# Stable and scalable multistage terahertz-driven particle accelerator

Heng Tang1,2∗, Lingrong Zhao1,2∗, Pengfei Zhu1,2, Xiao Zou1,2, Jia Qi3, Ya
Cheng3, Jiaqi Qiu4, Xianggang Hu5, Wei Song5, Dao Xiang1,2,6,7† and Jie
Zhang1,2† 1 Key Laboratory for Laser Plasmas (Ministry of Education), School
of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240,
China
2 Collaborative Innovation Center of IFSA (CICIFSA), Shanghai Jiao Tong
University, Shanghai 200240, China
3 State Key Laboratory of Precision Spectroscopy, East China Normal
University, Shanghai 200062, China
4 Nuctech Company Limited, Beijing, 100084, China
5 Science and Technology on High Power Microwave Laboratory, Northwest
Institute of Nuclear Technology, Xi’an, Shanxi 710024, China
6 Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240,
China
7 Zhangjiang Institute for Advanced Study, Shanghai Jiao Tong University,
Shanghai 200240, China

###### Abstract

Particle accelerators that use electromagnetic fields to increase a charged
particle’s energy have greatly advanced the development of science and
industry since invention. However, the enormous cost and size of conventional
radio-frequency accelerators have limited their accessibility. Here we
demonstrate a mini-accelerator powered by terahertz pulses with wavelengths
100 times shorter than radio-frequency pulses. By injecting a short
relativistic electron bunch to a 30-mm-long dielectric-lined waveguide and
tuning the frequency of a 20-period terahertz pulse to the phase-velocity-
matched value, precise and sustained acceleration for nearly 100% of the
electrons is achieved with the beam energy spread essentially unchanged.
Furthermore, by accurately controlling the phase of two terahertz pulses, the
beam is stably accelerated successively in two dielectric waveguides with
close to 100% charge coupling efficiency. Our results demonstrate stable and
scalable beam acceleration in a multistage mini-accelerator and pave the way
for functioning terahertz-driven high-energy accelerators.

Particle accelerators have played important roles in advancing science and
industry, with applications ranging from high-energy colliders, synchrotron
light sources, and free-electron lasers to x-ray imaging and cancer therapy
machines. However, the large size and cost of conventional accelerators based
on radio-frequency (rf) technology limit accessibility and are motivating
development of compact accelerators LPA1 ; LPA2 ; LPA3 ; LPA4 ; LPA5 ; PWFA1 ;
LPA6 ; LPA7 ; PWFA2 ; LPA8 ; PWFA3 ; LPA9 ; DLA1 ; DLA2 ; THz1 ; THz2 ; THz3 ;
THz4 ; THz5 ; THz6 ; THz7 . Among the techniques, terahertz (THz)-driven
acceleration has great potential THz1 ; THz2 ; THz3 ; THz4 ; THz5 ; THz6 ;
THz7 . Compared to the rf accelerator, the approximately 100-fold reduction in
wavelength for the THz accelerator promises significant reduction in size, but
increase in breakdown threshold to the gigaelectronvolt-per-meter regime GV1 ;
GV2 . Compared to the dielectric laser accelerator DLA1 ; DLA2 , the
approximately 100-fold increase in wavelength facilitates the fabrication of
the accelerating structure and relaxes the tolerance on electron beam
transverse size, pulse width, and timing for stable acceleration. Compared to
plasma acceleration LPA1 ; LPA2 ; LPA3 ; LPA4 ; LPA5 ; PWFA1 ; LPA6 ; LPA7 ;
PWFA2 ; LPA8 ; PWFA3 ; LPA9 , the accelerating medium, i.e., dielectric-lined
waveguide (DLW), is more stable and it is easier to control the beam size
during staged acceleration.

Figure 1: Mini-accelerator powered by THz pulses. (a) Picture of DLW assembled
with a horn; (b) Schematic of two-stage THz acceleration experiment; Measured
THz waveform (c) and spectrum (d); Measured profiles of the radially polarized
THz pulse before (e) and after (f) a polarizer (the arrow representing the
direction of the wire).

Despite extensive efforts in the past few years, challenges, such as proof-of-
principle THz acceleration to a functioning THz accelerator, remain. In a
functioning accelerator, the beam is stably accelerated in successive modules
to attain high energy since only finite energy gain can be reached in each
accelerating structure. Therefore, a functioning THz accelerator requires
stable whole-bunch acceleration, i.e., all the electrons in a bunch are
accelerated at approximately the same phase to reach approximately the same
energy, as well as high charge coupling efficiency and accurate phase control
for staging. However, with the electron beam being longer than half of the THz
wavelength, only a fraction of the electrons are accelerated by a laser-
generated THz pulse, while others are decelerated THz1 ; THz2 ; THz3 ; THz4 ;
THz5 . Though recently, the staging of acceleration for relativistic electrons
has been reported THz7 , the short acceleration length from an accelerator-
produced single-cycle transition radiation THz pulse and the limited beam
quality is not sufficient to address the question of whether an electron bunch
can be stably accelerated in successive long structures with high charge
coupling efficiency and high stability. We note that while whole-bunch
acceleration for keV electron beam in two stages has been reported in THz6 ,
the small energy gain and large energy spread of the beam makes it difficult
to answer the question of whether beam energy spread can be preserved in
staged acceleration with considerable energy gain.

In this Letter, we demonstrate sustained and stable whole-bunch acceleration
of a relativistic electron beam in two stages driven by 20-period THz pulses.
The accelerating structure is a 30-mm-long DLW with 0.86 mm aperture, about
100 times smaller than a conventional rf structure. The DLW consists of a
dielectric layer of fused silica with a wall thickness of 43 $\mu$m and a
gold-coated external surface. Our analysis shows that such a structure
supports an accelerating mode at 0.54 THz with a phase velocity of
approximately $v_{p}=0.989c$ and a group velocity of $v_{g}=0.72c$ (see
Supplementary Materials Fig.S1). A horn with an opening angle of 10 degrees is
attached to the DLW to increase the coupling efficiency THz1 ; THz6 ; THz7 ,
which is measured to be about 70% in this experiment. An optical image of the
mini-accelerator is shown in Fig. 1(a). The layout of the experiment is
depicted in Fig. 1(b). A 3 MeV, 10 fC relativistic electron beam with a
velocity of $v_{e}=v_{p}=0.989c$, is successively accelerated in two identical
modules consisting of an off-axis parabolic (OAP) mirror for focusing the THz
pulse, a horn for adiabatically guiding the THz pulse, and a DLW for the THz-
electron interaction. A solenoid is used to control the electron beam size and
steering magnets are used to control the beam trajectory during staged
acceleration. The beam energy distribution is measured with a spectrometer
magnet (see Fig.S2).

Figure 2: Visualization of THz-electron interaction. (a) Schematic depiction
of the relative position between the electron beam (purple) and THz pulse
(red) at various time delays (the arrow represents the direction of travel);
Measured electron energy changes when the THz frequency equals to (b) and
deviates from (c) that of the accelerating mode; Simulated electron energy
change when the THz frequency equals to (d) and deviates from (e) that of the
accelerating mode.

