# Renormalization group improvement

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

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

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

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 $\gamma=\mu\frac{\partial\sqrt{Z}}{\partial\mu}$.

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

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

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

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

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.

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

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

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.

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

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

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

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