Gradient Flow of O(N) nonlinear sigma model at large N

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.


Introduction
In recent years, the gradient flow equation [1] has been the focus of attention. The gradient flow equation is originally proposed in the context of the SU(N) lattice gauge theory [2] and the SU(N) Yang-Mills theory [1]. In Ref. [3], Lüsher also gave the matter fields version of the gradient flow equation. The gradient flow can also be viewed as a nice way of smearing the bare field respecting the gauge symmetry which could tame fluctuations of the operator arising from the contributions at high momentum scale. For this reason, the gradient flow can give a useful physical quantities which are numerically very stable and well-defined in the continuum.
In a recent paper [23], the generalization of the gradient flow equation for field theory with non-linearly realized symmetry are proposed. This equation gives a unified method to construct a gradient flow equation of the action with non-linearly realized symmetry, for example, the supersymmetric Yang-Mills theory, the O(N) nonlinear sigma model in two dimensions. Of course the equation can reconstruct the equation of the Yang-Mills theory and the lattice gauge theory.
In this paper, we focus on this O(N) nonlinear sigma model in two dimensions [24][25][26]. Since this theory is known to be a good toy model of the Yang-Mills theory with asymptotic freedom and non-perturbative generation of the mass gap, and exactly solvable, it can be an ideal laboratory for the theoretical study of the gradient flow. Recently, using the gradient flow method, the ultraviolet finiteness of the O(N) nonlinear sigma model in two dimensions was proved to all order in perturbation theory [27].
The most interesting point to study the O(N) nonlinear sigma model in two dimensions is that the model is solvable at large N limit [28]. Therefore, one could also expect that the finiteness proof may be possible at the non-perturbative level, while such a non-perturbative proof of the finiteness for correlation function of the operators constructed from the solution to the gradient flow equation seems difficult for the Yang-Mills theory or QCD, despite its importance, since these theories are not exactly solvable. It would therefore be important to give a non-perturbative proof of finiteness and also carry out various applications in an exactly solvable model in order to get a deeper theoretical insight.
In this paper, 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 . Due to the interaction terms in the flow equation, the single scalar field solution to the gradient flow equation is given by the infinite sum of the convolutions in n-th order multiple bare fields, where n = 1, 3, · · · . We show that at large N after dropping the subleading contributions a drastic reduction takes place and one obtains a closed set of equations for n = 1, 3, · · · , which can in principle be solved iteratively. In particular, we give an explicit solution for n = 1, from which we can construct the exact expression of the two point function at finite flow time t.
From the exact expression, one can show that the two point function is finite at finite t. In the discussion, we also give a formal solution to the n = 3 case, from which one can obtain the connected four point function.
In Section 2, we introduce the O(N) nonlinear sigma model in two dimensions and solve the gap equation to determine the vacuum in the large N limit. In Section 3, we introduce the gradient flow equation of this model, and solve it for n = 1 in Section 4. The finiteness of the two point function for nonzero flow time is shown in Section 5. As an application, the non-perturbative running coupling is discussed in Section 6. In Section 7, the four point function for nonzero flow time is briefly considered, though the discussion on the finiteness is left to future studies. In Appendix A, the four point function in the two dimensional model is calculated, and the solution to the gradient flow equation for n = 3 is presented in Appendix B. We also give an alternative way of solving the gradient flow equation using the Schwinger-Dyson equation of the two and four point functions in Appendix C.

Model and the gap equation
We consider the O(N) nonlinear sigma model in two dimensional Euclidean space. The generating functional with source J is given as where ϕ α (α = 1, · · · , N ) is the scalar fields in the vector representation of O(N) with unit length, whose action S is given by 2) and the inner product is understood as A · B = N α=1 A α B α . After integrating ϕ field, we obtain where we have defined the rescaled field β(x) and rescaled coupling λ as (2.6) In this paper, according to the context, the repeated coordinate index is understood as its integration such that (2.7) In the large N limit, the path integral over β is dominated by the stationary point determined by the following gap equation.
where m 2 = −2iλ β with β being the vacuum expectation value of β. Introducing momentum cutoff Λ, the solution is given as Since m is the nonperturbative physical mass of the scalar field, we impose the renormalization condition that which implies that the coupling λ vanishes in the Λ → ∞ limit as This shows that the theory is asymptotically free.

Power-counting in the large N expansion
We expand S eff around the stationary point β as where {z} n = z 1 , · · · , z n and the first few terms are given by 14) Now let us consider the power counting in the large N expansion. We define where F 2k corresponds to the connected 2k-pt function. Since the propagator, (N D 0 ) −1 , is O(1/N ), a diagram which contains v n vertices of the type V n , t n vertices of the type T n and I internal propagators, behaves as N ν , where while the number of J 2k is given by Therefore the leading power of N for F 2k , denoted N ν 2k , is given by (2.20) which corresponds to the tree level diagrams. For k = 1 and 2, for example, we have Thus the 2-pt function is O(1/N ) and is given by at the leading order of the large N expansion.

