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Linking Efimov physics, few-fermion
universality, and the 3n and 4n systems
Chris Greene, Purdue University
with Michael Higgins, Alejandro Kievsky and Michele Viviani
(on the 3n and 4n systems)
Thanks to the
NSF for support!
and on the 3 unitary fermions,
with Yu-Hsin Chen
Today’s Talk Outline
1.Theoretical search for a low energy bound or
resonant trineutron or tetraneutron, and
connections with Efimov physics and
universality (see arXiv: 2005.04714, now in
press at PRL)
2. Investigation of alternative examples of three-
fermion systems with short range forces at
unitarity, and manifestations of Efimov physics
(preliminary, unpublished)
The premise which got me interested in studying the 4n and 3n
systems, after I learned about the fascinating 4n problem from Emiko
Hiyama at a 2016 KITP Program we ran on “Universality in Few-Body
Physics”, was the following thought, based on extensive experience
with atomic & molecular systems:
WE KNOW RESONANCES
and
WE KNOW UNITARITY of 4 FERMIONS
And
We have a hyperspherical coordinate toolkit that makes
existence (or nonexistence) of resonances visually clear
Next…
1. Review some earlier successful predictions of shape
resonances in few-body systems using the adiabatic
hyperspherical coordinate framework
2. Discuss the collision problem with N=4 or N=3 particles in the
open continuum above the total breakup threshold
3. Present quantitative calculations of hyperspherical potentials
for the 4-neutron and 3-neutron systems, based on the several
state of the art inter-nucleon potentials (plus three-body
terms), to explore the possible existence of resonances or
bound states in those systems.
MAIN CONCLUSIONS
a. There are no low energy 3n or 4n bound states or
resonances that are consistent with known n-n force fields
b. There is a strong connection with Efimov
physics/universality which guarantees a strong enhancement of
the 3n and 4n density of states at E→0
Fossez, Rotureau, Michel, Ploszajczak Phys. Rev. Lett. 119, 032501 (2017)
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
Gandolfi, Hammer, Klos, Lynn, Schwenk Phys. Rev. Lett. 118, 232501 2017
conclusion: a three-neutron resonance exists below a four-neutron
resonance in nature and is potentially measurable
First: 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…”
Shirokov,Papadimitriou,Mazur,Mazur, Roth,Vary Phys. Rev. Lett. 117, 182502
(2016) conclusion: 4n resonance, E=0.8 MeV,
NO!
YES!
YES!
YES!
Other recent (2018) work on the 3n and 4n systems from
a collision physics point of view:
Tetraneutron: Rigorous continuum calculation
A.Deltuva → PhysicsLetters B 782(2018) 238–241
“... This indicates the absence of an observable 4n
resonance...”
Tri-neutron: Three-neutron resonance study using
transition operators
A. Deltuva, PRC 97, 034001 (2018)
“...There are no physically observable three-neutron
resonant states consistent with presently accepted
interaction models”
NO!
NO!
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
also r2
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.
• 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:
Notes
Next: a few previous results about
shape resonances and
hyperspherical potential curves
PRL 35, 1150 (1975)
Hyperspherical potential
curves for H- 1PoScattering phaseshift versus E
While we normally think of a strong resonance as having its phaseshift
increase by p over a narrow energy range, here is an example, where the
rise versus E is only about 2 radians, which is not unusual.
Botero&CHG, 1986 PRL
C D Lin, 1975 PRL
2016 Nature
Commun. by
Michisio et al.
An expt on Ps-
photodetachment
H-
H-
Ps-
Bryant et al, LAMPF
experiment, 1977
PRL H-
photodetachment
compared with Broad
& Reinhardt’s
calculation
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
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
Some results about N-body elastic scattering,
from Mehta et al., PRL 103, 153201 (2009)
and this simplifies, if a single scattering
matrix eigenchannel dominates, to:
This quantity is in principle an observable, which would
be relevant for the thermalization of a gas of neutrons that
are out of thermal equilibrium, even if it is not a typical
observable that could be readily seen in any standard
nuclear physics experiment today....
And an important tool for resonance analysis on the real
energy axis is the collisional time delay:
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
Hyperspherical potential curves
Dimer-dimer scattering length
(Re and Im parts)
D’Incao et al, PRA 79, 030501 (2009)
a>0
For the 4-neutron 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
Rakshit and Blume, Phys. Rev. A 86, 062513 (2012) found
that as a-> - infinity, the hyperspherical potentials are
entirely repulsive, at |a|>>r :
First Conclusion: The true potential for 4n in this symmetry is
expected to be less attractive than the lower of these two
potential curves, making the possibility of a bound state for this
symmetry unlikely.
Whereas in the noninteracting limit the asymptotic
potential for two spin-up and two spin-down identical
fermions is known to be:
Considerations about the UNITARY limit
Aside: This reduction of the coefficient of the asymptotic 1/r2
potential in the unitary limit is analogous to the Efimov effect...
a(nn)=-18.98 fm for the Argonne
AV8’ potential, similar for AV18.
