Direct CP violation in internal W-emission dominated baryonic B decays

Abstract

The observation of CP violation has been experimentally verified in numerous B decays but is yet to be confirmed in final states with half-spin particles. We focus our attention on baryonic B-meson decays mediated dominantly through internal W-emission processes and show that they are promising processes to observe for the first time the CP violating effects in B decays to final states with half-spin particles. Specifically, we study the $\bar{B}^0\rightarrow p\bar{p}\pi ^0(\rho ^0)$ and $\bar{B}^0\rightarrow p\bar{p}\pi ^+\pi ^-$ decays. We obtain $\mathcal{B}(\bar{B}^0\rightarrow p\bar{p}\pi ^0)=(5.0\pm 2.1)\times 10^{-7}$, in agreement with current data, and $\mathcal{B}(\bar{B}^0\rightarrow p\bar{p}\rho ^0)\simeq \mathcal{B}(\bar{B}^0\rightarrow p\bar{p}\pi ^0)/3$. Furthermore, we find $\mathcal{A}_{CP}(\bar{B}^0\rightarrow p\bar{p}\pi ^0,p\bar{p}\rho ^0,p\bar{p}\pi ^+\pi ^-) =(-16.8\pm 5.4,-12.6\pm 3.0,-11.4\pm 1.9)\%$. With measured branching fractions $\mathcal{B}(\bar{B}^0\rightarrow p\bar{p}\pi ^0,p\bar{p}\pi ^+\pi ^-)\sim \mathcal{O}(10^{-6})$, we point out that $\mathcal{A}_{CP}\sim -(10-20)\%$ can be new observables for CP violation, accessible to the Belle II and/or LHCb experiments.

1 Introduction

The investigation of CP violation (CPV) has been one of the most important tasks in hadron weak decays. In the Standard Model (SM), CPV arises from a unique phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix; however, it is insufficient to explain the matter and antimatter asymmetry of the Universe. To try and shed light on solving the above puzzle, a diverse set of observations related to CPV is necessary. So far, direct CP violation has been observed in B and D decays [1, 2]. With $Re(\epsilon '/\epsilon )$, it is also found in kaon decays [3]. Although the decays involving half-spin particles offer an alternative route, evidence for CP violation is not richly provided [4, 5].

Baryonic B decays can be an important stage to investigate CPV within the SM and beyond. With $M^{(*)}$ denoting a pseudoscalar (vector) meson such as $K^{(*)},\pi ,\rho ,D^{(*)}$, the $B\rightarrow p\bar{p} M^{(*)}$ decays have been carefully studied by the B factories and the LHCb experiment [5,6,7,8,9,10,11]. Experimental information includes measurements of branching fractions, angular distribution asymmetries, polarization of vector mesons in $B\rightarrow p\bar{p} K^*$, Dalitz plot information, and $p\bar{p}$ ($M^{(*)}p$) invariant mass spectra. This helps to improve the theoretical understanding of the di-baryon production in $B\rightarrow \mathbf{B}\bar{\mathbf{B}}'M$ [12,13,14,15,16], such that the data can be well interpreted. Predictions are confirmed by recent measurements. For example, one obtains $\mathcal{B}(\bar{B}^0_s\rightarrow p\bar{\Lambda }K^- + \Lambda \bar{p} K^+) =(5.1\pm 1.1)\times 10^{-6}$ [17], in excellent agreement with the value of $(5.46\pm 0.61\pm 0.57\pm 0.50\pm 0.32)\times 10^{-6}$ measured by LHCb [18]. Moreover, the theoretical extension to four-body decays allows to interpret $\mathcal{B}(\bar{B}^0\rightarrow p\bar{p} \pi ^+ \pi ^-)$ [19,20,21]. The same can be said for CP asymmetries.

In this report we focus our attention on the baryonic B-meson decays mediated dominantly through the internal W-emission diagrams. Although the internal W-emission decays are regarded as suppressed processes, the measured branching fractions of the baryonic B decays

$$\begin{aligned} \mathcal{B}(\bar{B}^0\rightarrow p\bar{p}\pi ^0)= & {} (5.0\pm 1.8\pm 0.6)\times 10^{-7}\,,\nonumber \\ \mathcal{B}(\bar{B}^0\rightarrow p \bar{p} \pi ^+\pi ^-)= & {} (2.7\pm 0.1\pm 0.1\pm 0.2)\times 10^{-6}\,,\nonumber \\ \end{aligned}$$

(1)

are not small [19, 22], which make these modes an ideal place to observe for the first time CP violation in B decays to final states with half-spin particles. Therefore, we will study the branching fractions for the decays of $\bar{B}^0\rightarrow p\bar{p} \pi ^0(\rho ^0),\,p\bar{p} \pi ^+\pi ^-$, and predict their direct CP violating asymmetries.

