Establishing CP violation in $b$-baryon decays
Ji-Xin Yu Jia-Jie Han Ya Li Hsiang-nan Li Zhen-Jun Xiao Fu-Sheng Yu









^1MOE Frontiers Science Center for Rare Isotopes, and School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000,   People’s Republic of China 

^2Department of Physics, College of Sciences, Nanjing Agricultural University, Nanjing 210095, People’s Republic of China


^3Institute of Physics, Academia Sinica, Taipei, Taiwan 115, Republic of China


^4Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing 210023, People’s Republic of China


§ ABSTRACT 
The CP violation (CPV) in the baryon system has not yet been definitively established. We demonstrate that individual partial-wave CPV in the Λ_b→ pπ^-,pK^- decays can exceed 10%, but the destruction between different partial waves results in small net direct CPV as observed in current experiments. There is thus high possibility of identifying CPV in b-baryon decays through measurements of partial-wave CPV. The above observation is supported by the first full QCD calculation of two-body hadronic Λ_b baryon decays with controllable uncertainties in the perturbative QCD formalism. 



xxxx

Establishing CP violation in b-baryon decays
     Ji-Xin Yu^1, 
Jia-Jie Han^1 [Corresponding author, Email: hanjj@lzu.edu.cn],
Ya Li^2 [Corresponding author, Email: liyakelly@163.com],
Hsiang-nan Li^3,
Zhen-Jun Xiao^4,
Fu-Sheng Yu^1 [Corresponding author, Email: yufsh@lzu.edu.cn] 
    
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Introduction.—
The CP violation (CPV) plays a crucial role in explaining the matter-antimatter asymmetry in the Universe and in searching for New Physics. 
The CPVs in K<cit.>, B<cit.> and D<cit.> meson decays, which are attributed to an irreducible phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix, have been well established and found to be consistent with Standard Model (SM) predictions. By contrast, the CPV in the baryon system has not yet been identified, and 
numerous experiments have been conducted to search for baryon CPV. Recent efforts by BESIII yielded the most precise hyperon decay asymmetry A_CP^α (Λ→ p π^-) = -0.002±0.004<cit.>. LHCb  achieved the most precise measurement of CPV in charm baryon decays, A_CP (Λ_c→ pK^+K^-) - A_CP (Λ_c→ pπ^+π^-) = 0.003± 0.011<cit.>. Nevertheless, the SM predictions for CPVs in hyperons and charm baryons are one or two orders of magnitude lower than current experimental sensitivities.







Bottom hadron decays involving a relatively large weak phase allow CPV at order of 10%, which has been confirmed in B meson decays. On the contrary, measurements of CPV in two-body Λ_b baryon decays gave<cit.>

    A_CP(Λ_b→ pπ^-)   =-0.025±0.029,
    
    	A_CP(Λ_b→ pK^-)   =-0.025±0.022,

compatible with null asymmetries within precision of 1%. That is,
The CPV in Λ_b baryon decays is much lower than in similar B meson decays, although both are induced by the b→ uu̅q transition, q=d,s.
The discrepancy remains a puzzle in heavy flavor physics. 

It seems that the dynamics in baryon and meson processes differs significantly, but there is a lack of convincing explanations for this distinction. As a consequence, CPV in other baryon decay modes cannot be predicted accurately either.

A Λ_b baryon decay is a multi-scale process, and involves more diagrams owing to an additional spectator quark compared to a B meson decay. 
This results in lots of W-exchange topological diagrams and abundant sources of strong phases required for direct CPV.
A precise evaluation of the strong phases in these topological diagrams poses a challenge in theory. 

