# BaryonicB Meson Decays

Authors: Xiaotao Huang; Yu-Kuo Hsiao; Jike Wang; Liang Sun
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

We review the two- and three-body baryonicB decays with the dibaryon (BB′¯) as the final states. Accordingly, we summarize the experimental data of the branching fractions, angular asymmetries, and CP asymmetries. Using the W-boson annihilation (exchange) mechanism, the branching fractions of B⟶BB′¯ are shown to be interpretable. In the approach of perturbative QCD counting rules, we study the three-body decay channels. In particular, we review the CP asymmetries of B⟶BB′¯M, which are promising to be measured by the LHCb and Belle II experiments. Finally, we remark the theoretical challenges in interpreting BB−⟶pp¯ρ− and BB−⟶pp¯μ−ν¯μ.


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## Body

## 1. Introduction

The baryonicB meson decays have been richly measured with the branching fractions, angular asymmetries, and CP asymmetries in two- and three-body decay channels [1–13], as summarized in Table 1. Typically, BB⟶BB′¯ is as small as 10−8−10−7. Nonetheless, it is observed that BB⟶BB′¯M~10−100×BB⟶BB′¯, due to a sharply rising peak in B⟶BB′¯M observed around the threshold area of mBB′¯~mB+mB′¯ in the dibaryon invariant mass spectrum [4]. Known as the threshold effect, it enhances BB⟶BB′¯M as large as 10−6. While the BB′¯ production shows the tendency to occur around mBB′¯~mB+mB′, B⟶BB′¯ proceeds at mB scale, far from the threshold area. This interprets the suppressed BB⟶BB′¯ [14, 15].Table 1
The measured branching fractions, forward-backward asymmetries (AFB), and CP asymmetries (ACP) for the baryonic B decays, where the notation † is for AFB with mpp¯<2.85 GeV.
Decay modeBranching fractionAFBACPRef.B¯0⟶pp¯1.25±0.32×10−8[1]B¯0⟶ΛΛ¯<3.2×10−7[1]B−⟶Λp¯2.4−0.9+1.0×10−7[1]B¯s0⟶pp¯<1.5×10−8[1]B¯0⟶pp¯π05.0±1.9×10−7[1]B¯0⟶pp¯K¯02.66±0.32×10−6[1]B¯0⟶Λp¯π+3.14±0.29×10−6−0.41±0.11±0.030.04±0.07[1, 12]B¯0⟶Σ0p¯π+<3.8×10−6[1]B¯0⟶Λp¯K+<8.2×10−7[1]B¯0⟶ΛΛ¯K¯04.8−0.9+1.0×10−6[1]B−⟶pp¯π−1.62±0.20×10−6−0.409±0.033±0.006†0.00±0.04[1, 2]B−⟶pp¯K−5.9±0.5×10−60.495±0.012±0.007†0.00±0.04[1, 2]B−⟶Λp¯π03.0−0.6+0.7×10−6−0.16±0.18±0.030.01±0.17[1, 12]B−⟶ΛΛ¯π−<9.4×10−7[1]B−⟶ΛΛ¯K−3.4±0.6×10−6[1]B¯s0⟶p¯ΛK++c.c.5.5±1.0×10−6[1]B¯0⟶pp¯K¯∗01.24−0.25+0.28×10−60.05±0.12[1]B¯0⟶ΛΛ¯K¯∗02.5−0.8+0.9×10−6[1]B−⟶pp¯K∗−3.6−0.7+0.8×10−60.21±0.16[1]B−⟶Λp¯ρ0,ρ0⟶π+π−4.8±0.9×10−6[1]B−⟶Λp¯ϕ8.0±2.2×10−7[1]B−⟶ΛΛ¯K∗−2.2−0.9+1.2×10−6[1]B¯0⟶Λc+p¯1.54±0.18×10−5[1]B¯0⟶Σc+p¯<2.4×10−5[1]B−⟶Σc0p¯2.9±0.7×10−5[1]B−⟶pp¯D−<1.5×10−5[1]B−⟶pp¯D∗−<1.5×10−5[1]B−⟶Λp¯D01.43±0.32×10−5[1]B−⟶Λp¯D∗0<5×10−5[1]B¯0⟶np¯D∗+1.4±0.4×10−3[1]B¯0⟶pp¯D01.04±0.07×10−4[1]B¯0⟶pp¯D∗00.99±0.11×10−4[1]B¯0⟶Λp¯Ds+2.8±0.9×10−5[1]B¯0⟶Λp¯D+2.5±0.4×10−5−0.08±0.10[1, 13]B¯0⟶Λp¯D∗+3.4±0.8×10−5+0.55±0.17[1, 13]B¯0⟶ΛΛ¯D¯01.00−0.26+0.30×10−5[1]B¯0⟶Σ0Λ¯D¯0+c.c.