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!
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
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.
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
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 Hphotodetachment
compared with Broad
& Reinhardt’s
calculation
H-
HBotero&CHG, 1986 PRL
-
And expt on Ps
photodetachment
Ps-
2016 Nature
Commun. by
Michisio et al.
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
U((R)
MeV
Nuclear physics
Few-Body Syst
(2015) 56:753–759
3-atom hyperspherical
potential curves for
He+He+He on a 10-3 eV
scale, looks very similar to
the 3-nucleon potentials
For a review, see Yujun Wang,
Jose D’Incao, Brett Esry, Adv. At.
Molec.Opt. Phys. Vol.62 (2013)
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
a<0
Esry & CHG, Nature News & Views 2006
Other examples of potential wells with a barrier: 4body and 5-body recombination in a Cs gas
Zenesini et al., New Journal of Physics 15 (2013) 043040
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
3n potential curves
Potential curve
convergence study
as the maximum
value of K is
increased.
Note the strong spin-orbit
effect at small hyperradii,
with the (3/2)- minimum far
lower than (1/2)-
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.)
n+n+n adiabatic hyperspherical potential curve
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
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
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
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 3body forces are important
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.
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
Expected validity is probably only for about R>50 fm
Rakshit and Blume, Phys. Rev. A 86, 062513 (2012)
found that as a-> - infinity, the hyperspherical
potentials are entirely repulsive, namely:
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?
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)
2.256
2.006
1.683
1.301
1.132
1.043
1.016
1.027
Exp[p/s0]
4.050
4.876
6.847
15.2
36.2
123
355
3.52x105
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. 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. 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.