The scalar $f_0(500)$ and $f_0(980)$ resonances and vector mesons in the single Cabibbo-suppressed decays $\Lambda_c \to p K^+K^-$ and $p\pi^+\pi^-$

In the chiral unitary approach, we have studied the single Cabibbo-suppressed decays $\Lambda_c\to pK^+K^-$ and $\Lambda_c \to p \pi^+\pi^-$ by taking into account the $s$-wave meson-meson interaction as well as the contributions of the intermediate vectors $\phi$ and $\rho^0$. Our theoretical results for the ratios of the branching fractions of $\Lambda_c\to p \bar{K}^{*0}$ and $\Lambda_c\to p \omega$ with respect to the one of $\Lambda_c\to p \phi$ are in agreement with the experimental data. Within the picture that the scalar resonances $f_0(500)$ and $f_0(980)$ are dynamically generated from the pseudoscalar-pseudoscalar interaction, we have calculated the $K^+K^-$ and $\pi^+\pi^-$ mass distributions respectively for the decays $\Lambda_c\to pK^+K^-$ and $\Lambda_c\to p\pi^+\pi^-$. One can find a broad bump structure for the $f_0(500)$ and a narrow peak for the $f_0(980)$ in the $\pi^+\pi^-$ mass distribution of the decay $\Lambda_c\to p\pi^+\pi^-$, which is compatible with the BESIII measurement. For the $K^+K^-$ mass distribution, in addition to the narrow peak for the resonance $\phi$, one can see an enhancement structure near the $K^+K^-$ threshold. We encourage our experimental colleagues to measure these two decays, which would be helpful to understand the nature of the $f_0(500)$ and $f_0(980)$.

In this work, we perform the calculations of the decays Λ c → pK + K − and Λ c → pπ + π − using the chiral unitary approach and the final state interactions of the meson-meson interaction in coupled channels. The two pions in the final states of the decay Λ c → pπ + π − can propagate in s-wave, which will generate the f 0 (500) and f 0 (980) resonances, and for the decay Λ c → pK + K − , the f 0 (980) resonance dynamically generated from the s-wave K + K − final state interaction will result in an enhancement structure close to the K + K − threshold.
The paper is organized as follows. In Section II, we present the formalism and ingredients for the decay amplitudes of the Λ c → pK + K − and pπ + π − decays. Numerical results for invariant mass distributions of the K + K − and π + π − and discussions are given in Section III, followed by a short summary in the last section.

