# Formation of Dimers in Axion-Like Dark Matter Using the Feshbach Resonance

A.M. Gavrilik111e-mail<EMAIL_ADDRESS>A.V. Nazarenko222e-mail:
<EMAIL_ADDRESS>Bogolyubov Institute for Theoretical Physics of NAS of
Ukraine,
14b, Metrolohichna Str., Kyiv 03143, Ukraine

###### Abstract

Within the model of self-gravitating Bose–Einstein condensate (BEC) dark
matter (DM) it is argued that the axion-like self-interaction of ultralight
bosons provides the existence of rarefied and dense phases, which are
predicted earlier on the base of the models with polynomial-like self-
interactions. Associating the very short scattering length in BEC DM with the
predominant participating composites of few DM particles, we attempt to form a
dimer of two particles at a quantum mechanical level, using a smooth
$\mu$-deformation of the axion cosine-like potential and replacing the field-
dependent argument with the distance between particles. Part of the obtained
results concerns potential two-particle scattering with $\mu$-deformed
interaction, and they allow us to focus on a special option with unique values
of the deformation parameter $\mu=1$ and the coupling constant. In this case
of the potential with an infinite scattering length, we get a rather simple
solution for the dimer in the ground state. We involve two-channel scattering
and Feshbach resonance to describe the formation of a dimer in space.
Specifying the parameters of interactions, we reveal a long-lived resonance
that occurs when a pair of particles jumps between the open and closed
scattering channels with close energy values. This indicates the possibility
of participation of such dimers in forming BEC DM halo of galaxies.

dark matter, axions, BEC, deformed potentials, potential scattering, two-
channel scattering, Feshbach resonance

###### pacs:

95.35.+d, 03.75.Hh, 03.65.Nk

##  I Introduction

Axion-like bosons belong to most popular candidates of the Bose–Einstein
condensate (BEC) dark matter (DM), the properties of which are studied in many
theoretical works PWW ; Sin ; Lee ; Hu ; Sahni ; Ferreira ; Bohmer ; SY09 ;
Harko2011 ; axion3 ; axion4 ; axion5 ; Chavanis2 . Their being involved in the
description of astrophysical phenomena should borrow the chiral cosine-like
self-interaction PQ77 ; Wit80 , and also requires engaging the gravity
(usually treated separately of the unification theory), the account of which
leads to breaking the inherent symmetry. Note that there are also a large
number of works, including experimental ones Brad03 , that study the
interaction of axions with other substances and their transformation (due to
the Primakoff effect, see Raff ), as well as their role in cosmology SY09 .

The condensate properties of the axion-like DM, taking into account both the
leading pairwise contributions to the self-action Bohmer ; Harko2011 ; axion5
and the next three-particle corrections Chavanis2 ; GKN20 ; GN21 , are
promising for further consideration and application of axion-like particles.
Having got a number of characteristics, the dilute and dense phases along with
the phase transitions are revealed in BEC Chavanis2 ; GKN20 ; GN21 . Besides,
analysis of these effects and ways to better describe the observables suggests
the existence of molecule-like composites GN22  and the relevance of
deformation-based description muBose1 ; muBose2 ; Nazar . Theoretically, these
possibilities are considered as very important, when dealing with the dark
sector.

Moreover, there are predictions concerning both the first-order phase
transition at zero temperature with changing the interaction parameters and
its influence on the rotation curves of the DM halo of dwarf galaxies GN21 .
Physically, this is associated with quantum fluctuations in regions with a
relatively high number density of ultralight particles, where the three-
particle effects become significant. There are indications that quantum
entanglement may be involved Qent ; GN21 . At first glance, composites would
be also produced under these conditions as well. However, as was shown earlier
GN22 , the appearance of particle complexes (“molecules”) is not supported at
high density because of disintegration stimulated by frequent collisions, but
may be caused by a large scattering length of particle interaction. It is
clear that this information is able to shed light on the characteristics and
behavior of initial DM particles, the nature of which have not yet been
identified.

In general, it is natural to assume that the DM consists of particles of
different sorts, including composites. What we observe and describe now is
mainly the result of self-interaction, that is, a steady state with a
vanishingly small scattering length, confirmed by numerous models based on the
Gross–Pitaevskii equation Harko2011 ; Chavanis2 ; GN21 ; GN22 . Therefore, we
need to explain these peculiarities which can be associated with the particle
states separated by a large energy gap.

Heuristically, formation of the simplest molecules of two and three particles
is explored in Ref. GN22 . Here we have in mind some analogy with scattering
processes in BEC studied in the laboratory Grimm10 . Assuming this to be
admissible, we appeal to the quantum mechanical formation of a molecule
(dimer) of two particles, borrowing the ideas of the Feshbach resonance and
using two scattering channels Fesh62 ; Joa75 ; Yam93 ; Grimm10 . Although it
is appropriate to include auxiliary influences in our consideration, a more
detailed analysis of which is an independent task, our goal is to disclose a
plausible mechanism for the formation of bound states during the Universe
evolution. Thereby, we emphasize the fact that one good potential is not
sufficient to form DM molecules in space.

Anyway, the choice of the self-interaction potential is decisive. Having
gained an idea of the nontrivial phase structure of DM with two- and three-
particle interactions and its manifestations in observables GKN20 ; GN21 , we
want to show here that the model with the cosine-like interaction mentioned
above should also lead to similar consequences. Obviously, the already used
self-interactions of the polynomial form become treated as the expansion terms
of the total potential. It is important that the cosine-like generalization
not only complicates the form of interaction, but also reduces the number of
independent parameters. We focus on different phases of dark matter with
axion-like interaction in the spherically symmetric case, when the main
function we find is the spatial distribution of particles in BEC.

Further, what concerns the problem of the formation of dimers in DM, we follow
our strategy, according to which the structured (composite) particles are
described by deformed functions GM1 ; GM2 ; GM3 . The approach based on
$\mu$-deformation has already been successfully tested in our works muBose1 ;
muBose2 ; Nazar . Here, we replace the ordinary cosine potential of axion-like
bosons with the $\mu$-deformed cosine introduced earlier muBose2 . Of course,
when $\mu=0$, the usual form of the cosine function is restored. The physical
attractivity of such a modification lies in the possibility of smooth
transforming the periodic function into a typical pairwise potential, the form
of which is often encountered in classical and quantum mechanics.

Despite the violation of the symmetry of the corresponding field theory, the
expediency of this deformation can be justified by its efficiency. This indeed
follows by developing the quantum mechanical picture of two-particle
scattering, in which the distance between particles is used as a variable for
the potential instead of the axion field. Devoting a part of our study to the
scattering properties, when the deformation parameter $\mu$ strongly affects
the scattering length, we discover the possibility of the dimer formation from
a pair of DM particles. Namely, we use the deformed potential with suitable
value $\mu=1$ in the two-channel resonance scattering to describe the creation
of a single molecule, leaving aside the many-particle condensate in the
present treatment.

The paper is organized as follows. In Sec. II we show the existence of two
phases of BEC DM on the base of a pair of different initial conditions for the
stationary Gross–Pitaevskii equation with cosine-like and gravitational
interactions at fixed values of the coupling constants and chemical potential.
The $\mu$-deformed self-interaction is introduced in Sec. III instead of the
typical cosine-like one for axions to account for the structured (composite)
particles of DM. The scattering properties of such potentials, which we treat
as the functions of distance between particles, are studied in the Born
approximation for $s$-waves. In the case of $\mu=1$, we find an exact solution
to the problem with attractive potential which corresponds to a two-particle
molecule (dimer) of DM particles. To describe the dimer formation, we appeal
to the resonance scattering in Sec. IV, where the two-channel mechanism is
applied in accordance with Feshbach. This allows to analyze the conditions of
forming DM composites in the Universe. The final Section V is devoted to
discussion of implications and concluding remarks.

##  II Gross–Pitaevskii Equation for Axion-Like DM and Its Solution

We start with formulating stationary macroscopic model of gravitating
Bose–Einstein condensate (BEC) of ultralight bosons with axion-like
interaction, restricting ourselves to the spherical symmetry and the absence
of hydrodynamic flows. Let the BEC with a constant chemical potential
$\tilde{\mu}$ be described by real function $\psi(r)$ of radial variable
$r=|\mathbf{r}|$, and $\psi^{2}(r)$ defineing a local particle density. The
development of $\psi(r)$ in the ball
$B=\\{\mathbf{r}\in\mathbb{R}^{3}|\,|\mathbf{r}|\leq R\\}$ follows from both
vanishing variation of the energy functional $\Gamma$ and the Poisson equation
for the gravitational potential $V(r)$:

$\displaystyle\Gamma=4\pi\int_{0}^{R}\left\\{\frac{\hbar^{2}}{2m}(\partial_{r}\psi)^{2}-\tilde{\mu}\psi^{2}+mV\psi^{2}\right.$
$\displaystyle\hskip
17.07164pt\left.+\frac{U}{v}\left[1-\cos{\left(\sqrt{v}\psi\right)}\right]-\frac{U}{2}\psi^{2}\right\\}\,r^{2}\,\mathrm{d}r;$
(1) $\displaystyle\Delta_{r}V=4\pi Gm\psi^{2},$ (2)

where the radial part of Laplace operator $\Delta_{r}$ and its inverse
$\Delta^{-1}_{r}$ of variable $r$ are

$\displaystyle\Delta_{r}f(r)=\partial^{2}_{r}f(r)+\frac{2}{r}\,\partial_{r}f(r),$
(3)
$\displaystyle\Delta^{-1}_{r}f(r)=-\frac{1}{r}\int_{0}^{r}f(s)\,s^{2}\,\mathrm{d}s-\int_{r}^{R}f(s)\,s\,\mathrm{d}s;$
(4)

$R$ is the radius of the ball, where the matter is located.

The axion-like cosine-interaction is characterized by two constants $U$ and
$v$, which have the dimensions of energy and volume, respectively. In the
particle physics, they are related with the axion mass and decaying constant
$f_{\mathrm{a}}$ as $U=mc^{2}$, $v=\hbar^{3}c/(mf^{2}_{\mathrm{a}})$. Note
their relativistic nature, which is exploited in Chavanis2 .

To analyze the general properties of the model, we reformulate it in
dimensionless variables:

$\displaystyle\xi=\frac{\sqrt{mU}}{\hbar}\,r,\quad\chi(\xi)=\sqrt{v}\,\psi(r),$
(5) $\displaystyle\xi_{B}=\frac{\sqrt{mU}}{\hbar}\,R,\quad
A=4\pi\frac{G\hbar^{2}m}{U^{2}v},$ (6)
$\displaystyle\nu=2u+2A\int_{0}^{\xi_{B}}\chi^{2}(\xi)\,\xi\,\mathrm{d}\xi+1,\quad
u=\frac{\tilde{\mu}}{U},$ (7)

where $\nu$ plays the role of effective chemical potential, which absorbs the
constant term of axion interaction and the gravitational potential at the
center $\xi=0$:

$V_{\mathrm{grav}}(0)=-A\int_{0}^{\xi_{B}}\chi^{2}(\xi)\,\xi\,\mathrm{d}\xi.$
(8)

In our study, $\nu$ is regarded as a free parameter.

