1 Introduction

Very recently, the LHCb Collaboration observed the CP violation in the charm sector [1], with the value of

$$\begin{aligned} \Delta A_{CP}&\equiv A_{CP}(D^0\rightarrow K^+K^-) - A_{CP}(D^0\rightarrow \pi ^+\pi ^-) \nonumber \\&= (-1.54\pm 0.29)\times 10^{-3}, \end{aligned}$$
(1)

in which the dominated direct CP violation is defined by

$$\begin{aligned} A_{CP}^\mathrm{dir}(i\rightarrow f) = \frac{|\mathcal {A}(i\rightarrow f)|^2-|\mathcal {A}({\overline{i}}\rightarrow {\overline{f}})|^2}{|\mathcal {A}(i\rightarrow f)|^2+|\mathcal {A}({\overline{i}}\rightarrow {\overline{f}})|^2} . \end{aligned}$$
(2)

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 devoted efforts were made 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 [23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81]. However, only a few publications studied the CP asymmetries in charmed baryon decays [62, 73, 82]. The difference between CP asymmetries of \(\Lambda ^+_c\rightarrow pK^+K^-\) and \(\Lambda ^+_c\rightarrow p\pi ^+\pi ^-\) modes has been measured by the LHCb Collaboration [21] and no signal of CP violation is found:

$$\begin{aligned} \Delta A_{CP}^\mathrm{baryon}&\equiv A_{CP}(\Lambda ^+_c\rightarrow pK^+K^-) \nonumber \\&\quad -A_{CP}(\Lambda ^+_c\rightarrow p\pi ^+\pi ^-) \nonumber \\&= (0.30\pm 0.91\pm 0.61)\%. \end{aligned}$$
(3)

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 \(D^0\rightarrow K^+K^-\) and \(D^0\rightarrow \pi ^+\pi ^-\) decays have following relation in the U-spin limit [86, 87]:

$$\begin{aligned} A^\mathrm{dir}_{CP}(D^0\rightarrow K^+K^-) + A^\mathrm{dir}_{CP}(D^0\rightarrow \pi ^+\pi ^-) = 0. \end{aligned}$$
(4)

The two CP asymmetries in \(\Delta A_{CP}\) have opposite sign and hence are constructive in \(\Delta A_{CP}\). But the two CP asymmetries in \(\Delta A_{CP}^\mathrm{baryon}\), as pointed out in [73], do not have a relation like the ones in \(\Delta A_{CP}\). 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 future measurements or provide a guide to find better observables for experiments. Besides, the branching fraction \(\mathcal {B}r(\Xi ^+_c\rightarrow pK^-\pi ^+)\) and fragmentation–fraction ratio \(f_{\Xi _b}/f_{\Lambda _b}\) 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 CP violation sum rules for charmed meson and baryon decays and sum up a general law for CP violation sum rules in charm. In Sect. 4, we list some results of the CP 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.

2 Effective Hamiltonian of charm decay

The effective Hamiltonian in charm quark weak decay in the Standard Model (SM) can be written as [97]

$$\begin{aligned} {\mathcal {H}}_\mathrm{eff}= & {} {\frac{G_F}{\sqrt{2}} } \left[ \sum _{q=d,s}V_{cq_1}^*V_{uq_2}\left( \sum _{i=1}^2C_i(\mu )O_i(\mu )\right) \right. \nonumber \\&\left. -V_{cb}^*V_{ub}\left( \sum _{i=3}^6C_i(\mu )O_i(\mu )+C_{8g}(\mu )O_{8g}(\mu )\right) \right] ,\nonumber \\ \end{aligned}$$
(5)

where \(G_F\) is the Fermi coupling constant, \(C_{i}\) is the Wilson coefficient of operator \(O_i\). The tree operators are

$$\begin{aligned}&O_1=(\bar{u}_{\alpha }q_{2\beta })_{V-A} (\bar{q}_{1\beta }c_{\alpha })_{V-A},\quad \nonumber \\&O_2=(\bar{u}_{\alpha }q_{2\alpha })_{V-A} (\bar{q}_{1\beta }c_{\beta })_{V-A}, \end{aligned}$$
(6)

in which \(\alpha ,\beta \) are color indices, \(q_{1,2}\) are d and s quarks. The QCD penguin operators are

$$\begin{aligned} O_3&=\sum _{q'=u,d,s}({\bar{u}}_\alpha c_\alpha )_{V-A}({\bar{q}}'_\beta q'_\beta )_{V-A},\nonumber \\ O_4&=\sum _{q'=u,d,s}({\bar{u}}_\alpha c_\beta )_{V-A}({\bar{q}}'_\beta q'_\alpha )_{V-A}, \nonumber \\ O_5&=\sum _{q'=u,d,s}({\bar{u}}_\alpha c_\alpha )_{V-A}({\bar{q}}'_\beta q'_\beta )_{V+A},\nonumber \\ O_6&=\sum _{q'=u,d,s}({\bar{u}}_\alpha c_\beta )_{V-A}({\bar{q}}'_\beta q'_\alpha )_{V+A}, \end{aligned}$$
(7)

and the chromomagnetic-penguin operator is

$$\begin{aligned} O_{8g}=\frac{g}{8\pi ^2}m_c{\bar{u}}\sigma _{\mu \nu }(1+\gamma _5)T^aG^{a\mu \nu }c. \end{aligned}$$
(8)

The magnetic-penguin contributions can be included into the Wilson coefficients for the penguin operators following the substitutions [98,99,100]:

$$\begin{aligned} C_{3,5}(\mu )\rightarrow&C_{3,5}(\mu ) + \frac{\alpha _s(\mu )}{8\pi N_c} \frac{2m_c^2}{\langle l^2\rangle }C_{8g}^\mathrm{eff}(\mu ),\nonumber \\ C_{4,6}(\mu )\rightarrow&C_{4,6}(\mu ) - \frac{\alpha _s(\mu )}{8\pi } \frac{2m_c^2}{\langle l^2\rangle }C_{8g}^\mathrm{eff}(\mu ), \end{aligned}$$
(9)

with the effective Wilson coefficient \(C_{8g}^\mathrm{eff}=C_{8g}+C_5\) and \(\langle l^2\rangle \) 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

$$\begin{aligned} c\rightarrow s{\bar{d}} u, \quad c\rightarrow d{\bar{d}}/ s {\bar{s}}u,\quad c\rightarrow d{\bar{s}} u, \end{aligned}$$
(10)

respectively. In the SU(3) picture, the operators in charm decays embed into the four-quark Hamiltonian,

$$\begin{aligned} \mathcal {H}_\mathrm{eff}= \sum _{i,j,k=1}^{3}H^k_{ij}O^{ij}_k=\sum _{i,j,k=1}^{3}H^k_{ij}(\bar{q}^iq_k)(\bar{q}^jc). \end{aligned}$$
(11)

