Universal properties of Wilson loop operators in large N

Universal properties of Wilson loop operatorsin large N QCDLarge N transition in the 2D SU(N)xSU(N)nonlinear sigma model

• Collaborators: Herbert Neuberger (Rutgers), Ettore Vicari (Pisa)

Rajamani NarayananDepartment of Physics

Florida International University

LATTICE 2008

Williamsburg, July 15


LATTICE 2008

Wilson loop operator

• Unitary operator for SU(N) gauge theories.

• A probe of the transition from strong coupling to weak coupling.

• Large (area) Wilson loops are non-perturbative and correspond to strong coupling.

• Small (area) Wilson loops are perturbative and correspond to weak coupling.

Rajamani Narayanan 2


LATTICE 2008

Definition of the probeWN(z, b, L) = 〈det(z −W )〉

• W is the Wilson loop operator.

• z is a complex number.

• N is the number of colors.

• b = 1g2N

is the lattice gauge coupling.

• L is the linear size of the square loop.

• 〈· · · 〉 is the average over all gauge fields with the standard gauge action.



LATTICE 2008

Multiplicative matrix model – Janik-Wieczorekmodel

WN(z, b, L) = 〈det(z −W )〉

• W =∏n

i=1 Ui; Uis are the transporters around the individual plaquettes that make up theloop and n = L2 is equal to the area of the loop.

• Two dimensional gauge theory on an infinite lattice can be gauge fixed so that the onlyvariables are the individual plaquettes and these will be independently and identicallydistributed.

• Set Uj = eiεHj and set P (Uj) = N e−N2 Tr H2

j .

• t = ε2n is the dimensionless area which is kept fixed as one takes the continuum limit,n →∞ and ε → 0.

• The parameters b and L get replaced by one parameter, t in the model.

WN(z, b, L) → QN(z, t)

• Note that N can take on any value.



LATTICE 2008

Average characteristic polynomial

QN(z, t) =

√

Nτ2π

∫∞−∞ dνe−

N2 τν2 [

z − e−τν−τ2]N

SU(N)√Nt2π

∫∞−∞ dνe−

N2 tν2 [

z − e−tν−τ2]N

U(N)

QN(z, t) =

∑N

k=0

(N

k

)zN−k(−1)ke−

τk(N−k)2N SU(N)

∑Nk=0

(N

k

)zN−k(−1)ke−

tk(N+1−k)2N U(N)

τ = t

(1 +

1

N

)

• Result is exact for the multiplicative matrix model and QCD in two dimensions.

• Both forms are useful in understanding the physics.



LATTICE 2008

Heat-kernel measure

The result for QN(z, t) is consistent with

P (W, τ )dW =∑R

dRχR(W )e−τC2(R)dW

• R denotes the representation.

• dR is the dimension of the representation R.

• C2(R) is the second order Casimir in thr representation R.

QN(z, t) = 〈N∏

j=1

(z − eiθj)〉 =

N∑k=0

zN−k(−1)kMk(t)

Mk(t) = 〈∑

1≤j1<j2<j3....<jk≤N

ei(θj1+θj2+...+θjk)〉 = 〈χk(W )〉 = dke

−τC2(k) =

(N

k

)e−

τk(N−k)2N



LATTICE 2008

Zeros of QN (z, t)

We can rewrite QN(z, t) for SU(N) as

ZN(z, t) = QN(z, t)(−1)Ne(N−1)τ

8 (−z)−N2 =

∑σ1,σ2,...σN=±1

2

eln(−z)∑

i σieτN

∑i>j σiσj

• Ferromagnetic interaction for positive τ .

• ln(−z) is a complex external magnetic field.

Conditions for Lee-Yang theorem are fulfilled.

All roots of QN(z, t) lie on the unit circle for SU(N).

This is not the case for U(N).



LATTICE 2008

Weak coupling vs strong coupling

QN(z, t) =

N∑k=0

(Nk

)zN−k(−1)ke−

t(1+ 1N )k(N−k)

2N

• Weak coupling; small area; t = 0

QN(z, t) = (z − 1)N

All roots at z = 1 on the unit circle.

• Strong coupling; large area; t = ∞

QN(z, t) = zN + (−1)N

Roots uniformly distributed on the unit circle.

QN(z, t) is analytic in z for all t at finite N . This is not the case as N →∞.



LATTICE 2008

Phase transition in an observable –Durhuus-Olesen transition

There is a critical area, t = 4, such that the distribution of zeros of Q∞(z, t) on the unit circlehas a gap around z = −1 for t < 4 and has no gap for t > 4.

The integral

QN(z, t) =

√Nτ

2π

∫ ∞

−∞dνe−

N2 τν2 [

z − e−τν−τ2]N

is domimated by the saddle point, ν = λ(t, z), given by

λ = λ(t, z) =1

zet(λ+12) − 1

With z = eiθ and w = 2λ + 1, ρ(θ) = − 14πRe w gives the distribution of the eigenvalues of W

on the unit circle.

