Open charm mesons in variational scheme and HQET
K. K. Vishwakarma Alka Upadhyay

An Efficient Two-Dimensional Functional Mixed-Effect Model Framework for Repeatedly Measured Functional Data
    CHENG CAO 

    Department of Data Science, City University of Hong Kong 

    and 

    JIGUO CAO 

    Department of Statistics and Actuarial Science, Simon Fraser University 

    and 

    HAO PAN,  YUNTING ZHANG, FAN JIANG 

    Department of Developmental and Behavioral Pediatrics, 
 Shanghai Children’s Medical Center, 
School of Medicine, Shanghai Jiao Tong University 
 
    and 

    XINYUE LI  Corresponding author: Xinyue Li, Email: xinyueli@cityu.edu.hk

    Department of Data Science, City University of Hong Kong
    September 9, 2024
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§ ABSTRACT  
The charm (D) and charm-strange (D_s) mesons are investigated in a variational scheme using  Gaussian trial wave functions. The Hamiltonian contains Song and Lin potential with a constant term dependent on radial and orbital quantum numbers. The Gaussian wave function used has a dependence on radial distance r, radial quantum number n, orbital quantum number l and a trial parameter μ. These wave functions are used to compute the expectation of Song and Lin potential dependent on r. The expectation of relativistic kinetic energy of both quarks is computed from the Fourier transform of the above Gaussian wave function in the linear momentum (p) space. The total energy (expectation of Hamiltonian) for each state is minimized with a parameter μ from the wavefunctions. The obtained spectra of D and D_s mesons are in good agreement with other theoretical models and available experimental masses. The mass spectra of D and D_s mesons are also used to plot Regge trajectories in the (J^P, M^2) and (n_r, M^2) planes. In (J^P, M^2) plane, both natural and unnatural parity states of D and D_s mesons are plotted. The trajectories are parallel and equidistant from each other. The two-body strong decays of D and D_s are analyzed in the framework of heavy quark effective theory using computed masses. The strong decay widths are given in terms of strong coupling constants. These couplings are also estimated by comparing them with available experimental values for observed states. Also, the partial decay width ratios of different states are analyzed and used to suggest assignments to the observed states. We have assigned the spin-parity to newly observed D^*_s2(2573) as the strange partner of D^*_2(2460) identified as 1^3P_2, D_1^*(2760) and D^*_s1(2860) as 1^3D_1, D^*_3(2750) and D^*_s3(2860) as 1^3D_3, D_2(2740) as 1D_2, D_0(2550) as 2^1S_0, D^*_1(2660) and D^*_s1(2700) as 2^3S_1, D^*_J(3000) as 2^3P_0 D_J(3000) as 2P_1, D^*_2(3000) as 1^3F_2 states. 



§ INTRODUCTION

The abundance of experimental investigation of hadrons containing heavy quarks has fueled many theoretical explorations in the past decade. A multitude of new findings are anticipated in this field with the current third run of LHC, which makes a busy path for the theoretical journey of heavy hadrons. The experimentally observed D- and D_s- states are listed in the review of particle physics (RPP) by Particle Data Group (PDG) <cit.>. Some of these states have confirmed J^P, while many others require more data to be confirmed. The S and P waves of non-strange and strange charm mesons are well-established states.  The J^P of states D_0(2550), D^*_1(2600), D_2(2740) and D^*_3(2750) reported in the RPP <cit.> are determined to be 0^-, 1^-, 2^- and 3^-, respectively, by LHCb Collaboration <cit.>. The D_J(3000) and D_J^*(3000) were observed for the first time by LHCb <cit.> in a study of D^+π^-, D^0π^+ and D^*+π^- final states. The D_J(3000) meson was suggested to have unnatural parity and decaying in the D^*+π^- channel final state. The other D meson D_J^*(3000) was reported to be decaying in the D^+π^- and D^0π^+ final states, which suggests it to have natural parity. Later, in 2016, LHCb collaboration <cit.> also reported another new resonance D_2^*(3000) in the study of B^-→ D^+π^-π^- resonant substructures. The D_2^*(3000) propounded to be inconsistent with the previously reported D_J^*(3000), inferring they are different states. However, the LHCb collaboration also acknowledged that both resonances may have the same origin regardless of different masses and decay widths. In the strange charm (D_s) sector, the states D_sJ(2700)^+ and D_sJ(2860)^+ were observed by BABAR Collaboration <cit.> in 2006. The state D_sJ(2700)^+ was confirmed by Belle Collaboration <cit.> in 2008 with J^P=1^-. Also, the decays D^*_s1(2700)^+→ D^* K and D^*_sJ(2860)^+ → D^* K were observed and branching fractions relative to DK were measured by BABAR Collaboration <cit.> in 2009. The states  D_sJ(2700)^+ and D_sJ(2860)^+ were later reconfirmed by LHCb <cit.> in 2012.  The same study reported a new broad structure D_sJ(3040)^+ in the excited D_s region. The LHCb Collaboration <cit.> in 2014 by analyzing the B_s^0→D^0K^-π^+ decays observed two resonance states D^*_s1(2860)^- with J^P=1^- and D^*_s3(2860)^- with J^P=3^- in the final state of D^0K^-. Therefore, the reported state D_sJ(2860) by BABAR <cit.> and LHCb <cit.> consists of two states with J^P values 1^- and 3^-. In 2016, LHCb Collaboration <cit.> reported the first observation of D_s2^*(2573)^+→ D^*+K_s^0 decay with a significance of 6.9σ and measured its branching fraction relative to D^+K^0_s final state. Also, the presence of D^*_s3(2860)^+→ D^*+K_s^0 decay was reported by the LHCb. A new D_s meson state D_s(2590)^+ is observed by LHCb Collaboration <cit.> in B^0→ D^- D^+ K^+ π^- decay at a center-of-mass energy of 13 TeV. The D_s(2590)^+ observed decaying into the D^+K^+π^- final state with J^P=0^-. The LHCb assigns this state as a strong candidate to be the D_s(2^1S_0)^+ state. 