Equation (5) implies that the tensor components of \(H_{ij}^k\) can be obtained from the map \(({\bar{u}}q_1)({\bar{q}}_2c)\rightarrow V^*_{cq_2}V_{uq_1}\) in current–current operators and \(({\bar{q}}q)(\bar{u}c)\rightarrow -V^*_{cb}V_{ub}\) in penguin operators and the others are zero. The non-zero components of the tensor \(H_{ij}^k\) corresponding to tree operators in Eq. (5) are

$$\begin{aligned} H_{13}^2&= V_{cs}^*V_{ud}, \quad H^{2}_{12}=V_{cd}^*V_{ud},\nonumber \\ H^{3}_{13}&= V_{cs}^*V_{us}, \quad H^{3}_{12}=V_{cd}^*V_{us}, \end{aligned}$$
(12)

and the non-zero components of the tensor \(H_{ij}^k\) corresponding to penguin operators in Eq. (5) are

$$\begin{aligned}&H_{11}^1 = -V_{cb}^*V_{ub}, \quad H_{21}^2=-V_{cb}^*V_{ub}, \quad H_{31}^3=-V_{cb}^*V_{ub}. \end{aligned}$$
(13)

The operator \(O^{ij}_k\) is a representation of the SU(3) group, which is decomposed as four irreducible representations: \({\overline{3}} \otimes 3 \otimes {\overline{3}} = {\overline{3}}\oplus \overline{3}^\prime \oplus 6 \oplus \overline{15}\). The explicit decomposition is [86]

$$\begin{aligned} O_k^{ij}=&\, \delta ^j_k\Big (\frac{3}{8}O({\overline{3}})^i-\frac{1}{8}O({\overline{3}}^{\prime })^i\Big )+ \delta ^i_k\Big (\frac{3}{8}O({\overline{3}}^{\prime })^j-\frac{1}{8}O({\overline{3}})^j\Big )\nonumber \\&+\,\epsilon ^{ijl}O( 6)_{lk}+O(\overline{15})_k^{ij}. \end{aligned}$$
(14)

All components of the irreducible representations are listed in Appendix A. The non-zero components of \(H_{ij}^k\) corresponding to tree operators in the SU(3) decomposition are

$$\begin{aligned} H( 6)^{22}&=-\frac{1}{2}V_{cs}^*V_{ud},\quad H( 6)^{23}=\frac{1}{4}(V_{cd}^*V_{ud}-V_{cs}^*V_{us}), \nonumber \\ H(6)^{33}&= \frac{1}{2}V_{cd}^*V_{us},\nonumber \\ H(\overline{15})^{1}_{11}&=-\frac{1}{4}(V_{cd}^*V_{ud}+V_{cs}^*V_{us}) = \frac{1}{4}V_{cb}^*V_{ub}, \nonumber \\ H(\overline{15})^{2}_{13}&= \frac{1}{2}V_{cs}^*V_{ud}, \nonumber \\ H(\overline{15})^{3}_{12}&=\frac{1}{2}V_{cd}^*V_{us},\nonumber \\ H(\overline{15})^{2}_{12}&= \frac{3}{8}V_{cd}^*V_{ud}-\frac{1}{8}V_{cs}^*V_{us},\nonumber \\ H(\overline{15})^{3}_{13}&=\frac{3}{8}V_{cs}^*V_{us}-\frac{1}{8}V_{cd}^*V_{ud},\nonumber \\ H({\overline{3}})_1&=V_{cd}^*V_{ud}+V_{cs}^*V_{us} = -V_{cb}^*V_{ub}. \end{aligned}$$
(15)

The non-zero components of \(H_{ij}^k\) corresponding to penguin operators in the SU(3) decomposition are

$$\begin{aligned} H^P({\overline{3}})_1=-V_{cb}^*V_{ub}, \quad H^P(\overline{3}^\prime )_1=-3V_{cb}^*V_{ub}. \end{aligned}$$
(16)

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 \(H(\overline{15})^{1}_{11}\) and \(H(\overline{3})_1\) 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 CP asymmetries because they ignore the two 3-dimensional irreducible representations and make the approximation of \(V^*_{cs}V_{us}\simeq -V^*_{cd}V_{ud}\) in the 15- and 6-dimensional irreducible representations, leading to the vanishing of the contributions proportional to \(\lambda _b=V_{cb}^*V_{ub}\). If the contributions proportional to \(\lambda _b\) are included, the SU(3) irreducible representation amplitude approach then can be used to investigate CP asymmetries in the charmed baryon decays.

3 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 CP violation sum rules in charm decays.

3.1 CP violation sum rules in charmed meson decays

Two CP asymmetry sum rules for \(D\rightarrow PP\) decays in the flavor SU(3) limit have been given in [86, 87]:

$$\begin{aligned} A^\mathrm{dir}_{CP}(D^0\rightarrow K^+K^-) + A^\mathrm{dir}_{CP}(D^0\rightarrow \pi ^+\pi ^-)&= 0, \end{aligned}$$
(17)
$$\begin{aligned} A^\mathrm{dir}_{C P}(D^+\rightarrow K^+{\overline{K}}^0) + A^\mathrm{dir}_{CP}(D^+_s\rightarrow \pi ^+K^0)&=0. \end{aligned}$$
(18)

To see why the two sum rules are correct, we express the decay amplitudes of \(D^0\rightarrow K^+K^-\), \(D^0\rightarrow \pi ^+\pi ^-\), \(D^+\rightarrow K^+{\overline{K}}^0\) and \(D^+_s\rightarrow \pi ^+K^0\) modes in the SU(3) irreducible representation amplitude (IRA) approach. The charmed meson anti-triplet is

$$\begin{aligned} D^{i}= (D^0, D^+, D_s^+). \end{aligned}$$
(19)

The pseudoscalar meson nonet is

$$\begin{aligned} P^i_j= & {} \left( \begin{array}{ccc} \frac{1}{\sqrt{2}} \pi ^0+ \frac{1}{\sqrt{6}} \eta _8 &{} \pi ^+ &{} K^+ \\ \pi ^- &{} - \frac{1}{\sqrt{2}} \pi ^0+ \frac{1}{\sqrt{6}} \eta _8 &{} K^0 \\ K^- &{} {\overline{K}}^0 &{} -\sqrt{2/3}\eta _8 \\ \end{array}\right) \nonumber \\&+\, \frac{1}{\sqrt{3}} \left( \begin{array}{ccc} \eta _1 &{} 0 &{} 0 \\ 0 &{} \eta _1 &{} 0 \\ 0 &{} 0 &{} \eta _1 \\ \end{array}\right) . \end{aligned}$$
(20)

