Study of the nonleptonic decay $\Xi^0_c \to \Lambda^+_c \pi^-$ in the covariant confined quark model

The nonleptonic decay $\Xi^0_c \to \Lambda^+_c \pi^-$ with $\Delta C=0$ is systematically studied in the framework of the covariant confined quark model with accounting for both short and long distance effects. The short distance effects are induced by four topologies of external and internal weak $W^\pm$ exchange, while long distance effects are saturated by an inclusion of the so-called pole diagrams with an intermediate $\frac12^+$ and $\frac12^-$ baryon resonances. The contributions from $\frac12^+$~resonances are calculated straightforwardly by accounting for single charmed $\Sigma^0_c$ and $\Xi^{'\,+}_c$~baryons whereas the contributions from $\frac12^-$~resonances are calculated by using the well-known soft-pion theorem in the current-algebra approach. It allows to express the parity-violating S-wave amplitude in terms of parity-conserving matrix elements. It is found that the contribution of external and internal $W$-exchange diagrams is significantly suppressed by more than one order of magnitude in comparison with data. The pole diagrams play the major role to get consistency with experiment.


I. INTRODUCTION
The study of the heavy-flavor-conserving nonleptonic weak decays of heavy baryons has received a lot of attention due to their observation and measurement of branching fractions by the LHCb and Belle collaborations.The decay Ξ 0 c → Λ + c +π − was first observed at LHCb experiment and the branching fraction was measured to be B = (0.55 ± 0.02 ± 0.18)% [1].classification of the diagrams appearing in these decays and give the analytical expressions for matrix elements.In Sec.IV we present numerical results for the amplitudes and branching fractions.We compare them with those available in the literature.Finally, in Sec.V we make conclusions and summarize the main results obtained in this paper.

II. THE SINGLY CHARMED BARYONS
The masses of singly charmed baryons have been predicted in one gluon exchange model developed in Ref. [33].The comprehensive review on heavy baryons, their spectroscopy, semileptonic and nonleptonic decays may be found in Ref. [34].In Tables I we display the names, quark contents and interpolating currents of the low-lying multiplets of singly charmed baryons with spin 1/2.For singly charmed baryons the flavor decomposition of the diquark, made of (u, d, s)-quarks is 3 ⊗ 3 = 3A + 6 S (A=antisymmetric, S=symmetric).The values of masses with errors are taken from particle data group (PDG) [35].
We are aiming to study the two-body nonleptonic decay Ξ 0 c → Λ + c π − which branching fraction was measured for the first time by LHCb collaboration [1].The effective Hamiltonian relevant for this purpose is written as where Q 1 and Q 2 is the set of flavor-changing effective four-quark operators given by Here ) is the left-handed chiral weak matrix.One has to note that we adopt the numeration of the operators from Ref. [36] where the C 2 Q 2 means the leading order whereas the C 1 Q 1 is for subleading order.The numerical values of the Wilson coefficients C 1 and C 2 from Ref. [36] are being equal to We do not include penguin operators because their Wilson coefficients are small compare with those from current-current operators.
In the standard model (SM) the relation V * cs V cd = −V * us V ud holds to an excellent approximation.For instance, in the Wolfenstein parametrization of the Cabibbo-Kobayashi- ).The global fit in the SM for the Wolfenstein parameter gives λ = 0.22500 ± 0.00067.
In what follows, we introduce the short notations The numerical values of the CKM matrix elements needed in our calculations are taken from PDG [35]: that approximately give V   H eff The pole diagrams which effectively account for the long-distance contributions.

III. MATRIX ELEMENTS AND DECAY WIDTHS
We are going to calculate the matrix elements of nonleptonic decays of Ξ 0 c -baryon in the framework of the CCQM developed in our previous papers.The starting point is the Lagrangian describing couplings of the baryon field with its interpolating quark current.
where the coupling constant g B is determined from the so-called compositeness condition, which was proposed by Salam and Weinberg [37,38] and extensively used in the literature (see, e.g., Refs.[39,40]).
The nonlocal extension of the interpolating currents shown in Table I reads where ) and m i is the mass of the quark at the space-time point x i .The matrices Γ 1 , Γ 2 are the Dirac strings of the initial and final baryon states as specified in Table I.The vertex function Φ B is written as For simplicity and calculational advantages we mostly adopted a Gaussian form for the functions Φ B .Here Λ B is the size parameter for a given baryon.The size parameter phenomenologically describes the distribution of the constituent quarks in the given baryon.
In our approach the matrix elements contributing to the baryon transitions Ξ 0 c → Λ + c π − are represented by a set of the quark diagrams shown in Fig. 3.They describe the so-called short distance contributions.

