Universal properties of Wilson loop operators in large N ...
Transcript of Universal properties of Wilson loop operators in large N ...
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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
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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.
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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.
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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.
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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.
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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
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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).
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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 →∞.
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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.
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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.
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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)
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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)
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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 →∞.
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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.
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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
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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
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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).
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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
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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
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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
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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
fit with 2-by-2 continuum value is 0.124(3), χ2
/d.o.f=3.72
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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
fit with 2-by-2 continuum value is 4.5(9), χ2
/d.o.f=0.171
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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
fit with 2-by-2 continuum value is 0.96(4), χ2
/d.o.f=0.301
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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.
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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)
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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.
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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)
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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.
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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)
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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)
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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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