Semileptonic decays $D\to\eta\pi e^+\nu_e$ in the $a_0(980)$ region

The mechanism of the four-quark production of the light scalar isovector four-quark state $a_0(980)$ in the $D\to\eta\pi e^+\nu_e$ decays is discussed. It is shown that the characteristic features of the shape of the $\eta\pi$ mass spectra expected in our scheme can serve as the indicator of the production mechanism and internal structure of the $a_0(980)$ resonance.

The mechanism of the four-quark production of the light scalar isovector four-quark state a0(980) in the D → ηπe + νe decays is discussed. It is shown that the characteristic features of the shape of the ηπ mass spectra expected in our scheme can serve as the indicator of the production mechanism and internal structure of the a0(980) resonance.
We continue the study of the decays D 0 → dūe + ν e → a 0 (980) − e + ν e → ηπ − e + ν e and D + → dde + ν e → a 0 (980) 0 e + ν e → ηπ 0 e + ν e started in Ref. [16]. These decays are interesting in that they provide direct probing of constituent two-quark dū and (uū − dd)/ √ 2 components in the wave functions of the a 0 (980) − and a 0 (980) 0 mesons, respectively [5]. In Ref. [16] the mass spectra of ηπ pairs were presented for the case when the a 0 (980) states have no two-quark components at all. It was assumed that the production of the four-quark a 0 (980) state occurs via its mixing with the heavy qq state a ′ 0 (1400), due to their common decay channels into ηπ, η ′ π, and KK. Here we also assume that the a 0 (980) meson has a four-quark structure [23,24]. The difference is that we consider the mechanism of four-quark fluctuations of dū and dd sources, dū → dqqū and dd → dqqd, which in the language of two-body hadronic states means the creation of the ηπ, η ′ π, and KK pairs, which are then dressed by strong interactions in the final state. We elucidate the characteristic features of this mechanism and do not involve in consideration a heavy state of the a ′ 0 (1400) type. A similar approach was used in Refs. [20,25] to describe the CLEO [1] and BESIII [10] data on the decays D + s → f 0 (980)e + ν e → π + π − e + ν e and D + → f 0 (500)e + ν e → π + π − e + ν e as well as the BESIII [26] data on the J/ψ → γπ 0 π 0 decay in the π 0 π 0 invariant mass region from the threshold up to 1 GeV. This paper is organized as follows. In Sec. II after a brief discussion of the experimental situation, the general formulas are given for the widths of semileptonic decays D 0 → ηπ − e + ν e and D + → ηπ 0 e + ν e with the production of the ηπ system in the S wave. Section III is devoted to a discussion of the four-quark mechanism of the a 0 (980) resonance production in the D → ηπe + ν e decays. We show that the characteristic features of the shape of the ηπ − and ηπ 0 mass spectra expected in our scheme can serve as an indicator of the production mechanism and internal structure of the a 0 (980) state. In Sec. IV we discuss the decays D 0 → K 0 K − e + ν e , D + → K 0K 0 e + ν e , and D + → K + K − e + ν e which also are of interest in connection with the production of light scalar mesons. However, these decays are strongly suppressed by the phase space near the KK thresholds and their experimental investigations are little realistic at present. In Sec. V the conclusions are briefly formulated.
When obtaining Eq. (3), it was assumed that B(a 0 (980) − → ηπ − ) = B(a 0 (980) 0 → ηπ 0 ). In the measured ηπ mass spectra in the range of the invariant mass of the ηπ system, m ηπ ≡ √ s, from 0.7 GeV to 1.3 GeV, there is a complex configuration of background contributions [9] (see Fig. 1). A small number of signal events, a noticeable background and a wide step in √ s (equal to 50 MeV) do not allow us to clearly see in the ηπ − and ηπ 0 mass spectra the line shape of the a 0 (980) resonance. It is clear that the high statistics on the decays D → a 0 (980)e + ν e is highly demanded by the physics of light scalar mesons.
