Sum rules for CP asymmetries of charmed baryon decays in

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Sum rules for CP asymmetries of charmed baryon decays

in the SU(3)F limit

Di Wang∗

School of Nuclear Science and Technology, Lanzhou University,

Lanzhou 730000, People’s Republic of China

Abstract

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

Ξ+c → pK−π+ is predicted to be (1.7 ± 0.5)% in the U -spin limit and the fragmentation-fraction ratio is

determined to be fΞb/fΛb

= 0.065± 0.020 using the LHCb data.

∗Electronic address: [email protected]

1

http://arxiv.org/abs/1901.01776v3

mailto:[email protected]

I. INTRODUCTION

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

the value of

∆ACP ≡ ACP (D0 → K+K−)−ACP (D

0 → π+π−) = (−1.54 ± 0.29) × 10−3, (1)

in which the dominated direct CP violation is defined by

AdirCP (i→ f) ≡ |A(i→ f)|2 − |A(i→ f)|2

|A(i→ f)|2 + |A(i→ f)|2. (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 [3–5] and the progress of singly charmed baryon measurements [6–

22], plenty of theoretical interests focus on charmed baryon decays [23–81]. However, only a few

publications 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 the

LHCb Collaboration [21] and no signal of CP violation is found:

∆AbaryonCP ≡ ACP (Λ

+c → pK+K−)−ACP (Λ

+c → pπ+π−) = (0.30 ± 0.91 ± 0.61)%. (3)

In charmed and bottomed meson decays, some relations for CP asymmetries in (or beyond) the

flavor SU(3) limit are found [83–94]. For example, the direct CP asymmetries in D0 → K+K−

and D0 → π+π− decays have following relation in the U -spin limit [86, 87]:

AdirCP (D

0 → K+K−) +AdirCP (D

0 → π+π−) = 0. (4)

The two CP asymmetries in ∆ACP have opposite sign and hence are constructive in ∆ACP . But

the two CP asymmetries in ∆AbaryonCP , as pointed out in [73], do not have a relation like the ones

in ∆ACP . 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

2

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 Br(Ξ+c → pK−π+) and fragmentation-fraction ratio fΞb

/fΛbare estimated in

the U -spin limit.

The rest of this paper is organized as follows. In Sect. II, the effective Hamiltonian of charm

decay is decomposed into the SU(3) irreducible representations. In Sect. 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 Sect. IV, we list some results of the CP violation sum rules in

charmed baryon decays. Section V is a brief summary. 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 Standard Model (SM) can be

written as [97]

Heff =GF√2

∑

q=d,s

V ∗cq1Vuq2

(

2∑

i=1

Ci(µ)Oi(µ)

)

− V ∗cbVub

(

6∑

i=3

Ci(µ)Oi(µ) + C8g(µ)O8g(µ)

)

, (5)

where GF is the Fermi coupling constant, Ci is the Wilson coefficient of operator Oi. The tree

operators are

O1 = (uαq2β)V−A(q1βcα)V−A, O2 = (uαq2α)V−A(q1βcβ)V−A, (6)

in which α, β are color indices, q1,2 are d and s quarks. The QCD penguin operators are

O3 =∑

q′=u,d,s

(uαcα)V−A(q′βq

′β)V −A, O4 =

∑

q′=u,d,s

(uαcβ)V−A(q′βq

′α)V −A,

O5 =∑

q′=u,d,s

(uαcα)V−A(q′βq

′β)V +A, O6 =

∑

q′=u,d,s

(uαcβ)V−A(q′βq

′α)V +A, (7)

and the chromomagnetic-penguin operator is

O8g =g

8π2mcuσµν(1 + γ5)T

aGaµνc. (8)

The magnetic-penguin contributions can be included into the Wilson coefficients for the penguin

operators following the substitutions [98–100]:

C3,5(µ) → C3,5(µ) +αs(µ)8πNc

2m2c

〈l2〉 Ceff8g (µ), C4,6(µ) → C4,6(µ)−

αs(µ)

8π

2m2c

〈l2〉 Ceff8g (µ), (9)

3

with the effective Wilson coefficient Ceff8g = C8g + C5 and 〈l2〉 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

c→ sdu, c→ dd/ssu, c→ dsu, (10)

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

Hamiltonian,

Heff =

3∑

i,j,k=1

HkijO

ijk =

3∑

i,j,k=1

Hkij(q

iqk)(qjc). (11)

Equation (5) implies that the tensor components of Hkij can be obtained from the map (uq1)(q2c) →

