Sum rules for CP asymmetries of charmed baryon decays in the SU(3)F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SU(3)_F$$\end{document} limit

Motivated by the recent LHCb observation of CP violation in charm, we study CP violation in the charmed baryon decays. A simple method to search for the CP violation relations in the flavor SU(3) limit, which is associated with a complete interchange of d and s quarks, is proposed. With this method, hundreds of CP violation sum rules in the doubly and singly charmed baryon decays can be found. As examples, the CP violation sum rules in two-body charmed baryon decays are presented. Some of the CP violation sum rules could help the experiment to find better observables. As byproducts, the branching fraction of Ξc+→pK-π+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi ^+_c\rightarrow pK^-\pi ^+$$\end{document} is predicted to be (1.7±0.5)%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1.7\pm 0.5)\%$$\end{document} in the U-spin limit and the fragmentation–fraction ratio is determined to be fΞb/fΛb=0.065±0.020\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{\Xi _b}/f_{\Lambda _b}=0.065\pm 0.020$$\end{document} using the LHCb data.


Introduction
Very recently, the LHCb Collaboration observed the C P violation in the charm sector [1], with the value of in which the dominated direct C P violation is defined by It is a milestone of particle physics, since C P violation has been well established in the kaon and B systems for many years [2], while the last piece of the puzzle, C P violation in the charm sector, has not been observed until now. To find C P violation in charm, many theoretical and experimental devoted efforts were made in the past decade. On the other a e-mail: dwang15@lzu.edu.cn hand, with the discovery of doubly charmed baryon [3][4][5] and the progress of singly charmed baryon measurements [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], plenty of theoretical interests focus on charmed baryon decays . However, only a few publications studied the C P asymmetries in charmed baryon decays [62,73,82]. The difference between C P asymmetries of + c → pK + K − and + c → pπ + π − modes has been measured by the LHCb Collaboration [21] and no signal of C P violation is found: In charmed and bottomed meson decays, some relations for C P asymmetries in (or beyond) the flavor SU (3) limit are found [83][84][85][86][87][88][89][90][91][92][93][94]. For example, the direct C P asymmetries in D 0 → K + K − and D 0 → π + π − decays have following relation in the U -spin limit [86,87]: The two C P asymmetries in A C P have opposite sign and hence are constructive in A C P . But the two C P asymmetries in A baryon C P , as pointed out in [73], do not have a relation like the ones in A C P . Prospects of measuring the C P asymmetries of charmed baryon decays on LHCb [95], as well as Belle II [96], are bright. It is significative to study the relations for C P asymmetries in the charmed baryon decays and then help to find some promising observables in experiments.
In Ref. [73], three C P violation sum rules associated with a complete interchange of d and s quarks are derived. In this work, we illustrate that the complete interchange of d and s quarks is a universal law to search for the C P violation sum rules of two charmed hadron decay channels in the flavor SU (3) limit. With the universal law, hundreds of C P violation sum rules can be found in the doubly and singly charmed baryon decays. The C P violation sum rules could be tested in future measurements or provide a guide to find better observables for experiments. Besides, the branching fraction Br ( + c → pK − π + ) and fragmentation-fraction ratio f b / f b are estimated in the U -spin limit.
The rest of this paper is organized as follows. In Sect. 2, the effective Hamiltonian of charm decay is decomposed into the SU (3) irreducible representations. In Sect. 3, we derive the C P violation sum rules for charmed meson and baryon decays and sum up a general law for C P violation sum rules in charm. In Sect. 4, we list some results of the C P violation sum rules in charmed baryon decays. Section 5 is a brief summary. The explicit SU (3) decomposition of the operators in charm decays is presented in Appendix A.

