Pion Physics at Finite Volume

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Pion Physics at Finite Volume

• Jie Hu, Fu-Jiun Jiang, Brian Tiburzi
• Duke University
• Lattice 2008, William Mary, VA
• July 16, 2008

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QCD
Lattice QCD
Systematic Errors Quenching Large
pion mass Volume effects
Chiral Perturbation Theory
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Compton Scattering Electromagnetic
Polarizabilities

• At low energy the infinite volume Compton
  scattering amplitude for a real photon to scatter
  off a pion can be parametrized as

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Predictions from ChiPT

• One loop result
• Two loop result

B. R. Holstein, Comments Nucl. Part. Phys. 19,
239 (1990)
Gasser, Ivanov and Sainio, Nulc. Phys. B745, 84
(2006)
?
Experimental
J.Ahrens et al. Eur. Phys. J. A23 (2005)
Dispersion relation calculation
L.V. Filkov and V. L. Kashevarov, Phys. Rev. C
73, 035210 (2006)
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Can This be Calculated on Lattice?
Lattice four point function? Not now.
Background field method measure energy shift in
classical background electromagnetic fields.
Talk by A. Alexandru, C. Aubin, B. Tiburzi, S.
Moerschbacher A lot of systematic errors
Quenching Large pion mass Volume
effects ...
F.X. Lee, L. M. Zhou, W. Wilcox, and J.
Christensen, Phys. Rev. D 73, 034503 (2006)
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Chiral Lagrangian
J. F. Domoghue, E. Golowich, and B.R. Holstein
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Finite Volume ChPT

• We choose time direction to be continuous and
  finite spatial volume with periodic boundary
  condition.
• The momentum modes are .
• Power counting
• Same Chiral lagrangian and same diagrams with
• Observable X at finite volume

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Pion Current Renormalizations at Finite Volume
(?0)
J. H., F.-J. Jiang and B. C. Tiburzi, Phys. Lett.
B653, 350 (2007)
At infinite volume
At finite volume
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Ward-Takahashi Identity at Finite Volume
J. H., F.-J. Jiang and B. C. Tiburzi, Phys. Lett.
B653, 350 (2007)
Ward-Takahashi identity is valid at finite
volume.
Ward identity is not achieved at finite volume
since pion momentum is not differentiable at
finite volume.
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Charged Pion Current at Finite Volume
J. H., F.-J. Jiang and B. C. Tiburzi, Phys. Lett.
B653, 350 (2007)
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Compton Scattering at Zero Photon Energy
At infinite volume T(?0) 0 for p0
At finite volume ?T(L)
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Ward-Takahashi Identity at Finite Volume
J. H., F.-J. Jiang and B. C. Tiburzi, Phys. Lett.
B653, 350 (2007)
Ward-Takahashi identity is valid at finite
volume.
Ward identity is not achieved at finite volume
since pion momentum is not differentiable at
finite volume.
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Neutral Pion Compton Scattering at Zero Photon
Energy at Finite Volume
J. H., F.-J. Jiang and B. C. Tiburzi, Phys. Lett.
B653, 350 (2007)
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Effective Lagrangian at Finite Volume
J. H., F.-J. Jiang and B. C. Tiburzi, Phys. Lett.
B653, 350 (2007)

• Under gauge symmetry, write down the general
  form of the ultra low energy effective theory for
  a simple f field coupled to zero frequency
  photons.
• The new couplings are determined from finite
  volume calculations and are exponentially small
  in asymptotically large volume.

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Compton Scattering at Finite Volume

• The amplitude for a real photon to scatter off
  a pion in infinite volume

The amplitude for a real photon to scatter off
a pion in finite volume? Volume corrections to
pion polarizabilities?
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Volume Corrections to Pion Polarizabilities?

• All terms are form factors in . Because of
  momentum quantization , these form
  factors cannot be expanded in for the
  smallest modes.
• Thus we cannot use the finite volume Compton
  tensor to deduce finite volume corrections to
  polarizabilities.

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Summary

• Recent discrepancies between ChPT measurements
  of pion polarizabilities motivate lattice
  calculations.
• Use ChPT to get finite volume effects.
• Demonstrate that the conserved current are
  additively renormalized at finite volume.
• Ward-Takahashi identity is valid for all volume.
• Single particle effective theory is derived.
• Finite volume corrections to the Compton
  scattering tensor of pions are determined.
• Finite volume effects to polarizabilities are not
  achieved.

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Thank You.
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Compton Tensor at Low Energy

• Typical diagrams to one-loop order

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The Most Recent Experimental Measurements

• MAMI at Mainz results
• More new results?
• COMPASS at CERN increased statistics for
• Jefferson lab has plans to measure pion
  polarizabilities

J.Ahrens et al. Eur. Phys. J. A23 (2005)
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Other Experimental Data for Pion Polarizabilities
J.Ahrens et al. Eur. Phys. J. A23 (2005)
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Pion Polarizabilities to One-loop
For both neutral and charged pions
B. R. Holstein, Comments Nucl. Part. Phys. 19,
221 (1990)
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Volume Corrections to Pion Polarizabilities?

• All terms are form factors in . Because of
  momentum quantization, these form factors cannot
  be expanded in for the smallest modes.
• Thus we cannot use the finite volume Compton
  tensor to deduce finite volume corrections to
  polarizabilities.

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Finite Volume Corrections to Neutral Pion Compton
Amplitude
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Finite Volume Corrections to Charged Pion Compton
Amplitude
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Gasser, Ivanov and Sainio, Nulc. Phys. B745, 84
(2006)
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Finite Volume ChPT

• Cubic box of with periodic boundary
  condition.

Matching terms
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PQChPT Lagrangian
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Gauge Invariance on a Torus
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Effective Lagrangian at Finite Volume
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Finite corrections to pion polarizabilities?

• Matching terms

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Polarizabilities

• Atomic polarizabilities are well described
  theoretically
• Pion polarizabilities involve non-perturbative
  effects