Chiral dynamics, S-wave contributions and angular analysis in $D\to ππ\ell\barν$
Yu-Ji Shi Wei Wang Shuai Zhao






INPAC, Shanghai Key Laboratory for Particle Physics and Cosmology, Department of Physics and Astronomy, Shanghai Jiao-Tong University, Shanghai, 200240,   China



We present a theoretical analysis of  the D^-→π^+π^-
ℓν̅ and  D̅^0→π^+π^0 ℓν̅ decays. We
construct a general  angular distribution which can include
arbitrary partial waves of ππ. Retaining the S-wave and P-wave
contributions we study the branching ratios, forward–backward
asymmetries and a few other observables. The P-wave contribution is
dominated by ρ^0 resonance, and the S-wave contribution is
analyzed using the unitarized chiral perturbation theory.  The
obtained branching fraction for D→ρℓν,  at the order
10^-3, is consistent with the available experimental data. The
S-wave contribution has a branching ratio  at the order of
10^-4, and this prediction can be tested  by experiments like
BESIII and LHCb. Future measurements can also be used to examine the
π–π scattering phase shift.


Chiral dynamics,  S-wave contributions  and angular analysis in  D→ππℓν̅ 
    
Yu-Ji Shi, Wei Wang [Email:wei.wang@sjtu.edu.cn] and Shuai Zhao [Email:shuai.zhao@sjtu.edu.cn] 
    December 30, 2023
====================================================================================================









§ INTRODUCTION



 The Cabbibo–Kobayashi–Maskawa (CKM) matrix elements are key
parameters in the Standard Model (SM). They are essential to
understand CP violation within the SM and search for new physics
(NP). Among these matrix elements,
 |V_cd| can
be determined from either exclusive or inclusive weak D decays,
which are governed by c→ d transition, for example, c→ dℓν transitions. However, for a general D decay process it is
difficult to extract CKM matrix elements, because strong and weak
interactions may be entangled.

 The semi-leptonic D decays are ideal
channels to determine |V_cd|, not only because the weak and
strong dynamics can be separated in these process, but also the
clean experimental signals. Moreover, one can study the dynamics in
the heavy-to-light transition from semi-leptonic D decays. For
leptons do not participate in the strong interaction, all the strong
dynamics is included in the form factors; thus it provides a good
platform to measure the form factors. The D→ρ form factors
have been measured from D^0→ρ^- e^+ ν_e and D^+→ρ^0
e^+ ν_e at the CLEO-c experiment for both charged and neutral
channels <cit.>. Because of the large width of the
ρ meson, D→ρℓν̅_ℓ is in fact a quasi-four
body process D→ππℓν̅_ℓ.  The  ρ can be
reconstructed from the P-wave ππ mode.  However,  other
ππ resonant or non-resonant states may interfere with the
P-wave ππ pair, and thus  it is necessary to analyze the
S-wave contribution to D→ππℓν̅_ℓ.


In addition, the internal structure of light mesons is an important
issue in hadron physics. It is difficult to study light mesons by
QCD perturbation theory due to the large strong coupling in the low
energy region. On the other hand, because of the large mass scale,
one can establish factorization for many heavy meson decay
processes, thus heavy mesons like B and D can be used to probe
the internal structure of light
mesons <cit.>.  As mentioned above,
D→ππℓν̅_ℓ can receive contributions from
various partial waves of ππ. ρ(770) dominant for D to
P-wave ππ decay, at the same time, D meson can decay into
S-wave ππ through f_0(980). The structure of f_0(980) is
not fully understood  yet. Analysis of D→ππℓν̅_ℓ may shed more light on understanding the nature of
f_0(980). The BESIII collaboration has collected 2.93
fb^-1 data in e^+e^- collisions at the energy around
3.773 GeV <cit.>, which can be used to study the
semi-leptonic D decays. Thus it presently is mandatory to make
reliable theoretical predictions. Some analyses of multi-body heavy
meson decays can be found in
Refs. <cit.>,
where the final state interactions between the light pseudoscalar
mesons are taken into account.

In this paper we present a theoretical analysis of
D^-→π^+π^-ℓν̅_ℓ and D^0→π^+π^0ℓν̅_ℓ decays.  In Sec. II, we will present the results of
D→ f_0 (980) and D→ρ form factors. We also calculate D
to S-wave ππ pair form factors in non-resonance region, the
ππ form factor will be calculated by using unitarized chiral
perturbation theory. Based on these results, we present a full
analysis on the angular distribution of D→ππℓν̅_ℓ. We explore various distribution observables, including
the differential decay width, the S-wave fraction, forward–backward
asymmetry, and so on. These results will be collected in Sec. III.
The conclusion of this paper will be given in Sec. IV. The details
of the coefficients in angular distributions are relegated to the
appendix.








 


§ HEAVY-TO-LIGHT TRANSITION FORM FACTORS


 







Feynman diagram for the D→ππℓ^-ν̅_ℓ decay is
shown in Fig. <ref>. The lepton can  be an  electron or
a muon, ℓ=e,μ. The spectator quark could be the u or d
quark, corresponding to D^0→π^+π^0 ℓ^-ν̅_ℓ
and D^-→π^+π^- ℓ^-ν̅_ℓ. Integrating out the
virtual W-boson,  we  obtain  the effective Hamiltonian describing
the c→  d transition

    ℋ_eff = G_F/√(2) V_cd [d̅γ_μ(1-γ_5) c][ ν̅γ^μ(1-γ_5) ℓ] +h.c.,

where G_F is the Fermi constant and V_cd is the CKM
matrix element. The leptonic part is calculable using the
perturbation theory, while the hadronic effects are encoded into the
transition form factors.







 §.§ D →ρ form factors

For the P-wave ππ state, the dominant contribution is from
the ρ(770) resonance. The  D→ρ form factors are
parametrized  by <cit.>
 
    ⟨ρ(p_2,ϵ)|d̅γ^μc|D(p_D)⟩   =   -2V(q^2)/m_D+m_ρϵ^μνρσϵ^*_ν  p_Dρp_2σ, 
    ⟨ρ(p_2,ϵ)|d̅γ^μγ_5 c|
      D(p_D)⟩   =   2im_ρ A_0(q^2)ϵ^*·  q / q^2q^μ
        +i(m_D+m_ρ)A_1(q^2)[ ϵ^*μ
        -ϵ^*·  q /q^2q^μ] 
          -iA_2(q^2)ϵ^* ·  q /  m_D+m_ρ[ P^μ-m_D^2-m_ρ^2/q^2q^μ],

 with q=p_D-p_2, and P=p_D+p_2.  The V(q^2), and
A_i(q^2) (i=0,1,2) are nonperturbative form factors.


These form factors have been computed in many different approaches
  <cit.>,
and here we quote the results from the light-front quark model
(LFQM) <cit.> and light-cone sum rules (LCSR)
 <cit.>. To access the momentum distribution in the full
kinematics region,  the following parametrization has been used:

    F_i(q^2)=F_i(0)/1-a_iq^2/m_D^2+b_i(q^2/m_D^2)^2.

Their results are  collected in Tab. <ref>. We
note that a different parametrization is adopted in
Ref. <cit.>, where A_3 appears instead of A_0. The
relation between A_0 and A_3 is given by

    A_0(q^2)=1/2m_ρ(m_D+m_ρ)[A_1(q^2)
    (m_D+m_ρ)^2+A_2(q^2)(m_ρ^2-m_D^2)-A_3(q^2)q^2].