Three popular theoretical approaches to studies of two-body hadronic B meson decays have been developed, known as the QCD 
factorization (QCDF)<cit.>, the soft-collinear-effective theory (SCET)<cit.> and the perturbative QCD (PQCD) factorization<cit.>. 
The QCDF and SCET are based on the collinear factorization theorem, in which B meson transition form factors develop an endpoint singularity if they were computed perturbatively. 
The PQCD is based on the k_T factorization theorem, in which the endpoint contribution is absorbed into a transverse-momentum-dependent distribution amplitude (DA) or resummed into a Sudakov factor.

The factorizable and nonfactorizable emission, W-exchange and annihilation diagrams are calculable in this framework free of the endpoint singularities.
The CPV of two-body hadronic B meson decays has been successfully predicted in PQCD<cit.>.
Recently, the Λ_b→ p transition form factors with reasonable high-twist hadron DAs are reproduced in PQCD, and the results agree with those from lattice QCD and other nonperturbative methods<cit.>.
Various exclusive heavy baryon decays can thus be analyzed systematically.

We will extend the above well-established PQCD formalism to hadronic Λ_b decays.
Our full QCD calculation, including all the factorizable and nonfactorizable topological diagrams, demonstrates the presence of large partial-wave CPV, greater than 10%, in the Λ_b→ pπ^- decay. This amount is close to that in the corresponding B meson decay, but the cancellation between different partial waves turns in small net direct CPV.
The P-wave CPV in the penguin-dominant Λ_b→ pK^- decay can also exceed 10%. However, its CPV is governed by the S-wave, which is only at the percent level.
We further predict the CPVs in the Λ_b→ pρ^-, pK^∗ -, pa_1^-(1260), pK_1^-(1270) and  pK_1^-(1400) decays, examining their partial-wave CPVs. Overall speaking, the partial-wave CPV can reach 10%. Our investigation sheds light on the dynamical distinction between CPVs in bottom baryon and meson decays, and suggests high possibility of detecting baryon CPV through partial-wave CPV measurements.











Λ_b decay in the PQCD.—
Unlike meson decays, the decay amplitude of a baryon with non-zero spin is decomposed into two different structures. 
For the Λ_b→ p h decays, h=π^-, K^-, the amplitudes can be expressed as,

    ℳ(Λ_b→ ph)=iu̅_p (f_1+f_2γ_5)u_Λ_b.

where u_p and u_Λ_b represent the proton and Λ_b baryon spinors, respectively. 
The partial-wave amplitudes f_1 and f_2 correspond to the parity-violating S-wave and parity-conserving P-wave, associated with the terms 1 and γ_5, respectively.



The partial-wave amplitudes f_1,2 receive contributions from tree operators and penguin operators,

    f_1   =|f_1^T|e^iϕ^Te^iδ_1^T + |f_1^P|e^iϕ^Pe^iδ_1^P,
    
    		f_2   =|f_2^T|e^iϕ^Te^iδ_2^T + |f_2^P|e^iϕ^Pe^iδ_2^P,

where the superscripts T,P denote the tree and penguin contributions,
the weak phase ϕ from the CKM matrix takes the same value for the S- and P-waves, and
the strong phase δ varies with different partial-wave amplitudes. The direct CPV in the Λ_b→ pπ^-,pK^- decays is then defined as

    A_CP(Λ_b→ ph)≡Br(Λ_b→ ph) - Br(Λ̅_b→p̅h̅)/Br(Λ_b→ ph) + Br(Λ̅_b→p̅h̅)
       =-2{A|f_1^T|^2 r_1 sinΔϕsinΔδ_1 + B|f_2^T|^2 r_2 sinΔϕsinΔδ_2 }
       / {A|f_1^T|^2 (1+r_1^2+2r_1 cosΔϕcosΔδ_1) 
       + B|f_2^T|^2 (1+r_2^2+2r_2 cosΔϕcosΔδ_2)}.

Here r_1,2≡ |f_1,2^P|/|f_1,2^T| denote the ratios of penguin over tree contributions, A=((M_Λ_b+M_p)^2-M_h^2)/M_Λ_b^2, B=((M_Λ_b-M_p)^2-M_h^2)/M_Λ_b^2, Δϕ≡ϕ^P-ϕ^T, Δδ_1,2≡δ^P_1,2-δ^T_1,2.