Fig. 1

The $\bar{B}^0\rightarrow p\bar{p}\pi ^0(\rho ^0)$ decay processes, depicted as (a,b,c) for the $\bar{B}^0\rightarrow \pi ^0(\rho ^0)$ transition with $0\rightarrow p\bar{p}$ production, and (d,e,f) for the $\bar{B}^0\rightarrow p\bar{p}$ transition with recoiled $\pi ^0(\rho ^0)$ meson

2 Formalism

For the tree-level dominated B meson decays, the relevant effective Hamiltonian is given by [23]

$$\begin{aligned} \mathcal{H}_{eff}= & {} \frac{G_F}{\sqrt{2}}\bigg [ V_{ub}V_{ud}^*\bigg (\sum _{i=1,2} c_i O_i\bigg )\nonumber \\&-V_{tb}V_{td}^*\bigg (\sum _{j=3}^{10} c_j O_j\bigg )\bigg ]+h.c., \end{aligned}$$

(2)

where $G_F$ is the Fermi constant, $c_{i(j)}$ the Wilson coefficients, and $V_{ij}$ the CKM matrix elements. The four-quark operators $O_{i(j)}$ for the tree (penguin)-level contributions are written as

$$\begin{aligned}&O_1=(\bar{d}_\alpha u_\alpha )_{V-A}(\bar{u}_\beta b_\beta )_{V-A}\,,\;\nonumber \\&O_2=(\bar{d}_\alpha u_\beta )_{V-A}(\bar{u}_\beta b_\alpha )_{V-A}\,,\nonumber \\&O_{3(5)}=(\bar{d}_\alpha b_\alpha )_{V-A}\sum _q(\bar{q}_\beta q_\beta )_{V\mp A}\,,\;\nonumber \\&O_{4(6)}=(\bar{d}_\alpha b_\beta )_{V-A}\sum _q(\bar{q}_\beta q_\alpha )_{V\mp A}\,,\nonumber \\&O_{7(9)}={3\over 2}(\bar{d}_\alpha b_\alpha )_{V-A}\sum _q e_q(\bar{q}_\beta q_\beta )_{V\pm A}\,,\;\nonumber \\&O_{8(10)}={3\over 2}(\bar{d}_\alpha b_\beta )_{V-A}\sum _q e_q(\bar{q}_\beta q_\alpha )_{V\pm A}\,, \end{aligned}$$

(3)

where $q=(u,d,s)$, $(\bar{q}_1 q_2)_{V\pm A}=\bar{q}_1\gamma _\mu (1\pm \gamma _5)q_2$, and the subscripts $(\alpha ,\beta )$ denote the color indices. With the identity of $\delta _{\beta \beta '}\delta _{\alpha \alpha '} =\delta _{\alpha \beta }\delta _{\alpha '\beta '}/N_c+2T^a_{\alpha \beta }T^a_{\alpha '\beta '}$, where $N_c=3$ is the color number, $O_i$ and $O_{i+1}$ can be related. For example, we have $O_1=O_2/N_c+2\bar{d}\gamma _\mu (1-\gamma _5)T^a u\bar{u}\gamma ^\mu (1-\gamma _5)T^a b$ with $T^a$ the Gell-Mann matrices.

Fig. 2

The tree-level $b\rightarrow u\bar{u}d$ weak transition, where the blue blob represents the short-distance internal W-boson emission

In the factorization ansatz [24, 25], one is able to express $\langle h_1 h_2|O|B\rangle$ as a product of two factors, $\langle h_1 |J_1|0\rangle$ and $\langle h_2 |J_2|B\rangle$, where $O=J_1\cdot J_2$ is the product of the two color singlet quark currents $J_1$ and $J_2$ and $h_{1,2}$ denote the hadron states. The matrix elements $\langle h_1 |J_1|0\rangle$ and $\langle h_2 |J_2|B\rangle$ are obtained in such a way that the flavor quantum numbers of $J_{1,2}$ match the hadron states in the separate matrix elements. We hence decompose $\langle p\bar{p} \pi ^0|O_2|\bar{B}^0\rangle$ as [15, 16]

$$\begin{aligned} \langle O_2\rangle _{a}= & {} \langle \pi ^0|(\bar{u}_\beta u_\beta )_{V-A}|0\rangle \langle p\bar{p}|(\bar{d}_\alpha b_\alpha )_{V-A}|\bar{B}^0\rangle \,,\nonumber \\ \langle O_2\rangle _{d}= & {} \langle p\bar{p}|(\bar{u}_\beta u_\beta )_{V-A}|0\rangle \langle \pi ^0|(\bar{d}_\alpha b_\alpha )_{V-A}|\bar{B}^0\rangle \,, \end{aligned}$$

(4)

