Asymmetries of anti-triplet charmed baryon decays
aa r X i v : . [ h e p - ph ] M a y Asymmetries of anti-triplet charmed baryon decays
C.Q. Geng , , , Chia-Wei Liu and Tien-Hsueh Tsai Chongqing University of Posts & Telecommunications, Chongqing 400065 Department of Physics, National Tsing Hua University, Hsinchu 300 Physics Division, National Center for Theoretical Sciences, Hsinchu 300 (Dated: May 17, 2019)
Abstract
We analyze the decay processes of B c → B n M with the SU (3) F flavor symmetry and spin-dependent amplitudes, where B c ( B n ) and M are the anti-triplet charmed (octet) baryon andnonet meson states, respectively. In the SU (3) F approach, it is the first time that the decayrates and up-down asymmetries are fully and systematically studied without neglecting the O (15)contributions of the color anti-symmetric parts in the effective Hamiltonian. Our results of the up-down asymmetries based on SU (3) F are quite different from the previous theoretical values in theliterature. In particular, we find that the up-down symmetry of α (Λ + c → Ξ K + ) SU (3) = 0 . +0 . − . ,which is consistent with the recent experimental data of 0 . ± .
78 by the BESIII Collaboration,but predicted to be zero in the literature. We also examine the K S − K L asymmetries betweenthe decays of B c → B n K S and B c → B n K L with both Cabibbo-allowed and doubly Cabibbo-suppressed transitions. . INTRODUCTION Recently, the Belle collaboration has measured the absolute branching ratio of Λ + c → pK − π + with high precision [1], resulting in the world average value of B (Λ + c → pK − π + ) =(6 . ± . + c decay branching fractions are presented relative to it.Subsequently, this golden mode, along with many other Λ + c ones, has also been observed bythe BESIII Collaboration [3–12] with Λ + c ¯Λ − c pairs, produced by e + e − collisions at a center-of-mass energy of √ s = 4 . + c . In particular, the decay of Λ + c → Σ + η ′ has been seen for thefirst time with η ′ in the final states for the charmed baryon decays [12]. In addition, theabsolute decay branching fraction of Ξ c → Ξ − π + , which involves the anti-triplet charmedbaryon state of Ξ c , has also been measured by the Belle collaboration [13]. Clearly, a newexperimental physics era for charmed baryons has started.On the other hand, the theoretical study of the charmed baryon decays has faced sev-eral difficulties. The most serious one is that the factorization approach in the non-leptonicdecays of charmed baryons is not working. For example, the Cabibbo-allowed decays ofΛ + c → Σ π + and Λ + c → Σ + π do not receive any factorizable contributions, whereas theexperimental data show that their branching fractions are all close to O (10 − ) [2], indicat-ing the failure of the factorization method. In addition, the complication of the charmedbaryon structure makes us impossible to directly evaluate the decay amplitude in a model-independent way. It is known that the most reliable and simple way to examine the charmedbaryon processes is to use the flavor symmetry of SU (3) F [14–26]. Indeed, it has been re-cently demonstrated that the results for the charmed baryon decays based on the SU (3) F ap-proach [17–26] are consistent with the current experimental data. Nevertheless, the charmedbaryon decays have been extensively studied in various dynamical models [27–37], partic-ularly, the recent dynamical calculations of the singly Cabibbo-suppressed Λ + c decays byCheng, Kang and Xu (CKX) [37] based on current algebra.For the two-body charmed baryon decay of B c → B n M , with B c ( B n ) and M the anti-triplet charmed (octet) baryon and nonet meson states, respectively, beside its decay branch-ing fraction, there exits another interesting physical observable, the up-down asymmetry α ,which is related to the longitudinal polarization of B n . Currently, there are three experi-2ental measurements of the up-down asymmetries in the charmed baryon decays [2], alongwith the recent one by BESIII [11], given by α (Λ + c → Ξ K + ) exp = 0 . ± . , (1)which has been suggested to be approximately zero in the previous theoretical studies withthe dynamical models [27–34] as well as the SU (3) F approach [16]. However, the up-downasymmetries in B c → B n M were not discussed in the previous studies with SU (3) F inRefs. [17–26].In addition, it has been noticed that the physical Cabibble-allowed dominated decay pro-cesses of B c → B n K L and B c → B n K S are the same when the doubly Cabibbo-suppressedcontributions are taking to be zero [19]. However, in some of these processes, the doublyCabibbo-suppressed transitions are not negligible, which can be examined by defining the K S − K L asymmetries between the K S and K L modes [19] to track the interferences.In this work, we will systematically analyze the decay processes of B c → B n M with the SU (3) F symmetry with all operators under SU (3) F . We will also include the effect of the η − η ′ mixing. There are two different ways to link the amplitudes among the processesby SU (3) F . The first one is a purely mathematical consideration. By imposing the SU (3)group, we are able to write down the amplitude by tensor contractions. The second oneis the diagrammatic approach, in which one draws down all the possible diagrams for thedecay process with ascertaining that the amplitude from each diagram shall be the sameby interchanging up, down and strange quarks. Both ways have their own advantages. Thetensor method is easier to cooperate with the other symmetry and it allows us to estimate theorder of the contribution from the amplitude with the Wilson coefficients. Explicitly, it couldcooperate with the SU (3) color symmetry and take account of the strange quark mass asthe source of the SU (3) F symmetry breaking [15, 22]. On the other hand, the diagrammaticapproach can distinguish the factorizable and non-factorizable amplitudes [35]. The closerelations between the two methods have been examined in Ref. [38]. In Ref. [23], it has beenproved to be useful if one combines both methods.This paper is organized as follows. In Sec. II, we give the formalism for the two-bodycharmed baryon decays of B c → B n M , in which we first write the decay amplitudes in termsof parity conserved and violated parts under the SU (3) F flavor symmetry, and then displaythe decay rates and asymmetries. In Sec. III, we show our numerical results and present3iscussions. We conclude in Sec. IV. In Appendix A, we list the all decay amplitudes of theanti triplet baryon states in terms of the SU (3) F parameters. We give the definitions of theup-down and longitudinal polarization asymmetries in Appendix B. II. FORMALISM
To study the two-body decays of the anti-triplet charmed baryon ( B c ) to octet baryon( B n ) and nonet pseudoscalar meson ( M ) states, we write the hadronic state representationsunder the SU (3) F flavor symmetry to be B c = (Ξ c , − Ξ + c , Λ + c ) , B n = √ Λ + √ Σ Σ + p Σ − √ Λ − √ Σ n Ξ − Ξ − q Λ ,M = √ ( π + c φ η + s φ η ′ ) π + K + π − √ ( − π + c φ η + s φ η ′ ) K K − ¯ K − s φ η + c φ η ′ , (2)respectively, where ( c φ , s φ ) = (cos φ, sin φ ) and φ = 39 . ◦ [39] to describe the mixing between η and η of the octet and nonet sates for η .From c → u ¯ ds , c → u and c → u ¯ sd transitions at tree level, the effective Hamiltonian isgiven by [40] H eff = X i =+ , − G F √ c i (cid:0) V cs V ud O dsi + V cd V ud O qqi + V cd V us O sdi (cid:1) , (3)with O q q ± = 12 [(¯ uq ) V − A (¯ q c ) V − A ± (¯ q q ) V − A (¯ uc ) V − A ] , (4)where ( | V cs V ud | , | V cd V ud | , | V cd V us | ) ≃ (1 , s c , s c ) with s c ≡ sin θ c ≈ .
225 [2] and θ c the Cabibboangle, c i (i=+,-) represent the Wilson coefficients, G F is the Fermi constant, O q q ± and O qq ± ≡ O dd ± − O ss ± are the four-quark operators, and (¯ q q ) ≡ ¯ q γ µ (1 − γ ) q . In Eq. (3), thedecays associated with O ds ± , O qq ± and O sd ± are the so-called Cabibbo-allowed (favored), singlyCabibbo-suppressed and doubly Cabibbo-suppressed processes, respectively.4ote that O +( − ) , corresponding to the O (15(6)) representation, is (anti)symmetric inflavor and color indices. The tensor forms of H (15) and H (6) under SU (3) F are given by H (15) ijk = , s c s c , − s c − s c − s c − s c ,H (6) ij = − s c − s c s c , (5)respectively, where we have used the conversion of V cd = − V us = s c . In general, we writethe spin-dependent amplitude of B c → B n M as M ( B c → B n M ) = iu B n ( A − Bγ ) u B c , (6)where A and B are the s -wave and p -wave amplitudes, corresponding to the parity violatingand conserving ones, and u B n,c are the baryon Dirac spinors, respectively. From Eqs. (3)and (6), we can decompose A in terms of the tensor forms under SU (3) F as A ( B c → B n M ) = a H (6) ij ( B ′ c ) ik ( B n ) jk ( M ) ll + a H (6) ij ( B ′ c ) ik ( B n ) lk ( M ) jl + a H (6) ij ( B ′ c ) ik ( M ) lk ( B n ) jl + a H (6) ij ( B n ) ik ( M ) jl ( B ′ c ) kl + a ′ ( B n ) ij ( M ) ll H (15) jki ( B c ) k + a H (15) lik ( B c ) j ( M ) ji ( B n ) kl + a ( B n ) ij ( M ) li H (15) jkl ( B c ) k + a ( B n ) ji ( M ) ml H (15) lim ( B c ) j + a ( B n ) li ( M ) ij H (15) jkl ( B c ) k ,B ( B c → B n M ) = A ( B c → B n M ) { a ( ′ ) i → b ( ′ ) i } (7)where ( B ′ c ) ij ≡ ǫ ijk ( B c ) k . Here, we have assumed that the mass dependence of A and B arenegligible, while the Wilson coefficients of c i have been absorbed into the SU (3) F parameters a ( ′ ) i and b ( ′ ) i . Note that we treat the SU (3) F flavor symmetry to be exact. To obtain moreprecise results, one has to include the SU (3) F breaking terms in the amplitudes as shown inRefs. [15, 22]. Note that the analysis with SU (3) F breaking effect can be done when moreexperimental data are available in the future. The expansions of A ( B c → B n M ) are listed inAppendix