link.springer.com … · eur. phys. j. c (2020) 80:359 regular article - theoretical physics...
TRANSCRIPT
Eur. Phys. J. C (2020) 80:359https://doi.org/10.1140/epjc/s10052-020-7862-5
Regular Article - Theoretical Physics
Unification of flavor SU(3) analyses of heavy Hadron weak decays
Xiao-Gang He1,2,3,4,a, Yu-Ji Shi1,b, Wei Wang1,c
1 INPAC, SKLPPC, MOE KLPPC, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China2 T.-D. Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China3 Department of Physics, National Taiwan University, Taipei 106, Taiwan4 National Center for Theoretical Sciences, Hsinchu 300, Taiwan
Received: 10 February 2020 / Accepted: 19 March 2020 / Published online: 5 May 2020© The Author(s) 2020
Abstract Analyses of heavy mesons and baryons hadroniccharmless decays using the flavor SU(3) symemtry can beformulated in two different forms. One is to construct theSU(3) irreducible representation amplitude by decomposingeffective Hamiltonian, and the other is to draw the topo-logical diagrams. In the flavor SU(3) limit, we study vari-ous B/D → PP, V P, VV , Bc → DP/DV decays, andtwo-body nonleptonic decays of beauty/charm baryons, anddemonstrate that when all terms are included these two waysof analyzing the decay amplitudes are completely equiv-alent. Furthermore we clarify some confusions in draw-ing topological diagrams using different ways of describingbeauty/charm baryons.
1 Introduction
Weak decays of heavy mesons and baryons carrying a bot-tom and/or a charm quark are of great interests and havebeen studied extensively on both experimental and theoreti-cal sides. These decays provide useful information about thestrong and electroweak interactions in the standard model(SM). Rare decays are ideal to look for new physics effectsbeyond SM, and recent measurements of lepton flavor uni-versality have shown notable deviations from the standardmodel (see Ref. [1] for a recent brief review on the anoma-lies in B decays). Quite a number of physical observableslike branching fractions, CP asymmetries and polarizationshave been precisely measured by experiments [2–4]. On theother hand, due to our limited understanding of QCD at lowenergy regions, theoretical calculations of decay amplitudesare not well understood. Most of the current calculations relyon the factorization methods. Among them, many available
a e-mail: [email protected] e-mail: [email protected] (corresponding author)c e-mail: [email protected]
studies are conducted at leading power in 1/mb, while recentanalyses of semileptonic and radiative processes have indi-cated the importance of next-to-leading power corrections[5,6].
Apart from factorization approaches, the flavor SU(3)symmetry is a powerful tools frequently used in two-bodyand three-body heavy meson decays [7–20]. Although fla-vor SU(3) symmetry is approximate, yet it still provides veryuseful information about the decays. Since the SU(3) invari-ant amplitudes can be determined by fitting the data, theSU(3) analysis bridges experimental data and the dynami-cal approaches.
Among different realizations of carrying out SU (3) analy-sis for decay amplitudes there are two popular methods. Oneof them is topological diagram amplitude (TDA) method,where decay amplitudes are represented by connecting quarklines flows in different ways and then relate them by SU (3)
symmetry, and another way is to construct the SU(3) irre-ducible representation amplitude (IRA) by decomposingeffective Hamiltonian. The TDA approach gives a betterunderstanding of decay dynamics especially in the bottomhadron decays, where the power expansion of 1/mb canbe properly conducted so that each TDA amplitude can berelated with matrix elements of SCET operators [21]. On theother hand, the IRA approach shows a convenient connec-tion with the SU(3) symmetry. These two methods looks verydifferent in formulations, one may wonder whether they willobtain the same results. This equivalence for some considereddecays has been discussed in Refs. [7,16,20], in particular forcharmed meson decays, Ref. [20] has shown the equivalencein the presence of SU(3) symmetry breaking effects. In Ref.[22], two of us have explored two-body B/D meson decays,B → PP and D → PP and pointed out that in the exactSU(3) symmetry limit, the two methods are consistent whenall contributions are included. However this equivalence isnontrivial: a few amplitudes are suppressed and thus were not
123
359 Page 2 of 36 Eur. Phys. J. C (2020) 80 :359
included in some TDA analysis; among the known diagram-matic amplitudes, one of them is not SU(3) independent, andshould be absorbed into other amplitudes. Actually, in charmmeson decays, the fact that one of the known amplitudes,T, A,C, E , is not independent has already been noticed inRef. [20].
In this work, we extend our analysis to several othertypes of two body decays of B/D and Bc mesons and alsobeauty/charm baryons to show the equivalence of the TDAand IRA methods. We will also work in the exact flavorSU(3) limit throughout this work. For two-body decays ofbeauty/charm baryons, we clarify some subtleties includingthe description of baryon representation, one or two indicesfor 3, in relation to TDA. As we will show, it is easy to deter-mine the independent amplitudes in IRA while TDA givessome redundancy. A few amplitudes are not independent andtherefore should be absorbed into other amplitudes. Despitethis disadvantage, the topological nature of TDA is still help-ful for understanding the internal dynamics underlying in bdecays in a more intuitive way.
The rest of this paper is arranged as follows. In Sect. 2, webriefly summarize the SU(3) properties of various inputs. InSect. 3, we give the results for B → PP in the TDA and IRAmethods to set up the notation. Then we provide results forB → PV, VV and discuss some points specialized to thesedecays. In Sect. 4, we give the results for D → PP, VV, PVin the TDA and IRA methods. In Sect. 5, we carry out a simi-lar analysis for Bc → DP, DV . In Sects. 6 and 7, we respec-tively discuss beauty and charm baryon decays into an octetbaryon and an octet pseudo-scalar meson, while for beautybaryon decays we also consider the final states containinga decuplet baryon. The expanded amplitudes and relationsgiven these sections are useful for a global analysis whenenough data is available in the future. In Sect. 8 we sum-marize our results. In the “Appendix”, we give the relationsfor different parametrizations in TDA and IRA methods forbottom and charmed baryon decays.
2 SU(3) properties of Hamiltonian and Hadron states
2.1 Hadron multiplets
Several classes of heavy hadron, containing at heavy quark bor c, will be considered in this work. The involved processesinclude decays of heavy SU(3) triplet mesons B and D intoPP, PV, VV , and the Bc meson into DP, DV . For heavybaryons, the decay processes include a heavy anti-triplet Tc3or a Tb3 decays into a baryon in the decuplet T10 plus a lightmeson, and decay into a baryon in the octet T8 plus a lightmeson. We display the hadron SU (3) properties and theircomponent fields in this section.
The Bc meson contains no light quark and it is a singlet.Heavy mesons containing one heavy quark
(Bi ) = (B−(bu), B0(bd), B
0s (bs)) ,
(Di ) = (D0(cu), D+(cd), D+s (cs)) , (1)
are flavor SU(3) anti-triplets.Light pseudoscalar P and vectorV mesons are mixtures of
octets and singlets so that each of them contain nine hadrons:
P =
⎛⎜⎜⎝
π0√2
+ η8√6
+ η1√3
π+ K+
π− η8√6
− π0√2
+ η1√3
K 0
K− K0 η1√
3− 2 η8√
6
⎞⎟⎟⎠ ,
V =
⎛⎜⎜⎝
ρ0+ω√2
ρ+ K ∗+
ρ− −ρ0+ω√2
K ∗0
K ∗− K∗0
φ
⎞⎟⎟⎠ , (2)
where ω and φ mix in an ideal form. The η and η′ are mixturesof η8 and η1 with the mixing angle θ :
η = cos θη8 + sin θη1,
η′ = − sin θη8 + cos θη1. (3)
Since η8 and η1 are not physical states, optionally one canchoose the ηq and ηs basis for the η mixing, which aredefined so that the pseudoscalar octets P has the same formof parametrization as vector octets V :
P =
⎛⎜⎜⎝
π0+ηq√2
π+ K+
π− −π0+ηq√2
K 0
K− K0
ηs
⎞⎟⎟⎠ . (4)
with
η8 = 1√3ηq −
√2
3ηs, η1 =
√2
3ηq + 1√
3ηs (5)
An advantage of the parametrization in Eq. (4) is that there isa one-to-one correspondence between the decay amplitudesof channels with vector final state and that of channels withpseudoscalar final state in the SU(3) limit.
A charmed or a bottom baryons with two light quarkscan form an anti-triplet or sextet. Most members of the sex-tet can decay through strong interaction or electromagneticinteractions. The only exceptions are �b and �c [24]. We willconcentrate on anti-triplet weak decays. For the anti-tripletbottom and charmed baryons, we have the following matrixexpressions:
(T i jc3
) =
⎛⎜⎜⎝
0 +c +
c
−+c 0 0
c
−+c −0
c 0
⎞⎟⎟⎠ ,
123
Eur. Phys. J. C (2020) 80 :359 Page 3 of 36 359
(T i jb3
) =
⎛⎜⎜⎝
0 0b 0
b
−0b 0 −
b
−0b −−
b 0
⎞⎟⎟⎠ . (6)
One can also contract the above matrix with the anti-symmetric tensor εi jk (ε123 = +1) to have T3,i = εi jkT
jk3
with
((Tc3)i ) = (0
c −+c +
c
), ((Tb3)i ) = (
−b −0
b 0b
).
(7)
The lowest-lying baryon octet is given by:
((T8)ij ) =
⎛⎜⎜⎝
1√2�0 + 1√
60 �+ p
�− − 1√2�0 + 1√
60 n
− 0 −√
230
⎞⎟⎟⎠ .
(8)
One can also contract the above with εi jk to have (T8)i jk ≡εi jn(T8)
nk .
The light baryon decuplet is given as:
T 11110 = ++, T 112
10 = T 12110 = T 211
10 = 1√3 +,
T 22210 = −, T 122
10 = T 21210 = T 221
10 = 1√3 0,
T 11310 = T 131
10 = T 31110 = 1√
3�′+,
T 22310 = T 232
10 = T 32210 = 1√
3�′−,
T 12310 = T 132
10 = T 21310 = T 231
10 = T 31210 = T 321
10 = 1√6�′0,
T 13310 = T 313
10 = T 33110 = 1√
3′0,
T 23310 = T 323
10 = T 33210 = 1√
3′−, T 333
10 = �−. (9)
2.2 SU (3) properties of effective Hamiltonian
Effective Hamiltonian for charmless b decaysIn the SM weak decays of charmless b decays are induced
by the following electroweak effective Hamiltonian [25–27]:
Hbe f f = GF√
2
{VubV
∗uq [C1O1 + C2O2] − VtbV
∗tq
10∑i=3
Ci Oi
}+ h.c..
(10)
Here GF is the Fermi constant, and the Vuq and Vtq are CKMmatrix elements. The Oi is a four-quark operator with Ci asits Wilson coefficient. The explicit forms of Oi s are given asfollows:
O1 = (qi u j )V−A(u j bi )V−A,
O2 = (qu)V−A(ub)V−A,
O3 = (qb)V−A
∑q ′
(q ′q ′)V−A,
O4 = (qi b j )V−A
∑q ′
(q ′ j q ′i )V−A,
O5 = (qb)V−A
∑q ′
(q ′q ′)V+A,
O6 = (qi b j )V−A
∑q ′
(q ′ j q ′i )V+A,
O7 = 3
2(qb)V−A
∑q ′
eq ′(q ′q ′)V+A,
O8 = 3
2(qi b j )V−A
∑q ′
eq ′(q ′ j q ′i )V+A,
O9 = 3
2(qb)V−A
∑q ′
eq ′(q ′q ′)V−A,
O10 = 3
2(qi b j )V−A
∑q ′
eq ′(q ′ j q ′i )V−A. (11)
q = d, s and q ′ = u, d, s. Here the V − A and V + Acorresponds a left-handed γμ(1 − γ5) and a right-handedcurrent γμ(1 + γ5) respectively.
In the SU(3) group for light flavors, tree operators O1,2 andelectroweak penguin operators O7−10 can be decomposed interms of a vector Hi
3, a traceless tensor antisymmetric in
upper indices, (H6)[i j]k , and a traceless tensor symmetric in
upper indices, (H15){i j}k . For the S = 0(b → d) decays,
the non-zero components of the effective Hamiltonian are[8,11,12]:
(H3)2 = 1, (H6)
121 = −(H6)
211 = (H6)
233 = −(H6)
323 = 1,
2(H15)121 = 2(H15)
211 = −3(H15)
222
= −6(H15)233 = −6(H15)
323 = 6. (12)
For the S = −1(b → s) decays the nonzero entries in theH3, H6, H15 can be obtained from Eq. (12) with the exchange2 ↔ 3 corresponding to the d ↔ s exchange.
QCD penguin operators O3−6 behave as the 3 representa-tion. For the magnetic moment operators, the color magneticmoment operator O8g = (gsmb/4π)sσμνT aGa
μν(1+γ5)b isan SU(3) triplet, while the electromagnetic moment operatorO7γ = emb
4πsσμνFμν(1 + γ5)b can be effectively incorpo-
rated into the O7−10. Thus both of them are not included inEq. (10) and the above decomposition is complete.
The irreducible representation amplitude (IRA) method ofdescribing related decays is to decompose effective Hamil-tonian according to the above mentioned representations andconstruct the amplitudes accordingly. On the other hand thethe topological diagrams (TDA) method is to take the effec-tive Hamiltonian with two light anti-quarks and a light quark
123
359 Page 4 of 36 Eur. Phys. J. C (2020) 80 :359
Hi jk to represent quub with i = u, k = u and j = q (omit-
ting the Lorentz indicies), and then contract the indices withinitial and final hadron states. In this way the decays are rep-resented by diagrams following the quark line flows. Notethat in the TDA method, the indices i and j ordering matterswhich are neither symmetry nor anti-symmetric. They arenot traceless neither.
The effective Hamiltonian have both tree and loop con-tributions. When strong penguin and electroweak penguinare all included the tree and loop contributions have 3, 6and 15 representations. The independent amplitudes havethe same numbers, except that one can make one of thetree or penguin amplitude real and the rest all in principlecomplex. Using the unitarity property of the CKM matrixVubV ∗
uq + VcbV ∗cq + VtbV ∗
tq = 0, one can also rewrite c loopinduced penguin contributions into amplitude proportionalto VubV ∗
uq and VtbV ∗tq ,
A = VubV∗uqAu + VtbV
∗tqAt . (13)
For simplicity, we refer to Au as “tree” amplitude since it isdominated by tree contributions with modifications from uand c loop contributions. At is “penguin” amplitude with cand t loop contributions. It is necessary to stress that not allcontributions in Au are tree diagrams in topology, and thesame for At .
In Ref. [22], two of us have shown that both Au and At
have similar form of amplitudes with SU (3) representations.Thus in our later discussions we will concentrate on the Au
amplitudes. One can easily obtain the At amplitudes by justchanging the amplitude labels.
Effective electroweak Hamiltonian for c hadronic decaysFor hadronic decays of charmed hadrons, the effective
Hamiltonian with C = 1 is given as:
Hce f f = GF√
2
{VcsV
∗ud [C1O
sd1 + C2O
sd2 ]
+VcdV∗ud [C1O
dd1 + C2O
dd2 ]
+VcsV∗us[C1O
ss1 + C2O
ss2 ]
+VcdV∗us[C1O
ds1 + C2O
ds2 ]}, (14)
where we have neglected the highly suppressed penguin con-tributions, and
Osd1 = [siγμ(1 − γ5)c
j ][uiγ μ(1 − γ5)dj ],
Osd2 = [sγμ(1 − γ5)c][uγ μ(1 − γ5)d], (15)
while other operators can be obtained by replacing the d, squark fields. Tree operators transform under the flavor SU(3)symmetry as 3 ⊗ 3 ⊗ 3 = 3 ⊕ 3 ⊕ 6 ⊕ 15.
For the Cabibbo allowed c → sud transition, we haveamplitudes proportional to VcsV ∗
ud and the Hamiltonians are:
(H6)312 = −(H6)
132 = 1, (H15)
312 = (H15)
132 = 1. (16)
For the doubly Cabbibo suppressed c → dus transition, wehave amplitudes to be proportional to VcdV ∗
us and the Hamil-tonians are:
(H6)213 = −(H6)
123 = 1, (H15)
213 = (H15)
123 = 1. (17)
For decays proportional to VcsV ∗us , we have:
(H6)313 = −(H6)
133 = 1, (H15)
313 = (H15)
133 = 1, (18)
and for decays proportional to VcdV ∗ud , we have:
(H6)122 = −(H6)
212 = 1, (H15)
122 = (H15)
212 = −1. (19)
For singly Cabbibo suppressed decays, c → udd andc → uss transitions have approximately equal magnitudesbut opposite signs:VcdV ∗
ud = −VcsV ∗us−VcbV ∗
ub ≈ −VcsV ∗us
(with 10−3 deviation). As a result, the contributions from the3 representation vanish, and one has the nonzero componentscontributed only by 6 and 15 representations.
For the singly Cabibbo-suppressed transition, there arealso loop contributions proportional to VcbV ∗
ub. Such loopcontributions are small so that we will concentrate on thedominant amplitude proportional to VcsV ∗
us . However, onecan include these contributions by adding a 3 representationin the Hamiltonian.
Again, we use the above SU (3) decompositions for IRAanalysis and use the effective Hamiltonian Hi j
k with i = s,j = u and k = q for TDA analysis to trace the quark lineflows.
3 Charmless two-body B decays
3.1 B → PP decays
Let us start with the B → PP decays. The generic amplitudeis decomposed according to CKM matrix elements:
A = VubV∗uqAI RA
u + VtbV∗tqAI RA
t
= VubV∗uqAT DA
u + VtbV∗tqAT DA
t , (20)
where the amplitudes expressed by IRA and TDA should beequivalent. Although in fact this equivalence is concrete andobvious, as we argued in [22], it has not been thoroughlydiscussed in the literature.
To obtain IRA, one takes various representations inEq. (12) and contracts all indices in Bi and light meson Pi
jwith various combinations:
AI RAu = AT
3 Bi (H3)i P j
k Pkj + CT
3 Bi (H3)k Pi
j Pjk
+BT3 Bi (H3)
i Pkk P
jj + DT
3 Bi (H3)j Pi
j Pkk
+AT6 Bi (H6)
[i j]k Pl
j Pkl + CT
6 Bi (H6)[ jl]k Pi
j Pkl
+BT6 Bi (H6)
[i j]k Pk
j Pll
123
Eur. Phys. J. C (2020) 80 :359 Page 5 of 36 359
Fig. 1 Topological diagrams for the amplitudes with CKM factorVubV ∗
uq in B → PP and B → VV decays. We work in the effec-tive field theory at mb scale, and thus there exists four-quark interactionoperators as shown in the above. The quark flavors corresponding to
these operators are explicitly given. In these diagrams, when the uuannihilate, two or more gluons are needed to create one pair of quarkswith flavor u, d, s, which are denoted by the unspecified lines
+AT15Bi (H15)
{i j}k Pl
j Pkl + CT
15Bi (H15){ jk}l Pi
j Plk
+BT15Bi (H15)
{i j}k Pk
j Pll . (21)
For the TDA decomposition, one can classify the differenttopologies of diagram as [18]:
(i) T , denoting the color-allowed tree amplitude with Wemission;
(ii) C , denoting the color-suppressed tree diagram;(iii) E denoting the W -exchange diagram;(iv) P , corresponding to the QCD penguin contributions;(v) S, being the flavor singlet QCD penguin;
(vi) A, annihilation diagrams.
With each coefficient denoted by the above topologies, onehas the TDA decomposition as:
AT DAu = T Bi H
jlk Pi
j Pkl + CBi H
l jk Pi
j Pkl
+ABi Hilj P
jk P
kl + EBi H
lij P
jk P
kl
+Su Bi Hl jl Pi
j Pkk + Pu Bi H
lkl Pi
j Pjk
+PuABi H
lil P j
k Pkj + SuS Bi H
lil P j
j Pkk
+EuS Bi H
jil Pl
j Pkk + Au
S Bi Hi jl Pl
j Pkk . (22)
According to this decomposition, topological diagrams forB → PP decays can be found in Fig. 1. Since we workin the effective field theory at mb scale, as shown in thisfigure there exists four-quark interaction operators. In thesediagrams, the quark flavors corresponding to these operatorsare explicitly given. When the uu annihilate, two or moregluons are needed to create one pair of quarks with flavoru, d, s, which are denoted by the unspecified lines. Apart
123
359 Page 6 of 36 Eur. Phys. J. C (2020) 80 :359
from the ordinary T,C, A, E , we have also included the otherSU(3) irreducible amplitudes, most of which come from loopdiagrams, and/or the flavor singlet diagram.
Expanding Eqs. (21,22), one obtains B → PP ampli-tudes in Table 1, where the IRA amplitudes are consistentwith Ref. [12] and an earlier work [23]. Since we have decom-posed the effective Hamiltonian into irreducible representa-tions, one may expect that there are 10 independent ampli-tudes for Au and similarly 10 amplitudes for At . A carefulexamination shows that the AT
6 can be absorbed into BT6 and
CT6 with a redefinition:
CT ′6 = CT
6 − AT6 , BT ′
6 = BT6 + AT
6 . (23)
This combination can also be found explicitly from Table 1.After eliminating the redundant amplitude, there are actuallyonly 18 (Au and At contribute 9 each) SU(3) independentamplitudes. Since one overall phase can be chosen free, thereare 35 independent real independent parameters. If one con-sider η1 − η8 mixing, the mixing angle θ should also beintroduced.
We list all TDA amplitudes in Table 1. It is necessary topoint out that the last six diagrams in Fig. 1 are omitted insome SU(3) TDA analysis [16,18], while other amplitudesare consistent with Refs. [16,18]. However only by includingthem the complete equivalence of IRA and TDA can be estab-lished. One of the 10 TDA amplitudes must be redundant.Such redundancy can be understood through the followingrelations between the IRA and TDA amplitudes:
T + E = 4AT15 + 2C ′T
6 + 4CT15,
C − E = −4AT15 − 2C ′T
6 + 4CT15,
A + E = 8AT15,
Pu − E = −5AT15 + CT
3 − C ′T6 − CT
15,
PuA + E
2= AT
3 + AT15,
EuS + E = 4AT
15 − 2B ′T6 + 4BT
15,
AuS − E = −4AT
15 + 2B ′T6 + 4BT
15,
SuS − E
2= −2AT
15 + BT3 + B ′T
6 − BT15,
Su + E = 4AT15 − B ′T
6 − BT15 + C ′T
6 − CT15 + DT
3 . (24)
We have adopted the choice in which E is always in com-panion with another amplitude. It is also possible to replacethe role of E by one of the amplitudes A, C or even T . Onecan also reversely obtain:
AT3 = − A
8+ 3E
8+ Pu
A,
BT3 = SuS + 3Eu
S − AuS
8,
CT3 = 1
8(3A − C − E + 3T ) + Pu,
DT3 = Su + 1
8(3C − Eu
S + 3AuS − T ),
B ′T6 = 1
4(A − E + Au
S − EuS),
C ′T6 = 1
4(−A − C + E + T ),
AT15 = A + E
8,
BT15 = Au
S + EuS
8,
CT15 = C + T
8. (25)
Similar analysis for the At contributions can be obtainedwith the replacement for the IRA:
ATi → AP
i , BTi → BP
i , CTi → CP
i , DTi → DP
i . (26)
while for TDA, we have:
T → PT , C → PC , A → PT A, Pu → P, E → PT E ,
PuA → PA, Eu
S → PAS, AuS → PES, SuS → PSS, Su → S.
