The semileptonic baryonic decay $D_s^+\to p\bar p e^+ \nu_e$

The decay $D_s^+\to p \bar p e^+\nu_e$ with a proton-antiproton pair in the final state is unique in the sense that it is the only semileptonic baryonic decay which is physically allowed in the charmed meson sector. Its measurement will test our basic knowledge on semileptonic $D_s^+$ decays and the low-energy $p\bar p$ interactions. Taking into account the major intermediate state contributions from $\eta, \eta', f_0(980)$ and $X(1835)$, we find that its branching fraction is at the level of $10^{-9} \sim 10^{-8}$. The location and the nature of $X(1835)$ state are crucial for the precise determination of the branching fraction. We wish to trigger a new round of a careful study with the upcoming more data in BESIII as well as the future super tau-charm factory.


I. INTRODUCTION
A great deal of effort has been devoted to the baryonic decay modes of B mesons [1,2] due to the fact that the B meson is heavy enough to allow a baryonantibaryon pair production in the final state. Concerning the semileptonic decay involving a baryon-antibaryon pair, B − → ppℓ −ν ℓ (ℓ = e, µ) is the only measurement that has been done by the Belle Collaboration in 2014 [3]. Its branching fraction was reported to be (5.8 +2. 4 −2.1 ± 0.9) × 10 −6 with the upper limit 9.6 × 10 −6 at the 90% confidence level. In the charmed meson sector, D + s → pn is the only hadronic baryonic D decay mode which is physically allowed. Its branching ratio is naively expected to be very small, of order 10 −6 , due to chiral suppression [4]. Hence, the observation of this mode by CLEO with B(D + s → pn) = (1.30 ± 0.36 +0.12 −0.16 ) × 10 −3 [5] is indeed a surprise. Nevertheless, it can be explained by the final-state rescattering of π + η ( ′ ) and K +K 0 into pn [6]. Besides the channel D + s → pn, we notice that there is another physically allowed one, D + s → ppe + ν e . The mass difference m D + s − 2m p ≈ 82 MeV prohibits the emission of π + or even the lepton µ + , thus only the electron mode is permissible. Moreover, the pp pair stays in the near-threshold region, i.e, the invariant mass squared s = (p p +pp) 2 is not far from 4m 2 p . The future experimental measurement can rectify the description of D + s → pp hadronic transition form factors as well as the low-energy pp interaction. If this channel can be observed, it renders a preponderant possibility to access the pp bound state due to the low-energy pp region.
Below we will calculate the branching fraction of the decay channel D + s → ppe + ν e . We first consider the Cabibbo-favored decay D + s → M(ss) + e + ν e with M being the meson containing a sizable ss quark component. Since such meson decaying to a pp pair is an OZI suppressed process, we shall focus on the intermediate mesons M with comparable amount of qq = 1 √ 2 (uū + dd) and ss components in order to alleviate the OZI suppression. Combining the existing knowledge on the D + s → M transition form factors and Mpp couplings fixed by the pp scattering data, we are able to take into account the η, η ′ , f 0 (980) and X(1835) meson exchanges, and find that the branching fraction of D + s → ppe + ν e is at the level of 10 −9 ∼ 10 −8 .

II. KINEMATICS AND DECAY RATE
The four-body decay kinematics can be described in terms of five variables: the invariant mass squared of the pp pair, s = (p p +pp) 2 = M 2 pp , the invariant mass squared of the dilepton pair, s ℓ = (p ℓ + p ν ) 2 , the angles θ p , θ ℓ and φ, where θ p (θ ℓ ) is formed by the proton p (e + ) direction in the diproton (dilepton) center-of-mass (CMS) frame with respect to the diproton (dilepton) line of flight in the D + s frame, and φ is the dihedral angle between the diproton and dilepton planes. Their physical ranges are One may refer to e.g., Ref. [7] for an illustration of the four-body decay kinematics. Instead of the separate momenta p p , pp, p e , p ν , it is more convenient to use the following kinematic variables It follows that where the function X is defined by with m p being the proton mass. The term P · N can be derived by expressing the four momenta of p,p, e + , ν e in the rest frame of D + s via the Lorentz transformation, see e.g., [8]. Note that we have neglected the electron mass over most of the available phase space (although pp sits in the low energy region), i.e., m 2 e /s l ≪ 1. This has also been checked numerically 1 .