 §.§ Scalar ππ form factor and D to S-wave ππ





We first give the D→ f_0(980) form factor parametrized as

    ⟨ f_0(p_2) |d̅γ_μγ_5  c|
        D^-(p_D)⟩   =   -i{    F_+^D→ f_0 (q^2)
        [P_μ - m_D^2-m_f_0^2/q^2q_μ ]
        +F_0^D→ f_0 (q^2)m_D^2-m_f_0^2/q^2q_μ},

where    F^D→ f_0_+ and F^D→ f_0_0 are D→ f_0 form
factors. We will use LCSR to compute the   D → f_0(980)
transition  form factors with some inputs, and we refer the reader
to Ref. <cit.> for a detailed derivation in LCSR.
The meson masses are fixed to the PDG values   m_D=1.870 GeV and
m_f_0=0.99  GeV <cit.>. For quark masses we use
 m_c=1.27 GeV <cit.> and m_d=5 MeV. As for
decay constants, we use f_D=0.21 GeV <cit.> and
f_f_0=0.18 GeV <cit.>. The threshold s_0 is
fixed at s_0=4.1 GeV^2, which should correspond to the squared
mass of the first radial excitation of D.  The parameters
F_i(0), a_i and b_i are fitted in the region
-0.5 GeV^2<q^2<0.5 GeV^2, and the Borel
parameter M^2 is taken to be (6±1) GeV^-2. With
these parametrizations, we give the numerical results in
Tab. <ref>.









In the region where the two pseudo-scalar mesons strongly interacts, the resonance approximation fails and thus has to be abandoned. One of the such examples is the S-wave partial wave under 1 GeV, for which we can use the form factors as defined  in Ref. <cit.>:

    ⟨ (ππ)_S(p_ππ)|u̅γ_μγ_5 c| D (p_D)
     ⟩   =    -i  1/m_ππ{[P_μ
     -m_D^2-m_ππ^2/q^2 q_μ]  F_1^D→ππ(m_ππ^2, q^2) 
          
     +m_D^2-m_ππ^2/q^2 q_μ F_0^D→ππ(m_ππ^2, q^2)  }.




The Watson theorem implies that phases measured in ππ elastic
scattering and in a decay channel in which the ππ system has
no strong interaction with other hadrons  are equal modulo π
radians. In the process we consider here, the lepton pair ℓν̅ indeed decouples from the ππ final state, and thus
the phases of D to scalar ππ  decay amplitudes  are equal to
ππ scattering with the same isospin. It is plausible that

    ⟨ (ππ)_S|d̅Γ c|D⟩∝ F_ππ(m_ππ^2),

where  the scalar form factor is defined as

    ⟨ 0| d̅d |π^+π^-⟩ = B_0
     F_ππ(m_ππ^2),

 where B_0=(1.7±0.2) GeV <cit.> is the QCD condensate
parameter. 


An explicit calculation of these quantities  requires knowledge of
generalized light-cone distribution amplitudes
(LCDAs) <cit.>. The twist-3 one  has the same
asymptotic form with the LCDAs  for a scalar
resonance <cit.>.   Inspired by this similarity, we may
plausibly  introduce an intuitive matching between the D→ f_0
and D→ (ππ)_S form factors <cit.>:

    F_i^D→ππ(m_ππ^2, q^2) ≃  B_01/ f_f_0  F_ππ(m_ππ^2) F_i^D→ f_0
     (q^2).


It is necessary to stress at this stage that the Watson theorem does
not strictly guarantee that one may use Eq. (<ref>).
Instead it indicates that, below the opening of inelastic channels
the strong phases in the D→ππ form factor and ππ
scattering are the same. First above the 4π or KK̅
threshold, additional inelastic channels will also contribute. The
KK̅ contribution can be incorporated in a coupled-channel
analysis.  As a process-dependent study, it has been demonstrated
that states with two additional pions may not give sizable
contributions to the physical observables <cit.>.
Secondly, some polynomials with nontrivial  dependence on
m_ππ have been neglected in Eq. (<ref>). In
principle, once the generalized LCDAs for the (ππ)_S system
are known,  the D→ππ form factor can be straightforwardly
calculated in LCSR and thus this approximation in the matching
equation can be avoided. On the one side,   the space-like
generalized parton distributions for the pion have been calculated
at one-loop level  in the chiral perturbation theory
(χPT) <cit.>. The analysis of time-like
generalized LCDAs in χPT and the  unitarized framework is in
progress.  On the other  side, the γγ^*→π^+π^-
reaction is helpful to extract the generalized LCDAs for the
(ππ)_S system <cit.>. The
experimental prospects at BEPC-II and BELLE-II in the near future
are very promising.



In the kinematic region where the π is soft, the crossed channel
from D+π→π will contribute as well and this crossed channel
would modify Eq. (<ref>) by an inhomogeneous part. For
the analogous decay of K or B mesons, it has been taken into
account either dynamically in terms of phase shifts (in the case of
the kaon decay) <cit.> or approximately in terms
of a pole contribution (in the case of the B meson
decay) <cit.>. However, if both pions move fast, the
D–π invariant mass is far from the D^* pole and this
contribution is negligible. In this case,  the transition amplitude
for the D to 2-pion form factor can be calculated in light-cone
sum rules <cit.>. This will lead to the conjectured
formula in Eq. (<ref>).




The scalar ππ form factor can be handled using the unitarized chiral perturbation theory. In the following, we will give a brief description of this approach.
In terms of  the isoscalar S-wave states

    |ππ⟩_I=0^    = 1/√(3)|π^+π^-⟩ + 1/√(6)|π^0π^0⟩, 
       |KK̅⟩_I=0^    =
    1/√(2)|K^+K^-⟩ +
    1/√(2)|K^0K̅^0⟩,

the scalar form factors for the π and K  mesons are defined as

    √(2)B_0  F^n/s_1(s)    =   ⟨ 0|n̅n /s̅s|ππ⟩_I=0^,
    
    √(2)B_0 F^n/s_2(s)    =   ⟨ 0|n̅n/s̅s|KK̅⟩_I=0^,

where s=m_ππ^2. The n̅n = (u̅u+d̅d)/√(2)
denotes the non-strange scalar current, and  the notation (π =
1, K = 2) has been introduced  for simplicity. With the above
notation, we have

    F_ππ(m_ππ^2)  = √(2/3) F_1^n(m_ππ^2).

 Expressions  have already been derived
in χPT  up to next-to-leading
order <cit.>:

    F_1^n(s)   =    . √(3/2)[ 1 + μ_π -
    μ_η/3 + 16 m_π^2/f^2(2L_8^r-L_5^r)
    + 8(2L_6^r-L_4^r)2m_K^2 + 3m_π^2/f^2 +
    8s/f^2 L_4^r + 4s/f^2 L_5^r 
           + .
    (2s - m_π^2/2f^2) J^r_ππ(s) +
    s/4f^2 J^r_KK(s) + m_π^2/18f^2 J^r_ηη(s)
    ],   
       F_1^s(s)   =    √(3)/ 2[ 16
    m_π^2/f^2(2L_6^r-L_4^r) + 8s/f^2 L_4^r +
    s/2f^2 J^r_KK(s) + 2/9m_π^2/f^2
    J^r_ηη(s)
    ],
       F_2^n(s)   =    . 1/√(2)[ 1 +
    .8 L_4^r/f^2(2s - m_π^2 - 6 m_K^2) +
    4 L_5^r/f^2(s - 4 m_K^2) + 16 L_6^r/f^2(6 m_K^2 + m_π^2) + 32 L_8^r/f^2 m_K^2 +
    2/3μ_η. 
           + . (9s
    - 8 m_K^2/36f^2) J^r_ηη(s) + 3s/4f^2
    J^r_KK(s) + 3s/4f^2 J^r_ππ(s)
    ], 
       F_2^s(s)   =     1 + 8 L_4^r/f^2(s - m_π^2 - 4
    m_K^2) + 4 L_5^r/f^2(s - 4 m_K^2) +
    16 L_6^r/f^2(4 m_K^2 + m_π^2) + 32
    L_8^r/f^2 m_K^2 + 2/3μ_η
           +
    (9s - 8 m_K^2/18f^2) J^r_ηη(s) +
    3s/4f^2 J^r_KK(s).

