[Paper Review] (2+1)D Noncommutative CP$^{N-1}$ Model

E. A. Asano, M. Gomes+2 more|arXiv (Cornell University)|Jul 11, 2003

Noncommutative and Quantum Gravity Theories3 citations

TL;DR

This paper investigates the (2+1)-dimensional noncommutative CP^{N-1} model, demonstrating that when the fundamental field transforms in the left fundamental representation of the gauge group, the model is renormalizable and free of dangerous infrared divergences up to the next-to-leading order in the 1/N expansion. In contrast, the adjoint representation exhibits quadratic infrared divergences in the gauge field two-point function and self-energy corrections, threatening the validity of the 1/N expansion at higher orders.

ABSTRACT

We investigate possible extensions of the (2+1) dimensional $CP^{N-1}$ model to the noncommutative space. Up to the leading nontrivial order of 1/N, we prove that the model restricted to the left fundamental representation of the gauge group is renormalizable and does not have dangerous infrared divergences. In contrast, if the basic field $ϕ$ transforms in accord with the adjoint representation, infrared singularities are present in the two point function of the auxiliary gauge field and also in the leading correction to the self-energy of the $ϕ$ field. These infrared divergences may produce nonintegrable singularities leading at higher orders to a breakdown of the 1/N expansion. Gauge invariance of the renormalization procedure is also discussed.

Motivation & Objective

• To investigate the renormalizability and infrared behavior of the (2+1)-dimensional noncommutative CP^{N-1} model under different gauge group representations.
• To determine whether the 1/N expansion remains valid in noncommutative field theories, particularly in the presence of IR/UV mixing.
• To analyze the role of noncommutative phases and planar vs. nonplanar diagrams in UV and IR divergences.
• To examine the impact of gauge invariance on the renormalization procedure in noncommutative settings.
• To compare the behavior of the model in the fundamental (left) representation versus the adjoint representation, especially regarding divergences.

Proposed method

• Employing the 1/N expansion up to next-to-leading order, the authors analyze quantum corrections to propagators and vertices in the noncommutative CP^{N-1} model.
• Using dimensional regularization, the paper evaluates ultraviolet divergences and checks their cancellation via the BPHZ subtraction scheme.
• The analysis includes explicit computation of Feynman diagrams, distinguishing between planar and nonplanar graphs based on noncommutative phase factors.
• The authors use graphical identities and momentum-space Taylor expansions to demonstrate cancellation of divergences, particularly for overlapping subgraphs.
• The noncommutative phase factor, exp[i(p₁∧p₂ − p₁∧p₃ + p₂∧p₃)], is factorized in summing diagrams to analyze infrared behavior.
• The study examines both the fundamental (left) and adjoint representations of the gauge group, comparing their divergent structures.

Experimental results

Research questions

• RQ1 Is the noncommutative (2+1)D CP^{N-1} model renormalizable when the matter field transforms in the left fundamental representation of the gauge group?
• RQ2 Do dangerous infrared divergences appear in the two-point functions of the auxiliary gauge field and the scalar field self-energy when the field transforms in the adjoint representation?
• RQ3 Can ultraviolet divergences in the 1/N expansion be systematically canceled in noncommutative field theories using the BPHZ scheme?
• RQ4 How do nonplanar diagrams with distinct noncommutative phases affect the renormalization and infrared behavior of the model?
• RQ5 To what extent does the absence of charge conjugation symmetry in noncommutative theories lead to new divergent contributions not present in the commutative case?

Key findings

• The noncommutative CP^{N-1} model with the scalar field in the left fundamental representation is renormalizable and free of dangerous infrared divergences up to next-to-leading order in the 1/N expansion.
• In the fundamental representation, subleading contributions to propagators do not introduce new UV divergences, and linear divergences cancel pairwise due to symmetry and Lorentz invariance.
• The mass counterterm insertions cancel pairwise due to a graphical identity that enforces cancellation in the BPHZ subtraction scheme.
• For the three-loop diagram G, the sum of all unsubtracted amplitudes is finite because all subtraction terms cancel exactly, even for overlapping divergent subgraphs.
• Nonplanar diagrams with opposite charge flow exhibit linear infrared divergences individually, but these cancel upon summation due to phase factor factorization and first-order Taylor expansion.
• In the adjoint representation, quadratic infrared divergences appear in the gauge field two-point function and in the scalar self-energy, which could disrupt the 1/N expansion at higher orders.