The decay amplitude of D + s → ppe + ν e can be written as where the currents V µ and A µ denote the vector and axial-vector ones, respectively, and their hadronic matrix elements will be discussed in Sec. III. We then have the differential decay rate with The four-body phase space was studied very early in 1960s within the context of K l4 analysis [9], see also Ref. [10] for a modern compilation. More details of derivation can be found in e.g., Refs. [11,12]. Equation (6) is in agreement with Refs. [13,14], as has been checked. 1 As a cross check, we may first keep the electron mass and retain the factor of z l = m 2 e /s l . Letting z l → 0, we then recover Eq. (29) below. The numerical results remain stable irrespective of the tiny electron mass.

III. HADRONIC MATRIX ELEMENTS AND RESULTS
We begin with the hadronic matrix elements [15] pp|V µ |D + Note the spinors u and v have a relative opposite sign under parity transformation. Various form factors f i and g i will be evaluated below. As mentioned in the Introduction, to alleviate the OZI suppression for the intermediate meson exchange that leads to the decay D + s → ppe + ν e , we shall focus on the intermediate states which have comparable qq and ss components. The twoand multi-meson exchanges are expected to be loop suppressed, and also the direct D + s ppW production vertex without any meson exchange can be safely neglected. We then concentrate on one-meson exchange denoted by M. The combination of the existing knowledge of D + s → M transition and the coupling ppM constitutes our basic strategy. In Ref. [16], we have explored the form factors and branching fractions for the semileptonic D s → M transition. As for the ppM part, we shall stick to the Jülich nucleon-antinucleon model [17] 2 which provides a fair description of pp total, elastic, charge-exchange and annihilation cross sections. In such a pp model, the exchanged mesons with mass up to 1.5 GeV were considered. We first include the spin-0 boson, η, η ′ , f 0 (980) in our study. The decay mechanism is shown in Fig. 1, where the upper panel describes the mechanism at the quark level with the bulk denoting the meson with the qq and ss components, and the lower one from the viewpoint of effective meson theory with the dashed line denoting the exchanged mesons.
We will calculate the Feynman diagram to single out the contributions of η, η ′ , f 0 (980) to the D + s → pp transition form factors. We have 2 The pp interaction within the framework of chiral effective field theory involving pion degrees of freedom and contact terms was recently explored in Ref. [18] and Ref. [19], see also a short review [20]. A similar method has been recently applied to charmed baryon scattering [21]. which amounts to inserting the intermediate meson with the momentum p and mass m, and V is the vertex of ppM coupling. The f 0 (980) has a large width which may remind us of replacing p 2 − m 2 by p 2 − m 2 − im f0 Γ f0 . However, it is not necessary to do so since the mass of f 0 (980) is still far from the pp invariant mass. The induced difference by including Γ f0 is only of order 0.1%. The D + s → M transition 3 can be described by where P denotes the pseudoscalars η and η ′ , S the scalar 3 The semileptonic Ds → η decay can be also treated in SU (3) heavy meson chiral perturbation theory [22][23][24], but the expression of form factors there is valid only in the soft η region. The pole model employed in the current work is applicable to the whole phase space.
thus q 2 = s l . The form factors F 1 (q 2 ) and F 0 (q 2 ) for D + s → η(η ′ ) have been investigated using the covariant light-front quark model [25,26], for i = 0 or 1. φ is the mixing angle between η and η ′ defined by [27] |η = cos φ|η q − sin φ|η s , It is determined to be 39.3 • ±1.0 • in the Feldmann-Kroll-Stech mixing scheme [27], which is consistent with the recent result φ = 42 • ± 2.8 • extracted from the CLEO data [28]. In Ref. [16] it has been shown that such a description of form factors gives a rather good description of the branching fraction compared to experiment, and that replacing m Ds by m D in the denominator does not make significant difference for the result. As we have already commented in Ref. [16], the D + s → f 0 (980) transition form factor cannot be appropriately treated by the covariant light-front model since i) f 0 (980) is widely believed to be a tetraquark state (see e.g., [29]) or a KK molecular (see e.g. Refs. [30,31]) rather than a pure quark-antiquark meson; and ii) the decay constant of f 0 (980) vanishes due to the charge conjugation invariance and thus there is no reliable constraint on the parameter in its wave function within the light-front quark model. However, the information of F 1 (q 2 ) for D + s → f 0 (980) is directly accessible by experiment, that is 4 with M pole = 1.7 +4.5 −0.7 GeV from the CLEO Collaboration [32]. This situation is different from Refs. [34,35], where only the form factor F 0 (q 2 ) enters in the factorization scheme of the two-body nonleptonic decay. In 4 The value F 1 (0) = 0.4 is not shown explicitly in Ref. [32], but can be obtained using the masses m f 0 , M pole and B(D + s → f 0 (980)e + νe) ≈ 0.4% reported there. The slope of F 1 (q 2 ), namely, M pole , is fitted to the measured event distribution, which differs from the dΓ/dq 2 only by an overall constant, so F 1 (0) cannot be constrained by the event distribution and is left as a float in Ref. [32]. the decay rate of the semileptonic decay for D or D + s to spin-0 boson, the form factor F 0 (q 2 ) is accompanied by the electron mass and thus negligible. In other words, F 0 (q 2 ) can be constrained by the corresponding nonleptonic decay rate based on factorization, but not from a direct experimental measurement. That is [34], For the part of the pp interaction, we have the Lagrangian [17] for the nucleon-nucleon-pseudoscalar (NNP ) coupling, and for the nucleon-nucleon-scalar coupling (NNS), with ψ(x) denoting the nucleon field and φ(x) the meson field. The dimensionless couplings read [17] 5 g η = 2.87, g η ′ = 3.72, g f0 = 8.48.