(27)
It should be pointed out that the amplitudes generatedQ7,8,9,10 operators have the form (2/3)uu− (1/3)(dd + ss)and these amplitudes can be can again be as a sum of a tree-like operator uu and a penguin-like operator −(1/3)(uu +dd + ss). Penguin-like contributions have been absorbedinto P, PA, S, PSS , while tree-like amplitudes are denotedas PT , , PC , PT A, PT E , PES . It should be noticed thatthe tree-like amplitudes PT and PC are also denoted asPEW , PEW,C in some references.
3.1.1 Impact of the new TDA amplitudes
The new TDA amplitudes in Fig. 1 may play an important rolein understanding CP violation (CPV) phenomena. Withoutthe new TDA amplitudes, some decays only have terms pro-
portional to V ∗tqVtb, such as B
0 → K 0 K 0 and B0s → K 0 K 0.
For instance, in Ref. [18], the amplitudes for B0 → K 0 K 0
read:
A(B0 → K 0 K 0) = VtbV
∗td
(P − 1
2PCEW + 2PA
). (28)
This would imply the CP violating asymmetry is identicallyzero. However, as we have shown, these two decays receivecontributions from the Pu + 2Pu
A multiplied by V ∗uqVub:
A(B0 → K 0 K 0) = VubV
∗ud(P
u + 2PuA)
+VtbV∗td(P + 2Pu
A). (29)
123
Eur. Phys. J. C (2020) 80 :359 Page 7 of 36 359
Table 1 Decay amplitudes for two-body B → PP decays. Only the amplitudes with CKM factor VubV ∗uq are shown in this table and the following
ones. “Penguin” amplitudes with VtbV ∗tq can be obtained with the replacement Eq. (26) and Eq. (27)
b → d IRA TDA
B− → π0π− 4√
2CT15 (C + T)/
√2
B− → π−ηq√
2(AT
6 + 3AT15 + BT
6 + 3BT15 + CT
3 + 2CT15 + DT
3
)(2Au
s + 2A + C + 2Pu + 2Su + T)/√
2
B− → π−ηs BT6 + 3BT
15 + CT6 − CT
15 + DT3 Au
s + Su
B− → K 0K− AT6 + 3AT
15 + CT3 − CT
6 − CT15 A + Pu
B0 → π+π− 2AT
3 − AT6 + AT
15 + CT3 + CT
6 + 3CT15 2Pu
A + E + Pu + T
B0 → π0π0 2AT
3 − AT6 + AT
15 + CT3 + CT
6 − 5CT15 2Pu
A − C + E + Pu
B0 → π0ηq −AT
6 + 5AT15 − BT
6 + 5BT15 − CT
3 + 2CT15 − DT
3 EuS + E − Pu − Su
B0 → π0ηs −(BT
6 − 5BT15 + CT
6 − CT15 + DT
3 )/√
2 (EuS − Su)/
√2
B0 → K+K− 2
(AT
3 + AT15
)2Pu
A + E
B0 → K 0K
02AT
3 + AT6 − 3AT
15 + CT3 − CT
6 − CT15 2Pu
A + Pu
B0 → ηqηq 2AT
3 − AT6 + AT
15 + 4BT3 − 2BT
6 + 2BT15 + CT
3 − CT6 + CT
15 + 2DT3 2Pu
A + C + 2EuS + E + Pu + 2Su + 4SuS
B0 → ηqηs (4BT
3 + BT6 − BT
15 + CT6 − CT
15 + DT3 )/
√2 (Eu
S + Su + 4SuS )/√
2
B0 → ηsηs 2(AT
3 + AT6 − AT
15 + BT3 + BT
6 − BT15) 2(Pu
A + SuS )
B0s → π0K 0 (AT
6 + AT15 − CT
3 − CT6 + 5CT
15)/√
2 (C − Pu)/√
2
B0s → π−K+ −AT
6 − AT15 + CT
3 + CT6 + 3CT
15 Pu + T
B0s → K 0ηq −(AT
6 + AT15 + 2BT
6 + 2BT15 − CT
3 + CT6 − CT
15 − 2DT3 )/
√2 (C + Pu + 2Su)/
√2
B0s → K 0ηs −AT
6 − AT15 − BT
6 − BT15 + CT
3 − 2CT15 + DT
3 Pu + Su
b → s IRA TDA
B− → π0K− (AT6 + 3AT
15 + CT3 − CT
6 + 7CT15)/
√2 (A + C + Pu + T)/
√2
B− → π−K0
AT6 + 3AT
15 + CT3 − CT
6 − CT15 A + Pu
B− → K−ηq (AT6 + 3AT
15 + 2BT6 + 6BT
15 + CT3 + CT
6 + 5CT15 + 2DT
3 )/√
2 (2Aus + A + C + Pu + 2Su + T)/
√2
B− → K−ηs AT6 + 3AT
15 + BT6 + 3BT
15 + CT3 − 2CT
15 + DT3 Au
s + A + Pu + Su
B0 → π+K− −AT
6 − AT15 + CT
3 + CT6 + 3CT
15 Pu + T
B0 → π0K
0(AT
6 + AT15 − CT
3 − CT6 + 5CT
15)/√
2 (C − Pu)/√
2
B0 → K
0ηq −(AT
6 + AT15 + 2BT
6 + 2BT15 − CT
3 + CT6 − CT
15 − 2DT3 )/
√2 (C + Pu + 2Su)/
√2
B0 → K
0ηs −AT
6 − AT15 − BT
6 − BT15 + CT
3 − 2CT15 + DT
3 Pu + Su
B0s → π+π− 2
(AT
3 + AT15
)2Pu
A + E
B0s → π0π0 2(AT
3 + AT15) 2(Pu
A + (E)/2)
B0s → π0ηq −2
(AT
6 − 2AT15 + BT
6 − 2BT15
)EuS + E
B0s → π0ηs −√
2(BT
6 − 2BT15 + CT
6 − 2CT15
)(C + Eu
S)/√
2
B0s → K+K− 2AT
3 − AT6 + AT
15 + CT3 + CT
6 + 3CT15 2Pu
A + E + Pu + T
B0s → K 0K
02AT
3 + AT6 − 3AT
15 + CT3 − CT
6 − CT15 2Pu
A + Pu
B0s → ηqηq 2(AT
3 + AT15 + 2
(BT
3 + BT15
)) 2(Pu
A + EuS + (E)/2 + 2SuS )
B0s → ηqηs
√2
(2BT
3 − BT15 + CT
15 + DT3
)(C + Eu
S + 2Su + 4SuS )/√
2
B0s → ηsηs AT
3 − 2AT15 + BT
3 − 2BT15 + CT
3 − 2CT15 + DT
3 PuA + Pu + Su + SuS
123
359 Page 8 of 36 Eur. Phys. J. C (2020) 80 :359
Therefore a non-vanishing direct CP asymmetry is obtained,as noticed in many references for instance Refs. [28–30].This would certainly affect the search for new physics in aprecise CP violation measurement.
Most new TDA amplitudes in Fig. 1 arise from higherorder loop corrections, and thus they are likely small inmagnitude. However, sometimes they can not be completelyneglected. In Ref. [12], the authors have performed a fit ofB → PP decays in the IRA framework. Depending on dif-ferent choices of data, four cases were considered in theiranalysis [12]. Here for illustration, we give their results incase 4:
|CT3| = −0.211 ± 0.027, δT
3= (−140 ± 6)◦,
|BT15
| = −0.038 ± 0.016, δBT15
= (78 ± 48)◦, (30)
where the magnitudes and strong phases are defined relativeto the amplitude CP
3. From Eq. (25), one can find that the
CT3
is a mixture of T , C and others, while the BT15
equals(Eu
S + AuS)/8. The fitted results in Eq. (30) indicate, com-
pared to CT3
, the BT15
could reach 20% in magnitude, andmore notably, the strong phases are sizably different. Thefact that the BT
15, namely Eu
S and AuS , have non-negligible
contributions supports our call for a complete analysis.
3.1.2 Comparison with QCDF amplitudes
The topological amplitudes in B → PP decays can be com-pared to the QCDF amplitude in Ref. [31]. Such a compari-son requests two remarks. Firstly, in our decomposition, weadopt the CKM matrix elements VubV ∗
uq and VtbV ∗tq , while
Ref. [31] used VubV ∗uq and VcbV ∗
cq . The unitarity of CKMmatrix guarantees the equivalence of the two approaches.So we will directly compare the “tree” Au and “penguin”At amplitudes, though some of them might be recombinedin order to have the same CKM factors. Secondly, we havedecomposed one part of the electroweak penguin into theQCD penguin as shown in Sect. 2, and we will do so forQCDF amplitudes too.
We have the following correspondence between the SU(3)TDA amplitudes and the QCDF amplitudes for “tree” ampli-tudes:
T → α1, Pu → αu4 + βu
3 ,
C → α2, Su → αu3 + βu
S3, A → β2,
E → β1, PuA → βu
4 , AuS → βS2,
EuS → βS1, SuS → βu
S4. (31)
where the notations αi and βi are from Ref. [31]. The corre-spondence for “penguin” ones is given as:
PT → αc4,EW , P → αc
4 + βc3,
PC → αc3,EW , S → αc
3 + βcS3,
PT A → βc3,EW ,
PT E → βc4,EW , PA → βc
4,
PES → βcS3,EW , PAS → βc
S4,EW , PSS → βcS4. (32)
3.1.3 U-Spin relations
Some decay channels shown in Table 1 with S = 0 and S = 1 are related by U-spin, the d ↔ s exchange sym-metry. The relations will be discussed explicitly in the fol-lowing. These pairs of channels include: B− → K 0K−
and B− → π−K0; B
0 → π+π− and B0s → K+K−;
B0 → K 0K
0and B
0s → K 0K
0; B
0 → K−K+ and B0s →
π+π−; B0s → π0K 0 and B
0 → π0K0; B
0s → π−K+ and
B0 → π+K−; B
0s → K 0π0 and B
0 → π0K0.
In the past years, there have been extensive examinationson the U-spin symmetry. One of the interesting features ofthese U-spin pairs is that there are CP violating relationamong them. Here we consider two U -spin related decayswith the same “tree” Au and and “penguin” At
1:
A(Bi → PP, S = 0) = VubV∗udAu + VtbV
∗tdAt ,
A(Bi → PP, S = 1) = VubV∗usAu + VtbV
∗tsAt . (33)
Through the relation Im(VubV ∗udV
∗tbVtd) = −Im(VubV ∗
usV ∗tbVts), one can obtain the CP violating rate difference
(Bi → PP, S) = �( S) − �( S) [9,10,32]
(Bi → PP, S = 0) = − (Bj → PP, S = 1) . (34)
This leads to a relation between branching ratio and CP asym-metry Ai
CP ( S) = (Bi → PP, S)/B(Bi → PP):
AiCP ( S = 0)
A jCP ( S = 1)
= −τ jB( S = 1)
τiB( S = 0). (35)
Here B(Bi → PP) is the branching ratio of Bi → PP andτi is the lifetime of Bi .
One of the most prominent example is the case of the U-
spin pair B0s → π−K+ and B
0 → π+K−. Their CP asym-metry have been studied in Ref. [33]. Here we will commenton the experimental situation for this case and introduce aparameter rc to account for the deviation from SU(3) sym-
1 For decay modes to be discussed in the following, there are non-trivialClebsch-Gordon coefficients, such that Eq. (33) is modified as
A( S = 0) = r(VubV∗udAu + VtbV
∗tdAt ) ,
A( S = 1) = VubV∗usAu + VtbV
∗tsAt .
The relation in Eq. (35) is changed to:
AiCP ( S = 0)
A jCP ( S = 1)
= −r2 τ jB( S = 1)
τiB( S = 0).
123
Eur. Phys. J. C (2020) 80 :359 Page 9 of 36 359
metry.
ACP (B0 → π+K−)
ACP (B0s → π−K+)
+ rcτBB(B
0s → π−K+)
τBsB(B0 → π+K−)
= 0. (36)
In the SU(3) symmetry limit rc = 1.Using the experimental data from PDG [3,4]:
B(B0s → π−K+) = (5.7 ± 0.6) × 10−6,
ACP (B0s → π−K+) = (0.26 ± 0.04),
B(B0 → π+K−) = (19.6 ± 0.5) × 10−6,
ACP (B0 → π+K−) = −0.082 ± 0.006, (37)
one finds:
rc = 1.084 ± 0.219 (38)
where all errors have been added in quadrature. The resultingrc value indicates that the U-spin symmetry is well in thecase of this decay pair. The exploration in more decay pairsis helpful for further investigation on this symmetry.
Similar U-spin relations existing in other decays will bestudied in the following sections. We will comment on themwhen specific decay channels are be discussed.
3.2 B → VV decays
Decay amplitudes for B → VV channels can be obtainedsimilarly by replacing the pseudo-scalar multiplet P by thevector multiplet V in Eq. (21) and in Eq. (22).
• Since we have chosen the same parametrization for pseu-doscalar and vector mesons, the expanded amplitudes forthe B → VV channels can be obtained directly from theB → PP .
• There are three sets of amplitudes for B → VVdecays, corresponding to different polarizations. Forconvenience, one can choose the helicity amplitudesA0, A+, A− defined as:
A = S1ε∗V1
· ε∗V2
+ S21
m2B
ε∗V1
· pBε∗V2
· pB−i S3εμνρσ p
μV1pνV2
ε∗ρV1
ε∗σV2
, (39)
with ε0123 = 1, and
A0 = m2B
2mV1mV2
(S1 + S2
2
), A± = S1 ∓ S3. (40)
Thus there are in total 3 × 9 = 27 complex ampli-tudes for both tree and penguin, where “9” is the num-ber of the polarization combination of final two vectors.These amplitudes correspond to 2 × 54 − 1 = 107
real parameters in theory. Two phases can not be mea-sured through direct measurements of individual B andB decays, but one of the two can be obtained through thetime-dependent analysis.
• In principle, all these 107 parameters could be determinedthrough the angular distribution studies in experiment.Each B → V (→ P1P2)V (→ P3P4) channel can pro-vide 10 observables. The angular distribution is given as:
d�
d cos θ1d cos θ2dφ
∝ |A0|2 cos2 θ1 cos2 θ2
+1
4sin2 θ1 sin2 θ2
(|A+|2 + |A−|2
)
+1
2sin2 θ1 sin2 θ2Re(e2iφ A+A∗−)
− cos θ1 sin θ1 cos θ2 sin θ2[Re(e−iφ A0A∗+
+Re(eiφ A0A∗−)].
Here θ1 (θ2) is defined by the flight direction of P1(P3) inthe rest frame of V1(V2) and the flight direction of V1(V2)
in the B meson rest frame. φ is the relative angle betweenthe two decay planes.
• Unfortunately, due to the large amount of input param-eters, it is a formidable task to perform a global fit, andin particular only limited data is available [3]. A realisticanalysis at this stage will pick up only a limited amount ofamplitudes. In this direction, the weak annihilations andhard scattering amplitudes were extracted by fitting rele-vant data in Ref. [34], while the authors in Ref. [35] haveperformed a factorization-assisted TDA analysis. Thisallows one to remove some suppressed amplitudes at theleading order approximation. In Ref. [36], the authorshave adopted the dynamical analysis in the SCET andperformed a flavor SU(3) fit of B → VV decays. Onthe other hand, recent dynamical improvements exist inRefs. [37,38] using the perturbative QCD approach andRef. [39] in QCDF.
3.3 B → V P decays
We now study the B → V P decays, whose amplitudes canbe obtained by replacing one of the P in Eq. (21) and inEq. (22) by V to obtain the IRA and TDA amplitudes. Thereare two ways to replace one of the P , therefore the amplitudeswill be doubled compared with B → PP . We have IRA andTDA for B → V P decays as follows:
AI RAu = AT
3 Bi (H3)i P j
k Vkj + CT 1
3 Bi (H3)k Pi
j Vjk
+CT 23 Bi (H3)
kV ij P
jk + BT
3 Bi (H3)i Pk
k Vjj
+DT 13 Bi (H3)
j Pij V
kk + DT 2
3 Bi (H3)j V i
j Pkk
123
359 Page 10 of 36 Eur. Phys. J. C (2020) 80 :359
+AT 16 Bi (H6)
[i j]k Pl
j Vkl + AT 2
6 Bi (H6)[i j]k V l
j Pkl
+CT 16 Bi (H6)
[ jl]k Pi
j Vkl + CT 2
6 Bi (H6)[ jl]k V i
j Pkl
+BT 16 Bi (H6)
[i j]k Pk
j Vll + BT 2
6 Bi (H6)[i j]k V k
j Pll
+AT 115 Bi (H15)
{i j}k Pl
j Vkl + AT 2
15 Bi (H15){i j}k V l
j Pkl
+CT 115 Bi (H15)
{ jk}l Pi
j Vlk + CT 2
15 Bi (H15){ jk}l V i
j Plk
+BT 115 Bi (H15)
{i j}k Pk
j Vll + BT 2
15 Bi (H15){i j}k V k
j Pll ,
(41)
AT DAu = T1Bi H
jlk Pi
j Vkl + T2Bi H
jlk V i
j Pkl
+C1Bi Hl jk Pi
j Vkl + C2Bi H
l jk V i
j Pkl
+A1Bi Hilj P
jk V
kl + A2Bi H
ilj V
jk Pk
l
+E1Bi Hlij P
jk V
kl + E2Bi H
lij V
jk Pk
l
+Su1Bi Hl jl Pi
j Vkk + Su2Bi H
l jl V i
j Pkk
+Pu1Bi Hlkl Pi
j Vjk + Pu2Bi H
lkl V i
j Pjk
+PuABi H
lil P j
k Vkj + SuS Bi H
lil P j
j Vkk
+Eu1S Bi H
jil Pl
j Vkk + Eu2
S Bi Hjil V l
j Pkk
+Au1S Bi H
i jl Pl
j Vkk + Au2
S Bi Hi jl V l
j Pkk . (42)
The expanded amplitudes are given in Table 2 and Tab. 3.Relations between the two sets of amplitudes are derived as:
AT3 = −1
8(A1 + A2 − 3E1 − 3E2) + Pu
A,
BT3 = SuS + 1
8(3Eu1
S + 3Eu2S − Au1
S − Au2S )
CT 13 = 1
8(3T1 − C1 + 3A1 − E1) + Pu1,
CT 23 = 1
8(3T2 − C2 + 3A2 − E2) + Pu2
DT13 = 1
8
(3C1 − T1 − Eu1
S + 3Au1S
)+ Su1,
DT 23 = 1
8
(3C2 − T2 − Eu2
S + 3Au2S
)+ Su2
AT 16 = 1
4(A2 − E2), AT 2
6 = 1
4(A1 − E1),
CT 16 = 1
4(T1 − C1), CT 2
6 = 1
4(T2 − C2)
AT 115 = 1
8(A2 + E2), AT 2
15 = 1
8(A1 + E1),
CT 115 = 1
8(T1 + C1), CT 2
15 = 1
8(T2 + C2),
BT 16 = 1
4(Au1
S − Eu1S ), BT 2
6 = 1
4(Au2
S − Eu2S ),
BT 115 = 1
8(Eu1
S + Au1S ), BT 2
15 = 1
8(Eu2
S + Au2S ). (43)
The inverse relations are solved as:
A1 = 4AT 215 + 2AT 2
6 , A2 = 2(2AT 115 + AT 1
6 ),
T1 = 2(
2CT 115 + CT 1
6
), T2 = 2
(2CT 2
15 + CT 26
),
C1 = 2(
2CT 115 − CT 1
6
), C2 = 2
(2CT 2
15 − CT 26
),
E1 = 2(
2AT 215 − AT 2
6
), E2 = 2
(2AT 1
15 − AT 16
),
Au1S = 2
(2BT 1
15 + BT 16
), Au2
S = 2(
2BT 215 + BT 2
6
),
Eu1S = 2
(2BT 1
15 − BT 16
), Eu2
S = 2(
2BT 215 − BT 2
6
),
PuA = −AT 1
15 + AT 16 − AT 2
15 + AT 26 + AT
3 ,
SuS = −BT 115 + BT 1
6 − BT 215 + BT 2
6 + BT3 ,
Pu1 = −AT 215 − AT 2
6 − CT 115 + CT 1
3 − CT 16 ,
Pu2 = −AT 115 − AT 1
6 − CT 215 + CT 2
3 − CT 26 ,
Su1 = −BT 115 − BT 1
6 − CT 115 + CT 1
6 + DT 13 ,
Su2 = −BT 215 − BT 2
6 − CT 215 + CT 2
6 + DT 23 . (44)
Unlike the B → PP and B → VV case, we are not ableto find any redundant amplitude. Thus in total, we have 18complex amplitudes for “tree” and “penguin”, respectively.It corresponds to 2×36−1 = 71 real parameters in theory. Afit with all parameters is not available again, and most of thecurrent analyses have made approximations by neglectingsome suppressed amplitudes [18,19,40,41].
Decay amplitudes for B → V P decays in Eqs. (42)and (43) are organized in a symmetric way. Taking the B− →ρ− K 0 and B− → K
∗0π− as an example, the IRA ampli-
tudes are related with the replacement AT 16 ↔ AT 2
6 , AT 115 ↔
AT 215 ,CT 1
3 ↔ CT 23 ,CT 1
6 ↔ CT 26 , andCT 1
15 ↔ CT 215 , while for
the TDA amplitudes, the correspondence is: A1 ↔ A2 andPu1 ↔ Pu2. Other relations can be found in a similar way.
