construction of SUGRA GUT models. One scale corrections from supergravity. Of course, in of the immediate problems one faces in construct- the limit MP lanck one recovers the result of ing such models is that of protecting the gauge →∞ global supersymmetry. However, the terms pro- hierarchy. Since <z> O(MP lanck) one finds ∼ portional to 1/MPLanck are essential in generat- that terms involving zizjz would develop masses ing an acceptable spontaneous breaking of super- which are size O(MP lanck) and one has to sup- symmetry.
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