V ∗cq2Vuq1 in current-current operators and (qq)(uc) → −V ∗

cbVub in penguin operators and the others

are zero. The non-zero components of the tensor Hkij corresponding to tree operators in Eq. (5)

are

H213 = V ∗

csVud, H212 = V ∗

cdVud, H313 = V ∗

csVus, H312 = V ∗

cdVus, (12)

and the non-zero components of the tensor Hkij corresponding to penguin operators in Eq. (5) are

H111 = −V ∗

cbVub, H221 = −V ∗

cbVub, H331 = −V ∗

cbVub. (13)

The operator Oijk is a representation of the SU(3) group, which is decomposed as four irreducible

representations: 3⊗ 3⊗ 3 = 3⊕ 3′ ⊕ 6⊕ 15. The explicit decomposition is [86]

Oijk = δjk

(3

8O(3)i − 1

8O(3

′)i)

+ δik

(3

8O(3

′)j − 1

8O(3)j

)

+ ǫijlO(6)lk +O(15)ijk . (14)

All components of the irreducible representations are listed in Appendix A. The non-zero compo-

nents of Hkij corresponding to tree operators in the SU(3) decomposition are

H(6)22 = −1

2V ∗csVud, H(6)23 =

1

4(V ∗

cdVud − V ∗csVus), H(6)33 =

1

2V ∗cdVus,

H(15)111 = −1

4(V ∗

cdVud + V ∗csVus) =

1

4V ∗cbVub, H(15)213 =

1

2V ∗csVud, H(15)312 =

1

2V ∗cdVus,

H(15)212 =3

8V ∗cdVud −

1

8V ∗csVus, H(15)313 =

3

8V ∗csVus −

1

8V ∗cdVud,

H(3)1 = V ∗cdVud + V ∗

csVus = −V ∗cbVub. (15)

The non-zero components of Hkij corresponding to penguin operators in the SU(3) decomposition

are

HP (3)1 = −V ∗cbVub, HP (3

′)1 = −3V ∗

cbVub. (16)

4

Here we use the superscript P to differentiate penguin contributions from tree contributions. Equa-

tions (15) and (16) were derived in [86] for the first time. But the non-zero components H(15)111

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 CP asymmetries be-

cause they ignore the two 3-dimensional irreducible representations and make the approximation

of V ∗csVus ≃ −V ∗

cdVud in the 15- and 6-dimensional irreducible representations, leading to the van-

ishing of the contributions proportional to λb = V ∗cbVub. If the contributions proportional to λb 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 CP violation sum rules in

charm decays.

A. CP violation sum rules in charmed meson decays

Two CP asymmetry sum rules for D → PP decays in the flavor SU(3) limit have been given

in [86, 87]:

AdirCP (D

0 → K+K−) +AdirCP (D

0 → π+π−) = 0, (17)

AdirCP (D

+ → K+K0) +Adir

CP (D+s → π+K0) = 0. (18)

To see why the two sum rules are correct, we express the decay amplitudes of D0 → K+K−,

D0 → π+π−, D+ → K+K0and D+

s → π+K0 modes in the SU(3) irreducible representation

amplitude (IRA) approach. The charmed meson anti-triplet is

Di = (D0,D+,D+s ). (19)

The pseudoscalar meson nonet is

P ij =

1√2π0 + 1√

6η8 π+ K+

π− − 1√2π0 + 1√

6η8 K0

K− K0 −

√

2/3η8

+1√3

η1 0 0

0 η1 0

0 0 η1

. (20)

5

To obtain the SU(3) irreducible representation amplitude of D → PP decay, one takes various

representations in Eqs. (15) and (16) and contracts all indices in Di and light meson P ij with

various combinations:

AtreeD→PP = a15D

iH(15)kijPjl P

lk + b15D

iH(15)kijPjkP

ll + c15D

iH(15)kjlPji P

lk

+ a6DiH(6)kijP

jl P

lk + b6D

iH(6)kijPjkP

ll + c6D

iH(6)kjlPji P

lk

+ a3DiH(3)iP

jkP

kj + b3D

iH(3)iPkk P

jj + c3D

iH(3)kPki P

jj

+ d3DiH(3)kP

ji P

kj . (21)

ApenguinD→PP =Pa3D

iHP (3)iPjkP

kj + Pb3D

iHP (3)iPkk P

jj + Pc3D

iHP (3)kPki P

jj

+ Pd3DiHP (3)kP

ji P

kj + Pa′3D

iHP (3′)iP

jkP

kj + Pb′3D

iHP (3′)iP

kk P

jj

+ Pc′3DiHP (3

′)kP

ki P

jj + Pd′3D

iHP (3′)kP

ji P

kj . (22)

