Abstract
We study the flow equation of the O($N$) $\varphi^4$ model in $d$ dimensions at the next-to-leading order (NLO) in the $1/N$ expansion. Using the Schwinger-Dyson equation, we derive 2-pt and 4-pt functions of flowed fields. As the first application of the NLO calculations, we study the running coupling defined from the connected 4-pt function of flowed fields in the $d+1$ dimensional theory. We show in particular that this running coupling has not only the UV fixed point but also an IR fixed point (Wilson-Fisher fixed point) in the 3 dimensional massless scalar theory. As the second application, we calculate the NLO correction to the induced metric in $d+1$ dimensions with $d=3$ in the massless limit. While the induced metric describes a 4-dimensional Euclidean Anti-de-Sitter (AdS) space at the leading order as shown in the previous paper, the NLO corrections make the space asymptotically AdS only in UV and IR limits. Remarkably, while the AdS radius does not receive a NLO correction in the UV limit, the AdS radius decreases at the NLO in the IR limit, which corresponds to the Wilson-Fisher fixed point in the original scalar model in 3 dimensions.
Highlights
In the previous paper (Ref. [1]), the present authors studied the proposal (Ref. [2]) that a (d + 1)dimensional induced metric can be constructed from a d-dimensional field theory using gradient flow (Refs. [3–6]), applying the method to the O(N ) φ 4 model
Equations (A.36) and (A.37) tell us that μ21 is finite for all u including u = ∞, and there is no divergence at d = 1 up to the
Z1 (Q + m2 )[2] which is manifestly finite at d = 3
Summary
In the previous paper (Ref. [1]), the present authors studied the proposal (Ref. [2]) that a (d + 1)dimensional induced metric can be constructed from a d-dimensional field theory using gradient flow (Refs. [3–6]), applying the method to the O(N ) φ 4 model. [2] may provide an alternative way to understand the AdS/CFT (or more generally gravity/gauge theory) correspondence N limit, some systematic way to solve the flow equation in the 1/N expansion is needed. Flow equation for the scalar model in the large N expansion and its applications. 1/N expansion for the O(N ) invariant φ 4 model in d dimensions. Using this method we explicitly calculate the 2-pt and 4-pt functions at the NLO
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
