1. Are there three-body forces between point particles ?
July 8, 2019
Kimiko Sekiguchi Workshop
Are there three-body forces between point particles ?
H. Sakai
RIKEN Nishina Center

2. N+N’ Three-Nucleon Force (3NF) Fujita•Miyazawa type 3NF
(Prog. Theor. Phys. 17(1957)360.)
J. Fujita
H. Miyazawa
2 exchange type
N+N’ p n
Nucleon (finite size) has internal structure
N : essential for the origin of 3NF
My longstanding (20 y) question is :
Is there a three-body force between point particles ?
Looking for a genuine three-body force.

3. Three point(?) particles
Graviton (mass=0, Q=0, s=2) not found
Photon (mass=0, Q=0, s=1) gauge particle
Neutrino (mass<0.12 eV, Q=0, s=1/2) ferimion, Dirac/Majorana?
Electron/Positron (mass=0.511 MeV, Q=1, s=1/2, ), Muon, Tauon
Three-body force between electron/positron system :
 Electro-magnetic three-body force
Interactions :
Coulomb
electron-photon int.  higher order effect(3BF)
Appears as a correction to hyperfine interaction effects (mag-mag-mag)
 three-body effect will be very very small !
 needs three-body QED calculations.
Are there three electron/positron (e- e - e) system ?
 Yes! (First predicted by John Wheeler in 1946)

4. Positronium and its negative ion
Positronium (Ps) Ps
B.E.= 6.80 eV (B.E.(H)=13.6 eV)
<r(e-e+)>=2a0 (a0=0.529A)
p-Ps : 1S0(J=0),  =125 ps, 2-decay
o-Ps : 3S1(J=1),  =142 ns, 3-decay e- e+ r
Ps-
Positronium negative ion (Ps-) e- e+ r
B.E.(e-Ps)= 0.33 eV (B.E.(H-)=0.775 eV)
<r(e-Ps)>=5.5a0 (a0=0.529A)
Ps- :mainly o-Ps?,  =0.5 ns, 2-decay
J=1/2
lightest three-body bound state e- e+
p-Ps
o-Ps

5. Positronium negative ion

6. Production of positronium negative ions (Ps-)
Cs coated
Ps- production rate = 1.25 % !!!
E(e+) = thermal
W=3 kV
E() : Doppler shifted  540 kV

7. Spectroscopy of Ps- (photo ionization)

8. Ps- beam station at KEK PF

9. Muonium negative ion : Mu-  (e)e
Negative Muonium (Mu-) e- + r
B.E.(e-Mu)= 0.75 eV (B.E.(Mu)=13.54 eV)
<r(e-Mu)>=1.005a0 (a0=0.529A)
=2.2 s (same as + decay):
J=1/2 (or 3/2 ?)
Kuang et al. PRA 39(1989) 6109
Almost H- ion
T=0.5 MeV
Mu- production rate = % !!!

10. Mu- produced and accelerated at JPARC
Kitamura et al., Proc. PASA(2018) 239 (in Japanese)
89 keV
Mu- production rate = 0.05 % ???

11. QED corrections in Ps- binding energy1 2 3
r23
r12
rel.
QED

12. One photon-decay of Ps- and 3-body force?
GSI
E(e+)=316keV
e+e+e- e++
 𝟑𝟐𝟏 = 𝜳 𝟎,𝟎,𝟎 𝟐
𝝌𝑷𝑻
3BInteraction? or
3BCorrolation?

13. Effective 3-body forces everywhere
Two virtual photon emission 3BF
But usually

14. Summary of my dream (may be nonsense !)
Does 3BF among 3 point particles exist or not?
Ps- and Mu- systems, available as a beam.
3BF under long-range electro-magnetic force (expanded system).
Cf. 3NF works in compact nuclear system under short-range
nuclear forces.
3BF effects seems to be very tiny, may be an order of 6
(Cf. 5 calculations are done for g-2)
How to detect such tiny 3BF effects?
B.E. effect? Or scattering effects or …. It sounds impossible.
If observed as an interference effect, it may be possible.
(Cf. LIGO, gravitational wave detection, △L/L=10-24)
So my dream stays as a ‘dream’ for a while.