where the Fierz reordering has been used to exchange $(\bar{d}_\alpha ,\bar{u}_\beta )$. The amplitudes $\langle O_2\rangle _{a,d}$ correspond to the two configurations depicted in Fig. 1a, d, respectively. As depicted in Fig. 2 for the $b\rightarrow u\bar{u} d$ transition, dynamically, the d-quark moves collinearly with the spectator quark $\bar{d}$ from $\bar{B}^0(b\bar{d})$, so that in Fig. 1d the $d\bar{d}$ for the $p\bar{p}$ formation can be seen as a consequence of the B meson transition, which is in accordance with the matrix element of $\langle p\bar{p}|(\bar{d} b)|\bar{B}^0\rangle$. Moreover, since $u\bar{u}$ and $d\bar{d}$ in the $\bar{B}^0$ rest frame can be seen to move in opposite directions, we take $\pi ^0(u\bar{u})$ in Fig. 1d as the recoiled state, in accordance with $\langle \pi ^0|(\bar{u} u)|0\rangle$ with $|0\rangle$ representing the vacuum. On the other hand, $\langle p\bar{p} \pi ^0|O_1|\bar{B}^0\rangle$ is expressed as $\langle O_1\rangle _{a(d)}=\langle O_2\rangle _{a(d)}/N_c+\langle \chi _1\rangle$ with $\langle \chi _1\rangle \equiv \langle p\bar{p} \pi ^0|2\bar{u}\gamma _\mu (1-\gamma _5)T^a d \bar{u}\gamma ^\mu (1-\gamma _5)T^a b|\bar{B}^0\rangle$. The $T^a$ in $\langle \chi _1\rangle$ correspond to the gluon exchange between the two currents, which causes an inseparable connection between the final states. Hence, $\langle \chi _1\rangle$ is regarded as the non-factorizable QCD corrections. Subsequently, we note that $\langle p\bar{p} \pi ^0|c_1 O_1+c_2 O_2 |\bar{B}^0\rangle = a_2 \langle O_2\rangle _{a,d}$ with $a_2=c_2^{eff}+c_1^{eff}/N_c$, where $c_i^{eff}$ represents the effective Wilson coefficient for $c_i$ to receive the next-to-leading-order contributions [25]. In the generalized edition of the factorization, one varies $N_c$ between 2 and infinity in order to estimate $\langle \chi _1\rangle$ [15, 24, 25]. This makes $N_c$ a phenomenological parameter determined by data.

To complete the amplitudes, we extend our calculation for $\langle p\bar{p} \pi ^0|c_1 O_1+c_2 O_2 |\bar{B}^0\rangle$ to the penguin-level diagrams, as depicted in Fig. 1b, c, e, f. Moreover, with $\pi ^0$ replaced by $\rho ^0$ and $\pi ^+\pi ^-$, we get the amplitudes of $\bar{B}^0\rightarrow p\bar{p}\rho ^0$ and $\bar{B}^0\rightarrow p\bar{p}\pi ^+\pi ^-$, respectively. Hence, the decay amplitudes of $\bar{B}^0\rightarrow p\bar{p} X_M$ with $X_M\equiv (\pi ^0(\rho ^0),\pi ^+\pi ^-)$ can be written as [16, 17, 20]

$$\begin{aligned} \mathcal{A}(\bar{B}^0\rightarrow p\bar{p} X_M)= & {} \mathcal{A}_1(X_M)+\mathcal{A}_2(X_M)\,, \end{aligned}$$

(5)

with $\mathcal{A}_{1,2}(X_M)$ corresponding to Fig. 1a–c and d–f, respectively. Explicitly, $\mathcal{A}_{1,2}$ are given by [15,16,17, 25,26,27]

$$\begin{aligned} \mathcal{A}_1(X_M)= & {} \frac{G_F}{\sqrt{2}}\bigg \{ \bigg [ \langle p\bar{p}|\bar{u}\gamma ^\mu (\alpha _2^+ -\alpha _2^-\gamma _5) u|0\rangle \nonumber \\&+ \langle p\bar{p}|\bar{d}\gamma ^\mu (\alpha _3^+ -\alpha _3^-\gamma _5) d|0\rangle \bigg ]\nonumber \\&\times \langle X_M|\bar{d} \gamma _\mu (1-\gamma _5) b|\bar{B}^0\rangle \nonumber \\&+\alpha _6\langle p\bar{p}|\bar{d}(1+\gamma _5) d|0\rangle \langle X_M|\bar{d}(1-\gamma _5) b|\bar{B}^0\rangle \bigg \}\;,\nonumber \\ \mathcal{A}_2(X_M)= & {} \frac{G_F}{\sqrt{2}}\bigg \{ \bigg [\langle X_M|\bar{u}\gamma ^\mu (\alpha _2^+ -\alpha _2^-\gamma _5) u|0\rangle \nonumber \\&+ \langle X_M|\bar{d}\gamma ^\mu (\alpha _3^+ -\alpha _3^-\gamma _5) d|0\rangle \bigg ]\nonumber \\&\times \langle p\bar{p}|\bar{d} \gamma _\mu (1-\gamma _5) b|\bar{B}^0\rangle \nonumber \\&+ \alpha _6\langle X_M|\bar{d}(1+\gamma _5) d|0\rangle \langle p\bar{p}|\bar{d}(1-\gamma _5) b|\bar{B}^0\rangle \bigg \}.\nonumber \\ \end{aligned}$$

(6)

The parameters $\alpha _i$ are defined as

$$\begin{aligned} \alpha _2^\pm= & {} V_{ub}V_{ud}^* a_2-V_{tb}V_{td}^* (a_3\pm a_5\pm a_7+a_9)\,,\nonumber \\ \alpha _3^\pm= & {} -V_{tb}V_{td}^*\left( a_3+a_4\pm a_5\mp \frac{a_7}{2}-\frac{a_9}{2}-\frac{a_{10}}{2}\right) ,\nonumber \\ \alpha _6= & {} V_{tb}V_{td}^*(2a_6-a_8)\,, \end{aligned}$$

(7)

with $a_i\equiv c^{eff}_i+c^{eff}_{i\pm 1}/N_c$ for $i=$odd (even) [25]. We note that $\mathcal{A}_2(\pi ^+\pi ^-)$ is neglected since $\mathcal{A}_1(\pi ^+\pi ^-)\gg \mathcal{A}_2(\pi ^+\pi ^-)$ [20].