<3.1×10−5[1]The partial branching fraction ofB⟶BB′¯M can be a function of cosθB, where θB is the angle between the baryon and meson moving directions in the dibaryon rest frame. One hence defines the forward-backward angular asymmetry,
(1)AFB≡BcosθB>0−BcosθB<0BcosθB>0+BcosθB<0.In Table1, AFBB−⟶pp¯π−,pp¯K−=−40.9±3.4,49.5±1.4% [2] indicate that one of the dibaryons favors to move collinearly with the meson.We search for the theoretical approach to interpret the threshold effect, branching fractions, and angular asymmetries of the baryonicB decays. We find that the factorization approach can be useful [16], where one factorizes (decomposes) the amplitude of the decay as two separate matrix elements. In our case, we present
(2)MB⟶BB′¯∝BB′¯q¯q′00q¯bB,M1B⟶BB′¯M∝BB′¯q¯q′0Mq¯bB,M2B⟶BB′¯M∝Mq¯q′0BB′¯q¯bB,where q¯q′ and q¯b stand for the quark currents and the matrix element of BB′¯q¯q′0BB′¯q¯bB can be parameterized as the timelike baryonic (B to BB′¯ transition) form factors FBB′¯. Moreover, one derives FBB′¯∝1/tn in perturbative QCD (pQCD) counting rules [17–24], where t≡pB+pB′¯2 and n accounts for the number of the gluon propagators that attach to the baryon pair. It results in dB/dmBB′¯∝1/t2n, which shapes a peak around mBB′¯~mB+mB′ in the mBB′¯ spectrum, and then, the threshold effect can be interpreted. In the B⟶pp¯ transition, there exists the term of pp¯−ppμu¯γ5v for FBB′¯ [24], which is reduced as Ep¯−Epu¯γ5v in the pp¯ rest frame. Since Ep¯−Ep∝cosθp, the term for FBB′¯ can be used to describe the highly asymmetric AFBB−⟶pp¯π−,pp¯K−. Alternatively, the baryonic B decays is studied with the pole model, where the nonfactorizable contributions can be taken into account [25–28].We have explainedBB⟶BB′¯~10−8−10−7 [29]. We have studied B⟶BB′¯M and explained the branching fractions and CP asymmetries [30–38]. In addition, we have predicted BB¯s0⟶pΛ¯K−+Λp¯K+=5.1±1.1×10−6 [37], in excellent agreement with the value of 5.46±0.61±0.57±0.50±0.32×10−6 measured by LHCb [8]. This demonstrates that the theoretical approach can be reliable. Therefore, we would like to present a review, in order to illustrate how the approach of pQCD counting rules based on the factorization can be applied to the baryonic B decays. We will also review our theoretical results that have explained the branching fractions of B⟶BB′¯ and B⟶BB′¯M, particularly, the CP asymmetries, promising to be observed by future measurements.


