An overview of $\Lambda_c$ decays

The decays of the ground-state charmed baryon $\Lambda_c$ are now close to being completely mapped out. In this paper we discuss some remaining open questions, whose answers can help shed light on weak processes contributing to those decays, on calculations of such quantities as transition form factors in lattice QCD, and on missing decay modes such as $\Lambda_c \to \Lambda^* \ell^+ \nu_\ell$, where $\Lambda^*$ is an excited resonance. The discussion is in part a counterpart to a previous analysis of inclusive $D_s$ decays.


I INTRODUCTION
The lowest-lying charmed baryon Λ c was discovered more than 40 years ago [1] , but its decays have not yet been fully mapped out due to the many available modes. Significant progress toward this goal has been made in the past few years, thanks to advances in particle identification, tracking, and collider luminosity. In the present paper we identify some missing modes of interest, the questions associated with them, and ways of filling the gaps in our knowledge. Some modes involving neutrons cannot be identified directly, so one must resort to models such as isospin statistical models [2,3,4]. These techniques also apply to modes with many neutral pions. Even when isospin multiplets have been filled, however, there remains a gap. Some of this gap arises from unreported modes with η or η ′ . In addition, we propose that some of it be filled with semileptonic decays Λ c → Λ * ℓ + ν ℓ , where Λ * is either an excited resonance such as Λ(1405) or Λ(1520) [8] number in parentheses denotes the mass in MeV) or a continuum I = 0 state such as Σπ or NK.
A global analysis of Λ c decays is particulary timely now that Belle [6] and BESIII [7] have significantly improved the accuracy of the branching fraction B(Λ c → pK − π + ), which has been used to normalize other Λ c branching fractions. Their results and the resulting Particle Data group's [5] "fit" value are summarized in Table I. BESIII [7] quotes updated absolute branching fractions for a dozen Λ c modes, incorporated into the latest PDG averages [8]. Also new are a set of updated branching fractions for Λ c → Σππ, including the first observation of the mode Λ c → Σ + π 0 π 0 [9]. The situation has greatly improved in the past six years since a Table I: Values of B(Λ c → pK − π + ).

Source
Ref.
Value (%) Belle [6] 6.84 ± 0.24 +0.21 −0.27 BESIII [7] 5.84 ± 0.27 ± 0.23 PDG "fit" [8] 6.23 ± 0.33 plea was issued for improvement of Λ c absolute branching fractions [10]. The present paper is devoted in part to an update of that analysis. For an early model-dependent discussion of Cabibbo-favored two-body Λ c decays and for a recent study of singly Cabibbo-suppressed decays, quoting other papers using a similar approach, see Refs. [11] and [12], respectively. We review the isospin statistical method in Sec. II, giving examples of its predictions for the NKπ modes in Sec. III, for the Σ2π modes in Sec. IV, for the NK2π modes in Sec. V, and for the Σ3π modes in Sec. VI. Some other modes are treated in Sec. VII. We apply the method to identify missing charge modes in Λ c final states in Sec. VIII. This approach mirrors one applied to D s decays [4]. Section IX is devoted to suggestions for placing these estimates on a firmer footing, such as identifying missing neutrons and taking account of decays involving η and η ′ . Section X is devoted to systematic errors associated with possible deviations from the statistical isospin model, such as the dominance of resonant substructure. Part of the remaining shortfall is proposed in Sec. XI to be filled by Λ c semileptonic decays to excited states. Section XII concludes. An Appendix discusses details of obtaining branching fractions not quoted by Ref. [5].

