Studies of large-n reduction with adjoint fermions

Studies of large-n reduction with adjoint fermions Barak Bringoltz University of Washington Non-perturbative volume-reduction of large-n QCD with adjoint fermions BB and S.R.Sharpe, 0906.3538 Large-N volume reduction of lattice QCD with adjoint Wilson fermions at weak coupling BB, JHEP, 0905.2406

Main motivation Study a large-n limit of QCD where quarks are back-reacting on gauge fields Armoni-Veneziano-Shifman: Natural to put quarks in two-antisymmetric : QCD(AS) Corrigan-Ramond, Armoni-Shifman-Veneziano, Sannino et al. QCD(AS) infinite volume Orientifold QCD(Adj) infinite volume Kovtun-Unsal-Yaffe: QCD(Adj) infinite volume Orbifold QCD(Adj) zero volume Eguchi-Kawai volume reduction works if have light adjoint quarks around Study QCD(Adj) on a single-site gives physics of QCD(AS) at infinite volume

More motivations QCD + adjoints is an interesting theory: Nf=1/2 is softly broken N=1 SUSY Nf=2 is in (or close by to) the conformal window. Any value of Nf=2 and heavy enough quarks is YM. Can study large-n limit of all these with method I will describe. Does it really provide a workable Eguchi-Kawai? (finally, after 27 years...) Original Eguchi-Kawai. Quenched Eguchi-Kawai. Twisted Eguchi-Kawai. Jan `82. Bhanot Heller Neuberger, Feb `82 Gonzalez-Arroyo Okawa, July `82 But Bhanot-Heller-Neuberger Feb `82. But BB-Sharpe, 2008. But, 2006. Teper-Vairinhos, Hanada-et-al, Bietenholtz et al. Adjoint Eguchi-Kawai. Deformed Eguchi-Kawai. Kovtun- Unsal- Yaffe Unsal- Yaffe, 2007, 2008 This talk. BB, Vairinhos, 2008 No But s?

I. What is the Eguchi-Kawai equivalence. Given SU(N) gauge theory on an L1xL2xL3xL4 lattice, defined by g 2 N, am, aµ,... Then if: Translation symmetry is intact. ZN center symmetry is intact. large-n factorization holds. at N=oo Wilson loops, Hadron spectra, condensates, etc. are independent of L1,2,3,4. This talk: fate of ZN. First at weak coupling, then non-perturbatively

I. What is the Eguchi-Kawai equivalence. Given SU(N) gauge theory on an L1xL2xL3xL4 lattice, defined by g 2 N, am, aµ,... Then if: Translation symmetry is intact. ZN center symmetry is intact. Not academic requirements: breakdown of EK equivalence by formation of a baryon crystal BB, PRD,0811.4141, 0901.4035 large-n factorization holds. QEK model BB+Sharpe 2008 at N=oo Wilson loops, Hadron spectra, condensates, etc. are independent of L1,2,3,4. This talk: fate of ZN. First at weak coupling, then non-perturbatively

II. Fields: Weak coupling analysis: L2,3,4 = oo, L1=1. U µ=0 ( x, x 0 = 1) Ω x BB, JHEP, 0905.2406 U µ=1,2,3 ( x, x 0 = 1) U µ=1,2,3 ( x) ψ( x, x 0 = 1) ψ x Action: Gauge: Wilson, b =1/g 2 N Fermions: Wilson, κ = 1 8+2am 0 kappa=1/8 : chiral quarks kappa=0 : infinitely massive quarks Weak coupling expansion: g 2 N 0: Ω ab x = δab e iθa + δω ab ( x)+... U x,µ = 1 + ia µ ( x)+...

II. Weak coupling analysis: L2,3,4 = oo, L1=1. BB, JHEP, 0905.2406 Get familiar form V (θ) = a b ( ) 3 { ( dp θ log [ˆp 2 + 4 sin 2 a θ b )] 2π 2 2N f log [ˆp 2 + sin 2 ( θ a θ b) ] } + m 2 W (θ,p) ˆp 2 =4 3 i=1 sin 2 p i /2 a 0 p 2 ˆp 2 = 3 i=1 sin 2 p i a 0 p 2 m W = am 0 +2 [ sin 2 (p/2) + sin 2 ((θ a θ b )/2) ] θ a Z N Z N Ø Z N Z 2 Calculate V (θ) potential for different corresponding to,,. Chiral quarks κ =1/8 These are good news: κ =0 YM model (original EK) Reduction works!

