1. Satoshi Nakamura (Osaka University)
Extracting low-energy h-nucleon scattering amplitude from g d  h n p data
Satoshi Nakamura (Osaka University)
Collaborators : H. Kamano (KEK), T. Ishikawa (Tohoku Univ.)

2. Introduction

3. h N scattering length (ahN)
Fundamental quantity in hadron physics
Important relevance to the existence of h-mesic nuclei
B.E. of various h-mesic nuclei from theoretical calculations
Q. Haider and L. Liu, Int.J.Mod.Phys. E (2015)

4. h N scattering length (ahN)
Fundamental quantity in hadron physics
Important relevance to the existence of h-mesic nuclei
But not well-known
Current status
Several combined analyses of p N  h N and g N  h N data
Re[ahN] = 0.2 ~ 1.1 fm
Im[ahN] = 0.2 ~ 0.3 fm (optical theorem)
pn  dh data Eur. Phys. J. A 38, 209 (2008)
Re[ahN] = 0.4 ~ 0.6 fm
ahN determined with indirect information  model dependence
Need process that sensitively probes h N  h N scattering but not contaminated by others

5. Proposed experiment at ELPH@Tohoku Univ.
g d  h n p at Eg ~ 0.95 GeV and proton detected at 0o
An ideal kinematical setting for extracting h N scattering length
h is produced almost at rest  strong h n  h n rescattering is expected
p n  h n and NN rescatterings are expected to be suppressed
Data still need to be analyzed with reliable model to extract ahN

6. Dynamical Coupled-Channels (DCC) model for meson productions
Kamano, SXN, Lee, Sato, PRC 88, (2013)
PRC 94, (2016)
Developed through fully combined analysis of gN, pN  pN, hN, KL, KS data, W < 2.1 GeV
This talk
Show DCC model meets requirements to extract ahN from ELPH data
Develop g d  h n p reaction model with DCC model as building block
Demonstrate model prediction agrees well with g d  h n p data
Predict g d  h n p cross sections at ELPH kinematics
Study sensitivity of ELPH exp. to ahN and rhN

7. Dynamical Coupled-Channels model for meson productions

8. Both on- and off-shell amplitudes are calculated
Kamano et al., PRC 88, (2013)
Coupled-channel Lippmann-Schwinger equation for meson-baryon scattering ,
By solving the LS equation, coupled-channel unitarity is fully taken into account
Both on- and off-shell amplitudes are calculated

9. Kamano et al., PRC 88, (2013)
Coupled-channel Lippmann-Schwinger equation for meson-baryon scattering
1 or 2 bare N* in
each partial wave ,

10. In addition, gN channels are included perturbatively
Kamano et al., PRC 88, (2013)
Coupled-channel Lippmann-Schwinger equation for meson-baryon scattering T V ,
By solving the LS equation, coupled-channel unitarity is fully taken into account
In addition, gN channels are included perturbatively T

11. DCC analysis of gN, pN  pN, hN, KL, KS

12. DCC analysis of meson production data
Kamano, Nakamura, Lee, Sato, PRC 88 (2013)
Fully combined analysis of gN, pN  pN, hN, KL, KS data and polarization observables
(W ≤ 2.1 GeV)
~ 23,000 data points are fitted
by adjusting parameters (N* mass, N*  MB couplings, cutoffs)

13. Eta production reactions
Kamano, Nakamura, Lee, Sato, PRC 88 (2013)
Relevant to p n  h n rescattering
in g d  h n p reaction

14. Relevant to p n  h n rescattering in g d  h n p reaction
γp  π0p
dσ/dΩ for W < 2.1 GeV
Kamano, Nakamura, Lee, Sato, PRC 88 (2013)
Kamano, Nakamura, Lee, Sato, 2012
Relevant to p n  h n rescattering
in g d  h n p reaction

15. Tested important elementary amplitudes for g d  h n p
g p  h p
Kamano, Nakamura, Lee, Sato, PRC 88 (2013)
Tested important elementary amplitudes for g d  h n p
relevant to impulse and h n  h n rescattering mechanisms

16. Application of DCC model to g d reactions

17. Model for g d  h n p Impulse NN rescattering pN & hN rescattering TNN
p, h g d d d
g N  p N, h N amplitude  DCC model
p N, h N  h N amplitude  DCC model
TNN , deuteron w.f  CD-Bonn potential (PRC 63, (2001) )
Off-shell effects are taken into account

18. Numerical results for g d  h n p

19. Comparison with data for h angular distribution in g d  h X
Very near threshold
Data: Hejny et al.
Eur. Phys. J. A (2002)
Significant underestimate of data
 Missing contributions
Mechanisms
Small IA contribution (momentum mismatch)
Very large NN rescattering contribution
h rescattering effect is also significant
Coherent process
3-body hNN correlations
 possible virtual pole