Recent experimental developments have allowed studying higher radially and orbitally excited D- and D_s- mesons. Theoretical understanding of these recently observed mesons is an ongoing endeavor, and many phenomenological models are being used to investigate their different properties. To study masses and decays of D- and D_s- mesons, the heavy quark effective theory <cit.>, relativistic quark model <cit.>, Regge-phenomenology <cit.>, sum rules <cit.>, lattice QCD <cit.>, effective chiral Lagrangian approach <cit.>, and potential models <cit.> are used. A detailed discussion of the theoretical understanding of D and D_s mesons will be given in later sections. The observed states D^*(2300), D_1(2430), D_1(2420) and D^*_2(2460) are considered to be 1P states with J^P=0^+, 1^+, 1^+ and 2^+, respectively; however other interpretations <cit.> are also possible. 



The present paper uses Song and Lin's potential <cit.> to study the spectroscopy of the D and D_s mesons. The potential has previously been used to successfully study the quarkonium structures in mass spectroscopy, and their decays <cit.>. This potential is not a directly QCD-inspired potential; it contains a vector term inspired by the leptonic decay width of vector mesons and a scalar term for quark confinement at large distances.  Many other potentials are also used to study the spectroscopy and decays of heavy-light mesons <cit.>. This paper extends the potential to the heavy-light system from the heavy-heavy system to which Song and Lin introduced this potential. Further, to study the strong decays of D and D_s mesons, we have used the heavy quark effective theory (HQET). This phenomenological effective theory combines heavy quark symmetry with chiral symmetry to analyze the strong decays of heavy hadrons through the pseudoscalar Goldstone bosons. The details are given in the further sections. This paper is organized in the following way: In section <ref>, we present the framework of variational method with the employed potential and the spin-dependent potential for the hyperfine splitting. The formalism of heavy quark effective theory to study the strong decay is also given in this section. The results of the calculations are discussed in the section <ref>. The conclusions of the present analysis are given in the last section <ref>.



§ FRAMEWORK



 §.§ Song and Lin's Potential in variational scheme

The Hamiltonian of a system consisting of a heavy quark of mass m_Q and a light anti-quark of mass m_q is given below in Eq. (<ref>) with kinetic energies of both quarks and potential energy between them. This Hamiltonian treats the kinetic energy of both heavy and light quarks relativistically.

    H=√(p^2+m_Q^2)+√(p^2+m_q^2)+V(r)

The Gaussian trial wave function is used to solve for the eigen-energy of the Hamiltonian. 

    R_nl(μ,r)=   μ^3/2(2 Γ(n)/Γ(n+l+1/2))^1/2 (μ r)^l 
       × e^-μ^2 r^2/2 L_n-1^l+1/2(μ^2 r^2)

The wave function in the momentum space is the Fourier transform of Eq. (<ref>) given as

    R_nl(μ,p)=   (-1)^n/μ^3/2(2 Γ(n)/Γ(n+l+1/2))^1/2(p/μ)^l 
       × e^r^2/2μ^2 L_n-1^l+1/2(p^2/μ^2)

The wave functions are normalized, and a variational parameter μ is used, which will be estimated for all states.
The Song and Lin's potential <cit.> with a constant potential depending on radial and orbital angular momentum quantum numbers is given as 

    V(r)=-a/r^1/2+b r^1/2+V_0

where, V_0=a_0+cn+dl. The a, b, a_0, c and d are potential parameters. n and l are the usual radial and orbital quantum numbers. In the present study, the above potential in Eq.(<ref>) is applied on the D- and D_s- mesons in a variational method. The variational method we have employed to study the heavy-light mesons in the present work is inspired by Ref. <cit.>. Later, this method was used by Ref. <cit.> to study the masses and decay constants of heavy-light mesons. We have extended from these studies in the sense that we used a more general wave function to study higher radial and orbital excited states. The total energy of the system is given by the expectation of the Hamiltonian H as

    ⟨ψ | H | ψ⟩ = E(μ)

The variational parameter μ is estimated by minimizing the total energy d E(μ)/dμ=0 at μ=μ̅.  The masses taken from <cit.> of 1S, 1P, and 2S states of D-mesons and 1S state of D_s-mesons are used to fix the potential parameters and masses of quarks (m_Q and m_q). The parameters are a=1.17  GeV^1/2, b=0.11 GeV^-1/2, a_0=-0.401 GeV, c=0.33 GeV, d=0.20 GeV, quark masses m_c=2.31 GeV, m_u/d=0.36 GeV and m_s=0.51 GeV. The heavy quark mass m_c is higher than other models <cit.>. The present potential is not much explored for heavy-light systems, and the higher heavy quark mass may hint that this potential is better suited for bottom mesons. This will be analyzed in a future study. The obtained spin-averaged masses for D- and D_s- mesons and corresponding variational parameter μ are given in Table <ref> and <ref> respectively. The spin-averaged masses M_SA,J of mesons are given as

    M_SA,J=∑_J(2J+1)M_J/∑_J(2J+1)

The obtained masses are in fair agreement with the other theoretical and available experimental masses in Tables <ref> and <ref>. This is a good sign for the application of Song and Lin potential in heavy-light systems. We also appreciate the consistency of obtained masses by plotting Regge-trajectories, which gives good results. The variational parameter μ for all masses is also in good agreement with other models shown in Tables <ref> and <ref>.
The hyperfine splitting is added perturbatively using a potential distinguishing between electric and magnetic parts of the interaction between quark and anti-quark pair. Also known as the one-gluon exchange potential (OGEP) <cit.>.

    V_SD(r)=4/3α_s2/3m_Q m_q̅S_Q.S_q 4πδ(r)+4/3α_s1/m_Q m_q
       ×L.S/r^3
        +(L.S_Q/2 m_Q^2+L.S_q/2 m_q^2)(-1/rdV(r)/dr+8/3α_s1/r^3) 
       +4/3α_s1/m_Q m_q1/r^3[3(S_Q.r̂)(S_q.r̂)-S_Q.S_q]

The 1^st term in spin-dependent potential V_SD(r) describes the spin-spin interaction between quark and light anti-quark pair. The 2^nd and 3^rd terms consist of the interactions of orbital angular momentum (L) with spin of heavy quark (S_Q) and anti-quark (S_q). The last term describes the tensor interactions, where tensor operator S_12=2[3(S_Q.r̂)(S_q.r̂)-S_Q.S_q] is given by <cit.>, 

    ⟨S_12⟩=   -1/(2l-1)(2l+3)
       ×[12⟨L.S⟩^2+6⟨L.S⟩-4⟨S^2⟩⟨L^2⟩]
  