## 2. Formalism

To review the two-body baryonicB decays, we take B¯0⟶pp¯ as our example. According to Figure 1, B¯0⟶pp¯ is regarded as an W-boson exchange process [29, 39–41]. In the factorization, we derive the amplitude as [29]
(3)MB¯0⟶pp¯=GF2VubVud∗a2pp¯u¯γμ1−γ5u00d¯γμ1−γ5bB¯0,where GF is the Fermi constant and Vubd∗ the Cabibbo-Kobayashi-Maskawa (CKM) matrix element. One has defined 0d¯γμ1−γ5bB¯0=−ifBqμ for the B¯0 meson annihilation, where fB is the decay constant and qμ the four-momentum. For the pp¯ production, the matrix elements read [22, 23]
(4)BB′¯Vμ0=u¯F1γμ+F2mB+mB′¯iσμνqμv,BB′¯Aμ0=u¯gAγμ+hAmp+mp¯qμγ5v,with the (axial-)vector current VAμ=q¯γμγ5q′, where F1,2, gA, and hA are the timelike baryonic form factors.Figure 1
Feynman diagram forB¯0⟶pp¯.At a very large momentum transfer (Q2⟶∞), the approach of pQCD counting rules results in [17–21]
(5)F1,gA∝αs2Q2Q4lnQ2Λ2−4/3β,where β=11−2nf/3 is the β function of QCD to one loop, nf=3 the flavor number, and Λ=0.3 GeV the scale factor. Moreover, αsQ2≡4π/βlnQ2/Λ2−1 is the running coupling constant in the strong interaction [20]. Interestingly, αs2/Q4 reflects the fact that one needs two hard gluon propagators to attach to the baryons as drawn in Figure 1 [42], whereas lnQ2/Λ2−4/3β is caused by the wave function.AsVμ and Aμ are combined as the right- or left-handed chiral current, that is, JμR,L=Vμ±Aμ/2, one obtains B′R+LJμRLBR+L for the spacelike B⟶B′ transition. With the right-handed current, the matrix elements can be written as [19, 29]
(6)BR+L′JμRBR+L=u¯γμ1+γ52FR+γμ1−γ52FLu,where BR+L=BR+BL and FR,L are the chiral form factors. With qi (i=1,2,3) denoting one of the valence quarks in B, Q≡Jμ=0R known as the chiral charge is able to change the flavor for qi, such that B is transformed as B′. Note that the chirality is regarded as the helicity at Q2⟶∞. Since the helicity of qi can be (anti-)parallel ∥∥¯ to the helicity of B, we define Q∥∥¯i that is responsible for acting on qi. Thus, the approach of pQCD counting rules leads to [19, 29]
(7)FR=e∥RF∥+e∥¯RF∥¯,FL=e∥LF∥+e∥¯LF∥¯,with e∥∥¯R=BR′Q∥∥¯BR, e∥∥¯L=BL′Q∥∥¯BL, and Q∥∥¯=∑iQ∥∥¯i, where the SU3 flavor (SU3f) and SU2 spin symmetries are both respected. In the crossing symmetry, the spacelike form factors behave as the timelike ones, such that one can relate F1 and gA with the chiral form factors in Equations (6) and (7) derived in the spacelike region, leading to F1gA=e∥R±e∥LF∥+e∥¯R±e∥¯LF‖¯. In addition to the momentum dependence of Equation (5), F1 and gA are presented as [22, 23, 29]
(8)F1=CF1t2lntΛ2−γ,gA=CgAt2lntΛ2−γ,where γ=2+4/3β=2.148.Forpp¯u¯u0, we obtain [23]
(9)CF1=53C∥+13C∥¯,CgA=53C∥−13C∥¯,where C∥∥¯ is from F∥∥¯≡C∥∥¯/t2lnt/Λ2−γ, and we have used e∥R,e∥L=5/3,0 and e∥¯R,e∥¯L=0,1/3 [23, 29]. In Ref. [43], the pQCQ calculation causes F2=F1/tlnt/Λ2, indicating that F2 has a suppressed contribution. Taking the form factors as the inputs, we reduce the amplitude of B¯0⟶pp¯ as
(10)M∝1mp+mp¯u¯mp+mp¯2gA+mB2hAγ5v,where F12 has been vanishing, in accordance with the conservation of vector current (CVC) qμpp¯u¯γμu0=0. Using the partial conservation of axial-vector current (PCAC), where qμBB′¯Aμ0=0, it is obtained that [23, 29, 39–41, 44]