Here the L_i^r are the renormalized low-energy constants, and f
is the pion decay constant at tree level. The μ_i and J_ii^r
are defined as follows:

    μ_i    =   m_i^2/32π^2 f^2lnm_i^2/μ^2,
    
    J_ii^r(s)    =   1/16π^2[ 1 -
    log(m_i^2/μ^2) - σ_i(s)log(
    σ_i(s)+1/σ_i(s)-1)],

with  σ_i(s) = √(1- 4m_i^2/s). It is interesting to
note that the next-to-next-to-leading order results can also be
found in Refs. <cit.>. Imposing  the
unitarity constraints, the scalar form factor can be expressed  in
terms of the algebraic  coupled-channel equation

    F(s)    =    [I+K(s) g(s)]^-1 R(s) 
       =    [I-K(s) g(s)] R(s)  + 𝒪(p^6),

where R(s) has no right-hand cut and   in the second line, the
equation has been expanded up to NLO  in the  chiral expansion.
K(s) is the S-wave  projected  kernel of meson-meson scattering
amplitudes that can be derived from the  leading-order chiral
Lagrangian:

    K_11 = 2s - m_π^2/2f^2,   
       K_12 = K_21 = √(3)s/4f^2,     
       K_22 = 3s/4f^2 .

 The  loop integral can be  calculated either in the
cutoff-regularization scheme with q_ max∼ 1GeV being the
cutoff (cf. Erratum of Ref. <cit.> for an explicit
expression) or  in dimensional regularization with the
MS subtraction scheme. In the latter scheme,  the
meson loop function g_i(s) is  given by

    J_ii^r(s)    ≡   1/16π^2[ 1 -
    log(m_i^2/μ^2) - σ_i(s)log(
    σ_i(s)+1/σ_i(s)-1)] 
       =    -g_i(s).






The expressions for the R_i are obtained  by matching the
unitarization and chiral perturbation
theory <cit.>:

    R_1^n(s)   =    √(3/2){ 1 + μ_π -
    μ_η/3 + 16 m_π^2/f^2(2L_8^r-L_5^r)
    + 8(2L_6^r-L_4^r)2m_K^2 + 3m_π^2/f^2
    + 8s/f^2 L_4^r + 4s/f^2 L_5^r  
           -  m_π^2/288π^2 f^2[1 +
    log(m_η^2/μ^2)]
    }, 
       R_1^s(s)   =    √(3)/ 2{16
    m_π^2/f^2(2L_6^r-L_4^r) + 8s/f^2 L_4^r -
    m_π^2/72π^2 f^2[1 +
    log(m_η^2/μ^2)]
    },
       R_2^n(s)   =    1/√(2){ 1 + 8 L_4^r/f^2(2s - 6m_K^2 - m_π^2) + 4 L_5^r/f^2(s -
    4m_K^2) + 16 L_6^r/f^2(6m_K^2 + m_π^2)
    + 32 L_8^r/f^2 m_K^2 + 2/3μ_η.
           + .m_K^2/72π^2 f^2[1 +
    log(m_η^2/μ^2)]
    }, 
       R_2^s(s)   =     1 + 8 L_4^r/f^2(s - 4m_K^2 -
    m_π^2) + 4 L_5^r/f^2(s - 4m_K^2) +
    16 L_6^r/f^2(4m_K^2 + m_π^2)
    + 32 L_8^r/f^2 m_K^2 + 2/3μ_η
           + m_K^2/36π^2 f^2[1 +
    log(m_η^2/μ^2)].



With the above formulas and the fitted results for the low-energy
constants L_i^r in Ref. <cit.> (evolved  from
M_ρ to the scale μ= 2q_ max/√(e)), we show the
non-strange  ππ   form factor  in Fig. <ref>.  The
modulus, real part and imaginary part are shown as solid, dashed and
dotted curves. As the figure shows, the chiral unitary ansatz
predicts a form factor F^n_1 with a zero close to the K̅K
threshold. This feature has been extensively discussed in
Ref. <cit.>.






§ FULL ANGULAR DISTRIBUTION OF D→ΠΠℓΝ̅








In this section, we will derive a full angular distribution of D→ππℓν̅. For the literature, one may consult Refs. <cit.>.
We set up the kinematics  for the D^-→π^+π^-  ℓν̅
as shown in Fig. <ref>, which can also be used for   D^0→π^+π^0  ℓν̅. The  ππ moves
along the z axis in the D^- rest frame.
θ_π^+(θ_ℓ) is defined in the ππ (lepton
pair) rest frame as the angle between z-axis and the flight
direction of π^+ (ℓ^-), respectively. The azimuth angle
ϕ is the angle between the ππ decay and lepton pair
planes.


Decay amplitudes   for D→ππℓν̅_ℓ can
be divided into  several individual pieces and each of them can be
expressed in terms of the  Lorentz invariant helicity amplitudes.
The amplitude for the hadronic part can be obtained by   the
evaluation of the matrix element:

    A_λ  =   √( N_f_0/ρ)iG_F/√(2) V_cd^* ϵ_μ^*(h) ⟨ππ |c̅γ^μ(1-γ_5) d |D⟩,

where ϵ_μ(h) is an auxiliary polarization vector for the
lepton pair system and h= 0, ±, t,    N_f_0/ρ=
√(λ) q^2β_l /(96 π^3m_D^3), β_l=1-m̂_l^2 and m̂_l= m_l/√(q^2).   |V_cd| is taken to
be 0.22 <cit.>. The functions A_i can be
decomposed into different partial waves,

    A_0/t (q^2, m_ππ^2,θ_π^+)   =   ∑_J=0,1,2...  A^J_0/t (q^2, m_ππ^2)Y_J^0(θ_π^+,0),
    
     A_||/⊥(q^2, m_ππ^2,θ_π^+)   =   ∑_J= 1,2...  A^J_||/
     ⊥(q^2, m_ππ^2)Y_J^-1(θ_π^+,0),
    
     A^J_0/t (q^2, m_ππ^2)   =   √( N_f_0/ρ) M_D(f_0/ρ, 0/t )(q^2)
     L_f_0/ρ(m_ππ^2) ≡ | A^J_ 0/t |
     e^iδ^J_ 0/t ,
    
     A^J_ ||/⊥(q^2, m_ππ^2)   =   √( N_f_0/ρ) M_D(f_0/ρ,  ||/⊥)(q^2)
     L_f_0/ρ(m_ππ^2)≡ | A^J_||/⊥| e^iδ^J_ ||/
     ⊥.

 Here J denotes the partial wave of the ππ system and   the script t denotes the time-like component of a virtual vector/axial-vector meson decays into a lepton pair.  The  L_f_0/ρ(m_ππ) is the   lineshape and  for
the P-wave ρ we use the Breit–Wigner distribution:

    L_ρ(m_ππ^2)= √( m_ρΓ_ρ→ππ/π)1/m_ππ^2 -m_ρ^2+ i m_ρΓ_ρ.

Considering the  momentum dependence of the ρ decay,  we have
the running width as

    Γ_ρ (m_ππ^2) =  Γ_ρ^0 ( |q⃗ |/ |
     q⃗_0|)^3   m_ρ/m_ππ1+ (R|q⃗_0|)^2/1+ (R|q⃗ |)^2,

and the Blatt–Weisskopf parameter R=(2.1± 0.5± 0.5) 
GeV^-1 <cit.>.


The spin-0 final state has only one polarization state and the
amplitudes are

    i M_D(f_0,0)   =    N_1 i[   √(λ)/√( q^2) F_1(q^2)  ],   
     i M_D(f_0,t)=N_1 i[ m_D^2-m_f_0^2/√(q^2) F_0(q^2) ],

with  N_1=  iG_FV_cd^*/√(2). For mesons with
spin J≥1, the π^+π^- system can be either longitudinally
or transversely polarized and thus we have the following form:

    i M_D(ρ,0)   =   -α_L^J N_1  i/2m_ρ√(q^2)[   (m_D^2-m_ρ^2-q^2)(m_D+m_ρ)A_1
     -λ/m_D+m_ρA_2], 
    
     i M_D(ρ,±)
        =    -β_T^J N_1  i [  (m_D+m_ρ)A_1±√(λ)/m_D+m_ρV ],
    
     i M_D(ρ,  t)   =   -α_L^J i N_1√(λ)/√(q^2)A_0.

