1. Hyperspherical adiabatic description of 3n and 4n states
Chris Greene, Purdue University
with Alejandro Kievsky and Michele Viviani (Pisa)
Thanks to the
NSF for support!

2. Relationship to Efimov physics
Thoughts on weakly bound states or low energy resonances from an adiabatic hyperspherical viewpoint
Some background on other weakly bound systems and low energy resonances studied using this approach
Relationship to Efimov physics
Quantitative 3n calculation and analysis
Preliminary ideas about the 4n system emerging from this point of view

3. A brief review of the recent 3n, 4n literature
Expt: a 4n candidate published in PRL 116, (2016), Kisamori et al.
conclusion: energy is
And an upper limit on its width is quoted to be
And a Nature News & Views by Bertulani & Zelevinsky, > 2000 page views
Theory: Hiyama, Lazauskas, Carbonell, Kamimura 2016 Phys. Rev. C.
conclusion: “…a remarkably attractive 3N force would be required…”
Gandolfi, Hammer, Klos, Lynn, Schwenk
conclusion: a three-neutron resonance exists below a four-neutron
resonance in nature and is potentially measurable
Fossez, Rotureau, Michel, Ploszajczak
conclusion: while the energy (4n) …may be compatible with expt… its width must be larger than the reported upper limit  (probably) …reaction process too short to form a nucleus
Shirokov,Papadimitriou,Mazur,Mazur, Roth,Vary
conclusion: 4n resonance, E=0.8 MeV,

4. Our main theoretical tool: formulate the problem in hyperspherical coordinates, treating the hyperradius R adiabatically
And then the rest of the problem comes down to calculating energy levels as a function of R, which we call “hyperspherical potential curves”, and their mutual couplings, which can then be used to compute bound state and resonance properties, scattering and photoabsorption behavior, nonperturbatively
The hyperradius R (squared) is a coordinate proportional to total moment of inertia of any N-particle system, i.e.:
Here ri is the distance of the i-th particle from the center-of-mass. All other coordinates of the system are 3N-4 hyperangles.
This follows the formulation of the N-body problem in the adiabatic hyperspherical representation, as pioneered by Macek, Fano, Lin, Klar, and others

5. Strategy of the adiabatic hyperspherical representation: FOR ANY NUMBER OF PARTICLES, convert the partial differential Schroedinger equation into an infinite set of coupled ordinary differential equations:
To solve:
First solve the fixed-R Schroedinger equation, for eigenvalues Un(R):
Next expand the desired solution into the complete set of eigenfunctions with unknowns F(R)
And the original T.I.S.Eqn. is transformed into the following set which can be truncated on physical grounds, with the eigenvalues interpretable as adiabatic potential curves, in the Born-Oppenheimer sense.

6. Notes
Various methods can be used to compute the needed potential curves and couplings: diagonalization in a basis set of hyperspherical harmonics, or correlated Gaussians, or Monte Carlo techniques, etc.
A theorem exists about the truncation to a single potential curve, i.e. the adiabatic approximation, namely:

7. Next: a few previous results about shape resonances and hyperspherical potential curves

8. C D Lin, 1975 PRL
Bryant et al, LAMPF experiment, 1977 PRL H- photodetachment compared with Broad & Reinhardt’s calculation H- H-
Botero&CHG, 1986 PRL
And expt on Ps- photodetachment
2016 Nature Commun. by Michisio et al.
Ps-

9. Universality, from nuclear scale energies to the chemical
adiabatic potential curves for n+n+p, in collaboration with Alejandro Kievsky and Kevin Daily, nuclear physics on 106 eV scale
Atomic physics
3-atom hyperspherical potential curves for He+He+He on a 10-3 eV scale, looks very similar to the 3-nucleon potentials
U((R)
MeV
Nuclear physics
Few-Body Syst (2015) 56:753–759
For a review, see Yujun Wang, Jose D’Incao, Brett Esry, Adv. At. Molec.Opt. Phys. Vol.62 (2013)

10. 3-body interaction term included
Hyperspherical potential curve for the triton
3-body interaction term included
This graph documents the lowering effect of the 3-body term for the triton system
Daily, Kievsky, CHG, Few-Body Syst (2015) 56:753–759