(11)hA=−mB+mB′2gAt−mM2,by which gA and hA cancel each other and then M≃0. This seems that the W-exchange (annihilation) mechanism based on the factorization fails to explain BB⟶BB′¯. As a consequence, one turns to think of the nonfactorizable effects as the main contributions [25, 44–50].In Equation (11), 1/t−mM2 describes a meson pole, so that B⟶BB′¯ can be regarded to receive the contribution from the intermediate process of B⟶M⟶BB′¯, which is much suppressed. On the other hand, there might exist a QCD-based contribution to hA, by which MB⟶BB′¯≠0, and PCAC is violated. Here, we choose to parameterize hA with slightly violated PCAC in the timelike region. To this end, we derive hA+gA≃0 with qμBB′¯Aμ0≃0 at the threshold area of t≃mB+mB′¯2, where the meson pole is supposed to be inapplicable [40, 41]. Since the QCD-based calculation of hA is still lacking, besides hA+gA≃0 suggesting hA∝gA, we are allowed to present hA=ChA/t2 for its momentum dependence [29].To describe the three-body baryonicB decays, we take B⟶pp¯V with V=ρ−0 or K∗−K¯∗0 as our examples. According to Figure 2, the amplitudes are given by [30–32]
(12)MB⟶pp¯V≃GF2αVVq¯γμ1−γ5q′0pp¯q′¯γμ1−γ5bB,with q=s,d and q′=u for B−⟶pp¯K∗−,ρ−, and q=s,d and q′=d for B¯0⟶pp¯K¯∗0,ρ0. For αV, we define
(13)αρ−=VubVud∗a1−VtbVtd∗a4,αρ0=VubVud∗a2+VtbVtd∗a4−32a9,αK∗−=VubVus∗a1−VtbVts∗a4,αK¯∗0=−VtbVts∗a4.Figure 2
Feynman diagrams forB⟶pp¯V.
(a)(b)(c)(d)In Equation (12), one presents that Vq¯γμ1−γ5q′0=fVmVεμ∗, where fV and εμ∗ are, respectively, the decay constant and polarization four-vector of the vector meson. The amplitudes are both associated with the matrix elements of the B⟶BB′¯ transition, and we parameterize them as [24]
(14)BB′¯VμbB=iu¯g1γμ+g2iσμνpν+g3pμ+g4pB′¯+pBμ+g5pB′¯−pBμγ5v,BB′¯AμbB=iu¯f1γμ+f2iσμνpν+f3pμ+f4pB′¯+pBμ+f5pB′¯−pBμv,where VμbAμb=q′¯γμγ5b, p=pB−pB′¯+pB, and fi,gi (i=1,2,⋯,5) are the B⟶BB′¯ transition form factors. Inspired by pQCD counting rules [17, 18, 21, 23, 24], the momentum dependences for fi and gi are given by
(15)fi=Dfitn,gi=Dgitn,with Dfigi to be extracted by the data. According to the gluon lines in Figure 2, n in 1/tn should be 2+1, which accounts for two gluon propagators attaching to the valence quarks in BB′¯ and an additional one for kicking (speeding up) the spectator quark in B [23]. For the gluon kicking, it is similar to the meson transition form factor derived as FM∝1/q2 in pQCD counting rules [20, 51], where 1/q2 is for a hard gluon to transfer the momentum to the spectator quark in the meson.Like the case ofF1 and gA, we relate fi,gi to the B⟶BB′¯ chiral form factors, which is in terms of [23, 24]
(16)BR+LB′¯R+LVμb+Aμb/2B=imbu¯γμ1+γ52GR+1−γ52GLu+iu¯γμpb1+γ52GRj+1−γ52GLju,where Bq~b¯γ5q0 has been used. In addition, we obtain GRL=e∥RLG∥+e∥¯RLG∥¯ and GRLj=e∥RLG∥j+e∥¯RLG∥¯j, which are similar to FR,L in Equation (7). Under the SU3 flavor and SU2 spin symmetries, together with G∥∥¯j≡D∥∥¯j/tnj=2,3,⋯,5, it is derived that [23, 24]
(17)Dg1=53D∥−13D∥¯,Df1=53D∥+13D∥¯,Dgj=53D∥j=−Dfj,Dg1=13D∥−23D∥¯,Df1=13D∥+23D∥¯,Dgj=−13D∥j=−Dfj,for pp¯u¯bB− and pp¯d¯bB¯0, respectively. We also review the direct CP asymmetry, defined by
(18)ACPB⟶BB′¯M≡ΓB⟶BB′¯M−ΓB¯⟶B¯B′M¯ΓB⟶BB′¯M+ΓB¯⟶B¯B′M¯,where Γ denotes the decay width and B¯⟶B¯B′M¯ the antiparticle decay.