FIG. 3: Quark diagrams describing the SD-contributions
The diagrams describing the building blocks of the LD-contributions are shown in Fig. 4.

FIG. 4: Feynman diagrams describing the building blocks of the LD-contributions
First, we discuss the matrix elements corresponding to the SD-contributions.One has Here, the factor ξ = 1/N c where N c is the number of colors.This factor is set to zero in the numerical calculations according to the widely accepted phenomenology of the nonleptonic decays.
The contribution from the tree diagram factorizes into two pieces where The expression for Ω 2 is given by Eq. ( 8).Hereafter we adopt the brief notations B 1 for the ingoing baryon with the momentum p 1 , B 2 for the outgoing baryon with the momentum p 2 and M for the outgoing meson with the momentum q.The minus sign in front of f M appears because the momentum q flows in the opposite direction from the decay of M-meson.
The calculation of the three-loop W -exchange diagrams is much more involved because the matrix element does not factorize.One has The calculation of the three-loop integrals proceeds in two steps, first, one has to perform the loop integration by using Fock-Schwinger representation for the quark propagators and Gaussian form for the vertex functions.This allows one to do tensor loop integrals in a very efficient way since one can convert loop momenta into derivatives of the exponent function.
The calculations are done by using a FORM code which works for any numbers of loops and propagators.Second, one has to calculate the obtained integrals numerically over Fock-Schwinger variables by adopting the quark confinement anzatz.The numerical calculations are done by using the FORTRAN codes which include the output from the FORM code written in the format of double precision accuracy.Since the files with the output from FORM contain several thousand lines we are unable to show them in the paper.The details of such calculations may be found in our recent papers [22,24].The calculation is quite time consuming both analytically and numerically.
Finally, the matrix element describing the SD-contributions are written as where Now, we discuss the matrix elements corresponding to the LD contributions.The contribution coming from the pole diagram in Fig. 2 with the Σ 0 c -resonance is written as where S Σ 0 c (p 1 ) = 1/(m Σ 0 c − p 1 ).The explicit form of D-functions are written down as where Here the notations are "res ′′ = Σ 0 c , "out ′′ = Λ + c and M = π − . where )p 1 , r 2 = k 3 + w res 2 p 1 and "in ′′ = Ξ 0 c .By using the mass-shell conditions, one obtains where where where where and "in ′′ = Ξ 0 c .By using the mass-shell conditions, one obtains ).The final expression for the second pole diagram is written as where where g where where where By using the mass-shell conditions, one obtains It appears that the strong transition Ξ 0 c → Ξ + c + π − is identically equal to zero due to the chosen form of the interpolating quark current as shown in Table I: ǫ abc c a (u b Cγ 5 s c ).As a result, this transition is described by the diagram which contains the trace of a string with three quark propagators and three γ 5 matrices that gives zero contribution.Explicitly we have In Ref. [4] it was shown that the vanishing strong couping for Ξ 0 c → Ξ + c π − transition is a consequence of heavy quark and chiral symmetries.Hence it is a model-independent statement.Here, one has to comment that there are two kinds of the interpolating currents for the Λ-type baryons (Λ Q , Ξ Q ) where Q = b, c.They are written as ǫ abc Q a (u b Cγ 5 s c ) (scalar diquark) and ǫ abc γ α Q a (u b Cγ α γ 5 s c ) (vector diquark).For the details, see Refs [41][42][43][44].
Ordinarily, their contributions are calculated by using the well-known soft-pion theorem in the current-algebra approach.It allows one to express the parity-violating S-wave amplitude in terms of parity-conserving matrix elements.In our case, one has The quantities a are defined by Eqs. ( 26)- (28).
Finally, the transition Ξ 0 c → Λ + c +π − amplitude is written in terms of invariant amplitudes as where A and B are given by It is more convenient to use helicity amplitudes H λ 1 λ M instead of invariant ones A and B as described in [51].One has where Finally, the two-body decay width reads where 2 , q 2 )/(2m 1 ).