The B → V P channels related by the U-spin include:
B− → K∗0
π− and B− → K ∗0K−; B− → ρ−K0
and
B− → K ∗−K 0; B0s → K ∗+K− and B
0 → ρ+π−;
B0s → K ∗−K+ and B
0 → ρ−π+; B0s → ρ+π− and B
0 →K ∗+K−; B
0s → K
∗0K 0 and B
0 → K ∗0K0; B
0s → K ∗0K
0
and B0 → K
∗0K 0; B
0s → ρ−π+ and B
0 → K ∗−K+;
B0 → K ∗−π+ and B
0s → ρ−K+; B
0 → ρ+K− and
B0s → K ∗+π−. However on the experimental side, there
are not enough measurements to examine these relations, inparticular the CPV in Bs sector has received less considera-tion. We expect the situation will be improved when a largeamount of data is available at LHCb, and Belle-II.
4 D → P P, VV, PV decays
Using the effective Hamiltonian in Eqs. (16) and (17), onecan easily obtain the SU (3) decay amplitudes in a similarfashion as that for B → PP, VV, PV . For D decayswe will only discuss tree contributions which are writtenas A = Vcs/dV ∗
ud/sAI RA,T DAu . The penguin amplitudes are
123
Eur. Phys. J. C (2020) 80 :359 Page 11 of 36 359
Table 2 B → V P decays induced by the b → d transition
Channel IRA TDA
B− → ρ0π− (AT 16 + 3AT 1
15 − AT 26 − 3AT 2
15 − CT 13
−CT 16 + 5CT 1
15 + CT 23 + CT 2
6 + 3CT 215 )/
√2
(−A1 + A2 + C1 + Pu2 − Pu1 + T2)/√
2
B− → ρ−π0 (−AT 16 − 3AT 1
15 + AT 26 + 3AT 2
15 + CT 13
+CT 16 + 3CT 1
15 − CT 23 − CT 2
6 + 5CT 215 )/
√2
(A1 − A2 + C2 − Pu2 + Pu1 + T1)/√
2
B− → ρ−ηq (1/√
2)(AT 16 + 3AT 1
15 + AT 26 + 3AT 2
15 + 2BT 26 + 6BT 2
15+CT 1
3 +CT 16 + 3CT 1
15 +CT 23 −CT 2
6 +CT 215 + 2DT 2
3 )
(2Au2S + A1 + A2 + C2 + Pu2
+Pu1 + 2Su2 + T1)/√
2
B− → ρ−ηs BT 26 + 3BT 2
15 + CT 26 − CT 2
15 + DT 23 Au2
S + Su2
B− → K ∗0K− AT 26 + 3AT 2
15 + CT 13 − CT 1
6 − CT 115 A1 + Pu1
B− → K ∗−K 0 AT 16 + 3AT 1
15 + CT 23 − CT 2
6 − CT 215 A2 + Pu2
B− → ωπ− (1/√
2)(AT 16 + 3AT 1
15 + AT 26 + 3AT 2
15 + 2BT 16 + 6BT 1
15+CT 1
3 −CT 16 +CT 1
15 +CT 23 +CT 2
6 + 3CT 215 + 2DT 1
3 )
(2Au1S + A1 + A2 + C1 + Pu2
+Pu1 + 2Su1 + T2)/√
2
B− → φπ− BT 16 + 3BT 1
15 + CT 16 − CT 1
15 + DT 13 Au1
S + Su1
B0 → ρ+π− AT
3 − 2AT 115 − AT 2
6 + 3AT 215 + CT 2
3 + CT 26 + 3CT 2
15 PuA + E1 + Pu2 + T2
B0 → ρ0π0 (1/2)(2AT
3 − AT 16 + AT 1
15 − AT 26 + AT 2
15 + CT 13
+CT 16 − 5CT 1
15 + CT 23 + CT 2
6 − 5CT 215 )
(1/2)(2PuA − C1 − C2 + E1 + E2 + Pu2 + Pu1)
B0 → ρ0ηq (1/2)(−AT 1
6 + 5AT 115 − AT 2
6 + 5AT 215 − 2BT 2
6 + 10BT 215
−CT 13 −CT 1
6 + 5CT 115 −CT 2
3 +CT 26 −CT 2
15 − 2DT 23 )
(1/2)(C1 − C2 + 2Eu2S + E1
+E2 − Pu2 − Pu1 − 2Su2)
B0 → ρ0ηs −(BT 2
6 − 5BT 215 + CT 2
6 − CT 215 + DT 2
3 )/√
2 (Eu2S − Su2)/
√2
B0 → ρ−π+ AT
3 − AT 16 + 3AT 1
15 − 2AT 215 + CT 1
3 + CT 16 + 3CT 1
15 PuA + E2 + Pu1 + T1
B0 → K ∗+K− AT
3 + AT 16 − AT 1
15 − AT 26 + 3AT 2
15 PuA + E1
B0 → K ∗0K
0AT
3 + AT 16 − AT 1
15 − 2AT 215 + CT 1
3 − CT 16 − CT 1
15 PuA + Pu1
B0 → K
∗0K 0 AT
3 − 2AT 115 + AT 2
6 − AT 215 + CT 2
3 − CT 26 − CT 2
15 PuA + Pu2
B0 → K ∗−K+ AT
3 − AT 16 + 3AT 1
15 + AT 26 − AT 2
15 PuA + E2
B0 → ωπ0 (1/2)(−AT 1
6 + 5AT 115 − AT 2
6 + 5AT 215 − 2BT 1
6 +10BT 1
15 − CT 13
+CT 16 − CT 1
15 − CT 23 − CT 2
6 + 5CT 215 − 2DT 1
3 )
(1/2)(−C1 + C2 + 2Eu1S + E1
+E2 − Pu2 − Pu1 − 2Su1)
B0 → ωηq (1/2)(2AT
3 − AT 16 + AT 1
15 − AT 26 + AT 2
15 +4BT3 −2BT 1
6+2BT 1
15 − 2BT 26 + 2BT 2
15 +CT 13 −CT 1
6 +CT 115 +CT 2
3−CT 2
6 + CT 215 + 2DT 1
3 + 2DT 23 )
(1/2)(2PuA + C1 + C2 + 2Eu1
S + 2Eu2S + E1
+E2 + Pu2 + Pu1 + 2Su2 + 2Su1 + 4SuS )
B0 → ωηs (2BT
3 + 2BT 16 − 2BT 1
15 − BT 26 + BT 2
15 + CT 26 − CT 2
15 +DT 2
3 )/√
2(Eu2
S + Su2 + 2SuS )/√
2
B0 → φπ0 −(BT 1
6 − 5BT 115 + CT 1
6 − CT 115 + DT 1
3 )/√
2 (Eu1S − Su1)/
√2
B0 → φηq (2BT
3 − BT 16 + BT 1
15 + 2BT 26 − 2BT 2
15 + CT 16 − CT 1
15 +DT 1
3 )/√
2(Eu1
S + Su1 + 2SuS )/√
2
B0 → φηs AT
3 + AT 16 − AT 1
15 + AT 26 − AT 2
15 + BT3 + BT 1
6−BT 1
15 + BT 26 − BT 2
15
PuA + SuS
B0s → ρ0K 0 (AT 2
6 + AT 215 − CT 1
3 − CT 16 + 5CT 1
15 )/√
2 (C1 − Pu1)/√
2
B0s → ρ−K+ −AT 2
6 − AT 215 + CT 1
3 + CT 16 + 3CT 1
15 Pu1 + T1
B0s → K ∗+π− −AT 1
6 − AT 115 + CT 2
3 + CT 26 + 3CT 2
15 Pu2 + T2
123
359 Page 12 of 36 Eur. Phys. J. C (2020) 80 :359
Table 2 continued
Channel IRA TDA
B0s → K ∗0π0 (AT 1
6 + AT 115 − CT 2
3 − CT 26 + 5CT 2
15 )/√
2 (C2 − Pu2)/√
2
B0s → K ∗0ηq −(AT 1
6 + AT 115 + 2BT 2
6 + 2BT 215 − CT 2
3 + CT 26 −
CT 215 − 2DT 2
3 )/√
2(C2 + Pu2 + 2Su2)/
√2
B0s → K ∗0ηs −AT 2
6 − AT 215 − BT 2
6 − BT 215 + CT 1
3 − CT 16
−CT 115 + CT 2
6 − CT 215 + DT 2
3
Pu1 + Su2
B0s → ωK 0 −(AT 2
6 + AT 215 + 2BT 1
6 + 2BT 115 − CT 1
3 + CT 16 −
CT 115 − 2DT 1
3 )/√
2(C1 + Pu1 + 2Su1)/
√2
B0s → φK 0 −AT 1
6 − AT 115 − BT 1
6 − BT 115 + CT 1
6 − CT 115
+CT 23 − CT 2
6 − CT 215 + DT 1
3
Pu2 + Su1
suppressed and thus neglected in this work, however it isnecessary to stress penguin amplitudes are mandatory forthe CP violation. The SU(3) amplitudes for D → PP areparametrized as
AI RAu = AT
6 Di (H6)[i j]k Pl
j Pkl + CT
6 Di (H6)[ jl]k Pi
j Pkl
+BT6 Di (H6)
[i j]k Pk
j Pll
+AT15Di (H15)
{i j}k Pl
j Pkl + CT
15Di (H15){ jk}l Pi
j Plk
+BT15Di (H15)
{i j}k Pk
j Pll , (45)
AT DAu = T Di H
jlk Pi
j Pkl + CDi H
l jk Pi
j Pkl
+ADi Hilj P
jk P
kl + EDi H
lij P
jk P
kl
+EuS Di H
jil Pl
j Pkk + Au
SDi Hi jl Pl
j Pkk . (46)
The expanded amplitudes are given in Table 4. The ampli-tudes AT
6 can be incorporated in BT ′6 and CT ′
6 , and then wehave five independent amplitudes for D → PP:
AT15 = A + E
2, BT
15 = AuS + Eu
S
2, CT
15 = T + C
2,
B ′T6 = Au
S − EuS + A − E
2, C ′T
6 = T − C − A + E
2,
(47)
with the inverse relation:
T + E = AT15 + C ′T
6 + CT15,
C − E = −AT15 − C ′T
6 + CT15, A + E = 2AT
15,
AuS − E = −AT
15 + B ′T6 + BT
15,
EuS + E = AT
15 − B ′T6 + BT
15. (48)
Since one amplitude is redundant, fits with all six complexamplitudes should not be resolved in principle. This has beenindicated by the strong correlation of parameters in the fitsin Ref. [42].
Again for D → VV decays there are three sets of ampli-tudes similar as the D → PP , and thus we have 15 indepen-dent amplitudes in total (Fig. 2).
The IRA and TDA for D → V P decays are given as:
AI RAu = AT 1
6 Di (H6)[i j]k Pl
j Vkl + AT 2
6 Di (H6)[i j]k V l
j Pkl
+CT 16 Di (H6)
[ jl]k Pi
j Vkl + CT 2
6 Di (H6)[ jl]k V i
j Pkl
+BT 16 Di (H6)
[i j]k Pk
j Vll + BT 2
6 Di (H6)[i j]k V k
j Pll
+AT 115 Di (H15)
{i j}k Pl
j Vkl + AT 2
15 Di (H15){i j}k V l
j Pkl
+CT 115 Di (H15)
{ jk}l Pi
j Vlk + CT 2
15 Di (H15){ jk}l V i
j Plk
+BT 115 Di (H15)
{i j}k Pk
j Vll + BT 2
15 Di (H15){i j}k V k
j Pll ,
(49)
AT DAu = T1Di H
jlk Pi
j Vkl + T2Di H
jlk V i
j Pkl
+C1Di Hl jk Pi
j Vkl + C2Di H
l jk V i
j Pkl
+A1Di Hilj P
jk V
kl + A2Di H
ilj V
jk Pk
l
+E1Di Hlij P
jk V
kl + E2Di H
lij V
jk Pk
l
+Eu1S Di H
jil Pl
j Vkk + Eu2
S Di Hjil V l
j Pkk
+Au1S Di H
i jl Pl
j Vkk + Au2
S Di Hi jl V l
j Pkk . (50)
The above amplitudes are also expanded in a symmetric way.The expanded amplitudes are collected in Tables 5, 6 and 7for the different transitions. Relations between the two setsof amplitudes are derived as:
AT 16 = 1
2(A2 − E2), AT 2
6 = 1
2(A1 − E1),
BT 16 = 1
2(Au1
S − Eu1S ), BT 2
6 = 1
2(Au2
S − Eu2S )
CT 16 = 1
2(T1 − C1), CT 2
6 = 1
2(T2 − C2),
AT 115 = 1
2(A2 + E2), AT 2
15 = 1
2(A1 + E1)
BT 115 = 1
2(Eu1
S + Au1S ), BT 2
15 = 1
2(Eu2
S + Au2S ),
CT 115 = 1
2(T1 + C1), CT 2
15 = 1
2(T2 + C2). (51)
123
Eur. Phys. J. C (2020) 80 :359 Page 13 of 36 359
Table 3 B → V P decays induced by the b → s transition
Channel IRA TDA
B− → ρ0K− (AT16 + 3AT1
15 − 2CT16 + 4CT1
15 + CT23 + CT2
6 +3CT2
15 )/√
2
(A2 + C1 + Pu2 + T2)/√
2
B− → ρ−K0
AT16 + 3AT1
15 + CT23 − CT2
6 − CT215 A2 + Pu2
B− → K∗0
π− AT26 + 3AT2
15 + CT13 − CT1
6 − CT115 A1 + Pu1
B− → K ∗−π0 (AT26 + 3AT2
15 + CT13 + CT1
6 + 3CT115 − 2CT2
6 +4CT2
15 )/√
2
(A1 + C2 + Pu1 + T1)/√
2
B− → K ∗−ηq (AT26 + 3AT2
15 + 2BT26 + 6BT2
15 + CT13
+CT16 + 3CT1
15 + 2CT215 + 2DT2
3 )/√
2
(2Au2S + A1 + C2 + Pu1 + 2Su2 + T1)/
√2
B− → K ∗−ηs AT16 + 3AT1
15 + BT26 + 3BT2
15 + CT23 − 2CT2
15 + DT23 Au2
S + A2 + Pu2 + Su2
B− → ωK− (AT16 + 3AT1
15 + 2BT16 + 6BT1
15 + 2CT115
+CT23 + CT2
6 + 3CT215 + 2DT1
3 )/√
2
(2Au1S + A2 + C1 + Pu2 + 2Su1 + T2)/
√2
B− → φK− AT26 + 3AT2
15 + BT16 + 3BT1
15 + CT13 − 2CT1
15 + DT13 Au1
S + A1 + Pu1 + Su1
B0 → ρ+K− −AT1
6 − AT115 + CT2
3 + CT26 + 3CT2
15 Pu2 + T2
B0 → ρ0K
0(AT1
6 +AT115−2CT1
6 +4CT115 −CT2
3 +CT26 +CT2
15 )/√
2 (C1 − Pu2)/√
2
B0 → K
∗0π0 (AT2
6 +AT215−CT1
3 +CT16 +CT1
15 −2CT26 +4CT2
15 )/√
2 (C2 − Pu1)/√
2
B0 → K
∗0ηq −(AT2
6 + AT215 + 2BT2
6 + 2BT215 − CT1
3
+CT16 + CT1
15 − 2CT215 − 2DT2
3 )/√
2
(C2 + Pu1 + 2Su2)/√
2
B0 → K
∗0ηs −AT1
6 − AT115 − BT2
6 − BT215 + CT2
3 − 2CT215 + DT2
3 Pu2 + Su2
B0 → K ∗−π+ −AT2
6 − AT215 + CT1
3 + CT16 + 3CT1
15 Pu1 + T1
B0 → ωK
0 −(AT16 + AT1
15 + 2BT16 + 2BT1
15 − 2CT115
−CT23 + CT2
6 + CT215 − 2DT1
3 )/√
2
(C1 + Pu2 + 2Su1)/√
2
B0 → φK
0 −AT26 − AT2
15 − BT16 − BT1
15 + CT13 − 2CT1
15 + DT13 Pu1 + Su1
B0s → ρ+π− AT
3 + AT16 − AT1
15 − AT26 + 3AT2
15 PuA + E1
B0s → ρ0π0 AT
3 + AT115 + AT2
15 (1/2)(2PuA + E1 + E2)
B0s → ρ0ηq 2(AT1
15 + AT215 − BT2
6 + 2BT215 ) − AT1
6 − AT26 (1/2)(2Eu2
S + E1 + E2)
B0s → ρ0ηs −√
2(BT26 − 2BT2
15 + CT16 − 2CT1
15 ) (C1 + Eu2S )/
√2
B0s → ρ−π+ AT
3 − AT16 + 3AT1
15 + AT26 − AT2
15 PuA + E2
B0s → K ∗+K− AT
3 − 2AT115 − AT2
6 + 3AT215 + CT2
3 + CT26 + 3CT2
15 PuA + E1 + Pu2 + T2
B0s → K ∗0K
0AT
3 − 2AT115 + AT2
6 − AT215 + CT2
3 − CT26 − CT2
15 PuA + Pu2
B0s → K
∗0K 0 AT
3 + AT16 − AT1
15 − 2AT215 + CT1
3 − CT16 − CT1
15 PuA + Pu1
B0s → K ∗−K+ AT
3 − AT16 + 3AT1
15 − 2AT215 + CT1
3 + CT16 + 3CT1
15 PuA + E2 + Pu1 + T1
B0s → ωπ0 2(AT1
15 + AT215 − BT1
6 + 2BT115 ) − AT1
6 − AT26 (1/2)(2Eu1
S + E1 + E2)
B0s → ωηq AT
3 + AT115 + AT2
15 + 2(BT3 + BT1
15 + BT215 ) Pu
A + Eu1S + Eu2
S + (E1)/2 + (E2)/2 + 2SuS
B0s → ωηs
√2(BT
3 − 2BT115 + BT2
15 + CT115 + DT1
3 ) (C1 + Eu2S + 2(Su1 + SuS ))/
√2
B0s → φπ0 −√
2(BT16 − 2BT1
15 + CT26 − 2CT2
15 ) (C2 + Eu1S )/
√2
B0s → φηq
√2(BT
3 + BT115 − 2BT2
15 + CT215 + DT2
3 ) (C2 + Eu1S + 2(Su2 + SuS ))/
√2
B0s → φηs AT
3 − 2AT115 − 2AT2
15 + BT3 − 2BT1
15 − 2BT215
+CT13 − 2CT1
15 + CT23 − 2CT2
15 + DT13 + DT2
3
PuA + Pu2 + Pu1 + Su2 + Su1 + SuS
123
359 Page 14 of 36 Eur. Phys. J. C (2020) 80 :359
Table 4 Decay amplitudes for two-body D → PP decays. Decay amplitudes for two-body D → PP decays. The CKM factor should bemultiplied: VcsV ∗
ud for Cabibblo-Allowed decays; VcsV ∗us for singly Cabibbo-suppressed modes and VcdV ∗
us for doubly Cabibbo-suppressed modes
VcsV ∗ud IRA TDA
D0 → π+K− −AT6 + AT
15 + CT6 + CT
15 E + T
D0 → π0K0
(AT6 − AT
15 − CT6 + CT
15)/√
2 (C − E)/√
2
D0 → K0ηq (−AT
6 + AT15 − 2BT
6 + 2BT15 − CT
6 + CT15)/
√2 (C + 2Eu
S + E)/√
2
D0 → K0ηs −AT
6 + AT15 − BT
6 + BT15 Eu
S + E
D+ → π+K0
2CT15 C + T
D+s → π+ηq
√2
(AT
6 + AT15 + BT
6 + BT15
) √2
(AuS + A
)
D+s → π+ηs BT
6 + BT15 + CT
6 + CT15 Au
S + T
D+s → K+K
0AT
6 + AT15 − CT
6 + CT15 A + C
VcsV ∗us IRA TDA
D0 → π+π− AT6 − AT
15 − CT6 − CT
15 −E − T
D0 → π0π0 AT6 − AT
15 − CT6 + CT
15 C − E
D0 → π0ηq −AT6 + AT
15 − BT6 + BT
15 EuS + E
D0 → π0ηs (−BT6 + BT
15 − CT6 + CT
15)/√
2 (C + EuS)/
√2
D0 → K+K− −AT6 + AT
15 + CT6 + CT
15 E + T
D0 → ηqηq AT6 − AT
15 + 2BT6 − 2BT
15 + CT6 − CT
15 −C − 2EuS − E
D0 → ηqηs (−BT6 + BT
15 − CT6 + CT
15)/√
2 (C + EuS)/
√2
D0 → ηsηs −AT6 + AT
15 − BT6 + BT
15 EuS + E
D+ → π+π0√
2CT15 (C + T)/
√2
D+ → π+ηq −√2
(AT
6 + AT15 + BT
6 + BT15 + CT
15
) −(2AuS + 2A + C + T)/
√2
D+ → π+ηs −BT6 − BT
15 − CT6 + CT
15 C − AuS
D+ → K+K0 −AT
6 − AT15 + CT
6 + CT15 T − A
D+s → π+K 0 AT
6 + AT15 − CT
6 − CT15 A − T
D+s → π0K+ (AT
6 + AT15 − CT
6 + CT15)/
√2 (A + C)/
√2
D+s → K+ηq (AT
6 + AT15 + 2BT
6 + 2BT15 + CT
6 − CT15)/
√2 (2Au
S + A − C)/√
2
D+s → K+ηs AT
6 + AT15 + BT
6 + BT15 + 2CT
15 AuS + A + C + T
VcdV ∗us IRA TDA
D0 → π0K 0 (AT6 − AT
15 − CT6 + CT
15)/√
2 (C − E)/√
2
D0 → π−K+ −AT6 + AT
15 + CT6 + CT
15 E + T
D0 → K 0ηq (−AT6 + AT
15 − 2BT6 + 2BT
15 − CT6 + CT
15)/√
2 (C + 2EuS + E)/
√2
D0 → K 0ηs −AT6 + AT
15 − BT6 + BT
15 EuS + E
D+ → π+K 0 AT6 + AT
15 − CT6 + CT
15 A + C
D+ → π0K+ (AT6 + AT
15 − CT6 − CT
15)/√
2 (A − T)/√
2
D+ → K+ηq (AT6 + AT
15 + 2BT6 + 2BT
15 + CT6 + CT
15)/√
2 (2AuS + A + T)/
√2
D+ → K+ηs AT6 + AT
15 + BT6 + BT
15 AuS + A
D+s → K+K 0 2CT
15 C + T
123
Eur. Phys. J. C (2020) 80 :359 Page 15 of 36 359
Fig. 2 Topological diagramsfor tree amplitudes inCabibbo-allowed D → PP andD → VV decays. The penguinamplitudes are suppressed byboth CKM matrix elements andWilson coefficients, and thusneglected in this work, howeverit is necessary to point outpenguin amplitudes aremandatory to study the CPviolation. The s can be replacedby d and d can be replaced by sto obtain the singly-Cabibbosuppressed and doubly-Cabibbosuppressed decays
The inverse relations are solved as:
A1 = AT 215 + AT 2
6 , A2 = AT 115 + AT 1
6 ,
T1 = CT 115 + CT 1
6 , T2 = CT 215 + CT 2
6
C1 = CT 115 − CT 1
6 , C2 = CT 215 − CT 2
6 ,
E1 = AT 215 − AT 2
6 , E2 = AT 115 − AT 1
6
Au1S = BT 1
15 + BT 16 , Au2
S = BT 215 + BT 2
6 ,
Eu1S = BT 1
15 − BT 16 , Eu2
S = BT 215 − BT 2
6 . (52)
It is interesting to explore the useful relations for decaywidths from the amplitudes listed in Tables 5, 6 and 7. ForCabibblo Allowed channels, we find �(D+
s → ρ+π0) =�(D+
s → ρ0π+). For singly Cabibblo suppressed channels,one has:
�(D0 → ρ+π−)
= �(D0 → K ∗+K−)
,
�(D0 → ρ−π+)
= �(D0 → K ∗−K+)
,
�(D+ → K ∗+K
0)
= �(D+s → ρ+K 0
),
�(D+ → K
∗0K+)
= �(D+s → K ∗0π+)
,
�(D0 → K
∗0K 0
)= �
(D0 → K ∗0K
0)
. (53)
We refer the reader to Refs. [42–45] for some explorationsof the implications on decay rates and CP asymmetries, andRefs. [46,47] for the experimental analyses.
