# A prediction of $D^{*}$-multi-$\rho$ states

C. W. Xiao 1, M. Bayar 1, 2 and E. Oset 1 1Departamento de Física Teórica and
IFIC, Centro Mixto Universidad
de Valencia-CSIC, Institutos de Investigación de Paterna, Apartado 22085,
46071 Valencia, Spain
2 Department of Physics, Kocaeli University, 41380 Izmit, Turkey

###### Abstract

We present a study of the many-body interaction between a $D^{*}$ and
multi-$\rho$. We use an extrapolation to SU(4) of the hidden gauge formalism,
which produced dynamically the resonances $f_{2}(1270)$ in the $\rho\rho$
interaction and $D^{*}_{2}(2460)$ in the $\rho D^{*}$ interaction. Then let a
third particle, $\rho$, $D^{*}$, or a resonance collide with them, evaluating
the scattering amplitudes in terms of the Fixed Center Approximation of the
Faddeev equations. We find several clear resonant structures above
$2800\textrm{ MeV}$ in the multibody scattering amplitudes. They would
correspond to new charmed resonances, $D^{*}_{3}$, $D^{*}_{4}$, $D^{*}_{5}$
and $D^{*}_{6}$, which are not yet listed in the PDG, which would be analogous
to the $\rho_{3}(1690)$, $f_{4}(2050)$, $\rho_{5}(2350)$, $f_{6}(2510)$ and
$K^{*}_{3}(1780)$, $K^{*}_{4}(2045)$, $K^{*}_{5}(2380)$ described before as
multi-$\rho$ and $K^{*}$-multi-$\rho$ states respectively.

##  I Introduction

One of the important aims in the study of the strong interaction is to
understand the nature and structure of hadronic resonances. The search for new
resonances is another goal both in theories and experiments. At low energy,
using the input of chiral Lagrangians Gasser:1984gg ; Meissner:1993ah ;
Pich:1995bw ; Ecker:1994gg ; Bernard:1995dp  and implementing unitarity in
coupled channels, one develops a theoretical tool, chiral unitarity theory,
which explains the two-body interaction very successfully Kaiser:1995eg ;
Oller:1997ti ; Oset:1997it ; Oller:1998hw ; Oller:1998zr ; Oller:2000fj ;
Jido:2003cb ; Guo:2006fu ; Guo2006wp ; GarciaRecio:2002td ; GarciaRecio:2005hy
; Hyodo:2002pk . For the three-body interaction, the pioneer work of Ref.
alberone  combined Faddeev equations and chiral dynamics and reported several
S-wave $J^{P}=\frac{1}{2}^{+}$ resonances which qualify as two mesons-one
baryon molecular states. One more step, using the Fixed Center Approximation
Faddeev:1960su ; Barrett:1999cw ; Deloff:1999gc ; Kamalov:2000iy  to Faddeev
equations for multi-$\rho(770)$ states, was given in Ref. Roca2010  in which
the resonances $f_{2}(1270)$, $\rho_{3}(1690)(3^{--})$, $f_{4}(2050)(4^{++})$,
$\rho_{5}(2350)(5^{--})$, and $f_{6}(2510)(6^{++})$ were explained as
basically molecules of an increasing number of $\rho(770)$ particles with
parallel spins. Similarly, it was also found in Ref. Yamagata2010  that the
resonances $K^{*}_{2}(1430)$, $K^{*}_{3}(1780)$, $K^{*}_{4}(2045)$,
$K^{*}_{5}(2380)$ and a new $K^{*}_{6}$ could be understood as molecules made
of an increasing number of $\rho(770)$ and one $K^{*}(892)$ meson. The
$\Delta_{\frac{5}{2}^{+}}(2000)$ puzzle was explained in Ref. Xie:2011uw  with
a $\pi-(\Delta\rho)$ system with the same method Xie:2010ig ; Bayar:2011qj ;
Xiao:2011rc ; Bayar:2012rk .

The Fixed Center Approximation to Faddeev equations is technically very simple
and fairly accurate when dealing with bound states, as discussed in Ref.
Bayar:2011qj . It should also be avoided when dealing with states which have
enough energy to excite its components in intermediate states
MartinezTorres:2010ax . It should be noted that the results of the Fixed
Centre Approximation to Faddeev equations have been confirmed by the
variational method approach with the effective one-channel Hamiltonian in a
more recent work Bayar:2012dd  which predicts a narrow $DNN$ quasi-bound state
in the energy range of about $3500\textrm{ MeV}$.

In our present work, the main ideas of Refs. Roca2010 ; Yamagata2010  are
followed to search $D^{*}$-multi-$\rho$ resonances in the charm sector. Using
effective Lagrangians of the local hidden gauge theory hidden1 ; hidden2 ;
hidden3 ; hidden4 , the $\rho\rho$ interaction was studied in Ref.
Molina:2008jw  with on-shell factorized Bethe-Salpeter equations. It was found
that the $\rho\rho$ interaction was attractive in the isospin zero, spin 0 and
2 channels, particularly in the tensor channel where it led to the formation
of a $\rho\rho$ quasibound state or molecule that could be associated to the
$f_{2}(1270)$ ( $I(J^{PC})=0(2^{++})$ ). With the same formalism, the
composite systems of light ($\rho$ and $\omega$) and heavy ($D^{*}$) vector
mesons were studied in Ref. Molina:2009eb . In that work, a strong attraction
was found in the isospin, spin channels $(I,S)=(1/2,0),(1/2,1)$ and $(1/2,2)$,
with positive parity, leading to bound $\rho(\omega)D^{*}$ states, one of them
identified as the $D^{*}_{2}(2460)\;(\;I(J^{P})=\frac{1}{2}(1^{-})\;)$.
Therefore, the resonance $D^{*}_{2}(2460)$ was generated as a $\rho D^{*}$
quasibound state or molecule by the strong and attractive $\rho D^{*}$
interaction. As discussed in Ref. Roca2010 , because of the large binding
energy per $\rho$ meson in spin 2, it is possible to obtain bound systems with
several $\rho$ mesons as building blocks. As mentioned in Ref. Molina:2009eb ,
the $\rho D^{*}$ interaction is also very strong and can bind the system. Thus
the main aim in the present work is, first, to study the three-body
interaction of $D^{*}$ and two $\rho$ mesons, for which we have two options,
the clusters $D^{*}-f_{2}(1270)(\rho\rho)$ and $\rho-D^{*}_{2}(2460)(\rho
D^{*})$, in order to see if there are some resonance structures in the
scattering amplitudes. If this is the case, one can predict a not-yet-
discovered $D^{*}_{3}$ resonance, and we could continue our study extending
these ideas to include more $\rho$ mesons as building blocks of the many-body
system. Then we repeat the test in the four-body system and so forth. In
analogy to the $K^{*}$-multi-$\rho$ systems Yamagata2010 , we find clear peaks
in the scattering amplitudes for system with increasing number of $\rho$
mesons. Thus we predict new resonances in our work which are basically
quasibound states or molecules made of an increasing number of $\rho(770)$
mesons and one $D^{*}$ meson which are not yet reported in the list of the PDG
pdg2012 .

##  II $\rho\rho$ and $\rho D^{*}$ two-body interactions

To evaluate the Faddeev equations under the Fixed Centre Approximation, we
need to define the two-body cluster and then let the third particle collide
with the cluster. Thus the starting point of our work is the two-body
$\rho\rho$ and $\rho D^{*}$ interactions, which were studied in Refs.
Molina:2008jw  and Molina:2009eb  with the local hidden gauge formalism
hidden1 ; hidden2 ; hidden3 ; hidden4  and the unitary coupled channels method
Kaiser:1995eg ; Oller:1997ti ; Oset:1997it ; Oller:1998hw ; Oller:1998zr ;
Oller:2000fj ; Jido:2003cb ; Guo:2006fu ; GarciaRecio:2002td ;
GarciaRecio:2005hy ; Hyodo:2002pk . We briefly summarize the model of Refs.
Molina:2008jw  and Molina:2009eb  here to explain how to obtain the unitarized
$\rho\rho$ and $\rho D^{*}$ scattering amplitudes.

To evaluate the $\rho\rho$ and $\rho D^{*}$ scattering amplitudes with the
coupled channels unitary approach, the Bethe-Salpeter equation in coupled
channels is used:

$t=[1-VG]^{-1}V,$ (1)

where the kernel $V$ is a matrix of the interaction potentials in each
channel, which is calculated from the hidden gauge Lagrangian. $G$ is a
diagonal matrix of the loop function of every channel. The details can be seen
in Refs. Molina:2008jw ; Molina:2009eb .