Gradient Flow Equation
The gradient flow equation of the O(N ) nonlinear sigma model is defined for the field φ α (t, x), where an additional parameter t corresponds to the flow time, with an initial condition that φ α (0, x) = ϕ α (x). Since the field ϕ α (x) is subject to the constraint N α=1 (ϕ α ) 2 = 1, we impose the same constraint for φ α (t, x), so that the N -th component can be expressed as Substituting eq. (3.1), the action can be rewritten as where a = 1, 2, · · · , N − 1. Here, the metric for the O(N ) nonlinear sigma model is given by In Ref. [23], it was shown that the gradient flow equation for the field theory with nontrivial metric in the field space is given by .
We then obtain the gradient flow equation of the O(N ) nonlinear sigma model in two dimensions as Here we rescaled as t → g 2 t and the following notation for the summation over the indices are introduced

Solution to the gradient flow equation in the large N expansion
In this section, we propose a method to solve the gradient flow equation non-perturbatively in the large N expansion, and explicitly give a non-perturbative solution needed for the two point function of the φ field.

Ansatz for the solution
For the solution to the gradient flow equation, we take the following form where X 2n+1 only contains 2n + 1-th order of ϕ, and : O : represents the "normal ordering", where self-contractions within the operator O are prohibited. Formally we can define the normal ordering recursively in the perturbation theory around the large N vacuum as for an arbitrary operator O. From the initial condition for ϕ, we have The gradient flow equation in the momentum space is written as where we define The left hand side can be expressed in term of the solution eq. (4.1) as In the present approach, we are looking for the solution of the field φ(t, p) itself. As an alternative approach, one could also solve the 2n-point correlation function This will be given in Appedix C.

Solution for O 1
Taking O 1 as the order ϕ operator, we evaluate L a and R a at the leading order of the large N expansion as where (4.11) The gradient flow equation that L a (t, which can easily be solved as where x + m 2 (4.14) and E i (x) is the exponential integral function defined by In this section, we show the finiteness of the two point function in terms of the gradient field φ α non-perturbatively at the leading order of large N expansion, without the field renormalization.
Since the leading behavior of ϕ a ϕ b (ϕϕ) n c is N −(2n+1)+n = N −(n+1) , the leading contribution to the two point function is simply given by as long as t a t b = 0, without renormalization factor for the field φ. This is the main result of this paper. At small t a , t b , we have which diverges as 1/ √ log t a log t b in the t a , t b → 0 limit.

Applications
One At leading order in perturbation theory, it can be evaluated as Then, one can define the non-perturbative running coupling constant λ R (µ) with the renormalization scale µ = 8π/t as From our non-perturbative result, the left hand side of eq. (6.3) is evaluated as Combing eqs. (6.3) and (6.4), one finds Recalling the gap equation (2.8), the renormalized coupling reduces to the bare coupling at t = 0, and for finite t it is a UV finite coupling at the fully nonpertubative level defined by a momentum integral in which the cutoff Λ is replaced with an effective cutoff of order 1/t.

Discussions: four point function
In this paper we calculate the two point function of the gradient flow field non-perturbatively at the leading order of the large N expansion, by solving the gradient flow equation necessary for the calculation. We then show the finiteness of the two point function without renormalization, as is given in eq. (5.4). It is thus interesting to ask whether the finiteness hold for higher point correlation functions. Let us consider the four point function as a concrete example. To calculate the four point function, we have to determine X a 3 (ϕ, p, t) in eq. (4.1) by the gradient flow equation, which leads to where X(p 1 , p 2 , p 3 , t) is given in eq. (B.14) and ϕ · ϕ = N −1 a=1 (ϕ a ) 2 . See Appendix B for the detail of the derivation.
We then consider the power counting in the large N expansion for the four point function of the gradient flow fields. There are three types of contributions for ϕ correlation functions.
• (ϕϕ) n c : the leading behavior is N −(2n−1)+n = N −(n−1) . Therefore, contributions which have the leading behavior of the connected four point function, N −3 , are the following three types.
3. ϕ a ϕ b c ϕ c ϕ d c (ϕϕ) 2 c where one (ϕϕ) comes from ϕ a or ϕ b and the other from ϕ c or ϕ d .

A Connected four point function in the two dimensional O(N) nonlinear sigma model
The connected four point function at the leading order of the large N expansion can be calculated as with F 4 in eq. (2.22), which leads to In the momentum space, we obtain We then finally obtain G (4) a 1 a 2 a 3 a 4 (p 1 , p 2 , p 3 , p 4 ) = −δ(p 1234 ) The connected four point function is