To obtain the SU(3) irreducible representation amplitude of \(D\rightarrow PP\) 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:

$$\begin{aligned} \mathcal{A}^\mathrm{tree}_{D\rightarrow PP} =&\, a_{15}D^i H(\overline{15})_{ij}^kP^j_lP^l_k + b_{15}D^i H(\overline{15})_{ij}^kP^j_kP^l_l \nonumber \\&+\, c_{15}D^i H(\overline{15})_{jl}^kP^j_iP^l_k \nonumber \\&+ a_6D^i H( 6)_{ij}^kP^j_lP^l_k + b_6D^i H(6)_{ij}^kP^j_kP^l_l \nonumber \\&+\, c_6D^i H(6)_{jl}^kP^j_iP^l_k\nonumber \\&+\,a_3 D^i H({\overline{3}})_i P_k^jP_j^k +b_3 D^iH({\overline{3}})_i P_k^kP_j^j\nonumber \\&+\,c_3 D^i H({\overline{3}})_k P_i^kP_j^j\nonumber \\&+\,d_3 D^i H({\overline{3}})_k P_i^jP_j^k. \end{aligned}$$
(21)
$$\begin{aligned} \mathcal{A}^\mathrm{penguin}_{D\rightarrow PP} =&\, Pa_3 D^i H^P({\overline{3}})_i P_k^jP_j^k +Pb_3 D^i H^P({\overline{3}})_i P_k^kP_j^j\nonumber \\&+\,Pc_3 D^i H^P({\overline{3}})_k P_i^kP_j^j\nonumber \\&+\,Pd_3 D^i H^P({\overline{3}})_k P_i^jP_j^k+Pa^\prime _{3} D^i H^P({\overline{3}}^\prime )_i P_k^jP_j^k \nonumber \\&+\,Pb^\prime _{3} D^i H^P({\overline{3}}^\prime )_i P_k^kP_j^j\nonumber \\&+\,Pc^\prime _{3} D^i H^P({\overline{3}}^\prime )_k P_i^kP_j^j+Pd^\prime _{3} D^i H^P({\overline{3}}^\prime )_k P_i^jP_j^k. \end{aligned}$$
(22)

Notice that only the first components of \({\overline{3}}\) and \({\overline{3}}^{\prime }\) irreducible representations are non-zero. Some amplitudes, for example, \(a_3\), \(Pa_3\) and \(Pa^\prime _{3}\) are always appear simultaneously since they correspond to the same contraction. Noting that \(H^P({\overline{3}}^\prime )_1 = 3 H(\overline{3})_1= 3 H^P({\overline{3}})_1 =-3V_{cb}^*V_{ub}\), if we define

$$\begin{aligned} Pa=a_3 + Pa_3+ 3Pa^\prime _{3},\quad Pb=b_3 + Pb_3+ 3Pb^\prime _{3},\quad \ldots , \end{aligned}$$
(23)

the amplitude of \(D\rightarrow PP\) decay will be reduced to

$$\begin{aligned} \mathcal{A}^\mathrm{tree+penguin}_{D\rightarrow PP} =&\, a_{15}D^i H(\overline{15})_{ij}^kP^j_lP^l_k + b_{15}D^i H(\overline{15})_{ij}^kP^j_kP^l_l \nonumber \\&+\, c_{15}D^i H(\overline{15})_{jl}^kP^j_iP^l_k \nonumber \\&+ a_6D^i H( 6)_{ij}^kP^j_lP^l_k + b_6D^i H(6)_{ij}^kP^j_kP^l_l \nonumber \\&+\,c_6D^i H(6)_{jl}^kP^j_iP^l_k\nonumber \\&+\,Pa D^i H({\overline{3}})_i P_k^jP_j^k +Pb D^iH({\overline{3}})_i P_k^kP_j^j\nonumber \\&+\,Pc D^i H({\overline{3}})_k P_i^kP_j^j\nonumber \\&+\,Pd D^i H({\overline{3}})_k P_i^jP_j^k. \end{aligned}$$
(24)

With Eq. (24), the decay amplitudes of the \(D^0\rightarrow K^+K^-\), \(D^0\rightarrow \pi ^+\pi ^-\), \(D^+\rightarrow K^+{\overline{K}}^0\) and \(D^+_s\rightarrow \pi ^+K^0\) modes read

$$\begin{aligned} \mathcal {A}(D^0\rightarrow K^+K^-) =&-\lambda _d\left( \frac{1}{8}a_{15}+\frac{1}{8}c_{15}+\frac{1}{4}a_6-\frac{1}{4}c_6\right) \nonumber \\&+\lambda _s\left( \frac{3}{8}a_{15}+\frac{3}{8}c_{15}+\frac{1}{4}a_6-\frac{1}{4}c_6\right) \nonumber \\&-\, \lambda _b(2Pa+Pd- a_{15}/4), \end{aligned}$$
(25)
$$\begin{aligned} \mathcal {A}(D^0\rightarrow \pi ^+\pi ^-) =&\lambda _d\left( \frac{3}{8}a_{15}+\frac{3}{8}c_{15}+\frac{1}{4}a_6-\frac{1}{4}c_6\right) \nonumber \\&-\lambda _s\left( \frac{1}{8}a_{15}+\frac{1}{8}c_{15}+\frac{1}{4}a_6-\frac{1}{4}c_6\right) \nonumber \\&-\, \lambda _b(2Pa+Pd- a_{15}/4), \end{aligned}$$
(26)
$$\begin{aligned} \mathcal {A}(D^+\rightarrow K^+{\overline{K}}^0) =&\lambda _d\left( \frac{3}{8}a_{15}-\frac{1}{8}c_{15}-\frac{1}{4}a_6+\frac{1}{4}c_6\right) \nonumber \\&-\lambda _s\left( \frac{1}{8}a_{15}-\frac{3}{8}c_{15}-\frac{1}{4}a_6+\frac{1}{4}c_6\right) \nonumber \\&-\, \lambda _bPd, \end{aligned}$$
(27)
$$\begin{aligned} \mathcal {A}(D^+_s\rightarrow \pi ^+K^0) =&-\lambda _d\left( \frac{1}{8}a_{15}-\frac{3}{8}c_{15}-\frac{1}{4}a_6+\frac{1}{4}c_6\right) \nonumber \\&+\lambda _s\left( \frac{3}{8}a_{15}-\frac{1}{8}c_{15}-\frac{1}{4}a_6+\frac{1}{4}c_6\right) \nonumber \\&-\, \lambda _bPd, \end{aligned}$$
(28)

in which \(\lambda _d = V^*_{cd}V_{ud}\), \(\lambda _s = V^*_{cs}V_{us}\), \(\lambda _b = V^*_{cb}V_{ub}\). Equations (25)–(28) are consistent with [86] except for the last terms in Eqs. (25) and (26) because of the non-vanishing \(H(\overline{15})^{1}_{11}\) component in Eq. (15). From above formulas, the CP violation sum rules listed in Eqs. (17) and (18) are derived if the approximation of

$$\begin{aligned} \lambda _b\lambda _d= -\lambda _b(\lambda _s+\lambda _b)=-(\lambda _b\lambda _s+\lambda ^2_b)\simeq -\lambda _b\lambda _s \end{aligned}$$
(29)

is used. Besides, the decay amplitude of \(D^0\rightarrow K^0{\overline{K}}^0\) is expressed as

$$\begin{aligned} \mathcal {A}(D^0\rightarrow K^0{\overline{K}}^0) = -\lambda _b\left( 2Pa+\frac{1}{4}a_{15}\right) . \end{aligned}$$
(30)

The direct CP asymmetry in \(D^0\rightarrow K^0{\overline{K}}^0\) decay is zero in the flavor SU(3) limit:

$$\begin{aligned}&A^\mathrm{dir}_{CP}(D^0\rightarrow K^0{\overline{K}}^0) = 0. \end{aligned}$$
(31)

For the CP violation relations (17) and (18), the decay amplitudes of two channels are connected by the interchange of \(\lambda _d \leftrightarrow \lambda _s\), and their initial and final states are connected by the interchange of \(d \leftrightarrow s\):

$$\begin{aligned} D^+&\leftrightarrow D^+_s,\quad D^0 \leftrightarrow D^0,\quad K^+ \leftrightarrow \pi ^+, \nonumber \\ K^-&\leftrightarrow \pi ^-, \quad K^0\leftrightarrow {\overline{K}}^0. \end{aligned}$$
(32)