To construct the three-body system we start from the clusters $f_{2}(1270)$ (
$I(J^{PC})=0(2^{++})$ ) and
$D^{*}_{2}(2460)\;(\;I(J^{P})=\frac{1}{2}(1^{-})\;)$ and add to them a $D^{*}$
or a $\rho$ respectively. The new particles are introduced with their spins
aligned with that of the cluster such that the total spin adds one unity. Thus
we only need to take into account the potential of spin $S=2$ for $\rho\rho$
and $\rho D^{*}$ interactions,

$\displaystyle V^{(I=0,S=2)}_{\rho\rho}(s)$ $\displaystyle=$
$\displaystyle-4g^{2}-8g^{2}\Big{(}\frac{3s}{4m_{\rho}^{2}}-1\Big{)},$ (2)
$\displaystyle V^{(I=2,S=2)}_{\rho\rho}(s)$ $\displaystyle=$ $\displaystyle
2g^{2}+4g^{2}\Big{(}\frac{3s}{4m_{\rho}^{2}}-1\Big{)},$ (3) $\displaystyle
V^{(I=1/2,S=2)}_{\rho D^{*}(11)}(s)$ $\displaystyle=$
$\displaystyle-\frac{5}{2}g^{2}-2\frac{g^{2}}{m_{\rho}^{2}}(k_{1}+k_{3})\cdot(k_{2}+k_{4})-\frac{1}{2}\frac{\kappa
g^{2}}{m_{\rho}^{2}}(k_{1}+k_{4})\cdot(k_{2}+k_{3}),$ (4) $\displaystyle
V^{(I=1/2,S=2)}_{\rho D^{*}(12)}(s)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{2}g^{2}+\frac{\sqrt{3}}{2}\frac{\kappa
g^{2}}{m_{\rho}^{2}}(k_{1}+k_{4})\cdot(k_{2}+k_{3}),$ (5) $\displaystyle
V^{(I=1/2,S=2)}_{\rho D^{*}(22)}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{2}g^{2}+\frac{1}{2}\frac{\kappa
g^{2}}{m_{\rho}^{2}}(k_{1}+k_{4})\cdot(k_{2}+k_{3}),$ (6) $\displaystyle
V^{(I=3/2,S=2)}_{\rho D^{*}}(s)$ $\displaystyle=$ $\displaystyle
2g^{2}+\frac{g^{2}}{m_{\rho}^{2}}(k_{1}+k_{3})\cdot(k_{2}+k_{4})+\frac{\kappa
g^{2}}{m_{\rho}^{2}}(k_{1}+k_{4})\cdot(k_{2}+k_{3}),$ (7)

where $g=M_{V}/2f_{\pi}$ with $M_{V}$ the vector meson mass and $f_{\pi}$ the
pion decay constant. In these equations $k_{i},\;i=1,2,3,4$ are the initial,
$(1,2)$, and final $(3,4)$ momenta of the particles. The quantity
$\kappa=m_{\rho}^{2}/m^{2}_{D^{*}}$ appears because in some transitions one is
exchanging a heavy vector instead of a light one. Note that in isospin $I=1/2$
there are two coupled channels, 1 is $\rho D^{*}$ and 2 is $\omega D^{*}$.

As mentioned in Refs. Molina:2008jw  and Molina:2009eb , we also should take
into account the contribution of the box diagram with two pseudoscalar mesons
in the intermediate state. We only add the imaginary part of the box diagram
contribution to the potential $V$, which is not accounted for by the coupled
channels Wu:2010jy ; Wu:2010vk , and neglect the real part which is very
small. We finally note that we take into account the $\rho$ mass distribution
by replacing the $G$ function by its convoluted form by the mass distribution
of the $\rho$ mesons in the loop.

Finally, considering the box diagram contribution to the potential $V$ and the
convolution of the $\rho$ mass distribution in the loop function $G$, we show
the evaluated results of $\rho\rho$ and $\rho D^{*}$ scattering amplitudes in
Fig. 1, which are consistent with Refs. Molina:2008jw  and Molina:2009eb . The
structure of the resonances $f_{2}(1270)$ and $D^{*}_{2}(2460)$ are clear in
the peak of the modulus squared of the amplitudes. The nonresonant amplitudes
$t_{\rho\rho}^{(I=0,S=2)}$ and $t_{\rho D^{*}}^{(I=3/2,S=2)}$ are not shown
here.

Figure 1: Modulus squared of the scattering amplitudes. Left:
$|t_{\rho\rho}^{I=0}|^{2},\;f_{2}(1270)$; Right: $|t_{\rho
D^{*}}^{I=1/2}|^{2},\;D^{*}_{2}(2460)$.

##  III Multi-body interaction formalism

The Faddeev equations under the Fixed Centre Approximation are an effective
tool to deal with multi-hadron interaction Roca2010 ; Yamagata2010 ;
Xie:2011uw ; Xie:2010ig ; Bayar:2011qj ; Xiao:2011rc ; Bayar:2012rk . They are
particularly suited to study systems in which a pair of particles cluster
together and the cluster is not much modified by the third particle. The Fixed
Centre Approximation to Faddeev equations assumes a pair of particles (1 and
2) forming a cluster. Then particle 3 interacts with the components of the
cluster, undergoing all possible multiple scattering with those components.
This is depicted in Fig. 2.

Figure 2: Diagrammatic representation of the FCA to Faddeev equations.

With this basic idea of the Fixed Centre Approximation, we could write the
Faddeev equations easily. For this one defines two partition functions
$T_{1}$, $T_{2}$ which sum all diagrams of the series of Fig. 2 which begin
with the interaction of particle 3 with particle 1 of the cluster ($T_{1}$),
or with the particle 2 ($T_{2}$). The equations then read

$\displaystyle T_{1}$ $\displaystyle=t_{1}+t_{1}G_{0}T_{2},$ (8)
$\displaystyle T_{2}$ $\displaystyle=t_{2}+t_{2}G_{0}T_{1},$ (9)
$\displaystyle T$ $\displaystyle=T_{1}+T_{2},$ (10)

where $T$ is the total three-body scattering amplitude that we are looking
for. The amplitudes $t_{1}$ and $t_{2}$ represent the unitary scattering
amplitudes with coupled channels for the interactions of particle 3 with
particle 1 and 2, respectively. Besides, $G_{0}$ is the propagator of particle
3 between the components of the two-body system.

For the unitary amplitudes corresponding to single-scattering contribution,
one must take into account the isospin structure of the cluster and write the
$t_{1}$ and $t_{2}$ amplitudes in terms of the isospin amplitudes of the (3,1)
and (3,2) systems. Details will be discussed in next section. Besides, because
of the normalization of Mandl and Shaw mandl  which has different weight
factors for the particle fields, we must take into account how these factors
appear in the single scattering and double scattering and in the total
amplitude. This is easy and is done in detail in Yamagata2010 ; Xie:2010ig .
We show below the details for the present case of a meson cluster (also
particle 1 and 2) and a meson as scattering particle (the third particle). In
this case, following the field normalization of mandl  we find for the $S$
matrix of single scattering,

$\displaystyle\begin{split}S^{(1)}_{1}=&-it_{1}\frac{1}{{\cal
V}^{2}}\frac{1}{\sqrt{2\omega_{3}}}\frac{1}{\sqrt{2\omega^{\prime}_{3}}}\frac{1}{\sqrt{2\omega_{1}}}\frac{1}{\sqrt{2\omega^{\prime}_{1}}}\\\
&\times(2\pi)^{4}\,\delta(k+k_{R}-k^{\prime}-k^{\prime}_{R}),\\\ \end{split}$
(11) $\displaystyle\begin{split}S^{(1)}_{2}=&-it_{2}\frac{1}{{\cal
V}^{2}}\frac{1}{\sqrt{2\omega_{3}}}\frac{1}{\sqrt{2\omega^{\prime}_{3}}}\frac{1}{\sqrt{2\omega_{2}}}\frac{1}{\sqrt{2\omega^{\prime}_{2}}}\\\
&\times(2\pi)^{4}\,\delta(k+k_{R}-k^{\prime}-k^{\prime}_{R}),\\\ \end{split}$
(12)

where, $k,\,k^{\prime}$ ($k_{R},\,k^{\prime}_{R}$) refer to the momentum of
initial, final scattering particle ($R$ for the cluster), $\cal V$ is the
volume of the box where the states are normalized to unity and the subscripts
1, 2 refer to scattering with particle 1 or 2 of the cluster.