It is necessary to notice that since the QCD scale is com-parable to charm quark mass, finite quark mass difference ins and d quarks may lead to sizable SU(3) symmetry break-ing effect. A notable example to explore SU(3) symmetrybreaking effects is the D0 → K 0 K 0. This channel has van-ishing branching fraction if exact SU(3) symmetry holds and
the 3 contribution proportional to VcbV ∗ub is neglected, which
can also been seen from Table 4. The experimental data forbranching ratios of D0 → K 0 K 0 and D0 → K+K− aregiven as [3,4]:
B(D0 → K+K−) = (3.97 ± 0.07) × 10−3,
B(D0 → K 0 K 0) = (3.40 ± 0.12) × 10−4. (54)
The decay amplitude of D0 → K+K− is E + T , whereT is color-allowed and could be estimated using the factor-ization framework. As |VcbV ∗
ub/VcsV∗sd | < 1.3 × 10−3, it is
unlikely that the above non-zero branching ratio is caused bythe 3 penguin contribution. The above data shows that theSU(3) symmetry breaking effects in some channels can be aslarge as 30% at the amplitude level. Though the above esti-mate is channel-dependent, it indicates that symmetry break-ing effects must carefully treated in D meson and charmedbaryon decays in a systematic way [20,42,44,45,48,49].
5 Bc → DP, DV decays
The effective Hamiltonian for b quark decays can induceBc → DP, DV transitions. The corresponding topologicaldiagrams are given in Fig. 3. The IRA and TDA for Bc →DP decays are given as:
AI RAu = AT
3 BcDi Hi3P jj + BT
3 BcDi Hj
3Pij
+AT6 BcDi (H6)
[ik]j P j
k + AT15BcDi (H15)
{ik}j P j
k ,
(55)
AT DAu = Su BcDi H
lil P j
j + Pu BcDi Hl jl Pi
j
+T BcDi Hikl Pl
k + CBcDi Hkil Pl
k . (56)
123
359 Page 16 of 36 Eur. Phys. J. C (2020) 80 :359
Table 5 Decay amplitudes for two-body Cabibbo-Allowed D → V P decays
Channel IRA TDA
D0 → ρ+K− −AT16 + AT1
15 + CT16 + CT1
15 T1 + E2
D0 → ρ0K0
(AT16 − AT1
15 − CT26 + CT2
15 )/√
2 (C2 − E2)/√
2
D0 → K∗0
π0 (AT26 − AT2
15 − CT16 + CT1
15 )/√
2 (C1 − E1)/√
2
D0 → K∗0
ηq (−AT26 + AT2
15 − 2BT26 + 2BT2
15 − CT16 + CT1
15 )/√
2 (C1 + 2Eu2S + E1)/
√2
D0 → K∗0
ηs −AT16 + AT1
15 − BT26 + BT2
15 Eu2S + E2
D0 → K ∗−π+ −AT26 + AT2
15 + CT26 + CT2
15 T2 + E1
D0 → ωK0
(−AT16 + AT1
15 − 2BT16 + 2BT1
15 − CT26 + CT2
15 )/√
2 (C2 + 2Eu1S + E2)/
√2
D0 → φK0 −AT2
6 + AT215 − BT1
6 + BT115 Eu1
S + E1
D+ → ρ+K0
CT16 + CT1
15 − CT26 + CT2
15 C2 + T1
D+ → K∗0
π+ −CT16 + CT1
15 + CT26 + CT2
15 C1 + T2
D+s → ρ+π0 (AT1
6 + AT115 − AT2
6 − AT215 )/
√2 (A2 − A1)/
√2
D+s → ρ+ηq (AT1
6 + AT115 + AT2
6 + AT215 + 2
(BT2
6 + BT215
))/
√2 (2Au2
S + A1 + A2)/√
2
D+s → ρ+ηs BT2
6 + BT215 + CT1
6 + CT115 Au2
S + T1
D+s → ρ0π+ (−AT1
6 − AT115 + AT2
6 + AT215 )/
√2 (A1 − A2)/
√2
D+s → K ∗+K
0AT2
6 + AT215 − CT2
6 + CT215 A1 + C2
D+s → K
∗0K+ AT1
6 + AT115 − CT1
6 + CT115 A2 + C1
D+s → ωπ+ (AT1
6 + AT115 + AT2
6 + AT215 + 2
(BT1
6 + BT115
))/
√2 (2Au1
S + A1 + A2)/√
2
D+s → φπ+ BT1
6 + BT115 + CT2
6 + CT215 Au1
S + T2
The expanded amplitudes can be found in Table 8. Relationsbetween the two sets of amplitudes are given as:
AT3 = Su − 1
8T + 3
8C, BT
3 = Pu + 3
8T − 1
8C,
AT6 = 1
4T − 1
4C, AT
15 = 1
8T + 1
8C. (57)
“Penguin” amplitudes are obtained similarly:
AT3,6,15 → AP
3,6,15, BT3 → BP
3 , Su → S,
Pu → P, T → PT , C → PC . (58)
Including the “penguins”, one has 8 complex amplitudes intotal.
Again decay amplitudes for Bc → DV can be obtained byreplacing the pseudoscalars by their vector counterparts. The
U-spin related channels include: B−c → D
0K− and B−
c →D
0π−; B−
c → D−K 0 and B−c → D−
s K0; B−c → D
0K ∗−
and B−c → D
0ρ−; B−
c → D−K∗0
and B−c → D−
s K∗0
.In Ref. [50], the LHCb collaboration has measured the
product:
f (Bc)
f (B+)× B(B+
c → D0K+) =(
9.3+2.8−2.5 ± 0.6
)× 10−7,
(59)
where the f (Bc) and f (B+) are the production rates of B+c
and B+, respectively. With the measured ratio [51]:
f (Bc)
f (B+)∼ 0.004 − 0.012, (60)
one can obtain an estimated branching fraction:
B(B+c → D0K+) ∼ 7.8 × 10−5 − 2.3 × 10−4. (61)
On theoretical side, model-dependent analyses give 1.3 ×10−7 [52], and 6.6 × 10−5 [53], while a phenomenologicalstudy implies the B(B+
c → D0K+) ∼ [4.4 − 9] × 10−5
[54]. Since this transition is induced by b → s, the largebranching fraction may imply a large penguin amplitude P .Such a scenario can be tested by measuring the correspond-ing (B+
c → D+K 0), which has the same penguin ampli-tude. Model-dependent calculations of other Bc decays canbe found in Refs. [55–58]. Some recent SU(3) analyses oftwo-body Bc decays can be found in Refs. [59,60]. Comparedto these studies, we have included all penguin amplitudes.
For the B−c meson, the charm quark can also decays, with
the final state BP or BV [61]. Since the heavy bottom quarkplays as a spectator, the decay modes are simpler. For exam-ple, for Cabibbo-allowed decay modes, there are only two
channels: B−c → π−B
0s and B−
c → ρ−B0s . Thus we expect
that the SU(3) symmetry will not provide much informationin these decays.
123
Eur. Phys. J. C (2020) 80 :359 Page 17 of 36 359
Table 6 Decay amplitudes for two-body Singly Cabibblo-Suppressed D → V P decays
Channel IRA TDA
D0 → ρ+π− AT16 − AT1
15 − CT16 − CT1
15 −T1 − E2
D0 → ρ0π0 (1/2)(AT1
6 − AT115 + AT2
6 − AT215 − CT1
6 + CT115 − CT2
6 + CT215
)(1/2) (C1 + C2 − E1 − E2)
D0 → ρ0ηq (1/2)(−AT 16 + AT 1
15 − AT 26 + AT 2
15 − 2BT 26
+2BT 215 − CT 1
6 + CT 115 + CT 2
6 − CT 215 )
(1/2)(C1 − C2 + 2Eu2s + E1 + E2)
D0 → ρ0ηs (−BT26 + BT2
15 − CT26 + CT2
15 )/√
2 (C2 + Eu2S )/
√2
D0 → ρ−π+ AT26 − AT2
15 − CT26 − CT2
15 −T2 − E1
D0 → K ∗+K− −AT16 + AT1
15 + CT16 + CT1
15 T1 + E2
D0 → K ∗0K0 −AT1
6 + AT115 + AT2
6 − AT215 E2 − E1
D0 → K∗0K 0 AT1
6 − AT115 − AT2
6 + AT215 E1 − E2
D0 → K ∗−K+ −AT26 + AT2
15 + CT26 + CT2
15 T2 + E1
D0 → ωπ0 (1/2)(−AT 16 + AT 1
15 − AT 26 + AT 2
15 − 2BT 16
+2BT 115 + CT 1
6 − CT 115 − CT 2
6 + CT 215 )
(1/2)(−C1 + C2 + 2Eu1s + E1 + E2)
D0 → ωηq (1/2)(AT 16 − AT 1
15 + AT 26 − AT 2
15 + 2BT 16 − 2BT 1
15+2BT 2
6 − 2BT 215 + CT 1
6 − CT 115 + CT 2
6 − CT 215 )
(1/2)(−C1 − C2 − 2Eu1s − 2Eu2
s − E1 − E2)
D0 → ωηs (−2BT16 + 2BT1
15 + BT26 − BT2
15 − CT26 + CT2
15 )/√
2 (C2 + 2Eu1S − Eu2
S )/√
2
D0 → φπ0 (−BT16 + BT1
15 − CT16 + CT1
15 )/√
2 (C1 + Eu1S )/
√2
D0 → φηq (BT16 − BT1
15 − 2BT26 + 2BT2
15 − CT16 + CT1
15 )/√
2 (C1 − Eu1S + 2Eu2
S )/√
2
D0 → φηs −AT16 + AT1
15 − AT26 + AT2
15 − BT16 + BT1
15 − BT26 + BT2
15 Eu1S + Eu2
S + E1 + E2
D+ → ρ+π0 (−AT16 − AT1
15 + AT26 + AT2
15 + CT16 + CT1
15 − CT26 +
CT215 )/
√2
(A1 − A2 + C2 + T1)/√
2
D+ → ρ+ηq −(AT16 + AT1
15 + AT26 + AT2
15 + 2BT26
+2BT215 + CT1
6 + CT115 − CT2
6 + CT215 )/
√2
−(2Au2S + A1 + A2 + C2 + T1)/
√2
D+ → ρ+ηs −BT26 − BT2
15 − CT26 + CT2
15 C2 − Au2S
D+ → ρ0π+ (AT16 +AT1
15 −AT26 −AT2
15 −CT16 +CT1
15 +CT26 +CT2
15 )/√
2 (−A1 + A2 + C1 + T2)/√
2
D+ → K ∗+K0 −AT2
6 − AT215 + CT1
6 + CT115 T1 − A1
D+ → K∗0K+ −AT1
6 − AT115 + CT2
6 + CT215 T2 − A2
D+ → ωπ+ −(AT16 + AT1
15 + AT26 + AT2
15 + 2BT16
+2BT115 − CT1
6 + CT115 + CT2
6 + CT215 )/
√2
−(2Au1S + A1 + A2 + C1 + T2)/
√2
D+ → φπ+ −BT16 − BT1
15 − CT16 + CT1
15 C1 − Au1S
D+s → ρ+K 0 AT2
6 + AT215 − CT1
6 − CT115 A1 − T1
D+s → ρ0K+ (AT2
6 + AT215 − CT1
6 + CT115 )/
√2 (A1 + C1)/
√2
D+s → K ∗+π0 (AT1
6 + AT115 − CT2
6 + CT215 )/
√2 (A2 + C2)/
√2
D+s → K ∗+ηq (AT1
6 + AT115 + 2BT2
6 + 2BT215 + CT2
6 − CT215 )/
√2 (2Au2
S + A2 − C2)/√
2
D+s → K ∗+ηs AT2
6 + AT215 + BT2
6 + BT215 +CT1
6 +CT115 −CT2
6 +CT215 Au2
S + A1 + C2 + T1
D+s → K ∗0π+ AT1
6 + AT115 − CT2
6 − CT215 A2 − T2
D+s → ωK+ (AT2
6 + AT215 + 2BT1
6 + 2BT115 + CT1
6 − CT115 )/
√2 (2Au1
S + A1 − C1)/√
2
D+s → φK+ AT1
6 + AT115 + BT1
6 + BT115 −CT1
6 +CT115 +CT2
6 +CT215 Au1
S + A2 + C1 + T2
It is necessary to point out that the charmless two-bodyBc decays are purely annihilation, and the typical branchingfractions are below the order 10−6 [62–64]. Since there arenot too many channels, it is less useful to apply the flavorSU(3) symmetry to these modes.
6 Antitriplet bottom Baryon decay into a Baryon and aMeson
In this and next sections, we discuss weak decays of baryonswith a heavy b and c quark. Charmed or bottom baryons withtwo light quarks can form an anti-triplet or a sextet. Mostmembers of the sextet can decay via strong interactions orelectromagnetic interactions. The only exceptions are �b and
123
359 Page 18 of 36 Eur. Phys. J. C (2020) 80 :359
Table 7 Decay amplitudes for two-body Doubly Cabibblo-Suppressed D → V P decays
Channel IRA TDA
D0 → ρ0K 0 (AT26 − AT2
15 − CT26 + CT2
15 )/√
2 (C2 − E1)/√
2
D0 → ρ−K+ −AT26 + AT2
15 + CT26 + CT2
15 T2 + E1
D0 → K ∗+π− −AT16 + AT1
15 + CT16 + CT1
15 T1 + E2
D0 → K ∗0π0 (AT16 − AT1
15 − CT16 + CT1
15 )/√
2 (C1 − E2)/√
2
D0 → K ∗0ηq (−AT16 + AT1
15 − 2BT26 + 2BT2
15 − CT16 + CT1
15 )/√
2 (C1 + 2Eu2S + E2)/
√2
D0 → K ∗0ηs −AT26 + AT2
15 − BT26 + BT2
15 Eu2S + E1
D0 → ωK 0 (−AT26 + AT2
15 − 2BT16 + 2BT1
15 − CT26 + CT2
15 )/√
2 (C2 + 2Eu1S + E1)/
√2
D0 → φK 0 −AT16 + AT1
15 − BT16 + BT1
15 Eu1S + E2
D+ → ρ+K 0 AT26 + AT2
15 − CT26 + CT2
15 A1 + C2
D+ → ρ0K+ (AT26 + AT2
15 − CT26 − CT2
15 )/√
2 (A1 − T2)/√
2
D+ → K ∗+π0 (AT16 + AT1
15 − CT16 − CT1
15 )/√
2 (A2 − T1)/√
2
D+ → K ∗+ηq (AT16 + AT1
15 + 2BT26 + 2BT2
15 + CT16 + CT1
15 )/√
2 (2Au2S + A2 + T1)/
√2
D+ → K ∗+ηs AT26 + AT2
15 + BT26 + BT2
15 Au2S + A1
D+ → K ∗0π+ AT16 + AT1
15 − CT16 + CT1
15 A2 + C1
D+ → ωK+ (AT26 + AT2
15 + 2BT16 + 2BT1
15 + CT26 + CT2
15 )/√
2 (2Au1S + A1 + T2)/
√2
D+ → φK+ AT16 + AT1
15 + BT16 + BT1
15 Au1S + A2
D+s → K ∗+K 0 CT1
6 + CT115 − CT2
6 + CT215 C2 + T1
D+s → K ∗0K+ −CT1
6 + CT115 + CT2
6 + CT215 C1 + T2
�c. In the following we will concentrate on the anti-tripletbaryons, whose weak decays are induced by the effectiveHamiltonian Hb
ef f and Hcef f .
6.1 Tb : (b, 0b,
−b ) decay into a decuplet baryon T10
and a light meson
The IRA amplitudes for the Tb decays into a decuplet baryonand a light meson can be parametrized as:
AI RAu = AT
3 T[i j]b3
Hk3(T 10)ikl P
lj
+AT6 T
[i j]b3
(H6)[kl]j (T 10)ikm Pm
l
+AT15T
[i j]b3
(H15){kl}j (T 10)ikm Pm
l
+BT15T
[i j]b3
(H15){kl}m (T 10)ikl P
mj
+CT15T
[i j]b3
(H15){kl}j (T 10)ikl P
mm
+DT15T
[i j]b3
(H15){kl}j (T 10)klm Pm
i . (62)
The TDA amplitudes are shown in Fig. 4 with the parametriza-tion:
AT DAu = a1T
[i j]b3
Hmkm (T 10)ikl P
lj + b1T
[i j]b3
Hklj (T 10)ikm Pm
l
+b2T[i j]b3
Hlkj (T 10)ikm Pm
l
+b3T[i j]b3
Hklm (T 10)ikl P
mj + b4T
[i j]b3
Hklj (T 10)ikl P
mm
+b5T[i j]b3
Hklj (T 10)klm Pm
i . (63)
We find relations between the two sets of amplitudes as:
a1 = AT3 + AT
6 − AT15 − 2BT
15 + 2DT15,
b1 = 4AT15 + 2AT
6 ,
b2 = 4AT15 − 2AT
6 , b3 = 8BT15,
b4 = 8CT15, b5 = 8DT
15. (64)
The expanded amplitudes for individual decay modes can befound in Table 9.
A few remarks are given in order.
• As the two light quarks in the initial state are antisym-metric in the flavor space while they are symmetric inthe final state. An overlap of wave functions of the ini-tial and final baryons is zero [65], which lead to vanish-ing decay amplitudes unless hard scattering interactionsoccur [66]. In other words, there is no “factorizable” con-tribution in the transition. In addition, all diagrams inFig. 4 are suppressed by powers of 1/Nc compared tothe Tb → T8P . This will indicate that branching frac-tions for these decays are likely smaller than the relevantB decays and Tb → T8P decays, where T8 representsthe octet baryon.
