The two-dimensional CP(N-1) sigma model we consider is an effective theory on the world-sheet of a non-Abelian string with a special mass deformation parameter. As the model is a non-supersymmetric, it may contain, the study of which is one of the purposes of this work.

CP(N-1) model is asymptotically free [1], dynamic scale in this Λ arises from the dimensional transmutation, similar to QCD. Furthermore, as a result of the renormalization procedure, a new parameter with the dimension of mass arises, Λ_σ. In the limit of vanishing mass deformation parameter, m≪Λ,Λ_σ, the model is at a strong coupling. For the m=0 it was solved by Witten [2] in the large-N limit: photon interacts with N “quarks” --- scalar fields with charges ∼1/√N which appear in the spectrum only in pairs n^*n, since between them there is a confining potential that grows linearly with distance. We call this phase Coulomb/confining since a photon field is massless in it. In this massless limit Z_N symmetry remains unbroken. When m≫Λ, the theory is at weak coupling, Z_N symmetry is spontaneously broken, because scalar fields n develop non-zero vacuum expectation value. That is why we call this regime the Higgs phase.

In this work we generalize the result, obtained by Witten in [2] to the case with m≠0, we also work in the large-N limit. The first step is to calculate an effective potential, from which we derive vacuum equations and solve them in two different phases. Then we find a phase transition point and explore a vacuum energy behavior near it, which allows us to determine its type. In the last step we study the dynamics of fields and make some conclusions about applicability of using large-N method.

One of the important results of this work is the need to modify the original gauge-invariant formulation of the model --- as will be shown, there is a need to introduce an additional term into the original Lagrangian to carry out a self-consistent renormalization procedure, as a result of which another mass dimension parameter appears in the theory, in addition to Λ, that is, Λ_σ. There is a paper on this topic [3], in which the absence of an additional term in the original Lagrangian leads to the fact that the minimum of the potential becomes a maximum at sufficiently small masses, that is, the theory becomes unstable.

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E. Witten. Instantons, The Quark Model, And The 1/N Expansion, Nucl. Phys. B 149, 285 (1979).
A. Gorsky, M. Shifman and A. Yung, The Higgs and Coulomb/confining phases in ‘twisted-mass’ deformed CP(N-1) model, Phys. Rev. D 73, 065011 (2006) [hep-th/0512153v2].
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