Nonleptonic two-body weak decays of charmed baryons

We analyze the two-body nonleptonic weak decays of charmed baryons, employing the pole approximation in tandem with the $SU(3)_F$ symmetry. We are able to make novel predictions for decay channels of $\Omega_c^0 \to {\bf B}_n P$ and ${\bf B}_{cc}\to {\bf B}_c^{A,S} P$ based on the experimental data of ${\bf B}_c^A \to {\bf B}_n P$. Here, ${\bf B}_n$, ${\bf B}_{c}^A$, ${\bf B}_c^S$ and ${\bf B}_{cc}$ are the low-lying octet, antitriplet charmed, sextet charmed and doubly charmed baryons, respectively, and $P$ is the pseudoscalar meson. Our findings reveal that the fitted effective Wilson coefficient ${\cal C}_+=0.469$ is notably smaller than the naive expectation, and the low-lying pole approximation fails to account for ${\cal B}(\Lambda_c^+ \to n \pi^+ , \Xi^0 K^+)$, despite consistencies with the soft-meson limit. We further recommend the decay channel $\Xi_{cc}^+ \to \Xi_c^0 \pi^+ \to \Xi^- \pi^+\pi^+\pi^+\pi^-$ for exploring evidence of $\Xi_{cc}^+$, estimating the branching fraction at $(1.1\pm 0.6)\times 10^{-3}$.