11. Hyperradial potential barrier for 3 identical bosons and Efimovian shape resonances
Esry & CHG, Nature News & Views 2006

12. Other examples of potential wells with a barrier: 4-body and 5-body recombination in a Cs gas
Zenesini et al., New Journal of Physics 15 (2013)

13. Langer-corrected centrifugal barrier term, suitable for WKB analysis
Schematic potential curve structure for N identical bosons with a<0
Langer-corrected centrifugal barrier term, suitable for WKB analysis
(Mehta, Rittenhouse, D’Incao, CHG PRL 2009)

14. An important aside: The d-dimensional Laplacian operator is:
This Laplacian operator acts on the full wavefunction, so like one normally does in d=3, we can rescale the radial wavefunction, i.e. set
Nonadiabatic coupling terms
Note: d= dimension of the relative Jacobi coordinates of the system, i.e. d=6 for 3 particles, d=9 for 4 particles, etc. d=3N-3
For the 4n problem, d=9, , so for this symmetry, the centrifugal barrier at large R in the lowest channel is

15. Recall that the eigenvalues of
3n potential curves
Potential curve convergence study as the maximum value of K is increased.
Recall that the eigenvalues of
are known analytically to have the form K(K+4) with K=1,3,5,… for this odd-parity system
(Kievsky & Viviani collab.)
Preliminary Results!!
Note the strong spin-orbit effect at small hyperradii, with the (3/2)- minimum far lower than (1/2)-

16. n+n+n adiabatic hyperspherical potential curve
Preliminary Results!!
Notes: The short-range minimum is accurately converged. The region near the long range barrier converges more slowly
BUT (a big but): This is not the full potential. We also must include the rest of the potential curve, namely, add

17. Preliminary Results!!
Again:This is not the full potential. We also must include the rest of the potential curve, namely, add
Why is this necessary?
Because only then do we obtain the full potential relevant for a purely 2nd derivative radial kinetic energy operator.
We can see a net attraction. 2 attractive V terms in H + 3Body, and 2 repulsive KE terms

18. Utot, MeV
Non-interacting potential curve for this symmetry of the n-n-n system
Long range attraction due primarily to the large negative singlet n-n scattering length, virtually identical for the J=1/2- and 3/2- symmetries
In this region, the large and negative n-n singlet scattering length causes most of the attraction compared to the pure centrifugal barrier
Preliminary Results!!
R(fm)

19. The preceding graph showed that there is insufficient attraction at large R to make a bound state or shape resonance. Here we look at the region where the raw potentials showed a minimum, and clearly there is none once we’ve included the 15/8mR2 term.
So this calculation suggests strongly that in this symmetry, which appears to be the most attractive for the 3n system, it is extremely far from having enough attraction to produce a resonant or bound state
Preliminary Results!!

20. Utot, MeV
Non-interacting potential curve for this symmetry of the n-n-n system
Long range attraction due primarily to the large negative singlet n-n scattering length, virtually identical for the J=1/2- and 3/2- symmetries
In this region, the large and negative n-n singlet scattering length causes most of the attraction, far more than the shorter range region where 3-body forces are important
Preliminary Results!!
R(fm)

21. Next, consider the 4-fermion problem

22. Some previous work from our group on 4-body hyperspherical studies of 4 equal mass fermions or bosons (reasonable agreement with 2004 Petrov, Salomon, Shlyapnikov results)
The system of two spin up, two spin down fermions at large 2-body scattering lengths a is important for the theory of the BCS-BEC crossover
Dimer-dimer scattering length (Re and Im parts)
Hyperspherical potential curves
PRA 79, (2009)

23. For the 4n system, since there are no bound subsystems, this is simplest to treat in the H-type Jacobi tree:
 2 spin up neutrons, p-wave Y1m(13)
 2 spin down neutrons, p-wave Y1m’(24)
 s-wave Y00 in the motion of the two pairs about each other
So we consider the L=0, S=0, even parity symmetry, which corresponds to K=2, 4, 6, … and the lowest channel asymptotically should have a zeroth order potential curve

24. To find the lowest order effect of the attractive n-n scattering length (a= fm) in the long wavelength limit, we can find the expectation value of the Fermi pseudopotential, i.e.
Preliminary Results!!

