# Controlling the atom-sphere interaction with an external electric field

P. P. Abrantes<EMAIL_ADDRESS>Instituto de Física, Universidade
Federal do Rio de Janeiro, 21941-972, Rio de Janeiro, Brazil  V. Pessanha
<EMAIL_ADDRESS>Instituto de Matemática, Universidade Federal do
Rio de Janeiro, 21941-909, Rio de Janeiro, Brazil  Reinaldo de Melo e Souza
<EMAIL_ADDRESS>Instituto de Física, Universidade Federal Fluminense,
24210-346, Rio de Janeiro, Brazil  C. Farina<EMAIL_ADDRESS>Instituto de
Física, Universidade Federal do Rio de Janeiro, 21941-972, Rio de Janeiro,
Brazil

###### Abstract

We investigate the system constituted by a polarizable atom near a nanosphere
under the influence of an external electrostatic field, showing that the
attractive dispersive force between them can be overcome by the electrostatic
interaction. Therefore, in addition to the advantageous possibility of
actively tuning the resultant force with an external agent without the
requirement of physical contact, this force may also become repulsive. We
analyze this situation in different physical regimes of distance and explore
the interaction of different atoms with both metallic and dielectric spheres,
discussing which cases are easier to control. Furthermore, our results reveal
that these repulsive forces can be achieved with feasible field intensities in
the laboratory.

##  I Introduction

Quantum electrodynamics (QED) describes light-matter interactions with
unprecedented success, presenting an excellent agreement between theoretical
predictions and experimental results. One of the many situations in which QED
plays a key role is in the understanding of electromagnetic interactions
between neutral bodies with no permanent multipoles, the so-called dispersive
forces [1], that arise from quantum fluctuations in their charge and current
distributions. It is a multidisciplinary field that has sparked much interest
for decades in different areas, such as chemistry, biology, colloid science,
quantum field theory, and material science (see Refs. [2, 3] and references
therein). These ubiquitous forces typically present an attractive character, a
consequence of the fact that a fluctuating dipole in a given body induces a
dipole in the bodies in its vicinity, which, in most cases, favors attraction,
although this is not always the case. However, when the interaction is
attractive, this may lead to undesirable effects in nano and micromechanical
systems, such as adhesion and stiction [4, 5, 6, 7, 8]. It has been an
important issue and gave rise to a significant search for engineering
configurations that exhibit repulsive forces. For a detailed overview of
theoretical and experimental efforts regarding dispersive forces, see Refs.
[9, 10, 11, 12, 13].

Unfortunately, we have very few general recipes that enable us to architect
systems endowed with repulsive Casimir forces, and most situations require a
complete calculation before we conclude their sign. Some examples of
configurations in which repulsion can already be ruled out beforehand were
stated by Kenneth and Klich [14]. The authors proved that the interaction of
(nonmagnetic) dielectric bodies or conductors is always attractive whenever
they constitute a mirror-symmetric setup, independently of the objects’ shape
or their local dielectric functions. Interestingly, it was recently shown that
a strategy to circumvent this restriction and achieve repulsion can be
accomplished with the insertion of an intermediate chiral medium between the
two materials [15].

Actually, it was discovered long ago that repulsive Casimir force may be
reached. In 1961, Dzyaloshinskii, Lifshitz, and Pitaevskii [16] investigated
systems constituted of three different nonmagnetic media, namely, two parallel
semi-infinite homogeneous dielectrics of permittivities
$\varepsilon_{1}(\omega)$ and $\varepsilon_{2}(\omega)$ separated by an
infinite slab of a third dispersive homogeneous medium of permittivity
$\varepsilon_{3}(\omega)$. If the relation
$\varepsilon_{1}~{}(i\xi)~{}<~{}\varepsilon_{3}~{}(i\xi)~{}<~{}\varepsilon_{2}~{}(i\xi)$
holds for a wide range of frequencies, repulsion occurs according to their
theory. Experimentally, it had not been observed until 1996, when measurements
were performed with an atomic force microscope in the van der Waals limit
[17]. In this same regime, subsequent evidence of repulsive interaction was
also presented by other groups [18, 19, 20, 21]. In 2009, a direct measurement
of long-range repulsive forces between a gold-covered sphere and a large
silica plate mediated by bromobenzene was reported [22].

Another possible way of achieving repulsion concerns dielectric-magnetic
materials. As shown by Feinberg and Sucher [23, 24], this is exactly the case
for two atoms if one of them is electrically polarizable while the other is
magnetically polarizable. Based on these works, but in the context of
stochastic QED, Boyer [25] verified repulsive forces when two parallel plates
are placed close to each other with vacuum in between, if one of them is
perfectly conducting (electric permittivity $\varepsilon\rightarrow\infty$)
and the other one is perfectly permeable (magnetic permeability
$\mu\rightarrow\infty$). Results involving dielectric-magnetic materials have
also been discussed [3, 26, 27]. In recent years, the search for repulsive
forces has been expanded to topological materials [28, 29, 30, 31, 32, 33, 34]
and metamaterials. In the latter case, attempts were carried out [35, 36, 37],
but it turned out that repulsion in such an assembly is exceedingly difficult
[38, 39]. Likewise, systems out-of-thermal equilibrium have received special
attention. Non-equilibrium Casimir forces arise in cases where the objects and
the environment are kept at different temperatures, and some situations may be
a possible source of repulsive interactions [12, 40, 41, 42].

Some predictions have demonstrated that bodies with nontrivial geometry can
present a great opportunity to generate repulsive Casimir interaction in
vacuum. Interesting examples consist of a polarizable particle centered above
an infinitely conducting [43, 44] or dielectric [45] plate with a circular
hole. Particularly, in the case of the conducting plate, this repulsive
nonretarded force was also analytically investigated [46, 47]. In addition,
for sufficient anisotropy, repulsion can be obtained between an atom and some
conducting bodies as, for instance, a semi-infinite plate, a wedge [48], and a
cylinder, provided that the atom moves on a trajectory nonintersecting with
the cylinder [49]. Recently, the nonretarded force between a polarizable
particle and a grounded conducting toroid has been investigated and repulsion
was predicted depending on the values of its two radii and the distance from
the particle to the geometrical center of the toroid [50].

However, if one has to deal with specific geometries and materials, the
aforementioned routes will not help. To obtain repulsion in this scenario, we
must tailor light-matter interactions at the nanoscale. The idea of tuning QED
effects can be traced back to the pioneering work of Purcell in 1946 [51],
where it was shown that the environment of a quantum emitter may substantially
affect its spontaneous emission rate. Controlling the dispersive interaction
resorting to the application of tunable external electric or magnetic fields
has proved to be a profitable venue [15, 33, 52, 53]. It was recently shown
that an electrostatic field could generate a repulsive interaction between
atoms [54]. The interpretation of their result was disputed [55, 56], and
subsequently, a thorough and rigorous study was carried out to settle the
issue [57, 58]. However, regardless of the interpretation, Ref. [54] has
correctly and interestingly shown that realistic values of electrostatic
fields can be used to tune the interaction between atoms, changing it from
attractive to repulsive.

A natural question is whether this holds for interactions between atoms and
surfaces. In this regard, here we investigate the effect of applying an
external electrostatic field in the system composed of a neutral and isolated
sphere and an atom in its ground state with no permanent dipole moment. We
analyze both metallic and dielectric spheres and show that, by varying the
intensity and the orientation of the electric field, the component of the
resultant force on the atom along the line containing the center of the sphere
and the atom, given by the sum of dispersive and electrostatic contributions,
can become repulsive, meaning that the electrostatic force may overcome the
dispersive one. Our findings reveal that such a control can be performed using
the electric field as an effective knob that allows for the manipulation of
the attractive or repulsive character of the atom-sphere interaction,
requiring no physical contact. Most importantly, this tunability can be
obtained even for feasible values of the external field, and can easily be
implemented in laboratories. Furthermore, we show that our results are robust
against the size of the sphere, enabling the same electrostatic field to
control a system composed of different bodies interacting with the atom.

This paper is organized as follows. In Sec. II we introduce the relevant
forces that take place in the setup and establish the validity conditions of
our calculations. Section III is dedicated to the discussions of our results
and Sec. IV is left for final remarks and conclusions. Additionally, we
included an Appendix containing important information on the mathematical
description of materials and atoms studied in this paper.

##  II Atom-sphere interaction

In this paper, we are concerned with the physical system depicted in Fig. 1.
It consists of a neutral and isolated sphere of radius $R$ and a polarizable
atom in the ground state with no permanent dipole moment (nor higher-order
multipoles). For convenience, we choose the Cartesian axes in such a way that
the sphere is centered at the origin and the atom is located at a generic
point $z_{a}=R+a$ of the positive semi-axis $Oz$, with $a$ being the distance
from the atom to the surface of the sphere. This system is exposed to an
external uniform electrostatic field $\bm{E}_{0}$ applied at an arbitrary
angle $\theta_{0}$ with respect to the $Oz$ axis. The atom interacts with the
sphere through a dispersive force and the applied electric field also induces
an electrostatic coupling between them.

Figure 1: A neutral and isolated sphere of radius $R$ and a polarizable atom
in its ground state at a distance $a$ from the surface of the sphere. The $Oz$
axis is chosen parallel to the line connecting the atom to the center of the
sphere. An external uniform electrostatic field $\bm{E}_{0}$ is applied at an
angle $\theta_{0}$ with respect to the $Oz$ axis.

In the following subsections, we separately present the evaluation of each
contribution to analyze which effect is dominant.

###  II.1 Dispersive force

Before presenting the expression of the dispersive force exerted by the sphere
on the atom, a comment is in order. In principle, this dispersion force is
modified by the external electrostatic field since both energy levels and
eigenstates of an atom are affected by the electrostatic field. As a
consequence, the atomic polarizability of the ground state of the atom is also
altered. However, for the field intensities and range of distances between the
atom and the sphere to be considered in this paper, it can be shown that such
a variation is negligible. A heuristic way of realizing this is to consider
the atom as a two-level oscillating system with frequency $\omega_{0}$ and
reduced mass $\mu$. A constant force, such as the one exerted by an
electrostatic uniform field, only shifts the equilibrium position, leaving the
oscillations unaltered. Hence, the polarizability is unaffected as long as the
harmonic approximation holds during the whole motion. This requires
$eE_{0}/(\mu\omega_{0}^{2})\ll a_{0}$, where $e$ is the electron charge,
$E_{0}$ stands for the applied electrostatic field, and $a_{0}$ is the Bohr
radius. Taking $a_{0}\sim 0.5$ Å, $\mu$ as the electron mass, and
$\omega_{0}^{2}\sim 10^{31}$ Hz2 (see Appendix A), it is possible to confirm
that we may neglect changes in the atomic polarizability for fields satisfying
$E_{0}\ll 10^{10}$ V/m, a condition met by the fields employed in this paper.
A full quantum-mechanical calculation yields an equivalent conclusion. As
shown in Ref. [55], the relative change in the static polarizability is given
by the Stark effect, $\delta\omega_{0}/\omega_{0}\sim\alpha
E_{0}^{2}/(\hbar\omega_{0})$, which shows that we may neglect variations in
the polarizability for all cases in which perturbation theory remains valid.
For cesium, the atom with the greatest polarizability considered here, the
relation $\delta\omega_{0}/\omega_{0}\ll 1$ requires $E_{0}\ll 10^{10}$ V/m.
Hence, we may comfortably disregard any effect of $\bm{E}_{0}$ in the
dispersive interaction, as can also be shown by a full quantum electrodynamics
analysis [57, 58].