The saddle point equation at z = −1 is

w = tanht

4w

showing that w admits a non-zero solution for t > 4.



LATTICE 2008

Double scaling limit

t =4

1 + α√3N

; z = −e(4

3N )34ξ

limN→∞

(4N

3

)14

(−1)Ne(N−1)τ

8 (−z)−N2 QN(z, t) =

∫ ∞

−∞due−u4−αu2+ξu ≡ ζ(ξ, α)

Claim

The behavior in the double scaling limit is universal and should be seen in the large N limit of3D QCD, 4D QCD, 2D PCM ....

The modified Airy function, ζ(ξ, α), is a universal scaling function.



LATTICE 2008

Large N universality hypothesis

Let C be a closed non-intersecting loop: xµ(s), s ∈ [0, 1].

Let C(m) be a whole family of loops obtained by dialation: xµ(s, m) = 1mxµ(s),with m > 0.

Let W (m, C(∗)) = W (C(m)) be the family of operators associated with the family of loopsdenoted by C(∗) where m labels one member in the family.

DefineON(y, m, C(∗)) = 〈det(e

y2 + e−

y2W (m, C(∗))〉

Then our hypothesis is

limN→∞

N (N, b, C(∗))ON

(y =

(4

3N 3

)14 ξ

a1(C(∗)), m = mc

[1 +

α√3Na2(C(∗))

])= ζ(ξ, α)



LATTICE 2008

Numerical test of the universality hypothesis –3D large N QCD• Use standard Wilson gauge action

• The lattice coupling b = 1g2N

has dimensions of length.

• Use square Wilson loops and use the linear length, L, to label C(∗).

• Change b to generate a family of square loops labelled by L.

• Need to keep b > bB = 0.43 to be in the continuum phase.

• Need to keep b < b1 where b1 depends on the lattice size in order to be in the confinedphase.

• Need to use smeared links in the construction of the Wilson loop operator to avoid cornerand perimeter divergences.

• Need to obtain bc(L), a1(L) and a2(L) such that

limN→∞

N (b, N)ON

(y =

(4

3N 3

)14 ξ

a1(L), b = bc(L)

[1 +

α√3Na2(L)

])= ζ(ξ, α)



LATTICE 2008

Numerical test of the universality hypothesis –3D large N QCD

• Fix N and L.

• Obtain estimates for bc(L, N), a1(L, N) and a2(L, N).

• Check that there is a well defined limit as N →∞.

• Check that bc(L), a1(L) and a2(L) have proper continuum limits as L →∞.



LATTICE 2008

Binder cumulant

WithON(y, b) = C0(b, N) + C1(b, N)y2 + C2(b, N)y4 + · · ·

define

Ω(b, N) =C0(b, N)C2(b, N)

C21(b, N)

.

As N →∞, Ω(b,∞) is a step function with

• Strong coupling; b < bc(L); Ω = 16.

• Weak coupling; b > bc(L); Ω = 12.



LATTICE 2008

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2b=4/t

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Ω(b

,N)

Multiplicative matrix model



LATTICE 2008

-30 -20 -10 0 10 20 30α

0.2

0.3

0.4

0.5

Ω(α

)

Scaling limit of the multiplicative matrix modelΩ(0)=0.364739936



LATTICE 2008

Extraction of bc(L, N), a2(L, N) and a1(L, N)

We useΩ(bc(L, N), N) = 0.364739936

to extract bc(L, N).

We usedΩ(b, N)

dα

∣∣∣∣α=0

=1

a2(L, N)√

3N

dΩ

db

∣∣∣∣b=bc(L,N)

= 0.0464609668

to extract a2(L, N).

We use √4

3N 3

1

a21(L, N)

C1(bc(L, N), N)

C0(bc(L, N), N)= 0.16899456

to extract a1(L, N).



LATTICE 2008

0 0.01 0.02 0.03 0.04 0.05 0.061/N

0.17

0.18

0.19

0.2

b c(L,N

)/L

Linear fit using ΩFitted values0.174(3)Nonlinear simultaneous fitFitted values0.173

L=6, 83 lattice



LATTICE 2008

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

1/N0.5

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6a 2(N

)

Linear fit using ΩFitted values2.46(25)Nonlinear simultaneous fitFitted values2.27

L=6, 83 lattice



LATTICE 2008

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

1/N0.5

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

a 1(L,N

)

Linear fit using ΩResults from the linear fit0.99(4)Nonlinear simultaenous fitFitted values0.955