A, while those of B ( B c → B n M ) can be derived by replacing a i in A ( B c → B n M ) with b i . Since the operator O (15) ∼ (¯ uq )(¯ q c ) + (¯ q q )(¯ uc ) is symmetric in color index, whereasthe baryon states are antisymmetric, the contributions of O (15) from the nonfactorizable5 IG. 1. Topological diagram related to factorizable processes with the bubble representing thefour-quark interaction. part to the amplitude vanish, so that we only need to consider the factorizable amplitudefrom O (15) [23]. The factorizable diagram is shown in Fig. 1 with the bubble representingthe four-quark interaction, which corresponds to the factorized amplitude, given by [27] G F √ V cq V uq m χ ± h B n | (¯ q i c ) V − A | B c ih M | (¯ q l q m ) V − A | i , (8)where q = q i ( q l ) and χ ± are related to the effective Wilson coefficients for the charged(neutral) meson in the final states.From the topological diagram in Fig. 1, one concludes that only a and b terms inEq. (7) contain the factorizable contributions in O (15), in which the octet meson state M is directly given by the weak interaction alone as demonstrated in Ref. [23]. As a result, inour calculations we will neglect the terms associated with a ′ , a , a and a and b ′ , b , b and b in Eq. (6).The decay angular distribution of the direction ˆ p B n = ~p B n /p B n ( p B n ≡ | ~p B n | ) of B n inthe rest frame of B c is found to be d Γ dθ ∝ α ~P B n · ˆ p B n = 1 + α cos θ, (9)where ~P B n is the polarization vector of B n with the longitudinal component being P B n = α , θ is the angle between ~P B n and ˆ p B n and α is the so-called up-down asymmetry parameter,given by α = 2 κ Re( A ∗ B ) | A | + κ | B | , κ = p B n E B n + m B n (10)with E B n and ~p B n the energy and three momentum of B n . The definitions of the up-downand longitudinal asymmetries can be found in Appendix B. Consequently, we obtain thedecay rate asΓ = p B n π (cid:18) ( m B c + m B n ) − m M m B c | A | + ( m B c − m B n ) − m M m B c | B | (cid:19) (11)6o extract the doubly Cabibbo-suppressed contributions in the Cabibbo-allowed dominatingdecays of B c → B n K L /K S , we also define the K S − K L asymmetry parameter as [19] R K S,L ( B c → B n ) = Γ( B c → B n K S ) − Γ( B c → B n K L )Γ( B c → B n K S ) + Γ( B c → B n K L ) . (12) III. NUMERICAL RESULTS AND DISCUSSIONS
We now determine the SU F parameters through the experimental data [2, 11–13, 41],listed in Table I, where we have also shown the reproduced values for the observables.In the following analysis, we take the amplitudes of A and B as real by using the factthat CP is mainly conserved in charmed decays and assuming the final state interaction isnegligible [42] . Note that in our fit, we have used the original data point of B (Λ + c → pπ ) =(0 . ± . × − from the BESIII Collaboration [41], but the result of α (Λ + c → Ξ K + ) =0 . ± .
78 [11] is not included. Consequently, there are 16 experimental data inputs to fitwith 10 SU (3) F parameters in Eq. (7), given by( a , a , a , a , ˜ a, b , b , b , b , ˜ b ) , (13)resulting in the degree of freedom (d.o.f) to be 6. In order to separate the amplitudes from η and octet meson states, we define ˜ a and ˜ b by˜ a ≡ a + 13 ( a + a − a ) , ˜ b ≡ b + 13 ( b + b − b ) (14)respectively. As a result, the η amplitude depends only on ˜ a and ˜ b . By performing theminimal χ fitting as shown in Ref. [21], we obtain ( a , a , a , a , ˜ a ) = (4 . ± . , − . ± . , . ± . , − . ± . , . ± .
83) 10 − G F GeV , ( b , b , b , b , ˜ b ) = ( − . ± . , − . ± . , . ± . , − . ± . , . ± .
56) 10 − G F GeV . (15) We note that A and B are relative real if CP is conserved and the final state interactions are negligible.This statement has been given in many textbooks, such as those in Refs. [43, 44]. ABLE I. Comparisons of the decay branching ratios and asymmetries between the experimentaldata [2, 11–13, 41] and theoretical reproductions with SU (3) F .Channel B exp α exp B SU (3) F α SU (3) F Λ + c → Λ π + (13 . ± . × − − . ± .
15 (13 . ± . × − − . ± . + c → pK S (15 . ± . × − (15 . ± . × − − . +0 . − . Λ + c → Σ π + (12 . ± . × − (12 . ± . × − − . ± . + c → Σ + π (12 . ± . × − − . ± .
32 (12 . ± . × − − . ± . + c → Σ + η (4 . ± . × − (3 . ± . × − − . ± . + c → Σ + η ′ (13 . ± . × − (14 . ± . × − . +0 . − . Λ + c → Ξ K + (5 . ± . × − ∗ . ± .
78 (5 . ± . × − . +0 . − . Λ + c → pπ (0 . ± . × − [41] (1 . ± . × − − . ± . + c → pη (12 . ± . × − (11 . ± . × − − . +0 . − . Λ + c → Λ K + (6 . ± . × − (6 . ± . × − . ± . + c → Σ K + (5 . ± . × − (5 . ± . × − − . +0 . − . Ξ c → Ξ − π + (1 . ± . × − − . ± . . ± . × − − . +0 . − . Ξ c → Λ K S (5 . ± . × − − . ± . ∗∗ R Ξ c . ± . ∗ This value is not included in the data input. ∗∗ R Ξ c ≡ B (Ξ c → Λ K S ) / B (Ξ c → Ξ − π + ). SU (3) F parameters in Eq. (13) are givenby R = .
64 0 . − . − .
30 0 . − .
47 0 . − .
66 0 . .
64 1 − . − . − .
38 0 . − .
59 0 . − .
12 0 . . − .