The model equations in terms of the wave-function $\chi(\xi)$ and auxiliary
gravitational potential $\Phi(\xi)$ read

$\displaystyle\left(\Delta_{\xi}+\nu\right)\,\chi-2A\Phi\chi-\sin{\chi}=0,$
(9)
$\displaystyle\Phi(\xi)=-\frac{1}{\xi}\int_{0}^{\xi}\chi^{2}(s)\,s^{2}\,\mathrm{d}s+\int_{0}^{\xi}\chi^{2}(s)\,s\,\mathrm{d}s,$
(10)

where $\Delta_{\xi}\Phi(\xi)=\chi^{2}(\xi)$ is satisfied, and $\Phi(0)=0$.

To obtain a finite and stable solution $\chi(\xi)$, the non-linear Eqs.
(9)-(10) should be numerically integrated under the following conditions:
$\chi(0)<\infty$, $\chi^{\prime}(0)=0$, $\chi^{\prime\prime}(0)<0$. For given
$A$ and $\nu$, the finite initial value $\chi(0)=z$ should be positive
solution of the equation:

$2Az^{2}+\left(\nu-\frac{\sin{z}}{z}\right)(\nu-\cos{z})=0,$ (11)

which is derived by substituting
$\chi(\xi)=\chi(0)+\chi^{\prime\prime}(0)\xi^{2}/2$ into (9)-(10) at $\xi\to
0$, and by determining first $\chi^{\prime\prime}(0)$. Note that the
requirement $\chi^{\prime\prime}(0)<0$ is equivalent to imposing
$\nu>\sin{z}/z$.

The absence of a solution $z$ at given pair $(A,\nu)$ means that $\chi(\xi)=0$
everywhere. It happens at $\nu>\nu_{\mathrm{max}}$, where
$\nu_{\mathrm{max}}(A)$ is also found numerically from (11). For
$\nu<\nu_{\mathrm{max}}$, two branches of $\chi_{0}(\nu)$ can occur, which
witness the possibility of two regimes existence and a first-order phase
transition in the model.

Figure 1: The initial value $\chi_{0}=\chi(0)$ (a) and the first zero
$\xi_{B}$ (b) of $\chi(\xi)$ versus parameter $\nu$ for different $A$. The
presented curves are limited by
$\nu^{\mathrm{black}}_{\mathrm{max}}=0.841954176$,
$\nu^{\mathrm{red}}_{\mathrm{max}}=0.818967072$,
$\nu^{\mathrm{blue}}_{\mathrm{max}}=0.795228679$. At $\nu>\nu_{\mathrm{max}}$
one has $\chi_{0}=0$ and $\chi(\xi)=0$. Figure 2: A particular form of the
wave function $\chi(\xi)$ (a) and the effective potential
$V_{\mathrm{eff}}(\xi)$ (b) for the fixed parameters $A$ and $\nu$. Green and
orange lines are obtained with the same $A$ and $\nu$, but differ in the
initial values $\chi_{0}$, see Fig. 1(a). The value of $\chi_{0}$ for green
curves belongs to the upper branch of $\chi_{0}(\nu)$ in Fig. 1(a), while the
orange lines are plotted using $\chi_{0}$ of the lower branch of
$\chi_{0}(\nu)$.

Assuming that gravity is weaker than self-interaction, we consider the
parameter $A\sim 10^{-3}$. Then, analyzing Eq. (11), the two-phase structure
of DM appears for $z>\pi$. Indeed, choosing $A$ as in Fig. 1(a), the upper
branch of $\chi_{0}(\nu)$ is represented by $\chi_{0}\in[2\pi;5\pi/2]$, while
the lower branch of $\chi_{0}(\nu)$ gives us $\chi_{0}$ in the interval
$[3\pi/2;2\pi]$. In the both cases, one has $\nu>\sin{\chi_{0}}/\chi_{0}$.

Although the axion field in QCD is usually assumed to be weak ($\chi_{0}<\pi$)
near the true vacuum, the situation looks different in a non-relativistic
model of DM with Newtonian interaction. Also, there is no invariance there
under the global transformation $\chi\to\chi+2\pi$, violated by gravitation.
We associate such a discrepancy with the consideration of the condensate in a
finite volume (galactic halo of DM), and we admit its correction when
describing gravity as the geometry of space-time.

Indeed, Fig. 1(b) demonstrates the value of (first) zero $\xi_{B}$ of function
$\chi(\xi)$ (that is $\chi(\xi_{B})=0$), which limits the system size in our
model. This is found by integrating the Eqs. (9)–(10) for given $A$ and $\nu$.

The typical solutions for the wave function $\chi(\xi)$ are presented in Fig.
2(a), where $\xi\in[0;\xi_{B}]$. They describe a core with a finite value of
the particle (and mass) density in the center $\xi=0$. Besides, an additional
information may be extracted from an effective potential $V_{\mathrm{eff}}$
(see Fig. 2(b)) as a function of radial variable $\xi$:

$\displaystyle
V_{\mathrm{eff}}(\xi)=V_{\mathrm{grav}}(0)+A\Phi(\xi)+\frac{1}{2}\left(\frac{\sin{\chi(\xi)}}{\chi(\xi)}-1\right),$
(12)
$\displaystyle\left[-\frac{1}{2}\Delta_{\xi}+V_{\mathrm{eff}}(\xi)\right]\chi(\xi)=u\chi(\xi),$
(13)

where Eq. (9) is re-written in the form of Schrödinger equation with the
chemical potential $u$ from (7).

Fig. 2(b) indicates the presence of a minimum of the attractive potentials
colored in green and orange. Since these minima have different depths for the
same $A$ and $\nu$, two macroscopic phases become possible in the system. The
being of the particles in one of the two phases is conditioned by the applied
factors e.g. pressure GKN20 ; GN21 .

Outside the system at $\xi>\xi_{B}$, where there is no the matter,
$V_{\mathrm{eff}}(\xi)$ is continuously extended by the gravitational
potential $-A{\cal N}/\xi$, where ${\cal N}$ is the total number of particles
in the ball $\xi\leq\xi_{B}$.

To estimate the characteristics of our model, we relate them with the
dimensionless parameters (5)-(7). Following the model from Ref. GKN20 , which
allows to separate approaches to describing the core and tail of the DM halo
of (dwarf) galaxies due to the different role of self-interaction, we intend
to estimate the size scale $r_{0}$ and the central mass density $\rho_{0}$ of
the core, so that $r=r_{0}\xi$ and
$\rho(\xi)=\rho_{0}\chi^{2}(\xi)/\chi^{2}_{0}$. At the same time, we need to
control the parameters $U$ and $v$ of axion-like interaction.

As it was stated in Ref. Chavanis2 , the models involving axions for
describing compact objects (such as axion stars) and for cosmology lead to
different parametrizations to be taken into account. First of all, this
concerns the different mass ranges. While cosmological models constrain the
axion mass as $10^{-7}\,\text{eV}<mc^{2}<10^{-2}\,\text{eV}$, ultralight
particle models suggest that $mc^{2}\sim 10^{-22}\ \text{eV}$, typical for the
fuzzy DM. Physically, the choice of a smaller particle mass affects the
formation of certain structures in the Universe FMT08 , which we are trying to
account for. It stimulates us to fix $m\sim 10^{-22}\ \text{eV}\,c^{-2}$.
Although the parameter $U$ must be proportional to $mc^{2}$ to ensure
transmutation, we have no reason to declare identity in the non-relativistic
case. We only admit that the value of $U$ at relatively large magnitude of the
axion field allows the dominance of repulsion, caused by self-interaction,
over gravitation.

Relating the volume $v=\hbar^{3}c/(mf^{2}_{\mathrm{a}})$ with the axion decay
constant $f_{\mathrm{a}}$, we note that
$10^{18}\,\text{eV}<f_{\mathrm{a}}<10^{21}\,\text{eV}$ in cosmology. On the
other hand, its value may be larger in the models with ultralight particles
Chavanis2 .

In our model, it is easy to estimate that

$\displaystyle f_{\mathrm{a}}$ $\displaystyle\simeq$ $\displaystyle
3.304\times
10^{19}\,\text{eV}\,\left[\frac{\rho_{0}}{10^{-19}\,\text{kg}\,\text{m}^{-3}}\right]^{1/2}$
(14)
$\displaystyle\times\left[\frac{mc^{2}}{10^{-22}\,\text{eV}}\right]^{-1}\left[\frac{\chi_{0}}{2\pi}\right]^{-1}$

for $\rho_{0}\sim 10^{-19}\,\text{kg}\,\text{m}^{-3}$ while a mean mass
density is usually assumed to be of the order
$10^{-20}\,\text{kg}\,\text{m}^{-3}$.

The characteristic scale is evaluated to be

$\displaystyle r_{0}$ $\displaystyle\simeq$ $\displaystyle
0.138\,\text{kpc}\,\left[\frac{\rho_{0}}{10^{-19}\,\text{kg}\,\text{m}^{-3}}\right]^{-1/4}\left[\frac{mc^{2}}{10^{-22}\,\text{eV}}\right]^{-1/2}$
(15) $\displaystyle\times\left[\frac{A}{2\times
10^{-3}}\right]^{1/4}\left[\frac{\chi_{0}}{2\pi}\right]^{1/2}.$

Indeed, $r_{0}$ is applicable for measuring the size of the central part of
the DM halo as $R=r_{0}\xi_{B}$, but should be fixed together with the total
mass $M$.

To confirm the presence of a first-order phase transition in the model under
consideration, it is needed to develop a statistical approach GKN20 ; GN21 ,
which is omitted here. It is expected that the discontinuous change in density
at zero temperature will be caused by a change in pressure/compression. The
effect of different phases of DM on the observables, as well as on the
rotation curves, is an interesting problem for a separate study.

Contrary to the background of opening prospects, the results of which can be
predicted using the known models Chavanis2 ; GKN20 ; GN21 , the cosine self-
interaction is worth to study due to additional possibility of smooth
modification, which suggests a number of new effects in DM.

##  III $\mu$-Deformed Interaction

Although we left aside the statistical description of the condensate, it is
foreseen there that the scattering length in the model with cosine-like self-
interaction will be vanishingly small (smaller than the Planck one), as in
similar models with few-particle interactions, even in the Thomas–Fermi
approximation Harko2011 ; Chavanis2 ; GKN20 ; GN21 . It can be assumed that
such a steady state is due to the interaction of the initial DM particles with
a long scattering length leading to the formation of complexes GN22 . In
addition, various macroscopic phases imply the ability of DM structuring, that
is, its various fractions. Taking this into account, we deduce that the
initial interaction between DM particles can be more complicated.

An effective way to extend the abilities of the used interaction is its smooth
deformation. Choosing the kind of deformation to achieve our goals, we rely on
existing experience that indicates the relationship between the so-called
$\mu$-deformation in particle systems and forming composites GM3 .
Technically, this needs introducing a numerical parameter $\mu$, while the
original form of interaction is recovered at $\mu=0$, without involving other
physical entities. In this way we intend to introduce $\cos_{\mu}{x}$ instead
of ordinary function $\cos{x}$.

