Predictions for nonleptonic Lambda_b and Theta_b decays

We study nonleptonic Lambda_b ->Lambda_c pi, Sigma_c pi and Sigma_c^* pi decays in the limit m_b, m_c, E_pi>>Lambda_{QCD} using the soft-collinear effective theory. Here Sigma_c = Sigma_c(2455) and Sigma_c^* = Sigma_c(2520). At leading order the Lambda_b ->Sigma_c^{(*)} pi rates vanish, while the Lambda_b ->Lambda_c pi rate is related to Lambda_b ->Lambda_c\ell\bar\nu, and is expected to be larger than Gamma(B ->D^{(*)} pi). The dominant contributions to the Lambda_b ->Sigma_c^{(*)} pi rates are suppressed by Lambda_{QCD}^2/E_pi^2. We predict Gamma(Lambda_b ->Sigma_c^* pi) / Gamma(Lambda_b ->Sigma_c pi) = 2 + O[Lambda_{QCD}/m_Q, alpha_s(m_Q)], and the same ratio for Lambda_b ->Sigma_c^{(*)} rho and for Lambda_b ->Xi_c^{(',*)}K. ``Bow tie'' diagrams are shown to be suppressed. We comment on possible discovery channels for weakly decaying pentaquarks, Theta_{b,c} and their nearby heavy quark spin symmetry partners, Theta_{b,c}^*.

Heavy baryon decays are interesting for many reasons. Heavy quark symmetry [1] is more predictive in semileptonic Λ b → Λ c ℓν decay than in B → D ( * ) ℓν, and may eventually give a determination of |V cb | competitive with meson decays [2]. In this paper we concentrate on the more complicated case of nonleptonic b → cūd baryon transitions, as shown in Table I. These channels provide a testing ground for our understanding of QCD in nonleptonic decays. Our analysis is based on heavy quark symmetry and the soft-collinear effective theory (SCET) [4].
There is considerable experimental interest in these decays. Recently the CDF Collaboration measured [5] f where f Λ b and f d are the fragmentation fractions of b quarks to Λ b and B 0 , respectively.
The part of the weak Hamiltonian relevant for this paper is Notation s l I(J P ) Mass (MeV) Decays considered where both the Wilson coefficients, C i , and the four-quark operators depend on the renormalization scale which we take to be m b , and P L = (1 − γ 5 )/2. The Weak nonleptonic decays are sometimes characterized by diagrams corresponding to different Wick contractions. As shown in Fig. 1, there are more possibilities in baryon than in meson decays. In particular, a "Bow tie" contraction is unique to baryons. The color structure for baryons also differs from mesons: we find that the C diagram is of the same order in the large N c limit as the T diagram. 1 Nonleptonic meson decay amplitudes are sometimes estimated using naive factorization, which would set Λ c π|O 1 |Λ b = Λ c |cγ µ P L b|Λ b π|dγ µ P L u|0 . In baryon decays the extra light quark implies that this procedure is ill-defined for all but the tree diagram. In naive factorization the Λ b → Σ ( * ) c π decays are very suppressed, since the T contribution vanishes separately in the isospin and heavy quark limits [7] (just like the semileptonic Λ b → Σ ( * ) c ℓν decays), the C contribution vanishes after doing a Fiertz transformation on the four-quark operator, and the E and B amplitudes are identically zero since the u and b fields are in different quark bilinears.
In this letter we show that more rigorous techniques can still be applied to make reasonable predictions for all these decays. By expanding in m b , m c , E π ≫ Λ QCD we show that for Λ b → Λ + c π − the amplitudes corresponding to the diagrams in Fig. 1 satisfy T ≫ C ∼ E ≫ B, and we find that the experimental result in Eq. (1) is consistent with theoretical expectations. Next we consider Λ b → Σ ( * ) c π decays, and show the leading contributions to these nonleptonic rates are suppressed by Λ 2 QCD /E 2 π , much like in B 0 → D 0 π 0 . Using heavy quark symmetry we derive a relation between the decay rate to Σ c and Σ * c and comment on decays to Ξ c . Finally we consider the detection of possible weakly decaying heavy pentaquarks, Θ b and Θ c , with nonleptonic decays.