Notice that only the first components of 3 and 3′irreducible representations are non-zero. Some

amplitudes, for example, a3, Pa3 and Pa′3 are always appear simultaneously since they correspond

to the same contraction. Noting that HP (3′)1 = 3H(3)1 = 3HP (3)1 = −3V ∗

cbVub, if we define

Pa = a3 + Pa3 + 3Pa′3, P b = b3 + Pb3 + 3Pb′3, . . . , (23)

the amplitude of D → PP decay will be reduced to

Atree+penguinD→PP = a15D

iH(15)kijPjl P

lk + b15D

iH(15)kijPjkP

ll + c15D

iH(15)kjlPji P

lk

+ a6DiH(6)kijP

jl P

lk + b6D

iH(6)kijPjkP

ll + c6D

iH(6)kjlPji P

lk

+ PaDiH(3)iPjkP

kj + PbDiH(3)iP

kk P

jj + PcDiH(3)kP

ki P

jj

+ PdDiH(3)kPji P

kj . (24)

6

With Eq. (24), the decay amplitudes of the D0 → K+K−, D0 → π+π−, D+ → K+K0and

D+s → π+K0 modes read

A(D0 → K+K−) = −λd(1

8a15 +

1

8c15 +

1

4a6 −

1

4c6) + λs(

3

8a15 +

3

8c15 +

1

4a6 −

1

4c6)

− λb(2Pa+ Pd− a15/4), (25)

A(D0 → π+π−) = λd(3

8a15 +

3

8c15 +

1

4a6 −

1

4c6)− λs(

1

8a15 +

1

8c15 +

1

4a6 −

1

4c6)

− λb(2Pa+ Pd− a15/4), (26)

A(D+ → K+K0) = λd(

3

8a15 −

1

8c15 −

1

4a6 +

1

4c6)− λs(

1

8a15 −

3

8c15 −

1

4a6 +

1

4c6)

− λbPd, (27)

A(D+s → π+K0) = −λd(

1

8a15 −

3

8c15 −

1

4a6 +

1

4c6) + λs(

3

8a15 −

1

8c15 −

1

4a6 +

1

4c6)

− λbPd, (28)

’in which λd = V ∗cdVud, λs = V ∗

csVus, λb = V ∗cbVub. Equations (25)-(28) are consistent with [86]

except for the last terms in Eqs. (25) and (26) because of the non-vanishing H(15)111 component in

Eq. (15). From above formulas, the CP violation sum rules listed in Eqs. (17) and (18) are derived

if the approximation of

λbλd = −λb(λs + λb) = −(λbλs + λ2b) ≃ −λbλs (29)

is used. Besides, the decay amplitude of D0 → K0K0is expressed as

A(D0 → K0K0) = −λb(2Pa+

1

4a15). (30)

The direct CP asymmetry in D0 → K0K0decay is zero in the flavor SU(3) limit:

AdirCP (D

0 → K0K0) = 0. (31)

For the CP violation relations (17) and (18), the decay amplitudes of two channels are connected

by the interchange of λd ↔ λs, and their initial and final states are connected by the interchange

of d↔ s:

D+ ↔ D+s , D0 ↔ D0, K+ ↔ π+, K− ↔ π−, K0 ↔ K

0. (32)

For D0 → K0K0decay, its corresponding mode in the interchange of d ↔ s is itself. So all CP

violation relations in Eqs. (17), (18) and (31) are associated with U -spin transformation.

On the other hand, Eqs. (17), (18) and (31) include all the SCS modes without π0, η(′) in the

final states in D → PP decays. Mesons π0 and η(′) do not have definite U -spin quantum numbers.

7

Under the interchange of d ↔ s, there are no mesons corresponding to π0 and η(′). For example,

π0 has the quark constituent of (dd − uu)/√2. Under the interchange of d ↔ s, (dd − uu)/

√2

turns into (ss − uu)/√2. No meson has the quark constituent of (ss − uu)/

√2. So those decay

channels involving π0, 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 → PP decays, there are also some CP violation sum rules in the

D → PV decays [87]

AdirCP (D

0 → π−ρ+) +AdirCP (D

0 → K−K∗+) = 0, (33)

AdirCP (D

0 → π+ρ−) +AdirCP (D

0 → K+K∗−) = 0, (34)

AdirCP (D

+ → K0K∗+) +Adir

CP (D+s → K0ρ+) = 0, (35)

AdirCP (D

+ → K+K∗0) +Adir

CP (D+s → π+K∗0) = 0, (36)

AdirCP (D

0 → K0K∗0) +Adir

CP (D0 → K

0K∗0) = 0. (37)

The detailed derivation of these sum rules is similar to D → PP and can be found in Ref. [86].