## 3. Numerical Analysis

For the numerical analysis, we adopt the CKM matrix elements as [1]
(19)Vub=Aλ3ρ−iη,Vud=1−λ2/2,Vus=λ,Vtb=1,Vtd=Aλ3,Vts=−Aλ2,with λ=0.22453±0.00044, A=0.836±0.015, ρ¯=0.122−0.017+0.018, and η¯=0.355−0.011+0.012 in the Wolfenstein parameterization, where ρ¯,η¯=1−λ2/2×ρ,η. The decay constants are given by fB,fρ,fK∗=0.19,0.21,0.22 GeV [1, 52]. With the global fit to the data, we obtain [29, 37]
(20)C∥,C∥¯,ChA=−102.4±7.3,210.9±85.2,8.4±1.4GeV4,D∥,D∥¯=45.7±33.8,−298.2±34.0GeV5,D∥2,D∥3,D∥4,D∥5=33.1±30.7,−203.6±133.4,6.5±18.1,−147.1±29.3GeV4.We takeai≡cieff+ci±1eff/Nc for i=odd (even) with Nc the color number, where the effective Wilson coefficients ci come from Ref. [16]. For a2 in B¯0⟶pp¯ρ0 and B¯0⟶pp¯, we use Nc=2 to take into account the nonfactorizable QCD corrections. We can hence present our theoretical calculations for B⟶pp¯V in Table 2, along with the other B⟶BB′¯M results. Besides, we present our studies of the angular and CP asymmetries.Table 2
Theoretical results of the two- and three-body baryonicB decays, in comparison with the experimental data.
Decay modeExperimental data [1, 2, 12]TheoryBB¯0⟶pp¯1.25±0.32×10−81.4±0.5×10−8 [29]BB¯0⟶ΛΛ¯<3.2×10−70.3±0.2×10−8 [29]BB−⟶Λp¯2.4−0.9+1.0×10−73.5−0.5+0.7×10−8 [29]BB¯s0⟶pp¯<1.5×10−83.0−1.2+1.5×10−8 [29]BB¯0⟶pp¯π05.0±1.9×10−75.0±2.1×10−7 [31]BB¯0⟶ΛΛ¯K¯04.8−0.9+1.0×10−62.5±0.3×10−6 [33]BB−⟶pp¯π−1.62±0.20×10−61.60±0.18×10−6 [35]BB−⟶ΛΛ¯π−<9.4×10−71.7±0.7×10−7 [33]BB−⟶ΛΛ¯K−3.4±0.6×10−62.8±0.2×10−6 [33]B(B¯s0⟶p¯ΛK++c.c.)5.5±1.0×10−65.1±1.1×10−6 [37]BB¯0⟶pp¯K¯∗01.24−0.25+0.28×10−60.9±0.3×10−6 [32]BB¯0⟶ΛΛ¯K¯∗02.5−0.8+0.9×10−61.76±0.18×10−6 [36]BB−⟶pp¯K∗−3.6−0.7+0.8×10−66.0±1.3×10−6 [32]BB−⟶Λp¯ρ0,ρ0⟶π+π−4.8±0.9×10−63.28±0.31×10−6 [36]BB−⟶Λp¯ϕ8.0±2.2×10−71.51±0.28×10−6 [36]BB−⟶ΛΛ¯K∗−2.2−0.9+1.2×10−61.91±0.20×10−6 [36]AFBB¯0⟶Λp¯π+−0.41±0.11±0.03−14.6−1.5+0.9±6.9×10−2 [53]AFBB−⟶Λp¯π0−0.16±0.18±0.03−14.6−1.5+0.9±6.9×10−2 [53]AFBB−⟶pp¯π−−0.409±0.033±0.006−49.1−0.3+0.6±1.0±6.3×10−2 [54]AFBB−⟶pp¯K−0.495±0.012±0.00746.9−0.041+0.043±0.2±4.7×10−2 [54]ACPB−⟶pp¯π−0.00±0.04−0.06 [30]ACPB−⟶pp¯K−0.00±0.040.06±0.01 [30]ACPB−⟶pp¯K∗−0.21±0.160.22±0.04 [30]ACPB¯0⟶pp¯π0−16.8±5.4×10−2 [31]ACPB¯0⟶pp¯ρ0−12.6±3.0×10−2 [31]