Effective Hamiltonian of charm decay
The effective Hamiltonian in charm quark weak decay in the Standard Model (SM) can be written as [97] where G F is the Fermi coupling constant, C i is the Wilson coefficient of operator O i . The tree operators are in which α, β are color indices, q 1,2 are d and s quarks. The QCD penguin operators are and the chromomagnetic-penguin operator is The magnetic-penguin contributions can be included into the Wilson coefficients for the penguin operators following the substitutions [98][99][100]: with the effective Wilson coefficient C eff 8g = C 8g + C 5 and l 2 being the averaged invariant mass squared of the virtual gluon emitted from the magnetic-penguin operator.
The charm quark decays are categorized into three types, Cabibbo-favored (CF), singly Cabibbo-suppressed (SCS), and doubly Cabibbo-suppressed (DCS) decays, with the flavor structures of respectively. In the SU (3) picture, the operators in charm decays embed into the four-quark Hamiltonian, Equation (5) implies that the tensor components of H k i j can be obtained from the map (ūq 1 )(q 2 c) → V * cq 2 V uq 1 in currentcurrent operators and (qq)(ūc) → −V * cb V ub in penguin operators and the others are zero. The non-zero components of the tensor H k i j corresponding to tree operators in Eq. (5) are and the non-zero components of the tensor H k i j corresponding to penguin operators in Eq. (5) are The operator O i j k is a representation of the SU (3) group, which is decomposed as four irreducible representations: All components of the irreducible representations are listed in Appendix A. The non-zero components of H k i j corresponding to tree operators in the SU (3) decomposition are The non-zero components of H k i j corresponding to penguin operators in the SU (3) decomposition are Here we use the superscript P to differentiate penguin contributions from tree contributions. Equations (15) and (16) were derived in [86] for the first time. But the non-zero components H (15) 1 11 and H (3) 1 in the tree operator contributions are missing in [86].
Recent studies for charmed baryon decays in the SU (3) irreducible representation amplitude (IRA) approach [29,40,42,54,60,61,64,68,70,71,74,81] do not analyze C P asymmetries because they ignore the two 3-dimensional irreducible representations and make the approximation of V * cs V us −V * cd V ud in the 15-and 6-dimensional irreducible representations, leading to the vanishing of the contributions proportional to λ b = V * cb V ub . If the contributions proportional to λ b are included, the SU (3) irreducible representation amplitude approach then can be used to investigate C P asymmetries in the charmed baryon decays.

C P violation sum rules in charmed meson/baryon decays
In this section, we discuss the method to search for the relations for C P asymmetries of charm decays in the flavor SU (3) limit. We first analyze the C P violation sum rules in charmed meson and baryon decays, respectively, and then sum up a general law for the C P violation sum rules in charm decays.

C P violation sum rules in charmed meson decays
Two C P asymmetry sum rules for D → P P decays in the flavor SU (3) limit have been given in [86,87]: To see why the two sum rules are correct, we express the decay amplitudes of D 0 → K + K − , D 0 → π + π − , D + → K + K 0 and D + s → π + K 0 modes in the SU (3) irreducible representation amplitude (IRA) approach. The charmed meson anti-triplet is The pseudoscalar meson nonet is To obtain the SU (3) irreducible representation amplitude of D → P P decay, one takes various representations in Eqs. (15) and (16) and contracts all indices in D i and light meson P i j with various combinations: Notice that only the first components of 3 and 3 irreducible representations are non-zero. Some amplitudes, for example, a 3 , Pa 3 and Pa 3 are always appear simultaneously since they correspond to the same contraction. Noting that the amplitude of D → P P decay will be reduced to With Eq. (24), the decay amplitudes of the [86] except for the last terms in Eqs. (25) and (26) because of the non-vanishing H (15) 1 11 component in Eq. (15). From above formulas, the C P violation sum rules listed in Eqs. (17) and (18) are derived if the approximation of is used. Besides, the decay amplitude of D 0 → K 0 K 0 is expressed as The direct C P asymmetry in D 0 → K 0 K 0 decay is zero in the flavor SU (3) limit: For the C P violation relations (17) and (18), the decay amplitudes of two channels are connected by the interchange of λ d ↔ λ s , and their initial and final states are connected by the interchange of d ↔ s: For D 0 → K 0 K 0 decay, its corresponding mode in the interchange of d ↔ s is itself. So all C P violation relations in Eqs. (17), (18) and (31) are associated with the U -spin transformation.
On the other hand, Eqs. (17), (18) and (31) include all the SCS modes without π 0 , η ( ) in the final states in D → P P decays. Mesons π 0 and η ( ) do not have definite U -spin quantum numbers. Under the interchange of d ↔ s, there are no mesons corresponding to π 0 and η ( ) . For example, π 0 has the quark constituent of No meson has the quark constituent of (ss −ūu)/ √ 2. So those decay channels involving π 0 , η ( ) do not have their corresponding modes in the interchange of d ↔ s, and then have no simple C P violation sum rules with two channels.
In fact, not only the D → P P decays, there are also some C P violation sum rules in the D → PV decays [87] The detailed derivation of these sum rules is similar to D → P P and can be found in Ref. [86]. Again, all the C P violation sum rules in D → PV decays are associated with a complete interchange of d and s quarks, and all the singly Cabibbosuppressed D → PV modes with all final states having definite U -spin quantum numbers are included in Eqs. (33)-(37).