II STATISTICAL ISOSPIN MODEL
A multiparticle amplitude may be decomposed into a series of invariant isospin amplitudes depending on particle momenta, as we shall show by examples in the next three sections. The statistical isospin model [2,3] parametrizes one's ignorance of underlying dynamics by assuming that each invariant amplitude contributes equally and incoherently to each decay mode, with relative branching fractions determined only by Clebsch-Gordan coefficients. The squares of each invariant amplitude's coefficients then sum to 1, and for each mode the branching fraction is the sum of squares of each contributing amplitude, divided by the number of invariant amplitudes. The answer does not depend on how the isospin decomposition is performed.
We illustrate this process for a three-body final state ABC produced in a state of definite isospin I and third component I 3 . We may first decompose the BC system into isospin amplitudes I BC with I B − I C ≤ I BC ≤ I B + I C . We then combine the amplitudes I BC with I A in such a way that the final isospin is the desired value I, while I BC 3 + I A 3 = I 3 . This process then reduces to manipulation of Clebsch-Gordan coefficients.

III DECAYS Λ c → N Kπ
The normalizing branching fraction for many Λ c decays is B(Λ c → pK − π + ). The isospinpartner modes are nK 0 π + and pK 0 π 0 . The initial Λ c has isospin zero, while the ∆S = 1 satisfying the sum rule as noted by BESIII, the observers of the nK 0 π + mode [18]. The statistical isospin model postulates equality and incoherence of A 0 and A 1 , so that the branching fractions of Λ c to the above NKπ modes are in the ratio 3/8:3/8:1/4. This prediction is compared with data [5] in Table II. The ratios of branching fractions for the two modes with a proton are underestimated by 2.4σ and 1.5σ with respect to measurements, while the branching fraction for the nK 0 π + mode is slightly overestimated by 2.9σ. This gives an idea of the degree to which we can trust the statistical model. Deviations from its predictions will be discussed in Sec. X. One could, if desired, decompose the final states into ones labeled by the isospin of the Kπ system, with invariant amplitudes A 1/2 and A 3/2 . Assuming these two amplitudes are equal in magnitude and incoherent, one arrives at the same result.

IV DECAYS Λ c → Σππ
The final state in Λ c → Σππ decays must have I = I 3 = 1, as noted in the previous subsection. One way to count invariant amplitudes is to designate them by the isospin of the two-pion system, I ππ = 0, 1, 2. For each such isospin there is a unique coupling with the Σ (whose isospin is 1) to the I = 1 final state. Thus there are three invariant amplitudes A 0 , A 1 , A 2 . The Λ c decay amplitudes are expressed in terms of them as: A(Σ 0 π + π 0 ) = − 1 2 A(Σ 0 π 0 π + ) = − 1 2 Here we quote amplitudes for both orders of differing pion charges, needed for the sum of squares of coefficients of each isospin amplitude to add up to 1. The statistical-model predictions are obtained by assuming that invariant amplitudes are equal in magnitude and incoherent. They are compared with experiment [5,9] in Table III. The parentheses around pion pairs denote the sum of both orders. The agreement with the statistical isospin model is quite good. In Ref. [3] bounds were placed on Λ c decays to Σππ final states consisting of all charged particles, with the result The quotient in Eq. (9) has the value 3/5 in the statistical isospin model.

V DECAYS Λ c → N Kππ
One convenient way to define invariant amplitudes for the NKππ final state is to couple the NK pair to isospin I N K = 0, 1 and the pion pair to I ππ = 0, 1, 2. When I N K = 0, only I ππ = 1 can lead to a final state with I = 1; we call the corresponding reduced amplitude A 1a . When I N K = 1, the I = 1 final state receives contributions from I ππ = 0, 1, 2; we call the corresponding reduced amplitudes A 0 , A 1b , A 2 . The decomposition of Λ c decay amplitudes in terms of these reduced amplitudes is A(pK − π 0 π + ) = − 1 2 (a) Error to be multiplied by a scale factor of 5.67 in the final total.
A(nK 0 π 0 π + ) = − 1 2 The implications of this decomposition for the statistical isospin model are summarized in Table IV. Only two modes are measured, and they imply quite different values of the total B(Λ c → NKππ). We shall bear this uncertainty in mind when evaluating the accuracy of our predictions. Using the average value of the total branching fraction, we predict the branching fractions for as yet unseen decay modes in brackets in the table.
The average of implied total branching fractions involves two very disparate values, so we apply a scale factor of 5.67 to the error, giving 3.92 to be quoted in the summary table. Possible deviations from the statistical isospin model will be noted in Sec. X.