II. FORMALISM
In this section, we will present the formalism for the decays Λ c → pK + K − and Λ c → pπ + π − . The three-body decays of Λ c can preform in s-wave, where the final state interactions of π + π − or K + K − will dynamically generate the scalar resonances f 0 (500) and f 0 (980). In addition, the three-body decays can happen via the intermediate vector mesons ρ 0 or φ. We first introduce the formalism for the mechanism of final state interactions of π + π − or K + K − in s-wave in Subsect. II A, then we show the details for the mechanism of the Λ c decay via the intermediate vector A. s-wave final state interactions of K + K − and π + π − Following Refs. [37][38][39][40], we take the decay mechanism of the internal W emission mechanism for the decays Λ c → pK + K − and Λ c → pπ + π − as depicted in Figs. 1(a) and (b). For the weak decays of Λ c , the c quark decays into a W + boson and a s (or d) quark, then the W + boson decays into asu (ordu) pair. In order to give rise to the final states of pK + K − (or pπ + π − ), the ss (or dd) quark pair need to hadronize together with theqq (=ūu +dd +ss) produced in the vacuum, H (a) or H (b) , which are given by, where V (a) and V (b) are the weak interaction strengths. We use |p = 1 The matrix M in terms of pseudoscalar mesons can be written as, Then, we have, where we neglect the η ′ because of its large mass. V P is the meson-meson production vertex which contains all dynamical factors. In this work we take V cs = V ud = −sinθ c = 0.22534, V cd = V us = cosθ c = 0.97427 [41].
After the production of a meson-meson pair, the final state interaction in the s-wave between the mesons takes place, which can be parameterized by the re-scattering in the hadron level, as show in Figs. 2 and 3, where we will take into account both contributions from the diagrams of Fig. 1.
On the other hand, the decays Λ c → pK + K − and Λ c → pπ + π − can also proceed with the following steps: i) the charmed quark turns into W + and the s or d quark, with the K + or π + emission from the W + ; ii) the remaining quarks s or d and ud in the Λ c hadronize to the K − p or π − p. Although this mechanism of the external W emission is color favored, we do not know any information about the relative weight and phase between the external W emission and internal W emission. Furthermore, the external W emission process provides the contributions for the pK The mechanisms of the decay Λc → pK + K − , left) tree diagram, right) the s-wave final state interactions.
The mechanisms of the decay Λc → pπ + π − , left) tree diagram, right) the s-wave final state interactions.
and pπ + π − as background, and do not affect much the invariant mass distributions of the final state K + K − and π + π − [40]. Since the purpose of this work is to study the scalar mesons, dynamically generated from the s-wave meson-meson interactions, we will leave the contributions from the external W diagrams in future studies, when more accurate experimental data are available.
Finally, the amplitudes of the decays Λ c → pK + K − and Λ c → pπ + π − in s-wave can be expressed as, where we include the factor 1/2 in the intermediate loops involving a pair of identical mesons [28]. The scattering matrix t i→j has been calculated within the chiral unitary approach in Refs. [22,29,34,42,43], and we takẽ [42]. G l is the loop function for the two mesons propagator in the lth channel, as given by, where √ s is the invariant mass of the meson-meson pair, and the meson energies ω i = ( q ) 2 + m 2 i (i = 1, 2). The integral on q in Eq. (13) is performed with a cutoff | q max | = 600 MeV, as used in Refs. [29,34,42]. The transition amplitude t ij is obtained by solving the Bethe-Salpeter equation in coupled channels, where five channels π + π − , π 0 π 0 , K + K − , K 0K 0 , and ηη are included. The elements of the diagonal matrix G are given by the loop function of Eq. (13), and V is the matrix of the interaction kernel corresponding to the tree level transition amplitudes obtained from phenomenological Lagrangians [22] and can be expressed as [42], where f is the pion decay constant, f = f π = 93 MeV, and m π , m K , and m η are the averaged masses of the pion, kaon, and η mesons, respectively [41].
With the amplitudes of Eqs. (11) and (12), we can write the differential decay width for the decays Λ c → pK + K − and Λ c → pπ + π − in s-wave, where M inv is the invariant mass of the K + K − or π + π − , p p is the momentum of the proton in the Λ c rest frame, and k is the momentum of the K + (or π + ) in the rest frame of the K + K − (or π + π − ) system, with the Källen function λ(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2yz − 2zx. The masses of the baryons and mesons involved in our calculations are taken from PDG [41].

B. Λc decays via the intermediate vector mesons φ and ρ 0
In this section, we will present the formalism for the decays Λ c → pK + K − and Λ c → pπ + π − via the intermediate mesons φ and ρ 0 . The quark level diagrams for the two-body decays of Λ c into a proton and a vector meson are shown in Fig. 4.
At the quark level, the quark components of the vector mesons are, The amplitudes can be written as, where V ′ p is a normalization factor for the Λ c decay into proton and a vector meson. The factor of 1/ √ 2 in the above amplitudes comes from the quark component of the ρ 0 and ω. With those amplitudes, the decay width for the two-body decay of Λ c into proton and a vector meson in s-wave is, where V stands for the vector mesons ρ 0 , φ, ω, andK * 0 .
The K + K − and π + π − mass distributions respectively for the φ and ρ 0 mesons can be obtain by converting the total rate for vector production into a mass distribution as Refs. [28,44], where we have considered that the K + K − decay accounts for 1/2 of the KK decay width of the φ meson. Since ρ 0 → π + π − and φ → K + K − are in p-wave, we takẽ andΓ Λc→pV = Γ Λc→pV

