S-wave contributions in $\bar B_s^0\to (D^0,\bar D^0)\pi^+\pi^- $ within perturbative QCD approach

The $\bar B_s^0\to (D^0,\bar D^0) \pi^+\pi^-$ is induced by the $b\to c \bar us$/$b \to u\bar cs$ transition, and can interfere if a CP-eigenstate $D_{\rm CP}$ is formed. The interference contribution is sensitive to the CKM angle $\gamma$. In this work, we study S-wave $\pi^+\pi^-$ contributions to the process in the perturbative QCD factorization. Under the factorization framework, we adopt two-meson light-cone distribution amplitudes, whose normalization is parametrized by the S-wave time-like two-pion form factor with the resonance contributions from $f_0(500)$, $f_0(980),f_0(1500),f_0(1790)$. We find the branching ratios of $\bar B_s^0\to (D^0,\bar D^0) (\pi^+\pi^-)_S$ can reach the order of $10^{-6}$, and significant interferences exist in $\bar B_s^0\to D_{CP} (\pi^+\pi^-)_S$. The future measurement can not only provide useful constraints on the CKM angle $\gamma$ but is also helpful to explore the multi-body decay mechanism of heavy mesons.

with f Bs as the decay constant of B s meson. Since the contribution ofφ Bs (k 1 ) is numerically small [27], we neglect it and keep only φ Bs (k 1 ) in the above equation. In momentum space the light cone matrix of B s meson can be expressed as follows: Usually the hard part is independent of k + or/and k − , thus one can integrate one of them out from φ Bs (k + , k − , k ⊥ ).
With b as the conjugate space coordinate of k ⊥ , we can express φ Bs (x, k ⊥ ) in b-space by where x is the momentum fraction of the light quark in B s meson. In this paper, we adopt the following expression for φ Bs (x, b) with N Bs the normalization factor, which is determined by equation at b = 0. In our calculation, we adopt ω b = (0.50 ± 0.05)GeV and f Bs = (0.23 ± 0.03)GeV [9], from which we determine the N Bs = 63. 58. The wave function of the charmed D meson, treated as the heavy-light system, is defined by the light cone matrix element as follows [10]: which satisfies the normalization Here f D is the decay constant, and the chiral D meson mass is taken as For the numerical calculation, we adopt the parametrization [47], with the free shape parameter C D taken as C D = 0.5 ± 0.1, f D , ω D read as f D = 0.221 ± 0.018 and ω D = 0.1, respectively [13].
Then the S-wave two-pion distribution amplitudes is given as [44] Φ S−wave where z is the momentum fraction carried by the spectator positive quark, Φ ππ , Φ s ππ and Φ T ππ are twist-2 and twist-3 distribution amplitudes. m ππ is the invariant mass of the pion pair. We consider the two-pion system move in the n direction. ξ as the momentum fraction of π + in pion pair. The asymptotic forms are parameterized as [48][49][50] Here, F s (m 2 ππ ) and a 2 are the timelike scalar form factor and the Gegenbauer coefficient respectively. As a first approximation, the S-wave resonances are used to parametrized F s (m 2 ππ ), to include both resonant and nonresonant contributions into the S-wave two-pion distribution amplitudes. Therefore, we take into account f 0 (980), f 0 (1500) and f 0 (1790) for the ss density operator, f 0 (500) for the uū density operator: c 0 , c i and θ i , i = 1, 2, 3, are tunable parameters. m S is the pole mass of the resonance, and Γ S (m ππ ) is the energydependent width for a S-wave resonance decaying into two pions. For the contribution of f 0 (980), the Flatté model has been used, and the phase space factors ρ ππ and ρ KK are given as [45]