Jp =0+ Hyperspherical potentials for 4 fermions:
both noninteracting and unitary limit a→-infinity
Non-interacting
unitary limit
a(nn)=-18.98 fm for the Argonne
AV8’ potential, similar for AV18.
First attempt to compute the 4n 0+ hyperspherical potential curves
using a hyperspherical harmonic basis set, for several different
K(max) up to 140 (from Kievsky and Viviani HH codes in Pisa)
Potential curves are still not
converged, but they keep getting
more attractive at r>20 fm as
K(max) is increased in the basis
set
After failing to get satisfactory
convergence at very long range, we
decided to implement our stochastic
correlated Gaussian hyperspherical
basis set method, using the AV8’
potential, which had been fitted to
Gaussians by Emiko Hiyama and
provided to us.
Side note: in order to be thorough, we also treated other nuclear force models,
i.e. AV8’, AV18, Minnesota potential, a chiral EFT potential (NV2-Ia) , and with
or without the 3-body n-n-n term from the Urbana and Illinois (e.g. IL)
interaction. Minimal differences are observed with these different interactions
Improved convergence is apparent in this black curve, obtained for the
AV8’ interaction using the correlated Gaussian hyperspherical basis
set method
Lowest 0+ adiabatic hyperspherical potential for 4 neutrons
0+
The most attractive hyperspherical potential curves for the 4n and 3n
systems, obtained using the AV8’ n-n interaction potentials (magenta)
The converged potentials are clearly totally repulsive, with no sign
of a local maximum that can trap probability in a resonance.
HH expansion, unconverged at
large r
Next consider the scattering phaseshift
in the lowest 4n potential
Note the very sharp rise of the
phaseshift close to zero energy, which
might lead some to interpret this as a
resonance! However, I now show that
this is a consequence of the long-
range potential curve near unitarity.
A key point: The lowest energy behavior
is controlled by the longest range
portions of the potential curveRecall what we learned from Efimov physics,
treated in the adiabatic hyperspherical framework
(e.g. Zhen & Macek, 1988 Phys. Rev. A):
For three identical bosons, with finite particle-
particle scattering length a, the asymptotic
hyperradial potential was shown to behave for
large |a| as
Noninteracting term,
For 3 identical bosons
But in the UNITARITY LIMIT,
a→infinity, as Efimov taught
us, the potential changes to
the following form:
Here are the values of these parameters for our 4n and 3n systems:
Our
numerical
result
Yin and
Blume,
2015 PRA
Here is a test of our asymptotic hyperspherical potentials for the 4n system
Implications of the attractive a/r3 potential at long range
One can readily derive (e.g. using the Born approximation)
that the limiting low energy phaseshift in such a potential of
the form,
is:
4n 0+
3n 3/2-
Next, consider further implications of the low
energy phaseshift behavior from the perspective
of a Wigner-Smith time delay analysis.
→(in the single-
channel limit)
Moreover, Q(E) divided by is the density of states
We can conclude that both the 3n and 4n
systems have a divergent density of states
proportional to 1/sqrt(E) at E→0, because
the phaseshift is proportional to sqrt(E).
E. P. Wigner, Lower limit for the energy derivative of the scattering phase shift, Phys. Rev. 98, 145 (1955).F. T. Smith, Lifetime matrix in collision theory, Phys.Rev. 118, 349 (1960).
4n 0+
4n 0+
3n 3/2-
3n 3/2-
Rescaled time delays for the 3n and 4n
systems, showing that they are finite at
E→0 when multiplied by sqrt(E)
Main conclusions about the tetraneutron and trineutron study:
(a) There is strong attraction in the system at each hyperradius that lowers the
potential energy, associated with a(nn) ~ -19 fm (in the singlet channel)
(b) The attraction diminishes with increasing hyperradius, such that the
tetraneutron always experiences an outward force, as shown in the cartoon
(c) The enhanced density of states associated with the long range attractive potential could
help to explain the enhanced low energy events observed by Kisamori et al.
4n →4n elastic scatteringDerivation: Mehta et al. PRL 103,
153201 (2009), Eq.6
Rescaled elastic
cross section4n density of
states in the
lowest channel
Recall: textbook applications of
the “Fermi golden rule”, include a
density of final states factor (from
integration over the energy-
conserving Dirac delta function).
The transition probability per unit
time is the familiar formula:
Kisamori et al.
