# Recombination in the universal four-fermion system

A. Deltuva<EMAIL_ADDRESS>Institute of Theoretical Physics and
Astronomy, Vilnius University, Saulėtekio al. 3, LT-10257 Vilnius, Lithuania

###### Abstract

In the systems of spin $\frac{1}{2}$ fermions with resonant $S$-wave
interactions supporting only weakly bound dimers the antisymmetry forbids
recombination of three (or more) fermions at zero energy. However, the
fermion-fermion-dimer recombination is only partially suppressed. It is
studied in the framework of momentum-space integral equations for the four-
particle transition operators. In the vicinity of the unitary limit the
fermion-fermion-dimer recombination rate, rescaled to build dimensionless
quantity, is found to be linear in the effective range parameter, enabling a
simple and accurate parametrization as well as evaluation of finite-range
effects for any potential model. This feature makes the present results very
useful in benchmarking different methods for three-cluster breakup and
recombination calculations in four-particle systems. The interplay of the
three-fermion and fermion-fermion-dimer recombination processes and their
consequences for ultracold mixtures of fermions and dimers is discussed.

###### keywords:

Four-particle scattering , fermionic dimer , recombination , universality

##  1 Introduction

Universality in few-body systems with resonant $S$-wave interactions,
characterized by a large scattering length, has been investigated in numerous
works, where the considered systems range from cold atoms and molecules to
nuclear and particle physics [1, 2, 3, 4, 5]. The qualitative behavior of
those physical systems depends critically on the permutation symmetry they
obey, i.e., the bosonic/fermionic nature. While bosons (or distinguishable
particles) exhibit a rich spectrum of few-body bound states, the most famous
example being the three-body Efimov effect [6], identical spin $\frac{1}{2}$
fermions (unpolarized, i.e., in both spin states) can form only one shallow
bound state, a ${}^{1}S_{0}$ dimer with the binding energy
$b_{d}\approx\hbar^{2}/ma^{2}$, where $m$ is the fermion mass and $a$ is the
two-fermion scattering length. Although there are no three- and four-fermion
bound states111This work assumes the absence of non universal deeply bound
two-body states, implying also the absence of the associated many-body states,
the three- and four-body physics is important for the properties of cold
dilute atomic and molecular gases that are determined by the parameters of
low-energy collisions [7, 8, 9, 10]. For example, the fermionic dimer-dimer
scattering at low energy, well below the breakup threshold, has been
investigated in a number of works [7, 8, 9, 11, 12, 13, 15], achieving a good
agreement between the different methods for the dimer-dimer scattering length
and for the effective range, except for the lattice effective field theory
approach [13] for the latter observable. The present work focuses on a
different aspect of the four-fermion system near unitarity, namely, the
production of fermionic dimers in few-body collisions. In terms of initial
state complexity, the most straightforward path is the collision of two
fermions in different spin states; however, the formation of a dimer
necessitates an additional interaction with an external field, e.g., with the
electromagnetic field where a photon is radiated thereby ensuring the energy
and momentum conservation. If this interaction is suppressed for some reason
and the only forces retained are those acting between the particles
themselves, the creation of a dimer may proceed via recombination process
involving at least three colliding particles or clusters. However, the
antisymmetry requirement suppresses the recombination of three spin
$\frac{1}{2}$ fermions in the ultracold limit since the zero kinetic energy
state of three (or more) fermions can not be fully antisymmetric; the dominant
recombination channel has total orbital momentum $\mathcal{L}=1$ where the
recombination rate vanishes at the threshold since it scales with the energy
[14]. The above antisymmetry restrictions do not apply and zero-energy
recombination is possible in the collision of two fermions with a third
distinct particle. However, such a system loses its fermionic character
becoming effectively a system of three distinguishable particles; it is
therefore out of the interest for the present study, except for a very special
choice of the third particle, namely a dimer consisting of the same kind of
fermions as the first two. This choice preserves the purely fermionic
character of the system but implies right away a four-body problem. The zero
kinetic energy state in the two-fermion plus dimer system is not precluded
such that the recombination rate does not vanish in the ultracold limit.
However, one may expect partial suppression due to the antisymmetry of the
system, since the wave function must be antisymmetric also under the exchange
of the free fermions with those building the dimer, a condition that puts
restrictions on the three-cluster system as well. It is therefore interesting
to study the fermion-fermion-dimer recombination as a four-fermion problem, to
evaluate its importance relative to other possible channels and consequences
to ultracold mixtures of fermionic atoms and diatomic molecules. Furthermore,
the results in the unitary limit are expected to be universal, i.e.,
independent of the short-range interaction details, and therefore perfectly
suitable for the future benchmark calculations of the three-cluster
recombination or its time-reverse dimer-dimer breakup reaction in the four-
fermion system.

