Volume reduction through perturbative Wilson loops Margarita Garca - - PowerPoint PPT Presentation

Volume reduction through perturbative Wilson loops

Margarita García Pérez In collaboration with Antonio González-Arroyo, Masanori Okawa

Instituto de Física

Teórica

UAM-CSIC

Eguchi-Kawai volume reduction

O∞(b) = lim

N→∞ lim L→∞ O(b, N, L)

b = β 2N 2 = λ−1

L

lattice

L4 Large N observable on a

fixed

L4 lattice

Eguchi & Kawai 82

Eguchi-Kawai volume reduction

O∞(b) = lim

N→∞ lim L→∞ O(b, N, L)

b = β 2N 2 = λ−1

L

lattice

L4 Large N observable on a

fixed

L4 lattice

Eguchi & Kawai 82

Eguchi-Kawai volume reduction

O∞(b) = lim

N→∞ lim L→∞ O(b, N, L)

Eguchi-Kawai reduction

O∞(b) = lim

N→∞ O(b, N, L = 1)

• ne-point lattice

Uµ ∈ SU(N)

Thermodynamic limit irrespective of L

b = β 2N 2 = λ−1

L

lattice

L4 Large N observable on a

fixed

L4 lattice

Eguchi & Kawai 82

Conditions Tr ( ) = 0 Center symmetry preserved Depends on boundary conditions Depends on matter content For pbc

L > Lc

Narayanan & Neuberger

Pbc with adjoint fermions

Kotvun, Unsal & Yaffe

Z(N)d

Amber, Basar, Cherman, Dorigoni, Hanada, Koren, Poppitz, Sharpe,…

Volume independence of single trace observables if For tbc

González-Arroyo & Okawa

k, ¯ k ∝ N

Bhanot, Heller & Neuberger

✦

In this talk: Test volume reduction for Wilson loops in lattice perturbation theory with twisted boundary conditions lattice L4

SU(N) gauge theory on a

log W(b, N, L) = −W1(N, L)λ − W2(N, L)λ2

Compare with pbc

Heller&Karsch

Compare with infinite volume

Weisz, Wetzel & Wohlert

lattice

L4

b = β 2N 2 = λ−1

L

S = bN X

n

X

µν

[N − Zµν(n)Tr(Uµ(n)Uν(n + ˆ µ)U †

µ(n + ˆ

ν)U †

ν(n))]

k and co-prime

√ N

λ = g2N

’t Hooft coupling

Zµν =

n

1

exp n 2⇡i k √ N ✏µν

• nµ = nν = L − 1

Twisted boundary conditions

Twist González-Arroyo & Okawa symmetric twist

k, ¯ k ∝ N

Luscher&Weisz, Gonzalez-Arroyo & Korthals-Altes, Snippe

Uµ(n) = e−igAµ(n)Γµ(n) ΓµΓν = ZνµΓνΓµ

n

Γµ(n) = 1 1 for nµ 6= L 1 Γµ for nµ = L − 1

with Note: zero momentum not compatible with the boundary conditions Periodic links Uµ(n) = Uµ(n + Lˆ

ν)

Perturbation theory

twist eaters

To satisfy b.c. momentum is quantised in units of Effective box - size momentum dependent basis for the SU(N) Lie algebra

ˆ Γ(p) ∝ Γs1

1 Γs2 2 · · · Γsd d

Aa

µ(p)Ta

To implement boundary conditions Aµ(n) = 1 L2 X

p

eip(n+ 1

2 ) ˆ

Aµ(p)ˆ Γ(p) pµ = 2πmµ Leff Leff = L √ N leff = a √ N N → ∞, a fixed leff = ∞

TEK thermodynamic limit

L = 1

Aµ(x + l ˆ ν) = ΓνAµ(x)Γ†

ν

Perturbation theory

• Free propagator identical that on a finite lattice

Leff F(p, q, −p − q) = − r 2 N sin ✓θµν 2 pµqν ◆ ✓µν = L2

eff

4⇡2 × ˜ ✏µν ˜ ✓

Momentum dependent phases in the vertices

González-Arroyo, Korthals Altes, Okawa

• Momentum quantized in units of

Γ(p)

Links to non-commutative gauge theories

Leff

˜ θ = 2π¯ k √ N

¯ kk = 1 (mod √ N)

• Group structure constants

In perturbation theory,

˜ θ, λ, Leff

Volume independence

For fixed , volume and N dependence encoded in the effective size

˜ θ r 2λ Veff sin ⇣θµν 2 pµqν ⌘

Vertices α

Comment ✦ Certain momenta excluded by the twist in SU(N)

nµ = 0 (mod √ N) ∀µ

Exclude Reintroduces N dependence - gives correct number of degrees of freedom

Tr ˆ Γ(p) = 0 Aµ(n) = 1 L2 X

p

eip(n+ 1

2 ) ˆ

Aµ(p)ˆ Γ(p) pµ = 2πnµ Leff

degrees of freedom

L4

eff − L4 = L4(N 2 − 1)

p ∈ ΛLeff\ΛL ΛLeff

Lattice of momenta

pµ = 2πnµ L , ∀µ

˜ W (R⇥T )