reliable method derived from first principles to address the W -exchange diagrams in exclusive decays, leading to the need for several approximations [24][25][26][27][28][29][30][31][32][33][34][35][36][37].One less model-dependent approach is to perform a global fit using the SU(3) flavor (SU(3) F ) symmetry, which has become popular .Nevertheless, even in the simplest case of B A c → B n P , where B n and P represent the octet baryon and pseudoscalar meson respectively, this method requires dozen one-time parameters.While the results of the global fit often align with the experimental data used for fitting, the predictive accuracy is disputable.The predicted branching fractions significantly diverge across various theoretical studies relying on the SU(3) F symmetry, illustrating that the free parameters are not tightly constrained by the existing experimental data.
In an effort to reduce the number of free parameters in the SU(3) F global fit, Geng, Tsai, and the author of this work considered the pole approximation in 2019 [46,47].This approach, grounded in the Körner-Pati-Woo (KPW) theorem [59], enables the exclusion of six parameters from O qq ′ + .Here O qq ′ + is the four-quark operator in the effective Hamiltonian [60] H ef f = q,q ′ =d,s with where G F is the Fermi constant and V qq ′ is the Cabibbo-Kobayashi-Maskawa matrix element.After considering the factorizable contributions of O qq ′ + , the smallness of B(Λ + c → pπ 0 ) is explained [46].More importantly, Ref. [47] predicted that which were not measured at that time.In particular, Eq. ( 3) is an critical prediction stemming from the KPW theorem and the modest ratio in Eq. ( 4) is quite surprising as both of them are Cabibbo favored (CF).These theoretical benchmarks have since been found consistent with recent experimental results [7,11].real.The decay width Γ and up-down asymmetry α are calculated by where M i,f and M P are the masses of B i,f and P , respectively and p f and E f are the magnitudes of the 3-momentum and energy of B f at the rest frame of B i .
To relate the decays with the SU(3) F symmetry, one has to write down the hadron representations in the SU(3) F group.We start with the low-lying pseudoscalar mesons.
The responsible SU(3) F tensor is given by which is related to the flavor part of wave functions according to Here, the superscript and subscript of P i j describe the quark and antiquark flavors with i, j ∈ {1, 2, 3} and (q 1 , q 2 , q 3 ) = (u, d, s).
We exclusively consider the SU(4) F 20 multiplets, where the low-lying 1 where (1,2) interchange the first and second elements and (2, 3) the second and third.
For instance, we have e 23 |q a q b q c = |q a q b q c + |q b q a q c − |q a q c q b − |q b q c q a .(10) It is clear that after operating e 23 , states are antisymmetric in regard to the second and third quarks.The idempotent in Eq. ( 9) generates a subspace in the sense that e 23 e A = e 23 e S = 0 , where e S(A) are the totally (anti)symmetric idempotent, given by We stress that throughout this work the SU(4) F representations are merely bookkeeping tools to unify the expressions and we do not take advantage of the SU(4) F symmetry.
If a light quark (u, d, s) pair is in antisymmetric, we utilize that the totally antisymmetric tensor ǫ ijk is invariant under the SU(3) F transformation to simplify the indices, i.e. two antisymmetric quarks transform as an antiquark.As a result, the light quarks of B A c are presented by one lower index as Eq. ( 12) can be translated back to a tensor with three quarks by with q 4 = c.Here, Eq. ( 13) is derived by where we have used Λ + c as an instance.We start with |cud − |cdu to make sure its isospin vanishes.One arrives at |Σ + c if |cud + |cdu is used instead.
On the other hand, the other low-lying baryons with spin-parity where B S c and B cc are the singly charmed sextet and doubly charmed baryons, respectively.Similarly, they are translated to tensors with three quark indices by which would lead us to the convention in Ref.
[3] up to some unphysical overall phase factors.In the quark model, the spin-flavor wave functions are obtained by with The effective Hamiltonian can be written in a compact way of where the nonzero elements are The factors of 1/c − and 1/2c + are included to match the convention.Comparing to Eq. ( 1), it is clear that H(6) and H(15) take account O − and O + in the effective Hamiltonian.
By far we have only considered the quark flavors and here is an appropriate place to further consider their colors also.With the Fierz transformation, it is straightforward to show that the color structure of q and u in O qq ′ + is symmetric, and the same also applies to c and q ′ .Recall that baryons are antisymmetric in color, we arrive at where the initial and final states are an arbitrary baryon and three quark state, respectively.The same also applies to B f |O qq ′ + |q a q b q c = 0 with B f the final state baryon.
In the decays B i → B f P , the nonfactorizable contributions can be approximated by the pole diagrams shown in FIG. 2, where the symbol × marks the insertion of the effective Hamiltonian.This approximation results in the well-known KPW theorem, which states that O + contributes solely to the factorizable amplitudes.Notably, Eq. ( 21) is scale-independent, as O ± do not undergo mixing in the renormalization group evolution [60].While a hard gluon exchange could challenge the KPW theorem, any breaking effect is likely below 10%.For a deeper dive into this topic, readers can consult Ref. [61].There, the small branching fraction of B(B 0 → pp) is attributed to a violation against the KPW theorem1 .Since this deviation is even less significant than that of the SU(3) F breaking, we uphold the KPW theorem in this study.
To identify the factorizable contributions of O + , we observe the direct product of H(15) ij k and (P † ) l m has the representation of Hermitian conjugate is taken in P as it appears in the final states.The factorizable condition demands that the quark lines of P originate from O + exclusively.In other word, all the indices of (P † ) l m shall contract to the ones of H(15 where δ is the Kronecker delta, F i := H(15) ij k (P † ) k j and the other linear combinations do not contribute to B i → B f P .It shows that only the 3 representation in Eq. (22) contributes, reducing numbers of free parameters.
By identifying the factorizable contribution, we reduce the number of free parameters from 14 to 8 for B A c → B n P and arrive at [47] where T ij ≡ (B A c ) k ǫ kij and a 1,2,3,6 are free parameters in general.We note that we do not consider η ′ as its mass differs largely from the other pseudoscalar mesons.On the other hand, the B S c and B cc decays are parameterized by for B S c → B n P , for B cc → B A c P , and for B cc → B S c P .The P -wave amplitudes share the same flavor structures with the S-wave ones and are obtained by Please note that the same symbols are used to denote the parameters in Eqs. ( 24), ( 25), (26), and (27).Although the symbols are the same, it is important to recognize that they do not represent identical values in each equation.We have used the same symbols in these different contexts due to a limitation in the available symbols.The above parameterizaions with (24), (25), (26), and ( 27) would be referred to as the general pole (GP) scenario.
One of the shortcoming of the GP scenario is that there are too many parameters.
As there are few available input for B S c and B cc decays, the GP scenario does not have concrete predictions except for several direct relations.To overcome this problem, we assume that the intermediate baryons B I depicted in FIG. 2 are dominated by the low-lying ones, which would be referred to as the low-lying pole (LP) scenario.It allows us to infer the baryon matrix elements exhibited in B S c → B n P and B cc → B A,S c P from B A c → B n P .To this end, the next section is devoted to calculating the factorizable contributions, and in the section following the next one, we relate the fourquark operator matrix element in the decays of B cc and B A c for evaluating the pole diagrams.

