1. Phases of planar QCD1 July 26, Lattice 05, Dublin Phases of planar QCD on the torus H. Neuberger Rutgers On the two previous occasions that I gave plenary talks (higgs triviality, lattice chirality) enough time had passed that the main conceptual progress was well behind us. Not so today: we have new results, but, I feel that there are major new discoveries left to be made. My talk will reflect this by devoting a larger fraction of time to work in progress, things you can’t yet find on the archive. More than the news, I wish to convey that large N is a new exciting research direction.

2. Phases of planar QCD2 July 26, Lattice 05, Dublin Why work on large N ? Contribute to the search for a string representation. There is a shortcut to N= 1 : reduction. Even for massless quarks quenching OK (At finite N the quenched massless theory is divergent At infinite N the order of limits has to be: ) This is a feasible problem for PC clusters of today, and can be useful both phenomenologically and as a case study. Communication/Computation favorable.

3. Phases of planar QCD3 July 26, Lattice 05, Dublin N= 1 ¼ N=3 M. Teper and associates have shown that extrapolate smoothly to the planar limit. L. Del Debbio, H. Panagopoulos, P. Rossi, E. Vicari. Theta dependence at large N goes as proposed by Witten. The above extensive work ought to be reviewed in a future plenary talk, because I can’t do it justice today.

4. Phases of planar QCD4 July 26, Lattice 05, Dublin Collaborators on various projects R. Narayanan J. Kiskis A. Gonzalez-Arroyo L. Del Debbio E. Vicari

5. Phases of planar QCD5 July 26, Lattice 05, Dublin Plan and Summary Planar lattice QCD on an L 4 torus has 6 phases, 0h, [0-4]c, 5 of which survive in the continuum, [0-4]c. In each phase one has a certain amount of large N reduction, the 0x’s have the most. Planar QCD breaks chiral symmetry in 0c spontaneously and RMT works very well:

6. Phases of planar QCD6 July 26, Lattice 05, Dublin Plan and Summary – cont’d Large N reduction extends to mesons in 0c and the pion can be separated from the higher stable resonances:

7. Phases of planar QCD7 July 26, Lattice 05, Dublin Plan and Summary cont’d Chiral symmetry is restored in 1c and the Dirac operator develops a temperature dependent spectral gap. Twisting tricks will reduce computation time significantly. Speculations.

8. Phases of planar QCD8 July 26, Lattice 05, Dublin Phase Structure on a finite Torus: Lattice and Continuum at N= 1 Figure is about the lattice in 4D In the continuum, 0h disappears and boundaries obeying AF become of critical size (exponentials) values: take a b=constant line at some large b. L=1: no 0c. Situation at higher L was missed in past work on reduction. 0c extends by metastability into 0h 3D: similar situation.

9. Phases of planar QCD9 July 26, Lattice 05, Dublin Polyakov loop opens a gap: schematic In 0c all Polyakov Loops are Uniform. In Xc some open gaps.

10. Phases of planar QCD10 July 26, Lattice 05, Dublin Polyakov loop opens a gap: data Perimeter UV divergence forces trace to zero and wipes out gap. Smearing eliminates UV fluctuations and restores gap. Will massless fermions also see an effective gap via dependence on boundary conditions ?

11. Phases of planar QCD11 July 26, Lattice 05, Dublin Characteristics of the phases 0h is a “hot” lattice phase. One has exact reduction: W.L’s are independent of L at infinite N. Open plaquette loop has no gap in its spectrum. Space of gauge fields is connected. 0c is the first [as b=1/λ(‘t Hooft) increases] continuum phase. In it all infinite N W.L’s are independent of the physical torus size. Open plaquette loop has a gap and space of gauge fields dynamically splits into disconnected components. Good news for overlap fermions: no need to project out small modes for sign funct. In Xc=[1-4]c the Z(N)’s in x directions break spontaneously, and independence of the size in the corresponding directions is lost, but preserved in the other directions. Hence, 1c represents finite temperature planar, deconfined, QCD.

12. Phases of planar QCD12 July 26, Lattice 05, Dublin AF of L c (b) in 4D Data for 0c  1c AF using “Tadpole Improvement”.