The double scattering diagram, Fig. 2 (b), is given by

$\begin{split}S^{(2)}=&-i(2\pi)^{4}\delta(k+k_{R}-k^{\prime}-k^{\prime}_{R})\frac{1}{{\cal
V}^{2}}\frac{1}{\sqrt{2\omega_{3}}}\frac{1}{\sqrt{2\omega^{\prime}_{3}}}\frac{1}{\sqrt{2\omega_{1}}}\frac{1}{\sqrt{2\omega^{\prime}_{1}}}\frac{1}{\sqrt{2\omega_{2}}}\frac{1}{\sqrt{2\omega^{\prime}_{2}}}\\\
&\times\int\frac{d^{3}q}{(2\pi)^{3}}F_{R}(q)\frac{1}{{q^{0}}^{2}-\vec{q}\,^{2}-m_{3}^{2}+i\,\epsilon}t_{1}t_{2},\end{split}$
(13)

where $F_{R}(q)$ is the cluster form factor that we shall discuss below.
Similarly the full $S$ matrix for scattering of particle 3 with the cluster
will be given by

$\begin{split}S=&-i\,T\,(2\pi)^{4}\delta(k+k_{R}-k^{\prime}-k^{\prime}_{R})\frac{1}{{\cal
V}^{2}}\\\
&\times\frac{1}{\sqrt{2\omega_{3}}}\frac{1}{\sqrt{2\omega^{\prime}_{3}}}\frac{1}{\sqrt{2\omega_{R}}}\frac{1}{\sqrt{2\omega^{\prime}_{R}}}.\end{split}$
(14)

In view of the different normalization of these terms by comparing Eqs. (11),
(12), (13) and (14), we can introduce suitable factors in the elementary
amplitudes,

$\tilde{t_{1}}=\frac{2m_{R}}{2m_{1}}~{}t_{1},~{}~{}~{}~{}\tilde{t_{2}}=\frac{2m_{R}}{2m_{2}}~{}t_{2},$
(15)

where we have taken the approximations, suitable for bound states,
$\frac{1}{\sqrt{2\omega_{i}}}=\frac{1}{\sqrt{2m_{i}}}$, and sum all the
diagrams by means of

$T=T_{1}+T_{2}=\frac{\tilde{t_{1}}+\tilde{t_{2}}+2~{}\tilde{t_{1}}~{}\tilde{t_{2}}~{}G_{0}}{1-\tilde{t_{1}}~{}\tilde{t_{2}}~{}G_{0}^{2}}.$
(16)

The function $G_{0}$ in Eq. (16) is given by

$G_{0}(s)=\int\frac{d^{3}\vec{q}}{(2\pi)^{3}}F_{R}(q)\frac{1}{q^{02}-\vec{q}^{~{}2}-m_{3}^{2}+i\,\epsilon}.$
(17)

where $F_{R}(q)$ is the form factor of the cluster of particles 1 and 2. We
must use the form factor of the cluster consistently with the theory used to
generate the cluster as a dynamically generated resonance. This requires to
extend into wave functions the formalism of the chiral unitary approach
developed for scattering amplitudes. This work has been done in gamerjuan ;
yamajuan ; Aceti:2012dd  for $s$-wave bound states, $s$-wave resonant states
and states with arbitrary angular momentum respectively, here we are only need
the expressions for $s$-wave bound states, and then the expression for the
form factors is given in section 4 of yamajuan , which we use in the present
work and reproduce below

$\displaystyle\begin{split}F_{R}(q)&=\frac{1}{\mathcal{N}}\int_{|\vec{p}|<\Lambda^{\prime},|\vec{p}-\vec{q}|<\Lambda^{\prime}}d^{3}\vec{p}\frac{1}{2E_{1}(\vec{p})}\frac{1}{2E_{2}(\vec{p})}\frac{1}{M_{R}-E_{1}(\vec{p})-E_{2}(\vec{p})}\\\
&\quad\frac{1}{2E_{1}(\vec{p}-\vec{q})}\frac{1}{2E_{2}(\vec{p}-\vec{q})}\frac{1}{M_{R}-E_{1}(\vec{p}-\vec{q})-E_{2}(\vec{p}-\vec{q})},\end{split}$
(18) $\displaystyle\mathcal{N}$
$\displaystyle=\int_{|\vec{p}|<\Lambda^{\prime}}d^{3}\vec{p}\Big{(}\frac{1}{2E_{1}(\vec{p})}\frac{1}{2E_{2}(\vec{p})}\frac{1}{M_{R}-E_{1}(\vec{p})-E_{2}(\vec{p})}\Big{)}^{2},$
(19)

where $E_{1}$ and $E_{2}$ are the energies of the particles 1, 2 and $M_{R}$
the mass of the cluster. The parameter $\Lambda^{\prime}$ is a cut off that
regularizes the integral of Eqs. (18) and (19). This cut off is the same one
needed in the regularization of the loop function of the two particle
propagators in the study of the interaction of the two particles of the
cluster yamajuan . We take in the present work $\Lambda^{\prime}=875\textrm{
MeV}$, the same as used to generate the bound states Yamagata2010 ; Xie:2010ig
of $f_{2}(1270)$ in the two-body interaction, and
$\Lambda^{\prime}=1200\textrm{ MeV}$ for getting the $D^{*}_{2}(2460)$
cluster. Thus we do not introduce any free parameters in the present
procedure.

In addition, $q^{0}$, the energy carried by particle 3 in the rest frame of
the three particle system, is given by

$q^{0}(s)=\frac{s+m_{3}^{2}-M_{R}^{2}}{2\sqrt{s}}.$ (20)

Note also that the arguments of the amplitudes $T_{i}(s)$ and $t_{i}(s_{i})$
are different, where $s$ is the total invariant mass of the three-body system,
and $s_{i}$ are the invariant masses in the two-body systems. The value of
$s_{i}$ is given by Yamagata2010

$s_{i}=m_{3}^{2}+m_{i}^{2}+\frac{(M_{R}^{2}+m_{i}^{2}-m_{j}^{2})(s-m_{3}^{2}-M_{R}^{2})}{2M_{R}^{2}},(i,j=1,2,\;i\neq
j)$ (21)

where $m_{l},(l=1,2,3)$ are the masses of the corresponding particles in the
three-body system and $M_{R}$ the mass of two body resonance or bound state
(cluster).

##  IV Results

The $D^{*}$-multi-$\rho$ interactions that we investigate in the present work
are listed in Table 1, and are explained as follows. For the three-body
interaction, we have two options: particle $3=D^{*}$, cluster or resonance
$R=f_{2}$ (particle $1=\rho,\;2=\rho$) and $3=\rho$, $R=D^{*}_{2}$
($1=\rho,\;2=D^{*}$). For four-body, we also have two cases: $3=f_{2}$,
$R=D^{*}_{2}$ ($1=\rho,\;2=D^{*}$) and $3=D^{*}_{2}$, $R=f_{2}$
($1=\rho,\;2=\rho$). For five-body, $3=D^{*}$, $R=f_{4}$ ($1=f_{2},\;2=f_{2}$)
and $3=\rho$, $R=D^{*}_{4}$ ($1=f_{2},\;2=D^{*}_{2}$). For six-body,
$3=D^{*}_{2}$, $R=f_{4}$ ($1=f_{2},\;2=f_{2}$) and $3=f_{2}$, $R=D^{*}_{4}$
($1=f_{2},\;2=D^{*}_{2}$). We describe all these cases in detail below.

Table 1: The cases considered in the $D^{*}$-multi-$\rho$ interactions. particles: |  3 |  R (1,2) |  amplitudes
---|---|---|---
Two-body | $\rho$ | $D^{*}$ | $t_{\rho D^{*}}$
| $\rho$ | $\rho$ | $t_{\rho\rho}$
Three-body | $D^{*}$ | $f_{2}\;(\rho\rho)$ | $T_{D^{*}-f_{2}}$
| $\rho$ | $D^{*}_{2}\;(\rho D^{*})$ | $T_{\rho-D^{*}_{2}}$
Four-body | $D^{*}_{2}$ | $f_{2}\;(\rho\rho)$ | $T_{D^{*}_{2}-f_{2}}$
| $f_{2}$ | $D^{*}_{2}\;(\rho D^{*})$ | $T_{f_{2}-D^{*}_{2}}$
Five-body | $D^{*}$ | $f_{4}\;(f_{2}f_{2})$ | $T_{D^{*}-f_{4}}$
| $\rho$ | $D^{*}_{4}\;(f_{2}D^{*}_{2})$ | $T_{\rho-D^{*}_{4}}$
Six-body | $D^{*}_{2}$ | $f_{4}\;(f_{2}f_{2})$ | $T_{D^{*}_{2}-f_{4}}$
| $f_{2}$ | $D^{*}_{4}\;(f_{2}D^{*}_{2})$ | $T_{f_{2}-D^{*}_{4}}$

###  IV.1 Three-body interaction

For three-body interaction, we have two options of structure:
$D^{*}-f_{2}(\rho\rho)$ and $\rho-D^{*}_{2}(\rho D^{*})$, which means
$3=D^{*}$, $R=f_{2}$ ($1=\rho,\;2=\rho$) and $3=\rho$, $R=D^{*}_{2}$
($1=\rho,\;2=D^{*}$). Thus, to evaluate these scattering amplitudes, we need
as input the $t_{1}$ and $t_{2}$ amplitudes of the (3,1) and (3,2) systems,
$t_{1}=t_{2}=t_{\rho D^{*}}$ for $D^{*}-f_{2}(\rho\rho)$ and
$t_{1}=t_{\rho\rho},\;t_{2}=t_{\rho D^{*}}$ for $\rho-D^{*}_{2}(\rho D^{*})$.
We should calculate the two-body $\rho\rho$ and $\rho D^{*}$ amplitudes.

