Physical Origin of the Universal Three-body Parameter in Atomic Efimov Physics
Pascal Naidon$^{1}$
Shimpei Endo$^{2}$
and Masahito Ueda$^{2}$
We address the microscopic origin of the universal three-body parameter that fixes the spectrum of few-atom systems in the Efimov regime. We identify it with a nonadiabatic deformation of the three-atom system which occurs when three atoms come within the distance of the van der Waals length. This deformation explains the universal ratio of the scattering length at the triatomic resonance to the van der Waals length observed in several experiments and confirmed by numerical calculations.
Paper18e-bib.tex
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\begin{document}

\title{Physical Origin of the Universal Three-body Parameter in Atomic Efimov
Physics}


\author{Pascal Naidon$^{1}$, Shimpei Endo$^{2}$, and Masahito Ueda$^{2}$}


\affiliation{$\,^{1}$RIKEN Nishina Centre, RIKEN, Wak\={o} 351-0198, Japan, }


\affiliation{$\,^{2}$Department of Physics, University of Tokyo, 7-3-1 Hong\={o},
Bunky\={o}-ku, T\={o}ky\={o} 113-0033, Japan}


\date{\today}
\begin{abstract}
We address the microscopic origin of the universal three-body parameter
that fixes the spectrum of few-atom systems in the Efimov regime.
We identify it with a nonadiabatic deformation of the three-atom system
which occurs when three atoms come within the distance of the van
der Waals length. This deformation explains the universal ratio of
the scattering length at the triatomic resonance to the van der Waals
length observed in several experiments and confirmed by numerical
calculations. 
\end{abstract}
\maketitle

\section{Introduction}

In recent years, the investigation of Efimov physics~\cite{Efimov1971,Ferlaino2010},
the universal physics of few particles interacting via nearly resonant
short-range interactions, has developed tremendously, both on the
experimental~\cite{Kraemer2006,Ottenstein2010,Huckans2009,Williams2009,Zaccanti2009,Barontini2009,Wenz2009,Pollack2009,Gross2009,Knoop2009,Lompe2010PRL,Lompe2010Science,Nakajima2010,Gross2010,Nakajima2011,Berninger2011,Ferlaino2009,Ferlaino2010,Machtey2012,Zenesini2012,Knoop2012}
and theoretical fronts~\cite{Petrov2004,Braaten2007,Hammer2007,vonStecher2009}.
The essence of this physics is the appearance of a universal $1/R^{2}$
attraction between three particles at an average separation $R$.
This long-range three-body attraction, discovered by V.~Efimov~\cite{Efimov1971},
emerges from the pairwise interactions, despite their finite range.
It can be interpreted as an interaction between two particles mediated
by a third particle. Its strength is universally determined by the
masses and quantum statistics of the particles. The Efimov attraction
extends from distances on the order of the range of the interaction
$b$ to distances on the order of $\vert a\vert$, where $a$ is the
scattering length of the pairwise interaction. It therefore requires
$\vert a\vert>b$, a condition well satisfied for resonant pairwise
interactions. At the unitarity limit $a\to\infty$, the Efimov attraction
extends to infinity. Decaying as $1/R^{2}$, it supports an infinite
number of bound states known as the Efimov trimers. Furthermore, each
bound state is related to the neighbouring state by a scale transformation,
due to the scale invariance of $1/R^{2}$ potentials~\cite{Efimov1971,Braaten2007},
so that the energy spectrum forms a geometric series. This constitutes
the most remarkable and characteristic feature of the Efimov trimers.
However, since the three particles are attracted to each other, the
physics at short separations comparable to the range $b$ fixes the
wave functions and spectrum of the Efimov trimers.

Until recently, little had been known about this short-distance physics.
Efimov's original investigation made use of the asymptotic two-body
behaviour (or zero-range potential limit) to derive the three-body
attraction, but did not address the short-distance region directly.
Its effect on the longer-distance region was accounted for by a three-body
boundary condition, expressed either as a phase in the three-body
wavefunction or a log-periodic inverse length $\Lambda$ known as
the Efimov three-body parameter~\cite{Efimov1971}. This long-distance
picture is equivalent to a zero-range low-energy picture, where $\Lambda$
plays the role of the parameter that renormalises the low-energy effective
field theory~\cite{Hammer2007}. The Efimov effect is the only known
physical example of the renormalisation-group limit cycle~\cite{Wilson1971}.
Since the short-distance region involves the short-range details of
the interaction potentials, $\Lambda$ has long been thought to be
a non-universal quantity that is strongly dependent on the system.

Later, it was found that $\Lambda$ is universally determined in cases
where a length scale larger than $b$ arises in the problem, most
notably in the case of a narrow Feshbach resonance in the pairwise
interaction~\cite{Chin2010}, which entails a large and negative
effective range setting the value of $\Lambda$~\cite{Petrov2004},
and the case of particles with additional dipolar interactions, whose
strength also sets the value of $\Lambda$~\cite{YujunWang2011PRL}.
In the absence of such large length scales, however, it was believed
that $\Lambda\sim1/b$, but its precise value would vary by a factor
spanning its log-period $e^{\pi/s_{0}}\approx22.7$ from one system
to another, or even from one Feshbach resonance to another within
the same system~\cite{DIncao2009}.

However, several recent experiments with identical ultra-cold atoms~\cite{Kraemer2006,Ottenstein2010,Huckans2009,Williams2009,Zaccanti2009,Barontini2009,Wenz2009,Pollack2009,Gross2009,Knoop2009,Lompe2010PRL,Lompe2010Science,Nakajima2010,Gross2010,Nakajima2011,Berninger2011,Ferlaino2009,Ferlaino2010,Machtey2012,Zenesini2012,Knoop2012}
have revealed Efimov trimers and thereby determined their three-body
parameters. In these experiments, rather broad Feshbach resonances
are used, implying that the range $b$ of the interactions between
atoms is typically the van der Waals length $r_{\mbox{\tiny vdW}}=\frac{1}{2}(mC_{6}/\hbar^{2})^{1/4}$
associated with the $-C_{6}/r^{6}$ tail of the open-channel potential~\cite{Chin2010}.
The measured value of the three-body parameter expressed in units
of $r_{\tiny\mbox{vdW}}$ turned out to stay fairly constant for different
atomic species~\cite{Ferlaino2010}, nuclear spin states~\cite{Gross2010},
or even resonances of the same species~\cite{Berninger2011}. This
indicates that the three-body parameter is universally determined
by the van der Waals length, and relatively insensitive to other short-range
details specific to each species. It was first suggested that this
van der Waals universality was due to the very deep well of the potentials
for these species, which support many two-body bound states. When
the three atoms enter the short-range region of the potential, they
feel such a deep potential well that for all these species it would
result in the same effect on the phase of the wavefunction, and lead
to the same three-body parameter~$\Lambda$. However, Efimov features
for helium atoms, which interact through a shallow potential supporting
only one two-body bound state, were shown to also follow the van der
Waals universality, both theoretically \cite{Naidon2012a} and experimentally~\cite{Knoop2012}.