Again, all the CP violation sum rules inD → PV decays are associated with a complete interchange

of d and s quarks, and all the singly Cabibbo-suppressedD → PV modes with all final states having

definite U -spin quantum numbers are included in Eqs. (33)-(37).

B. 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 ↔ s is still valid for the CP violation

sum rules in charmed baryon decays. The charmed anti-triplet baryon is expressed as

Bc3 =

0 Λ+c Ξ+

c

−Λ+c 0 Ξ0

c

−Ξ+c −Ξ0

c 0

. (38)

8

TABLE I: SU(3) irreducible representation amplitudes in Bc3 → B10M decays, in which only those modes

that all initial and final states have definite U -spin quantum numbers are listed.

Channel Amplitude

Λ+c → ∆0π+ 1

8√3λd(6e1 − 6e2 + 5e3 − 2e4)− 1

8√3λs(2e1 − 2e2 − e3 − 2e4) +

1√3λbPe

Λ+c → Σ∗+K0 1

8√3λd(2e1 − 2e2 + 3e3 + 2e4) +

1

8√3λs(2e1 + 6e2 − e3 − 2e4)− 1√

3λbPe

Λ+c → Σ∗0K+ 1

8√6λd(6e1 + 2e2 + 5e3 − 2e4)− 1

8√6λs(2e1 + 6e2 − e3 − 2e4) +

1√6λbPe

Λ+c → ∆++π− 1

8λd(2e1 − 2e2 + 3e3 + 2e4) +

1

8λs(2e1 − 2e2 − e3 − 2e4)− λbPe

Ξ+c → Ξ∗0K+ − 1

8√3λd(2e1 − 2e2 − e3 − 2e4) +

1

8√3λs(6e1 − 6e2 + 5e3 − 2e4) +

1√3λbPe

Ξ+c → ∆+K

0 1

8√3λd(2e1 + 6e2 − e3 − 2e4) +

1

8√3λs(2e1 − 2e2 + 3e3 + 2e4)− 1√

3λbPe

Ξ+c → Σ∗0π+ − 1

8√6λd(2e1 + 6e2 − e3 − 2e4) +

1

8√6λs(6e1 + 2e2 + 5e3 − 2e4) +

1√6λbPe

Ξ+c → ∆++K− 1

8λd(2e1 − 2e2 − e3 − 2e4) +

1

8λs(2e1 − 2e2 + 3e3 + 2e4)− λbPe

Ξ0c → Σ∗−π+ − 1

2√3λd(e3 − e4) +

1

2√3λs(e3 − e4)

Ξ0c → Ξ∗−K+ − 1

2√3λd(e3 − e4) +

1

2√3λs(e3 − e4)

Ξ0c → ∆0K

0 − 1

8√3λd(6e1 − 6e2 + e3 + 2e4) +

1

8√3λs(2e1 − 2e2 + 3e3 + 2e4)− 1√

3λbPe

Ξ0c → Ξ∗0K0 − 1

8√3λd(2e1 − 2e2 + 3e3 + 2e4) +

1

8√3λs(6e1 − 6e2 + e3 + 2e4) +

1√3λbPe

Ξ0c → Σ∗+π− − 1

8√3λd(2e1 − 2e2 + 3e3 + 2e4) +

1

8√3λs(6e1 + 2e2 + e3 + 2e4) +

1√3λbPe

Ξ0c → ∆+K− − 1

8√3λd(6e1 + 2e2 + e3 + 2e4) +

1

8√3λs(2e1 − 2e2 + 3e3 + 2e4)− 1√

3λbPe

The light baryon decuplet is given as

∆++ = B11110 , ∆− = B222

10 , Ω− = B33310 ,

∆+ =1√3(B112

10 + B12110 + B211

10 ), ∆0 =1√3(B122

10 + B21210 + B221

10 ),

Σ∗+ =1√3(B113

10 + B13110 + B311

10 ), Σ∗− =1√3(B223

10 + B23210 + B322

10 ),

Ξ∗0 =1√3(B133

10 + B31310 + B331

10 ), Ξ∗− =1√3(B233

10 + B32310 + B332

10 ),

Σ∗0 =1√6(B123

10 + B13210 + B213

10 + B23110 + B312

10 + B32110 ). (39)