The α_L^J and β_T^J are products of the Clebsch–Gordan
coefficients

    α_L^J    =    C^J,0_1,0;J-1,0 C^J-1,0_1,0; J-2,0⋯ C^2,0_1,0;1,0,   β_T^J = C^J,1_1,1;J-1,0 C^J-1,0_1,0; J-2,0⋯ C^2,0_1,0;1,0.


For the sake of convenience, we define

    i M_D(ρ, ⊥/||)   =   1/√(2)[i M_D(ρ, +) ∓ i M_D(ρ, -)],
    
    i M_D(ρ,  ⊥)    =    -iβ_T^J √(2) N_1[
     √(λ)V/m_D+m_ρ],    
    i M_D(ρ,||)= -iβ_T^J√(2) N_1[
    (m_D+m_ρ)A_1 ].



Using the generalized form factor, the  matrix elements for D
decays into the spin-0 non-resonating  ππ  final state  are
given as

    A_0^0    =   √( N_2) i1/m_ππ[   √(λ)/√( q^2) F_1^ππ(m_ππ^2, q^2)  ] ,   
     A_t^0 =√( N_2) i 1/m_ππ[ m_D^2-m_ππ^2/√(q^2) F_0^ππ(m_ππ^2, q^2)],

N_2=N_1 N_ρρ_π/(16π^2), with ρ_π=
√(1-4m_π^2/m^2_ππ).




The above  quantities      can lead to the full angular distributions

    d^5Γ/dm_ππ^2dq^2dcosθ_π^+dcosθ_l dϕ   =   3/8[I_1(q^2, m_ππ^2, θ_π^+)   
           +I_2 (q^2, m_ππ^2, θ_π^+)
     cos(2θ_ℓ)  
           + I_3(q^2, m_ππ^2, θ_π^+) sin^2θ_ℓcos(2ϕ) 
          +I_4(q^2, m_ππ^2, θ_π^+)  sin(2θ_ℓ)cosϕ
           +I_5 (q^2, m_ππ^2, θ_π^+)  sin(θ_ℓ) cosϕ
          +I_6 (q^2, m_ππ^2, θ_π^+)  cosθ_ℓ
           +I_7 (q^2, m_ππ^2, θ_π^+)
     sin(θ_ℓ) sinϕ
           +I_8(q^2, m_ππ^2, θ_π^+)  sin(2θ_ℓ)sinϕ
          +I_9(q^2, m_ππ^2, θ_π^+)  sin^2θ_ℓsin(2ϕ)].

For the general expressions of I_i, we refer the reader to the
appendix and to Refs. <cit.> for the
formulas with the S-, P- and D-waves. In the following, we shall
only consider the S-wave and P-wave contributions and thus the above
general expressions are reduced to:

    I_1     =   1/4π[(1+m̂_l^2) |A^0_0|^2
     +2 m̂_l^2  |A_t^0|^2] + 3/4πcos^2θ_π^+[(1+m̂_l^2) |A^1_0|^2
     +2 m̂_l^2  |A_t^1|^2]   
          + 2√(3)cosθ_π^+/4π[  (1+m̂_l^2)  Re[A^0_0 A^1*_0]   + 2m̂_l^2  Re[A^0_t A^1*_t] ]
    
    
    +   3+m̂_l^2/23/8πsin^2θ_π^+    [|A^1_⊥|^2+|A^1_|||^2 ],
    
     
    
     I_2      =    -β_l     {1/4π |A^0_0|^2 + 3/4πcos^2θ_π^+    |A^1_0|^2   + 2√(3)cosθ_π^+/4π Re[A^0_0 A^1*_0]   }+
     
    
     
    
     
    
     1/2β_l    3/8πsin^2θ_π^+    (|A^1_⊥|^2+|A^1_|||^2),
     
    
     
    
     
     I_3   =   β_l    3/8πsin^2θ_π^+    (|A^1_⊥|^2-|A^1_|||^2),
     
    
     
    
     
     I_4
         =   2   β_l     [ √(3)sinθ_π^+/4√(2)π Re[A^0_0A^1*_|| ] + 3sinθ_π^+cosθ_π^+/4√(2)π Re[A^1_0A^1*_|| ]  ],
      
    
     
    
     I_5
         =   4{√(3)sinθ_π^+/4√(2)π  ( Re[A^0_0A^1*_⊥ ] -m̂_l^2   Re[A^0_tA^1*_|| ] )      +   3 sinθ_π^+cosθ_π^+/4√(2)π (  Re[A^1_0A^1*_⊥ ] -m̂_l^2      Re[A^1_tA^1*_|| ]) } ,
       
    
     
    
      
     I_6   =    4{3/8πsin^2θ_π^+ Re[ A^1_||A^1*_⊥ ]  +  m̂_l^2 1/4π Re[A_t^0 A_0^0*] + m̂_l^2 3/4πcos^2θ_π^+ Re[A_t^1 A_0^1*] }
    
     
    
      I_7
        =    4{√(3)/4√(2)πsinθ_π^+   ( Im[A^0_0A^1*_||] - m̂_l^2  Im[ A_t^0 A_⊥^1*] )   
           +    3/4√(2)πsinθ_π^+cosθ_π^+  ( Im[A^1_0A^1*_||] - m̂_l^2  Im[ A_t^1 A_⊥^1*] )}
    
     
    
     
     I_8    =   
     2 β_l  {√(3)/4√(2)πsinθ_π^+ Im [A_0^0 A_⊥^1*]+   3/4√(2)πsinθ_π^+cosθ_π^+ Im [A_0^1 A_⊥^1*]},
     
    
     
    
     
     I_9
        =   2 β_l  3/8πsin^2θ_π^+ Im[A_⊥^1 A_||^1* ].

Since the phase in P-wave contributions arise from the lineshape
which is the same for different polarizations, the I_9 term  and
the second line   in the I_7 are zero.



 §.§ Differential and integrated decay widths



Using the narrow width approximation, we obtain  the integrated
branching fraction:

    B(D^-→ρ^0 e^-ν̅)    =    (2.24±0.09)× 10^-3/(2.16±0.36)× 10^-3(LFQM/LCSR),
     B(D^-→ρ^0 μ^-ν̅)    =    (2.15± 0.08)× 10^-3/(2.06± 0.35)× 10^-3(LFQM/LCSR),
     B(D̅^0→ρ^+ e^-ν̅)    =    (1.73± 0.07)× 10^-3/(1.67± 0.27)× 10^-3(LFQM/LCSR),

where theoretical errors are from the heavy-to-light transition form factors.
These theoretical results are in good agreement with the
data <cit.>:

    B(D^-→ρ^0 e^-ν̅)    =    (2.18^+0.17_-0.25)× 10^-3,
     B(D^-→ρ^0 μ^-ν̅)    =    (2.4±0.4)× 10^-3,
     B(D̅^0→ρ^+ e^-ν̅)    =    (1.77±0.16)× 10^-3.



The starting point for detailed analysis of D→ππℓν̅
is to obtain the double-differential distribution
d^2Γ/dq^2 dm_ππ^2 after
performing integration over all the angles

    d^2Γ/dq^2 dm_ππ^2   =   (1+m̂_l^2/2)( |A_0^0|^2 +  |A_0^1|^2  +
    |A_||^1|^2 + |A_⊥^1|^2 )   + 3/2m̂_l^2
    (|A_t^1|^2 + |A_t^0|^2 ),

where apparently in the massless limit for the involved lepton,  the
total normalization for angular distributions changes to the sum of
the S-wave and P-wave amplitudes

    d^2Γ/dq^2 dm_ππ^2 =
    |A_0^0|^2 + |A_0^1|^2 + |A_||^1|^2 +  |A_⊥^1|^2.



In  Fig. <ref>, we give
the dependence of branching fraction  on m_ππ in the D^-→π^+π^-e^-ν̅_e process.    The solid, dashed, and dotted    curves correspond to the total,  S-wave and P-wave contributions.
For the S-wave contribution, there is no resonance around 0.98 GeV, and theoretically, this should be a dip.


Due to the quantum number constraint, the process D̅^0→π^+π^0 ℓν̅ receives only a P-wave contribution and  D^-→π^0π^0 ℓν̅ is generated by the S-wave term.












