Radiative pion capture in 2 H, 3 He and 3 H J. Golak , R. Skibiski, - - PowerPoint PPT Presentation

Radiative pion capture in 2H, 3He and 3H

• J. Golak,
• R. Skibiński, K. Topolnicki, H. Witała, A. Grassi,
• H. Kamada, A. Nogga, L.E. Marcucci

JAGIELLONIAN UNIVERSITY

15th International Workshop on Meson Physics KRAKÓW, POLAND, 7th - 12th June 2018

Outline



Introduction: elements of formalism



Radiative pion capture on 2H, 3He and 3H



Conclusions and outlook

MESON2018, Cracow, 11 June 2018

A very efficient momentum space framework to deal with nucleon-nucleon scattering, nucleon-deuteron scattering and many electroweak processes has been constructed and tested:

• Phys. Rept. 274, 107 (1996); Phys. Rept. 415, 89 (2005);
• Eur. Phys. J. A24, 31 (2005)

Limitations: nonrelativistic character and lack of the Coulomb force in the 3N continuum Calculations performed with semi-phenomenological 2N and 3N potentials: Bonn B, AV18, Nijmegen I and II, CD Bonn, Urbana IX, older chiral potentials from the Bonn/Bochum group and recently with the improved chiral potentials from E. Epelbaum et al.

Introduction

MESON2018, Cracow, 11 June 2018

Methods developed originally for elastic and inelastic electron scattering, photodisintegration, and applied later to neutrino induced reactions and muon capture are now used to investigate the following processes

n n n n n p e n d e e n n d                     

    

          H H H H H

3 3 3 3 3

Introduction

MESON2018, Cracow, 11 June 2018

These processes combine information from several areas (pion absorption ↔ pion photoproduction, weak processes, nuclear interactions) and should be ultimately studied within ChEFT

Introduction

General strategy in the few-nucleon systems:

• use (consistent) dynamical ingredients (2N and 3N potentials,

electroweak current operators)

• solve the dynamical equations (Schrödinger equation, Lippmann-

Schwinger equation, Faddeev equations)

• give predictions for nuclear structure and reaction observables
• confront results of theoretical calculations with experimental data to

improve your input

MESON2018, Cracow, 11 June 2018

Presented here results have been calculated with the AV18 2N and Urbana IX 3N potentials

Formalism

MESON2018, Cracow, 11 June 2018

initial nuclear system final nuclear system external field nuclear current

• perator

i f

j N   

 

crucial dynamical quantity interaction of nuclear system with external probe

Pion capture from the lowest K-shell of the pionic atom followed by photon emission studied with impulse approximation:

2 ' ) ' ( ) ( ) (

2 2 1 ' 3 100

m Z E m m m m m e m Z r r

Z Z r m Z K

    

  

      



negligible for Z=1,2 when compared to the pion or nucleon mass reduced mass

Formalism

p

n





MESON2018, Cracow, 11 June 2018

i A f fi

j T       

final photon polarization vector



 



A i A

i i j

1

) ( ) (    

nuclear axial current

Radiative pion capture on 2H

MESON2018, Cracow, 11 June 2018

n n d    



 

kinematically allowed region 

p 

1

p 

2

p 

Radiative pion capture on 2H

MESON2018, Cracow, 11 June 2018

n n d    



 



p 

1

p 

2

p 

p



best way to get Γnn from dΓnn /dEγ

Radiative pion capture on 2H

MESON2018, Cracow, 11 June 2018

n n d    



 

Γnn = 0.318 × 1015 1/s (PW) Γnn = 0.328 × 1015 1/s (FSI) FSI PW FSI PW similar results but completely different physics !

Radiative pion capture on 2H

MESON2018, Cracow, 11 June 2018

n n d    



 

this contribution: 3.28 × 1014 1/s Earlier theoretical predictions for Γnn:

• A. Reitan, Nucl. Phys. 87, 232 (1966):

3.32 × 1014 1/s → 4 × 1014 1/s (corrected by ST)

• M. Sotona and E. Truhlik, Nucl. Phys. A262, 400 (1976):

3.75 × 1014 1/s (based on pion photoproduction data) 3.83 × 1014 1/s (based on soft-pion limit+ corrections)