A first attempt~\cite{Chin2011} to explain this universality suggested
that it could be due to quantum reflection in the sum of pairwise
$-1/r^{6}$ potentials. Particles coming from large distances to separations
on the order of the van der Waals length would experience a sudden
drop in the resulting effective three-body potential, which would
reflect the particles before they start to probe shorter regions.
However, a numerical study~\cite{JiaWang2012} showed that the relevant
three-body potential in the van der Waals region does not exhibit
a sudden drop, but instead a sudden repulsive barrier. The numerical
results indicate that this three-body repulsion is universally located
around $R\approx2\, r_{\tiny\mbox{vdW}}$ for several model potentials
and occurs whenever a pairwise interaction potential features a deep
well supporting many two-body bound states or a short-range hardcore
repulsion, which is the case for all atomic species. Reference~\cite{JiaWang2012}
attributes the appearance of the repulsion to an increase of kinetic
energy due to the squeezing of the hyperangular wavefunction into
a smaller volume caused by the suppression of two-body probability
inside the two-body potential well. However, it has remained to be
clarified how precisely this mechanism occurs, what physical picture
it corresponds to, and why it is universal.

The purpose of this work is to present a simple physical account of
the three-body repulsion and universality of the three-body parameter.
This paper is organised as follows. In Section II, we review how the
three-body repulsion arises in the hyperspherical formalism. In Sect.~III,
we discuss how the universal three-body repulsion occurs as a consequence
of nonadiabatic deformation due to pair correlation. In Sect.~IV,
we confirm this scenario by using two simple models. In Sect.~V,
we give the conclusion of this work.\\





\section{The three-body repulsion}

The three-body repulsion is observed in the hyperspherical formalism,
where the three-body wave function $\Psi$ is expressed in terms of
the hyperradius $R=\sqrt{\frac{2}{3}(r_{12}^{2}+r_{23}^{2}+r_{31}^{2})}$
which corresponds to the global size of the three-body system~%
\footnote{There are several conventions for the definition of the hyperradius.
We choose the definition of Ref.~\cite{Efimov1971}, which differs
only by a factor $\sqrt{\sqrt{3}/2}\approx0.93$ from that of Ref.~\cite{JiaWang2012}
for the case of three identical bosons.%
}, and the hyperangles, which describe the shape of the three-body
system and are collectively denoted by $\Omega$. We present here
a simple approximation that captures the bare essentials of both the
Efimov attraction and the universal three-body repulsion. 

The three-body wave function can be decomposed over a basis of hyperangular
wave functions $\tilde{\Phi}_{n}(\Omega;R)$ which are normalised
to unity:
\begin{equation}
\Psi(R,\Omega)=\frac{1}{R^{5/2}}\sum_{n}f_{n}(R)\tilde{\Phi}_{n}(\Omega;R).\label{eq:TotalWaveFunction}
\end{equation}
The hyperangular wavefunction $\tilde{\Phi}_{n}$ itself can be decomposed
into three Faddeev components $\phi_{n}^{(i)}$, 
\begin{equation}
\tilde{\Phi}_{n}(\Omega;\, R)=\sum_{i=1,2,3}\frac{\phi_{n}^{(i)}(\Omega;R)}{\sin2\alpha_{i}}.\label{eq:FaddeevDecomposition}
\end{equation}
Here,  $\alpha_{i}=\arctan\frac{\sqrt{3}r_{jk}}{2r_{i,jk}}$ is the
Delves hyperangle in the $i$th Jacobi coordinate system $(\vec{r}_{jk},\vec{r}_{i,jk})$,
where $(i,j,k)$ denotes the cyclic permutations of (1,2,3). For identical
bosons, the functional forms of all Faddeev components are the same,
$\phi_{n}^{(i)}=\phi_{n}$. The advantage of this Faddeev decomposition
is that it treats the three particles on equal footing. In the low-energy
Faddeev approximation \cite{Fedorov1993}, the dependence of $\phi_{n}$
is reduced to the Delves hyperangle $\alpha$. By doing so, one ignores
the dependence of $\phi_{n}$ on the directions of $\vec{r}_{jk}$
and $\vec{r}_{i,jk}$, \emph{i.e.} higher angular-momentum partial
waves. This excludes accidental resonances with higher partial waves
\cite{DIncao2009,JiaWang2012}, and gives less accurate results.
Nevertheless, this approximation is good enough for our purpose, as
we shall see below.

The Faddeev component $\phi_{n}$ is chosen to be the eigensolution
with the eigenvalue $\lambda_{n}$ of the Faddeev equation
\begin{multline}
\left(-\frac{\partial^{2}}{\partial\alpha^{2}}-\lambda_{n}\right)\phi_{n}(\alpha)=\\
-R^{2}\frac{m}{\hbar^{2}}V(R\sin\alpha)\left(\phi_{n}(\alpha)+\frac{4}{\sqrt{3}}\int_{\alpha_{\tiny\mbox{min}}}^{\alpha_{\tiny\mbox{max}}}\phi_{n}(\alpha^{\prime})d\alpha^{\prime}\right),\label{eq:HyperangularEquation}
\end{multline}
where $m$ is the particle mass, $V$ is the pairwise interaction
potential, $\alpha_{\tiny\mbox{min}}=\vert\frac{\pi}{3}-\alpha\vert$,
and $\alpha_{\tiny\mbox{max}}=\frac{\pi}{2}-\vert\frac{\pi}{6}-\alpha\vert$.
We assume that three-body interactions are negligible~%
\footnote{This is justified to the extent that the range of the physical three-body
force does not exceed the range of the three-body repulsion.%
}.