For \(D^0\rightarrow K^0{\overline{K}}^0\) decay, its corresponding mode in the interchange of \(d \leftrightarrow s\) is itself. So all CP 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 \(\pi ^0\), \(\eta ^{(\prime )}\) in the final states in \(D\rightarrow PP\) decays. Mesons \(\pi ^0\) and \(\eta ^{(\prime )}\) do not have definite U-spin quantum numbers. Under the interchange of \(d\leftrightarrow s\), there are no mesons corresponding to \(\pi ^0\) and \(\eta ^{(\prime )}\). For example, \(\pi ^0\) has the quark constituent of \(({\bar{d}} d- {\bar{u}}u)/\sqrt{2}\). Under the interchange of \(d\leftrightarrow s\), \(({\bar{d}} d- \bar{u}u)/\sqrt{2}\) turns into \(({\bar{s}} s- {\bar{u}}u)/\sqrt{2}\). No meson has the quark constituent of \(({\bar{s}} s- {\bar{u}}u)/\sqrt{2}\). So those decay channels involving \(\pi ^0\), \(\eta ^{(\prime )}\) do not have their corresponding modes in the interchange of \(d\leftrightarrow s\), and then have no simple CP violation sum rules with two channels.

In fact, not only the \(D\rightarrow PP\) decays, there are also some CP violation sum rules in the \(D\rightarrow PV\) decays [87]

$$\begin{aligned} A_{CP}^\mathrm{dir}(D^0\rightarrow \pi ^-\rho ^+) + A_{CP}^\mathrm{dir}(D^0\rightarrow K^-K^{*+})&= 0, \end{aligned}$$
(33)
$$\begin{aligned} A_{CP}^\mathrm{dir}(D^0\rightarrow \pi ^+\rho ^-) + A_{CP}^\mathrm{dir}(D^0\rightarrow K^+K^{*-})&=0,\end{aligned}$$
(34)
$$\begin{aligned} A_{CP}^\mathrm{dir}(D^+\rightarrow {\overline{K}}^0K^{*+}) + A_{CP}^\mathrm{dir}(D^+_s\rightarrow K^0\rho ^+)&=0,\end{aligned}$$
(35)
$$\begin{aligned} A_{CP}^\mathrm{dir}(D^+\rightarrow K^+{\overline{K}}^{*0}) + A_{CP}^\mathrm{dir}(D^+_s\rightarrow \pi ^+K^{*0})&=0,\end{aligned}$$
(36)
$$\begin{aligned} A_{CP}^\mathrm{dir}(D^0\rightarrow K^0{\overline{K}}^{*0}) + A_{CP}^\mathrm{dir}(D^0\rightarrow {\overline{K}}^0K^{*0})&=0. \end{aligned}$$
(37)

The detailed derivation of these sum rules is similar to \(D\rightarrow PP\) and can be found in Ref. [86]. Again, all the CP violation sum rules in \(D\rightarrow PV\) decays are associated with a complete interchange of d and s quarks, and all the singly Cabibbo-suppressed \(D\rightarrow PV\) modes with all final states having definite U-spin quantum numbers are included in Eqs. (33)–(37).

3.2 CP 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 \(d \leftrightarrow s\) is still valid for the CP violation sum rules in charmed baryon decays. The charmed anti-triplet baryon is expressed as

$$\begin{aligned} \mathcal {B}_{c{\overline{3}}}= \left( \begin{array}{ccc} 0 &{} \Lambda _c^+ &{} \Xi _c^+ \\ -\Lambda _c^+ &{} 0 &{} \Xi _c^0 \\ -\Xi _c^+ &{} -\Xi _c^0 &{} 0 \\ \end{array}\right) . \end{aligned}$$
(38)

The light baryon decuplet is given as

$$\begin{aligned} \Delta ^{++}&= \mathcal {B}_{10}^{111}, \quad \Delta ^{-}= \mathcal {B}_{10}^{222},\quad \Omega ^-= \mathcal {B}_{10}^{333}, \nonumber \\ \Delta ^{+}&= \frac{1}{\sqrt{3}}( \mathcal {B}_{10}^{112} + \mathcal {B}_{10}^{121} + \mathcal {B}_{10}^{211}),\nonumber \\ \Delta ^{0}&= \frac{1}{\sqrt{3}}( \mathcal {B}_{10}^{122} + \mathcal {B}_{10}^{212} + \mathcal {B}_{10}^{221}),\quad \nonumber \\ \Sigma ^{*+}&= \frac{1}{\sqrt{3}}( \mathcal {B}_{10}^{113} + \mathcal {B}_{10}^{131} + \mathcal {B}_{10}^{311}), \nonumber \\ \Sigma ^{*-}&=\frac{1}{\sqrt{3}} ( \mathcal {B}_{10}^{223} + \mathcal {B}_{10}^{232} + \mathcal {B}_{10}^{322}), \nonumber \\ \Xi ^{*0}&=\frac{1}{\sqrt{3}}( \mathcal {B}_{10}^{133} + \mathcal {B}_{10}^{313} + \mathcal {B}_{10}^{331}),\nonumber \\ \Xi ^{*-}&=\frac{1}{\sqrt{3}}( \mathcal {B}_{10}^{233} + \mathcal {B}_{10}^{323}+ \mathcal {B}_{10}^{332}),\nonumber \\ \Sigma ^{*0}&=\frac{1}{\sqrt{6}}( \mathcal {B}_{10}^{123} + \mathcal {B}_{10}^{132} + \mathcal {B}_{10}^{213}+ \mathcal {B}_{10}^{231} + \mathcal {B}_{10}^{312} + \mathcal {B}_{10}^{321}). \end{aligned}$$
(39)

The SU(3) irreducible representation amplitude of \(\mathcal {B}_{c{\overline{3}}}\rightarrow \mathcal {B}_{10} M\) decay can be written as

$$\begin{aligned} \mathcal {A}^\mathrm{tree}_{\mathcal {B}_{c{\overline{3}}}\rightarrow \mathcal {B}_{10}M} =&\, e_1(\mathcal {B}_{c{\overline{3}}})_{ij}H(\overline{15})^j_{kl}M^i_m\overline{\mathcal {B}}_{10}^{klm}\nonumber \\&+\,e_2(\mathcal {B}_{c{\overline{3}}})_{ij}H(\overline{15})^k_{lm}M^j_k\overline{\mathcal {B}}_{10}^{ilm}\nonumber \\&+\,e_3({\mathcal {B}}_{c{\overline{3}}})_{ij}H(\overline{15})^j_{kl}M^l_m\overline{\mathcal {B}}_{10}^{ikm}\nonumber \\&+\, e_4({\mathcal {B}}_{c{\overline{3}}})_{ij}H( 6)^j_{kl}M^l_m\overline{\mathcal {B}}_{10}^{ikm}\nonumber \\&+\,e_5({\mathcal {B}}_{c{\overline{3}}})_{ij}H(\overline{15})^j_{kl}M^m_m\overline{\mathcal {B}}_{10}^{ikl}\nonumber \\&+\,e_6({\mathcal {B}}_{c{\overline{3}}})_{ij}H(\overline{3})_{k}M^j_m\overline{\mathcal {B}}_{10}^{ikm}, \end{aligned}$$
(40)
$$\begin{aligned} \mathcal {A}^\mathrm{penguin}_{\mathcal {B}_{c{\overline{3}}}\rightarrow \mathcal {B}_{10}M} =&\, Pe_6({\mathcal {B}}_{c{\overline{3}}})_{ij}H^P(\overline{3})_{k}M^j_m\overline{\mathcal {B}}_{10}^{ikm}\nonumber \\&+\,Pe_7({\mathcal {B}}_{c{\overline{3}}})_{ij}H^P(\overline{3}^\prime )_{k}M^j_m\overline{\mathcal {B}}_{10}^{ikm}. \end{aligned}$$
(41)

