A global analysis of charmless two body hadronic decays for anti-triplet charmed baryons

Recently Belle collaboration reported new measurements for the branching fractions with the first observing two processes of $\mathcal{B}(\Xi_c^0\to\Lambda K^0_S)$, $\mathcal{B}(\Xi_c^0\to\Sigma^0 K^0_S)$ and updating data for $\mathcal{B}(\Xi_c^0\to\Sigma^+ K^-)$. Combined with other known data on charmless two body decays of anti-triplet charmed baryons, a lot of information can be derived with the assistance of $SU(3)$ flavour symmetry. Using $SU(3)$ relations between different decay modes, we can give some predictions based on the new measurements which can be tested with the high luminosity experiments in the future. More interestingly, we find that a global fit is now possible with the addition of new Belle data. In general, there are 18 complex $SU(3)$ invariant amplitudes. We find that a scenario of all amplitudes being real can fit the data well with a $\chi^2/d.o.f$ only $0.773$. This indicates that neglecting the phases of the amplitudes is a reasonable assumption. When more data become available, one may be able to get more information for phases in the amplitudes. We give several comments on the feature of global fit regarding the branching fractions, relations between different decays, and decays involving $K^0$ and $\bar K^0$. Many of the unknown branching fractions and polarization asymmetry parameters of anti-triplet charmed baryon for charmless two body decays are predicted to be accessible by experiments at Belle, Belle~II, BES-III, and LHCb. The validity of $SU(3)$ for charmless two body hadronic decays can be more accurately tested.

(1) SU (3) symmetry predicts relations between different decays and it can be used to test the model.Existing data show that the predicted relation Γ(Λ + c → Σ 0 π + ) = Γ(Λ + c → Σ + π 0 ) holds well, which gives some confidence for the model.It is tempting to make a more elaborated analysis using new data in combination with existing data to test other possible relations predicted by SU (3).It sparked our interest in systematic SU (3) analysis of charmless two body hadronic decays of anti-triplet charmed baryons.In this work, we carry out such an analysis.We find that SU (3) flavor symmetry can describe data well, unlike the case in semileptonic anti-triplet charmed baryon decays [37].Tests of SU (3) symmetry not only using certain relations but in wider scope also become possible.In general there are 18 complexes SU (3) invariant amplitudes for charmless two body hadronic decays of anti-triplet charmed baryons.We find the subset of all amplitudes is real can fit the data well with a χ 2 /d.o.fonly 0.773.This indicates that neglecting the phases of the amplitudes is a reasonable assumption.When more data become available, one may be able to get more information for phases in the amplitudes.We await more data to come to do such an analysis.We give several comments on the features of global fit regarding the branching fractions, relations between different decays and decays involving K 0 and K0 .Many of the unknown branching fractions and asymmetry parameters are predicted to be accessible by experiments at Belle, Belle II, BESIII, and LHCb.The validity of SU (3) for charmless two body hadronic decays can be more accurately tested.
The paper is organized as follows.In Sec.II, we give the theoretical framework of SU (3) symmetry analysis using the irreducible representation amplitude (IRA) method.In Sec.III, we discuss possible relations among different decay modes by SU (3) symmetry to see how predictions for other decays can be obtained.In Sec.IV, we use experimental data to carry out a global fit to obtain SU (3) irreducible amplitudes and to predict some unknown branching fractions for charmless two body hadronic decays of anti-triplet charmed baryons.In Sec.V, the K 0 S − K 0 L mixing effects on some of the decay modes are discussed.A brief conclusion will be given in the last section.