VI DECAYS Λ c → Σ3π
The number of invariant amplitudes may be counted by noting the number of 3π amplitudes with each isospin and then coupling them up with the I = 1 Σ to a final state with I = 1. The multiplicities of three-pion amplitudes are 1 for I = 0, 3 for I = 1, 2 for I = 2, and 1 for I = 3. Each of these except the I = 3 amplitude can couple up with the Σ to form final isospin 1. Thus there are a total of six reduced amplitudes.
Error to be multiplied by a scale factor of 2.0 in the final total.
The statistical model's predictions of relative branching fractions for Cabibbo-favored decays have been given in Ref. [3]. For Σ3π final states of Λ c we show the results in Table  V. In averaging the two values leading to different implied total fractions we multiply the uncertainty of 1.2% by a scale factor of 2 to give a final uncertainty of 2.4%. Deviations from the statistical isospin model will be discussed in Sec. X. Bounds on Λ c decays to Σ3π final states with three charged particles [3] are The quotient in Eq. (18) has the value 4/5 in the statistical isospin model.

VII SOME OTHER CABIBBO-FAVORED MODES
We use a standard format, extracting estimates of branching fraction to the sum of all charge states for a given mode and averaging where there is more than one measured charge state. The statistical model fractions are taken from Table 4 of Ref. [3].

A N K3π
Only one mode (K − p2π + π − ) is used in estimating the total, as the branching fraction for K − pπ + 2π 0 is suspiciously large in comparison with the all-charged-particle mode. It bears watching, however. (See Table VI.)

B Λ3π
So far only the mode with no neutral pions has been detected. Ref. [3] obtains the bounds where the value of the quotient in the statistical isospin model is 3/5 (see Table VII).
The PDG value of 1.0 ± 0.5 is ignored but bears watching.

C Λ4π
The mode with a single neutral pion is the only one detected. With three neutral pions the missing mode is unlikely to be confirmed soon. Ref. [3] finds the bounds with the statistical isospin model giving 4/5 for the quotient (see Table VIII).

VIII IDENTIFYING MISSING MODES
In the previous sections we have used the isospin statistical model to estimate missing charge states for Λ c decay modes due to the Cabibbo-favored process c → sud, populating final states with strangeness S = −1 and isospin I = I 3 = 1. The results are shown in Table IX. Also shown are much rougher estimates of branching fractions to S = 0 final states, populated by the singly-Cabibbo-suppressed transitions c → dud and c → sus.
The sum of the two sets of branching fractions is (89.6 ± 5.0)%. Thus there is a hint, though not statistically compelling at present, that about 10% of Λ c decays remain to be accounted for. We shall suggest that this could be due to semileptonic Λ c decays to excited states such as Λ(1405), Λ(1520), or continuum Σπ and/or NK states. The estimates for the ∆S = 0 transitions are very rough, as many of them rely on the assumption that each charge mode is equally populated. What one sees in the statistical model, instead, is that the modes with the most neutral pions tend to be populated the least. Thus the total branching fraction for ∆S = 0 decays may in fact be an upper bound.

IX SUGGESTIONS FOR IMPROVEMENT A Modes needing further attention
The uncertainty on the ∆S = −1 transitions is dominated by the disagreement with the isospin statistical model in the NKππ modes. The ratio B(Λ c → pK − π + π 0 )/B(Λ c → pK 0 π + π − ) is measured to be 1.39 ± 0.11, whereas in the isospin statistical model it is predicted to be (9/40)/(3/10) = 3/4. The corresponding total NK2π branching ratio is very different depending on which mode one uses to estimate it. Measurement of further NK2π modes might help to resolve the ambiguity. The singly-Cabibbo-suppressed (∆S = 0) modes are not readily amenable to a statistical treatment, as the final states are a mixture of I = 1/2 and I = 3/2. Thus measuring their branching fractions in the widest possible cases is called for instead.