The THz pulse is produced by sending a linearly polarized femtosecond laser
(about 3 mJ) through a cryogenically cooled periodically poled lithium niobate
(PPLN) crystal PPLN  with a size of 3 mm by 3 mm. The PPLN crystal has a total
length of 4.4 mm and a period of 0.22 mm for producing a 20-period narrowband
THz pulse centered around 0.54 THz. The THz waveform measured with electro-
optic sampling is shown in Fig. 1(c), and the corresponding spectrum is shown
in Fig. 1(d). The duration of the THz pulse is $\tau_{THz}=40~{}$ps, matching
the length of the DLW for maximal energy gain. The temperature of the PPLN is
reduced to below 200 K for reducing THz absorption. It should be noted that
the frequency of the THz pulse depends on the refractive index of the laser
and THz pulse in the crystal. Therefore, the THz pulse’s central frequency may
be varied by changing the crystal temperature, and in this experiment this
offers about 6% tunability for the THz frequency while keeping the THz energy
at a similar level (see Fig.S3).

To allow efficient excitation of the TM01 mode in DLW, the linearly polarized
narrowband THz pulse is converted into a radially polarized pulse with a
spatially variable retardation plate (see Fig.S4). This conversion ensures the
best matching of the Poynting vector for optimally transferring energy from
the incident THz pulse to the accelerating mode in the DLW. In general, the
coupling efficiency in converting the energy of a THz pulse into that of a
specific mode in the DLW depends on the similarity of the Poynting vector
distribution. Therefore, a linearly polarized THz pulse can efficiently excite
TM11-like deflecting mode THzO  and a radially polarized THz pulse is required
to excite the TM01 accelerating mode. The radially polarized THz pulse profile
is measured with a THz camera and shows the expected donut shape (Fig. 1(e)).
The THz energy is measured to be about 100 nJ at the entrance to the horn. It
should be noted that an azimuthally polarized pulse may also take a donut
shape, so a wire grid polarizer is used to confirm the polarization. For a
radially polarized pulse, the two lobes after the polarizer line up in the
perpendicular direction of the wire (Fig. 1(f)), while those for an
azimuthally polarized pulse should line up in the same direction of the wire
(see Fig.S4).

In addition to exciting the accelerating mode, sustained acceleration also
requires the THz phase velocity to be equal to the electron velocity, or, in
other words, requires the THz frequency to be equal to that of the
accelerating mode. In a conventional rf accelerator, this is achieved through
precision machining of the rf cavity and utilization of a low-level rf system.
Here we vary the crystal temperature to tune the frequency of the THz pulse in
order to match that of the accelerating mode. With both the electron beam and
THz pulse propagating in the same direction, this THz accelerator works in the
traveling-wave mode, and the acceleration process can be visualized through
measurement of beam energy change as a function of time delay ($\Delta t$)
between the THz pulse and electron beam.

To understand the acceleration process, we schematically depict the relative
position of the electron and THz pulses at various time delays in Fig. 2(a).
With the THz pulse frequency matching that of the accelerating mode, optimal
acceleration should be achieved at $\Delta t=\tau_{THz}=40~{}$ps when the
electron beam catches up with the tail of the THz pulse at the entrance to the
DLW. With the beam velocity higher than the group velocity of the THz pulse in
the DLW, the electron beam interacts across the full THz envelope while
remaining phase matched and achieves maximal energy exchange. Deviating from
this time delay causes the beam to miss part of the THz pulse, and the energy
exchange is reduced. This is exactly what has been observed in the experiment
(Fig. 2(b)), where the envelope of the energy modulation monotonically
increases for $\Delta t<40~{}$ps and then decreases for $\Delta t>40~{}$ps.
Within one wavelength window, the energy change depends on the delay time,
which translates to the THz phase seen by the electron beam.

When the THz frequency deviates from that of the accelerating mode, the
effective interaction length and energy exchange should be reduced (see
Fig.S5). We tuned the THz frequency to 0.51 THz (corresponding dephasing
length, i.e. the distance over which the phase slippage increases to $\pi/2$,
is about 6.7 mm) to see this effect, and the result is shown in Fig. 2(c). In
contrast to the matching case, here at $\Delta t=40~{}$ps, the energy exchange
is close to zero. This is because the electron beam slips from the
accelerating phase to the decelerating phase, and the energy gain from
interaction with the first half of the THz pulse is canceled by the energy
reduction from the second half of the THz pulse, similar to a detuned rf
cavity. The simulated electron energy change for the matched and unmatched
cases is in excellent agreement with the measurements shown in Fig. 2(d) and
2(e), respectively. Furthermore, no noticeable deflection to the beam has been
measured, which indicates that most of the THz energy is converted into a
single accelerating mode, demonstrating high efficiency in converting THz
energy to accelerating mode and finally to the energy of the electron beam.
Here an energy conversion efficiency of about 1.5 keV per nJ1/2 has been
achieved, much higher than previous results (see Fig.S6). It also allows
comparison and confirmation of predictive models and scaling laws and is
critical for understanding the physics and engineering issues behind THz
acceleration in long structures.

A functioning THz accelerator requires shot-to-shot stability and the staging
of the accelerating structures for reaching higher energy. Here a second
module 80 cm downstream of the first is used to demonstrate the feasibility,
as well as the repeatability of the module. With similar tuning procedures,
the energy gain of about 16 keV (for a 110 nJ THz pulse) in the second DLW is
achieved. We then vary the time delay of the second THz pulse to change the
interaction phase (see Fig.S2), and the corresponding beam energy
distribution, as well as transverse beam size (measured at a screen just
before the spectrometer magnet), is shown in Fig. 3(a) and 3(b), respectively.
From Fig. 3(a), one can see that when the electron bunch interacts with THz
pulse at the on-crest phases, its energy is upshifted ($\phi=\pi/2$,
acceleration) or downshifted ($\phi=3\pi/2$, deceleration) with minimal
broadening. In contrast, at the zero-crossing phases (0 and $\pi$), the beam
energy distribution receives maximal broadening with the centroid energy
unchanged. Due to the initial beam energy chirp, the broadening at the two
zero-crossing phases is slightly different (see Fig.S7). Similar to the zero-
phasing method with rf structure zero-phasing  and laser streaking technique
from attosecond metrology AS , electron’s time information is mapped into its
energy distribution when the interaction is around the zero-crossing phases.
Accordingly, the electron bunch length is estimated to be about 100 fs RMS and
the beam arrival time jitter is determined to be about 50 fs RMS.

Figure 3: Image of the electron beam on the spectrometer screen (a) and the
transverse beam size (b) for various phases of the second THz pulse.

The beam size in Fig. 3(b) shows the effect of the transverse force in the DLW
(see, e.g. DLW ). Similar to an rf accelerator, the beam experiences a small
focusing force at $\phi=\pi$ (bunch head is accelerated more than bunch tail)
leading to a reduction in beam size, and a defocusing force is exerted at
$\phi=0$ where the bunch head is decelerated more, which is also the phase for
bunch compression RFC1 ; RFC2 . The transverse force is negligibly small at
the accelerating phase (electron beam size is similar to the case with THz
off), which is crucial for maximizing the charge coupling efficiency in
staging. Due to the absence of transverse force, nearly 100% charge coupling
efficiency is achieved (see Fig.S8).

Figure 4: Stable and staged THz-driven interaction. (a) Consecutive
measurement of beam energy distribution with both THz pulses off (the first 50
shots), with one-stage acceleration (the middle 50 shots), and with two-stage
acceleration (the final 50 shots); (b) Statistical results of the beam energy
distribution for the data in (a).

With the phases of the two THz pulses both set for maximal acceleration,
consecutive measurements of the beam energy distribution after one-stage and
two-stage acceleration are shown in Fig. 4(a). The corresponding statistical
results for beam energy distribution are shown in Fig. 4(b). The beam energy
spread (about 1.3 keV) and energy jitter (about 0.6 keV) remain unchanged
within the measurement accuracy after staged acceleration, setting a clear
route to higher energy with more stages. The shot-to-shot fluctuation of the
central beam energy is mainly from fluctuations of rf amplitude in the
electron gun, and the fluctuation from the electron beam injection timing
jitter is estimated to be about 0.2 keV.

