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

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 B¯0→pp¯π0(ρ0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{B}^0\rightarrow p\bar{p}\pi ^0(\rho ^0)$$\end{document} and B¯0→pp¯π+π-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{B}^0\rightarrow p\bar{p}\pi ^+\pi ^-$$\end{document} decays. We obtain B(B¯0→pp¯π0)=(5.0±2.1)×10-7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{B}(\bar{B}^0\rightarrow p\bar{p}\pi ^0)=(5.0\pm 2.1)\times 10^{-7}$$\end{document}, in agreement with current data, and B(B¯0→pp¯ρ0)≃B(B¯0→pp¯π0)/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{B}(\bar{B}^0\rightarrow p\bar{p}\rho ^0)\simeq \mathcal{B}(\bar{B}^0\rightarrow p\bar{p}\pi ^0)/3$$\end{document}. Furthermore, we find ACP(B¯0→pp¯π0,pp¯ρ0,pp¯π+π-)=(-16.8±5.4,-12.6±3.0,-11.4±1.9)%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{A}_{CP}(\bar{B}^0\rightarrow p\bar{p}\pi ^0,p\bar{p}\rho ^0,p\bar{p}\pi ^+\pi ^-) =(-16.8\pm 5.4,-12.6\pm 3.0,-11.4\pm 1.9)\%$$\end{document}. With measured branching fractions B(B¯0→pp¯π0,pp¯π+π-)∼O(10-6)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{B}(\bar{B}^0\rightarrow p\bar{p}\pi ^0,p\bar{p}\pi ^+\pi ^-)\sim \mathcal{O}(10^{-6})$$\end{document}, we point out that ACP∼-(10-20)%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{A}_{CP}\sim -(10-20)\%$$\end{document} can be new observables for CP violation, accessible to the Belle II and/or LHCb experiments.


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( / ), 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]. a e-mail: yukuohsiao@gmail.com b e-mail: shangyuu@gmail.com (corresponding author) c e-mail: eduardo.rodrigues@cern.ch 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 ( * ) , π, ρ, D ( * ) , the B → ppM ( * ) 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 → ppK * , Dalitz plot information, and pp (M ( * ) p) invariant mass spectra. This helps to improve the theoretical understanding of the di-baryon production in B → BB M [12][13][14][15][16], such that the data can be well interpreted. Predictions are confirmed by recent measurements. For example, one obtains B(B 0 s → p¯ K − + pK + ) = (5.1±1.1)×10 −6 [17], in excellent agreement with the value of (5.46 ± 0.61 ± 0.57 ± 0.50 ± 0.32) × 10 −6 measured by LHCb [18]. Moreover, the theoretical extension to four-body decays allows to interpret B(B 0 → ppπ + π − ) [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 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 ofB 0 → ppπ 0 (ρ 0 ), ppπ + π − , and predict their direct CP violating asymmetries.

Formalism
For the tree-level dominated B meson decays, the relevant effective Hamiltonian is given by [23] where G F is the Fermi constant, c i( j) the Wilson coefficients, and V i j the CKM matrix elements. The four-quark operators O i( j) for the tree (penguin)-level contributions are written as where q = (u, d, s), (q 1 q 2 ) V ±A =q 1 γ μ (1 ± γ 5 )q 2 , and the subscripts with T a the Gell-Mann matrices.
In the factorization ansatz [24,25], one is able to express h 1 h 2 |O|B as a product of two factors, h 1 |J 1 |0 and h 2 |J 2 |B , where O = J 1 · 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 h 1 |J 1 |0 and h 2 |J 2 |B 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 ppπ 0 |O 2 |B 0 as [15,16] where the Fierz reordering has been used to exchange (d α ,ū β ). The amplitudes O 2 a,d correspond to the two configurations depicted in Fig. 1a, d, respectively. As depicted in Fig. 2 for the b → uūd transition, dynamically, the dquark moves collinearly with the spectator quarkd from B 0 (bd), so that in Fig. 1d the dd for the pp formation can be seen as a consequence of the B meson transition, which is in accordance with the matrix element of pp|(db)|B 0 . Moreover, since uū and dd in theB 0 rest frame can be seen to move in opposite directions, we take π 0 (uū) in Fig. 1d as the recoiled state, in accordance with π 0 |(ūu)|0 with |0 representing the vacuum. On the other hand, The T a in χ 1 correspond to the gluon exchange between the two currents, which causes an inseparable connection between the final states. Hence, χ 1 is regarded as the non-factorizable QCD corrections. Subsequently, we note that ppπ 0 where c e f f i 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 χ 1 [15,24,25]. This makes N c a phenomenological parameter determined by data.