C P violation sum rules in charmed baryon decays
In this subsection, we take charmed baryon decays into one pseudoscalar meson and one decuplet baryon as examples to show the complete interchange of d ↔ s is still valid for the C P violation sum rules in charmed baryon decays. The charmed anti-triplet baryon is expressed as The light baryon decuplet is given as Similar to D → P P decay, if we define The first four terms are the same with the formula given in [101]. The fifth term is the decay amplitude associated with singlet η 1 , and the six term is the amplitude proportional to λ b . With Eq. (43), the SU (3) irreducible representation amplitudes of B c3 → B 10 M decays are obtained. The results are listed in Table 1. From Table 1, seven C P violation sum rules in the SU (3) F limit for the charmed baryon decays into one pseudoscalar meson and one decuplet baryon are found: Similar to the charmed meson decays, all the C P violation sum rules are associated with a complete interchange of d and s quarks in the initial and final states. For charmed anti-triplet baryons, For light decuplet baryons, that the contributions proportional to λ b are neglected in this literature. To get a complete expression of decay amplitude and then analyze the C P asymmetries, the neglected terms must be found back, just like we have done in this work. One can check that the C P violation sum rules associated with the complete interchange of d and s quarks works in various types of decay.

A universal law for C P violation sum rules in the charm sector
From the above discussions, one can find the C P violation sum rules in the SU (3) F limit are always associated with a complete interchange of d and s quarks. In this subsection, we illustrate that it is a universal law in the charm sector.
According to Eq. (15), Eq. (55) equals One can find that Eq. (56) is equivalent to the interchange of λ d ↔ λ s . The contributions proportional to λ b in the SCS decays are induced by the following operators: Form Appendix A, it is found that these operators are invariable under the interchange of d ↔ s, as are the corresponding CKM matrix elements. Thirdly, if two decay channels have the relations that their decay amplitudes proportional to λ d /λ s are connected by the interchange of λ d ↔ λ s and the decay amplitudes proportional to λ b are the same, the sum of their direct C P asymmetries is zero in the SU (3) F limit under the approximation in Eq. (29). For one decay mode with amplitude of its C P asymmetry in the order of O(λ b ) is derived as For one decay mode with amplitude of which is connected to Eq. (58) by λ d ↔ λ s , its C P asymmetry in the order of O(λ b ) is derived as It is apparent that Based on the above analysis, a useful method to search for the C P violation sum rules with two charmed hadron decay channels is proposed: • For one type of charmed hadron decay, write down all the SCS decay modes in which the associated hadrons have definite U -spin quantum numbers; • For each decay mode, find the corresponding decay mode in the complete interchange of d ↔ s; • If there are two decay modes connected by the interchange of d ↔ s, the sum of their direct C P asymmetries is zero in the SU (3) F limit; • If the corresponding decay mode is itself, the direct C P asymmetry in this mode is zero in the SU (3) F limit.