In summary, we have demonstrated stable and scalable acceleration of a
relativistic electron beam in successive DLWs powered by multicycle THz
pulses. The energy gain achieved in each module of this mini-accelerator is
about 15 keV, limited by the THz pulse energy (about 100 nJ). First, we note
that keV electrons have been widely used in ultrafast electron diffraction
(UED) for probing nonequilibrium states and structural dynamics keV1 ; keV2 .
Our technique may immediately find applications in UED, e.g., enabling all-
optical UED with matched or even enhanced performance. Second, the DLW may be
used as an add-on module for compressing both relativistic and sub-
relativistic beams BaumC ; SJTUC ; SLACC , enhancing the performance of UED
and 4D electron microscopy 4DEM . Third, narrowband THz pulse with millijoule-
level energy has been demonstrated HETHz , so MeV energy gain could be
obtained with such a THz source, which opens up new opportunities in fields as
portable medical and industrial imaging, and cancer therapy. Furthermore, with
the demonstrated stability and scalability, a functioning miniaturized high
energy THz accelerator with dozens of stages can be foreseen.

This work was supported by the National Natural Science Foundation of China
(Grants No. 11925505, 12005132, 11504232, and 11721091), the office of Science
and Technology, Shanghai Municipal Government (No. 16DZ2260200 and
18JC1410700), and China National Postdoctoral Program for Innovative Talents
(No. BX20200220).
* These authors contributed equally to this work.
${\dagger}$<EMAIL_ADDRESS>
${\dagger}$<EMAIL_ADDRESS>

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# Strong Decay Widths and Coupling Constant of Recent Charm Meson States

Meenakshi Batra, Alka Upadhayay
School of Physics and Material Science
Thapar University, Patiala-147004, India
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Open charm hadrons with strange and non-strange mesons have been discovered in
recent years. We study the spectra of several newly observed resonances by
different collaborations like BaBar [2]and LHCb [3] etc. Using an effective
Lagrangian approach based on heavy quark symmetry and chiral dynamics, we
explore the strong decay widths and branching ratios of various resonances and
suggest their $J^{p}$ values. We try to fit the experimental data to find the
coupling constants involved in the strong decays through pseudoscalar mesons.
The present work also discusses about the possible spin-parity assignments of
recently observed states by LHCb collaboration. The tentative assignment of
newly discovered state $D_{J}^{\ast}(3000)$ can be natural parity states
$(0^{-},1^{+},2^{-},3^{+}....)$ while $D_{J}(3000)$ can be identified with
unnatural parity states like $(0^{+},1^{-},2^{+},3^{-}....)$. Therefore, the
missing doublets 2S,1D,1F,2P and 3S can be thought of filled up with these
states. We study the two-body strong decay widths and branching ratios of
missing doublets and plot branching ratios vs mass of decaying particle. These
plots are used to analyze all assignments to $D_{J}(3000)$ deeply and various
possibilities for $J^{P}$ values.
PACS:12.39.Hg,12.39.Pn,12.40.Yx,14.40.Nd
Keywords:Heavy quark effective theory,spectroscopy, heavy-light mesons

##  1 Introduction

The hadrons containing a single heavy quark can be analyzed in a framework
which is formulated for $N_{f}$ heavy quarks with mass $m_{Q}>>\Lambda_{QCD}$
using heavy quark effective theory [1]. This theory assumes heavy quark to act
as static color source and its spin is $s_{Q}$ which can be thought of coupled
to spin of light degrees of freedom $s_{l}$. Secondly, heavy quark flavor
symmetry leads to interaction of heavy quark with light degrees of freedom
through exchange of soft gluons only. The heavy quark spin-flavor symmetry can
be exploited further to explore several hadronic properties. The motivation
for present work arises due to recently observed charm and bottom meson states
by experimental collaborations like BaBar, LHCb and CDF [4]. The heavy meson
spectrum is one of the recent interest to place for the various particles at
different resonances and energies. Recently, some excited charm meson states
were observed which are D(2550), D(2600), D(2750) and D(2760)in the decay
channels $D^{0}(2550)\rightarrow D^{\ast+}\pi^{-}$,$D^{0}(2600)\rightarrow
D^{\ast+}\pi^{-},D^{+}\pi^{-}$,$D^{0}(2750)\rightarrow
D^{+}\pi^{-}$,$D^{+}(2600)\rightarrow D^{0}\pi^{+}$ and
$D^{+}(2760)\rightarrow D^{0}\pi^{+}$ in the inclusive $e^{+}e^{-}\rightarrow
c\bar{c}$ interactions by BaBar Collaboration [2]. The most suitable spin-
parity assignments for D(2750),D(2760)is $(2^{-},3^{-})$ or 1D and for
D(2550), D(2600)is $(0^{-},1^{-})$ i.e 2S state respectively. LHCb
collaboration [3] observed some new resonances in addition to above i.e.
$D_{J}(3000)^{+,0}$ and $D_{J}^{\ast}(3000)^{0}$ around 3GeV in association
with $D_{2}^{\ast}(2460)^{0}$ and $D_{J}^{\ast}(2760)^{0}$ and exists in the
$D^{+}\pi^{-}$ invariant mass spectrum. The states,
$D_{J}^{\ast}(3000)^{+}$,$D_{2}^{*}(2460)^{+}$ and $D_{J}^{\ast}(2760)^{+}$
were observed in $D_{0}\pi^{+}$ spectrum whereas the states measured in
$D^{*+}\pi^{-}$ spectrum were $D_{1}(2420)^{0}$,$D_{2}^{\ast}(2460)^{0}$,
$D_{J}^{\ast}(2760)^{0}$,$D_{J}(2580)^{0}$, $D_{J}(2740)^{0}$ and
$D_{J}(3000)^{0}$ respectively. A table for the properties of recently
observed states has been shown below. Similar is the case in beauty sector.
Very recently CDF Collaboration [4] has found evidence for a new resonance
B(5970) simultaneously in $B^{0}\pi^{+}$ and $B^{+}\pi^{-}$ mass distributions
with a significance of 4.4$\sigma$ standard deviations and further reported
the first study of resonances with orbitally excited $B^{+}$ mesons and
updated measurement of orbitally excited $B^{0}$ and $B_{s}^{0}$ mesons. The
branching ratio for $B_{s2}^{\ast 0}\rightarrow B^{\ast+}K^{-}$ decays is also
measured. The masses of new B(5970) measured resonances are $5978\pm
5(stat)\pm 12(syst)MeV/c^{2}$ for neutral state and $5961\pm 5(stat)\pm
12(syst)MeV/c^{2}$ for charged asymmetry into $B\pi$ states. This state may be
proposed as belonging to radially excited bottom meson family. Therefore, in
past decades, we faced several ground as well as excited states of charm meson
family such as discovery of $D_{sJ}(2460)$, $D_{sJ}(2632)$ and $D_{sJ}(3040)$
etc. In bottom meson family we witness some new states such as B(5279),
$B^{\ast}(5325)$ for n=1 family in $(0^{-},1^{-})$ doublet. In the infinite
heavy quark mass limit, a heavy light system $Q\overline{q}$ can be classified
into doublets depending upon their quantum numbers. A heavy hadronic system
containing heavy quark with spin quantum number $S_{Q}$ and light degrees of
freedom $s_{l}$ that include light quark and gluons interacting through quark-
antiquark pairs. It should have the quantum number of light quark that is
$S_{l}$ in order to have total conserved quantum number J where
$J=S_{Q}+S_{l}$. Defining J as $J^{2}=j(j+1)$ and $S_{Q}^{2}=(s_{Q})(s_{Q}+1)$
and $S_{l}^{2}=(s_{l})(s_{l}+1)$,the total spin $j\pm=s_{l}\pm\frac{1}{2}$ can
be obtained by combining the spin of heavy quark spin $\frac{1}{2}$ with spin
of light degrees of freedom. The heavy mesonic system form a degenerate
doublet of ground state with $J=\pm 1$ and negative parity denoted as D and
$D^{\ast}$ for charm meson. The first excited states $0^{+}$ and $1^{+}$ heavy
mesons are the quantum numbers of the $s_{l}^{p}=\frac{1}{2}^{+}$ doublet.
There is also an excited doublet of heavy meson with $J^{P}=1^{+}$ and
$2^{+}$. Similarly, other excited mesonic states have their $J^{P}$ states. In
this article, we identify the recent charmed meson states $D_{J}(2550)$,
$D_{J}^{\ast}(2600)$, $D_{J}(2740)$, $D_{J}^{\ast}(2760)$, $D_{J}(3000)$ and
$D_{J}^{\ast}(3000)$ with their $J^{P}$ assignment. These states were observed
by LHCb collaboration and predicting their decay widths and masses. We study
strong decays of these charmed mesons to ground states heavy mesons along with
the emission of pseudo-scalar pions in heavy quark effective theory in the
leading order approximations. Although the same work has been studied by [10]
but we extend their predictions by fitting the experimental data to find the
coupling constants in various strong decays. Also, we include two additional
possibilities for assignment of $J^{P}$ states to $D_{J}^{\ast}(3000)$ and
$D_{J}(3000)$. In the end, we also try to justify all the possible assignments
to $D_{J}^{\ast}(3000)$ and $D_{J}(3000)$ by analyzing their branching ratios
graphically.

