Delbrück scattering as a High Precision Vacuum Probe J. K. Koga1 and T. Hayakawa2

1Quantum Beam Science Directorate, Japan Atomic Energy Agency, Kizugawa, Kyoto 619-0215 Japan

2Quantum Beam Science Directorate, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195 Japan

From Fundamental Physics to Societal Applications at the

Embassy of France – Tokyo – Japan November 18 & 19, 2013

Delbrück Scattering? Scattering of a photon by Coulomb field of nucleus

L. Meitner, H. Kosters (and M. Delbrück),Z.Phys. 84 (1933) 137

Virtual electron-positron pairs

Lowest order theoretically calculated

Experimentally measured by square of amplitude of coherent sum of all processes

Atomic Rayleigh (R)

Nuclear Thomson (T)

Delbrück (D)

Coherent Nuclear Resonance (N)

e+

e-

Why Now? Activities on Delbrück scattering mostly stopped 1990’s

New high flux g-ray sources ELI–NP linear Compton back-scattering γ source1

maximum g energy 13.2 MeV

g-energy spread (FWHM) 10-3

total flux (ph/s) 1013

Almost 100% polarized

tunable in energy

possible to evaluate only Delbrück scattering component

Precision measurements possible muon anomalous magnetic moment →deviations from the standard model2

prompted higher precision theoretical calculations3

1D. Habs, et al., Nuclear Physics News 21, 23 (2011) 2G. W. Bennett et al., Phys. Rev. D, 73, 072003 (2006) 3T. Aoyama et al., Phys. Rev. Lett., 109, 111808 (2012)

Revival D. Habs, Phys. Rev. Lett. 108, 18402 (2012)

High intensity LCS γ-ray sources＋γ-ray spectroscopy leads to

precision measurement

How to Calculate? Feynman diagrams

• Lowest order diagrams

• k, k’ incoming and outgoing g

• i, j polarization

• x’s Coulomb field

D momentum transfer

• Due to the complex nature of the cross section calculations in the1980’s took over one solar year to perform for 128 points1and in the early 1990’s 40 minutes per point 2

1S. Turrini, G. Maino, and A. Ventura: Phys. Rev. C 39 (1989) 824. 2S. Kahane: Nucl. Phys. A542 (1992) 341.

Explicit expression

following notation of Cheng et al., Phys. Rev. D 26 (1982) 908

Explicit form of Traces 1 and 2 First two traces of Numerators after cancellations

Explicit form of Trace 3

Last trace of Numerators after cancellations

 

D0 =2p( )

4-D

ip 2

dD p

2p( )4

1

p2 - m0

2+ ie( ) p + q1( )

2- m1

2+ ie[ ] p + q2( )

2- m2

2+ ie[ ] p + q3( )

2- m3

2+ ie[ ]

ò

 

Dmnrs =2p( )

4-D

ip 2

dD p

2p( )4

pm pn pr ps

p2 - m0

2+ ie( ) p + q1( )

2- m1

2+ ie[ ] p + q2( )

2- m2

2+ ie[ ] p + q3( )

2- m3

2+ ie[ ]

ò

Integrals over p Tensor loop integrals

Scalar loop integrals

Dilogarithm functions

Numerical instabilities

Special techniques to evaluate accurately

G. J. van Oldenborgh and J. A. M. Vermaseren, Z. Phys. C 46 (1990) 425

 

Dm = pim D i

i ,=1

3

å

Dmn = gmn D00 + pim p jn D i j

i , j ,k,l =1

3

å

Dmnr = gmn pir + gnr pim + gmr pin( )D00i

i , j =1

3

å + pim p jn pkr D i jk

i , j ,k=1

3

å

Dmnrs = gmn grs + gmrgns + gms gnr( )D0000 + gmn pir p js + gnr pim p js + gmr pin p js + gns pim p jr + grs pim p jn( )D00 i j

i , j =1

3

å + pim p jn pkr pls D i jkl

i , j ,k,l =1

3

å

 

D0 =2pm( )

4-D

ip 2d Dq

1

q2 - m0

2+ ie( ) q + p1( )

2- m1

2+ ie[ ] q + p2( )

2- m2

2+ ie[ ] q + p3( )

2- m3

2+ ie[ ]

ò

LoopTools package* D-dimensional Scalar loop integrals

Tensor loop integral re-expressed

Coefficients calculated from Scalar loop integrals

http://www.feynarts.de/looptools/ *T. Hahn, Comput. Phys. Commun. 118 (1999) 153

A. Denner, Fortschr. Phys. 41 (1993) 307

 

D0000 D00i j D i jkl

Integral over q Multi-dimensional integration

CUBA library

Four algorithms

T. Hahn, Comput. Phys. Commun. 168 (2005) 78

Routine Integration method Algorithm type Variance reduction

Vegas Sobol quasi-random sample or Monte Carlo importance sampling

pseudo-random sample Monte Carlo

Suave Sobol quasi-random sample or Monte Carlo globally adaptive subdivision

pseudo-random sample Monte Carlo

Divonne Korobov quasi-random sample or Monte Carlo stratified sampling, aided by methods from numerical optimization

Sobol quasi-random sample or Monte Carlo

pseudo-random sample or Monte Carlo

cubature rules deterministic

Cuhre cubature rules deterministic globally adaptive subdivision

Scattering geometry

Scattering matrix of first trace

for 2.09 MeV photon at q=24.6o

|qx,qy,qz| < 10 MeV

Numerical instabilities

Also at other energies and angles

Under investigation

Preliminary results

 

M 0 k,k',q( )trace 1

 

qx

 

qy

 

qz

y

z

x

q

 

k

 

k '

 

ˆ e

 

ˆ e '^

Experimental Measurements→Need for Higher Order

M. Schumacher, Radiation Physics and Chemistry 56 (1999) 101

Z = 92

R+N+D

R+N

a R+N+D

b R+N+D+Coulomb correction

R+N

Higher order Feynman Diagrams (Coulomb corrections)

k → incoming photon

k’ → outgoing photon

x → Coulomb field k’ k

k’ k

x x

x x

x x

x x x

x

Higher order calculations:

1.Delbrück scattering in a strong

external field

• Scherdin, et. al, PRD 45 (1992)

2982

2.Coulomb corrections to Delbrück

scattering

• Scherdin, et. al, Z. Phys. A 353

(1995) 273

3.Next order

• projected number of terms is so

large, that it has not been

completed

Conclusions Delbrück scattering calculation

Necessary due to development of new g-ray sources high flux

almost 100% polarized

tunable

Delbrück scattering component only possible

For calculation take advantage of Packaged routines

Speed of computers

Preliminary results

Numerical instabilities need to be suppressed

Higher order corrections may be calculated via packages