As mentioned before, the isospin structure of the cluster should be considered
for the $t_{1}$ and $t_{2}$ amplitudes. For the case of
$D^{*}-f_{2}(\rho\rho)$, the cluster of $f_{2}$ has isospin $I=0$. Therefore
the two $\rho$ mesons are in an $I=0$ state, and we have

$|\rho\rho>^{(0,0)}=\frac{1}{\sqrt{3}}\Big{(}|(1,-1)>+|(-1,1)>-|(0,0)>\Big{)},$
(22)

where $|(1,-1)>$ denote $|(I_{z}^{1},I_{z}^{2})>$ which shows the $I_{z}$
components of particles 1 and 2, and $|\rho\rho>^{(0,0)}$ means
$|\rho\rho>^{(I,I_{z})}$. The third particle is a $D^{*}$ meson taken
$|I_{z}^{3})>=|\frac{1}{2}>$. Then we obtain

$\begin{split}T^{(\frac{1}{2},\frac{1}{2})}_{D^{*}-f_{2}}=&<D^{*}\rho\rho|\,\hat{t}\,|D^{*}\rho\rho>^{(\frac{1}{2},\frac{1}{2})}\\\
=&(<D^{*}|^{(\frac{1}{2},\frac{1}{2})}\otimes<\rho\rho|^{(0,0)})\,(\hat{t}_{31}+\hat{t}_{32})\,(|D^{*}>^{(\frac{1}{2},\frac{1}{2})}\otimes|\rho\rho>^{(0,0)})\\\
=&\Big{[}<\frac{1}{2}|\otimes\frac{1}{\sqrt{3}}\Big{(}<(1,-1)|+<(-1,1)|-<(0,0)|\Big{)}\Big{]}\,(\hat{t}_{31}+\hat{t}_{32})\,\Big{[}|\frac{1}{2}>\\\
&\otimes\frac{1}{\sqrt{3}}\Big{(}|(1,-1)>+|(-1,1)>-|(0,0)>\Big{)}\Big{]}\\\
=&\frac{1}{3}\Big{[}<(\frac{3}{2},\frac{3}{2}),-1|+\sqrt{\frac{1}{3}}<(\frac{3}{2},-\frac{1}{2}),1|+\sqrt{\frac{2}{3}}<(\frac{1}{2},-\frac{1}{2}),1|-\sqrt{\frac{2}{3}}<(\frac{3}{2},\frac{1}{2}),0|\\\
&-\sqrt{\frac{1}{3}}<(\frac{1}{2},\frac{1}{2}),0|\Big{]}\hat{t}_{31}\Big{[}|(\frac{3}{2},\frac{3}{2}),-1>+\sqrt{\frac{1}{3}}|(\frac{3}{2},-\frac{1}{2}),1>+\sqrt{\frac{2}{3}}|(\frac{1}{2},-\frac{1}{2}),1>\\\
&-\sqrt{\frac{2}{3}}|(\frac{3}{2},\frac{1}{2}),0>-\sqrt{\frac{1}{3}}|(\frac{1}{2},\frac{1}{2}),0>\Big{]}+\frac{1}{3}\Big{[}\sqrt{\frac{1}{3}}<(\frac{3}{2},-\frac{1}{2}),1|+\sqrt{\frac{2}{3}}<(\frac{1}{2},-\frac{1}{2}),1|\\\
&+<(\frac{3}{2},\frac{3}{2}),-1|-\sqrt{\frac{2}{3}}<(\frac{3}{2},\frac{1}{2}),0|-\sqrt{\frac{1}{3}}<(\frac{1}{2},\frac{1}{2}),0|\Big{]}\hat{t}_{32}\Big{[}\sqrt{\frac{1}{3}}|(\frac{3}{2},-\frac{1}{2}),1>\\\
&+\sqrt{\frac{2}{3}}|(\frac{1}{2},-\frac{1}{2}),1>+|(\frac{3}{2},\frac{3}{2}),-1>-\sqrt{\frac{2}{3}}|(\frac{3}{2},\frac{1}{2}),0>-\sqrt{\frac{1}{3}}|(\frac{1}{2},\frac{1}{2}),0>\Big{]},\end{split}$
(23)

where the notation of the states followed in the terms is
$|(\frac{3}{2},\frac{3}{2}),-1>\equiv|(I^{31},I_{z}^{31}),I_{z}^{2}>$ for
$t_{31}$, and $|(I^{32},I_{z}^{32}),I_{z}^{1}>$ for $t_{32}$. Then we find

$t_{1}=t_{\rho
D^{*}}=\frac{1}{3}\big{(}2t_{31}^{I=3/2}+t_{31}^{I=1/2}\big{)},\quad
t_{2}=t_{1}.$ (24)

But for the case of $\rho-D^{*}_{2}(\rho D^{*})$, the situation is different.
Because the isospins of $\rho$ and $D^{*}_{2}$ are $I_{\rho}=1$ and
$I_{D^{*}_{2}}=\frac{1}{2}$, the total isospin of the three-body system are
$I_{total}\equiv I_{\rho\rho D^{*}}=\frac{1}{2}$ or $I_{total}\equiv
I_{\rho\rho D^{*}}=\frac{3}{2}$, and then we have

$\begin{split}&|\rho D^{*}_{2}>^{(\frac{1}{2},\frac{1}{2})}=|\rho\rho
D^{*}>^{(\frac{1}{2},\frac{1}{2})}=\sqrt{\frac{2}{3}}|(1,-\frac{1}{2})>-\sqrt{\frac{1}{3}}|(0,\frac{1}{2})>,\\\
&|\rho D^{*}_{2}>^{(\frac{3}{2},\frac{1}{2})}=|\rho\rho
D^{*}>^{(\frac{3}{2},\frac{1}{2})}=\sqrt{\frac{1}{3}}|(1,-\frac{1}{2})>+\sqrt{\frac{2}{3}}|(0,\frac{1}{2})>,\end{split}$
(25)

where we have taken $I_{z}=\frac{1}{2}$ for convenience. Therefore the $|\rho
D^{*}>$ states inside the $D^{*}_{2}$ for the $I_{z}=-\frac{1}{2}$ and
$I_{z}=+\frac{1}{2}$ are given by

$\begin{split}&|\rho
D^{*}>^{(\frac{1}{2},-\frac{1}{2})}=\sqrt{\frac{1}{3}}|(0,-\frac{1}{2})>-\sqrt{\frac{2}{3}}|(-1,\frac{1}{2})>,\\\
&|\rho
D^{*}>^{(\frac{1}{2},\frac{1}{2})}=\sqrt{\frac{2}{3}}|(1,-\frac{1}{2})>-\sqrt{\frac{1}{3}}|(0,\frac{1}{2})>.\end{split}$
(26)

For the two possibilities, combining Eqs. (25) and (26) and performing a
similar derivation of Eq. (23), we obtain

$\begin{split}&T_{\rho-D^{*}_{2}}^{(I=1/2)}:\quad
t_{1}=t_{\rho\rho}=\frac{2}{3}t_{31}^{(I=0)},\quad t_{2}=t_{\rho
D^{*}}=\frac{1}{9}\big{(}8t_{32}^{I=3/2}+t_{32}^{I=1/2}\big{)};\\\
&T_{\rho-D^{*}_{2}}^{(I=3/2)}:\quad
t_{1}=t_{\rho\rho}=\frac{5}{6}t_{31}^{(I=2)},\quad t_{2}=t_{\rho
D^{*}}=\frac{1}{9}\big{(}5t_{32}^{I=3/2}+4t_{32}^{I=1/2}\big{)}.\end{split}$
(27)