II. Weak coupling analysis: L2,3,4 = oo, L1=1. BB, JHEP, 0905.2406 Aside : Comparing some weak coupling calculations. Kovtun et al. 07 Lattice calculation Bedaque et al. 08 4D YM + adjoints continuum 4D YM+Wilson adjoints and L1=1 Think about L1=1 as 3D theory defined on spatial continuum Z N Z N Z N Z 2 Useful to understand the reason for the difference

II. Weak coupling analysis: L2,3,4 = oo, L1=1. BB, JHEP, 0905.2406 Bedaque et al. (focus on gauge dynamics first) S gauge = 2N λ Re x i<j Tr + 2N λ Re x i ( ) U x,i U x+i,j U x+j,i U x,j Tr ( ) U x,i Ω x+i U x,i Ω x a s 0 S one site = d 3 x 1 g 2 Tr Fij 2 + f 2 Tr i<j [1,3] i D i Ω(x) = i Ω(x)+i[A i, Ω] Ω SU(N) D i Ω 2 V (θ) = a b ( ) 3 ( dp θ log [a 2t p 2 + sin 2 a θ b )] 2π 2 = Λ 3 + a b ( ) 3 [ dp log 1+ 1 2π a 2 t p 2 sin2 ( θ a θ b )] 2 = Λ 3 + Λ a b ( θ sin 2 a θ b ) 2 +... = Λ 3 + Λ tr Ω classical 2 +..., Ω ab classical = e iθa Bernard&Appelquist `80, Longhitano `80, is non-renormalizable... S one site Banks&Ukawa `84, Gasser-Leutwyler 84, need counter-terms... Arkani-Hamed-Cohen-Georgi `01, Pisarski `06,

II. Weak coupling analysis: L2,3,4 = oo, L1=1. BB, JHEP, 0905.2406 What does this teach us? Continuum limit in space with L1=1 (or for any D > 2L1 ). need counter-terms new Low Energy Constants (LEC). means treating theory as an Effective Field Theory (EFT), and at one-loop: V (θ) V (θ)+b 1 tr Ω classical 2 + b 2 tr Ω 2 classical 2 Amusing: these are (Mithat and Larry) s terms Dim-reg hides this and implicitly sets b1=b2=0 b1,2 > 0 fixes ZN Z2 breaking. EFT point of view is not necessary In any case, large-n reduction defined at fix lattice cutoff: But ZN-realization may depend on lattice action... Results I got cannot be anticipated in advance (from Kovtun et al.)

In any case, we saw that : Wilson fermions at weak coupling obey reduction But really need a non-perturbative lattice study Really interested in L1,2,3,4=1, but IR div s. g 2 N 1 3 What happens at? Non-perturbative effect (e.g. QEK and TEK model). Wilson gauge + Nf=1 Wilson fermions, Metropolis. BB+S.Sharpe, 0906.3538 SU(N) with N=8,10,11,13,15. A variety of of spacings and masses. Goal : Map single-site theory in κ and g 2 N Look for intact ZN.

III. Results of non-perturbative MC lattice simulations What should we expect? L1,2,3,4=oo : Continuum physics BB+S.Sharpe, 0906.3538 quarks are light along line strong-to-weak lattice transition strong-coupling/ lattice physics YM

III. Results of non-perturbative MC lattice simulations BB+S.Sharpe, 0906.3538 What should we expect? L1,2,3,4=1: Continuum physics ~0.04 quarks are light along line strong-to-weak lattice transition strong-coupling/ lattice physics YM

III. Results of non-perturbative MC lattice simulations BB+S.Sharpe, 0906.3538 What should we expect? L1,2,3,4=1: Continuum physics ~0.04 3 quarks are light along line strong-to-weak lattice transition strong-coupling/ lattice physics YM