Note for the NNP coupling there is another form, namely, the so-called pseudovector coupling, The pseudoscalar coupling and the pseudovector one are related by for free nucleon satisfying the Dirac equation. One may refer to Ref. [36] for more details. The pseudoscalar coupling was used in Ref. [17], although the pseudovector form of the Lagrangian appeared in the appendix of the paper. Another essential and "dominant" piece should be the X(1835) (J P C = 0 −+ ) exchange since i) it locates near the pp threshold such that the propagator can enhance the contribution 6 , and ii) the strong connection/relation between X(1835) and the pp state. The first observation of X(1835) (denoted by the X particle below) was reported by BESII from the channel J/ψ → γpp [38], where the mass reads 1859 +3 +5 −10 −25 MeV with the statistic and systematic errors, in order, by using the S−wave Breit-Wigner function. The huge enhancement of the event distribution near the pp threshold was interpreted as the effect due to the pp final-state interaction (FSI) [39], where the Watson-Migdal approach is exploited, i.e., the amplitude for J/ψ → γpp is expressed by a normalization constant multiplied by the pp scattering T -matrix. A refit with the inclusion of the fixed FSI factor introduced in Ref. [39] has been carried out in a subsequent publication [40]. The resulting mass is slightly changed and reads 1826.5 +13.0 −3.4 MeV [1]. From the state-of-the-art viewpoint, such FSI treatment has been superseded by the outcome of Ref. [18], where the total amplitude A is written as A = A 0 + A 0 G 0 T with A 0 , G 0 , T denoting bare production amplitude without FSI, free Green function and pp scattering T -matrix, respectively. One can refer to the review in Ref. [20] for more details. It has been shown that the threshold enhancement could be indeed a pp bound state [18]. However, one should be cautious that the pure FSI explanation proposed in Ref. [39] reproduces the data very well and thus cannot be excluded. To date, the nature or even its existence of X(1835) still remains mysterious. However, X(1835) can be viewed, at least, as a poor man's approach or an effective way to incorporate the strong FSI of pp, and in this respect, we include it as a subthreshold resonance in our meson-exchange model calculation.
Note that the X(1835) has also been observed in γη ′ ππ channel with a statistical significance of 7.7σ [40]. So, it could be most likely a mixing state of pp and ss and this idea has been investigated in e.g., Ref. [41]. Then one may write with |c 1 | 2 + |c 2 | 2 = 1. The maximum production for D + s → ppe + ν e corresponds to c 1 = c 2 = 1/ √ 2. The Lagrangian for the X(1835)pp coupling is of the same form as ppη, and we will take the coupling constant g Xpp ≈ 3.5 [42] provided that X(1835) is a pure baryonium. This value agrees with the one given in Ref. [43] after applying the Weinberg compositeness theorem [12,[44][45][46], i.e., the coupling g Xpp is obtained from the vanishing wave function renormalization. In the case of Eq. (21), the true Xpp coupling will be multiplied by a factor of 1/ √ 2, while the transition D + s → X(1835) will proceed via the form factors F Ds→ηs Combining all these ingredients together, we obtain and all other form factors vanish, where To evaluate the amplitude modulus squared, we introduce the hadronic and leptonic tensor currents given by (27) and respectively, with the convention ǫ 0123 = 1. The amplitude modulus squared reads 7 Under these discussions, we can also obtain an approximate branching fraction D + s → X(1835)e + νe ≈ (1.6 +0.2 −0.7 ) × 10 −6 using the averaged mass m X = 1826.5 +13.0 −3.4 MeV obtained by PDG [1].