We show our results in Fig. 3. In Fig. 3 (left) we show the modulus squared of
the amplitudes for $|T_{D^{*}-f_{2}}^{I=1/2}|^{2}$ and
$|T_{\rho-D^{*}_{2}}^{I=1/2}|^{2}$, and we find that there are clear peaks
around the energy $2800-2850\textrm{ MeV}$ which is about $400\textrm{ MeV}$
lower than the $D^{*}-f_{2}$ threshold. The bindings are large because they
scale with the mass of the mesons and we have now a $D^{*}$ interacting with
two $\rho$ mesons. The strength of the peak of $|T_{D^{*}-f_{2}}^{I=1/2}|^{2}$
is two times bigger than for $|T_{\rho-D^{*}_{2}}^{I=1/2}|^{2}$, and we see
that the $D^{*}-f_{2}$ component is a bit more bound than the $\rho-D^{*}_{2}$
one. We expect that a real state would be an admixture of both with a binding
in between that of the individual components. In Fig. 3 (right) we show
$|T_{\rho-D^{*}_{2}}^{I=3/2}|^{2}$, and there is a clear resonant structure
about $3120\textrm{ MeV}$, the strength of which is 30 times smaller than that
of $|T_{\rho-D^{*}_{2}}^{I=1/2}|^{2}$ in the left figure and less bound. We
are concerned with the lowest lying states and hence we concentrate on the
predicted new $D^{*}_{3}$ state with a structure formed by a mixture of
$D^{*}-f_{2}$ and $\rho-D^{*}_{2}$, with a mass about $2800-2850\textrm{ MeV}$
and a width about $60-100\textrm{ MeV}$.

Figure 3: Modulus squared of the $T_{D^{*}-f2}$ and $T_{\rho-D^{*}_{2}}$
scattering amplitudes. Left: $I_{total}=\frac{1}{2}$; Right:
$I_{total}=\frac{3}{2}$.

###  IV.2 Four-body interaction

There are also two possibilities in the four-body interaction as we have shown
in Table 1: particle $3=f_{2}$, $R=D^{*}_{2}$ ($1=\rho,\;2=D^{*}$) or particle
$3=D^{*}_{2}$, $R=f_{2}$ ($1=\rho,\;2=\rho$). Because $I_{f_{2}}=0$ and
$I_{D^{*}_{2}}=\frac{1}{2}$, the total isospin of the four-body system is only
$I_{total}=\frac{1}{2}$. In the first case, $f_{2}$ collides with the
$D^{*}_{2}$, the amplitudes $t_{1}=t_{f_{2}\rho}=T_{\rho-f_{2}}$ has been
evaluated in Ref. Roca2010  and is reproduced in our work, and
$t_{2}=t_{f_{2}D^{*}}=T_{D^{*}-f_{2}}$ which has been evaluated in the former
subsection IV.1. For the second case, $D^{*}_{2}$ collides with the $f_{2}$,
and the amplitudes $t_{1}=t_{2}=t_{D^{*}_{2}\rho}=T_{\rho-D^{*}_{2}}$ have
been evaluated in the former subsection IV.1. We must now consider that the
three-body amplitude $T_{\rho-D^{*}_{2}}$ is also combined with different
isospins as mentioned in subsection IV.1. This situation is similar to the
case when the $D^{*}$ collides with the $f_{2}$, because the isospins of both
the $D^{*}_{2}$ and $D^{*}$ are $I=\frac{1}{2}$, thus from Eq. (24) we have

$t_{1}=T_{\rho
D^{*}_{2}}=\frac{1}{3}\big{(}2T_{31}^{I=3/2}+T_{31}^{I=1/2}\big{)},\quad
t_{2}=t_{1}.$ (28)

The results are shown in Fig. 4. The left of Fig. 4 is
$|T_{D^{*}_{2}-f_{2}}^{I=1/2}|^{2}$. We find that there is a clear peak at an
energy of $3200\textrm{ MeV}$, the width of which is about $200\textrm{ MeV}$.
The right of Fig. 4 shows $|T_{f_{2}-D^{*}_{2}}^{I=1/2}|^{2}$ and there is a
resonant peak around the energy $3075\textrm{ MeV}$ with a large width of
nearly $400\textrm{ MeV}$. The strength of the peak of
$|T_{f_{2}-D^{*}_{2}}^{I=1/2}|^{2}$ is about two times bigger than the one of
$|T_{D^{*}_{2}-f_{2}}^{I=1/2}|^{2}$ and the energy of the peak is more bound
too. But from the former results, subsection IV.1, we found that
$|T_{D^{*}_{2}-f_{2}}^{I=1/2}|^{2}$ has more strength and is more bound than
$|T_{f_{2}-D^{*}_{2}}^{I=1/2}|^{2}$. We have investigated that this is because
of the contribution of $|T_{\rho-D^{*}_{2}}^{I=3/2}|^{2}$, even though the
strength of $|T_{\rho-D^{*}_{2}}^{I=3/2}|^{2}$ is much smaller and less bound
than the one of $|T_{\rho-D^{*}_{2}}^{I=1/2}|^{2}$ from the former results.
When we removed the contribution of $|T_{\rho-D^{*}_{2}}^{I=3/2}|^{2}$ in Eq.
(28), the strength of the peak was enhanced by a factor five and was more
bound. Therefore, within the uncertainty of the theory, we find a new
$D^{*}_{4}$ resonance, of a mass about $3075-3200\textrm{ MeV}$ and a width
about $200-400\textrm{ MeV}$.

Figure 4: Modulus squared of the $T_{D^{*}_{2}-f2}$ (left) and
$T_{f_{2}-D^{*}_{2}}$ (right) scattering amplitudes.

###  IV.3 Five-body interaction

For the five-body interaction, we also have two options for the cluster, one
of which is the particle $f_{4}$ studied in Ref. Roca2010  and the other one
the resonance $D^{*}_{4}$ obtained in the four-body interaction, subsection
IV.2. Thus letting the third particle ($D^{*}$ or $\rho$) collide with them,
we have $3=D^{*}$, $R=f_{4}$ ($1=f_{2},\;2=f_{2}$) or $3=\rho$, $R=D^{*}_{4}$
($1=f_{2},\;2=D^{*}_{2}$). Because the isospin $I_{f_{4}}=0$ and
$I_{D^{*}_{4}}=\frac{1}{2}$, the total isospin of the five-body system is only
$I_{total}=\frac{1}{2}$ in the $D^{*}-f_{4}$ structure, but
$I_{total}=\frac{1}{2}$ or $I_{total}=\frac{3}{2}$ in the $\rho-D^{*}_{4}$
structure. Thus the situation is similar to the three-body interaction
discussed before, $D^{*}$ (or $\rho$) collide with $f_{2}$ (or $D^{*}_{2}$).
Therefore in the first case, the $D^{*}$ collides with the $f_{4}$, and the
amplitudes $t_{1}=t_{2}=t_{D^{*}f_{2}}=T_{D^{*}-f_{2}}^{(I=1/2)}$ have been
evaluated in subsection IV.1. For the second case, the $\rho$ collides with
the $D^{*}_{4}$, which is similar to $\rho-D^{*}_{2}$ in the three-body
interaction, thus, after doing a similar derivation as in Eq. (23), we have

$\begin{split}&T_{\rho-D^{*}_{4}}^{(I=1/2)}:\quad t_{1}=t_{\rho
f_{2}}=T_{31}^{(I=1)},\quad t_{2}=t_{\rho D^{*}_{2}}=T_{32}^{I=1/2};\\\
&T_{\rho-D^{*}_{4}}^{(I=3/2)}:\quad t_{1}=t_{\rho f_{2}}=T_{31}^{(I=1)},\quad
t_{2}=t_{\rho D^{*}_{2}}=T_{32}^{I=3/2},\end{split}$ (29)

where the $T_{31}^{(I=1)}$ is the same as $T_{\rho-f_{2}}$ in the subsection
IV.2 reproducing the results of Ref. Roca2010 , and
$T_{\rho-D^{*}_{2}}^{I=1/2}$ and $T_{\rho-D^{*}_{2}}^{I=3/2}$ have also been
evaluated in subsection IV.1.

In Fig. 5 we show our results. The left of Fig. 5 is
$|T_{D^{*}-f_{4}}^{I=1/2}|^{2}$ and we observe a resonant peak around the
energy $3375\textrm{ MeV}$ with a width of less than $200\textrm{ MeV}$. The
right of Fig. 5 is $|T_{\rho-D^{*}_{4}}^{I=1/2}|^{2}$ and
$|T_{\rho-D^{*}_{4}}^{I=3/2}|^{2}$. We find that there is a resonant structure
in $|T_{\rho-D^{*}_{4}}^{I=1/2}|^{2}$ at the energy $3360\textrm{ MeV}$, the
width of which is about $400\textrm{ MeV}$, and the position is very close to
the one of $|T_{D^{*}-f_{4}}^{I=1/2}|^{2}$. But the strength of
$|T_{\rho-D^{*}_{4}}^{I=1/2}|^{2}$ is one order smaller than the one
$|T_{D^{*}-f_{4}}^{I=1/2}|^{2}$. For $|T_{\rho-D^{*}_{4}}^{I=3/2}|^{2}$ there
is no resonant structure. Therefore, within uncertainties, we also find a new
$D^{*}_{5}$ resonance, with a mass about $3360-3375\textrm{ MeV}$ and a width
about $200-400\textrm{ MeV}$.