IV. NUMERICAL RESULTS
Our covariant constituent quark model contains a number of model parameters which have been determined by a global fit to a multitude of decay processes.The values of the constituent quark masses m q are taken from the last fit in [17].In the fit, the infrared cutoff parameter λ of the model has been kept fixed as found in the original paper [14].Table II shows as below: The size parameters of light meson were fixed by fitting the data on the leptonic decay constant.The numerical values of the size parameters and the leptonic decay constants for pion is shown in Table III.Since the experimental data of the single charm baryon decays become to appear recently, we will assume for the time being that the size parameters of all single charm baryons are the same.In Fig. 5 we plot the dependence on this parameter denoted as Λ c of branching fractions Ξ 0 c → Λ + c + π − .One can see that the measured branching fraction can be accommodated in the framework of this work by having Λ c ≈ 0.61 GeV.In addition to the line describing the central value of the experimental data, we also display the strip corresponding to experimental uncertainties.In order to estimate the uncertainty caused by the choice of the size parameter we allow the size parameter to vary from Λ c min = 0.54 to Λ c max = 0.66 GeV that correspond to the intersections of the theoretical curve for branching fraction with the experimantal lower and upper error bars.
For comparison, we plot in Fig. 5 both the separate SD-contributions coming from the diagrams with topologies Ia, IIa, IIb, and III and the LD-contributions coming from the pole diagrams.It is readily seen that the SD-contributions are much smaller than those coming from the pole LD-diagrams.The numerical results for the SD, LD and full amplitudes are shown in Table IV  Also it would be instructive to evaluate the asymmetry parameter defined by where κ = |p 2 |/(E 2 + m 2 ) and E 2 = (m 2 1 + m 2 2 − q 2 )/(2m 1 ).The numerical value of the asymmetry parameter is found to be equal to α = −0.751.
Finally, we compare our results obtained for the branching fraction and the asymmetry parameter with other the data and other approaches in Table V.We have studied two-body nonleptonic ∆C = 0 decay Ξ 0 c → Λ + c + π − in the framework of the covariant confined quark model (CCQM) with account for both short and long distance effects.The short distance effects are induced by four topologies of external and internal weak W -interactions, while long distance effects are saturated by an inclusion of the so-called pole diagrams.Pole diagrams are generated by resonance contributions of the low-lying spin We can get consistency with the experimental data for the value of size parameter being equal to Λ ≈ 0.61 GeV.

1 . 2 .
≈ 0.218 and V (c) CKM ≈ −0.215.The quark diagrams that contribute to the Cabibbo-favored decay are shown in Fig.After hadronizarion, the diagram Ia factorizes out into two parts: the weak transition Ξ 0 c → Λ + c via the W -emission and the matrix element describing the pion leptonic decay.The W -exchange diagrams IIa, IIb and III contribute into both the pure quark diagrams called the short distance (SD) contributions and effectively into the pole diagrams shown in Fig.They describe the so-called long distance (LD) contributions.For instance, the diagrams IIa and III effectively generate the Σ 0 c -resonance diagram, whereas the diagram IIb effectively generates the Ξ + c and Ξ ′ + c -resonance diagrams.

FIG. 5 :
FIG.5: Dependence of the branching fractions on the size parameter.

1 2 + 2 − 2 + and 1 2 −
(Σ 0 c and Ξ ′ + c ) and spin 1 baryons.The last contributions are calculated by using the well-known soft-pion theorem.It is found that the contribution of the SD diagrams is significantly suppressed, by more than one order of magnitude in comparison with data.The most significant contributions are coming from the intermediate 1 resonances.

TABLE II :
Constituent quark masses and infrared cutoff parameter λ.

TABLE III :
Size parameter and leptonic decay constant of pion.
. One can see that |A LD | > |A SD |.

TABLE IV :
SD, LD and full amplitudes in units of GeV 2 .

TABLE V :
Comparison of our findings with other approaches.