Chiral dynamics, S-wave contributions and angular analysis in \(D\rightarrow \pi \pi \ell \bar{\nu }\)

The European Physical Journal C, Jul 2017

We present a theoretical analysis of the \(D^-\rightarrow \pi ^+\pi ^- \ell \bar{\nu }\) and \(\bar{D}^0\rightarrow \pi ^+\pi ^0 \ell \bar{\nu }\) decays. We construct a general angular distribution which can include arbitrary partial waves of \(\pi \pi \). 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 \(\rho ^0\) resonance, and the S-wave contribution is analyzed using the unitarized chiral perturbation theory. The obtained branching fraction for \(D\rightarrow \rho \ell \nu \), 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 \(\pi \)–\(\pi \) scattering phase shift.

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Chiral dynamics, S-wave contributions and angular analysis in \(D\rightarrow \pi \pi \ell \bar{\nu }\)

Eur. Phys. J. C Chiral dynamics, S-wave contributions and angular analysis in D → π π ν¯ Yu-Ji Shi 0 Wei Wang 0 Shuai Zhao 0 0 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 − ν¯ and D¯ 0 → π +π 0 ν¯ decays. We construct a general angular distribution which can include arbitrary partial waves of π π . 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 ρ0 resonance, and the S-wave contribution is analyzed using the unitarized chiral perturbation theory. The obtained branching fraction for D → ρ ν, 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. 1 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, |Vcd | can be determined from either exclusive or inclusive weak D decays, which are governed by c → d transition, for example, c → d ν 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 |Vcd |, 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 D0 → ρ−e+νe and D+ → ρ0e+νe at the CLEO-c experiment for both charged and neutral channels [ 1 ]. Because of the large width of the ρ meson, D → ρ ν¯ is in fact a quasifour body process D → π π ν¯ . The ρ can be reconstructed from the P-wave π π mode. However, other π π resonant or non-resonant states may interfere with the P-wave π π pair, and thus it is necessary to analyze the S-wave contribution to D → π π ν¯ . 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 [ 2,3 ]. As mentioned above, D → π π ν¯ can receive contributions from various partial waves of π π . ρ(770) dominant for D to P-wave π π decay, at the same time, D meson can decay into S-wave π π through f0(980). The structure of f0(980) is not fully understood yet. Analysis of D → π π ν¯ may shed more light on understanding the nature of f0(980). The BESIII collaboration has collected 2.93 fb−1 data in e+e− collisions at the energy around 3.773 GeV [4], 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. [ 5–19 ], 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 − ν¯ and D0 → π +π 0 ν¯ decays. In Sect. 2, we will present the results of D → f0 (980) and D → ρ form factors. We also calculate D to S-wave π π pair form factors in non-resonance region, the π π 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 → π π ν¯ . 