17 1 − .
67 0 .
01 0 .
55 0 .
03 0 . − . − . − . − . − .
67 1 0 . − .
65 0 . − .
59 0 . − . − . − .
38 0 .
01 0 .
11 1 − .
31 0 . − .
29 0 . − . .
96 0 .
61 0 . − . − .
31 1 − .
51 0 . − .
70 0 . − . − .
59 0 .
03 0 .
21 0 . − .
51 1 − .
59 0 . − . .
67 0 .
38 0 . − . − .
29 0 . − .
59 1 − .
69 0 . − . − . − .
75 0 .
93 0 . − .
70 0 . − .
69 1 − . .
25 0 . − . − . − .
35 0 . − .
29 0 . − .
10 1 . (16) In our fit, we find that χ /d.o.f = 0 .
5, which indicates that our results with the SU (3) F symmetry can well explain all current existing experimental data for the decay branchingratios and up-down asymmetries. Indeed, as seen in Table I, our reproductions based on SU (3) F are all consistent with the corresponding experimental measurements. However, itis important to pointed out that our values of B (Ξ c → Ξ − π + ) = (2 . ± . × − and | α (Ξ c → Ξ − π + ) | = 0 . +0 . − . are consistent with, but higher than, the corresponding data of(1 . ± . × − [13] and 0 . ± . H (15)only contributes to the factorization amplitudes, which can be parametrized only in terms of a and b terms, corresponding to the vector and axial-vector currents in the baryonic matrixelements, respectively. Our result of b ≫ a in Eq. (15) suggests that the axial-vector partof the factorization contribution is much larger that the vector one. This can be understoodas follows. In the decay of B c → B n M , the pseudoscalar meson part of the factorizationapproach is given by h | j µ | M i = if M q µ , (17)where f M is the meson decay constant, while q µ is the four-momentum of M , which is alsoequal to the four-momentum difference between the initial and final baryons of B c and B n .Consequently, we get that q µ h B n | ¯ qγ µ γ c | B c i ≫ q µ h B n | ¯ qγ µ c | B c i = i h B n | ∂ µ (¯ qγ µ c ) | B c i , (18)9here q stands for the light quarks. In the case of the SU (4) flavor symmetry, in whichthe charm quark is also treated as q , Eq. (18) is automatically satisfied as the right-handedpart is zero. It is clear that the inequality in Eq. (18) depends on the parameters a and b ,which are not quite determined yet, particularly a . In fact, from Table IX in Appendix A,we have that A (Λ + c → pπ ) = √ (cid:16) a + a − a (cid:17) , (19)in which a and a get almost canceled out each other, resulting in that it could be dominatedby the a terms. In this case, the experimental search for the up-down asymmetry as wellas the future measurement on the branching ration of Λ + c → pπ will be helpful to obtainthe precise value of a .In Tables II, III and IV, we list our predictions of the branching ratios and up-down asym-metries for the Cabibbo-allowed, singly Cabibbo-suppressed and doubly Cabibbo-suppresseddecays, respectively. In the tables, we have also presented the values of A and B, which areuseful to understand the up-down asymmetries as well as the comparisons with those givenby specific theoretical models. We note that some of our results for the up-down asymme-tries have been discussed for the first time in the literature, while the decay branching ratiosare almost the same as those in Refs. [17–23]. In particular, we find that B (Λ + c → pπ ) =(1 . ± . × − , which is consistent with our previous value of (1 . ± . × − in Ref. [23]and 0 . × − calculated by the pole model with current algebra in Ref. [37] as well as thecurrent experimental upper limit of 2 . × − [2]. In addition, the decay branching ratio forthe related Cabibbo-suppressed mode of Λ + c → nπ + is predicted to be (8 . ± . × − , incomparison with (6 . ± . × − in Ref. [23] and 2 . × − in Ref. [37]. We remark thatmost of the branching ratios in the present work with the spin-dependent amplitudes havesmall uncertainties comparing to those of our previous study with SU (3) F in Ref. [23] exceptthe decay of Λ + c → pπ due to the cancellation effect as well as the correlations in Eq. (16).Explicitly, as shown Table III, the sign in A (Λ + c → pπ ) = ( − . ± . θ c × − G F GeV is not well determined, resulting in a large error in α (Λ + c → pπ ) SU (3) = − . ± .
72. Todetermine the asymmetry precisely, the experiment with a smaller uncertainty is clearlyneeded.To compare our predictions of the up-down asymmetries with those in the literature, wesummarize the values of α for the Cabibbo-allowed and singly Cabibbo-suppressed decays10 ABLE II. Predictions of the branching ratios and up-down asymmetries for the Cabibbo-alloweddecays, where we have also listed the values of A and B in the unit of 10 − G F GeV .channel A B B α Λ + c → Λ π + − . ± .
06 1 . ± .
12 13 . ± . − . ± . + c → p ¯ K − . ± .
15 1 . ± .
62 31 . ± . − . +0 . − . Λ + c → Σ π + − . ± .
02 0 . ± .
29 12 . ± . − . ± . + c → Σ + π . ± . − . ± .
29 12 . ± . − . ± . + c → Σ + η − . ± .
07 0 . ± .
44 3 . ± . − . ± . + c → Σ + η ′ − . ± . − . ± .
54 14 . ± . . +0 . − . Λ + c → Ξ K + . ± .
06 1 . ± .
24 5 . ± . . +0 . − . Ξ + c → Σ + ¯ K . ± .