1

(N, L, k) = 1 4Veff X

q

sin2(Rqµ/2) sin2(Tqν/2) b q 2

µ b

q 2

ν

b q 2

µ + b

q 2

ν

b q2

The Wilson loop at

✦

O(λ)

The same as with pbc but with different set of momenta Zero momentum excluded in all cases

ΛL 1 Veff X

q

1 L4

eff

X

q∈Λ0

Leff

− 1 N 2L4 X

q∈Λ0

L

Exclude momenta in Momenta in

Λ0

Leff

Λ0

L

Momenta in L

Leff = L √ N

N 2

The Wilson loop at O(λ)

MGP , González-Arroyo & Okawa

N → ∞

For PBC

N → ∞

retains L dependence Volume independence

For TBC

W1(N, L) = F1(L √ N) − 1 N 2 F1(L) − → F1(∞)

W1(N, L) = F1(L) N 2 − 1 N 2 − → F1(L)

Heller&Karsch

Effective size correction

1 N 2

The Wilson loop at O(λ2)

W pbc

2

(L, N, k = 0) = (1 − 1 N 2 )F2(L) + (1 − 1 N 2 )2FW (L)

With periodic boundary conditions

Heller&Karsch

Tadpole

FW (L) = 1 8 ⇣ 1 − 1 V ⌘ F1(L)

For N → ∞

W pbc

2

(L, N = ∞, k = 0) = F2(L) + FW (L)

retains L dependence

✦

Non-abelian terms containing the structure constant

NF 2(p, q, −p − q) = 1 − cos(θµνpµqν)

1 NL4 X

q

F 2

The Wilson loop at with tbc

O(λ2)

It is zero for momenta in ΛL

1 L4

eff

X

q∈Λ0

Leff

− 1 L4

eff

X

q∈Λ0

Leff

cos(θµνpµpν)

Planar diagrams Non-planar diagrams Contain all the dependence

• n the twist

The same structure as pbc

With twisted boundary conditions

W tbc

2

(L, N, k) = ⇣ F2(L √ N) − 1 N 2 F2(L) ⌘ +1 8 ⇣ 1 − 1 N 2 ⌘⇣ F1(L √ N) − F1(L) N 2 ⌘ +F2T (L, N, k)

} }

Planar diagrams Non-planar diagrams Effective size corrections

1 N 2

twist dependence

˜ θ = 2π¯ k √ N ¯ kk = 1 (mod √ N) + 1 N 2 F NA

2

(L) − 1 N 2 F NA

2

(L)

Volume independence

For TBC

W tbc

2

(L, N = ∞, k) = F2(∞) + FW (∞)

lim

N→∞ F2T = 0

large N limit Correct thermodynamic limit

+ F2T (L, N = ∞, k)

For volume independence to hold it is essential that

Non-planar diagrams

0.05 0.1 0.15 0.2 0.25 0.3 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Veff F2T kbar/Lhat Plaquette correction L=1 k=1 L=1 k=2 L=1 k=3 L=1 k=4 L=2 k=1 L=2 k=2 L=2 k=3 L=3 k>0 L=4 k>0 L>4 L=1 k>4 L=2 k>3

¯ k √ N

Suppressed as 1/Veff Goes to zero both for N infinity or L infinity Plaquette

Veff F2T (L, N, k)

slower rate

1/L4−α

eff

For TBC large V limit

W tbc

2

(L = ∞, N, k) = ⇣ 1 − 1 N 2 ⌘ F2(∞) + ⇣ 1 − 1 N 2 ⌘2 FW (∞)

The formula reproduces the correct infinite volume limit We have used that F2T goes to zero in the thermodynamic limit

• 0.18
• 0.175
• 0.17
• 0.165
• 0.16
• 0.155
• 0.15
• 0.145
• 0.14
• 0.135
• 0.13

20 30 10 L4 { F2(L) - F2( ∞) } L 2x2 loop

F2(L) = F2(∞) − R2T 2(γ2 + γ0

2 log(L))

L4 + . . .

Bali e.a.

F2(∞)

F1(∞)

LOOP F1(∞) F2(∞) ˜ W2(∞, ∞) 1 × 1 0.125

• 0.0027055703(3)

0.0129194297(3) 2 × 2 0.34232788379

• 0.00101077(1)

0.04178022(1) 3 × 3 0.57629826424 0.00295130(2) 0.07498858(2) 4 × 4 0.81537096352 0.0076217(1) 0.1095431(1)

F2(L) = F NA

2

(L) + Fmeas(L)

Numerically evaluated Consistent with

• B. Alles e.a

For Twisted Eguchi-Kawai L=1

lim

N→∞ F2T = 0

W tbc

2

(L = 1, N, k) = W pbc

2

(L = √ N, ∞, 0) + F2T (L = 1, N, k) Non-planar contribution Effective size Effective colour

L = √ N N = ∞

W tbc

1

(L = 1, N, k) = W pbc

1

(L = √ N, ∞, 0) Fi(L = 1) = 0

Simplification

Summary

• We have analysed the PT expansion of Wilson loops with tbc
• The expansion is expressed in terms of 3 functions:

F1(L), F2(L), F2T (L, N.k)

• Volume independence holds as far as

lim

N→∞ F2T = 0

• Our analysis shows that this holds, also for TEK on the one-site lattice
• The code developed can be applied to other twists and number
• f dimensions