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 Sect. 3. The conclusion of this paper will be given in Sect. 4. The details of the coefficients in angular distributions are relegated to the appendix. GF Heff = √ 2 2 Heavy-to-light transition form factors Feynman diagram for the D → π π −ν¯ decay is shown in Fig. 1. The lepton can be an electron or a muon, = e, μ. The spectator quark could be the u or d quark, corresponding to D0 → π +π 0 −ν¯ and D− → π +π − −ν¯ . Integrating out the virtual W -boson, we obtain the effective Hamiltonian describing the c → d transition Vcd d¯γμ(1 − γ5)c ν¯ γ μ(1 − γ5) + h.c., (1) where GF is the Fermi constant and Vcd 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. 2.1 D → ρ form factors For the P-wave π π state, the dominant contribution is from the ρ(770) resonance. The D → ρ form factors are parametrized by [ 20 ] 2V (q2) ρ( p2, )|d¯γ μc|D( pD) = − m D + mρ ρ( p2, )|d¯γ μγ5c|D( pD) = 2i mρ A0(q2) ∗ · q qμ μνρσ ν∗ pDρ p2σ , q2 ¯ c l− ν¯l ¯ d u/d Fig. 1 Feynman diagram for the D → ππ −ν decays. The lep¯ ton could be an electron or a muon, = e, μ. Depending on the D meson, the spectator could be a u or a d quark, corresponding to D0 → π+π0 −ν¯ and D− → π+π− −ν ¯ +i (m D + mρ ) A1(q2) ∗μ − −i A2(q2) ∗ · q with q = pD − p2, and P = pD + p2. The V (q2), and Ai (q2)(i = 0, 1, 2) are nonperturbative form factors. These form factors have been computed in many different approaches [ 21–25 ], and here we quote the results from the light-front quark model (LFQM) [ 23,24 ] and light-cone sum rules (LCSR) [25]. To access the momentum distribution in the full kinematics region, the following parametrization has been used: Fi (q2) = Fi (0) 1 − ai mq22D + bi q2 m2D Their results are collected in Table 1. We note that a different parametrization is adopted in Ref. [ 25 ], where A3 appears instead of A0. The relation between A0 and A3 is given by 1 A0(q2) = 2mρ (m D + mρ ) A1(q2)(m D + mρ )2 + A2(q2)(m2ρ − m2D) − A3(q2)q2 . 2.2 Scalar π π form factor and D to S-wave π π We first give the D → f0(980) form factor parametrized as f0( p2)|d¯γμγ5c|D−( pD) = −i F+D→ f0 (q2) × Pμ − m2D − m2f0 qμ + F0D→ f0 (q2) m2D − m2f0 qμ , q2 q2 where F+D→ f0 and F0D→ f0 are D → f0 form factors. We will use LCSR to compute the D → f0(980) transition form factors with some inputs, and we refer the reader to Ref. [ 26 ] for a detailed derivation in LCSR. The meson masses are fixed to the PDG values m D = 1.870 GeV and m f0 = 0.99 GeV [ 27 ]. For quark masses we use mc = 1.27 GeV [ 27 ] and md = 5 MeV. As for decay constants, we use f D = 0.21 GeV [ 27 ] and f f0 = 0.18 GeV [ 28 ]. The threshold s0 is fixed at s0 = 4.1 GeV2, which should correspond to the squared mass of the first radial excitation of D. The parameters Fi (0), ai and bi are fitted in the region −0.5 GeV2 < q2 < 0.5 GeV2, and the Borel parameter M 2 is taken to be (6 ± 1) GeV−2. With these parametrizations, we give the numerical results in Table 2. 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 + m2D −q2m2ππ qμF0D→ππ (m2ππ , q2) . The Watson theorem implies that phases measured in π π elastic scattering and in a decay channel in which the π π system has no strong interaction with other hadrons are equal modulo π radians. In the process we consider here, the lepton pair ν¯ indeed decouples from the π π final state, and thus the phases of D to scalar π π decay amplitudes are equal to π π scattering with the same isospin. It is plausible that (π π )S|d¯ c|D ∝ Fππ (m2ππ ), where the scalar form factor is defined as 0|d¯d|π +π − = B0 Fππ (m2ππ ), where B0 = (1.7 ± 0.2) GeV [ 10 ] is the QCD condensate parameter. An explicit calculation of these quantities requires knowledge of generalized