L=6, 83 lattice



LATTICE 2008

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51/L

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24b c/L

bb

I

fit without 2-by-2 continuum value is 0.113(5), χ2

/d.o.f=0.102

fit with 2-by-2 continuum value is 0.126(3), χ2

/d.o.f=4.63

fit without 2-by-2 continuum value is 0.112(5), χ2

/d.o.f=0.078


/d.o.f=3.72



LATTICE 2008

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51/L

0

1

2

3

4

5

6

7

8

a 2

bb

I

fit without 2-by-2 continuum value is 6.2(2.0), χ2

/d.o.f=0.065

fit with 2-by-2 continuum value is 6.3(1.4), χ2

/d.o.f=0.034

fit without 2-by-2 continuum value is 5.1(1.6), χ2

/d.o.f=0.116


/d.o.f=0.171



LATTICE 2008

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51/L

0.85

0.86

0.87

0.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

a 1

datafit without 2-by-2 continuum value is 0.97(6), χ2

/d.o.f=0.456


/d.o.f=0.301



LATTICE 2008

Two dimensional SU(N) X SU(N) principal chiralmodel

• Similar to four dimensional SU(N) gauge theory in many respects.

•S =

N

T

∫d2xTr∂µg(x)∂µg

†(x)

g(x) ∈ SU(N).

• The global symmetry group SU(N)L× SU(N)R reduces down to a single SU(N) “diagonalsubgroup” if we make a translation breaking “gauge choice”, g(0) = 1.

• Model is asymptotically free and there are N − 1 particle states with masses

MR = Msin(Rπ

N )

sin( πN )

, 1 ≤ R ≤ N − 1.

The states corresponding to the R-th mass are a multiplet transforming as an R componentantisymmetric tensor of the diagonal symmetry group.



LATTICE 2008

Connection to multiplicative matrix model• W = g(0)g†(x) plays the role of Wilson loop with the separation x playing the role of area.

• One expects

GR(x) = 〈χR(g(0)g†(x))〉 ∼ CR

(N

R

)e−MR|x|

where χR is the trace in the R-antisymmetric representation.

• Comparison with the multiplicative matrix model suggests that M |x| plays the role of thedimensionless area.

• Numerical measurement of the correlation length using the lattice action

SL = −2Nb∑x,µ

<Tr[g(x)g†(x + µ)]

and

ξ2G =

1

4

∑x x2G1(x)∑x G1(x)

yields the following continuum result:

MξG = 0.991(1)



LATTICE 2008

Setting the scale

• ξG will be used to set the scale and it is well described by

ξG = 0.991

[e

2−π4

16π

]√

E exp(π

E

)in the range 11 ≤ ξG ≤ 20 with

E = 1− 1

N<〈Tr[g(0)g†(1)]〉 =

1

8b+

1

256b2+

0.000545

b3− 0.00095

b4+

0.00043

b5

The above equations will be used to find a b for a given ξ.



LATTICE 2008

Smeared SU(N) matrices

One needs to smear to defined well defined operators.

• Start with g(x) ≡ g0(x).

• One smearing step takes us from gt(x) to gt+1(x).

• Define Zt+1(x) by:

Zt+1(x) =∑±µ

[g†t (x)gt(x + µ)− 1]

• Construct antihermitian traceless SU(N) matrices At+1(x)

At+1(x) = Zt+1(x)− Z†t+1(x)− 1

NTr(Zt+1(x)− Z†t+1(x)) ≡ −A†

t+1(x)

• SetLt+1(x) = exp[fAt+1(x)]

• gt+1(x) is defined in terms of Lt+1(x) by:

gt+1(x) = gt(x)Lt+1(x)



LATTICE 2008

Numerical details

• We need L/ξG > 7 to minimize finite volume effects.

• Since we want 11 ≤ ξG ≤ 20, we chose L = 150.

• We used a combination of Metropolis and over-relaxation at east site x for our updates. Thefull SU(N) group was explored.

• 200-250 passes of the whole lattices was sufficient to thermalize starting from g(x) ≡ 1.

• 50 passes were enough to equilibriate if ξG was increased in steps of 1.



LATTICE 2008

Test of the universality hypothesis

The test of the universality hypothesis proceeds in the same manner as for three D large Ngauge theory.

Given an N and a ξ, we find the the dc the makes the Binder cumulantΩ(dc, N) = 0.364739936.

We look at dc as a function of ξ for a given N . This gives us the continuum value of dc/ξ forthat N .

We then take the large N limit and it gives us

dc

ξG|N=∞ = 0.885(3)



LATTICE 2008

2 3 4 5 6 7 8

x 10−3

0.65

0.7

0.75

0.8

1/ξ2

d c/ξ

N=30

d

c/ξ at finite ξ

fitted 2−nd deg polynomial

fit: dc/ξ =1543/ξ4−26.10/ξ2+0.7682

With error estimation: (d/ξ)critical, continuum

=0.768(2)



LATTICE 2008

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07

0.65

0.7

0.75

0.8

0.85

1/N

(d/ξ

G) cr

it co

nt

Extrapolation of continuum critical dc/ξ

G to N=∞

d

c/ξ

G data

linear fit

dc (N) /ξ

G = − 3.535/N + 0.885

dc/ξ

G |

N=∞ = 0.885(3)


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Universal properties of Wilson loop operators in large N