AI 中文总结
研究任意质量\(m\)和曲率耦合\(\xi\)的量子标量场应力张量关联函数，给出紧凑形式，强调\(m\to\infty\)解耦及\(m\to0\)共形反常红外起源，验证Ward恒等式，分析\(\xi = 1/6\)情况及相关特性与贡献。
AI 中文摘要
本文给出了任意质量\(m\)和曲率耦合\(\xi\)的量子标量场应力张量\(\langle T^{ab}T^{cd}\rangle\)关联函数的紧凑形式，特别强调了\(m\to\infty\)极限下的解耦以及\(m\to0\)时共形反常的红外起源。验证了协变守恒的Ward恒等式对于单圈有效作用量的全二阶度规变化成立，其中\(\langle T^{ab}T^{cd}\rangle\)是一部分，包括局部接触项。该恒等式由两个张量满足，一个自旋为2（无迹），另一个自旋为0（有迹），每个都乘以在\(n\)维正则化中计算的洛伦兹不变形状因子。\(n = 4\)时极点项的最小减法产生的形状因子对于一般\(\xi\neq1/6\)不满足解耦，但可以简单修正。特别关注\(\xi = 1/6\)的情况，其中\(n = 4\)时自旋为0的形状因子完全有限，不需要紫外正则化或减法，并且直接满足解耦。其虚部定义了一个谱函数，对于任何\(m\)都服从紫外有限和规则，其实部明确确定了无质量极限下的\(\square R\)。其\(m^2\)依赖性描述了在大距离下到有效场论极限的威尔逊重整化群流。在\(m = 0\)时，自旋为0的谱函数在零能量处变为狄拉克\(\delta\)函数，证明了由于引力低能有效理论中的共形反常而存在无质量标量戈德斯通集体激发。
英文摘要
A compact form of the stress tensor $\langle T^{ab}T^{cd}\rangle$ correlation function of a quantum scalar field of arbitrary mass $m$ and curvature coupling $ξ$ is presented, with particular emphasis on decoupling in the $m\to\infty$ limit and infrared origin of the conformal anomaly as $m \to 0$. The Ward Identity (WI) of covariant conservation is verified for the full second order metric variation of the one-loop effective action, of which $\langle T^{ab}T^{cd}\rangle$ is part, including local contact terms. This WI is satisfied by two tensors, one spin-2 (traceless) and the second spin-0 (traceful), each multiplied by a Lorentz invariant form factor computed in $n$-dimensional regularization. Minimal subtraction of pole terms at $n =4$ produces form factors that fail to satisfy decoupling for general $ξ\neq 1/6$, but can be simply amended to do so. Particular interest attaches to the $ξ=1/6$ case, in which the spin-0 form factor at $n =4$ is completely finite, requires no UV regularization or subtractions, and satisfies decoupling directly. Its imaginary part defines a spectral function that obeys a UV finite sum rule for any $m$, while its real part unambiguously determines the $\square R$ in the massless limit. Its $m^2$ dependence describes a Wilsonian renormalization group flow to an effective field theory limit at large distances. At $m= 0$ the spin-0 spectral function becomes a Dirac $δ$-function at zero energy, demonstrating the existence of a massless scalar Goldstone collective excitation due to the conformal anomaly in the low energy effective theory of gravity.
Comments39 pages, 5 figures
