平移不变非对易汤川理论的单圈重整化群流
One-loop renormalization group flow of the translation-invariant noncommutative Yukawa theory
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- University of Medea（梅迪亚大学）
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中文总结 AI 辅助
本文研究四维欧几里得Moyal空间上平移不变非对易赝标量汤川理论的单圈重整化群流，解析求解其耦合系统，分析流的动力学行为并与对易理论对比，确立单圈可重整性。
中文摘要 AI 辅助
我们在四维欧几里得Moyal空间上研究平移不变非对易赝标量汤川理论的单圈重整化群流。Moyal星汤川顶点的两种不等价序给出两个独立耦合常数$g_1$和$g_2$，它们的β函数构成耦合非线性系统；所有单圈发散均被作用量中已存在的抵消项吸收，确立了该理论的单圈可重整性。我们解析求解单圈系统：该系统在变量$u=g_1^2+g_2^2$和$v=g_1^2-g_2^2$中退耦，四次耦合通过对称面$g_1=g_2$上的Riccati约化得到，一般非对称流通过精确不变量$\udcdev{\text{I}}=(g_1g_2)^3/(g_1^2-g_2^2)^4$约化为单次积分。由此得到两个动力学结果：比值$r=v/u$具有UV吸引对称面$r=0$和IR吸引非对称方向$r=\udcdev{\text{±}}1$，因此流在紫外方向恢复两种Moyal序之间的对称性，在红外方向则以对数的分数次幂放大任意初始序不对称性。结果，非平面$1/(\theta^2p^2)$结构的汤川诱导系数在$p\rightarrow0$时被动力学抑制；该奇点本身未被移除，仍需由红外改进项处理。我们还在对称临界轨迹上闭合求解有量纲部分（$M^2$、$m$、$a^2$），并全程与对易理论进行比较。
英文摘要
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 $1/(θ^{2}p^{2})$ 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 ($M^{2}$, $m$, $a^{2}$ and $b$) in closed form on the symmetric critical trajectory, and compare throughout with the commutative theory.
