Progress towards calcula/ng higher

order Delbrück sca7ering and

prospects for measurements

J. K. Koga1 and T. Hayakawa2

1Na/onal Ins/tutes for Quantum and Radiological Science and Technology, Kizugawa,

Kyoto 619-0215, JAPAN2Na/onal Ins/tutes for Quantum and Radiological Science and Technology, Tokai,

Ibaraki 319-1195, JAPAN

Nuclear Photonics 2018, June 24-29, 2018, Brasov, Romania

Measuring Vacuum for new physicse+

e-

Low Energy Precision measurements →devia;ons from the Standard Model1. Muon anomalous magne;c moment G. W. BenneC et al., PRD, 73, 072003 (2006)Precise theore;cal calcula;onsT. Aoyama et al., PRL 109, 111808 (2012)2. 8Be AnomalyKrasznahorkay, et al., PRL 116, 042501 (2016)J. L. Feng et al., PRL 117, 071803 (2016)

ΔEΔt ~ !

High Energy

ATLAS detector (LHC) heavy-ion collisions photon-photon scattering so far agrees with Theory 5.02 TeV: 13 candidate events, Nat. Physics 2017

Studied Delbrück scattering

J. K. Koga and T. Hayakawa, PRL 118 (2017) 204801

Delbrück Sca+ering

• Sca+ering of a photon by Coulomb field of nucleus– L. Meitner, H. Kösters (and M.

Delbrück),Z.Phys. 84 (1933) 137

• Virtual electron-positron pairs• Lowest order theorePcally

calculated

• Experimentally measured• Most research stopped 1990’s

Advantages

• Cross section scales as (Za)4• Previous sources unpolarized• New Polarized Laser Compton scattering sources (LCS)• ELI–NP Gamma Beam System (GBS)1

– g energy 0.2-20 MeV– Bandwidth (rms) < 0.5%– Linear polarization >99%– 8.3 x 108 photons/s

• Energy Recovery Linac (ERL) LCS2– g energy 1-2 MeV– Total flux photons/s ~1013

• NESTOR3– g energy 6-900 keV– photons/s ~1013

• UVSOR-III (See Poster by Hayakawa et al. P.25)– g energy Less than 1.022 MeV

• Precision measurements possible

1D. Filipescu et al., Eur. Phys. J. A (2015) 51: 1852R. Hajima, et al., NIMA 608, S57 (2009)3V. Androsov, et al., NIMA 543, 58 (2005)

• 4 COHERENT CONTRIBUTIONS TO THE ELASTIC SCATTERING– atomic Rayleigh (R)– nuclear Thomson (T)– Giant Dipole Resonance (GDR)– Delbrück (D)

Disadvantage

€

A||⊥

= A||⊥

R + A||⊥

T + A||⊥

D + A||⊥

GDR

Total coherent elastic scattering amplitude

€

dσdΩ

=12A⊥

2+ A||

2( )

Differential cross section(unpolarized photons)

COHERENT SUM

How to Isolate Delbrück?

• Look where other components small• nuclear Thomson (T)

• Giant Dipole Resonance (GDR)B. Kasten et al., Phys. Rev. C 33, 1606 (1986)

€

A⊥T = −

Z 2e2

Mc2(1− 1

3k2 r2 )

rigid spin-zero nucleus, charge radius r, photon wave number k

€

Eν ,σν ,ΓνGiant Dipole Resonance Lorentz parameters

At 90o GDR,T → 0 for polarizaEons parallel to scaFering plane

De Tollis et al., Il Nuovo Cim A Series 11 32, 227 (1976)

P. Rullhusen et al., Phys. Rev. C 23, 1375 (1981)

AGDR? =E2

4⇡~c

2X

⌫=1

�⌫�⌫E2⌫ � E2 + iE�⌫

(E2⌫ � E2)2 + E2�2⌫

AT,GDRk = AT,GDR? cos ✓

q

€

A||

€

! k

€

A⊥

Delbrück (D)

• Lowest order Feynman diagrams• k, k’ incoming and outgoing g• i, j polariza>on• x’s Coulomb field• D momentum transfer• Complex calcula>on

• 1980’s took over one solar year to perform for 128 points1

• 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) 341.3Kirilin and Terekhov PRA 77 032118 (2008)

We use the formulae obtained by:B. De Tollis, M. Lusignoli, and G. Pistoni, Il Nuovo Cimento A Series 11 32, 227 (1976)B. De Tollis and G. Pistoni, Il Nuovo Cimento A Series 11 42, 499 (1977) B. De Tollis and L. Luminari, Il Nuovo Cimento A Series 11 81, 633 (1984).

