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!

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

A brief review of the recent 3n, 4n literature http://online.kitp.ucsb.edu/online/fbs16/hiyama/pdf/Hiyama_FBS16_KITP.pdf Expt: a 4n candidate published in PRL 116, 052501 (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,

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

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.

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:

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

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-

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)

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

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

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) 043040

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)

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

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)-

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

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

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)

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!!

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)

Next, consider the 4-fermion problem

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, 030501 (2009)

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

To find the lowest order effect of the attractive n-n scattering length (a= -18.7 fm) in the long wavelength limit, we can find the expectation value of the Fermi pseudopotential, i.e. Preliminary Results!!

A brief review of the recent 3n, 4n literature Expt: a 4n candidate published in PRL 116, 052501 (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,

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

Rakshit and Blume, Phys. Rev Rakshit and Blume, Phys. Rev. A 86, 062513 (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.

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. 4.88 (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?

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

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

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

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= -1400 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)

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

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

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.