C Inclusive η, η ′ branching fractions
Although some portion of decay modes involving η or η ′ appears in multi-pion final states, the inclusive η and η ′ branching fractions have not been reported. It would be very helpful to have them, in the same manner that inclusive measurements were very helpful in sorting out D s decays [4].

X DEVIATIONS FROM STATISTICAL MODEL
In all decays involving three or more final-state particles, pairwise associations in resonant substructures can lead to deviations from the statistical isospin model. However, in their high-statistics studies of Λ c decays, neither BESIII [7] nor Belle [9] show Dalitz plots or onedimensional plots of pairwise effective masses. Consequently, we have to anticipate possible deviations from the statistical isospin model without the help of experiment. We hope this situation will change in the near future.
In Figure 1 we give three examples of processes contributing to Λ c decays. These have the potential of populating final states in a manner differing from the statistical isospin model, giving rise to characteristic resonant substructures. We shall estimate the corresponding uncertainties for a series of final states. In cases where all charge states are allowed, we will comment on how well the statistical isospin model is obeyed, but will not assign any uncertainty to the branching fractions.

A N Kπ
All branching fractions have been observed. Defining R 1 = B(pK − π + ), R 2 = B(nK 0 π + ), R 3 = B(pK 0 π 0 ), amplitudes in Eq. (1) (up to a common factor) are related by The two amplitudes A 0 and A 1 are unequal in magnitude and have some degree of coherence, in contrast to the statistical isospin model which would have them equal in magnitude and out of phase with one another. One can qualitatively anticipate the violation of the statistical isospin model by reference to the three diagrams of Fig. 1. The relatively short lifetime of the Λ c , about 0.2 ps [5], can be ascribed in large part to the contribution of Fig. 1(a), which leads to a final state suu resembling a J = 1/2 excited Σ * + . This can hadronize by production of either a uū or dd pair. In the former case one produces a configuration (sū)(uuu) which can materialize as K − ∆ ++ = K − pπ + . The latter case leads to a configuration (sd)(duu) which can materialize to K 0 nπ + or K 0 pπ 0 . In this example, K − π + accounts for 50% of Λ c → NKπ decays, exhibiting the likely direction of deviation from the statistical isospin model. What if the spectator diagram (b) were dominant? The spectator ud pair would remain in an isospin-zero final state, implying equal branching fractions for nK 0 π + and pK − π + , and no contribution to pK 0 π 0 , far from the observed situation.

B Σππ
Here the statistical isospin model works surprisingly well. Both the internal conversion and spectator processes can contribute. They both can excite resonances such as Λ(1405)(J P = 1/2 − ) and Λ(1520)(J P = 3/2 − ), which in turn can couple to all charge states of Σπ and NK. (The contribution to the NK amplitude of the Λ(1405) would involve an intermediate off-shell state.) Either hyperon resonance decays to an I = 0 final state, so Λ(1405) → (π + Σ − ), (π 0 Σ 0 ), (π − Σ + ) in equal proportions. The spectator process does not contribute to Λ c → Σ + π 0 π 0 , which is the smallest Σππ branching fraction, both predicted by the isospin statistical model and observed.

XI SEMIILEPTONIC Λ c DECAYS
The only semileptonic Λ c decays that have been reported are those to a Λℓ + ν ℓ final state. A hint that there may be other semileptonic final states is provided by a calculation assuming pointlike Λ c and Λ, whose rate is predicted to be where f (x) ≡ 1 − 8x + 8x 3 − x 4 + 12x 2 ln(1/x) = 0.1763 for M(Λ) = 1115.68 GeV and M(Λ c ) = 2286.46 GeV. The total decay rate of Λ c for a lifetime of 200 fs is 3.29×10 −12 GeV, so the pointlike prediction corresponds to a branching fraction of 7.6%, about twice the observed rate. This suggests that a form factor is present, which can be interpreted as indicating the presence of excited final hadronic states. A similar conclusion can be drawn from a quark model cartoon of charm semileptonic decay [19]. With (m c , m s , m u,d ) = (1710, 536, 364) MeV [20], a slightly higher branching fraction is obtained (see Appendix), again indicating that semileptonic Λ c decays are not saturated by the Λℓ + ν ℓ final state. A calculation in lattice QCD [21] finds B(Λ c → Λe + ν e ) = (3.80 ± 0.19 ± 0.11)% and B(Λ c → Λµ + ν µ ) = (3.69 ± 0.19 ± 0.11)%, where the first error comes from lattice QCD and the second from the uncertainty in the Λ c lifetime. The agreement with experiment further confirms the need for a form factor and, indirectly, hints at a role for excited final states in Λ c semileptonic decays.
The detection of Λ(1405) or Λ(1520) in the final state of Λ c semileptonic decays may not be straightforward. The former decays only to Σπ, while the latter decays both to Σπ and to NK. Thus branching fractions are spread over many final states. There is also a measurement [22] B(Λ c → e + + anything) = (4.5 ± 1.7)% from 1982 which needs to be re-examined if our proposal is to account for a significant portion of the missing Λ c decays.