Figure 5: Modulus squared of the $T_{D^{*}-f_{4}}$ (left) and
$T_{\rho-D^{*}_{4}}$ (right) scattering amplitudes.

###  IV.4 Six-body interaction

Similarly to the five-body interaction, we also have two options of the
cluster for the six-body interaction, the particle $f_{4}$ studied in Ref.
Roca2010  and the resonance $D^{*}_{4}$ obtained in subsection IV.2. Now
letting a resonance ($D^{*}_{2}$ or $f_{2}$) be the third particle and collide
with them, we have $3=D^{*}_{2}$, $R=f_{4}$ ($1=f_{2},\;2=f_{2}$) or
$3=f_{2}$, $R=D^{*}_{4}$ ($1=f_{2},\;2=D^{*}_{2}$). Because
$I_{f_{2}}=I_{f_{4}}=0$ and $I_{D^{*}_{2}}=I_{D^{*}_{4}}=\frac{1}{2}$, the
total isospin of the six-body system is only $I_{total}=\frac{1}{2}$. Thus, in
the first case, the $D^{*}_{2}$ collides with the $f_{4}$, the amplitudes
$t_{1}=t_{2}=t_{D^{*}_{2}f_{2}}=T_{D^{*}_{2}-f_{2}}^{(I=1/2)}$ have been
evaluated in subsection IV.2. For the second case, the $f_{2}$ collides with
the $D^{*}_{4}$, the amplitudes $t_{1}=t_{f_{2}f_{2}}=T_{f_{2}-f_{2}}$
reproduce the results from Ref. Roca2010 , and
$t_{2}=t_{f_{2}D^{*}_{2}}=T_{f_{2}-D^{*}_{2}}$ has been calculated in
subsection IV.2.

Our results are shown in Fig. 6. The left of Fig. 6 is
$|T_{D^{*}_{2}-f_{4}}^{I=1/2}|^{2}$ where we see a peak around the energy
$3775\textrm{ MeV}$ with a large width of nearly $400\textrm{ MeV}$. The right
of Fig. 6 is $|T_{f_{2}-D^{*}_{4}}^{I=1/2}|^{2}$, and there we find that there
is not a clear peak at the energy $3550\textrm{ MeV}$. It looks like the
resonant structure of $f_{2}-D^{*}_{4}$ is not as stable as the
$D^{*}_{2}-f_{4}$ one. From these results, we could predict a new $D^{*}_{6}$
resonance with more uncertainty, with a mass of about $3775\textrm{ MeV}$ and
a width about $400\textrm{ MeV}$.

Figure 6: Modulus squared of the $T_{D^{*}_{2}-f_{4}}$ (left) and
$T_{f_{2}-D^{*}_{4}}$ (right) scattering amplitudes.

##  V Conclusions

In the present work, we show the results of our investigation of the
$D^{*}$-multi-$\rho$ systems. Our idea is based on the fact that the two-body
interactions of $\rho\rho$ and $\rho D^{*}$ in spin $S=2$ are so strong as to
bind the particles forming the resonances$f_{2}(1270)$ Molina:2008jw  and
$D^{*}_{2}(2460)$ Molina:2009eb  respectively. So we could study the many-body
$D^{*}$-multi-$\rho$ systems in an iterative way looking at the structure of
the amplitudes and observing clear peaks that become wider as the number of
$\rho$ mesons increase. The work proceeded analogously to the study of
multi-$\rho$ system in Roca2010 , where the $\rho_{3}(1690)(3^{--})$,
$f_{4}(2050)(4^{++})$, $\rho_{5}(2350)(5^{--})$, and $f_{6}(2510)(6^{++})$
were described as basically molecules of multi-$\rho(770)$ states, and
similarly to the work of Yamagata2010  where the $K^{*}_{2}(1430)$,
$K^{*}_{3}(1780)$, $K^{*}_{4}(2045)$, $K^{*}_{5}(2380)$ and $K^{*}_{6}$ could
be interpreted as molecules made of one $K^{*}(892)$ meson and an increasing
number of $\rho(770)$ mesons. The $D^{*}$-multi-$\rho$ states with spins
aligned combined to give some new charmed resonances, $D^{*}_{3}$,
$D^{*}_{4}$, $D^{*}_{5}$ and $D^{*}_{6}$, which are basically made of one
$D^{*}$ meson and an increasing number of $\rho(770)$ mesons and are not found
in the list of PDG pdg2012 . Their masses are predicted around
$2800-2850\textrm{ MeV}$, $3075-3200\textrm{ MeV}$, $3360-3375\textrm{ MeV}$
and $3775\textrm{ MeV}$ respectively. And their widths are about
$60-100\textrm{ MeV}$, $200-400\textrm{ MeV}$, $200-400\textrm{ MeV}$ and
$400\textrm{ MeV}$ respectively. The analogy with the states already known in
the strange and non-strange sector, together with the stronger interaction of
the $D^{*}$ mesons, make our predictions solid within the uncertainties
admitted. We are, thus, reasonably confident that such states can be found in
the future in coming facilities like FAIR and others.

## Acknowledgements

We thank R. Molina for useful discussions. This work is partly supported by
DGICYT contract number FIS2011-28853-C02-01, and the Generalitat Valenciana in
the program Prometeo, 2009/090. We acknowledge the support of the European
Community-Research Infrastructure Integrating Activity Study of Strongly
Interacting Matter (acronym HadronPhysics3, Grant Agreement n. 283286) under
the Seventh Framework Programme of the EU. One of us, M. Bayar acknowledges
support through the Scientific and Technical Research Council (TUBITAK)
BIDEP-2219 grant.

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# A Machine Learning Method for Predicting Traffic Signal Timing from Probe
Vehicle Data

Juliette Ugirumurera
National Renewable Energy Laboratory
Joseph Severino
National Renewable Energy Laboratory
Erik A. Bensen
Carnegie Mellon University
Qichao Wang
National Renewable Energy Laboratory
Jane Macfarlane
UC Berkeley

###### Abstract

Traffic signals play an important role in transportation by enabling traffic
flow management, and ensuring safety at intersections. In addition, knowing
the traffic signal phase and timing data can allow optimal vehicle routing for
time and energy efficiency, eco-driving, and the accurate simulation of
signalized road networks. In this paper, we present a machine learning (ML)
method for estimating traffic signal timing information from vehicle probe
data. To the authors best knowledge, very few works have presented ML
techniques for determining traffic signal timing parameters from vehicle probe
data. In this work, we develop an Extreme Gradient Boosting (XGBoost) model to
estimate signal cycle lengths and a neural network model to determine the
corresponding red times per phase from probe data. The green times are then be
derived from the cycle length and red times. Our results show an error of less
than 0.56 sec for cycle length, and red times predictions within 7.2 sec error
on average.

_K_ eywords Traffic Signal Timing, Machine Learning, Probe Vehicle Data

##  1 INTRODUCTION

In traffic networks, traffic signals play a key role in determining and
managing vehicular traffic flow. They control the flow of traffic, ensure
safety by regulating the flow of competing movements through intersections and
can reduce traffic congestion when optimized [1, 2]. However, they can also
induce stop-and-go traffic, which increases vehicles’ delay and fuel
consumption. In addition, not knowing the traffic signals’ timings and
switching patterns makes it very challenging to accurately determine the most
time-efficient or energy-efficient routes, to inform driving decisions to
maximize arrival on green or minimize engine idling at intersections [3], and
to correctly simulate traffic for signal-controlled regions.

Usually, traffic lights are managed by different local agencies. For example,
in the US, hundreds of agencies are in charge of the more than 320,000 traffic
lights installed [4]. This makes direct access to traffic signal timing and
phase data a very challenging task. In this paper, we present a machine
learning method that uses probe vehicle data to estimate the timings for pre-
timed traffic signals. We demonstrate this algorithm on probe data generated
from a well-calibrated microscopic simulation using the SUMO simulator[5] and
the NEMA Type controller in SUMO [6]. We focus on fixed-time traffic signals
since they represent the majority of traffic lights in the US [7].

