Phases of planar QCD on the torus

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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 cant yet
find on the archive. More than the news, I wish
to convey that large N is a new exciting
research direction.
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Why work on large N ?

• Contribute to the search for a string
  representation.
• There is a shortcut to N1 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.

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N1 ¼ N3

• 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 cant do it justice today.
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Collaborators on various projects

• R. Narayanan
• J. Kiskis
• A. Gonzalez-Arroyo
• L. Del Debbio
• E. Vicari

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Plan and Summary

• Planar lattice QCD on an L4 torus has 6 phases,
  0h, 0-4c, 5 of which survive in the continuum,
  0-4c. In each phase one has a certain amount of
  large N reduction, the 0xs have the most.
• Planar QCD breaks chiral symmetry in 0c
  spontaneously and RMT works very well

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Plan and Summary contd

• Large N reduction extends to mesons in 0c and the
  pion can be separated from the higher stable
  resonances

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Plan and Summary contd

• Chiral symmetry is restored in 1c and the Dirac
  operator develops a temperature dependent
  spectral gap.
• Twisting tricks will reduce computation time
  significantly.
• Speculations.

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Phase Structure on a finite Torus Lattice and
Continuum at N1
Figure is about the lattice in 4D In the
continuum, 0h disappears and boundaries obeying
AF become of critical size (exponentials) values
take a bconstant line at some large b. L1 no
0c. Situation at higher L was missed in past work
on reduction. 0c extends by metastability into
0h 3D similar situation.
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Polyakov loop opens a gap schematic
In 0c all Polyakov Loops are Uniform. In Xc some
open gaps.
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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 ?

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Characteristics of the phases

• 0h is a hot lattice phase. One has exact
  reduction W.Ls 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 b1/?(t Hooft) increases
  continuum phase. In it all infinite N W.Ls 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 Xc1-4c 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.

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AF of Lc (b) in 4D
Data for 0c ?1c AF using Tadpole Improvement
.
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Spontaneous chiral symmetry breaking

• S?SB at N1 is described by the random matrix
• model (RMT) of Shuryak and Verbaarschot.

Make C random, with enhanced symmetry
n is proportional to Ld Nc, where d is the
dimension.
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RMT determination of ?-condensate
Universal ratio of smallest to next smallest
eigenvalue. RMT holds when Nc is large
enough. Fits to individual eigenvalue
distributions permit the extraction of the chiral
condensate.
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Distributions of smallest eigenvalues
0c b0.35. All the data with Nc 23 fits RMT
with S (0.14)3
next smallest ev
smallest ev
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Reduction for Mesons L independence
Assume you have two degenerate fermion flavors.
We have flavor non-singlet, gauge singlet mesons
We are interested in the F. T. of
The gauge covariant, background dependent,
fermion propagator is
Momentum is force-fed by the following
prescription
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Pion mass vs. quark mass
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Pion mass parameterization

• Parameterization in terms of ? reminiscent of
  mass formulae in cases where the AdS/CFT
  correspondence holds and of explicit formulae in
  planar 2D QCD.
• In principle, m?2 (mq) contains enough
  information to determine the warp factor in a
  hypothetical 5D string background.

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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.

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0c?1c latent heat scaling J. Kiskis
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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.

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The fermion gap in 1c - preliminary
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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
  T4 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.

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Tricks with twists

• The 1c phase can be eliminated by minimal
  twisting switch the sign of ß , take L odd and
  N4(prime).
• This twist preserves CP and extends the range
  of the 0c phase to smaller Ls at fixed bß/(2N2
  ).
• 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 N2(prime).
  Twist has
• a similar effect in 3D.

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More reduction by twists preliminary

• Almost same physics on a 34 as on a 94 at the
• same coupling.
• Similar effects in 3D.
• Add quarks ?

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A few speculations

• 0c twists may be adapted to simulations of the
  Veneziano limit ( N!1 , Nf!1 , ?Nf/N fixed)
  perhaps at µ?0.
• Nonequilibrium reduction (real time OK) RHIC ?
• The phases 2-4c need to be studied 3c likely
  is Bjorkens 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.