AI 中文总结
本文对Holdom提出的四导数标量场理论的重整化进行图论分析，证明其欧氏关联函数红外有限，推导非重整化定理，揭示其β函数与特定φ⁴理论的对应关系，确定相关量至六圈。
AI 中文摘要
我们对Holdom在arXiv:2303.06723和arXiv:2402.09223中提出的一族渐近自由、 shift对称的四导数标量场理论的重整化进行了详细的图论分析。我们使用R*方法和动量的渐近展开，将该理论的重整化从一圈扩展到三圈。我们证明，欧氏关联函数（或离壳振幅）在微扰论的所有阶都是红外有限的，并推导了描述重整化常数全阶结构的非重整化定理。特别地，纯立方相互作用是RG不变的，且完全平方拉格朗日密度在重整化下保持不变。后一结果源于相关引力理论中的Ward恒等式——二次引力（CFQG）的共形平坦极限。我们表明，完全平方理论的β函数与负耦合下O(2)对称、两导数、无质量的φ⁴理论的β函数完全对应。我们明确验证了这种对应关系直至三圈，从而确定了完全平方理论和CFQG的β函数及反常维数到六圈。
英文摘要
We perform a detailed diagrammatic analysis of the renormalisation of a family of asymptotically free, shift-symmetric four-derivative scalar field theories introduced by Holdom in arXiv:2303.06723 and arXiv:2402.09223. We extend the renormalisation of the theory from one to three loops using both an $R^*$ method and an asymptotic expansion in momenta. We prove that the Euclidean correlators (or off-shell amplitudes) are IR finite, to all orders in perturbation theory, and derive a non-renormalisation theorem describing the all-order structure of the renormalisation constants. In particular, a purely cubic interaction is RG invariant and a perfect square Lagrangian density is preserved under renormalisation. The latter result is due to a Ward identity in a related $\textit{ gravitational}$ theory $-$ the conformally flat limit of quadratic gravity (CFQG). We show that the beta function for the perfect square theory maps exactly to that of an $O(2)$-symmetric, two-derivative, massless $ϕ^4$ theory at negative coupling. We verify this relationship explicitly up to three loops and thus determine the beta function and anomalous dimension for both the perfect square theory and CFQG to six loops.
Comments28 pages