III. RESULTS AND DISCUSSION
We first extract the factors V p and V ′ p from the branching fractions of the Λ c decays. Our results for the ratios of the branching fractions of the decays Λ c → pK * 0 , Λ c → pω, Λ c → pρ 0 with respect to the decay Λ c → pφ are, where R th 1 and R th 2 are consistent with the experimental results [41], By fitting to the branching fractions of the decays Λ c → pK * 0 , Λ c → pφ, and Λ c → pω, we can obtain the (V ′ P ) 2 /Γ Λc = (4.5 ± 0.4) × 10 3 MeV. With this value, the branching fraction of the decay Λ c → pρ 0 is estimated to be B(Λ c → pρ 0 ) = (6.3 ± 0.6) × 10 −4 .
On the other hand, in order to extract the value of the V p , we calculate the branching fraction for the decay Λ c → pK + K − in s-wave with Eq. (16) , Based on the measured branching fraction of the B(Λ c → pK + K − ) non−φ = (5.3 ± 1.2) × 10 −4 [41], we can obtain Then the branching fraction of the decay Λ c → pπ + π − in s-wave can be given as, With the obtained values of (V p ) 2 /Γ Λc and (V ′ p ) 2 /Γ Λc , we show the K + K − mass distribution for the decay Λ c → pK + K − in Fig. 5, where we can see that the peak of the φ is clear. In addition, there is an enhancement structure close to the K + K − threshold, which is the reflection of the resonance f 0 (980). Although the BESIII Collaboration has reported the K + K − mass distribution, it is difficult to confirm this enhancement structure because of the large uncertainties of the experimental data [18]. It is worth to mention that, in the K + K − mass distribution of the decay χ cJ → ppK + K − measured by the BESIII Collaboration [45], one can find an enhancement structure close to the threshold, which can be associated to the resonance f 0 (980). The similar structure can also be found in the decay D + s → K + K − π + measured by the BABAR Collaboration [46]. The theoretical results for the π + π − invariant mass distributions of the decay Λ c → pπ + π − are shown in Fig. 6, from where one can see a clear peak around 770 MeV, corresponding to the vector meson ρ, and a broad peak around 500 MeV, which can be associated to the scalar meson f 0 (500), dynamically generated from the meson-meson interactions in s-wave. In addition, there is a narrow sharp around 980 MeV, which can be associated to the scalar state f 0 (980). For comparison, the experimental data [18] has been adjusted to the strength of our theoretical calculations at the peak of ρ 0 . We can see that the broad peak for f 0 (500), the peak for ρ 0 , and a narrow sharp for f 0 (980) of our results are compatible with the BESIII measurement. Note that the BESIII data include also the background in the sideband region [18]. The π + π − invariant mass distributions of the Λc → pπ + π − decay compared with the experimental data from Ref. [18].
The green dotted curve stands for the contribution from the meson-meson interaction in s-wave, the blue dashed curve corresponds to the results for the intermediate vector ρ, and the red solid line shows the total contributions.

IV. CONCLUSIONS
In this work, we have studied the decays Λ c → pK + K − and Λ c → pπ + π − , by taking into account contributions of the intermediate vector mesons, and the s-wave meson-meson interactions within the chiral unitary approach, where the f 0 (500) and f 0 (980) resonances are dynamically generated.
The K + K − and π + π − invariant mass distributions for these two decays are calculated. In the K + K − mass distribution, one can find a narrow peak for the φ, and an enhancement structure close to the K + K − threshold, which should be the reflection of the f 0 (980) resonance. Although there is a hint of the enhancement structure in the Belle measurement, the signal of the f 0 (980) is still needed to be confirmed with more accurate measurements in future. For the Λ c → pπ + π − mass distribution, in addition to the broad peak of the ρ 0 , one can find a bump structure around 500 MeV for the f 0 (500), and a narrow sharp around 980 MeV for the f 0 (980), in agreement with the BESIII measurement. We encourage our experimental colleagues to measure these two decays, which can be used to test the molecular nature of the scalar resonances f 0 (500) and f 0 (980).