From $\Xi_b \to \Lambda_b \pi$ to $\Xi_c \to \Lambda_c \pi$

Using a successful framework for describing S-wave hadronic decays of light hyperons induced by a subprocess $s \to u (\bar u d)$, we presented recently a model-independent calculation of the amplitude and branching ratio for $\Xi^-_b \to \Lambda_b \pi^-$ in agreement with a LHCb measurement. The same quark process contributes to $\Xi^0_c \to \Lambda_c \pi^-$, while a second term from the subprocess $cs \to cd$ has been related by Voloshin to differences among total decay rates of charmed baryons. We calculate this term and find it to have a magnitude approximately equal to the $s \to u (\bar u d)$ term. We argue for a negligible relative phase between these two contributions, potentially due to final state interactions. However, we do not know whether they interfere destructively or constructively. For constructive interference one predicts ${\cal B}(\Xi_c^0 \to \Lambda_c \pi^-) = (1.94 \pm 0.70)\times 10^{-3}$ and ${\cal B}(\Xi_c^+ \to \Lambda_c \pi^0) = (3.86 \pm 1.35)\times 10^{-3}$. For destructive interference, the respective branching fractions are expected to be less than about $10^{-4}$ and $2 \times 10^{-4}$.


I INTRODUCTION
Most decays of charmed and beauty baryons observed up to now occur by c and b quark decays. In strange heavy flavor baryons an s quark may decay instead via the heavy flavor conserving subprocess s → u(ūd) or su → ud, with the c or b quark acting as a spectator. In strange charmed baryons an additional Cabibbo-suppressed subprocess cs → cd can contribute. Early investigations of heavy flavor conserving two body hadronic decays of charmed and beauty baryons involving a low energy pion have been performed in Ref. [1][2][3][4][5][6]. In these studies a soft pion limit, partial conservation of the axial-vector current (PCAC) and current algebra have implied expressions for decay amplitudes in terms of matrix elements of four-fermion operators between initial and heavy baryon states. These matrix elements are difficult to estimate and depend strongly on models for heavy baryon wave functions.
Recently we proposed a model-independent approach for studying the decay Ξ − b → Λ b π − [7] which had just been observed by the LHCb collaboration at CERN [8]. In the heavy b quark limit this decay by s → u(ūd) proceeds purely via an S-wave. Assuming that properties of the light diquark in Ξ − b are not greatly affected by the heavy nature of the spectator b quark, the decay amplitude for Ξ − b → Λ b π − may be related to amplitudes for S-wave nonleptonic decays of Λ, Σ, and Ξ which have been measured with high precision [9]. We calculated a branching fraction for Ξ − b → Λ b π − consistent with the range allowed in the LHCb analysis. Our purpose now is to extend this calculation to charmed baryon decays Ξ 0 c → Λ c π − and Ξ + c → Λ c π 0 . Sec. II summarizes the result of Ref. [7] for the amplitude of Ξ − b → Λ b π − , in which the underlying quark transition is s → u(ūd). This result is then applied to a contribution of the same quark subprocess to Ξ 0 c → Λ c π − . A second term in this amplitude due to the subprocess cs → cd is studied in Sec. III. The total amplitude and the branching ratios for Ξ 0 c → Λ c π − and Ξ + c → Λ c π 0 are calculated in Sec. IV while Section V concludes.
We will use notations which are common for describing hadronic hyperon decays [9]. The effective Lagrangian for B 1 → B 2 π given by involves two dimensionless parameters A and B describing S-wave and P-wave amplitudes, respectively. Here G F = 1.16638 × 10 −5 GeV −2 is the Fermi decay constant. The partial width is where q is the magnitude of the final three-momentum of either particle in the B 1 rest frame. Consider first Ξ − b → Λ b π − studied in Ref. [7]. In the heavy b quark limit the light quarks s and d in Ξ − b = bsd are in an S-wave state anstisymmetric in flavor with total spin S = 0. The light quarks u and d in the Λ b = bud are also in an S-wave state with I = S = 0. In the decay Ξ − b → Λ b π − which proceeds via s → u(ūd) the b quark acts as a spectator. The transition among light quarks is thus one with J P = 0 + → 0 + π, and hence is purely a parity-violating S wave. Thus it may be related to parity-violating S-wave amplitudes in nonleptonic decays of the hyperons Λ, Σ, and Ξ.