The SU(3) irreducible representation amplitude of Bc3 → B10M decay can be written as

AtreeBc3→B10M

= e1(Bc3)ijH(15)jklMimBklm

10 + e2(Bc3)ijH(15)klmMjkB

ilm10 + e3(Bc3)ijH(15)jklM

lmBikm

10

+ e4(Bc3)ijH(6)jklMlmBikm

10 + e5(Bc3)ijH(15)jklMmmBikl

10

+ e6(Bc3)ijH(3)kMjmBikm

10 , (40)

ApenguinBc3→B10M

= Pe6(Bc3)ijHP (3)kM

jmBikm

10 + Pe7(Bc3)ijHP (3

′)kM

jmBikm

10 . (41)

9

Similar to D → PP decay, if we define

Pe = e6 + Pe6 + 3Pe7, (42)

the amplitude of Bc3 → B10M decay will be reduced to

Atree+penguinBc3→B10M

= e1(Bc3)ijH(15)jklMimBklm

10 + e2(Bc3)ijH(15)klmMjkB

ilm10 + e3(Bc3)ijH(15)jklM

lmBikm

10

+ e4(Bc3)ijH(6)jklMlmBikm

10 + e5(Bc3)ijH(15)jklMmmBikl

10

+ Pe(Bc3)ijH(3)kMjmBikm

10 . (43)

The first four terms are the same with the formula given in [101]. The fifth term is the decay

amplitude associated with singlet η1, and the six term is the amplitude proportional to λb. With

Eq. (43), the SU(3) irreducible representation amplitudes of Bc3 → B10M 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:

AdirCP (Λ

+c → ∆0π+) +Adir

CP (Ξ+c → Ξ∗0K+) = 0, (44)

AdirCP (Λ

+c → Σ∗+K0) +Adir

CP (Ξ+c → ∆+K

0) = 0, (45)

AdirCP (Λ

+c → Σ∗0K+) +Adir

CP (Ξ+c → Σ∗0K+) = 0, (46)

AdirCP (Λ

+c → ∆++π−) +Adir

CP (Ξ+c → ∆++K−) = 0, (47)

AdirCP (Ξ

0c → Σ∗−π+) +Adir

CP (Ξ0c → Ξ∗−K+) = 0, (48)

AdirCP (Ξ

0c → ∆0K

0) +Adir

CP (Ξ0c → Ξ∗0K0) = 0, (49)

AdirCP (Ξ

0c → Σ∗+π−) +Adir

CP (Ξ0c → ∆+K−) = 0. (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,

Λ+c ↔ Ξ+

c , Ξ0c ↔ Ξ0

c . (51)

For light decuplet baryons,

∆0 ↔ Ξ∗0, Σ∗+ ↔ ∆+, Σ∗0 ↔ Σ∗0, ∆++ ↔ ∆++, Ξ∗− ↔ Σ∗−, ∆− ↔ Ω−. (52)

Also, Eqs. (44)-(50) include all the SCS modes with all associated particles having definite U -spin

quantum numbers in Bc3 → B10M decays.

10

For other types of charm baryon decay, for example Bc3 → B8M decay, multi-body decay and

doubly charmed baryon decay, the treatments of their SU(3) irreducible representation amplitudes

are similar to Bc3 → B10M . Related discussions can be found in Refs. [29, 40, 42, 101]. But

notice that the contributions proportional to λb 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.

C. 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 ↔ s quarks in initial and final states leads to the in-

terchange of d ↔ s in operators Oijk . 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↔ s must be

performed in tensor Hkij in order to keep all covariant and contravariant indices contracted. From

Eq. (11), one can find Hkij corresponds to Oij

k one by one. So the d and s quark constituents in

operators Oijk must be interchanged. In physics, if the quark constituents of all initial and final

particles in one decay channel are replaced by d → s and s → d, the quark constituents in the

effective weak vertexes should be replaced by d → s and s → d also. The operators Oijk are ab-

stracted from the effective weak vertexes, so the quark constituents of operators Oijk transform as

a complete interchange of d↔ s.