##  2 The Lagrangian for Strong Decays to Heavy Mesons

A single field $H_{a}$ where it annihilates the $s_{l}=\frac{1}{2}^{-}$ meson
doublet, pseudoscalar and vector mesons can be mentioned as[5]

$H_{a}=\frac{1+\not{v}}{2}(P_{a}^{\ast\mu}\gamma_{\mu}-P_{a}\gamma_{5})$ (1)

Here a is the SU(3) index. In charm mesons sector, $H_{a}$ consists of the
$D^{0},D^{+},D_{s}^{+}$ pseudo-scalar mesons and $D^{*0},D^{*+},D_{s}^{*+}$
vector mesons. The lowest lying excited states are the $J^{P}=0^{+}$ and
$1^{+}$ i.e. $s_{l}^{P}=\frac{1}{2}^{+}$ doublet and represented by the fields
$S_{a}$ [6]. The fields for excited spin doublets are mentioned below:

$S_{a}=\frac{1+\not{v}}{2}(P_{1a}^{\mu}\gamma_{\mu}\gamma_{5}-P_{0a}^{\ast})$
(2)
$T_{a}^{\mu}=\frac{1+\not{v}}{2}(P^{\ast\mu\nu}_{2a}\gamma_{\mu}-P_{1a\nu}\sqrt{\frac{3}{2}}\gamma_{5}[g^{\mu\nu}-\frac{1}{3}\gamma^{\nu}(\gamma^{\mu}-v^{\mu})])$
(3)
$X_{a}^{\mu}=\frac{1+\not{v}}{2}(P^{\ast\mu\nu}_{2a}\gamma_{5}\gamma_{\nu}-P^{\prime\ast}_{1a\mu}\sqrt{\frac{3}{2}}[g^{\mu\nu}-\frac{1}{3}\gamma^{\nu}(\gamma^{\mu}+v_{\mu})])$
(4)
$Y_{a}^{\mu\nu}=\frac{1+\not{v}}{2}(P_{3a}^{\ast\mu\nu\sigma}\gamma_{5}\gamma_{\sigma}-P_{2a}^{\alpha\beta}\frac{5}{3}\gamma_{5}[g_{\alpha}^{\mu}g_{\beta}^{\nu}-\frac{g_{\beta}^{\nu}\gamma_{\alpha}(\gamma^{\mu}-v^{\mu})}{5}-\frac{g_{\alpha}^{\mu}\gamma_{\beta}(\gamma^{\nu}-v^{\nu})}{5}]$
(5)
$Z_{a}^{\mu\nu}=\frac{1+\not{v}}{2}(P_{3a}^{\ast\mu\nu\sigma}\gamma_{5}\gamma_{\sigma}-P_{2a}^{\ast\alpha\beta}\sqrt{\frac{5}{3}}[g_{\alpha}^{\mu}g_{\beta}^{\nu}-\frac{g_{\beta}^{\nu}\gamma_{\alpha}(\gamma^{\mu}+v^{\mu})}{5}-\frac{g_{\alpha}^{\mu}\gamma_{\beta}(\gamma^{\nu}+v^{\nu})}{5}])$
(6)
$R_{a}^{\mu\nu\rho}=\frac{1+\not{v}}{2}(P_{4a}^{\ast\mu\nu\rho\sigma}\gamma_{5}\gamma_{\sigma}-P_{3a}^{\alpha\beta\tau}\sqrt{\frac{7}{4}}[g_{\alpha}^{\mu}g_{\beta}^{\nu}g_{\tau}^{\rho}-\frac{g_{\beta}^{\nu}g_{\tau}^{\rho}\gamma_{\alpha}(\gamma^{\mu}-v^{\mu})}{7}-\frac{g_{\alpha}^{\nu}g_{\tau}^{\rho}\gamma_{\beta}(\gamma^{\mu}-v^{\mu})}{7}-\frac{g_{\alpha}^{\mu}g_{\beta}^{\nu}\gamma_{\tau}(\gamma^{\rho}-v^{\rho})}{7}])$
(7)

The super fields $H_{a}$ contain s-wave mesons whereas $S_{a},T_{a}$ contain
the p-wave mesons. The light pseudoscalar mesons are described by the fields
$\xi=\exp^{\frac{iM}{f}}$. The pion octet is introduced by the vector and
axial combinations
$V^{\mu}=\frac{1}{2}{\xi\partial^{\mu}\xi^{{\dagger}}+\xi^{{\dagger}}\partial^{\mu}\xi}$
and
$A^{\mu}=\frac{1}{2}{\xi\partial^{\mu}\xi^{{\dagger}}-\xi^{{\dagger}}\partial^{\mu}\xi}$
We choose $f_{\pi}=130MeV$. Here, all traces are taken over Dirac spinor
indices, light quark $SU(3)_{V}$ flavor indices $a=u,d,s$ and heavy quark
flavor indices $Q=c,b$. The Dirac structure of chiral Lagrangian has been
replaced by velocity vector $v$. At the leading order, the heavy meson chiral
chiral Lagrangian terms
$L_{H}$,$L_{S}$,$L_{T}$,$L_{X}$,$L_{Y}$,$L_{Z}$,$L_{R}$ for the strong decays
to the $D^{(\ast)}\pi$,$D^{(\ast)}\eta$ and $D_{s}^{(\ast)}K$ states can be
written as:

$L_{H}=g_{H}Tr\\{\bar{H_{a}}H_{b}\gamma_{\mu}\gamma_{5}\emph{A}^{\mu}_{ba}\\}$
(8)
$L_{S}=g_{S}Tr\\{\bar{H_{a}}S_{b}\gamma_{\mu}\gamma_{5}\emph{A}^{\mu}_{ba}\\}+h.c.,$
(9)
$L_{T}=\frac{g_{T}}{\Lambda}Tr(\bar{H_{a}}T_{b}^{\mu}(\emph{D}_{\mu}\not{\emph{A}}+i\not{D}\emph{A}_{\mu})_{ba}\gamma_{5})+h.c.,$
(10)
$L_{X}=\frac{g_{X}}{\Lambda}Tr(H_{a}X_{b}^{\mu}(i\emph{D}_{\mu}\not{\emph{A}}+i\not{\emph{D}}\emph{A}_{\mu})_{ba}\gamma_{5})+h.c.,$
(11)
$L_{Y}=\frac{1}{\Lambda^{2}}Tr(\bar{H_{a}}Y_{b}^{\mu\nu}[k_{1}^{Y}\\{\emph{D}_{\mu},\emph{D}_{\nu}\\}\emph{A}_{\lambda}+k_{2}^{Y}(\emph{D}_{\mu}\emph{D}_{\lambda}\emph{A}_{\mu}+\emph{D}_{\nu}\emph{D}_{\lambda}\emph{A}_{\mu})]_{ba}\gamma^{\lambda}\gamma_{5})+h.c.,$
(12)
$L_{Z}=\frac{1}{\Lambda^{2}}Tr(\bar{{H_{a}}}Z_{b}^{\mu\nu}[k_{1}^{z}\\{\emph{D}_{\mu},\emph{D}_{\nu}\\},\emph{D}_{\nu}\\}\emph{A}_{\lambda}+k_{2}^{Z}(\emph{D}_{\mu},\emph{D}_{\lambda}\emph{A}_{\nu}+\emph{D}_{\nu}\emph{D}_{\lambda}\emph{A}_{\mu})]_{ba}\gamma^{\lambda}\gamma_{5})+h.c.,$
(13)

where $\emph{D}_{\mu}=\partial_{\mu}+\nu_{\mu}$,
$\\{\emph{D}_{\mu},\emph{D}_{\nu}\\}=\emph{D}_{\mu}\emph{D}_{\nu}+\emph{D}_{\mu}\emph{D}_{\nu}$,$\\{\emph{D}_{\mu},\emph{D}_{\nu},\emph{D}_{\rho}\\}=\emph{D}_{\mu}\emph{D}_{\nu}\emph{D}_{\rho}+\emph{D}_{\mu}\emph{D}_{\rho}\emph{D}_{\nu}+\emph{D}_{\nu}\emph{D}_{\rho}\emph{D}_{\mu}+\emph{D}_{\rho}\emph{D}_{\mu}\emph{D}_{\nu}+\emph{D}_{\rho}\emph{D}_{\nu}\emph{D}_{\mu}$,
These terms describe the transitions of positive and negative parity mesons
with the emission of light pseudo-scalar mesons. The mixing angle between two
states are determined by including spin symmetry violating corrections in the
Lagrangian. The term should respect parity and time reversal and may be of
generic form as written below.

$\emph{L}_{d1}=\frac{h_{1}}{2m\Lambda}Tr[\bar{H}\sigma^{\mu\nu}T^{\alpha}\sigma_{\mu\nu}\gamma^{k}\gamma^{5}(iD_{\alpha}A_{\kappa}+iD_{\kappa}A_{\alpha})]+h.c.$
(14)

The corresponding operator for the mixing of $1^{+}$ in 2S and 1D respectively
is due to spin symmetry violating effect and can be written as:
$\emph{L}_{mix}=g_{1}Tr[\bar{\tilde{H}}\phi^{\mu\nu}_{s}X_{\mu}\sigma_{\nu\alpha}\emph{v}^{\alpha}]+h.c.$.

##  3 Strong Decay Width Formula and Coupling Constants

From the chiral Lagrangian terms, we can obtain the decay widths $\Gamma$ for
strong decays to final states $D^{(\ast)}\pi$, $D^{(\ast)}\eta$,
$D_{s}^{(\ast)}K$ where the symmetry breaking scale $\Lambda_{X}=1GeV$. The
expression for decay widths if we consider various doublets which the decaying
meson belongs to are as follows where M denotes the emission of pseudoscalar
mesons i.e. $\pi$, K and $\eta$ fields.
$(0^{-},1^{-})$ to $(0^{-},1^{-})+M$

$\Gamma(1^{-}\rightarrow 0^{-})=C_{M}\frac{g_{H}^{2}}{6\pi
f_{\pi}^{2}}\frac{M_{f}}{M_{i}}\mid\vec{p_{M}}\mid^{3}$ (15)
$\Gamma(1^{-}\rightarrow 1^{-})=C_{M}\frac{g_{H}^{2}}{3\pi
f_{\pi}^{2}}\frac{M_{f}}{M_{i}}\mid\vec{p_{M}\mid}^{3}$ (16)

$(0^{+},1^{+})$ to $(0^{-},1^{-})+M$

$\Gamma(1^{+}\rightarrow 1^{-})=C_{M}\frac{g_{S}^{2}}{2\pi
f_{\pi}^{2}}\frac{M_{f}}{M_{i}}\mid\vec{p_{M}\mid}[m_{M}^{2}+\mid\vec{p_{M}}\mid^{2}]$
(17) $\Gamma(1^{+}\rightarrow 0^{-})=C_{M}\frac{g_{S}^{2}}{2\pi
f_{\pi}^{2}}\frac{M_{f}}{M_{i}}\mid\vec{p_{M}\mid}[m_{M}^{2}+\mid\vec{p_{M}}\mid^{2}$
(18)

$(1^{+},2^{+})$ to $(0^{-},1^{-})+M$

$\Gamma(1^{+}\rightarrow 1^{-})=C_{M}\frac{2g_{T}^{2}}{3\pi
f_{\pi}^{2}\Lambda^{2}}\frac{M_{f}}{M_{i}}\mid\vec{p_{M}}^{5}\mid$ (19)
$\Gamma(2^{+}\rightarrow 0^{-})=C_{M}\frac{4g_{T}^{2}}{15\pi
f_{\pi}^{2}\Lambda^{2}}\frac{M_{f}}{M_{i}}\mid\vec{p_{M}}^{5}\mid$ (20)
$\Gamma(2^{+}\rightarrow 1^{-})=C_{M}\frac{2g_{T}^{2}}{5\pi
f_{\pi}^{2}\Lambda^{2}}\frac{M_{f}}{M_{i}}\mid\vec{p_{M}}^{5}\mid$ (21)

$(1^{-},2^{-})$ to $(0^{-},1^{-})+M$

$\Gamma(1^{-}\rightarrow 0^{-})=C_{M}\frac{4g_{X}^{2}}{9\pi
f_{\pi}^{2}\Lambda^{2}}\frac{M_{f}}{M_{i}}\mid\vec{p_{M}}^{3}\mid[m_{M}^{2}+\mid\vec{p_{M}}\mid^{2}]$
(22) $\Gamma(1^{-}\rightarrow 1^{-})=C_{M}\frac{2g_{X}^{2}}{9\pi
f_{\pi}^{2}\Lambda^{2}}\frac{M_{f}}{M_{i}}\mid\vec{p_{M}}^{3}\mid[m_{M}^{2}+\mid\vec{p_{M}}\mid^{2}]$
(23) $\Gamma(2^{-}\rightarrow 1^{-})=C_{M}\frac{2g_{X}^{2}}{3\pi
f_{\pi}^{2}\Lambda^{2}}\frac{M_{f}}{M_{i}}\mid\vec{p_{M}}^{5}\mid[m_{M}^{2}+\mid\vec{p_{M}}\mid^{2}]$
(24)