S-wave hadronic decays of hyperons, B 1 → B 2 π, where the baryons B 1 and B 2 belong to the lowest SU(3) octet baryons, have been known for fifty years to be described well by using PCAC and current algebra and assuming octet dominance [10,11]. An equivalent and somewhat more compact parametrization of these amplitudes based on duality has been suggested a few years later [12]. All hyperon S-wave amplitudes may be expressed in terms of an overall normalization parameter x 0 and a parameter F describing the ratio of antisymmetric and symmetric three-octet coupling. (In the soft pion limit the commutator of the axial charge with the weak Hamiltonian represents a third octet in addition to the two baryons.) Thus one finds [7,12] while amplitudes involving a neutral pion are related to these amplitudes by isospin. Using best fit values F = 1.652, x 0 = 0.861, one finds good agreement between predicted and measured amplitudes as shown in Table I (see [7]). The relative signs of S-wave amplitudes are convention-dependent and differ from those in Ref. [9]. An overall sign change is also permitted, associated with two possible signs of x 0 . In the decay Ξ − b → Λ b π − , which also proceeds by s → u(ūd), the light diquarks sd and ud in the initial and final baryons form each a spinless antisymmetric 3 * of flavor SU(3). The weak transition occurs between this pair of diquarks while the b quark acts as a spectator. Neglecting the effect of the heavy b quark on relevant properties of the light diquarks, this amplitude is expected to be equal to an amplitude for a transition between light hyperons, Λ → Λ(ūu), in which the diquarks in initial and final hyperons are also in an antisymmetric 3 * while the s quark acts as a spectator. Thus one finds [7] A Using the best fit values of x 0 and F one obtains A(Ξ − b → π − Λ b ) = ±1.796. One may improve this calculation somewhat by including SU(3) breaking. We note that the measured S-wave amplitudes for Λ → pπ − and Σ − → nπ − alone determine a slightly different value for x 0 , x 0 = 0.835 having practically no effect on F . The relation and experimental values of the amplitudes on the right-hand side imply In the three amplitudes occurring in (5) an s quark occurs in the decaying baryons taking part in the transition but not as a spectator. This leads to a common redefinition of x 0 which now includes SU(3) breaking. While the value (6) includes this effect of SU(3) breaking we have attributed to it an uncertainty of 15% caused by assuming octet dominance and by neglecting the effect of the heavy b quark on properties of the light diquarks. The considerations and calculation leading to (6) apply also to the contribution of the transition s → u(ūd) to the S-wave amplitude for Ξ 0 c → Λ c π − . Here one replaces a spectator b quark in Ξ − b and Λ b by a c quark in Ξ 0 c = csd and Λ c = cud, assuming that the c quark mass is much heavier than the light u, d and s quarks. In this approximation we have The S-wave amplitude for Ξ 0 c → Λ c π − obtains a second contribution from an "annihilation" subprocess cs → cd involving an interaction between the c and s quarks in the Ξ 0 c . We will now present in some detail a method proposed by Voloshin [3,13,14] for calculating this amplitude in the heavy c-quark limit in terms of differences among measured total widths of charmed baryons.
The effective weak Hamiltonian responsible for this Cabibbo-suppressed strangenesschanging transition is given by In the following we will use values C + = 0.80 and C − = 1.55 for Wilson coefficients calculated in a leading-log approximation at a scale µ = m c = 1.4 GeV corresponding to α s (m c )/α(m W ) = 2.5. Applying a soft pion limit and using PCAC, the amplitude due to cs → cd is given in our normalization (1) [which is related to that of Ref. [3] by a factor ξ/(G F m 2 π )] by Here f π = 0.130 GeV, ξ ≡ 2m Ξ 0 c / (m Ξ 0 c + m Λc ) 2 − m 2 π − = 1.04 [15]. In the above one defines two matrix element x and y (of dimension GeV 3 ) in which the contribution of the axial-current vanishes for a heavy c quark, where i, k are color indices. Using flavor SU(3) one may write these two terms as differences of diagonal matrix elements of four fermion operators, O ψ−φ ≡ ψ|O|ψ − φ|O|φ , for charmed baryon states belonging to V-spin and U-spin doublets: Within a heavy quark expansion the quantities x and y can be used to describe differences of inclusive decay rates among the above three charmed baryons. Adding contributions of hadronic and semileptonic Cabibbo-favored and singly Cabibbo-suppressed decays one finds in the flavor SU(3) limit [3,13,14]: Substituting the above values of C + , C − and cos θ C = 0.97424, sin θ C = 0.2253 [15] one has Eliminating x and y in these equations Eq. (9) now implies Using the measured charmed baryon lifetimes [15] τ (Ξ 0 c ) = 0.112 +0.013 −0.010 ps , τ (Ξ + c ) = 0.442 ± 0.026 ps , τ (Λ c ) = 0.200 ± 0.006 ps , we calculate The first (symmetrized) error corresponds to errors in lifetime measurements, while the second one is associated with uncertainties due to SU(3) breaking and due to a finite cquark mass. We checked that replacing the Wilson coefficients C ± by values calculated beyond the leading-log approximation, C + = 0.80, C − = 1.63 [16], has a negligible effect on the central value.