XII CONCLUSIONS
We have investigated Λ c decays from a global standpoint, finding impressive progress in mapping out branching fractions. Nevertheless, there remains the possibility of a shortfall of about 10%. We have suggested that this could be filled by semileptonic decays to excited final states, not just the Λ. To reduce the uncertainty in the total observed branching fraction, we urge more studies of modes contining neutrons, greater investigation of the singly-Cabibbosuppressed modes, and inclusive studies of η and η ′ . Determination of resonant substructure is a crucial ingredient in filling gaps only partially addressed by an imperfect isospin statistical model. The fact that such progress has already been made for charmed meson decays [5] should serve as an encouragement for similar advances in our understanding of charmed baryon decays.

ACKNOWLEDGMENTS
We thank Roy Briere, Alexander Gilman, Hai-Bo Li, Xiao-Ryu Lyu, Stefan Meinel, Hajime Muramatsu, Ruth Van de Water, and Charles Wohl for useful communications. J.L.R. wishes to thank the Technion for generous hospitality during the inception of this work.

APPENDIX
We give some details of how branching fractions are quoted in Table IX if they are not taken directly from Ref. [5].
Modes with K 0 are inferred from those quoted in [5] for K 0 S by multiplying by 2. We renormalize B(Λ c → Σ + π 0 π 0 ), first measured by Belle [9] using a PDG (2016) value B(Λ c → pK − π + ) = 6.35 ± 0.33%, by the slightly smaller value of this normalizing branching ratio given in Table I. Using the same normalization for decays to Σ + π + π − and Σ 0 π + π 0 we also include early measurements of these branching ratios.
For the NKπ and Σππ modes, all charge states have been measured, so the experimental totals in Tables II and III are transcribed in Table IX. For the NK2π and Σ3π modes, not all charge states are measured, so totals implied by the statistical model are averaged and quoted (with a scale factor for the NK2π average) in Table IX. Finally, the modes described in Sec. VII have only one charge state, which is used to estimate the missing modes, with the inferred total quoted in Table IX.
An alternative way of estimating the total contribution of singly-Cabibbo-suppressed decays to Λ c branching fractions is to use free-quark estimates. For a crude calculation we may consider only the subprocess c → dud, neglecting contributions from c → sus by virtue of phase space suppression. We take effective quark masses from Ref. [20]: m c = 1710 Mev, m s = 536 MeV, m u,d = 364 MeV, implying a phase space enhancement of 1.46 for c → dud relative to c → sud. The corredponding ratio of squared CKM matrix elements is |V cd /V cs | 2 = (0.2265/0.974) 2 = 0.0541, implying a total branching fraction for subprocesses dominated by c → dud of 88.6% × 1.46 × 0.0541 = 7.0%. Thus we could be missing a few ∆S = 0 modes, not to mention those governed by c → sus.
In parallel with the lattice QCD calculations mentioned earlier for Λ c → Λℓ + ν ℓ [21], there appeared recently one for the Cabibbo-suppressed ∆S = 0 processes Λ c → nℓ + ν ℓ [13]. The corresponding branching fractions are displayed in the right-hand column of Table IX.