• For the Tb → T10P , one can construct the amplitudeswith the spinors, and a general form is:
A = pTb,μuμ(pT10)(A + Bγ5)u(pTb ), (65)
123
Eur. Phys. J. C (2020) 80 :359 Page 19 of 36 359
Fig. 3 Feynman diagrams for tree amplitudes in the Bc → DP, DV decays
Table 8 Decay amplitudes for Bc → DP decays
b → d IRA TDA b → s IRA TDA
B−c → D
0π− AT
6 + 3AT15 + BT
3 Pu + T B−c → D
0K− AT
6 + 3AT15 + BT
3 Pu + T
B−c → D−π0 (−AT
6 + 5AT15 − BT
3 )/√
2 (C − Pu)/√
2 B−c → D−K
0 −AT6 − AT
15 + BT3 Pu
B−c → D−ηq (2AT
3 − AT6 + AT
15 + BT3 )/
√2 (C + Pu + 2Su)/
√2 B−
c → D−s π0
√2
(2AT
15 − AT6
)C/
√2
B−c → D−ηs AT
3 + AT6 − AT
15 Su B−c → D−
s ηq√
2(AT
3 + AT15
)(C + 2Su)/
√2
B−c → D−
s K 0 −AT6 − AT
15 + BT3 Pu B−
c → D−s ηs AT
3 − 2AT15 + BT
3 Pu + Su
Fig. 4 Topology diagrams for the bottom baryon decays into a decuplet baryon and a light meson. We have explicitly specified the quark flavorsfor the four-quark interaction vertex, while the unspecified quarks can be u, d, s
123
359 Page 20 of 36 Eur. Phys. J. C (2020) 80 :359
Table 9 Decay amplitudes for Tb → T10P decays. Only those amplitudes proportional to VubV ∗uq are shown, while “penguin” amplitudes
proportional to VtbV ∗tq are similar
b → d IRA TDA
0b → +π− (AT
3 − AT6 − 5AT
15 + 6BT15 − 6DT
15)/√
3 (a1 − b1 + b3 − b5)/√
3
0b → 0π0 √
2/3(−AT
3 + AT6 + AT
15 − 2BT15 + 2DT
15
)(−2a1 + b1 − b2 − b3 + b5)/
√6
0b → 0ηq −4
√2/3
(AT
15 + BT15 + 2CT
15 + DT15
) −(b1 + b2 + b3 + 2b4 + b5)/√
6
0b → 0ηs −(8CT
15)/√
3 −b4/√
3
0b → −π+ −AT
3 + AT6 − 3AT
15 + 2BT15 − 2DT
15 −a1 − b2
0b → �′0K 0 −(−AT
3 + AT6 + 5AT
15 + 2BT15 + 6DT
15)/√
6 (a1 − b1 − b5)/√
6
0b → �′−K+ (−AT
3 + AT6 − 3AT
15 + 2BT15 − 2DT
15)/√
3 −(a1 + b2)/√
3
0b → +K− (AT
3 + AT6 − AT
15 + 6BT15 − 6DT
15)/√
3 (a1 + b3 − b5)/√
3
0b → 0K
0(AT
3 + AT6 − AT
15 − 2BT15 − 6DT
15)/√
3 (a1 − b5)/√
3
0b → �′+π− −(2
(AT
6 + 2AT15
))/
√3 −b1/
√3
0b → �′0π0 (−AT
3 + 3AT6 + AT
15 − 6BT15 − 2DT
15)/2√
3 −(a1 − b1 + b2 + b3)/2√
3
0b → �′0ηq −(AT
3 + AT6 + 7AT
15 + 6BT15 + 16CT
15 + 2DT15)/2
√3 −(a1 + b1 + b2 + b3 + 2b4)/2
√3
0b → �′0ηs (AT
3 + AT6 − AT
15 − 2BT15 − 8CT
15 − 6DT15)/
√6 (a1 − b4 − b5)/
√6
0b → �′−π+ (−AT
3 + AT6 − 3AT
15 + 2BT15 − 2DT
15)/√
3 −(a1 + b2)/√
3
0b → ′0K 0 −(2
(AT
6 + 2AT15
))/
√3 −b1/
√3
0b → ′−K+ (−AT
3 + AT6 − 3AT
15 + 2BT15 − 2DT
15)/√
3 −(a1 + b2)/√
3
−b → 0K− (AT
3 + AT6 − AT
15 + 6BT15 + 2DT
15)/√
3 (a1 + b3)/√
3
−b → −K
0AT
3 + AT6 − AT
15 − 2BT15 + 2DT
15 a1
−b → �′0π− (−AT
3 − AT6 + AT
15 − 6BT15 − 2DT
15)/√
6 −(a1 + b3)/√
6
−b → �′−π0 (AT
3 + AT6 − AT
15 − 2BT15 + 2DT
15)/√
6 a1/√
6
−b → �′−ηq (−AT
3 − AT6 + AT
15 + 2BT15 − 2DT
15)/√
6 −a1/√
6
−b → �′−ηs (AT
3 + AT6 − AT
15 − 2BT15 + 2DT
15)/√
3 a1/√
3
−b → ′−K 0 (−AT
3 − AT6 + AT
15 + 2BT15 − 2DT
15)/√
3 −a1/√
3
b → s IRA TDA
0b → +K− −(2
(AT
6 + 2AT15
))/
√3 −b1/
√3
0b → 0K
0 −(2(AT
6 + 2AT15
))/
√3 −b1/
√3
0b → �′+π− (AT
3 + AT6 − AT
15 + 6BT15 − 6DT
15)/√
3 (a1 + b3 − b5)/√
3
0b → �′0π0 −(AT
3 + AT15 + 2BT
15 − 2DT15)/
√3 −(2a1 + b2 + b3 − b5)/2
√3
0b → �′0ηq (AT
6 − 2AT15 − 4
(BT
15 + 2CT15 + DT
15
))/
√3 −(b2 + b3 + 2b4 + b5)/2
√3
0b → �′0ηs −√
2/3(AT
6 + 2AT15 + 4CT
15
) −(b1 + b4)/√
6
0b → �′−π+ (−AT
3 + AT6 − 3AT
15 + 2BT15 − 2DT
15)/√
3 −(a1 + b2)/√
3
0b → ′0K 0 (AT
3 + AT6 − AT
15 − 2BT15 − 6DT
15)/√
3 (a1 − b5)/√
3
0b → ′−K+ (−AT
3 + AT6 − 3AT
15 + 2BT15 − 2DT
15)/√
3 −(a1 + b2)/√
3
0b → �′+K− (AT
3 − AT6 − 5AT
15 + 6BT15 − 6DT
15)/√
3 (a1 − b1 + b3 − b5)/√
3
0b → �′0K 0 −(−AT
3 + AT6 + 5AT
15 + 2BT15 + 6DT
15)/√
6 (a1 − b1 − b5)/√
6
0b → ′0π0 −(AT
3 − AT6 + 3AT
15 + 6BT15 + 2DT
15)/√
6 −(a1 + b2 + b3)/√
6
0b → ′0ηq −(AT
3 − AT6 + 3AT
15 + 6BT15 + 16CT
15 + 2DT15)/
√6 −(a1 + b2 + b3 + 2b4)/
√6
0b → ′0ηs −(−AT
3 + AT6 + 5AT
15 + 2BT15 + 8CT
15 + 6DT15)/
√3 (a1 − b1 − b4 − b5)/
√3
0b → ′−π+ (−AT
3 + AT6 − 3AT
15 + 2BT15 − 2DT
15)/√
3 −(a1 + b2)/√
3
0b → �−K+ −AT
3 + AT6 − 3AT
15 + 2BT15 − 2DT
15 −a1 − b2
−b → �′0K− (AT
3 + AT6 − AT
15 + 6BT15 + 2DT
15)/√
6 (a1 + b3)/√
6
123
Eur. Phys. J. C (2020) 80 :359 Page 21 of 36 359
Table 9 continued
b → s IRA TDA
−b → �′−K
0(AT
3 + AT6 − AT
15 − 2BT15 + 2DT
15)/√
3 a1/√
3
−b → ′0π− (−AT
3 − AT6 + AT
15 − 6BT15 − 2DT
15)/√
3 −(a1 + b3)/√
3
−b → ′−π0 (AT
3 + AT6 − AT
15 − 2BT15 + 2DT
15)/√
6 a1/√
6
−b → ′−ηq (−AT
3 − AT6 + AT
15 + 2BT15 − 2DT
15)/√
6 −a1/√
6
−b → ′−ηs (AT
3 + AT6 − AT
15 − 2BT15 + 2DT
15)/√
3 a1/√
3
−b → �−K 0 −AT
3 − AT6 + AT
15 + 2BT15 − 2DT
15 −a1
where A and B are two nonperturbative coefficients con-taining the CKM factors, and have the same flavor struc-ture with Au,t . Thus in total, one has 6 × 2 × 2 = 24complex amplitudes in theory.
• Since the initial baryon and final baryons can be polar-ized, it is convenient to express the decays with helicityamplitudes:
A(Sin → S f 1 S f 2), (66)
where Sin and S f 1, S f 2 are polarizations of initialand final states. The two sets of helicity amplitudes forTb → T10P can be derived using the parametrization inEq. (65):
A(
1
2→ 1
20
)=
√2
3
mTb
mT10
pcmNT10 NTb
(A − B
pcmET10 + mT10
), (67)
A(
−1
2→ −1
20
)=
√2
3
mTb
mT10
pcmNT10 NTb
(A + B
pcmET10 + mT10
). (68)
Here ET10 and pcm are the energy and 3-momentummagnitude of T10 in the rest frame of Tb. NT10 and NTbare normalization factors of T10 and Tb spinors:
pcm = 1
2mTb
√(m2
Tb− (
mT10 + mP)2
) (m2
Tb− (
mT10 − mP)2
),
ET10 = m2T10
+ m2Tb
− m2P
2mTb,
NT10 =√
(mT10 + mTb )2 − m2
P
2mTb, NTb = √
2mTb . (69)
• For Tb → T10V , one can construct the amplitudes withthe spinors and polarization vector:2
A = ε∗ · pTb pTb,μuμ(pT10)(A′ + B ′γ5)u(pTb )
+ε∗ν pTb,μuμ(pT10)(C
′γν + D′γνγ5)u(pTb )
+ε∗μu
μ(pT10)(E′ + F ′γ5)u(pTb ). (70)
There are six different polarization configurations. Thehelicity amplitudes are given as:
A(
1
2→ 1
20
)=
√2
3
mTb
mT10
pcmNT10 NTb
×[−mTb pcm
mV
(A′ − B ′ pcm
ET10 + mT10
)
+C ′ pcmmV
(mTb − ET10
ET10 + mT10
− 1
)
+D′(mTb − ET10
mV− p2
cm
mV (ET10 + mT10 )
)
+(
− pcmmVmTb
+ ET10 (mTb − ET10 )
mVmTb pcm
)
×(E ′ − F ′ pcm
ET10 + mT10
)],
A(−1
2→ −1
20) =
√2
3
mTb
mT10
pcmNT10 NTb
×[−mTb pcm
mV
(A′ + B ′ pcm
ET10 + mT10
)
+C ′ pcmmV
(mTb − ET10
ET10 + mT10
− 1
)
−D′(mTb − ET10
mV− p2
cm
mV (ET10 + mT10 )
)
+(
− pcmmVmTb
+ ET10 (mTb − ET10 )
mVmTb pcm
)
×(E ′ + F ′ pcm
ET10 + mT10
)],
A(
1
2→ −1
21
)= 1√
3NTb NT10
2 One may expect a term which looks like εμναβε∗μuν(pT10 )(G′σαβ +
H ′σαβγ5)u(pb). Actually such term cam be absorbed into the termε∗μu
μ(pT10 )(E′ + F ′γ5)u(pb) by using the fact that the spinor-vector
uμ(pT10 ), as a irreducible representation of 1/2 ⊗ 1, must satisfyγμuμ(pT10 ) = 0.
123
359 Page 22 of 36 Eur. Phys. J. C (2020) 80 :359
×[
2pcmmTb
mT10
(D′ − C ′ p
ET10 + mT10
)
+(E ′ − F ′ p
ET10 + mT10
)],
A(
−1
2→ 1
2− 1
)= 1√
3NTb NT10
×[−2pcmmTb
mT10
(D′ + C ′ p
ET10 + mT10
)
+(E ′ + F ′ p
ET10 + mT10
)],
A(
1
2→ 3
2− 1
)= NTb NT10
(E ′ − F ′ pcm
ET10 + mT10
),
A(
−1
2→ −3
21
)= NTb NT10
(E ′ + F ′ pcm
ET10 + mT10
). (71)
The definitions of ET10 , pcm , NT10 and NTb are the sameas Eq. (69) except replacing mP by mV . Again all theseamplitudes can be determined from the angular distribu-tions of the four-body decays Tb → T10(→ T8P1)V (→P2P3).
• Branching fractions for Tb decays into a proton withthree charged pion/kaons are found at the order 10−5
in Ref. [67]. A plausible scenario is that the Tb → T10Vcontribute significantly to the Tb decaying into a protonand three charged light mesons. If this is true, we expectthat with more data in future, a detailed analysis willdetermine the decay widths of Tb → T10V . Then the fla-vor SU(3) symmetry can be examined, and meanwhile itwill also shed light on the CP and T violation in bary-onic transitions by using the triplet product asymmetries[68,69].
• Through the results in Table 9, we can find the relationsboth for decays into T8P and T8V . Here only the channelswith one vector octet in final states can be listed (72),(73). For channels with one pseudoscalar in final statesthe relations are almost the same, obtained by replacingthe vector multiplets V by the pseudo-scalar multipletsP . However, ηq and ηs are unphysical states so that thedecay width relations involving them should be removed.For b → d transitions, one has:
�(0b → −ρ+) = 3�(0
b → �′−K ∗+),
�(−b → −K
∗0) = 6�(−
b → �′−ω),
�(0b → −ρ+) = 3�(0
b → �′−ρ+),
�(−b → −K
∗0) = 3�(−
b → �′−φ),
�(0b → −ρ+) = 3�(0
b → ′−K ∗+),
�(−b → −K
∗0) = 3�(−
b → ′−K ∗0),
�(0b → �′−K ∗+) = �(0
b → �′−ρ+),
�(−b → �′−ρ0) = �(−
b → �′−ω),
�(0b → �′−K ∗+) = �(0
b → ′−K ∗+),
�(−b → �′−ρ0) = 1
2�(−
b → �′−φ),
�(0b → �′+ρ−) = �(0
b → ′0K ∗0),
�(−b → �′−ρ0) = 1
2�(−
b → ′−K ∗0),
�(0b → �′−ρ+) = �(0
b → ′−K ∗+),
�(−b → �′−ω) = 1
2�(−
b → �′−φ),
�(−b → 0K ∗−) = 2�(−
b → �′0ρ−),
�(−b → �′−ω) = 1
2�(−
b → ′−K ∗0),
�(−b → −K
∗0) = 6�(−
b → �′−ρ0),
�(−b → �′−φ) = �(−
b → ′−K ∗0). (72)
For b → s transition, we have the relations for decaywidths:
�(0b → +K ∗−) = �(0
b → 0K∗0
),
�(−b → �′−K
∗0) = 2�(−
b → ′−ω),
�(0b → �′−ρ+) = �(0
b → ′−K ∗+),
�(−b → �′−K
∗0) = �(−
b → ′−φ),
�(0b → �′−ρ+) = �(0
b → ′−ρ+),
�(−b → �′−K
∗0) = 1
3�(−
b → �−K ∗0),
�(0b → �′−ρ+) = 1
3�(0
b → �−K ∗+),
�(−b → ′−ρ0) = �(−
b → ′−ω),
�(0b → ′−K ∗+) = �(0
b → ′−ρ+),
�(−b → ′−ρ0) = 1
2�(−
b → ′−φ),
�(0b → ′−K ∗+) = 1
3�(0
b → �−K ∗+),
�(−b → ′−ρ0) = 1
6�(−
b → �−K ∗0),
�(0b → ′−ρ+) = 1
3�(0
b → �−K ∗+),
�(−b → ′−ω) = 1
2�(−
b → ′−φ),
�(−b → �′0K ∗−) = 1
2�(−
b → ′0ρ−),
�(−b → ′−ω) = 1
6�(−
b → �−K ∗0),
�(−b → �′−K
∗0) = 2�(−
b → ′−ρ0),
�(−b → ′−φ) = 1
3�(−
b → �−K ∗0). (73)
• As discussed in the previous section, charmless b → dand b → s transitions can be connected by U-spin. InTable 10, we collect the Tb → T10V decay pairs related
123
Eur. Phys. J. C (2020) 80 :359 Page 23 of 36 359
Table 10 U-spin relations for Tb → T10V . If the final state contains a light pseudoscalar meson, the U-spin relations can be obtained similarlyexcept that ηq and ηs mix
b → d b → s r b → d b → s r
A(0b → +ρ−) A(0
b → �′+K ∗−) 1 A(−b → �′−ω) A(−
b → �′−K∗0
) − 1√2
A(0b → −ρ+) A(0
b → �′−ρ+)√
3 A(−b → �′−ω) A(−
b → ′−ρ0) −1
A(0b → −ρ+) A(0
b → ′−K ∗+)√
3 A(−b → �′−ω) A(−
b → ′−ω) 1
A(0b → −ρ+) A(0
b → ′−ρ+)√
3 A(−b → �′−ω) A(−
b → ′−φ) − 1√2
A(0b → −ρ+) A(0
b → �−K ∗+) 1 A(−b → �′−ω) A(−
b → �−K ∗0) 1√6
A(0b → �′0K ∗0) A(0
b → �′0K ∗0) 1 A(−
b → �′−φ) A(−b → �′−K
∗0) 1
A(0b → �′−K ∗+) A(0
b → �′−ρ+) 1 A(−b → �′−φ) A(−
b → ′−ρ0)√
2
A(0b → �′−K ∗+) A(0
b → ′−K ∗+) 1 A(−b → �′−φ) A(−
b → ′−ω) −√2
A(0b → �′−K ∗+) A(0
b → ′−ρ+) 1 A(−b → �′−φ) A(−
b → ′−φ) 1
A(0b → �′−K ∗+) A(0
b → �−K ∗+) 1√3
A(−b → �′−φ) A(−
b → �−K ∗0) − 1√3
A(0b → +K ∗−) A(0
b → �′+ρ−) 1 A(−b → ′−K ∗0) A(−
b → �′−K∗0
) −1
A(0b → 0K
∗0) A(0
b → ′0K ∗0) 1 A(−b → ′−K ∗0) A(−
b → ′−ρ0) −√2
A(0b → �′+ρ−) A(0
b → +K ∗−) 1 A(−b → ′−K ∗0) A(−
b → ′−ω)√
2
A(0b → �′+ρ−) A(0
b → 0K∗0
) 1 A(−b → ′−K ∗0) −1A(−
b → ′−φ) −1
A(0b → �′−ρ+) A(0
b → �′−ρ+) 1 A(−b → ′−K ∗0) A(−
b → �−K ∗0) 1√3
A(0b → �′−ρ+) A(0
b → ′−K ∗+) 1 A(−b → �′0ρ−) A(−
b → �′0K ∗−) −1
A(0b → �′−ρ+) A(0
b → ′−ρ+) 1 A(−b → �′0ρ−) A(−
b → ′0ρ−) 1√2
A(0b → �′−ρ+) A(0
b → �−K ∗+) 1√3
A(−b → �′−ρ0) A(−
b → �′−K∗0
) 1√2
A(0b → ′0K ∗0) A(0
b → +K ∗−) 1 A(−b → �′−ρ0) A(−
b → ′−ρ0) 1
A(0b → ′0K ∗0) A(0
b → 0K∗0
) 1 A(−b → �′−ρ0) A(−
b → ′−ω) −1
A(0b → ′−K ∗+) A(0
b → �′−ρ+) 1 A(−b → �′−ρ0) A(−
b → ′−φ) 1√2
A(0b → ′−K ∗+) A(0
b → ′−K ∗+) 1 A(−b → �′−ρ0) A(−
b → �−K ∗0) − 1√6
A(0b → ′−K ∗+) A(0
b → ′−ρ+) 1 A(−b → −K
∗0) A(−
b → �′−K∗0
)√
3
A(0b → ′−K ∗+) A(0
b → �−K ∗+) 1√3
A(−b → −K
∗0) A(−
b → ′−ρ0)√
6
A(−b → 0K ∗−) A(−
b → �′0K ∗−)√
2 A(−b → −K
∗0) A(−
b → ′−ω) −√6
A(−b → 0K ∗−) A(−
b → ′0ρ−) −1 A(−b → −K
∗0) A(−
b → ′−φ)√
3
A(−b → −K
∗0) A(−
b → �−K ∗0) −1
by U -spin, while results for the final state with a lightpseudoscalar meson can be obtained similarly. CP asym-metries for these pairs satisfy relation in Eq. (35). Inspiredfrom B decay data [3,4], we expect CP asymmetries forthese decays are at the order 10%. Experimental mea-surements of these relations are important to test flavorSU (3) symmetry and the CKM description of CP viola-tion in SM.
6.2 Tb(b, 0b,
−b ) decay into an octet baryon and a
meson
If the final state contains a baryon octet, the topological dia-grams are shown in Fig. 5 where ten diagrams can be found.
However unlike the decuplet baryon, the octet baryon is notfully symmetric or antisymmetric in flavor space. Thus eachof the diagrams can provide more than one amplitudes. Intotal, one can have 26 independent TDA amplitudes:
AT DAu = a1T
[i j]b3
Hklm (T 8)i jk P
ml + a2T
[i j]b3
Hklm (T 8)i jl P
mk
+a3T[i j]b3
Hkli (T 8) jkl P
mm + a4T
[i j]b3
Hkli (T 8) jkm Pm
l
+a5T[i j]b3
Hkli (T 8) jlk P
mm + a6T
[i j]b3
Hkli (T 8) jmk P
ml
+a7T[i j]b3
Hkli (T 8) jlm Pm
k + a8T[i j]b3
Hkli (T 8) jml P
mk
+a9T[i j]b3
Hkli (T 8)kl j P
mm + a10T
[i j]b3
Hkli (T 8)kmj P
ml
+a11T[i j]b3
Hkli (T 8)lm j P
mk + a12T
[i j]b3
Hkli (T 8)klm Pm
j
123
359 Page 24 of 36 Eur. Phys. J. C (2020) 80 :359
Fig. 5 Topology diagrams for the bottom baryon decays into an octetbaryon and a light meson. Since the octet baryon is not fully symmetricor antisymmetric in flavor space, there are more than one amplitudes
corresponding to one topological diagram. Actually the ten topologicaldiagrams correspond to 26 amplitudes shown in Eq.(74). As in Fig. 4,the unspecified quark flavors can be u, d, s
+a13T[i j]b3
Hkli (T 8)kml P
mj + a14T
[i j]b3
Hkli (T 8)lmk P
mj
+a15T[i j]b3
Hklm (T 8)ik j P
ml + a16T
[i j]b3
Hklm (T 8)il j P
mk
+a17T[i j]b3
Hklm (T 8)ikl P
mj + a18T
[i j]b3
Hklm (T 8)ilk P
mj
+a19T[i j]b3
Hklm (T 8)kl j P
mi
+b1T[i j]b3
Hlkl (T 8)i jk P
mm + b2T
[i j]b3
Hlkl (T 8)i jm Pm
k
+b3T[i j]b3
Hlkl (T 8)ik j P
mm + b4T
[i j]b3
Hlkl (T 8)im j P
mk
+b5T[i j]b3
Hlkl (T 8)ikm Pm
j + b6T[i j]b3
Hlkl (T 8)imk P
mj
+b7T[i j]b3
Hlkl (T 8)kmi P
mj . (74)
In the IRA approach, one can construct 14 amplitudes:
AI RAu = AT
3 (Tb3)i Hj
3(T 8)
ij P
kk + BT
3 (Tb3)i Hj
3(T 8)
ik P
kj
+CT3 (Tb3)i H
i3(T 8)
kl P
lk + DT
3 (Tb3)i Hj
3(T 8)
kj P
ik
+AT6 (Tb3)i (H6)
[ik]j (T 8)
jk P
ll
+BT6 (Tb3)i (H6)
[ik]j (T 8)
lk P
jl
+CT6 (Tb3)i (H6)
[ik]j (T 8)
jl P
lk
+ET6 (Tb3)i (H6)
[ jk]l (T 8)
ij P
lk
+DT6 (Tb3)i (H6)
[ jk]l (T 8)
lj P
ik
+AT15(Tb3)i (H15)
{ik}j (T 8)
jk P
ll
+BT15(Tb3)i (H15)
{ik}j (T 8)
lk P
jl
+CT15(Tb3)i (H15)
{ik}j (T 8)
jl P
lk
+ET15(Tb3)i (H15)
{ jk}l (T 8)
ij P
lk
+DT15(Tb3)i (H15)
{ jk}l (T 8)
lj P
ik . (75)
It should be noticed that in the above hadrons have been writ-ten in different forms for the same SU(3) multiplet. For illus-tration, we use the heavy bottom anti-triplet as the example.
123
Eur. Phys. J. C (2020) 80 :359 Page 25 of 36 359
Since it is an anti-triplet, it is most straightforward to use the(Tb3)i to represent this particle multiplet, as in IRA approach.The advantage of this form is its compactness, however, withthis form it is not easy to understand the quark flows in thedecays. Instead there are two anti-symmetric light quarks inthe anti-triplet heavy bottom baryon, and thus it is viable touse (Tb3)
i j (with i j anti-symmetric) to denote this multiplet.The second form contains two SU(3) indices, less compact,but it can reflect the quark flows in the decays. Thus thesecond form is suitable for drawing Feynman diagrams, andthus adopted in the above TDA amplitude. These two formsare equal, and one can establish relations between the SU(3)parameters corresponding to these two forms. The explicitdiscussions in IRA approach are given in “Appendix A”.