The $B\rightarrow X_M$ transition matrix elements in $\mathcal{A}_1(X_M)$ are written as [28,29,30,31]

$$\begin{aligned}&\langle M(p)|\bar{q}\gamma _\mu b|B(p_B)\rangle \nonumber \\&\quad =\bigg [(p_B+p)^\mu -\frac{m^2_B-m^2_M}{q^2}q^\mu \bigg ] F_1^{BM}\nonumber \\&\qquad +\frac{m^2_B-m^2_M}{q^2}q^\mu F_0^{BM}\,,\nonumber \\&\langle M^*(p)|\bar{q} \gamma _\mu b|B(p_B)\rangle =\epsilon _{\mu \nu \alpha \beta } \varepsilon ^{*\nu }p_{B}^{\alpha }p^{\beta }\frac{2V_1}{m_{B}+m_{M^*}}\;,\nonumber \\&\langle M^*(p)|q\gamma _\mu \gamma _5 b|B(p_B)\rangle \nonumber \\&\quad =i\bigg [\varepsilon ^*_\mu -\frac{\varepsilon ^*\cdot q}{q^2}q_\mu \bigg ](m_B+m_{M^*})A_1\nonumber \\&\qquad + i\frac{\varepsilon ^*\cdot q}{q^2}q_\mu (2m_{M^*})A_0\nonumber \\&\qquad -i\bigg [(p_B+p)_\mu -\frac{m^2_{B}-m^2_{M^*}}{q^2}q_\mu \bigg ] (\varepsilon ^*\cdot q)\frac{A_2}{m_{B}+m_{M^*}}\;,\nonumber \\&\langle M_1(p_1) M_2(p_2)|\bar{q}\gamma _\mu (1-\gamma _5)b|B(p_B)\rangle \nonumber \\&\quad =\epsilon _{\mu \nu \alpha \beta }p_B^\nu (p_2+p_1)^\alpha (p_2-p_1)^\beta h \nonumber \\&\qquad +iw_+ (p_2+p_1)_\mu +iw_-(p_2-p_1)+irq_\mu \,, \end{aligned}$$

(8)

where $\varepsilon _\mu$ is the polarization vector of $M^*$, $q_\mu =(p_B-p)_\mu =(p_B-p_1-p_2)_\mu$ as the momentum transfer for the $B\rightarrow X_M$ transition, $(F^{BM}_{0,1},V_1,A_{0,1,2})$ the $B\rightarrow M^{(*)}$ transition form factors and $(h,r,w_{\pm })$ the $B\rightarrow M_1 M_2$ transition form factors.

The matrix elements of $0\rightarrow \mathbf{B}\bar{\mathbf{B}}'$ are expressed as [27]

$$\begin{aligned} \langle \mathbf{B}\bar{\mathbf{B}}'|\bar{q}\gamma _\mu q'|0\rangle= & {} \bar{u}\bigg [F_1\gamma _\mu +\frac{F_2}{m_{\mathbf{B}}+m_{\bar{\mathbf{B'}}}}i\sigma _{\mu \nu }q^\nu \bigg ]v\;,\nonumber \\ \langle \mathbf{B}\bar{\mathbf{B}}'|\bar{q}\gamma _\mu \gamma _5 q'|0\rangle= & {} \bar{u}\bigg [g_A\gamma _\mu +\frac{h_A}{m_{\mathbf{B}}+m_{\bar{\mathbf{B}}'}}q_\mu \bigg ]\gamma _5 v\,,\nonumber \\ \langle \mathbf{B}\bar{\mathbf{B}}'|\bar{q} q'|0\rangle= & {} f_S\bar{u}v\;, \langle \mathbf{B}\bar{\mathbf{B}}'|q\gamma _5 q'|0\rangle =g_P\bar{u} \gamma _5 v\,, \end{aligned}$$

(9)

where u(v) is the (anti-)baryon spinor, and $F_{1,2}$, $g_A$, $h_A$, $f_S$, $g_P$ the timelike baryonic form factors.

In $\mathcal{A}_2(\bar{B}^0\rightarrow p\bar{p} M^{(*)0})$, the $0\rightarrow M^{(*)}$ matrix elements are written as [1]

$$\begin{aligned} \langle M(p)|\bar{q}\gamma _\mu \gamma _5 q'|0\rangle =-if_M p_\mu , \ \langle M^*|\bar{q}\gamma _\mu q'|0\rangle =m_{M^*} f_{M^*}\varepsilon _\mu ^*,\nonumber \\ \end{aligned}$$