The hyperfine splitting of nS states is given by

    M(^3 S_1)-M(^1 S_0)= 8 α_s | R(0)|^2/9 m_1 m_2

The sum of individual spins of heavy quark and light anti-quark gives the total spin momentum S= S_Q + S_q. Also, the sum of the orbital angular momentum and the spin of the light antiquark makes the light angular momentum s_l = L + S_q. The total angular momentum is the sum of light angular momentum and heavy-quark spin, J = s_l + S_Q. 
The notations for orbital and spin angular momentum are usual ⟨L^2⟩=l(l+1), and ⟨S^2⟩=s(s+1). For l=1 the states with s_l=l+s_q=3/2 and J=s_l+s_Q=2 are denoted as ^2s+1l_J= ^3P_2 and the states with s_l=l-s_q=1/2 and J=s_l-s_Q=0 are denoted as ^1P_0. States with different total spins (s) and the same total angular momentum (J) get mixed through the spin-orbit interaction potential given in Eq. (<ref>). The physical states for P-wave (l=1) with J=1 are given by the linear combinations of the states with J=s_l+s_Q=1 for s_l=l-s_q=1/2 (^3P_1) and J=s_l-s_Q=1 for s_l=l+s_q=3/2 (^1P_1) as

    P_1    = ^1P_1 cosθ_nP +  ^3P_1 sinθ_nP
    
        P'_1    = - ^1P_1 sinθ_nP+ ^3P_1 cosθ_nP

Similar mixing expressions can be obtained for higher orbitally excited states (D and F waves). The higher mass state is denoted as P'_1, and the lower mass state is P_1. In the heavy-quark limit m_Q→∞, the light angular momentum s_l and s_Q are good quantum numbers. This mixing occurs due to the non-diagonal spin-orbit and tensor terms in Eq. (<ref>). The readers may go through these Ref <cit.> for more clarification. The mass shift due to the spin potential terms for P and D waves are given in <cit.>. For the mass shift in F wave, we give the relations below

    Δ E_nF_4=α_s 34/9 m_Q m_q̅⟨1/r^3⟩+3/4(1/m_q̅^2+1/m_Q^2)
    ⟨-1/rdV(r)/dr+8/31/r^3⟩
    Δ E_nF_2=-α_s 88/15 m_Q m_q̅⟨1/r^3⟩-(1/m_q̅^2+1/m_Q^2)⟨-1/rdV(r)/dr+8/31/r^3⟩

The expectation values are taken with the radial wavefunction R_nl(r) given in Eqn (<ref>) with μ=μ̅ given in Tables <ref> and <ref>. The mixing elements for F wave are given below:

    ⟨ ^7F_3|Δ E| ^7F_3⟩   =-α_s 6/21m_Q m_q̅⟨1/r^3⟩+(3/2m_q̅^2-27/14m_Q^2)⟨-1/rdV(r)/dr+8/31/r^3⟩
    ⟨ ^5F_3|Δ E| ^5F_3⟩   =-α_s 24/21m_Q m_q̅⟨1/r^3⟩+(-1/m_q̅^2-3/7m_Q^2)⟨-1/rdV(r)/dr+8/31/r^3⟩
    ⟨ ^7F_3|Δ E| ^5F_3⟩   =-α_s 2√(12)/21m_Q m_q̅⟨1/r^3⟩-√(21)/7m_q̅^2⟨-1/rdV(r)/dr+8/31/r^3⟩

Here the notation ^2s_l+1F_J is followed.  
We have computed the mixing elements and their eigenvalues and mixing angles through equations like (<ref>), (<ref>), and (<ref>) for l=1, 2, and 3. The physical state masses for D and D_s mesons are enumerated in the Tables <ref>, <ref>, <ref>, and <ref> for l=0, 1, 2 and 3 states with their mixing angles mentioned in the caption for the corresponding states. The masses are in good agreement with the available experimental and other theoretical masses. 
























 §.§ Strong Decays of charm- and charm-strange mesons

We use the heavy quark effective theory (HQET) to study the strong decays of charm (D) and charm-strange (D_s) mesons. In the heavy quark limit m_Q→∞, the heavy quark spin s_Q and light angular momentum s_l. Thus, these angular momenta (s_Q, s_l) and the total angular momentum J are separately conserved. This enables the classification of heavy-light mesons as doublets with different s_l. As discussed above, the total angular momentum is given as J=s_l± s_Q, and each doublet contains two states, called spin partners. The parity of the doublets is given as P=(-1)^l+1, where l is the orbital angular momentum. For l=0 and  s_l^P=1/2^-, the doublet comprises two states with J^P=(0^-, 1^-), denoted as (P, P^*). For l=1, we have two doublets with s_l^P=1/2^+ and 3/2^+. The two doublets having J^P_s_l=(0^+,1^+)_1/2 and J^P=(1^+,2^+)_3/2 are denoted as (P^*_0, P'_1) and (P_1, P_2^*), respectively. For l=2, we have s_l^P=3/2^- and 5/2^-. The doublets with J^P=(1^-,2^-)_3/2 are denoted as (P^*_1,P_2) and with J^P_s_l=(2^-,3^-)_5/2 are denoted as (P'_2,P_3^*). Carrying on for l=3, the s_l^P=5/2^+ and 7/2^+. The first doublet is denoted as (P^*_2,P_3) with J^P_s_l=(2^+,3^+)_5/2. The second doublet is denoted as (P'_3, P^*_4) with J^P_s_l=(3^+,4^+)_7/2. The classification of higher radial excitation is done in the same manner; to denote the radial excitation, we use the above notations with a tilde (P̃,P̃^*, ...). The effective Lagrangians can be constructed by introducing effective fields in each doublet to study the strong decay dynamics of the heavy-light mesons doublets. The field H_a corresponds to the s_l=1/2^+ doublet (a=u,d,s), S_a and T_a to s_l=1/2^+ and s_l3/2^+, X_a and Y_a to s_l=3/2^- and s_l=5/2^-, Z_a and R_a to the doublets s_l=5/2^+ and s_l=7/2^+, where a=u,d,s is the index for light quarks. The effective fields of the mentioned doublets are given below as 

    H_a   =1+v/2[P^*_aμγ^μ-P_aγ_5]
    
        S_a   =1+v/2[P'^μ_1aγ_μγ_5-P^*_0a]
    