12 0 . ± .
52 8 . +9 . − . . +0 . − . Ξ + c → Ξ π + − . ± .
06 0 . ± .
23 3 . ± . − . ± . c → Ξ − π + . ± . − . ± . . ± . − . +0 . − . Ξ c → Λ ¯ K − . ± .
07 0 . ± .
39 10 . ± . − . ± . c → Σ ¯ K − . ± .
10 0 . ± .
33 0 . ± . − . ± . c → Σ + K − − . ± . − . ± .
24 5 . ± . . ± . c → Ξ π − . ± .
06 1 . ± .
23 7 . ± . − . +0 . − . Ξ c → Ξ η . ± .
08 1 . ± .
55 10 . ± . . +0 . − . Ξ c → Ξ η ′ . ± .
25 4 . ± .
51 9 . ± . . +0 . − . ABLE III. Legend is the same as Table II but for the singly Cabibbo-suppressed decays with anoverall factor of sin θ c for A and B omitted.channel A B B α Λ + c → pπ . ± . − . ± .
33 1 . ± . − . ± . + c → pη − . ± .
18 1 . ± .
77 12 . ± . − . +0 . − . Λ + c → pη ′ . ± .
27 4 . ± .
91 24 . ± . . +0 . − . Λ + c → nπ + − . ± . − . ± .
20 8 . ± . . ± . + c → Λ K + . ± .
06 0 . ± .
35 6 . ± . . ± . + c → Σ K + . ± . − . ± .
23 5 . ± . − . +0 . − . Λ + c → Σ + K . ± . − . ± .
33 10 . ± . − . +0 . − . Ξ + c → Λ π + . ± . − . ± .
24 12 . ± . − . ± . + c → p ¯ K . ± . − . ± .
33 43 . ± . − . +0 . − . Ξ + c → Σ π + . ± . − . ± .
34 25 . ± . − . ± . + c → Σ + π − . ± . − . ± .
53 26 . ± . . ± . + c → Σ + η . ± .
16 0 . ± .
85 15 . ± . . +0 . − . Ξ + c → Σ + η ′ . ± .
29 4 . ± .
84 34 . ± . . +0 . − . Ξ + c → Ξ K + − . ± . − . ± .
20 8 . ± . . ± . c → Λ π − . ± .
06 1 . ± .
21 2 . ± . − . ± . c → Λ η − . ± .
15 2 . ± .
65 6 . ± . − . ± . c → Λ η ′ . ± .
33 4 . ± .
28 16 . ± . . +0 . − . Ξ c → pK − . ± .
06 1 . ± .
24 5 . ± . . ± . c → n ¯ K . ± . − . ± .
42 7 . ± . − . ± . c → Σ π . ± . − . ± .
28 3 . ± . − . +0 . − . Ξ c → Σ η − . ± . − . ± .
60 1 . ± . . ± . c → Σ η ′ − . ± . − . ± .
30 3 . ± . . +0 . − . Ξ c → Σ + π − − . ± . − . ± .
24 3 . ± . . ± . c → Σ − π + . ± . − . ± .
30 13 . ± . − . +0 . − . Ξ c → Ξ K − . ± .
03 0 . ± .
42 7 . ± . − . ± . c → Ξ − K + − . ± .
08 2 . ± .
30 9 . ± . − . +0 . − . ABLE IV. Legend is the same as Table II but for the doubly Cabibbo-suppressed decays with anoverall factor of sin θ c for A and B omitted.channel A B B α Λ + c → pK . ± .
12 0 . ± .
52 1 . +1 . − . . +0 − . Λ + c → nK + − . ± .
06 0 . ± .
23 0 . ± . − . +0 . − . Ξ + c → Λ K + − . ± . − . ± .
24 3 . ± . . ± . + c → pπ . ± .
05 1 . ± .
17 6 . ± . . ± . + c → pη . ± . − . ± .
53 20 . ± . − . ± . + c → pη ′ − . ± . − . ± .
87 40 . ± . . +0 . − . Ξ + c → nπ + . ± .
06 1 . ± .
24 12 . ± . . ± . + c → Σ K + − . ± .
06 1 . ± .
21 11 . ± . − . +0 . − . Ξ + c → Σ + K − . ± .
15 1 . ± .
62 19 . ± . − . ± +0 . − . Ξ c → Λ K − . ± . − . ± .
26 0 . ± . . ± . c → pπ − . ± .
06 1 . ± .
24 3 . ± . . ± . c → nπ − . ± . − . ± .
17 1 . ± . . ± . c → nη . ± . − . ± .
53 5 . ± . − . ± . c → nη ′ − . ± . − . ± .
87 10 . ± . . +0 . − . Ξ c → Σ K . ± . − . ± .
44 2 . ± . − . +0 . − . Ξ c → Σ − K + − . ± .
08 2 . ± .