III. FACTORIZATION CONTRIBUTIONS
The amplitude is decomposed into the factorizable and nonfactorizable parts as followed by A = A fac + A pole and B = B fac + B pole .The factorizable amplitude reads Expressing the baryon matrix element with the SU(3) F symmetry, we find where C +,0 are the effective Wilson coefficients with the subscript denoting the charge of P , f P is the meson decay constant, F V and G V are the leading vector and axialvector form factors, respectively, H is obtained by substituting C +,0 for c 1,2 in Eq. ( 19), and B and B † are the tensors of B i and B f , respectively, given in Eqs. ( 13) and ( 16).
From Eq. ( 17), we have and arrive at At the limit of the SU(3) F symmetry, the form factors of c → s and c → u/d would be numerically the same.Here we see that they deviate roughly 15%, which is a common size of the SU(3) F breaking.
The form factors of B cc → B A,S c from LQCD are not available yet.Nonetheless, we utilize the approximation that the form factors are independent of the spectator quark flavors, which allows us to infer them from Λ + c → Λ/n.We match the form factors of representing the 4-velocity of B i(f ) .By using the form factors provided in Refs.[62,63], we arrive at for B cc → B A,S c .The main difference between Eqs. ( 33) and ( 34) arises from the ω dependencies of the form factors. Specifically, the values of (ω − 1) for the transitions Ξ ++ cc → Ξ + c π + and Λ + c → Λπ + are 0.074 and 0.269, respectively, which deviate significantly from each other.
In this work, we fix C 0 = −0.36 ± 0.04 by B exp (Λ + c → pφ) from the experiment as shown in Appendix A while C + is treated as a free parameter in general.

IV. POLE CONTRIBUTIONS
The amplitude of the s-channel can be illustratively represented in the form: where B I and M I denote the intermediate baryon and its corresponding mass, respectively.The coupling of B I −B f −P is represented by g B I B f P .The u-channel amplitude can be parameterized in a manner akin to the above expression.
In this work, the baryon-meson couplings of g BB ( * ) P are extracted by the generalized Goldberg-Treiman relations where The symbols B ′ and B * denote the intermediate baryons with spin-parity 1 2 + and 1 2 − , respectively.The corresponding masses of B (′) and B * are represented by M (′) and M * .
The Goldberg-Treiman relations are derived by operating q µ on both sides of Eq. ( 37) and impose the equation of motion.The actual values of g 2 would be irrelevant to this work and g 3 is mainly contributed by the baryon-meson couplings.
The baryon matrix elements of the effective Hamiltonian with ∆c = −1 are decomposed as In the following, b B ′ B will be dropped as it is tiny [65].Collecting Eqs. ( 35), ( 36) and (38), we are led to and where the mass ratios are defined by and Here, M n,c,d (′, * ) represent the masses of B (′, * ) n,c,d , respectively.
Up to the present, there is no ample data to accurately fit the unknown hadronic parameters for Ω 0 c and B cc decays.In the subsequent analysis, we will utilize two essential approximations, as delineated in the Introduction: • The intermediate states B I are exclusively confined to the low-lying 20 multiplets of the SU(4) F group.Here, 20 = 8 ⊕ 3 ⊕ 6 ⊕ 3 in the SU(3) F group.
• The baryon matrix elements are independent of the spectator quarks, implying that the amplitudes shown in Fig. 3 do not depend on q (′) .
The reliability of our predictions hinges on the validity of these two approximations.
The first approximation emphasizes that B ′ ∈ {B A,S c , B n , B cc } and B * belong to the representation of 20 also.
On the other hand, we have already used the second approximation to extract the form factors of B cc → B A,S c in Eq. (34), which are essentially two-quark operator baryon matrix elements.For the four-quark operators, it facilitates the parameterization expressed in and Furthermore, by implementing Eq. ( 17), we obtain the ratio g 2 /g 1 = 5/4, leading to the vanishing of g P FIG. 3: The topological diagrams for the baryon matrix elements of the two-quark and fourquark operators.We use the approximation of that their magnitudes do not depend on q (′) .
where we have taken the baryons with spin-parity To calculate R Bs,u c,cc , the masses of B (′) are readily available from experimental measurements [1].However, the masses of B * are not fully available yet.For the charmless octet baryons, we consider the states N(1535) and Σ(1750), taking the average mass value of M n * = 1643 MeV.For the charmed baryons with negative parity B * c , we identify the candidates as Λ + c (2595), Ξ + c (2790), and Σ + c (2792), from which we calculate the average masses M c * = 2700 MeV and M c * = 2900 MeV for the 3 and 6 representations, respectively.In the case of the doubly charmed baryons with J = 1 2 − , we adopt the value M cc * = 3932 MeV [67].Summarizing, the mass ratios related to the J = 1 2 − baryons utilized in this work are expressed as: where the parenthesis denotes the representation of M * c .We note that focusing solely on B A c decays, the uncertainties in R As,u c,cc would be incorporated into the baryon matrix elements of g ′ 1,2 and b(′) .Consequently, the uncertainties in Eq. ( 46) would only influence the predictions for the Ω 0 c and B cc decays.