Each solution $\phi_{n}$ defines a channel $n$, and one finds in
general that the hyperradial functions $f_{n}(R)$ are solutions of
the coupled equations,
\begin{multline}
\left(-\frac{\partial^{2}}{\partial R^{2}}+\frac{\lambda_{p}(R)}{R^{2}}-\frac{1}{4R^{2}}-\frac{m}{\hbar^{2}}E\right)f_{p}(R)\\
+\sum_{n}\left(Q_{pn}f_{n}(R)+2P_{pn}\frac{\partial f_{n}(R)}{\partial R}\right)=0,\label{eq:HyperradialEquation}
\end{multline}


with the nonadiabatic couplings
\begin{equation}
Q_{pn}=-\int d\Omega\tilde{\Phi}_{p}^{*}\frac{\partial^{2}\tilde{\Phi}_{n}}{\partial R^{2}}\quad;\quad P_{pn}=-\int d\Omega\tilde{\Phi}_{p}^{*}\frac{\partial\tilde{\Phi}_{n}}{\partial R}.\label{eq:DefinitionOfPandQ}
\end{equation}


If we restrict our consideration to a particular channel $n$ (neglecting
couplings to other channels) and note that $P_{nn}=0$ due to the
normalisation of $\tilde{\Phi}_{n}$, we arrive at a simple Schr�dinger
equation
\begin{equation}
\left(-\frac{\partial^{2}}{\partial R^{2}}-\frac{1}{4R^{2}}+U_{n}(R)-\frac{m}{\hbar^{2}}E\right)f_{n}(R)\approx0,\label{eq:SingleChannelEquation}
\end{equation}
with the three-body potential 
\begin{equation}
U_{n}(R)=\frac{\lambda_{n}(R)}{R^{2}}+Q_{nn}(R).\label{eq:ThreeBodyPotential}
\end{equation}
This potential is the sum of an adiabatic (first term) and nonadiabatic
(second term) contributions. The Efimov attraction manifests itself
in this framework as the appearance in a particular channel $n=0$
of a negative eigenvalue $\lambda_{0}(R)\to-s_{0}^{2}$ at large hyperradii,
with $s_{0}\approx1.00624$. One can show that $Q_{00}(R)\to O(1/R^{3})$,
so that the potential $U_{0}(R)$ tends to the $1/R^{2}$ Efimov attraction
at large $R$. At shorter distance, the potential becomes repulsive.
This is illustrated in Fig.~\ref{fig:ThreeBodyPotential}, where
the three-body potentials $U_{n}(R)$ obtained by solving Eq.~(\ref{eq:HyperangularEquation})
are represented for several two-body potentials with a van der Waals
tail. Namely, we used the following soft-core van der Waals and Lennard-Jones
potentials,
\begin{eqnarray}
V_{\mbox{\tiny soft}}(r) & = & -C_{6}\frac{1}{r^{6}+\sigma^{6}},\label{eq:SoftVanderWaals}\\
V_{\tiny\mbox{LJ}}(r) & = & C_{6}\left(\frac{\sigma^{6}}{r^{12}}-\frac{1}{r^{6}}\right),\label{eq:LennardJones}
\end{eqnarray}
where the length $\sigma$ is adjusted to produce a shape resonance
(divergence of the scattering length, leading to the unitarity limit
$a\to\infty$). There are several possible choices of $\sigma$, corresponding
to different depths of the potential well, or equivalently different
numbers of $s$-wave two-body bound states $n_{b}$ (including the
one at the breakup threshold). We also use a realistic helium potential~\cite{Aziz1991}
rescaled to reach unitarity~\cite{Naidon2012a}, which is qualitatively
similar to a Lennard-Jones potential at unitarity with one two-body
bound state.

Figure~\ref{fig:ThreeBodyPotential} shows that for all these two-body
potentials, the three-body potential $U_{0}(R)$ in the Efimov channel
$(n=0)$ exhibits both the Efimov attraction at large distance and
a repulsive barrier at short distance. Consistent with Ref.~\cite{JiaWang2012},
for all these pairwise potentials with the exception of the soft-core
van der Waals potential with one bound state, the repulsive barrier
is universally located around $R\approx2r_{\mbox{\tiny vdW}}$.

\begin{figure}
\includegraphics[bb=3bp 0bp 269bp 202bp,clip,scale=0.9]{Fig1-ThreeBodyPotentials}

\caption{\label{fig:ThreeBodyPotential}Three-body potentials $U_{0}(R)$ and
$U_{1}(R)$ for different pairwise interactions at unitarity: soft-core
van der Waals potential (blue) with $n_{b}=1$ up to 10 two-body bound
states, Lennard-Jones potential (green) with $n_{b}=1$ up to 10 two-body
bound states, helium potential (red) rescaled to reach unitarity with
$n_{b}=1$ two-body bound state. Note that only the case of the soft-core
van der Waals potential with no bound state is significantly different
from the other cases. The dashed curve shows the asymptotic Efimov
attraction.}
\end{figure}


\begin{figure}
\includegraphics[bb=0bp 0bp 250bp 174bp,clip]{Fig2-TwoBody}

\caption{\label{fig:TwoBody}Zero-energy two-body probability density distribution
$\vert\varphi\vert^{2}$ (normalised asymptotically to unity) as a
function of interparticle distance for different two-body potentials:
soft-core van der Waals potential (blue) with $n_{b}=1$ up to 8 bound
states, Lennard-Jones potential (green) with $n_{b}=1$ up to 8 bound
states, helium potential (red) rescaled to reach unitarity with 1
bound state. The corresponding potentials are shown in faded colours.
The probability density corresponding to the universal van der Waals
correlation given in Eq.~(\ref{eq:universalCorrelation}) is shown
by the dashed black curve.}
\end{figure}





\section{Interpretation of the three-body repulsion}


\subsection{Two-body suppression}

According to Ref.~\cite{JiaWang2012}, the three-body repulsive barrier
is caused by the suppression of two-body probability in the pairwise
potentials. In Fig.~\ref{fig:TwoBody}, we represent the two-body
probability density at zero scattering energy for each potential.
One can see that for all potentials except the soft core van der Waals
potentials with $n_{b}=1$ and 2 $s$-wave bound state, the probability
is indeed significantly suppressed below $r<r_{\tiny\mbox{vdW}}$.
This suppression is due to the presence of either a repulsive core
(as in the case of the helium potential or Lennard-Jones with one
bound state) or a deep well where the particles are accelerated and
therefore spend little time, as they would classically. Let us now
analyse how this two-body suppression explains the universal three-body
repulsion.