II. SU (3) SYMMETRY DECAY AMPLITUDES
The decays we study are one of the charmed baryons in anti-triplet T c 3 to a baryon in the ground octet T 8 plus a meson in the nonet pesudoscalar P .The component states in representations are written as Here the SU (3) flavour anti-triplet charmed baryons can be also represented as ).The η s and η q are the mixture of meson singlet η 1 and octet η 8 components: Moreover, the physical η and η ′ states are mixtures of η q = 1 √ 2 (uū + d d) and η s = ss given by with φ = (39.3± 1.0) • [38].
For the decay processes we are discussing, the effective Hamiltonian in the standard model (SM) can be divided into three groups: Cabibbo-allowed, Cabibbo-suppressed, and doubly Cabibbo-suppressed, with The tree operators transform under the SU (3) flavour symmetry as: 3 The Cabibbo-suppressed c → uss( dd) transitions are proportional to V * cs V us (−V * cd V ud ) ≈ sin θ = 0.2265 ± 0.00048 [38], one has (H6) 31  3 = −(H6) 13  3 = (H6) The doubly Cabibbo-suppressed c → dsu transitions are proportional to In the above, we have used the Hamiltonians for Cabibbo allowed and doubly suppressed decays without the H 3 representation.We also used the relation of us , which leads to cancellation H 3 contributions of the two Cabibbo suppressed terms to high accuracy.At the loop level, a small H 3 effect will be generated, but can also be neglected [5].Using the anti-symmetric tensor ǫ kmi to contract (H6) km j , we can define a new form for the H6 representation, The amplitudes of processes T c 3 → T 8 P in irreducible representation amplitude (IRA) can be written as By expanding out the above expression, we obtain SU (3) amplitudes for the charmless two body hadronic decay processes of T c 3 → T 8 P in Table I and Table II.
decays.This way of treating relations predicted will also ease the potential problem from the asymmetry parameters α for different decays in Eq.( 12) which will be discussed in the next section.In exact SU (3) symmetry, a similar analysis would show that the asymmetry parameters α for decay channels in Eq.( 12) are all equal.We find that the values for α are also very sensitive to the actual hadron masses of the decays.We, therefore, think that using actual masses for each process would give a more realistic prediction which partially takes into some obvious SU (3) breaking effects for particle masses into account.In our later analysis, we will follow the above procedure.There are several relations involving K 0 or K0 in the final states.These data need to be careful interpreted.For example, the measured branching fraction (1.59 ± 0.08)% for B(Λ + c → pK 0 S ) is for the final state K 0 S .But it is related to K 0 and K0 by where small CP violating effect has been neglected.Therefore one also predict related decay invovling K 0 L in the final state.The following SU (3) relations can be derived The above would imply the relations for K 0 L and K 0 S in the final states, It can be seen that the amplitude in the leading order of sin 2 θ.In this order, we obtain B(Λ + c → pK 0 L ) = (1.59 ± 0.08)% .But the convergence of this expansion will be violated if ) is in order of O(sin 4 θ).Therefore these interesting relations can give precise predictions only if at least one of the channels B(Ξ

More measurements of this channel will provide further information about SU (3) relations.
There are also several relations, whose branching fractions have not been measured on both sides of the relation.We hope some of these branching fractions can be measured to predict other branching fractions to test the model.Note that for the processes involving final state K 0 and K0 , we can not give any predictions until both final states involving K 0 S and K 0 L are measured in the experiment.We eagerly hope more experimental data will be available in the future to allow us to give more meaningful comments.

