Fun with Higgsless theories

Motivated by recent works on “Higgsless theories,” I discuss an SU(2)0 × SU(2) × U(1) gauge theory with arbitrary bifundamental (or custodial SU(2) preserving) symmetry breaking between the gauge subgroups and with ordinary matter transforming only under the U(1) and SU(2)0. When the couplings, gj, of the other SU(2)s are very large, this reproduces the standard model at the tree level. I calculate theW and Z masses and other electroweak parameters in a perturbative expansion in 1/g j , and give physical interpretations of the results in a mechanical analog built out of masses and springs. In the mechanical analog, it is clear that even for arbitrary patterns of symmetry breaking, it is not possible (in the perturbative regime) to raise the Higgs mass by a large factor while keeping the S parameter small. [email protected] 1 Higgsless theories, deconstruction, masses, springs So-called “Higgsless” theories [1, 2] make use of boundary conditions on an extra dimension to break the electroweak symmetry of the standard model. In a phenomenologically successful model of this kind (if such could be constructed), there would be no light scalars, but instead one would find additional massive vector bosons at the electroweak symmetry breaking scale, the Kaluza-Klein partners of the W and Z from the extra dimension. A number of groups (see for example [3] and [4]) have studied Higgsless theories using the technique of deconstruction [5, 6] to actualize the extra-dimensional metaphor in conventional four dimensional quantum field theory. These works are the primary motivation for this note. The approach here differs from that of previous works in several ways. I consider a more general pattern of symmetry breaking, preserving a custodial SU(2) symmetry [7], but otherwise completely arbitrary. I analyze these models in a power series expansion around a standard model limit. This is a strong-coupling expansion in the couplings of the “extra” SU(2) gauge groups. I also make use of what I think is an interesting trick to relate the W and Z properties in this general class of theories. Finally, I discuss a mechanical analog of the field theories in systems of masses and springs. I believe that this is extremely useful in developing intuition about the properties of these theories. In particular, I find physical interpretations of the two most critical issues facing theories of this kind: raising the scale of symmetry breaking and keeping the S parameter small. Sadly, I conclude, in agreement with previous analyses, that the promise of Higgsless theories is unlikely to be realizable, even in this more general class of theories. But I hope that the reader will find that this analysis is sufficiently unusual to justify the term “fun” in the title. In section 2, I introduce the class of models I discuss in this paper and briefly discuss the scalar sector. In section 3, I introduce the mechanical analog two systems of masses and springs one related to the Z mass matrix and the other to the W . In sections 4 and section 5, I study the W and Z mass matrices, respectively. The analysis of the light W mass is straightforward in a strong-coupling expansion of the inverse mass matrix around the standard model limit. A similar analysis of the Z is possible after a transformation of the inverse mass matrix. In section 6, I discuss the phenomenology of the class of models by calculating the electroweak parameters, S, T and U , of which S is the potential problem. Finally in section 7, I give a physical interpretation of S in the mechanical analog that makes it obvious that S is a very strong constraint for all models in the class. 2 Where is the Higgs? The class of theories that we consider in this paper are SU(2)0 × SU(2) × U(1)N+1 gauge theories with arbitrary bifundamental (or custodial SU(2) preserving) symmetry breaking Reference [4] generalizes the simple deconstruction in a different direction, including additional U(1) gauge groups, but still retaining the local structure of symmetry breaking associated with deconstruction.