Large N Lattice Results and Perspectives
•
1 like•413 views
Talk given at the SFB TR/55 Meeting, Regensburg, 1st of August 2014
Recommended
More Related Content
Similar to Large N Lattice Results and Perspectives
Similar to Large N Lattice Results and Perspectives (20)
Recently uploaded
Recently uploaded (20)
Large N Lattice Results and Perspectives
- 1. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Large N: Lattice Results and Perspectives Biagio Lucini SFB TR 55 Meeting, Regensburg, 1st August 2014 B. Lucini Large N
- 2. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Outline 1 The ’t Hooft large-N limit 2 Quenched orientifold planar equivalence 3 Reduced models 4 Conclusions and Perspectives B. Lucini Large N
- 3. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Outline 1 The ’t Hooft large-N limit 2 Quenched orientifold planar equivalence 3 Reduced models 4 Conclusions and Perspectives B. Lucini Large N
- 4. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Why a lattice study? In QCD some phenomena (confinement, chiral symmetry breaking) are non-perturbative Lattice calculations are successful in this regime, but more effective if some analytic guidance is available If we embed QCD in a larger context (SU(N) gauge theory with Nf quark flavours), the theory simplifies in the limit N → ∞, Nf and g2N = λ fixed. Yet, it is non-trivial. Perhaps QCD is physically close to this limit? However large-N QCD is still complicated enough that an analytic solution has not been found Lattice calculations can shed light on the existence of the limiting theory and on the proximity of QCD to it B. Lucini Large N
- 5. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Meson masses from gauge-string duality 0 0.25 0.5 0.75 1 (mπ / mρ0 ) 2 1 1.2 1.4 mρ /mρ0 Lattice extrapolation AdS/CFT computation N= 3 N= 4 N= 5 N= 6 N= 7 N=17 Ads/CFT data from Erdmenger et al., Eur.Phys.J. A35 (2008) 81-133 [arXiv:0711.4467] B. Lucini Large N
- 6. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives The spectrum from the topological string0 1 2 3 4 5 6 0 1 m 2 QCD 2 s 3 s 5 F i g u r e 1 . T h e g l u e b a l l a n d m e s o n s p e c t r u m o f l a r g e - N m a s s l e s s Q C D : T h e p o i n t s i n b l a c k , a n d t h e s t r a i g h t t r a j e c t o r i e s l a b e l l e d b y t h e s p i n s , r e p r e s e n t t h e s p e c t r u m i m p l i e d b y t h e l a w s E q s . ( 1 . 1 ) . T h e r e d a n d y e l l o w p o i n t s r e p r e s e n t r e s p e c t i v e l y g l u e b a l l s a n d m e s o n s a c t u a l l y f o u n d i n t h e l a t t i c e c o m p u t a t i o n s [ 1 – 4 ] . 0 1 2 3 4 5 6 0 1 2 3 4 5 m 2 QCD 2 s k 1 k 2 k 3k 4 k 1 k 2 n0 n1 n2 n3 F i g u r e 2 . T h e g l u e b a l l a n d m e s o n s p e c t r u m o f l a r g e - N m a s s l e s s Q C D : T h e p o i n t s i n b l a c k , a n d t h e s t r a i g h t R e g g e t r a j e c t o r i e s l a b e l l e d b y t h e i n t e r n a l q u a n t u m n u m b e r s , k f o r g l u e b a l l s a n d n f o r m e s o n s , r e p r e s e n t t h e s p e c t r u m i m p l i e d b y t h e l a w s E q s . ( 1 . 1 ) . – 2 – [M. Bochicchio, Glueball and meson spectrum in large-N massless QCD, arXiv:1308.2925 [hep-th]] B. Lucini Large N
- 7. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Glueball spectrum at aTc = 1/6 0 0.5 1 1.5 2 2.5 3 am Ground States Excitations [Lucini,Teper,Wenger 2004] A1 ++ A1 -+ A2 +- E ++ E +- E -+ T1 +- T1 -+ T2 ++ T2 -- T2 -+ T1 ++ E -- T2 +- Figure 20: The spectrum at N = ∞. The yellow boxes represent the large N extrapolation of masses obtained in ref. [38]. 3 [B. Lucini, A. Rago and E. Rinaldi, JHEP 1008 (2010) 119] B. Lucini Large N
- 8. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives The meson spectrum at large N [G. Bali, F. Bursa, L. Castagnini, S. Collins, L. Del Debbio, B. Lucini and M. Panero, JHEP 1306 (2013) 071] B. Lucini Large N
- 9. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Comparison with QCD 0 1 2 3 4 5 m/√σ Ground states Excited states Experiment, √σ = 395 MeV Experiment, √σ = 444 MeV mq = mud ^ Fπ ^ fρ π ρ a0 a1 b1 0 4.6 9.2 13.8 18.4 23.0 m/ ^ F∞ √ σ fixed from the condition ˆF∞ = 85.9 MeV, mud from mπ = 138 MeV B. Lucini Large N
