Low-mass enhancement of kaon pairs in $B^+\to\bar{D}^{(*)0}K^+\bar{K}^0$ and $B^0\to D^{(*)-}K^+\bar{K}^0$ decays

Very recently, the Belle~II Collaboration presented a measurement for the decays $B^+\to\bar{D}^{(*)0} K^+\bar{K}^0$ and $B^0\to D^{(*)-}K^+\bar{K}^0$, the bulk of observed $m(K^+ K_S^0)$ distributions showing low-mass structures in all four channels. In this work, we study the contributions of $\rho(770,1450)^+$, $a_2(1320)^+$ and $a_0(980,1450)^+$ resonances to these decay processes. The intermediate states $\rho(770,1450)^+$ are found to dominate the low-mass distribution of kaon pairs roughly contributing to half of the total branching fraction in each of the four decay channels. The contribution of the tensor $a_2(1320)^+$ meson is found to be negligible. Near the threshold of the kaon pair, the state $a_0(980)^+$ turns out to be much less important than expected, not being able to account for the enhancement of events in that energy region observed in the $B^+\to\bar{D}^{(*)0} K^+\bar{K}^0$ decays. Further studies both from the theoretical and experimental sides are needed to elucidate the role of the non-resonant contributions governing the formation of $K^+\bar{K}^0$ pairs near their threshold in these decay processes.


I. INTRODUCTION
Three-body hadronic B meson decay processes are regularly interpreted in terms of the contribution of various resonant states.The investigation of appropriate decay channels will help us to comprehend the properties and substructures of the related hadronic resonances involved in these decays.By employing the Dalitz plot amplitude analysis technique [1], the experimental efforts on relevant decay processes combined with the analysis within the isobar formalism have revealed valuable information on low-energy resonance dynamics [2,3].Very recently, the Belle II Collaboration presented a measurement for the decay channels B + → D( * )0 K + K 0 S and B 0 → D ( * )− K + K 0 S [4,5].In addition to the four branching fractions for these concerned decays, the m(K + K 0 S ) distribution of kaon pairs was also provided, showing relevant low-mass structures in all four channels [4].
Given the presence of an open charm meson D ( * ) in the final state, these four decay processes measured by Belle II, which have also been previously searched by the Belle experiment [6], are relatively simple and clear from a theoretical point of view.One only has to consider the contributions from the tree-level W exchange operators O 1 and O 2 in the effective Hamiltonian H eff [7] within the framework of the factorization method [8].In the low-mass region, the isospin I = 1 K + K0 kaon pair emitted in the B + → D( * )0 K + K0 and B 0 → D ( * )− K + K0 decays can be originated from the charged intermediate states, ρ(770) + , a 0 (980) + , a 2 (1320) + and their excited states, via the quasi-two-body mechanism shown schematically in Fig. 1.The intermediate state R + in the figure, which decays into the final kaon pair, is generated in the hadronization of the light quark-antiquark pair u d or can be formed as a dynamically generated state through the meson-meson interactions.
As for the contribution of the isovector tensor meson a 2 (1320), we note that it is the ground state of the a 2 family with quantum numbers I G J P C = 1 − 2 ++ and it can be reasonably understood as a constituent quark-antiquark pair within the quark-model [2].The transition form factors for the B meson to the a 2 (1320) state have been obtained in Refs.[73][74][75][76][77] within various methods.Moreover, the hadronic B meson decays involving a tensor meson a 2 (1320) in the final state have been studied in Refs.[78][79][80][81][82][83][84][85] in recent years.The tensor meson a 2 (1700), assigned as the first radial excitation of the a 2 (1320) [2,86] state, will not be considered in this work in view of the negligible branching fraction of the decay of the a 2 (1700) into K K pairs [2,87].
This paper is organized as follows.In Sec.II, we briefly describe the theoretical framework for obtaining the resonance contributions to the decay rates of the B + → D( * )0 R + → D( * )0 K + K0 and B 0 → D ( * )− R + → D ( * )− K + K0 processes, relegating to Appendices A and B the specific details of the calculation of the decay amplitudes.In Sec.III, we present our numerical results of the branching fractions for the concerned quasi-two-body decay processes along with some necessary discussions.To test our model, we will also present results for the branching ratios of the B decay processes into a D ( * ) meson and a pair of pions in the final state.A summary and the conclusions of this work are given in Sec.IV.