III. Results of non-perturbative MC lattice simulations BB+S.Sharpe, 0906.3538 Scan no. 1 : infinitely massive quarks X-axis: Real(Polyakov) Y-axis: Imag(Polyakov)

III. Results of non-perturbative MC lattice simulations BB+S.Sharpe, 0906.3538 Scan no. 1 : infinitely massive quarks X-axis: Real(Polyakov) Y-axis: Imag(Polyakov) b =0

III. Results of non-perturbative MC lattice simulations BB+S.Sharpe, 0906.3538 Scan no. 1 : infinitely massive quarks X-axis: Real(Polyakov) Y-axis: Imag(Polyakov) b =0 b 0.3

III. Results of non-perturbative MC lattice simulations BB+S.Sharpe, 0906.3538 Scan no. 1 : infinitely massive quarks X-axis: Real(Polyakov) Y-axis: Imag(Polyakov) b =0 b 0.3 b =0.5

III. Results of non-perturbative MC lattice simulations BB+S.Sharpe, 0906.3538 Scan no. 2 : decreasing the quark mass b=0.5,su(10) κ 0 κ =0.03 κ =0.06 κ =0.12 κ =0.1475 κ =0.16

III. Results of non-perturbative MC lattice simulations BB+S.Sharpe, 0906.3538 Scan no. 2 : decreasing the quark mass b=0.5,su(10),su(15) κ 0 κ =0.03 κ =0.06 κ =0 κ =0.06 κ =0.09 κ =0.1275 κ =0.1475 κ =0.155 κ =0.12 κ =0.1475 κ =0.16

III. Results of non-perturbative MC lattice simulations BB+S.Sharpe, 0906.3538 Scan no. 2 : looking for the critical line b=0.35: 1st transition at kappa~0.15

III. Results of non-perturbative MC lattice simulations BB+S.Sharpe, 0906.3538 Scan no. 2 : looking for the critical line 0.08 0.06 increasing kappa decreasing kappa 0.04 0.02 Im(Polyakov) 0-0.02-0.04-0.06-0.08 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.06 0.04 0.02 kappa increasing kappa decreasing kappa b=0.35: 1st transition at kappa~0.15 Re(Polyakov) 0-0.02-0.04-0.06-0.08 0.08 0.1 0.12 0.14 0.16 0.18 0.2 kappa

III. Results of non-perturbative MC lattice simulations BB+S.Sharpe, 0906.3538 Scan no. 2 : looking for the critical line 0.125 0.08 0.06 increasing kappa decreasing kappa 0.04 0.02 Im(Polyakov) 0-0.02-0.04 Re(Polyakov) -0.06-0.08 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.06 0.04 0.02 0-0.02 kappa increasing kappa decreasing kappa 1st b=0.35: transition structure 1st transition at all b. at Extending kappa~0.15 from kappa=0.25 to 0.125-0.04-0.06-0.08 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.25 kappa

III. Results of non-perturbative MC lattice simulations BB+S.Sharpe, 0906.3538 All results consistent with phase diagram and validity of reduction.

III. Results of non-perturbative MC lattice simulations BB+S.Sharpe, 0906.3538 All results consistent with phase diagram and validity of reduction. But: See long autocorrelation times there at very very weak coupling. More order parameters for nontrivial breaking of ZN. Perform long runs and measure order parameters of the form tr [P n 1 1 P n 2 2 P n 3 3 P n 4 4 ] with n i [ 5, 5] 14641 order parameters!

IV. Conclusions and future prospects. Weak coupling with L2,3,4=oo, L1=1 Large-N volume reduction works for YM+Wilson adjoints fermions. Non-perturbative lattice Monte-Carlo of Nf=1 case. Large-N volume reduction works for YM+Wilson adjoints fermions. This is exciting : Eguchi-Kawai finally works start extracting physics of QCD(Adj) and QCD(AS) Mesons, baryons (see Carlos Q. Hoyos s talk). Realization of chiral symmetry (of both adjoint sea-quarks and valence fundamental). Comparisons with RMT. Static potential, string tensions. Open to suggestions... A new UNEXPLORED physically relevant and rich theory