1. Pion Physics at Finite Volume
Jie Hu, Fu-Jiun Jiang, Brian Tiburzi
Duke University
Lattice 2008, William & Mary, VA
July 16, 2008

2. Chiral Perturbation Theory
Lattice QCD
QCD
Systematic Errors • Quenching • Large pion mass • Volume effects • …
Chiral Perturbation Theory

3. 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:

4. 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. Fil’kov and V. L. Kashevarov, Phys. Rev. C 73, (2006)

5. 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, (2006)

6. J. F. Domoghue, E. Golowich, and B.R. Holstein
Chiral Lagrangian:
J. F. Domoghue, E. Golowich, and B.R. Holstein

7. 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

8. Pion Current Renormalizations at Finite Volume (ω=0)
J. H., F.-J. Jiang and B. C. Tiburzi, Phys. Lett. B653, 350 (2007)
• At infinite volume:
• At finite volume:

9. 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.

10. Charged Pion Current at Finite Volume
J. H., F.-J. Jiang and B. C. Tiburzi, Phys. Lett. B653, 350 (2007)

11. Compton Scattering at Zero Photon Energy:
• At infinite volume: T(ω=0) = 0 for π0
• At finite volume: ΔT(L)

12. 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.

13. 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)

14. 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 φ field coupled to zero frequency photons.
• The new couplings are determined from finite volume calculations and are exponentially small in asymptotically large volume.

15. 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?

16. 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.

17. 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.

18. Thank You.

19. Compton Tensor at Low Energy
• Typical diagrams to one-loop order

20. 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)

21. Other Experimental Data for Pion Polarizabilities
J.Ahrens et al. Eur. Phys. J. A23 (2005)

22. Pion Polarizabilities to One-loop
For both neutral and charged pions
B. R. Holstein, Comments Nucl. Part. Phys. 19, 221 (1990)

23. 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.

24. Finite Volume Corrections to Neutral Pion Compton Amplitude

25. Finite Volume Corrections to Charged Pion Compton Amplitude

26. Gasser, Ivanov and Sainio, Nulc. Phys. B745, 84 (2006)

27. Finite Volume ChPT • Cubic box of with periodic boundary condition.
• Matching terms

28. PQChPT Lagrangian

29. Gauge Invariance on a Torus

30. Effective Lagrangian at Finite Volume

31. Finite corrections to pion polarizabilities?
Matching terms

32. Polarizabilities:
• Atomic polarizabilities are well described theoretically
• Pion polarizabilities involve non-perturbative effects