To match the kinematics constraints  implemented in experimental
measurements, one may explore the generic observable  with
m_ππ^2 integrated out:

    ⟨ O⟩ = ∫_(m_ρ-δ_m)^2^(m_ρ+δ_m)^2dm^2_ππdO/dm_ππ^2.

We use the following choice  in
our study of D→ππℓν̅:

    δ_m = Γ_ρ.

In the narrow width-limit, the integration of the lineshape is conducted as

    ∫dm^2_ππ|L_ρ(m^2_ππ)|^2 =  B(ρ^0→π^-π^+)=1.

However, with the explicit form given in
Eq. (<ref>), we find  that the integration

    ∫_(m_ρ-δ_m)^2^(m_ρ+δ_m)^2dm^2_ππ|L_ρ(m^2_ππ)|^2 =0.70

is below the expected value.
On the other hand,  the integrated  S-wave lineshape in this region is

    ∫_(m_ρ-δ_m)^2^(m_ρ+δ_m)^2dm^2_ππ|L_S(m^2_ππ)|^2 =  0.37,

which is smaller but  at the same order. Integrated from
m_ρ-Γ_ρ to m_ρ+Γ_ρ, we have

    B(D^-→ρ^0 (→π^+π^-) e^-ν̅) = (1.57± 0.07)× 10^-3/(1.51± 0.26)× 10^-3 (LFQM/LCSR),
     B(D^-→ρ^0 (→π^+π^-) μ^-ν̅) = (1.57± 0.07)× 10^-3/(1.51± 0.26)× 10^-3 (LFQM/LCSR).

The S-wave branching fractions for 2m_π<m_ππ<
1.0 GeV are given as

    B(D^-→    (π^+π^-)_S e^-ν̅) =   (6.99± 2.46)× 10^-4,
     B(D^-→  (  π^+π^-)_S μ^-ν̅) =  (7.20± 2.52)× 10^-4.

Above 1 GeV, the unitarized χPT will fail and thus we lack any
reliable prediction.


 Furthermore, one may   explore the q^2-dependent ratio

    R_ππ^μ/e(q^2)    =   ⟨dΓ(D→ππμν̅_μ)/dq^2 ⟩/⟨dΓ( D→ππ
    eν̅_e)/dq^2⟩.

Differential decay widths for  D→ππℓν̅_ℓ are
given in Fig. <ref>,  with ℓ= e in panel (a) and
ℓ=μ in panel (b). The q^2-dependent ratio
R_ππ^μ/e  is given in panel (c). Errors from the form
factors and QCD condensate parameter B_0 are shown as shadowed
bands, and most errors cancel in the ratio R_ππ^μ/e
given in panel (c).








 

 §.§ Distribution in θ_π^+ 







We explore the distribution in θ_π^+:

    d^3Γ/dq^2 dm_ππ^2 dcosθ_π^+   =   π/2 (3I_1-I_2) 
       =   1/8{ (4+2m̂_l^2) |A_0^0|^2 + 6m̂_l^2 |A_t^0|^2 
            + √(3) (8+4m̂_l^2) cosθ_π^+ Re[A_0^0 A_0^1*] + 12√(3)m̂_l^2 cosθ_π^+ Re[A_t^0 A_t^1*] 
          + (12+ 6m̂_l^2) |A_0^1|^2 cos^2θ_π^+ + 18m̂_l^2 cos^2θ_π^+ |A_t^1|^2   
            + (6+3m̂_l^2) sin^2θ_π^+  (|A_⊥^1|^2 +
    |A_||^1|^2)  }.





Compared to the distribution with only P-wave contribution, namely
D→ρ(→ππ)ℓν̅,  the first two lines   of
Eq. (<ref>) are new: the first one is the
S-wave ππ contribution,  while the second line  arises from
the interference of   S-wave and P-wave.  Based on this
interference,  one can define a forward–backward  asymmetry for the
involved pion,

    A_FB^π   ≡   [∫_0^1
     - ∫_-1^0] dcosθ_π^+d^3Γ/dq^2 dm_ππ^2
      dcosθ_π^+
       =   √(3)/2  (2+ m̂_l^2)   Re[A_0^0 A_0^1*] + 3√(3)/2m̂_l^2    Re[A_t^0 A_t^1*].















We define the polarization fraction  at a given value of q^2 and
m_ππ^2:

    F_S (q^2, m_ππ^2)  =  (1+m̂_l^2/2) |A_0^0|^2 + 3/2 m̂_l^2 |A_t^0|^2 /d^2Γ/(dq^2 dm_ππ^2) ,
     F_P  (q^2, m_ππ^2)  =   (1+m̂_l^2/2)(|A_0^1|^2  + |A_||^1|^2 +  |A_⊥^1|^2 ) + 3/2m̂_l^2 |A^1_t|^2 /d^2Γ/(dq^2 dm_ππ^2) ,

and also

    F_L (q^2, m_ππ^2)  =    (1+m̂_l^2/2)|A_0^1(q^2, m_ππ^2)|^2  + 3/2m̂_l^2 |A^1_t|^2  /   (1+m̂_l^2/2)(|A_0^1|^2  + |A_||^1|^2 +  |A_⊥^1|^2 ) + 3/2m̂_l^2 |A^1_t|^2 ,
    A_FB^π(q^2, m_ππ^2)=  √(3)/2  (2+m̂_l^2)   Re[A_0^0 A_0^1*] +  3√(3)/2m̂_l^2    Re[A_t^0 A_t^1*] /d^2Γ/(dq^2 dm_ππ^2) .

By definition, F_S+ F_P=1.

In Fig. <ref>, we give our results for the S-wave
fraction ⟨  F_S⟩ (panel (a)), longitudinal
polarization fraction ⟨  F_L⟩  in P-wave contributions
(panel (b)) and the asymmetry ⟨A_FB^π⟩ (panel (c)). Only  the curves  for the light lepton e are
shown since the results for the μ lepton are similar. These
observables and the following ones are defined by the integration
over m_ππ^2; for instance,

    ⟨  F_S (q^2) ⟩ = ∫dm_ππ^2 [ (1+m̂_l^2/2) |A_0^0|^2 + 3/2 m̂_l^2 |A_t^0|^2]  /∫dm_ππ^2  d^2Γ/(dq^2 dm_ππ^2) ,

and likewise for the others.





 §.§  Distribution in θ_l and forward–backward asymmetry









Integrating over θ_π^+ and ϕ, we have the
distribution:

    d^3Γ/dq^2 dm_ππ^2 dcosθ_l   =   3π/4∫dcosθ_π^+ (I_1 +I_2cos(2θ_l) +I_6 cosθ_l )
       =   3/4m̂_l^2 ( (|A_t^0|^2 +|A_t^1|^2 ))  + 3/2cosθ_l  (  Re[A_||^1 A_⊥^1*] + m̂_l^2   Re[A_t^0 A_0^0*+A_t^1 A_0^1*] )
    
           +3/4 [1 -(1-m̂_l^2)cos^2θ_l] (|A_0^0|^2 +|A_0^1|^2 ) + 3/8 [ (1+m̂_l^2) + (1-m̂_l^2) cos^2θ_l ]  (|A_||^1|^2 +|A_⊥^1|^2 ).

The forward–backward asymmetry is defined as

    A_FB^l   ≡   [∫_0^1
     - ∫_-1^0] dcosθ_l d^3Γ/dq^2 dm_ππ^2
      dcosθ_l   =  3/2   (  Re[A_||^1 A_⊥^1*] + m̂_l^2   Re[A_t^0 A_0^0*+A_t^1 A_0^1*] ) ,

and the results  for A_FB^l  are given in
Fig. <ref>.




