# Charmless Quasi-two-body $B$ Decays in Perturbative QCD Approach:
Taking $B\to K({\cal R}\to)K^{+}K^{-}$ As Examples

Wen-Feng Liu  Department of Physics, Yantai University, Yantai 264005, China
Zhi-Tian Zou  Department of Physics, Yantai University, Yantai 264005, China
Ying<EMAIL_ADDRESS>Department of Physics, Yantai University, Yantai
264005, China

###### Abstract

Three-body $B$ decays not only significantly broaden the study of $B$ meson
decay mechanisms, but also provide information of resonant particles. Because
of complicate dynamics, it is very hard for us to study the whole phase space
in a specific approach. In this review, we take $B\to K({\cal
R}\to)K^{+}K^{-}$ decays [1] as examples and show the application of the
perturbative QCD (PQCD) approach in studying the quasi-two-body $B$ decays,
where two particles move collinearly with large energy and the bachelor one
recoils back. To describe the dynamics of two collinear particles, the ($S$,
$P$ and $D$)-wave functions of kaon-pair with different waves are introduced.
By keeping the transverse momenta, all possible diagrams including the hard
spectator diagrams and annihilation ones can be calculated in PQCD approach.
Most results are well consistent with the current measurements from BaBar,
Belle and LHCb experiments. Moreover, under the narrow-width approximation we
can extract the branching fractions of the two-body decays involving the
resonant states, and also predict the branching fractions of the corresponding
quasi-two-body decays $B\to K(\cal{R}\to)\pi^{+}\pi^{-}$. All prediction are
expected to be tested in the ongoing LHCb and Belle-II experiments.

##  1 Introduction

It is well known that $B$ meson hadronic decays provide us unique information
on three fronts: $CP$ violation and the angles of the CKM matrix, the
structure of QCD in the presence of heavy quarks and energetic light
particles, and new particles beyond the standard model (SM) physics in the
quark sector. In past twenty years, experimental information on non-leptonic
$B$ decays has been collected progressively, at CLEO, two $B$-factories BaBar
and Belle [2], the Tevatron, and currently at the LHC, most prominently at
LHCb [3]. In addition, the running Belle II also includes serious plans for
analyses of non-leptonic $B$ decays [4] with higher precision. Because of
large combinatoric backgrounds, studies of charmless $B$ decays have
concentrated mainly on two-body decay processes. Although with complex
kinematics, three-body decays could not only significantly broaden the study
of $B$ meson decay mechanisms and provide additional possibilities for direct
$CP$ violation searches, but also provide information of resonant particles.
In past decades, more and more analysis of three-body decays have been
performed by the BaBar, Belle, CLEO and LHCb, and the branching fractions and
$CP$ violations have been measured with high precision. Motivated by the
accumulated experimental results, many theoretical studies of various three-
body $B$ hadronic decays have been performed in different frameworks, such as
approaches based on the symmetry principles [5, 6, 7, 8, 9], the QCD
factorization (QCDF) [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], the
perturbative QCD approach (PQCD) [21, 22, 23, 24, 25, 26, 1, 27, 28], and
other theoretical methods [29, 30, 31].

Figure 1: Phase space of the three-body decay $B^{-}\to K^{+}K^{-}\pi^{-}$ in
terms of the invariants $m_{12}$ and $m_{13}$ [32].

Differing from the two-body $B$ decays where the kinematics are fixed, the
momentum of each final state in three-body decays is variable. Moreover, both
resonant and non-resonant contributions are involved, therefore how to
disentangle the resonant and non-resonant contributions reliably is also very
important in studying the multi-body decays. For clarity, we define the
kinematics of the three-body decay as

$\displaystyle B(p_{B})\to M_{a}(p_{1})M_{b}(p_{2})M_{c}(p_{3}).$ (1)

It is customary to take these variables as two invariant masses of two pairs
of final state particles:

$\displaystyle m_{12}^{2}=(p_{1}+p_{2})^{2}\ ,\quad
m_{13}^{2}=(p_{1}+p_{3})^{2}.$ (2)

Thus, the amplitude of the three-body decay is a function of the two kinematic
variables, $m_{12}^{2}$ and $m_{13}^{2}$, and the corresponding distribution
in the phase-space region is called a Dalitz plot, which can be divided in
different regions with characteristic kinematics, as shown in Fig. 1. As
usual, we refer to the $B$-meson rest frame. In the centre of the Dalitz plot,
the three final states particles almost have a large energy ($E\sim m_{B}/3$)
and none of them flies collinearly to any others. The corners imply that one
final particle is approximately at rest, and the other two fly back-to-back
energetically, leading to that one invariant mass is large, and the other two
are small. Specially, the central part of the edges mean that two particles
move collinearly with large energy and the other particle recoils back. In the
naive factorization, the contribution at the centre of the Dalitz plot is
viewed to be both power-suppressed and $\alpha_{s}$ suppressed with respect to
that at the edge, because two hard gluons are need. However, recent studies
based on QCD factorization showed that the power corrections are large, which
means that the centre region maybe as important as the regions at the edge
[12, 11]. Assuming that these different regions can be well described by the
respective calculations, it is a very challenging work to determine how well
these descriptions merge at intermediate kinematical regimes. So, the theory
of three-body non-leptonic decays is still in an early stage of development.

In the past decades, the $B$ mesons two-body hadronic decays have been
explored systematically in the context of QCD factorization, soft-collinear
effective theory or PQCD approach. It is natural to ask if the accumulated
experience can be used to study the multi-body $B$ decays. Inspired by this,
we paid much attention to the edges regions of the Dalitz plot, in which the
two energetic particles are collinear and form a moving-fast meson-pair, and
the interactions between the meson-pair and the bachelor particle are power
suppressed naturally. This kind of process is also called quasi-two-body
process. The interactions in the meson-pair can be absorbed into the two-meson
wave function. In this case, the obvious generalization of the factorization
theorem for two-body decays applies. It is reasonable for us to assume the
validity of the factorization for these quasi two-body $B$ decays. In the PQCD
framework that is based on the $k_{\rm T}$ factorization, the decay amplitude
of quasi-two-body $B$ decays can be decomposed as the convolution [33]

$\mathcal{A}(B\to
M_{3}(R\to)M_{1}M_{2})=C(t)\otimes\Phi_{B}\otimes\mathcal{H}\otimes\Phi_{M_{1}M_{2}}\otimes\Phi_{M_{3}}\otimes\exp[-S(t)]$
(3)

where the $\Phi_{B}$, $\Phi_{M_{3}}$ are the wave functions of $B$ meson and
the light bachelor meson, respectively. The newly introduced
$\Phi_{M_{1}M_{2}}$ is the two-meson wave function, which describes how four
quarks are combined into two mesons. It should be emphasized that both
resonant and nonresonant contributions to the hadron-pair system are included
into the wave function. The Wilson coefficient $C(t)$ covers all physics above
$m_{b}$ scale. The exponential term is the so-called Sudakov form factor
caused by the additional scale introduced by the intrinsic transverse momenta
$k_{\rm T}$, which suppresses the soft dynamics effectively. The hard kernel
$\mathcal{H}$ for the $b$ quark decay, similar to the two-body case, starts
with the diagrams of single hard gluon exchange, and it can be calculated
perturbatively.

In the following, we will take $B\to K({\cal R}\to)K^{+}K^{-}$ decay as
examples to show how to study quasi-two-body processes in the PQCD approach at
leading order, where $K^{+}K^{-}$-pair move collinearly and combine into
resonances with different waves, such as $f_{0}(980)$, $\phi(1020)$ and
$f_{2}^{\prime}(1525)$. In Sec. 2, we will introduce the definitions of the
two-kaon wave functions with different waves, and present the typical feynmann
diagrams in PQCD. In Sec. 3, the numerical results and discussions will be
given. At last, we summarize this review in Sec. 4.

##  2 Framework

In the quasi-two-body region of phase space, the Dalitz plot analysis allows
one to describe the decay amplitude in the isobar model, where the decay
amplitude is represented by a coherent sum of amplitudes from $N$ individual
decay channels with different resonances,

$\displaystyle\mathcal{A}=\sum_{j=1}^{N}c^{j}\mathcal{A}^{j},$ (4)

where the $\mathcal{A}^{j}$ is the amplitude corresponding to certain
resonance and $c^{j}$ is the complex coefficient describing the relevant
magnitude and phase of the different decay channel.