To describe its properties that are important for physics, we first turn to
the quantum mechanical problem of potential scattering, where $\cos_{\mu}{x}$
plays the role of a pair interaction potential depended on the distance $r$
between particles. Then the useful task is to establish a connection between
the scattering characteristics and the deformation parameter $\mu$. Moreover,
we show that the quantum problem for $\mu=1$ reveals the bound state of a two-
particle molecule as a DM dimer.

###  III.1 Definitions

Using the $\mu$-bracket (see muBose2  and references therein),

$[x]_{\mu}=\frac{x}{1+\mu x},$ (16)

we define the $\mu$-deformed trigonometric functions as

$\sin_{\mu}{x}=\sum\limits_{n=0}^{\infty}(-1)^{n}\frac{x^{2n+1}}{[2n+1]_{\mu}!},\quad\cos_{\mu}{x}=\sum\limits_{n=0}^{\infty}(-1)^{n}\frac{x^{2n}}{[2n]_{\mu}!},$
(17)

where $[n]_{\mu}!=[1]_{\mu}\,[2]_{\mu}\ldots[n]_{\mu}$.

Contracting the series, we can write

$\displaystyle\sin_{\mu}{x}=I_{\mu}(x)\sin{\varphi_{\mu}(x)},\quad\cos_{\mu}{x}=I_{\mu}(x)\cos{\varphi_{\mu}(x)},$
(18) $\displaystyle
I_{\mu}(x)\equiv(1+\mu^{2}x^{2})^{-\frac{1+\mu}{2\mu}},\quad\varphi_{\mu}(x)\equiv\frac{1+\mu}{\mu}\arctan{(\mu
x)}.$

Figure 3: Basic $\mu$-deformed trigonometric functions.

Another equivalent form of $\cos_{\mu}{x}$ is also available:

$\cos_{\mu}{x}=\frac{1}{2}\left[\left(1+\mathrm{i}\mu
x\right)^{-1-\frac{1}{\mu}}+\left(1-\mathrm{i}\mu
x\right)^{-1-\frac{1}{\mu}}\right].$ (19)

Although we are interested in $\mu>0$, note that $\cos_{\mu}{x}=1$ at
$\mu=-1/2$ for any $x$.

Behavior of the deformed functions is shown in Fig. 3, where the
$\mu$-deformed cosine in Fig. 3(b) does imitate the behavior of the pair
interaction potential, associating $x$ with dimensionless distance between
particles, and it continuously changes shape as $\mu$ varies. This fact makes
it useful for studying the properties of particle systems.

Beyond our vision, we get the following asymptotic:

$\cos_{\mu}{x}\sim-\sin{\left(\frac{\pi}{2\mu}\right)}\,(\mu
x)^{-1-\frac{1}{\mu}},\hskip 25.0ptx\to\infty,$ (20)

where $\mu\not=1/(2n)$ for an integer number $n$. In the case of $\mu=1/(2n)$,
the term of the order $\sim(\mu x)^{-\frac{1}{\mu}}$ becomes significant.

Using the asymptotic, the interaction potentials $x^{-n}$ with $n\geq 3$ are
traditionally considered as short-range ones, while potentials with $n\leq 2$
belong to long-range ones LL . According this classification, the function
$\cos_{\mu}{x}$ asymptotically describes short-range potentials for $\mu<1/2$
and long-range potentials for $\mu>1/2$. In the latter case, a large
scattering length $a$ can be achieved, which is justified below using the
relation with the scattering phase shift $\delta$.

Authors of Sade  deduced the special properties of phase shift in $s$-channel
near a collision threshold when wave number $k$ vanishes. If the potential
varies as $x^{-n}$ at large $x$, then $\tan{\delta(k)}\sim k$ for $n\geq 3$,
and $\tan{\delta(k)}\sim k^{n-2}$ for $n\leq 3$. This tells us what happens,
when computing the scattering length $a$ for $n=1+1/\mu$:

$a=-\lim\limits_{k\to 0}\frac{\tan{\delta(k)}}{k}.$ (21)

Requiring a unitary regime at $a\to\pm\infty$, we single out the case $\mu=1$
when $\cos_{\mu}{x}\simeq-x^{-2}$ is similar to the Efimov hyperspherical
potential with interesting scaling properties BrH06 . We pay special attention
to this situation below.

###  III.2 Potential Scattering Properties

Let the interaction potential of two particles, which depends on the distance
$r$ between them, be equal to

$V_{\mu}(r)=U_{0}\cos_{\mu}{\frac{r}{r_{0}}},$ (22)

where $U_{0}=V_{\mu}(0)>0$ and $r_{0}$ are constant parameters.

Then, the stationary Schrödinger equation for two identical particles in the
center-of-mass reference frame takes on the form:

$-\frac{\hbar^{2}}{2m}\Delta\Psi+V_{\mu}(r)\Psi=E\Psi,$ (23)

where $\Delta$ is the Laplace operator in spherical coordinates
$(r,\theta,\varphi)$; $m$ is the reduced mass of two particles.

Focusing on the study of low-energy processes in the spherically symmetric
problem, we consider only the $s$-channel component $\psi(r)$ of the total
wave function $\Psi(r,\theta,\varphi)=\psi(r)+\Psi_{L>0}(r,\theta,\varphi)$,
discarding the series $\Psi_{L>0}$ in spherical harmonics with angular
momentum $L>0$ along with the corresponding centrifugal term in equation. This
leads to the one-dimensional equation:

$-\frac{\hbar^{2}}{2m}\Delta_{r}\psi+V_{\mu}(r)\psi=E\psi,\hskip
25.0pt\Delta_{r}=\frac{1}{r}\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}r,$ (24)

where $\Delta_{r}$ is the radial part of Laplace operator.

For convenience, we use dimensionless variables as $x$ (distance), $k$ (wave
number), and $g$ (coupling constant):

$x=\frac{r}{r_{0}},\quad E=\frac{\hbar^{2}k^{2}}{2mr^{2}_{0}},\quad
g=\frac{2mr^{2}_{0}U_{0}}{\hbar^{2}}.$ (25)

Substituting $\psi(r)=\chi(x)/x$, we arrive at

$\frac{\mathrm{d}^{2}\chi(x)}{\mathrm{d}x^{2}}+k^{2}\chi(x)=g\chi(x)\cos_{\mu}{x},$
(26)

where we can interpret the right hand side as the inhomogeneous term within
the scattering problem at large $x$.

To analyze the solution for an arbitrary deformation parameter $\mu>0$, we
turn to the general formalism, transforming the Schrödinger equation into a
matrix (integral) one. We apply the orthogonal basis of spherical Bessel
functions:

$\int_{0}^{\infty}j_{0}(ux)\,j_{0}(vx)\,x^{2}\mathrm{d}x=\frac{\pi}{2uv}\delta(u-v),$
(27)

where $u$ and $v$ are real; the function $j_{0}(x)=\sin{x}/x$ such that

$\Delta_{x}j_{0}(kx)+k^{2}j_{0}(kx)=0.$ (28)

Therefore, the following Hankel transform takes place:

$\displaystyle{\tilde{f}}(k)$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\int_{0}^{\infty}f(x)\,j_{0}(kx)\,x^{2}\mathrm{d}x,$
(29) $\displaystyle f(x)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}{\tilde{f}}(k)\,j_{0}(kx)\,k^{2}\mathrm{d}k.$
(30)

In dimensionless variables, Eq. (24) is reduced to the form:

$\displaystyle(q^{2}-k^{2}){\tilde{\psi}}_{k}(q)+g\int_{0}^{\infty}U_{\mu}(q,q_{*}){\tilde{\psi}}_{k}(q_{*})\,q^{2}_{*}\mathrm{d}q_{*}=0,$
(31) $\displaystyle
U_{\mu}(q,q_{*})=\frac{2}{\pi}\int_{0}^{\infty}j_{0}(qx)\,\cos_{\mu}(x)\,j_{0}(q_{*}x)\,x^{2}\mathrm{d}x,$
(32)

where $U(q,q_{*})$ is the matrix element of the potential $\cos_{\mu}{x}$ and
depends on absolute values of $q$ and $q_{*}$.

Neglecting the dependence on angle $\theta$ between wave vectors $\mathbf{q}$
and $\mathbf{q}_{*}$, $U(q,q_{*})$ takes into account the first term of
expansion:

$\displaystyle
j_{0}(xQ(\theta))=\sum\limits_{\ell=0}^{\infty}(2\ell+1)j_{\ell}(qx)\,j_{\ell}(q_{*}x)\,P_{\ell}(\cos{\theta}),$
(33) $\displaystyle Q(\theta)=\sqrt{q^{2}+q^{2}_{*}-2qq_{*}\cos{\theta}},$
(34)

where $P_{\ell}(z)$ is the Legendre polynomial; $P_{0}(z)=1$.

In fact, the matrix $U(q,q_{*})$ admits two fixed values of scattering angle
$\theta$ which give us $Q(0)=|q-q_{*}|$ and $Q(\pi)=|q+q_{*}|$.

Our calculations yield:

$\displaystyle U_{\mu}(q,q_{*})$ $\displaystyle=$
$\displaystyle\frac{1}{2qq_{*}}[T_{\mu}(q-q_{*})-T_{\mu}(q+q_{*})];$ (35)
$\displaystyle T_{\mu}(q)$ $\displaystyle\equiv$
$\displaystyle\frac{2}{\pi}\int_{0}^{\infty}\cos{(qx)}\,\cos_{\mu}{x}\,\mathrm{d}x$
(36) $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(1/\mu)}\left(\frac{|q|}{\mu}\right)^{\frac{1}{\mu}}\exp{\left(-\frac{|q|}{\mu}\right)}.$

where it is convenient to substitute the expression in terms of the Gauss
hypergeometric function:

$\cos_{\mu}{x}={}_{2}F_{1}\left(\frac{1}{2}+\frac{1}{2\mu},1+\frac{1}{2\mu};\frac{1}{2};-\mu^{2}x^{2}\right).$
(37)

The absolute value $|q|$ of a real $q$ in (36) provides
$T_{\mu}(-q)=T_{\mu}(q)$ in accordance with the definition. Note also that
$U_{\mu}(q,-q_{*})=U_{\mu}(q,q_{*})$.

Besides, it is useful to define

$u_{\mu}(q)\equiv\lim\limits_{q_{*}\to
0}U_{\mu}(q,q_{*})=\frac{q-1}{q^{2}\mu}\,T_{\mu}(q),$ (38)

in order to obtain an integral form of writing $\cos_{\mu}{x}$:

$\cos_{\mu}{x}=\int_{0}^{\infty}u_{\mu}(q)\,j_{0}(qx)\,q^{2}\mathrm{d}q.$ (39)

To solve Eq. (31), we substitute the expression LL :

${\tilde{\psi}}_{k}(q)=\frac{1}{q^{2}}\delta(q-k)+{\tilde{\eta}}_{k}(q),$ (40)

where, due to Eq. (30), the first term results in $j_{0}(kx)$ that corresponds
to the $s$-component of plane wave.