light-cone distribution amplitudes (LCDAs) [30]. The twist-3 one has the same asymptotic form with the LCDAs for a scalar resonance [31]. Inspired by this similarity, we may plausibly introduce an intuitive matching between the D → f0 and D → (π π )S form factors [ 7 ]: FiD→ππ (m2ππ , q2) Fππ (m2ππ )FiD→ f0 (q2). (9) B0 1 f f0 It is necessary to stress at this stage that the Watson theorem does not strictly guarantee that one may use Eq. (9). Instead it indicates that, below the opening of inelastic channels the strong phases in the D → π π form factor and π π (6) (7) (8) 0.78 ± 0.24 scattering are the same. First above the 4π or K K¯ threshold, additional inelastic channels will also contribute. The K K¯ 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 [32]. Secondly, some polynomials with nontrivial dependence on mππ have been neglected in Eq. (9). In principle, once the generalized LCDAs for the (π π )S system are known, the D → π π 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 (χ PT) [33]. The analysis of timelike generalized LCDAs in χ PT and the unitarized framework is in progress. On the other side, the γ γ ∗ → π +π − reaction is helpful to extract the generalized LCDAs for the (π π )S system [34,35]. The experimental prospects at BEPCII 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. (9) 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) [36] or approximately in terms of a pole contribution (in the case of the B meson decay) [ 15 ]. 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 [ 7 ]. This will lead to the conjectured formula in Eq. (9). The scalar π π 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 1 |π π I=0 = √3 1 |K K¯ I=0 = √2 1 π +π − + √ 6 1 K + K − + √ 2 0 0 , π π K 0 K¯ 0 , 2B0 F1n/s (s) = 0|n¯ n/s¯s|π π I=0, the scalar form factors for the π and K mesons are defined as √ (12) (10) (11) √2B0 F2n/s (s) = 0|n¯ n/s¯s|K K¯ I=0, where s = m2ππ . The n¯ n = (u¯u + d¯d)/√2 denotes the nonstrange scalar current, and the notation (π = 1, K = 2) has been introduced for simplicity. With the above notation, we have Fππ (m2ππ ) = Expressions have already been derived in χ PT up to next-toleading order [37–40]: μi = 32πm2i2 f 2 ln mμ2i2 , F (s) = [I + K (s) g(s)]−1 R(s) 1 Jiri (s) = 16π 2 m2 i μ2 1 − log − σi (s) log σi (s) + 1 σi (s) − 1 with σi (s) = 1 − 4mi2/s. It is interesting to note that the next-to-next-to-leading order results can also be found in Refs. [41,42]. Imposing the unitarity constraints, the scalar form factor can be expressed in terms of the algebraic coupled-channel equation = [I − K (s) g(s)] R(s) + O( p6), 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 mesonmeson scattering amplitudes that can be derived from the leading-order chiral Lagrangian: The loop integral can be calculated either in the cutoffregularization scheme with qmax ∼ 1 GeV being the cutoff (cf. Erratum of Ref. [ 43 ] for an explicit expression) or in dimensional regularization with the MS subtraction scheme. In the latter scheme, the meson loop function gi (s) is given by 1 − log − σi (s) log σi (s) + 1 σi (s) − 1 1 Jiri (s) ≡ 16π 2 = −gi (s). The expressions for the Ri are obtained by matching the unitarization and chiral perturbation theory [ 44,45 ]: R1n(s) = 23 1 + μπ − μ3η + 16fm22π 2Lr8 − Lr5 r 2m2K + 3m2π +8 2Lr6 − L4 f 2 + 8f s2 Lr4 + 4f s2 Lr5 (13) (14) (15) (16) (17) (18) (20) m2π − 288π 2 f 2 1 + log