(10)

with $f_{M^{(*)}}$ the decay constant. For the $B\rightarrow \mathbf{B}\bar{\mathbf{B}}'$ transitions we have [13, 26]

$$\begin{aligned} \langle \mathbf{B}\bar{\mathbf{B}}'|\bar{q}\gamma _\mu b|B\rangle= & {} i\bar{u}[ g_1\gamma _{\mu }+g_2i\sigma _{\mu \nu }\hat{p}^\nu +g_3 \hat{p}_{\mu }+g_4(p_{\bar{\mathbf{B}}'}\nonumber \\&+p_{\mathbf{B}})_\mu +g_5(p_{\bar{\mathbf{B}}'}-p_{\mathbf{B}})_\mu ]\gamma _5v\,,\nonumber \\ \langle \mathbf{B}\bar{\mathbf{B}}'|\bar{q}\gamma _\mu \gamma _5 b|B\rangle= & {} i\bar{u}[ f_1\gamma _{\mu }+f_2i\sigma _{\mu \nu }\hat{p}^\nu +f_3 \hat{p}_{\mu }\nonumber \\&+f_4(p_{\bar{\mathbf{B}}'}+p_{\mathbf{B}})_\mu +f_5(p_{\bar{\mathbf{B}}'}-p_\mathbf{B})_\mu ]v\,,\nonumber \\ \langle \mathbf{B}\bar{\mathbf{B}}'|\bar{q} b|B\rangle= & {} i\bar{u}[ \bar{g}_1{\hat{p}}\!\!/+\bar{g}_2(E_{\bar{\mathbf{B}}'}+E_{\mathbf{B}})\nonumber \\&+\bar{g}_3(E_{\bar{\mathbf{B}}'}-E_{\mathbf{B}})]\gamma _5v\,,\nonumber \\ \langle \mathbf{B}{\bar{\mathbf{B}}'}|\bar{q}\gamma _5 b|B\rangle= & {} i\bar{u}[ \bar{f}_1{\hat{p}}\!\!/+\bar{f}_2(E_{\bar{\mathbf{B}}'}+E_{\mathbf{B}}) \nonumber \\&+\bar{f}_3(E_{\bar{\mathbf{B}}'}-E_{\mathbf{B}})]v\,, \end{aligned}$$

(11)

where $\hat{p}_\mu =(p_B-p_{\mathbf{B}}-p_{\bar{\mathbf{B}}'})_\mu$, $g_i(f_i)$ $(i=1,2, \ldots ,5)$ and $\bar{g}_j(\bar{f}_j)$ $(j=1,2,3)$ are the $B\rightarrow \mathbf{B}\bar{\mathbf{B}}'$ transition form factors.

The mesonic and baryonic form factors have momentum dependencies. For $B\rightarrow M^{(*)}$, they are given by [32]

$$\begin{aligned} F_A(q^2)= & {} \frac{F_A(0)}{\left( 1-\frac{q^2}{M_A^2}\right) \left( 1-\frac{\sigma _{1} q^2}{M_A^2}+\frac{\sigma _{2} q^4}{M_A^4}\right) }\,,\nonumber \\ F_B(q^2)= & {} \frac{F_B(0)}{1-\frac{\sigma _{1} q^2}{M_B^2}+\frac{\sigma _{2} q^4}{M_B^4}}\,, \end{aligned}$$

(12)

where $F_A=(F_1^{BM},V_1,A_0)$ and $F_B=(F_0^{BM},A_{1,2})$. According to the approach of perturbative QCD counting rules, one presents the momentum dependencies of the form factors for $B\rightarrow \mathbf{B}\bar{\mathbf{B}}'$, $0\rightarrow \mathbf{B}\bar{\mathbf{B}}'$ and $B\rightarrow M_1 M_2$ as [13, 26, 33,34,35,36,37]

$$\begin{aligned}&F_1=\frac{\bar{C}_{F_1}}{t^2}\,,\;g_A=\frac{\bar{C}_{g_A}}{t^2}\,,\; f_S=\frac{\bar{C}_{f_S}}{t^2}\,,\;g_P=\frac{\bar{C}_{g_P}}{t^2}\,,\;\nonumber \\&f_i=\frac{D_{f_i}}{t^3}\,,\;g_i=\frac{D_{g_i}}{t^3}\,,\; \bar{f}_i=\frac{D_{\bar{f}_i}}{t^3}\,,\;\bar{g}_i=\frac{D_{\bar{g}_i}}{t^3}\,,\nonumber \\&h=\frac{C_h}{s^2}\,,\;w_-=\frac{D_{w_-}}{s^2}\,, \end{aligned}$$

(13)

where $t\equiv (p_{\mathbf{B}}+p_{\bar{\mathbf{B}}'})^2$, $s\equiv (p_1+p_2)^2$, and $\bar{C}_i=C_i [\text {ln}({t}/{\Lambda _0^2})]^{-\gamma }$ with $\gamma =2.148$ and $\Lambda _0=0.3$ GeV. In Ref. [38], $F_2=F_1/(t\text {ln}[t/\Lambda _0^2])$ is calculated to be much less than $F_1$; hence we neglect it. Since $h_A$ corresponds to the smallness of $\mathcal{B}(\bar{B}^0\rightarrow p\bar{p})\sim 10^{-8}$ [39,40,41], we neglect $h_A$ as well. The terms $(r,w_+)$ in Eq. (8) are neglected – following Refs. [36, 37] – due to the fact that their parity quantum numbers disagree with the experimental evidence of $J^P=1^-$ for the meson-pair production [42].