To soften the singularities of Eq. (31), we set the unknown function
${\tilde{\eta}}_{k}(q)$ as LL

${\tilde{\eta}}_{k}(q)=\frac{F(k,q)}{q^{2}-k^{2}}.$ (41)

The new function $F(k,q)$ must satisfy the equation:

$F(k,q)=-gU_{\mu}(k,q)-g\int_{0}^{\infty}\frac{F(k,q_{*})U_{\mu}(q_{*},q)}{q^{2}_{*}-k^{2}}\,q^{2}_{*}\mathrm{d}q_{*}.$
(42)

Restoring the dependence of wave function on space, let us substitute (41)
into (30) to obtain:

$\displaystyle\eta_{k}(x)$ $\displaystyle=$
$\displaystyle-\frac{\mathrm{i}}{2x}\int_{0}^{\infty}\frac{F(k,q)\mathrm{e}^{\mathrm{i}qx}-F(k,-q)\mathrm{e}^{-\mathrm{i}qx}}{q^{2}-k^{2}}\,q\mathrm{d}q$
(43) $\displaystyle=$
$\displaystyle-\frac{\mathrm{i}}{2x}\int_{-\infty}^{\infty}\frac{F(k,q)\mathrm{e}^{\mathrm{i}qx}}{q^{2}-k^{2}}\,q\mathrm{d}q,$

where we have used that $F(k,-q)=F(k,q)$.

Figure 4: Integration contours for calculating the wave function
$\eta_{k}(x)$. The contour $\Gamma_{+}$ takes into account a positive momentum
$k$ of outgoing wave, while $\Gamma_{-}$ is used for incoming wave.

It is useful to decompose $\eta_{k}(x)$ into a sum of two integrals:

$I_{\pm}(x)=-\frac{\mathrm{i}}{4x}\int_{-\infty}^{\infty}\frac{F(k,q)\mathrm{e}^{\mathrm{i}qx}}{q\mp
k}\,\mathrm{d}q.$ (44)

Computing the indefinite integrals (44), we treat $q$ as a complex variable
and choose the corresponding integration contours $\Gamma_{\pm}$ in the
complex plane, sketched in Fig. 4, to provide decaying factor $\exp{(-x\,{\rm
Im}\,q)}$. The integrals are then determined by two residues (multiplied by
$2\pi\mathrm{i}$), which give us

$\eta_{k}(x)=\frac{\cos{(kx)}}{x}\,\frac{\pi}{2}F(k,k).$ (45)

Thus, this expression also consists of two terms which correspond to the two
scattering angles $\theta=0$ and $\theta=\pi$ in $q=k\cos{\theta}$.

As it follows from Eq. (45), the $s$-scattering amplitude is given by
expression $f(k)=(\pi/2)F(k,k)$, permitting us to find the scattering length
$a$.

In an effort to obtain $F(k,k)$ as a series in powers of $g$, Eq. (42) can be
solved by using an iterative procedure with the initial value
$F^{(0)}(k,q)=-gU_{\mu}(k,q)$. Limiting ourselves by this initial stage, we
obtain that

$f^{(0)}(k)=g\frac{\pi\mu^{-\frac{1}{\mu}}}{4\Gamma(1/\mu)}(2|k|)^{\frac{1}{\mu}}k^{-2}\mathrm{e}^{-\frac{2|k|}{\mu}}.$
(46)

Note the scaling law $f^{(0)}\sim k^{\frac{1}{\mu}-2}$ at small $k$.

Using that $kf^{(0)}(k)=\tan{\delta^{(0)}(k)}$, where $\delta^{(0)}(k)$
denotes the phase shift, we get $\psi_{k}(x)$ for large $x$ by summing
$j_{0}(kx)$ and $\eta_{k}(x)$ (see (40)):

$\psi_{k}(x)=C_{k}\frac{\sin{\left[kx+\delta^{(0)}(k)\right]}}{kx},$ (47)

where $C_{k}$ is a normalization.

The phase shift is evaluated as

$\delta^{(0)}(k)=\arctan{\left[\frac{g\pi}{4\Gamma(1/\mu)}\left(\frac{2}{\mu}\right)^{\frac{1}{\mu}}k^{\frac{1}{\mu}-1}\mathrm{e}^{-\frac{2k}{\mu}}\right]}.$
(48)

We note a weak dependence of this on small $k$ at $\mu=1$:

$\delta^{(0)}_{\mu=1}(k)=\arctan{\left(\frac{g\pi}{2}\mathrm{e}^{-2k}\right)},$
(49)

when $V_{\mu=1}(r_{0}x)\simeq(-g)\,x^{-2}$ at $x\gg 1$, see Fig. 3(b).

Let us relate the real-valued amplitude $f^{(0)}(k)$ with the scattering
amplitude $f_{\rm B}(2k\sin{(\theta/2)})$ in the Born approximation, where
$f_{\rm B}(q)=-(g\pi/2)\,u_{\mu}(q)$ accordingly to its definition LL  and Eq.
(38). In fact, we need to extract $f^{(0)}(k)\sim P_{0}(\cos{\theta})$ from
$f_{\rm B}$, using the orthogonality property of the Legendre polynomials
$P_{\ell}(z)$.

One has

$f^{(0)}(k)=\frac{1}{2}\int_{0}^{\pi}f_{\rm
B}\left(2k\sin{\frac{\theta}{2}}\right)\,\sin{\theta}\mathrm{d}\theta,$ (50)

where we have substituted $P_{0}(\cos{\theta})=1$. This formula is also
confirmed numerically.

Thus, the amplitude $f^{(0)}(k)$ is the $s$-component of the Born amplitude,
and then we can evaluate the scattering length (in units of $r_{0}$) as a
function of $\mu$ using the relation:

$a^{(0)}=-\lim_{k\to 0}f^{(0)}(k).$ (51)

We have that $a^{(0)}\to 0$ for $\mu<1/2$; $a^{(0)}=-4g\pi$ at $\mu=1/2$;
$a^{(0)}\to-\infty$ for $\mu>1/2$. Therefore, the potential
$\cos_{\mu}(r/r_{0})$ demonstrates a long-range attraction for $\mu\geq 1/2$
and reveals the unitary regime with $1/a^{(0)}=0$ if $\mu>1/2$.

###  III.3 Case of $\mu=1$

Let us look at the case of $\mu=1$ in more detail, when

$\cos_{\mu=1}{x}=\frac{1-x^{2}}{(1+x^{2})^{2}}.$ (52)

This function varies within the range $[1;-1/8]$, see Fig. 5(a).

We turn again to the Schrödinger equation (26), imposing conditions
$\chi(0)=0$ and $\chi^{\prime}(0)=c_{g}$, in order to get $\psi=\chi(x)/x$.
Then the analytical unnormalized solution for $\mu=1$ is given by

$\displaystyle\chi_{k}(x)$ $\displaystyle=$ $\displaystyle
c_{g}\,x\,(1+x^{2})^{\frac{1}{2}+\frac{1}{2}\beta}\times$ (53)
$\displaystyle\times
H_{C}\left(0,\frac{1}{2},\beta,-\frac{k^{2}}{4},\frac{k^{2}}{4}-\frac{g}{4}+\frac{1}{2},-x^{2}\right),$

here $\beta=\sqrt{1-2g}$.

The Heun confluent function $H_{C}(\alpha,\beta,\gamma,\delta,\eta,z)$
(altogether with $z^{-\beta}H_{C}(\alpha,-\beta,\gamma,\delta,\eta,z)$) is a
local Frobenius solution of the problem:

$\displaystyle
y^{\prime\prime}+\left(\alpha+\frac{\beta+1}{z}+\frac{\gamma+1}{z-1}\right)y^{\prime}+\left(\frac{\nu}{z-1}+\frac{\mu}{z}\right)y=0,$
$\displaystyle\delta=\mu+\nu-\alpha\frac{\beta+\gamma+2}{2},$
$\displaystyle\eta=\alpha\frac{\beta+1}{2}-\frac{\beta+\gamma+\beta\gamma}{2}-\mu.$

Particularly, we would like to note special unnormalized solution
$\phi_{+}(x)$ at $g=3/8$:

$\displaystyle-\frac{\mathrm{d}^{2}\phi_{+}}{\mathrm{d}x^{2}}+\frac{3}{8}\frac{1-x^{2}}{(1+x^{2})^{2}}\phi_{+}=0,\quad\phi_{+}(0)=0,$
(54)
$\displaystyle\phi_{+}(x)=A_{+}(1+x^{2})^{1/4}\sin{\left[\frac{\sqrt{2}}{4}\ln{\left(x+\sqrt{1+x^{2}}\right)}\right]}.$

The total wave function is then $\psi_{+}=\phi_{+}(x)/x$.

For small positive $k$ (and $g$), we construct oscillating asymptotic solution
using phase shift (49):

$\psi^{\rm
as}_{k}(x)=\frac{A_{k}}{x}\sin{\left[kx+\arctan{\left(\frac{g\pi}{2}\mathrm{e}^{-2k}\right)}\right]},$
(55)

where $A_{k}$ is found from the equality $\psi^{\rm
as}_{k}(x_{0})=\chi_{k}(x_{0})/x_{0}$ for some fixed $x_{0}\gg 1$.

The case $k=0$ requires separate consideration. To describe the asymptotic in
this situation, we appeal to the auxiliary equation:

$-\frac{\mathrm{d}^{2}\chi^{\rm
as}_{0}}{\mathrm{d}x^{2}}-\frac{g}{x^{2}}\chi^{\rm as}_{0}=0.$ (56)

Its solutions are $x^{\alpha_{+}}$ and $x^{\alpha_{-}}$ with
$\alpha_{\pm}=(1\pm\sqrt{1-4g})/2$. Then a stable and finite solution can be
guaranteed at $g\geq 1/4$. Substituting $g=1/4+s^{2}$, $s\in\mathbb{R}_{+}$,
and combining linearly $x^{\alpha_{\pm}}$, we arrive at $\psi^{\rm
as}_{0}=\chi^{\rm as}_{0}/x$:

$\psi^{\rm as}_{0}(x)=\frac{A_{0}}{\sqrt{x}}\sin{(s\ln{x}+d_{0})},$ (57)

where $A_{0}$ and $d_{0}$ are arbitrary constants.

It is useful to compare (54) at $g=3/8$ with $\chi^{\rm as}_{0}$. Indeed,
function $\phi_{+}(x)$ for $x\gg 1$ tends to $\chi^{\rm as}_{0}(x)$ at
$s=\sqrt{2}/4$ and $d_{0}=s\ln{2}$.

Figure 5: Panel (a): Interparticle potential $\cos_{\mu=1}{x}$ (violet line)
and $-1/x^{2}$ (pink line). Panel (b): Wave functions $\psi_{k}$ and their
asymptotics at $\mu=1$ for different $k$: red and violet lines for $k=0$;
green and black lines for $k=1/4$. The fixed parameters are $g=1/2$ and
$c_{1/2}=1$.

We would like to test the asymptotic (57) by determining the parameters
$A_{0}$ and $d_{0}$ on the base of two fixed points of the exact curve
$\psi_{0}(x)$. It leads to the equations:

$\displaystyle\arcsin{\frac{\sqrt{x_{2}}\,\psi_{0}(x_{2})}{A_{0}}}-\arcsin{\frac{\sqrt{x_{1}}\,\psi_{0}(x_{1})}{A_{0}}}=s\ln{\frac{x_{2}}{x_{1}}},$
(58) $\displaystyle
d_{0}=\arcsin{\frac{\sqrt{x_{2}}\,\psi_{0}(x_{2})}{A_{0}}}-s\ln{x_{2}}.$ (59)

The first transcendent equation determines $A_{0}$, while the latter returns
the value of phase $d_{0}$. Note that $d_{0}$ and $A_{0}$ are not
independently fixed.