When we calculate the amplitude $\mathcal{A}^{j}$ in PQCD, the most important
inputs are the nonperturbative wave functions. For the decay $B^{0}\to
K^{0}({\cal R}\to)K^{+}K^{-}$, the wave functions of the $B$ meson and the
bachelor meson $K^{0}$ have been well determined by comparing the theoretical
predictions with those well measured experimental data of the two-body $B$
decays. For the sake of brevity, we shall not discuss them anymore, and their
explicit expressions can be found in refs.[34, 35, 36]. As aforementioned, in
the quasi-two-body decays, the newly introduced ingredient is the two-meson
wave function corresponding to different resonances with different spin [22].

We firstly discuss the $S$-wave two-meson wave function of the kaon-pair,

$\displaystyle\Phi_{KK,S}=\frac{1}{\sqrt{2Nc}}[P\mkern-11.5mu/\phi_{S}(z,\zeta,\omega^{2})+\omega\phi_{S}^{s}(z,\zeta,\omega^{2})+\omega(n\mkern-9.5mu/v\mkern-7.5mu/-1)\phi_{S}^{t}(z,\zeta,\omega^{2})],$
(5)

where $z$ is the momentum fraction of the spectator quark, and $\zeta$ is the
momentum fraction of one kaon in the pair.$N_{c}=3$ is the number of colors,
and $\omega$ and $P$ are the invariant mass and momentum of the kaon-pair,
respectively. $n=(1,0,{\bf 0_{\rm T}})$ and $v=(0,1,{\bf 0_{\rm T}})$ are two
dimensionless vectors. The $\phi_{S}$, $\phi_{S}^{s}$, $\phi_{S}^{t}$ are the
twist-2 and twist-3 distribution amplitudes, and they are parameterized as
[37]

$\displaystyle\phi_{S}(z,\zeta,\omega^{2})$ $\displaystyle=$
$\displaystyle\frac{9}{\sqrt{2Nc}}F_{S}(\omega^{2})a_{S}z(1-z)(2z-1),$ (6)
$\displaystyle\phi_{S}^{s}(z,\zeta,\omega^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2Nc}}F_{S}(\omega^{2}),$ (7)
$\displaystyle\phi_{S}^{t}(z,\zeta,\omega^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2Nc}}F_{S}(\omega^{2})(1-2z).$ (8)

The Gegenbauer moment $a_{S}=-0.8$ is determined with the experimental data
[38]. We have to adopt the asymptotic form within the SU(3) symmetry, as the
reliable theoretical studies are still absent.

Specially, the function $F_{S}(\omega^{2})$ is the $S$-wave time-like form
factor, which contains the interactions between the two kaons in the kaon-
pair. For most resonances, the form factors are usually taken to be
relativistic Breit-Wigner (RBW) line shapes [39]

$\displaystyle
F_{S}(\omega^{2})=\frac{m_{j}^{2}}{m_{j}^{2}-\omega^{2}-im_{j}\Gamma_{j}(\omega)},$
(9)

where $m_{j}$ is the mass of the resonance, and $\Gamma(\omega)$ is the mass-
dependent width [40, 39]. However, the case of $f_{0}(980)$ is complicated.
Because there is an anomalous structure corresponding to the enhancement from
the $KK$ system found around $980$ MeV in the $\pi^{+}\pi^{-}$ scattering [41,
42], it can be interpreted as a two-channel resonance combining the $\pi\pi$
and $KK$ channels. In order to include these effects, eq.(9) is modified to
the Flatté form [43, 44, 45] as

$\displaystyle
F_{S}(\omega^{2})=\frac{m_{f_{0}(980)}^{2}}{m_{f_{0}(980)}^{2}-\omega^{2}-im_{f_{0}(980)}(g_{\pi\pi}\rho_{\pi\pi}+g_{KK}\rho_{KK}F_{KK}^{2})},$
(10)

where $g_{\pi\pi}$ and $g_{KK}$ are the $f_{0}(980)$ coupling constants to the
$\pi\pi$ and $KK$ final states, respectively. The phase space factors
$\rho_{\pi\pi}$ and $\rho_{KK}$ are given as

$\displaystyle\rho_{\pi\pi}=\sqrt{1-\frac{4m_{\pi}^{2}}{\omega^{2}}},\,\,\,\,\rho_{KK}=\sqrt{1-\frac{4m_{K}^{2}}{\omega^{2}}}.$
(11)

The factor $F_{KK}=e^{-\alpha q^{2}}$ with $\alpha\approx 2.0~{}\rm GeV^{-2}$
is introduced to suppress the $K\overline{K}$ contribution [45], $q$ being the
momentum of each kaon in the $K\overline{K}$ rest frame.

It should be noted that we only consider the contributions from three scalar
resonances $f_{0}(980)$, $f_{0}(1500)$ and $f_{0}(1710)$, which have been well
analyzed by BaBar experiments [46, 38]. The coefficients of the coherence
summation of these three resonances in eq.(4) are set to be
$c^{f_{0}(980)}=2.9$, $c^{f_{0}(1500)}=1.0$, $c^{f_{0}(1710)}=0.5$, which have
been determined by the experimental measurements already [46, 38].

Next, we come to the $P$-wave two-kaon wave function. Because the third
bachelor kaon in $B^{0}\to K^{0}K^{+}K^{-}$ decay is a pseudoscalar one,
therefore only the longitudinal part contributes, and its form can be
expressed as

$\displaystyle\Phi_{KK,P}=\frac{1}{\sqrt{2N_{c}}}\left(\not\\!P\phi_{P}(z,\zeta,\omega)+\omega\phi_{P}^{s}(z,\zeta,\omega)+\frac{\not\\!p_{1}\not\\!p_{2}-\not\\!p_{2}\not\\!p_{1}}{\omega(2\zeta-1)}\phi_{P}^{t}(z,\zeta,\omega)\right),$
(12)

where $p_{1(2)}$ is the momentum of each kaon in the $K\bar{K}$-pair. The
corresponding twist-2 and 3 distribution amplitudes that can be decomposed as
Gegenbauer polynomials are given as

$\displaystyle\phi_{P}^{0}(z,\zeta,\omega)$ $\displaystyle=$
$\displaystyle\frac{3F_{P}^{\parallel}(\omega^{2})}{\sqrt{2N_{c}}}z(1-z)\Big{[}1+a_{P}^{0}C_{2}^{3/2}(2z-1)\Big{]}(2\zeta-1),$
(13) $\displaystyle\phi_{P}^{s}(z,\zeta,\omega)$ $\displaystyle=$
$\displaystyle\frac{3F_{P}^{\perp}(\omega^{2})}{2\sqrt{2N_{c}}}(1-2z)\Big{[}1+a_{P}^{s}(1-10z+10z^{2})\Big{]}(2\zeta-1),$
(14) $\displaystyle\phi_{P}^{t}(z,\zeta,\omega)$ $\displaystyle=$
$\displaystyle\frac{3F_{P}^{\perp}(\omega^{2})}{2\sqrt{2N_{c}}}(2z-1)^{2}\Big{[}1+a_{P}^{t}C_{2}^{3/2}(2z-1)\Big{]}(2\zeta-1),$
(15)

with $a_{P}^{0}=-0.6$, $a_{P}^{s}=-0.8$, and $a_{P}^{t}=-0.3$. Similarly, the
$P$-wave time-like form factor $F_{P}^{\parallel}(\omega^{2})$ is also taken
to be the RBW line shape in eq. (9). As for $F_{P}^{\perp}(\omega^{2})$, the
relation [47]

$\displaystyle\frac{F_{P}^{\parallel}(\omega^{2})}{F_{P}^{\perp}(\omega^{2})}\approx\frac{f_{V}}{f_{V}^{T}},$
(16)

will be used, where $f_{V}$ and $f_{V}^{T}$ are the vector and tensor decay
constants.