For the cascade decays , where the intermediate state R + stands for ρ(770, 1450) + , a 0 (980, 1450) + or a 2 (1320) + , the related effective weak Hamiltonian H eff accounting for the b → c transition is written as [7] where is the Fermi coupling constant, C 1,2 (µ) are the Wilson coefficients at scale µ, and V cb and V ud are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements.
The four-quark operators With the factorization ansatz, the decay amplitudes for B + → D( * )0 K + K0 and B 0 → D ( * )− K + K0 are given as [122] M( D( * )0 where the effective Wilson coefficients are expressed as The differential branching fraction (B) for the considered decays is written as [2,105,123] where the amplitudes M V and M S are related to the vector ρ(770, 1450) + and scalar a 0 (980, 1450) + intermediate states, respectively, with the help of the Eqs.( 2)-(3).Here, τ B (m B ) is the mean lifetime (mass) for the B meson, s = m 2 K + K0 is the invariant mass square, and correspond, respectively, to the magnitude of the momentum of each kaon and that of the bachelor meson D( * )0 or D ( * )− , with mass m D , in the rest frame of the intermediate resonance.By combining various contributions from the relevant Feynman diagrams at quark level in Fig. 2, the total decay amplitudes for the concerned quasi-two-body decays in the PQCD approach are written as where hh ′ ∈ {π + π 0 , K + K0 }.The label F (M ) denotes that the corresponding decay amplitude comes from the factorizable (nonfactorizable) Feynman diagrams, the subscripts T ρ and T D stand for the transition B → ρ and B → D, respectively, and the subscript aρ is related to the annihilation Feynman diagram of Fig. 2 (c).The specific expressions in the PQCD approach for these general amplitudes F and M in these decay amplitudes are found in Appendix A. One should note that the A's here have a constant factor (2/m B ) 2 different from M V in Eq. ( 4) because of the different definitions between PQCD and QCDF, see the corresponding expression for the differential branching fraction of the former in [38].
The quasi-two-body decay amplitudes (7)-( 8) are related to the corresponding two-body decay amplitude M 2B for the B → Dρ + transition via the relation where ⟨hh ′ |ρ + ⟩ stands for the coupling between the ρ + and the hh ′ pair.Note that the former equation is effectively incorporating the electromagnetic form factor associated to the subprocesses ρ(770, 1450) + → π + π 0 and ρ(770, 1450) + → K + K0 in the corresponding quasi-two-body decays, given by [124-127] where the label R represents the resonance, ρ(770) or ρ(1450), and the coefficient [124] depends on the corresponding decay constant f R , the coupling constant g Rππ and the mass m R .To obtain the coefficient c K R we relay on flavor SU(3) symmetry, which establishes [124].The function BW R (s) stands for a Breit-Wigner shape of the form [124,127,128] with the s-dependent width given by where h stands for the pion and kaon, respectively, in the π + π 0 and K + K0 final state pairs.The magnitude |p h0 | corresponds to the value of |p h | at s = m 2 R , while |p π | can be obtained from Eq. ( 5) with the replacement of m K +,0 by m π +,0 .The Blatt-Weisskopf barrier factor [129] with barrier radius r R BW = 4.0 GeV −1 [128] is given by As for the other two decay amplitudes corresponding to the B + and B 0 decays into final vector mesons D * 0 and D * − , respectively, they are obtained from Eqs. ( 7)-( 8) with the replacement of the D meson wave function by the D * one.As has been done in the study of B decays into two vector mesons in the final state, the two-body decay amplitudes for B → D * ρ + in this work can be decomposed as [130] with three kinds of polarizations of the vector meson, namely, longitudinal (L), normal (N ) and transverse (T ).According to the polarized decay amplitudes, one has the total decay amplitude

and a longitudinal polarization fraction
, where the amplitudes A L , A ∥ and A ⊥ are related to the two-body amplitudes M L , M N and M T , respectively, via Eq.( 9).For a detailed discussion, one is referred to Refs.[130][131][132][133][134].