The 14 IRA amplitudes and 26 TDA amplitudes are relatedas follows:
AT3 = 1
8(−2a1 + 6a2 − 5a3 − a5 + a6 − 3a8 + 4a9 + a10
−3a11 + 2a13 + 2a14 − a15 + 3a16 + 3a17 − a18 + 4a19)
+2b1 + b3 + b6 + b7,
BT3 = 1
8(6a1 − 2a2 − 5a4 − 3a6 − a7 + a8 + 2a10 + 2a11
+4a12 + a13 − 3a14 + 3a15 − a16 − a17 + 3a18 − 4a19)
+ (2b2 + b4 + b5 − b7
),
CT3 = 1
8(−a4 + 3a7 + a10 − 3a11 − 4a12 − a13 + 3a14
+a17 − 3a18 + 4a19 − 8b5) + b7,
DT3 = 1
8
(−4(a19 + 2
(b6 + b7
)) − a6
+3a8 − a10 + 3a11 − 2a13 − 2a14 − 3a17 + a18) ,
AT6 = 1
4(a3 − a5 − 2a9 − a13 + a14) ,
BT6 = 1
4(a13 − a14 − a17 + a18 − 2a19) ,
CT6 = 1
4(a4 − a7 − a10 + a11 − 2a12) ,
DT6 = 1
4(a6 − a8 + a10 − a11 − a13 + a14) ,
ET6 = 1
4(2a1 − 2a2 − a6 + a8 − a10 + a11 + a13
−a14 + a15 − a16 − a17 + a18 − 2a19) ,
AT15 = 1
8(a3 + a5 − a13 − a14) ,
BT15 = 1
8(a13 + a14 − a17 − a18) ,
CT15 = 1
8(a4 + a7 − a10 − a11) ,
DT15 = 1
8(a6 + a8 + a10 + a11 + a13 + a14) ,
ET15 = 1
8(2a1 + 2a2 − a6 − a8 − a10 − a11
−a13 − a14 + a15 + a16 + a17 + a18) . (76)
However, even after such reduction, there still exists oneindependent degree of freedom among the 14 IRA ampli-
tudes. The redundant amplitude can be made explicit withthe redefinitions:
AT ′6 = AT
6 + BT6 , BT ′
6 = BT6 − CT
6 ,
CT ′6 = CT
6 − ET6 , DT ′
6 = CT6 + DT
6 . (77)
In addition, this redundancy can be understood moreexplicitly. In this work as well as the previous work Ref.[22] we use the irreducible representation operators for IRAas (H6/15)
i jk . Actually there exists a simpler H6 represen-
tation introduced by Ref. [70], where H6 has only twolower indexes (H6)i j . With the use of (H6)i j we do haveonly 13 IRA amplitudes. However, Since the IRA operators(H6/15)
i jk have the same index structure as the TDA opera-
tors. They make the derivation of IRA/TDA correspondencemore directly so we will keep the use of them.
The expanded amplitudes can be found in Table 11 forthe b → d transition and Table 12 for the b → s transition,respectively. Again if the final state is a vector meson, theamplitudes can be derived similarly.
A few remarks are given in order.
• At a first sight, the diagrammatic approach, as depictedin Fig. 5, is more intuitive, however there are more TDAamplitudes than the corresponding diagrams. For an octetbaryon in the final state, there are three light quarks. In thesame diagram, the symmetry of the quarks in flavor spacecould be different. For example, in the third diagram ofFig. 5, the u quark and another light quark could be flavoranti-symmetric, or the two unspecified quarks could beflavor anti-symmetric. These different combinations willlead to different TDA amplitudes. Thus it is very hard todetermine the independent amplitudes in this approach,which will introduce subtleties to the global fit in the dia-grammatic approach. The mismatch between Feynmandiagrams and TDA amplitudes will not happen for thedecays into decuplet baryons, since all three quarks aresymmetric in flavor space.
• Without including the polarization, one can see fromthe IRA approach, there exist 13 independent complexamplitudes with CKM factor VubV ∗
uq and another 13amplitudes accompanied by VtbV ∗
tq .• Two polarization configurations exist for decays into a
pseudoscalar meson, while there are four possibilities fordecays into a vector meson.
• The U-spin related decay pairs are given in Table 13,which completely fits with the results given by Ref. [73].Here only the case for Tb → B8P is listed. Since nounphysical states ηq and ηs exist in Table 13. The U-spin pairs for Tb → B8V are similar by replacing pseu-doscalar octets by vector octets.
• Some theoretical analyses of nonleptonic bottom baryondecays based on either explicit modes or the flavor sym-
123
359 Page 26 of 36 Eur. Phys. J. C (2020) 80 :359
Table 11 Decay amplitudes for Tb → T8P decays governed by the b → d transition. Results for vector meson final state are similar
Channel IRA TDA
0b → 0K 0 (−2BT
3 + DT3 − BT
6 + 2CT6 + 2ET
6 + 3DT6
−BT15 + 2CT
15 + 2ET15 + 3DT
15)/√
6(2a4 + a6 − a10 − 2a12 − a13 + a14
−4b2 − 2b4 − 2b5 − b6 + b7)/√
6
0b → �0K 0 (−DT
3 + BT6 + DT
6 + BT15 + 5DT
15)/√
2 (a6 + a10 + a13 + a14 + b6 + b7)/√
2
0b → �−K+ DT
3 − BT6 − DT
6 − BT15 + 3DT
15 a8 + a11 − b6 − b7
0b → pπ− BT
3 − CT6 + ET
6 − CT15 + 3ET
15 2a1 − a4 − a6 + a12 − a14 + a15+a18 − a19 + 2b2 + b4 + b5 − b7
0b → nπ0 (−BT
3 + CT6 − ET
6 + CT15 + 5ET
15)/√
2 (2a2 + a4 − a8 − a10 − a11 − a12 − a13 + a16+a17 + a19 − 2b2 − b4 − b5 + b7)/
√2
0b → nηq (2AT
3 +BT3 −2AT
6 −CT6 −ET
6 −2AT15−CT
15+ET15)/
√2 (1/
√2)(2a2 − 2a3 − a4 − a8 + 2a9 + a10 − a11
+a12 + a13 + a16 + a17 + a19 + 4b1 + 2b2+2b3 + b4 + b5 + 2b6 + b7)
0b → nηs AT
3 + DT3 − AT
6 − BT6 + ET
6+DT
6 − AT15 − BT
15 − ET15 − DT
15
−a3 + a9 + 2b1 + b3
0b → 0π0 (BT
3 + DT3 + BT
6 + CT6 + ET
6 + 3DT6
−5BT15 − 5CT
15 − 5ET15 + 3DT
15)/2√
3(1/2
√3)(−2a2−a4+a6−a7+a8+2a10+2a11−a16
−a17 + a18 − 2a19 + 2b2 + b4 + b5 − b6 − 2b7)
0b → 0ηq −(2AT
3 + BT3 + 2CT
3 + DT3 − 6AT
6 − BT6 −CT
6 − ET6
+3DT6 + 6AT
15 + BT15 + CT
15 + ET15 + 3DT
15)/2√
3−(1/2
√3)(2a2 − 2a3 − a4 + 2a5 + a6 + a7 − a8
+4a9 + 2a10 − 2a11 + a16 + a17 − a18 + 2a19+4b1 + 2b2 + 2b3 + b4 − b5 + b6 + 2b7)
0b → 0ηs −(AT
3 − 2CT3 − 3AT
6 − 2BT6 − 2CT
6 + ET6
+3AT15 + 2BT
15 + 2CT15 − ET
15)/√
6(a3 − a5 − 2a9 − 2a12 − a13 + a14
−2b1 − b3 − 2b5 − b6 + b7)/√
6
0b → �+π− −BT
3 − CT3 + BT
6 − ET6 − 3BT
15 + 2CT15 − 3ET
15 −2a1 + a4 + a6 − a15 − 2b2 − b4
0b → �0π0 (1/2)(−BT
3 − 2CT3 − DT
3 + BT6 + CT
6−ET
6 + DT6 − BT
15 − CT15 + 5ET
15 + 5DT15)
(1/2)(2a2 + a4 + a6 − a7 − a8 + a16+a17 + a18 − 2b2 − b4 + b5 + b6)
0b → �0ηq (1/2)(2AT
3 + BT3 + DT
3 + 2AT6 + BT
6 + CT6 − ET
6−DT
6 − 10AT15 − 5BT
15 − 5CT15 + ET
15 − 5DT15)
(1/2)(2a2 − 2a3 − a4 − 2a5 − a6 − a7 − a8 + a16+a17 + a18 + 4b1 + 2b2 + 2b3 + b4 + b5 + b6)
0b → �0ηs (AT
3 + AT6 + ET
6 − 5AT15 − ET
15)/√
2 (−a3 − a5 + a13 + a14 + 2b1 + b3 + b6 + b7)/√
2
0b → �−π+ −CT
3 − DT3 + CT
6 + DT6 + 2BT
15 − 3CT15 − 3DT
15 −a7 − a8 + b5 + b6
0b → pK− −CT
3 + BT6 − CT
6 − 3BT15 + CT
15 a12 − a14 + a18 − a19 + b5 − b7
0b → nK
0 −CT3 − DT
3 − CT6 − DT
6 + 2BT15 + CT
15 + DT15 a12 + a13 + b5 + b6
0b → −K+ −CT
3 − BT6 + CT
6 + BT15 − 3CT
15 −a7 + a11 + b5 − b7
0b → 0K 0 −BT
3 − CT3 − BT
6 + ET6 + BT
15 + 2CT15 + ET
15 a4 − a10 − 2b2 − b4
−b → 0π− (BT
3 + DT3 + BT
6 + CT6 + ET
6 + 3DT6
+3BT15 + 3CT
15 + 3ET15 + 3DT
15)/√
6(2a1 + a15 − a17 + a18 − 2a19
+2b2 + b4 + b5 − b6 − 2b7)/√
6
−b → �0π− (BT
3 − DT3 − BT
6 + CT6 + ET
6 + DT6
−3BT15 + 3CT
15 + 3ET15 + 5DT
15)/√
2(2a1 + a15 + a17 + a18 + 2b2 + b4 + b5 + b6)/
√2
−b → �−π0 −(BT
3 − DT3 − BT
6 + CT6 + ET
6 + DT6
−3BT15 + 3CT
15 − 5ET15 − 3DT
15)/√
2(2a2 + a16 − 2b2 − b4 − b5 − b6)/
√2
−b → �−ηq (2AT
3 + BT3 + DT
3 + 2AT6 + BT
6 + CT6 − ET
6 − DT6
+6AT15 + 3BT
15 + 3CT15 + ET
15 + 3DT15)/
√2
(2a2 + a16 + 4b1 + 2b2 + 2b3 + b4 + b5 + b6)/√
2
−b → �−ηs AT
3 + AT6 + ET
6 + 3AT15 − ET
15 2b1 + b3 + b6 + b7
−b → nK− DT
3 + BT6 + DT
6 + 3BT15 − DT
15 −a17 − a19 − b6 − b7
−b → −K 0 BT
3 + CT6 − ET
6 + 3CT15 − ET
15 2b2 + b4 + b5 − b7
123
Eur. Phys. J. C (2020) 80 :359 Page 27 of 36 359
Table 12 Decay amplitudes for two-body Tb → T8P decays induced by the b → s transition
Channel IRA TDA
0b → 0π0 −(−2(ET
6 + BT15 + CT
15 − 2ET15) + BT
6 + CT6 )/
√3 (−4a2 + a7 + 2a8 + a11 + a12 + 2a13 + a14
−2a16 − 2a17 − a18 − a19)/2√
3
0b → 0ηq (−2AT
3 + CT3 + 6AT
15 + BT15 + CT
15 − 2ET15)/
√3 −1/2
√3(4a2 − 4a3 − 2a5 − a7 − 2a8 + 2a9 − a11
+a12 + 2a13 + a14 + 2a16 + 2a17 + a18+a19 + 8b1 + 4b3 + 2b5 + 4b6 + 2b7)
0b → 0ηs −
√23 (AT
3 + BT3 + CT
3 + DT3
−3AT15 − 2BT
15 − 2CT15 − 2ET
15 − 3DT15)
(2a3 + 2a4 + a5 + a6 − a9 − a10−4b1 − 4b2 − 2b3 − 2b4)/
√6
0b → �+π− CT
3 − BT6 + CT
6 + 3BT15 − CT
15 −a12 + a14 − a18 + a19 − b5 + b7
0b → �0π0 CT
3 + BT15 + CT
15 1/2(a7 − a11 − a12 + a14 − a18 + a19 − 2b5 + 2b7)
0b → �0ηq −2AT
6 − BT6 − CT
6 + 4AT15 + 2BT
15 + 2CT15 1/2(2a5 + a7 + 2a9 − a11 + a12 − a14 − a18 + a19)
0b → �0ηs
√2(−AT
6 + DT6 + 2(AT
15 + DT15)) (a5 + a6 + a9 + a10)/
√2
0b → �−π+ CT
3 + BT6 − CT
6 − BT15 + 3CT
15 a7 − a11 − b5 + b7
0b → pK− BT
3 + CT3 − BT
6 + ET6 + 3BT
15 − 2CT15 + 3ET
15 2a1 − a4 − a6 + a15 + 2b2 + b4
0b → nK
0BT
3 + CT3 + BT
6 − ET6 − BT
15 − 2CT15 − ET
15 −a4 + a10 + 2b2 + b4
0b → −K+ CT
3 + DT3 − CT
6 − DT6 − 2BT
15 + 3CT15 + 3DT
15 a7 + a8 − b5 − b6
0b → 0K 0 CT
3 + DT3 + CT
6 + DT6 − 2BT
15 − CT15 − DT
15 −a12 − a13 − b5 − b6
0b → 0K
0(−BT
3 + 2DT3 − 2BT
6 + CT6 + ET
6−2BT
15 + CT15 + ET
15 − 6DT15)/
√6
(a4 − a6 − 2a10 − a12 − 2a13 − a14−2b2 − b4 − b5 − 2b6 − b7)/
√6
0b → �+K− −BT
3 + CT6 − ET
6 + CT15 − 3ET
15 −2a1 + a4 + a6 − a12 + a14 − a15 − a18+a19 − 2b2 − b4 − b5 + b7
0b → �0K
0(BT
3 − CT6 − ET
6 − 2DT6 − CT
15 − ET15 − 4DT
15)/√
2 −(a4 + a6 − a12 + a14 − 2b2 − b4 − b5 + b7)/√
2
0b → −π+ −DT
3 + BT6 + DT
6 + BT15 − 3DT
15 −a8 − a11 + b6 + b7
0b → 0π0 (DT
3 − BT6 + 2ET
6 + DT6 − BT
15 − 4ET15 − DT
15)/√
2 −(2a2 − a8 − a11 + a16 + a17 + a19 + b6 + b7)/√
2
0b → 0ηq (−2AT
3 − DT3 + 2AT
6 + BT6 − DT
6+2AT
15 + BT15 − 2ET
15 + DT15)/
√2
−(2a2 − 2a3 − a8 + 2a9 − a11 + a16 + a17+a19 + 4b1 + 2b3 + b6 + b7)/
√2
0b → 0ηs −AT
3 − BT3 + AT
6 + CT6 + AT
15 + CT15 + 2ET
15 a3 + a4 − a9 − a10 − a12 − a13 − 2b1−2b2 − b3 − b4 − b5 − b6
−b → 0K− (BT
3 − 2DT3 − 2BT
6 + CT6 + ET
6−6BT
15 + 3CT15 + 3ET
15 + 6DT15)/
√6
(2a1 + a15 + 2a17 + a18 + a19+2b2 + b4 + b5 + 2b6 + b7)/
√6
−b → �0K− (BT
3 +CT6 + ET
6 +2DT6 +3CT
15 +3ET15 +4DT
15)/√
2 (2a1 + a15 + a18 − a19 + 2b2 + b4 + b5 − b7)/√
2
−b → �−K
0BT
3 + CT6 − ET
6 + 3CT15 − ET
15 2b2 + b4 + b5 − b7
−b → −π0 (DT
3 + BT6 −2ET
6 − DT6 +3BT
15 +4ET15 +3DT
15)/√
2 (2a2 + a16 − b6 − b7)/√
2
−b → −ηq (2AT
3 + DT3 + 2AT
6 + BT6 − DT
6+6AT
15 + 3BT15 + 2ET
15 + 3DT15)/
√2
(2a2 + a16 + 4b1 + 2b3 + b6 + b7)/√
2
−b → −ηs AT
3 + BT3 + AT
6 + CT6 + 3AT
15 + 3CT15 − 2ET
15 2b1 + 2b2 + b3 + b4 + b5 + b6
−b → 0π− DT
3 + BT6 + DT
6 + 3BT15 − DT
15 −a17 − a19 − b6 − b7
metry can be found in Refs. [74–79], while the experi-mental measurements can be found in Refs. [80–82]. Todate, the available measurements of two-body b branch-ing fractions are [3,4]:
B(b → pπ−) = (4.2 ± 0.8) × 10−6,
B(b → pK−) = (5.1 ± 0.9) × 10−6,
B(b → η) = (9+7−5) × 10−6,
B(b → η′) < 3.1 × 10−6,
B(b → pφ) = (9.2 ± 2.5) × 10−6. (78)
• The CP asymmetries for b → pπ−/pK− [82] havebeen measured:
Apπ−CP = −0.020 ± 0.013 ± 0.019,
ApK−CP = −0.035 ± 0.017 ± 0.020. (79)
123
359 Page 28 of 36 Eur. Phys. J. C (2020) 80 :359
Thus measuring the branching fractions and CP asym-metries for 0
b → π−�+ and 0b → K−�+ will help
us to understand the U-spin in baryonic decays.
7 Antitriplet charmed Baryon Tc(�c,�+c ,�0
c) decays
For the charmed baryon decays, the H3 contributions arevanishingly small, and thus we have 19 amplitudes in TDA:
AT DAu = a1T
[i j]c3
Hklm (T 8)i jk P
ml + a2T
[i j]c3
Hklm (T 8)i jl P
mk
+a3T[i j]c3
Hkli (T 8) jkl P
mm
+a4T[i j]c3
Hkli (T 8) jkm Pm
l
+a5T[i j]c3
Hkli (T 8) jlk P
mm + a6T
[i j]c3
Hkli (T 8) jmk P
ml
+a7T[i j]c3
Hkli (T 8)ilm Pm
k + a8T[i j]c3
Hkli (T 8) jml P
mk
+a9T[i j]c3
Hkli (T 8)kl j P
mm + a10T
[i j]c3
Hkli (T 8)kmj P
ml
+a11T[i j]c3
Hkli (T 8)lm j P
mk + a12T
[i j]c3
Hkli (T 8)klm Pm
j
+a13T[i j]c3
Hkli (T 8)kml P
mj + a14T
[i j]c3
Hkli (T 8)lmk P
mj
+a15T[i j]c3
Hklm (T 8)ik j P
ml + a16T
[i j]c3
Hklm (T 8)il j P
km
+a17T[i j]c3
Hklm (T 8)ikl P
mj + a18T
[i j]c3
Hklm (T 8)ilk P
mj
+a19T[i j]c3
Hklm (T 8)kl j P
mi . (80)
The corresponding Feynman diagrams are given in Fig. 6, inwhich seven Feynman diagrams can be found. The analysisfor independent amplitudes are almost the same as that ofbottom baryon decays. On the other side, ten IRA amplitudescan be constructed as:
AI RAu = AT
6 (Tc3)i (H6)[ik]j (T 8)
jk P
ll
+BT6 (Tc3)i (H6)
[ik]j (T 8)
lk P
jl
+CT6 (Tc3)i (H6)
[ik]j (T 8)
jl P
lk
+ET6 (Tc3)i (H6)
[ jk]l (T 8)
ij P
lk
+DT6 (Tc3)i (H6)
[ jk]l (T 8)
lj P
ik
+AT15(Tc3)i (H15)
{ik}j (T 8)
jk P
ll
+BT15(Tc3)i (H15)
{ik}j (T 8)
lk P
jl
+CT15(Tc3)i (H15)
{ik}j (T 8)
jl P
lk
+ET15(Tc3)i (H15)
{ jk}l (T 8)
ij P
lk
+DT15(Tc3)i (H15)
{ik}l (T 8)
lj P
ik . (81)
Only nine of them are independent, and one redundant ampli-tude can be made explicit with the redefinitions:
AT ′6 = AT
6 + BT6 , BT ′
6 = BT6 − CT
6 ,
CT ′6 = CT
6 − ET6 , DT ′
6 = CT6 + DT
6 , (82)
which is exactly the same as Eq. (77).After a careful examination, one can also find the relations:
AT6 = 1
2(a3 − a5 − 2a9 − a13 + a14) ,
BT6 = 1
2(a13 − a14 − a17 + a18 − 2a19) ,
CT6 = 1
2(a4 − a7 − a10 + a11 − 2a12) ,
DT6 = 1
2(a6 − a8 + a10 − a11 − a13 + a14) ,
ET6 = 1
2(2a1 − 2a2 − a6 + a8 − a10 + a11 + a13
−a14 + a15 − a16 − a17 + a18 − 2a19) ,
AT15 = 1
2(a3 + a5 − a13 − a14) ,
BT15 = 1
2(a13 + a14 − a17 − a18) ,
CT15 = 1
2(a4 + a7 − a10 − a11) ,
DT15 = 1
2(a6 + a8 + a10 + a11 + a13 + a14) ,
ET15 = 1
2(2a1 + 2a2 − a6 − a8 − a10 − a11
−a13 − a14 + a15 + a16 + a17 + a18) . (83)
Some further remarks are given in order.
• The flavor SU(3) symmetry in charmed baryon decaysand the symmetry breaking effects have been extensivelyexplored in Refs. [70,83–90], and we refer the reader tothese references for detailed discussions.
• On the experimental side, BESIII collaboration has giventhe first measurement of decay branching fractions for theW -exchange induced decays [91]:
B(c → 0K+) = (5.90 ± 0.86 ± 0.39) × 10−3,
(84)
B(c → (1530)0K+) = (5.02 ± 0.99 ± 0.31) × 10−3.
(85)
It indicates that the decays into a decuplet baryon mightnot be power suppressed compared to those decays intoan octet baryon. This introduces a theoretical difficultyto understand the charmed baryon decays.