\subsection{Trivial three-body suppression}

We first point out a possible misinterpretation of the relation between
the three-body repulsion and the two-body suppression. In Ref.~\cite{Sorensen2012},
it is argued that the repulsion results from the interatomic two-body
hard-core repulsion. Let us first note that this statement is not
general enough, since the three-body repulsion also occurs for purely
attractive potentials with no hard-core repulsion, as shown in Ref.~\cite{JiaWang2012}
and in Fig.~\ref{fig:ThreeBodyPotential}. However, the argument
presented by the authors to support their claim actually relies only
on the suppression of two-body probability induced by the hard-core
repulsion. It can thus be extended to all potentials leading to a
sufficiently strong suppression of probability shown in Fig.~\ref{fig:TwoBody},
irrespective of the presence or absence of a hard core.

The argument given in Ref.~\cite{Sorensen2012} then proceeds by
noticing that the three-body configurations for which two particles
are within the van der Waals length must have near zero probability
because of the two-body probability suppression~%
\footnote{In the special case of hard core interactions considered in Ref.~\cite{Sorensen2012},
this is equivalent to saying that these configurations should not
exist because they would probe an infinitely large potential.%
}. Therefore, for small enough hyperradius, all three-body configurations
are trivially suppressed since there are always at least two particles
within the suppression radius, no matter what the shape of the three-body
system is. It is tempting to attribute this trivial three-body suppression
by two-body suppression to the three-body repulsion of Ref.~\cite{JiaWang2012}.
We will now show, however, that this is not the case.

Let us assume for simplicity that the two-body probability is completely
suppressed for separations less than $r_{\tiny\mbox{vdW}}$ and let
us fix the global size of the system to a certain hyperradius $R$.
It is straightforward to show that the shape that maximises the smallest
two-particle distance is the equilateral shape $r_{ij}=R/\sqrt{2}$.
Thus, if the hyperradius is smaller than $\sqrt{2}r_{\tiny\mbox{vdW}}$,
at least two particles are separated by less than $r_{\tiny\mbox{vdW}}$,
whatever the shape of the system. Therefore, the two-body probability
suppression implies with certainty a suppression of the three-body
probability at hyperradii less than $\sqrt{2}r_{\tiny\mbox{vdW}}$.
This is shown in Fig.~\ref{fig:Suppression} by the solid red region
indicating the trivially suppressed three-body configurations. For
other shapes, the hyperradius below which three-body configurations
are suppressed is larger. For elongated shapes, (corresponding to
$\alpha<\pi/4$) the smallest two-particle distance is given by $R\sin\alpha$,
so that configurations with $R<r_{\tiny\mbox{vdW}}/\sin\alpha$ are
suppressed. For squeezed shapes, (corresponding to $\alpha>\pi/4$)
the smallest two-particle distance is given by $\frac{R}{2}\sqrt{1+2\cos^{2}\alpha}$,
so that configurations with $R<2r_{\tiny\mbox{vdW}}/\sqrt{1+2\cos^{2}\alpha}$
are suppressed. The particular case of the linear shape ($\alpha=\pi/2$)
considered in Ref.~\cite{Sorensen2012} gives a suppression hyperradius
$R=2r_{\tiny\mbox{vdW}}$. Unlike what is stated in Ref.~\cite{Sorensen2012},
this shape does not guarantee that configurations at larger hyperradius
are not suppressed by the two-body suppression (i.e. that ``no regions
with infinite potentials are sampled'' in the hard-core potential
picture), because some configurations with elongated shapes are clearly
suppressed at larger hyperradii, as can be seen in Fig.~\ref{fig:Suppression}.
The only statement one can make from these purely geometrical considerations
is that three-body configurations with a hyperradius shorter than
$\sqrt{2}r_{\tiny\mbox{vdW}}$ are suppressed with certainty.

It is thus clear that this trivial three-body suppression does not
explain the three-body repulsion of Ref.~\cite{JiaWang2012} which
manifests itself at $R\approx2r_{\tiny\mbox{vdW}}$. What is more,
neither does it require nor explain the presence of a three-body repulsion
in the hyperradial potential given in Eq.~(\ref{eq:ThreeBodyPotential}).

\begin{figure}
\includegraphics[scale=0.48]{Fig3-Regions}

\caption{\label{fig:Suppression}Regions of three-body configurations plotted
against the hyperradius and one hyperangle, where the molecules illustrate
the corresponding configurations. The region in solid red corresponds
to the region trivially excluded when at least two particles are within
the suppression distance $r_{\tiny\mbox{vdW}}$. The hatched region
corresponds to the region excluded by the nonadiabaticity of the deformation
occuring when the hyperradius is smaller than $\sim2r_{\tiny\mbox{vdW}}$.
The shades of blue represent the integrated three-body probability
obtained by numerically solving the three-body problem with a separable
potential Eq.~(\ref{eq:SeparablePotential}) whose two-body correlation
reproduces that of a Lennard-Jones potential at the appearance of
the first two-body bound state. It shows that the probability is indeed
excluded from the red and hatched regions.}
\end{figure}



\subsection{Suppression due to nonadiabatic deformation}

\begin{figure}
\includegraphics[width=8.6cm]{Fig4-DensityPlot}

\caption{\label{fig:ProbabilityDensity}Three-dimensional contour plots of
the probability distribution in Eq.~(\ref{eq:ProbabilityDensity})
of finding a particle for a given separation of the two other particles
(which are indicated by a pair of small gray balls connected by a
black line). For clarity, we only show the cross sections of a plane
containing the two particles and the probability density behind this
plane. The darker, the higher the probability of finding the third
particle. The top figures correspond to a separation of $6.0\, r_{\tiny\mbox{vdW}}$,
while the bottom ones to a separation of $1.4r_{\tiny\mbox{vdW}}$.
In order to appreciate the change in configuration between the figures,
a typical location of the third particle is indicated by a small green
ball connected to the other two particles by green lines. The left
figures were computed from the zero-range Efimov theory at unitarity;
they show the invariance of the Efimov configuration distribution
with the size of the system. The right figures were computed for a
Lennard-Jones pairwise potential at unitarity supporting four two-body
bound states. At large separations, the probability is consistent
with the Efimov configuration probability, but around each of the
two particles there is a noticeable sphere of radius $\sim r_{\tiny\mbox{vdW}}$
in which the probability is significantly suppressed. This suppression
leads to an abrupt change in configuration probability when the particles
come close.}
\end{figure}


\begin{figure}[tph]
\includegraphics[bb=10bp 0bp 312bp 136bp,clip,width=8.5cm]{Fig5-RepulsiveBarrier}