 

 §.§ Distribution in the azimuth  angle  ϕ 








The angular distribution in ϕ is derived  as

    d^3Γ/dq^2 dm_ππ^2
    dϕ   =    a_ϕ +b_ϕ^c cosϕ + b_ϕ^s sinϕ +
    c_ϕ^c cos(2ϕ) + c_ϕ^s sin(2ϕ)

with

    a_ϕ   =   1/2πd^2Γ/dq^2 dm_ππ^2  ,
    
     b_ϕ^c    =   3/16π∫  I_5 dcosθ_π^+ =  3√(3)/32√(2)π( Re[A_0^0 A_⊥^1*]-m̂_l^2  Re[A_t^0 A_⊥^1*])
    
     b_ϕ^s    =   3/16π∫  I_7  dcosθ_π^+=  3√(3)/32√(2)π( Im[A_0^0 A_⊥^1*]-m̂_l^2  Im[A_t^0 A_⊥^1*])
    
     c_ϕ^c    =   1/2∫ I_3 dcosθ_π^+ =  1/4πβ_l (|A_⊥^1|^2-|A_||^1|^2),    
     c_ϕ^s =1/2∫  I_9 dcosθ_π^+=  1/ 2 πβ_l  Im[A_⊥^1 A_||^1*].

Since the complex phase in the P-wave amplitudes comes from the
Breit–Wigner lineshape, the coefficient c_ϕ^s vanishes.

Numerical results for the normalized coefficients  using the two
sets of form factors   are shown in Fig. <ref>. The
coefficients b_ϕ^c and b_ϕ^s contain a very small
prefactor, 3√(3)/(32√(2)π )∼ 0.037, and thus are
numerically tiny, as shown in this figure. The c_ϕ^c is also
small due to the cancellation between the |A_⊥|^2 and
|A_|||^2.











 §.§ Polarization of μ lepton








In this work, we also give the polarized angular distributions as

    d^5Γ(λ_μ)/dm_ππ^2dq^2dcosθ_π^+dcosθ_l dϕ   =   3/8[I_1^(λ_μ)   +I_2 ^(λ_μ)cos(2θ_l)   + I_3^(λ_μ)sin^2θ_l
     cos(2ϕ) 
          +I_4^(λ_μ)sin(2θ_l)cosϕ  +I_5^(λ_μ)sin(θ_l) cosϕ  +I_6 ^(λ_μ)cosθ_l 
           +I_7^(λ_μ)sin(θ_l) sinϕ  +I_8^(λ_μ)sin(2θ_l)sinϕ  +I_9^(λ_μ)sin^2θ_l
     sin(2ϕ)],

with the coefficients

    I_1^(-1/2)   =     |A_0|^2  +3/2 (|A_⊥|^2
     +|A_|||^2),     
     I_2^(-1/2)   = -  |A_0|^2+ 1/2  (|A_⊥|^2
     +|A_|||^2),
     
    
      I_3^(-1/2)   =      |A_⊥|^2-|A_|||^2 ,   
     I_4^(-1/2)    =  2 Re(A_0A_||^*),   
     I_5^(-1/2)
      =4  Re(A_0A_⊥^*), 
    
     I_6^(-1/2)    =     4
       Re(A_||A^*_⊥),    
       I_7^(-1/2)   = 4  Im(A_0A^*_||)  ,    
     I_8^(-1/2)     =  2    Im(A_0A^*_⊥),     
     I_9^(-1/2)  =2     Im(A_⊥A^*_||).

The coefficients for the λ_μ=1/2 are easily obtained by
comparing Eqs. (<ref>) and
(<ref>). For instance, the lepton
polarization fraction is defined as

    A^λ_μ(q^2, m_ππ^2 )    =   d^2Γ^(1/2)/dq^2dm_ππ^2 - d^2Γ^(-1/2)/dq^2dm_ππ^2 /d^2Γ/dq^2dm_ππ^2 
       =   (-1+ m̂_l^2/2)( |A_0^0|^2 +  |A_0^1|^2  + |A_||^1|^2 +  |A_⊥^1|^2 )   + 3/2m̂_l^2 (|A_t^1|^2 + |A_t^0|^2 )/d^2Γ/dq^2dm_ππ^2 ,

and we show the numerical results in   Fig. <ref>.





 §.§ Theoretical uncertainties



Before closing this section, we will briefly  discuss the
theoretical uncertainties in this analysis.  The parametric errors
in  heavy-to-light transition form factors and   QCD condensate
parameter B_0 have been included in the above. As one can see,
these uncertainties are sizable to  branching fractions and other
related observables, but are negligible in the ratios like
R_ππ^μ/e. This is understandable, since most
uncertainties will cancel in the ratio.


For the heavy-to-light form factors, we have used the LCSR and LFQM
results. In LCSR,  the theoretical accuracy  for most form factors
is at leading order in α_s. An analysis of B_s→
f_0 <cit.> has indicated the NLO radiative
corrections to form factors may reach 20%. The radiative
corrections are, in general, channel-dependent but  should be
calculated in a high precision study. It should be pointed out that
radiative corrections in the light-front quark model is not
controllable.


A third type of uncertainties resides in the scalar ππ form
factor. In this work, we have used the unitarized results from
Refs. <cit.>, where the low-energy
constants (L_is) are obtained by fitting the J/ψ decay data.
A Muskhelishvili–Omnès formalism has been developed for the
scalar ππ form factor in Ref. <cit.>. Compared to
the results in Ref. <cit.>, we find  an overall
agreement in  the shape of the non-strange ππ form factor, but
the modulus from Ref. <cit.> is about 20% larger.
This would  induce  about 40% uncertainties to the branching
ratios of the D→ππℓν̅_ℓ, while the results for
the ratio observables are not affected.



Finally, the Watson theorem does not always  guarantee the use of
Eq. (<ref>), the matching of D→ππ form factor
and D→ f_0 form factors. As we have discussed in Sec. II, such
an approximation might be improved in the future.






§ CONCLUSIONS





In summary, we have presented a theoretical analysis of  the D^-→π^+π^- ℓν̅ and  D̅^0→π^+π^0 ℓν̅
decays. We have   constructed  a general  angular distribution which
can include arbitrary partial waves of ππ. Retaining the
S-wave and P-wave contributions we have  studied  the branching
ratios, forward–backward asymmetries and a few other observables.
The P-wave contribution is dominated by ρ^0 resonance, and the
S-wave contribution is analyzed using the unitarized chiral
perturbation theory.  The obtained branching fraction for D→ρℓν,  at the order 10^-3, is consistent with the
available experimental data, while  the  S-wave contribution is
found to have  a branching ratio  at the order of 10^-4, and
this prediction can be tested  by experiments like BESIII and LHCb.
The BESIII collaboration  has accumulated about 10^7 events of the
D^0 and will collect about  3 fb^-1 data  at the
center-of-mass √(s)= 4.17 GeV to produce the
D_s^+D_s^- <cit.>. All these data can
be used to study the charm decays into the f_0 mesons. In
addition, sizable branching fractions  also indicate a promising
prospect at the ongoing  LHC experiment <cit.>, the
forthcoming Super-KEKB factory <cit.> and the
under-design Super Tau-Charm factory. Future measurements can   be
used to study the π–π scattering phase shift.





§ ACKNOWLEDGEMENTS

We thank Jian-Ping Dai,  Liao-Yuan Dong, Hai-Bo Li and Lei Zhang for useful discussions.
This work is supported  in part   by National  Natural
Science Foundation of China under Grant
 Nos.11575110, 11655002,  Natural  Science Foundation of Shanghai under Grant  No.
15DZ2272100 and No. 15ZR1423100,  by the Young Thousand Talents Plan,   and  by    Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education.







§ ANGULAR COEFFICIENTS


In the angular distribution, the coefficients have the form

    I_1      =    (1+m̂_l^2) |A_0|^2
     +2 m̂_l^2  |A_t|^2 + (3+m̂_l^2)/2(|A_⊥|^2
     +|A_|||^2)
     
    
     I_2      =    -β_l   |A_0|^2+  β_l /2 (|A_⊥|^2
     +|A_|||^2),           
      I_3    = β_l (|A_⊥|^2-|A_|||^2),           
     I_4    =  2 β_l  [ Re(A_0A_||^*)],
    
     I_5      =   4  [ Re(A_0A_⊥^*) -m̂_l^2  Re(A_tA_||^*) ],             
     I_6     =  4  [ Re(A_||A^*_⊥)+ m̂_l^2  Re(A_tA^*_0)],
    
     I_7      =    4[ Im(A_0A^*_||)-m̂_l^2  Im(A_tA^*_⊥)],             
     I_8   =  2 β_l   [ Im(A_0A^*_⊥)],               
     I_9   = 2β_l   [ Im(A_⊥A^*_||)].