• One can find some relations between the different chan-nels listed in Tables 14, 15 and 16. For the charmedbaryon two-body decay, there is only one relation fordecay width:
�(+
c → �+π0)
= �(+c → �0π+). (86)
123
Eur. Phys. J. C (2020) 80 :359 Page 29 of 36 359
Table 13 U-spin relations for Tb → T8P
b → d b → s r b → d b → s r
A(−b → K−n) A(−
b → π−0) +1 A(0b → K 0n) A(0
b → K 00) −1
A(−b → K 0−) A(−
b → K 0�−) +1 A(0b → K 00) A(0
b → K 0n) −1
A(0b → π−�+) A(0
b → K− p) −1 A(0b → π− p) A(0
b → K−�+) −1
A(0b → π+�−) A(0
b → K+−) −1 A(0b → K+�−) A(0
b → π+−) −1
A(0b → K− p) A(0
b → π−�+) −1 A(0b → K+−) A(0
b → π+�−) +1
Fig. 6 Topology diagrams for the charmed baryon decays into an octet baryon and a light meson. Due to the same reason as that of bottom baryondecays, one topological diagrams correspond to more than one TDA amplitudes. Here the 7 topological diagrams correspond to 19 TDA amplitudesas given in Eq. (80)
This relation fits well with the data in Ref. [3]:
B(+
c → �+π0)
= 1.24 ± 0.10%,
B(+
c → �0π+)= 1.28 ± 0.07%. (87)
• In Ref. [70], a global fit was conducted for charmedbaryon decays, particularly inspired by the recent BESIIIdata [71,72,91]. In that work the sextet contribution wasexpressed in a different representation. Relating the fourcoefficients in [70] with our notations, we have:3
− AT6 + DT
6 = h = (0.105 ± 0.073) GeV3,
−BT6 + ET
6 = a1 = (0.244 ± 0.006) GeV3, (88)
−CT6 − DT
6 = a2 = (0.115 ± 0.014) GeV3,
3 In Ref. [70], these parameters are denoted as a1, a2, a3, h. Here weadd primes in order to distinguish them with the parameters used in thiswork.
ET6 + DT
6 = a3 = (0.088 ± 0.019) GeV3. (89)
Such a fit was conducted with the neglect of the H15terms, which might be challenged in interpreting thec → pπ0 [84,87,89].
8 Discussions and conclusions
In this work, we have carried out a comprehensive analy-sis comparing two different realizations of the flavor SU(3)symmetry, the irreducible operator representation amplitudeand topological diagram amplitude, to study various bot-tom/charm meson and baryon decays.
We find that previous analyses in the literature using thesetwo methods do not match consistently in several ways. TheTDA approach provides a more intuitive understanding of thedecays, however it also suffers from a few subtleties. Usingtwo-body B/D meson decays, we have demonstrated that a
123
359 Page 30 of 36 Eur. Phys. J. C (2020) 80 :359
Table 14 Decay amplitudes for two-body Cabibblo-Allowed charmed baryon decays
Channel IRA TDA
+c → 0π+ (BT
6 + CT6 − 2ET
6 + BT15 + CT
15 − 2ET15)/
√6 (−4a1 + a4 + 2a6 + a10 − a12 + a13 + 2a14
−2a15 − a17 − 2a18 + a19)/√
6
+c → �+π0 (−BT
6 + CT6 − BT
15 + CT15)/
√2 (a4 − a10 − a12 − a13 + a17 + a19)/
√2
+c → �+ηq (2AT
6 + BT6 + CT
6 + 2AT15 + BT
15 + CT15)/
√2 (2a3 + a4 − 2a9 − a10 − a12 − a13 − a17 − a19)/
√2
+c → �+ηs AT
6 − DT6 + AT
15 + DT15 a3 + a8 − a9 + a11
+c → �0π+ (BT
6 − CT6 + BT
15 − CT15)/
√2 (−a4 + a10 + a12 + a13 − a17 − a19)/
√2
+c → pK
0BT
6 − ET6 + BT
15 + ET15 2a2 − a8 − a11 + a16
+c → 0K+ CT
6 + DT6 + CT
15 + DT15 a4 + a6 − a12 + a14
+c → �+K
0ET
6 + DT6 − ET
15 − DT15 −2a2 − a16 − a17 − a19
+c → 0π+ −ET
6 − DT6 − ET
15 − DT15 −2a1 − a15 − a18 + a19
0c → 0K
0(2BT
6 − CT6 − ET
6 − 2BT15 + CT
15 + ET15)/
√6 (2a2 + a7 − a8 − 2a11 + a12 − a13 − 2a14
+a16 + a17 + 2a18 − a19)/√
6
0c → �+K− −CT
6 − DT6 + CT
15 + DT15 a7 + a8 + a12 + a13
0c → �0K
0(CT
6 − ET6 − CT
15 + ET15)/
√2 (2a2 − a7 − a8 − a12 − a13 + a16 + a17 + a19)/
√2
0c → −π+ −BT
6 + ET6 + BT
15 + ET15 2a1 − a6 − a10 + a15
0c → 0π0 (BT
6 + DT6 − BT
15 + DT15)/
√2 (a6 + a10 + a18 − a19)/
√2
0c → 0ηq (−2AT
6 − BT6 + DT
6 + 2AT15 + BT
15 + DT15)/
√2 (2a5 + a6 + 2a9 + a10 − a18 + a19)/
√2
0c → 0ηs −AT
6 − CT6 + AT
15 + CT15 a5 + a7 + a9 − a11 + a12 − a14
Table 15 Decay amplitudes for two-body Singly Cabibblo-Suppressed charmed baryon decays
Channel IRA TDA
+c → 0K+ (BT
6 − 2CT6 − 2ET
6 − 3DT6 + BT
15−2CT
15 − 2ET15 − 3DT
15)/√
6−(4a1 + 2a4 + a6 − a10 − 2a12 − a13
+a14 + 2a15 + a17 + 2a18 − a19)/√
6
+c → �+K 0 BT
6 + DT6 + BT
15 − DT15 −a8 − a11 − a17 − a19
+c → �0K+ (BT
6 + DT6 + BT
15 + DT15)/
√2 (a6 + a10 + a13 + a14 − a17 − a19)/
√2
+c → pπ0 (CT
6 − ET6 + CT
15 + ET15)/
√2 (2a2 + a4 − a8 − a10 − a11
−a12 − a13 + a16 + a17 + a19)/√
2
+c → pηq (2AT
6 + CT6 + ET
6 + 2AT15 + CT
15 − ET15)/
√2 −(2a2 − 2a3 − a4 − a8 + 2a9 + a10 − a11
+a12 + a13 + a16 + a17 + a19)/√
2
+c → pηs AT
6 + BT6 − ET
6 − DT6 + AT
15 + BT15 + ET
15 + DT15 2a2 + a3 − a9 + a16
+c → nπ+ CT
6 − ET6 + CT
15 − ET15 −2a1 + a4 + a6 − a12 + a14 − a15 − a18 + a19
+c → 0π+ (BT
6 +CT6 +ET
6 +3DT6 +BT
15+CT15+ET
15+3DT15)/
√6 (2a1 + a4 + 2a6 + a10 − a12 + a13 + 2a14
+a15 − a17 + a18 − 2a19)/√
6
+c → �+π0 (−BT
6 +CT6 +ET
6 +DT6 −BT
15+CT15−ET
15−DT15)/
√2 −(2a2 − a4 + a10 + a12 + a13 + a16)/
√2
+c → �+ηq (2AT
6 + BT6 + CT
6 − ET6 − DT
6 + 2AT15
+BT15 + CT
15 + ET15 + DT
15)/√
2(2a2 + 2a3 + a4 − 2a9 − a10 − a12 − a13 + a16)/
√2
+c → �+ηs AT
6 + ET6 + AT
15 − ET15 −2a2 + a3 + a8 − a9 + a11 − a16 − a17 − a19
+c → �0π+ (BT
6 −CT6 −ET
6 −DT6 +BT
15 −CT15 −ET
15 −DT15)/
√2 −(2a1 + a4 − a10 − a12 − a13 + a15 + a17 + a18)/
√2
+c → pK
0BT
6 + DT6 + BT
15 − DT15 −a8 − a11 − a17 − a19
123
Eur. Phys. J. C (2020) 80 :359 Page 31 of 36 359
Table 15 continued
Channel IRA TDA
+c → 0K+ CT
6 − ET6 + CT
15 − ET15 −2a1 + a4 + a6 − a12 + a14 − a15 − a18 + a19
0c → 0π0 −(BT
6 + CT6 + ET
6 + 3DT6 − BT
15 − CT15 − ET
15 +3DT
15)/2√
3(2a2 − 3a6 + a7 − a8 − 3a10 − 2a11 + a12
−a13 − 2a14 + a16 + a17 − a18 + 2a19)/2√
3
0c → 0ηq (6AT
6 + BT6 + CT
6 + ET6 − 3DT
6 − 6AT15
−BT15 − CT
15 − ET15 − 3DT
15)/2√
3−(1/2
√3)(2a2 + 6a5 + 3a6 + a7 − a8 + 6a9 + 3a10
−2a11 + a12 − a13 −2a14 + a16 + a17 − a18 +2a19)
0c → 0ηs (3AT
6 + 2BT6 + 2CT
6 − ET6 − 3AT
15−2BT
15 − 2CT15 + ET
15)/√
6(2a2 − 3a5 − 2a7 − a8 − 3a9 + a11 − 2a12
−a13 + a14 + a16 + a17 + 2a18 − a19)/√
6
0c → �+π− CT
6 + DT6 − CT
15 − DT15 −a7 − a8 − a12 − a13
0c → �0π0 1/2(BT
6 +CT6 − ET
6 +DT6 − BT
15 −CT15 + ET
15 +DT15) (1/2)(2a2 + a6 − a7 − a8 + a10 − a12
−a13 + a16 + a17 + a18)
0c → �0ηq 1/2(−2AT
6 − BT6 − CT
6 + ET6 + DT
6+2AT
15 + BT15 + CT
15 − ET15 + DT
15)
(1/2)(−2a2 + 2a5 + a6 + a7 + a8 + 2a9+a10 + a12 + a13 − a16 − a17 − a18)
0c → �0ηs (−AT
6 − ET6 + AT
15 + ET15)/
√2 (2a2 + a5 − a8 + a9 − a11 − a13−
a14 + a16 + a17 + a19)/√
2
0c → �−π+ BT
6 − ET6 − BT
15 − ET15 −2a1 + a6 + a10 − a15
0c → pK− −CT
6 − DT6 + CT
15 + DT15 a7 + a8 + a12 + a13
0c → nK
0BT
6 − CT6 − BT
15 + CT15 a7 − a11 + a12 − a14 + a18 − a19
0c → −K+ −BT
6 + ET6 + BT
15 + ET15 2a1 − a6 − a10 + a15
0c → 0K 0 −BT
6 + CT6 + BT
15 − CT15 −a7 + a11 − a12 + a14 − a18 + a19
Table 16 Decay amplitudes for two-body Doubly Cabibblo-Suppressed charmed baryon decays
Channel IRA TDA
+c → pK 0 −ET
6 − DT6 + ET
15 + DT15 2a2 + a16 + a17 + a19
+c → nK+ ET
6 + DT6 + ET
15 + DT15 2a1 + a15 + a18 − a19
+c → 0K+ −(BT
6 − 2CT6 + ET
6 + BT15 − 2CT
15 + ET15)/
√6 −(2a1 − 2a4 − a6 + a10 + 2a12 + a13 − a14 +a15 − a17 + a18 − 2a19)/
√6
+c → �+K 0 −BT
6 + ET6 − BT
15 − ET15 −2a2 + a8 + a11 − a16
+c → �0K+ (−BT
6 + ET6 − BT
15 + ET15)/
√2 (2a1 − a6 − a10 − a13 − a14 + a15 + a17 + a18)/
√2
+c → pπ0 −(CT
6 + DT6 + CT
15 − DT15)/
√2 (−a4 + a8 + a10 + a11 + a12 + a13)/
√2
+c → pηq −(2AT
6 + CT6 − DT
6 + 2AT15 + CT
15 + DT15)/
√2 (−2a3 − a4 − a8 + 2a9 + a10 − a11 + a12 + a13)/
√2
+c → pηs −AT
6 − BT6 − AT
15 − BT15 −a3 + a9 + a17 + a19
+c → nπ+ −CT
6 − DT6 − CT
15 − DT15 −a4 − a6 + a12 − a14
0c → 0K 0 (−BT
6 + 2CT6 − ET
6 + BT15 − 2CT
15 + ET15)/
√6 (2a2 − 2a7 − a8 + a11 − 2a12 − a13 + a14 +a16 + a17 − a18 + 2a19)/
√6
0c → �0K 0 (BT
6 − ET6 − BT
15 + ET15)/
√2 (2a2 − a8 − a11 − a13 − a14 + a16 + a17 + a18)/
√2
0c → �−K+ −BT
6 + ET6 + BT
15 + ET15 2a1 − a6 − a10 + a15
0c → pπ− −CT
6 − DT6 + CT
15 + DT15 a7 + a8 + a12 + a13
0c → nπ0 (CT
6 + DT6 − CT
15 + DT15)/
√2 (a6 − a7 + a10 + a11 − a12 + a14)/
√2
0c → nηq (−2AT
6 − CT6 + DT
6 + 2AT15 + CT
15 + DT15)/
√2 (2a5 + a6 + a7 + 2a9 + a10 − a11 + a12 − a14)/
√2
0c → nηs −AT
6 − BT6 + AT
15 + BT15 a5 + a9 − a18 + a19
few SU(3) independent amplitudes (the last six diagrams inFig. 1) are sometimes overlooked in TDA (for instance Refs.[16,18]). Most of these amplitudes arises from higher orderloop corrections, but they are irreducible in the flavor SU(3)space, and thus can not be neglected in principle. Taking thesenew amplitudes into account, we find a consistent description
in both approaches. In addition, using B and D decays wehave found that one of the A,C, E, T amplitudes should beabsorbed into others, which has been pointed out in Ref. [20].For heavy baryon decays, we pointed out though the TDAapproach is very intuitive, it suffers the difficulty in providing
123
359 Page 32 of 36 Eur. Phys. J. C (2020) 80 :359
the independent amplitudes. On this point, the IRA approachis more helpful.
All results derived in this paper can be used to study theheavy meson and baryon decays in the future when sufficientdata become available. Then one can have a better under-standing of the role of flavor SU(3) symmetry in heavy mesonand baryon decays.
For charm quark decays, we did not include the penguincontributions, which can also be studied in a similar manner.It is also necessary to notice that the flavor SU(3) symme-try has been applied to study weak decays of doubly heavybaryons [92–94], and multi-body c decays [90]. The equiv-alence between the TDA and IRA approaches in these decaymodes can be studied similarly.
Acknowledgements The authors are grateful to Hai-Yang Cheng,Chao-Qiang Geng, Di Wang, Fu-Sheng Yu and Ruilin Zhu for use-ful discussions. This work is supported in part by National NaturalScience Foundation of China under Grant Nos. 11575110, 11575111,11735010, and Natural Science Foundation of Shanghai under GrantNo. 15DZ2272100. X.G. He was also supported in part by MOST(Grant Nos. MOST104-2112-M-002-015-MY3 and 106-2112-M-002-003-MY3).
Data Availability Statement This manuscript has no associated dataor the data will not be deposited. [Authors’ comment: There is no asso-ciated data for our paper.]
Open Access This article is licensed under a Creative Commons Attri-bution 4.0 International License, which permits use, sharing, adaptation,distribution and reproduction in any medium or format, as long as yougive appropriate credit to the original author(s) and the source, pro-vide a link to the Creative Commons licence, and indicate if changeswere made. The images or other third party material in this articleare included in the article’s Creative Commons licence, unless indi-cated otherwise in a credit line to the material. If material is notincluded in the article’s Creative Commons licence and your intendeduse is not permitted by statutory regulation or exceeds the permit-ted use, you will need to obtain permission directly from the copy-right holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.Funded by SCOAP3.
Appendix A: Relations for bottom antitriplet baryondecay amplitudes
In Sect. 5, we have argued that due to different forms to rep-resent the hadron SU(3) multiplet, the IRA amplitudes canbe constructed in different ways. In Eq. 75, the antitripletbaryons and octet baryons are used as (Tb3)i and (T8)
i j .Instead it is possible to use another set of forms, (Tb3)
i j and(T8)i jk , which gives the IRA amplitudes with 26 terms:
AI RAu = A1T
[i j]b3
Hk3εi jn(T8)
nk P
ll + A2T
[i j]b3
Hk3εi jn(T8)
nl P
lk
+ A3T[i j]b3
Hk3εikn(T8)
nj P
lk + A4T
[i j]b3
Hk3εiln(T8)
nj P
lk
+ A5T[i j]b3
Hk3εikn(T8)
nl P
lj + A6T
[i j]b3
Hk3εiln(T8)
nk P
lj
+ A7T[i j]b3
Hk3εkln(T8)
ni P
lj
+B1T[i j]b3
(H6)[kl]m εi jn(T8)
nk P
ml
+B2T[i j]b3
(H6)[kl]m εikn(T8)
nj P
ml
+B3T[i j]b3
(H6)[kl]m εikn(T8)
nl P
mj
+B4T[i j]b3
(H6)[kl]m εkln(T8)
ni P
mj
+B5T[i j]b3
(H6)[kl]i ε jkn(T8)
nm Pm
l
+B6T[i j]b3
(H6)[kl]i εkln(T8)
nm Pm
j
+B7T[i j]b3
(H6)[kl]i ε jkn(T8)
nl P
mm
+B8T[i j]b3
(H6)[kl]i εkln(T8)
nj P
mm
+B9T[i j]b3
(H6)[kl]i εmjn(T8)
nk P
ml
+B10T[i j]b3
(H6)[kl]i εmkn(T8)
nj P
ml
+B11T[i j]b3
(H6)[kl]i εmkn(T8)
nl P
mj
+C1T[i j]b3
(H15){kl}m εi jn(T8)
nk P
ml
+C2T[i j]b3
(H15){kl}m εikn(T8)
nj P
ml
+C3T[i j]b3
(H15){kl}m εikn(T8)
nl P
mj
+C4T[i j]b3
(H15){kl}i ε jkn(T8)
nm Pm
l
+C5T[i j]b3
(H15){kl}i ε jkn(T8)
nl P
mm
+C6T[i j]b3
(H15){kl}i εmjn(T8)
nk P
ml
+C7T[i j]b3
(H15){kl}i εmkn(T8)
nj P
ml
+C8T[i j]b3
(H15){kl}i εmkn(T8)
nl P
mj . (A1)
The relation between the two sets of baryon SU(3) represen-tation is:
T [i j]b3
= 1
2εi jk(Tb3)k, (T8)i jk = εi jl(T8)
lk . (A2)
These two sets of IRA are related to each other by:
AT3 = 2 A1 + A3 + A6 + A7, BT
3 = 2 A2 + A4 + A5 − A7,
CT3 = A7 − A5, DT
3 = − A6 − A7
AT6 = B7 + B11 − 2B8, BT
6 = −(B3 + B11) + 2B4,
CT6 = B5 + B10 − 2B6, DT
6 = −(B9 + B10 − B11),
AT15 = C5 + C8, BT
15 = −(C3 + C8),
CT15 = C4 + C7, DT
15 = −(C6 + C8 + C7),
ET6 = 2B1 + 2B4 + B2 − B3 + B9 + B10 − B11,
ET15 = 2C1 + C2 + C3 + C6 + C7 + C8. (A3)
In addition, the relation between the coefficients of 26amplitudes in AI RA
Tb→PT8(u) and AT DATb→PT8(u) is:
A1 = 1
8(−a1 + 3a2 − 3a3 + a5) + b1,
A2 = 1
8(3a1 − a2 − 3a4 + a7) + b2,
123
Eur. Phys. J. C (2020) 80 :359 Page 33 of 36 359
A3 = 1
8(a3 − 3a5 + 4a9) + b3,
A4 = 1
8(−3a6 + a8 + 3a10 − a11 + 3a15 − a16) + b4,
A5 = 1
8(4a12 − a17 + 3a18 + a4 − 3a7) + b5,
A6 = 1
8(3a13 − a14 + 3a17 − a18 + a6 − 3a8) + b6,
A7 = 1
8(a10 − 3a11 − a13 + 3a14 + 4a19) + b7,
B1 = 1
4(a1 − a2), C1 = 1
8(a1 + a2),
B2 = 1
4(a15 − a16), C2 = 1
8(a15 + a16),
B3 = 1
4(a17 − a18), C3 = 1
8(a17 + a18), B4 = −1
4a19,
B5 = 1
4(a4 − a7), C4 = 1
8(a4 + a7), B6 = 1
4a12,
B7 = 1
4(a3 − a5), C5 = 1
8(a3 + a5),
B8 = 1
4a9, B9 = 1
4(a8 − a6), C6 = −1
8(a6 + a8),
B10 = 1
4(a11 − a10), C7 = −1
8(a10 + a11),
B11 = 1
4(a14 − a13), C8 = −1
8(a13 + a14). (A4)
Combination of Eqs. (A3) and (A4) leads to Eq. (76).The authors of Ref. [95] have given another parametriza-
tion of IRA amplitudes, in which they have focused on theflavor non-singlet. Comparing with their results, one findsthe following relations:
BT3 = 2b(3)2 + d(3)1 − e(3)2 + c(3),
CT3 = 2a(3) − c(3),
DT3 = 2b(3)1 + d(3)2 − e(3)1 + c(3),
BT6 = 2(a(6)2 − g(6) − n(6)1) + d(6)2 − e(6)1 − e(6)2,
CT6 = 2(a(6)1 + f (6) − n(6)2) + d(6)1,
ET6 = 2(b(6)2 − g(6)) − c(6) + d(6)1
+d(6)2 − e(6)1 − e(6)2 − g(6),
DT6 = −2(b(6)1 + f (6)) + c(6) − d(6)1
−d(6)2 + e(6)1 + e(6)2,
BT15 = 2a(15)2 + d(15)2 − e(15)1,
CT15 = 2a(15)1 + d(15)1 − e(15)2,
ET15 = 2b(15)2 + c(15) + d(15)1
−d(15)2 + e(15)1 − e(15)2,
DT15 = 2b(15)1 − c(15) − d(15)1
+d(15)2 − e(15)1 + e(15)2. (A5)
Appendix B: Relations for charmed baryon decays
Similar with the previous section, one can construct anotherset of IRA for charmed antitriplet baryon decays with 19amplitudes:
AI RAu = B1T
[i j]c3
(H6)klm εi jn(T8)
nk P
ml
+B2T[i j]c3
(H6)klm εikn(T8)
nj P
ml
+B3T[i j]c3
(H6)klm εikn(T8)
nl P
mj
+B4T[i j]c3
(H6)klm εkln(T8)
ni P
mj
+B5T[i j]c3
(H6)kli ε jkn(T8)
nm Pm
l
+B6T[i j]c3
(H6)kli εkln(T8)
nm Pm
j
+B7T[i j]c3
(H6)kli ε jkn(T8)
nl P
mm
+B8T[i j]c3
(H6)kli εkln(T8)
nl P
mm
+B9T[i j]c3
(H6)kli εmjn(T8)
nk P
ml
+B10T[i j]c3
(H6)kli εmkn(T8)
nj P
ml
+B11T[i j]c3
(H6)kli εmkn(T8)
nl P
mj
+C1T[i j]c3
(H15)klm εi jn(T8)
nk P
ml
+C2T[i j]c3
(H15)klm εikn(T8)
nj P
ml
+C3T[i j]c3
(H15)klm εikn(T8)
nl P
mj
+C4T[i j]c3
(H15)kli ε jkn(T8)
nm Pm
l
+C5T[i j]c3
(H15)kli ε jkn(T8)
nl P
mm
+C6T[i j]c3
(H15)kli εmjn(T8)
nk P
ml
+C7T[i j]c3
(H15)kli εmkn(T8)
nj P
ml
+C8T[i j]c3
(H15)kli εmkn(T8)
nl P
mj . (A6)
The relation between the two sets of IRA is given as:
AT6 = 1
2(B7 + B11) − B8, BT
6 = −1
2(B3 + B11) + B4,
CT6 = 1
2(B5 + B10) − B6, DT
6 = −1
2(B9 + B10 − B11)
AT15 = 1
2(C5 + C8), BT
15 = −1
2(C3 + C8),
CT15 = 1
2(C4 + C7), D
T15 = −1
2(C6 + C8 + C7)
ET6 = B1 + B4 + 1
2(B2 − B3 + B9 + B10 − B11),
ET15 = C1 + 1
2(C2 + C3 + C6 + C7 + C8). (A7)
The relation between the newAI RATc→PT8(u) andAT DA
Tc→PT8(u)
can be obtained as:
B1 = 1
2(a1 − a2), C1 = 1
2(a1 + a2),
123
359 Page 34 of 36 Eur. Phys. J. C (2020) 80 :359
B2 = 1
2(a15 − a16), C2 = 1
2(a15 + a16)
B3 = 1
2(a17 − a18), C3 = 1
2(a17 + a18),
B4 = −1
2a19, B5 = 1
2(a4 − a7), C4 = 1
2(a4 + a7),
B6 = 1
2a12, B7 = 1
2(a3 − a5), C5 = 1
2(a3 + a5),
B8 = 1
2a9, B9 = 1
2(a8 − a6), C6 = −1
2(a6 + a8),
B10 = 1
2(a11 − a10), C7 = −1
2(a10 + a11),
B11 = 1
2(a14 − a13), C8 = −1
2(a13 + a14). (A8)
References
1. Y. Li, C.D. Lü, Sci. Bull. 63, 267 (2018). https://doi.org/10.1016/j.scib.2018.02.003. arXiv:1808.02990 [hep-ph]
2. Y. Amhis et al. [HFLAV Collaboration], Eur. Phys. J. C77, 895 (2017). https://doi.org/10.1140/epjc/s10052-017-5058-4.arXiv:1612.07233 [hep-ex]
3. C. Patrignani et al. [Particle Data Group], Chin. Phys. C 40(10),100001 (2016). https://doi.org/10.1088/1674-1137/40/10/100001
4. M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98(3),030001 (2018). https://doi.org/10.1103/PhysRevD.98.030001
5. Y.L. Shen, Y.B. Wei, C.D. Lü, Phys. Rev. D 97(5), 054004 (2018).https://doi.org/10.1103/PhysRevD.97.054004. arXiv:1607.08727[hep-ph]
6. C.D. Lü, Y.L. Shen, Y.M. Wang, Y.B. Wei, JHEP 1901, 024 (2019).https://doi.org/10.1007/JHEP01(2019)024. arXiv:1810.00819[hep-ph]
7. D. Zeppenfeld, Z. Phys. C 8, 77 (1981). https://doi.org/10.1007/BF01429835
8. M.J. Savage, M.B. Wise, Phys. Rev. D 39, 3346 (1989). https://doi.org/10.1103/PhysRevD.39.3346, https://doi.org/10.1103/PhysRevD.40.3127 [Erratum: Phys. Rev. D 40, 3127 (1989)]
9. N.G. Deshpande, X.G. He, Phys. Rev. Lett.75, 1703 (1995). https://doi.org/10.1103/PhysRevLett.75.1703. arXiv:hep-ph/9412393
10. X.G. He, Eur. Phys. J. C 9, 443 (1999). https://doi.org/10.1007/s100529900064. arXiv:hep-ph/9810397
11. X.G. He, Y.K. Hsiao, J.Q. Shi, Y.L. Wu, Y.F. Zhou, Phys. Rev. D64, 034002 (2001). https://doi.org/10.1103/PhysRevD.64.034002.arXiv:hep-ph/0011337
12. Y.K. Hsiao, C.F. Chang, X.G. He, Phys. Rev. D 93(11),114002 (2016). https://doi.org/10.1103/PhysRevD.93.114002.arXiv:1512.09223 [hep-ph]
13. L.L. Chau, H.Y. Cheng, Phys. Rev. Lett. 56, 1655 (1986). https://doi.org/10.1103/PhysRevLett.56.1655
14. L.L. Chau, H.Y. Cheng, Phys. Rev. D 36, 137 (1987). https://doi.org/10.1103/PhysRevD.39.2788, https://doi.org/10.1103/PhysRevD.36.137 [Addendum: Phys. Rev. D 39, 2788 (1989)]
15. L.L. Chau, H.Y. Cheng, W.K. Sze, H. Yao, B. Tseng, Phys. Rev.D 43, 2176 (1991). https://doi.org/10.1103/PhysRevD.43.2176,https://doi.org/10.1103/PhysRevD.58.019902 [Erratum: Phys.Rev. D 58, 019902 (1998)]
16. M. Gronau, O.F. Hernandez, D. London, J.L. Rosner, Phys. Rev.D 50, 4529 (1994). https://doi.org/10.1103/PhysRevD.50.4529.arXiv:hep-ph/9404283
17. M. Gronau, O.F. Hernandez, D. London, J.L. Rosner, Phys. Rev.D 52, 6356 (1995). https://doi.org/10.1103/PhysRevD.52.6356.arXiv:hep-ph/9504326
18. H.Y. Cheng, C.W. Chiang, A.L. Kuo, Phys. Rev. D, 2015,p. 014011. https://doi.org/10.1103/PhysRevD.91.014011.arXiv:1409.5026 [hep-ph]
19. S.H. Zhou, Q.A. Zhang, W.R. Lyu, C.D. Lü, Eur. Phys. J. C 77(2),125 (2017). https://doi.org/10.1140/epjc/s10052-017-4685-0.arXiv:1608.02819 [hep-ph]
20. S. Müller, U. Nierste, S. Schacht, Phys. Rev. D 92(1),014004 (2015). https://doi.org/10.1103/PhysRevD.92.014004.arXiv:1503.06759 [hep-ph]
21. C.W. Bauer, D. Pirjol, Phys. Lett. B 604, 183 (2004). https://doi.org/10.1016/j.physletb.2004.10.047. arXiv:hep-ph/0408161
22. X.G. He, W. Wang, Chin. Phys. C 42(10), 103108 (2018). https://doi.org/10.1088/1674-1137/42/10/103108. arXiv:1803.04227[hep-ph]
23. Y. Grossman, Z. Ligeti, Y. Nir, H. Quinn, Phys. Rev. D68, 015004 (2003). https://doi.org/10.1103/PhysRevD.68.015004.arXiv:hep-ph/0303171
24. W. Wang, R.L. Zhu, Phys. Rev. D 96(1), 014024 (2017). https://doi.org/10.1103/PhysRevD.96.014024. arXiv:1704.00179 [hep-ph]
25. G. Buchalla, A.J. Buras, M.E. Lautenbacher, Rev. Mod. Phys.68, 1125 (1996). https://doi.org/10.1103/RevModPhys.68.1125.arXiv:hep-ph/9512380
26. M. Ciuchini, E. Franco, G. Martinelli, L. Reina, Nucl. Phys. B415, 403 (1994). https://doi.org/10.1016/0550-3213(94)90118-X.arXiv:hep-ph/9304257
27. N.G. Deshpande, X.G. He, Phys. Lett. B 336, 471 (1994). https://doi.org/10.1016/0370-2693(94)90560-6. arXiv:hep-ph/9403266
28. R. Fleischer, R. Jaarsma, K.K. Vos, JHEP 1703, 055 (2017). https://doi.org/10.1007/JHEP03(2017)055. arXiv:1612.07342 [hep-ph]
29. S. Descotes-Genon, J. Matias, J. Virto, Phys. Rev. Lett. 97,061801 (2006). https://doi.org/10.1103/PhysRevLett.97.061801.arXiv:hep-ph/0603239
30. R. Fleischer, S. Recksiegel, Eur. Phys. J. C 38, 251 (2004). https://doi.org/10.1140/epjc/s2004-02023-0. arXiv:hep-ph/0408016
31. M. Beneke, M. Neubert, Nucl. Phys. B 675, 333 (2003). https://doi.org/10.1016/j.nuclphysb.2003.09.026. arXiv:hep-ph/0308039
32. M. Gronau, Phys. Lett. B 492, 297 (2000). https://doi.org/10.1016/S0370-2693(00)01119-9. arXiv:hep-ph/0008292
33. Y. Grossman, Z. Ligeti, D.J. Robinson, JHEP 1401, 066 (2014).https://doi.org/10.1007/JHEP01(2014)066. arXiv:1308.4143[hep-ph]
34. Q. Chang, X.N. Li, J.F. Sun, Y.L. Yang, J. Phys. G43(10), 105004 (2016). https://doi.org/10.1088/0954-3899/43/10/105004. arXiv:1610.02747 [hep-ph]
35. C. Wang, Q.A. Zhang, Y. Li, C.D. Lu, Eur. Phys. J. C 77(5),333 (2017). https://doi.org/10.1140/epjc/s10052-017-4889-3.arXiv:1701.01300 [hep-ph]
36. C. Wang, S.H. Zhou, Y. Li, C.D. Lu, Phys. Rev. D 96(7),073004 (2017). https://doi.org/10.1103/PhysRevD.96.073004.arXiv:1708.04861 [hep-ph]
37. Z.T. Zou, A. Ali, C.D. Lu, X. Liu, Y. Li, Phys. Rev. D91, 054033 (2015). https://doi.org/10.1103/PhysRevD.91.054033.arXiv:1501.00784 [hep-ph]
38. D.C. Yan, X. Liu, Z.J. Xiao, Nucl. Phys. B 935, 17 (2018). https://doi.org/10.1016/j.nuclphysb.2018.08.002. arXiv:1807.00606[hep-ph]
39. Q. Chang, X. Li, X.Q. Li, J. Sun, Eur. Phys. J. C 77(6),415 (2017). https://doi.org/10.1140/epjc/s10052-017-4980-9.arXiv:1706.06138 [hep-ph]
40. W. Wang, Y.M. Wang, D.S. Yang, C.D. Lu, Phys. Rev. D78, 034011 (2008). https://doi.org/10.1103/PhysRevD.78.034011.arXiv:0801.3123 [hep-ph]
123
Eur. Phys. J. C (2020) 80 :359 Page 35 of 36 359
41. Q. Chang, X. Hu, J. Sun, Y. Yang, Phys. Rev. D 91, 074026 (2015).https://doi.org/10.1103/PhysRevD.91.074026. arXiv:1504.04907[hep-ph]
42. H. Li, C.D. Lu, F.S. Yu, Phys. Rev. D 86, 036012 (2012). https://doi.org/10.1103/PhysRevD.86.036012. arXiv:1203.3120 [hep-ph]
43. H.Y. Cheng, C.W. Chiang, A.L. Kuo, Phys. Rev. D 93(11),114010 (2016). https://doi.org/10.1103/PhysRevD.93.114010.arXiv:1604.03761 [hep-ph]
44. H.Y. Cheng, C.W. Chiang, Phys. Rev. D 86, 014014 (2012). https://doi.org/10.1103/PhysRevD.86.014014. arXiv:1205.0580 [hep-ph]
45. Y. Grossman, D.J. Robinson, JHEP 1304, 067 (2013). https://doi.org/10.1007/JHEP04(2013)067. arXiv:1211.3361 [hep-ph]
46. M. Ablikim et al. [BESIII Collaboration], Phys. Rev. D 97(7),072004 (2018). https://doi.org/10.1103/PhysRevD.97.072004.arXiv:1802.03119 [hep-ex]
47. R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 116(19),191601 (2016). https://doi.org/10.1103/PhysRevLett.116.191601.arXiv:1602.03160 [hep-ex]
48. G. Hiller, M. Jung, S. Schacht, Phys. Rev. D 87(1),014024 (2013). https://doi.org/10.1103/PhysRevD.87.014024.arXiv:1211.3734 [hep-ph]
49. S. Müller, U. Nierste, S. Schacht, Phys. Rev. Lett. 115(25),251802 (2015). https://doi.org/10.1103/PhysRevLett.115.251802.arXiv:1506.04121 [hep-ph]
50. R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 118(11),111803 (2017). https://doi.org/10.1103/PhysRevLett.118.111803.arXiv:1701.01856 [hep-ex]
51. R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 114,132001 (2015). https://doi.org/10.1103/PhysRevLett.114.132001.arXiv:1411.2943 [hep-ex]
52. H.F. Fu, Y. Jiang, C.S. Kim, G.L. Wang, JHEP 1106, 015 (2011).https://doi.org/10.1007/JHEP06(2011)015. arXiv:1102.5399[hep-ph]
53. J. Zhang, X.Q. Yu, Eur. Phys. J. C 63, 435 (2009). https://doi.org/10.1140/epjc/s10052-009-1112-1. arXiv:0905.0945 [hep-ph]
54. C.H. Chen, Y.H. Lin. arXiv:1710.05531 [hep-ph]55. Z. Rui, Z.T. Zou, C.D. Lu, Phys. Rev. D 86, 074008 (2012). https://
doi.org/10.1103/PhysRevD.86.074008. arXiv:1112.1257 [hep-ph]56. Z.T. Zou, X. Yu, C.D. Lu, Phys. Rev. D 87, 074027 (2013). https://
doi.org/10.1103/PhysRevD.87.074027. arXiv:1208.4252 [hep-ph]57. W. Wang, Y.L. Shen, C.D. Lu, Phys. Rev. D 79, 054012 (2009).
https://doi.org/10.1103/PhysRevD.79.054012. arXiv:0811.3748[hep-ph]
58. X.X. Wang, W. Wang, C.D. Lu, Phys. Rev. D 79, 114018 (2009).https://doi.org/10.1103/PhysRevD.79.114018. arXiv:0901.1934[hep-ph]
59. B. Bhattacharya, A.A. Petrov, Phys. Lett. B774, 430 (2017). https://doi.org/10.1016/j.physletb.2017.10.004. arXiv:1708.07504 [hep-ph]
60. R. Zhu, X.L. Han, Y. Ma, Z.J. Xiao, Eur. Phys. J. C78, 740 (2018). https://doi.org/10.1140/epjc/s10052-018-6214-1.arXiv:1806.06388 [hep-ph]
61. R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 111(18),181801 (2013). https://doi.org/10.1103/PhysRevLett.111.181801.arXiv:1308.4544 [hep-ex]
62. X. Liu, Z.J. Xiao, C.D. Lu, Phys. Rev. D 81, 014022 (2010). https://doi.org/10.1103/PhysRevD.81.014022. arXiv:0912.1163 [hep-ph]
63. Z.J. Xiao, X. Liu, Chin. Sci. Bull. 59, 3748 (2014). https://doi.org/10.1007/s11434-014-0418-z. arXiv:1401.0151 [hep-ph]
64. Q. Chang, N. Wang, J. Sun, L. Han, J. Phys. G 44(8),085005 (2017). https://doi.org/10.1088/1361-6471/aa7bca.arXiv:1707.03691 [hep-ph]
65. T. Mannel, Y.M. Wang, JHEP 1112, 067 (2011). https://doi.org/10.1007/JHEP12(2011)067. arXiv:1111.1849 [hep-ph]
66. W. Wang, Phys. Lett. B 708, 119 (2012). https://doi.org/10.1016/j.physletb.2012.01.036. arXiv:1112.0237 [hep-ph]
67. R. Aaij et al. [LHCb Collaboration], JHEP 1802, 098 (2018).https://doi.org/10.1007/JHEP02(2018)098. arXiv:1711.05490[hep-ex]
68. R. Aaij et al. [LHCb Collaboration]. arXiv:1808.0886569. R. Aaij et al. [LHCb Collaboration], JHEP 1808, 039 (2018).
https://doi.org/10.1007/JHEP08(2018)039. arXiv:1805.03941[hep-ex]
70. C.Q. Geng, Y.K. Hsiao, C.W. Liu, T.H. Tsai, Phys. Rev.D 97(7), 073006 (2018). https://doi.org/10.1103/PhysRevD.97.073006. arXiv:1801.03276 [hep-ph]
71. M. Ablikim et al. [BESIII Collaboration], Phys. Rev. Lett. 116(5),052001 (2016). https://doi.org/10.1103/PhysRevLett.116.052001.arXiv:1511.08380 [hep-ex]
72. M. Ablikim et al. [BESIII Collaboration], Chin. Phys. C 43(8),083002 (2019). https://doi.org/10.1088/1674-1137/43/8/083002.arXiv:1811.08028 [hep-ex]
73. M. He, X.G. He, G.N. Li, Phys. Rev. D 92(3), 036010 (2015).https://doi.org/10.1103/PhysRevD.92.036010. arXiv:1507.07990[hep-ph]
74. J. Zhu, H.W. Ke, Z.T. Wei, Eur. Phys. J. C 76(5), 284 (2016). https://doi.org/10.1140/epjc/s10052-016-4134-5. arXiv:1603.02800[hep-ph]
75. C.D. Lu, Y.M. Wang, H. Zou, A. Ali, G. Kramer, Phys. Rev. D80, 034011 (2009). https://doi.org/10.1103/PhysRevD.80.034011.arXiv:0906.1479 [hep-ph]
76. Y.K. Hsiao, Y. Yao, C.Q. Geng, Phys. Rev. D 95(9), 093001 (2017).https://doi.org/10.1103/PhysRevD.95.093001. arXiv:1702.05263[hep-ph]
77. Y.K. Hsiao, C.Q. Geng, Phys. Rev. D 91(11), 116007 (2015).https://doi.org/10.1103/PhysRevD.91.116007. arXiv:1412.1899[hep-ph]
78. Z.T. Wei, H.W. Ke, X.Q. Li, Phys. Rev. D 80, 094016 (2009).https://doi.org/10.1103/PhysRevD.80.094016. arXiv:0909.0100[hep-ph]
79. Y.M. Wang, Y.L. Shen, JHEP 1602, 179 (2016). https://doi.org/10.1007/JHEP02(2016)179. arXiv:1511.09036 [hep-ph]
80. R. Aaij et al. [LHCb Collaboration], JHEP 1210, 037 (2012).https://doi.org/10.1007/JHEP10(2012)037. arXiv:1206.2794[hep-ex]
81. T.A. Aaltonen et al. [CDF Collaboration], Phys. Rev. Lett. 113(24),242001 (2014). https://doi.org/10.1103/PhysRevLett.113.242001.arXiv:1403.5586 [hep-ex]
82. R. Aaij et al. [LHCb Collaboration], Phys. Lett. B 787,124 (2018). https://doi.org/10.1016/j.physletb.2018.10.039.arXiv:1807.06544 [hep-ex]
83. C.D. Lü, W. Wang, F.S. Yu, Phys. Rev. D 93(5), 056008 (2016).https://doi.org/10.1103/PhysRevD.93.056008. arXiv:1601.04241[hep-ph]
84. C.Q. Geng, Y.K. Hsiao, Y.H. Lin, L.L. Liu, Phys. Lett. B776, 265 (2018). https://doi.org/10.1016/j.physletb.2017.11.062.arXiv:1708.02460 [hep-ph]
85. C.Q. Geng, Y.K. Hsiao, C.W. Liu, T.H. Tsai, JHEP 1711,147 (2017). https://doi.org/10.1007/JHEP11(2017)147.arXiv:1709.00808 [hep-ph]
86. D. Wang, P.F. Guo, W.H. Long, F.S. Yu, JHEP 1803, 066 (2018).https://doi.org/10.1007/JHEP03(2018)066. arXiv:1709.09873[hep-ph]
87. H.Y. Cheng, X.W. Kang, F. Xu, Phys. Rev. D 97(7), 074028 (2018).https://doi.org/10.1103/PhysRevD.97.074028. arXiv:1801.08625[hep-ph]
88. Z.X. Zhao, Chin. Phys. C 42(9), 093101 (2018). https://doi.org/10.1088/1674-1137/42/9/093101. arXiv:1803.02292 [hep-ph]
89. C.Q. Geng, Y.K. Hsiao, C.W. Liu, T.H. Tsai, Eur. Phys. J. C 78(7),593 (2018). https://doi.org/10.1140/epjc/s10052-018-6075-7.arXiv:1804.01666 [hep-ph]
123
359 Page 36 of 36 Eur. Phys. J. C (2020) 80 :359
90. C.Q. Geng, Y.K. Hsiao, C.W. Liu, T.H. Tsai. arXiv:1810.01079[hep-ph]
91. M. Ablikim et al. [BESIII Collaboration], Phys. Lett. B783, 200 (2018). https://doi.org/10.1016/j.physletb.2018.06.046.arXiv:1803.04299 [hep-ex]
92. W. Wang, Z.P. Xing, J. Xu, Eur. Phys. J. C 77(11), 800(2017). https://doi.org/10.1140/epjc/s10052-017-5363-y.arXiv:1707.06570 [hep-ph]
93. Y.J. Shi, W. Wang, Y. Xing, J. Xu, Eur. Phys. J. C 78(1),56 (2018). https://doi.org/10.1140/epjc/s10052-018-5532-7.arXiv:1712.03830 [hep-ph]
94. W. Wang, J. Xu, Phys. Rev. D 97(9), 093007 (2018). https://doi.org/10.1103/PhysRevD.97.093007. arXiv:1803.01476 [hep-ph]
95. X.G. He, G.N. Li, Phys. Lett. B 750, 82 (2015). https://doi.org/10.1016/j.physletb.2015.08.048. arXiv:1501.00646 [hep-ph]
123