• W. R. Gibbs, B. F. Gibson, and Q. J. Stephenson, Jr.,
• Phys. Rev. C16, 327 (1977); 17, 856 (1978) (E)

(4.2 ± 0.5) × 1014 1/s

Radiative pion capture on 2H

MESON2018, Cracow, 11 June 2018

n n d    



 



p 

1

p 

2

p 

1 



another way to get Γnn from neutron spectrum

Radiative pion capture on 2H

MESON2018, Cracow, 11 June 2018

n n d    



 

FSI QFS

) /(

1 1 5

dE d d d

nn

  



ann= -21.8 fm ann= -18.8 fm ann= -16.5 fm neutron-neutron potential is changed by 1 %

• nly in the 1S0

channel



179

1  



neutron energy spectra for

Radiative pion capture on 2H

MESON2018, Cracow, 11 June 2018

n n d    



 

FSI QFS



179

1  



neutron TOF spectra for normalized at QFS peak for s= 2.55 m neutron-neutron potential is changed by 1 %

• nly in the 1S0

channel

) /(

1 1 5

dt d d d

nn

  



ann= -21.8 fm ann= -21.8 fm ann= -21.8 fm

Radiative pion capture on 3He: triton channel

MESON2018, Cracow, 11 June 2018

H He

3 3

  



 

Γ3H = 2.059 × 1015 1/s (2NF) Γ3H = 2.132 × 1015 1/s (2NF+3NF) 

p 

H

p3 

two-body kinematics

 

H

p p

3

 



135.7 MeV/c

Radiative pion capture on 3He: triton channel

MESON2018, Cracow, 11 June 2018

this contribution: 2.132 × 1015 1/s Earlier theoretical predictions for Γ3H

• Fujii and D. Hall, Nucl. Phys. 32, 102 (1962).

(8.32 → 4.28) × 1015 1/s (corrected by Truöl 1974)

• P. Divakaran, Phys. Rev. 139, 3887 (1965).

(0.97 → 3.88) × 1015 1/s (corrected by Truöl 1974)

• D. Griffiths and C. Kim, Phys. Rev. 173, 1584(1968)

2.32 × 1015 1/s

• P. Pascual and A. Fujii, Nuovo Cimento 65, 411 (1970)

(3.37 → 2.25) × 1015 1/s (corrected by Truöl 1974)

• P. Truöl et al., Phys. Rev. Lett. 32, 1268 (1974)

3.60 × 1015 1/s

• A. C. Phillips and F. Roig, Nucl. Phys. A234, 378 (1974)

(3.1 - 3.7) × 1015 1/s

• W. R. Gibbs et al., Phys. Rev. C18, 1761 (1978)

3.30 × 1015 1/s

n d e    



  H

3

Radiative pion capture on 3He: two-body breakup

MESON2018, Cracow, 11 June 2018

kinematically allowed regions 

p 

n

p 

d

p 

Radiative pion capture on 3He: two-body breakup

MESON2018, Cracow, 11 June 2018

Γnd = 5.201 × 1015 1/s (PWIAS) Γnd = 2.013 × 1015 1/s (FSI 2NF) Γnd = 1.840 × 1015 1/s (FSI 2NF+3NF ) PWIAS FSI 2NF FSI 2NF+3NF

n d e    



  H

3

crucial importance

• f FSI !

Radiative pion capture on 3He: three-body breakup

MESON2018, Cracow, 11 June 2018

n n p e     



  H

3



p 

3

p 

2

p 

1

p 

kinematically allowed region

Radiative pion capture on 3He: three-body breakup

MESON2018, Cracow, 11 June 2018

Γnnp = 3.816 × 1015 1/s (PWIAS) Γnnp = 0.659 × 1015 1/s (FSI 2NF) Γnnp = 0.615 × 1015 1/s (FSI 2NF+3NF ) PWIAS FSI 2NF FSI 2NF+3NF

n n p e     



  H

3

crucial importance

• f FSI !

Radiative pion capture on 3He: comparison of two- and three-body breakup with best dynamics

MESON2018, Cracow, 11 June 2018

Γnd+nnp = 9.017 × 1015 1/s (PWIAS) Γnd+nnp = 2.672 × 1015 1/s (FSI 2NF) Γnd+nnp = 2.455 × 1015 1/s (FSI 2NF+3NF) nd nnp nd+nnp Γnnp = 0.615 × 1015 1/s Γnd = 1.840 × 1015 1/s Γnd+nnp = 2.455 × 1015 1/s crucial importance

• f FSI !

two-body breakup dominates !