III. RESULTS AND DISCUSSIONS
In this section we present our results for the branching ratios of the decay of B mesons into a charm D or D * meson and a pair of light pseudoscalar mesons.In the numerical calculations, we adopt the decay constants f ρ(770) = 0.216 ± 0.003 GeV [135] and f ρ(1450) = 0.185 +0.030 −0.035 GeV [92,136] for the ρ(770) and ρ(1450) resonances, respectively, and the mean lives τ B ± = 1.638×10 −12 s and τ B 0 = 1.519×10 −12 s for the initial states B ± and B 0 [2], respectively.The masses for the particles in the relevant decay processes, the decay constants for B, D and D * mesons (in units of GeV), and the Wolfenstein parameters for the CKM matrix elements, A and λ, are presented in Table I.We adopt the full widths Γ ρ(770) = 149.1 ± 0.8 MeV, Γ ρ(1450) = 400 ± 60 MeV, Γ a 0 (1450) = 265 ± 13 MeV, and Γ a 2 (1320) = 107 ± 5 MeV for the intermediate states involved in this work.TABLE I: Masses, decay constants (in units of GeV) for relevant states, as well as the Wolfenstein parameters for the CKM matrix elements from the Review of Particle Physics [2].The value of f D * is taken from [137].
With the help of the kaon form factor F K + K0 (s) discussed in detail in [38], we obtain the concerned branching fractions of the B mesons into a D or D * meson and a pair of kaons for the quasi-two-body processes ρ(770) + + ρ(1450) + → K + K0 .Our results are displayed in Table III. −0.06−0.12−0.06 In the results for the branching fractions shown in Tables II-III, the first source of the error corresponds to the uncertainties of the shape parameter ω B = 0.40 ± 0.04 of the B ±,0 wave functions, while the the Gegenbauer moments C D = 0.6 ± 0.15 or C D * = 0.5 ± 0.10 present in the D or D * wave functions [99] contribute to the second source of error.The third one is induced by the Gegenbauer moments a 0 R = 0.25±0.10,a t R = −0.60±0.20 and a s R = 0.75±0.25 [92] present in the wave functions of the intermediate states.The other errors for the PQCD predictions in this work, which come from the uncertainties of the masses and the decay constants of the initial and final states, from the uncertainties of the Wolfenstein parameters, etc., are small and have been neglected.Comparing our calculated branching rations of Table III with the measured results (in units of 10 −4 ) [4] and taking into account that half of the K0 or K 0 goes to K 0 S , we conclude that an important fraction of the decays B + → D( * )0 K + K0 and B 0 → D ( * )− K + K0 proceeds through the intermediate states ρ(770) + and ρ(1450) + , but there is still room for other contributions.
One could argue that the resonance ρ(770) + , as a virtual bound state [9,10], will not completely exhibit its properties in a quasi-two-body cascade decay like B 0 → D − [ρ(770) + →]K + K0 , since the invariant masses of the emitted kaon pairs exclude the region around the ρ(770) pole mass.However, as we will show below, the width of this resonance renders its contribution quite sizable in the energy region of interest.It is therefore important to consider explicitly the subthreshold resonances in the analysis of the branching ratios, even if they contribute via the tail of their mass distribution.In other words, experimental analyses or theoretical studies of three-body B meson decay process should not attribute as nonresonant K K invariant mass strength the specific known contribution from a certain resonant state like the ρ(770).
To make this point more evident, we show in Fig. 3 the differential branching fraction for the quasitwo-body decay B 0 → D − ρ + → D − K + K0 .The dashed line with a peak at about 1.465 GeV reveals the contribution from the ρ(1450) + , while the dash-dot line, depicting the contribution of the ρ(770) + , presents a bump around 1.2 GeV, which shall not be claimed experimentally as a resonant state with quite a large decay width.This bump is actually formed by the BW tail of the ρ(770) + along with the phase space factor of Eq. ( 4).The interference between the BW expressions for ρ(770) + and ρ(1450) + is constructive in the region before the pole mass of the ρ(1450) + and destructive after it as a result of the sign difference between c K ρ(770) = 1.247 ± 0.019 and c K ρ(1450) = −0.156± 0.015 [38] in Eq. (10).Note that the theoretical distribution has the same pattern in the low-mass region of the kaon pair as that shown in the bottom panel of Fig. 4 for the three-body decay B0 → D + K − K 0 S .This comparison reflects the dominant contributions for this decay coming from the intermediate states ρ(770) + and ρ(1450) + .