Let us write the differential width for the D 0 = cū decay into ηπ − e + ν e in the form (see, for example, Refs. [1,10,28]) where s and q 2 are the invariant mass squared of the virtual scalar state S − (or the S-wave ηπ − system) and the e + ν e system, respectively; G F is the Fermi constant, |V cd | = 0.221 ± 0.004 is a Cabibbo-Kobayshi-Maskawa matrix element [6]; p ηπ − is the magnitude of the three-momentum of the ηπ − system in the D 0 meson rest frame, and In a simplest pole approximation, the form factor f D 0 + (q 2 ) has the form [1,10,28] where we put m A = m D + 1 = 2.42 GeV [6] (in principle, m A can be extracted from the data by fitting). The amplitude s) describes the formation and decay into ηπ − of the virtual scalar isovector state S − produced in the D 0 → ηπ − e + ν e decay. The ηπ − invariant mass distribution integrated over the full q 2 region is given by where the function Φ( As can be seen from Fig. 2, it notably enhances the ηπ − mass spectrum as √ s decreases. The differential widths of the semileptonic decay of the D + = cd meson into the S-wave ηπ 0 system are described by the formulas similar to Eqs. (4)-(8).

III. FOUR-QUARK PRODUCTION MECHANISM OF THE FOUR-QUARK a0(980) RESONANCE
We start with the semileptonic decay of the D 0 = cū meson D 0 → dūe + ν e → (S − → ηπ − )e + ν e . Let, as a result of radiation of the lepton pair e + ν e by the valence c quark included in D 0 , a virtual system of d andū quarks in the scalar state is produced. Consider the four-quark fluctuations of such a dū source, dū → dqqū, corresponding to the diagram shown in Fig. 3. The limitation to this type of fluctuations is related to the Okubo-Zweiga-Iizuka (OZI) rule [29][30][31][32], according to which other fluctuations such as annihilation or creation of qq pairs, corresponding to the so-called "hair-pin" diagrams, are suppressed. In the language of hadronic states the seed four-quark fluctuations dū → dqqū imply the production of ηπ − , η ′ π − , and K 0 K − meson pairs. As a first approximation, we assume that the amplitude of the dū → dqqū transition, which we denote by g 0 , does not depend on the flavor of the light q quark. Then for the hadron production constants g dūηπ − , g dūη ′ π − , and g dūK 0 K − the following relations hold: Here θ i = 35.3 • is the so-called "ideal" mixing angle and θ p = −11.3 • is the mixing angle in the nonet of the light pseudoscalar mesons [6]. The first two relations in Eq. (9) are obtained taking into account the expansion of the state containing non-strange quarks η n = (uū + dd)/ √ 2, which is created in a pair with π − , in terms of physical states η and η ′ with definite masses: Thus, the production of the four-quark a 0 (980) − resonance can occur as a result of the fact that the seed four-quark fluctuations dū → ηπ − , η ′ π − , K 0 K − are dressed by strong interactions in the final state. The described picture of the a 0 (980) − production in the language of hadronic diagrams is shown in Fig. 4. According to this figure, we write   Figure 4: The production mechanism of the four-quark a0(980) − resonance in the decay D 0 → ηπ − e + νe.
where I ab (s) is the amplitude of the two-point loop diagram with ab intermediate state and T ab→ηπ − (s) is the amplitude of the S-wave ab → ηπ − transition; ab = ηπ − , η ′ π − , K 0 K − . The loop amplitude I ab (s) has the form where the function I ab (s) is the singly subtracted at s = 0 dispersion integral. Explicit expressions for I ab (s) in different regions of s are given in the Appendix. The real and imaginary parts of this function for different ab are shown in Fig. 5. The quantity C ab in Eq. (11) is a subtraction constant. The choice of these constants will be discussed below. We saturate the amplitudes T ab→ηπ − (s) with the contribution of the four-quark a 0 (980) − resonance. The flavor structure of its wave function has the form [23] a 0 (980 Considering that η s = ss = η ′ sin(θ i − θ p ) − η cos(θ i − θ p ), for the coupling constants of the a 0 (980) − with the superallowed decay channels into pairs of pseudoscalar mesons, the following relations hold: whereḡ is the overall coupling constant. Thus, where D a − 0 (s) is the inverse propagator of the a 0 (980) − resonance. It has the form where m a −  We now rewrite Eq. (10) using Eqs. (9), (13), and (14) and notation ϑ = θ i − θ p as follows: This expression shows that the contributions proportional to the loops I ηπ − (s) and I η ′ π − (s) enter with different signs and therefore partially cancel each other. This cancellation implements the OZI rule in the language of hadronic intermediate states [33,34]. Indeed, in the case under consideration, the sum of the contributions of the ηπ − and η ′ π − loops is due to the transition dū → η n π − and then η n π − into the η s π − component of the a 0 (980) − wave function, which is suppressed according to the OZI rule. Let us discuss the choice of the subtraction constants C ab in the loops I ab (s), see (11). The assumption that all these constants are equal, C ηπ − = C η ′ π − = C K 0 K − , is simplest and most economical in terms of the number of free parameters. Note that nothing changes if we put (17) does not depend on their value at all, due to the OZI reduction, and depends only on the parameter C K 0 K − . Below we utilize this choice for C ab . In so doing, the model is still quite flexible.
