Abstract
The standard wisdom on the origin of massless bosons in the spectrum of a Quantum Field Theory $(QFT)$ describing the interaction of gauge fields coupled to matter fields is based on two well known features: gauge symmetry, and spontaneous symmetry breaking of continuous global symmetries. However we will show in this article how the topological properties, that originate the $U(1)_A$ axial anomaly in a $QFT$ which describes the interaction of fermion matter fields and gauge bosons, are the basis of an alternative mechanism to generate massless bosons in the chiral limit, if the non-abelian $SU(N_f)_A$ chiral symmetry is fulfilled in the vacuum. We will also test our predictions with the results of a well known two-dimensional model, the two-flavour Schwinger model, which was analyzed by Coleman long ago, and will give a reliable answer to some of the questions he asked himself on the spectrum of the model in the strong-coupling (chiral) limit. We will also analyze what are the expectations for the $U(N)$ gauge-fermion model in two dimensions, and will discuss on the impact of our results in the chirally symmetric high temperature phase of $QCD$, which was present in the early universe, and is expected to be created in heavy-ion collision experiments.
Highlights
There are two well-known mechanisms in quantum field theory that allow us to understand the existence of massless bosons in the spectrum of a given model of gauge fields coupled to matter fields: gauge symmetry and spontaneous symmetry breaking of continuous global symmetries
There are some well-known examples, for instance, two-flavor quantum electrodynamics in (1 þ 1) dimensions, in which chiral quasimassless bosons appear in the spectrum of the model near the chiral limit [1] and in which the explanation of this phenomenon escapes the two aforementioned mechanisms to generate massless bosons
II to reviewing some results concerning the relation between vacuum expectation values of local and nonlocal operators computed in the Q 1⁄4 0 topological sector, with their corresponding values in the full theory, taking into account the contribution of all topological sectors, and will see how, notwithstanding that the Q 1⁄4 0 sector breaks spontaneously the Uð1ÞA axial symmetry and shows a divergent pseudoscalar susceptibility in the chiral limit, the associated pseudoscalar correlation length remains finite in this sector, and the Nambu-Goldstone theorem is not fulfilled
Summary
There are two well-known mechanisms in quantum field theory that allow us to understand the existence of massless bosons in the spectrum of a given model of gauge fields coupled to matter fields: gauge symmetry and spontaneous symmetry breaking of continuous global symmetries. Since the Q 1⁄4 0 topological sector will play a main role in our physical discussions, we will devote Sec. II to reviewing some results concerning the relation between vacuum expectation values of local and nonlocal operators computed in the Q 1⁄4 0 topological sector, with their corresponding values in the full theory, taking into account the contribution of all topological sectors, and will see how, notwithstanding that the Q 1⁄4 0 sector breaks spontaneously the Uð1ÞA axial symmetry and shows a divergent pseudoscalar susceptibility in the chiral limit, the associated pseudoscalar correlation length remains finite in this sector, and the Nambu-Goldstone theorem is not fulfilled. Summarizing, we have shown that, even if the Q 1⁄4 0 topological sector breaks spontaneously the Uð1ÞA axial symmetry to give account of the anomaly, the Goldstone theorem is not fulfilled because the divergence of the pseudoscalar susceptibility does not come from a divergent correlation length but from some peculiar features of the pseudoscalar correlation function which can emerge in systems with global constraints. The case in which the SUðNfÞ chiral symmetry is fulfilled in the vacuum will be discussed in detail
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