Secondly, the interchange of d↔ s in operators Oijk leads to the decay amplitudes proportional

to λd/λs are connected by the interchange of λd ↔ λs and the decay amplitudes proportional to

λb are the same in the flavor SU(3) symmetry. The contributions proportional to λd/λs in SCS

decays are induced by following operators in the SU(3) irreducible representation:

O(6)23, O(15)122 , O(15)133 . (53)

11

Under the interchange of d↔ s, these operators are transformed as

O(6)23 ↔ −O(6)23, O(15)122 ↔ O(15)133 . (54)

These properties can be read from the explicit SU(3) decomposition of Oijk ; see Appendix A. The

corresponding CKM matrix elements then transform as

H(6)23 ↔ −H(6)23, H(15)212 ↔ H(15)313. (55)

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

1

4(λd − λs) ↔ −1

4(λd − λs),

3

8λd −

1

8λs ↔ 3

8λs −

1

8λd. (56)

One can find that Eq. (56) is equivalent to the interchange of λd ↔ λs. The contributions propor-

tional to λb in the SCS decays are induced by the following operators:

O(15)111 , O(3)1, O(3′)1. (57)

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

λd/λs are connected by the interchange of λd ↔ λs and the decay amplitudes proportional to

λb 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

A(i→ f) =λdA+ λsB + λbC

= −(λs + |λb| eiφ) |A| eiδA + λs |B| eiδB + |λb| eiφ |C| eiδC , (58)

its CP asymmetry in the order of O(λb) is derived as

AdirCP (i→ f) =

|λdA+ λsB + λbC|2 − |λ∗dA+ λ∗sB + λ∗bC|2|λdA+ λsB + λbC|2 + |λ∗dA+ λ∗sB + λ∗bC|2

≃ 2|λb|λs

|AB| sin(δA − δB)− |AC| sin(δA − δC) + |BC| sin(δB − δC)

|A|2 + |B|2 − 2|AB| cos(δA − δB)sinφ. (59)

For one decay mode with amplitude of

A(i′ → f ′) = λdB + λsA+ λbC, (60)

which is connected to Eq. (58) by λd ↔ λs, its CP asymmetry in the order of O(λb) is derived as

AdirCP (i

′ → f ′) ≃ −2|λb|λs

|AB| sin(δA − δB)− |AC| sin(δA − δC) + |BC| sin(δB − δC)

|A|2 + |B|2 − 2|AB| cos(δA − δB)sinφ. (61)

12

It is apparent that

AdirCP (i→ f) +Adir

CP (i′ → f ′) ≃ 0. (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↔ 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 DISCUSSIONS

With the method proposed in Sect. 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 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↔ s, the light octet baryons are interchanged as

p↔ Σ+, n↔ Ξ0, Σ− ↔ Ξ−. (63)

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

one octet baryon are

AdirCP (Λ

+c → Σ+K0) +Adir

CP (Ξ+c → pK

0) = 0, (64)

AdirCP (Λ

+c → nπ+) +Adir

CP (Ξ+c → Ξ0K+) = 0, (65)

AdirCP (Ξ

0c → Σ−π+) +Adir

CP (Ξ0c → Ξ−K+) = 0, (66)

AdirCP (Ξ

0c → nK

0) +Adir

CP (Ξ0c → Ξ0K0) = 0, (67)

AdirCP (Ξ

0c → Σ+π−) +Adir

CP (Ξ0c → pK−) = 0. (68)

13

Under the complete interchange of d↔ s, the doubly charmed baryons are interchanged as

Ξ++cc ↔ Ξ++

cc , Ξ+cc ↔ Ω+

cc. (69)

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

and one charmed triplet baryon are

AdirCP (Ξ

++cc → Λ+

c π+) +Adir

CP (Ξ++cc → Ξ+

c K+) = 0, (70)

AdirCP (Ξ

+cc → Ξ+

c K0) +Adir

CP (Ω+cc → Λ+

c K0) = 0, (71)

AdirCP (Ξ

+cc → Ξ0

cK+) +Adir

CP (Ω+cc → Ξ0

cπ+) = 0. (72)

Under the complete interchange of d↔ s, the charmed sextet baryons are interchanged as

Σ+c ↔ Ξ∗+

c , Σ++c ↔ Σ++

c , Ξ∗0c ↔ Ξ∗0

c , Σ0c ↔ Ω0

c . (73)

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

and one charmed sextet baryon are

AdirCP (Ξ

++cc → Σ+

c π+) +Adir

CP (Ξ++cc → Ξ∗+

c K+) = 0, (74)

AdirCP (Ξ

+cc → Σ++

c π−) +AdirCP (Ω

+cc → Σ++

c K−) = 0, (75)

AdirCP (Ξ

+cc → Σ0

cπ+) +Adir

CP (Ω+cc → Ω0

cK+) = 0, (76)

AdirCP (Ξ

+cc → Ξ∗+

c K0) +AdirCP (Ω

+cc → Σ+

c K0) = 0, (77)

