Editing Renormalization group (section)

== Renormalization group improvement of the effective potential ==
The renormalization group can also be used to compute [[Effective action|effective potentials]] at orders higher than 1-loop. This kind of approach is particularly interesting to compute corrections to the Coleman–Weinberg<ref>{{Cite journal |last1=Coleman |first1=Sidney |last2=Weinberg |first2=Erick |date=1973-03-15 |title=Radiative Corrections as the Origin of Spontaneous Symmetry Breaking |url=https://link.aps.org/doi/10.1103/PhysRevD.7.1888 |journal=Physical Review D |language=en |volume=7 |issue=6 |pages=1888–1910 |doi=10.1103/PhysRevD.7.1888 |issn=0556-2821 |arxiv=hep-th/0507214 |bibcode=1973PhRvD...7.1888C |s2cid=6898114}}</ref> mechanism. To do so, one must write the renormalization group equation in terms of the effective potential. To the case of the <math>\varphi^4</math> model:
<math display="block">\left(\mu\frac{\partial}{\partial\mu} + \beta_\lambda\frac{\partial}{\partial\lambda} + \varphi\gamma_\varphi\frac{\partial}{\partial\varphi}\right) V_\text{eff} = 0.</math>

In order to determine the effective potential, it is useful to write <math>V_\text{eff}</math> as
<math display="block">V_\text{eff} = \frac{1}{4} \varphi^4 S_\text{eff}\big(\lambda, L(\varphi)\big),</math>
where <math>S_\text{eff}</math> is a [[power series]] in <math>L(\varphi) = \log \frac{\varphi^2}{\mu^2}</math>:
<math display="block">S_\text{eff} = A + BL + CL^2 + DL^3 + \cdots.</math>

Using the above [[ansatz]], it is possible to solve the renormalization group equation perturbatively and find the effective potential up to desired order. A pedagogical explanation of this technique is shown in reference.<ref>{{Cite journal |last1=Souza |first1=Huan |last2=Bevilaqua |first2=L. Ibiapina |last3=Lehum |first3=A. C. |date=2020-08-05 |title=Renormalization group improvement of the effective potential in six dimensions |journal=Physical Review D |volume=102 |issue=4 |pages=045004 |doi=10.1103/PhysRevD.102.045004 |arxiv=2005.03973 |bibcode=2020PhRvD.102d5004S |doi-access=free}}</ref>