Renormalization group improvement

【Note】Tommi Markkanen, Arttu Rajantie, Stephen Stopyra, Cosmological Aspects of Higgs Vacuum Metastability, arXiv: 1809.06923

我们知道，场在量子化后，其经典势将获得一个量子修正，从而变成有效势

As we known, the classical potential of a field after quantization receives quantum corrections and thus becomes the effective potential

$V_\mathrm{eff}(\phi; g_i)=V_\mathrm{cl}(\phi(\mu); g_i(\mu))+\delta V(\phi(\mu); g_i(\mu), \mu)$

其中 $\mu$ 即所谓重整化能标，跑动耦合参数 $g_i(\mu)$ 对重整化能标的依赖由各自的贝塔函数 $\beta_{g_i}=\mu\frac{\partial g_i}{\partial\mu}$ 给出，量子场的重整化一点函数与裸场的关系 $\phi_\mathrm{bare}=\sqrt{Z(\mu)}\phi(\mu)$ 则由场重整化因子 $Z(\mu)$ 给出，而场重整化因子则由反常量纲 $\gamma=\mu\frac{\partial\sqrt{Z}}{\partial\mu}$ 给出。

where $\mu$ is the so-called renormalization scale, and the dependences of running coupling parameters $g_i(\mu)$ on the renormalization scale are given by the corresponding beta function $\beta_{g_i}=\mu\frac{\partial g_i}{\partial\mu}$ , and the renormalized one-point function of a quantized field is related to the bare field $\phi_\mathrm{bare}=\sqrt{Z(\mu)}\phi(\mu)$ by the field-renormalization-factor $Z(\mu)$ determined by the anomalous dimension $\gamma=\mu\frac{\partial\sqrt{Z}}{\partial\mu}$ .

包含所有阶圈图量子修正 $\delta V_1, \delta V_2, \cdots$ 的完整的有效势，其本身并不依赖于重整化能标的选取，即所谓 Callan-Symazik 方程

The full effective potential including all orders of loop-diagram quantum-corrections $\delta V_1, \delta V_2, \cdots$ does not depend on the choice of renormalization scale, leading to so-called Callan-Symazik equation

$\mu\frac{\mathrm{d}}{\mathrm{d}}V_\mathrm{eff}(\phi; g_i)=\left(\mu\frac{\partial}{\partial\mu}+\beta_{g_i}\frac{\partial}{\partial g_i}-\gamma\phi\frac{\partial}{\partial\phi}\right)V_\mathrm{eff}(\phi; g_i)=0$

如果存在重整化能标的某种选取函数 $\mu=\mu^*(\phi)$ ，使得总的量子修正 $\delta V(\phi(\mu^*); g_i(\mu^*), \mu^*(\phi))$ 为零，那么我们就得到了一个经过重整化群提升的有效势

If there is such a choice of function for the renormalization scale $\mu=\mu^*(\phi)$ so that the total quantum correction $\delta V(\phi(\mu^*); g_i(\mu^*), \mu^*(\phi))$ is vanished, then we have a renormalization-group (RG) improved effective potential

$V_\mathrm{eff}^\mathrm{RGI}(\phi; g_i)=V_\mathrm{cl}(\phi(\mu^*(\phi)); g_i(\mu^*(\phi)))$ 。

但是实际计算一般只能包含有限阶圈图的量子修正，比如一圈修正 $\delta V_1$ 即所谓 Coleman-Weinberg 势，此时有效势将明显依赖重整化能标的选取，并影响微扰展开的有效性和精确度。那么一个重要的问题是如何选取重整化能标。

However, the practical calculations normally could only contain finite orders of loop-diagram quantum-corrections, for example, one-loop correction $\delta V_1$ is the dubbed Coleman-Weinberg potential, and the corresponding effective potential would explicitly depend on the choice of renormalization scale and impact the effectiveness and precision of perturbative expansion. Then an important question is how to choose the renormalization scale.

首先我们可以退而求其次，要求重整化能标的某种选取函数 $\mu=\mu^*_1(\phi)$ 使得 $\delta V_1$ 为零，此时有效势由经过重整化群提升的经典势

Firstly, we could be conservative by requiring some function form $\mu=\mu^*_1(\phi)$ for the renormalization scale so that $\delta V_1$ is vanished, thus the corresponding effective potential is given by the RG-improved classical potential

$V^\mathrm{classical}_\mathrm{RGI-eff}(\phi(\mu^*_1); g_i(\mu^*_1))=V_\mathrm{cl}(\phi(\mu^*_1); g_i(\mu^*_1))$

给出。然而，由 $\delta V_1$ 为零的条件出发通常无法显式求解该重整化能标的选取函数，即使侥幸能显式求解，该选取函数的形式也将颇为复杂。

Nevertheless, starting from the condition of a vanishing $\delta V_1$ usually could not solve for an explicit function form of renormalization scale. Even such a explicit solution is obtained, it would be considerably complicated.

因此我们再次退而求其次，要求 $\delta V_1$ 虽然不为零但是其对数项尽可能的小，比如我们可以选取重整化群能标为 $\mu^{\#}_1\propto\phi$ ，此时有效势是由经过重整化群提升的一圈修正有效势

Therefore, we can be even more conservative by requiring $\delta V_1$ to be nonzero but as small as possible for its logarithmic part, for example, we could choose the renormalization scale as $\mu^{\#}_1\propto\phi$ , and the corresponding effective potential is given by the RG-improved 1-loop-level effective potential

$V_\mathrm{RGI-eff}^\mathrm{1-loop}(\phi(\mu^{\#}_1); g_i(\mu^{\#}_1), \mu^{\#}_1)=V_\mathrm{cl}(\phi(\mu^{\#}_1); g_i(\mu^{\#}_1))+\delta V_1(\phi(\mu^{\#}_1); g_i(\mu^{\#}_1), \mu^{\#}_1)$

给出。甚至我们还可以再大胆一些，干脆把上式中的一圈修正项丢掉，从而得到经过重整化群提升的树图阶有效势

We could be even bolder by throwing away the whole 1-loop corrections and obtain the RG-improved tree-level effective potential

$V_\mathrm{RGI-eff}^\mathrm{tree}(\phi(\mu^{\#}_1); g_i(\mu^{\#}_1))=V_\mathrm{cl}(\phi(\mu^{\#}_1); g_i(\mu^{\#}_1))$

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