30 6 . ± . − . +0 . − . B c → B n M in Tables V and VI, respectively. In the tables, the data are taken from theexperimental values in Ref. [2], KK and Iva correspond to the calculations with the covariantquark models by Korner and Kramer (KK) [27] and Ivanov el al. (Iva) [33], XK, CT andZen are based on the pole models by Xu and Kamal (XK) [28], Cheng and Tseng (CT) [30]and Zenczykowski (Zen) [32], SV1, CT ′ , UVK ( ′ ) and CKX are related to the considerationsof current algebra by Sharma and Verma (SV1) [34], Cheng and Tseng (CT) [30], Uppal,Verma and Khanna (UVK) without (with) the baryon wave function scale variation [31]and Cheng, Kang and Xu (CKX) [37], and SV2 ( ′ ) represent the results with SU (3) F bySharma and Verma with two different signs of B (Λ + c → Ξ K + ) [16], respectively. As seenin Table V, our results of the up-down asymmetries are quite different from those in theliterature [16, 27–34]. In particular, it is interesting to see that we predict that α (Λ + c → Ξ K + ) SU (3) = 0 . +0 . − . (20)which is consistent with the current experimental data of 0 . ± .
78 in Eq. (1) [11], butdifferent from all theoretical predictions in the literature. For example, it has been suggestedthat this asymmetry is approximately zero in dynamical models [27–34], while the authorsin Ref. [16] have also taken it to be zero as a data input when the SU (3) F symmetry isimposed. In our fit, the value in Eq. (1) has not been included as an input in order to seeits value based on the SU (3) F approach. Since the error of our predicted result in Eq. (20)is small, we are confident that α (Λ + c → Ξ K + ) should be much lager than zero and close toone. Moreover, our result of α (Λ + c → Λ K + ) SU (3) = 0 . ± .
32 is different from the CKXone of α (Λ + c → Λ K + ) CKX = − .
96 in Ref. [37]. The reason for the difference is due to thesigns in the parity violated amplitudes of A (Λ + c → Λ K + ) SU (3) = (1 . ± . × − G F GeV inour calculation and A (Λ + c → Λ K + ) CKX = − . × − G F GeV in Ref. [37]. To clarify theseissues, further precision measurements on these asymmetries are highly recommended.In addition, due to the vanishing contributions to the decays from the a , a , a and a ′ terms of O (15), we get A (Λ + c → Σ K + ) = A (Λ + c → Σ + K S , Σ + K L ) = √ a − a ) s c ,B (Λ + c → Σ K + ) = B (Λ + c → Σ + K S , Σ + K L ) = √ b − b ) s c , (21)14 ABLE V. Summary of our results with SU (3) F and those in the literature for the up-downasymmetries of the Cabibno-allowed charmed baryon decays, where the data, KK, XK, CT, UVK,Zen, Iva, SV1, and SV2 are from the PDG [2], Korner and Kramer [27], Xu and Kamal [28], Chengand Tseng [30], Uppal, Verma and Khanna [31], Zenczykowski [32], Ivanov el al. [33], Sharma andVerma [34], and Sharma and Verma [16], respectively. channel our result data KK XK CT UVK Zen Iva SV1 SV2(CT ′ ) (UVK ′ ) (SV2 ′ )Λ + c → Λ π + − . ± . − . ± . − . − . − . − . − . − . − .
99 input( − .
95) ( − . + c → p ¯ K − . +0 . − . − . . − . − . − . − . − . − . ± . − .
49) ( − . + c → Σ π + − . ± .
27 0 .
70 0 . − . − .
32 0 .
39 0 . − . − . ± . .
78) ( − . + c → Σ + π − . ± . − . ± .
32 0 .
70 0 . − . − .
32 0 .
39 0 . − .
31 input(0 .
78) ( − . + c → Σ + η − . ± .
47 0 . − .
94 0 0.55 − .
99 0 . ± . − .
99) (0 . ± . + c → Σ + η ′ . +0 . − . − .
45 0 . − . − .
05 0 . − . ± . .
68) 0 .
44 ( − . ± . + c → Ξ K + . +0 . − . . ± .
78 0 0 0 0 0 0 0Ξ + c → Σ + ¯ K . +0 . − . − . .
24 0 .
43 1 . − . − .
38 0 . ± . − .
09) ( − . ± . + c → Ξ π + − . ± . − . − . − .
77 1 . − . − .
74 0 . ± . − .
77) ( − . ± . c → Ξ − π + − . +0 . − . − . ± . − . − . − . − . − . − . − . ± . − . c → Λ ¯ K − . ± . − .
76 1 . − . − . − . − . − . ± . − . c → Σ ¯ K − . ± . − . − .
99 0 . − . − . − .
15 0 . ± . − . c → Σ + K − . ± .
16 0 0 0 0 0 0Ξ c → Ξ π − . +0 . − . .
92 0 . − .
78 0 .
21 0 . − . − . ± . − . c → Ξ η . +0 . − . − . − . − . − . − . ± . . ± . c → Ξ η ′ . +0 . − . − . − . − .
32 0 . − . ± . − . ± . ABLE VI. Summary of our results with SU (3) F and those in the literature for the up-downasymmetries of the singly Cabibbo-suppressed charmed baryon decays, where UVK, SV2 and CKXare from Refs. [31], [16] and [37], respectively.channel our result UVK ( ′ ) SV2 ( ′ ) CKXΛ + c → pπ − . ± .
72 0 .
82 (0 .
85) 0 .
05 (0 . − . + c → pη − . +0 . − . − .
00 ( − . − .
74 ( − . − . + c → pη ′ . +0 . − . .
87 (0 . − .
97 ( − . + c → nπ + . ± . − .
13 (0 .
68) 0 .
05 (0 . − . + c → Λ K + . ± . − .
99 ( − . − .
54 (0 . − . + c → Σ K + − . +0 . − . − .
80 ( − .
80) 0 .
68 ( − . − . + c → Σ + K − . +0 . − . − .