Near the threshold of the kaon pair, one finds remarkable enhancements in the m(K − K 0 S ) distributions for the decays B + → D0 K + K 0 S and B + → D * 0 K + K 0 S from Belle II [4], but not for B 0 → D − K + K0 or B 0 → D * − K + K0 .The invariant kaon pair mass around 1 GeV is the energy region of the state a 0 (980), but we do not expect the same strength of the a 0 (980) contributions in the B + → D( * )0 K + K0 and the B 0 → D ( * )− K + K0 processes, since their decay mechanisms proceed through different quark-type Feynman diagrams, shown in Fig. 2, as explained in the following.The annihilation Feynman diagrams represented by Fig. 2 (c) will only contribute to the decays B 0 → D ( * )− R + → D ( * )− K + K0 , and the contributions are highly suppressed when comparing with the those from the emission diagrams of Figs.
Let us now proceed to the explicit numerical calculation.Within the naive factorization approach, the evaluation of the decay amplitude for the B + → D0 K + K0 decay with the subprocesses a 0 (980, 1450) + → K + K0 can be found in Appendix B. With Eq. (B5) and the inputs from the Review of Particle Physics [2], we obtain a branching fraction B = 1.56 × 10 −5 for the quasi-two-body decay B + → D0 a 0 (980) + → D0 K + K0 , which corresponds to a value B(B + → D0 a 0 (980) + ) = 1.07 × 10 −4 for the two-body decay.
Likewise, we obtain B = 0.72 × 10 −5 for B + → D0 a 0 (1450) + → D0 K + K0 , where we have employed F B→a 0 (1450) 0 (0) = 0.26 [146] and Γ(a 0 (1450) → K K)/Γ(a 0 (1450) → ωππ) ≈ 0.082 [2].In order to check the reliability of the method we adopted here, the measured channel B 0 → D + s a 0 (980) − is studied as a reference.This is a process with a B 0 → a 0 (980) − transition and an emitted D + s state.Within naive factorization, we find B(B 0 → D + s a 0 (980) − ) = 1.93 × 10 −5 .This branching fraction is very close to the upper limit 1.9 × 10 −5 at 90% C.L. presented by the BABAR Collaboration in Ref. [147] assuming B(a 0 (980) + → ηπ + ) to be 100%, but it is much smaller than the prediction B = 4.81 +2.19  −1.79 × 10 −5 in [148] within PQCD for the decay B 0 → D + s a 0 (980) − .However, taking into account Γ(a 0 (980) → K K)/Γ(a 0 (980) → πη) = 0.172 ± 0.019 [2], one has B(a 0 (980) + → ηπ + ) ≈ 0.85 and this will change the upper limit in [147] for B 0 → D + s a 0 (980) − up to 2.24 × 10 −5 at 90% C.L., which is still much smaller than the prediction in [148], hinting that the PQCD approach is possibly not appropriate for the study of the B 0 → D + s a 0 (980) − decay with the B → a 0 (980) transition.When we put the contributions from a 0 (980, 1450) + → K + K0 and ρ(770, 1450) + → K + K0 for the decay B + → D0 K + K0 together, the resulting differential branching fraction does not have the shape shown in the top panel of Fig. 4. The contribution from the scalar intermediate state a 0 (980) + is far from what would be required to overcome the peak of the ρ(770, 1450) + distribution in order to reproduce the enhancement near the threshold of K + K0 pairs measured experimentally.The shape of the measured B + → D0 K + K0 differential branching fraction would only be obtained with a branching fraction B ≈ 4.5 × 10 −4 for the quasi-two-body decay B + → D0 a 0 (980) + → D0 K + K0 , which is beyond the total branching fraction for B + → D0 K + K0 decay.This situation is probably indicating the existence of large nonresonant contributions to the B + → D0 K + K0 decay around the threshold of the kaon pair or other unknown sources.Note that the interference between ρ(770) + and ρ(1450) + could reduce the corresponding branching fractions in Table III through an appropriate complex phase difference between their respective BW expressions.This would alleviate the requirement of an enhanced contribution from the a 0 (980) + .For example, a factor of e iπ/4 before the BW of the ρ(1450) + will produce half of the B + → D0 [ρ + →]K + K0 branching fraction listed in Table III.But such an universal phase difference will also make the branching fractions of the decays B 0 → D ( * )− [ρ + →]K + K0 decrease by half in Table III, which is not desirable.