        T^μ_a   =1+v/2{P^*μν_2aγ_ν-P_1aν√(3/2)γ_5 [g^μν-1/3γ^ν(γ^μ-v^μ) ] }
    
        X^μ_a   =1+v/2{P^μν_2aγ_5γ_ν-P^*_1aν√(3/2)[g^μν-1/3γ^ν(γ^μ-v^μ) ] }
    
        Y^μν_a   =1+v/2{P^*μνσ_3aγ_σ-P^'αβ_2a√(5/3)γ_5[g_α^μg_β^ν-1/5g^ν_βγ_α(γ^μ-v^μ)-1/5g^μ_αγ_β(γ^ν-v^ν) ] }
    
        Z^μν_a   =1+v/2{P^μνσ_3aγ_5γ_σ-P^*αβ_2a√(5/3)[g^μ_αg^ν_β-1/5g^ν_βγ_α(γ^μ+v^μ)-1/5g_α^μγ_β(γ^ν+v^ν) ] }
    
        R^μνρ_a   =1+v/2{P^*μνσ_4aγ_5γ_σ-P'^αβτ_3a√(7/4) 
                       ×[g^μ_αg^ν_βg^ρ_τ-1/7g^ν_βg^ρ_τγ_α(γ^μ-v^μ)-1/7g^μ_αg^ρ_τγ_β(γ^ν-v^ν)-1/7g^μ_αg^ν_βγ_τ(γ^ρ-v^ρ) ] }
    
where v is the meson four-velocity, conserved in the strong interactions. The operators P annihilate the corresponding mesons with four-velocity v. ref <cit.>. The operators include a factor of √(m_Q) and have dimension 3/2. These effective fields of mesons interact with each other through the pseudoscalar Goldstone bosons. The octet of light pseudoscalar mesons is introduced by defining the ξ=e^ιℳ/f_π and Σ=ξ^2, where the matrix ℳ incorporating the π, K and η mesons fields given as

    ℳ=
        [  √(1/2)π^0+√(1/6)η                π^+                K^+;                π^- -√(1/2)π^0+√(1/6)η                K^0;                K^-               K̅^0           -√(2/3)η ]

The matrix accommodates the field of light pseudoscalar π, K, and η mesons. To describe the transitions F→ HM, where F = H, S, T, X, Y, Z, R are effective fields of heavy meson doublet and M is a light pseudoscalar meson, at the leading order approximation in the light meson momentum and heavy quark mass expansion, the interaction Lagrangian terms are given as

    ℒ_HH   =g_HH[ H̅_a H_b γ_μγ_5 𝒜^μ_ba]
    ℒ_SH   =g_SH[H̅_a S_b γ_μγ_5 𝒜^μ_ba]+H.c.
    ℒ_TH   =g_TH/Λ_χ[H̅_aT^μ_b(ι̇D_μ𝒜+ι̇D𝒜_μ)_baγ_5 ]+H.c.
    ℒ_XH   =g_XH/Λ_χ[H̅_a X^μ_b(ι̇D_μ𝒜+
        ι̇D𝒜_μ)_baγ_5 ]+H.c.
    ℒ_YH   =1/Λ^2_χ[H̅_aY^μν_b[k^Y_1{D_μ,D_ν}𝒜_λ+k^Y_2(D_μD_λ𝒜_ν+D_νD_λ𝒜_μ)]_baγ^λγ_5]+H.c.
    ℒ_ZH   =1/Λ_χ[H̅_a Z^μν_b[k^Z_1{D_μ,D_ν}𝒜_λ+k^Z_2(D_μD_λA_ν+D_νD_λ𝒜_mu)]_baγ^λγ_5]+H.c.
    ℒ_RH   =1/Λ^3_χ[H̅_a R^μνρ[k^R_1{D_μ,D_ν,D_ρ}𝒜_λ+k^R_2(D_μ,D_ρD_λ𝒜_νD_ν,D_ρD_λ𝒜_μ
                               +D_μ,D_νD_λ𝒜_ρ)]_baγ^λγ_5]+H.c.

The definitions of identities in the above equations are D_μ=∂_μ+𝒱_μ; {D_μ,D_ν}=D_μD_ν+D_νD_μ; {D_μ,D_ν,D_ρ}=D_μD_νD_ρ+D_μD_ρD_ν+D_νD_μD_ρ+D_νD_ρD_μ+D_ρD_μD_ν+D_ρD_νD_μ. The axial-vector (𝒜) and vector (𝒱) currents are defined as 

    𝒜_μ=1/2(ξ^†∂_μξ-ξ∂_μξ^†)
    𝒱_μ=1/2(ξ^†∂_μξ+ ξ∂_μξ^†)

The chiral symmetry-breaking scale Λ_χ is set to 1 GeV. The strong coupling constants involved in their corresponding transitions are g_HH, g_SH, g_TH, g_XH, g_YH=k^Y_1+k^Y_2, g_ZH=k^Z_1+k^Z_2, g_RH=k^R_1+k^R_2. These couplings can be estimated by the experimental decay widths. The f_π is taken to be 132 MeV. The decay widths from the above effective fields and the interaction Lagrangians are given below <cit.>. The strong decay width for two-body decay of heavy-light (D or D_s) mesons by emitting pseudoscalar light mesons (π, η, and K) according to the Lagrangians ℒ_HH, ℒ_SH, ℒ_TH, ℒ_XH, ℒ_YH, ℒ_ZH and ℒ_RH is given as 

    Γ   =1/2J+1∑p_M/8π M_i^2|𝒜|^2 
    
        p_M   = √((M_i^2-(M_f+M_M)^2)(M_i^2-(M_f-M_M)^2))/2M_i

where, 𝒜 is the amplitude of the transition, J is total angular momentum of initial heavy meson, ∑ represents the sum over all polarization vectors, M_i is the mass of initial heavy meson, M_f represents the mass of final heavy meson, M_M and p_M are mass and momentum of emitted light pseudoscalar meson. The explicit expressions for decay widths of transition of heavy-light mesons from different fields (H, S, T, X, Y, Z, and R) to H-field for different channels are given below <cit.>.