\caption{\label{fig:Comparison}Comparison between the Faddeev three-body calculations
(left) and the simple two-body correlation model described in the
main text (right). The dashed curves show the nonadiabatic kinetic
energy $Q_{00}$ for Lennard-Jones potentials of different depths,
corresponding to the unitarity limit with different numbers of two-body
bound states ranging from 1 to 5. The solid curves show the full three-body
potential $U_{0}(R)$ obtained by adding the adiabatic contribution
$\lambda_{0}/R^{2}$ obtained from Faddeev calculations, which is
shown by the dotted curves. }
\end{figure}


To explain the repulsive barrier, one needs an energetic argument.
Since the repulsion does not arise from the interaction energy (as
exemplified by the purely attractive pairwise potentials), it must
have a kinetic origin. Our calculation shows that the repulsion originates
from the nonadiabatic kinetic term $Q_{00}$ in Eq.~(\ref{eq:ThreeBodyPotential}),
while the first term remains attractive (it becomes repulsive only
at shorter hyperradii). The nonadiabatic kinetic term is indeed a
positive definite quantity, 
\begin{equation}
Q_{00}(R)=\int d\Omega\left|\frac{\partial\tilde{\Phi}_{0}(\Omega;R)}{\partial R}\right|^{2}\ge0,\label{eq:NonadiabaticKineticEnergy}
\end{equation}
owing to the normalisation of $\tilde{\Phi}_{0}$, and the fact that
$\tilde{\Phi}_{0}$ can be chosen to be real. Since $Q_{00}$ has
to vanish at large distance, it must be repulsive. This nonadiabatic
kinetic energy comes from a change in the hyperangular wavefunction
$\tilde{\Phi}_{0}$ with respect to the hyperradius. To visualise
this, we use $\tilde{\Phi}_{0}$ to plot in Fig.~\ref{fig:ProbabilityDensity}
the probability density of finding a particle 3 for a given separation
$r_{12}$ of the two other particles 1 and 2, 
\begin{equation}
P(\vec{r}_{12,3})=(\sin2\alpha_{3})^{2}\left|\tilde{\Phi}_{0}(\Omega;R)\right|^{2}.\label{eq:ProbabilityDensity}
\end{equation}
At large distance, the hyperangular wavefunction is given by the zero-range
limit (corresponding to the Efimov theory), which at unitarity admits
the following analytical expression~\cite{Efimov1971}, 
\begin{eqnarray}
\tilde{\Phi}_{0}^{\tiny\mbox{(ZR)}}(\Omega) & = & \sum_{i=1}^{3}\frac{\phi_{0}^{\tiny\mbox{(ZR)}}(\alpha_{i})}{\sin2\alpha_{i}},\nonumber \\
\mbox{with}\,\quad\phi_{0}^{\tiny\mbox{(ZR)}}(\alpha) & = & \sinh(s_{0}(\frac{\pi}{2}-\alpha)),\label{eq:ZeroRangeLimit}
\end{eqnarray}
that is independent of the hyperradius. The probability density therefore
remains the same up to a scale transformation. In other words, the
shape of the three-particle system remains the same, namely the third
particle is located close to one of the other two. This invariance
of the hyperangular wave function with respect to the hyperradius
results in $Q_{00}=0$. At shorter separations, however, this picture
becomes inadequate because the interaction region is no longer negligible.
In Fig.~\ref{fig:ProbabilityDensity}, one can clearly see two regions
of suppressed probability near the two particles, corresponding to
the two-body suppression seen in Fig.~\ref{fig:TwoBody}. As the
two particles come close, the two-body suppression pushes out the
Efimov configuration probability to a region forming a ring in between
the two particles, corresponding to an equilateral shape. The necessity
to deform to a more equilateral shape is consistent with the previous
discussion on the trivial three-body suppression, and can be seen
in Fig.~\ref{fig:Suppression} where it is indicated by a black arrow.
We find that this change of shape happens very suddenly, making it
difficult for the system to follow the Efimov channel adiabatically
(see the animations in the supplementary materials). This abrupt variation
results in a significant gain of nonadiabatic kinetic energy $Q_{00}$
in Eq.~(\ref{eq:NonadiabaticKineticEnergy}), thereby creating the
three-body repulsion.

The hyperradius at which the deformation occurs can easily be estimated
from the zero-range Efimov wave function and the radius of two-body
suppression $r_{\tiny\mbox{vdW}}$. The system has to deform when
the distance between the two closest particles is equal to $r_{\tiny\mbox{vdW}}$.
The distance $r_{ij}$ between two particles $i$ and $j$ is equal
to $R\sin\alpha_{k}$. At large hyperradii, the probability distribution
of $\alpha$ at unitarity follows from the Efimov hyperangular wavefunction
in ~Eq.~(\ref{eq:ZeroRangeLimit}). From this wavefunction, one
can calculate the average hyperangle $\langle\alpha\rangle\approx0.508$,
which leads to an approximate location of the onset of deformation
at $R=r_{\tiny\mbox{vdW}}/\sin\langle\alpha\rangle\approx2.05r_{\tiny\mbox{vdW}}$
\footnote{According to definitions of the hyperradius in Refs.~\cite{JiaWang2012}
and \cite{Sorensen2012}, one would find respectively $R\approx\sqrt{\sqrt{3}/2}\times2.05r_{0}=1.91r_{0}$
and $R\approx\frac{1}{\sqrt{2}}\times2.05r_{0}=1.45r_{0}$%
}, simply reproducing the numerical results of \cite{JiaWang2012}.
This is represented in Fig.~\ref{fig:Suppression} by the dashed
lines. Note that the knowledge of the location of the barrier is not
enough to determine precisely the three-body parameter, as it also
depends on the shape of the barrier, as well as the couplings to other
channels. In light of the previous results, we expect these to be
essentially determined by the form of the pair correlation.