Similar to \(D\rightarrow PP\) decay, if we define

$$\begin{aligned} Pe=e_6 + Pe_6+ 3Pe_7, \end{aligned}$$
(42)

the amplitude of \(\mathcal {B}_{c{\overline{3}}}\rightarrow \mathcal {B}_{10} M\) decay will be reduced to

$$\begin{aligned} \mathcal {A}^\mathrm{tree+penguin}_{\mathcal {B}_{c{\overline{3}}}\rightarrow \mathcal {B}_{10}M} =&\, e_1(\mathcal {B}_{c{\overline{3}}})_{ij}H(\overline{15})^j_{kl}M^i_m\overline{\mathcal {B}}_{10}^{klm}\nonumber \\&+\,e_2(\mathcal {B}_{c{\overline{3}}})_{ij}H(\overline{15})^k_{lm}M^j_k\overline{\mathcal {B}}_{10}^{ilm}\nonumber \\&+\,e_3({\mathcal {B}}_{c{\overline{3}}})_{ij}H(\overline{15})^j_{kl}M^l_m\overline{\mathcal {B}}_{10}^{ikm}\nonumber \\&+\, e_4({\mathcal {B}}_{c{\overline{3}}})_{ij}H( 6)^j_{kl}M^l_m\overline{\mathcal {B}}_{10}^{ikm}\nonumber \\&+\,e_5({\mathcal {B}}_{c{\overline{3}}})_{ij}H(\overline{15})^j_{kl}M^m_m\overline{\mathcal {B}}_{10}^{ikl}\nonumber \\&+\,Pe({\mathcal {B}}_{c{\overline{3}}})_{ij}H({\overline{3}})_{k}M^j_m\overline{\mathcal {B}}_{10}^{ikm}. \end{aligned}$$
(43)

The first four terms are the same with the formula given in [101]. The fifth term is the decay amplitude associated with singlet \(\eta _1\), and the six term is the amplitude proportional to \(\lambda _b\). With Eq. (43), the SU(3) irreducible representation amplitudes of \(\mathcal {B}_{c\overline{3}}\rightarrow \mathcal {B}_{10}M\) decays are obtained. The results are listed in Table 1.

Table 1 SU(3) irreducible representation amplitudes in \(\mathcal {B}_{c{\overline{3}}}\rightarrow \mathcal {B}_{10}M\) decays, in which only those modes that all initial and final states have definite U-spin quantum numbers are listed
Full size table

From Table 1, 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:

$$\begin{aligned} A^\mathrm{dir}_{CP}(\Lambda _{c}^{+}\rightarrow \Delta ^0\pi ^+) + A^\mathrm{dir}_{CP}(\Xi _{c}^{+}\rightarrow \Xi ^{*0}K^+)&=0, \end{aligned}$$
(44)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Lambda _{c}^{+}\rightarrow \Sigma ^{*+}K^0) + A^\mathrm{dir}_{CP}(\Xi _{c}^{+}\rightarrow \Delta ^{+}{\overline{K}}^0)&=0,\end{aligned}$$
(45)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Lambda _{c}^{+}\rightarrow \Sigma ^{*0}K^+) + A^\mathrm{dir}_{CP}(\Xi _{c}^{+}\rightarrow \Sigma ^{*0}K^+)&=0,\end{aligned}$$
(46)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Lambda _{c}^{+}\rightarrow \Delta ^{++}\pi ^-) + A^\mathrm{dir}_{CP}(\Xi _{c}^{+}\rightarrow \Delta ^{++} K^-)&=0,\end{aligned}$$
(47)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{c}^{0}\rightarrow \Sigma ^{*-}\pi ^+) + A^\mathrm{dir}_{CP}(\Xi _{c}^{0}\rightarrow \Xi ^{*-}K^+)&=0,\end{aligned}$$
(48)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{c}^{0}\rightarrow \Delta ^{0}{\overline{K}}^0) + A^\mathrm{dir}_{CP}(\Xi _{c}^{0}\rightarrow \Xi ^{*0} K^0)&=0,\end{aligned}$$
(49)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{c}^{0}\rightarrow \Sigma ^{*+}\pi ^-) + A^\mathrm{dir}_{CP}(\Xi _{c}^{0}\rightarrow \Delta ^{+}K^-)&=0. \end{aligned}$$
(50)

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,

$$\begin{aligned} \Lambda ^+_c\leftrightarrow \Xi ^+_c,\quad \Xi ^0_c\leftrightarrow \Xi ^0_c. \end{aligned}$$
(51)

For light decuplet baryons,

$$\begin{aligned}&\Delta ^0\leftrightarrow \Xi ^{*0},\quad \Sigma ^{*+}\leftrightarrow \Delta ^+,\quad \Sigma ^{*0}\leftrightarrow \Sigma ^{*0}, \quad \Delta ^{++}\leftrightarrow \Delta ^{++},\quad \nonumber \\&\Xi ^{*-}\leftrightarrow \Sigma ^{*-},\quad \Delta ^{-}\leftrightarrow \Omega ^{-}. \end{aligned}$$
(52)

Also, Eqs. (44)–(50) include all the SCS modes with all associated particles having definite U-spin quantum numbers in \(\mathcal {B}_{c{\overline{3}}}\rightarrow \mathcal {B}_{10}M\) decays.

For other types of charm baryon decay, for example \(\mathcal {B}_{c{\overline{3}}}\rightarrow \mathcal {B}_{8}M\) decay, multi-body decay and doubly charmed baryon decay, the treatments of their SU(3) irreducible representation amplitudes are similar to \(\mathcal {B}_{c{\overline{3}}}\rightarrow \mathcal {B}_{10}M\). Related discussions can be found in Refs. [29, 40, 42, 101]. But notice that the contributions proportional to \(\lambda _b\) are neglected in this literature. 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.

3.3 A universal law for CP violation sum rules in the charm sector

From the above discussions, one can find 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 the charm sector.