The proof of factorization at leading order for Λ b → Λ c π decay follows closely that for , so we do not review it here. In this case the nonperturbative expansion parameter for SCET is λ = Λ QCD /E π [9]. Since E π is set by the bottom and charm quark masses, we take this to be of the same order as the expansion parameter for the heavy quark effective theory (HQET), i.e., λ ∼ Λ QCD /m Q (Q = c, b). Working at leading order in λ and α s (m b ) and neglecting the pion mass, the Λ b → Λ c π matrix element factorizes in the standard way, where f π = 131 MeV is the pion decay constant, n is a light-like four-vector along the direction of the pion's four-momentum, p µ π = E π n µ , and the four-velocities of the Λ b and Λ c are v and v ′ , respectively. Perturbative corrections induce a multiplicative factor in Eq. (4), is computable and φ π is the nonperturbative lightcone pion distribution function [10,11], and a term proportional to the matrix element of c n /P R b. At leading order in α s (m Q ), we can set T (x) π = 1 and the term involvingc n /P R b to 0. This implies that the nonleptonic rate is related to the semileptonic differential decay rate at maximal recoil, where where the f i and g i are functions of w, and the relevant currents are V ν =cγ ν b and A ν = cγ ν γ 5 b. The spinors are normalized toū(p, s)γ µ u(p, s) = 2p µ . In the heavy quark limit, where ζ(w) is the Isgur-Wise function for ground state baryons. The differential decay rate is given by where in the m Q ≫ Λ QCD limit F Λ (w) is equal to the Isgur-Wise function, ζ(w), and in particular F Λ (1) = 1. In terms of the original form factors Combining the above results for Λ b → Λ + c π − decay with the analogous ones for B 0 → D ( * )+ π − we find that where ξ is the Isgur-Wise function for B → D ( * ) semileptonic decay, and r D ( * ) = m D ( * ) /m B .
When the Λ b → Λ + c ℓ −ν rate is measured, one can directly test factorization using Eq. (5) or Eq. (10). In the absence of this data, we have to resort to using model predictions for the baryon Isgur-Wise function. If the ratio of Isgur-Wise functions in Eq. (10) is unity then the prefactor in Eq. (10) implies that Γ(Λ b → Λ c π − )/Γ(B 0 → D ( * )+ π − ) = 1.6(1.8). This enhancement is in rough agreement with the data in Eq. (1). A similar result also follows from the small velocity limit (m Q ≫ m b − m c ≫ Λ QCD ), in which the nonleptonic rates The large N c limit provides some support for the ratio of baryon to meson Isgur-Wise functions being close to unity at maximal recoil. In the large N c limit the heavy baryons can be treated as bound states of chiral solitons and mesons containing a heavy quark. In this picture, the baryon Isgur-Wise function, ζ(w), is predicted to be ζ(w) = 0.99 e −1.3(w−1) [12]. This gives ζ(w Λ max = 1.4) = 0.57, which is indeed close to ξ(w D * max = 1.5) ≃ 0.55 [13]. Using this model for ζ(w), |V cb | = 0.04, τ Λ b = 1.23 ps, and Eqs. (5) and (8) yield the prediction that As expected, this is larger than B(B 0 → D ( * )+ π − ) ≃ 2.7×10 −3 . However, the uncertainty in this prediction is quite large, particularly given that large N c strictly only applies for w near 1. The same large N c inputs predict B(Λ b → Λ + c ℓ −ν ) ≈ 6%, i.e., this channel is expected to make up a large part of the inclusive Λ b → X c ℓ −ν rate, with the s P l = 1 − excited Λ c states making up a significant fraction of the remainder [14]. Order Λ QCD /m Q corrections to these predictions may be significant. The Λ b → Λ + c π − amplitude receives contributions from the T , C, E, and B classes of diagrams in Fig. 1. In SCET, |E/T | and |C/T | are of order Λ QCD /m Q [15], and we will show later that |B/T | is fur- that Λ QCD /m Q corrections affect the amplitudes at the 15 − 30% level. In particu- GeV. The ratio of these amplitudes can be reproduced by a power correction involving a hadronic parameter |s eff | ≃ 430 MeV, which is of natural size [15]. Since B s → D − s π + only has a T contribution, accurate measurement of this rate will improve our understanding of the size of E and C. CDF recently measured Now we turn to Λ b → Σ c π decays. As shown in Table I, there are two Σ c states with different spin which we refer to as Σ c and Σ * c . They form a heavy quark spin symmetry doublet with the spin and parity of the light degrees of freedom, s π l l = 1 + . Under isospin, the Λ b is I = 0, the Σ ( * ) c is I = 1, and the Hamiltonian is I = 1, so the Σ ( * ) c π final state must be I = 1 (it can not be I = 0 or 2). Therefore the rates to the two different charge channels are equal, Based on B decay data and the SCET power counting, we expect Γ(Λ b → Σ ( * ) c π) to be up to about an order of magnitude smaller than Γ(Λ b → Λ c π), since the leading contributions to Λ b → Σ ( * ) c π are power suppressed. Again, we use SCET to expand in Λ QCD /m Q , Λ QCD /E π , and α s (m Q ), keeping only the leading terms that cause the Λ b → Σ ( * ) c π transitions. These come from the C and E diagrams in Fig. 1 and their contributions can be studied following the analysis ofB 0 → D ( * )0 π 0 in Ref. [15]. The leading diagrams in SCET I that determine the matching onto power suppressed operators are shown in Fig. 2. To match the C and E diagrams, two insertions of the mixed usoft-collinear Lagrangian, L (1) ξq [17], is required, each yielding a suppression of Λ QCD /E π . This yields the power counting that |C/T | and |E/T | are O(Λ QCD /E π ). In contrast, matching the B diagram in Fig. 1 requires four insertions of L (1) ξq (or other higher dimensional terms in the Lagrangian), and B is therefore power suppressed compared to C and E by at least an additional Λ QCD /E π .
In addition there is a further matching onto SCET II . The resulting matrix element involves soft and collinear operators which factor [15]. 4 The matrix element of the weak , s) , can be written (neglecting α s (m Q ) corrections) as a convolution integral of a jet function, J(x, k + 1 , k + 2 ), with the matrix element involv-ing the collinear fields, π|O c (x)|0 which gives φ π (x), and that involving the soft fields, s) . In what follows we only need the form of the soft operator [15] where h (Q) v is an HQET heavy quark field, S is a soft Wilson line, and the subscripts denote the momentum carried by the fields. For our purposes the most important aspect of the analysis is that O s only involves the the combinationh Thus, by heavy quark symmetry where v and v ′ are the four-velocities of the Λ b and Σ ( * ) c respectively. The spinor field normalizations areū(v, s) u(v, s) = 1 for the Λ b and Σ c , andū α (v, s) u α (v, s) = −1 for the Σ * c . X µ is the most general vector compatible with the symmetries of QCD, Note that in Eq. (12) the part of O s involving the light quark fields is parity violating, so we need not worry about the fact that Λ b → Σ c is an "unnatural" transition. Using Σc v ′ + E π n to eliminate the term proportional to v µ in Eq. (14), it is easy to see that any term in X µ proportional to v ′ µ does not contribute, so only n µ remains. Hence the ratio of the rates for Λ b → Σ c π and Λ b → Σ * c π are determined model independently at leading order in Λ QCD /m Q and α s (m Q ), similar to the B 0 → D ( * )0 π 0 case. We find To evaluate the square of the matrix element in Eq. (13), we used the spin sums from Ref. [14] for the various spin Σ ( * ) c states. The explicit calculation shows that the rate to Σ * c with |s ′ | = 3/2 vanishes, as required by angular momentum conservation.