$(2^{-},3^{-})$ to $(0^{-},1^{-})+M$

$\Gamma(2^{-}\rightarrow 1^{-})=C_{M}\frac{4g_{Y}^{2}}{15\pi
f_{\pi}^{2}\Lambda^{4}}\frac{M_{f}}{M_{i}}\mid\vec{p_{M}}^{7}\mid$ (25)
$\Gamma(3^{-}\rightarrow 0^{-})=C_{M}\frac{4g_{Y}^{2}}{35\pi
f_{\pi}^{2}\Lambda^{4}}\frac{M_{f}}{M_{i}}\mid\vec{p_{M}}^{7}\mid$ (26)
$\Gamma(3^{-}\rightarrow 1^{-})=C_{M}\frac{16g_{Y}^{2}}{105\pi
f_{\pi}^{2}\Lambda^{4}}\frac{M_{f}}{M_{i}}\mid\vec{p_{M}}^{7}\mid$ (27)

The coefficients $C_{M}$ are different for the various light pseudoscalar
mesons:
$C_{\pi^{+}}=C_{K^{+}}=1,C_{\pi^{0}}=C_{K_{s}}=\frac{1}{2},C_{\eta}=\frac{1}{6}$.
$\vec{p_{M}}$ is the three momentum of M. The higher order corrections to
heavy quark limit can also be considered by adding terms of the order
$\frac{1}{m_{Q}}$ with some un-known constants. The decay rates depend upon
effective coupling constants. The parameters used in the above expressions for
decay widths are taken from the particle data group[7]. Thus the numerical
values of decay widths comes out in terms of coupling constants
$g_{H},g_{Y},g_{X}$ etc. Here the first radial excitation of $D^{\ast}$ is
represented as $\tilde{D}^{\ast}$. The first radially excited state of H is
governed by the decay constant $\tilde{g_{H}}$ which can be fitted to
experimental data within mass range of 2600-2700 MeV. Coupling constants can
either be determined theoretically or by fitting the experimental data.
However, various quark models [20, 21] and sum rule (eg. QCD sum rules)[15,
17, 19] techniques predict the coupling constants. Another possible method is
to use lattice QCD [18] which incorporate QCD in their first principle. Using
experimental data of decay widths and branching ratios as input, one can fit
the experimental data to find the effective coupling constants. The coupling
constants play an important role in heavy quark phenomenology. They are
directly related to charm meson strong decays and are further useful to
explore other decays of charm mesons involving pionic emissions.

##  4 Spin-parity Analysis for Non-strange Charm Meson States

The recent experimental data of charm meson states from LHCb and BaBar
collaboration motivates us to find the best fit values of coupling constants
in strong decays. The table 1 mentions the recent experimental data of non-
strange charm mesons.

Table 1:  Experimental Status of Latest Charm Mesons

Sr.No | Charm Meson state | Mass[MeV][LHCb] | Mass[MeV][BaBar] | Width[MeV][LHCb] | Width[MeV][BaBar] | Decay Channel
---|---|---|---|---|---|---
1 | $D_{J}^{\ast}(2650)^{0}$ | $2649.2\pm 3.5\pm 3.5$ | $2608.7\pm 2.4\pm 2.5$ | $140.2\pm 17.1\pm 18.6$ | $93\pm 6\pm 13$ | $D^{\ast+}\pi^{-}$
2 | $D_{J}^{\ast}(2760)^{0}$ | $2761.1\pm 5.1\pm 6.5$ | $2763.3\pm 2.3\pm 2.3$ | $74.4\pm 3.4\pm 37.0$ | $60.9\pm 5.1\pm 3.6$ | $D^{\ast+}\pi^{-}$
3 | $D_{J}(2580)^{0}$ | $2579.5\pm 3.4\pm 5.5$ | $2539.4\pm 4.5\pm 6.8$ | $177.5\pm 17.8\pm 46.0$ | $130\pm 12\pm 13$ | $D^{\ast+}\pi^{-}$
4 | $D_{J}(2740)^{0}$ | $2737.0\pm 3.5\pm 11.2$ | $2752.4\pm 1.7\pm 2.7$ | $73.2\pm 13.4\pm 25.0$ | $71\pm 6\pm 11$ | $D^{\ast+}\pi^{-}$
5 | $D_{J}(3000)^{0}$ | $2971.8\pm 8.7$ | $188.1\pm 44.8$ |  |  | $D^{\ast+}\pi^{-}$
6 | $D_{J}^{\ast}(2760)^{0}$ | $2760.1\pm 1.1\pm 3.7$ | $74.4\pm 3.4\pm 19.1$ |  |  | $D^{+}\pi^{-}$
7 | $D_{J}^{\ast}(3000)^{0}$ | $3008.1\pm 4.0$ | $110.5\pm 11.5$ |  |  | $D^{+}\pi^{-}$
8 | $D_{J}^{\ast}(2760)^{+}$ | $2771.7\pm 1.7\pm 3.8$ | $66.7\pm 6.6\pm 10.5$ |  |  | $D^{0}\pi^{+}$
9 | $D_{J}^{\ast}(3000)^{+}$ | 3008.1 | $66.7\pm 6.6\pm 10.5$ |  |  | $D^{0}\pi^{+}$

The states with $J^{P}=(0^{-},1^{-},0^{+},1^{+},1^{+},2^{+})$ are well known.
The doublets having spin-parity assignments $s_{l}^{P}=\frac{3}{2}^{+}$,
consists of $D_{1}(2420)$ and $D_{2}^{\ast}(2460)$ in non-strange sector. The
states $(D_{0}^{\ast}(2400),D_{1}^{\prime}(2430)$ belong to
$s_{l}^{P}=\frac{1}{2}^{+}$ charm doublet[2]. The experimental data on decay
widths suggest that states $(0^{+},1^{+})$ are quite broad, expecting to decay
via s-wave whereas the states belonging to $(1^{+},2^{+})$ doublets are quite
narrow and decay via d-wave. The measured branching ratio by BaBar
Collaboration is given as:

$\frac{BR(D_{2}^{\ast 0}(2460)\rightarrow D^{+}\pi^{-})}{BR(D_{2}^{\ast
0}(2460)\rightarrow D^{\ast+}\pi^{-})}=1.47\pm 0.03\pm 0.16$ (28)

There are few more recent states whose branching ratios as measured by BaBar
is mentioned below.