- 10. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Finite a corrections – SU(7) 0 0.5 1 1.5 2 2.5 3 0 0.05 0.1 0.15 0.2 0.25 0.3 X/ √ σ a √ σ 11% 3.3% 16% 4.2% 11% 5.5% a1 b1 a0 ρ ˆfρ ˆFπ [G. Bali, L. Castagnini, BL and M. Panero, in preparation] B. Lucini Large N
- 11. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Monte Carlo history of Q – SU(3) V=164, β = 6.0 B. Lucini Large N
- 12. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Monte Carlo history of Q – SU(5) V=164, β = 17.45 B. Lucini Large N
- 13. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives A first exploration of open boundary conditions Gauge group: SU(7) Lattice spacing: a √ σ ∼ 0.21 Lattice sizes: 163 × Nt, Nt = 32, 48, 64 Statistics: ∼ 500 configurations, separated by 200 composite sweeps (1 composite sweep is 1 hb + 4 overrelax) Purpose: comparing results with PBC and OBC at fixed lattice parameters in a case where there is a severe ergodicity problem Observables: Q, gluonic correlators in the 0++ and 0−+ channels and instantons Both cooling and Wilson Flow used to filter UV modes [A. Amato, G. Bali and BL, in progress] B. Lucini Large N
- 14. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Monte Carlo history of Q 0 50 100 150 200 250 300 350 400 450 CFG 0 1 2 3 4 5 6 7 HistoryofQ PERIODIC 0 100 200 300 400 500 600 CFG −10 −5 0 5 10 HistoryofQ OPEN Significantly better decorrelation for OBC B. Lucini Large N
- 15. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Distribution of Q PBC OBC OBC seem to cure the problem of ergodicity B. Lucini Large N
- 16. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Outline 1 The ’t Hooft large-N limit 2 Quenched orientifold planar equivalence 3 Reduced models 4 Conclusions and Perspectives B. Lucini Large N
- 17. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Orientifold planar equivalence The antisymmetric and the antifundamental representations coincide for SU(3) (but not in general for SU(N)) ⇒ different SU(N) generalizations of QCD. In the planar limit, the (anti)symmetric representation is equivalent to another gauge theory with the same number of Majorana fermions in the adjoint representation (in a common sector). In particular, QCD with one massless fermion in the antisymmetric representation is equivalent to N = 1 SYM in the planar limit ⇒ copy analytical predictions from SUSY to QCD. The orientifold planar equivalence holds if and only if the C-symmetry is not spontaneously broken in both theories ⇒ a calculation from first principles is mandatory. Assuming that planar equivalence works, how large are the 1/N corrections? A. Armoni, M. Shifman and G. Veneziano. SUSY relics in one-flavor QCD from a new 1/N expansion. Phys. Rev. Lett. 91, 191601, 2003. B. Lucini Large N
- 18. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Orientifold planar equivalence The antisymmetric and the antifundamental representations coincide for SU(3) (but not in general for SU(N)) ⇒ different SU(N) generalizations of QCD. In the planar limit, the (anti)symmetric representation is equivalent to another gauge theory with the same number of Majorana fermions in the adjoint representation (in a common sector). In particular, QCD with one massless fermion in the antisymmetric representation is equivalent to N = 1 SYM in the planar limit ⇒ copy analytical predictions from SUSY to QCD. The orientifold planar equivalence holds if and only if the C-symmetry is not spontaneously broken in both theories ⇒ a calculation from first principles is mandatory. Assuming that planar equivalence works, how large are the 1/N corrections? A. Armoni, M. Shifman and G. Veneziano. SUSY relics in one-flavor QCD from a new 1/N expansion. Phys. Rev. Lett. 91, 191601, 2003. B. Lucini Large N
- 19. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Orientifold planar equivalence The antisymmetric and the antifundamental representations coincide for SU(3) (but not in general for SU(N)) ⇒ different SU(N) generalizations of QCD. In the planar limit, the (anti)symmetric representation is equivalent to another gauge theory with the same number of Majorana fermions in the adjoint representation (in a common sector). In particular, QCD with one massless fermion in the antisymmetric representation is equivalent to N = 1 SYM in the planar limit ⇒ copy analytical predictions from SUSY to QCD. The orientifold planar equivalence holds if and only if the C-symmetry is not spontaneously broken in both theories ⇒ a calculation from first principles is mandatory. Assuming that planar equivalence works, how large are the 1/N corrections? M. Unsal and L. G. Yaffe. (In)validity of large N orientifold equivalence. Phys. Rev. D74:105019, 2006. A. Armoni, M. Shifman and G. Veneziano. A note on C-parity conservation and the validity of orientifold planar equivalence. Phys.Lett.B647:515-518,2007. B. Lucini Large N
