Lattice Gauge Theories beyond QCD: large N, Supersymmetry and Orientifold planar equivalence

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Among the fundamental interactions (strong, electroweak and gravitational). the strong interactions are probably the most elusive. Although they do not manifest themselves directly in the macroscopic world, strong interactions account for the stability of matter, since they are responsible for forces that bind neutrons and protons inside nuclei. Neutrons and protons are members of a wider family of particles, the hadrons, all of which interact strongly. The hadrons are not elementary particles: they are made of more fundamental particles, quarks and gluons. In electromagnetism particles interact through their electric charge; in strong interactions particles interact through a charge which this time is called pictorially colour. In nature there are three different colour charges; we speak then of the red, blue and green quarks, referring by the colour to the particular charge carried by a particle (which has nothing to do with the colours we see in our everyday experience). The theory that describes quark, gluons and their interaction is QCD. Currently, QCD (whose fine details are going to be tested at LHC) is not fully understood: while we have clear analytical predictions for the high energy regime (energies of 1 GeV or more), analytical computation techniques we currently know can not be used in the low energy regime. Unfortunately this is the most interesting regime, since e.g. quarks inside nucleons have low energies. In the absence of analytical predictions, a successful numerical approach has been developed: Monte Carlo based simulations of the theory discretised on a lattice. In this approach, the theory is formulated on a discrete and finite spacetime of spacing a and linear dimension L. This simplified theory is formulated in terms of a finite number of integrals, which can be performed on a (super)computer. The continuous theory is recovered by extrapolating to the limits L going to infinity and a going to zero. In order to fully exploit the predictive power of the calculations, an analytical understanding is still needed. Much progress has been achieved on this side by considering QCD as a special case of a more general class of theories. One possibility in this sense is to take the large N limit for a SU(N) gauge theory coupled to quarks. QCD is recovered when N=3. The large N theory is simpler than the original N=3 one, and corrections to it due to a finite value of N can be expressed as a power series in 1/N. Recently the lattice has proved to be an useful tool for determining the unknown coefficients of that power series in the absence of quarks. We plan to extend that work also when Nf quarks are included in the theory. Another interesting direction is the Orientifold planar equivalence. This is a different large N limit that relates theories with generalised quarks: a SU(N) gauge theory with an antisymmetric quark on one side and a SU(N) gauge theory with an adjoint quark on the other. The predictive power of this framework is due to the fact that the latter is related to N=1 Super Yang-Mills, for which much is know analytically, while the former reduces to QCD with one quark when N=3. Open questions remain, like the size of the corrections to recover QCD with the correct number of flavours or to go from the infinite N case to N=3. One of the central points of our project is to answer those questions. This will require a non-trivial generalisation of currently used lattice QCD algorithms and techniques. Once we have developed the techniques to simulate a generic number of fermions in a generic representation of a SU(N) gauge group, we would have the tools to attack other interesting problems, like N=1 Super Yang-Mills on the lattice and alternative scenarios for the electroweak symmetry breaking mechanism. Both those problems have a high phenomenological relevance, and will be studied at the upcoming LHC experiments.