III. PERTURBATIVE CALCULATIONS
According to factorization theorems, the amplitude for the process can be calculated as an expansion of α s (Q) and Λ/Q, Q denotes a large momentum transfer, and Λ is a small hadronic scale. Usually, the factorization formula for the nonleptonic b-meson decays can be expressed as where the Wilson coefficients C(t), organizing the large logarithms from the hard gluon corrections, is described by the renormalization-group summation of QCD dynamics between W boson m W and the typical scale t. The hard kernel , representing b-quark decay sub-amplitude, and the nonperturbative meson wave function φ i (x i , b i , t), describes the evolution from scale t to the lower hadronic scale Λ QCD . For a review of this approach, see Ref. [7]. The effective Hamiltonian forB 0 s → D 0 (D 0 )π + π − is given as with for the process ofB 0 s →D 0 π + π − . In particular, the penguin operators do not contribute to the processes. Using the above effective Hamiltonian, we obtain the typical Feynman diagrams for theB 0 s → D 0 π + π − process shown in Fig. 1, in which the first row represents the color-suppressed emission process, and the second row indicates the W-exchange process. In the factorization framework, the factorizable diagrams in Fig. 1(a,b,e,f) are relevant to a 2 , and the non-factorizable diagrams in Fig. 1 We will work in the light-cone coordinates. The momentum of the mesons are defined as follows: Accordingly, the transfer momentum and light-cone components can be achieved as Bs . In the heavy quark limit, the mass difference of bquark(c-quark) and B s (D) meson is negligible, m Bs,D = m b,c +Λ(Λ is the order of QCD scale). Since m Bs ≫ m D ≫Λ, we expand the amplitudes in terms of mD mB s ,Λ mD and high orderΛ mB s . At the leading order of expansion, ρ ∼ 1, q 2 ∼ 0. The momenta of the light quark in mesons (k 1 , k 3 represent the momentum of light quark in B s and D meson, k 2 is the momentum of positive quark in pion-pair system) are given as In the k T -factorization, the color-suppressed emission Feynman diagrams can be calculated out, with the formulas labelling as e x (x=1,2,3,4) in subscript. Thus factorization formulas for the color-suppressed D 0 -emission diagrams are given as where r 0 = mππ mB s , C F is the color factor. φ ππ (ss, x 2 ) represents the two-pion distribution amplitude defined by ss operator. The hard kernels E ex and h ex are given in the following.
The factorization formulas for the W-exchange D 0 diagrams M w12 and M w34 are given as where r D = mD mB s , φ ππ (uū, x 2 ) represents the distribution amplitude of the uū operator. Due to the helicity suppression, the contribution of factorizable diagrams M w12 is suppressed significantly. Therefore, the dominant contribution comes from the non-factorizable diagrams M w34 .
In theD 0 -emission process, the two factorizable diagrams have the same factorization M e12 = M e ′ 12 . Accordingly, we give the factorization formulas for the nonfactorizable emission diagrams M e ′ 34 , the factorizable W-exchange diagrams M w ′ 12 and the nonfactorizable W-exchange diagrams M w ′ 34 as follows: In the following, we give the forms for the offshellness of the intermediate gluon β ex /β wx and quarks α ex /α wx (x = 1, 2, 3, 4) in theB 0 s → D 0 π + π − process.
For the B 0 s →D 0 π + π − , we have The hard kernel functions h ex (h e ′ x ) and h wx (h w ′ x ) are written as where i, k = 1, 2 and j, l = 3, 4, the I 0 , K 0 and H 0 = J 0 + iY 0 are Bessel functions. The threshold resummation factor S t (x) follows the parametrization as with the parameter c = 0.4 in this paper. The evolution factors E x (t)s in the factorization formulas are given by where S Bs (t) = s(x 1 m Bs , b 1 ) with the quark anomalous dimension γ q = −α s /π. The explicit expression of s(Q, b) can be found, for example, in Appendix A of Ref [9]. The hard scales are chosen as Therefore, we obtain the total decay amplitudes, The differential branching ratio for theB 0 s → D 0 (D 0 )π + π − decay follows the formula given as [52,53] dB with the B s meson mean lifetime τ Bs . The kinematic variables | − → p 1 | and | − → p 3 | denote the magnitudes of the π + and D momenta in the center-of-mass frame of the pion pair,

IV. NUMERICAL RESULTS
We adopt the following inputs(in units of GeV) [52,53]  The parameters for the scalar form factor F s (m 2 ππ ) are extracted from the LHCb data in the process of B s → J/ψπ + π − , given as [45,54]

Resonances
Branching ratio (×10 −6 ) We calculate the branching ratios with the different resonances in S-wave pion-pair function shown in Tab I. In this table, the first uncertainties are from ω b = 0.50 ± 0.05 in the B s wave function, the second errors arise from a 2 = 0.2 ± 0.2 in the pion-pair wave function, and the third uncertainties come from QCD scale Λ = 0.25 ± 0.05. The errors from the parameter of D-meson function C D , the variations of CKM matrix elements and the mean lifetime of B s are tiny, and have been omitted. However the above results are sensitive to ω b and a 2 , namely the B s and S-wave two-pion wave functions. The future measurements of decay branching fractions will be valuable to understand the B s physics and the S-wave two-pion resonances.
For the comparison ofB s → D 0 (ππ) S andB s →D 0 (ππ) S , we determine the rate of their branching ratios with the quite different CKM ratio factor The CKM elements ofB 0 , in which V ub is sensitive to the γ. Therefore, we can achieve the dependence of our results about γ, by providing a parameter D CP ± defined as [55] √ Accordingly, the dependence curve of branching ratio B(B 0 s → D CP ± (π + π − ) S ) on γ is obtained in Fig. 2(a,b). In experimentally side, the corresponding physical observable measurement is defined as We give the dependencies of R CP ± on γ shown in Fig. 2(c,d). The current bound on γ is constrained as γ = (73.5 +4.2 −5.9 ) • [56]. The predicted dependencies of the differential branching ratios dB/dm ππ on the pion-pair invariant mass m ππ are presented in Fig. 3.(a) and Fig. 3.(b) for the resonances f 0 (500), f 0 (980), f 0 (1500) and f 0 (1790) in theB s → D 0 π + π − andB s →D 0 π + π − decay. The graphs show that the main contribution of the two decays lies in the region around the pole mass m f0(980) = 0.97, while the f 0 (500) lead to the primary contribution below the region m ππ = 1GeV .
The other resonances f 0 (1500) and f 0 (1790) still give the considerable contributions to the processes. Therefore, we expect that more precise data from the LHCb and the future KEKB may test our theoretical calculations.

V. CONCLUSIONS
In the past decades, two-body B decays have provided an ideal platform to extract the standard model parameters, and probe the new physics beyond the SM [57,58]. In this work, we have studied the three-bodyB 0 s → D 0 (D 0 )π + π − decay within the PQCD framework, and in particular the S-wave contribution is explicitly calculated. The S-wave two-pion light-cone distribution amplitudes can receives both resonant f 0 (500), f 0 (980), f 0 (1500), f 0 (1790) and nonresonant contributions. Furthermore, the processes proceed via the tree level operators, and branching ratios are found in the range from 10 −7 to 10 −6 . It is found that the branching ratios are sensitive to the parameters ω b and a 2 , in the B s and two-pion distribution amplitudes. Therefore, we expect that the future measurement can help us better understanding the multi-body processes, and S-wave two-pion resonance and B s distribution amplitudes.