- 20. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Orientifold planar equivalence The antisymmetric and the antifundamental representations coincide for SU(3) (but not in general for SU(N)) ⇒ different SU(N) generalizations of QCD. In the planar limit, the (anti)symmetric representation is equivalent to another gauge theory with the same number of Majorana fermions in the adjoint representation (in a common sector). In particular, QCD with one massless fermion in the antisymmetric representation is equivalent to N = 1 SYM in the planar limit ⇒ copy analytical predictions from SUSY to QCD. The orientifold planar equivalence holds if and only if the C-symmetry is not spontaneously broken in both theories ⇒ a calculation from first principles is mandatory. Assuming that planar equivalence works, how large are the 1/N corrections? B. Lucini Large N
- 21. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Orientifold planar equivalence The antisymmetric and the antifundamental representations coincide for SU(3) (but not in general for SU(N)) ⇒ different SU(N) generalizations of QCD. In the planar limit, the (anti)symmetric representation is equivalent to another gauge theory with the same number of Majorana fermions in the adjoint representation (in a common sector). In particular, QCD with one massless fermion in the antisymmetric representation is equivalent to N = 1 SYM in the planar limit ⇒ copy analytical predictions from SUSY to QCD. The orientifold planar equivalence holds if and only if the C-symmetry is not spontaneously broken in both theories ⇒ a calculation from first principles is mandatory. Assuming that planar equivalence works, how large are the 1/N corrections? Dynamical fermions difficult to simulate ⇒ start with the quenched theory B. Lucini Large N
- 22. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Condensates on the lattice Aim To measure the bare quark condensate with staggered fermions in the two-index representations of the gauge group, in the quenched lattice theory. Wilson action. Staggered Dirac operator. D = m − K. The two-index representations. The bare condensate. A. Armoni, B. Lucini, A. Patella and C. Pica, Phys. Rev. D78 (2008) 045019 B. Lucini Large N
- 23. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Condensates on the lattice Aim To measure the bare quark condensate with staggered fermions in the two-index representations of the gauge group, in the quenched lattice theory. Wilson action. Staggered Dirac operator. D = m − K. The two-index representations. The bare condensate. SYM = − 2N λ p e tr U(p) B. Lucini Large N
- 24. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Condensates on the lattice Aim To measure the bare quark condensate with staggered fermions in the two-index representations of the gauge group, in the quenched lattice theory. Wilson action. Staggered Dirac operator. D = m − K. The two-index representations. The bare condensate. Dxy = mδxy − Kxy = = mδxy + 1 2 µ ηµ(x) R[Uµ(x)]δx+ˆµ,y − R[Uµ(x − ˆµ)]† δx−ˆµ,y B. Lucini Large N
- 25. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Condensates on the lattice Aim To measure the bare quark condensate with staggered fermions in the two-index representations of the gauge group, in the quenched lattice theory. Wilson action. Staggered Dirac operator. D = m − K. The two-index representations. The bare condensate. tr Adj[U] = | tr U|2 − 1 tr S/AS[U] = (tr U)2 ± tr(U2) 2 B. Lucini Large N
- 26. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Condensates on the lattice Aim To measure the bare quark condensate with staggered fermions in the two-index representations of the gauge group, in the quenched lattice theory. Wilson action. Staggered Dirac operator. D = m − K. The two-index representations. The bare condensate. For S/AS representations: ¯ψψ q = 1 V Tr(m − K)−1 YM For the adjoint representation: λλ q = 1 2V Tr(m − K)−1 YM B. Lucini Large N
