Sum rules for $CP$ asymmetries of charmed baryon decays in the $SU(3)_F$ 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 $\Xi^+_c\to pK^-\pi^+$ is predicted to be $(1.7\pm 0.5)\%$ in the $U$-spin limit and the fragmentation-fraction ratio is determined as $f_{\Xi_b}/f_{\Lambda_b}=0.065\pm 0.020$ using the LHCb data.


I. INTRODUCTION
Very recently, the LHCb Collaboration observed the CP violation in the charm sector [1], with the value of It is a milestone of particle physics, since CP violation has been well established in the kaon and B systems for many years [2], while the last piece of the puzzle, CP violation in the charm sector, has not been observed until now. To find CP violation in charm, many theoretical and experimental efforts were devoted in the past decade. On the other 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 literatures studied the CP asymmetries in charmed baryon decays [62,73,82]. The difference between CP asymmetries of Λ + c → pK + K − and Λ + c → pπ + π − modes has been measured by LHCb collaboration [21] and no signal of CP violation is found: In charmed and bottomed meson decays, some relations for CP 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 CP asymmetries in D 0 → K + K − and D 0 → π + π − decays have following relation in the U -spin limit [86,87]: The two CP asymmetries in ∆A CP have opposite sign and hence constructive in ∆A CP . But the two CP asymmetries in ∆A baryon CP , as pointed out in [73], do not have such relation like the ones in ∆A CP . Prospects of measuring the CP asymmetries of charmed baryon decays on LHCb [95], as well as Belle II [96], are bright. It is significative to study the relations for CP asymmetries in the charmed baryon decays and then help to find some promising observables in experiments.
In Ref. [73], three CP 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 CP violation sum rules of two charmed hadron decay channels in the flavor SU (3) limit. With the universal law, hundreds of CP violation sum rules can be found in the doubly and singly charmed baryon decays. The CP violation sum rules could be tested in the future measurements or provide a guide to find better observables for experiments. Besides, the branching fraction Br(Ξ + c → pK − π + ) and fragmentation-fraction ratio f Ξ b /f Λ b are estimated in the U -spin limit.
The rest of this paper is organized as follows. In Sec. II, the effective Hamiltonian of charm decay is decomposed into the SU (3) irreducible representations. In Sec. III, we derive the CP violation sum rules for charmed meson and baryon decays and sum up a general law for CP violation sum rules in charm. In Sec. IV, we list some results of the CP violation sum rules in charmed baryon decays. Sec. V is a brief summary. And the explicit SU (3) decomposition of the operators in charm decays is presented in Appendix A.

II. EFFECTIVE HAMILTONIAN OF CHARM DECAY
The effective Hamiltonian in charm quark weak decay in the SM can be written as [97] where G F is the Fermi coupling constant, C i is the Wilson coefficients 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 [98-100] 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 Cabibbosuppressed(SCS), and doubly Cabibbo-suppressed(DCS) decays, with the flavor structures of c → sdu, c →dd/ssu, c → dsu, (9) respectively. In the SU (3) picture, the operators in charm decays embed into the four-quark Hamiltonian, Eq. (4) implies that the tensor components of H k ij can be obtained from the map (ūq 1 )(q 2 c) → V * cq 2 V uq 1 in current-current 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 ij corresponding to tree operators in Eq. (4) are and the non-zero components of the tensor H k ij corresponding to penguin operators in Eq. (4) are The operator O ij k is a representation of SU (3) group, which is decomposed as four irreducible representations: 3 ⊗ 3 ⊗ 3 = 3 ⊕ 3 ′ ⊕ 6 ⊕ 15. The explicit decomposition is [86] O ij k = δ j All components of the irreducible representations are listed in Appendix A. The non-zero components of H k ij corresponding to tree operators in the SU (3) decomposition are The non-zero components of H k ij corresponding to penguin operators in the SU (3) decomposition are Here we use superscript P to differentiate penguin contributions from thee contributions. Eqs. (14) and (15) were derived in [86] for the first time. But the non-zero components H(15) 1 11 and H(3) 1 in 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 CP 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 CP asymmetries in the charmed baryon decays.

III. CP VIOLATION SUM RULES IN CHARMED MESON/BARYON DECAYS
In this section, we discuss the method to search for the relations for CP asymmetries of charm decays in the flavor SU (3) limit. We first analyze the CP violation sum rules in charmed meson and baryon decays respectively, and then sum up a general law for the the CP violation sum rules in charm decays.