A strong phase arises from the on-shellness of internal particles in Feynman diagrams, which differs between the parity-conserving and parity-violating contributions. This allows us to define the partial-wave CPV,

    A_CP^S   =-2r_1 sinΔϕsinΔδ_1/1+r_1^2+2r_1cosΔϕcosΔδ_1,
    
    		A_CP^P   =-2r_2 sinΔϕsinΔδ_2/1+r_2^2+2r_2cosΔϕcosΔδ_2.




In the PQCD framework, a decay amplitude is expressed as a convolution of hadron DAs, hard scattering amplitudes H, Sudakov factors and jet functions as described in Fig. <ref>, and 
formulated as 

    ℳ(Λ_b→ ph)=∫_0^1[dx] [dx^'] dy ∫ [db] [db^'] db_q
       H([x],[x^'],y,[b],[b^'],b_q,μ) S_t([x],[x^'],y)
       ϕ_Λ_b([x],[b],μ) ϕ_p([x^'],[b^'],μ) ϕ_h(y,b_q,μ)
       e^-S_Λ_b([x],[b])  e^-S_p([x],[b^'])  e^-S_h(y,[b_q]).

The hadron DAs are inputted from Refs. <cit.> for the Λ_b baryon, Refs. <cit.> for the proton and Refs. <cit.> for the pesudoscalar mesons.



Compared to meson decays, more types of topological diagrams contribute to the Λ_b→ pπ^-,pK^- decays. The exchange of two hard gluons is necessary for H at leading order in α_s to ensure the two light spectator quarks in the Λ_b baryon to form the energetic final state. A typical diagram responsible for the Λ_b→ pπ^- decay is displayed in Fig. <ref>. We evaluate the contributions from all diagrams to the Λ_b→ pπ^-,pK^- decays, and summarize the outcomes in Table. <ref> and <ref>, respectively. For clarity, we list only central values.











Discussion.— 
Tables. <ref> and <ref> manifest the hierarchy r_1 ≫ r_2 in the Λ_b → ph decays, where the contributions from the factorizable penguin diagrams P_f^C_1 dominate. 

The S- and P-wave amplitudes P_f^C_1 are expressed as

    f_1(P_f^C_1)=   -G_F/√(2)f_hV_tbV_td^∗ (C_3/3+C_4+ C_9/3+C_10
       +R_1^h(C_5/3+C_6+C_7/3+C_8))
       [F_1(m_h^2)(M_Λ_b-M_p)+F_3(m_h^2)m_h^2]
    
        f_2(P_f^C_1)=   -G_F/√(2)f_hV_tbV_td^∗ (C_3/3+C_4+ C_9/3+C_10
       -R_2^h(C_5/3+C_6+C_7/3+C_8))
       [G_1(m_h^2)(M_Λ_b+M_p)-G_3(m_h^2)m_h^2]











where the form factors F_1,2,3 and G_1,2,3 are defined in terms of ⟨ p|u̅γ_μ b |Λ_b⟩ = p̅(F_1γ_μ +F_2iσ_μνq^ν + F_3q_μ)Λ_b and ⟨ p|u̅γ_μγ_5 b |Λ_b⟩ = p̅(G_1γ_μ +G_2iσ_μνq^ν + G_3q_μ)γ_5Λ_b, and the chiral factors are given by R_1=2m_h^2/[(m_b-m_u)(m_u+m_q)] and R_2=2m_h^2/[(m_b+m_u)(m_u+m_q)] with R_1^π≈ R_2^π≈ 1.01 and R_1^K≈ R_2^K≈ 0.89 .
Since the negative sign of R_2 in Eq. (<ref>) induces cancellations among different Wilson coefficients, the term f_2(P_f^C_1) and the ratio r_2 are suppressed.