V. NUMERICAL RESULTS
The numerical results of this study are organized into several subsections.In Subsec.A, we recall the experimental data of the B A c decays, and the free parameters in both the LP and GP scenarios are extracted accordingly.Although the GP scenario provides more reliable predictions for B A c decays, the LP scenario has broader applications, i.e., its parameters can be applied to both Ω 0 c and B cc decays.Subsec.B and C are devoted to the study of Ω 0 c and B cc decays in the LP scenario, respectively.39) and ( 40)), we absorb g 1 (g ′ 1 ) into ã( b(′) ) so that g For the nonfactorizable amplitudes, there remain (ã) and ( b, b′ , g ′ 2 ) to be fitted in the P -and S-waves, respectively.Comparing to the GP scenario, the parameters of the nonfactorizable amplitudes in the P -waves have been reduced from 3 to 1.It is due to that we have related g P BnB ′ n with g P BcB ′ c in Eq. ( 44) and demand 4g 2 = 5g 1 .On the other hand, due to a lack of knowledge of parity-odd baryons, we impose no further constraints on the S-waves in comparison to the GP scenario.
The experimental data of the B A c decays up to date [1,6,12] are collected in Table I.By adopting the minimal χ 2 fitting, we find ã, b, b′ , g ′ 2 , C + = (2.06 ± 0.25, 12.51 ± 1.03, −4.01 ± 1.13, 0.148 ± 0.075, 0.467 ± 0.034) , where (ã, b, b′ ) are in units of 10 −3 G F GeV 3 .In the limit of the SU(4) F symmetry, we would expect b = b′ , but we observe a significant SU(4) F breaking as they differ both in sign and magnitude.It indicates that the charm quark and the light quarks behave very differently in B * .We note that C + is twice smaller than the expected value of C + ≈ 1.2 from the effective color number approach, discussed in Appendix A.
For comparison, we also update the results of the GP scenario.The free parameters in Eq. ( 24) are found to be (a 1 , a 2 , a 3 , a 6 ) = (3.25 ± 0.11, 1.60 ± 0.07, 0.58 ± 0.12, 1.74 in units of 10 −2 G F GeV 2 .Comparing to the previous values 2 , we see that the parameters modify significantly.It is a hint of that the results shall not be trust fully.Since the SU(3) F symmetry is not exact and too many parameters are required, it is reasonable that the best fitting solutions are not stable along with the experimental update.
In regard to the results in Table I, several comments are in order: and especially B(Ξ 0 c → Ξ − π + ), good accordance is found in two scenarios but both suggest very different values against the current experimental data.It indicates that the short distance contributions may play a dominate role in these decays.Experimental revisits on these channels shall be welcome.
• The results of Λ + c → Ξ 0 K + , Λ + c → nπ + and Ξ 0 c → Σ + K − deviate largely between two scenarios.It implies that the excited states which do not belong to the 20 SU(4) F multiplets may play an important role in B I .
• In contrast to the P -wave, the S-wave does not vanish in general for Λ + c → Ξ 0 K + in the LP scenario.However, the current experimental data prefers a vanishing S-wave also, leading to contradiction against B exp (Λ + c → Ξ 0 K + ).• Continuing the above comment, we see that the LP scenario also fails to explain B exp (Λ + c → nπ + ), but B GP (Λ + c → Ξ 0 K + ) and B GP (Λ + c → nπ + ) are consistent with the experimental data.
• We do not include B exp (Λ + c → pη) and B exp (Λ + c → Σ + η) into the global fit as we do no consider the SU(3) F singlet in P .The results of this work are obtained by assuming the mixing between η 0 and η 8 is absent.Surprisingly, the numerical results turn out to be compatible with the current experimental data.
It is insightful to compare the LP scenario with Ref. [35] which computes the Swave amplitudes by the soft meson approximation.Comparisons for several chosen channels are collected in Table II.The factorizable amplitudes with the neutral P agree well as they are fixed by B exp (Λ + c → pφ).However, for Λ + c → Λπ + our A fac and B fac are roughly twice smaller than Ref. [35] as we adopt a much smaller C + , and we find a sizable A pole in contrast to A pole = 0 at the soft meson limit.One possible explanation to reconcile two approaches is that a sizable proportion from exited intermediate baryons is reabsorbed into C + , leading to a smaller value of C + = 0.469 against the naïve expectation of C + ≈ 1.We see that although our sizes of the S-and P -wave amplitudes differ with Ref. [35], the signs are consistent for most of the cases.
We point out that good agreements in Λ + c → pπ 0 and Λ + c → nπ + with Ref. [35] are found, where large destructive interference between factorizable and pole amplitudes occurs.It indicates that the current algebra approach with the soft meson limit is a good approximation for describing the low-lying poles.However, it shall be noted that the LP scenario and Ref. [35]   The numerical results of the B A c decay channels, for which there are no experimental references yet, are collected in Appendix B for use in future experiments as a basis for verification.