AdirCP (Ξ

+cc → Ξ∗0

c K+) +Adir

CP (Ω+cc → Ξ∗0

c π+) = 0. (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

AdirCP (Ξ

++cc → Σ+D+

s ) +AdirCP (Ξ

++cc → pD+) = 0, (79)

AdirCP (Ξ

+cc → pD0) +Adir

CP (Ω+cc → Σ+D0) = 0, (80)

AdirCP (Ξ

+cc → nD+) +Adir

CP (Ω+cc → Ξ0D+

s ) = 0. (81)

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

decuplet baryon are

AdirCP (Ξ

++cc → ∆+D+) +Adir

CP (Ξ++cc → Σ∗+D+

s ) = 0, (82)

AdirCP (Ξ

+cc → ∆+D0) +Adir

CP (Ω+cc → Σ∗+D0) = 0, (83)

AdirCP (Ξ

+cc → ∆0D+) +Adir

CP (Ω+cc → Ξ∗0D+

s ) = 0, (84)

AdirCP (Ξ

+cc → Σ∗0D+

s ) +AdirCP (Ω

+cc → Σ∗0D+) = 0. (85)

14

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:

AdirCP (Λ

+c → pK−K+) +Adir

CP (Ξ+c → Σ+π−π+) = 0, (86)

AdirCP (Λ

+c → pπ−π+) +Adir

CP (Ξ+c → Σ+K−K+) = 0, (87)

AdirCP (Λ

+c → Σ+π−K+) +Adir

CP (Ξ+c → pK−π+) = 0, (88)

AdirCP (Λ

+c → Σ−π+K+) +Adir

CP (Ξ+c → Ξ−K+π+) = 0, (89)

AdirCP (Λ

+c → nK+K

0) +Adir

CP (Ξ+c → Ξ0π+K0) = 0, (90)

AdirCP (Ξ

0c → Σ+K−K0) +Adir

CP (Ξ0c → pπ−K

0) = 0, (91)

AdirCP (Ξ

0c → Σ−K+K

0) +Adir

CP (Ξ0c → Ξ−π+K0) = 0, (92)

AdirCP (Ξ

0c → Ξ0π−K+) +Adir

CP (Ξ0c → nK−π+) = 0. (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:

π+ → ρ+, π− → ρ−, K+ → K∗+, K− → K∗−, K0 → K∗0, K0 → K

∗0. (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 ∆ACP ≡ ACP (D0 → K+K−) − ACP (D

0 → π+π−) 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:

∆Abaryon,1CP = ACP (Λ

+c → Σ+K∗0)−ACP (Ξ

+c → pK

∗0), (95)

∆Abaryon,2CP = ACP (Ξ

0c → Σ+π−)−ACP (Ξ

0c → pK−), (96)


+c → ∆++π−)−ACP (Ξ

+c → ∆++K−), (97)


+c → Σ+π−K+)−ACP (Ξ

+c → pK−π+). (98)

If the contributions proportional to λb 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

15

example, we estimate the branching fraction of Ξ+c → pK−π+. The integration over the phase

space of the three-body decay Bc → BM1M2 relies on the equation of [2]

Γ(Bc → BM1M2) =

∫

m2

12

∫

m2

23

|A(Bc → BM1M2)|232m3

Bc

dm212dm

223, (99)

where m212 = (pM1

+ pM2)2 and m2

23 = (pM2+ pB)2. With the experimental data given in [2],

Br(Λ+c → Σ+π−K+) = (2.1 ± 0.6) × 10−3, (100)

and the relation

|A(Ξ+c → pK−π+)| ≃ |A(Λ+

c → Σ+π−K+)|, (101)

the branching fraction of Ξ+c → pK−π+ decay is predicted to be

Br(Ξ+c → pK−π+) = (1.7 ± 0.5)%. (102)

One can find the branching fraction Br(Ξ+c → pK−π+) is larger than Br(Λ+

c → Σ+π−K+) because

of the larger phase space and the longer lifetime of Ξ+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

Λ+c → Σ+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 U -spin

breaking can be found in [66].

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

fΞb

fΛb

· Br(Ξ0b → Ξ+

c π−)

Br(Λ0b → Λ+

c π−)· Br(Ξ

+c → pK−π+)

Br(Λ+c → pK−π+)

= (1.88 ± 0.04 ± 0.03) × 10−2, (103)

the fragmentation-fraction ratio fΞb/fΛb

is determined to be

fΞb/fΛb

= 0.065 ± 0.020. (104)

Recent measurement confirmed this result [103]. Our result is consistent with the one obtained via

Λ0b → J/ψΛ0 [104], fΞb

/fΛb= 0.11±0.03, the one via Ξ−

b → J/ψΞ− [105], fΞb/fΛb

= 0.108±0.034,

and the one via the diquark model for Ξ−b → Λ0

bπ− [106] using the LHCb data [107], fΞb

/fΛb=

0.08 ± 0.03. The detailed comparison for different methods of estimating fΞb/fΛb

can be found in

[66].