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 The decay constants are f π,ρ = (130.4 ± 0.2, 210.6 ± 0.4) MeV [1], with ( We adopt the B → 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 ∞. With the nearly universal value for N c in the specific decays, the 5.0 ± 1.9 ± 0.3 ± 0.9 5 .0 ± 1.9 [22] 10 7 B(B 0 → ppρ 0 ) 1.8 ± 1.1 ± 0.1 ± 0.4 - factorization is demonstrated to be valid. For the tree-level internal W -emission dominated b-hadron decays, the extraction has given N c 2 that corresponds to [15,20,[45][46][47][48][49], where δ N c differs due to the experimental uncertainties. For example, one obtains N c = 2.15 ± 0.17 in b → BM c [47,48]. Here, we test if N c 2 can be used to explain the measured B(B 0 → ppπ 0 , ppπ + π − ).
The C h,w − forB 0 → π + π − and C i (D i ) for 0 → pp (B 0 → pp) have been determined to be [15,17,20] For α i in Eq. (7), the effective Wilson coefficients c e f f 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 A C P .
Our results for the branching fractions and CP violating asymmetries ofB 0 → ppX M decays are summarized in Table 2, where we have averaged the particle and antiparticle contributions for the total branching fractions.
Expressing the decay amplitude as A = T e iδ W + Pe iδ S , the CP asymmetry can be derived as where δ W and δ S are the weak and strong phases arising from the tree (T ) and penguin (P)-level contributions, and the ratio R ≡ P/T suggests that a more suppressed T amplitude is able to cause a more sizeable A C P . Although B 0 → ppX M involves complicated amplitudes, the relation in Eq. (18) can be used as a simple description for A C P (B 0 → ppX M ). Being external and internal Wemission decays, B − → ppπ − andB 0 → ppπ 0 proceed with a 1 ∼ O(1.0) and a 2 ∼ 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 |A C P (B 0 → ppπ 0 )| > |A C P (B − → ppπ − )|. In fact, we predict |A C P (B 0 → ppπ 0 )| = (16.8 ± 5.4)%, which is three times larger than |A C P (B − → ppπ − )| [43,44]. For the same reason, |A C P (B 0 → ppρ 0 , ppπ + π − )| can be as large as (10 − 20)%. Since B(B 0 → ppπ 0 , ppπ + π − ) are measured as large as 10 −6 , and well explained by the theory, with the predicted |A C P | > 10%, they become promising decays for measuring CP violation. By contrast,B 0 → ppρ 0 as well as the internal W -emission dominated b decays of 0 b → nπ 0 , nρ 0 have B (1 − 2) × 10 −7 , which make CP measurements a challenge even in the case of large |A C P | > 10% [49].
In summary, we have investigated the branching fractions and direct CP violating asymmetries of theB 0 → ppπ 0 (ρ 0 ) andB 0 → ppπ + π − 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 A C P = (−16.8 ± 5.4)%, which is accessible to the Belle II experiment,B 0 → ppπ 0 is particularly suited for a potential first observation of CP violation in baryonic B decays in the coming years. Furthermore, theB 0 → ppπ + π − decay, with its branching fraction of order 10 −6 and the large predicted direct CP asymmetry A C P ∼ −(10 − 20)%, is also in the realm of both Belle II and LHCb experiments.