To begin with, we consider the small sphere limit ($R\ll a$), so that the
dipole approximation holds. In this case, assuming a non-magnetic sphere, the
dispersive interaction energy is given by [59, 60]

$\displaystyle U^{\rm(disp)}_{\rm dip}(z_{a})$ $\displaystyle=-\frac{\hbar
R^{3}}{4\pi^{2}\varepsilon_{0}z_{a}^{6}}\int_{0}^{\infty}\\!\\!\\!d\xi\,\alpha_{a}(i\xi)\frac{\varepsilon(i\xi)-1}{\varepsilon(i\xi)+2}e^{-2\xi
z_{a}/c}$ $\displaystyle\times\left[3+\frac{6\xi z_{a}}{c}+\frac{5(\xi
z_{a})^{2}}{c^{2}}+\frac{2(\xi z_{a})^{3}}{c^{3}}+\frac{(\xi
z_{a})^{4}}{c^{4}}\right]\,,$ (1)

where $\varepsilon_{0}$ is the electric permittivity of vacuum, $c$ is the
light velocity in vacuum, $\alpha_{a}(i\xi)$ denotes the atomic polarizability
evaluated at imaginary frequencies $\xi$, and we used that the sphere
dynamical polarizability can be written as [61]

$\alpha_{s}(i\xi)=4\pi\varepsilon_{0}\frac{\varepsilon(i\xi)-1}{\varepsilon(i\xi)+2}R^{3}\,,$
(2)

where $\varepsilon(i\xi)$ is its electric permittivity (normalized by
$\varepsilon_{0}$). In the next section, we shall perform the integration in
Eq. (1) numerically for different materials and atoms. However, there are some
limits for which we may obtain simple analytical expressions, useful for a
first analysis, since they can already furnish a physical intuition about the
orders of magnitude of all quantities involved. In the case of a two-level
atom, we can employ a Lorentz oscillator model with a single resonance as
indicated in Eq. (24) of the Appendix. Furthermore, for a perfectly conducting
sphere, Eq. (2) simplifies to

$\alpha_{s}^{\rm(c)}=4\pi\varepsilon_{0}R^{3}\,.$ (3)

Substituting these results back into Eq. (1), we may arrive at closed formulas
both for the retarded regime ($z_{a}\gg c/\omega_{0}$, where $\omega_{0}$ is
the atomic transition frequency) and for the non-retarded one ($z_{a}\ll
c/\omega_{0}$). In the retarded regime, we may take
$\alpha_{a}(i\xi)\approx\alpha_{a}(0)$ in Eq. (1) [59], with $\alpha_{a}(0)$
denoting the static atomic polarizability, and, by performing the integration,
we obtain

$U^{\rm(disp)}_{\rm dip,R}=-\frac{23\hbar
c\alpha_{s}^{\rm(c)}\alpha_{a}(0)}{64\pi^{3}\varepsilon_{0}^{2}z_{a}^{7}}\,.$
(4)

On the other hand, in the non-retarded regime, Eq. (1) reduces to

$\displaystyle U^{\rm(disp)}_{\rm dip,NR}$
$\displaystyle=-\frac{3\hbar}{16\pi^{3}\varepsilon_{0}^{2}z^{6}}\int_{0}^{\infty}\\!\\!d\xi\,\alpha_{s}^{\rm(c)}\alpha_{a}(i\xi)$
$\displaystyle=-\frac{3\hbar\omega_{0}\alpha_{s}^{\rm(c)}\alpha_{a}(0)}{32\pi^{2}\varepsilon_{0}^{2}z_{a}^{6}}\,.$
(5)

This expression is usually not very realistic, since the non-retarded and the
perfectly conducting sphere approximations are not compatible in general. As a
matter of fact, in this distance regime, the distance $a$ must be much smaller
than the dominant transition wavelength $\lambda_{0}=2\pi c/\omega_{0}$ of the
atom such that retarded effects in the interaction can be disregarded. In
addition, the perfect conductor approximation assumes that the electric field
does not penetrate the material, holding, thereby, as long as $a$ is much
greater than the penetration length $l_{p}$. Therefore, these two
approximations together are valid for $l_{p}\ll a\ll\lambda_{0}$, which
restricts their applicability regime. However, it will be useful as a first
rough description to investigate orders of magnitude.

When $a\lesssim R$, the dipole approximation for the sphere breaks, implying
that Eq. (1) is no longer valid, and we must now sum over much more
multipoles. Assuming the non-retarded regime, we may obtain the dispersive
interaction from the expression

$\displaystyle U^{\rm(disp)}_{\rm NR}$ $\displaystyle=$
$\displaystyle-\frac{\hbar}{8\pi^{2}\varepsilon_{0}}\sum_{l=1}^{\infty}(2l+1)(l+1)$
(6) $\displaystyle\times$
$\displaystyle\frac{R^{2l+1}}{z_{a}^{2l+4}}\int_{0}^{\infty}\\!\\!d\xi\,\alpha_{a}(i\xi)\frac{\varepsilon(i\xi)-1}{\varepsilon(i\xi)+[(l+1)/l]}\,,$
(8)

as shown in Ref. [60]. Let us now analyze the electrostatic part of the
interaction.

###  II.2 Electrostatic force

In this subsection, we describe the electrostatic force that arises between
the atom and the sphere due to the application of an electrostatic field
${\bm{E}}_{0}$. In such a situation, the atom acquires an induced electric
dipole moment in response to the total field acting on it, namely, the
external field and the electrostatic field created by the sphere. By the same
token, in principle, the sphere also suffers the influence of a resultant
electric field given by the sum of the external field and the field created by
the polarized atom. However, the atomic polarizability of an atom in its
ground state scales with $4\pi\varepsilon_{0}a_{0}^{3}$, and the electric
field intensity of a point dipole of magnitude $p$ behaves as $\sim
p/(4\pi\varepsilon_{0}r^{3})$, with $r$ being the distance of the space point
under consideration to the dipole. Hence, the ratio between the magnitude of
the field created by the atom on the sphere and that of the external field is
on the order of $\sim(a_{0}/a)^{3}$, a negligible quantity in our
configuration. As a consequence, although the electric dipole moment induced
on the atom will be computed taking into account the external field and the
electrostatic field created by the sphere, it will be enough to consider only
the external field $\bm{E}_{0}$ when computing the response of the sphere to
the electrostatic field acting on it.

It is well-known that the electrostatic field created by the sphere when
exposed to an external uniform and constant field ${\bm{E}}_{0}$ is identical,
in its outer region, to that of a point dipole located at its center [61],
which we shall denote by ${\bm{p}}_{s}=\alpha_{s}(0){\bm{E}}_{0}$, with
$\alpha_{s}$ given in Eq. (2). The electrostatic field created by the sphere
at the atom’s position is given by

${\bm{E}}_{s}({\bm{r}}_{a})=\frac{\alpha_{s}}{4\pi\varepsilon_{0}}\frac{3E_{0}\,\mbox{cos}\theta_{0}\,\bm{\hat{z}}-{\bm{E}}_{0}}{z_{a}^{3}}\,,$
(9)

so that, the electric dipole moment induced on the atom by the resultant
field,
${\bm{p}}_{a}=\alpha_{a}\left[{\bm{E}}_{0}+{\bm{E}}_{s}({\bm{r}}_{a})\right]$,
takes the form

${\bm{p}}_{a}=\alpha_{a}\left(1-\frac{\alpha_{s}}{4\pi\varepsilon_{0}z_{a}^{3}}\right){\bm{E}}_{0}+\frac{3\alpha_{a}\alpha_{s}E_{0}\,\mbox{cos}\theta_{0}}{4\pi\varepsilon_{0}z_{a}^{3}}\,{\bm{\hat{z}}}\,.$
(10)

Hence, the electrostatic force exerted by the sphere on the atom is the same
as that exerted by a point dipole ${\bm{p}}_{s}$ at the origin on a point
dipole ${\bm{p}}_{a}$ at the atom’s position. It is worth emphasizing that we
are only interested in the $z$ component of this force. In this sense, we
initially write the general expression for the force exerted by a dipole
$\bm{p}^{\prime}$ at the origin on a dipole $\bm{p}$ at an arbitrary position
$\bm{r}$, given by

$\displaystyle{\bm{F}}^{\rm(el)}_{{\bm{p}}{\bm{p}}^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi\varepsilon_{0}r^{4}}\Bigl{[}3({\bm{p}}\cdot\bm{\hat{r}){\bm{p}}^{\prime}}\;+\;3({\bm{p}}^{\prime}\cdot\bm{\hat{r}}){\bm{p}}$
(11) $\displaystyle+$
$\displaystyle\;3({\bm{p}}\cdot{\bm{p}}^{\prime})\bm{\hat{r}}\;-\;15({\bm{p}}\cdot\bm{\hat{r}})({\bm{p}}^{\prime}\cdot\bm{\hat{r}})\bm{\hat{r}}\Bigr{]}\,.$
(13)

The $z$ component of the electrostatic force $F_{z}^{\rm(el)}$ on the atom can
be obtained by making the substitutions
${\bm{p}}^{\prime}\longrightarrow{\bm{p}}_{s}$,
${\bm{p}}\longrightarrow{\bm{p}}_{a}$,
${\bm{\hat{r}}}\longrightarrow{\bm{\hat{z}}}$, and taking the scalar product
of both sides of Eq. (13) with $\bm{\hat{z}}$, which leads us to

$F^{\rm(el)}_{z}=\frac{3}{4\pi\varepsilon_{0}z_{a}^{4}}\Bigl{[}{\bm{p}}_{a}\cdot{\bm{p}}_{s}\;-3({\bm{p}}_{a}\cdot\bm{\hat{z}})({\bm{p}}_{s}\cdot{\bm{\hat{z}}})\Bigr{]}\,.$
(14)

The last two scalar products readily follow from Eq. (10) and the fact that
${\bm{p}}_{s}=\alpha_{s}{\bm{E}}_{0}$, while the first one is given by

${\bm{p}}_{a}\cdot{\bm{p}}_{s}=\alpha_{a}\alpha_{s}E_{0}^{2}\left[1+\frac{\alpha_{e}}{4\pi\varepsilon_{0}z_{a}^{3}}(3\,\mbox{cos}^{2}\theta_{0}-1)\right]\,.$
(15)

After plugging these results back into Eq. (14), we are left with

$\displaystyle F_{z}^{\rm(el)}$
$\displaystyle=\frac{3\alpha_{a}\alpha_{s}E_{0}^{2}}{4\pi\varepsilon_{0}z_{a}^{4}}$
$\displaystyle\times\left[1-\frac{\alpha_{s}}{4\pi\varepsilon_{0}z_{a}^{3}}-3\cos^{2}\theta_{0}\left(1+\frac{\alpha_{s}}{4\pi\varepsilon_{0}z_{a}^{3}}\right)\right]\,.$
(16)

To check the consistency of this formula, let us consider two particular
configurations. First, we choose $\theta_{0}=0$ so that ${\bm{p}}_{s}$ and
${\bm{p}}_{a}$ are aligned and the force between them is attractive. Indeed,
we see that Eq. $\eqref{eqn: Fz_res_final}$ becomes

$F_{z}^{\rm(el)}=-\frac{6\alpha_{a}\alpha_{s}E_{0}^{2}}{4\pi\varepsilon_{0}z_{a}^{4}}\left(1+2\frac{\alpha_{s}}{4\pi\varepsilon_{0}z_{a}^{3}}\right)<0\,.$
(17)

If we now consider $\theta_{0}=\pi/2$, a situation in which the dipoles are
parallel to each other but perpendicular to the $Oz$ axis, we obtain a
repulsive force. In fact, it follows from Eq. (16) that

$F_{z}^{\rm(el)}=\frac{3\alpha_{a}\alpha_{s}E_{0}^{2}}{4\pi\varepsilon_{0}z_{a}^{4}}\left(1-\frac{\alpha_{s}}{4\pi\varepsilon_{0}z_{a}^{3}}\right)\,,$
(18)

which is clearly a positive expression since
$\alpha_{s}/(4\pi\varepsilon_{0}z_{a}^{3})<1$.

