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

Instituto de Física Teórica UAM-CSIC Volume reduction through perturbative Wilson loops Margarita García Pérez In collaboration with Antonio González-Arroyo, Masanori Okawa

Eguchi & Kawai 82 Eguchi-Kawai volume reduction β 2 N 2 = λ − 1 fixed b = L 4 lattice L Large N observable on a O ∞ ( b ) = lim N →∞ lim L →∞ O ( b, N, L ) L 4 lattice

Eguchi & Kawai 82 Eguchi-Kawai volume reduction β 2 N 2 = λ − 1 fixed b = L 4 lattice L Large N observable on a O ∞ ( b ) = lim N →∞ lim L →∞ O ( b, N, L ) L 4 lattice

Eguchi & Kawai 82 Eguchi-Kawai volume reduction β 2 N 2 = λ − 1 fixed b = L 4 lattice L Large N observable on a O ∞ ( b ) = lim N →∞ lim L →∞ O ( b, N, L ) L 4 lattice Eguchi-Kawai reduction O ∞ ( b ) = lim N →∞ O ( b, N, L = 1) Thermodynamic limit U µ ∈ SU ( N ) irrespective of L one-point lattice

Conditions Volume independence of single trace observables if Z ( N ) d Tr ( ) = 0 Center symmetry preserved Bhanot, Heller & Neuberger Depends on boundary conditions For tbc k, ¯ González-Arroyo & Okawa k ∝ N For pbc Narayanan & Neuberger L > L c Depends on matter content Pbc with adjoint fermions Kotvun, Unsal & Yaffe Amber, Basar, Cherman, Dorigoni, Hanada, Koren, Poppitz, Sharpe,…

✦ In this talk: Test volume reduction for Wilson loops in lattice perturbation theory with twisted boundary conditions SU ( N ) gauge theory on a lattice L 4 log W ( b, N, L ) = − W 1 ( N, L ) λ − W 2 ( N, L ) λ 2 Compare with pbc Heller&Karsch Compare with infinite volume Weisz, Wetzel & Wohlert

Twisted boundary conditions Twist L 4 lattice X X µ ) U † ν ) U † S = bN [ N − Z µ ν ( n )Tr( U µ ( n ) U ν ( n + ˆ µ ( n + ˆ ν ( n ))] n µ ν n β 1 2 N 2 = λ − 1 b = L Z µ ν = 2 ⇡ i k n o n µ = n ν = L − 1 exp ✏ µ ν √ N λ = g 2 N symmetric twist ’t Hooft coupling k and co-prime k, ¯ √ k ∝ N N González-Arroyo & Okawa

Perturbation theory U µ ( n ) = e − igA µ ( n ) Γ µ ( n ) Periodic links U µ ( n ) = U µ ( n + L ˆ ν ) n for n µ 6 = L � 1 1 1 Γ µ ( n ) = Γ µ for n µ = L − 1 with twist eaters Γ µ Γ ν = Z ν µ Γ ν Γ µ Note: zero momentum not compatible with the boundary conditions Luscher&Weisz, Gonzalez-Arroyo & Korthals-Altes, Snippe

To implement boundary conditions ν ) = Γ ν A µ ( x ) Γ † A µ ( x + l ˆ ν µ ( p ) T a A a 0 A µ ( n ) = 1 2 ) ˆ e ip ( n + 1 A µ ( p )ˆ X Γ ( p ) L 2 p ˆ Γ ( p ) ∝ Γ s 1 1 Γ s 2 2 · · · Γ s d momentum dependent d basis for the SU(N) Lie algebra To satisfy b.c. momentum is quantised in units of p µ = 2 π m µ L e ff Effective box - size √ L e ff = L N TEK L = 1 l e ff = ∞ √ l e ff = a N thermodynamic limit N → ∞ , a fixed

Perturbation theory • Momentum quantized in units of L e ff • Free propagator identical that on a finite lattice L e ff • Group structure constants Γ ( p ) r ✓ θ µ ν ◆ 2 F ( p, q, − p − q ) = − N sin 2 p µ q ν Momentum dependent phases in the vertices θ = 2 π ¯ k ˜ √ N ✓ µ ν = L 2 ✏ µ ν ˜ e ff ✓ 4 ⇡ 2 × ˜ ¯ √ kk = 1 (mod N ) Links to non-commutative gauge theories González-Arroyo, Korthals Altes, Okawa

Volume independence r 2 λ ⇣ θ µ ν ⌘ sin 2 p µ q ν Vertices α V e ff In perturbation theory, ˜ θ , λ , L e ff ˜ For fixed , volume and N dependence encoded in the effective size θ

Comment Certain momenta excluded by the twist in SU(N) ✦ Tr ˆ 0 Γ ( p ) = 0 A µ ( n ) = 1 2 ) ˆ e ip ( n + 1 A µ ( p )ˆ X Γ ( p ) L 2 p Exclude √ p µ = 2 π n µ n µ = 0 (mod N ) ∀ µ L e ff p µ = 2 π n µ Lattice of momenta , ∀ µ Λ L eff L Reintroduces N dependence - gives correct number of degrees of freedom e ff − L 4 = L 4 ( N 2 − 1) degrees of freedom L 4 p ∈ Λ L eff \ Λ L