A. CP violation sum rules in charmed meson decays
Two CP 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 correct, we express the decay amplitudes of The pseudoscalar meson nonet is To obtain the SU (3) irreducible representation amplitude of D → P P decay, one takes various representations in Eqs. (14) and (15) 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 , P a 3 and P a ′ 3 are always appear simultaneously since they correspond to the same contraction. Noting that H P (3 the amplitude of D → P P decay will be reduced to be With Eq. (23), the decay amplitudes of (27) are consistent with [86] except for the last terms in Eq. (24) and Eq. (25) because of the non-vanishing H(15) 1 11 component in Eq. (14). From above formulas, the CP violation sum rules listed in Eqs. (16) and (17) are derived if the approximation of is used. Besides, the decay amplitude of D 0 → K 0 K 0 is expressed as The direct CP asymmetry in D 0 → K 0 K 0 decay is zero in the flavor SU (3) limit: For the CP violation relations (16) and (17), the decay amplitudes of two channels are connected by the interchange of λ d ↔ λ s , and their initial and final 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 CP violation relations in Eqs. (16), (17) and (30) are associated with U -spin transformation.
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 (dd −ūu)/ √ 2. Under the interchange of d ↔ s, (dd −ūu)/ √ 2 turns into (ss −ūu)/ √ 2. 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 CP violation sum rules with two channels.
In fact, not only the D → P P decays, there are also some CP violation sum rules in the The detailed derivation of these sum rules is similar to D → P P and can be found in Ref. [86].
The light baryon decuplet is given as The SU (3) irreducible representation amplitude of B c3 → B 10 M decay can be written as Similar to D → P P decay, if we define P e = e 6 + P e 6 + 3P e 7 , the amplitude of B c3 → B 10 M decay will be reduced to be 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. (42), the SU (3) irreducible representation amplitudes of B c3 → B 10 M decays are obtained.
The results are listed in Table. I.
From Table. I, seven CP 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 CP 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, And for light decuplet baryons,  [29,40,42,101]. But notice that the contributions proportional to λ b are neglected in these literatures. To get a complete expression of decay amplitude and then analyze the CP asymmetries, the neglected terms must be found back, just like we have done in this work. One can check that the CP violation sum rules associated with the complete interchange of d and s quarks works in various types of decay.

C. A universal law for CP violation sum rules in charm sector
From above discussions, one can find whether for charmed meson or baryon decays, the CP 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 charm sector.
According to Eq. (14), Eq. (54) equals to One can find Eq. (55) is equivalent to the interchange of λ d ↔ λ s . The contributions proportional to λ b in the SCS decays are induced by following operators: Form Appendix A, it is found these operators are invariable under the interchange of d ↔ s, so as 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 CP asymmetries is zero in the SU (3) F limit under the approximation in Eq. (28). For one decay mode with amplitude of its CP asymmetry in the order of O(λ b ) is derived as For one decay mode with amplitude of which is connected to Eq. (57) by λ d ↔ λ s , its CP asymmetry in the order of O(λ b ) is derived as It is apparent that Based on above analysis, a useful method to search for the CP 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 CP asymmetries is zero in the SU (3) F limit; • If the corresponding decay mode is itself, the direct CP asymmetry in this mode is zero in the SU (3) F limit.

IV. RESULTS AND DISCUSSION
With the method proposed in Sec. III, one can find many sum rules for CP asymmetries in charm meson/baryon decays. There are hundreds of sum rules for CP asymmetries in the singly and doubly charmed baryon decays. We're not going to list all the CP violations 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 CP 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 The sum rules for CP 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 CP asymmetries in doubly charmed baryon decays into one pseudoscalar meson and one charmed sextet baryon are With those interchange rules mentioned above, the CP violation sum rules in doubly charmed baryon decays into one charmed meson and one octet baryon are The CP 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 CP 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 with [73]. In all above sum rules, the pseudoscalar mesons can be replaced by vector mesons by following correspondence: The CP violation sum rules are derived in the U -spin limit. Considering the U -spin breaking, the CP violation sum rules are no longer valid, as pointed out in [73]. Since the U -spin breaking is sizeable in charm sector, the CP violation sum rules might not be reliable. But they indicate that the CP 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. (3) makes the two Similarly, one can use the CP violation sum rules in charmed baryon decays to construct some obserbables in which two CP asymmetries are constructive. Some observables are selected for experimental discretion: ∆A baryon,2 ∆A baryon,4 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 Eq. (57) and Eq. (59)). One can use this relation to predict the branching fractions. As an example, we estimate the branching fraction of Ξ +

V. 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 CP asymmetries is zero in the flavor SU (3) limit. According to this conclusion, many CP 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(Ξ + c → pK − π + ) is predicted to be (1.7 ± 0.5)% in the U -spin limit, and the fragmentation-fraction ratio is determined Above results are consistent with Ref. [86].