At last, we will discuss the wave function of $D$-wave kaon-pair in which the
information of tensor meson resonances is included. In previous studies of $B$
meson decays involving a tensor [48, 49],it is found that the polarization
components $\pm 2$ of tensor meson do not contribute of the amplitudes, due to
the conservation of the angular momentum. Hence, the form of $D$-wave two-kaon
wave function is the same as that of the $P$-wave,

$\displaystyle\Phi_{KK,D}=\frac{1}{\sqrt{2N_{c}}}\left(p\mkern-8.5mu/\phi_{D}(z,\zeta,\omega)+\omega\phi_{D}^{s}(z,\zeta,\omega)+\frac{p\mkern-8.5mu/_{1}p\mkern-8.5mu/_{2}-p\mkern-8.5mu/_{2}p\mkern-8.5mu/_{1}}{\omega(2\zeta-1)}\phi_{D}^{t}(z,\zeta,\omega)\right).$
(17)

The distribution amplitudes are given as

$\displaystyle\phi_{D}(z,\zeta,\omega)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3}}\frac{9F_{D}^{\parallel}(\omega^{2})}{\sqrt{2N_{c}}}z(1-z)a_{D}^{0}\Big{[}2z-1\Big{]}P_{2}(\zeta),$
(18) $\displaystyle\phi_{D}^{s}(z,\zeta,\omega)$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{2}{3}}\frac{9F_{D}^{\perp}(\omega^{2})}{4\sqrt{2N_{c}}}a_{D}^{0}\Big{[}1-6z+6z^{2}\Big{]}P_{2}(\zeta),$
(19) $\displaystyle\phi_{D}^{t}(z,\zeta,\omega)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3}}\frac{9F_{D}^{\perp}(\omega^{2})}{4\sqrt{2N_{c}}}a_{D}^{0}(2z-1)\Big{[}1-6z+6z^{2}\Big{]}P_{2}(\zeta),$
(20)

with $a_{D}^{0}=0.6$ and $P_{2}(\zeta)=1-6\zeta+6\zeta^{2}$. Likewise, we use
the RBW line shape for describing the time-like form factor
$F_{D}^{\parallel}$ and determine the $F_{D}^{\perp}$ by the similar relation
as eq.(16).

Figure 2: Typical Feynman diagrams for the quasi-two-body decay $B\to K({\cal
R}\to)K^{+}K^{-}$ in PQCD, where the black squares stand for the weak
vertices, large (purple) spots on the quark lines denote possible attachments
of hard gluons, and the green ellipse represent Kaon-pair and the red one is
the light bachelor $K$ meson[1].

According to the effective Hamiltonian of $b$ quark decay, we can draw the
possible Feynman diagrams as shown in Fig. 2. In diagram (a), the spectator
quark enters to the bachelor particle, and the $KK$-pair is emitted. In order
to make the spectator quark from soft to hard, the hard gluon is needed, and
it can come from any quark participating in the four-quark weak interaction.
So, with three wave functions, we could calculate amplitude with different
operators in the PQCD approach. In practice, we keep the transverse momenta of
each quark for smearing the end-point singularities. For decays $B^{\pm}\to
K^{\pm}({\cal R}\to)K^{+}K^{-}$, the spectator quark can also enter into the
recoiled meson pair, as shown in diagram (b). Besides these two kinds of
contributions, the annihilation diagrams (c) and (d) also play important
roles, which can be calculated quantitatively without introducing new
parameters. Due to the space limit, we cannot list all amplitudes of these
diagrams, and one can find them in ref.[1].

##  3 Results and Discussions

With the calculated amplitude $\mathcal{A}$, we then obtain the differential
branching fraction

$\displaystyle\frac{d^{2}\mathcal{B}}{d\zeta
d\omega}=\frac{\tau_{B}\omega|\vec{p}_{1}||\vec{p}_{3}|}{32\pi^{3}m_{B}^{3}}|\mathcal{A}|^{2},$
(21)

where $\tau_{B}$ is the life-time of the $B$ meson. The magnitudes of three-
momenta of one kaon and the bachelor particle in the rest frame of the
$KK$-pair are given by in the rest frame of the $KK$-pair are given by

$\displaystyle|\vec{p}_{1}|=\frac{\sqrt{\lambda(\omega^{2},m_{K}^{2},m_{K}^{2})}}{2\omega},\quad|\vec{p}_{3}|=\frac{\sqrt{\lambda(m_{B}^{2},m_{K}^{2},\omega^{2})}}{2\omega},$
(22)

with the standard K$\ddot{a}$ll$\acute{e}$n function
$\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2(ab+ac+bc)$.

Integrating out the variables $\zeta$ and $\omega$, we could obtain the
branching fractions of the quasi-two-body decays $B\to K({\cal
R}\to)K^{+}K^{-}$, and the $CP$ averaged branching fractions are presented in
Table.1. As aforementioned, the theoretical studies of multi-body $B$ hadronic
decays are on the early stage, so there are many uncertainties in our
calculations. The first and foremost parameters uncertainties are from
nonperturbative parameters, including the shape parameter in the heavy $B$
wave function, the Gegenbauer moments in the distribution amplitudes of kaon-
pair and kaon. We call for the results with higher precision from
nonperturbative approaches, such as QCD sum rules and lattice QCD. The second
uncertainties origin from the PQCD approach itself, including the unknown QCD
radiative corrections and the power corrections. Although some higher order
and higher power corrections have been explored, the complete calculations are
still in absence. The last kind of uncertainties are caused by the CKM matrix
elements. In the table, the currently available experimental measurements are
also given for comparison. It is obvious that within the uncertainties most of
our results are in good agreement with experimental results, except decay mode
$B^{0}\to K^{0}(f_{0}(1710)\to)K^{+}K^{-}$

Table 1: $CP$ averaged branching fractions (in $10^{-6}$) of $B\to K({\cal R}\to)K^{+}K^{-}$ decays in PQCD [1] together with experimental data [39].  Decay Modes | PQCD |  EXP[39]
---|---|---
$B^{+}\to K^{+}(\phi(1020)\to)K^{+}K^{-}$ | $3.81^{+1.44+0.64+0.27}_{-1.03-0.33-0.00}$ | $4.48\pm 0.22^{+0.33}_{-0.24}$
$B^{+}\to K^{+}(f_{0}(980)\to)K^{+}K^{-}$ | $10.13^{+5.60+2.22+0.71}_{-4.38-2.44-0.00}$ | $9.4\pm 1.6\pm 2.8$
$B^{+}\to K^{+}(f_{0}(1500)\to)K^{+}K^{-}$ | $0.60^{+0.24+0.07+0.05}_{-0.24-0.06-0.02}$ | $0.74\pm 0.18\pm 0.52$
$B^{+}\to K^{+}(f_{0}(1710)\to)K^{+}K^{-}$ | $1.64^{+0.89+0.42+0.08}_{-0.70-0.46-0.02}$ | $1.12\pm 0.25\pm 0.50$
$B^{+}\to K^{+}(f_{2}^{\prime}(1525)\to)K^{+}K^{-}$ | $0.68^{+0.37+0.13+0.07}_{-0.29-0.14-0.00}$ | $0.69\pm 0.16\pm 0.13$
$B^{+}\to K^{+}(f_{2}(2010)\to)K^{+}K^{-}$ | $1.18^{+0.65+0.26+0.12}_{-0.50-0.19-0.00}$ |
$B^{0}\to K^{0}(\phi(1020)\to)K^{+}K^{-}$ | $3.22^{+1.36+0.48+0.18}_{-0.98-0.18-0.08}$ | $3.48\pm 0.28^{+0.21}_{-0.14}$
$B^{0}\to K^{0}(f_{0}(980)\to)K^{+}K^{-}$ | $9.10^{+5.12+2.19+0.69}_{-3.89-2.11-0.00}$ | $7.0^{+2.6}_{-1.8}\pm 2.4$
$B^{0}\to K^{0}(f_{0}(1500)\to)K^{+}K^{-}$ | $0.57^{+0.26+0.09+0.04}_{-0.22-0.15-0.00}$ | $0.57^{+0.25}_{-0.19}\pm 0.12$
$B^{0}\to K^{0}(f_{0}(1710)\to)K^{+}K^{-}$ | $1.48^{+0.82+0.39+0.11}_{-0.63-0.42-0.00}$ | $4.4\pm 0.7\pm 0.5$
$B^{0}\to K^{0}(f_{2}^{\prime}(1525)\to)K^{+}K^{-}$ | $0.58^{+0.31+0.12+0.05}_{-0.27-0.13-0.01}$ | $0.13^{+0.12}_{-0.08}\pm 0.16$
$B^{0}\to K^{0}(f_{2}(2010)\to)K^{+}K^{-}$ | $1.09^{+0.57+0.26+0.11}_{-0.48-0.23-0.00}$ |
Figure 3: The $\omega$-dependence of differential branching fractions for the
$B^{0}\to K^{0}K^{+}K^{-}$ decay [1].

In Fig. 3, we show the $K^{+}K^{-}$ invariant mass-dependent differential
branching fractions for the quasi-two-body decays $B^{0}\to
K^{0}(\phi(1020)\to)K^{+}K^{-}$. It can be seen that the contribution of
$\phi(1020)$ is dominate in the region near 1 GeV. Under the narrow-width
approximation, the three-body decay and corresponding two-body one satisfy the
relation

$\displaystyle\mathcal{B}(B\to P_{3}({\cal
R}\to)P_{1}P_{2})\approx\mathcal{B}(B\to P_{3}{\cal R})\times\mathcal{B}({\cal
R}\to P_{1}P_{2}),$ (23)

with ${\cal R}$ being the resonance. With the experimental results
$\mathcal{B}(\phi\to K^{+}K^{-})=(49.2\pm 0.5)\%$ [39], we obtain ${\cal
B}(B^{0}\to K^{0}\phi)=(6.4^{+2.9}_{-2.0})\times 10^{-6}$, which is in
agreement with above experimental results with uncertainties.

Figure 4: The $\omega$-dependence of differential branching fractions from
$f_{0}(980)$, $f_{0}(1500)$, $f_{0}(1710)$, $f_{2}^{\prime}(1525)$ and
$f_{2}(2010)$ for the $B^{0}\to K^{0}K^{+}K^{-}$ decay [1].

Although the conventional quark model has achieved great success, the quark
structures of scalar particles are still quite controversial, especially for
the light scalar ones. Although there are many hints that the light scalars
are four-quark states, the wave functions of these particle have not
established, and we left it as our future work. Currently, we here still
regard $f_{0}(980)$ as a two quark structure. Many experimental evidences
indicate that both $s\bar{s}$ and $q\bar{q}$ are involved in the $f_{0}(980)$,
so the mixing form is supposed to be

$\displaystyle|f_{0}(980)\rangle=|q\bar{q}\rangle\sin\theta+|s\bar{s}\rangle\cos\theta,$
(24)

with $q\bar{q}=(u\bar{u}+d\bar{d})/\sqrt{2}$ and the mixing angle
$\theta=40^{\circ}$ [50]. For the heavier scalar mesons, there are glueball
contents in isosinglet scalar mesons $f_{0}(1370)$, $f_{0}(1500)$ and
$f_{0}(1710)$. It is commonly accepted that $f_{0}(1710)$ is dominated by the
scalar glueball, while $f_{0}(1500)$ is an approximately SU(3) octet with
negligible glueball component. So, the glueball content of $f_{0}(1500)$ can
be neglected.

In Fig. 4, we show the dependencies of the differential branching ratios
$d{\cal B}(B^{+}\to K^{+}({\cal R}\to)K^{+}K^{-})/d\omega$ on the kaon-pair
invariant mass $\omega$ with the $S$-wave resonances $f_{0}(980)$,
$f_{0}(1500)$ and $f_{0}(1710)$ and the $D$-wave particles
$f_{2}^{\prime}(1525)$ and $f_{2}(2010)$. From the figure, it is obvious that
the $f_{0}(980)$ production is apparently dominant, and it is about ten times
larger than that of $f_{0}(1710)$. In fact, it has been shown that within
present experimental and theoretical uncertainties, the narrow peak at 980 MeV
in the $K\bar{K}$ system may be interpreted as a cusp phenomenon with large
width, and does not necessarily imply that there exists a narrow resonance in
that region. Because of large width, the contribution of the tail of
$f_{0}(980)$ with is still larger than the total effect of $f_{0}(1500)$. We
also find that the contributions of $f_{0}(1710)$ and $f_{0}(1500)$ overlap
with each other, so that it is very hard for us to disentangle two resonances
at the region about $1.5~{}\rm GeV$. Unlike $S$-wave, the contributions from
$D$-wave resonances do not overlap anymore because of the narrow width of
$f_{2}^{\prime}(1525)$. As for the $f_{2}(1270)$, although there are some
studies on it [51], the mixing between different quark components is quit
complicated, so its contribution has not been included.

As aforementioned, the narrow-width approximation is invalid in describing the
scalar particle $f_{0}(980)$, but it still holds in the processes $B\to
K(f_{0}(1500)\to)K^{+}K^{-}$. Again, we could obtain the branching fractions
of $B\to Kf_{0}(1500)$ as

$\displaystyle\mathcal{B}(B^{0}\to K^{0}f_{0}(1500))$ $\displaystyle=$
$\displaystyle(13.7\pm 6.1)\times 10^{-6},$ (25)
$\displaystyle\mathcal{B}(B^{+}\to K^{+}f_{0}(1500))$ $\displaystyle=$
$\displaystyle(13.9\pm 5.8)\times 10^{-6},$ (26)

within ${\cal B}(f_{0}(1500)\to K^{+}K^{-})\simeq 4.3\%$. For the decay
$B^{0}\to K^{0}f_{0}(1500)$, our result agrees with experimental data
$(1.3^{+0.7}_{-0.6})\times 10^{-5}$ [39]. As for the decay $B^{+}\to
K^{+}f_{0}(1500)$, the center value of prediction is about 3.7 times larger
than that averaged experimental data $(3.7\pm 2.2)\times 10^{-6}$ [39], but
both theoretical and experimental uncertainties are very larger.

Under the narrow-width approximation, we get a ratio as

$\displaystyle{R}_{1}=\frac{\mathcal{B}(f_{0}(1500)\to
K^{+}K^{-})}{\mathcal{B}(f_{0}(1500)\to\pi^{+}\pi^{-})}\approx\frac{\mathcal{B}(B\to
K(f_{0}(1500)\to)K^{+}K^{-})}{\mathcal{B}(B\to
K(f_{0}(1500)\to\pi^{+}\pi^{-}))}.$ (27)

Using the experimental data $\mathcal{B}(f_{0}(1500)\to K^{+}K^{-})=4.3\%$ and
$\mathcal{B}(f_{0}(1500)\to\pi^{+}\pi^{-})=23.27\%$ [39], we can get the
fraction ${R}_{1}=0.185$. Thereby, the branching fractions of $B\to
K(f_{0}(1500)\to\pi^{+}\pi^{-})$ decays are estimated to be

$\displaystyle\mathcal{B}(B^{+}\to K^{+}(f_{0}(1500)\to)\pi^{+}\pi^{-})$
$\displaystyle=$ $\displaystyle(3.24\pm 1.35)\times 10^{-6},$ (28)
$\displaystyle\mathcal{B}(B^{0}\to K^{0}(f_{0}(1500)\to)\pi^{+}\pi^{-})$
$\displaystyle=$ $\displaystyle(3.15\pm 1.40)\times 10^{-6}.$ (29)

Similarly, we also define another ratio as

$\displaystyle{R}_{2}=\frac{\mathcal{B}(f_{0}(1710)\to
K^{+}K^{-})}{\mathcal{B}(f_{0}(1710)\to\pi^{+}\pi^{-})}\approx\frac{\mathcal{B}(B\to
K(f_{0}(1710)\to)K^{+}K^{-})}{\mathcal{B}(B\to
K(f_{0}(1710)\to)\pi^{+}\pi^{-})}.$ (30)

With the experimental result
$\Gamma(f_{0}(1710)\to\pi\pi)/\Gamma(f_{0}(1710)\to K\overline{K})=0.23\pm
0.05$ [39], we then predict the branching fractions of $B\to
K(f_{0}(1710)\to)\pi^{+}\pi^{-}$ decays as

$\displaystyle\mathcal{B}(B^{+}\to K^{+}(f_{0}(1710)\to)\pi^{+}\pi^{-})$
$\displaystyle=$ $\displaystyle(5.0^{+3.9}_{-3.4})\times 10^{-7},$
$\displaystyle\mathcal{B}(B^{0}\to K^{0}(f_{0}(1710)\to)\pi^{+}\pi^{-})$
$\displaystyle=$ $\displaystyle(4.5^{+3.9}_{-3.4})\times 10^{-7}.$ (31)

All above predictions are expected to be measured in LHCb and Belle-II
experiments.

##  4 Summary

In this review, we took $B\to K({\cal R}\to)K^{+}K^{-}$ decays as examples and
showed the application of PQCD in studying the quasi-two-body $B$ decays. In
this approach, we here only discussed the region at the central part of the
edges, where two particles move collinearly with large energy and the other
particle recoils back. In order to describe the dynamics of two collinear
particles, the wave functions of kaon-pair with different spins are
introduced. By keeping the transverse momenta, all possible diagrams at
leading order were calculated, including the hard spectator diagrams and
annihilation ones. Most of our numerical results are well consistent with the
current experimental data. Under the narrow-width approximation we extracted
the branching fractions of the corresponding two-body decays involving the
resonances, such as the $B\to K\phi$, whose branching fractions agree with the
current experimental data well. We also predicted the branching fractions of
the corresponding quasi-two-body decays $B\to K(\cal{R}\to)\pi^{+}\pi^{-}$.
All prediction are expected to be tested in the ongoing LHCb and Belle-II
experiments.

