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Publication# A New Method to Explore Conformal Field Theories in Any Dimension

Résumé

The thesis represents an investigation into Conformal Field Theories (CFT's) in arbitrary dimensions. We propose an innovative method to extract informations about CFT's in a quantitative way. Studying the crossing symmetry of the four point function of scalar operators we derive consistency constraints on the CFT structure, in the form of functional sum rules. The technique we introduce allows to address the feasibility of the sum rule and translate it into restrictions on the CFT spectrum and interactions. Our analysis only assumes unitarity of the CFT, crossing symmetry of the four point function and existence of an Operator Product Expansion (OPE) for scalars. We demonstrate that a CFT satisfying the above hypothesis and containing a scalar operator is not compatible with arbitrary spectra of the operators nor with arbitrary large OPE coefficients. More specifically we prove two main results. First, the spectrum of the CFT must contain a second scalar operator with dimension smaller than a given value. Second, the value of the three point function of two scalars with equal dimension and a third arbitrary operator is bounded from above. As an application of the fist statement we present the bound on the smallest dimension operator entering the OPE of a real scalar with itself. We perform the analysis in two and four dimensions. The comparison of the two dimensional case with exactly solvable models shows a saturation of the bound. We repeat for CFT's with global symmetries and superconformal field theories in four dimensions. As a demonstration of the second result we provide a lower bound on the central charge for CFT in two and four dimensions without global symmetries and for superconformal field theories. We also discuss in the potentialities of the method and possible future research lines. Finally, we discuss possible implications for model building beyond the Standard Model of particle physics.

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En physique la notion de symétrie, qui est intimement associée à la notion d'invariance, renvoie à la possibilité de considérer un même système physique selon plusieurs points de vue distincts en te

Symmetries are omnipresent and play a fundamental role in the description of Nature. Thanks to them, we have at our disposal nontrivial selection rules that dictate how a theory should be constructed. This thesis, which is naturally divided into two parts, is devoted to the broad physical implications that spacetime symmetries can have on the systems that posses them. In the first part, we focus on local symmetries. We review in detail the techniques of a self-consistent framework -- the coset construction -- that we employed in order to discuss the dynamics of the theories of interest. The merit of this approach lies in that we can make the (spacetime) symmetry group act internally and thus, be effectively separated from coordinate transformations. We investigate under which conditions it is not needed to introduce extra compensating fields to make relativistic as well as nonrelativistic theories invariant under local spacetime symmetries and more precisely under scale (Weyl) transformations. In addition, we clarify the role that the field strength associated with shifts (torsion) plays in this context. We also highlight the difference between the frequently mixed concepts of Weyl and conformal invariance and we demonstrate that not all conformal theories (in flat or curved spacetime), can be coupled to gravity in a Weyl invariant way. Once this ``minimalistic'' treatment for gauging symmetries is left aside, new possibilities appear. Namely, if we consider the Poincar'e group, the presence of the compensating modes leads to nontrivial particle dynamics. We investigate in detail their behavior and we derive constraints such that the theory is free from pathologies. In the second part of the thesis, we make clear that even when not gauged, the presence of spontaneously broken (global) scale invariance can be quite appealing. First of all, it makes possible for the various dimensionful parameters that appear in a theory to be generated dynamically and be sourced by the vacuum expectation value of the Goldstone boson of the nonlinearly realized symmetry -- the dilaton. If the Standard Model of particle physics is embedded into a scale-invariant framework, a number of interesting implications for cosmology arise. As it turns out, the early inflationary stage of our Universe and its present-day acceleration become linked, a connection that might give us some insight into the dark energy dynamics. Moreover, we show that in the context of gravitational theories which are invariant under restricted coordinate transformations, the dilaton instead of being introduced ad hoc, can emerge from the gravitational part of a theory. Finally, we discuss the consequences of the nontrivial way this field emerges in the action.

We continue the study of model-independent constraints on the unitary conformal field theories (CFTs) in four dimensions, initiated in [R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, J. High Energy Phys. 12 (2008) 031]. Our main result is an improved upper bound on the dimension Delta of the leading scalar operator appearing in the operator product expansion (OPE) of two identical scalars of dimension d: phi(d)x phi(d)=1+O-Delta+.... In the interval 1 < d < 1.7 this universal bound takes the form Delta < 2+0.7(d-1)(1/2)+2.1(d-1)+0.43(d-1)(3/2). The proof is based on prime principles of CFT: unitarity, crossing symmetry, OPE, and conformal block decomposition. We also discuss possible applications to particle phenomenology and, via a 2D analogue, to string theory.

2009We introduce a new numerical algorithm based on semidefinite programming to efficiently compute bounds on operator dimensions, central charges, and OPT coefficients in 4D conformal and N = 1 superconformal field theories. Using our algorithm, we dramatically improve previous bounds on a number of quantities, particularly for theories with global symmetries. In the case of SO(4) or SU(2) symmetry, our bounds severely constrain models of conformal technicolor. In N = 1 superconformal theories, we place strong bounds on dim(Phi(dagger)Phi), where Phi is a chiral operator. These bounds asymptote to the line dim(Phi(dagger)Phi)

2012