$\frac{BR(D^{0}(2600)\rightarrow D^{+}\pi^{-})}{BR(D^{0}(2600)\rightarrow
D^{\ast+}\pi^{-})}=0.32\pm 0.02\pm 0.09$ (29)

$\frac{BR(D^{0}(2760)\rightarrow D^{+}\pi^{-})}{BR(D^{0}(2750)\rightarrow
D^{\ast+}\pi^{-})}=0.42\pm 0.05\pm 0.11$ (30)

The information from the BaBar Collaboration and the quark model suggests that
$D^{0}(2550)$ state lies in $0^{-}$ state. The $D^{0}(2600)$ corresponds to
$1^{-}$ state either in the 2S or 1D spectrum respectively because this state
was observed in both $D{\pi}$ and $D^{\ast}\pi$ channels. If we find the mass
of these particular states using heavy quark symmetry and other theoretical
models [13],[14], it can be suggested that the state $D(2600)$ can be
identified as either radial excitation of heavy quark doublet $H$ or 1D. The
branching ratios for $D\pi$ and $D^{\ast}\pi$ for both the decay states are
calculated as $\frac{BR(D^{0}(2600)\rightarrow
D^{+}\pi^{-})}{BR(D^{0}(2600)\rightarrow D^{\ast+}\pi^{-})}=0.82$ for 2S and
$\frac{BR(D^{0}(2600)\rightarrow D^{+}\pi^{-})}{BR(D^{0}(2600)\rightarrow
D^{\ast+}\pi^{-})}=0.38$ The comparison with the experimental data results in
favor of 1D assignment. Therefore, the possible assignment for this particular
state can be 1D respectively. The theoretical estimation of coupling constant
for the strong decay width of mesons in this particular state is $0.53\pm
0.01$. The theoretical estimation of branching ratios from the heavy quark
effective theory [11] leads to the conclusion that there may be possibility of
violations in flavor and spin symmetry. In ref.[8], Sun et al. used the
${}^{3}P_{0}$ model to examine the strong decays of these states and they
concluded that $D^{0}(2600)$ state can be identified as the mixture of
$2^{3}S_{1}$ and $1^{3}D_{3}$ state. Therefore, the other possibility is that
D(2600) may be considered as a mixing state of 2S and 1D respectively. The
other two states $D(2750)$ and $D(2760)$ can be identified with
$J^{P}=(2^{-},3^{-})$ assignment. It is very interesting to point out that
non-strange partner of $D_{sJ}(2860)$ can be associated with D(2760) due to
mass gap which is about 150 MeV. There are also several references like
[10],[11] which suggest possible assignment for $D(2750)$ and $D^{\ast}(2760)$
state with the l=2, n=1 state. Moreover, the branching ratio measurement
$\frac{BR(D^{0}(2760)\rightarrow D^{+}\pi^{-})}{BR(D^{0}(2750)\rightarrow
D^{\ast+}\pi^{-})}$ gives value 0.80 from the leading order effective theory
which is found to be matching with the experimental data. Saturating the total
decay width with the ground state to two body decays, we can fit the
experimental data of LHCb and BaBar to estimate the coupling constant. We take
the experimental data of decay widths of recent states as mentioned in table 1
to find the hadronic coupling constants. The decay widths are calculated using
the decay formulae given above. We can fit the experimental data of BaBar and
LHCb collaborations to estimate the coupling constants. The observed radially
excited non-strange charm meson states in the heavy meson spectrum are the two
resonances (D(2550),$D^{\ast}(2600)$). The values of coupling constant is
obtained from the measured width of (D(2550)) and the computed value is
$0.35\pm 0.03$ [11]. The calculated value of coupling constants in our fitting
program for D(2550) comes out to be $0.40\pm 0.05$. The best fit value of
these coupling constants is estimated with in experimental error using chi-
square minimization technique. The errors are clearly dominated by the
statistical and systematic uncertainties. To check the consistency of our
fitting algorithm, the coupling constant estimation is being carried out for
the decays of $J^{P}=(0^{+},1^{+})$ and the fitted value comes out to be
$0.53\pm 0.04$. This value has been found to be matching well with predictions
from other theoretical approaches[11]. The value of coupling constants for
D(2750) and $D^{\ast}(2760)$ states are also estimated in our fitting program
which are $0.61\pm 0.01$ and $0.79\pm 0.03$. Let us consider new states
predicted by LHCb in the $D\pi$ and $D^{\ast}\pi$ spectrum and from the strong
decays,the states are labeled as $D_{J}^{\ast}(3000)^{0}\rightarrow
D^{+}\pi^{-}$ and $D_{J}^{\ast}(3000)^{+}\rightarrow D^{0}\pi^{+}$. On the
basis of LHCb data[2],the angular distribution of $D_{J}(3000)\rightarrow
D^{\ast}\pi$ is found to be consistent with unnatural parity. The possible
spin-parity assignment for $D_{J}(3000)^{0}$ can be
$J^{P}=0^{-},1^{+},2^{-},3^{+}...$ and $D_{J}^{\ast}(3000)$ have possible
spin-parity assignments $J^{P}=0^{+},1^{-},2^{+},3^{-},4^{+}.....$. Thus, it
can be stated that the two states can be higher radial excitations or can
belong to 1F states in the meson spectrum. Some possible indications about the
possible assignment of $J^{P}$ quantum numbers can be realized from the masses
of these states. One of the well known potential models [13] calculated the
masses of all possible excited mesonic states and we can suggest the following
possible spin and parities assignments of these newly discovered states. To
extract the detailed information about the newly observed states, we present
the summary of all D meson states and various possibilities of
$D_{J}^{\ast}(3000)$ and $D_{J}^{\ast}(3000)^{0}$ in the table 2 below.

Table 2:  Table showing all non-strange charm meson states

$s_{l}^{p}$ | $\frac{1}{2}^{-}$ | $\frac{1}{2}^{+}$ | $\frac{3}{2}^{+}$ | $\frac{3}{2}^{-}$ | $\frac{5}{2}^{+}$ | $\frac{7}{2}^{+}$ | $\frac{5}{2}^{-}$
---|---|---|---|---|---|---|---
n=1 |  |  |  |  |  |  |
| $D(1869)(J^{P}=0^{-})$ | $D_{0}^{\ast}(2400)(J^{P}=0^{+})$ | $D_{1}(2460)(J^{P}=1^{+})$ | $D(2750)\cite[cite]{[\@@bibref{}{Colangelo}{}{}]}(J^{P}=1^{-})$ | $D_{J}^{\ast}(3000)(J^{P}=2^{+})\star$ | $D_{J}(3000)(J_{P}=3^{+})\star$ | $D_{J}(3000)(J^{P}=2^{-})\star$
| $D^{\ast}(2010)(J^{P}=1^{-})$ | $D_{1}^{\backprime}(2430)(J^{P}=1^{+})$ | $D_{2}^{\ast}(2430)(J^{P}=2^{+})$ | $D(2760)\cite[cite]{[\@@bibref{}{Colangelo}{}{}]}(J^{P}=2^{-})$ | $D_{J}(3000)(J^{P}=3^{+})\star$ | $D_{J}^{\ast}(3000)(J_{P}=4^{+})\star$ | $D_{J}^{\ast}(3000)(J^{P}=3^{-})\star$
n=2 |  |  |  |  |  |  |
| $D(2550)(J^{P}=0^{-})$ | $D_{J}^{\ast}(3000)(J^{P}=0^{+})\star$ | $D_{J}(3000)(J^{P}=1^{+})\star$ | $D_{J}^{\ast}(3000)(J^{P}=1^{-})\star$ |  |  |
| $D(2600)(J^{P}=1^{-})$ | $D_{J}(3000)(J^{P}=1^{+})\star$ | $D_{J}^{\ast}(3000)(J^{P}=2^{+})\star$ | $D_{J}(3000)(J^{P}=2^{-})\star$ |  |  |
n=3 |  |  |  |  |  |  |
| $D_{J}(3000)(J^{P}=0^{-})\star$ | $D_{J}^{\ast}(3000)(J^{P}=1^{-})\star$ |  |  |  |  |

Wang et al.[10] also suggested the various possibilities of $D_{J}(3000)$
states and calculated the decay widths of these states in terms of relevant
coupling constants. We also suggest the same but we add the two more
possibilities i.e. $(1^{-},2^{-})$ and $(2^{-},3^{-})$ lying in 1D spectrum.
To analyze the spectrum of above $J^{P}$ states, we study the two body decay
behavior and calculate the branching ratios for the states for which decay to
$PM$ and $P^{\ast}M$ both are allowed.