## Acknowledgement

Y.Li thanks Q-X Li and X Liu for collaborations. This work was supported in
part by the National Natural Science Foundation of China under the Grant No.
11975195, by the Natural Science Foundation of Shandong province under the
Grant No.ZR2019JQ04, and by the Project of Shandong Province Higher
Educational Science and Technology Program under Grants No. 2019KJJ007.

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#  Infinitely Many Periodic Solutions for Some N-Body
Type Problems with Fixed Energies††thanks: Supported partially by NSF of
China.

Pengfei Yuan and Shiqing Zhang
Department of Mathematics, Sichuan University, Chengdu 610064, China

###### Abstract

In this paper, we apply the Ljusternik-Schnirelman theory with local Palais-
Smale condition to study a class of N-body problems with strong force
potentials and fixed energies. Under suitable conditions on the potential $V$,
we prove the existence of infinitely many non-constant and non-collision
symmetrical periodic solutions .

Key Words: Periodic solutions, N-body problems, Ljusternik-Schnirelman theory,
Local Palais-Smale condition.
2000 Mathematicals Subject Classification: 34C15, 34C25, 58F

##  1 Introduction and Main Results

The N-body problem is an old and very difficult problem. Many mathematicians
studied this problem using many different methods, here we only concern
variational methods. In 1975 and 1977, Gordon $([15],[16])$ firstly used
variational methods to study the periodic solutions of 2-body problems. In
$[16]$, he put forward to the strong force condition (SF for short) and got
the following Lemma:

######  Lemma 1.1.

(Gordon$[16]$) Suppose that $V(x)$ satisfies the so called Gordon’s Strong
Force condition: There exists a neighborhood $\mathcal{N}$ of 0 and a function
$U\in C^{1}(\mathcal{N}\setminus\\{0\\},\mathbb{R})$ such that:
$(i).\>\underset{|x|\rightarrow 0}{\lim}U(x)=-\infty;$
$(ii).\>-V(x)\geq|\nabla U(x)|^{2}$ for every
$x\in\mathcal{N}\,\backslash\,\\{0\\}.$

Let

$\displaystyle\Lambda$ $\displaystyle\triangleq\\{x|\,x(t)\in
H^{1}(\mathbb{R}/\mathbb{Z},\mathbb{R}^{n}),\,x(t)\neq
0,\forall\,t\,\in[0,1]\\}$ $\displaystyle\partial\Lambda$
$\displaystyle\triangleq\\{x|\,x(t)\in
H^{1}(\mathbb{R}/\mathbb{Z},\mathbb{R}^{n}),\,\exists\,t_{0}\in[0,1]\,s.t.\,x(t_{0})=0\\}$

Then we have

$\int_{0}^{1}V(x_{n})\,\mathrm{d}t\rightarrow-\infty,\quad\forall\>x_{n}\rightharpoonup
x\in\partial\Lambda.$

After Gordon, many researchers used variational methods to study N ($N\geq 3$)
body problems $([1]-[8],[10]-[19],[22]-[27],etc.)$.

In $[18]$, P.Majer used Ljusternik-Schnirelman theory (LS for short) with
local Palais-Smale condition to seek T-periodic solutions of the following
second order Hamiltonian system:

$\ddot{x}+ax+\nabla_{x}W(t,x)=0$ $None$

where $W$ is singular at $x=0,W(t+T,x)=W(t,x).$

He got the following theorem :

######  Theorem 1.2.

Suppose

$\quad(P_{1}).a<(\dfrac{\pi}{T})^{2};$

$\quad(P_{2}).W\in
C^{1}(S_{T}^{1}\times(\mathbb{R}^{n}\setminus\\{0\\}),\mathbb{R})$ satisfies
$(SF)$;

$\quad(P_{3}).\exists\,c,\,\theta<2,\,r>0$, such that $\forall\,|x|\geq
r,\forall\,t\in S_{T}^{1}$

$W(t,x)\leq c|x|^{\theta},\quad\langle\nabla W_{x}(t,x),x\rangle-2W(t,x)\leq
c|x|^{\theta}$

$\quad(P_{4}).W(t,x)\leq b.$

Then the system $(1.1)$ has infinitely many T-periodic non-collision solutions
.

After Majer , Zhang-Zhou $([27])$ studied a class of N-Body problems. They
considered the following system :

$m_{i}\ddot{x}(t)+\nabla_{x_{i}}V(t,x_{1}(t),\cdots,x_{N}(t))=0,\,x_{i}\in\mathbb{R}^{n},\,i=1,\cdots,N$
$None$

where $m_{i}>0$ for all $i$, and $V$ satisfies the following conditions :

$\quad(V1).V(t,x_{1},\cdots,x_{N})=\dfrac{1}{2}\underset{1\leq i\neq j\leq
N}{\sum}V_{ij}(t,x_{i}-x_{j});$

$\quad(V2).V_{ij}\in
C^{2}(\mathbb{R}\times(\mathbb{R}^{n}\setminus\\{0\\});\mathbb{R}),$ for all
$1\leq i\neq j\leq N$;

$\quad(V3).V_{ij}(t,\xi)\rightarrow-\infty$ uniformly on $t$ as
$|\xi|\rightarrow 0$, for all $1\leq i\neq j\leq N$;

$\quad(V4).V_{ij}(t,\xi)\leq 0$, for all
$t\in\mathbb{R},\,\xi\in\mathbb{R}^{n}\setminus\\{0\\}$;

$\quad(V5).V_{ij}$ satisfies Gordon’s strong force condition;

$\quad(V6)$. There exists an element $g$ of finite order $s$ in $SO(k)$ which
has no fixed point other than origin (i.e. 1 is not an eigenvalue of g), such
that

$V(t,x_{1},\cdots,x_{N})=V(t+T/s,gx_{1},\cdots,gx_{N}),\,\forall\,t\in[0,1],\,x_{i}\in\mathbb{R}^{n}.$

If the potential $V$ satisfies $(V1)-(V6)$, and
$x(t)=(x_{1}(t),\cdots,x_{N}(t))$ is a T-periodic non-collision solution of
system $(1.2)$ and satisfies
$x(t+T/s)=(x_{1}(t+T/s),\cdots,x_{N}(T+t/s))=(gx_{1}(t),\cdots,gx_{N}(t))=gx(t)$,
we say that $x(t)$ is a g-symmetric T-periodic non-collision solutions (we
also call $(g,T)$ is a non-collision solution for short).

Zhang-Zhou obtained the following theorem:

######  Theorem 1.3.

Suppose the potential $V$ satisfies $(V1)-(V6)$ and $T$ is any positive real
number. Then system $(1.2)$ has infinitely many $(g,T)$ non-collision
solutions.

Motivated by P.Majer, Zhang S.Q.- Zhou Q. ’s work , we consider the following
system:

$\left\\{\begin{array}[]{cc}&m_{i}\ddot{u}_{i}+\nabla_{u_{i}}V(u_{1},\ldots,u_{N})=0,\quad(1\leq
i\leq N),\quad(Ph.1)\\\ &\dfrac{1}{2}\sum
m_{i}|\dot{u}_{i}(t)|^{2}+V\left(u_{1}(t),\ldots,u_{N}(t)\right)=h.\quad\qquad(Ph.2)\end{array}\right.$
$None$

We have the following theorem :

######  Theorem 1.4.

Suppose $V(u_{1},\cdots,u_{N})=\dfrac{1}{2}\underset{1\leq i\neq j\leq
N}{\sum}V_{ij}(u_{i}-u_{j}),\,\,V_{ij}(\xi)\in
C^{1}(\mathbb{R}^{n}\setminus\\{0\\},\mathbb{R})$, and satisfies

$\quad(V5^{\prime}).V_{ij}(\xi)\leq
0,\forall\,\xi\in\mathbb{R}^{n}\setminus\\{0\\}$;

$\quad(V6^{\prime}).\exists\,\alpha>2$ and $r_{1}>0$ such that $\langle\nabla
V_{ij}(\xi),\xi\rangle\geq-\alpha V_{ij}(\xi),\forall 0<|\xi|\leq r_{1};$

$\quad(V7^{\prime}).\exists\,c\geq 0,-2<\theta<0,r_{2}>r_{1}$, such that when
$|\xi|\geq r_{2}$, there holds

$\langle\nabla_{\xi}V_{ij}(\xi),\xi\rangle\leq c|\xi|^{\theta};$

$\quad(V8^{\prime}).V_{ij}(\xi)=V_{ji}(\xi),\forall\,\xi\in\mathbb{R}^{n}\setminus\\{0\\}.$

Then for any $h>0$, the system $(Ph)$ has infinitely many non-constant and
non-collision periodic solutions .

