The $a_0(980)$ physics in semileptonic $D^0$ and $D^+$ decays

The decays $D^0\to d\bar u\,e^+\nu\to a^-_0(980)\, e^+\nu\to\pi^-\eta\, e^+\nu$ and $D^+\to d\bar d\,e^+\nu\to a^0_0(980)\, e^+\nu\to\pi^0\eta\, e^+\nu$ (and the charge conjugated ones) is the direct probe of the two-quark components in the $a^\pm_0(980)$ and $a^0_0(980)$ wave functions. Recent BESIII experiment is the first step in experimental study of these decays. We present a possible variant of $\eta\pi$ invariant mass distribution when $a_0(980)$ has no $q\bar q$ component at all.


I. INTRODUCTION
The a 0 (980) and f 0 (980) mesons are well-established parts of the proposed light scalar meson nonet [1]. From the beginning, the a 0 (980) and f 0 (980) mesons became one of the central problems of nonperturbative QCD, as they are important for understanding the way chiral symmetry is realized in the low-energy region and, consequently, for understanding confinement. Many experimental and theoretical papers have been devoted to this subject.
There is much evidence that supports the four-quark model of light scalar mesons [2,3].
It was shown in Ref. [8] that the production of a 0 0 (980) and f 0 (980) in φ → a 0 0 γ → ηπ 0 γ and φ → f 0 γ → π 0 π 0 γ decays is caused by the four-quark transitions, resulting in strong restrictions on the large-N C expansions of the decay amplitudes. The analysis showed that these constraints give new evidence in favor of the four-quark nature of the a 0 (980) and f 0 (980) mesons.
In Refs. [15,16] it was shown that the description of the φ → K + K − → γa 0 0 (980)/f 0 (980) decays requires virtual momenta of K(K) greater than 2 GeV, while in the case of loose molecules with a binding energy about 20 MeV, they would have to be about 100 MeV.
Besides, it should be noted that the production of scalar mesons in the pion-nucleon collisions with large momentum transfers also points to their compactness [17].
It was also shown in Refs. [18,19] that the linear S L (2) × S R (2) σ model [20] reflects all of the main features of low-energy ππ → ππ and γγ → ππ reactions up to energy 0.8 GeV and agrees with the four-quark nature of σ meson. This allowed for the development of a phenomenological model with the right analytical properties in the complex s plane that took into account the linear σ model, σ(600)−f 0 (980) mixing and the background [21]. This background has a left cut inspired by crossing symmetry, and the resulting amplitude agrees with results obtained using the chiral expansion, dispersion relations, and the Roy equation [22], and with the four-quark nature of the σ(600) and f 0 (980) mesons as well. This model well describes the experimental data on ππ → ππ scattering up to 1.2 GeV.
It is shown in Ref. [24] that the recent data on the K 0 S K + correlation in Pb-Pb interactions Ref. [25] agree with the data on the γγ → ηπ 0 and φ → ηπ 0 γ reactions and support the four-quark model of the a 0 (980) meson. It is shown that the data does not contradict the validity of the Gaussian assumption.
In Refs. [26,27] it was suggested the program of studying light scalars in semileptonic D and B decays. We studied production of scalars σ(600) and f 0 (980) in the D + s → π + π − e + ν decays, the conclusion was that the percentage of the qq components in σ(600) and f 0 (980) is small. This is the direct evidence in favor of exotic nature of these particles. Unfortunately, at the moment the statistics is rather poor, and thus new high-statistics data are highly desirable.

II. D-DECAYS INVOLVING SCALARS AND PSEUDOSCALARS
The amplitude of the D 0 → S(scalar) e + ν decay is of similar form to D + s decay [26] where G F is the Fermi constant, V cd is the Cabibbo-Kobayashi-Maskava matrix element, The influence of the f S − (q 2 ) form factor is negligible because of the small mass of the positron.
The decay rate into the stable S state is For the f S + (q 2 ) form factor we use the vector dominance model where Following Fig. 1 we write f S + (0) in the form ηπ 0 e + ν decays. Direct copy of Fig. 2 (a) and (c) in Ref. [28]. Green curves are signal, red ones represent total contribution, other ones represent backgrounds.
where g D 0 cū is the D 0 → cū coupling constant, g dūS is the dū → S coupling constant, F S is the loop integral assuming to be constant in the region of interest.
The amplitude of the  [25].
where m is the invariant mass of the taking into account the a ′ 0 scalar meson is where The D + → dd e + ν → S e + ν and D + → ηπ 0 e + ν decays are described the same way, see and π − → π 0 . Note g dda ′0 0 = g dūa ′− 0 / √ 2 with corresponding effect on matrix element and branching.
The key question is the size of a ′ 0 contribution. In Ref. [28] fits take into account only the a 0 (980) contribution, but the data prefer bigger signal contribution in the interval m ≡ M ηπ = 1.1 ÷ 1.3 GeV, i.e., on the right end of the distribution, see Fig. 3a. It can be a manifestation of sizeable a ′ 0 contribution.
Note also that the experimenters could obtain different results for total branching using the form of curve presented on Fig. 6.
Let us repeat that no less interesting is to probe the light scalars in semileptonic The approach of the given paper is valid for the B mesons decays B 0 → π − ηe + ν and IV. APPENDIX I: SCALAR PROPAGATORS AND POLARIZATION OPERA-

TORS
The matrix of the inverse propagators is where m = √ s, and the constant C a ′ 0 a 0 incorporates the subtraction constant for the transition a 0 (980) → (0 − 0 − ) → a ′ 0 and effectively takes into account the contributions of multiparticle intermediate states to the a 0 ↔ a ′ 0 transition. The inverse propagator of the scalar meson S [7,11,30,32] is where ab [ReΠ ab S (m 2 S ) − Π ab S (m 2 )] = ReΠ S (m 2 S ) − Π S (m 2 ) takes into account the finitewidth corrections of the resonance which are the one-loop contributions to the self-energy of the S resonance from the two-particle intermediate ab states. We take into account the intermediate states ηπ + , KK, and η ′ π + in the a + 0 (980) and a ′+ 0 propagators: and ηπ 0 , KK, and η ′ π 0 in the a 0 0 (980) and a ′0 0 propagators. For pseudoscalar mesons a, b and m a ≥ m b , m ≥ m + , one has where ρ ab (s) = 2p ab (s) and for m < m − , For completeness, we show in Table II the parameters that are not described above. One can find all of the details in Ref. [30]. Table II. Parameters not mentioned in Table I. In the given paper we take the formfactor G ω (s, t) = G ρ (s, t), differently from Refs. [6,30]. We take b ω (s) = b 0 ω + α ′ ω ln[1 + (s/s 0 )] , and obtain b 0 ω = 2.3 × 10 −3 GeV −2 , and s 0 = 1.005 GeV 2 . α ′ ω = 0.8 GeV −2 is the same. Form factors for the K * exchange are modified the same way. Besides, we obtain r a 2 = 1.2 GeV −1 instead of r a 2 = 1.9 GeV −1 in Refs. [6,30].
The πη scattering length agrees with the estimates based on current algebra and chiral perturbation theory, according to which a 1 0 ≈ 0.005 − 0.01 (in units of m −1 π ), see Ref. [6].