- 27. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Proof of the “quenched” equivalence Equivalence lim N→∞ 1 VN2 Tr(m − KS/AS)−1 = lim N→∞ 1 2VN2 Tr(m − KAdj)−1 Expand in m−1. Replace the two-index representations. Take the large-N limit. Mathematical details. The condensate is an analytical function of each real mass. The large-N limit can be exchanged with the series. B. Lucini Large N
- 28. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Proof of the “quenched” equivalence Equivalence lim N→∞ 1 VN2 Tr(m − KS/AS)−1 = lim N→∞ 1 2VN2 Tr(m − KAdj)−1 Expand in m−1. Replace the two-index representations. Take the large-N limit. Mathematical details. The condensate is an analytical function of each real mass. The large-N limit can be exchanged with the series. 1 VN2 Tr(m − K)−1 = 1 VN2 ∞ n=0 1 mn+1 Tr Kn = = 1 VN2 ω∈C c(ω) mL(ω)+1 tr R[U(ω)] B. Lucini Large N
- 29. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Proof of the “quenched” equivalence Equivalence lim N→∞ 1 VN2 Tr(m − KS/AS)−1 = lim N→∞ 1 2VN2 Tr(m − KAdj)−1 Expand in m−1. Replace the two-index representations. Take the large-N limit. Mathematical details. The condensate is an analytical function of each real mass. The large-N limit can be exchanged with the series. 1 VN2 Tr(m − KS/AS)−1 = 1 2V ω∈C c(ω) mL(ω)+1 [tr U(ω)]2 ± tr[U(ω)2] N2 1 2VN2 Tr(m − KAdj)−1 = 1 2V ω∈C c(ω) mL(ω)+1 | tr U(ω)|2 − 1 N2 B. Lucini Large N
- 30. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Proof of the “quenched” equivalence Equivalence lim N→∞ 1 VN2 Tr(m − KS/AS)−1 = lim N→∞ 1 2VN2 Tr(m − KAdj)−1 Expand in m−1. Replace the two-index representations. Take the large-N limit. Mathematical details. The condensate is an analytical function of each real mass. The large-N limit can be exchanged with the series. 1 VN2 Tr(m − KS/AS)−1 = 1 2V ω∈C c(ω) mL(ω)+1 [tr U(ω)]2 ± tr[U(ω)2] N2 1 2VN2 Tr(m − KAdj)−1 = 1 2V ω∈C c(ω) mL(ω)+1 | tr U(ω)|2 − 1 N2 B. Lucini Large N
- 31. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Proof of the “quenched” equivalence Equivalence lim N→∞ 1 VN2 Tr(m − KS/AS)−1 = lim N→∞ 1 2VN2 Tr(m − KAdj)−1 Expand in m−1. Replace the two-index representations. Take the large-N limit. Mathematical details. The condensate is an analytical function of each real mass. The large-N limit can be exchanged with the series. 1 VN2 Tr(m − KS/AS)−1 = 1 2V ω∈C c(ω) mL(ω)+1 tr U(ω) tr U(ω) N2 1 2VN2 Tr(m − KAdj)−1 = 1 2V ω∈C c(ω) mL(ω)+1 tr U(ω) tr U(ω)† N2 B. Lucini Large N
- 32. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Proof of the “quenched” equivalence Equivalence lim N→∞ 1 VN2 Tr(m − KS/AS)−1 = lim N→∞ 1 2VN2 Tr(m − KAdj)−1 Expand in m−1. Replace the two-index representations. Take the large-N limit. Mathematical details. The condensate is an analytical function of each real mass. The large-N limit can be exchanged with the series. B. Lucini Large N
- 33. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives A convenient parameterization 1 N2 ¯ψψ S/AS = 1 2V ω∈C c(ω) mL(ω)+1 [tr U(ω)]2 ± tr[U(ω)2] N2 1 N2 λλ Adj = 1 2V ω∈C c(ω) mL(ω)+1 | tr U(ω)|2 − 1 N2 B. Lucini Large N
- 34. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives A convenient parameterization 1 N2 ¯ψψ S/AS = f m, 1 N2 ± 1 N g m, 1 N2 1 N2 λλ Adj = ˜f m, 1 N2 − 1 2N2 ¯ψψ free Planar equivalence: f(m, 0) = ˜f(m, 0). Strategy 1 Simulate the condensates at various values of the mass. 2 Extract the functions f, g, ˜f. 3 Fit at fixed mass: ˜f = a0 + b0 N2 g = a1 + b1 N2 f − ˜f = a2 N2 + b2 N4 B. Lucini Large N
- 35. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Simulation details N = 2, 3, 4, 6, 8 β(N) chosen in such a way that (aTc)−1 = 5 (a 0.145 fm) 144 lattice, which corresponds to L 2.0 fm 22 values of the bare mass in the range 0.012 · · · 8.0 0 2 4 6 8 0 0.2 0.4 0.6 0.8 <ψψ> region I region II region III 0 0.005 0.01 0.015 0.02 m 0 0.04 0.08 0.12 <ψψ> B. Lucini Large N
- 36. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Function ˜f N=2 N=3 N=4 N=6 N=8 Infinite N 0 0.2 0.4 0.6 0.8 1 m 0.15 0.2 0.25f ~ For m ≤ 0.2 we get χ2/dof ≤ 0.53 (we use N = 4, 6, 8). B. Lucini Large N
- 37. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Function f N=2 N=3 N=4 N=6 N=8 Infinite N 0 0.2 0.4 0.6 0.8 1 m 0.15 0.2 0.25f For m ≤ 0.2 we get χ2/dof ≤ 0.37 (we are fitting here f − ˜f; we use N = 4, 6, 8). B. Lucini Large N
