The Abdus Salam International Centre for Theoretical Physics an M Theory Solution to the Hierarchy Problem

An old idea for explaining the hierarchy is strong gauge dynamics. We show that such dynamics also stabilises the moduli in M theory compactifications on manifolds of G2holonomy without fluxes. This gives stable vacua with softly broken susy, grand unification and a distinctive spectrum of TeV and sub-TeV sparticle masses. MIRAMARE – TRIESTE June 2006 1. Stabilising Hierarchies and Moduli M theory (and its weakly coupled string limits) is a consistent quantum theory including gravity, particle physics and much more. Although apparently unique, the theory has a large number of solutions. This problem is manifested by the appearance of moduli: massless scalar fields with classically undetermined vevs, whose values determine the masses and coupling constants of the low energy physics. In recent years, there has been substantial progress in understanding mechanisms which stabilise moduli in various corners of the M theory landscape. In particular, the stabilisation of all the moduli through the introduction of magnetic fields (fluxes) in the extra dimensions, perhaps also combined with other quantum effects, has been reasonably well understood in the context of Type IIB string theory [1, 2], M theory [3] and Type IIA string theory [4]. The effective potential of these compactifications fits into the framework of a low energy supergravity theory in four dimensions. A well known property of the latter is that there is a universal contribution to scalar masses of order the gravitino mass m3/2. Therefore, without miraculous cancellations, in theories in which m3/2 is large, the Higgs mass will also be large. In M theory and Type IIA flux vacua the vacuum superpotential is O(1) or larger in Planck units. This gives a large m3/2 (unless the volume of the extra dimensions is large, ruining standard unification). In heterotic flux vacua [5] m3/2 can be smaller, but only by a few orders of magnitude. Thus, in these vacua, stabilising the moduli using fluxes fails to generate and stabilise the hierarchy between the Planck and electroweak scales. In Type IIB theory, this is not so: m3/2 can be tuned small by choosing fluxes. One can also address the possibility of generating the hierarchy through warping [6] in this framework [1]. The hierarchy problem is less well understood in other corners of the landscape. Our focus will be M theory, and we will henceforth switch off all the fluxes else the hierarchy will be destroyed. Supersymmetry then implies that the seven extra dimensions form a space X with G2-holonomy. In these vacua, non-Abelian gauge fields are localised along three dimensional submanifolds Q ⊂ X at which there is an orbifold singularity [7] and chiral fermions are localised at points at which there are conical singularities [8–10]. These vacua can have interesting phenomenological features, independently of how moduli are stabilised: the Yukawa couplings are hierarchical; proton decay proceeds at dimension six with distinctive decays; grand unification is very natural; the μ-term is zero in the high scale lagrangian [8, 11–13]. Also, since the Q’s generically do not intersect each other, supersymmetry breaking will be gravity mediated in these vacua. Therefore, it is of considerable interest to understand whether or not there exist mechanisms which can a) stabilise the moduli of such compactifications, b) generate a hierarchy of scales, and if so, c) what is the resulting structure of the soft terms and their implications for LHC?