IV. SUMMARY
To sum up, the Belle II Collaboration presented a measurement for the B + → D( * )0 K + K0 and B 0 → D ( * )− K + K0 decays very recently, where the bulk of the observed m(K + K 0 S ) distribution was located in the low-mass region of the kaon pair, showing structures in all four decay channels.In this work we have presented a theoretical calculation of these decays within the factorization method.We have focused on exploring the region of kaon pair invariant masses m(K + K0 ) < 1.7 GeV.The resonance contributions from vector intermediate states ρ(770, 1450) + have been found to dominate the branching fractions for the three-body decays B + → D( * )0 K + K0 and B 0 → D ( * )− K + K0 , representing roughly half of the total branching fractions of the corresponding decay channels.The role of the tensor a 2 (1320) + was analyzed and found to give negligible contributions to the branching fractions of these four decay processes and the contribution of the state a 0 (980) + turned out to be less important than expected in the m(K + K 0 S ) region near the threshold of the kaon pair.As a result of our study, we conclude that the enhancement of events in the kaon pair distribution near threshold observed in the B + → D0 K + K0 and B + → D * 0 K + K0 decay processes can not be interpreted as the resonance contributions from the a 0 (980) + meson.The nonresonant contributions are probably governing the formation of the kaon pair in B + → D( * )0 K + K0 near the threshold of K + K0 , and hence deserve further examination both from the theoretical and the experimental sides.PID2020-118758GB-I00, financed by the Spanish MCIN/ AEI/10.13039/501100011033/, as well as by the EU STRONG-2020 project, under the program H2020-INFRAIA-2018-1 grant agreement no.824093.
The wave functions for B, D and D * mesons and the corresponding inputs are the same as they in Ref. [99].The kaon and pion timelike form factors are referred to the Section II of Ref. [38].With the subprocesses ρ + → K + K0 , where ρ is ρ(770) or ρ(1450), the specific expressions in PQCD approach for the Lorentz invariant decay amplitudes of these general amplitudes F and M for B → D( * ) ρ → D( * ) K + K0 decays are given as follows: The amplitudes from Fig. 2 (a) for the decays with a pseudoscalar D0 or D − meson in the final states are given as The amplitudes from Fig. 2 (c) the annihilation diagrams are written as Where the T ρ, T D and Aρ in the subscript of above expressions stand for B → ρ, B → D transitions and the annihilation Feynman diagrams, respectively.The longitudinal polarization amplitudes from Fig. 2 (a) for the decays with a vector D * 0 or D * − meson in the final state are written as The longitudinal polarization amplitudes from Fig. 2 (b) are The longitudinal polarization amplitudes from Fig. 2 (c) are The normal along with transverse polarization amplitudes from Fig. 2 for the decays with a vector D * 0 or D * − are written as The hard functions h i , the hard scales t i with i ∈ {a, b, c, d, e, f, g, h, m, n, o, p}, and the evolution factors E e,a,n have their explicit expression in the Appendix of Ref. [99].
The B → D ( * ) matrix element is described by the transition form factors [159] ⟨D(p ′ )|cγ µ b|B(p)⟩ = F BD 0 (q 2 ) where q = p − p ′ .We parameterize the matrix element for the B → a 0 transition in terms of form factors F Ba 0 and F Ba 1 as ⟨a 0 (p ′ )|qγ µ γ 5 b|B(p)⟩ = iF Ba 0 (q 2 ) With Eqs. ( 2)-( 3) and related transition form factors above, we have the decay amplitude The expressions and related parameters for F Ba 0 and F BD 0 (s) are found in Refs.[68,144,146,160].

FIG. 2 :
FIG.2: Typical Feynman diagrams for the decays B + → D( * )0 R + → D( * )0 K + K0 and B 0 → D ( * )− R + → D ( * )− K + K0 at quark level, where (a) and (b) are the emission diagrams, (c) is the annihilation one, the quark q = u and d for the B + and B 0 processes, respectively, and the symbol ⊗ stands for the weak vertex.
A2) where the symbol ζ = 1 − ζ, the mass ratios r = m D ( * ) /m B and r c = m c /m B .The amplitudes from Fig. 2 (b) are written as

TABLE II :
PQCD results for the branching fractions of the quasi-two-body decays B

TABLE III :
PQCD predictions for the branching fractions of the concerned quasi-two-body decays with the subprocess ρ