\subsection{Universality}

The pair correlation for two particles interacting with van der Waals
interactions is known to have a universal asymptotic form~\cite{Gao1998,Flambaum1999}.
In particular, the zero-energy radial two-body wavefunction $\varphi$
for a given scattering length $a$ has the following analytical form
in the van der Waals tail region:

\begin{equation}
\varphi(r)=\Gamma(5/4)\sqrt{x}J_{\frac{1}{4}}\!\!\left(2x^{-2}\right)-\frac{r_{\tiny\mbox{vdW}}}{a}\Gamma(3/4)\sqrt{x}J_{-\frac{1}{4}}\!\!\left(2x^{-2}\right),\label{eq:universalCorrelation}
\end{equation}
where $\Gamma$ and $J_{\alpha}$ denote the Gamma and Bessel functions,
and $x=\frac{r}{r_{\tiny\mbox{vdW}}}$. At large distance, $\varphi(r)$
asymptotes to the free wave form $1-\frac{r}{a}$. For $a\to\infty$,
$\varphi(r)$ asymptotes to unity and thus can be regarded as a correlation
function describing the deviations from the free wave. The corresponding
two-body probability density $\vert\varphi\vert^{2}$ is represented
in Fig.~\ref{fig:TwoBody} by the black dashed curve. One can see
that the probability densities obtained for all the considered potentials
nearly coincide with this analytical form for $r\gtrsim r_{\tiny\mbox{vdW}}$.
For potentials which strongly suppress the probability for $r\lesssim r_{\tiny\mbox{vdW}}$,
the whole pair correlation is thus very similar to the universal correlation.
This similarity holds even for shallow potentials with a short van
der Waals tail accompanied by a hard-core repulsion, such as that
of helium. The fact that the short-distance oscillations of the universal
correlation are not reproduced does not bring any major difference,
because the probability density is very small in this region. Since
the two-body correlation is nearly the same for these potentials,
the same nonadiabatic deformation occurs, leading to the same three-body
repulsion and three-body parameter. Conversely, the soft-core van
der Waals potential with one two-body bound state leads to a pair
correlation that deviates from the universal correlation more significantly,
with a less pronounced suppression of probability, as seen in Fig.~\ref{fig:TwoBody}.
According to our interpretation, this should create a softer three-body
repulsion at a shorter hyperradius, as can be checked in Fig.~\ref{fig:ThreeBodyPotential}.

In the present interpretation, the universality of the three-body
parameter is thus a direct consequence of the universality of the
two-body physics. 




\section{Check with simple models}

To verify our interpretation, we construct two simple models. The
first one verifies that the pair correlation does indeed create a
three-body repulsive barrier at $R\approx2r_{vdW}$ in the Efimov
channel through the nonadiabatic kinetic energy. The second one is
a more complete model verifying quantitatively that the pair correlation
fixes the three-body parameter to a value consistent with full numerical
calculations and experiments. 


\subsection{Pair correlation model}

To simply account for the two-body suppression, we consider a trial
hyperangular wavefunction of the Bijl-Jastrow form \cite{Bijl1940,Jastrow1955},
which is the uncorrelated hyperangular function $\tilde{\Phi}_{0}^{\tiny\mbox{(ZR)}}$
in the zero-range (Efimov) limit given by Eq.~(\ref{eq:ZeroRangeLimit}),
multiplied by a product of the universal two-body correlation $\varphi$
given by Eq.~(\ref{eq:universalCorrelation}), which causes the suppression
of probability in the two-body sector:
\begin{equation}
\tilde{\Phi}_{0}^{\tiny\mbox{(model)}}=\tilde{\Phi}_{0}^{\tiny\mbox{(ZR)}}\times\prod_{i<j}\varphi(r_{ij}).\label{eq:Ansatz}
\end{equation}


This simple ansatz leads to a probability density that is very similar
to the one calculated from the Faddeev equation (\ref{eq:HyperangularEquation}).
In particular, we have confirmed that it also leads to a sudden buildup
of probability in the ring-shaped region when two particles are close.
One can also calculate the nonadiabatic kinetic energy $Q_{00}$ from
Eq.~(\ref{eq:NonadiabaticKineticEnergy}). As expected, we find a
sudden increase of $Q_{00}$ at the hyperradius $R\approx2r_{\mbox{\tiny vdW}}$.
This model thus confirms our claim that the nonadiabatic change in
configuration originates from an interplay between the suppression
of two-body probability and the Efimov configuration.

Adding the adiabatic term $\frac{\lambda_{0}}{R^{2}}$ to $Q_{00}$,
we obtain the full potential $U_{0}(R)$. As shown in Fig.~\ref{fig:Comparison},
it reproduces very well the universal potential found using the solution
of the Faddeev equation (\ref{eq:HyperangularEquation}). Note that
this agreement is remarkable; although the adabiatic and nonadiabatic
terms taken separately vary signicantly for different numbers of two-body
bound states, their variations almost cancel out to give the universal
potential. 


\subsection{Separable model}

The hyperspherical formalism is useful to exhibit the three-body repulsion
mechanism, and the previous model satisfactorily reproduces the three-body
repulsion in the Efimov channel. However, this channel alone only
gives qualitative results for the actual trimer energies. To be more
quantitative, one would need to solve the many coupled equations in
Eq. (\ref{eq:HyperradialEquation}), as done in Ref.~\cite{JiaWang2012},
but that would defeat our purpose of using a simple model to reproduce
the physics. Hence we turn to another approach to get more quantitative
results, while keeping the central idea of the universal pair correlation
in Eq.~(\ref{eq:universalCorrelation}) being the essential ingredient
behind the universal three-body parameter.

One of the simplest pseudo-potentials that can reproduce the universal
pair correlation is the separable potential~\cite{Yamaguchi1954},
\begin{equation}
\hat{V}=\xi\vert\chi\rangle\langle\chi\vert,\label{eq:SeparablePotential}
\end{equation}
where the function $\chi$ in momentum space is chosen to be
\begin{equation}
\chi(q)=1-q\int_{0}^{\infty}dr\left(1-\frac{r}{a}-\varphi(r)\right)\sin(qr),\label{eq:Chi}
\end{equation}
and the coefficent $\xi$ is set to
\begin{equation}
\xi=4\pi\left(\frac{1}{a}-\frac{2}{\pi}\int_{0}^{\infty}dq\vert\chi(q)\vert^{2}\right)^{-1}.\label{eq:Xi}
\end{equation}
This potential has the advantage of being easily tractable because
of its separability, and one can show (see Appendix A) that the solution
of the two-body problem at zero energy for this potential is given
exactly by $\varphi(r)$, which is chosen to be the universal pair
correlation given by Eq.~(\ref{eq:universalCorrelation}). Numerically,
we find that this potential is an excellent substitute for the real
van der Waals interaction in the two-body problem at low energy: it
reproduces the low-energy scattering state and the two-body bound
state over energies on the order of $\hbar^{2}/(mr_{\tiny\mbox{vdW}}^{2})$
and scattering lengths $\vert a\vert\gtrsim2r_{\tiny\mbox{vdW}}$. 

