Non-Perturbative Large N Trans-series for the Gross-Witten-Wadia Beta Function

We describe the non-perturbative trans-series, at both weak- and strong-coupling, of the large N approximation to the beta function of the Gross-Witten-Wadia unitary matrix model. This system models a running coupling, and the structure of the trans-series changes as one crosses the large N phase transition. The perturbative beta function acquires a non-perturbative trans-series completion at large but finite $N$ in the 't Hooft limit, as does the running coupling.


I. INTRODUCTION
One of the big puzzles concerning resurgent asymptotics in QFT [1] is how it applies to the situation where the coupling is not fixed, but runs with the scale. In this short note, we explore this phenomenon in a simple solvable model, the Gross-Witten-Wadia (GWW) unitary matrix model [2,3], which mimics a running coupling through the dependence on the lattice plaquette scale. The form of the resurgent structure changes as one crosses the large N phase transition. The GWW unitary matrix model is a one-plaquette model of 2d Yang-Mills theory, and is defined by the partition function [2,3]: Here t ≡ N g 2 /2 is the 't Hooft coupling. The GWW model has a third-order phase transition at infinite N , as the specific heat develops a cusp at t = 1. This large N third order phase transition occurs in many related examples in physics and mathematics [4][5][6][7][8][9][10][11][12][13][14]. For any N , the partition function in (1) can be compactly expressed as a Toeplitz determinant [5]: where I j is the modified Bessel function. While this formula is explicit, the determinant structure makes it of limited use for studying the large N limit. Many alternative techniques have been developed to analyze the large N limit [4][5][6][7][8][9][10], including the double-scaling limit described by the universal Tracy-Widom form [11]. Resurgent asymptotics for the large N limit in matrix models was introduced in [15], using the pre-string difference equation. To study the analytic continuation of the large N trans-series structure, where N becomes complex, one can alternatively map the GWW model to a Painlevé III equation (in terms of the 't Hooft coupling t), in which N appears as a parameter [16]. The familiar double-scaling limit of the GWW model arises as the well-known coalescence limit reducing Painlevé III to Painlevé II [17]. In this paper, we extend this Painlevé-based approach to the analysis of the beta function of the GWW model, explaining the form of the large N trans-series, at both weak and strong coupling.

A. Running Coupling and Beta Function
The running coupling is defined [2] by reintroducing a length scale (the lattice spacing a) into the Wilson loop via the definition Keeping the string tension Σ fixed therefore defines t = t(a, N ) as a function of the scale a. This running coupling t(a, N ) can be obtained by inversion of the expression 1 The beta function is then defined [2]: From now on, we set the string tension Σ = 1, absorbing it into the units of a. At infinite N , the Wilson loop at strong and weak coupling is [2]: Therefore, at infinite N the running coupling t(a) is: and the beta function is: Gross and Witten observed that if one only had the infinite N expressions at either weak or strong coupling, one might erroneously deduce the existence of spurious zeros of the beta function. See Figures 1 and 2. Similarly for the running coupling, one might deduce the incorrect behavior at small or large a, starting from the other limit at N = ∞. See Figure 3. The resolution of course is that infinite N should be approached from finite N , with suitable large N corrections included. In the next Sections we show that these finite N corrections yield non-perturbative trans-series expressions both for the beta function and for the running coupling, and when these are included, the weak coupling expressions match consistently to the strong-coupling expressions. The kink in the beta function, indicating the third order phase transition, develops at N = ∞. See Figures 1 and 2.  (9). The red solid curve shows the exact beta function for N = 20. The dashed and dotted lines show the strong-coupling and weak-coupling approximations, respectively, at infinite N , from (8). The infinite N approximations show spurious zeros at t = 1/2 and t = 2, but in fact the true beta function has a single zero at t = 0. As N → ∞, the jump at t = 1 shown in the red curve becomes a cusp, indicating the N = ∞ third-order phase transition [2,3], and the beta function curve jumps from the infinite N strong-coupling form to the infinite N weak-coupling form as t decreases through the phase transition. See Fig. 2 for a close-up of the cusp at t = 1. The finite N corrections, which produce this jump, are described in Sec. II in the form of a large N trans-series.