As a final comment, we point out that
$\alpha_{s}/(4\pi\varepsilon_{0}z_{a}^{3})\sim(R/z_{a})^{3}$, according to Eq.
(2) and, therefore, whenever we work in the small sphere limit ($R\ll a$), we
may simply approximate Eq. (16) by

$F_{z}^{\rm(el)}\approx\frac{3\alpha_{a}\alpha_{s}E_{0}^{2}}{4\pi\varepsilon_{0}z_{a}^{4}}\left(1-3\cos^{2}\theta_{0}\right)\,.$
(19)

##  III Results and discussions

We start by considering the system composed of a hydrogen atom near a gold
sphere, demonstrating that an external electrostatic field can be applied to
control the attractive or repulsive character of the resultant force. Next, we
show that the same behavior also occurs when considering other atomic species,
such as Na, K, Rb, Cs, and Fe. Finally, to attest that the tuning mechanism we
are discussing here is robust under the change in the sphere’s material, we
analyze the case of a dielectric sphere made of silicon dioxide (SiO2). In
this situation, we verify that repulsive resultant forces can also be obtained
within the same approach with the only difference that, for a given distance
between the atom and the sphere, the values of necessary fields for switching
the attractive or repulsive character of the force are slightly different.

###  III.1 Hydrogen atom near a metallic sphere

In this subsection, we discuss the interaction between a hydrogen atom and a
gold sphere. The model for the atomic polarizability and the dielectric
function of the sphere can be found in the Appendix. To facilitate the
analysis of the change in sign of the resultant force, we define the ratio
$\Gamma$ between the $z$ component of the resultant force and the absolute
value of the dispersive force acting on the atom, namely,

$\Gamma=\frac{F_{z}^{\rm(net)}}{|F^{\rm(disp)}|}=\frac{F_{z}^{\rm(el)}+F^{\rm(disp)}}{|F^{\rm(disp)}|}\,,$
(20)

where $F^{\rm(disp)}=-\partial U^{\rm(disp)}/\partial z_{a}$, with
$U^{\rm(disp)}$ being the dispersive interaction energy. According to the
previous definition, $\Gamma>0$ means a repulsive force on the atom, while
$\Gamma<0$ means an attractive one. Throughout this paper, we will mainly
focus on discussions of the ratio $\Gamma$ calculated with the dispersive
interaction given in Eq. (1), which is valid for any distance $z_{a}$ as long
as obeying the dipole approximation. In a few situations, however, we will
also show results for $\Gamma$ considering distances $a$ within the range of
the non-retarded regime. In these cases, we shall resort to Eq. (8) to
evaluate the dispersive contribution. Nonetheless, we will call the reader’s
attention every time this change occurs to avoid any misinterpretation.

We begin assuming that the distance separating the atom and the sphere is much
larger than the sphere’s radius ($a\gg R$), and therefore we employ Eq. (1) in
order to evaluate the dispersive force. Note that, in doing so, some care must
be taken in choosing these parameters. To determine the direction of
application of the field that provides the strongest repulsion, Fig. 2
displays $\Gamma$ as a function of $\theta_{0}$ for feasible intensities
$E_{0}$ of the electrostatic field. We set the sphere’s radius $R=60$ nm and
the distance between the atom and the sphere $a=700$ nm, typical values in
experiments involving dispersive forces. First of all, a direct inspection of
Fig. 2 reveals that there are intervals of $\theta_{0}$ for which $\Gamma$ is
positive, which implies a repulsive force between the sphere and the atom. It
is also evident that the orientations of the electrostatic field that best
favor repulsion occur for $\theta_{0}=\pi/2$ or $\theta_{0}=3\pi/2$ as
expected, since the induced dipoles on the atom and the sphere are
perpendicular to the $Oz$ axis and parallel to each other in both cases, being
the configurations of maximal electrostatic repulsion. Another interesting
trait present in Fig. 2 is that the absolute value of $\Gamma$ at the peaks is
smaller than its absolute value at the valleys. This follows from the
combination of two intertwined features: (i) the repulsive electrostatic force
for $\theta_{0}=\pi/2$ is less intense than the attractive electrostatic force
for $\theta_{0}=0$, as can be seen from Eqs. (17) and (18), and (ii) the
electrostatic and dispersive contributions to the resultant force have
opposite directions for $\theta_{0}=\pi/2$, whereas they point in the same
direction for $\theta_{0}=0$. Finally, one should note that the intersection
of all curves occurs at $\Gamma=-1$, a direct consequence of the definition in
Eq. (20). $\Gamma=-1$ implies $F^{\rm(el)}_{z}=0$ and, since $F_{z}^{\rm(el)}$
is proportional to $E_{0}^{2}$ [see Eq. (16)], the solution of this condition
is independent of $E_{0}$.

Figure 2: Ratio $\Gamma$ as a function of the angle $\theta_{0}$ between the
applied electric field and the $Oz$-axis. Different colors refer to different
field intensities. We set $R=60$ nm and $a=700$ nm.

Let us now investigate the dependence of $\Gamma$ with the continuous
variation of the other parameters, keeping $\theta_{0}$ fixed at $\pi/2$. In
Fig. 3, we plot $\Gamma$ as a function of $E_{0}$ for different values of the
distance $a$ and $R=60$ nm. For a given value of $a$, as we increase the
magnitude of the external field $E_{0}$, the system undergoes from an
attractive to a repulsive resultant force. Moreover, this can be achieved with
feasible values of electrostatic fields. For instance, in the range of
distances considered in this figure, we see that the changes from attraction
to repulsion occur within the values of $0.4-1.1\times 10^{5}$ V/m. Notice
that increasing the distance $a$, the electrostatic field necessary to
overcome the dispersive force decreases, as expected, since the dispersive
contribution diminishes with $a$ faster than the electrostatic one. Figure 4
presents a profile that also follows the previous discussion, showing the
ratio $\Gamma$ as a function of the distance $a$ for different field
intensities and the same sphere’s radius $R=60$ nm and field orientation
$\theta_{0}=\pi/2$.

Figure 3: Ratio $\Gamma$ as a function of the field intensity $E_{0}$.
Different colors denote different distances $a$ from the atom to the sphere’s
surface. We set $R=60$ nm and $\theta_{0}=\pi/2$. Figure 4: Ratio $\Gamma$ as
a function of the distance $a$ between the atom and the sphere’s surface for
different field intensities. We set $R=60$ nm and $\theta_{0}=\pi/2$.

We call attention to the fact that $\Gamma$ depends only slightly on the
sphere’s radius. Indeed, in the small sphere limit ($R\ll a$), we may
approximate the electrostatic force by Eq. (19), as already discussed. A
comparison with Eq. (1) shows that the ratio $\Gamma$ is independent of $R$
for a given distance between the atom and the sphere’s center (i.e., for fixed
$z_{a}$). This is a remarkable feature as it holds regardless of the material
composing the sphere and implies that the electrostatic control of the atom-
sphere interaction is robust against the sphere’s size, as long as the dipole
approximation remains valid. However, when the atom-sphere distance is on the
order of the sphere’s radius or smaller, Eq. (1) no longer holds. Assuming
that this distance is short enough so that we can describe the dispersive
interaction in the non-retarded limit, we must change our approach and
calculate the ratio $\Gamma$, evaluating the dispersive contribution from Eq.
(8), instead of from Eq. (1). This situation is displayed in Fig. 5, in which
we plot $\Gamma$ as a function of shorter distances between the atom and the
sphere’s surface. Note that, for electric field intensities $30-100$ times the
values depicted in Fig. 4, we already overcome the dispersive force for
distance regimes smaller than $100$ nm.

Figure 5: Ratio $\Gamma$ as a function of the distance $a$ from the atom to
the sphere’s surface for different field intensities in the short-distance
regime. We set $R=60$ nm and $\theta_{0}=\pi/2$.

Analytical expressions both in the retarded and the non-retarded regimes can
be obtained, providing us with interesting physical insights for our results.
In this paragraph, we will not focus on hydrogen but rather on any atom
described by the two-level model given in Eq. (24). In the retarded regime
($z_{a}\gg c/\omega_{0}$), we may determine the dispersive force from Eq. (4)
and employ Eq. (19) with $\theta_{0}=\pi/2$ to obtain

$\Gamma=\frac{48\pi^{2}\varepsilon_{0}E_{0}^{2}z_{a}^{4}}{161\hbar c}-1\,.$
(21)

This expression has some notable characteristics. It is independent of the
atomic parameters and the material composing the sphere (but, to the latter,
we must also assume that the retarded limit is true for frequencies that
characterize the sphere’s material). This is an expected result since, in the
retarded regime, the dispersive force is proportional to the static
polarizability that cancels out when we evaluate $\Gamma$. We see that the
electrostatic force becomes relevant when the electrostatic energy
$U^{\rm(el)}\sim\varepsilon_{0}E_{0}^{2}z_{a}^{3}$, corresponding to the
electrostatic energy contained in a region of the size of the distance
separating the atom and the sphere, is on the order of the energy carried by
the photons relevant to the dispersive interaction, which is given by $U_{\rm
photon}\sim\hbar c/z_{a}$. To give a numerical estimate, if one takes the
distance to be ten times the hydrogen wavelength transition, it would be
necessary an electric field intensity around $10^{-3}$ V/m to overcome the
dispersive interaction. This small field reflects a feeble dispersive force.
However, situations of more practical interest to the applications mentioned
in Sec. I concern the short distance limit. In this non-retarded regime, when
calculating the dispersive force from Eq. (5) and using Eq. (19) with
$\theta_{0}=\pi/2$, we arrive at

$\Gamma=\frac{4\pi\varepsilon_{0}E_{0}^{2}z_{a}^{3}}{3\hbar\omega_{0}}-1\,.$
(22)

Therefore, the minimum field $E_{c}$ necessary for the electrostatic
attraction to overcome the dispersive force satisfies the relation

$\frac{4\pi\varepsilon_{0}E_{c}^{2}z_{a}^{3}}{3}=\hbar\omega_{0}\,,$ (23)

which means that repulsion is ensured once the atomic transition energy equals
the electrostatic energy contained in a sphere of radius corresponding to the
atom-sphere distance. We remind the reader that the results obtained in this
section did not assume the non-retarded regime nor a perfectly conducting
sphere. Nonetheless, the power-law present in Eq. (23) is valuable in order to
provide a first estimate regarding the necessary electric field to achieve
repulsion. For example, for a hydrogen atom and taking $z_{a}=800$ nm, we
obtain $E_{c}\sim 6\times 10^{4}$ V/m, a value on the same order of magnitude
and only $20\%$ above the predicted one using a more general expression, given
in Eq. (1). Expression (23) overestimates the necessary field for two reasons:
(i) the non-retarded limit overestimates the dispersive force and (ii) a
perfectly conductor demands a stronger field than a real metal (see Sec.
III.3).

