Pion Physics at Finite Volume

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Nov 16, 2013 (3 years and 11 months ago)

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1

Pion Physics at Finite Volume

Jie Hu,
Fu
-
Jiun Jiang, Brian Tiburzi


Duke University


Lattice 2008, William & Mary, VA

July 16, 2008

2

QCD

Lattice QCD

Chiral
Perturbation
Theory

Systematic Errors




Quenching




Large pion mass




Volume effects






,
L
,
a
2
n
L
p




,
m
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:


4 3
( ) 5.7 1.0(10 )
fm

 


  
4 3
2.7 0.4(10 )
fm
 
 
 

  



One loop result:






Two loop result:

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

 


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

Experimental:


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

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

Dispersion relation calculation:


2.6 4 3
1.9
( ) 13.0 (10 )
fm

 

 

 
L.V. Fil

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

4 3
( ) 0.16(10 )
fm

 


 
0.11 4 3
0.02
( ) 0.18 (10 )
fm

 

 

 
5

Can This be Calculated on Lattice?

'
,
(',) { ( ) ( )}
ik y ik x
x y
T k k e H T J x J y H
  
  




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)

M

6

Chiral Lagrangian:

2 2
2 2
9 10
( ) ( )
8 4
( ) ( )...
q q
f f
L tr D D tr m m
ie F tr QD D QD D e F tr Q Q


   


 
 
  
     
         
[,]
D ieA Q
  
    
0
0
1
2
1
2
 
 


 
 
 

 

 
 
exp(2/)
i f
  
F A A
    
  
9 10
Low energy constants: ,

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


( ) ( ) ( )
X L X X L
  
4 0
q
d q dq






3
L
2
n
p
L


p m
 
 
8

Pion Current Renormalizations at Finite
Volume (
ω
=0
)






2
eP




At infinite volume:





At finite volume:


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

2
0,( )
order p


9

Ward
-
Takahashi Identity at Finite Volume




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




Ward
-
Takahashi identity is valid at finite volume.


k
k


e






p
p
p
p k

p k

p k


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

10

Charged Pion Current at Finite Volume

2
0,( )
order p







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
π
0





At finite volume:
Δ
T(L)


2
Q

12

Ward
-
Takahashi Identity at Finite Volume




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




Ward
-
Takahashi identity is valid at finite volume.


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

13

Neutral Pion Compton Scattering at
Zero Photon Energy at Finite Volume

2
0,( )
order p


0

0



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

14

Effective Lagrangian at Finite Volume



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.

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

15

Compton Scattering at Finite Volume:




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

0

0



2 * 2 *
1 2
( )...( ) (') ( ) ('') ( )...
E M
T L C L C L k k

     
          



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

16



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?

L

L

2
n
L



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

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

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

 


    
J.Ahrens et al. Eur. Phys. J. A23 (2005)

p n
 


''
Z Z
 

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



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?

L

L

'
r k k
 
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.

( ) ( ) ( )
X L X X L
  



Matching terms

( )
X L

4 0
q
d q dq






3
L T

28

PQChPT Lagrangian

29

Gauge Invariance on a Torus

30

Effective Lagrangian at Finite Volume

31

Finite corrections to pion polarizabilities?


Matching terms

32




Atomic polarizabilities are well described theoretically








Pion polarizabilities involve non
-
perturbative effects

Polarizabilities
:

2
fs
E
N
m

 




2
0
fs
H
E
e
N
m E