B. Results of Ω 0 c decays
Lacking of experimental input, the GP scenario is not available for Ω 0 c decays.Based on the LP scenario, the predictions of Ω 0 c → B n P are collected in Table III, where the lifetime of Ω 0 c is taken to be (273±12) fs [1].It is interesting to see that B(Ω c → Ξ 0 K 0 S ) and B(Ω c → K 0 L ) deviate significantly, induced by the interference between the CF and DCS amplitudes.
Up to date, the measurements of the Ω 0 c decay ratios are performed in regard to Ω 0 c → Ω − π + .Fortunately, Ω 0 c → Ω − π + does not receive W -exchange contributions and is color-enhanced.The branching fraction is calculated by where H fac + and H fac − are the factorizable helicity amplitudes defined as q µ = (q 0 , 0, 0, −q 3 ) is the four-momentum of the pion, λ and J z are the helicity and angular momentum of Ω − and Ω 0 c , respectively, and C ′ + is the responsible effective Wilson coefficient.In this work, the baryonic matrix elements in Eq. ( 50) are evaluated from the homogeneous bag model [68].
As Ω − does not belong to the 20 SU(4) F multiplets, Ω 0 c → Ω − π + does not necessarily share the same effective Wilson coefficients with B A,S c → B n P .In Table IV, we compare the outcomes with various C ′ + , where with B(Ω c → B n P ) taken from Table III.We note that C ′ + = 1.2, 1 and 0.469 come from the effective color scheme, N c = 3 and B A c → B n P , respectively.The scheme of C ′ + = 0.469 is favored by the experiment of R(Ω 0 c → Ξ 0 K 0 S ) but disfavored by the others.On the other hand, R(Ω 0 c → Ω − e + ν e ) suggests C ′ + = 1.One shall bear in mind that these outcomes are based on the LP scenario and the inconsistencies may disappear in the GP scenario which is not available due to a lack of experimental input.The CF decays of B cc → B c P based on the LP scenario are collected in Table V, while the others in Appendix C. The lifetimes of the charmed baryons (Ξ ++ cc , Ξ + cc , Ω + cc ) are adopted as (256, 36, 136) fs, respectively.In analyzing the transition B A c → B n P , the fitted value of C + is found to be notably smaller than the naïve expectation.This TABLE IV: Comparisons of the evaluated branching fractions with the experiments [1].
All other parameters in this analysis are from Eq. ( 47).
The branching ratio of ) is calculated to be 1.19 ± 0.09 and 0.87 ± 0.06 for C + = 0.469 and 1, respectively.These results are roughly consistent with the experimental measurement of 1.41 ± 0.17 ± 0.10 [69].As R Ξcc is not included in the global fit, it is nontrivial for our outcome to agree with the experiment.Nevertheless, the calculated branching fraction B(Ξ cc → Ξ + c π + ) = (6.24± 0.21)% with C + = 1 exceeds the naïve expectation of (1.33 ± 0.74)%, referenced in [70,71].Note that Ξ ++ cc → Σ ++ cc K S/L do not receive pole contributions, and the ratio serves as an important prediction of the pole approximation.We emphasize that the differences between two cases only occur in A fac and B fac with charged P , related by a factor of 1/0.469.
Due to the smallness of the Ξ + cc lifetime, the branching fractions of Ξ + cc are systematically smaller, but the predicted B(Ξ 3)% is used.As the final state particles are all charged, searches of Ξ + cc → Ξ − π + π + π + π − are recommended.In addition B(Ω + cc → Ω 0 c π + ) consists solely of factorizable contributions and is predicted to be notably large.It is also recommended for future experimental investigations.
12 0 1.34(13) −0.99(0) 1.34(13) −0.99(0) 80 −1.59 −22.31 0 0.25(4) −0.99(0) 0.92(4) −1.00(0) at q 2 = M 2 p .Combing with B exp (Λ c → pφ) = (1.06 ± 0.14) × 10 −3 , we find where N ef f c is the effective color number.The formalism of the decay width and the definitions of f 1,2 and g 1,2 can be found in Ref. [35].In the effective color number approach, one assume In the naïve factorization approach, though C + behaves stably, C 0 varies heavily according to the energy scale and flip sign at the next-to-leading order (NLO).It is a sign that the naïve factorization approach cannot be trusted.On the other hand, the effective color approach provides a much stable value of C + .
Appendix B: Predictions of the LP scenario for B A c → B n P Tables VII, VIII, and IX compile the numerical predictions for the B A c decays from the LP scenario.These predictions can be used as a reference in future experiments for validation and testing.[60].[3] S. Groote and J. G.     [17] R.

