Lattice results on QCDstrings

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Lattice results on QCD-strings

• N.D. Hari Dass
• Institute of Mathematical Sciences
• Chennai
• In collaboration with Dr Pushan Majumdar, U.
  Muenster

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Understanding nuclear forces

• The challenge in the beginning was to understand
  the structure and stability of atomic nucleus.
• Initially it was thought that protons, neutrons
  and pions were the elementary constituents and
  that the exchange of pions between protons and
  neutrons gave rise to the attractive nuclear
  forces.

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Proliferation of particles.

• But with higher and higher energy collisions more
  and more particles other than the protons,
  neutrons and pions were produced.
• The simple picture of protons, neutrons and pions
  as the elementary constituents of nuclear matter
  was not viable.
• Two proposals came to be made almost concurrently
  to face the situation.

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The hadronic string

• Chew and Frautschi observed that all the new
  particles produced lay on parallel lines (Regge
  trajectories) when their masses(or squared
  masses) were plotted against their angular
  momenta.
• Veneziano proposed a simple formula for the
  scattering amplitudes for the new particles.
• Nambu, Nielsen and Susskind made the astonishing
  proposal that both the Veneziano formula and the
  Regge trajectories could be understood if the
  particles were states of excitation of a
  relativistic string.

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Problems with hadronic strings

• The full consistency of the hadronic string model
  required space-time to have 10 dimensions.
• Closer inspection of the spectrum of these
  strings revealed states that could not be
  hadronic states e.g massless spin-2 states.
• Such states would violate the so called Froissart
  bound for scattering amplitudes.
• Subsequently it was proposed not to use string
  theories to explain hadronic physics but as a
  means of unifying all interactions including
  gravity.

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Quark model

• At around the same time Gellmann and Zweig
  independently proposed the quark model.
• According to this protons are constituted by 3
  quarks, 2 u-quarks of 2/3 electric charge and
  one d-quark of -1/3 charge. Likewise neutron
  constituents were 2 d-quarks and one u-quark.
• Pions were made up of a quark and anti-quark
  pair.
• The quarks were taken to be spin ½ particles.

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Quark model..

• To overcome problems with Pauli exclusion
  principle, Nambu proposed that each quark comes
  with 3 different colours .
• This model was extremely succesful as
  book-keeping for the myriad of particles and
  could even explain some features like the
  observed magnetic moments.
• Attention then turned to constructing a theory
  for the interaction between quarks.
• Gellmann, Minkowski and Fritzsch proposed QCD
  with vector interactions motivated by the
  observed chiral symmetry among hadrons.

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What is QCD? More technically

• QCD is a non-Abelian Gauge Theory.
• The gauge group is SU(3).
• The Quarks carry the fundamental representation
  3.
• The Gluons, which transmit the forces between
  quarks, carry the adjoint representation 8.
• 3 has triality, 8 has no triality.
• The theory in its nonperturbative domain has
  defied analytical approach despite the best
  efforts for more than 25 years.
• It is an outstanding problem of theoretical
  physics.

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Quarks and Gluons of SU(3) QCD
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Quark Confinement

• Quarks are not liberated even in very high energy
  collisions!
• The theory of Quarks must be such that they can
  never be free!

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The dual superconductor idea
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Lattice
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Lattice QCD
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Wilson Loop
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Numerical Evidence For Flux Tubes
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Our Simulations

• We measure the qqbar-potential of d4 su(3) pure
  gauge theory extremely accurately on 243x32 and
  324 lattices at b 5.7
• Lattice spacing a 1/6 fm so temporal extent is
  nearly 5.3 fm while spatial extent is 4fm3.
• We use Polyakov Loop Correlation Function to
  measure the potential.
• We use the Luscher-Weisz Multilevel algorithm.
• We also use the analytical multihit method to
  achieve speedup(60).

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Type of accuracies needed

• The polyakov loop correlator is a stochastic
  variable of nearly unit magnitude.
• At a separation of r8 the average value of this
  is around 10 -26
• It roughly falls by two orders of magnitude with
  every increase in r by unity.
• This too needs to be measured at fraction of a
  percent if string-like behaviour is to be
  extracted.
• Without the multihit and multilevel methods this
  would be a hopeless task.