Radiative pion capture on 3He: other predictions and data for breakup channels

MESON2018, Cracow, 11 June 2018

nd nnp nd+nnp

• A. C. Phillips and F. Roig,
• Nucl. Phys. A234, 378 (1974).

Γnd+nnp /Γ3H=1.2 THEORY: Γnd+nnp /Γ3H in (0.84 -1.27)

• P. Truöl et al., Phys. Rev. Lett. 32, 1268 (1974)

EXPERIMENT: (1.12 ± 0.05)

Radiative pion capture on 3H: three-neutron breakup

MESON2018, Cracow, 11 June 2018



p 

3

p 

2

p 

1

p 

kinematically allowed region

n n n     



  H

3

Radiative pion capture on 3H: three-neutron breakup

MESON2018, Cracow, 11 June 2018

Γnnn = 0.117 × 1015 1/s (PWIAS) Γnnn = 0.141 × 1015 1/s (FSI 2NF) Γnnn = 0.128 × 1015 1/s (FSI 2NF+3NF ) PWIAS FSI 2NF FSI 2NF+3NF FSI work differently than for 3He !

Radiative pion capture on 3H: other predictions and data

MESON2018, Cracow, 11 June 2018

Γnnn = 0.128 × 1015 1/s PWIAS FSI 2NF FSI 2NF+3NF Calculations from

• A. C. Phillips and F. Roig,

AIP Conf. Proc. No. 26, (1975) Γnnn= 0.07 × 1015 1/s

• J. P. Miller et al., Nucl. Phys. A343, 347 (1980)

Conclusions and outlook

• A very robust momentum space framework to deal with many electroweak

processes has been applied to radiative pion capture processes

• First consistent results for 2H, 3He and 3H with realistic 2N and 3N potentials

have been obtained in impulse approximation

• Sensitivity to properties of neutron-neutron interaction in the 2H case has been

confirmed

• Comparisons with other theories yield a mixed picture
• Room for improvement: consistent 2N and 3N potentials as well as transition
• perators should be used for all radiative capture reactions
• New data are necessary to establish detailed relations among many processes

(3N scattering, weak proton-proton capture, neutrino scattering, muon capture, pion absorption, pion photoproduction) Investigations of double radiative pion capture are planned

MESON2018, Cracow, 11 June 2018

Selected references:

MESON2018, Cracow, 11 June 2018

• W. K. H. Panofsky et al., Phys. Rev. 81 (1951) 565
• K. M. Watson and R. N. Stuart, Phys. Rev. 82, 738 (1951)
• A. M. L. Messiah, Phys. Rev. 87, 639 (1952).
• N. M. Kroll and M. A. Ruderman, Phys. Rev. 93, 233 (1954)
• K. McVoy, Phys. Rev. 121, 1401 (1961)
• P. Truöl et al., Phys. Rev. Lett. 32, 1268 (1974)
• A. C. Phillips and F. Roig, Nucl. Phys. A 234, 378 (1974)
• J. A. Bistirlich et al., Phys. Rev. Lett. 36, 942 (1976)
• H. W. Baer, K. M. Crowe, and P. Truöl, Adv. Nucl. Phys. 9, 177 (1977)
• W. R. Gibbs, B. F. Gibson, and Q. J. Stephenson, Jr.,
• Phys. Rev. C 11, 90 (1975); 12, 2130(E) (1975);
• Phys. Rev. C 16, 322 (1977);
• Phys. Rev. C 16, 327 (1977); 17, 856(E) (1978);
• Phys. Rev. C 18, 1761 (1978)

Selected references:

MESON2018, Cracow, 11 June 2018

• G. F. de Teramond, Phys. Rev C 16, 1976 (1977); 36, 691 (1987)
• J. P. Miller et al., Nucl. Phys. A343, 341 (1980)
• V. Bernard, N. Kaiser, and U.-G. Meiβner, Phys. Lett. B 383, 116, (1996)
• H. W. Fearing et al., Phys. Rev. C 62, 054006 (2000)
• A. Gårdestig and D. R. Phillips, Phys. Rev. C 73, 014002 (2006)
• A. Gårdestig, Phys. Rev. C 74, 017001 (2006)
• Q. Chen et al., Phys. Rev. C 77, 054002 (2008)
• A. Gårdestig, J. Phys. G: Nucl. Phys. 36, 053001 (2009).
• J. Golak et al., Phys. Rept. 415, 89 (2005)
• J. Golak et al., Phys. Rev. C 90, 024001 (2014)
• J. Golak et al., Phys. Rev. C 94, 034002 (2016)

Thank you !