######  Example 1.5.

We take

$V_{ij}(\xi)=\left\\{\begin{array}[]{cc}-\dfrac{1}{|\xi|^{\alpha}},&\ \ {\rm
if}\ \ 0<|\xi|\leq r_{1},\\\ smooth\,\,connecting,&\ \ {\rm if}\ \
r_{1}<|\xi|<r_{2},\\\ -|\xi|^{\theta},&\ \ {\rm if}\ \ |\xi|\geq
r_{2}>r_{1}.\end{array}\right.$

##  2  Some Lemmas

In this section, we collect some known lemmas which are necessary for the
proof of Theorem 1.4.

Let us introduce the following notations:

$\displaystyle m^{*}=\min{\\{m_{1},\cdots,m_{N}\\}}\>;\qquad
H^{1}=W^{1,2}(\mathbb{R}/\mathbb{Z},\mathbb{R}^{n}).$ $\displaystyle
E=\\{u=(u_{1},\ldots,u_{N})\mid\ u_{i}\in
H^{1},\,u_{i}(t+\dfrac{1}{2})=-u_{i}(t)\\}.$ $\displaystyle\Lambda_{0}=\\{u\in
E\mid u_{i}(t)\neq u_{j}(t),\>\forall\,t,\,\forall\,i\neq j\\}.$
$\displaystyle\partial\Lambda_{0}=\\{u\in E\mid\exists\,t_{0},1\leq i_{0}\neq
j_{0}\leq N\,s.t.\,u_{i_{0}}(t_{0})=u_{j_{0}}(t_{0})\\}.$ $\displaystyle
p(u)=\underset{1\leq i\neq j\leq N,\,t\in[0,1]}{\min{}}|u_{i}(t)-u_{j}((t)|.$
$\displaystyle\\{f\leq c\\}=\\{u\in\Lambda_{0},\,f(u)\leq c\\}.$

######  Lemma 2.1.

$([1]-[4])$ Let
$f(u)=\dfrac{1}{2}\int_{0}^{1}\sum\limits_{i=1}^{N}m_{i}|u_{i}|^{2}\,\mathrm{d}t\int_{0}^{1}(h-V(u))\,\mathrm{d}t$
and $\tilde{u}\in\Lambda_{0}$ satisfy $f^{\prime}\left(\tilde{u}\right)=0$ and
$f\left(\tilde{u}\right)>0$. Set

$\dfrac{1}{T^{2}}=\dfrac{\int_{0}^{1}\left(h-V\left(\tilde{u}\right)\right)\,\mathrm{d}t}{\dfrac{1}{2}\int_{0}^{1}\sum\limits_{i=1}^{N}m_{i}|\dot{\tilde{u}}_{i}|^{2}\,\mathrm{d}t}.$
$None$

Then $\tilde{q}(t)=\tilde{u}\left(t/T\right)$ is a non-constant $T$-periodic
solution for (Ph).

######  Lemma 2.2.

$(Palais[20])$ Let $\sigma$ be an orthogonal representation of a finite or
compact group $G$ in the real Hilbert space $H$ such that for
$\forall\>\sigma\in\,G$,

$f(\sigma\cdot x)=f(x),$

where $f\in C^{1}(H,\mathbb{R})$.

Let $S=\\{x\in H\mid\sigma x=x,\>\forall\>\sigma\ in\ G\\}$, then the critical
point of $f$ in $S$ is also a critical point of $f$ in $H$.

By Lemma 2.1-2.2, we have

######  Lemma 2.3.

$([1]-[4])$ Assume $V_{ij}\in
C^{1}(\mathbb{R}^{n}\setminus\\{0\\},\mathbb{R})$ satisfies $(V8^{\prime}).$
If $\bar{u}\in\Lambda_{0}$ is a critical point of $f(u)$ and $f(\bar{u})>0$,
then $\bar{q}(t)=\bar{u}(t/T)$ is a non-constant $T$-periodic solution of
(Ph).

######  Lemma 2.4.

($[28]$) Let $q\in W^{1,2}(\mathbb{R}/T\mathbb{Z},\mathbb{R}^{n})$ and
$\int_{0}^{T}q(t)\,\mathrm{d}t=0$, then we have
$(i)$. Poincare-Wirtinger’s inequality:

$\int_{0}^{T}|\dot{q}(t)|^{2}\,\mathrm{d}t\geq{\Big{(}\frac{2\pi}{T}\Big{)}}^{2}\int_{0}^{T}|{q}(t)|^{2}\,\mathrm{d}t.$
$None$

$(ii)$. Sobolev’s inequality:

$\underset{0\leq t\leq
T}{\max}|q(t)|=|q|_{\infty}\leq\sqrt{\frac{T}{12}}\big{(}\int_{0}^{T}|\dot{q}(t)|^{2}\mathrm{d}t\big{)}^{1/2}.$
$None$

By the definition of $\Lambda_{0}$ and Lemma 2.3, for
$\forall\,u\in\Lambda_{0},(\int_{0}^{1}\sum\limits_{i=1}^{N}m_{i}|\dot{u}_{i}|^{2}\,\mathrm{d}t)^{1/2}$
is equivalent to the $(H^{1})^{N}=H^{1}\times\cdots H^{1}$ norm:

$\parallel
u\parallel_{(H^{1})^{N}}=\left(\int_{0}^{1}|\dot{u}|^{2}\,\mathrm{d}t\right)^{1/2}+\left(\int_{0}^{1}|u|^{2}\,\mathrm{d}t\right)^{1/2}.$

So we take norm $\parallel
u\parallel=(\int_{0}^{1}\sum\limits_{i=1}^{N}m_{i}|\dot{u}_{i}|^{2}\,\mathrm{d}t)^{1/2}$.

######  Lemma 2.5.

(Coti Zelati$\,[11]$) Let
$X=(x_{1},\ldots,x_{N})\in\mathbb{R}^{n}\times\cdots\mathbb{R}^{n},x_{i}\neq
x_{j},\,\forall\,1\leq i\neq j\leq N.$ Then

$\sum\limits_{1\leq i<j\leq N}\dfrac{m_{i}m_{i}}{|x_{i}-x_{j}|^{\alpha}}\geq
C_{\alpha}(m_{1},\ldots,m_{N})(\sum\limits_{i=1}^{N}m_{i}|x_{i}|^{2})^{-\alpha/2},$

where
$C_{\alpha}(m_{1},\ldots,m_{N})\overset{\triangle}{=}C_{\alpha}=(\sum\limits_{i=1}^{N}m_{i})^{-\alpha/2}(\sum\limits_{1\leq
i<j\leq N}m_{i}m_{j})^{\frac{2+\alpha}{2}}.$

######  Lemma 2.6.

$([18])$ Let $X$ be a Banach space with norm $\parallel\cdot\parallel$,
$\Lambda$ be an open subset of $X$, and suppose a functional
$f:\Lambda\rightarrow\mathbb{R}$ is given such that the following conditions
hold:

$\quad(i).Cat_{\Lambda}(\Lambda)=+\infty;$

$\quad(ii).f\in C^{1}(\Lambda)$ and $\forall
u_{n}\rightharpoonup\partial\Lambda,f(u_{n})\rightarrow+\infty;$

$\quad(iii).\forall\lambda\in\mathbb{R},Cat_{\Lambda}(\\{f\leq\lambda\\})<+\infty;$

Suppose in addition that there exist $g\in C^{1}(\Lambda),\beta\in(0,1)$ and
$\lambda_{0}\in\mathbb{R}$ such that

$\quad(iv).Cat_{\Lambda}(\\{f\leq g\\})<+\infty$;

$\quad(v).$ the PS condition holds in the set $\\{f\geq g\\};$

$\quad(vi).\beta\parallel f^{\prime}(u)\parallel\geq\parallel
g^{\prime}(u)\parallel,\forall u\in\\{f=g\geq\lambda_{0}\\}.$