- 38. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Function g N=2 N=3 N=4 N=6 N=8 Infinite N 0 0.2 0.4 0.6 0.8 1 m 0.2 0.3 0.4 0.5 g For m ≤ 0.2 we get χ2/dof ≤ 0.17 (we use N = 4, 6, 8). B. Lucini Large N
- 39. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Condensate in the adjoint representation 0 0.1 0.2 0.3 0.4 0.5 1/N 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 condensate/N^2 λλ Adj(m = 0.012) N2 = 0.23050(22) − 0.3134(72) N2 At N = 3, relative error 0.8%. B. Lucini Large N
- 40. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Condensate in the antisymmetric representation 0 0.1 0.2 0.3 0.4 0.5 1/N 0 0.05 0.1 0.15 0.2 condensata/N^2 ¯ψψ AS(m = 0.012) N2 = 0.23050(22) − 0.4242(11) N − 0.612(43) N2 − 0.811(25) N3 At N = 3, condensate < 0! B. Lucini Large N
- 41. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Condensate in the symmetric representation 0 0.1 0.2 0.3 0.4 0.5 1/N 0.22 0.24 0.26 0.28 0.3 0.32 0.34 condensate/N^2 ¯ψψ S(m = 0.012) N2 = 0.23050(22) + 0.4242(11) N − 0.612(43) N2 + 0.811(25) N3 At N = 3, relative error 4%. B. Lucini Large N
- 42. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Mesonic two-point functions on the lattice Aim To measure the mesonic two-point functions with Wilson fermions in the two-index representations of the gauge group, in the quenched lattice theory. Wilson action. Wilson Dirac operator. The two-index representations. The mesonic two-point correlation functions. B. Lucini, G. Moraitis, A. Patella and A. Rago, Phys. Rev. D82 (2010) 114510 B. Lucini Large N
- 43. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Mesonic two-point functions on the lattice Aim To measure the mesonic two-point functions with Wilson fermions in the two-index representations of the gauge group, in the quenched lattice theory. Wilson action. Wilson Dirac operator. The two-index representations. The mesonic two-point correlation functions. SYM = − 2N λ p e tr U(p) B. Lucini Large N
- 44. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Mesonic two-point functions on the lattice Aim To measure the mesonic two-point functions with Wilson fermions in the two-index representations of the gauge group, in the quenched lattice theory. Wilson action. Wilson Dirac operator. The two-index representations. The mesonic two-point correlation functions. Dxy;αβ = (m + 4r)δxyδαβ − Kxy;αβ Kxy;αβ = − 1 2 (r − γµ)αβ R Uµ(x) δy,x+ˆµ + (r + γµ)αβ R U† µ(y) δy,x−ˆµ B. Lucini Large N
- 45. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Mesonic two-point functions on the lattice Aim To measure the mesonic two-point functions with Wilson fermions in the two-index representations of the gauge group, in the quenched lattice theory. Wilson action. Wilson Dirac operator. The two-index representations. The mesonic two-point correlation functions. tr Adj[U] = | tr U|2 − 1 tr S/AS[U] = (tr U)2 ± tr(U2) 2 B. Lucini Large N
- 46. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Mesonic two-point functions on the lattice Aim To measure the mesonic two-point functions with Wilson fermions in the two-index representations of the gauge group, in the quenched lattice theory. Wilson action. Wilson Dirac operator. The two-index representations. The mesonic two-point correlation functions. CR Γ1Γ2 (x, y) = rR ¯ψR a (x)Γ† 1 ψR b (x) ¯ψR b (y)Γ2ψR a (y) YM = rR trR D−1 yx;αβΓγβ 1 D−1 xy;γδΓδα 2 YM rR = 1 R = S/AS rR = 1/2 R = Adj B. Lucini Large N
- 47. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Heuristic proof of the quenched equivalence Equivalence lim N→∞ 1 N2 C S/AS Γ1Γ2 (x, y) = lim N→∞ 1 N2 C Adj Γ1Γ2 (x, y) Expand in Wilson loops. Replace the two-index representations. Take the large-N limit. Use invariance under charge conjugation. B. Lucini Large N
- 48. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Heuristic proof of the quenched equivalence Equivalence lim N→∞ 1 N2 C S/AS Γ1Γ2 (x, y) = lim N→∞ 1 N2 C Adj Γ1Γ2 (x, y) Expand in Wilson loops. Replace the two-index representations. Take the large-N limit. Use invariance under charge conjugation. 1 N2 CR Γ1Γ2 (x, y) = rR N2 C⊃(x,y) αC trR WC B. Lucini Large N
- 49. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Heuristic proof of the quenched equivalence Equivalence lim N→∞ 1 N2 C S/AS Γ1Γ2 (x, y) = lim N→∞ 1 N2 C Adj Γ1Γ2 (x, y) Expand in Wilson loops. Replace the two-index representations. Take the large-N limit. Use invariance under charge conjugation. 1 N2 C S/AS Γ1Γ2 (x, y) = 1 2 C⊃(x,y) αC [tr WC]2 ± tr[W2 C] N2 1 N2 C Adj Γ1Γ2 (x, y) = 1 2 C⊃(x,y) αC | tr WC|2 − 1 N2 B. Lucini Large N