Another important aspect included in Eq. (23) is that the required
electrostatic field in the non-retarded regime scales with $\sqrt{\omega_{0}}$
and, consequently, atoms with lower transition frequencies are easier to
control. It can be well understood if we notice that when we increase
$\omega_{0}$, the dispersive and the electrostatic forces diminish since the
atomic polarizability is reduced. Nevertheless, as can be inferred from Eqs.
(1), (18), and (24), the integrand in the ratio
$F^{\rm(disp)}/F_{z}^{\rm(el)}$ depends on $\omega_{0}$ through the factor
$\omega_{0}^{2}/(\omega_{0}^{2}+\xi^{2})$. Therefore, increasing $\omega_{0}$
also increases the integrand for every value of $\xi$. As a consequence, as we
increase $\omega_{0}$, the dispersive force weakens more slowly than the
electrostatic one, thus explaining the aforementioned behavior. Physically, it
is related to the fact that the dispersive interaction is a fluctuating-
induced phenomenon that arises in the coupling between the dipole fluctuations
on each body. Although increasing the transition frequency reduces the
dispersive force (since it weakens the virtual excitations and reduces the
polarizability), it also allows for a faster atomic dipole fluctuation, which
enhances the dipole-dipole correlation and smoothes the decrease in the
dispersive force in comparison with the electrostatic force. This hallmark was
discussed for two-level atoms in the non-retarded regime but, in the following
subsection, we demonstrate that it remains accurate even for a more realistic
treatment of atoms and taking into account the complete dispersive energy
interaction, dropping the non-retarded interaction assumption.

###  III.2 Other atoms

Figure 6: (a) Ratio $\Gamma$ as a function of the distance $a$ between a given
atomic species and the sphere’s surface for $E_{0}=1\times 10^{5}$ V/m. (b)
Ratio $\Gamma$ as a function of the field intensity $E_{0}$ with $a=500$ nm.
In all panels, different colors refer to different atomic species and we set
$R=60$ nm and $\theta_{0}=\pi/2$.

Even though the two-level approximation suits well in the mathematical
description of the hydrogen polarizability, there is no good agreement with
experiments when assuming only one relevant transition for heavier atoms.
However, in such cases, a good solution is to implement the so-called two-
oscillator model, which consists of keeping two atomic transitions as the main
contributions to the dynamical atomic polarizability, as shown in the
Appendix. We now come back to the description of the dispersive interaction
through Eq. (1) and, by using Eqs. (18) and (25), we are guided to the results
presented in Fig. 6. The first panel exhibits $\Gamma$ as a function of the
distance $a$ for a fixed field intensity $E_{0}=1\times 10^{5}$ V/m, while the
second panel exposes the ratio $\Gamma$ as a function of the electric field at
a given distance $a=500$ nm from the sphere’s surface. In both of them, each
curve denotes a different atomic specimen. Note from Fig. 6(a) that all
$\Gamma$ curves eventually cross the zero value when $a$ ranges between
$508-530$ nm. It should also be noticed that it is possible to change the
profile of the net force on each atom just by varying its distance $a$. An
equivalent route of observing this same behavior is to investigate $\Gamma$ as
a function of the electric field and, accordingly, the outcome is the
possibility of changing the $\Gamma$ sign just by varying the intensity of
this external agent. Moreover, we stress that this modification occurs for
feasible values of the electric field, being within the scope of experimental
realization. The pattern presented in the curves of Figs. 6(a) and 6(b) can
also be understood. In the two-oscillator model [Eq. (25)], one of the
frequencies is generally much smaller than the other (typically,
$\omega_{01}/\omega_{02}\sim 0.01$) and, as a consequence, the lower
transition frequency ends up being responsible for the dominant response.
Hence, the electrostatic effect is more prominent for atoms exhibiting smaller
transition frequencies, exactly as illustrated in the panels. In other words,
for a given distance, atoms with smaller $\omega_{01}$ requires a smaller
electrostatic field to overcome the dispersive force.

Figure 7: Contour plot of the ratio $\Gamma$ for the cesium atom varying
$\theta_{0}$ and $E_{0}$. The dashed black line indicates combinations of
$\theta_{0}$ and $E_{0}$ for which $\Gamma=0$. Here, we chose $R=60$ nm and
$a=700$ nm.

Amid the atoms we chose to work with, cesium is the most easily controlled by
the electrostatic field, because it exhibits the smallest $\omega_{01}$
(although it has the highest $\omega_{02}$) – except for hydrogen, which is
well represented by the one-oscillator model, as we stated before. Therefore,
we have chosen this atomic species to explore the contour plots that follow.
We begin with Fig. 7 that shows $\Gamma$ as a function of $\theta_{0}$ and
$E_{0}$. In this figure, the dashed black line indicates $\Gamma=0$,
separating regions in the parameter space of $\theta_{0}$ and $E_{0}$ for
which the net force is attractive or repulsive. Note that the most intense
repulsion value (when it takes place) occurs for $\theta_{0}=\pi/2$, as
previously mentioned. Curves similar to the ones in Fig. 2, but to the cesium
atom, are found by taking horizontal lines of constant $E_{0}$ in Fig. 7.
Lastly, in Fig. 8, we consider $\Gamma$ as a function of $a$ and $E_{0}$, and,
once more, the dashed white line represents $\Gamma=0$. Likewise, by selecting
the horizontal lines, we obtain results similar to those in Fig. 4.

Figure 8: Contour plot of the ratio $\Gamma$ for the cesium atom varying $a$
and $E_{0}$. The dashed white line indicates combinations of $a$ and $E_{0}$
for which $\Gamma=0$. Here, we chose $R=60$ nm and $\theta_{0}=\pi/2$.

###  III.3 Atoms near a dielectric sphere

Figure 9: Ratio $\Gamma$ as a function of the distance $a$ between a cesium
atom and the sphere’s surface for (a) $E_{0}=7\times 10^{4}$ V/m, using Eq.
(1), and (b) $E_{0}=9\times 10^{6}$ V/m, using Eq. (8). In all panels,
different colors refer to spheres made of different materials, and we set
$R=60$ nm and $\theta_{0}=\pi/2$. The insets show the resultant force on the
atom as a function of $a$.

In our previous discussions, we have always assumed the interaction with a
metallic sphere. If the atoms are near a dielectric one instead, the general
expressions presented in Sec. II are still valid and we must only substitute
back on them the polarizability of the dielectric sphere in question. A model
for the electric permittivity of silica spheres is described in the Appendix.
For dielectrics, both the dispersive and the electrostatic interactions are
weaker than for the metallic case, when considering a given atom placed at a
fixed distance separating it from the sphere. Therefore, it is not clear at
first sight whether $\Gamma$ should be greater for a dielectric or a metallic
sphere (with all conditions being the same). In other words, is it easier to
generate repulsion when an atom is close to a dielectric instead of a metallic
sphere? To address this question, in Fig. 9, we plot $\Gamma$ for a cesium
atom next to a silica sphere as a function of the atom-sphere distance $a$ in
different regimes and compare it with $\Gamma$ for the same situation
replacing the silica sphere for a gold one with the same radius. We see that,
for a fixed distance, $\Gamma$ is always greater in the dielectric case. For
example, the distance for which the electrostatic force equalizes the
dispersive one for the gold case is a distance where the same field has
already generated repulsion in the dielectric situation. It is related to the
results obtained in Sec. III.1, where we explained the mechanism which makes
it easier to control atoms with smaller transition frequencies. Metals allow
for a faster response than dielectrics (analogous to atoms with higher
transition frequencies), implying greater polarizability for each imaginary
frequency, which, in turn, enhances the dispersive and the electrostatic
forces. However, the effect on the former is more pronounced due to its
fluctuating-induced nature, and, therefore, metals require stronger electric
field intensities than dielectrics so that the atom-sphere force becomes
repulsive.

##  IV Final remarks and conclusions

We have investigated the possibility of controlling the dispersive interaction
in the system composed of an atom and a neutral and isolated sphere when
exposed to an external electrostatic field. We have explored both metallic and
dielectric spheres as well as the implications of considering different atomic
species. Our results demonstrated that the electrostatic force, that arises
between the atom and the sphere, can provide active control of this
interaction without demanding physical contact. More specifically, the
electrostatic force enables the tunability of the sign of the resultant force
since this electrostatic contribution can overcome the dispersive one.
Moreover, we highlight that such a degree of control can be achieved for
feasible values of the electric field, being within the scope of experimental
realization. Concerning the outcomes of studying different atomic species and
materials composing the sphere, we concluded that larger field intensities are
required when dealing with metallic spheres and with atoms that exhibit larger
transition frequencies. We have also discussed in detail the dependence of our
results on the magnitude of the electric field, its orientation, and the atom-
sphere distance. Although our results assume a spherical surface, we expect
that the orders of magnitude of the electrostatic field required to overcome
the dispersive interaction are insensitive to small shape variations.
Nonetheless, it is still interesting to be able to investigate different
geometries. Whenever the dipole approximation remains valid, our treatment can
be immediately applied. Therefore, it suffices to substitute the
polarizability of the sphere with the appropriate dynamical polarizability of
the object under study. As the polarizability scales with the volume, the
orders of magnitude involved must hold regardless of the shape, replacing the
object with a sphere of the same volume. We expect that the results presented
above may open different routes to control dispersive forces and inspire other
configurations that may also exhibit repulsion.

###### Acknowledgements.

We are indebted to S. Y. Buhmann for valuable suggestions and for calling our
attention to Ref. [60] and to P. A. Maia Neto for enlightening discussions. We
also thank the Brazilian agencies FAPERJ, CAPES, and CNPq for financial
support (C.F. was supported under Grant No. CNPq, 310365/2018-0).