Decays of H-field (0^+,1^+) of S-wave

    Γ(1^-→ 0^-)= C_M g_HH^2/6π f_π^2M_f/M_i|p⃗_M|^3
    Γ(1^-→ 1^-)= C_M g_HH^2/3π f_π^2M_f/M_i|p⃗_M|^3 
    Γ(0^-→ 1^-)=C_M g_HH^2/2π f_π^2M_f/M_i|p⃗_M|^3
   

Decays of S-field (0^+,1^+) of P-wave

    Γ(0^+→0^-)=C_M  g_SH^2/2π f_π^2M_f/M_i[M^2_M+|p⃗_M|^2]|p⃗_M|
    Γ(1^+→ 1^-)=C_M  g_SH^2/2π f_π^2M_f/M_i[M^2_M+|p⃗_M|^2]|p⃗_M|


Decays of T-field (1^+,2^+) of P-wave

    Γ(1^+→ 1^-)=C_M2g_TH^2/3π f_π^2 Λ_χ^2M_f/M_i|p⃗_M|^5
    Γ(2^+→ 0^-)=C_M4g_TH^2/15π f_π^2Λ_χ^2M_f/M_i|p⃗_M|^5
    Γ(2^+→ 1^-)=C_M2g_TH^2/5π f_π^2Λ_χ^2M_f/M_i|p⃗_M|^5


Decays of X-field (1^-,2^-) of D-wave

    Γ(1^-→ 0^-)=C_M4g_XH^2/9π f_π^2 Λ_χ^2M_f/M_i[M_M^2+|p⃗_M|^2]|p⃗_M|^3
    Γ(1^-→ 1^-)=C_M2g_XH^2/9π f_π^2 Λ_χ^2M_f/M_i[M_M^2+|p⃗_M|^2]|p⃗_M|^3
    Γ(2^-→ 1^-)=C_M2g_XH^2/3π f_π^2 Λ_χ^2M_f/M_i[M_M^2+|p⃗_M|^2]|p⃗_M|^3


Decays of Y-field (2^-,3^-) of D-wave

    Γ(2^-→ 1^-)=C_M4g_YH^2/15π f_π^2 Λ_χ^4M_f/M_i|p⃗_M|^7
    Γ(3^-→ 0^-)=C_M4g_YH^2/35π f_π^2 Λ_χ^4M_f/M_i|p⃗_M|^7
    Γ(3^-→ 1^-)=C_M16g_YH^2/105π f_π^2 Λ_χ^4M_f/M_i|p⃗_M|^7


Decays of Z-field (2^+,3^+) of F-wave

    Γ(2^+→ 0^-)=C_M4g_ZH^2/25π f_π^2 Λ_χ^4M_f/M_i[M_M^2+|p⃗_M|^2]|p⃗_M|^5
    Γ(2^+→ 1^-)=C_M8g_ZH^2/75π f_π^2 Λ_χ^4M_f/M_i[M_M^2+|p⃗_M|^2]|p⃗_M|^5
    Γ(3^+→ 1^-)=C_M4g_ZH^2/15π f_π^2 Λ_χ^4M_f/M_i[M_M^2+|p⃗_M|^2]|p⃗_M|^5


Decays of R-field (3^+,4^+) of F-wave

    Γ(3^+→ 1^-)=C_M36g_RH^2/35π f_π^2 Λ_χ^6M_f/M_i|p⃗_M|^9
    Γ(4^+→ 0^-)=C_M16g_RH^2/35π f_π^2 Λ_χ^6M_f/M_i|p⃗_M|^9
    Γ(4^+→ 1^-)=C_M4g_RH^2/7π f_π^2 Λ_χ^6M_f/M_i|p⃗_M|^9

where C_M is the factor dependent on the light pseudoscalar mesons, C_π_+=C_K^+=1, C_π^0=C_K_s=1/2, C_η=1/6 or 2/3 (for initial meson cq̅ or cs̅ respectively). The decay widths of different candidate states for the experimentally observed D- and D_s- meson family are calculated and shown in Table <ref> for 1P and 1D states, Table <ref> for 2S and 2P states, Table <ref> for 3S and 2D states, and Table <ref> for 1F states. In the next section, we will discuss the results from this section, and some meaningful conclusions will be drawn later.







§ RESULTS AND DISCUSSION

The calculated spin-averaged masses using the Song and Lin's potential for D(cq̅) and D_s(cs) mesons are listed in Tables <ref> and <ref>. The masses with hyperfine splittings are also calculated and given for D mesons in the Tables <ref>, <ref> and for D_s mesons in the Tables <ref>, <ref>. The masses are compared with other theoretical models and experiments are in good agreement. The mixing angles for nP states are in the lower range of different models <cit.>, while for nD states are on the higher range. The mixing angles for nF states are very small. The hyperfine splitting of D meson for S-wave are Δ(1S)=145 MeV, Δ(2S)=62 MeV, Δ(3S)=42 MeV, Δ(4S)=32 MeV and Δ(5S)=28 MeV are in fair agreement available experimental <cit.> and theoretical <cit.> splitting, where Δ(1S)=M(^3S_1)-M(^1S_0). For the strange charm D_s meson, Δ(1S)=80 MeV, Δ(2S)=62 MeV, Δ(3S)=42 MeV, Δ(4S)=32 MeV and Δ(5S)=28 MeV are also in fare accordance with other available masses <cit.>. 
In the non-strange sector, 1S mesons D, D^* and 1P mesons D^*_0(2300), D_1(2430), D_1(2420) and D_2^*(2460) are well established as 1S_1/20^-, 1S_1/21^-, 1P_1/20^+, 1P_1/21^+, 1P_3/21^+ and 1P_3/22^+ states, respectively, given in the PDG <cit.>. In strange sector, 1S mesons D_s(1968), D_s^*(2112) and 1P mesons D_s0^*(2317), D_s1^'(2460) and D_s1(2536) are also well established. The calculated masses of mesons are used to construct the Regge trajectories in the (J, M^2) and (n_r, M^2) planes with the following definitions:

    J=α_(n) M^2+α_(n)(0) 
    
        n_r=β_(L) M^2+β_(L)(0)

The Eq. (<ref>) is in the (J, M^2) plane and Eq. (n_r, M^2) plane. α and β are slopes, and α_0 and β_0 are intercepts of the Regge trajectories. Also, n=1,2,3,4,5 are the radial quantum number, and L=S, P, D, F represents the orbital excitations. 

The Regge trajectories for unnatural (P=(-1)^J+1) and natural (P=(-1)^J) parity states are shown in Figs. <ref> and <ref> for D mesons and in Figs. <ref> and <ref> for D_s mesons are shown. The slopes and intercepts for corresponding Regge trajectories are also shown in Fig. <ref>. All trajectories are parallel and equidistant to each other, and the calculated masses also fit nicely with the linear trajectories. 

In the following sections, many properties like masses, decay width, partial width ratios, and strong coupling constants are discussed and compared with the available experimental and theoretical values. This discussion and analysis are used to give the assignment to experimentally observed states. Further, some properties that can be observed in the future by experimental facilities are discussed to clear the status of some observed states.



