FIG. 2 :
FIG. 2: The s-(left) and u-channels (right) of the pole contributions in B i → B f P , where × denotes the insertion of the effective Hamiltonian and B I is the intermediate state.
. Incorporating Eqs.(43) and(44) into Eqs.(39)and(40) and summing over B I , we eliminate the tensors of the intermediate states by employing the completeness relation[3]

TABLE I :
[1,ults of the low-lying and general pole scenarios, denoted with LP and GP in the subscripts, where the parameters are extracted from the current experimental data of B exp and α exp collected in the first column[1, 6, 12].Here, the numbers in the parentheses are the uncertainties counting backward in digits, for example, 1.59(8) = 1.59 ± 0.08.

TABLE III :
Predictions of the CF, Cabibbo suppressed (CS) and doubly Cabibbo suppressed (DCS) decays with Ω 0 c as the initial baryons, where A and B are in units of 10 −2 G F GeV 2 .

TABLE V :
Predictions of the CF decays in B cc → B A,S c P with C + = 0.469 and 1, where A and B are in units of 10 −2 G F GeV 2 .

TABLE VI :
The effective Wilson coefficient, where C + (N ef f c ) is fitted from Eq. (A2).The values of c 1,2 are from Ref.

TABLE VII :
Predictions of the LP scenario for the CF decays of B A c → B n P , where A and B are in units of 10 −2 G F GeV 2 .

TABLE VIII :
Predictions of the LP scenario for the CS decays of B A c → B n P , where A and B are in units of 10 −2 G F GeV 2 .

TABLE IX :
Predictions of the LP scenario for the DCS decays of B A c → B n P , where A and B are in units of 10 −2 G F GeV 2 .

TABLE X :
Predictions of the LP scenario for the CS decays of B cc → B A,S c P with C + = 0.469 and 1, where A and B are in units of 10 −2 G F GeV 2 .