One-loop renormalization group flow of the translation-invariant noncommutative Yukawa theory
We study the one-loop renormalization group flow of the translation-invariant noncommutative pseudoscalar Yukawa theory on four-dimensional Euclidean Moyal space. The two inequivalent orderings of the Moyal-star Yukawa vertex give rise to two independent couplings $g_{1}$ and $g_{2}$, whose beta functions form a coupled nonlinear system; every one-loop divergence is absorbed by a counterterm already present in the action, establishing one-loop renormalizability of the theory. We solve the one-loop system analytically: it decouples in the variables $u=g_{1}^{2}+g_{2}^{2}$ and $v=g_{1}^{2}-g_{2}^{2}$, the quartic coupling is obtained by a Riccati reduction on the symmetric surface $g_{1}=g_{2}$, and the generic asymmetric flow is reduced to a single quadrature by the exact invariant $\mathcal{I}=(g_{1}g_{2})^{3}/(g_{1}^{2}-g_{2}^{2})^{4}$. Two dynamical consequences follow. The ratio $r=v/u$ has a UV-attractive symmetric surface $r=0$ and IR-attractive asymmetric directions $r=\pm1$, so that the flow restores the symmetry between the two Moyal orderings towards the ultraviolet and amplifies any initial ordering asymmetry towards the infrared, as a fractional power of the logarithm. In consequence the Yukawa-induced coefficient of the non-planar $1/(\theta^{2}p^{2})$ structure is dynamically suppressed towards $p\rightarrow 0$; the singularity itself is not removed, and remains the task of the IR-improving term. We also solve the dimensionful sector ($M^{2}$, $m$, $a^{2}$) in closed form on the symmetric critical trajectory, and compare throughout with the commutative theory.