The constants $C_i$ ($D_i$) can be decomposed into sets of parameters that obey the SU(3) flavor and SU(2) spin symmetries. In Refs. [13, 15, 17, 20, 26, 27, 33, 34, 43, 44], they are derived as

$$\begin{aligned}&(C_{F_1},C_{g_A})\nonumber \\&\quad =\frac{1}{3} (5C_{||}+C_{\overline{||}},5C_{||}^*-C_{\overline{||}}^*)\,,\; \text {(for }\langle p\bar{p}|(\bar{u} u)_{V,A}|0\rangle )\,\nonumber \\&(C_{F_1},C_{g_A},C_{f_S},C_{g_P})\nonumber \\&\quad =\frac{1}{3} (C_{||}+2C_{\overline{||}},C_{||}^*-2C_{\overline{||}}^*, \bar{C}_{||},\bar{C}_{||}^*)\,,\;\nonumber \\&\text {(for }\langle p\bar{p}|(\bar{d} d)_{V,A,S,P}|0\rangle )\,\nonumber \\&(D_{g_1,f_1},D_{g_j},D_{f_j})\nonumber \\&\quad =\frac{1}{3} (D_{||}\mp 2D_{\overline{||}},-D_{||}^j,D_{||}^j)\,,\;\nonumber \\&\qquad \text {(for }\langle p\bar{p}|(\bar{d} d)_{V,A}|\bar{B}^0\rangle )\,\nonumber \\&(D_{\bar{g}_1,\bar{f}_1},D_{\bar{g}_{2,3}},D_{\bar{f}_{2,3}})\nonumber \\&\quad =\frac{1}{3} (\bar{D}_{||}\mp 2\bar{D}_{\overline{||}},-\bar{D}_{||}^{2,3},-\bar{D}_{||}^{2,3})\,,\; \nonumber \\&\qquad \text {(for }\langle p\bar{p}|(\bar{d} d)_{S,P}|\bar{B}^0\rangle )\, \end{aligned}$$

(14)

with $j=2, \ldots ,4,5$, $C_{||(\overline{||})}^*\equiv C_{||(\overline{||})}+\delta C_{||(\overline{||})}$ and $\bar{C}_{||}^*\equiv \bar{C}_{||}+\delta \bar{C}_{||}$. The direct CP violating asymmetry is defined as

$$\begin{aligned}&\mathcal{A}_{CP}(B\rightarrow \mathbf{B}\bar{\mathbf{B}}'X_M)\nonumber \\&\quad \equiv \frac{\Gamma (B\rightarrow \mathbf{B}\bar{\mathbf{B}}'X_M)-\Gamma (\bar{B}\rightarrow {\bar{\mathbf{B B}}'}\bar{X}_M)}{\Gamma (B\rightarrow \mathbf{B}\bar{\mathbf{B}}'X_M)+\Gamma (\bar{B}\rightarrow {\bar{\mathbf{B B}}'}\bar{X}_M)}\,, \end{aligned}$$

(15)

where $\bar{B}\rightarrow {\bar{\mathbf{B B}}'}\bar{X}_M$ denotes the anti-particle decay.

Table 1 The $\bar{B}^0\rightarrow M^{(*)0}$ transition form factors at zero-momentum transfer, with $(M_A,M_B)=(5.32,5.32)$ and (5.27, 5.32) GeV for $\pi$ and $\rho$, respectively

3 Numerical analysis

We use the following values for the numerical analysis. The CKM matrix elements are calculated via the Wolfenstein parameterization [1], with the world-average values

$$\begin{aligned} \lambda= & {} 0.22453\pm 0.00044\,, A=0.836\pm 0.015\,, \nonumber \\ \bar{\rho }= & {} 0.122^{+0.018}_{-0.017}\,, \bar{\eta }=0.355^{+0.012}_{-0.011}\,. \end{aligned}$$

(16)

Table 2 Decay branching fractions and direct CP asymmetries of $\bar{B}^0\rightarrow p\bar{p} X_M$, where the first errors come from the estimations of the non-factorizable effects, the second ones from the uncertainties of the CKM matrix elements, and the third ones from those of the decay constants and form factors

The decay constants are $f_{\pi ,\rho }=(130.4\pm 0.2,210.6\pm 0.4)$ MeV [1], with $(f_{\pi ^0},f_{\rho ^0})=(f_\pi ,f_\rho )/\sqrt{2}$. We adopt the $B\rightarrow M^{(*)}$ transition form factors in Ref. [32], listed in Table 1. In Sect. 2, $N_c$ has been presented as the phenomenological parameter determined by data. Empirically, one is able to determine $N_c$ between 2 and $\infty$. With the nearly universal value for $N_c$ in the specific decays, the factorization is demonstrated to be valid. For the tree-level internal W-emission dominated b-hadron decays, the extraction has given $N_c\simeq 2$ that corresponds to $a_2\sim \mathcal{O}(0.2-0.3)$ [15, 20, 45,46,47,48,49], where $\delta N_c$ differs due to the experimental uncertainties. For example, one obtains $N_c=2.15\pm 0.17$ in $\Lambda _b\rightarrow \mathbf{B}M_c$ [47, 48]. Here, we test if $N_c\simeq 2$ can be used to explain the measured $\mathcal{B}(\bar{B}^0\rightarrow p\bar{p}\pi ^0,p\bar{p}\pi ^+\pi ^-)$.