where the g 3 and f 3 terms are suppressed by the smallness of the electron mass. This leads to the branching fraction based on the mass of 1826.5 MeV for the X(1835) reported in PDG [1]. The uncertainties arise from various sources, for example, the coupling constants and the mass of X(1835). The dominant uncertainty should be ascribed to the precision on the X(1835) mass. If we use the mass 1859 MeV, which corresponds to the fit without the primitive treatment of FSI as we have already discussed above, the branching fraction will become B(D + s → ppe + ν e ) = 1.5 × 10 −8 .
As noticed in passing, the X(1835) exchange should dominate due to its proximity to the pp state. The X(1835) alone will contribute to 1.42 × 10 −8 for the branching fraction after turning off the η, η ′ , f 0 (980) effects. If the mass of X(1835) is closer to the pp threshold, the branching fraction will be further increased. Considering the width of X(1835) (around 80 MeV) [18], the branching fraction will become smaller by a few times. In this sense, we prefer to emphasize the importance of the "precision" on the X(1835) mass measurement. By the end of 2018, around 10 10 J/ψ data samples are going to be accumulated within the one-year running period [47] and both J/ψ → γpp and J/ψ → γη ′ ππ can be re-examined to improve the accuracy. The experimental situation will be further improved in the case of super tau-charm factory [48][49][50] with the planned luminosity of 100 times as much as BESIII.
In principle, the diagram with one-baryon exchange can be also considered. There is the process D + s → pn followed by the neutron beta decay,n →pe + ν e as depicted in Fig. 2. As noticed before, the decay rate of D + s → pn measured to be (1.3 ± 0.4) × 10 −3 [5] is unexpectedly large beyond the naive weak annihilation mechanism [6,51]. Moreover, both the antineutron and antiproton in Fig. 2 are soft and hence the contribution of this diagram may be possibly large due to the propagator of the antineutron. However, the net contribution is again highly suppressed because it involves two weak vertices proportional to G 2 F . We indeed calculated Fig. 2, and found that it contributes to the form factors g 1 , g 2 , g 3 and f 1 , f 2 , f 3 . Within such microscopic process, this gives a picture of how the general form factors (constructed from Lorentz structure) emerge. Here we comment on the possible OZI violation. The question may arise from the large φ production rate in pp collisions compared to the ω one, which is attributed to either the intrinsic ss component in the wave function of the proton [52] or the rescattering of kaons [53,54], see also the reviews in Refs. [55,56]. The strangeness content of the nucleon has also been revealed in several experimental observations, e.g., the strange quark spin polarization, σ πN term, magnetic moment of the proton and the ratio of strange and non-strange quark flavor distributions. However, the weight of the strange content is still small such that it is not expected to make large influence on the current results. On the other hand, the low-energy pp scattering data can be fairly well repro-duced without the inclusion of the φ meson exchange, as e.g., done in Ref. [17], for which we stick to the ppM coupling.
In the B meson sector, the semileptonic baryonic decay B − → ppℓ −ν ℓ has been studied in [57] where the form factors f i and g i with i = 1, · · · , 5 defined in analog to Eq. (8) were obtained by fitting them to the available data of B → ppM in conjunction with the pQCD counting rule for form factors. However, this pQCD argument is not applicable to our case as the energy release in D + s → pp transition is rather small. Moreover, we notice that the predicted branching fraction B(B − → ppℓ −ν ℓ ) = 1.04 × 10 −4 in [57] is too large compared to the experimental observation of order 6 × 10 −6 .

IV. CONCLUSION
In this work we have discussed the unique decay D + s → ppe + ν e with a proton-antiproton pair in the final state. It is the only semileptonic baryonic decay which is physically allowed in the charmed meson sector, besides the hadronic baryonic decay D + s → pn. There is abundant physics in this channel. Its measurement will test our knowledge on baryonic weak decays and the low-energy pp interactions. Taking into account the contributions from the intermediate states η, η ′ , f 0 (980) and X(1835), we find that its branching fraction is 10 −9 ∼ 10 −8 . Our prediction can be tested by BESIII/BEPCII data and its measurement is ongoing.