Section II outlines essential details of the four-fermion collision
calculations, and Sec. III presents the obtained results for the recombination
process. The summary is given in Sec. IV.

##  2 Theory

Elastic scattering of two fermionic dimers was considered in Ref. [15]
providing most accurate results for the dimer-dimer effective range and phase
shifts at finite energy below the breakup threshold. That work employed
rigorous Faddeev-Yakubovsky theory [16] for the four-particle scattering in
the integral form proposed by Alt, Grassberger, and Sandhas (AGS) [17]. The
AGS framework describes the four-particle scattering problem in terms of
subsystem transition operators, namely, the two-particle, i.e., 2+1+1,
transition matrix

$t=v+vG_{0}t,$ (1)

the three-particle, i.e., 3+1 with $\alpha=1$, and two-pair, i.e., 2+2 with
$\alpha=2$, transition operators

$U_{\alpha}=P_{\alpha}G_{0}^{-1}+P_{\alpha}tG_{0}U_{\alpha}.$ (2)

The resulting four-particle transition operators obey the symmetrized AGS
equations; the subset relevant for the present consideration of reactions
involving two dimers reads

$\displaystyle\mathcal{U}_{12}={}$
$\displaystyle(G_{0}tG_{0})^{-1}-P_{34}U_{1}G_{0}tG_{0}\mathcal{U}_{12}+U_{2}G_{0}tG_{0}\mathcal{U}_{22},$
(3a) $\displaystyle\mathcal{U}_{22}={}$
$\displaystyle(1-P_{34})U_{1}G_{0}tG_{0}\mathcal{U}_{12}.$ (3b)

In the above equations the subscripts $\alpha=1$ (2) label the clustering of
the 3+1 (2+2) type, $G_{0}=(E+i0-H_{0})^{-1}$ is the free four-particle
resolvent with the energy $E$ and the kinetic energy operator $H_{0}$, both in
the center-of-mass (c.m.) frame, $v$ is the two-particle potential, $P_{ab}$
is the permutation operator of particles $a$ and $b$, while $P_{\alpha}$ are
combinations of permutation operators [15] that, together with a proper choice
of basis states, ensure the desired antisymmetry of the four-fermion system as
explained in Ref. [15] and references therein.

The most efficient way to solve the AGS equations (3) with short-range forces
is by employing the partial-wave decomposition in the momentum-space, leading
to a system of integral equations with three continuous variables. Those are
the magnitudes of the Jacobi momenta $k_{x}$ , $k_{y}$, and $k_{z}$ describing
the relative motion between particles and/or clusters [18]. The corresponding
orbital angular momenta $l_{x}$, $l_{y}$, and $l_{z}$ and fermion spins
$s_{i}=\frac{1}{2}$ are coupled to build eigenstates of the total angular
momentum $\mathcal{J}$ and its projection $\mathcal{M}$. Different coupling
schemes are used for the 3+1 and 2+2 clustering,
$|k_{x}\,k_{y}\,k_{z}\rangle_{1}\otimes|\\{l_{z}[(l_{y}\\{[l_{x}(s_{1}s_{2})s_{x}]j_{x}\,s_{3}\\}S_{y})J_{y}s_{4}]S_{z}\\}\,\mathcal{JM}\rangle_{1}$
and
$|k_{x}\,k_{y}\,k_{z}\rangle_{2}\otimes|(l_{z}\\{[l_{x}(s_{1}s_{2})s_{x}]j_{x}\,[l_{y}(s_{3}s_{4})s_{y}]j_{y}\\}S_{z})\mathcal{JM}\rangle_{2}$,
respectively, where the remaining quantum numbers such as $j_{x}$ etc. are
angular momenta of the intermediate subsystems as explained in Ref. [15]. The
antisymmetry condition restricts $l_{x}+s_{x}$ (and $l_{y}+s_{y}$ for the
$2+2$ configuration) to even values; $l_{x}=s_{x}=j_{x}=0$ for the $S$-wave
dimer considered in the present work.

An important aspect of the solution is the treatment of integrable kernel
singularities in Eqs. (3) arising in $t$ and $U_{2}$ due to the one- and two-
dimer poles, respectively. Their treatment using the complex-energy method
with special integration weights is taken over from Ref. [19].

