Pion Physics at Finite Volume

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Pion Physics at Finite VolumPion Physics at Finite Volumee

Jie Hu, Jie Hu, Fu-Jiun Jiang, Brian TiburziFu-Jiun Jiang, Brian Tiburzi

Duke UniversityDuke University

Lattice 2008, William & Mary, VALattice 2008, William & Mary, VAJuly 16, 2008July 16, 2008

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QCDLattice QCD

Chiral Perturbation Theory

Systematic Errors • Quenching • Large pion mass • Volume effects • …

,L,a 2 nLp

,m

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Compton Scattering & Electromagnetic Compton Scattering & Electromagnetic Polarizabilities:Polarizabilities:

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

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Predictions from ChiPT:Predictions from ChiPT:

4 3( ) 5.7 1.0(10 )fm

4 32.7 0.4(10 )fm

•• One loop result: One loop result:

•• Two loop result: Two loop result:

4 3st sys model( ) 11.6 1.5 3.0 0.5 (10 )fm

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

Experimental:Experimental:

B. R. Holstein, Comments Nucl. Part. Phys. 19, 239 (1990)

J.Ahrens et al. Eur. Phys. J. A23 (2005)J.Ahrens et al. Eur. Phys. J. A23 (2005)

Dispersion relation calculation:Dispersion relation calculation: 2.6 4 31.9( ) 13.0 (10 )fm

L.V. Fil’kov and V. L. Kashevarov, Phys. Rev. C 73, 035210 (2006)L.V. Fil’kov and V. L. Kashevarov, Phys. Rev. C 73, 035210 (2006)

4 3( ) 0.16(10 )fm

0.11 4 30.02( ) 0.18 (10 )fm

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Can This be Calculated on Lattice?Can This be Calculated on Lattice?

'

,( ', ) { ( ) ( )}ik y ik x

x yT k k e H T J x J y H

•• Lattice four point function? Not now.Lattice four point function? Not now.

•• Background field method: measure energy shift in classical backgroBackground field method: measure energy shift in classical background electromagnetic fields.und electromagnetic fields.

Talk by A. Alexandru, C. Aubin, B. Tiburzi, S. MoerschbacherTalk by A. Alexandru, C. Aubin, B. Tiburzi, S. Moerschbacher

•• A lot of systematic errorsA lot of systematic errors

•• QuenchingQuenching

•• Large pion massLarge pion mass

•• Volume effectsVolume effects

•• ......

F.X. Lee, L. M. Zhou, W. Wilcox, and J. Christensen, Phys. Rev. D 73, 034503 (2006)

M

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Chiral Lagrangian:Chiral Lagrangian:

2 2

2 29 10

( ) ( )8 4

( ) ( ) ...

q q

f fL tr D D tr m m

ie F tr QD D QD D e F tr Q Q

[ , ]D ieA Q

0

0

1

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2

exp(2 / )i f

F A A

9 10Low energy constants: ,

J. F. Domoghue, E. Golowich, and B.R. HolsteinJ. F. Domoghue, E. Golowich, and B.R. Holstein

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Finite Volume ChPTFinite Volume ChPT

•• We choose time direction to be continuous and finite spatWe choose time direction to be continuous and finite spatial volume with periodic boundary condition. ial volume with periodic boundary condition. The momentum modes are . The momentum modes are .

•• Power counting:Power counting: •• Same Chiral lagrangian and same diagrams with Same Chiral lagrangian and same diagrams with

•• Observable X at finite volumeObservable X at finite volume

( ) ( ) ( )X L X X L

4 0

q

d q dq

3L2 n

pL

p m

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Pion Current Renormalizations at Finite VolPion Current Renormalizations at Finite Volume (ume (ωω=0=0))

2eP

•• At infinite volume: At infinite volume:

•• At finite volume: At finite volume:

J. H., F.-J. Jiang and B. C. Tiburzi, Phys. Lett. B653, 350 (2007)

20, ( )order p

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Ward-Takahashi Identity at Finite VolumeWard-Takahashi Identity at Finite Volume

•• Ward identity is not achieved at finite volume since pion momentum is not Ward identity is not achieved at finite volume since pion momentum is not differentiable at finite volume. differentiable at finite volume.