For the three-body problem, substituting the real potential by the
separable potential in the three-body Schr�dinger equation leads to
a one-dimensional integral equation \cite{Lee2007,Naidon2011} that
is similar to the Skorniakov--Ter-Martirosian equation obtained for
a contact potential~\cite{Skorniakov1957} and that can easily be
solved numerically (see the Appendix B for the derivation). We emphasise
that the only information contained in this model is the zero-energy
pair correlation.

From the numerical solution, we obtain the ground-state trimer spectrum
shown in Fig.~\ref{fig:SeparableSpectrum}. In particular, we extract
the binding wave number $\kappa$ at unitarity and the scattering
length $a_{-}$ at which this trimer disappears in the three-body
threshold, and find
\[
\kappa\, r_{\tiny\mbox{vdW}}=0.187(1)\qquad\mbox{and}\qquad\frac{a_{-}}{r_{\tiny\mbox{vdW}}}=-10.86(1).
\]
\begin{figure}
\includegraphics[scale=0.9]{Fig6-SeparableModel}

\caption{\label{fig:SeparableSpectrum}Lowest trimer and dimer energy for the
separable potential given by Eq.~(\ref{eq:SeparablePotential}),
as a function of its inverse scattering length $1/a$. For comparison,
the dotted and dashed curves represent the universal dimer energy
($E=-\frac{1}{ma^{2}}$) and the exact van der Waals dimer energy,
respectively.}
\end{figure}


Consistent with our interpretation, similar results are obtained for
pair correlations $\varphi(r)$ with a similar tail and low probability
at short distance. For instance, the pair correlation for a Lennard-Jones
potential with one two-body bound state leads to $\kappa\, r_{\tiny\mbox{vdW}}=0.205(1)$
and $\frac{a_{-}}{r_{\tiny\mbox{vdW}}}=-10.23(1)$. These values agree
within a few percents with the results of \cite{JiaWang2012}, and
are in fair agreement with the experimental value $\frac{a_{-}}{r_{\tiny\mbox{vdW}}}=-9.1(5)$~\cite{Ferlaino2011}.
Finally, one can check that this agreement is not accidental, as the
model also reproduces the deformation and repulsion effects. This
is demonstrated by the three-body probability density plot in Fig.~\ref{fig:Suppression}
where one can see the probability density being trivially excluded
by the two-body suppression (causing a deformation towards higher
values of $\alpha$), as well as the suppression at $R\approx2\, r_{\tiny\mbox{vdW}}$
due to the nonadiabaticity of this deformation, as explained previously.


\section{Conclusion}

We have shown that the universality of the three-body parameter revealed
in recent experiments with neutral atoms and numerical calculations
originate from a nonadiabatic deformation of the three-atom system.
This deformation is caused by the suppression of two-body probability
resulting from pair correlation. Because this suppression is universally
determined by the van der Waals length for atomic systems, this makes
the three-body parameter universal in these systems. More generally,
we expect to find such universality in any class of systems where
the two-body suppression has a universal form. This work also suggests
that for other systems, in general, pair correlations and their associated
length scale, the effective range, should play an essential role.
These points will be addressed in more detail in a separate paper.








\section*{Appendix A}

In this Appendix, we show that the radial wave function $\varphi$
in Eq.~(\ref{eq:Chi}) is the solution of the two-body problem at
zero energy for the separable potential given by Eq.~(\ref{eq:SeparablePotential}).

The two-body Schr�dinger equation at zero energy in momentum space
reads:
\begin{equation}
\frac{\hbar^{2}p^{2}}{m}\tilde{\psi}(\vec{p})+\int\frac{d^{3}\vec{q}}{(2\pi)^{3}}\tilde{V}(q)\tilde{\psi}(\vec{q})=0,\label{eq:AppendixA1}
\end{equation}
where $\tilde{V}$ is the Fourier transform of the pairwise potential~$V$,
and $\tilde{\psi}$ is the Fourier transform of the two-body wave
function. Replacing $\tilde{V}$ by the separable potential in Eq.~(\ref{eq:SeparablePotential}),
one obtains: 
\begin{equation}
p^{2}\tilde{\psi}(\vec{p})-f\,\chi(p)=0,\label{eq:AppendixA2}
\end{equation}
with
\begin{equation}
f=-\int\frac{q^{2}dq}{2\pi^{2}}\tilde{\psi}(\vec{q})\xi\chi^{*}(q).\label{eq:AppendixA3}
\end{equation}
Inserting Eq.~(\ref{eq:AppendixA2}) into Eq.~(\ref{eq:AppendixA3}),
one obtains the explicit expression for $f$:
\begin{equation}
f=-\left(\frac{1}{\xi}+\int\frac{dq}{2\pi^{2}}\vert\chi(q)\vert^{2}\right)^{-1},\label{eq:AppendixA4}
\end{equation}
and using the chosen form of $\xi$ given by Eq.~(\ref{eq:Xi}),
one obtains $f=-4\pi a$. Inserting this value in Eq.~(\ref{eq:AppendixA2}),
and inverting the resulting equation with the proper boundary conditions,
one finds:
\begin{equation}
\tilde{\psi}(\vec{p})=(2\pi)^{3}\delta^{3}(\vec{p})-4\pi a\frac{\chi(p)}{p^{2}}.\label{eq:AppendixA5}
\end{equation}
This translates in space coordinates as
\begin{equation}
\psi(\vec{r})=1-4\pi a\int\frac{d^{3}p}{(2\pi)^{3}}\frac{\chi(p)}{p^{2}}e^{i\vec{p}\cdot\vec{r}},\label{eq:AppendixA6}
\end{equation}
which after angular integration yields:
\begin{equation}
\psi(\vec{r})=1-a\frac{2}{\pi}\int_{0}^{\infty}\chi(p)\frac{\sin pr}{pr}dp.\label{eq:AppendixA7}
\end{equation}
Inserting the chosen form of $\chi$ given by Eq.~(\ref{eq:Chi}),
and using the closure relation $\int_{0}^{\infty}dp\sin(pr^{\prime})\sin(pr)=\frac{\pi}{2}\delta(r-r^{\prime})$,
one obtains
\begin{equation}
\psi(\vec{r})=\frac{a}{r}\varphi(r),\label{eq:AppendixA8}
\end{equation}
which shows that the form of the zero-energy radial wavefunction $r\psi(\vec{r})$
is indeed given by $\varphi(r)$.


\section*{Appendix B}

In this Appendix, we derive the equation we use to solve the three-body
problem with the separable potential given by Eq.~(\ref{eq:SeparablePotential}).