13. Phases of planar QCD13 July 26, Lattice 05, Dublin Spontaneous chiral symmetry breaking Make C random, with enhanced symmetry: n is proportional to L d N c, where d is the dimension. S  SB at N= 1 is described by the random matrix model (RMT) of Shuryak and Verbaarschot.

14. Phases of planar QCD14 July 26, Lattice 05, Dublin RMT determination of χ-condensate Universal ratio of smallest to next smallest eigenvalue. RMT holds when N c is large enough. Fits to individual eigenvalue distributions permit the extraction of the chiral condensate.

15. Phases of planar QCD15 July 26, Lattice 05, Dublin Distributions of smallest eigenvalues 0c: b=0.35. All the data with N c ≥ 23 fits RMT with Σ ~ (0.14) 3 smallest ev next smallest ev

16. Phases of planar QCD16 July 26, Lattice 05, Dublin Reduction for Mesons: L independence Assume you have two degenerate fermion flavors. We have flavor non-singlet, gauge singlet mesons: The gauge covariant, background dependent, fermion propagator is: We are interested in the F. T. of: Momentum is force-fed by the following prescription:

17. Phases of planar QCD17 July 26, Lattice 05, Dublin Pion mass vs. quark mass

18. Phases of planar QCD18 July 26, Lattice 05, Dublin Pion mass parameterization Parameterization in terms of Δ reminiscent of mass formulae in cases where the AdS/CFT correspondence holds and of explicit formulae in planar 2D QCD. In principle, m  2 (m q ) contains enough information to determine the warp factor in a hypothetical 5D string background.

19. Phases of planar QCD19 July 26, Lattice 05, Dublin The 1c phase The 0c phase corresponds to infinite volume planar QCD. In 1c, one direction is selected dynamically to play the role of a finite temperature direction. Tests that 1c indeed is planar QCD at infinite space volume and finite temperature: (a) Latent heat scales. (b) Chiral symmetry is restored, with physical temperature dependence.

20. Phases of planar QCD20 July 26, Lattice 05, Dublin 0c→1c latent heat scaling: J. Kiskis

21. Phases of planar QCD21 July 26, Lattice 05, Dublin Chiral and axial-U(1) symmetry restoration at finite temperature At infinite N the finite temperature gauge transition drags the fermions to restore chiral symmetry and the axial U(1). This works by the massless overlap Dirac operator developing a gap.

22. Phases of planar QCD22 July 26, Lattice 05, Dublin The fermion gap in 1c - preliminary

23. Phases of planar QCD23 July 26, Lattice 05, Dublin Polyakov loop and fermion gap The effective Polyakov loop has a spectrum with a support extending over a fraction of the circle: this could cause a reduction in the coefficient of T 4 in the pressure relative to Stefan-Boltzmann. Simple RMT does not describe the spectrum of the Dirac operator here: the correlations between eigenvalues are strong, indicative of the Polyakov loop influence.

24. Phases of planar QCD24 July 26, Lattice 05, Dublin Tricks with twists The 1c phase can be eliminated by minimal twisting: switch the sign of β, take L odd and N=4*(prime). This twist preserves CP and extends the range of the 0c phase to smaller L’s at fixed b=β/(2N 2 ). Fermions can be added in flavor multiplets of 4. One can twist only in spatial directions to produce 1c; now fermions can be taken in flavor multiplets of 2, L in space directions is odd and N=2*(prime). Twist has a similar effect in 3D.

25. Phases of planar QCD25 July 26, Lattice 05, Dublin More reduction by twists: preliminary Almost same physics on a 3 4 as on a 9 4 at the same coupling. Similar effects in 3D. Add quarks ?

26. Phases of planar QCD26 July 26, Lattice 05, Dublin A few speculations 0c: twists may be adapted to simulations of the Veneziano limit ( N !1, N f !1,  =N f /N =fixed) perhaps at μ≠0. Nonequilibrium reduction (real time OK) : RHIC ? The phases [2-4]c need to be studied: 3c likely is Bjorken’s femptouniverse at infinite N and 4c is the same at high temperature. There might be large N phase transitions in 4D Wilson loops and in the 2D nonlinear chiral model: nontrivial eigenvalue dynamics survive in the continuum limit.