Firstly, the complete interchange of \(d \leftrightarrow s\) quarks in initial and final states leads to the interchange of \(d\leftrightarrow s\) in operators \(O^{ij}_k\). It can be understood in following argument. In the IRA approach, each decay amplitude connects to one invariant tensor [in which all covariant indices are contracted with contravariant indices; see Eq. (24) for example], no matter charm meson or baryon decays and two- or multi-body decays. If a complete interchange of d and s quarks is performed in the tensors corresponding to initial and final states, the complete interchange of \(d \leftrightarrow s\) must be performed in tensor \(H_{ij}^{k}\) in order to keep all covariant and contravariant indices contracted. From Eq. (11), one can find \(H_{ij}^{k}\) corresponds to \(O^{ij}_{k}\) one by one. So the d and s quark constituents in operators \(O^{ij}_k\) must be interchanged. In physics, if the quark constituents of all initial and final particles in one decay channel are replaced by \(d \rightarrow s\) and \(s \rightarrow d\), the quark constituents in the effective weak vertexes should be replaced by \(d \rightarrow s\) and \(s \rightarrow d\) also. The operators \(O^{ij}_k\) are abstracted from the effective weak vertexes, so the quark constituents of operators \(O^{ij}_k\) transform as a complete interchange of \(d \leftrightarrow s\).

Secondly, the interchange of \(d\leftrightarrow s\) in operators \(O^{ij}_k\) leads to the decay amplitudes proportional to \(\lambda _d/\lambda _s\) are connected by the interchange of \(\lambda _d \leftrightarrow \lambda _s\) and the decay amplitudes proportional to \(\lambda _b\) are the same in the flavor SU(3) symmetry. The contributions proportional to \(\lambda _d/\lambda _s\) in SCS decays are induced by following operators in the SU(3) irreducible representation:

$$\begin{aligned} O(6)_{23}, \quad O(\overline{15})^{12}_2,\quad O(\overline{15})^{13}_3. \end{aligned}$$
(53)

Under the interchange of \(d \leftrightarrow s\), these operators are transformed as

$$\begin{aligned} O(6)_{23}\,\,\leftrightarrow \,\, -O(6)_{23} , \quad O(\overline{15})^{12}_2\,\, \leftrightarrow \,\, O(\overline{15})^{13}_3. \end{aligned}$$
(54)

These properties can be read from the explicit SU(3) decomposition of \(O^{ij}_k\); see Appendix A. The corresponding CKM matrix elements then transform as

$$\begin{aligned} H(6)^{23}\,\,\leftrightarrow \,\, -H(6)^{23} , \quad H(\overline{15})_{12}^2\,\, \leftrightarrow \,\, H(\overline{15})_{13}^3. \end{aligned}$$
(55)

According to Eq. (15), Eq. (55) equals

$$\begin{aligned} \frac{1}{4}(\lambda _d-\lambda _s)\,\,\leftrightarrow \,\, - \frac{1}{4}(\lambda _d-\lambda _s),\quad \frac{3}{8}\lambda _d-\frac{1}{8}\lambda _s\,\, \leftrightarrow \,\, \frac{3}{8}\lambda _s-\frac{1}{8}\lambda _d. \end{aligned}$$
(56)

One can find that Eq. (56) is equivalent to the interchange of \(\lambda _d\leftrightarrow \lambda _s\). The contributions proportional to \(\lambda _b\) in the SCS decays are induced by the following operators:

$$\begin{aligned} O(\overline{15})^{11}_1, \quad O(\overline{3})^{1},\quad O(\overline{3}^\prime )^{1}. \end{aligned}$$
(57)

Form Appendix A, it is found that these operators are invariable under the interchange of \(d \leftrightarrow s\), as are the corresponding CKM matrix elements.

Thirdly, if two decay channels have the relations that their decay amplitudes proportional to \(\lambda _d/\lambda _s\) are connected by the interchange of \(\lambda _d \leftrightarrow \lambda _s\) and the decay amplitudes proportional to \(\lambda _b\) are the same, the sum of their direct CP asymmetries is zero in the \(SU(3)_F\) limit under the approximation in Eq. (29). For one decay mode with amplitude of

$$\begin{aligned} \mathcal {A}(i\rightarrow f)=&\,\lambda _d A + \lambda _s B +\lambda _b C \nonumber \\ =&\,-\, (\lambda _s+|\lambda _b|\, e^{i\phi }) \,|A|\,e^{i\delta _A} + \lambda _s \,|B|\,e^{i\delta _B} \nonumber \\&+\,|\lambda _b|\,e^{i\phi } \,|C|\,e^{i\delta _C}, \end{aligned}$$
(58)

its CP asymmetry in the order of \(\mathcal {O}(\lambda _b)\) is derived as

$$\begin{aligned} A^\mathrm{dir}_{CP}(i\rightarrow f)&= \frac{|\lambda _d A + \lambda _s B +\lambda _b C|^2-|\lambda ^*_d A + \lambda ^*_s B +\lambda ^*_b C|^2}{|\lambda _d A + \lambda _s B +\lambda _b C|^2+|\lambda ^*_d A + \lambda ^*_s B +\lambda ^*_b C|^2}\nonumber \\&\simeq 2 \frac{|\lambda _b|}{\lambda _s}\frac{|AB|\sin (\delta _A-\delta _B)-|AC|\sin (\delta _A-\delta _C)+|BC|\sin (\delta _B-\delta _C)}{|A|^2+|B|^2-2|AB|\cos (\delta _A-\delta _B)}\sin \phi . \end{aligned}$$
(59)

For one decay mode with amplitude of

$$\begin{aligned} \mathcal {A}(i^\prime \rightarrow f^\prime )&= \lambda _d B + \lambda _s A +\lambda _b C, \end{aligned}$$
(60)

which is connected to Eq. (58) by \(\lambda _d\leftrightarrow \lambda _s\), its CP asymmetry in the order of \(\mathcal {O}(\lambda _b)\) is derived as

$$\begin{aligned}&A^\mathrm{dir}_{CP}(i^\prime \rightarrow f^\prime ) \nonumber \\&\quad \simeq -2 \frac{|\lambda _b|}{\lambda _s}\frac{|AB|\sin (\delta _A-\delta _B)-|AC|\sin (\delta _A-\delta _C)+|BC|\sin (\delta _B-\delta _C)}{|A|^2+|B|^2-2|AB|\cos (\delta _A-\delta _B)}\sin \phi . \end{aligned}$$
(61)

It is apparent that

$$\begin{aligned} A^\mathrm{dir}_{CP}(i\rightarrow f)+A^\mathrm{dir}_{CP}(i^\prime \rightarrow f^\prime )\simeq 0. \end{aligned}$$
(62)

Based on the 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 \(d\leftrightarrow s\);

  • If there are two decay modes connected by the interchange of \(d\leftrightarrow s\), 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.

4 Results and discussions

With the method proposed in Sect. 3, 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 are not going to list all the CP violation sum rules, but only present some of them as examples.