Then, the admissible behavior of two particles is described by the wave
functions as in Fig. 5(b), where $g=1/2$ and $s=1/2$ for definiteness. It is
seen that their maximum values are located near the potential minimum (see
Fig. 5(a)) and depends on the energy (or wave number $k$). A higher energy
value allows particles to approach each other, overcoming repulsion at short
distances.

We see in Fig. 5(b) that the asymptotics (55) and (57) are valid at $x\gg 1$.
In Eq. (55) we have used the phase shift (49) which is derived analytically.
The amplitude $A_{1/4}=3.538$ for $k=1/4$ is determined at $x_{0}=20$. To fix
the parameters of (57), we choose the points $x_{1}=20$, $x_{2}=30$ and obtain
$A_{0}=-2.275$ and $d_{0}=-2.867$.

We provide the applicability of the phase shift $\delta^{(0)}(k)$ in the Born
approximation by the smallness of the coupling $g<1$. However, dealing with a
problem with an infinite scattering length $a$, the dependence of phase shift
on running $g$ can be more complicated BrH06 .

Besides, the accurate derivation of the phase shift $d_{0}$ still requires a
detailed study. We only note the connection between the phase of the wave
function (57) and the phase in semiclassical approximation (with a non-zero
orbital number $l$ in general) LL :

$\int_{x_{0}}^{x}\sqrt{k^{2}-\frac{(l+1/2)^{2}}{x^{2}}-g\cos_{\mu}{x}}\,\mathrm{d}x+\frac{\pi}{4},$
(60)

where $x_{0}$ is the zero of the expression under square root.

Setting $l=0$, $k=0$, and $\cos_{\mu=1}{x}\simeq-x^{-2}$ for large $x$, this
phase takes the form

$\displaystyle s\ln{x}-s\ln{x_{0}}+\frac{\pi}{4},$ (61) $\displaystyle
x_{0}=\sqrt{\frac{2g+1+2\sqrt{g^{2}+2g}}{4g-1}},\quad s=\sqrt{g-1/4}.$

Although it may need some improvement to apply.

Let us consider a problem with a negative coupling constant $g=-|g|$, when
there are an attractive well in the range $x\in[0;1]$ and a positive-valued
barrier (with a maximum $|g|/8$) for $x>1$. At zero energy ($k=0$), the
barrier width and positive scattering length $a$ become infinite.

Figure 6: Interparticle potential $V=-12\cos_{\mu=1}{x}$ (pink line) and
normalized wave function (violet line) as functions of interparticle distance
$x$. Dashed line for $x\in[0;1]$ indicates classically admissible region of
the problem (62).

Analyzing the solution (53), we reveal a stable, physically correct solution
$\phi_{-}(x)$ in the case of $|g|=12$. This unique value of $g$ simplifies the
problem considerably:

$\displaystyle-\frac{\mathrm{d}^{2}\phi_{-}}{\mathrm{d}x^{2}}-12\,\frac{1-x^{2}}{(1+x^{2})^{2}}\,\phi_{-}=0,\quad\phi_{-}(0)=0,$
(62)
$\displaystyle\phi_{-}(x)=\sqrt{\frac{2}{\pi}}\frac{4x}{(1+x^{2})^{2}},\quad\int_{0}^{\infty}|\phi_{-}(x)|^{2}\mathrm{d}x=1.$
(63)

Function $\phi_{-}(x)$ determines $\psi_{-}=\phi_{-}(x)/x$.

The dependence presented in Fig. 6 allows us to interpret the obtained
solution as a two-particle “molecule” in the ground state without engaging any
chemical bond in the usual sense. We also have $\langle x^{2}\rangle=1$ that
corresponds to classically admissible region.

We get also unnormalized solution for non-zero energy $E=k^{2}\geq 0$ in the
terms of Heun function:

$\chi_{k}(x)=\frac{x}{(1+x^{2})^{2}}\,H_{C}\left(0,\frac{1}{2},-5,-\frac{k^{2}}{4},\frac{7}{2}+\frac{k^{2}}{4},-x^{2}\right),$
(64)

which generalizes $\phi_{-}(x)$ and satisfies the equation:

$-\frac{\mathrm{d}^{2}\chi_{k}}{\mathrm{d}x^{2}}-12\,\frac{1-x^{2}}{(1+x^{2})^{2}}\,\chi_{k}=k^{2}\chi_{k},\quad\chi_{k}(0)=0.$
(65)

In the energy range $0<E<E_{\rm thr}$, where the threshold $E_{\rm thr}=3/2$
corresponds to the potential maximum, the wave function
$\psi_{k}(x)=\chi_{k}(x)/x$ has a long-wavelength tail with a large but finite
amplitude, and its shape in the interval $x\in[0;1]$ does not change radically
in comparison with Fig. 6. In fact, it describes an excited system.

On the other hand, the function $\psi_{k}(x)$ at $E>E_{\rm thr}$ has an
oscillating damped tail which is typical for scattering.

Thus, we identify the problem (62)–(63) with the description of DM dimer. We
admit that a similar solution exists for $\mu<1$. However, in the case of
$\mu=1$, it has the simplest form suitable for further use.

##  IV Dimer Creation Due to Resonance Scattering

Actually, resonance scattering implies the detection of (quasi)discrete energy
levels admitted by the interaction potential. Indeed, it would be interesting
to take into account (quasi)bound states in scattering for a number of
deformed potentials $\cos_{\mu}(x)$, but we do not touch on this. Here the
(quasi)discrete level arise differently.

This study is addressed to the last problem of the previous section, which
suggests the existence of a “molecule” of two DM particles. We would like to
consider in detail the mechanism of its formation, because one good potential
is not enough to form a bound state of DM particles.

Our approach inherits ideas of the Feshbach resonance Fesh62 , which uses at
least two channels of scattering, one open and one closed channels. In fact,
it allows one to create (quasi)discrete level and turn a pair of scattered
particles into a “molecule” and vice versa under auxiliary influence. Despite
questions about the nature of interactions in DM, such an approach seems to be
appropriate, since it looks difficult to bring directly a particle with zero
energy into the scope of potential $-12\cos_{\mu=1}(x)$ of another particle.

We assume that a pair of spinless particles in a collision jumps between open
and closed channels. When the total energy exceeds the open channel threshold
($E=0$), the open channel in such a system is both an incoming and an outgoing
channel. Scattering reveals the Feshbach resonance when the energy of the
bound state of the closed channel is close to the threshold of the open
channel. Due to the coupling of the channels, the unperturbed bound state of
the closed channel becomes dressed, taking into account an interaction. This
dressed state is treated as a (quasi)bound state of the entire scattering
system. Scattered particles temporarily pass into a (quasi)bound state and
return to the open channel after a characteristic time $\tau=2/\Gamma$,
determined by the decay width $\Gamma$ of the (quasi)bound state.

Formulating our model based on the stationary Schrödinger equation for
$s$-wave functions, we fix one particle at the point $x=0$, and the partner
particle is moving and is characterized by the relative radial coordinate
$x\geq 0$. We assume that this particle is in one of two channels. A closed
channel in the absence of external interaction describes a molecule in the
ground state and has energy $E^{(1)}=0$, and an open channel corresponds to
elastic wave scattering due to another interaction and initially has a small
energy $E^{(2)}=k^{2}$ determined by the relative momentum $k$. Between these
channels there is an energy gap $Q=E^{(2)}-E^{(1)}>0$, which can be reduced
with the help of external influence. In general, there are three different
interactions in such a system, and $E^{(1)}$ and $E^{(2)}$ are not energy
levels of the same potential. For this reason, the interaction parameters must
be tuned to obtain the desired effect.

We describe the relative motions of a particle in two channels, adopting the
Fock-space representation Yam93 :

$\mathbb{H}X=EX,\hskip 25.0pt\mathbb{H}=\left(\begin{array}[]{cc}H_{\rm
m}&W\\\ W^{\dagger}&H_{\rm wv}\end{array}\right),$ (66)

where $H_{\rm m}$ and $H_{\rm wv}$ are the Hamiltonians of the molecule
(closed channel) and the scattered wave (open channel), respectively. The
coupling between channels is represented by $W$, and it is associated with
extra force.

We account for the energy gap $Q$ in Eq. (66) by defining the Hamiltonian of
molecule as $H_{\rm m}=H_{1}+Q$, where

$H_{1}=-\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}-12\,\frac{1-x^{2}}{(1+x^{2})^{2}}.$
(67)

In a sense, the system is doubly degenerate at $W=0$ due to existing two
independent wave functions for the same eigenvalue $E$:

$X^{(1)}=\left(\begin{array}[]{c}\chi^{(1)}\\\ 0\end{array}\right),\hskip
25.0ptX^{(2)}=\left(\begin{array}[]{c}0\\\ \chi^{(2)}\end{array}\right),$ (68)

which are evidently orthogonal in this representation. We can immediately
identify $\chi^{(1)}$ with $\phi_{-}$ from Eq. (63). In principle, we should
write $X=X^{(1)}\cos{\alpha}+X^{(2)}\sin{\alpha}$ with some $\alpha$ to
normalize the total wave function $X$ with the respect to Fock representation.

As shown in the previous section, the essential spatial interval for the
existence of a molecule is $x\in[0;1]$. Therefore, significant processes take
place in this region, which we call the resonance zone. For this reason, we
concentrate the external force, parametrized by $\omega$, there:

$W(x)=-\omega^{2}\,\theta(1-x),\hskip 25.0ptW^{\dagger}=W.$ (69)

Similarly, we describe an interaction in the open channel by using the square
well potential for simplicity:

$V_{\rm sq}(x)=-V\,\theta(1-x),\quad V>0.$ (70)

In fact, all quantities in such a system are divided into two components
belonging either to interval $x\in[0;1]$ or to interval $x\in[1;\infty)$,
which we label as “$<$” and “$>$” with the respect to separating point $x=1$,
respectively. Then, the wave functions in the channels numbered by
$\alpha=1,2$ are decomposed as

$\chi^{(\alpha)}(x)=\theta(1-x)\,\chi^{(\alpha)}_{<}(x)+\theta(x-1)\,\chi^{(\alpha)}_{>}(x).$
(71)

We require that $\chi^{(\alpha)}_{<}(0)=0$, and connect functions in
separating point $x=1$ by the merging condition:

$\left.\frac{\mathrm{d}}{\mathrm{d}x}\,\ln{\chi^{(\alpha)}_{<}(x)}\right|_{x=1}=\left.\frac{\mathrm{d}}{\mathrm{d}x}\,\ln{\chi^{(\alpha)}_{>}(x)}\right|_{x=1},$
(72)

to guarantee an equality of derivatives and proportionality of the functions
in the left and right sides of (72).