IV. DETERMINATION OF SU (3) AMPLITUDES BY A GLOBAL DATA FITTING
In this section, we discuss a global analysis of charmless two body hadronic decays of anti-triplet charmed baryons based on SU (3) symmetry.To carry out such an analysis we should have enough data points to have some handle on the SU (3) irreducible amplitudes.From Eq.( 11), we see that there are 9 such amplitudes.In fact, each of these amplitudes should be considered to have both the combination of scalar and pseudoscalar form factors. Generically, we can express them as according to the (V − A) current nature of the effective Hamiltonian due to W exchange.In the above equations, each of these amplitudes has scalar and pseudoscalar form factors corresponding to the parity violating and conserving amplitude respectively.Therefore there are total 18 amplitudes.Including the two newly measured branching fractions by Belle, we see that there are 16 branching fractions in Table III that have been measured.It is not possible to make a meaningful global data fitting if we only consider the 16 branching fractions.In refs.[6][7][8] under the approximation neglecting a part of amplitudes due to H 15 or omitting some amplitudes, global fitting was carried out.However, several predicted branching fractions do not agree with data well as can be seen in Table III.For example, the branching fraction of process Ξ 0 c → Σ + K − is measured as 0.221 ± 0.068%, which has more than 5σ deviation in previous works [6,7] and 4σ deviation in [8].Therefore a global fit keeping all amplitudes is necessary for a more realistic study.To this end, we notice that polarization parameter α for 5 of the decays have been measured.They provide additional constraints.If these amplitudes are real, the number of parameters of 18 is less than the total data points of 21 making a global fit possible.Therefore, we take this as a scenario to do our analysis and a more general analysis will be available with more and more experimental data is coming in the future.Whether this scenario is a good one to work with will be judged by the fitting results.We find the subset of all amplitudes is real can fit the data well with a χ 2 /d.o.fonly 0.773.This gives us confidence in the results obtained.This will be our working assumption in our following analysis.
For the process of anti-triplet charmed baryon decays B c → B n M , the polarization angular distribution of decay width is easily written as where ωi and pBn are the unit vector of initial state spin and final state momentum, respectively.Depending on the specific processes, the F and G linear functions of f i and g i are the scalar and peseudoscalar form factors, respectively.The parameter α [40] is given by With data from measurements of α one can constraint separately the G and F form factors.The parameter κ writing in terms of masses is given by Here M Bc , M Bn , and M M are the masses of anti-triplet charmed baryons, octet baryons, and mesons, respectively.This introduces additional sensitivity to relation predictions for each decay process mentioned before.Before carrying out our global fitting, we point out a special feature for the SU (3) amplitudes that the SU (3) amplitudes a 6 , a 15 only contribute to the processes with final state η and η ′ and the corresponding relations and data shown in Table II.The a 6 and a 15 actually have four form factors f a 6 , g a 6 , f a 15 and g a 15 .Experiments only Λ + c → pη, Λ + c → Σ + η and Λ + c → Σ + η ′ three decay modes have been measured.This shows that completely determining the SU (3) decay amplitudes is not possible.But we note that the three measured processes involve only a 6 − a 15 , therefore we can do a meaningful global fit to include them to predict branching fractions not yet measured whose decay amplitudes only depend on a 6 − a 15 .For this purpose we define which corresponds to the new amplitude Since the three experimental data which involve η and η ′ only rely on the form factors f a and g a , therefore processes depend on f a′ and g a′ cannot be determined from our fit which are indicated by "−" in Table II Using known experimental data on branching fractions and polarization asymmetry parameters α in Table III, we have performed our global fitting.We use the Nonlinear Least Squares Fitting (lsqfit) package in reference [39] for our analysis.With its definition of statistical data in the package, we get the χ 2 /d.o.f = 0.744 results, which indicates that the fit is a reasonable one.We conclude that SU (3) symmetry gives a reasonable description of charmless two body hadronic decays of anti-triplet charmed baryons.The predictions are shown in Tables I and II can be used to further test SU (3) symmetry for charmless two body hadronic decays of anti-triplet charmed baryons.
As mentioned earlier that the amplitude a 6 +a 15 corresponding to the form factors f a′ and g a′ cannot be determined in our fit without new inputs of experimental data involve a 6 + a 15 .Therefore new experimental data on this type of charmless two body anti-triplet charmed baryons having η and η ′ in the final states is crucial for a complete test of SU (3) symmetry.We hope in the near future some of these decays can be measured experimentally.
In this section, we study asymmetry between a B C → B n K 0 S and B C → B n K 0 L decay.In this type of decays, due to relations in Eq.( 12), for some of the decays there are interference between the Cabibbo-favored and doubly Cabibbo-suppressed amplitudes the asymmetry defined below does not vanish [8] After using relations in Eq.( 12), the asymmetry can be further written as the following Here H CA and H DC are respectively the Cabibbo allowed and doubly Cabibbo suppressed SU (3) amplitude for the processes of final states K 0 S,L mixed by K 0 and K0 , which are shown in the Table I.Using Table IV, we predict the asymmetries for several processes which are shown in the last column in Table V.The K 0 S − K 0 L asymmetries could be used to search for the doubly Cabibbo suppressed charmless two body hadronic decays of anti-triplet charmed baryon decays.

VI. CONCLUSIONS
In this work, we have carried out an analysis for charmless two body decays of anti-triplet charmed baryons based on SU (3) flavour symmetry.Combined with other known data on such decays and those newly measured ones from Belle collaboration, a lot of information can be obtained, in particular making a meaningful global analysis possible.
SU (3) symmetry predicts some relations among different processes.One needs to be careful in using these relations because the predictions are sensitive to particle masses in individual decays.Using physical masses for relevant particles we find that these relations are in good with our global analysis.For the scenario that all amplitudes are real, our global fit gives a reasonable with χ 2 /d.o.f = 0.744.This indicates that data fit SU (3) predictions with real amplitudes well.We await more data to come to do a more general analysis.We made predictions for 29 processes for their branching fractions and polarization parameters without η or η ′ decays which provide further tests for the SU (3) model.For processes involve η and η ′ in the final states, only the class of decay with amplitudes proportional to a 6 −a 15 have 3 decay branching fractions for data available for fitting.Even with this limitation, their polarization parameters have been predicted.We also predicted 5 decay processes with η or η ′ in the final states and their associated polarization parameters to test the model.
There is another class of decays proportional to a 6 + a 15 has no data available to constraint the decays.Thus no predictions can be made.Measurements of some of thesw types of decays are crucial to test SU (3) symmetry.We urge our experimental colleagues to measure some of this class of decays.We gave several comments on the feature of global fit regarding the branching fractions, relations between different decays involving 0 and K0 .These can also be tested with future data.We eagerly waiting for data from future experimental data from Belle, Belle II, BESIII, and LHCb to test the validity of SU (3) for charmless two body hadronic decays to better precision.

TABLE I :
SU(3)amplitudes and predicted branching fractions (the third column) and polarization parameters (the fourth column) of anti-triplet charmed baryons decays into an octet baryon and an octet meson.

TABLE III :
Experimental data on branching fractions and polarization parameters (the second column), previous theoretical predictions (the third column) and our fitting results (the fourth column).