B Solution to the gradient flow equation for X 3
Taking O 3 as the order ϕ 3 operator, we evaluate L a and R a at the leading order of the large N expansion as where in the last term, . Sinceḟ (t) = f 3 (t)I(t), the first terms in both sides agree. Therefore, the equation we have to solve at the leading order becomes Since the above equation is difficult to solve directly, we introduce the expansion in λ as where X 3,n is independent on λ. From eq. (B.3), we can easily obtain which, after a little algebra, leads to It is then not so difficult to guess the solution for a general n as which can be proven by the mathematical induction as follows. The solution for n = 1 is correct by eq. (B.6). If eq. (B.8) is correct for n = k, eq. (B.3) giveṡ with s k+1 = t. By integrating the above equation in t, we show that eq. (B.8) is correct for n = k + 1. This completes the proof. We now introduce the integral operator F (p 23 ) and a function H(p 2 2 + p 2 3 ) as Using these notations, X 3,n for all n can be expressed as Combining this with eq. (B.4), we finally obtain (B.14) C An alternative way to solve the flow equation In this appendix, we present an alternatively way to solve the flow equation. Instead of solving the field at flow time t in terms of bare fields, we derive the differential equation on the correlation function for the fields at finite time t.

C.1 Schwinger-Dyson equation
General correlation function φ a (t, x)O , where O is an arbitrary operator constructed from φ, satisfies the following differential equation.

C.2 Leading Contribution for two point function
Let us consider two point function. Setting t = t a , p = p a and choosing O = φ(t b , p b ), we obtain the differential equation for the two point function as Let us now consider the leading order contribution at large N . In Section 2, we have shown that the leading order contribution to the two point function is O(1/N ). Therefore, we should only consider O(1/N ) contribution on the right hand side of eq. (C.3). The four point function in the second term on the right hand side can be decomposed as where . . . c denotes the connected parts.
In Section 2, we have also shown that the leading order connected parts in the 2n-point function is of O(1/N 2n−1 ). Dut to the O(N) symmetry, the two point function φ a φ b is proportional to δ ab the four point function φ a φ b φ c φ d can be decomposed into the sum of three functions which are proportional to δ ab δ cd , δ ac δ bd , δ ad δ bc , respectively. From this fact, one can see that the first, the second and the third terms on the right hand side of eq. (C.4) are O(1/N 2 ), whereas the fourth term is O(1/N ).
Similar argument can be applied to the third term of eq. (C.3). For example, six point function in the term with n = 0 can be decomposed as follows (C.5) The first and second terms on the right hand side of eq. (C.5) are O(1/N 3 ) , O(1/N 2 ) or higher. In the third term only the first contribution gives O(1/N ) and "other products of 2pt" give only higher order contributions. It is found that the O(1/N ) in the third term contains a factor (p 2 + p 3 ) · (p 4 + p 5 ). Due to the momentum conservation for the two point functions, they only give vanishing contribution. From similar observation, one finds that there is no contribution from the third term of eq. (C.3).
From this consideration, one finally finds that the gradient flow equation at the leading order reduces to Substituting this into eq. (C.6), we obtain

C.3 Exact solution of two point function at large N
We employ the following ansatz for the two point function where f (t) is some function of t. In order to reproduce the propagator at t = 0, f (t) must satifsy the initial condition f (0) = 1. Substituing eq. (C.9) into eq. (C.8), one finds that Note that J 0 (t) is finite at finite t owing to the suppression factor exp(−2q 2 t) in the momentum integration, while at t = 0 it is logarithmically divergent. Solving eq. (C.10), one obtains the following solution for f (t) Using the Gap equation λJ 0 (0) = 1, f (t) is determined as Therefore the two point function is given as . (C.14) One can easily see that the two point function at nonzero flow time t is free from divergence.

C.4 Leading contribution to the connected four point function
We consider the connected four point function defined as Applying the gradient flow equation, one obtains the following differential equation for the four point function.
As shown in Section 2, the left hand side is O (1/N 3 ). What is the O(1/N 3 ) contribution on the right hand side? In the first term, there appear six point function, which can be decomposed into connected and disconnected contributions as In this decomposition, the first term on the right hand side of eq. (C.17) is O(1/N 4 ) and the second and third terms are O(1/N 3 ). The fourth term on the right hand side of eq. (C.17) is also O(1/N 3 ), but it is cancelled with the subtraction terms in eq. (C.16). The fifth term on the right hand side of eq. (C.5) is O(1/N 4 ) or higher since the O(N) invariant pair φ(t a , p 2 ) · φ(t a , p 3 ) is split into different correlation functions. Out of the product of three two point functions, the sixth term of eq. (C.17) is O(1/N 3 ) whereas others (seventh term) are cancelled with the subtraction terms in eq. (C.16). One therefore finds φ a (t a , p 1 ) φ(t a , p 2 ) · φ(t a , p 3 ) e=b,c,d φ e (t e , p e ) − ( subtraction terms ) where q 23 = q 2 + q 3 and p 23 = p 2 + p 3 and p abcd = p a + p b + p c + p d .
We can see that the gradient flow gives a closed equation also for the four point function. Note that the coefficients of this differential equation are finite, since they are expressed by the combination of λ times the product of two f (t)'s and f (t) = [λJ(t)] −1/2 so that the bare coupling λ dependence is explicitly cancelled. This means that the differential equation is consistent with the case that the connected four point function would be finite.