MESON2018, Cracow, 11 June 2018

Auxiliary slides

MESON2018, Cracow, 11 June 2018

Formalism (cont.)

MESON2018, Cracow, 11 June 2018

i f

m i m f

j N   

  from ab initio calculations in momentum space

Dynamical ingredients (1): 2N and 3N Hamiltonians

      

4

) 3 ( 4 ) 2 ( 4 ) 1 ( 4 3 2 1 3 4 3 2 1 3 123 12 13 23 3 3 12 2 2 V N N N N N N

V V V V V V H V V V V H V V V V H H V H H                   

used to generate nuclear bound and scattering states contain 2N and 3N potentials

12 2 1 2

j j j j N   

Formalism (cont.)

MESON2018, Cracow, 11 June 2018

Dynamical ingredients (2): nuclear single-nucleon, 2N and 3N current

• perators

                     

) 3 ( ) 3 ( 123 12 3 ) 2 ( ) 2 ( 123 13 2 ) 1 ( ) 1 ( 123 23 1 ) 3 ( 123 ) 2 ( 123 ) 1 ( 123 12 3 13 2 23 1 123 13 23 12 3 2 1 3

123

j j j j N

j j j j j j j j j j j j j j j j j j j j j j j j j j                         

describe interactions of an external probe with a nuclear system

deuteron state with Ed < 0

Formalism (reactions with 2H)

MESON2018, Cracow, 11 June 2018

d d d N

E H   

2

d d j

N  

 

' 

 

d N N

• a

d N

j G t p j N   

   2 2 12 2 ) (

1  





elastic channel break-up channel

 

12 2 12 12 12

V i E G t V t

N

    ,

2 ) ( ) ( 2

  

 

m p E E H N  

free 2N propagator Lippmann-Schwinger equation N N

H i E lim E G

2 2

1 ) (   









f N f

H i E i lim   

 3 ) (

    



 

formal definition including the channel state

Formalism (reactions with 3He and 3H)

two-body or three-body break-up channel with final scattering states

MESON2018, Cracow, 11 June 2018

  

  N

j N

3

'

elastic or quasielastic channel with initial and final bound states

  

b N

E H3

3N bound state with Eb < 0 generated by the Faddeev equation

i N f

j N   

   3 ) (

(1) 3N force decomposed as V4

(i) is symmetric under the exchange of nucleons j and k, i≠j≠k≠i

(3) 2N off-shell t-matrix generated via LSE: (2) free 3N propagator ) 3 ( 4 ) 2 ( 4 ) 1 ( 4 4

V V V V   

Operators in 3N space:

1 3 1 1 1

t G V V t

N

 

(4) permutation operator:

23 13 23 12

P P P P P  

MESON2018, Cracow, 11 June 2018

Formalism (reactions with 3He and 3H)

N N

H i E lim E G

3 3

1 ) (   









 

  

  

 

  

U P G t G V P P G t j P G t G V P G t U

N N N i N N N

                     

3 1 3 ) 1 ( 4 3 1 3 1 3 ) 1 ( 4 3 1

1 1 2 1 ) 1 ( 1 1 1 2 1

MESON2018, Cracow, 11 June 2018

) , , (

. .

Q E j U U

m c   

3N internal energy magnitude of the three momentum transfer

Auxiliary equation for

Formalism (reactions with 3He and 3H)

     

       

      U P G t U P j P G t j P N U P j P N

N N N i N N i N N Nd i Nd Nd 3 1 3 3 3 1 3 3 3

) 1 ( 1 ) 1 ( 1 ) 1 ( 1            

MESON2018, Cracow, 11 June 2018

Quadratures

to obtain nuclear matrix elements for arbitrary exclusive kinematics ! Semi-exclusive and inclusive observables are calculated by suitable integrations over the phase space domains.

Formalism (reactions with 3He and 3H)