##  2 RELATED WORKS

Probe vehicle data has enabled many important tasks in transportation
research, including estimating link-level hourly traffic volume [8], queue
length estimation [9], and traffic signal control [10]. The availability of
and opportunities to exploit probe vehicle data will only continue to grow due
to the predicted increase in connected and automated vehicle (CAV) adoption
and the increased availability of high-speed communication infrastructure.
Probe vehicle data has also been used broadly for estimating traffic signal
timings. In [3], the authors uses statistical patterns in probe data from
public buses in San Fransisco, California, USA, to estimate the cycle times,
the duration of red times and the greens start times for fixed-timed traffic
lights. Yu and Lu estimated cycle length for fixed time intersections using
low frequency taxi trajectory data[11]. They formulate the cycle length
estimation problem as a general approximate greatest common divisor (AGCD)
problem and solved it with a most frequent AGCD algorithm. The work in [12]
presents an optimization-based traffic signal state estimation algorithm on
vehicle speed profiles to determine traffic signal phase schedules. Wang and
Jiang [13] determine the cycle length and green times of signalized
intersection by analyzing the velocity-time curves of all vehicles going
through the intersection. Chuan et al. calculated cycle length and green times
using shockwave theory [14]. Finally, [15] and [16] focus on using
probabilistic prediction of traffic signal timings.

To our best knowledge, there are only a few works that study machine learning
(ML) methods to estimate signal timing parameters from probe vehicles. Unlike
mathematical methods, such as statistical, optimization and probabilistic
based-methods, which require the explicit mathematical representation of the
desired problem to solve, ML techniques can derive models from data to produce
meaningful insights and predictions[17]. Advances in sensing and communication
have lead to unprecedentedly large amounts of data, which, when combined with
ML’s spontaneous model-learning capability, explains the recent surge in ML
applications across domains from health care to self-driving cars and traffic
control. For example, the work in [18] uses a Bayesian learning approach to
determine the cycle length from historic trajectory data, but without
estimating the red or green times per phase. In [19], a Random Survival Forest
and a Long-Short-Term-Memory (LSTM) neural network are used to predict the
residual time per intersection phase; however, they do not estimate the cycle
length or the green and red phase times; additionally, these models require
vehicle detection data as input. In [20], the authors presents a deep learning
based LSTM methodology to predict actuated traffic lights’ switching time from
green to red and vice versa, but require signal timing parameters as input
rather than estimating them.

In this work, we showcase the potential of ML for estimating pre-timed traffic
signal timing by using an Extreme Gradient Boosting (XGBoost) model [21] to
determine the cycle length and a dense neural network model [22] to estimate
the red times from probe vehicle data. We choose XGBoost for estimating cycle
length because it has been known to perform well on tabulated data sets [23].
For predicting the red times, we tried several simpler models like XGBoost,
Random Forest and Logistic Regression models, but none performed as well as
the DNN due to the feature complexity. Traffic signals’ green times are
obtained by subtracting the red times from the cycle length for each
intersection’s phase. The yellow times are counted as part of the green times.
Hence, our paper’s contributions can be summarized as follows:

  * •

Development of a novel ML-based method for estimating pre-timed traffic signal
timing parameters including cycle length, red times and green times, that is
applicable to any location that has high-fidelity vehicle probe data.

  * •

Demonstration of the developed ML-based method on a sizeable realistic network
with 39 intersections, with results showing an error of less than 0.56 sec for
cycle length, and red times predictions within 7.2 sec error on average.

##  3 METHODOLOGY

###  3.1 Simulation Description

In order to train and evaluate our ML models, we use simulated probe data from
a realistic road network consisting of 39 intersections in the city of
Chattanooga in Tennessee, USA, shown in Figure 1. The network geometry was
validated using satellite images of the real-world network and its demand was
calibrated using historic, link-level, traffic volume estimates [8], as shown
in Figure 2. The City of Chattanooga’s department of transportation also
provided signal timing information for the 39 intersections, which served as
the ground truth for our models.

We used SUMO [5] to run our simulation for a day. SUMO gives us the advantage
of controlling and observing all aspects of the traffic and signal timings.
This eliminates any uncertainty that might be observed in the field when using
signal timing sheets that might be out of date or data that is difficult to
process because various formats. Additionally, SUMO provides a second by
second record of vehicle movements via the FCD (floating car device) listener.
Thus, we could store all vehicle trajectories in simulation and their speeds
for each simulation run. We also ran simulations with different percentiles of
the traffic demand in Figure 2 (for example, we ran simulation with 50% of the
hourly demand) to simulate demand variability. After running the simulations
with various demand levels, we recorded FCD data for up to 50% of the
vehicles. We then aggregated each trip and filtered only trips that served our
purpose of estimating cycle lengths, red times and green times.

Figure 1: Road network in Chattanooga, Tennessee, with 39 intersections used
for traffic signal timing estimation Figure 2: Hourly historic, link-level,
traffic volume estimates in vehicle per hour (vph) for the area of interest in
Chattanooga, Tennessee, used for demand calibration

###  3.2 Data Processing

To process the data for feature engineering, we used the FCD trajectory data
as inputs and the Traffic Light Scheme settings (tls) data as our target
variable. The first step of processing the FCD data was to create a bounding
box around each signalized intersection and filter out all the trajectories
that passed within the bounding box. We set the bounding box to be 500 feet
from the center of the intersection. This gave us enough room to develop
aggregated metrics of each trajectory journeying through the intersection.
After extracting the data points around each intersection, we then filtered
only trips that stopped for any given length of time before the intersection
and traversed through the center of the intersection with any vehicle movement
and approach.

Figure 3: Vehicle trajectory near an intersection, including a speed profile,
a heading profile, distance profile, and at the bottom, the actual way points
of the trajectory

In the top panel of Figure 3, we see one trip that approaches an intersection
and stops for a duration of approximately 48 seconds shown by the shaded area
in the top graphs. We can also see the way-points in the bottom panel showing
the deceleration in red and acceleration in green of the vehicle before,
through, and after the center of the signalized intersection. We filtered
trips using four criteria. These criteria were originally built using real-
world probe trajectories. Thus, this method translates easily to real-world
probe data for future implementation.

The first criteria is to find a pattern of deceleration, zero speed and
acceleration of a vehicle trip. We select this pattern because if the vehicle
never stops, we are unable to accurately estimate the initial onset of a green
light given by a vehicle accelerating from zero. Thus, if a vehicle just
passes through a light, we can only say the signal was green or yellow for the
few seconds that the vehicle approaches and passes through the center of the
intersection. This would not be sufficient information if we assume a low
penetration rate of probe vehicles.

The second criteria is based on sparsity of trace way points. For the
simulation, this criteria did not matter since all data was recorded without
sensing error. However, real-world data contain trips with missing way-points.
Thus, we dropped any vehicle trip that was missing data for a period of 10
seconds or more.

Third, we wanted to exclude any trips that were within 500 feet of the
intersection, but did not pass through the center of the intersection. In the
real-world, vehicles may be visiting a location at the corner of the
intersection, such as a gas station or fast food restaurant, but never
actually traverse the intersection. Thus, we only include vehicles that have a
50 feet minimum distance away from the center of the intersection. You can see
in the top panel of Figure 3 on the Distance graph that the minimum distance
can be easily calculated for each trip.

The last criteria we filter on is total duration of trip. Considering that the
majority of signalized intersections have a cycle length under 120 seconds, we
exclude any trip with a duration of over 2 minutes around the intersection.
With real-world probes, we came across some trips where the vehicle turned
around after passing the intersection and passed back through the light. These
types of trips were rare but added complexity and noise to our processing, and
this required extracting out these acceleration times.

We use acceleration times to estimate cycle lengths since, given enough
vehicle data, there will be a periodicity between consecutive green lights if
the signal timings are fixed. To get the acceleration start times, we use a
simple algorithm that looks at each vehicle trip and first finds the first
positive change in speed after the vehicle has stopped. This timestamp is then
recorded and assigned to an intersection and a phase so that it can be later
matched to the ground truth cycle length for a given hour.

###  3.3 Cycle Length Feature Engineering

To create ML features from the acceleration start times, we hypothesized that
the largest magnitude Fourier Frequencies from the Fourier Transform of the
distribution of acceleration start times will be a good predictor of cycle
lengths. This is because the change in signal phases throughout a cycle should
cause vehicles to start accelerating at periodic intervals. To create these
Fourier input features for the XGBoost model, we first group the acceleration
start times by the intersection they were located at, the signal phase of the
vehicle trajectory, and the time of day. This grouping ensures that all
acceleration start times come from vehicles passing through a specific
intersection with the same target cycle length. After binning, we perform the
following to create the input features:

  1. 1.

Bin all start times within the first grouping into 1 hour time windows.

  2. 2.