m2π − 72π 2 f 2 1 + log m2K + 72π 2 f 2 1 + log m2 i μ2 m2 η μ2 m2 η μ2 m2 η μ2 , 16Lr6 4m2K + m2π + 32fL2r8 m2K + 3 μη 2 + f 2 9s − 8m2K 18 f 2 Jηrη(s) + 43fs2 J Kr K (s). Here the Lri are the renormalized low-energy constants, and f is the pion decay constant at tree level. The μi and Jiri are defined as follows: (21) (22) (23) (24) (25) R1s(s) = √ 23 16fm22π 2Lr6 − Lr4 + 8f s2 Lr4 R2n(s) = √12 1 + 8fL2r4 2s − 6m2K − m2π + 4fL2r5 s − 4m2K 16Lr6 6m2K + m2π + 32fL2r8 m2K + 3 μη 2 + f 2 R2s(s) = 1 + f 2 4Lr5 s − 4m2K 8Lr4 s − 4m2K − m2π + f 2 16Lr6 4m2K + m2π + 32fL2r8 m2K + 3 μη 2 + f 2 m2K + 36π 2 f 2 1 + log m2 η μ2 With the above formulas and the fitted results for the lowenergy constants Lri in Ref. [ 45 ] (evolved from Mρ to the scale μ = 2qmax/√e), we show the non-strange π π form factor in Fig. 2. 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 1.5 1.0 with a zero close to the K¯ K threshold. This feature has been extensively discussed in Ref. [ 46 ]. 3 Full angular distribution of D → π π ν¯ In this section, we will derive a full angular distribution of D → π π ν¯ . For the literature, one may consult Refs. [ 47, 48 ]. We set up the kinematics for the D− → π +π − ν¯ as shown in Fig. 3, which can also be used for D0 → π +π 0 ν¯ . The π π moves along the z axis in the D− rest frame. θπ + (θ ) is defined in the π π (lepton pair) rest frame as the angle between z-axis and the flight direction of π + ( −), respectively. The azimuth angle φ is the angle between the π π decay and lepton pair planes. Decay amplitudes for D → π π ν¯ 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: Aλ = i GF N f0/ρ √2 Vc∗d μ∗(h) π π |c¯γ μ(1 − γ5)d| D , (26) where μ(h) is an auxiliary polarization vector for the lepton pair system and h = 0, ±, t , N f0/ρ = √λq2βl /(96π 3m3D ), βl = 1−mˆ l2 and mˆ l = ml / q2. |Vcd | is taken to be 0.22 [ 27 ]. The functions Ai can be decomposed into different partial waves, mmπρπ 11 ++ ((RR||qq0||))22 , and the Blatt–Weisskopf parameter R = (2.1 ± 0.5 ± 0.5) GeV−1 [ 49 ]. The spin-0 final state has only one polarization state and the amplitudes are i MD ( f0, 0) = N1i i MD ( f0, t ) = N1i √λ q2 F1(q2) , q2 m2D − m2f0 F0(q2) , (29) (30) with N1 = i GF Vc∗d /√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 MD (ρ , 0) = − αLJ N1i 2mρ q2 × (m2D − m2ρ − q2)(m D + mρ ) A1 A2 , i MD(ρ , ±) = −βTJ N1i (m D + mρ ) A1 ± m D√+λmρ V , i MD(ρ , t ) = −αLJ i N1 √λ q2 A0. The αLJ and βTJ are products of the Clebsch–Gordan coefficients αLJ = C1J,,00;J −1,0C1J,−0;1J,0−2,0 · · · C12,,00;1,0, βTJ = C1J,,11;J −1,0C1J,−0;1J,0−2,0 · · · C12,,00;1,0. For the sake of convenience, we define 1 i MD(ρ , ⊥ /||) = √ [i MD(ρ , +) ∓ i MD(ρ , −)], 2 i MD(ρ , ⊥) = −iβTJ √2N1 √λV , i MD(ρ , ||) = −iβTJ √2N1 (m D + mρ ) A1 . (34) Using the generalized form factor, the matrix elements for D decays into the spin-0 non-resonating π π final state are given as A00 = At0 = 1 N2i mππ 1 N2i mππ √λ q2 F1ππ (m2ππ , q2) , m2D − m2ππ F0ππ (m2ππ , q2) , q2 N2 = N1 Nρ ρπ /(16π 2), with ρπ = 1 − 4m2π /m2ππ . The above quantities can lead to the full angular distributions d5 dm2ππ dq2d cos θπ+ d cos θl dφ 3 = 8 I1(q2, m2ππ , θπ+ ) +I2(q2, m2ππ , θπ+ ) cos(2θ ) +I3(q2, m2ππ , θπ+ ) sin2 θ cos(2φ) +I4(q2, m2ππ , θπ+ ) sin(2θ ) cos φ +I5(q2, m2ππ , θπ+ ) sin(θ ) cos φ +I6(q2, m2ππ , θπ+ ) cos θ +I7(q2, m2ππ , θπ+ ) sin(θ ) sin φ +I8(q2, m2ππ , θπ+ ) sin(2θ ) sin φ +I9(q2, m2ππ , θπ+ ) sin2 θ sin(2φ) . For the general expressions of Ii , we refer the reader to the appendix and to Refs. [ 48,50 ] 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: I6 = 4 83π sin2 θπ+ Re[ A|1| A1⊥∗] + mˆ l2 41π Re[ At0 A00∗] +mˆ l2 43π cos2 θπ+ Re[ At1 A01∗] √3 I7 = 4 4√2π 3 + 4√2π I8 = 2βl 4√√23π sin θπ+ Im A00 A1⊥∗ 3 + 4√2π I9 = 2βl 83π sin2 θπ+ Im A1⊥ A|1|∗ . sin θπ+ cos θπ+ Im A10 A1⊥∗ , sin θπ+ Im[ A00 A|1|∗] − mˆ l2Im[ At0 A1⊥∗] sin θπ+ cos θπ+ Im[ A01 A|1|∗] − mˆ l2Im[ At1 A1⊥∗] Since the phase in P-wave contributions arise from the lineshape which is the same for different polarizations, the I9 term and the second line in the I7 are zero. 3.1 Differential and integrated decay widths Using the narrow width approximation, we obtain the integrated branching fraction: B(D− → ρ0e−ν¯) = (2.24 ± 0.09) × 10−3/(2.16 ± 0.36) ×10−3(LFQM/LCSR), (37) (38) B(D− → ρ0μ−ν¯) = (2.15 ± 0.08) × 10−3/(2.06 ± 0.35) × 10−3(LFQM/LCSR), (39) B(D¯ 0 → ρ+e−ν¯) = (1.73 ± 0.07) × 10−3/(1.67 ± 0.27) × 10−3(LFQM/LCSR), (40) where theoretical errors are from the heavy-to-light transition form factors. These theoretical results are in good agreement with the data [ 27 ]: B(D− → ρ0e−ν¯ ) = (2.18+−00..1275) × 10−3, B(D− → ρ0μ−ν¯ ) = (2.4 ± 0.4) × 10−3, B(D¯ 0 → ρ+e−ν¯ ) = (1.77 ± 0.16) × 10−3. The starting point for detailed analysis of D → π π ν¯ is to obtain the double-differential distribution d2 /dq2dm2ππ after performing integration over all the angles 0.4 0.6 1.0 0.8 GeV 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 dq2dm2ππ = | A00|2 + | A01|2 + | A||| + | A1⊥|2. 1 2 In Fig. 4, we give the dependence of branching fraction on mππ in the D− → π +π −e−ν¯e process. The solid, dashed, and dotted curves correspond to the total, S-wave and Pwave 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 D¯ 0 → π +π 0 ν¯ receives only a P-wave contribution and D− → π 0π 0 ν¯ is generated by the S-wave term. To match the kinematics constraints implemented in experimental measurements, one may explore the generic observable with m2ππ integrated out: O = (mρ +δm )2 (mρ −δm )2 dO dm2ππ dm2ππ . We use the following choice in our study of D → π π ν¯ : δm = ρ . In the narrow width-limit, the integration of the lineshape is conducted as dm2ππ |Lρ (m2ππ )|2 = B(ρ0 → π −π +) = 1. (41) However, with the explicit form given in Eq. (29), we find that the integration is below the expected value. On the other hand, the integrated S-wave lineshape in this region is dm2ππ |Lρ (m2ππ )|2 = 0.70 dm2ππ |L S(m2ππ )|2 = 0.37, which is smaller but at the same order. Integrated from mρ − ρ to mρ + ρ , we have B(D− → ρ0(→ π +π −)e−ν¯) = (1.57 ± 0.07) × 10−3/(1.51 ± 0.26) × 10−3 (LFQM/LCSR), (51) Differential decay widths for D → π π ν¯ are given in Fig. 5, with = e in panel (a) and = μ in panel (b). The q2dependent ratio Rπμπ/e is given in panel (c). Errors from the form factors and QCD condensate parameter B0 are shown (49) (50) (55) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 q2 GeV2 (a) Fig. 5 Differential decay widths for the D− → π +π − ν¯ with = e in (a) and = μ in (b). The q2-dependent ratio Rπμπ/e as defined in Eq. (55) is given in (c). The dashed and dotted curves are produced using the LFQM and LCSR results for D → ρ form factors. Errors from the form factors are shown as shadowed bands, and most errors cancel in the ratio Rπμπ/e given in (c) 2 1.5 V e G 30 1.0 1 2 dq 0.5 B d 0.0 2 1.5 V e 3G1.0 0 1 2 dq 0.5 B d 0.0 as shadowed bands, and most errors cancel in the ratio Rπμπ/e given in panel (c). 