Table 3:  Ratios of decay width for $D_{J}^{\ast}(3000)$ state $D_{J}^{\ast}(3000)$ | $D_{J}^{\ast}(3000)\rightarrow D^{\ast}\pi$ | $\frac{BR(D_{J}^{\ast}\rightarrow D^{\ast+}\pi^{-})}{BR(D_{J}^{\ast}\rightarrow D^{+}\pi^{-})}$
---|---|---
$s_{l}^{p}=\frac{5}{2}^{+},J^{P}=2^{+}$ | f-wave | 0.343
$s_{l}^{p}=\frac{7}{2}^{+},J^{P}=4^{+}$ | f-wave | 0.52
$s_{l}^{p}=\frac{1}{2}^{+},J^{P}=0^{+}$ | s-wave | 0
$s_{l}^{p}=\frac{3}{2}^{+},J^{P}=2^{+}$ | d-wave | 0.955
$s_{l}^{p}=\frac{1}{2}^{-},J^{P}=1^{-}$ | p-wave | 1.57
$s_{l}^{p}=\frac{5}{2}^{-},J^{P}=3^{-}$ | f-wave | 0.68
$s_{l}^{p}=\frac{3}{2}^{-},J^{P}=1^{-}$ | p-wave | 0.32

The table 3 collects the ratios of decay width for $D_{J}^{\ast}(3000)$ state.
Predicted ratios along with graphs can be analyzed to exclude some of the
assignments.

Figure 1:  Graph showing branching ratios Vs mass of decaying particle

The graphs show the variation of branching ratios Vs mass of decaying
particle. From the graphs, it can be stated that the states lying in 3S
doublet around the mass values of 3000 MeV does not produce ratios less than
or around 1. The spin partner of $D_{J}^{\ast}(3000)$ is $D_{J}(3000)$ which
is observed to decay via $D^{+}\pi^{-}$ channel. The other possible decay
modes are $DK$ and $D\eta$. If $D_{J}(3000)$ and $D_{J}^{\ast}(3000)$ belongs
to $2P(0^{+},1^{+})$ doublet then the allowed decay modes for
$D_{J}^{\ast}(3000)$ does not include $D^{\ast+}\pi^{-}$ decay channels which
agrees well with the experiments. In all other cases, decays to both the
channels $D\pi$ and $D^{\ast}\pi$ are observed. Moreover, the calculation of
branching ratios ($\frac{D_{J}^{\ast}(3000)\rightarrow
D^{+}\pi^{-}}{D_{J}^{\ast}(3000)\rightarrow D^{\ast+}\pi^{-}}$) for all
possible doublets suggests that the possible assignment of these two states
can also be either ($J^{P}=(1^{-},2^{-})$) or ($J^{P}=(2^{+},3^{+})$) for
$D_{J}^{\ast}(3000)$ and $D_{J}(3000)$ mesons. The state $3^{3}S_{1}$ state
decays to $D^{\ast}\pi$,$D\pi$,$D_{s}^{\ast}K$,$D_{s}K$ and $D\eta$ and the
coupling constant for this particular state lies near the value of $\simeq
0.1$. Also, the similar state appears to be a narrow D meson state. But its
partial decay width to $D^{\ast}\pi$ appears to be almost double than that of
$D\pi$ mode which is completely inconsistent with predictions by LHCb
collaboration. If we assign $2P(J^{P}=(1^{+},2^{+}))$ doublet to these mesons
than the coupling constant for strong decays to $\pi,K,\eta$ should lie
between 0.12-0.15. For ($2^{3}D_{3}$) state, $(D\pi,DK,D\eta)$ are the allowed
decay modes. Additional decay channels may include $D(2460)\pi$ and
$D(2420)\pi$. The additional information about these decay channels may help
to estimate coupling constant precisely. In addition to this, $2^{3}D_{1}$
state has most prominent decay mode is $D\pi$ therefore the ideal decay mode
to search for this state is $D\pi$. The ratio of total decay widths for
$D_{J}^{\ast}(3000)$ and $D_{J}(3000)$ can be calculated from the experimental
data of LHCb collaboration.

$\frac{\Gamma(D_{J}^{\ast}(3000))}{\Gamma(D_{J}(3000))}=0.587\pm 0.083$ (31)

The strong decay width ratios when calculated in HQET with leading order for
all the possible assignments produce values very far from the experimental
values. Thus, only $\frac{1}{m_{Q}}$ corrections when included, more authentic
results can be produced. Similar analysis can be carried out for the
$D_{J}(3000)$ mesons. The various possibilities include $3^{1}S_{0}$,
$2P(1^{+})$ and $1D(2^{-})$ and 1F etc.. A closer look at the decay width of
all the above states in the $D^{\ast}\pi$ spectrum suggests that the most
possible assignment can belong to 2P state. The most prominent decay mode for
$2P(1^{+})$ in $(1^{+},2^{+})$ doublet is found to be $D^{\ast}\pi$ which
matches well with the experimental data on decay width for $D_{J}(3000)$.
However, if we consider the $D_{J}(3000)$ as the spin partner of
$D_{J}^{\ast}(3000)$ then the only possibility can be that 2P$(1^{+})$ in
$(0^{+},1^{+})$ overlaps with that with $1^{+}$ in $(1^{+},2^{+})$ doublet.

##  5 Conclusion

We study the heavy meson decay width in the framework of heavy quark effective
theory that represents heavy quark and chiral symmetry at $\Lambda_{QCD}\simeq
1GeV$. We studied the recent charm meson states with their $J^{p}$ assignment.
The upcoming results at collaborations like LHCb produces the data for
branching ratios that is used to calculate the decay width, coupling constants
and suitable $J^{p}$ states. The coupling constants and their studies are
important to study heavy meson phenomenology. The accurate prediction of
coupling constants help to study the detailed interaction of heavy mesons. The
present work calculates the coupling constant for strong decays of non-strange
charm meson states (D(2550),D(2600),D(2750)and D(2760)) by using Chi-square
minimization techniques. The numerical values of decay widths from the
collaborations like LHCb, BaBaR and CDF are used to extract the values of
coupling constants. The various assignments of $J^{p}$ values to the above
mentioned states are also analyzed. The $J^{p}$ assignment for $D^{0}(2550)$
state is $0^{-}$ while $D^{0}(2600)$ is identified as mixture of $2^{3}S_{1}$
and $1^{3}D_{3}$ state with $J^{p}=1^{-}$. The states $D(2750)$ and $D(2760)$
are identified with $J^{P}=(2^{-},3^{-})$ assignment. All assignments to
$D_{J}(3000)$ are analyzed deeply and various possibilities for $J^{P}$ states
has been checked. Two more possibilities i.e. $(1^{-},2^{-})$ and
$(2^{-},3^{-})$ lying in 1D spectrum have also been included for analysis.
Most possible assignment in the present work is favored in 2P$(1^{+})$ for
$D_{J}(3000)$ state. While investigating for decays, it is concluded that the
results on decay widths are further helpful to search for un-known resonances
so that the excited meson spectrum for D meson family is clear to theorists as
well as experimentalists.

##  6 Acknowledgement

Part of the work is done under UGC Major Research project NO.SR/ 41-959/2012.

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