Mesonic Decay of Charm Hypernuclei $\Lambda^+_c$

$\Lambda^+_c$ hypernuclei are expected to have binding energies and other properties similar to those of strange hypernuclei in view of the similarity between the quark structures of the strange and charmed hyperons, namely $\Lambda(uds)$ and $\Lambda^+_c (udc)$. One striking difference however occurs in their mesonic decays, as there is almost no Pauli blocking in the nucleonic decay of a charm hypernucleus because the final-state nucleons leave the nucleus at high energies. The nuclear medium nevertheless affects the mesonic decays of charm hypernucleus because the nuclear mean fields modify the masses of the charm hyperon. In the present communication we present results of a first investigation of the effects of finite baryon density on different weak mesonic decay channels of the $\Lambda^+_c$ baryon. We found a non-negligible reduction of the decay widths as compared to their vacuum values.


Introduction
The possible existence of charm hypernuclei has been suggested [1] right after the discovery of the charm quark and a first explicit calculation [2] of their binding energies was performed using a meson exchange model with coupling constants determined by SU(4) flavor symmetry. More recently, calculations were performed [3] using the quark meson coupling model at the Hartree level -the predicted binding energies per baryon in Λ + c , Σ c , Ξ c hypernuclei ranging from O to Pb are quite substantial, on average of the order of 7.5 MeV. Such binding energies are of similar magnitude to those of J/Ψ binding in nuclear matter [4] and finite nuclei [5]. Vacuum effects [6] and Fock terms [7], although crucial for capturing pionic effects in matter [8], are not expected to have a great impact on the binding energies. Up to now there is no experimental evidence of the existence of a charm hypernucleus, apart from an emulsion experiment at Fermilab with 250 GeV protons that indicated that Λ + c hypernuclei have been produced in the collisions [9]. New experimental possibilities were explored [10,11] at the beginnings of the 1990's, but concrete hopes for the production of such hypernuclei comes with the completion of the FAIR facility (where Λ ± c can be produced inpp annihilation processes [12]) and the installation of a 50 GeV high-intensity proton beam at the J-PARC complex.
The study of charm hypernuclei is interesting for several reasons; among the most interesting in our view are: (1) they bring the opportunity to learn about the poorly known interactions of charmed baryons with nucleons [13,14]; (2) they offer also the opportunity to learn about medium effects on a e-mail: sabyaphy@ift.unesp.br b e-mail: eduardo@ift.unesp.br c e-mail: gkrein@ift.unesp.br diquark correlations, as the masses of the light u and d quarks are affected by the medium while the mass of the heavy c quark is not affected; and (3) few-nucleon (typically one or two) charmed states have the potential of having a rich spectroscopy similar to the X,Y,Z mesons.
One particular observable that can be affected by the medium is the weak-decay lifetime of the charmed baryon because of the change in its mass. There are no Pauli blocking effects on the decaying baryon -for the case of Λ + c , for instance, the decays Λπ + and Σ + π 0 are not blocked, and in the decay pK 0 the proton leaves the nucleus with a large momentum. In the present communication we present the results of a first investigation [15] of the effects of finite baryon density on different weak mesonic decay channels of the Λ + c baryon.

In-medium mesonic decays of Λ + c
We evaluate the decay rate in infinite nuclear matter via the imaginary part of the self-energy diagram depicted in Fig. 1. To evaluate the diagram, we employ the effective weak-decay Lagrangian density where For the density dependence of Λ + c we use the results from the QMC model [3]. Finite nuclei can be treated in the local density approximation. We start examining the role of the Pauli principle in the Λ + c → pK 0 decay, neglecting all other possible medium effects. The in-medium decay width Γ T (q) for a generic baryon-meson (BM) decay Λ + c → BM, where q = |q| with q the momentum of the Λ + c , can be written as the difference Γ T (q) = Γ V (q) − Γ D (q), where Γ V (q) is vacuum decay width and Γ D (q) is a density dependent width due to Pauli-blocking: where m Λ c , m B and m M are the masses of Λ + c , baryon B and meson M, respectively, and θ is the step function with Λ c /R 4 ) 1/2 , and k F is the nuclear matter Fermi momentum given in terms of the baryon density ρ as k F = (3π 2 /2 ρ) 1/3 .
There is no Pauli blocking in the Λ + c → Λπ + and Λ + c → Σ + π 0 decays and Γ D (q) is nonzero only in the nucleonic decay Λ + c → pK 0 . Notice that Γ D (q) vanishes for ρ B = 0 as ω th (q) > ω + (q) in this case and the step function in Eq. (2) gives zero for the integral. The dot-dashed line in Fig. 2 presents the momentum dependence of the ratio Γ T /Γ V = 1 − Γ D /Γ V for the Λ + c → pK 0 decay channel in nuclear matter (ρ = ρ 0 = 0.16 fm −3 ). Clearly, the effect of the Pauli principle is negligible. Physically, this is due to the fact that the momentum of the outgoing proton is of the order of, or larger than the Fermi momentum -this is transparent in the plot of ω th (q) that shows that it barely takes values within the region delimited by the curves of ω + (q) and ω − (q) as demanded by the theta function in Eq. (2) for a Γ D (q) 0. Next, we consider the effect of the in-medium mass shift of Λ + c on the decay widths of the different channels -as the proton leaves the nucleus with high momentum, its mass shift is negligible. The density dependence on the mass is taken from the QMC model [3], which can be parametrized as with a 0 = 0.121, a 1 = 0.565 and a 2 = 0.878. The in-medium decay width is denoted by Γ ρ and is given by the same expression as Γ V in Eq. (2) but with m Λ c replaced by m * Λ c . In Fig. 3 we present the ratio Γ ρ /Γ V as function of the baryon density, where Γ ρ is evaluated at q = m * Λ c and Γ V is evaluated at q = m Λ c . At the nuclear matter saturation density, m * Λ c = 2.146 GeV. The in-medium reduction of the decay widths is not small. Specifically, at the normal nuclear matter density, the reductions are 14.5%, 12% and 6.5% for Λ c → Λπ + , Σ + π 0 , and pK 0 decay channels, respectively. A local density approximation leads to similar reductions for medium to heavy nuclei [15].
Several issues need careful examination before definite conclusions can be drawn. Specifically, SU(4) flavor symmetry breaking in meson-baryon couplings have been shown to be significant [16] and its impact on the results needs to be assessed -the subject is also of crucial importance for the DN interaction [14] in connection to D-mesic nuclei. The consequences of medium modifications on the mass of Λ + c on non-mesonic weak decays of charmed hypernuclei is under investigation [17].