##  Appendix A Atomic polarizabilities and dielectric functions

In order to characterize the hydrogen atomic polarizability in the dispersive
interaction, we employed a single-oscillator model, given by

$\alpha_{a}(i\xi)=\frac{\omega_{0}^{2}\alpha_{a}(0)}{\omega_{0}^{2}+\xi^{2}}\,,$
(24)

with $\alpha_{a}(0)=4.5$ a.u. and $\omega_{0}=11.65$ eV [62]. In the case of
heavier atoms, such as Na, K, Fe, Rb, and Cs, we employed a two-oscillator
model, written as

$\alpha_{a}(i\xi)=\frac{\omega_{01}^{2}\alpha_{a1}(0)}{\omega_{01}^{2}+\xi^{2}}+\frac{\omega_{02}^{2}\alpha_{a2}(0)}{\omega_{02}^{2}+\xi^{2}}\,.$
(25)

The fitted parameters for each of these atomic specimen also analyzed here are
reported in Table 1.

Atom |  |  $\alpha_{a1}(0)$ (a.u.) |  |  $\omega_{01}$ (eV) |  |  $\alpha_{a2}(0)$ (a.u.) |  |  $\omega_{02}$ (eV)
---|---|---|---|---|---|---|---|---
Na |  | 162.1 |  | 2.12 |  | 0.547 |  | 116.4
K |  | 288.4 |  | 1.66 |  | 1.754 |  | 87.0
Fe |  | 307.8 |  | 1.75 |  | 9.972 |  | 42.8
Rb |  | 316.7 |  | 1.65 |  | 1.85 |  | 119.6
Cs |  | 397.3 |  | 1.53 |  | 2.597 |  | 123.8
Table 1: Data for Na, K, Fe, Rb, and Cs atoms. This table contains parameters of the two-oscillator model to be used in Eq. (25) [62] ($1$ a.u. $=1.648\times 10^{-41}$ C2m2J-1). Parameter |  | Value (Hz) |  | Parameter |  | Value (Hz)
---|---|---|---|---|---|---
$\omega_{p0}$ |  | $1.37\times 10^{16}$ |  | $\gamma_{0}$ |  | $4.05\times 10^{13}$
$\omega_{p1}$ |  | $1.75\times 10^{14}$ |  | $\gamma_{1}$ |  | $4.28\times 10^{13}$
$\omega_{p2}$ |  | $2.96\times 10^{16}$ |  | $\gamma_{2}$ |  | $8.09\times 10^{15}$
$\omega_{T1}$ |  | $1.32\times 10^{14}$ |  | $\omega_{T2}$ |  | $2.72\times 10^{16}$
Table 2: Data for gold and silicon dioxide spheres. This table contains
parameters for each model in Eqs. (26) [63] and (27) [60].

The mathematical description of the different materials composing the spheres
is based on their dielectric functions. We assume that the expressions for the
metallic (gold – Au) and dielectric (silicon dioxide – SiO2) spheres are
written using a Drude model and a Drude-Lorentz model, respectively, according
to

$\displaystyle\varepsilon_{\rm Au}(i\xi)$
$\displaystyle=1+\frac{\omega_{p0}^{2}}{\gamma_{0}\xi+\xi^{2}}\,,$ (26)
$\displaystyle\varepsilon_{\textrm{SiO${}_{2}$}}(i\xi)$
$\displaystyle=1+\frac{\omega_{p1}^{2}}{\omega_{T1}^{2}+\gamma_{1}\xi+\xi^{2}}+\frac{\omega_{p2}^{2}}{\omega_{T2}^{2}+\gamma_{2}\xi+\xi^{2}}\,.$
(27)

The fitted parameters for these materials are reported in Table 2.

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# Partial wave analysis of decays with arbitrary spins

Xiao-Yu Li<EMAIL_ADDRESS>Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing, 100049, China  School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China  Xiang-Kun
Dong<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics,
Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China  School of Physical
Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Hao-Jie Jing<EMAIL_ADDRESS>School of Physical Sciences, University of
Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

In this paper, we propose a method to construct the decay amplitudes in the
orbital ($L$) and spin ($S$) coupling scheme for particles with arbitrary
spins. For the $1\to 2$ decay with only massive particles involved, the
angular dependence is completely encoded in the angular momentum part, and the
spins of daughter particles are coupled in the rest frame of the mother
particle, which contributes only a constant factor. For the sequential decay,
the total amplitude is constructed by the two $1\to 2$ amplitudes evaluated in
the rest frame of their own mother particles, and then they are transformed to
the common frame, usually chosen as the laboratory frame, by certain Lorentz
transformations. In this way, it is easy to add the amplitudes of possible
different decay chains coherently. If massless particles show up in the final
states, the polarizations are expressed in helicity basis and the amplitudes
are modified correspondingly.

##  I Introduction

The underlying theory describing the strong interaction, the quantum
chromodynamics (QCD), is nonperturbative at low energy, and thus it still
remains a challenge to describe the low energy interaction between hadrons
from the first principle. On the other hand, the high quality and a vast
amount of data on hadron productions and decays are or will be available in
the worldwide high energy experiments. Hence rigorous amplitude analysis is
necessary to extract the parameters of resonances. In some cases, the
resonances have apparent signals in the data, and simple fit models, e.g.,
Breit-Wigner or Flattè, work well to extract the resonance parameters.
However, there are many other cases where several resonances live in the same
energy region and overlap. In such cases, the partial wave analysis is
necessary.

Several methods have been used to construct amplitudes that are used to fit
the experimental data. By taking the hadrons as elementary particles and
writing down the effective Lagrangian, one can obtain the corresponding
amplitudes, see, e.g., Refs Olsson and Osypowski (1975); Benmerrouche  _et
al._ (1995); Nakayama  _et al._ (2002); Liang  _et al._ (2002). This method
keeps the Lorentz invariance, but it is hard to obtain individual amplitudes
with certain orbital angular momentum. In the helicity-coupling amplitudes
proposed in Ref Jacob and Wick (1959) and developed in Refs Chung (1971, 1993,
1998), the amplitudes are constructed to represent the processes where final
states are in the helicity basis, and the angular dependence is encoded in the
Wigner $D$ matrix for rotations. The helicity-coupling amplitudes for
sequential decays with several decay chains have been derived in Ref Chen and
Ping (2017); Marangotto (2020) so that they can be added coherently. Another
way to write the amplitudes for a certain spin and orbital angular momentum is
called covariant $L$-$S$ scheme Zou and Bugg (2003); Zou and Hussain (2003)
where the blocks are the spin wave functions of initial and final particles
and the tensors for orbital angular momenta. In this scheme, the independent
amplitudes for a certain decay are characterized by the total spin of final
states and the orbital angular momentum. This method holds the Lorentz
covariance of each block, and different decay chains can be added coherently.
However, it is very complicated to construct the total spin wave functions if
high spin particles are involved. The comparisons between some of these
methods can be consulted in, e.g., Refs Mikhasenko  _et al._ (2018); Pilloni
_et al._ (2018)

In this paper, we overcome the drawbacks of the covariant $L$-$S$ coupling
method and provide a systematic way to construct the decay amplitudes with
certain orbital angular momentum and total spin of the final particles. Unlike
the treatments in Refs Zou and Bugg (2003); Zou and Hussain (2003) where
explicit couplings of spin wavefunctions are needed, which are usually
complicated when high spin particles are involved, we introduce an abstract
coupling function that couples two spins into one total spin. Besides, the
spins are defined in their “standard” frame, and the angular dependence is
totally absorbed into the angular momentum tensors for massive particles,
contrary to those in Refs Zou and Bugg (2003); Zou and Hussain (2003) where
the spin wave functions still depend on angles.

This paper is organized as follows: The spin wave functions for arbitrary
spins in arbitrary frames are introduced in Sect. II; The covariant $L$-$S$
coupling amplitudes for $1\to 2$ decays are constructed in Sect. III and the
sequential decays are considered in Sect. IV, followed by several illustrative
examples presented in Sect. V; We finally give the discussion in Sect. VI

##  II Spin wave functions

Let us begin with the spin wave function in the “standard” frame where the
particle has the “standard” momentum Weinberg (2005), namely,

$\displaystyle k^{\mu}=\left\\{\begin{array}[]{ll}(M,0,0,0)&\text{for mass
$M$}\\\ (\kappa,0,0,\kappa)&\text{for mass $0$}\end{array}\right.,$ (3)

with $\kappa$ a positive energy. In such a frame, the spin wave function reads

$\displaystyle\chi_{S}^{\mu_{1}\cdots\mu_{n}}(k,m)=\left\\{\begin{array}[]{ll}\epsilon^{\mu_{1}\cdots\mu_{n}}(k,m)&\text{for
a boson with }S=n\\\ u/v^{\mu_{1}\cdots\mu_{n}}(k,m)&\text{for a
fermion/antifermion with }S=n+1/2\end{array}\right.,$ (6)

with $m$ the polarization. For a massive particle with spin-1, we have

$\displaystyle\chi_{1}^{\mu}(k,0)=(0,0,0,1)\quad\chi_{1}^{\mu}(k,\pm
1)=\frac{1}{\sqrt{2}}(0,\mp 1,-i,0),$ (7)

while for spin-1/2, they take the following form in the Weyl basis

$\displaystyle\chi_{1/2}(k,1/2)=\sqrt{M}(1,0,\pm 1,0)^{\rm
T},\quad\chi_{1/2}(k,-1/2)=\sqrt{M}(0,1,0,\pm 1)^{\rm T},$ (8)

where $\pm$ correspond to fermion and antifermion, respectively. For a
massless spin-1 particle, say photon, the spin wave function takes the same
form as those in Eq.(7) except that $m=0$ is not allowed. The wave functions
for particles with higher spins in their “standard” frame can be constructed
via $\chi_{1}^{\mu}(k,m)$ and $\chi_{1/2}^{\mu}(k,m)$ together with Clebsch-
Gordan (CG) coefficients Rarita and Schwinger (1941); Zou and Hussain (2003).