The amplitude for the elastic dimer-dimer scattering, its relation to the
phase shift, and effective-range expansion parameters can be found in Ref.
[15]. The amplitude for the fermion-fermion-dimer recombination into two
dimers equals to the three-cluster breakup amplitude

$\begin{split}\langle\Phi_{3}|T_{32}|\Phi_{2}\rangle={}&2\langle\Phi_{3}|[(1-P_{34})U_{1}G_{0}\,t\,G_{0}\,\mathcal{U}_{12}\\\
&{}+U_{2}G_{0}\,t\,G_{0}\,\mathcal{U}_{22}]|\phi_{2}\rangle,\end{split}$ (4)

where $|\Phi_{3}\rangle$ abbreviates the three-cluster channel state and
$|\Phi_{2}\rangle$ the two-dimer channel state whose Faddeev component is
$|\phi_{2}\rangle=G_{0}v|\Phi_{2}\rangle$ obeying also the Faddeev equation
$|\phi_{2}\rangle=G_{0}tP_{2}|\phi_{2}\rangle$.

The amplitude (4) determines the three-cluster breakup and recombination
observables. The definition of the recombination rate $K_{4}$ follows from the
number of recombination events $K_{4}\rho_{d}\rho_{+}\rho_{-}$ per volume and
time, where $\rho_{d}$, $\rho_{+}$, and $\rho_{-}$ are densities of dimers and
of fermions in spin-up and spin-down states, respectively. Of special interest
is the fermion-fermion-dimer recombination rate $K_{4}^{0}$ at the threshold,
i.e., at vanishing three-cluster kinetic energy $E_{3}=E+b_{d}\to 0$. This
implies $k_{y}=k_{z}=0$ and only $l_{y}=l_{z}=0$ states contribute to this
particular channel state $|\Phi_{3}^{0}\rangle$. Furthermore, all the discrete
quantum numbers take their minimum possible values of 0 or $\frac{1}{2}$,
i.e., the angular momentum part of this state in the 2+2 basis reduces to a
single component
$|\Phi_{\mathcal{J}=0}^{0}\rangle=|(0\\{[0(\frac{1}{2}\frac{1}{2})0]0\,[0(\frac{1}{2}\frac{1}{2})0]0\\}0){00}\rangle_{2}$.
Since $\mathcal{J}$ is conserved and the dimer has zero spin, the final two-
dimer state $|\Phi_{2}^{0}\rangle$ is restricted to have exactly the same
angular momentum part $|\Phi_{\mathcal{J}=0}^{0}\rangle$, while the relative
momentum between the dimers is $k_{z}=p_{dd}^{0}=\sqrt{2mb_{d}}$. Under these
conditions the zero-energy fermion-fermion-dimer recombination rate is given
by a single amplitude of the breakup operator in the partial-wave
representation, i.e.,

$K_{4}^{0}={4\pi^{5}}mp_{dd}^{0}|\langle\Phi_{3}^{0}|T_{32}|\Phi_{2}^{0}\rangle|^{2}.$
(5)

It is important to note that, despite angular momentum limitations in the
initial and final channel states, the solution of the AGS equations (3) and
the calculation of the amplitude (4) necessitates the inclusion of higher
partial waves to achieve the convergence. In fact, $l_{y},l_{z}>0$ waves
contribute about 25% to $K_{4}^{0}$ results in the next section. A good
convergence is achieved by including states with $l_{y},l_{z}<4$; it was found
that those with $l_{y},l_{z}=3$ contribute about 0.2%, implying that higher
waves are negligible.

At finite $E_{3}$ a continuum of states contributes to the recombination, they
have to be integrated over either explicitly or implicitly, via the optical
theorem. Below the four-particle threshold the same integral determines also
the dimer-dimer breakup cross section $\sigma_{b}$, one of the standard
observables in the scattering processes. It is thus convenient to express the
fermion-fermion-dimer recombination rate as

$K_{4}=\frac{8\sqrt{2}\pi p_{dd}^{2}}{m^{3}\,E_{3}^{2}}\,\sigma_{b}.$ (6)

The expressions (5) and (6) take into account the identity of two fermions
(dimers) in the initial (final) states as well as the weight factors related
to the spin averaging.