Differential Cross section

Linear polarization(perpendicular/parallel to scattering plane)

Circular polarization

Real Part

Formulae in:B. De Tollis, M. Lusignoli, and G. Pistoni, Il Nuovo Cimento A Series 11 32, 227 (1976)B. De Tollis and G. Pistoni, Il Nuovo Cimento A Series 11 42, 499 (1977) B. De Tollis and L. Luminari, Il Nuovo Cimento A Series 11 81, 633 (1984).

For photon Energy > 1.022 MeV

Imaginary Part

Formulae in:B. De Tollis, M. Lusignoli, and G. Pistoni, Il Nuovo Cimento A Series 11 32, 227 (1976)B. De Tollis and G. Pistoni, Il Nuovo Cimento A Series 11 42, 499 (1977) B. De Tollis and L. Luminari, Il Nuovo Cimento A Series 11 81, 633 (1984).

Confirmed by De Tollis and C- normal integral De Tollis priv. comm.

D± ! =(a+++�

(d, p))

For less than 1.022 MeV found that Imaginary part can be used

Calcula&on

• For large Z Higher order (Coulomb) correc&ons become large– Z=40

Differen'al cross sec'on for Tin

from J. K. Koga and T. Hayakawa, PRL 118 (2017) 204801

R+T+GDR Minimum near ~70o

Eg=1.1 MeV

q

€

A||

€

! k

€

A⊥

Next-to-Leading-order• Higher order (Coulomb) correc9ons• lowest order (Za)2

• Higher order (Za)2n n=2,3,4 not done• Even though Experimental Data

shows it [1]• Only Empirical Formulae [1]• All orders done for E

Automa'c Calcula'on

• Particle Physics high precision experiments– Next-to-Leading-Order (NLO), Next-to-next-to-

Leading-Order (NNLO) calculations needed• Integrals have divergences• Various techniques to handle them:– Dimensional regularization, Sector Decomposition,

…• Automatic calculation packages created

Sector Decomposi-on1-4

I =

Z 1

0dx x�1�(a+b)✏

Z 1

0dt t�b✏(1 + (1� x)t)�1

+

Z 1

0dy y�1�(a+b)✏

Z 1

0dt t1�a✏(1 + (1� y)t)�1

I =

Z 1

0dx

Z 1

0dy x�1�a✏y�b✏(x+ (1� x)y)�1[⇥(x� y) +⇥(y � x)]

I =

Z 1

0dx

Z 1

0dy x�1�a✏y�b✏(x+ (1� x)y)�1

1.K.Hepp,Comm.Math.Phys.2(1966)301–326.2.M. Roth, A. Denner, Nuclear Phys. B 479 (1996) 495–5143.T. Binoth, G. Heinrich, Nuclear Phys. B 585 (2000) 741–759. 4.G.Heinrich,Internat.J.ModernPhys.A23(2008)1457–1486.

x

y

x

t

t

y

+

pySecDec 1

• Automa.c calcula.on• Python modules and FORM2 for Algebraic

manipula.on• C++ code generated for numerical calcula.on• CUBA library of Integral Evalua.on rou.nes3

1. S. Borowka et al., Computer Physics Communications 222 (2018) 3132. J.A.M.Vermaseren, arXiv:math-ph/0010025. J.Kuipers,et

al. ,Comput.Phys.Comm.189(2015)13. T.Hahn,Comput.Phys.Comm.168(2005)78.

T.Hahn,J.Phys.Conf.Ser.608(1)(2015)012066

Progress so far • Program is running• Can run examples• Next step: Reproduce previous Delbrück

calcula=onsp1

p2

p3

p4

m

0

0

0

p1

p2

p3

p4

0

0

0

0

p1

p2

0 0

Conclusions

• Measuring the vacuum precisely: possible new physics• Photons useful• Delbrück scattering: large cross section• Nearly Isolated: linear polarization in scattering plane,

specific angles

• Larger Z need higher order calculations• Next-to-leading-order using Particle Physics package:

PySecDec

• Next Step: lowest order Delbrück then next order

For details see :

J. K. Koga and T. Hayakawa, PRL 118 (2017) 204801