- 50. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Heuristic proof of the quenched equivalence Equivalence lim N→∞ 1 N2 C S/AS Γ1Γ2 (x, y) = lim N→∞ 1 N2 C Adj Γ1Γ2 (x, y) Expand in Wilson loops. Replace the two-index representations. Take the large-N limit. Use invariance under charge conjugation. 1 N2 C S/AS Γ1Γ2 (x, y) = 1 2 C⊃(x,y) αC [tr WC]2 ± tr[W2 C] N2 1 N2 C Adj Γ1Γ2 (x, y) = 1 2 C⊃(x,y) αC | tr WC|2 − 1 N2 B. Lucini Large N
- 51. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Heuristic proof of the quenched equivalence Equivalence lim N→∞ 1 N2 C S/AS Γ1Γ2 (x, y) = lim N→∞ 1 N2 C Adj Γ1Γ2 (x, y) Expand in Wilson loops. Replace the two-index representations. Take the large-N limit. Use invariance under charge conjugation. 1 N2 C S/AS Γ1Γ2 (x, y) = 1 2 C⊃(x,y) αC tr WC tr WC N2 1 N2 C Adj Γ1Γ2 (x, y) = 1 2 C⊃(x,y) αC tr WC tr W† C N2 B. Lucini Large N
- 52. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Heuristic proof of the quenched equivalence Equivalence lim N→∞ 1 N2 C S/AS Γ1Γ2 (x, y) = lim N→∞ 1 N2 C Adj Γ1Γ2 (x, y) Expand in Wilson loops. Replace the two-index representations. Take the large-N limit. Use invariance under charge conjugation. tr W† C = tr WC ⇒ lim N→∞ 1 N2 C S/AS Γ1Γ2 (x, y) = lim N→∞ 1 N2 C Adj Γ1Γ2 (x, y) B. Lucini Large N
- 53. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Heuristic proof of the quenched equivalence Equivalence lim N→∞ 1 N2 C S/AS Γ1Γ2 (x, y) = lim N→∞ 1 N2 C Adj Γ1Γ2 (x, y) Expand in Wilson loops. Replace the two-index representations. Take the large-N limit. Use invariance under charge conjugation. A more formal proof of the equivalence exists which does not use the expansion in Wilson loops, but is much more involved B. Lucini Large N
- 54. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Simulation strategy Simulations performed for N = 2, 3, 4, 6 β(N) chosen in such a way that (aTc)−1 = 5 (a 0.145 fm) Calculations on a 32 × 163 lattice, which corresponds to L 2.3 fm CR Γ1Γ2 determined for Γ1 = Γ2 = γ5 (π channel) and Γ1 = Γ2 = γi (ρ channel) Mass extracted from the ansatz CR Γ1Γ2 (t) = A cosh (m(t − T/2)) Chiral extrapolation of mρ using mρ(mπ) = cm2 π + mρ(mπ = 0) Extrapolation to large N B. Lucini Large N
- 55. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Simulation strategy Simulations performed for N = 2, 3, 4, 6 β(N) chosen in such a way that (aTc)−1 = 5 (a 0.145 fm) Calculations on a 32 × 163 lattice, which corresponds to L 2.3 fm CR Γ1Γ2 determined for Γ1 = Γ2 = γ5 (π channel) and Γ1 = Γ2 = γi (ρ channel) Mass extracted from the ansatz CR Γ1Γ2 (t) = A cosh (m(t − T/2)) Chiral extrapolation of mρ using mρ(mπ) = cm2 π + mρ(mπ = 0) Extrapolation to large N B. Lucini Large N
- 56. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Simulation strategy Simulations performed for N = 2, 3, 4, 6 β(N) chosen in such a way that (aTc)−1 = 5 (a 0.145 fm) Calculations on a 32 × 163 lattice, which corresponds to L 2.3 fm CR Γ1Γ2 determined for Γ1 = Γ2 = γ5 (π channel) and Γ1 = Γ2 = γi (ρ channel) Mass extracted from the ansatz CR Γ1Γ2 (t) = A cosh (m(t − T/2)) Chiral extrapolation of mρ using mρ(mπ) = cm2 π + mρ(mπ = 0) Extrapolation to large N B. Lucini Large N
- 57. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Simulation strategy Simulations performed for N = 2, 3, 4, 6 β(N) chosen in such a way that (aTc)−1 = 5 (a 0.145 fm) Calculations on a 32 × 163 lattice, which corresponds to L 2.3 fm CR Γ1Γ2 determined for Γ1 = Γ2 = γ5 (π channel) and Γ1 = Γ2 = γi (ρ channel) Mass extracted from the ansatz CR Γ1Γ2 (t) = A cosh (m(t − T/2)) Chiral extrapolation of mρ using mρ(mπ) = cm2 π + mρ(mπ = 0) Extrapolation to large N B. Lucini Large N
- 58. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Simulation strategy Simulations performed for N = 2, 3, 4, 6 β(N) chosen in such a way that (aTc)−1 = 5 (a 0.145 fm) Calculations on a 32 × 163 lattice, which corresponds to L 2.3 fm CR Γ1Γ2 determined for Γ1 = Γ2 = γ5 (π channel) and Γ1 = Γ2 = γi (ρ channel) Mass extracted from the ansatz CR Γ1Γ2 (t) = A cosh (m(t − T/2)) Chiral extrapolation of mρ using mρ(mπ) = cm2 π + mρ(mπ = 0) Extrapolation to large N B. Lucini Large N