Then $f$ has a sequence $\\{u_{n}\\}\subset\Lambda$ of critical points such
that $f(u_{n})\rightarrow+\infty$ and $f(u_{n})\geq g(u_{n})-1.$

##  3 The Proof of Theorem 1.4

######  Lemma 3.1.

Let $\\{u_{n}\\}\subset\Lambda_{0}$ and $u_{n}\rightharpoonup
u\in\partial\Lambda_{0}.$ Then $f(u_{n})\rightarrow+\infty.$

###### Proof.

First of all , we recall that

$f(u_{n})=\dfrac{1}{2}\int_{0}^{1}\sum_{i=1}^{N}m_{i}|\dot{u}_{n}^{i}|^{2}dt\int_{0}^{1}(h-V(u_{n}))dt$

(1).If $u=$ constant, we can deduce $u=0$ by
$u_{i}(t+\dfrac{1}{2})=-u_{i}(t)$. By Sobolev’s embedding theorem, we obtain

$|u_{n}|_{\infty}\rightarrow 0,\quad n\rightarrow\infty$ $None$

So when $n$ is large enough, $0\leq|u_{n}^{i}-u_{n}^{j}|\leq r_{1}$. By
$(V6^{\prime})$, there exists an $A>0$ such that

$V_{ij}(\xi)\leq-\dfrac{A}{|\xi|^{\alpha}},\quad\forall 0<|\xi|\leq r_{1}$
$None$

By $h>0$, Sobolev’s inequality $(2.3)$ , $(3.2)$ and Lemma 2.5, we have

$\displaystyle f(u_{n})$ $\displaystyle=\dfrac{1}{2}\parallel
u_{n}\parallel^{2}\int_{0}^{1}(h-\sum_{1\leq i<j\leq
N}V_{ij}(u_{n}^{i}-u_{n}^{j}))dt$ $\displaystyle\geq\dfrac{1}{2}\parallel
u_{n}\parallel^{2}\int_{0}^{1}\\{-\sum_{1\leq i<j\leq
N}V_{ij}(u_{n}^{i}-u_{n}^{j})\\}dt$ $\displaystyle\geq\dfrac{1}{2}\parallel
u_{n}\parallel^{2}\int_{0}^{1}\sum_{1\leq i<j\leq
N}\dfrac{A}{|u_{n}^{i}-u_{n}^{j}|^{\alpha}}dt$
$\displaystyle\geq\dfrac{1}{2}\parallel
u_{n}\parallel^{2}\int_{0}^{1}A[\dfrac{N(N-1)}{2}]^{\frac{2+\alpha}{2}}N^{\frac{-\alpha}{2}}|u_{n}|^{-\alpha}dt$
$\displaystyle\geq
3m^{*}A2^{-\frac{\alpha}{2}}N(N-1)^{1+\frac{\alpha}{2}}|u_{n}|_{\infty}^{2-\alpha}$

Then by $(3.1)$ and $\alpha>2$, we can deduce

$f(u_{n})\rightarrow+\infty,\quad n\rightarrow\infty$

(2). If $u\not\equiv$ constant, by the weakly lower-semi-continuity property
for norm, we have

$\displaystyle\underset{n\rightarrow\infty}{\liminf}\int_{0}^{1}\sum\limits_{i=1}^{N}m_{i}|\dot{u}_{n}^{i}|^{2}\,\mathrm{d}t$
$\displaystyle\geq\int_{0}^{1}\sum\limits_{i=1}^{N}m_{i}|\dot{u}_{i}|^{2}\,\mathrm{d}t>0.$
(3.3)

Since $u\in\partial\Lambda_{0}$, there exist $t_{0},1\leq i_{0}\neq j_{0}\leq
N$ s.t. $u_{i_{0}}(t_{0})=u_{j_{0}}(t_{0}).$ Set

$\displaystyle\xi_{n}(t)=u_{n}^{i_{0}}(t)-u_{n}^{j_{0}}(t)$
$\displaystyle\xi(t)=u_{i_{0}}(t)-u_{j_{0}}(t)$

By $u_{n}\rightharpoonup u$, we have $\xi_{n}(t)\rightharpoonup\xi(t)$. Then
by $(V6^{\prime})$ and Lemma 1.1, we have

$\int_{0}^{1}V_{i_{0}j_{0}}(u_{n}^{i_{0}}-u_{n}^{j_{0}})\,\mathrm{d}t\rightarrow-\infty.$

Recalling that

$V(u_{n})=\sum\limits_{i<j}V_{ij}(u_{n}^{i}-u_{n}^{j}).$

So we have

$f(u_{n})\rightarrow+\infty,\quad n\rightarrow\infty.$

∎

######  Lemma 3.2.

For every $\lambda\in\mathbb{R}$ there exists a constant $k=k(\lambda)$ such
that

$\parallel u\parallel\leq k(\lambda)p(u),\quad\forall u\in\\{f\leq\lambda\\}$
$None$

###### Proof.

Recall that

$f(u)=\dfrac{1}{2}\parallel u\parallel^{2}\int_{0}^{1}(h-V(u))dt$

Since $V_{ij}\leq 0$, we have $f(u)\geq\dfrac{1}{2}h\parallel u\parallel^{2}$.
Since $u\in\\{f\leq\lambda\\}$, we can deduce that

$\parallel u\parallel\leq C.$ $None$

If $(3.4)$ is not true, then there exists a sequence $\\{u_{n}\\}$ such that
$\parallel u_{n}\parallel\geq np(u_{n})$. By $(3.5)$, we have

$0<p(u_{n})\leq\dfrac{C}{n}.$ $None$

Let $n\rightarrow\infty$ in (3.6), we have $p(u_{n})\rightarrow 0.$ Then there
exists a subsequence $\\{u_{n}\\}\rightharpoonup u\in\partial\Lambda_{0}$, by
Lemma 3.1, $f(u_{n})\rightarrow+\infty$, which is a contradiction since
$\\{u_{n}\\}\subset\\{f\leq\lambda\\}.$

∎

######  Lemma 3.3.

For every $\lambda\in\mathbb{R}$, the set
$\Lambda_{\lambda}=\\{u\in\Lambda_{0}:\dfrac{\parallel
u\parallel}{p(u)}\leq\lambda\\}$ is of finite category in $\Lambda_{0}$.

###### Proof.

The proof is almost the same as the proof of Lemma 4.3 of [27]. For the
covenience of the readers, we write the complete proof.

It is sufficient that we give a homotopy
$h:[0,1]\times\Lambda_{c}\subset\subset\Lambda_{0}.$

Take $\delta\in(0,1)$ s.t. $2\dfrac{1}{m^{*}}\sqrt{\delta}c<1$, and define
$\phi(t)=\dfrac{p(u}{\delta}$, if $t\in[0,\delta]$,otherwise $\phi(t)=0.$

Define

$h(u,\lambda)=(1-\lambda)u(t)+\lambda\dfrac{(u\ast\phi)(t)}{p(u)},\quad\forall
u\in\Lambda_{c},\,0\leq\lambda\leq 1$

where the convolution
$(u\ast\phi)(t)=(\int_{0}^{1}u_{1}\phi(s)ds,\cdots,\int_{0}^{1}u_{N}\phi(s)ds)$,
then $h(u,0)$ is an inclusion and $h_{1}(\Lambda_{c})$ is paracompact since
$h(u,1)$ is a convolution operator , so it’s a compact operator. We need to
prove $h(\Lambda_{c}\times I)\subset\Lambda_{0}.$

Suppose this is not true , then
$\exists\lambda_{0}\in(0,1],\,u\in\Lambda_{c},\,1\leq i_{0}\neq j_{0}\leq
N,\,t_{0}\in[0,1]$, such that

$(1-\lambda_{0})(u_{i_{0}}-u_{j_{0}})+\lambda_{0}\dfrac{1}{p(u)}((u_{i_{0}}-u_{j_{0}})\ast\phi)(t_{0})$

Then

$(u_{i_{0}}-u_{j_{0})}\ast\phi(t_{0})=p(u)(1-\dfrac{1}{\lambda_{0}})(u_{i_{0}}(t_{0})-u_{j_{0}}(t_{0}))$

So we have

$\displaystyle|p(u)(u_{i_{0}}(t_{0})-u_{j_{0}}(t_{0})-((u_{i_{0}}-u_{j_{0}}\ast\phi))(t_{0}))|$
$\displaystyle=\dfrac{p(u)}{\lambda_{0}}\geq p(u)p(u)=p^{2}(u).$