25. A brief review of the recent 3n, 4n literature
Expt: a 4n candidate published in PRL 116, (2016), Kisamori et al.
conclusion: energy is
And an upper limit on its width is quoted to be
And a Nature News & Views by Bertulani & Zelevinsky, > 2000 page views
Theory: Hiyama, Lazauskas, Carbonell, Kamimura 2016 Phys. Rev. C.
conclusion: “…a remarkably attractive 3N force would be required…”
Gandolfi, Hammer, Klos, Lynn, Schwenk
conclusion: a three-neutron resonance exists below a four-neutron
resonance in nature and is potentially measurable
Fossez, Rotureau, Michel, Ploszajczak
conclusion: while the energy (4n) …may be compatible with expt… its width must be larger than the reported upper limit  (probably) …reaction process too short to form a nucleus
Shirokov,Papadimitriou,Mazur,Mazur, Roth,Vary
conclusion: 4n resonance, E=0.8 MeV,

26. Expected validity is probably only for about R>50 fm
Preliminary Results!!

27. Rakshit and Blume, Phys. Rev
Rakshit and Blume, Phys. Rev. A 86, (2012) found that as a-> - infinity, the hyperspherical potentials are entirely repulsive, namely:
Preliminary Results!!
Conclusion: The true potential for 4n in this symmetry should be no more attractive than the higher of these two potential curves, making the possibility of a resonance state for this symmetry very unlikely.

28. Next, an experimental quest to see multiple Efimov states and verify the universal scaling features: look at heavy-heavy-light Efimov systems, where Efimov himself and others (D’Incao and Esry) stressed that the scaling between successive resonance features is much better, e.g (for Cs-Cs-Li) instead of 22.7:
Question: can we predict the first value of the scattering length a-(LH) for any a(HH), i.e. where one will observe an Efimov resonance in 3-body recombination?

30. Tung, Chin, et al. (University of Chicago, PRL 2014) and
Since 2014, two experimental groups have independently observed a series of 3 Efimov resonances in 133Cs+133Cs+6Li :
Tung, Chin, et al. (University of Chicago, PRL 2014)
and
Pires, Ulmanis, Weidemueller et al. (University of Heidelberg, PRL 2014 & 2016)
And both experiments confirm the expected universal ratio of approximately 5 between successive resonances
Heidelberg
Chicago

31. Zero-range theory (regularized delta function interactions)
U(R)1/3

32. Note that the typical experimental scenario (as in the Heidelberg and Chicago expts) scans the vertical axis for a fixed value of the horizontal axis

33. Predictions of first Efimov resonance (negative a) and destructive interference Stueckelberg minimum (positive a)
Exp[p/s0]
4.050
4.876
6.847
15.2
36.2
123
355
3.52x105
2.256
2.006
1.683
1.301
1.132
1.043
1.016
1.027
Our expectation (2012 PRL) for the first Cs-Cs-Li resonance is either at a= or else -1400/4.88 = -287 a.u. The new experiments observe a_(expt)= -337(9) a.u. (Chicago) or -320(10) a.u. (Heidelberg)

34. Again:This is not the full potential
Again:This is not the full potential. We also must include the rest of the potential curve, namely, add
Why is this necessary?
Because only then do we obtain the full potential relevant for a purely 2nd derivative radial kinetic energy operator.
We can see a net attraction. 2 attractive V terms in H + 3Body, and 2 repulsive KE terms

35. Again:This is not the full potential
Again:This is not the full potential. We also must include the rest of the potential curve, namely, add
Why is this necessary?
Because only then do we obtain the full potential relevant for a purely 2nd derivative radial kinetic energy operator.
We can see a net attraction. 2 attractive V terms in H + 3Body, and 2 repulsive KE terms
So tetraneutron has one more KE but two more V terms,… to be continued

36. Summary:
Quantitative calculations based on AV18+UIX suggest that it is highly unlikely that a tri-neutron resonance exists
First qualitative analysis of the 4n system suggests that the long range attraction associated with the large, negative singlet nn scattering length is not sufficient to produce a shape resonance. So if one exists, it would have to be of very short range binding character.