 §.§ TEXT

The calculated mass of D(1^3P_2) state is 2482 MeV, which is close to mass 2461.1 ± 0.7 MeV of D_2^*(2460) given in PDG <cit.>. Using the total decay width Γ(D^*_2(2460))=47.3 ± 0.8 MeV, we get g_TH=0.385±0.003, which is in accordance with g_TH computed in other Refs. 0.43±0.01<cit.>, 0.40±0.003<cit.>, 0.40±0.01<cit.>, 0.43±0.05 <cit.>. We found the ratio
    Γ(D^*_2(2460)→ D^+π^-)/Γ(D^*_2(2460)→ D^*+π^-)=2.14,
which is in agreement with the average of experimental value 1.52±0.14 <cit.> and other theoretical estimates 2.29 <cit.>, 2.26 <cit.>, 1.96 <cit.>, 1.70 <cit.>. For D^*_2(2460), BaBar collaboration <cit.> measured the ratio Γ(D^+π^-)/Γ(D^+π^-)+Γ(D^*+π^-)=0.62±0.03±0.02, we have estimated this ratio to be 0.68. In the D_s sector, the calculated mass of D_s(1P_3/22^+) state is 2581 MeV, which is very close to D_s2^*(2573) state mass 2569.1±0.8 given in PDG <cit.>. Using the calculated coupling g_TH, we find the total decay width 19.26 MeV is also close to 16.9±0.7 from PDG <cit.>. The ratio Γ(D_s2^*(2573)→ D^*0K^+)/Γ(D_s2^*(2573)→ D^0K^+)=0.11 is in well accordance with the experimental value Γ(D_s2^*(2573)→ D^*0K^+)/Γ(D_s2^*(2573)→ D^0K^+)<0.33 <cit.>. D_2^*(2460) and D_s2^*(2573) are spin partners of each other, and the analyzed ratios support the given assignment.


 §.§ TEXT

The calculated mass of D(1^3D_1) state is 2759 MeV. In Table <ref>, the Dπ is the dominant decay mode for this state. The candidate available for this state is D^*_1(2760)^0 with mass 2781±18±13 MeV and decay width 177±32±21 MeV reported by LHCb <cit.>. The total decay width of D^*_1(2760)^0 estimates the coupling g_XH to be 0.243±0.026. The D^*_1(2760)^0 is also studied as a mixing of 2^3S_1-1^3D_1 states by other theoretical models <cit.>. In the D_s family, the mass of D_s(1^3D_1) state is 2868 MeV, which is in good agreement with other theoretical models given in Table <ref>. The reported D^*_s1(2860)^± state by LHCb <cit.> with mass 2859±12±6±23 MeV and decay width 159±23± 27±72 MeV is a candidate of D_s(1^3D_1) state. The value of coupling g_XH from total decay width D^*_s1(2860)^± we get is 0.194±0.022. The average of coupling g_XH estimated from D^*_1(2760)^0 and D^*_s1(2860)^± states is 0.219±0.017, which is close to the previous prediction 0.19±0.049 <cit.>, 0.12 <cit.>, 0.41±0.02 <cit.> from the bottom meson studies.  


 §.§ TEXT

The estimated mass of D(1^3D_3) state is 2785 MeV. The D^*_3(2750) resonance can be interpreted as D(1^3D_3) state with observed mass 2763.1±3.2 MeV and decay width 66±5 MeV <cit.>. The decay width of D(1^3D_3) from Table <ref>, 493.89 g_YH^2 is used to estimate the coupling g_YH=0.366±0.014. The computed ratio R_π(D^*_3(2750))=1.88 for D^*_3(2750) is in agreement with other HQET studies like <cit.> but not in agreement with other models like R_π(D^*_3(2750))=1.1 <cit.>. The ratio R_π is defined as

    R_π(D(1^3D_3))=Γ(D(1^3D_3)→ Dπ)/Γ(D(1^3D_3)→ D^*π)

Further experimental observations are needed to understand the nature of D^*_3(2750). In the strange sector, the mass computed for D_s(1^3D_3) state is 2895 MeV. The D^*_s3(2860)^± has a mass 2860.5±2.6±6.5 MeV which makes it a candidate for the D_s(1^3D_3) state. After putting the value g_YH the decay width of D_s(1^3D_3) state is estimated to be 66±5 MeV, which is close to the experimental value 53±7±7 MeV. We computed the ratio R_K(D^*_s3(2860)) given as,

    R_K(D^*_s3(2860))=Γ(D^*_s3(2860))→ D^*K)/Γ(D^*_s3(2860))→ DK)=0.43

which suggests that DK channel is more dominant than D^*K. The experimental value of the ratio R_K(D^*_s3(2860))=1.04±0.17±0.20 <cit.>, which is higher than our result. The results of the other calculations of R_K(D^*_s3(2860))=0.73 <cit.> and R(K)=0.802 <cit.> also suggest that further experimental observation of D^*_s3(2860)^± with the ratio R_K(D^*_s3(2860)) is needed to clarify the behaviour. By above agreement in mass and total decay width of D_s3^*(2860), we tentatively assign it as 1^3D_3 states of Y- field in HQET.



 §.§ TEXT and TEXT

The masses of D(1D_2) and D(1D^'_2) are 2773 MeV and 2776 MeV, respectively, given in Table <ref>, which are the physical states from the mixing of states 1^1D_2 and 1^3D_2 as discussed above. From the above-computed values of g_XH and g_YH, the decay widths of 1D_2 and 1D^'_2 are found to be 110±23 MeV and 35.7±2.7 MeV respectively. The D_2(2740)^0 may be a candidate for D(1^1D_2) or D(1D^'_2) state with mass 2747±6 MeV, and decay width 88±19 MeV <cit.>. 

From Table <ref>, the 1D^'_2 state is the compact state which is also supported by <cit.>. Taking D_2(2740)^0 as the 1D^'_2 state, we get the coupling g_YH=0.57±0.06. Using the earlier value of g_YH, we get an average of g_YH=0.468±0.031 is close to the estimated values 0.42 <cit.>, 0.70 <cit.>, 0.49±0.039 <cit.>, 0.61±0.05 <cit.>.
In the strange sector, D_s(1D_2) and D_s(1D^'_2) states have masses 2881 MeV and 2886 MeV, and decay widths estimated at 126±20 MeV and 50±7 MeV. The experimental observation of these states in strange sector is currently missing, which we hope to see in the future. 