Results and discussions
With the method proposed in Sect. 3, one can find many sum rules for C P asymmetries in charm meson/baryon decays. There are hundreds of sum rules for C P asymmetries in the singly and doubly charmed baryon decays. We are not going to list all the C P violation sum rules, but only present some of them as examples.
Under the complete interchange of d ↔ s, the light octet baryons are interchanged as The sum rules for C P asymmetries in charmed baryon decays into one pseudoscalar meson and one octet baryon are Under the complete interchange of d ↔ s, the doubly charmed baryons are interchanged as ++ cc ↔ ++ cc , The sum rules for C P asymmetries in doubly charmed baryon decays into one pseudoscalar meson and one charmed triplet baryon are Under the complete interchange of d ↔ s, the charmed sextet baryons are interchanged as The sum rules for C P asymmetries in doubly charmed baryon decays into one pseudoscalar meson and one charmed sextet baryon are With the interchange rules mentioned above, the C P violation sum rules in doubly charmed baryon decays into one charmed meson and one octet baryon are The C P violation sum rules in doubly charmed baryon decays into one charmed meson and one decuplet baryon are For three-body decays, we only list the C P violation sum rules in charmed baryon decays into one octet baryon and two pseudoscalar mesons as examples: The first three sum rules are the same as in [73]. In all above sum rules, the pseudoscalar mesons can be replaced by vector mesons by the following correspondence: The C P violation sum rules are derived in the U -spin limit. Considering the U -spin breaking, the C P violation sum rules are no longer valid, as pointed out in [73]. Since the U -spin breaking is sizable in the charm sector, the C P violation sum rules might not be reliable. But they indicate that the C P asymmetries in some decay modes have opposite sign and then can be used to find some promising observables in experiments. In charmed meson decays, Eq. (4) makes the two C P asymmetries in observable A C P ≡ A C P (D 0 → K + K − ) − A C P (D 0 → π + π − ) constructive. Similarly, one can use the C P violation sum rules in charmed baryon decays to construct some observables in which two C P asymmetries are constructive. Some observables are selected for experimental discretion: A baryon,4 C P If the contributions proportional to λ b are neglected, the decay amplitudes of the two channels connected by the interchange of d ↔ s are the same (except for a minus sign) in the SU (3) F limit [see Eqs. (58) and (60)]. One can use this relation to predict the branching fractions. As an example, we estimate the branching fraction of + c → pK − π + . The integration over the phase space of the three-body decay B c → BM 1 M 2 relies on the equation of [2] where m 2 12 = ( p M 1 + p M 2 ) 2 and m 2 23 = ( p M 2 + p B ) 2 . With the experimental data given in [2], and the relation the branching fraction of + c → pK − π + decay is predicted to be One can find the branching fraction Br ( + c → pK − π + ) is larger than Br ( + c → + π − K + ) because of the larger phase space and the longer lifetime of + c . But it is still smaller than the predictions given in [66,70]. In the above estimation, only the decay amplitude is obtained by the U -spin symmetry. The phase space is calculated without approximation. It is plausible since the global fit in Refs. [60,61,64,68,70,74,81] give the reasonable estimations for branching fractions of charmed baryon decays. The uncertainty in Eq. (102) is dominated by the branching fraction of + c → + π − K + decay and does not include the U -spin breaking effects. It is not available to estimate the U -spin breaking effects at the current stage since the understanding of the dynamics of charmed baryon decay is still a challenge. Some discussions of the uncertainty induced by Uspin breaking can be found in [66].
With the method introduced in [66], and the LHCb data [102] the fragmentation-fraction ratio f b / f b is determined to be Recent measurement confirmed this result [103]. Our result is consistent with the one obtained via 0 b → J/ψ 0 [104], using the LHCb data [107], f b / f b = 0.08 ± 0.03. The detailed comparison for different methods of estimating f b / f b can be found in [66].

Summary
In summary, we find that if two singly Cabibbo-suppressed decay modes of charmed hadrons are connected by a complete interchange of d and s quarks, the sum of their direct C P asymmetries is zero in the flavor SU (3) limit. According to this conclusion, many C P violation sum rules can be found in the doubly and singly charmed baryon decays. Some of them could help to find better observables in experiments. As byproducts, the branching fraction Br ( + c → pK − π + ) is predicted to be (1.7 ± 0.5)% in the U -spin limit, and the fragmentation-fraction ratio is determined as f b / f b = 0.065 ± 0.020.  (A4) The above results are consistent with Ref. [86].