On the other hand , $\forall u=(u_{1},\cdots,u_{N})\in\Lambda_{c},\,i\neq
j,\,t\in[0,1]$, we have

$\displaystyle|p(u)(u_{i}(t)-u_{j}(t))-((u_{i}-u_{j})\ast\phi)(t)|$
$\displaystyle=|\dfrac{p(u)}{\delta}\int_{0}^{\delta}(u_{i}(t)-u_{j}(t))ds-\dfrac{p(u)}{\delta}(u_{i}(t-s)-u_{j}(t-s))ds|$
$\displaystyle\leq\dfrac{p(u)}{\delta}\int_{0}^{\delta}|u_{i}(t)-u_{j}(t)-(u_{i}(t-s)-u_{j}(t-s))|ds$
$\displaystyle\leq
p(u)\cdot\underset{|s|\leq\delta}{\sup{}}|u_{i}(t)-u_{j}(t)-(u_{i}(t-s)-u_{j}(t-s))|$
$\displaystyle\leq
p(u)\sqrt{\delta}\parallel\dot{u}_{i}-\dot{u}_{j}\parallel_{L^{2}}\leq
p(u)\sqrt{\delta}(\parallel\dot{u}_{i}\parallel_{L^{2}}+\parallel\dot{u}_{j}\parallel_{L^{2}})$
$\displaystyle\leq p(u)2\sqrt{\delta}\parallel\dot{u}\parallel_{L^{2}}\leq
p(u)\cdot(2\dfrac{1}{m^{*}}\sqrt{\delta}c)\cdot p(u)<p^{2}(u)$

Which is a contradiction. ∎

######  Lemma 3.4.

The functional $f$ verifies the Palais-Smale condition on $\Lambda_{0}$.

###### Proof.

Let $\\{u_{n}\\}\subset\Lambda_{0}$ be a P.S. sequence, then up to a
subsequence, it converges weakly in $(H^{1})^{N}$ and uniformly in
$|u|_{\infty}$ to an element $u\in\Lambda_{0}$ by Lemma 3.1. Hence
$\langle\nabla_{\xi}V_{ij}(u_{i}^{n}-u_{j}^{n}),(u_{i}-u_{i}^{n}-u_{j}-u_{j}^{n})\rangle$
converges uniformly to zero. Since $f^{\prime}(u_{n})\rightarrow 0$, and
$u-u_{n}$ is $(H^{1})^{N}$-bounded, and

$\langle f^{\prime}(u),v\rangle=\int_{0}^{1}\sum_{1\leq i\leq
N}m_{i}\langle\dot{u_{i}},\dot{v_{i}}\rangle
dt\int_{0}^{1}(h-V(u))dt-\dfrac{1}{2}\parallel
u\parallel^{2}\int_{0}^{1}\langle\nabla_{u}V(u),v\rangle dt$

So we have

$\displaystyle\parallel
u\parallel^{2}-\underset{n\rightarrow\infty}{\lim{}}\parallel
u_{n}\parallel^{2}$
$\displaystyle=\underset{n\rightarrow\infty}{\lim{}}\int_{0}^{1}m_{i}\langle\dot{u}_{n},(\dot{u}-\dot{u_{n}})\rangle
dt$ $\displaystyle=\underset{n\rightarrow\infty}{\lim}{}\dfrac{\langle
f^{\prime}(u_{n}),u-u_{n}\rangle}{\int_{0}^{1}(h-V(u_{n}))dt}+\underset{n\rightarrow\infty}{\lim}{}\dfrac{\frac{1}{2}\parallel
u_{n}\parallel^{2}\int_{0}^{1}\langle\nabla_{u}V(u_{n}),(u-u_{n})\rangle
dt}{\int_{0}^{1}(h-V(u_{n}))dt}=0$

∎

######  Lemma 3.5.

$Cat_{\Lambda_{0}}(\Lambda_{0})=+\infty$

###### Proof.

See the Corollary 3.4 of [27]. ∎

######  Lemma 3.6.

There exist a functional $g\in C^{1}(\Lambda_{0}),\beta\in(0,1)$ and
$\lambda_{0}\in\mathbb{R}$ such that:

(iv)$Cat_{\Lambda_{0}}\\{f\leq g\\}<+\infty;$

(v) the P.S. condition holds in the set $\\{f\leq g\\};$

(vi)$\beta\parallel f^{\prime}(u)\parallel\geq\parallel
g^{\prime}(u)\parallel,\quad\forall u\in\\{f=g\geq\lambda_{0}\\}.$

###### Proof.

By Sobolev’s embedding theorem, we know there is a $k_{\infty}>0$ s.t.

$\parallel u\parallel_{\infty}\leq k_{\infty}\parallel
u\parallel,\forall\,u\in\Lambda_{0}.$

Take $\beta$ such that $\beta>\dfrac{\theta+2}{2}$, take $\gamma>0$ such that

$\beta\cdot[2\gamma-N(N-1)c2^{\theta-2}k_{\infty}^{\theta}]>\gamma(\theta+2).$

Let $g(u)=\gamma\parallel u\parallel^{\theta+2}$, we shall show that $\\{f\leq
g\\}$ is a set of finite category. We take $0<\varepsilon<\dfrac{h}{2}$ and
$M>0$ such that $\forall s\in\mathbb{R},\gamma|s|^{\theta+2}\leq\varepsilon
s^{2}+M$.

Define

$f_{\varepsilon}(u)=f(u)-\varepsilon\parallel u\parallel^{2}$

Then

$\\{f\leq g\\}\subset\\{f_{\varepsilon}(u)\leq M\\}$

We can use the similar proof of Lemma 3.2 to show that there exists
$k_{1}\in\mathbb{R}$ such that

$\parallel u\parallel\leq k_{1}p(u),\forall u\in\\{f_{\varepsilon}\leq M\\}$

then by Lemma 3.3,

$Cat_{\Lambda_{0}}\\{f\leq g\\}\leq Cat_{\Lambda_{0}}\\{f_{\varepsilon}\leq
M\\}<+\infty$

From Lemma 3.4, we know that $(v)$ is satisfied.

Since $\parallel u\parallel\leq k_{1}p(u)$, we take
$\lambda_{0}=:\gamma(k_{1}r_{2})^{\theta+2}$. Then if
$u\in\\{f=g\geq\lambda_{0}\\}$, so

$\parallel
u\parallel\geq[\dfrac{\lambda_{0}}{\gamma}]^{\frac{1}{\theta+2}}=k_{1}r_{2}$

So we can obtain :

$r_{2}\leq p(u)\leq|u_{i}(t)-u_{j}(t)|\leq 2|u|_{\infty}\leq
2k_{\infty}\parallel u\parallel$

$\displaystyle\int_{0}^{1}\langle\nabla_{u}V(u),u\rangle dt$
$\displaystyle=\int_{0}^{1}\underset{1\leq i<j\leq N}{\sum}\langle\nabla
V_{ij}(u_{i}-u_{j}),(u_{i}-u_{j})\rangle dt$
$\displaystyle\leq\int_{0}^{1}\sum_{1\leq i<j\leq N}c|u_{i}-u_{j}|^{\theta}dt$
$\displaystyle\leq c\dfrac{N(N-1)}{2}(2k_{\infty}\parallel
u\parallel)^{\theta}$ (3.7)

Since

$\displaystyle\parallel f^{\prime}(u)\parallel\parallel u\parallel$
$\displaystyle\geq\langle f^{\prime}(u),u\rangle$
$\displaystyle=2f(u)-\dfrac{1}{2}\parallel
u\parallel^{2}\int_{0}^{1}\langle\nabla_{u}(u),u\rangle dt$ (3.8)

Then by $(3.7)$ and $(3.8)$, we have

$\parallel
f^{\prime}(u)\parallel\geq[\,2\gamma-N(N-1)c2^{\theta-2}k_{\infty}^{\theta}\,]\parallel
u\parallel^{\theta+1}$ $None$

Since $\parallel g^{\prime}(u)\parallel=\gamma(\theta+2)\parallel
u\parallel^{\theta+1}$, from our choice of $\gamma$, we have

$\beta\parallel f^{\prime}(u)\parallel-\parallel g^{\prime}(u)\parallel\geq
0,\forall u\in\\{f=g\geq\lambda_{0}\\}$ $None$

That is, $(vi)$ holds. ∎

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