Under the complete interchange of \(d \leftrightarrow s\), the light octet baryons are interchanged as

$$\begin{aligned} p\leftrightarrow \Sigma ^+,\quad n\leftrightarrow \Xi ^0,\quad \Sigma ^-\leftrightarrow \Xi ^-. \end{aligned}$$
(63)

The sum rules for CP asymmetries in charmed baryon decays into one pseudoscalar meson and one octet baryon are

$$\begin{aligned} A^\mathrm{dir}_{CP}(\Lambda _{c}^{+}\rightarrow \Sigma ^+K^0) + A^\mathrm{dir}_{CP}(\Xi _{c}^{+}\rightarrow p{\overline{K}}^0)&=0, \end{aligned}$$
(64)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Lambda _{c}^{+}\rightarrow n\pi ^+) + A^\mathrm{dir}_{CP}(\Xi _{c}^{+}\rightarrow \Xi ^{0}K^+)&=0,\end{aligned}$$
(65)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{c}^{0}\rightarrow \Sigma ^{-}\pi ^+) + A^\mathrm{dir}_{CP}(\Xi _{c}^{0}\rightarrow \Xi ^{-}K^+)&=0, \end{aligned}$$
(66)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{c}^{0}\rightarrow n{\overline{K}}^0) + A^\mathrm{dir}_{CP}(\Xi _{c}^{0}\rightarrow \Xi ^{0} K^0)&=0,\end{aligned}$$
(67)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{c}^{0}\rightarrow \Sigma ^{+}\pi ^-) + A^\mathrm{dir}_{CP}(\Xi _{c}^{0}\rightarrow pK^-)&=0. \end{aligned}$$
(68)

Under the complete interchange of \(d \leftrightarrow s\), the doubly charmed baryons are interchanged as

$$\begin{aligned} \Xi ^{++}_{cc}\leftrightarrow \Xi ^{++}_{cc},\quad \Xi ^{+}_{cc}\leftrightarrow \Omega ^{+}_{cc}. \end{aligned}$$
(69)

The sum rules for CP asymmetries in doubly charmed baryon decays into one pseudoscalar meson and one charmed triplet baryon are

$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{cc}^{++}\rightarrow \Lambda ^+_c\pi ^+) + A^\mathrm{dir}_{CP}(\Xi _{cc}^{++}\rightarrow \Xi ^+_cK^+)&=0,\end{aligned}$$
(70)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{cc}^{+}\rightarrow \Xi ^+_cK^0) + A^\mathrm{dir}_{CP}(\Omega _{cc}^{+}\rightarrow \Lambda ^+_c{\overline{K}}^0)&=0,\end{aligned}$$
(71)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{cc}^{+}\rightarrow \Xi ^0_cK^+) + A^\mathrm{dir}_{CP}(\Omega _{cc}^{+}\rightarrow \Xi ^0_c\pi ^+)&=0. \end{aligned}$$
(72)

Under the complete interchange of \(d \leftrightarrow s\), the charmed sextet baryons are interchanged as

$$\begin{aligned} \Sigma _c^+&\leftrightarrow \Xi ^{*+}_{c},\quad \Sigma ^{++}_{c}\leftrightarrow \Sigma ^{++}_{c},\nonumber \\ \Xi ^{*0}_{c}&\leftrightarrow \Xi ^{*0}_{c}, \quad \Sigma ^{0}_{c}\leftrightarrow \Omega ^{0}_{c}. \end{aligned}$$
(73)

The sum rules for CP asymmetries in doubly charmed baryon decays into one pseudoscalar meson and one charmed sextet baryon are

$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{cc}^{++}\rightarrow \Sigma ^+_c\pi ^+) + A^\mathrm{dir}_{CP}(\Xi _{cc}^{++}\rightarrow \Xi ^{*+}_cK^+)&=0,\end{aligned}$$
(74)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{cc}^{+}\rightarrow \Sigma ^{++}_c\pi ^-) + A^\mathrm{dir}_{CP}(\Omega _{cc}^{+}\rightarrow \Sigma ^{++}_cK^-)&=0,\end{aligned}$$
(75)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{cc}^{+}\rightarrow \Sigma ^0_c\pi ^+) + A^\mathrm{dir}_{CP}(\Omega _{cc}^{+}\rightarrow \Omega ^0_cK^+)&=0,\end{aligned}$$
(76)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{cc}^{+}\rightarrow \Xi ^{*+}_cK^0) + A^\mathrm{dir}_{CP}(\Omega _{cc}^{+}\rightarrow \Sigma ^{+}_c {\overline{K}}^0)&=0,\end{aligned}$$
(77)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{cc}^{+}\rightarrow \Xi ^{*0}_cK^+) + A^\mathrm{dir}_{CP}(\Omega _{cc}^{+}\rightarrow \Xi ^{*0}_c\pi ^+)&=0. \end{aligned}$$
(78)

With the interchange rules mentioned above, the CP violation sum rules in doubly charmed baryon decays into one charmed meson and one octet baryon are

$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{cc}^{++}\rightarrow \Sigma ^+D^+_s) + A^\mathrm{dir}_{CP}(\Xi _{cc}^{++}\rightarrow pD^+)&=0,\end{aligned}$$
(79)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{cc}^{+}\rightarrow pD^0) + A^\mathrm{dir}_{CP}(\Omega _{cc}^{+}\rightarrow \Sigma ^{+}D^0)&=0,\end{aligned}$$
(80)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{cc}^{+}\rightarrow nD^+) + A^\mathrm{dir}_{CP}(\Omega _{cc}^{+}\rightarrow \Xi ^{0}D^+_s)&=0. \end{aligned}$$
(81)

The CP violation sum rules in doubly charmed baryon decays into one charmed meson and one decuplet baryon are

$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{cc}^{++}\rightarrow \Delta ^+D^+) + A^\mathrm{dir}_{CP}(\Xi _{cc}^{++}\rightarrow \Sigma ^{*+}D^+_s)&=0,\end{aligned}$$
(82)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{cc}^{+}\rightarrow \Delta ^{+}D^0) + A^\mathrm{dir}_{CP}(\Omega _{cc}^{+}\rightarrow \Sigma ^{*+}D^0)&=0,\end{aligned}$$
(83)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{cc}^{+}\rightarrow \Delta ^{0}D^+) + A^\mathrm{dir}_{CP}(\Omega _{cc}^{+}\rightarrow \Xi ^{*0}D^+_s)&=0,\end{aligned}$$
(84)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{cc}^{+}\rightarrow \Sigma ^{*0}D^+_s) + A^\mathrm{dir}_{CP}(\Omega _{cc}^{+}\rightarrow \Sigma ^{*0}D^+)&=0. \end{aligned}$$
(85)

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:

$$\begin{aligned} A^\mathrm{dir}_{CP}(\Lambda _{c}^{+}\rightarrow pK^-K^+) + A^\mathrm{dir}_{CP}(\Xi _{c}^{+}\rightarrow \Sigma ^+\pi ^-\pi ^+)&=0,\end{aligned}$$
(86)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Lambda _{c}^{+}\rightarrow p\pi ^-\pi ^+) + A^\mathrm{dir}_{CP}(\Xi _{c}^{+}\rightarrow \Sigma ^+K^-K^+)&=0,\end{aligned}$$
(87)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Lambda _{c}^{+}\rightarrow \Sigma ^+\pi ^-K^+) + A^\mathrm{dir}_{CP}(\Xi _{c}^{+}\rightarrow pK^-\pi ^+)&=0,\end{aligned}$$
(88)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Lambda _{c}^{+}\rightarrow \Sigma ^-\pi ^+K^+) + A^\mathrm{dir}_{CP}(\Xi _{c}^{+}\rightarrow \Xi ^-K^+\pi ^+)&=0,\end{aligned}$$
(89)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Lambda _{c}^{+}\rightarrow nK^+{\overline{K}}^0) + A^\mathrm{dir}_{CP}(\Xi _{c}^{+}\rightarrow \Xi ^0\pi ^+K^0)&=0,\end{aligned}$$
(90)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{c}^{0}\rightarrow \Sigma ^{+}K^-K^0) + A^\mathrm{dir}_{CP}(\Xi _{c}^{0}\rightarrow p\pi ^-{\overline{K}}^0)&=0,\end{aligned}$$
(91)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{c}^{0}\rightarrow \Sigma ^-K^+{\overline{K}}^0) + A^\mathrm{dir}_{CP}(\Xi _{c}^{0}\rightarrow \Xi ^{-}\pi ^+ K^0)&=0,\end{aligned}$$
(92)
$$\begin{aligned} A^\mathrm{dir}_{CP}(\Xi _{c}^{0}\rightarrow \Xi ^{0}\pi ^-K^+) + A^\mathrm{dir}_{CP}(\Xi _{c}^{0}\rightarrow nK^-\pi ^+)&=0. \end{aligned}$$
(93)