## 4. Discussions and Conclusions

The theoretical results in Table2 can agree well with the data, which is based on the factorization, pQCD counting rules, and baryonic form factors. In particular, several CP asymmetries are predicted as large as 10-20%, promising to be measured by LHCb and Belle II [55, 56]. It is reasonable to extend the theoretical approach to B⟶BcB′¯ and B⟶BB′¯Mc [34, 38, 57], where BcMc denotes a baryon (meson) containing a charm quark. As a consequence, B,AFB can also be expalined (see Table 3). Nonetheless, BB−⟶pp¯ρ−,ρ−⟶π−π0=28.8±2.1×10−6 and BB−⟶pp¯μ−ν¯μ=1.04±0.24±0.12×10−4 we have predicted are not verified by the observations [1, 32, 58, 59],
(21)BB−⟶pp¯π−π0=4.6±1.3×10−6,BB−⟶pp¯μ−ν¯μ=5.27−0.24+0.23±0.21±0.15×10−6,where the amplitude of B−⟶pp¯μ−ν¯μ is given by
(22)MB−⟶pp¯ℓν¯ℓ=GF2Vubpp¯u¯γμ1−γ5bB−ℓ¯γμ1−γ5νℓ,since B⟶pp¯ρ and B−⟶pp¯μ−ν¯μ are seen to be associated with the B⟶pp¯ transition form factors, which are inferred to cause the overestimations. Besides, BB−⟶pp¯μ−ν¯μ inconsistent with the data can be partly due to the inconsistent determination of Vub between the inclusive and exclusive B decays.Table 3
Theoretical results of the two- and three-bodyB decays with the baryon (meson) containing a charm quark, which are compared with the experimental data.
Decay modeExperimental data [1, 13]TheoryBB¯0⟶Λc+p¯1.54±0.18×10−51.0−0.3+0.4×10−5 [57]BB¯0⟶Σc+p¯<2.4×10−52.9−0.9+0.8×10−6 [57]BB−⟶Λp¯D01.43±0.32×10−51.14±0.26×10−5 [34]BB−⟶Λp¯D∗0<5×10−53.23±0.32×10−5 [34]BB¯0⟶np¯D∗+1.4±0.4×10−31.45±0.14×10−3 [34]BB¯0⟶pp¯D01.04±0.07×10−41.04±0.12×10−4 [38]BB¯0⟶pp¯D∗00.99±0.11×10−40.99±0.09×10−4 [38]BB¯0⟶Λp¯D+2.5±0.4×10−51.85±0.30×10−5 [38]BB¯0⟶Λp¯D∗+3.4±0.8×10−52.75±0.24×10−5 [38]BB¯0⟶Σ0Λ¯D¯0+c.c.<3.1×10−51.8±0.5×10−5 [34]AFBB¯0⟶Λp¯D+−0.08±0.10−0.030±0.002 [38]AFBB¯0⟶Λp¯D∗++0.55±0.17+0.150±0.000 [38]As the final remark, since the predictions ofBB−⟶pp¯ρ− and BB−⟶pp¯μ−ν¯μ are shown to deviate from the observations by the factors of 6 and 20, respectively, the theoretical approach is facing some difficulties. Therefore, the reexamination should be performed elsewhere.In summary, to review the baryonicB decays, we have summarized the experimental data, which includes branching fractions and angular and CP asymmetries. We have taken B¯0⟶pp¯ and B⟶pp¯V with V=ρ−0 or K∗−K¯∗0 for theoretical illustration. We have also reviewed the CP asymmetries of B⟶pp¯M, which can be used to compare with future measurements by LHCb and Belle II. With the theoretical results listed in the tables, we have demonstrated that the theoretical approach can be used to interpret most observations. Finally, we have also remarked that the theoretical approach has currently encountered some challenges in interpreting BB−⟶pp¯ρ− and BB−⟶pp¯μ−ν¯μ.



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