✦ The Wilson loop at O ( λ ) X 0 q 2 q 2 sin 2 ( Rq µ / 2) sin 2 ( Tq ν / 2) b µ + b 1 W ( R ⇥ T ) ˜ ν ( N, L, k ) = 1 q 2 q 2 q 2 4 V e ff b µ b b ν q The same as with pbc but with different set of momenta 0 1 1 1 X X X − N 2 L 4 N 2 L 4 V e ff e ff q ∈ Λ 0 q ∈ Λ 0 q L eff L Exclude momenta in Momenta in Momenta in Λ 0 Λ 0 Λ L L eff L L √ L e ff = L N Zero momentum excluded in all cases

The Wilson loop at O ( λ ) 1 Effective size correction For TBC N 2 N ) − 1 √ W 1 ( N, L ) = F 1 ( L N 2 F 1 ( L ) − → F 1 ( ∞ ) N → ∞ Volume independence MGP , González-Arroyo & Okawa For PBC W 1 ( N, L ) = F 1 ( L ) N 2 − 1 → F 1 ( L ) − N 2 N → ∞ Heller&Karsch retains L dependence

✦ The Wilson loop at O ( λ 2 ) With periodic boundary conditions Heller&Karsch ( L, N, k = 0) = (1 − 1 N 2 ) F 2 ( L ) + (1 − 1 W pbc N 2 ) 2 F W ( L ) 2 Tadpole F W ( L ) = 1 1 − 1 ⇣ ⌘ F 1 ( L ) 8 V For N → ∞ W pbc ( L, N = ∞ , k = 0) = F 2 ( L ) + F W ( L ) 2 retains L dependence

The Wilson loop at with tbc O ( λ 2 ) Non-abelian terms containing the structure constant NF 2 ( p, q, − p − q ) = 1 − cos( θ µ ν p µ q ν ) It is zero for momenta in Λ L 0 1 1 1 X F 2 X X cos( θ µ ν p µ p ν ) − NL 4 L 4 L 4 e ff e ff q q ∈ Λ 0 q ∈ Λ 0 L eff L eff Planar diagrams Non-planar diagrams The same structure Contain all the dependence as pbc on the twist

With twisted boundary conditions 1 Effective size corrections N 2 } N ) − 1 ⇣ √ ⌘ W tbc ( L, N, k ) = F 2 ( L N 2 F 2 ( L ) 2 Planar diagrams +1 1 − 1 N ) − F 1 ( L ) √ ⇣ ⌘⇣ ⌘ F 1 ( L − 1 8 N 2 N 2 N 2 F NA ( L ) 2 } Non-planar + F 2 T ( L, N, k ) diagrams θ = 2 π ¯ + 1 k ¯ twist dependence ˜ √ N 2 F NA kk = 1 (mod N ) ( L ) 2 √ N

For TBC large N limit Volume independence Correct W tbc ( L, N = ∞ , k ) = F 2 ( ∞ ) + F W ( ∞ ) thermodynamic limit 2 + F 2 T ( L, N = ∞ , k ) For volume independence to hold it is essential that N →∞ F 2 T = 0 lim

Non-planar diagrams Suppressed as 1 /V e ff Goes to zero both for N infinity or L infinity Plaquette correction 0.3 L=1 k=1 L=1 k=2 Plaquette L=1 k=3 L=1 k=4 0.25 L=2 k=1 L=2 k=2 L=2 k=3 V e ff F 2 T ( L, N, k ) L=3 k>0 L=4 k>0 0.2 L>4 L=1 k>4 L=2 k>3 slower rate V eff F 2T 0.15 1 /L 4 − α e ff 0.1 0.05 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ¯ k kbar/Lhat √ N

For TBC large V limit 1 − 1 1 − 1 ⌘ 2 ⇣ ⌘ ⇣ W tbc ( L = ∞ , N, k ) = F 2 ( ∞ ) + F W ( ∞ ) 2 N 2 N 2 The formula reproduces the correct infinite volume limit We have used that F 2 T goes to zero in the thermodynamic limit

˜ Numerically LOOP F 1 ( ∞ ) F 2 ( ∞ ) W 2 ( ∞ , ∞ ) evaluated 1 × 1 0.125 -0.0027055703(3) 0.0129194297(3) 2 × 2 0.34232788379 -0.00101077(1) 0.04178022(1) Consistent with 3 × 3 0.57629826424 0.00295130(2) 0.07498858(2) B. Alles e.a 4 × 4 0.81537096352 0.0076217(1) 0.1095431(1) F 1 ( ∞ ) F 2 ( ∞ ) F 2 ( L ) = F NA ( L ) + F meas ( L ) 2 2x2 loop -0.13 -0.135 -0.14 -0.145 L 4 { F 2 (L) - F 2 ( ∞ ) } -0.15 -0.155 -0.16 -0.165 -0.17 -0.175 -0.18 10 20 30 L F 2 ( L ) = F 2 ( ∞ ) − R 2 T 2 ( γ 2 + γ 0 2 log( L )) + . . . Bali e.a. L 4

For Twisted Eguchi-Kawai L=1 Simplification F i ( L = 1) = 0 √ ( L = 1 , N, k ) = W pbc W tbc ( L = N, ∞ , 0) 1 1 √ ( L = 1 , N, k ) = W pbc W tbc ( L = N, ∞ , 0) + F 2 T ( L = 1 , N, k ) 2 2 Non-planar Effective size √ L = N contribution Effective colour N = ∞ N →∞ F 2 T = 0 lim

Summary • We have analysed the PT expansion of Wilson loops with tbc • The expansion is expressed in terms of 3 functions: F 1 ( L ) , F 2 ( L ) , F 2 T ( L, N.k ) • Volume independence holds as far as N →∞ F 2 T = 0 lim • 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 of dimensions