content_image/375273.jpg content_image/375274.jpg content_image/375275.jpg content_image/375272.jpg
phase minimum nθ$_{min}$ = π is then given by [89]: 

<img src='content_image/375273.jpg'>

where m$_{0}$ , A$_{0}$ , and λ$_{n,}$$_{0}$ are the values of m$_{φ}$ , A and λ$_{n}$ given at a scale µ$_{0}$ . Here A$_{0}$ is chosen to be real and positive (this can always be done by re-parameterizing the phase of the complex scalar field φ ), and | K$_{i}$ | < 1 are coefficients determined by the one-loop renormalization group equations. 

In the limit when | K$_{i}$ | ≪ 1, one finds a simple relationship [89] 

For n = 6, the coefficient is A 2 ∼ 40 m 2 $_{φ}$, where φ$_{0}$ denotes the point of inflection. The coefficients K$_{i}$ need to be solved from the renormalization group equations at the inflationary scale µ = φ$_{0}$ . Since K$_{i}$ are already one loop corrections, taking the tree-level value as the renormalization scale is sufficient $^{75}$. 

<img src='content_image/375274.jpg'>

<img src='content_image/375275.jpg'>

The radiative corrections do not remove the inflection point nor shift it to unreasonable values. The existence of an inflection point is thus insensitive to radiative corrections. 

One can analytically obtain the values of K$_{i}$ for the LLe flat direction. For LLe the one-loop RG equations governing the running of m 2 $_{φ}$, A , and λ with the scale µ are given by [31, 32, 89] 

<img src='content_image/375272.jpg'>