The $C_{h,w_-}$ for $\bar{B}^0\rightarrow \pi ^+\pi ^-$ and $C_i(D_i)$ for $0\rightarrow p\bar{p}$ ($\bar{B}^0\rightarrow p\bar{p}$) have been determined to be [15, 17, 20]

$$\begin{aligned}&(C_h,C_{w_-})=(3.6\pm 0.3,0.7\pm 0.2)\;\mathrm{GeV}^3\,, \nonumber \\&(C_{||},C_{\overline{||}},\bar{C}_{||})= (154.4\pm 12.1,18.1\nonumber \\&\qquad \pm 72.2,537.6\pm 28.7)\;\mathrm{GeV}^{4}\,,\nonumber \\&(\delta C_{||},\delta C_{\overline{||}},\delta \bar{C}_{||})\nonumber \\&\quad = (19.3\pm 21.6,-477.4\pm 99.0,-342.3\pm 61.4)\;\mathrm{GeV}^{4}\,,\nonumber \\&(D_{||},D_{\overline{||}})=(45.7\pm 33.8,-298.2\pm 34.0)\;\mathrm{GeV}^{5}\,,\nonumber \\&(D_{||}^2,D_{||}^3,D_{||}^4,D_{||}^5) =(33.1\pm 30.7,-203.6\pm 133.4,6.5\nonumber \\&\qquad \pm 18.1,-147.1\pm 29.3)\;\mathrm{GeV}^{4}\,,\nonumber \\&(\bar{D}_{||},\bar{D}_{\overline{||}},\bar{D}_{||}^2,\bar{D}_{||}^3)\nonumber \\&\quad =(35.2\pm 4.8,-38.2\pm 7.5,-22.3\pm 10.2, 504.5\nonumber \\&\qquad \pm 32.4)\;\mathrm{GeV}^{4}\,. \end{aligned}$$

(17)

For $\alpha _i$ in Eq. (7), the effective Wilson coefficients $c^{eff}_i$ are calculated at the $m_b$ scale in the NDR scheme, see Ref. [25]. They are related to the size of the decay, where the strong phases, together with the weak phase in $V_{ub}$ and $V_{td}$, play the key role in $\mathcal{A}_{CP}$.

Our results for the branching fractions and CP violating asymmetries of $\bar{B}^0\rightarrow p\bar{p} X_M$ decays are summarized in Table 2, where we have averaged the particle and antiparticle contributions for the total branching fractions.

4 Discussions and conclusions

The improved theoretical approaches such as QCD factorization (QCDF) and soft-collinear effective theory have been applied to two-body mesonic B decays [50,51,52]. Hence, the non-factorizable corrections of order $1/N_c^n$ with $n=1,2$ have been considered by calculating the vertex corrections from the hard gluon exchange and the hard spectator scattering. Unfortunately, there exist no similar approaches well applied to the $B\rightarrow M_1M_2M_3$, $\mathbf{B}\bar{\mathbf{B}}'M$ and $\mathbf{B}\bar{\mathbf{B}}'MM'$ decays, due to the wave functions of $B\rightarrow \mathbf{B}\bar{\mathbf{B}}'(MM')$ not as clear as those of $B\rightarrow M$. By varying $N_c$ from 2 to $\infty$, one can still estimate the non-factorizable QCD effects with the corrections of order $1/N_c$. This relies on the generalized factorization, demonstrated to work well in $B\rightarrow M_1M_2M_3$, $B\rightarrow \mathbf{B}\bar{\mathbf{B}}'$, $B\rightarrow \mathbf{B}\bar{\mathbf{B}}'M(\mathbf{B}\bar{\mathbf{B}}'MM')$, $B\rightarrow D\pi$ and $\Lambda _b\rightarrow \mathbf{B}M(\Lambda _c^+\pi ^-)$ [26, 36, 37, 39, 53,54,55,56,57]. We determine $N_c=(2.15\pm 0.20,1.90\pm 0.03)$ to interpret $\mathcal{B}(\bar{B}^0\rightarrow p\bar{p}\pi ^0,p\bar{p} \pi ^+\pi ^-)$ with $\delta N_c$ receiving the experimental uncertainties, which are indeed close to $N_c\simeq 2$ used in $B\rightarrow \mathbf{B}\bar{\mathbf{B}}'M$ and $\Lambda _b\rightarrow \mathbf{B}M_{(c)}$ [15, 47,48,49].

In Table 2, $\mathcal{B}({\bar{B}}^0\rightarrow p\bar{p}\pi ^0)=5.0\times 10^{-7}$ receives the contributions from $\mathcal{A}_1,\mathcal{A}_2$ and their interference, denoted by $\mathcal{A}_{1\times 2}$, which give $\mathcal{B}(\bar{B}^0\rightarrow p\bar{p}\pi ^0)=\mathcal{B}_1+\mathcal{B}_2+\mathcal{B}_{1\times 2}$ with $(\mathcal{B}_1,\mathcal{B}_2,\mathcal{B}_{1\times 2})=(3.82,0.33,0.85)\times 10^{-7}$. The $\mathcal{B}_{1\times 2}>0$ indicates constructive interference between $\mathcal{A}_{1,2}$. By adopting $N_c$ from $\bar{B}^0\rightarrow p\bar{p}\pi ^0$, we predict $\mathcal{B}(\bar{B}^0\rightarrow p\bar{p}\rho ^0)$. We find $\mathcal{B}(\bar{B}^0\rightarrow p\bar{p}\rho ^0)\approx \mathcal{B}(\bar{B}^0\rightarrow p\bar{p}\pi ^0)/3$ with $(\mathcal{B}_1,\mathcal{B}_2,\mathcal{B}_{1\times 2})=(2.00,0.04,-0.24)\times 10^{-7}$. The minus sign of $\mathcal{B}_{1\times 2}$ indicates destructive interference.