Before proceeding, we recall the known results for the open channel
($\alpha=2$) in the absence of coupling $W$. The scattering characteristics
follow from the equation:

$H_{\rm
wv}\chi^{(2)}\equiv\left(-\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+V_{\rm
sq}(x)\right)\chi^{(2)}=E\chi^{(2)}.$ (73)

As mentioned above, solution of Eq. (73) is given by

$\displaystyle\chi^{(2)}(x)=\theta(1-x)\,\chi^{(2)}_{<}(x)+\theta(x-1)\,\chi^{(2)}_{>}(x),$
(74) $\displaystyle\chi^{(2)}_{<}(x)=\frac{\sin{Kx}}{K},\quad\
\chi^{(2)}_{>}(x)=\frac{\sin{K}}{K}\,\frac{\sin{(kx+\delta)}}{\sin{(k+\delta)}},$
(75)

where $K=\sqrt{E+V}$ and $k=\sqrt{E}$. When $E<0$, $\chi^{(2)}_{>}$ behaves as
$\exp{(-x\sqrt{|E|})}$. The phase shift $\delta$ is derived from the merging
condition (72):

$K\cot{K}=k\cot{(\delta+k)}.$ (76)

The scattering length $a_{V}$ of potential $V_{\rm sq}(x)$ is

$a_{V}=1-\frac{\tan{\sqrt{V}}}{\sqrt{V}}.$ (77)

Note that $a_{V}$ demonstrates discontinuous behavior at $\sqrt{V}=\pi/2$. We
exclude from our consideration this “zero energy resonance”.

The scattering matrix $S=\mathrm{e}^{2\mathrm{i}\delta}$ and amplitude $f$ are

$S=\mathrm{e}^{-2\mathrm{i}k}\,\frac{K\cot{K}+\mathrm{i}k}{K\cot{K}-\mathrm{i}k},\hskip
25.0ptf=\frac{\mathrm{e}^{2\mathrm{i}\delta}-1}{2\mathrm{i}k}.$ (78)

Thus, we admit a single bound state in the closed channel and a continuum of
spherical waves of relative momentum $k$ between the two particles in the open
channel. Taking into account the complexity of the molecule potential for
calculations, we describe the Feshbach resonance in the first approximation.

The set of equations in whole admissible space is

$\displaystyle(H_{1}+Q-E)\chi^{(1)}+W\chi^{(2)}=0,$ (79) $\displaystyle(H_{\rm
wv}-E)\chi^{(2)}+W^{\dagger}\chi^{(1)}=0.$ (80)

For convenience, we further use the bra- and ket-vectors to simplify the
notation of matrix elements.

In the first approximation, we put Joa75 :

$|\chi^{(1)}\rangle=\lambda|\phi_{-}\rangle,$ (81)

where $\lambda$ is a complex constant which should be found.

Taking into account (81) and acting by $\langle\phi_{-}|$ on Eq. (79), one has

$\lambda=\frac{\langle\phi_{-}|W|\chi^{(2)}_{<}\rangle}{E-Q},$ (82)

where we have used the equation $H_{1}|\phi_{-}\rangle=0$, the normalization
$\langle\phi_{-}|\phi_{-}\rangle=1$ and the equality
$\langle\phi_{-}|W|\chi^{(2)}\rangle=\langle\phi_{-}|W|\chi^{(2)}_{<}\rangle$
due to the form of $W(x)$. At this stage, the coefficient $\lambda$ depends on
the unknown function $\chi^{(2)}_{<}$.

Introducing the Hamiltonian:

$H_{2}=-\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}-K^{2},\hskip 25.0ptK^{2}=E+V,$
(83)

the equations for the open channel takes on the form:

$\displaystyle H_{2}|\chi^{(2)}_{<}\rangle+W^{\dagger}|\chi^{(1)}\rangle=0,$
(84)
$\displaystyle-\frac{\mathrm{d}^{2}\chi^{(2)}_{>}}{\mathrm{d}x^{2}}=k^{2}\chi^{(2)}_{>}.$
(85)

Solution to Eq. (84) can be written as

$\displaystyle|\chi^{(2)}_{<}\rangle$ $\displaystyle=$
$\displaystyle|\tau_{0}\rangle-G^{(+)}_{2}W^{\dagger}|\chi^{(1)}\rangle$ (86)
$\displaystyle=$ $\displaystyle|\tau_{0}\rangle-\lambda
G^{(+)}_{2}W^{\dagger}|\phi_{-}\rangle,$

where the unperturbed wave function $\tau_{0}(x)=\langle x|\tau_{0}\rangle$ is
equal to $\chi^{(2)}_{<}(x)$ from Eq. (75) and is found as

$H_{2}\,\tau_{0}(x)=0,\hskip 25.0pt\tau_{0}(x)=\frac{\sin{Kx}}{K}.$ (87)

The Hamiltonian $H_{2}$ determines also the Green’s operator
$G^{(+)}_{2}=(H_{2}-\mathrm{i}\epsilon)^{-1}$ that contains the shifted energy
$E+\mathrm{i}\epsilon$ at $\epsilon\to 0$. The corresponding Green’s function
is

$\displaystyle
G^{(+)}_{2}(x,y;K)=\frac{\sin{Kx_{1}}\cos{Kx_{2}}}{K}+\mathrm{i}\,\frac{\sin{Kx}\sin{Ky}}{K},$
(88) $\displaystyle x_{1}={\rm min}(x,y),\hskip 25.0ptx_{2}={\rm max}(x,y),$

which serves for finding the outgoing wave under the boundary condition
$G^{(+)}_{2}(0,y;K)=0$ to obtain a finite radial solution for the $s$-wave.
Accounting for other partial waves in the Green’s function may require the use
of expression $\exp{(\mathrm{i}K|{\bf x}-{\bf y}|)}/K$ resulted from the
Helmholtz equation.

To express $\lambda$ in the terms of known solutions $\tau_{0}$ and
$\phi_{-}$, let us act by $\langle\phi_{-}|W$ on Eq. (86). Then, it is easy to
obtain that

$\lambda=\frac{\langle\phi_{-}|W|\tau_{0}\rangle}{E-Q+\langle\phi_{-}|WG^{(+)}_{2}W^{\dagger}|\phi_{-}\rangle}.$
(89)

The condition of zeroing the denominator indicates the resonant state of the
compound system, i.e., the energy of a dressed (quasi)discrete level Fesh62 ;
Joa75 .

It is useful to introduce the notations:

$\displaystyle\omega^{4}\Delta(K)$ $\displaystyle=$ $\displaystyle{\rm
Re}\,\langle\phi_{-}|WG^{(+)}_{2}W^{\dagger}|\phi_{-}\rangle,$ (90)
$\displaystyle\omega^{4}\gamma(K)$ $\displaystyle=$ $\displaystyle{\rm
Im}\,\langle\phi_{-}|WG^{(+)}_{2}W^{\dagger}|\phi_{-}\rangle.$ (91)

To sketch how this works for a fixed $Q>0$, let us imagine the situation when
$V\gg E$ at $E\to 0$, and the denominator of $\lambda$ vanishes at some
complex value of the energy $E_{0}-\mathrm{i}\Gamma_{0}/2$, thereby making the
magnitude of the wave functions extremely large. The positive resonance energy
$E=E_{0}$ and decay width $\Gamma_{0}$ may be simply determined from the
relations:

$E_{0}=Q-\omega^{4}\Delta(\sqrt{V}),\hskip
25.0pt\Gamma_{0}=2\omega^{4}\gamma(\sqrt{V}).$ (92)

This shows that $Q>E_{0}>0$ due to two additional interactions, which are
controlled by $V$ and $\omega^{2}$. Besides, the lifetime of a dimer in such a
scenario would be $\tau=2/\Gamma_{0}$ (in dimensionless units).

In general, we write that

$\lambda=\frac{\langle\phi_{-}|W|\tau_{0}\rangle}{E-Q+\omega^{4}\Delta(K)+\mathrm{i}\omega^{4}\gamma(K)}.$
(93)

Let us pay attention to finding exact expressions, starting with the overlap
integral that defines $\langle\phi_{-}|W|\tau_{0}\rangle$:

$\displaystyle B(K)$ $\displaystyle\equiv$
$\displaystyle\int_{0}^{1}\frac{4x}{(1+x^{2})^{2}}\,\frac{\sin{Kx}}{K}\,\mathrm{d}x$
$\displaystyle=$ $\displaystyle[\pi+\mathrm{i}\,{\rm
Ci}(K+\mathrm{i}K)-\mathrm{i}\,{\rm Ci}(K-\mathrm{i}K)]\cosh{K}$
$\displaystyle-$ $\displaystyle[{\rm Si}(K+\mathrm{i}K)+{\rm
Si}(K-\mathrm{i}K)]\sinh{K}-\frac{\sin{K}}{K}.$

Here we have used the sine and cosine integrals:

${\rm Si}(z)=\int_{0}^{z}\frac{\sin{t}}{t}\mathrm{d}t,\ \ {\rm
Ci}(z)=\gamma+\ln{z}+\int_{0}^{z}\frac{\cos{t}-1}{t}\mathrm{d}t,$

$\gamma=0.57721...$ is the Euler’s constant.

We obtain that

$\langle\phi_{-}|W|\tau_{0}\rangle=-\sqrt{\frac{2}{\pi}}\,\omega^{2}B(K).$
(95)

Note that $B(K)$ is a decreasing oscillating function that vanishes at
$K_{1}=5.359982109$, $K_{2}=7.917546061$, $K_{3}=11.09524951$ and so on. At
these points one gets $\lambda=0$ and $\chi^{(2)}_{<}(x)=\tau_{0}(x)$ with
$\kappa_{n}=-K_{n}\cot{K_{n}}>0$ on the boundary. Index $n$ corresponds to the
number of zeros of the wave function $\tau_{0}(x)$ in interval $x\in[0;1]$.
This indicates the possibility of appearing virtual bound states in the
system. However, we ignore them here, focusing on the Feshbach resonance.

Figure 7: The functions $\Delta(K)$ and $\gamma(K)$ which determine
$\langle\phi_{-}|WG^{(+)}_{2}W^{\dagger}|\phi_{-}\rangle$. In the given
interval there is zero $\Delta(K_{0})=0$ for $K_{0}=2.107331071$, while
$\gamma(K_{0})=0.200330$.