Thus, there are two parameters characterizing the mechanism of the a 0 (980) − production in D 0 → a 0 (980) − e + ν e → ηπ − e + ν e in our model. They are: the product f D 0 + (0)g 0 , it determines the general normalization of the decay width, and the constant C K 0 K − which essentially influences on the shape of the ηπ − mass spectrum. The actual parameters of the a 0 (980) − resonance are its mass m a − 0 and coupling constantḡ, see (13). Figure 6 shows as a guide the shapes for the solitary a 0 (980) − resonance in ηπ − , K 0 K − , and η ′ π − decay channels. They are plotted using Eqs. (13), (15), (16), and (18) at m a − 0 = 0.985 GeV and g 2 is approximately equal to 132 MeV, while the visible width of the a 0 (980) − peak at its half-maximum in ηπ − channel is ≈ 55 MeV (see Fig. 6). This narrowing is the consequence of the proximity of m a − 0 to the K 0 K − threshold and strong coupling of the a 0 (980) − to both ηπ − and K 0 K − decay channel [35,36].
Recently, the BESIII Collaboration observed in the high-statistics experiment on the decay χ c1 → ηπ + π − the impressive peak from a 0 (980) resonance in the ηπ mass spectra [37]. There is a possibility that the a 0 (980) resonance will manifest itself in the D → ηπe + ν e decays in a similar way. Below we discuss the conditions under which such a possibility is realized in our model.
The solid curve in Fig. 7 shows an example of the shape of the ηπ − mass spectrum in the decay D 0 → dūe + ν e → (S − → ηπ − )e + ν e with a peak in the region of 1 GeV due to the creation of the a 0 (980) − resonance. The calculation was done using Eqs. (7), (13), and (17) at the above values of m a − 0 and g 2 a − 0 ηπ − /(16π) and C K 0 K − = 0.6. The value of C K 0 K − for this example we took to be comparable with the value of Re I K 0 K − (s) in the region near the K 0 K − threshold, see Fig. 5(a). The difference between the solid curves in Figs. 6 and 7 demonstrates the possible influence of the production mechanism on the shape of the a 0 (980) − peak in the ηπ − channel. The dotted curve in Fig. 7 shows the contribution from the last term in Eq. (17) corresponding, according to diagram (b) in Fig. 4, to creation of the a 0 (980) − resonance via the K 0 K − intermediate state. The dash-dotted curve shows the contribution due to the seed pointlike diagram (a) in Fig. 4. In Eq. (17), it corresponds to the term equal to g 0 √ 2 sin ϑ. The long dashed curve shows the total contribution to the a 0 (980) − production from the ηπ − and η ′ π − intermediate states (see Fig.  4). In Eq. (17), this contribution is proportional to I ηπ − (s) − I η ′ π − (s) . Recall that we consider the case when C ηπ − = C η ′ π − . At last, the short dashed curve in Fig. 7 shows the total contribution due to the terms in square brackets in Eq. (17). It is seen that these terms strongly compensate each other in the region √ s ≈ m a − 0 . First, this happens because in this region Re I ηπ − (s) − I η ′ π − (s) ≈ 0, as shown in Fig. 5(a). Second, the other contributions for √ s < m K 0 + m K − can be represented as e iδ ηπ − (s) cos δ ηπ − (s), where δ ηπ − (s) is the phase of the S-wave elastic ηπ − scattering amplitude. In this case, δ ηπ − (s) is a purely resonant phase and therefore cos δ ηπ − (m 2 a − 0 ) = 0. Our statement follows from the chain of equalities: Thus, the phase of the amplitude F D 0 dū→S − →ηπ − (s), see Eq. (17), for √ s < m K 0 + m K − coincides with the phase of the elastic ηπ − scattering in accordance with the requirement of the unitarity condition [38]. Equation (10), which has a more general form, also satisfies the unitarity requirement, since in the elastic region the phases of the amplitudes T η ′ π − →ηπ − (s) and T K 0 K − →ηπ − (s) coincide with the phase of the amplitude T ηπ − →ηπ − (s) and the functions I η ′ π − (s) and I K 0 K − (s) are real. The phenomenon of compensation of the pointlike production amplitude [g 0 √ 2 sin ϑ in Eq. (17)] by the resonant one at √ s ≈ m res is also well known, see, for example, [39,40].