Substituting the  expressions for A_i into the above equation, we obtain the general  expressions

    I_1(q^2, m_ππ^2, θ_π^+)     =   ∑_J=0,...{ |Y_J^0(θ_π^+, 0)|^2 [(1+m̂_l^2) |A^J_0|^2
     +2 m̂_l^2  |A_t^J|^2]   
          + 2∑_ J'=J+1, ... Y_J^0(θ_π^+, 0)Y_J'^0(θ_π^+, 0) [ cos(δ_0^J -
     δ^J'_0)|A^J_0||A^J'*_0|   + 2m̂_l^2
     cos (δ_t^J -δ_t^J' )|A^J_t||A^J'_t|]}
            +  3+m̂_l^2/2∑_J=1,...{ |Y_J^-1(θ_π^+, 0)|^2 [   [|A^J_⊥|^2+|A^J_|||^2 ]  ] 
           +  ∑_ J'=J+1, ... Y_J^-1(θ_π^+, 0)Y_J'^-1(θ_π^+, 0) [    2cos(δ_⊥^J
     - δ_⊥^J')|A_⊥^J||A_⊥^J'| ]  },

 
  
    I_2(q^2, m_ππ^2, θ_π^+)      =    -β_l   ∑_J=0,...{ |Y_J^0|^2   |A^J_0(θ_π^+, 0)|^2   + 2∑_ J'=J+1, ... Y_J^0(θ_π^+, 0)Y_J'^0(θ_π^+, 0)
      cos(δ_0^J - δ^J'_0)|A^J_0 A^J'_0|  }
    
     
    
     
            +
     1/2β_l  ∑_J=1,...{  |Y_J^-1(θ_π^+, 0)|^2  (|A^J_⊥|^2+|A^J_|||^2) 
                 +2∑_ J'=J+1 Y_J^-1(θ_π^+, 0)Y_J'^-1(θ_π^+, 0)[ cos(δ_⊥^J
     - δ^J'_⊥)|A^J_⊥ A^J'_⊥| + cos(δ_||^J
     - δ^J'_||)|A^J_||A^J'_||| ] },

 
  
    I_3(q^2, m_ππ^2, θ_π^+)     =   β_l ∑_J=1,...{  |Y_J^-1(θ_π^+, 0)|^2  (|A^J_⊥|^2-|A^J_|||^2)         
           +2∑_ J'=J+1,... Y_J^-1(θ_π^+, 0)Y_J'^-1(θ_π^+, 0)[ cos(δ_⊥^J
     - δ^J'_⊥)|A^J_⊥ A^J'_⊥| - cos(δ_||^J
     - δ^J'_||)|A^J_||A^J'_||| ] },

 
  
    I_4(q^2, m_ππ^2, θ_π^+)
         =   2 β_l   ∑_J=0, ...∑_J'=1, ..[  Y_J^0(θ_π^+, 0) Y_J'^-1(θ_π^+, 0) | A^J_0A^J'*_|| | cos(δ_0^J -δ_||^J')  ],

 
  
    I_5(q^2, m_ππ^2, θ_π^+)
         =   4   ∑_J=0, ...∑_J'=1, ..Y_J^0(θ_π^+, 0) Y_J'^-1 (θ_π^+, 0) [   | A^J_0A^J'*_⊥ | cos(δ_0^J -δ_⊥^J') -m̂_l^2  | A^J_tA^J'*_|| | cos(δ_t^J -δ_||^J')],

 
  
    I_6(q^2, m_ππ^2, θ_π^+)     =    4 ∑_J,J'=1,...{   Y_J^-1 (θ_π^+, 0)Y_J'^-1 (θ_π^+, 0)| A^J_||A^J'*_⊥ | cos(δ_||^J -δ_⊥^J')  }
           +  m̂_l^2 ∑_J,J'=0,...{   Y_J^0(θ_π^+, 0) Y_J'^0(θ_π^+, 0) | A^J_tA^J'*_0 | cos(δ_t^J -δ_0^J')  },

 
  
    I_7(q^2, m_ππ^2, θ_π^+)
        =   4 ∑_J=0, ...∑_J'=1, .. Y_J^0(θ_π^+, 0) Y_J'^-1(θ_π^+, 0)  [  | A^J_0A^J'*_|| | sin(δ_0^J -δ_||^J') -m̂_l^2 | A^J_tA^J'*_⊥ | sin(δ_t^J -δ_⊥^J')],

 
  
    I_8 (q^2, m_ππ^2, θ_π^+)   =   
     2 β_l  ∑_J=0, ...∑_J'=1, ..[  Y_J^0 (θ_π^+, 0)Y_J'^-1(θ_π^+, 0) | A^J_0A^J'*_⊥ | sin(δ_0^J -δ_⊥^J')  ],

 
 
    I_9(q^2, m_ππ^2, θ_π^+)
        =    2β_l ∑_J=1, ...∑_J'=1, ..[  Y_J^-1(θ_π^+, 0) Y_J'^-1(θ_π^+, 0) | A^J_⊥A^J'*_|| | sin(δ_⊥^J -δ_||^J')  ].






11

CLEO:2011ab
  S. Dobbs et al. [CLEO Collaboration],
  
  Phys. Rev. Lett.  110, no. 13, 131802 (2013)
  
  [arXiv:1112.2884 [hep-ex]].
  
  



Wang:2009azc
  W. Wang and C. D. Lü,
  
  Phys. Rev. D 82, 034016 (2010)
  
  [arXiv:0910.0613 [hep-ph]].
  
  



Achasov:2012kk
  N. N. Achasov and A. V. Kiselev,
  
  Phys. Rev. D 86, 114010 (2012)
  
  [arXiv:1206.5500 [hep-ph]].
  
  



Ablikim:2015orh
  M. Ablikim et al. [BESIII Collaboration],
  
  Phys. Lett. B 753, 629 (2016)
  
  [arXiv:1507.08188 [hep-ex]].
  
  



Lu:2011jm
  C. D. Lü and W. Wang,
  
  Phys. Rev. D 85, 034014 (2012)
  
  [arXiv:1111.1513 [hep-ph]].
  
  



Meissner:2013pba
  U. G. Meißner and W. Wang,
  
  JHEP 1401, 107 (2014)
  
  [arXiv:1311.5420 [hep-ph]].
  
  



Meissner:2013hya
  U. G. Meißner and W. Wang,
  
  Phys. Lett. B 730, 336 (2014)
  
  [arXiv:1312.3087 [hep-ph]].
  
  



Wang:2015uea
  W. F. Wang, H. n. Li, W. Wang and C. D. Lü,
  
  Phys. Rev. D 91, no. 9, 094024 (2015)
  
  [arXiv:1502.05483 [hep-ph]].
  
  



Wang:2015paa
  W. Wang and R. L. Zhu,
  
  Phys. Lett. B 743, 467 (2015)
  
  [arXiv:1502.05104 [hep-ph]].
  
  



Shi:2015kha
  Y. J. Shi and W. Wang,
  
  Phys. Rev. D 92, no. 7, 074038 (2015)
  
  [arXiv:1507.07692 [hep-ph]].
  
  



Xie:2014tma
  J. J. Xie, L. R. Dai and E. Oset,
  
  Phys. Lett. B 742, 363 (2015)
  
  [arXiv:1409.0401 [hep-ph]].
  
  



Sekihara:2015iha
  T. Sekihara and E. Oset,
  
  Phys. Rev. D 92, no. 5, 054038 (2015)
  
  [arXiv:1507.02026 [hep-ph]].
  
  



Oset:2016lyh
  E. Oset et al.,
  
  Int. J. Mod. Phys. E 25, 1630001 (2016)
  
  [arXiv:1601.03972 [hep-ph]].
  