The first correction to the wave function $\chi^{(2)}_{<}(x)$ is determined by
$G^{(+)}_{2}W^{\dagger}|\phi_{-}\rangle$. To find the complex function
$\langle x|G^{(+)}_{2}W^{\dagger}|\phi_{-}\rangle=-\omega^{2}\tau_{1}(x)$, we
turn to the solution of the inhomogeneous equation:

$\displaystyle H_{2}\,\tau_{1}(x)=\phi_{-}(x),$ (96)
$\displaystyle\tau_{1}(x)=\int_{0}^{1}G^{(+)}_{2}(x,y;K)\,\phi_{-}(y)\,\mathrm{d}y.$
(97)

Computing, the components of $\tau_{1}(x)=\tau^{\rm
R}_{1}(x)+\mathrm{i}\,\tau^{\rm I}_{1}(x)$ are written as

$\displaystyle\sqrt{\frac{\pi}{2}}\,\tau^{\rm R}_{1}(x)$ $\displaystyle=$
$\displaystyle\mathrm{i}[\zeta(Kx+\mathrm{i}K)-\zeta(Kx-\mathrm{i}K)]$ (98)
$\displaystyle-\left[\varsigma_{+}(K)+\varsigma_{-}(K)+\frac{\cos{K}}{K}\right]\sin{Kx}$
$\displaystyle+\pi\cosh{K}\cos{Kx},$
$\displaystyle\sqrt{\frac{\pi}{2}}\,\tau^{\rm I}_{1}(x)$ $\displaystyle=$
$\displaystyle B(K)\sin{Kx},$ (99)

where we denote

$\displaystyle\zeta(z)$ $\displaystyle=$ $\displaystyle{\rm Ci}(z)\cos{z}+{\rm
Si}(z)\sin{z},$ $\displaystyle\varsigma_{\pm}(K)$ $\displaystyle=$
$\displaystyle{\rm Ci}(K\pm\mathrm{i}K)\sinh{K}\pm\mathrm{i}\,{\rm
Si}(K\pm\mathrm{i}K)\cosh{K}.$

Taking into account the form of $\tau^{\rm I}_{1}(x)$ and the definition of
$B(K)$, we specify the function $\gamma(K)$ (see Eq. (91)):

$\displaystyle\gamma(K)$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\,KB^{2}(K);$ (100) $\displaystyle\gamma(K)$
$\displaystyle\simeq$ $\displaystyle\frac{(\pi-2)^{2}}{2\pi}\,K,\quad K\to 0.$
(101)

At this time, the real part $\tau^{\rm R}_{1}(x)$ determines also a deviation
$\Delta(K)$ of the resonance energy:

$\Delta(K)=\int_{0}^{1}\phi_{-}(x)\,\tau^{\rm R}_{1}(x)\,\mathrm{d}x.$ (102)

This integral is not simple to be calculated analytically, and we present the
numerical result in Fig. 7.

Combining, the wave function in the first approximation for the closed channel
is

$\chi^{(1)}(x)=-\frac{\omega^{2}}{D(K)}\,\sqrt{\frac{\gamma(K)}{K}}\,\phi_{-}(x),$
(103)

where we have introduced the notation:

$\displaystyle D(K)$ $\displaystyle=$ $\displaystyle
K^{2}-V-Q+\omega^{4}\Delta(K)+\mathrm{i}\omega^{4}\gamma(K)$ (104)
$\displaystyle=$ $\displaystyle D_{\rm R}(K)+\mathrm{i}D_{\rm I}(K).$

Solution (103) describes the short-living state of dimer and vanishes at
$\omega=0$. Beyond the resonance zone we arrive at equation
$(H_{1}+Q-E)\chi^{(1)}_{>}=0$ for $Q=E$ as it was assumed.

Due to proportionality $\tau^{\rm I}_{1}(x)\sim\tau_{0}(x)$, we present the
open channel wave function in the resonance zone as

$\chi^{(2)}_{<}(x)=\frac{D_{\rm
R}(K)}{D(K)}\,\tau_{0}(x)-\frac{\omega^{4}}{D(K)}\,\sqrt{\frac{\gamma(K)}{K}}\,\tau^{{\rm
R}}_{1}(x).$ (105)

To find the phase shift $\delta$ for the wave function $\chi^{(2)}_{>}(x)$:

$\displaystyle\chi^{(2)}_{>}(x)$ $\displaystyle=$
$\displaystyle\chi^{(2)}_{<}(1)\,\frac{\sin{(kx+\delta)}}{\sin{(k+\delta)}},$
(106) $\displaystyle\chi^{(2)}_{<}(1)$ $\displaystyle=$
$\displaystyle\frac{D_{\rm R}(K)\sin{K}-D_{\rm I}(K)\cos{K}}{K\,D(K)},$ (107)

we appeal to the merging condition (72) which gives us

$K\,\frac{D_{\rm R}(K)\cos{K}+D_{\rm I}(K)\sin{K}}{D_{\rm R}(K)\sin{K}-D_{\rm
I}(K)\cos{K}}=k\cot{(k+\delta)}.$ (108)

This relation can be re-written as

$\displaystyle K\cot{(K-\delta_{\rm rs})}=k\cot{(k+\delta)},$ (109)
$\displaystyle\delta_{\rm rs}(K)=\arctan{\frac{D_{\rm I}(K)}{D_{\rm R}(K)}},$
(110)

where $\delta_{\rm rs}$ is the phase shift which is caused by complex
interaction in the resonance zone. At $\delta_{\rm rs}\equiv 0$, only
potential scattering with $V_{\rm sq}(x)$ remains.

It is easy to derive that

$\displaystyle\delta$ $\displaystyle=$
$\displaystyle-k+\arctan{\left[\frac{k}{K}\,\tan{(K-\delta_{\rm
rs})}\right]},$ (111) $\displaystyle\simeq$ $\displaystyle
k\left[\frac{\tan{(K-\delta_{\rm rs})}}{K}-1\right],$ (112)

where the last expression is valid for small $k$.

The scattering matrix $S=\mathrm{e}^{2\mathrm{i}\delta}$ for the open channel
is

$S=\mathrm{e}^{-2\mathrm{i}k}\,\frac{K\cot{(K-\delta_{\rm
rs})}+\mathrm{i}k}{K\cot{(K-\delta_{\rm rs})}-\mathrm{i}k},\quad
K=\sqrt{k^{2}+V}.$ (113)

Its pole, resulted from zero of denominator $F(k)=0$,

$F(k)=K\cot{(K-\delta_{\rm rs})}-\mathrm{i}k,$ (114)

defines a resonance point in the system. However, not every pole of $S$ is
related with the compound system existence.

The resonance is usually observed in narrow region of energy $E$, which covers
the resonant value $E_{\rm res}=E_{0}-\mathrm{i}\Gamma_{0}/2$ with some
$E_{0}>0$ and $\Gamma_{0}>0$. Positivity of $E_{0}$ makes this (quasi)discrete
level unstable, whose lifetime is $\tau=2/\Gamma_{0}$.

In our model, we find $E_{\rm res}=K^{2}_{\rm res}-V$ for given $Q$, $V$, and
$\omega$ by solving the equation $F(\sqrt{K^{2}_{\rm res}-V})=0$ in an
appropriate form. This equation is identically transformed and is solved
numerically, assuming $\omega^{4}\ll 1$ and using an iterative procedure with
the initial value $K=\sqrt{V+Q}$:

$\displaystyle K^{2}_{m+1}$ $\displaystyle=$ $\displaystyle V+Q$
$\displaystyle-\omega^{4}\left[\Delta(K_{m})+\mathrm{i}\gamma(K_{m})\,\frac{k_{m}\cot{K_{m}}-\mathrm{i}K_{m}}{K_{m}\cot{K_{m}}-\mathrm{i}k_{m}}\right],$
$\displaystyle k_{m}$ $\displaystyle=$ $\displaystyle\sqrt{K^{2}_{m}-V}.$
(115)

Omitting the indexes $m$ and $m+1$, one gets the equation equivalent to
$F(\sqrt{K^{2}-V})=0$.

To extract the resonance part of $S$-matrix, let us expand the complex
function $F(k)$ of a real $k$ in vicinity of complex $k_{\rm res}=\sqrt{E_{\rm
res}}=k_{r}-\mathrm{i}\kappa_{r}$ as

$\displaystyle F(k)$ $\displaystyle\simeq$ $\displaystyle F(k_{\rm
res})+F^{\prime}(k_{\rm res})(k-k_{\rm res})$ (116) $\displaystyle=$
$\displaystyle C\,(k-k_{r}+\mathrm{i}\kappa_{r}),$

where $C=F^{\prime}(k_{\rm res})$ is a complex constant.

Since $k=k^{*}_{\rm res}$ solves conjugate equation $F^{*}(k)=0$, and
$F^{*}(k)\simeq C^{*}(k-k_{r}-\mathrm{i}\kappa_{r})$ near the resonance, we
define the phase shift $\delta_{0}$ associated with potential scattering so
that

$\mathrm{e}^{2\mathrm{i}\delta_{0}}=\mathrm{e}^{-2\mathrm{i}k}\,\frac{C^{*}}{C}.$
(117)

The calculated ingredients allow us to write down the scattering matrix and
the total phase shift in vicinity of the resonance:

$\displaystyle S$ $\displaystyle\simeq$
$\displaystyle\mathrm{e}^{2\mathrm{i}\delta_{0}}\,\frac{k-k_{r}-\mathrm{i}\kappa_{r}}{k-k_{r}+\mathrm{i}\kappa_{r}},$
(118) $\displaystyle\delta$ $\displaystyle=$
$\displaystyle\delta_{0}-\arctan{\frac{\kappa_{r}}{k-k_{r}}}.$ (119)

Presenting these quantities in the conventional form, depended on $E$, we
expand $F(\sqrt{E})$ in powers of $E$:

$F(\sqrt{E})\simeq\frac{C}{2k_{\rm res}}\,(E-E_{\rm res}),$ (120)

where $C$ is as above. It leads to redefinition of phase shift $\delta_{0}$
because of the relation:

$\arctan{\frac{\kappa_{r}}{k-k_{r}}}-\frac{1}{2}\arctan{\frac{\kappa_{r}}{k_{r}}}=\arctan{\frac{\Gamma_{0}}{2(E-E_{0})}}.$
(121)

Figure 8: Phase shift $\delta$ (a) and cross section $\sigma$ (b) as functions
of incident momentum $k$. Jumps and peaks in the graphs indicate resonance and
appear at $k_{r}={\rm Re}\,\sqrt{E_{\rm res}}$. Black dashed line in (b)
corresponds to maximal admissible value $\sigma_{\rm max}=4\pi/k^{2}$.

Thus, we can see that the phase shift $\delta$ experiences a jump
$\delta(k_{r}-0)-\delta(k_{r}+0)=\pi$, which reveals a resonance. Besides,
this determines the scattering cross section $\sigma$ in accordance with the
optic theorem LL :

$\sigma(k)=\frac{4\pi}{k^{2}}\sin^{2}{\delta(k)}.$ (122)

The behavior of $\delta(k)$ and $\sigma(k)$ (in dimensionless units) is
demonstrated in Fig. 8 and indicates that an excited two-particle molecule can
be created by an incident particle with a relative momentum
$k\in(k_{r}-\kappa_{r}/2;k_{r}+\kappa_{r}/2)$, where $k_{r}={\rm
Re}\,\sqrt{E_{\rm res}}>0$ and $\kappa_{r}=-{\rm Im}\,\sqrt{E_{\rm res}}$.

The obtained formulas describe, in general, the resonance mechanism of the
short-term production of a two-particle molecule (dimer) without specifying
the interactions used. This means that the model still needs to be refined in
accordance with physical conditions, and our formalism is applicable to the
study of different scenarios. In principle, even the energy gap $Q$ can be
supported by a spatially homogeneous interaction to create an energy
predominance of one particle configuration over another.

In this connection, we mention experiments in the laboratory, where the
configurations depend on the spin, and the energy gap is associated with the
applied magnetic field Grimm10 . Adjustment of the gap $Q$ by a magnetic field
leads to the Feshbach phenomenon Grimm10 ; BrH06 , that is a resonant
transition between configurations, accompanied by a jump in the scattering
length in open channel.