 §.§ TEXT and TEXT

The radially excited states are an essential part of the spectrum of mesons to understand the quark and anti-quark pair dynamics. The D(2^1S_0) and D(2^3S_1) states have calculated mass of 2563 MeV and 2625 MeV, respectively. The experimental candidate for D(2^1S_0) state is D_0(2550)^0 with mass 2549±19 MeV and decay width 165±24 MeV <cit.>. The candidate for D(2^3S_1) state is D^*_1(2600) with mass 2627±10 MeV and decay width 141±23 MeV. In Table <ref>, the state 2S_1/20^- decay in D^*π^- with ℬℛ=66.13% supports the experimental observation of D_0(2550)^0 in the D^*π^- channel by LHCb collaboration <cit.> and BaBar collaboration <cit.>. The coupling g̃_HH is found to be 0.375±0.027, considering the D_0(2550)^0 as D(2^1S_0) state. The D^*_1(2600) is reported in both Dπ and D^*π channels by LHCb collaboration <cit.> and BaBar collaboration <cit.>. From the experimental decay width of D^*_1(2600) and the total width in Table <ref> of 2S_1/21^-, we get the coupling g̃_HH=0.262±0.021. The theoretical ratio R_π(2^3S_1) is higher than the reported value 0.32±0.02±0.09 <cit.>. Many theoretical models have studied the D^*_1(2600) as the state of 2S-1D mixing <cit.>. In the D_s-meson sector, the computed masses of D_s(2^1S_0) and D_s(2^3S_1) are 2682 MeV and 2730 MeV with splitting (m(2^3S_1)-m(2^1S_0)) of 48 MeV is in agreement with the other theoretical models given in Table <ref>. The D_s1^*(2700)^± is a candidate for 2^3S_1 state with mass 2714 MeV and decay width 122±10 MeV.  Using the decay width of D_s1^*(2700)^±, the coupling g̃_HH is calculated as 0.255±0.010. The theoretically calculated ratio R_K(D_s1^*(2700))=0.96 is in excellent agreement with the experimental value 0.91±0.13±0.12 reported by BaBar collaboration <cit.>. 
The average g̃_HH from the above-mentioned values is 0.269±0.009. The total decay width of D_s(2^1S_0) is calculated as 63±4 MeV using the average value of g_HH. The newly reported D_s0(2590)^+ with mass 2591±6±7 MeV and decay width 89±16±12 MeV is suggested to be a strong candidate for the D_s(2^1S_0) state from LHCb collaboration <cit.>. Our calculated decay width for this state is close to the lower limits of the experimental value. We tentatively assign D_s0(2590)^+ as the candidate for D_s(2^1S_0) state. The D_s0(2590)^+ is observed to be decaying in the D^+K^+π^- final state. Further experiment observations may shed some light on the other decay channels and ratios of the D_s0(2590)^+.


 §.§ TEXT, TEXT and TEXT

The D^*_J(3000)^0 and D_J(3000)^0 states were observed by LHCb collaboration in 2013 <cit.>. The D^*_J(3000)^0 is reported to be decaying in the Dπ channel with mass 3008.1±4.0 MeV and decay width of 110.5±11.5 MeV. The D_J(3000)^0 is observed in the D^*+π^- channel with mass 2971.8±8.7 MeV and decay width 188.1±44.8 MeV. 
As LHCb collaboration <cit.> recommended, the D_J(3000)^0 with unnatural parity and D^*_J(3000) with natural parity. The suitable states for D_J(3000)^0 in the mass range are 3^1S_0(3S_1/20^-), 2P_1(2P_1/21^+), 2P^'_1(2P_3/21^+), 1F_3(1F_5/23^+), 1F^'_3(1F_7/23^+). While, the D^*_J(3000)^0 resonance is analysed as 3^3S_1(3S_1/21^-), 2^3P_0(2P_1/20^+), 2^3P_2(2P_3/22^+), 1^3F_2(1F_5/22^+), and 1^3F_4(1F_7/24^+). 


The masses of 3^1S_0 and 3^3S_1 are 3070 MeV and 3112 MeV, respectively. The mass difference Δ(3S-2S) between states 3^1S_0 and 2^1S_0 is about 500 MeV, consistent with other models. Considering both experimental resonances D_J^*(3000) and D_J(3000) as spin partners, the splitting Δ(3S)=m(3^3S_1)-m(3^1S_0)=42 MeV is close to the mass splitting of 36±10 MeV between the experimental masses of D_J^*(3000) and D_J(3000). But the calculated masses of 3^1S_0 and 3^3S_1 states are about 100 MeV higher than the measured masses of D_J^*(3000) and D_J(3000). Also, the ratio R_π(3^3S_1)=0.62 suggests that for 3^3S_1 state,  is in conflict with the observation of D^*_J(3000) is the Dπ channel. Considering D_J(3000) state as a candidate of 3^1S_0 state, the coupling g̃̃̃_HH = 0.150±0.018 is estimated.


The masses of  2^3P_0, 2P_1, 2P^'_1 and 2^3P_2 are 2881 MeV, 2947 MeV, 2956 MeV, and 2968 MeV, respectively. The 2^3P_0 state decay width given in Table <ref>, supports the assignment of D^*_J(3000) as 2^3P_0. The branching fraction of both D^+π^- and D^0π^0 channels are more than 60% and using the total decay width, we can estimate the coupling g̃_SH to be 0.115±0.006. Analyzing the D^*π channel for the D_J(3000), we can assign it as 2P_1 state, and the coupling g̃_SH using the total decay width is 0.162±0.019.  Among the 2P states, the 2^3P_0 and 2P_1 states are broad, and 2P^'_1 and 2^3P_2 are the narrow states. The D_J(3000) may also be 2P^'_1 state, and the coupling g̃_TH is calculated to be 0.205±0.024 by using the total decay widths. Also, the D^*_J(3000) state can be considered as the 2^3P_2 state, and this gives the value of the coupling g̃_TH=0.134±0.007. The average of this coupling is 0.139±0.007. The ratio R_π(2^3P_2)=1.08 may be a good indication of reported Dπ channel for D_J^*(3000) to be 2^3P_2 state. If the D_J^*(3000) is 2^3P_2 state then we propose that D_J^*(3000) should be searched in D^*π channel. Also, the branching ratio BR(D_s K) is about 30% for both 2^3P_0 and 2P_1 whereas the branching ratio BR(D_s K) is only about 15% for both 2P^'_1 and 2^3P_2. This can also be verified by future experiments.