The three-body Schr�dinger equation in momentum space reads:
\begin{multline}
\left(\frac{3}{4}\frac{\hbar^{2}}{m}P^{2}+\frac{\hbar^{2}}{m}p^{2}-E\right)\tilde{\Psi}(\vec{P},\vec{p})\\
+\sum_{i=1,2,3}\int\!\!\frac{d^{3}\vec{p}_{i}}{(2\pi)^{3}}\tilde{V}(p_{i})\tilde{\Psi}(\vec{P},\vec{p})=0,\label{eq:AppendixB1}
\end{multline}
where $\tilde{V}$ is the Fourier transform of the pairwise potential~$V$,
and $\tilde{\Psi}$ is the Fourier transform of the three-body wave
function $\Psi$ in Eq.~(\ref{eq:TotalWaveFunction}) expressed in
a particular Jacobi coordinate set $(\vec{P},\vec{p})$ chosen among
the three possible sets $(\vec{P}_{i},\vec{p}_{i})$ with $i=1,2,3$.
Because of the bosonic exchange symmetry, $\tilde{\Psi}(\vec{P},\vec{p})$
can be replaced by $\tilde{\Psi}(\vec{P}_{i},\vec{p}_{i})$ inside
the integral of Eq.~(\ref{eq:AppendixB1}).

Substituting $\tilde{V}$ by the separable potential in Eq.~(\ref{eq:SeparablePotential}),
one obtains: 
\begin{equation}
\left(\frac{3}{4}P^{2}+p^{2}-\frac{m}{\hbar^{2}}E\right)\tilde{\Psi}(\vec{P},\vec{p})+\sum_{i=1,2,3}F(\vec{P}_{i})\chi(p_{i})=0,\label{eq:AppendixB2}
\end{equation}
where 
\begin{equation}
F(\vec{P})=\xi\int\!\!\frac{d^{3}\vec{p}}{(2\pi)^{3}}\chi^{*}(p)\tilde{\Psi}(\vec{P},\vec{p}).\label{eq:AppendixB3}
\end{equation}


For $E<0$, Eq.~(\ref{eq:AppendixB2}) can be inverted as
\begin{equation}
\tilde{\Psi}(\vec{P},\vec{p})=-\sum_{i=1,2,3}\frac{F(\vec{P}_{i})\chi(p_{i})}{\frac{3}{4}P^{2}+p^{2}-\frac{m}{\hbar^{2}}E},\label{eq:AppendixB4}
\end{equation}
Inserting Eq.~(\ref{eq:AppendixB4}) into Eq.~(\ref{eq:AppendixB3})
gives:
\begin{equation}
\frac{1}{\xi}F(\vec{P})=-\sum_{i=1,2,3}\int\!\!\frac{d^{3}\vec{p}}{(2\pi)^{3}}\chi^{*}(p)\frac{F(\vec{P}_{i})\chi(p_{i})}{\frac{3}{4}P^{2}+p^{2}-\frac{m}{\hbar^{2}}E}.\label{eq:AppendixB5}
\end{equation}
Making the choice $(\vec{P},\vec{p})=(\vec{P}_{3},\vec{p}_{3})$,
one can factorise one of the terms in the sum with the left-hand side
of Eq.~(\ref{eq:AppendixB5}) as follows:
\begin{multline}
\left(\frac{1}{\xi}+\int\!\!\frac{d^{3}\vec{p}}{(2\pi)^{3}}\frac{\vert\chi(p)\vert^{2}}{\frac{3}{4}P^{2}+p^{2}-\frac{m}{\hbar^{2}}E}\right)F(\vec{P})\\
+\sum_{i=1,2}\int\!\!\frac{d^{3}\vec{p}}{(2\pi)^{3}}\chi^{*}(p)\frac{F(\vec{P}_{i})\chi(p_{i})}{\frac{3}{4}P^{2}+p^{2}-\frac{m}{\hbar^{2}}E}=0.\label{eq:AppendixB6}
\end{multline}
The two remaining terms are equal due to bosonic exchange symmetry,
and expressing one Jacobi coordinate set in terms of another, one
finally arrives at the integral equation for $F$: 
\begin{multline}
\left(\frac{1}{\xi}+\int\!\!\frac{d^{3}\vec{q}}{(2\pi)^{3}}\frac{\vert\chi(q)\vert^{2}}{q^{2}-(\frac{mE}{\hbar^{2}}-\frac{3}{4}P^{2})}\right)F(\vec{P})\\
+2\int\frac{d^{3}\vec{q}}{(2\pi)^{3}}\frac{\chi^{*}(\left|\vec{q}+\frac{\vec{P}}{2}\right|)\chi(\left|\frac{\vec{q}}{2}+\vec{P}\right|)}{P^{2}+q^{2}+\vec{q}\cdot\vec{P}-\frac{mE}{\hbar^{2}}}F(\vec{q})=0.\label{eq:AppendixB7}
\end{multline}
Using rotational symmetry, it can be reduced to an equation in which
$F$ depends only on the one-dimensional variable $P=\vert\vec{P}\vert$:
\begin{equation}
D(P)F(P)+\int_{0}^{\infty}\frac{q^{2}dq}{2\pi^{2}}H(P,q)F(q)=0,\label{eq:AppendixB8}
\end{equation}
 with
\begin{equation}
D(P)=\frac{1}{\xi}+\int_{0}^{\infty}\frac{dq}{2\pi^{2}}\frac{q^{2}\vert\chi(q)\vert^{2}}{q^{2}-(\frac{mE}{\hbar^{2}}-\frac{3}{4}P^{2})},\label{eq:AppendixB9}
\end{equation}
\begin{equation}
H(P,q)=\int_{-1}^{1}\! du\frac{\chi^{*}(\sqrt{q^{2}\!+\!\frac{1}{4}P^{2}\!+\! qPu})\chi(\sqrt{P^{2}\!+\!\frac{1}{4}q^{2}\!+\! qPu})}{P^{2}+q^{2}+qPu-\frac{mE}{\hbar^{2}}}.\label{eq:AppendixB10}
\end{equation}


Solving for the eigenvalues of the linear operator in the left-hand
side of Eq.~(\ref{eq:AppendixB8}) and looking for the energies $E$
that make one of these eigenvalues equal to zero, consistent with
the right-hand side of Eq.~(\ref{eq:AppendixB8}), yields the energies
of three-body bound states. The corresponding eigenvectors $F$ give
the three-body wave functions $\Psi$ through Eq.~(\ref{eq:AppendixB4}).

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\end{document}

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