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(preliminary, unpublished)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