With the theoretical approach reasonably well established for the branching fractions, one can have reliable predictions for CP violation. For example, $\mathcal{A}_{CP}(B^-\rightarrow p\bar{p} M^{(*)-})$ with $M^{(*)-}=(K^{*-},K^-,\pi ^-)$ were predicted as $(22\pm 4,6\pm 1,-6\pm 1)\%$ [43, 44], agreeing with the experimental values of $(21\pm 16,2.1\pm 2.0\pm 0.4,-4.1\pm 3.9\pm 0.5)\%$ [1, 5]. Here, our predictions for $\mathcal{A}_{CP}(\bar{B}^0\rightarrow p\bar{p}\pi ^0(\rho ^0),p\bar{p}\pi ^+\pi ^-)$ are around $-(10-20)\%$. With $\delta \mathcal{A}_{CP}$ denoting the uncertainty for $\mathcal{A}_{CP}$, we present $\delta \mathcal{A}_{CP}\simeq (0.2-0.3)\mathcal{A}_{CP}$, which receives the theoretical uncertainties from the non-factorizable strong interaction, CKM matrix elements, form factors and decay constants.

Expressing the decay amplitude as $\mathcal{A}=T e^{i\delta _W}+Pe^{i\delta _S}$, the CP asymmetry can be derived as

$$\begin{aligned} \mathcal{A}_{CP}=\frac{2R\sin \delta _W \sin \delta _S}{1+2R\cos \delta _W\cos \delta _S+R^2}\,, \end{aligned}$$

(18)

where $\delta _W$ and $\delta _S$ are the weak and strong phases arising from the tree (T) and penguin (P)-level contributions, and the ratio $R\equiv P/T$ suggests that a more suppressed T amplitude is able to cause a more sizeable $\mathcal{A}_{CP}$. Although $\bar{B}^0\rightarrow p\bar{p} X_M$ involves complicated amplitudes, the relation in Eq. (18) can be used as a simple description for $\mathcal{A}_{CP}(\bar{B}^0\rightarrow p\bar{p} X_M)$. Being external and internal W-emission decays, $B^-\rightarrow p\bar{p}\pi ^-$ and $\bar{B}^0\rightarrow p\bar{p}\pi ^0$ proceed with $a_1\sim \mathcal{O}(1.0)$ and $a_2\sim \mathcal{O}(0.2-0.3)$ in the tree-level amplitudes [43, 44], respectively. Consequently, the more suppressed T amplitude with $a_2$ causes more interfering effect with the penguin diagrams, which corresponds to $|\mathcal{A}_{CP}(\bar{B}^0\rightarrow p\bar{p}\pi ^0)|>|\mathcal{A}_{CP}(B^-\rightarrow p\bar{p}\pi ^-)|$. In fact, we predict $|\mathcal{A}_{CP}(\bar{B}^0\rightarrow p\bar{p}\pi ^0)|=(16.8\pm 5.4)\%$, which is three times larger than $|\mathcal{A}_{CP}(B^-\rightarrow p\bar{p}\pi ^-)|$ [43, 44]. For the same reason, $|\mathcal{A}_{CP}(\bar{B}^0\rightarrow p\bar{p}\rho ^0,p\bar{p}\pi ^+\pi ^-)|$ can be as large as $(10-20)\%$.

Since $\mathcal{B}(\bar{B}^0\rightarrow p\bar{p}\pi ^0,p\bar{p}\pi ^+\pi ^-)$ are measured as large as $10^{-6}$, and well explained by the theory, with the predicted $|\mathcal{A}_{CP}|>10\%$, they become promising decays for measuring CP violation. By contrast, $\bar{B}^0\rightarrow p\bar{p}\rho ^0$ as well as the internal W-emission dominated $\Lambda _b$ decays of $\Lambda _b^0\rightarrow n\pi ^0,n\rho ^0$ have $\mathcal{B}\simeq (1-2)\times 10^{-7}$, which make CP measurements a challenge even in the case of large $|\mathcal{A}_{CP}|>10\%$ [49].

In summary, we have investigated the branching fractions and direct CP violating asymmetries of the $\bar{B}^0\rightarrow p\bar{p}\pi ^0(\rho ^0)$ and $\bar{B}^0\rightarrow p\bar{p} \pi ^+\pi ^-$ decays. We have shown that these baryonic B-meson decays mediated dominantly through internal W-emission processes are promising processes to observe for the first time the CP violating effects in B decays to final states with half-spin particles.

With a large predicted CP asymmetry $\mathcal{A}_{CP}=(-16.8\pm 5.4)\%$, which is accessible to the Belle II experiment, $\bar{B}^0\rightarrow p\bar{p}\pi ^0$ is particularly suited for a potential first observation of CP violation in baryonic B decays in the coming years. Furthermore, the $\bar{B}^0\rightarrow p\bar{p}\pi ^+\pi ^-$ decay, with its branching fraction of order $10^{-6}$ and the large predicted direct CP asymmetry $\mathcal{A}_{CP}\sim -(10-20)\%$, is also in the realm of both Belle II and LHCb experiments.