- 59. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Simulation strategy Simulations performed for N = 2, 3, 4, 6 β(N) chosen in such a way that (aTc)−1 = 5 (a 0.145 fm) Calculations on a 32 × 163 lattice, which corresponds to L 2.3 fm CR Γ1Γ2 determined for Γ1 = Γ2 = γ5 (π channel) and Γ1 = Γ2 = γi (ρ channel) Mass extracted from the ansatz CR Γ1Γ2 (t) = A cosh (m(t − T/2)) Chiral extrapolation of mρ using mρ(mπ) = cm2 π + mρ(mπ = 0) Extrapolation to large N B. Lucini Large N
- 60. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Simulation strategy Simulations performed for N = 2, 3, 4, 6 β(N) chosen in such a way that (aTc)−1 = 5 (a 0.145 fm) Calculations on a 32 × 163 lattice, which corresponds to L 2.3 fm CR Γ1Γ2 determined for Γ1 = Γ2 = γ5 (π channel) and Γ1 = Γ2 = γi (ρ channel) Mass extracted from the ansatz CR Γ1Γ2 (t) = A cosh (m(t − T/2)) Chiral extrapolation of mρ using mρ(mπ) = cm2 π + mρ(mπ = 0) Extrapolation to large N B. Lucini Large N
- 61. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives mρ vs. mπ in SU(3) 0 0.1 0.2 0.3 0.4 0.5 mπ 2 0 0.2 0.4 0.6 0.8 1 mρ Adj S AS B. Lucini Large N
- 62. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives mρ vs. mπ in SU(6) 0 0.1 0.2 0.3 0.4 0.5 mπ 2 0 0.2 0.4 0.6 0.8 1 mρ Adj S AS B. Lucini Large N
- 63. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives mρ vs. mπ (Antisymmetric) 0 0.1 0.2 0.3 0.4 0.5 mπ 2 0 0.2 0.4 0.6 0.8 1 mρ N=3 N=4 N=6 B. Lucini Large N
- 64. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives mρ vs. mπ (Symmetric) 0 0.1 0.2 0.3 0.4 0.5 mπ 2 0 0.2 0.4 0.6 0.8 1 mρ N=3 N=4 N=6 B. Lucini Large N
- 65. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives mρ vs. mπ (Adjoint) 0 0.1 0.2 0.3 0.4 0.5 mπ 2 0 0.2 0.4 0.6 0.8 1 mρ N=2 N=3 N=4 N=6 B. Lucini Large N
- 66. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Chiral extrapolation of mρ 0 0.05 0.1 0.15 0.2 0.25 1/N 2 0 0.2 0.4 0.6 0.8 1 mρ Adj S A B. Lucini Large N
- 67. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Order of corrections The correlator in the adjoint representation decays with a mass m Adj ρ that can be expressed as a power series in 1/N2, while mAS ρ and mS ρ have 1/N corrections that are related: mAdj ρ (N) = F 1 N2 ; mS ρ(N) = M 1 N2 + 1 N µ 1 N2 ; mAS ρ (N) = M 1 N2 − 1 N µ 1 N2 . M = mS ρ + mAS ρ /2 and µ = N mS ρ − mAS ρ /2 can be expressed as a power series in 1/N2 Orientifold planar equivalence is the statement F(N = ∞) = M(N = ∞) B. Lucini Large N
- 68. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Order of corrections The correlator in the adjoint representation decays with a mass m Adj ρ that can be expressed as a power series in 1/N2, while mAS ρ and mS ρ have 1/N corrections that are related: mAdj ρ (N) = F 1 N2 ; mS ρ(N) = M 1 N2 + 1 N µ 1 N2 ; mAS ρ (N) = M 1 N2 − 1 N µ 1 N2 . M = mS ρ + mAS ρ /2 and µ = N mS ρ − mAS ρ /2 can be expressed as a power series in 1/N2 Orientifold planar equivalence is the statement F(N = ∞) = M(N = ∞) B. Lucini Large N
- 69. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Order of corrections The correlator in the adjoint representation decays with a mass m Adj ρ that can be expressed as a power series in 1/N2, while mAS ρ and mS ρ have 1/N corrections that are related: mAdj ρ (N) = F 1 N2 ; mS ρ(N) = M 1 N2 + 1 N µ 1 N2 ; mAS ρ (N) = M 1 N2 − 1 N µ 1 N2 . M = mS ρ + mAS ρ /2 and µ = N mS ρ − mAS ρ /2 can be expressed as a power series in 1/N2 Orientifold planar equivalence is the statement F(N = ∞) = M(N = ∞) B. Lucini Large N
- 70. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Chiral extrapolation of mρ 0 0.05 0.1 0.15 0.2 0.25 1/N 2 0 0.2 0.4 0.6 0.8 1 mρ Adj S A M µ B. Lucini Large N
- 71. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Chiral extrapolation of mρ 0 0.05 0.1 0.15 0.2 0.25 1/N 2 0 0.2 0.4 0.6 0.8 1 mρ Adj M µ B. Lucini Large N
- 72. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Large-N fits -0.05 0 0.05 0.1 0.15 0.2 0.25 1/N 2 0 0.2 0.4 0.6 0.8 1 mρ Adj M µ B. Lucini Large N
- 73. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Large-N fits -0.05 0 0.05 0.1 0.15 0.2 0.25 1/N 2 0 0.2 0.4 0.6 0.8 1 mρ Adj S A B. Lucini Large N
- 74. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Fit results mAdj ρ = 0.6819(51) − 0.202(67) N2 ; mS ρ = 0.701(25) + 0.28(12) N − 0.85(24) N2 + 1.4(1.0) N3 ; mAS ρ = 0.701(25) − 0.28(12) N − 0.85(24) N2 − 1.4(1.0) N3 . Orientifold planar equivalence verified within 3.5% B. Lucini Large N