80 ( − .
80) 0 .
68 ( − . − . leading to the fitted values of B (Λ + c → Σ K + , Σ + K S , Σ + K L ) = (5 . ± . × − ,α (Λ + c → Σ K + , Σ + K S , Σ + K L ) = − . +0 . − . , (22)as given in Table III. Note that the decay branching ratio of Λ + c → Σ K + has been measuredto be (5 . ± . × − [2], which agrees with with that in Eq. (22). Future measurementson Λ + c → Σ + K S and Λ + c → Σ + K L are important as they can tell us if Eqs. (21) and (22),which can also be derived through the isospin symmetry, are right or wrong.We now concentrate on the decay processes of B c → B n K L and B c → B n K S , whichinvolve both Cabibbo-allowed and doubly suppressed transitions, as shown in Table VII.If we ignore the later contributions associated with sin θ c , B ( B c → B n K S ) = B ( B c → B n K L ). Clearly, the K S − K L asymmetry depends on the doubly Cabibbo-suppressed partsof the decays. As shown in Table VII, the central values for the first three asymmetriesare predicted to be around 10% or more, which are consistent with those in Ref. [19]. ForΞ c → Σ K S /K L , the up-down asymmetry of R K S,L (Ξ c → Σ ) has different sign, indicatingthat the effect of the doublyCabibbo-suppressed transition is not ignorable in these decayprocesses. Explicitly, we find out that B (Ξ c → Λ K L ) can be a little larger than B (Ξ c → Λ K S ), in which the K S − K L asymmetry is predicted to be − (4 . ± . ABLE VII. Irreducible amplitudes, decay branching ratios and up-down and K S − K L asymmetriesof B c → B n K L /K S with both Cabibbo-allowed and doubly Cabibble-suppressed transitions, wherethe B amplitudes can be obtained directly by substituting a i with b i .channel Irreducible amplitude for A 10 B SU (3) F α SU (3) F R K S,L Λ + c → pK S √ (cid:0) ( a − a ) + ( a − a ) s c (cid:1) . ± . − . +0 . − . . ± . + c → pK L −√ (cid:0) ( a − a ) − ( a − a ) s c (cid:1) . ± . − . +0 . − . Ξ + c → Σ + K S −√ (cid:0) ( a − a ) + ( a − a ) s c (cid:1) . +5 . − . . +0 . − . . ± . + c → Σ + K L √ (cid:0) ( a − a ) − ( a − a ) s c (cid:1) . +5 . − . . +0 . − . Ξ c → Σ K S ( a + a − a ) + ( a − a ) s c . ± . − . +0 . − . . ± . c → Σ K L − ( a + a − a ) + ( a − a ) s c . +0 . − . . ± . c → Λ K S √ ((2 a − a − a − a ) 5 . ± . − . ± . − . ± . − ( a − a − a + a ) s c )Ξ c → Λ K L − √ ((2 a − a − a − a ) 5 . ± . − . ± . a − a − a + a ) s c ) uncertainty, which agrees well with − (3 . ± . IV. CONCLUSIONS
We have studied the two-body decays of B c → B n M with the SU (3) F flavor symmetrybased on the spin-dependent s and p -wave amplitudes of A and B , respectively. Theseamplitudes, which have been decomposed in terms of the SU (3) F parameters a ( ′ ) i and b ( ′ ) i ,allow us to examine the longitudinal polarization of P B n , which is related to the up-downasymmetry of α . We have obtained a good χ fit for the ten SU (3) F parameters in Eq. (15)from the all possible contributions of O (6) and O (15) with 16 data points in Table I inthe SU (3) F approach, in which all experimental data for the decay branching ratios andup-down asymmetries can be explained. Consequently, we have systematically predicted alldecay branching ratios and up-down asymmetries of the Cabibbo-allowed, singly Cabibbo-suppressed and doubly Cabibbo-suppressed charmed baryon decays. In particular, our re-sults of B (Ξ c → Ξ − π + ) = (2 . ± . × − and α (Ξ c → Ξ − π + ) = − . +0 . − . are17onsistent with the data of (1 . ± . × − [13] and − . ± . B (Λ + c → pπ ) = (1 . ± . × − , which is consistent with the cur-rent experimental upper limit of 2 . × − [2]. In addition, we have gotten that B (Λ + c → Σ K + , Σ + K S , Σ + K L ) = (5 . ± . × − and α (Λ + c → Σ K + , Σ + K S , Σ + K L ) = − . +0 . − . ,which are also guaranteed by the isospin symmetry.We have shown in Table V that our predictions of the up-down asymmetries are quitedifferent from the theoretical values in the literature for most of the decay modes. In par-ticular, we have found that α (Λ + c → Ξ K + ) SU (3) = 0 . +0 . − . in Eq. (20), which is consistentwith the current experimental data of 0 . ± .