For each one hour window perform the following:

    1. (a)

Convert the date-time representation of each start time to the seconds after
the first start time within the current time window.

    2. (b)

Approximate the distribution of start times within the current time window
using a Gaussian Kernel Density Estimator (KDE) with bandwidth 6.

    3. (c)

Take the fast Fourier Transform (fft) of the KDE and save the 30 frequencies
that had the largest magnitude of their Fourier amplitudes.

  3. 3.

Repeat for all groupings

In doing this process, one data point for training and testing the ML model
consists of information derived from a 1 hour time window of vehicles moving
through a specific intersection. We chose the bandwidth (standard deviation)
of the Gaussian KDE to be 6 seconds so that it was large enough to ensure
starts within the same cycle blend together. Second, we saved the top 30 most
significant Fourier frequencies arbitrarily; however, we optimize the number
of frequencies used by the model along with the other XGBoost hyperparameters.
Finally, we discard all time windows with less than two starts because having
no starts in a window provides no information and only one start in a window
only provides artificial noise in the Fourier frequencies from the length of
the time window and the frequencies that compose the single Gaussian in the
KDE. We also optimize and analyze the minimum number of starts per hour as
part of a multistage tuning process. Finally, to normalize the inputs, we use
the StandardScalar transformation provided by Sci-kit Learn[24].

###  3.4 Cycle Length Training Methods

Table 1: Optimized hyperparameters for the XGBoost model. Parameter | Search Space | Optimal Value
---|---|---
n_estimators |  $100$ – $2000$ | $2000$
learning_rate |  $10^{-4}$ – $10^{0}$ | $10^{0.0}$
max_depth |  $2$ – $20$ | $17$
gamma |  $10^{-5}$ – $10^{0}$ | $10^{-5.0}$
min_child_weight |  $1$ – $10$ | $1.499$
subsample |  $0.5$ – $1.0$ | $1.0$
colsample_by_tree |  $0.5$ – $1.0$ | $1.0$
n_fourier |  $2$ – $30$ | $1$
min starts |  $2$ – $250$ | 20

For training the XGBoost model, we first randomly selected $20\%$ of the
points to set aside as the test set. Then, we randomly divide the training
dataset into 5 folds for cross validation (CV), which is used during the
optimization process. To optimize the XGBoost hyperparameters, the number of
Fourier frequencies and minimum starts per hour, we used the following
multistage optimization process:

  1. 1.

Optimize the hyperparameters and number of Fourier frequencies using Bayesian
Optimization from the bayes_opt package in Python[25] with 10 initialization
points and 50 iterations.

  2. 2.

Compute the CV predictions using the optimized hyperparameters and use them to
compute the MAE of all points with more than $c$ acceleration starts for
$c=2,3,...,249,250$.

  3. 3.

Choose the optimal cutoff to be the minimum value of $c$ such that the
associated fraction of CV predictions with an error less than 2 seconds is at
least $95\%$.

  4. 4.

Down select the training and testing data to only include points with $>c$
acceleration starts and re-optimize the hyperparameters using the same
Bayesian Optimization scheme.

###  3.5 Red Times Feature Engineering

To predict the red times, we start with the vehicle stop durations instead of
the acceleration start times used for cycle estimation. We bin the vehicle
stop times by intersection, direction of travel (North, South, East, West) and
time of day (AM, PM, Off Peak). With this binning strategy, all vehicle stop
times in a bin correspond to the same unique phase and target red times, and
this means we can incorporate distributional information about the stop times
into the input features.

Next, we perform up-sampling to improve the granularity of the examples and
evenly 4 weight all target values in the training process. To do this, we
create groupings of vehicle stop times from each bin by sampling 50 stop times
with replacement from each bin to create a grouping and repeating this 40
times. The sampling ensures empirical cumulative distribution functions
(described below) were created from 50 stop times on average and the 40
repetitions presents distinct permutations of the data to the network for
training and validation. This is useful to improve network interpolation
between training points. We chose 40 repetitions to balance the increased
number of unique permutations with the training time of the network.

We can then encode the distributional information as input features by using
empirical quantiles. Let $s_{i}$ be the stop time of the $i^{th}$ vehicle,
$\alpha$ be the quantile of interest, and $n$ be the number of vehicle stop
times in the grouping. Then the $\alpha$ quantile $Q_{\alpha}$ is given by
Equation 1.

$Q_{\alpha}=\max\left\\{s_{i}\ \Bigg{|}\ \frac{\\#\left\\{s_{j}|s_{j}\leq
s_{i}\right\\}}{n}<\alpha\right\\}$ (1)

Where $\\{\\}$ denotes a set, the notation $\\{s_{i}\ |\ X\\}$ means the set
of $s_{i}$ values that satisfy condition $X$ and $\\#A$ indicates the number
of elements in the set $A$. We use a $100\times 1$ vector of the
$1\%,2\%,...,99\%,100\%$ quantiles as the input to the neural network.
Finally, we also use the StandardScalar transform from Sci-kit Learn[24] to
normalize the inputs before training. Note that the $100\%$ quantile is just
the maximum vehicle stop time from that grouping.

Figure 4: Cycle Length Parity Plot

###  3.6 Red Times Training Methods

To predict the red times, we use a Dense Neural Network (DNN). The DNN inputs
a $100\times 1$ vector, outputs a scalar prediction and has 11 hidden layers
of sizes: 550, 1000, 900, 800, 700, 600, 500, 400, 300, 200, 100 respectively.
All layers used a Leaky ReLU activation function with $\alpha=0.01$. To train
the model, we divided the data randomly into a 20% validation set and 80%
training set. Then we trained the model with minibatch size set to 32 until we
reached an early stopping trigger with patience set to 50. The Neural Network
was constructed using the Tensorflow[26] and Keras[27] packages in Python.

##  4 RESULTS AND DISCUSSION

###  4.1 Cycle Length Results

After the first round of the multi step optimization routine for the XGBoost
model, we determined the minimum starts points per hour cutoff to be 20. We
chose this cutoff because it is the minimum number of starts per hour for the
model to produce 95% of it’s estimations with an error less than 2 seconds.
Table 1 shows the results from the final Bayesian Optimization to optimize the
hyper parameters and number of Fourier frequencies.

The results for the cycle length model have a mean absolute error (MAE) less
than 0.7 seconds for both training and testing sets, along with a R-squared
values of .97 and .99 for training and testing respectively. These results are
shown in figure 4 with green being the test set and blue being the training
set.

Figure 5: Red and Green Times Parity Plots

###  4.2 Red Times and Green Times Results

After optimizing, the neural network model was able to estimate red time
within 7.2 seconds on average with an R-squared score of 0.85. The green times
are derived by subtracting the red times from the cycle length for each
intersection’s phase. We count the yellow time as part of the green time. The
accuracy of the green times was very similar to the red times accuracy. In
Figure 5 we show parity plots for the red and green times on the left side
with nearly 2,000 test points. We also include a point density to better
understand where the concentration plotted points. Darker shaded areas show
the highest concentration of estimates. As you can see, two large
concentration of points cluster around the true diagonals. We can also see the
spread of error residuals in the histograms on the right side of the Figure 5.
This model does moderately well at estimating the red times per phase at each
intersection, giving promise to estimating red times with real world probe
data. Since this model uses the distribution of stop times over an arbitrary
period of time, we do not conduct any penetration rate analysis since we can
compensate for a low penetration rate by collecting stop time distributional
data for a longer period of time.

##  5 CONCLUSION

This paper presented an ML approach to estimating signal timing for pre-timed
traffic lights from probe data. We used an XGBoost model to estimate the cycle
length and a neural network to estimate the red times per phase. The green
times are determined by substracting red times from cycle length durations.
Our results demonstrated highly accurate estimations for cycle lengths given
only 20 points of data or more and good accurate estimations for red times and
green times. Our work complements existing literature that has significantly
focused on using mathematical methods, such as statistical, probabilistic and
optimization based methods, for determining signal timing parameters from
vehicle probe data. Future research directions include doing a detailed
sensitivity analysis of the performance of the ML-approach given different
penetration rates of probe vehicle data, and extending our approach to
estimating the signal timing for actuated signals.

## ACKNOWLEDGMENT

This work was authored by the National Renewable Energy Laboratory, operated
by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy
(DOE) under Contract No. DE-AC36-08GO28308. Funding provided by U.S.
Department of Energy Vehicle Technology Office. The views expressed in the
article do not necessarily represent the views of the DOE or the U.S.
Government. The U.S. Government retains and the publisher, by accepting the
article for publication, acknowledges that the U.S. Government retains a
nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce
the published form of this work, or allow others to do so, for U.S. Government
purposes. This work was only made possible through the close cooperation of
City of Chattanooga Department of Transportation.

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