##  3 Results

A fermionic system with large scattering length is realized by neutrons, but
$a\approx-19$ fm is negative and therefore no bound dineutron exists.
Nevertheless, for the present investigation I take a fictitious four-neutron
system with slightly enhanced two-neutron force that supports a bound two-
particle state. Starting from the realistic CD Bonn potential [20] and fermion
mass $m=938.9$ MeV, enhancement factors from 1.35 to 1.105 lead to scattering
length $a$ from 9.1 to 176.1 fm, while the two-particle effective range
$r_{e}$ stays between 2.14 and 2.56 fm. Thus, within this relatively narrow
variation of the potential strength one can explore rather broad range of the
scattering length as well as of the ratio $r_{e}/a$ that quantifies the
importance of finite-range effects. Since the present study is devoted to the
$S$-wave interacting four-fermion universality, for the computational
efficiency the potential in all higher two-fermion partial waves with
$l_{x}>0$ is assumed to be zero.

The results of the above-described calculations for the fermion-fermion-dimer
recombination rate $K_{4}^{0}$ at the threshold are presented in Fig. 1. Since
the three-cluster recombination rate scales as $a^{4}$, a dimensionless
quantity $K_{4}^{0}m/\hbar a^{4}$ is build in order to demonstrate the
universal behavior of this observable. To a good accuracy, better than 1%
below $r_{e}/a=0.1$, it appears to be linear in the finite-range correction
parameter $r_{e}/a$, but the data points beyond this limit slightly violate
this linear dependence. Nevertheless, all data points are well described
including finite-range corrections of a higher order, quadratic in $r_{e}/a$,
resulting in an accurate analytical representation

$K_{4}^{0}\approx\left[10.20-18.27\frac{r_{e}}{a}+3.47\left(\frac{r_{e}}{a}\right)^{2}\right]\,\frac{\hbar
a^{4}}{m}.$ (7)

Obviously, in the considered regime the $(r_{e}/a)^{2}$ term yields only a
minor correction. Note that also the dimer-dimer scattering length and the
effective range parameter show a linear dependence on $r_{e}/a$ near the
unitary limit, independently of the short-range details of the employed
potential [15]. A tentative calculation using the separable interaction model
from Ref. [15] indicates this kind of independence also for $K_{4}^{0}m/\hbar
a^{4}$. Thus, the relation (7) yields not only an accurate value for the
ultracold fermion-fermion-dimer recombination rate in the unitary limit but
also reliably evaluates the importance of the finite-range effects.

Figure 1:  (Color online) Fermion-fermion-dimer recombination rate at zero
energy in dimensionless form as a function of the finite-range parameter
$r_{e}/a$. The symbols are the results of the calculation with the enhanced CD
Bonn potential, the solid curve represents Eq. (7), while the dashed-dotted
curve is a linear approximation.

Another question to be addressed is the energy dependence of the fermion-
fermion-dimer recombination rate. It is shown as a solid curve in Fig. 2 for
$r_{e}/a=0.0456$, since the shape of the energy dependence is quite
insensitive to the $r_{e}/a$ value provided it is below 0.1. The recombination
rate depends weakly on the energy, indicating the absence of resonances as
well as rather insignificant contribution of $l_{y},l_{z}>0$ waves in the
initial three-cluster channel.

It is interesting to compare the three-cluster recombination rates $K_{N}$ in
three- and four-fermion systems. In the $N=3$ case the recombination rate
$K_{3}$ vanishes at threshold as discussed in Sec. I, but the process is
dominated by the total orbital momentum $\mathcal{L}=1$ state and therefore
$K_{3}$ increases with energy as shown by the dashed-dotted curve in Fig. 2.
At very low energy the increase of $K_{3}$ is linear as predicted in Ref. [14]
by analytical considerations, but starts to slow down for $E_{3}>0.05\,b_{d}$.
Only the $\mathcal{L}=1$ component is included in these results, since other
components are suppressed even more strongly, at least as $E_{3}^{2}$ [14],
and were verified numerically to be negligible in the low-energy regime.
Nevertheless, near $E_{3}=0.2\,b_{d}$ the three-fermion recombination rate
$K_{3}$ already exceeds $K_{4}$. Note that the fermion-fermion-dimer
recombination in the $\mathcal{L}=1$ (equivalent to $\mathcal{J}=1$) state is
forbidden, since the final two-dimer state must be symmetric under the
exchange of dimers, which for zero-spin dimers is only possible for even
$\mathcal{L}$ and $\mathcal{J}$ values.

Figure 2:  (Color online) Three-cluster recombination rates in three- and
four-fermion systems, displayed by the dashed-dotted and solid curves,
respectively, as functions of the relative three-cluster kinetic energy. All
quantities are shown in the dimensionless form and correspond to the finite-
range parameter $r_{e}/a=0.0456$.