A practical complication in testing this prediction is that the Σ ( * ) c states decay to Λ c π, and so both decay channels Λ b → Σ ( * )0 c π 0 → Λ c π − π 0 and Λ b → Σ ( * )+ c π − → Λ c π 0 π − contain a π 0 that makes the reconstruction hard at hadron colliders. This can be circumvented by studying Λ b → Σ ( * )0 c ρ 0 decays. In this case the final states are Λ c π − π + π − . Decays to a vector meson are potentially more complicated due to the fact that "long-distance" contributions can induce transverse polarizations at the same order in Λ QCD /E π . However, at leading order in α s (m Q ) these long-distance contributions vanish for the ρ 0 final state [15] and we It is also worth noting that where in contrast to Eqs. (15) and (16)  The decays Λ b → Ξ c K decays are also Cabibbo-allowed. (These decays involve "ss popping" so only Ξ 0 c K 0 is allowed, not Ξ + c K − ). They are similar to Λ b → Σ ( * ) c π in the sense that the leading contribution in the heavy quark limit vanishes. As shown in Table I there are three Ξ c "ground states", Ξ c , Ξ ′ c , and Ξ * c . The Ξ c and Ξ ′ c can mix, but the former is expected to be mostly the state that transforms as 3 under flavor SU(3), while the latter is mostly a 6. The Ξ * c also transforms as a 6, and forms a heavy quark spin symmetry doublet with the Ξ ′ c . Thus, a relation similar to Eq. (15) also holds in this case, i.e., . This prediction may be hard to test since Ξ ′ c decays to Ξ c γ. One can also consider Cabibbo-suppressed Λ b decays, e.g., Λ b → Ξ c π, and the weak decays of other baryons containing a heavy bottom quark.
Perhaps the most exciting possibility is the existence of heavy baryonic pentaquark states.
Recently several experiments claimed to observe a baryon Θ + (1540) with the quantum numbers of K + n. A possible explanation is to consider the Θ + as a bound state of two spinzero ud diquarks in a P-wave with ans antiquark [19]. If diquarks play an important role in these exotic states then the analogous heavy flavor states, Θ c =c [ud] 2 and Θ b =b [ud] 2 , may be below threshold for strong decays by ∆E ≃ −100 MeV and ∆E ≃ −160 MeV respectively [19]. 5 Since the spin of the light degrees of freedom is s l = 1, we expect from heavy quark symmetry that Θ Q come with a doublet partner of similar mass, Θ * Q , as shown in Table I also be stable with respect to the strong interactions and decay to Θ Q γ. Since the splitting for Θ ( * ) c is larger, it is possible that the Θ * c is just above the strong decay threshold, making the spectroscopy even more interesting (like in D * decays).
It may be possible to discover the Θ b,c via the decay chains that are Cabibbo-allowed and lead to all charged final states. The most interesting aspect of the Θ + b → Θ 0 c decay is that in the diquark picture the correlation is maintained, as shown in Fig. 3, and so no additional suppression factor is expected. In weak Θ b decays to ordinary baryons this would not be the case. While we do not know the Θ Q production rates, we can estimate the branching ratios in Eq. (18). The lifetime of a weakly decaying Θ b,c is expected to be comparable with other weakly decaying hadrons that contain a charm or a bottom quark. The Θ + b → Θ 0 c π + amplitude factorizes, and is related to Θ + b → Θ 0 c ℓν via a formula identical to Eq. (5). For the nonleptonic rate we obtain in the heavy quark limit, where r Θ = m Θc /m Θ b , and η 1,2 are the two Isgur-Wise functions that parameterize the weak Θ b → Θ ( * ) c matrix elements where η 1 (1) = 1. In particular Θ c (v ′ , s ′ )|cΓb |Θ b (v, s) = 1 3 g αβ η 1 (w) − v α v ′β η 2 (w) ū(v ′ , s ′ ) γ 5 (γ α + v ′ α )Γ(γ β + v β )γ 5 u(v, s) .
Thus, B(Θ + b → Θ 0 c π + ) is expected to be similar to B(Λ b → Λ c π). If the Θ Q states exist then an analysis of the Λ QCD /m Q corrections would be warranted, as the mass of the light degrees of freedom is sizable. We expect B(Θ 0 c → Θ + π − ) to be at the few percent level, while the other branching ratios in Eq. (18) may be of order unity.
In summary, we studied nonleptonic Λ b decays to Λ c π, Σ c π and Σ * c π. Eqs. (10), (15), (16), and (19) are our main results. In the m Q ≫ Λ QCD limit the Λ b → Λ c π rate is related to Λ b → Λ c ℓν, and we found that Γ(Λ b → Λ c π) is expected to be larger than Γ(B → D ( * ) π), as observed by CDF. At leading order in Λ QCD /m Q the Λ b → Σ ( * ) c π rates vanish, but an analysis of the leading contributions suppressed by Λ QCD /m Q was still possible. We predict . We also discussed properties of pentaquarks with ab orc, including a possible discovery channel if they decay weakly.