The spin wave function of a particle with arbitrary momentum $p$ is defined by

$\displaystyle\chi_{S}(p,m)=U(\tilde{\Lambda}_{k\to p})\chi_{S}(k,m),$ (9)

where $\tilde{\Lambda}_{k\to p}$ is a Lorentz transformation that take $k$ to
$p$ and $U(\Lambda)$ is the corresponding representation matrix acting on
$\chi_{S}$. Note that all the Lorentz transformations $\Lambda_{k\to p}$ that
take $k$ to $p$ form a coset of the corresponding little group $H_{k}$ for the
“standard” momentum $k$ (SO(3) for massive particles and ISO(2) for the
massless ones). $\tilde{\Lambda}_{k\to p}$ belongs to this coset and it is
convenient to choose

$\displaystyle\tilde{\Lambda}_{k\to
p}=\left\\{\begin{array}[]{ll}R(\hat{\bm{p}})B\left(\sqrt{\frac{\bm{p}^{2}}{M^{2}+\bm{p}^{2}}}\right)R^{-1}(\hat{\bm{p}})&\text{
for massive particles,}\\\
R(\hat{\bm{p}})B\left(\frac{\bm{p}^{2}-\kappa^{2}}{\bm{p}^{2}+\kappa^{2}}\right)&\text{
for massless particles,}\end{array}\right.$ (12)

where $R(\hat{\bm{p}})$ rotates $z$ axis to the direction $\hat{\bm{p}}$ and
$B(v)$ is the boost along $z$-axis by velocity $v$. A general Lorentz
transformation can be decomposed as

$\displaystyle\Lambda_{k\to p}=\tilde{\Lambda}_{k\to p}\Lambda_{k\to k},$ (13)

where $\Lambda_{k\to k}$ belongs to $H_{k}$. Under this Lorentz
transformation, the spin state is transformed as

$\displaystyle\chi_{S}(p,m)$
$\displaystyle=\sum_{m^{\prime}}D_{mm^{\prime}}(\Lambda_{k\to
k})U(\tilde{\Lambda}_{k\to p})\chi_{S}(k,m^{\prime})$
$\displaystyle=\sum_{m^{\prime}}D_{mm^{\prime}}(\Lambda_{k\to
k})\chi_{S}(p,m^{\prime}),$ (14)

with $D_{mm^{\prime}}$ the representation matrix for $H_{k}$Weinberg (2005).

##  III $L-S$ coupling for $1\to 2$ process

Consider a two-body decay

$\displaystyle a^{S_{a}}(p_{a},m_{a})\to
b^{S_{b}}(p_{b},m_{b})\overbrace{+}^{L,S}c^{S_{c}}(p_{c},m_{c}),$ (15)

where $S_{i},m_{i}$ and $p_{i}$ are the spin, polarization and momentum of
particle $i$, respectively, defined in an arbitrary frame, say the laboratory
frame. $L$ is the orbital angular momentum between the final two particles in
this frame, and $S$ is the total spin of the final state. Note that $m_{i}$ is
the same as that in the “standard” frame of particle $i$. It is convenient to
construct the amplitude in the “standard” frame of the mother particle $a$,
denoted by $a^{*}$-frame, which is obtained by the Lorentz transformation
$\tilde{\Lambda}_{p_{a}\to k_{a}}$ from the laboratory frame. In this frame,
the momenta and polarizations are labeled by $p_{i}^{\prime}$ and
$m_{i}^{\prime}$, respectively, with $p_{i}=\tilde{\Lambda}^{\mu}_{k_{a}\to
p_{a}}p_{i\mu}^{\prime}$, $p_{a}^{\prime}=k_{a}$ and

$\displaystyle\chi_{S}(k_{i},m_{i})=\sum_{m_{i}^{\prime}}D_{m_{i},m_{i}^{\prime}}(\Lambda^{a}_{k_{i}\to
k_{i}})\chi_{S}(k_{i},m_{i}^{\prime}),$ (16)

with $\Lambda^{a}_{k_{i}\to k_{i}}=\tilde{\Lambda}_{p_{i}\to
k_{i}}\tilde{\Lambda}_{k_{a}\to p_{a}}\tilde{\Lambda}_{k_{i}\to
p^{\prime}_{i}}$. The spatial wave function of particle $b$ and $c$ can be
decomposed into angular and radial parts and the former, which can couple to
the spins, will be represented by the orbital angular momentum tensor
introduced in Refs Chung (1993, 1998),

$\displaystyle
T_{L}^{\mu_{1}\cdots\mu_{L}}(p^{\prime}_{b},p^{\prime}_{c},m_{L})={q^{\nu_{1}}\cdots
q^{\nu_{L}}\epsilon^{*}_{\nu_{1}\cdots\nu_{L}}(p^{\prime},m’_{L})}\epsilon_{L}^{\mu_{1}\cdots\mu_{L}}(p^{\prime},m^{\prime}_{L})B_{L}(Q_{abc}),$
(17)

with
$p^{\prime}=p^{\prime}_{b}+p^{\prime}_{c}=k_{a},q^{\prime}=p^{\prime}_{b}-p^{\prime}_{c}$
and $m^{\prime}_{L}$ the $z$-component of orbital angular momentum. We have
introduced a Blatt-Weisskopf barrier factor $B_{L}$ as in Refs Von Hippel and
Quigg (1972); Chung (1993); Zou and Hussain (2003); Zou and Bugg (2003) to
suppress the high energy tail of the orbital angular momentum tensor with
$Q_{abc}$ the magnitude of the 3-momentum of particle $b$ in the rest frame of
$a$.

In general, the spin wave functions of a particle, say $\alpha$, depends on
the momentum $p_{\alpha}$ and in turn introduces additional angular dependence
to the amplitude. To absorb such angular dependence into the orbital momentum,
one can express the $\chi_{S}(p,m)$ as that in a “standard” frame transformed
by a Lorentz transformation, see Eq.(9). When constructing the amplitudes,
these Lorentz transformation can be absorbed into the angular momentum tensors
defined in Eq.(17) and the spin part constructed with the spin wavefunctions
with standard momenta are now independent of angle.

The general expression of the amplitude $\mathcal{M}$ can be constructed as a
Lorentz scalar function of the momentum $p_{i}$ and spin wave functions and
can be decomposed into different partial-wave contributions,

$\displaystyle\mathcal{M}_{a\to
bc}(p^{\prime}_{a},m^{\prime}_{a};p^{\prime}_{b},m^{\prime}_{b},p^{\prime}_{c},m^{\prime}_{c};m^{\prime}_{L})=\sum_{L,S}A_{L,S}^{S_{a}}\mathcal{M}^{S_{a}}_{L,S},$
(18)

where $A^{S_{a}}_{LS}$ are unknown, usually complex, dynamic parameters
related to certain spatial angular momentum and total spin and
$\mathcal{M}^{S_{a}}_{LS}$ is the partial component with certain $L$ and $S$,
which can be expressed as

$\mathcal{M}^{S_{a}}_{L,S}\left(p^{\prime}_{a},m^{\prime}_{a};p^{\prime}_{b},m^{\prime}_{b},p^{\prime}_{c},m^{\prime}_{c};m^{\prime}_{L}\right)=J^{S_{a};L,S*}_{\mu_{1}\cdots\mu_{n}}\left(p^{\prime}_{a},m^{\prime}_{a};p^{\prime}_{b},m^{\prime}_{b},p^{\prime}_{c},m^{\prime}_{c};m^{\prime}_{L}\right){\chi}_{S_{a}}^{\mu_{1}\cdots\mu_{n}}(p^{\prime}_{a},m^{\prime}_{a}),$
(19)

where $J^{S_{a};L,S}$ denotes the total angular momentum ($S_{a}$) wave
function of the final states and it should be constructed by wave functions of
$L$, $S_{b}$, and $S_{c}$.

We introduce the coupling function, which is linear on its arguments and
defined by

$\displaystyle F^{[S,m]}\left(\chi^{\ }_{S_{1}}(p,m_{1}),\chi^{\
}_{S_{2}}(p,m_{2})\right)=C_{S_{1}m_{1}S_{2}m_{2}}^{Sm}\chi_{S}(p,m),$ (20)

to compose and project wave functions of two massive particles into a combined
wave function. It is straightforward to modified the above coupling function
for two massless particles. In the following we will focus on the case where
all particles are massive. It is easy to realize that this coupling function
is similar to $C^{jm}_{j_{1}m_{1}j_{2}m_{2}}$, which combines two states with
definite spin $j_{1}$ and $j_{2}$ into a state with spin $j$; here the
coupling function $F^{[S,m]}\left(\chi^{\ }_{S_{1}}(p,m_{1}),\chi^{\
}_{S_{2}}(p,m_{2})\right)$ combines two Lorentz covariant wave functions
$\chi^{\ }_{S_{1}}(p,m_{1})$ and $\chi^{\ }_{S_{2}}(p,m_{2})$ into another
Lorentz covariant wave function $\chi_{S}(p,m)$. Therefore, such coupling
functions correspond to the Lorentz covariant CG coefficients of the group
SO(3). Through group theory, especially by using irreducible tensors, one can
find the explicit forms of these coupling functions, so as to construct the
$L$-$S$ coupling satisfying the Lorentz covariance.The strict mathematical
basis of this function will be discussed in Ref. Jing  _et al._ and here just
keep in mind that it does yield the correct coupling. If one particle, say
particle 2, is massless, the coupling function reads

$\displaystyle F^{[S,m]}\left(\chi^{\ }_{S_{1}}(p,m_{1}),\chi^{\
}_{S_{2}}(p_{\gamma},m_{2})\right)=\sum_{m_{2}^{\prime}}D^{S_{2}}_{m_{2},m_{2}^{\prime}}(R(\hat{\bm{p}}_{\gamma}))C_{S_{1}m_{1}S_{2}m_{2}^{\prime}}^{Sm}\chi_{S}(p,m),$
(21)

where $C^{jm}_{j_{1}m_{1}j_{2}m_{2}}\equiv\langle
j_{1}m_{1};j_{2}m_{2}|jm\rangle$ is the CG coefficients of SU$(2)$,
$D^{S_{2}}_{m_{2},m_{2}^{\prime}}$ is the representation matrix for SU(2) and
$R(\hat{\bm{p}}_{\gamma})$ rotates $z$-axis to the direction of
${\bm{p}}_{\gamma}$.

In terms of this coupling function, the total spin wave function of the final
two particles reads

$\displaystyle\tilde{\chi}_{S}(p^{\prime}_{a},m^{\prime}_{S})=F^{[S,m^{\prime}_{S}]}\left(\chi_{S_{b}}(p^{\prime}_{a},m^{\prime}_{b}),\chi_{S_{c}}(p^{\prime}_{a},m^{\prime}_{c})\right),$
(22)

where $m^{\prime}_{S}=m^{\prime}_{b}+m^{\prime}_{c}$. Similarly, the total
wave function of the final states in Eq.(19) can be expressed as

$\displaystyle
J^{S_{a};L,S}\left(p^{\prime}_{a},m^{\prime}_{a};p^{\prime}_{b},m^{\prime}_{b},p^{\prime}_{c},m^{\prime}_{c}\right)$
$\displaystyle=F^{[S_{a},m^{\prime}_{a}]}\left(T_{L}(p^{\prime}_{b},p^{\prime}_{c},m^{\prime}_{L}),\tilde{\chi}_{S}(p^{\prime}_{a},m^{\prime}_{S})\right)$
$\displaystyle={q^{{}^{\prime}\nu_{1}}\cdots
q^{{}^{\prime}\nu_{L}}\epsilon^{*}_{\nu_{1}\cdots\nu_{L}}(p^{\prime}_{a},m^{\prime}_{L})}F^{[S_{a},m^{\prime}_{a}]}\left(\chi_{L}(p^{\prime}_{a},m^{\prime}_{L}),\tilde{\chi}_{S}(p^{\prime}_{a},m^{\prime}_{S})\right)B_{L}(Q_{abc})$
$\displaystyle={q^{{}^{\prime}\nu_{1}}\cdots
q^{{}^{\prime}\nu_{L}}\epsilon^{*}_{\nu_{1}\cdots\nu_{L}}(p^{\prime}_{a},m^{\prime}_{L})}C_{Lm^{\prime}_{L}Sm^{\prime}_{S}}^{S_{a}m^{\prime}_{a}}C_{S_{b}m^{\prime}_{b}S_{c}m^{\prime}_{c}}^{Sm^{\prime}_{S}}\chi_{S_{a}}(p^{\prime},m^{\prime}_{a})B_{L}(Q_{abc}),$
(23)

where $m^{\prime}_{L}$ is the $z$-component of the orbital angular momentum in
$a^{*}$-frame. Finally, we have