20. Comparison with data for h angular distribution in g d  h X
Data: Krusche et al.,
Phys. Lett. B (1995)
Prediction is good agreement with data
Mechanisms
Coherent contribution is small
Soundness of model demonstrated !
IA term dominates
NN rescattering contribution is small
h rescattering effect is not large but important

21. Numerical results for g d  h n p at ELPH kinematics

22. g d  h n p at proposed ELPH experiment
Kinematics :
Eg = 950 MeV, proton at 0o
One-to-one relation between
Mh n and proton momentum
Impulse current dominates
h n rescattering (mostly s-wave) gives sizable contribution
pN  hN rescattering are small but visible; NN rescattering negligible for Mh n< 1.5 GeV
h production suppressed in intermediate Mh n　 deuteron wave function

23. g d  h n p at proposed ELPH experiment
Kinematics : Eg = 950 MeV, proton at 0o
hn  hn rescattering (mostly s-wave) gives sizable (-40% ~ +20% ) contribution
pN  hN rescattering are smaller but visible; data exist  controllable contribution
NN rescattering negligible for Mh n< 1.5 GeV
 more multiple rescattering expected for Mh n> 1.5 GeV
Data for Mh n< 1.5 GeV will be useful to study hN scattering

24. h n  h n s-wave amplitude
Definition
Effective range expansion (ERE)
a : scattering length
r : effective range
a = i fm
r = –1.9 – 0.5 i fm
(DCC model)
ERE is valid for W < 1550 MeV where h n rescattering
is important
off-shell-ness of h n  h n s-wave amplitude turns out
to be small effect on g d  h n p at ELPH kinematics
hn  hn amplitude in g d  h n p can be replaced with
ERE and study ER parameter dependence ~

25. Re[ahN]-dependence of g d  h n p at ELPH exp.
Kinematics : Eg = 950 MeV, proton at 0o
Im[a]=0.25 fm, r = 0
g d  h n p at ELPH exp. kinematics has a good sensitivity to Re[ahN]
D (Re[ahN]) ~ 0.2 fm seems possible with 5% precision cross section measurement
Well-tested elementary amplitudes for g p  h p and p N  hN are essential

26. Im[ahN]-dependence of g d  h n p at ELPH exp.
Kinematics : Eg = 950 MeV, proton at 0o
Re[a]=0.6 fm, r = 0
Im[ahN] seems difficult to determine better than before with g d  h n p
at ELPH kinematics ; Im[ahN] has been already determined better than 0.1 fm

27. Re[rhN]-dependence of g d  h n p at ELPH exp.
Kinematics : Eg = 950 MeV, proton at 0o
a = i fm, Im[r] = 0
g d  h n p at ELPH exp. kinematics has some sensitivity to Re[rhN]
D (Re[rhN]) ~ 1 fm seems possible with 5% precision cross section measurement

28. Conclusion

29. Conclusion ✓ Dynamical coupled-channels (DCC) model developed
 analysis of g N, pN  pN, hN, KL, KS data
✓ DCC model applied to photo-reactions on the deuteron
* impulse, NN and meson-nucleon rescattering mechanisms
* g d  h n p data well reproduced for Eg = 700 ~ 800 MeV
✓ g d  h n p at ELPH exp. kinematics studied with the DCC model
* impulse dominates; g p  h p elementary amplitude needs solid validation
* h n  h n rescattering effect is sizable (-40% ~ +20%) at low Mh n
* p n  h n effect is small and NN rescattering effect is negligible for Mh n < 1.5 GeV
* g d  h n p at ELPH setup has sensitivity to h n scattering length & effective range
better than the current range ~

30. BACKUP

31. Partial wave amplitudes of p N scattering
Real part
Kamano, Nakamura, Lee, Sato, PRC 88 (2013)
Previous model
(fitted to pN  pN data only)
[PRC (2007)]
Imaginary part
Data: SAID pN amplitude

32. g d  p N N Purpose : test the soundness of the model
Data: EPJA 6, 309 (1999)
Data: NPB 65, 158 (1973)
Model prediction is reasonably consistent with data
Large NN (small pN) rescattering effect for p0 production
orthogonality between deuteron and pn scattering wave functions
Small rescattering effect for p- production

33. h n  h n s-wave amplitude
Re[a]-dependence of FhN Im Re
Q : Can we discriminate h n scattering length with g d  h n p data from ELPH ?

34. Resonance region (single nucleon)
gN  X
2nd
3rd D
(MeV)
Multi-channel reaction
 2p production is comparable to 1p
 h, K productions (n case: background of proton decay exp.)

35. “Δ” resonances (I=3/2) “N” resonances (I=1/2) JP(L2I 2J)
Kamano, Nakamura, Lee, Sato, PRC 88 (2013)
JP(L2I 2J)
-2Im(MR)
(“width”)
Re(MR)
PDG: 4* & 3* states assigned by PDG2012
AO : ANL-Osaka
J : Juelich (DCC)
[EPJA49(2013)44, Model A]
BG : Bonn-Gatchina (K-matrix)
[EPJA48(2012)5]