The fact that the $\mathcal{L}=1$ process, quantified by $K_{3}$, already at
relatively low energy exceeds the $\mathcal{L}=0$ dominated process,
quantified by $K_{4}$, may appear surprising, but has its explanation in the
partial suppression of the fermion-fermion-dimer recombination due to the
four-fermion antisymmetry. Although no antisymmetry restrictions apply to the
dimer directly, the wave function must be antisymmetric under the exchange of
free fermions with those inside the dimer, thereby putting the restrictions on
the dimer indirectly. This is well reflected in the partial cancellation, up
to 90%, of the 3+1 and 2+2 configuration contributions to the breakup or
recombination amplitude (4). This cancellation constitutes also a challenge in
numerical calculations, that require more dense momentum grids and smaller
values for the imaginary part of the energy as compared to the elastic
scattering. Another consequence of the antisymmetry restriction is that the
fermion-fermion-dimer recombination rate significantly smaller than typical
values in other three-cluster systems [1].

Finally, there are important implications of the three-cluster recombination
for the ultracold mixtures of spin $\frac{1}{2}$ fermions and dimers. Let’s
consider an idealized system at zero temperature. If the system consists of
unbound fermions, recombination processes are suppressed and the system
remains stable. However, even if very few dimers are added to this system,
they initiate fermion-fermion-dimer recombination process producing more
dimers, thereby increasing density of dimers and further enhancing dimer
production. Apart from this, the energy $b_{d}$ released in each recombination
event is initially taken by the two outgoing dimers, but after a series of
elastic collisions with fermions is transfered to fermions, that acquire small
but finite energy as well. This implies that also the three-fermion
recombination becomes possible, contributing to further creation of dimers.
Which of these mechanisms is more important, depends on the initial fermion
and dimer densities. If they are of a comparable size, the four-particle
process will dominate, since $K_{4}>>K_{3}$ for vanishing energy. On the
contrary, if the starting point is an ultracold fermionic gas with a very
small admixture of dimers $\rho_{d}^{0}$, the fermion-fermion-dimer
recombination would “ignite” the dimerization process, however, the three-
fermion recombination takes over once the fermions absorb the released energy.
A simple estimation is as follows: At the time point when $\rho_{d}$ exceeds
$\rho_{d}^{0}$ significantly, the released energy is roughly $b_{d}\rho_{d}$
per unit volume, thus, once dissipated between all particles through elastic
two-cluster collisions, it amounts to the kinetic energy per particle of
roughly $b_{d}\rho_{d}/(\rho_{+}+\rho_{-})$. This leads to the number of
three-fermion recombination events per volume and time
$K_{3}\rho_{+}\rho_{-}(\rho_{+}+\rho_{-})\approx(\partial K_{3}/\partial
E)|_{E=0}\,b_{d}\rho_{+}\rho_{-}\rho_{d}$. It is evident from Fig. 2 that
$(\partial K_{3}/\partial E)|_{E=0}\,b_{d}$ exceeds $K_{4}$, thereby
indicating the superiority of the three-fermion recombination under these
conditions.

##  4 Summary

The fermion-fermion-dimer recombination was studied using exact scattering
equations for the four-particle transition operators. They were solved
numerically in the momentum-space partial-wave representation, employing the
complex-energy method with special integration weights for the treatment of
kernel singularities.

In the systems of spin $\frac{1}{2}$ fermions with resonant $S$-wave
interactions (and no dimers in other waves) the recombination of three (or
more) fermions at zero energy is forbidden by the antisymmetry requirement.
However, the fermion-fermion-dimer recombination is only partially suppressed,
having finite rate at the threshold. In the present work it was calculated for
a rather broad range of the two-fermion scattering length and to a good
accuracy found to be linear in the effective range parameter $r_{e}/a$. This
allowed for a reliable extrapolation to the unitary limit as well as the
evaluation of finite-range effects. The energy-dependence of the fermion-
fermion-dimer recombination rate was shown to be weak, in contrast to the
three-fermion recombination. Their interplay was demonstrated to be important
for the dimer production in ultracold mixtures of fermions and dimers.

One of the achievements of the present work, namely, obtained universal result
for the zero energy fermion-fermion-dimer recombination rate, has important
impact for the microscopic scattering description in general. Being
independent of the short-range interaction details, it is well suitable for
benchmarking three-cluster breakup and recombination calculations in four-
particle systems using different methods, that often have their preferred type
of the potential.

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