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.
July 26, Lattice 05, Dublin
Phases of planar QCD
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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.
July 26, Lattice 05, Dublin
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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.
July 26, Lattice 05, Dublin
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Collaborators on various projects
•
•
•
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R. Narayanan
J. Kiskis
A. Gonzalez-Arroyo
L. Del Debbio
E. Vicari
July 26, Lattice 05, Dublin
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Plan and Summary
• Planar lattice QCD on an L4 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:
July 26, Lattice 05, Dublin
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Plan and Summary – cont’d
• Large N reduction extends to mesons in 0c
and the pion can be separated from the
higher stable resonances:
July 26, Lattice 05, Dublin
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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.
July 26, Lattice 05, Dublin
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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.
July 26, Lattice 05, Dublin
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Polyakov loop opens a gap: schematic
In 0c all Polyakov Loops are Uniform. In Xc some open gaps.
July 26, Lattice 05, Dublin
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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 ?
July 26, Lattice 05, Dublin
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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.
July 26, Lattice 05, Dublin
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AF of Lc (b) in 4D
Data for 0c 1c
AF using “Tadpole
Improvement”.
July 26, Lattice 05, Dublin
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Spontaneous chiral symmetry breaking
• SSB at N=1 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.
July 26, Lattice 05, Dublin
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RMT determination of χ-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.
July 26, Lattice 05, Dublin
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Distributions of smallest eigenvalues
0c: b=0.35. All the data with Nc ≥ 23 fits RMT with Σ ~ (0.14)3
smallest ev
July 26, Lattice 05, Dublin
next 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:
July 26, Lattice 05, Dublin
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Pion mass vs. quark mass
July 26, Lattice 05, Dublin
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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 (mq) contains enough
information to determine the warp factor in a
hypothetical 5D string background.
July 26, Lattice 05, Dublin
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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.
July 26, Lattice 05, Dublin
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0c→1c latent heat scaling: J. Kiskis
July 26, Lattice 05, Dublin
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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.
July 26, Lattice 05, Dublin
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The fermion gap in 1c - preliminary
July 26, Lattice 05, Dublin
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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.
July 26, Lattice 05, Dublin
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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/(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 N=2*(prime). Twist has
a similar effect in 3D.
July 26, Lattice 05, Dublin
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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 ?
July 26, Lattice 05, Dublin
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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 [2-4]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.
July 26, Lattice 05, Dublin
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