$\displaystyle\mathcal{M}_{a\to
bc}(p^{\prime}_{a},m^{\prime}_{a};p^{\prime}_{b},m^{\prime}_{b},p^{\prime}_{c},m^{\prime}_{c})=\sum_{L,S}A^{S_{a}}_{L,S}C_{Lm^{\prime}_{L}Sm^{\prime}_{S}}^{S_{a}m^{\prime}_{a}}C_{S_{b}m^{\prime}_{b}S_{c}m^{\prime}_{c}}^{Sm^{\prime}_{S}}{{\cal
Y}_{L,m_{L}^{\prime}}(p_{a}^{\prime},q^{\prime})}B_{L}(Q_{abc}),$ (24)

where

${\cal Y}_{L,m}(p,q)=q^{\nu_{1}}\cdots
q^{\nu_{L}}\epsilon^{*}_{\nu_{1}\cdots\nu_{L}}(p,m)$ (25)

is the covariant form of spherical harmonics.

##  IV Sequential decays

Consider the sequential decays

$\displaystyle a^{S_{a}}(p_{a},m_{a})$ $\displaystyle\to
b^{S_{b}}(p_{b},m_{b})\overbrace{+}^{L,S}c^{S_{c}}(p_{c},m_{c}),$ (26)
$\displaystyle c^{S_{c}}(p_{c},m_{c})$ $\displaystyle\to
d^{S_{d}}(p_{d},m_{d})\overbrace{+}^{L_{1},S_{1}}e^{S_{e}}(p_{e},m_{e}),$ (27)

where $p_{i}$ and $m_{i}$ are again defined in the laboratory frame and
$L_{1}$ and $S_{1}$ are the orbital angular momentum and total spin of $d,e$
system. The amplitude can be constructed using the $1\to 2$ decay amplitudes
in last section, which reads

$\displaystyle\mathcal{M}(p_{a}^{\prime},m_{a}^{\prime},p^{\prime}_{b},m^{\prime}_{b},p^{\prime\prime}_{d},m^{\prime\prime}_{d},p^{\prime\prime}_{e},m^{\prime\prime}_{e})$
(28) $\displaystyle=\sum_{m^{\prime}_{c}}\mathcal{M}_{a\to
bc}(p^{\prime}_{a},m^{\prime}_{a};p^{\prime}_{b},m^{\prime}_{b},p^{\prime}_{c},m^{\prime}_{c})\mathcal{M}_{c\to
de}(p^{\prime\prime}_{c},m^{\prime\prime}_{c}=m^{\prime}_{c};p^{\prime\prime}_{d},m^{\prime\prime}_{d},p^{\prime\prime}_{e},m^{\prime\prime}_{e})f(p_{c}),$
(29)

where $p^{\prime\prime}_{i}=(\tilde{\Lambda}_{p_{c}^{\prime}\to
k_{c}}\tilde{\Lambda}_{p_{a}\to k_{a}})_{\mu}p^{\mu}_{i}$ and
$m_{i}^{\prime\prime}(i=c,d,e)$ is the polarization of particle $i$ in its
“standard” frame, denoted by $a^{*}c^{*}$-frame, which is obtained from the
laboratory frame by sequential Lorentz transformations
$\tilde{\Lambda}_{p_{i}^{\prime\prime}\to
k_{i}}\tilde{\Lambda}_{p_{c}^{\prime}\to k_{c}}\tilde{\Lambda}_{p_{a}\to
k_{a}}$. $M_{a\to bc}$ and $M_{c\to de}$ are defined in Eq.(24). Note that
$m^{\prime\prime}_{c}=m^{\prime}_{c}$ since they are defined in the same
frame. We have introduced $f(p_{c})$ as the propagator of resonance $c$, which
can be parameterized in the Breit-Wigner form, Flattè form Flatte (1976),
Gounatis-Sakurai form Gounaris and Sakurai (1968) or any other reasonable
forms.

If the process only has one decay chain, Eq.(29) is enough since averaging the
initial states and summing over the final states in terms of
$m_{i}^{\prime},m_{i}^{\prime\prime}$ is equivalent to that in terms of
$m_{i}$. However, if more than one decay chains are possible, we have to
transform the amplitude expressed in Eq.(29) to the one evaluated in the
common frame, i.e., in terms of $m_{i}$, so that the amplitudes for different
decay chains can be added coherently. In this way the amplitude reads

$\displaystyle\mathcal{M}(p_{a},m_{a},p_{b},m_{b},p_{d},m_{d},p_{e},m_{e})=\sum_{m_{b}^{\prime},m_{d}^{\prime},m_{e}^{\prime}}$
$\displaystyle D_{m_{b},m_{b}^{\prime}}(\Lambda^{a}_{k_{b}\to
k_{b}})D_{m_{d},m_{d}^{\prime}}(\Lambda^{ac}_{k_{d}\to
k_{d}})D_{m_{e},m_{e}^{\prime}}(\Lambda^{ac}_{k_{e}\to k_{e}})$
$\displaystyle\times\mathcal{M}(p_{a}^{\prime},m_{a}^{\prime},p^{\prime}_{b},m^{\prime}_{b},p^{\prime\prime}_{d},m^{\prime\prime}_{d},p^{\prime\prime}_{e},m^{\prime\prime}_{e}),$
(30)

with $\Lambda^{a}_{k_{i}\to k_{i}}=\tilde{\Lambda}^{-1}_{k_{i}\to
p_{i}}\tilde{\Lambda}_{k_{a}\to p_{a}}\tilde{\Lambda}_{k_{i}\to
p^{\prime}_{i}}$ and $\Lambda^{ac}_{k_{i}\to
k_{i}}=\tilde{\Lambda}^{-1}_{k_{i}\to p_{i}}\tilde{\Lambda}_{k_{a}\to
p_{a}}\tilde{\Lambda}_{k_{c}\to p_{c}^{\prime}}\tilde{\Lambda}_{k_{i}\to
p^{\prime\prime}_{i}}$. Note that the amplitude in Eq.(30) is expressed in
terms of the momenta and polarizations in laboratory frame and hence it can be
used directly in the cases where there are different decay chains and
interference. The generalization to more decay chains is straightforward.

##  V Examples

In this section, we give several examples to illustrate how the method we
proposed above works.

###  V.1 $\phi\to K^{+}K^{-}$

Since the final particles are spinless, the CG coefficients in Eq.(24) are all
equal to 1 and the amplitude simply reads

$\displaystyle\mathcal{M}_{(}p^{\prime}_{a},m^{\prime}_{a};p^{\prime}_{b},m^{\prime}_{b}=0,p^{\prime}_{c},m^{\prime}_{c}=0)=q^{\prime\,\nu_{1}}\epsilon^{*}_{\nu_{1}}(p^{\prime}_{a},m^{\prime}_{a})B_{1}(Q_{abc}),$
(31)

which is exactly the same as that in covariant $L$-$S$ scheme.

###  V.2 $J/\psi\to\phi\eta$ or $\gamma\eta$

Let us label $J/\psi,\phi(\gamma),\eta_{c}$ as particle $a,b,c$, respectively.
In this case, $(L,S)=(1,1)$, and thus from Eq.(24), the amplitude in the
$a^{*}$-frame reads

$\displaystyle\mathcal{M}_{J/\psi\to\phi\eta_{c}}(p^{\prime}_{a},m^{\prime}_{a};p^{\prime}_{b},m^{\prime}_{b},p^{\prime}_{c},m^{\prime}_{c}=0)$
$\displaystyle=A^{1}_{1,1}C_{1(m^{\prime}_{a}-m^{\prime}_{b})1m^{\prime}_{b}}^{1m^{\prime}_{a}}{q^{\prime\,\nu_{1}}\epsilon^{*}_{\nu_{1}}(p^{\prime}_{a},m^{\prime}_{a}-m^{\prime}_{b})}B_{1}(Q_{abc}),$
(32)
$\displaystyle\mathcal{M}_{J/\psi\to\gamma\eta_{c}}(p^{\prime}_{a},m^{\prime}_{a};p^{\prime}_{b},m^{\prime}_{b},p^{\prime}_{c},m^{\prime}_{c}=0)$
$\displaystyle=A^{1}_{1,1}\sum_{m_{b}^{\prime\prime}}{D_{m_{b}^{\prime},m_{b}^{\prime\prime}}(R(\hat{\bm{p}}_{b}^{\prime}))}C_{1(m^{\prime}_{a}-m^{\prime\prime}_{b})1m^{\prime\prime}_{b}}^{1m^{\prime}_{a}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times{q^{\prime\,\nu_{1}}\epsilon^{*}_{\nu_{1}}(p^{\prime}_{a},m^{\prime}_{a}-m^{\prime\prime}_{b})}B_{1}(Q_{abc}).$
(33)

Summing over the initial and final spins we obtain the angular distribution.
If the $J/\psi$ is produced in the $e^{+}e^{-}$ annihilation, its polarization
can only takes the value $\pm 1$ and in turn

$\displaystyle\sum_{m_{a}^{\prime}=\pm 1}\sum_{m_{b}^{\prime}=0,\pm
1}|\mathcal{M}_{J/\psi\to\phi\eta_{c}}(p^{\prime}_{a},m^{\prime}_{a};p^{\prime}_{b},m^{\prime}_{b},p^{\prime}_{c},m^{\prime}_{c}=0)|^{2}$
$\displaystyle=$ $\displaystyle\sum_{m_{a}^{\prime}=\pm
1}\sum_{m_{b}^{\prime}=\pm
1}|\mathcal{M}_{J/\psi\to\gamma\eta_{c}}(p^{\prime}_{a},m^{\prime}_{a};p^{\prime}_{b},m^{\prime}_{b},p^{\prime}_{c},m^{\prime}_{c}=0)|^{2}\propto
1+\cos^{2}\theta,$ (34)

If the $J/\psi$ has equal possibilities for three polarizations, there should
be no special direction and thus the angular distribution is expected to be
flat, which is verified by the following explicit calculations,

$\displaystyle\sum_{m_{a}^{\prime}=0,\pm 1}\sum_{m_{b}^{\prime}=0,\pm
1}|\mathcal{M}_{J/\psi\to\phi\eta_{c}}(p^{\prime}_{a},m^{\prime}_{a};p^{\prime}_{b},m^{\prime}_{b},p^{\prime}_{c},m^{\prime}_{c}=0)|^{2}$
$\displaystyle=$ $\displaystyle\sum_{m_{a}^{\prime}=0,\pm
1}\sum_{m_{b}^{\prime}=\pm
1}|\mathcal{M}_{J/\psi\to\gamma\eta_{c}}(p^{\prime}_{a},m^{\prime}_{a};p^{\prime}_{b},m^{\prime}_{b},p^{\prime}_{c},m^{\prime}_{c}=0)|^{2}\propto
1,$ (35)

One can easily check that they agree with the distributions using the
amplitudes constructed in Ref Zou and Hussain (2003).