16

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

as fΞb/fΛb

= 0.065 ± 0.020.

Acknowledgments

We are grateful to Fu-Sheng Yu for useful discussions. This work was supported in part by the

National Natural Science Foundation of China under Grants no. U1732101 and the Fundamental

Research Funds for the Central Universities under Grant no. lzujbky-2018-it33.

Appendix A: SU(3) decomposition of operator Oijk

All components of the SU(3) decomposition in Eq. (14) are listed in the following.

3 presentation:

O(3)1 = (uu)(uc) + (ud)(dc) + (us)(sc), O(3)2 = (du)(uc) + (dd)(dc) + (ds)(sc),

O(3)3 = (su)(uc) + (sd)(dc) + (ss)(sc). (A1)

3′presentation:

O(3′)1 = (uu)(uc) + (dd)(uc) + (ss)(uc), O(3

′)2 = (uu)(dc) + (dd)(dc) + (ss)(dc),

O(3′)3 = (uu)(sc) + (dd)(sc) + (ss)(sc). (A2)

6 presentation:

O(6)11 =1

2[(du)(sc)− (su)(dc)], O(6)22 =

1

2[(sd)(uc)− (ud)(sc)],

O(6)33 =1

2[(us)(dc)− (ds)(uc)],

O(6)12 =1

4[(su)(uc)− (uu)(sc) + (dd)(sc)− (sd)(dc)],

O(6)23 =1

4[(ud)(dc)− (dd)(uc) + (ss)(uc)− (us)(sc)],

O(6)31 =1

4[(ds)(sc)− (ss)(dc) + (uu)(dc)− (du)(uc)]. (A3)

17

15 presentation:

O(15)111 =1

2(uu)(uc)− 1

4[(us)(sc) + (ud)(dc) + (dd)(uc) + (ss)(uc)],

O(15)222 =1

2(dd)(dc)− 1

4[(du)(uc) + (ds)(sc) + (uu)(dc) + (ss)(dc)],

O(15)333 =1

2(ss)(sc)− 1

4[(su)(uc) + (sd)(dc) + (uu)(sc) + (dd)(sc)],

O(15)231 =1

2[(du)(sc) + (su)(dc)], O(15)132 =

1

2[(sd)(uc) + (ud)(sc)],

O(15)123 =1

2[(us)(dc) + (ds)(uc)],

O(15)112 = (ud)(uc), O(15)113 = (us)(uc), O(15)221 = (du)(dc),

O(15)223 = (ds)(dc), O(15)331 = (su)(sc), O(15)332 = (sd)(sc),

O(15)211 =3

8[(uu)(dc) + (du)(uc)] − 1

4(dd)(dc)− 1

8[(ds)(sc) + (ss)(dc)],

O(15)122 =3

8[(ud)(dc) + (dd)(uc)]− 1

4(uu)(uc)− 1

8[(us)(sc) + (ss)(uc)],

O(15)311 =3

8[(uu)(sc) + (su)(uc)]− 1

4(ss)(sc)− 1

8[(sd)(dc) + (dd)(sc)],

O(15)133 =3

8[(us)(sc) + (ss)(uc)]− 1

4(uu)(uc)− 1

8[(ud)(dc) + (dd)(uc)],

O(15)322 =3

8[(dd)(sc) + (sd)(dc)]− 1

4(ss)(sc)− 1

8[(su)(uc) + (uu)(sc)],

O(15)233 =3

8[(ds)(sc) + (ss)(dc)]− 1

4(dd)(dc)− 1

8[(du)(uc) + (uu)(dc)]. (A4)

The above results are consistent with Ref. [86]. The operators O(6)ij are symmetric in the in-

terchange of their two indices and can be written as O(6)ijk by contracting with the totally an-

tisymmetric Levi-Civita tensor, O(6)ijk = ǫijlO(6)lk. There are 18 operators in the irreducible

representation 15, but only 15 of them are independent because of the following equations:

O(15)111 = −[O(15)122 +O(15)133 ], O(15)222 = −[O(15)211 +O(15)233 ],

O(15)333 = −[O(15)311 +O(15)322 ]. (A5)

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Sum rules for CP asymmetries of charmed baryon decays in