The 1F states are also candidates for both D_J(3000) and D^*_J(3000). 
The calculated masses of 1F states are  3070 MeV, 3070 MeV, 3077 MeV, and 3078 MeV of 1^3F_2, 1F_3, 1F^'_3, and 1^3F_4 states, respectively. The 1^3F_2 and 1^3F_4 states have a dominant Dπ decay channel, and the ratio R_π(1^3F_2)=2.77 is greater than the R_π(1^3F_4)=1.79. Thus, the D^*_J(3000)^0 is tentatively preferred to be 1^3F_2 state, and using its experimental decay width, we calculate the coupling g_ZH=0.173±0.009. The possibility of D^*_J(3000)^0 being 1^3F_4 state may not be ignored. The 1F_3 and 1F^'_3 states are possible candidates of D_J(3000) as they have D^*π decay channels with branching ratio BR(D^*π)≈ 80%. Using the total decay widths, we estimate the coupling g_ZH=0.284±0.034 and the coupling g_RH=0.178±0.021. The branching fraction BR(D_s K) for 1^3F_2 and 1F_3 are 15%. The branching fraction BR(D_s K) for 1F^'_3 and 1^3F_4 states are about 8%. These fractions may be observed in future experiments to clear the properties of both D_J(3000) and D^*_J(3000). 


In the year 2016, LHCb collaboration <cit.> reported D^*_2(3000)^0 with mass 3214±29 MeV and decay width 186±38 MeV. The LHCb proposed D^*_2(3000)^0 to be a different state than D^*(3000) reported previously <cit.>, but it may not be ruled out that both states are the same. The mass and decay width of D^*_2(3000)^0 are higher than the D^*(3000) resonance. The preferred assignments suggested for D^*_2(3000)^0 resonance by LHCb collaboration are of 2P or 1F family. The masses for these states are already discussed above, and treating the D^*_2(3000)^0 and D^*(3000)^0 as different resonances, we assign the D^*_2(3000)^0 as a 1^3F_2 state. This assignment is favored as the mass of the 1^3F_2  state is higher than the mass of the 2^3P_2 state. Using the total decay width of D^*_2(3000)^0 and the 1^3F_2 state from the table <ref>, we compute the coupling g_ZH to be 0.225±0.023. In the view of D^*_2(3000)^0 assigned as 1^3F_2, we prefer the D^*(3000)^0 as a 2P state. To distinguish between the 1^3F_2 and 2^3P_2 states, we estimate ratio Γ(D_sK)/Γ(D^*_sK) for  1^3F_2 state to be 3.73 and for 2^3P_2 state to be 1.60. Also, the ratio R_π(1^3F_2)≈ 2.5 and R_π(2^3P_2)≈1 may be helpful in distinguishing between states in the experimental reports. 

We tentatively assign D^*_2(3000)^0 as 1^3F_2 state on the basis of masses and decay ratio R_π. Considering D^*_J(3000) and D_J(3000) as spin partners, we tentatively assign them as 2P(0^+,1^+) states with the coupling g̃_SH estimated to be 0.115±0.006. If D^*_J(3000) and D_J(3000) resonances are not spin partners, then the assignment of D_J(3000) as 3S(0^-) state can be favored with coupling g̃̃̃_HH=0.150±0.018.



 §.§ TEXT

The BaBar collaboration <cit.> in 2009 reported a resonance D_sJ(3040)^± in the inclusive production of D^*K in the e^+e^- annihilation with mass 3044±8^+30_-5 MeV and decay width of 239±35^+46_-42 MeV. The  D_sJ(3040) is a candidate for 2P_1 and 2P^'_1 states. The masses of these states are 3054 MeV and 3062 MeV, given in Table <ref>, which looks close to the measured mass. The decay widths of 2P states are given in Table  Ref tab:2s2ppwidth with the possible channels. The decay width of 2P_1 state is broader than the 2P^'_1 state. Considering the D_sJ(3040) as 2P_1 state and using the total decay widths, we estimate the g̃_SH=0.159±0.012. The average from all the calculated values of g̃_SH is 0.127±0.005 is in accordance with other estimations 0.12±0.03 <cit.>, 0.10±0.015 <cit.>. The coupling g̃_SH from the non-strange 2P states and their strange partner are in very good agreement with other theoretical calculations, as shown above. We assign the D_sJ(3040) resonance as the D_s(2P_1) state of S- field. Table <ref> shows the couplings calculated in the present study. 





§ CONCLUSIONS

We have analyzed the heavy-light charm mesons in the potential of Song and Lin with calculations and results of masses and decay widths in the above sections. The 1S, 1P, and 2S masses of charm mesons are fitted to estimate the potential parameters. The spectrum of the charm meson is given in Tables <ref>, <ref>, <ref>, and <ref>. This potential was proposed as an improvement over Cornell's potential by modifying the vector term from the leptonic decay width of vector mesons. The application of Song and Lin's potential to the heavy-light system studied in the present study can improve the understanding of quark-antiquark confining. The charm quark mass (m_c) estimated in the present study is on the higher side. This characteristic of the present potential will also be explored for bottom mesons. The masses obtained for different states are compared with the results of other theoretical models and experimental masses. The hyperfine splittings are computed with the mixing angles for the corresponding states. The strong decay widths are analyzed in the framework of HQET. The fields of the doublet states of mesons with interacting Lagrangians, as well as the final decay width formulae, are given. The total decay widths of different states with the possible channels are computed in terms of the corresponding couplings and shown in the tables <ref>, <ref>, <ref>, and <ref>. The assignments of observed resonances in different experimental facilities like LHCb and BaBar are done. The total decay widths of observed states are used to estimate the strong couplings. The branching ratios are also estimated and compared with available theoretical and experimental results. The tentative assignments of states are also discussed in the present study. We hope for further observations of the resonances like D_2(2740), D_J(3000), D^*_J(3000), D_2^*(3000), and D_sJ(3040) in different decay channels and properties.


§ ACKNOWLEDGMENT

The authors gratefully acknowledge the financial support by the
Department of Science and Technology 
(SERB/F/9119/2020), New
Delhi and for Senior Research Fellowship (09/0677(11306)/2021-EMR-I) by Council of Scientific and Industrial Research, New Delhi.



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