###  V.3 $J/\psi\to\gamma\eta_{c}\to\gamma\Lambda\bar{\Lambda}$

Let us label $J/\psi,\gamma,\eta_{c},\Lambda,\bar{\Lambda}$ as particle
$a,b,c,d,e$, respectively. For $J/\psi\to\gamma\eta_{c}$, the amplitude is the
same as Eq.(33). For $\eta_{c}\to\Lambda\bar{\Lambda}$, we have $(L,S)=(0,0)$
and the amplitude in the $a^{*}c^{*}$-frame reads

$\displaystyle\mathcal{M}_{\eta_{c}\to\Lambda\bar{\Lambda}}(p^{\prime\prime}_{c},m^{\prime\prime}_{c};p^{\prime\prime}_{d},m^{\prime\prime}_{d},p^{\prime\prime}_{e},m^{\prime\prime}_{e})=A^{0}_{0,0}C_{\frac{1}{2}m^{\prime\prime}_{d}\frac{1}{2}m^{\prime\prime}_{e}}^{0,0},$
(36)

being flat as expected. Then the amplitude for the sequential decay from
Eq.(29) is

$\displaystyle\mathcal{M}(p_{a}^{\prime},m_{a}^{\prime},p^{\prime}_{b},m^{\prime}_{b},p^{\prime\prime}_{d},m^{\prime\prime}_{d},p^{\prime\prime}_{e},m^{\prime\prime}_{e})=A\,C_{\frac{1}{2}m^{\prime\prime}_{d}\frac{1}{2}m^{\prime\prime}_{e}}^{0,0}C_{1(m^{\prime}_{a}-m^{\prime}_{b})1m^{\prime}_{b}}^{1m^{\prime}_{a}}{q^{\prime\,\nu_{1}}\epsilon^{*}_{\nu_{1}}(p^{\prime}_{a},m^{\prime}_{a}-m^{\prime}_{b})}B_{1}(Q_{abc})f(p_{c}),$
(37)

Since this process has only one decay chain, summing over initial and final
polarizations can be performed directly to Eq.(37).

###  V.4 $\psi(2S)\to\Omega\bar{\Omega}$

In this case, $(L,S)=(0,1),(2,1),(2,3)$ and $(4,3)$. The amplitudes are
expressed as

$\displaystyle\mathcal{M}_{0,1}^{1}$
$\displaystyle=A_{0,1}^{1}C^{1m_{a}^{\prime}}_{\frac{3}{2}m_{b}^{\prime}\frac{3}{2}m_{c}^{\prime}}$
(38) $\displaystyle\mathcal{M}_{2,1}^{1}$
$\displaystyle=A_{2,1}^{1}C^{1m_{a}^{\prime}}_{2m_{L}^{\prime}1m_{S}^{\prime}}C^{1m_{S}^{\prime}}_{\frac{3}{2}m_{b}^{\prime}\frac{3}{2}m_{c}^{\prime}}q^{\prime\nu_{1}}q^{\prime\nu_{2}}\epsilon^{*}_{\nu_{1}\nu_{2}}(p_{a}^{\prime},m_{L}^{\prime})B_{2}(Q_{abc})$
(39) $\displaystyle\mathcal{M}_{2,3}^{1}$
$\displaystyle=A_{2,1}^{1}C^{1m_{a}^{\prime}}_{2m_{L}^{\prime}3m_{S}^{\prime}}C^{3m_{S}^{\prime}}_{\frac{3}{2}m_{b}^{\prime}\frac{3}{2}m_{c}^{\prime}}q^{\prime\nu_{1}}q^{\prime\nu_{2}}\epsilon^{*}_{\nu_{1}\nu_{2}}(p_{a}^{\prime},m_{L}^{\prime})B_{2}(Q_{abc})$
(40) $\displaystyle\mathcal{M}_{4,3}^{1}$
$\displaystyle=A_{4,3}^{1}C^{1m_{a}^{\prime}}_{4m_{L}^{\prime}3m_{S}^{\prime}}C^{3m_{S}^{\prime}}_{\frac{3}{2}m_{b}^{\prime}\frac{3}{2}m_{c}^{\prime}}q^{\prime\nu_{1}}q^{\prime\nu_{2}}q^{\prime\nu_{3}}q^{\prime\nu_{4}}\epsilon^{*}_{\nu_{1}\nu_{2}\nu_{3}\nu_{4}}(p_{a}^{\prime},m_{L}^{\prime})B_{4}(Q_{abc}),$
(41)

Summing over all possible polarizations will of course yield flat angular
distribution and if only $m_{a}^{\prime}=\pm$ are considered, we have

$\displaystyle\sum_{m_{a}^{\prime}=\pm
1}\sum_{m_{b}^{\prime},m_{c}^{\prime}=-3/2}^{3/2}|\mathcal{M}_{0,1}^{1}|^{2}$
$\displaystyle\propto 1,$ (42) $\displaystyle\sum_{m_{a}^{\prime}=\pm
1}\sum_{m_{b}^{\prime},m_{c}^{\prime}=-3/2}^{3/2}|\mathcal{M}_{2,1}^{1}|^{2}$
$\displaystyle\propto 5-3\cos^{2}\theta,$ (43)
$\displaystyle\sum_{m_{a}^{\prime}=\pm
1}\sum_{m_{b}^{\prime},m_{c}^{\prime}=-3/2}^{3/2}|\mathcal{M}_{2,3}^{1}|^{2}$
$\displaystyle\propto 5-\cos^{2}\theta,$ (44)
$\displaystyle\sum_{m_{a}^{\prime}=\pm
1}\sum_{m_{b}^{\prime},m_{c}^{\prime}=-3/2}^{3/2}|\mathcal{M}_{4,3}^{1}|^{2}$
$\displaystyle\propto 6-5\cos^{2}\theta,$ (45)

Let us emphasis again that the spin wavefunctions of $\Omega$ and
$\bar{\Omega}$ are not necessary here, contrary to the covariant $L$-$S$
scheme Zou and Hussain (2003) where the explicit expressions for spin coupling
is needed. If fermions with higher spins are included, such explicit
expressions will be much complicated, difficult to construct and in these
cases, the advantages of our method are revealed.

###  V.5 $D_{s}^{+}\to\phi e^{+}\nu_{e}$

The proposed method can also be applied to leptonic and semi-leptonic decays.
Let us take the semi-leptonic process $D_{s}^{+}\to\phi e^{+}\nu_{e}$ via
intermediate $W^{+}$ exchange as example where
$D_{s}^{+},\phi,W^{+},e^{+},\nu_{e}$ are denoted by particle $a,b,c,d,f$,
respectively. For the first vertex, $D_{s}^{+}\to\phi W^{+}$,
$(L,S)=(0,0),(1,1)$ or $(2,2)$ and the corresponding amplitudes read

$\displaystyle\mathcal{M}_{0,0}^{0}$
$\displaystyle=A_{0,0}^{0}C^{0,0}_{1m_{b}^{\prime}1m_{c}^{\prime}},$ (46)
$\displaystyle\mathcal{M}_{1,1}^{0}$
$\displaystyle=A_{2,1}^{0}C^{0,0}_{1m_{L}^{\prime}1m_{S}^{\prime}}C^{1m_{S}^{\prime}}_{1m_{b}^{\prime}1m_{c}^{\prime}}q^{\prime\nu_{1}}\epsilon^{*}_{\nu_{1}}(p_{a}^{\prime},m_{L}^{\prime})B_{1}(Q_{abc}),$
(47) $\displaystyle\mathcal{M}_{2,2}^{1}$
$\displaystyle=A_{2,1}^{0}C^{0,0}_{2m_{L}^{\prime}2m_{S}^{\prime}}C^{2m_{S}^{\prime}}_{1m_{b}^{\prime}1m_{c}^{\prime}}q^{\prime\nu_{1}}q^{\prime\nu_{2}}\epsilon^{*}_{\nu_{1}\nu_{2}}(p_{a}^{\prime},m_{L}^{\prime})B_{2}(Q_{abc}),$
(48)

The second vertex $W^{+}\to e^{+}\nu_{e}$ can be precisely described by the
electro-weak theory and the amplitudes reads

$\displaystyle\mathcal{M}(p_{c}^{\prime\prime},m_{c}^{\prime\prime};p_{d}^{\prime\prime},m_{d}^{\prime\prime},p_{f}^{\prime\prime},m_{f}^{\prime\prime})=\frac{e}{\sqrt{2}\sin\theta_{W}}\chi_{1}^{\mu}(p_{c}^{\prime\prime},m_{c})\bar{\chi}_{1/2}(p_{f}^{\prime\prime},m_{f})\gamma_{\mu}\frac{1-\gamma^{5}}{2}\chi_{1/2}(p_{d}^{\prime\prime},m_{d}).$
(49)

Note that the amplitude for the above weak interaction can also be constructed
using the method we proposed above but we need not to do that since we have
the explicit and accurate expression.

##  VI Summary

The covariant $L$-$S$ scheme to construct the decay amplitudes is very
convenient and useful for extracting the resonance parameters from the high-
quality data. In this scheme, the amplitudes with different total spin and
orbital angular momentum in the final states are constructed separately and
orthogonal to each other. In this work, we overcome some drawbacks in the
pioneer works Zou and Bugg (2003); Zou and Hussain (2003) and proposed a
modified version of the amplitude construction. Our method holds the Lorentz
invariance, avoids the complication of the explicit spin couplings, and if
only massive particles involved, separates the orbital and spin angular
momenta thoroughly.

We also would like to emphasis that the proposed amplitude construction method
only depends on the masses and spins of involved particles and has nothing to
do with the detailed dynamics. Therefore, it will also be useful for leptonic
or semi-leptonic decays.

We are grateful to the fruitful discussions with Liao-Yuan Dong, Feng-Kun Guo,
Bai-Cian Ke, Hai-Bo Li, Jia-Jun Wu and Bing-Song Zou. This work is supported
by the NSFC and the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) through the funds provided to the Sino German Collaborative
Research Center TRR110 Symmetries and the Emergence of Structure in QCD (NSFC
Grant No.12070131001, DFG Project-ID 196253076-TRR 110), by the NSFC Grants
Nos. 11835015, 11875054, 11935018, 12047503, 12192263, by the Chinese Academy
of Sciences (CAS) under Grant No.XDB34030000, by Joint Large-Scale Scientific
Facility Fund of the NSFC and CAS under Grant No.U2032104, and by China
Postdoctoral Science Foundation under Grant No. 2022M723115.

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