1. Nils A. Törnqvist University of Helsinki
Talk at Frascati, January 2006
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

2. EPR Lambda anti-Lambda N.A. Törnqvist+ L e - e L + p
10000 events at Daphne2? p
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

3. EPR Lambda anti-Lambda N.A. Törnqvist
In c.m.s. of L
The L-> pp decay works as a spin analyser!
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

4. EPR Lambda anti-Lambda N.A. Törnqvist
Resonance decay into
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

5. EPR Lambda anti-Lambda N.A. Törnqvist
In words this means that L’s coming from a singlet L anti-L state are polarized just like L’s prepared to be polarized in a tagged direction given by the direction of the p in the anti-L decay.
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

6. EPR Lambda anti-Lambda N.A. Törnqvist
This is a demonstration of the conceptual peculiarities involved in the EPR problem: Knowledge of how one of the L decayed, or will decay (time ordering is not relevant here) tells an observer that the second L decayed, or will decay, as if it had a definite polarization. p + p L L h c p p
In L cms
In L cms
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

7. EPR Lambda anti-Lambda N.A. Törnqvist
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

8. EPR Lambda anti-Lambda N.A. Törnqvist
Bell’s inequalities
The violation of Bell’s inequalities by quantum mechanics has been historically of great importance in removing any doubt that a local theory, in the EPR sense, is incompatible with quantum mechanics. These inequalities are usually written in terms of correlations, such that for the case of a spin 0 state decaying into two spin ½ particles the spin corellation function E obeys the inequality
Here
denote unit vectors along which the spin components are measured
in the classic Bohm variant of the EPR spin 0 decay to two spin ½ particles.
spin 0
spin 1/2
spin 1/2
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

9. EPR Lambda anti-Lambda N.A. Törnqvist
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

10. EPR Lambda anti-Lambda N.A. Törnqvist
However quantum mechanics deals with relations between amplitudes, and the amplitudes related to the cross sections in these inequalities form a triangle in the complex plane. Thus QM implies triangle inequalities for the square roots of the cross sections,
Or equivalently:
and not for the cross sections as in the Bell inequalities.
It is instructive to plot the domains separated by these inequalities in a barycentric coordinate system (Figure 4) in which ine plots the normalized ratios
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

11. EPR Lambda anti-Lambda N.A. Törnqvist
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

12. EPR Lambda anti-Lambda N.A. Törnqvist
Spin 1 decay to
helicity l=+1, transverse polarization
helicity l= -1, transverse polarization ( ) +
helicity l= 0, longitudinal polarization
Only the l=0 case is interesting (entangled) from the point of view of EPR correlations
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

13. EPR Lambda anti-Lambda N.A. Törnqvist=
i.e. it factorizes and one has no interesting EPR corellations
On the other hand for l=0 or longitudinal polarization one has
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

14. EPR Lambda anti-Lambda N.A. Törnqvist
Thus in e+e- to
one should look for situations where the initial photon is longitudinally polarized with respect to the LL axis.
This means not in the forward direction, but near 90 degrees in the center of mass.
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

15. EPR Lambda anti-Lambda N.A. Törnqvist
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

16. EPR Lambda anti-Lambda N.A. Törnqvist
A more general formula for the correlations:
Uninteresting factorized piece
Interesting EPR correlations
at 2.5 GeV k /E = 0.46
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

17. EPR Lambda anti-Lambda N.A. Törnqvist
Up til now only one experiment by the DM2 collaboration:
M. H. Tixier et al. Physics Letters B212 (1988) 523
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

18. EPR Lambda anti-Lambda N.A. Törnqvist
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

19. EPR Lambda anti-Lambda N.A. Törnqvist
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

20. EPR Lambda anti-Lambda N.A. Törnqvist
Concluding remarks
EPR correlations can be tested at in
e+e- ->
It would be a test involving weak interactions
Strongest effects with lambda pair at near 90 degrees and highest possible cms energy
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist

21. EPR Lambda anti-Lambda N.A. Törnqvist
Frascati January 2006
EPR Lambda anti-Lambda N.A. Törnqvist