Abstract
We study the gradient flow equation for the O(N) nonlinear sigma model in two dimensions at large N. We parameterize solution of the field at flow time t in powers of bare fields by introducing the coefficient function X_n for the n-th power term (n=1,3,...). Reducing the flow equation by keeping only the contributions at leading order in large N, we obtain a set of equations for X_n's, which can be solved iteratively starting from n=1. For n=1 case, we find an explicit form of the exact solution. Using this solution, we show that the two point function at finite flow time t is finite. As an application, we obtain the non-perturbative running coupling defined from the energy density. We also discuss the solution for n=3 case.
Highlights
Can give a useful physical quantities which are numerically very stable and well-defined in the continuum
We study the gradient flow equation for the O(N) nonlinear sigma model in two dimensions at large N
We study the finiteness of the solution to the gradient flow equation in the O(N) nonlinear sigma model in two dimensions at large N
Summary
(x, y), T1(x, y, z1) = −iλ2K0−1(x, z1)K0−1(y, z1), T2(x, y, z1, z2) = 2λ3K0−1(x, z1)K0−1(z1, z2)K0−1(z2, y). Where F2k corresponds to the connected 2k-pt function. Since the propagator, (N D0)−1, is O(1/N ), a diagram which contains vn vertices of the type Vn, tn vertices of the type Tn and I internal propagators, behaves as N ν , where ν = vn − tn − I = vn(1 − n/2) − tn(1 + n/2), n=3 n=0 n=3 n=0. While the number of J2k is given by k = tn. Which corresponds to the tree level diagrams. For k = 1 and 2, for example, we have. K0−1(x y) d2p eip(x−y) (2π) p2 + m2. At the leading order of the large N expansion
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