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Polyakov Loop Correlators
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Luscher-Weisz Multilevel Algorithm
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Some string actions and potentials
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Results of Luscher and Weisz
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Initial Conclusions of Luscher and Weisz

• The coefficient of the 1/r term in the qqbar
  potential agrees to within 15 from that of the
  free bosonic string theory in the distance range
  0.5-1.0 fm.
• They argued that the discrepancy could be due to
  boundary terms and other interaction terms.
• They found that a boundary term with b.04 could
  accommadate the data well.
• They found it surprising that even at 0.5fm there
  was evidence for string behaviour.

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The Work of Kuti et al

• Kuti et al have undertaken very detailed studies
  of the spectrum of string excitations.
• They use the extended Wilson loops for this
  purpose.
• They use 242x30x60 lattice with as 0.2 fm and
  at .04 fm so that their longest time extent is
  2.4 fm.

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Wilson Loops for Spectrum Studies
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Results of Kuti et al
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Spectrum according to Kuti et al
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Summary of Results of Kuti et al

• The ordering of the spectrum does not agree with
  that of the bosonic string even upto nearly 3fm.
• There is a systematic overshooting of the string
  results as one goes to larger and larger
  distances.
• The data fits the free bosonic string ground
  state energy very well in the range 0.5-1.0 fm.

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Our Simulation Results
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Force vs Distance
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Summary of Our Results

• We do not see any obvious convergence to either
  Nambu-Goto or free bosonic string to even upto
  0.75 fm.
• As r approaches 1 fm we see clear convergence to
  the Nambu-Goto potential.
• We see very clear discrepancy with the free
  bosonic string expectations. In fact a fit to
  this behaviour produces very poor c2

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Luscher-weisz Revisited

• How then do we reconcile our conclusions with
  those of Luscher-Weisz who had suggested onset of
  string behaviour as early as 0.5 fm by ascribing
  the differences over free string behaviour to
  boundary terms?
• In 2004 they showed that Open String Closed
  String duality forbids boundary terms of the type
  considered before.
• It is better to interpret the results as string
  behaviour not setting so soon.

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But Nambu-Goto string is inconsistent in d4

• Though the data picks the Nambu-Goto potential so
  accurately, there is a serious problem!
• This string theory can be consistently quantised
  only in d26
• The Luscher term is a consequence of such
  quantisations, so why should one place too much
  weight on its occurrence?

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Effective string theories

• Polchinski and Strominger in 1992 set out to
  formulate effective string theories and their
  quantisations.
• This is similar in spirit to effective theories
  like chiral models for the description of pions.
• Non-polynomial, non-renormalisable.
• Designed so that quantisation preserves Lorentz
  Invariance.
• Leading correction to the linear potential is
  again exactly the old Luscher term.
• What are the corrections and how well does data
  fit them? Tachyons?

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Are string and gauge theories dual to each

• Polyakov and others believe that gauge and string
  descriptions are dual to each other.
• This is also the content of Maldacenas AdS/CFT
  correspondence.
• This is expected to hold even for
  non-supersymmetric theories like QCD.
• Satchi Naik (HRI) showed how all these theories
  give the conventional Luscher term.
• The d3 case is more straightforward in this
  picture.

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What Next?

• Push the simulations to larger and larger
  distances.
• Very time consuming! With each dr 1 the
  simulation time increases by 2 and already at r7
  it takes a day for 3 meas and for good statistics
  one needs about 500 meas!
• Probe effective string theories as well as
  AdS/CFT correspondence deeper.
• Extrinsic curvature strings.
• Investigate the Center issue investigate adjoint
  strings (large memory and large simulation time).
• Investigate Center-less groups like SO(3), G2 etc
• Investigate z(3) gauge theories.
• Investigate baryons and effective string
  coupling. Construct an effective QCD String
  Theory.
• The eventual goal is solving the Quark
  Confinement Problem