3.2 Distribution in θπ + We explore the distribution in θπ + : d3 dq2dm2π π d cos θπ + π = 2 (3I1 − I2) Compared to the distribution with only P-wave contribution, namely D → ρ (→ π π ) ν¯ , the first two lines of Eq. (56) are new: the first one is the S-wave π π 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, π AFB ≡ 0 √3 = 2 1 − 0 −1 (2 + mˆ l2)Re[ A00 A10∗] + d3 d cos θπ + dq2dm2π π d cos θπ + 3√3 2 mˆ l2Re[ At0 At1∗]. We define the polarization fraction at a given value of q2 and m2π π : (1 + mˆ l2/2)| A00|2 + 3/2mˆ l2| At0|2 , d2 /(dq2dm2ππ ) (1 + mˆ l2/2)(| A01|2 + | A||| + | A1⊥|2) + 3/2mˆ l2| At1|2 1 2 d2 /(dq2dm2ππ ) FS (q2, m2ππ ) = FP (q2, m2ππ ) = , (58) and also FL (q2, m2π π ) (1 + mˆ l2/2)| A01(q2, m2π π )|2 + 3/2mˆ l2| At1|2 = (1 + mˆ l2/2)(| A0| + | A||| + | A⊥| ) + 3/2mˆ l2| At1|2 1 2 1 2 1 2 . By definition, FS + FP = 1. In Fig. 6, we give our results for the S-wave fraction FS (panel (a)), longitudinal polarization fraction FL in P-wave contributions (panel (b)) and the asymmetry AπF B (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 m2π π ; for instance, FS (q2) = dm2π π [(1 + mˆ l2/2)| A0| + 3/2mˆ l2| At0|2] 0 2 dm2π π d2 /(dq2dm2π π ) (59) (60) and likewise for the others. 3.3 Distribution in θl and forward–backward asymmetry Integrating over θπ + and φ, we have the distribution: d3 dq2dm2ππ d cos θl = 3 m2 = 4 ˆ l | At0|2 + | At1|2 3π 4 d cos θπ+ × (I1 + I2 cos(2θl ) + I6 cos θl ) 3 + 2 cos θl × Re[ A|1| A1⊥∗] + mˆ l2Re[ At0 A00∗ + At1 A01∗] 3 + 4 3 + 8 (1 + mˆ l2) + (1 − mˆ l2) cos2 θl 1 − (1 − mˆ l2) cos2 θl | A00|2 + | A01|2 ×(| A||| + | A1⊥|2). 1 2 (61) FL0.6 1.2 1.0 0.8 0.4 0.2 0.0 0.00 0.05 LFQM LCSR B 0.2 π AF Fig. 7 Same as Fig. 5 but for the asymmetry AlF B in the D → π π ν¯ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 q2 GeV2 (a) Fig. 6 Same as Fig. 5 but for the S-wave contributions (a) and the longitudinal polarizations in P-wave contributions (b) to the D → π π ν¯ , and the forward–backward asymmetry AπF B (c). Notice that, for the AπF B , there is a sign ambiguity arising from the use of Watson theorem. These diagrams are for the light lepton e, while the results for the μ lepton are similar LFQM LCSR Since the complex phase in the P-wave amplitudes comes from the Breit–Wigner lineshape, the coefficient cφs vanishes. Numerical results for the normalized coefficients using the two sets of form factors are shown in Fig. 8. The coefficients bφc and bφs contain a very small prefactor, 3√3/(32√2π ) ∼ 0.037, and thus are numerically tiny, as shown in this figure. The cφc is also small due to the cancellation between the | A⊥|2 and | A|||2. 3.5 Polarization of μ lepton In this work, we also give the polarized angular distributions as 0.0020 Fig. 8 Same as Fig. 5 but for the normalized coefficients in the φ distributions of the D− → π +π − ν¯ . The left panels (a, c, e) are for the light lepton e, while the right panels (b, d, f) are for the μ lepton 0.0 – 0.2 – 0.4 μ A – 0.6 Fig. 9 Same as Fig. 5 but for the polarization distribution of D → π π μν¯μ. Theoretical errors are negligible + I2(λμ) cos(2θl ) + I3(λμ) sin2 θl cos(2φ) = Aλμ (q2, m2ππ ) = and we show the numerical results in Fig. 9. 