  



Wang:2016rlo
  W. F. Wang and H. n. Li,
  
  Phys. Lett. B 763, 29 (2016)
  
  [arXiv:1609.04614 [hep-ph]].
  
  



Kang:2013jaa
  X. W. Kang, B. Kubis, C. Hanhart and U. G. Meißner,
  
  Phys. Rev. D 89, 053015 (2014)
  
  [arXiv:1312.1193 [hep-ph]].
  
  



Faller:2013dwa
  S. Faller, T. Feldmann, A. Khodjamirian, T. Mannel and D. van Dyk,
  
  Phys. Rev. D 89, no. 1, 014015 (2014)
  
  [arXiv:1310.6660 [hep-ph]].
  
  



Niecknig:2015ija
  F. Niecknig and B. Kubis,
  
  JHEP 1510, 142 (2015)
  
  [arXiv:1509.03188 [hep-ph]].
  
  



Daub:2015xja
  J. T. Daub, C. Hanhart and B. Kubis,
  
  JHEP 1602, 009 (2016)
  
  [arXiv:1508.06841 [hep-ph]].
  
  



Albaladejo:2016mad
  M. Albaladejo, J. T. Daub, C. Hanhart, B. Kubis and B. Moussallam,
  
  JHEP 1704, 010 (2017)

  [arXiv:1611.03502 [hep-ph]].
  
  



Wirbel:1985ji
  M. Wirbel, B. Stech and M. Bauer,
  
  Z. Phys. C 29, 637 (1985).
  
  
  



Scora:1995ty
  D. Scora and N. Isgur,
  
  Phys. Rev. D 52, 2783 (1995)
  
  [hep-ph/9503486].
  
  


Fajfer:2005ug
  S. Fajfer and J. F. Kamenik,
  
  Phys. Rev. D 72, 034029 (2005)
  
  [hep-ph/0506051].
  
  

Verma:2011yw
  R. C. Verma,
  
  J. Phys. G 39, 025005 (2012)
  
  [arXiv:1103.2973 [hep-ph]].
  
  


Cheng:2003sm
  H. Y. Cheng, C. K. Chua and C. W. Hwang,
  
  Phys. Rev. D 69, 074025 (2004)
  
  [hep-ph/0310359].
  
  



Wu:2006rd
  Y. L. Wu, M. Zhong and Y. B. Zuo,
  
  Int. J. Mod. Phys. A 21, 6125 (2006)
  
  [hep-ph/0604007].
  
  



Colangelo:2010bg
  P. Colangelo, F. De Fazio and W. Wang,
  
  Phys. Rev. D 81, 074001 (2010)
  
  [arXiv:1002.2880 [hep-ph]].
  
  



Olive:2016xmw
  C. Patrignani et al. [Particle Data Group],
  
  Chin. Phys. C 40, no. 10, 100001 (2016).
  
  
  



DeFazio:2001uc
  F. De Fazio and M. R. Pennington,
  
  Phys. Lett. B 521, 15 (2001)
  
  [hep-ph/0104289].
  
  



Doring:2013wka
  M. Döring, U. G. Meißner and W. Wang,
  
  JHEP 1310, 011 (2013)
  
  [arXiv:1307.0947 [hep-ph]].
  
  



Diehl:2003ny
  M. Diehl,
  
  Phys. Rept.  388, 41 (2003)
  
  [hep-ph/0307382].
  
  


Cheng:2005nb
  H. Y. Cheng, C. K. Chua and K. C. Yang,
  
  Phys. Rev. D 73, 014017 (2006)
  
  [hep-ph/0508104].
  
  

Bar:2012ce
  O. Bär and M. Golterman,
  
  Phys. Rev. D 87, no. 1, 014505 (2013)
  
  [arXiv:1209.2258 [hep-lat]].
  
  



Diehl:2005rn
  M. Diehl, A. Manashov and A. Schäfer,
  
  Phys. Lett. B 622, 69 (2005)
  
  [hep-ph/0505269].
  
  




Diehl:1998dk
  M. Diehl, T. Gousset, B. Pire and O. Teryaev,
  
  Phys. Rev. Lett.  81, 1782 (1998)
  
  [hep-ph/9805380].
  
  



Diehl:2000uv
  M. Diehl, T. Gousset and B. Pire,
  
  Phys. Rev. D 62, 073014 (2000)
  
  [hep-ph/0003233].
  
  


Colangelo:2015kha
  G. Colangelo, E. Passemar and P. Stoffer,
  
  Eur. Phys. J. C 75, 172 (2015)
  doi:10.1140/epjc/s10052-015-3357-1
  [arXiv:1501.05627 [hep-ph]].
  
  



Gasser:1983yg
  J. Gasser and H. Leutwyler,
  
  Annals Phys.  158, 142 (1984).
  
  
  



Gasser:1984gg
  J. Gasser and H. Leutwyler,
  
  Nucl. Phys. B 250, 465 (1985).
  
  
  



Gasser:1984ux
  J. Gasser and H. Leutwyler,
  
  Nucl. Phys. B 250, 517 (1985).
  
  
  



Meissner:2000bc
  U. G. Meißner and J. A. Oller,
  
  Nucl. Phys. A 679, 671 (2001)
  
  [hep-ph/0005253].
  
  



Bijnens:1998fm
  J. Bijnens, G. Colangelo and P. Talavera,
  
  JHEP 9805, 014 (1998)
  
  [hep-ph/9805389].
  
  



Bijnens:2003xg
  J. Bijnens and P. Dhonte,
  
  JHEP 0310, 061 (2003)
  
  [hep-ph/0307044].
  
  



Oller:1998hw
  J. A. Oller, E. Oset and J. R. Peláez,
  
  Phys. Rev. D 59, 074001 (1999)
  Erratum: [Phys. Rev. D 60, 099906 (1999)]
  Erratum: [Phys. Rev. D 75, 099903 (2007)]
  
  [hep-ph/9804209].
  
  



Oller:1997ti
  J. A. Oller and E. Oset,
  
  Nucl. Phys. A 620, 438 (1997)
  Erratum: [Nucl. Phys. A 652, 407 (1999)]
  
  [hep-ph/9702314].
  
  



Lahde:2006wr
  T. A. Lähde and U. G. Meißner,
  
  Phys. Rev. D 74, 034021 (2006)
  
  [hep-ph/0606133].
  
  



Oller:2007xd
  J. A. Oller and L. Roca,
  
  Phys. Lett. B 651, 139 (2007)
  
  [arXiv:0704.0039 [hep-ph]].
  
  



Cabibbo:1965zzb
  N. Cabibbo and A. Maksymowicz,
  
  Phys. Rev.  137, B438 (1965)
  Erratum: [Phys. Rev.  168, 1926 (1968)].
  
  
  



Pais:1968zza
  A. Pais and S. B. Treiman,
  
  Phys. Rev.  168, 1858 (1968).
  
  
  




delAmoSanchez:2010fd
  P. del Amo Sanchez et al. [BaBar Collaboration],
  
  Phys. Rev. D 83, 072001 (2011)
  
  [arXiv:1012.1810 [hep-ex]].
  
  





Lee:1992ih
  C. L. Y. Lee, M. Lu and M. B. Wise,
  
  Phys. Rev. D 46, 5040 (1992).
  
  
  






Ablikim:2014cea
  M. Ablikim et al. [BESIII Collaboration],
  
  Phys. Rev. D 89, no. 5, 052001 (2014)
  
  [arXiv:1401.3083 [hep-ex]].
  
  



Asner:2008nq
  D. M. Asner et al.,
  
  Int. J. Mod. Phys. A 24, S1 (2009)
  [arXiv:0809.1869 [hep-ex]].
  
  



Bediaga:2012py
  R. Aaij et al. [LHCb Collaboration],
  
  Eur. Phys. J. C 73, no. 4, 2373 (2013)
  
  [arXiv:1208.3355 [hep-ex]].
  
  



Aushev:2010bq
  T. Aushev et al.,
  
  arXiv:1002.5012 [hep-ex].
  
  

 