- 75. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Fit results mAdj ρ = 0.6819(51) − 0.202(67) N2 ; mS ρ = 0.701(25) + 0.28(12) N − 0.85(24) N2 + 1.4(1.0) N3 ; mAS ρ = 0.701(25) − 0.28(12) N − 0.85(24) N2 − 1.4(1.0) N3 . Orientifold planar equivalence verified within 3.5% B. Lucini Large N
- 76. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Towards two-index fermion unquenched simulations In the context of strongly interacting dynamics beyond the Standard Model, unquenched lattice studies exist for SU(2) with Nf = 1, 2 adjoint (symmetric) Dirac fermions SU(3) with Nf = 2 symmetric Dirac fermions SU(3) with Nf = 2 adjoint Dirac fermions SU(4) with Nf = 6 antisymmetric Dirac fermions . . . The lesson we learn is that at small N the equivalence can have strong deviations due to the dependence on N (and on the representation) of the lower edge of the conformal window. B. Lucini Large N
- 77. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Outline 1 The ’t Hooft large-N limit 2 Quenched orientifold planar equivalence 3 Reduced models 4 Conclusions and Perspectives B. Lucini Large N
- 78. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Large-N reduction At N = ∞ the SU(N) gauge theory can be formulated on a single lattice point (Eguchi-Kawai model) Non-perturbatively, centre symmetry needs to be preserved Earlier proposals (Bhanot, Heller and Neuberger; González-Arroyo and Okawa) disproved by recent simulations (Bringoltz and Sharpe; Teper and Vairinhos) New proposal by González-Arroyo and Okawa currently being tested, with encouraging results Another possibility is partial volume reduction (Narayanan and Neuberger) B. Lucini Large N
- 79. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Test of TEK (González-Arroyo and Okawa) The expected large-N result seems to be obtained B. Lucini Large N
- 80. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Centre stabilised Yang-Mills theories (Unsal and Yaffe) The centre can be stabilised by adding a deformation to the action that counteracts the centre-breaking effective Polyakov loop potential SW → SW + 1 L3 [N/2] n=1 an |Tr (Pn )|2 for appropriate values of the an The deformed theory (which is large-N equivalent to the original Yang-Mills theory) can then be shrunk to a single point Preliminary lattice studies in progress (Vairinhos) B. Lucini Large N
- 81. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Reduction with adjoint fermions The action with adjoint fermions is centre-symmetric Adjoint fermions with periodic boundary conditions helps in stabilising the centre symmetry (Kovtun, Unsal and Yaffe) This can be used to to establish a chain of large-N equivalences from QCD to a reduced adjoint model Non-perturbative lattice tests exist, but their are inconclusive B. Lucini Large N
- 82. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Test of adjoint reduction (Bringoltz and Sharpe) A region of preserved centre symmetry seems to exist in the continuum limit However, a study by Hietanen and Narayanan reaches the opposite conclusion B. Lucini Large N
- 83. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Phases with a compactified direction The phase structure is non-trivial at finite mass (Cossu and D’Elia) (Also confirmed analytically by Myers and Ogilvie and Unsal and Yaffe) B. Lucini Large N
- 84. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Outline 1 The ’t Hooft large-N limit 2 Quenched orientifold planar equivalence 3 Reduced models 4 Conclusions and Perspectives B. Lucini Large N
- 85. Large N B. Lucini The ’t Hooft large-N limit Quenched orientifold planar equivalence Reduced models Conclusions and Perspectives Where from here? 1 The large-N limit à la ’t Hooft is a mature field of investigation, but more should be done: Better control over continuum limit Unquenching effects Scattering and decays Veneziano limit? 2 Orientifold/Orbifold equivalences are an almost unexplored territory Unquenched simulations Link with supersymmetries Lessons from lattice studies of near-conformal gauge theories? 3 There have been several studies of reduced models (with partial or total reduction) Important questions about viability remain Need of efficient algorithms B. Lucini Large N