In the case of dark matter, until the interactions involved are clarified, we
minimize the number of interactions and demonstrate the capabilities of our
model.

For simplicity, let us put $V=0$, when $K=k$, and reduce Eq. (115) to the
form:

$k^{2}_{m+1}=Q-\omega^{4}\left[\Delta(k_{m})+\mathrm{i}\gamma(k_{m})\right],\quad
k_{0}=\sqrt{Q}.$ (123)

This case describes the transformation of a pair of asymptotically free
particles (scattered waves) into a dimer in the resonance zone due to the
potential $-12\cos_{\mu=1}{x}$ and the channel coupling $\omega^{2}$.

Iterating Eq. (123), we obtain the solutions:

$E^{(11)}_{\rm res}=0.07317639856-\mathrm{i}\,0.005600193629,$ (124)

for $Q_{1}=0.1$ and $\omega^{4}_{1}=0.1$; and

$E^{(22)}_{\rm res}=0.09638881959-\mathrm{i}\,0.01294943156,$ (125)

for $Q_{2}=0.15$ and $\omega^{4}_{2}=0.2$.

Having additionally calculated the resonance energies for the sets of
parameters $(Q_{1},\omega^{4}_{2})$ and $(Q_{2},\omega^{4}_{1})$:

$\displaystyle E^{(12)}_{\rm res}$ $\displaystyle=$ $\displaystyle
0.04538086735-\mathrm{i}\,0.008987063657,$ (126) $\displaystyle E^{(21)}_{\rm
res}$ $\displaystyle=$ $\displaystyle
0.1236649388-\mathrm{i}\,0.007215615748,$ (127)

and comparing them with (124) and (125), we deduce that increasing
$\omega^{4}$ leads to an increase in the decay width of the resonance.

We also note that the resonance energy $E_{0}={\rm Re}\,E_{\rm res}$ is less
than the gap $Q$ for all cases at hand.

To complete the resonance description for two sets of parameters $Q$ and
$\omega^{4}$, leading to (124) and (125), we obtain numerically the potential
phase shifts $\delta^{(11)}_{0}(k;Q_{1},\omega_{1})\simeq-k+0.54331$ and
$\delta^{(22)}_{0}(k;Q_{2},\omega_{2})\simeq-k+0.62674$, respectively. The
results of calculations are shown in Fig. 8.

Since Eq. (123) is identical to the condition $D(k)=0$ for the denominator of
the wave functions at $V=0$, an incident particle is capable to penetrate into
the resonance zone at resonance energy $E_{0}$ and is actually reflected at
other energies.

To convert dimensionless characteristics into physical units, we use formulas
(25). It can be seen that the scale $r_{0}$ is an essential parameter here,
the determination of which requires additional conditions that are beyond the
scope of this consideration. Although $r_{0}$ characterizes the interaction
radius, we constrain its value by the relativistic limit of a non-relativistic
model, noting that the Compton wavelength $\hbar/(mc)$ is less than the de
Broglie wavelength $\hbar/(mv)$ of slow particles at $v\to 0$. Thus, replacing
the reduced mass in (25) with half the particle mass, we write the following
scales for distance, energy and time:

$r_{0}=\frac{\hbar}{mc},\quad\varepsilon_{0}=mc^{2},\quad\tau_{0}=\frac{\hbar}{\varepsilon_{0}},$
(128)

where the particle mass $m$ is used. Kinematically, $\tau_{0}=r_{0}/c$ is the
time taken for a light to travel the distance $r_{0}$. Besides, the wave
number $k$ determines the momentum value $p=\hbar k/r_{0}=mck$.

For $\varepsilon_{0}\sim 10^{-22}\,\text{eV}$, one has

$\displaystyle r_{0}\simeq 1.973\times
10^{15}\,\text{m}\,\left[\frac{mc^{2}}{10^{-22}\,\text{eV}}\right]^{-1},$
(129) $\displaystyle\tau_{0}\simeq 6.582\times
10^{6}\,\text{s}\,\left[\frac{mc^{2}}{10^{-22}\,\text{eV}}\right]^{-1},$ (130)

where $r_{0}$ is shorter than the typical scale of BEC DM (healing length
$\sim 10^{19}$ m) in Sec. II (see also GN21 ). Although the shape of the dimer
wave function in Fig. 6 indicates the localization of one particle near
another, the influence between particles at large $r_{0}$ can be maintained by
an infinite scattering length of the long-range potential used. Note that the
interparticle distance in BEC DM is of the order $(m/\rho)^{1/3}\sim 10^{-13}$
m, where $\rho\sim 10^{-20}\,\text{kg}\,\text{m}^{-3}$ is a mean mass density.

Defining the dimer lifetime as $t=2\tau_{0}/\Gamma_{0}$ and accounting for
$\tau_{0}\sim 0.2$ year and $\Gamma_{0}\sim 0.01$ as above, we deduce dimer
stability over a period $\sim 40$ years. The long-term decay of the dimer in
this scenario suggests to treat DM as multicomponent because of involving
composites, whose presence affects also the BEC properties. This fact
stimulates a detailed study of aspects of the proposed mechanism and the
formation of complexes from several particles, such as trimers.

##  V Concluding Remarks

The main message of this work is to show that the axion-like self-interaction
as applied to ultralight DM bosons provides two phases (dilute and dense) of
the self-gravitating BEC DM, and its $\mu$-modification at the quantum
mechanical level ensures the appearance of composites, namely, dimers, using
the Feshbach resonance between two scattering channels. In our treatment, the
creation of individual complexes from initial particles with a large
scattering length precedes the BEC state and promotes its formation. This
assumption can be used to explain the very small scattering length (repeatedly
confirmed by calculations Harko2011 ; Chavanis2 ; GN22 ) of effective
particles in BEC, which then should be a mixture of composites. Such a
scenario is induced by the experiments with atomic BEC Grimm10 ; BrH06  and
the resonant transformations under strong interaction. In particular, the
connection between the observed intercepts of the $\pi$-meson correlations and
the deformation parameter $\mu$ used for their description is revealed in GM3
by accounting for a quark structure and suggests a theoretical basis of
describing composites of the particles governed by various statistics.

The results shown here indicate the presence of two phases of the gravitating
BEC DM using the cosine-like self-interaction derived for QCD axions. This
type of interaction generalizes non-linear interactions in previous studies
Harko2011 ; Chavanis2 ; GKN20 ; GN21  that help us interpret innovations. In
the present case, the existence of rarefied (gaseous) and dense (liquid)
states immediately follows from the initial conditions for the
Gross–Pitaevskii equation at zero temperature. A first-order phase transition
between these states should be stimulated by quantum fluctuations that
contribute to the pressure/compression, according to outcomes of GKN20 . The
influence of the two-phase structure on the rotation curves of galaxies is
expected to be similar to the results of GN21 , but requires yet a detailed
study. We admit that the contribution of each phase depends on the formation
stage of galaxy and DM halo. However, as shown earlier in GN22 , the dense
phase of BEC appears unfavorable for composites of DM particles because of
their probable destruction caused by collisions.

To describe two-particle composites, in contrast to the heuristic approach in
GN22 , we turn to a quantum mechanical model with the smoothly deformed
cosine-like axion potential, which allows us to control the scattering length
changing the deformation parameter $\mu\geq 0$. Such potentials as functions
of the distance $r$ between two particles are studied for the first time and
turn out to be similar to typical atomic (molecular) potentials. In other
words, we have actually replaced the interaction of QCD axions with an
interaction inherent in atoms and molecules, and we argue the constructiveness
of this change by the results obtained.

Thus, having got characteristics of $s$-wave potential scattering in the Born
approximation for the $\mu$-deformed potentials $g\cos_{\mu}(r/r_{0})$ with
dimensionless coupling $g$, we treat these potentials as short-range ones for
$\mu<1/2$, while the potentials at $\mu>1/2$ behave as long-range ones with
infinite scattering length. It follows from the small-$k$ asymptotic $f\sim
k^{\frac{1}{\mu}-2}$ of the scattering amplitude $f$ as function of wave
number $k$ in the linear approximation in the coupling constant $|g|<1$. The
strong coupling case requires the use of alternative approach beyond the Born
approximation BrH06 .

Analyzing, we deduce that the problem with attractive potential for $g=-12$
and $\mu=1$ is suitable for describing a dimer, that is, a composite of two
particles. In the case of $\mu=1$, one has $\cos_{\mu}{x}\sim-x^{-2}$ at large
$x$ that allows one to associate this problem to the Efimov’s physics BrH06 .
Although the general solution of the stationary Schrödinger equation for fixed
$\mu=1$, but arbitrary $g$ and some energy $E$ of relative motion, is
formulated in terms of the confluent Heun function, the case of $g=-12$ and
$E=0$ is unique because it leads to a particularly simple solution for a dimer
in the ground state. In this case, a quantum mean squared displacement of
moving particle in vicinity of $r=0$ is equal to parameter $r_{0}$ of the
potential used and corresponds to classically admissible region. Strictly
speaking, the range of $r_{0}$ is unknown, and its value can be large enough,
dealing with the potential of infinite scattering length.

Our vision of forming such a dimer in space is based on the Feshbach resonance
and the use of two channels of two-particle scattering, when a single bound
state of dimer represents evidently a closed channel, while the asymptotically
free scattering of two particles corresponds to an open channel. There is no
tunneling within the one potential problem here, but it rather appears a
possibility to form an intermediate compound system (dressed state), when a
pair of particles jumps between these channels with close energies by acting
external perturbation. Having positive energy of relative motion, the dressed
state is characterized by a finite lifetime and exhibits the resonance
properties to decay.

We have formulated a solution to the problem of two-channel scattering in the
first approximation, taking into account an additional potential scattering in
the open channel, which makes it possible to use the results obtained to
analyze various situations. In particular, the presence of this type of
scattering provides a nonvanishing resonance decay width at zero energy, see
(92). Moreover, in the overall picture, one would also have to take into
account the possibility of resonant states in the open channel Marc04 .
However, in an effort to minimize the number of interactions between DM
particles, we discarded the supplement interaction in the open channel when
obtaining the resonance energy $E_{\rm res}$ numerically. This means that
apart from the interaction potential of the dimer, we only need the coupling
between the channels. In our approach, both the dimer potential and the extra
influence (69) are assumed to be characterized by the same range $r_{0}$.
Without additional research beyond the scope of this work, it is impossible to
estimate $r_{0}$, which determines the resonance lifetime at the conventional
particle mass $m\sim 10^{-22}~{}\text{eV}/c^{2}$. Limiting $r_{0}$ from above
by the Compton length, one obtains a long-lived dimer (about 40 years), see
(130). Its fate depends on the properties of the open channel. Anyway, the
case of large $r_{0}$ may indicate its applicability for describing larger
structures of DM particles.

Thus, we are able to interpret dimers as constituents of DM, which participate
and contribute to forming BEC DM halo of galaxies. There is a reasonings that
the appearance of composites enhances the formation of condensate TBM .
Justifying this hypothesis requires the use of quantum field and statistical
approach to a system of a large number of particles, which can serve as a
prospect for forthcoming research.

###### Acknowledgements.

Both authors acknowledge support from the National Academy of Sciences of
Ukraine by its Project No. 0122U000888.

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