II. LARGE N TRANS-SERIES FOR THE BETA FUNCTION
From the definition (4), for any N , we compute ∂ t a and invert, in order to express the beta function in terms of the Wilson loop: This implies that the beta function β(t, N ) inherits its non-perturbative trans-series structure directly from the transseries structure of the Wilson loop W(t, N ). The large N trans-series for W(t, N ) was studied in [16], showing how the form of the trans-series changes across the phase transition at t = 1. Related changes therefore occur for the beta function. For other discussions of non-perturbative effects for the GWW Wilson loop, see [18,19]. We briefly review some relevant results from [16]. The non-perturbative trans-series form of W(t, N ) at any N is efficiently expressed in terms of a solution to a Painlevé III equation. Define ∆(t, N ) as the expectation value of the determinant in the Gross-Witten-Wadia model: Then W(t, N ) is related to ∆(t, N ), for any N , as: The expectation value ∆(t, N ) satisfies the following nonlinear ordinary differential equation, as a function of the 't Hooft coupling t, for any value of N [5,16,20]: Notice that N appears as a parameter in this equation, thereby enabling a simple analysis of the large N limit, including analytic continuation in N . The equation (12) is directly related to the Painlevé III equation, and standard resurgent asymptotic techniques [21] permit the development of explicit trans-series expansions in various limits: for example, weak or strong 't Hooft coupling [16].
Combining (9) and (11), the GWW beta function can also be expressed in terms of ∆(t, N ): For example, from (12) we see that at infinite N from which follows the infinite N beta function in (8).
The correspondence (13) means that we can use the trans-series structure of ∆(t, N ) to study the trans-series structure of β(t, N ). And since the trans-series expansions of ∆(t, N ) were shown in [16] to display concrete resurgence relations between different non-perturbative sectors in the trans-series, it follows that the same is true for the beta function β(t, N ).
We can also use the relation (13) to plot the beta function as a function of coupling, for various values of N : see Figures 1 and 2. These figures illustrate the fact that for any given N , the weak coupling dependence merges consistently with the strong coupling dependence, with a cusp developing at the critical 't Hooft coupling only at N = ∞. In particular, it is clear that the zeros of the infinite N beta function at t = 1/2 and t = 2 (see Fig. 1) are indeed spurious.
It is instructive to study the leading trans-series corrections to the infinite N beta functions in (8). The form of the trans-series changes across the phase transition, so we illustrate this change of structure by considering the leading contributions at large but finite N . Express the Wilson loop for any finite N as Keeping the leading power of the non-perturbative term, we obtain the following expression for the beta function: where the dots refer to higher powers of W non−pert .
A. Large N expansion at strong 't Hooft coupling In the strong coupling limit, ∆ pert is identically zero, so ∆(t, N ) is purely non-perturbative [16]. Consequently, from (11) we deduce that the Wilson loop W(t, N ) has only one perturbative term, W pert = 1 2t , which is independent of N , and equal to the familiar infinite N Wilson loop in (8). At finite N , the further corrections are all non-perturbative.
Keeping the leading such non-perturbative correction [16,18,19], (17) where the large N instanton action at strong coupling is This translates into a non-perturbative large N instanton correction to the infinite N beta function in (8): Note the appearance of further terms involving ln(t) in the fluctuations about the leading large N instanton term, consistent with general trans-series structure [21][22][23]. At any finite N , the expression (19) has an unphysical divergence at t = 1, arising from use of the Debye expansion for the Bessel functions [24]. In [16], the leading large N correction for the Wilson loop at strong coupling was calculated more precisely to be: This leading correction, in terms of Bessel J functions, is exponentially small at large N , and represents a resummation of all fluctuations about the leading large N instanton exponential factor in (17). At finite N , expression (20) is therefore much more accurate than the conventional large N expression (17) in the vicinity of the large N phase transition, at t = 1, where instantons and their fluctuations condense [16,25]. A uniform large N instanton expression is obtained by using the uniform large N approximation [26] for the Bessel functions appearing in (20). This is a nonlinear analogue of the uniform WKB approximation, smooth through the transition point for any finite N , and expressed in terms of an Airy function rather than an exponential [16,26]. Physically, this uniform large N approximation arises from the merging of two saddles at the large N phase transition. A similar expression, along with a corresponding uniform approximation, can be deduced for the beta function at large N , in the strong coupling regime:

B. Large N expansion at weak 't Hooft coupling
In the weak coupling regime, the infinite N expression in (14), ∆ ∼ √ 1 − t, receives both perturbative and nonperturbative corrections at finite N : This structure flows through to the Wilson loop and to the beta function.
where the large N instanton action at weak coupling is The corresponding large N trans-series expansion for the beta function has the form

C. Large N Double-scaling Limit
It is well known that the double-scaling limit is described by the Painlevé II equation [2,3,15]. In our approach this can be seen as follows. In the double-scaling limit, zoomed in to the immediate vicinity of the GWW phase transition at t = 1, the Rossi equation (12) reduces to a Painlevé II equation in terms of the scaled variable κ which measures the scaled deviation from t = 1: Here V (κ) is the real Hastings-McLeod solution of the Painlevé II equation [15,16]. In this double-scaling limit, the Wilson loop behaves as and the beta function as This matches smoothly to the strong-and weak-coupling sides of the phase transition, as shown for the double-scaling limit of ∆(t, N ) in [16].

III. LARGE N TRANS-SERIES FOR THE RUNNING COUPLING
At infinite N , the running coupling has the form in (7). The finite N corrections, described in the previous section for the beta function, lead also to trans-series structures for t(a, N ). At strong coupling, where the scale a is large, the corrections are naturally expressed in terms of the Wilson loop, W = exp[−a 2 ]; while at weak coupling, where the scale a is small, the corrections are naturally expressed in terms of 1 − W = 1 − exp[−a 2 ]. The infinite N phase transition occurs at W = 1/2. At any finite N , the running coupling, t(a, N ) solves the scaling equation which is both non-linear and non-perturbative. It is convenient to consider the coupling as a function of the Wilson loop W. At N = ∞ we have: The actionsŜ strong (W) andŜ weak (W) are the strong and weak coupling actions S strong (t) and S weak (t), evaluated at the infinite N values of t as given in (30): The leading terms in the strong coupling trans-series (31) read: understood as being expanded in W = exp[−a 2 ]. At weak coupling

IV. CONCLUSIONS
The Gross-Witten-Wadia unitary matrix model is a one-plaquette model of 2 dimensional lattice Yang-Mills theory, which has the interesting feature of a third-order phase transition at infinite N , in addition to a running coupling [2,3]. The perturbative beta function for this model acquires a non-perturbative trans-series completion at large but finite N in the 't Hooft limit, as does the running coupling. The 't Hooft coupling runs with the scale a, and the trans-series rearranges itself across the phase transition. Physically, this transition is identified with the condensation of instantons [25], with different kinds of instantons dominating at weak-and strong-coupling [15,27,28]. Technically, the beta function β(t, N ) can be expressed explicitly in terms of the expectation value ∆(t, N ) ≡ det U , whose resurgent trans-series structure was studied in detail in [16]. The beta function β(t, N ) inherits its trans-series structure from that of ∆(t, N ), and therefore the beta function trans-series also has full resurgent properties, including concrete relations between different instanton sectors. It would be interesting to study further this trans-series structure directly in the renormalization group approach to matrix models [29][30][31][32].