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:

$$\begin{aligned} \pi ^+&\rightarrow \rho ^+,\quad \pi ^-\rightarrow \rho ^-, \quad K^+\rightarrow K^{*+},\quad K^-\rightarrow K^{*-}, \nonumber \\&K^0\rightarrow K^{*0}, \quad {\overline{K}}^0\rightarrow \overline{K}^{*0}. \end{aligned}$$
(94)

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 sizable in the 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. (4) makes the two CP asymmetries in observable \(\Delta A_{CP} \equiv A_{CP}(D^0\rightarrow K^+K^-) - A_{CP}(D^0\rightarrow \pi ^+\pi ^-)\) constructive. Similarly, one can use the CP violation sum rules in charmed baryon decays to construct some observables in which two CP asymmetries are constructive. Some observables are selected for experimental discretion:

$$\begin{aligned} \Delta A_{CP}^\mathrm{baryon,1}&= A_{CP}(\Lambda _{c}^{+}\rightarrow \Sigma ^+K^{*0}) - A_{CP}(\Xi _{c}^{+}\rightarrow p{\overline{K}}^{*0}),\end{aligned}$$
(95)
$$\begin{aligned} \Delta A_{CP}^\mathrm{baryon,2}&=A_{CP}(\Xi _{c}^{0}\rightarrow \Sigma ^{+}\pi ^-) - A_{CP}(\Xi _{c}^{0}\rightarrow pK^-),\end{aligned}$$
(96)
$$\begin{aligned} \Delta A_{CP}^\mathrm{baryon,3}&=A_{CP}(\Lambda _{c}^{+}\rightarrow \Delta ^{++}\pi ^-) - A_{CP}(\Xi _{c}^{+}\rightarrow \Delta ^{++} K^-),\end{aligned}$$
(97)
$$\begin{aligned} \Delta A_{CP}^\mathrm{baryon,4}&=A_{CP}(\Lambda ^+_c\rightarrow \Sigma ^+\pi ^-K^+) -A_{CP}(\Xi ^+_c\rightarrow pK^-\pi ^+). \end{aligned}$$
(98)

If the contributions proportional to \(\lambda _b\) are neglected, the decay amplitudes of the two channels connected by the interchange of \(d \leftrightarrow s\) 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 \(\Xi ^+_c\rightarrow pK^-\pi ^+\). The integration over the phase space of the three-body decay \(\mathcal {B}_c\rightarrow \mathcal {B}M_1M_2\) relies on the equation of [2]

$$\begin{aligned}&\Gamma (\mathcal {B}_c\rightarrow \mathcal {B}M_1M_2)\nonumber \\&\quad =\int _{m_{12}^2}\int _{m_{23}^2}\frac{|\mathcal {A}(\mathcal {B}_c\rightarrow \mathcal {B}M_1M_2)|^2}{32m^3_{\mathcal {B}_c}}\mathrm{{d}}m^2_{12}\mathrm{{d}}m^2_{23}, \end{aligned}$$
(99)

where \(m_{12}^2=(p_{M_1} + p_{M_2})^2\) and \(m_{23}^2=(p_{M_2} + p_{\mathcal {B}})^2\). With the experimental data given in [2],

$$\begin{aligned} \mathcal {B}r(\Lambda ^+_c\rightarrow \Sigma ^+\pi ^-K^+) = (2.1\pm 0.6)\times 10^{-3}, \end{aligned}$$
(100)

and the relation

$$\begin{aligned} |\mathcal {A}(\Xi ^+_c\rightarrow pK^-\pi ^+)| \simeq |\mathcal {A}(\Lambda ^+_c\rightarrow \Sigma ^+\pi ^-K^+)|, \end{aligned}$$
(101)

the branching fraction of \(\Xi ^+_c\rightarrow pK^-\pi ^+\) decay is predicted to be

$$\begin{aligned} \mathcal {B}r(\Xi ^+_c\rightarrow pK^-\pi ^+) = (1.7\pm 0.5)\%. \end{aligned}$$
(102)

One can find the branching fraction \(\mathcal {B}r(\Xi ^+_c\rightarrow pK^-\pi ^+)\) is larger than \(\mathcal {B}r(\Lambda ^+_c\rightarrow \Sigma ^+\pi ^-K^+)\) because of the larger phase space and the longer lifetime of \(\Xi ^+_c\). 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 \(\Lambda ^+_c\rightarrow \Sigma ^+\pi ^-K^+\) 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 U-spin breaking can be found in [66].

With the method introduced in [66], and the LHCb data [102]

$$\begin{aligned}&\frac{f_{\Xi _b}}{f_{\Lambda _b}}\cdot \frac{\mathcal {B}r(\Xi ^0_b\rightarrow \Xi ^+_c\pi ^-)}{\mathcal {B}r(\Lambda ^0_b\rightarrow \Lambda ^+_c\pi ^-)}\cdot \frac{\mathcal {B}r(\Xi ^+_c\rightarrow pK^-\pi ^+)}{\mathcal {B}r(\Lambda ^+_c\rightarrow pK^-\pi ^+)} \nonumber \\&\quad = (1.88\pm 0.04\pm 0.03 )\times 10^{-2}, \end{aligned}$$
(103)

the fragmentation–fraction ratio \(f_{\Xi _b}/f_{\Lambda _b}\) is determined to be

$$\begin{aligned} f_{\Xi _b}/f_{\Lambda _b}=0.065\pm 0.020. \end{aligned}$$
(104)

Recent measurement confirmed this result [103]. Our result is consistent with the one obtained via \(\Lambda ^0_b\rightarrow J /\psi \Lambda ^0\) [104], \(f_{\Xi _b}/f_{\Lambda _b}=0.11\pm 0.03\), the one via \(\Xi ^-_b\rightarrow J /\psi \Xi ^-\) [105], \(f_{\Xi _b}/f_{\Lambda _b}=0.108\pm 0.034\), and the one via the diquark model for \(\Xi ^-_b\rightarrow \Lambda ^0_b\pi ^-\) [106] using the LHCb data [107], \(f_{\Xi _b}/f_{\Lambda _b}=0.08\pm 0.03\). The detailed comparison for different methods of estimating \(f_{\Xi _b}/f_{\Lambda _b}\) can be found in [66].

5 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 \(\mathcal {B}r(\Xi ^+_c\rightarrow pK^-\pi ^+)\) is predicted to be \((1.7\pm 0.5)\%\) in the U-spin limit, and the fragmentation–fraction ratio is determined as \(f_{\Xi _b}/f_{\Lambda _b}=0.065\pm 0.020\).