The relative role of the K 0 K − intermediate state in the a 0 (980) − production increases with increasing the parameter C K 0 K − . The width of the resonance peak in the ηπ − mass spectrum narrows, while its height increases and it becomes more pronounced. However, the characteristic enhancement of the left wing of the resonance spectrum and a sharp jump of its right wing (see Fig. 7), caused by interference between different contributions, persist when C K 0 K − changes over a wide range. The specific form of the ηπ − mass spectrum is directly related to the considered mechanism of the a 0 (980) − production and therefore can serve as its indicator.
Decays of D + → a 0 (980) 0 e + ν e → ηπ 0 e + ν e and D 0 → a 0 (980) − e + ν e → ηπ − e + ν e have the same mechanisms, as is clear from Figs each other (within a constant phase factor which can be chosen equal to one) by the relation Figure 9: The production mechanism of the four-quark a0(980) 0 resonance in the decay D + → ηπ 0 e + νe.
which follows from the assumption of equality of the a 0 (980) 0 and a 0 (980) − masses and isotopic symmetry for the coupling constants. It is easy to check using the following relations together with Eqs. (9) and (13). Relations in (21) take into account that the flavor structure of the a 0 (980) 0 wave function has the form [23] a 0 (980 Of course, for m a − 0 = m a 0 0 Eq. (19) may be slightly violated in the region of the KK thresholds, but the relation for the total decay widths should hold very well. However, it should be noted here that the shapes of the ηπ − and ηπ 0 mass spectra near the KK thresholds are very sensitive to the a 0 (980) mass and this fact can be used to estimate the mass difference of the Production of the subthreshold a 0 (980) − resonance in the decay D 0 → K 0 K − e + ν e is strongly (at least an order of magnitude) suppressed by the phase space in comparison with its production in D 0 → ηπ − e + ν e . Certainly, experimental investigations of the D → KKe + ν e decays near the KK thresholds is a difficult problem. A detailed theoretical analysis of these decays will be urgent as soon as it becomes possible to carry out the corresponding measurements. Therefore, here we only briefly describe the characteristic features of the KK mass spectra associated with the manifestation of the a 0 (980) resonance in our model.
The corresponding K 0K 0 and K + K − mass spectra are shown in Fig. 11, together with the K 0 K − one. Here we pay attention to the dominance of the decay channel into K 0K 0 over the K + K − channel. In fact, the decays D + → K 0K 0 e + ν e and D + → K + K − e + ν e are more complicated than the decay D 0 → K 0 K − e + ν e , as they can also contain the contribution from the isoscalar f 0 (980) resonance.

V. CONCLUSION
A simple model of the four-quark mechanism of the a 0 (980) resonance production in the decays D 0 → ηπ − e + ν e and D + → ηπ 0 e + ν e is constructed. It is shown that the characteristic features of the shape of the ηπ − and ηπ 0 mass spectra can serve as the indicator of the production mechanism and internal structure of the a 0 (980) state. Future experiments with high statistics on the decays D → a 0 (980)e + ν e → ηπe + ν e are highly demanded by the physics of light scalar mesons.