78 in Eq. (1) [11], but much larger than zeropredicted in the literature. A future precision measurement on this asymmetry is clearlyvery important as our prediction based on SU (3) F is close to one with a small uncertainty,which can be viewed as a benchmark for the SU (3) F approach.We have also explored the K S − K L asymmetries in the decays of B c → B n K L /K S withboth Cabibbo-allowed and doubly Cabibbo-suppressed transitions. The asymmetries dependstrongly on the contributions from the doubly Cabibbo-suppressed contributions. Clearly,the measurements of these asymmetries are good tests for the doubly Cabibbo-suppressedtransitions.In conclusion, we give a systematic consideration of the up-down asymmetries in thetwo-body charmed baryon decays of B c → B n M as well as the K S − K L asymmetries inthe decays of B c → B n K L /K S in the SU (3) F approach. Some of our predictions based on SU (3) F are different from those in the dynamical models, can be tested by the experimentsat BESIII and Belle. Appendix A: Irreducible Amplitudes
In this Appendix, we provide the irreducible amplitudes A B c → B n M from Eq. (7) basedon the flavor SU (3) F symmetry, while those of B B c → B n M can be obtained by substituting b i with a i in A B c → B n M . Note that in the limits of η = η and η ′ = η , one has that s φ = √ c φ ,resulting in the η ′ = η modes only contain ˜ a . In Tables VIII, IX and X, we show theCabibbo-allowed, singly Cabibbo-suppressed and doubly Cabibbo-suppressed amplitudes of A B c → B n M , respectively. Here, we have only considered the factorizable amplitudes from O (15), so that the terms associated with a , , , and b , , , are set to be zero.18 ABLE VIII. Cabibbo-allowed amplitudes for A B c → B n M Channel A Λ + c → Λ π + √ ( − a − a − a − a ) → p ¯ K − a + a → Σ π + √ − a + a + a ) → Σ + π √ a − a − a ) → Σ + η √ c φ ( − a − a + a − a ) + s φ ( − a − a + a + 3˜ a ) → Σ + η ′ c φ ( a + a − a − a ) + √ s φ ( − a − a + a − a ) → Ξ K + − a Ξ + c → Σ + ¯ K a − a → Ξ π + − a − a Ξ c → Λ ¯ K √ ( − a + a + a + a ) → Σ ¯ K √ − a − a + a ) → Σ + K − a → Ξ π √ − a + a ) → Ξ η √ c φ ( a − a − a + 6˜ a ) + s φ ( a − a − a − a ) → Ξ η ′ c φ ( − a + 2 a + a + 3˜ a ) + √ s φ ( a − a − a + 6˜ a ) → Ξ − π + a + a Appendix B: Up-down and Longitudinal Polarization Asymmetries
From Eq. (9), the up-down is defined by α = d Γ( ~P B n · ˆ p B n = +1) − d Γ( ~P B n · ˆ p B n = − d Γ( ~P B n · ˆ p B n = +1) + d Γ( ~P B n · ˆ p B n = − , (B1)which is equal to the longitudinal polarization asymmetry, i.e. P B n = α .19 ABLE IX. Singly Cabibbo-suppressed amplitudes for A B c → B n M Channel sin − θ c A Λ + c → Λ K + √ ( a − a + a + a ) → pπ √ a + a − a ) → pη √ c φ ( − a + a + a + a + 2˜ a ) + s φ ( − a + 2 a + 2 a + 3 a − a ) → pη ′ c φ (4 a − a − a − a + 6˜ a ) + √ s φ ( − a + a + a + a + 2˜ a ) → nπ + a + 2 a + a → Σ K + √ a − a ) → Σ + K S √ a − a ) → Σ + K L √ a − a )Ξ + c → Λ π + √ ( a + a − a − a ) → pK S √ − a + a ) → pK L √ a − a ) → Σ π + √ a − a + a ) → Σ + π √ − a + a + a ) → Σ + η √ c φ ( a + a + a − a + 2˜ a ) + s φ (2 a + 2 a + 4 a − a − a ) → Σ + η ′ c φ ( − a − a − a + 3 a + 6˜ a ) + √ s φ ( a + a + a − a + 2˜ a ) → Ξ K + a + 2 a + a Ξ c → Λ π √ ( − a − a + 2 a − a ) → Λ η √ c φ ( − a − a + a + 6˜ a ) + √ s φ ( − a − a + a − a ) → Λ η ′ √ c φ ( a + a − a + 3˜ a ) + √ s φ ( − a − a + a + 6˜ a ) → pK − − a → nK S √ − a + a + a ) → nK L √ a − a − a ) → Σ π a + a − a → Σ η c φ ( − a − a − a + a − a ) + √ s φ ( − a − a − a + a + ˜ a ) → Σ η ′ √ c φ ( a + a + a − a − ˜ a ) + s φ ( − a − a − a + a − a ) → Σ + π − a → Σ − π + a + a → Ξ K S √ − a + a + a ) → Ξ K L √ − a + a + a ) → Ξ − K + − a − a ABLE X. Doubly Cabibbo-suppressed amplitudes for A B c → B n M Channel sin − θ c A Λ + c → pK a − a → nK + − a − a Ξ + c → Λ K + √ ( − a + 2 a + 2 a + a ) → pπ −√ a → pη √ c φ (2 a − a − a − a ) + s φ (2 a − a − a + 3˜ a ) → pη ′ c φ ( − a + a + 2 a − a ) + √ s φ (2 a − a − a − a ) → nπ + − a → Σ K + √ − a − a ) → Σ + K − a + a Ξ c → Λ K √ ( − a + 2 a + 2 a − a ) → pπ − − a → nπ √ a → nη √ c φ ( − a + 2 a − a − a ) + s φ (3˜ a + 2 a − a − a ) → nη ′ c φ ( − a − a + a + 2 a ) + √ s φ ( − a + 2 a − a − a ) → Σ K √ a − a ) → Σ − K + − a − a CKNOWLEDGMENTS
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