Large NC strongly coupled lattice QCD

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Large NC strongly coupled lattice QCD

• Donatella Marmottini
• Università di Perugia INFN- Perugia

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• Low lying meson spectrum of large N(C) strongly
  coupled QCD.
• by G. Grignani, D. Marmottini and P. Sodano
• Phys. Rev. D68 076003, 2003
  hep-lat/0306004
• QCD meson spectrum in the large N(C) limit.
• by G. Grignani, D. Marmottini and P. Sodano
• Contributed to the 20th International
  Symposium on Lattice Field Theory
• (LATTICE 2002), MIT Boston
• Nucl. Phys. Proc. Suppl. 119 281-283, 2003
• The chiral condensate in the t Hooft limit of
  strongly coupled
• QCD
• by G. Grignani, D. Marmottini and P. Sodano
• hep-lat/0310060

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Contents

• Calculation of the low energy meson spectrum of
  massless QCD in the large NC limit (t Hooft
  limit) and
• evaluation of the QCD chiral condensate.
• Main results
• Development of a procedure to extend the famous
  calculation of the hadron spectrum by Banks et
  al. to a general color number NC
• Construction of the gauge invariant eigenstates
  of the unperturbed Hamiltonian
• The t Hooft limit is smooth and the ratios
  between the meson masses are in good agreement
  with experiment
• Estimate of the lattice light velocity from the
  chiral condensate.

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QCD in the continuum
Gausss law
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Hamiltonian formulation oflattice QCD
The time is continuous and the space is
discretized in a 3-dimensional cubic lattice
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Staggered fermions
Free latticized Dirac equation
The degeneracy is removed reducing the number of
degrees of freedom by using a single component
field on each site of the lattice
(Susskind,Phys.Rev.D16,3031)
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Introducing the fields ?1,?2,?3,?4 we obtain the
equations of motion for fermion fields which for
long wavelenghts are equivalent to the
conventional Dirac equation
Two complete and independent Dirac fields (u and
d) can be found which exhaust the finite-energy
spectrum of the continuum theory
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Gauge field
We now require that the free Dirac Hamiltonian on
the lattice be invariant under SU(NC)
transformations
We introduce a gauge field defined on the lattice
link
The Hamiltonian for staggered fermions invariant
under SU(NC) transformations has the form
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Magnetic field Hamiltonian
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Strong-coupling
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Simmetries

• HE,Hq and HB are gauge invariant
• where Ga is the Generator of the Static
  Gauge Transformation
• The Hamiltonian is invariant under
• Traslation by a single link
• Shift along a face diagonal
• Shift along a large diagonal
• Parity
• G-parity

Discrete chiral symmetry
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The vacuum state energy

• The vacuum state 0gt must be
• A singlet of electric field algebra

• A gauge invariant state

• It must conserve charge

m of lattice sites
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Second order
Ea generates the left-action of the Lie algebra
on Ur,i
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After the integration over the link variables U
where ? has eigenvalues NC/2. HE has two
degenerate ground states (Antiferromagnetic
Ising model, spin NC/2)
Spontaneous breaking of chiral symmetry is
realized by choosing one of the two vacua
e empty f full
N of lattice links
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Large NC limit (t Hooft,1974)
Weak coupling
N.B. There is a correspondence between gauge and
string theories in the large NC limit
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The low-mass meson spectrum
In the strong-coupling limit the lowest lying
states are those consisting of a quark and an
antiquark at opposite ends of a single link
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Second order
Dependence on the lattice links
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Fourth order
The projection operator in the middle does not
allow patterns in which there are fermion
operators creating and destroying the same quark
at the same lattice site
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Meson masses
Meson masses have smooth t Hooft limit for large
NC and they go as a constant in this limit
After rescaling the coupling constant according
to the t Hooft prescription (large NC with g2NC
fixed)
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Lattice versus continuum
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Irrelevant operators
At the fourth order there is a divergence for
NC3. According to (Banks et al.) one can choose

• In the large NC limit the series for the meson
  masses are finite and do not depend on
  arbitrary parameter.

There is no need of irrelevant operators.
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Spontaneous breaking of chiral symmetry
In the staggered fermion formalism there is a
remnant of chiral symmetry on the lattice that is
the invariance of the theory under translation by
a single link
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QCD chiral condensate
t lattice light velocity
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Using the 0,2 Padé approximant one gets
Giusti et al. (1999) Giusti et al.
(2002) Hernandez et al. (2001)
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Values of the lattice light velocity, t,
obtained using different numerical determinations
of the lattice chiral condensate
Giusti et al. (1999)
Giusti et al. (2002)
Hernandez et al. (2001)
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Conclusions and outlook

• The large NC is very effective also in
  strong-coupling limit calculations
• The results are in very good agreement with
  experiments and can be obtained also without
  numerical computations. Computer calculations
  should give higher orders in the perturbative
  expansion
• The baryon masses can also be computed with this
  procedure
• The method could be extended to other types of
  lattice fermions.