Physics Letters B 785 (2018) 342–346
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Non-perturbative large N trans-series for the Gross–Witten–Wadia
beta function
Anees Ahmed, Gerald V. Dunne
∗
Department of Physics, University of Connecticut, Storrs, CT 06269, USA
a r t i c l e i n f o a b s t r a c t
Article history:
Received
20 August 2018
Accepted
28 August 2018
Available
online 3 September 2018
Editor: M.
Cveti
ˇ
c
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.
© 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. 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 cou-
pling
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]:
Z(t, N) =
U (N)
DU exp
N
2t
tr
U + U
†
(1)
Here t ≡ Ng
2
/2is 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 transi-
tion
occurs in many related examples in physics and mathematics
[4–14].
For
any N, the partition function in (1)can be compactly ex-
pressed
as a Toeplitz determinant [5]:
Z(t, N) = det
I
j−k
N
t
j,k=1,...,N
(2)
where I
j
is the modified Bessel function. While this formula is ex-
plicit,
the determinant structure makes it of limited use for study-
*
Corresponding author.
E-mail
address: gerald.dunne@uconn.edu (G.V. Dunne).
ing the large N limit. Many alternative techniques have been de-
veloped
to analyze the large N limit [4–10], 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 struc-
ture,
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.
1.1. 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
W(t, N) ≡exp
−
a
2
(3)
Keeping the string tension fixed therefore defines t = t(a, N) as
a function of the scale a. This running coupling t(a, N) can be ob-
tained
by inversion of the expression
1
a
2
≡−
1
ln W (t, N) (4)
The beta function is then defined [2]:
1
Note that for any finite N, the relation between the ’t Hooft coupling t and the
lattice scale a is monotonic.
https://doi.org/10.1016/j.physletb.2018.08.072
0370-2693/
© 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.