•• Ward-Takahashi identity is valid at finite volume. Ward-Takahashi identity is valid at finite volume.

k

k e p

p

p

p k

p k

p k


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Charged Pion Current at Finite VolumeCharged Pion Current at Finite Volume

20, ( )order p


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Compton Scattering at Zero Photon Compton Scattering at Zero Photon Energy:Energy:

•• At infinite volume: T(At infinite volume: T(ωω=0) = 0 for =0) = 0 for ππ00

•• At finite volume: At finite volume: ΔΔT(L)T(L)

2Q

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Ward-Takahashi Identity at Finite VolumeWard-Takahashi Identity at Finite Volume

•• Ward identity is not achieved at finite volume since pion momentum is not Ward identity is not achieved at finite volume since pion momentum is not differentiable at finite volume. differentiable at finite volume.

•• Ward-Takahashi identity is valid at finite volume. Ward-Takahashi identity is valid at finite volume. k


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Neutral Pion Compton Scattering at ZerNeutral Pion Compton Scattering at Zero Photon Energy at Finite Volumeo Photon Energy at Finite Volume

20, ( )order p

0 0


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Effective Lagrangian at Finite VolumeEffective Lagrangian at Finite Volume

•• Under gauge symmetry, write down the general form of tUnder gauge symmetry, write down the general form of the ultra low energy effective theory for a simple he ultra low energy effective theory for a simple φφ field co field coupled to zero frequency photons.upled to zero frequency photons.

•• The new couplings are determined from finite volume calThe new couplings are determined from finite volume calculations and are exponentially small in asymptotically larculations and are exponentially small in asymptotically large volume.ge volume.


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Compton Scattering at Finite Volume:Compton Scattering at Finite Volume:

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

0 0

2 * 2 *1 2( ) ... ( ) ( ' ) ( ) ( ' ') ( ) ...E MT L C L C L k k

•• The amplitude for a real photon to scatter off a pion in finThe amplitude for a real photon to scatter off a pion in finite volume? Volume corrections to pion polarizabilities?ite volume? Volume corrections to pion polarizabilities?

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• 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.

Volume Corrections to Pion Polarizabilities?Volume Corrections to Pion Polarizabilities?

L

L2 n

L

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SummarySummary

• 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 reno

rmalized 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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Compton Tensor at Low EnergyCompton Tensor at Low Energy

•• Typical diagrams to one-loop orderTypical diagrams to one-loop order

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The Most Recent Experimental The Most Recent Experimental MeasurementsMeasurements

•• MAMI at Mainz results:MAMI at Mainz results:

•• More new results?More new results?

COMPASS at CERN increased statistics for COMPASS at CERN increased statistics for

Jefferson lab has plans to measure pion polarizabilitiesJefferson lab has plans to measure pion polarizabilities

4 3st sys model( ) 11.6 1.5 3.0 0.5 (10 )fm


p n

' 'Z Z

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Other Experimental Data for Pion PolariOther Experimental Data for Pion Polarizabilitieszabilities


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Pion Polarizabilities to One-loopPion Polarizabilities to One-loop

For both neutral and charged pions

B. R. Holstein, Comments Nucl. Part. Phys. 19, 221 (1990)B. R. Holstein, Comments Nucl. Part. Phys. 19, 221 (1990)

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• 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.

Volume Corrections to Pion Polarizabilities?Volume Corrections to Pion Polarizabilities?

L

L

'r k k

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Finite Volume Corrections to Neutral PiFinite Volume Corrections to Neutral Pion Compton Amplitudeon Compton Amplitude

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Finite Volume Corrections to Charged Finite Volume Corrections to Charged Pion Compton AmplitudePion Compton Amplitude

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

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Finite Volume ChPTFinite Volume ChPT

•• Cubic box of with periodic boundary condition. Cubic box of with periodic boundary condition.

( ) ( ) ( )X L X X L •• Matching termsMatching terms ( )X L

4 0

q

d q dq

3L T

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PQChPT LagrangianPQChPT Lagrangian

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Gauge Invariance on a TorusGauge Invariance on a Torus

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Effective Lagrangian at Finite VolumeEffective Lagrangian at Finite Volume

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Finite corrections to pion polarizabilities?Finite corrections to pion polarizabilities?

• Matching termsMatching terms

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•• Atomic polarizabilities are well described theoreticallyAtomic polarizabilities are well described theoretically

•• Pion polarizabilities involve non-perturbative effectsPion polarizabilities involve non-perturbative effects

PolarizabilitiesPolarizabilities:

2

fsE N

m

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fsHE

e

Nm E

Pion Physics at Finite Volume

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