2016 PRL
Evidence that these 3n and 4n systems are in
the universal regime→ i.e. where the scattering length a is much larger than every
other length scale in the system
To address universality in cold atom systems, one typically
chooses a very short-range two-body potential, like a Gaussian
or a square well or a delta function, and tunes the potential depth
to get any desired scattering length.
e.g. This is the approach that was used in the important study by
Petrov, Salomon, and Shlyapnikov (2004 PRL) who first showed
that: Re[ a(dimer-dimer) ] ~ 0.6 a(atom-atom)
and that Im[ a(dimer-dimer) ] ~ a-2.55 very important for
studies of the BEC-BCS crossover studied extensively in the
ultracold atom experiments and theory
To see how well the 3n and 4n systems are described at low energy by
universality, we have adopted a single Gaussian n-n interaction potential
of short range, which looks VERY DIFFERENT from the true n-n
interaction potentials, but which gives the same singlet scattering length.
Next, we recalculated everything, i.e. hyperspherical adiabatic potential
curves, scattering phaseshifts, time delays, using the SG(single Gaussian)
potential, and compared to the full AV8’ results
Comparison between the central AV8’
potentials and the single Gaussian potentials
0+
The most attractive hyperspherical potential curves for the 4n and 3n
systems, obtained using the AV8’ n-n interaction potentials (magenta), and
the open circles are the potential curves for the SINGLE GAUSSIAN model
This agreement between calculated results with the AV8’ potential and the SINGLE
GAUSSIAN potential shows that the 3n and 4n systems are clearly in the universal
regime, insensitive to details of the short range forces
HH expansion, unconverged at
large r
4n 0+
4n 0+
3n 3/2-
3n 3/2-
This graph (shown earlier) also confirms that phaseshifts and
time delays in the 3n and 4n systems are controlled almost
exclusively by the singlet n-n scattering length, and are not
dependent on the short range details of the n-n forces
Magenta = AV8’
Open Circles=SG
Another conclusion from this universality study:
Even if the n-n force is made significantly more
attractive, such that the singlet scattering length
diverges to infinity, there would still be no bound or
resonant trimer, and no bound or resonant tetramer
Other tests we have made:
1. Inclusion of 3-body force terms, often highly important in few-nucleon
systems: for the 3n and 4n systems, they are weak and repulsive,
and have only a small effect on our results, so most of our
calculations have not included them.
2. Inclusion of multichannel coupling of more than one adiabatic
hyperspherical potential curve: These also make only a small change
in the computed eigenphaseshifts
Lowest 5 adiabatic hyperspherical potential curves for the
4n 0+ system (excluding parity-unfavored channels)
Note that all potential curves are fully repulsive
Channel coupling test for the trineutron system
Channel coupling test for the 4n system
Comparison of the rescaled 4n 0+
lowest potential with and without the
IL7 3-body force term included
Gandolfi, et al. PRL 118, 232501 (2017), extrapolated resonance positions
4n, 2.2 MeV
3n, 1 MeV
Our conclusion: Extrapolations of bound state calculations to predict resonance
existence/position is dangerous if you don’t have a full theory of the analytic
continuation
PRL 2016Our results
using the AV8’
potential
Interestingly, the Shirokov et al. phaseshift resembles our calculated
4n-4n phaseshift, but they reached a different conclusion. We
conclude from this that a rapid rise of the phaseshift is not enough to
establish existence of a resonance, because such a rise can also be
contributed non-resonantly by a long range potential in the system
In summary, we have a new treatment, quite complementary to
other theoretical approaches, to address the question of whether
the tetraneutron has a low lying bound state or resonance state.
Here is the tally so far:
NO - Hiyama et al. 2016 PRC:
NO - Deltuva et al. 2018 PRC, PL
NO - Our work (Higgins et al. 2020 PRL in press)
YES – Gandolfi et al. 2017 PRL
YES -- Fossey et al. 2017 PRL
YES -- Shirokov et al. 2016 PRL
While the results in this table alone might not seem to strongly
support our conclusion that there is no tetraneutron low-lying
resonance or bound state, we believe that our visual evidence of
total repulsion settles the issue once and for all.
It remains desirable to understand the very low energy 4n events
observed by Kisamori et al., and here we believe that the
divergent zero energy density of states enhancement could
provide a helpful clue.
Other recent work, not discussed today:
Fractional Quantum Hall Studies with the few-body hyperspherical toolkit:
K. Daily, R. Wooten, CHG Phys. Rev. B 92, 125427 (2015)
R. Wooten, Bin Yan, CHG Phys. Rev. B 95, 035150 (2017)
Bin Yan, R. Biswas, CHG, Phys. Rev. B 99, 035153 (2019)
Few-body physics with spinor systems
Three-Body Physics in Strongly Correlated Spinor Condensates
(with V. Colussi and J. P. D’Incao) PRL 113, 045302 (2014);
& J. Phys. B 2016
Spin current generation and relaxation in a
quenched spin-orbit-coupled Bose-Einstein
condensate:
Nature Commun. 10, 375 (2019); Chuan-Hsun Li, Chunlei Qu,
Robert J. Niffenegger, Su-Ju Wang, Mingyuan He,
David B. Blasing, Abraham J. Olson, CHG, Yuli Lyanda-Geller,
Qi Zhou, Chuanwei Zhang & Yong P. Chen