3.6 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 B0 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 Rπμπ/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 αs . An analysis of Bs → f0 [ 26 ] 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 π π form factor. In this work, we have used the unitarized results from Refs. [ 44,45 ], where the low-energy constants (Li s) are obtained by fitting the J /ψ decay data. A Muskhelishvili– Omnès formalism has been developed for the scalar π π form factor in Ref. [18]. Compared to the results in Ref. [ 18 ], we find an overall agreement in the shape of the non-strange π π form factor, but the modulus from Ref. [ 18 ] is about 20% larger. This would induce about 40% uncertainties to the branching ratios of the D → π π ν¯ , while the results for the ratio observables are not affected. Finally, the Watson theorem does not always guarantee the use of Eq. (9), the matching of D → π π form factor and D → f0 form factors. As we have discussed in Sect. 2, such an approximation might be improved in the future. 4 Conclusions In summary, we have presented a theoretical analysis of the D− → π +π − ν¯ and D¯ 0 → π +π 0 ν¯ decays. We have constructed a general angular distribution which can include arbitrary partial waves of π π . Retaining the S-wave and Pwave contributions we have studied the branching ratios, forward–backward asymmetries and a few other observables. The P-wave contribution is dominated by ρ0 resonance, and the S-wave contribution is analyzed using the unitarized chiral perturbation theory. The obtained branching fraction for D → ρ ν, 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 107 events of the D0 and will collect about 3 fb−1 data at the center-ofmass √s = 4.17 GeV to produce the Ds+ Ds− [ 51,52 ]. All these data can be used to study the charm decays into the f0 mesons. In addition, sizable branching fractions also indicate a promising prospect at the ongoing LHC experiment [53], the forthcoming Super-KEKB factory [ 54 ] and the underdesign 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, HaiBo 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. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP3. Appendix A: Angular coefficients In the angular distribution, the coefficients have the form I1 = (1 + mˆ l2)| A0|2 + 2mˆ l2| At |2 + (3 + mˆ l2)/2(| A⊥|2 + | A|||2) I2 = −βl | A0|2 + βl /2(| A⊥|2 + | A|||2), I3 = βl (| A⊥|2 − | A|||2), I4 = 2βl [Re( A0 A|∗|)], I5 = 4[Re( A0 A∗⊥) − mˆ l2Re( At A|∗|)], I6 = 4 Re( A|| A∗⊥) + mˆ l2Re( At A0∗) , I7 = 4 Im( A0 A|∗|) − mˆ l2Im( At A∗⊥) , I8 = 2βl [Im( A0 A∗⊥)], I9 = 2βl [Im( A⊥ A|∗|)]. (A1) Substituting the expressions for Ai into the above equation, we obtain the general expressions I1(q2, m2ππ , θπ+ ) |YJ0(θπ+ , 0)|2 J=0,... = (1 + mˆ l2)|A0J |2 + 2mˆ l2|AtJ |2 +2 +2mˆ l2 cos(δtJ − δtJ )| AtJ || AtJ | + × 2 cos(δ ⊥J − δ ⊥J )| A ⊥J|| A ⊥J | , × | A0J A|J| ∗| sin(δ0J − δ|J| ) − mˆ l2| AtJ A ⊥J ∗| sin(δtJ − δ ⊥J ) , I8(q2, m2ππ , θπ+ ) = 2βl I9(q2, m2ππ , θπ+ ) = 2βl J =1,... J =1,... × YJ−1(θπ+ , 0)YJ−1(θπ+ , 0)| A ⊥J A|J| ∗| sin(δ ⊥J − δ|J| ) . (A10) (A2) (A3) (A4) (A5) (A6) (A7) (A8) (A9) 29. M. Döring, U.G. Meißner, W. 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Yu-Ji Shi, Wei Wang, Shuai Zhao. Chiral dynamics, S-wave contributions and angular analysis in \(D\rightarrow \pi \pi \ell \bar{\nu }\), The European Physical Journal C, 2017, 452, DOI: 10.1140/epjc/s10052-017-5016-1