The calculated branching fractions and CPVs of the Λ_b→ pπ^-,pK^- decays are presented in Table. <ref>.

It is worth mentioning that the magnitudes of CPV are small, consistent with the experimental measurements.

Note that the partial-wave CPV of the Λ_b→ pπ^- decay can exceed 10%, similar to those in B meson decays. 
However, the opposite signs of the partial-wave contributions leads to the small direct CPV in this mode. 

The topology P^C^', which contains
40 Feynman diagrams, gives the most significant penguin contributions. Among these Feynman diagrams, Fig. <ref> is the largest, whose strong phases exhibit an almost 180^∘ difference between the S- and P-wave as indicated in Table. <ref>. 



For the Λ_b→ pK^- mode, the ratios 
r_1=4.94 and r_2=0.33 imply that
the direct CPV is determined by the S-wave. 
Unlike the Λ_b→ pπ^- decay, the Λ_b→ pK^- decay lacks the P^C^' topology, such that the total penguin contributions are dominated by the factorizable penguin diagrams. These diagrams generate a small strong phase difference for the S-wave, i.e., a small S-wave CPV A_CP^S(Λ_b→ pK^-)=-0.05, and consequently a small direct CPV.

As indicated in Table <ref>, the partial-wave CPV can be large in magnitude, with A_CP^S(Λ_b→ pπ^-)=0.17 and A_CP^P(Λ_b→ pK^-)=-0.23. These large partial-wave CPVs closely resemble the corresponding processes in B meson decays.
The partial-wave CPVs of baryon decays are directly related to the asymmetry parameters α, β and γ <cit.>, which can be probed experimentally to search for baryon CPVs.
Table <ref> also provides our predictions for the decay asymmetry parameters and their associated CPVs for further measurements at LHCb.


The cancellation between partial-wave CPVs as a differentiation between b-baryon and b-meson decays is the main highlight of the Letter. 
In order to explore the potential enhancements of partial-wave CPVs, we have also analyzed the decays Λ_b→ pρ^-,pK^∗ - with vector final states, and Λ_b→ pa_1^-(1260),pK_1^-(1270),pK_1^-(1400) with axial-vector final states in the PQCD approach. 
These modes involve four independent partial-wave amplitudes or helicity amplitudes. 
They share the same topological diagrams as the Λ_b→ pπ^-,pK^- decays, but with different meson DAs.




The predictions for the CPVs in the above decays are shown in Table. <ref>. 
It is found that the CPVs of Λ_b→ pρ^-,pa_1^-(1260) are small, while the others are relatively large.
These modes are actually three-body or four-body decays Λ_b→ p π^- π^0, p K_S^0 π^- or p K^- π^0, p π^+ π^- π^-, and p K^- π^+ π^-,  all of which have large data sample at LHCb; the three-body decays have about 4000 events, and the four-body decays have about 20000 and 90000 events, respectively. 



Furthermore, multi-body decays through two or more intermediate resonances may produce substantial interference effects, resulting in notable regional CPVs. Hence, there is a big chance to observe CPVs higher than 20% in these modes at LHCb. The rich data samples and complicated dynamics in multi-body decays offer promising opportunities to establish CPVs in bottom baryon decays.


Conclusions.— 
This Letter presented the first full QCD dynamical analysis on two-body hadronic Λ_b baryon decays in the PQCD approach. Our study elucidates the reason for the observed small CPVs in the Λ_b→ pπ^-,pK^- decays, in contrast to the sizable CPVs in the similar B meson decays. The partial-wave CPVs in the Λ_b→ pπ^- decay could reach 10% potentially, but the destruction between them leads to the small CPV. The direct CPV of the Λ_b→ pK^- mode is primarily attributed to the modest S-wave CPV.
We have also extended our analysis by investigating the CPVs in the channels with vector and axial-vector final states. 
Our predictions suggest that certain partial-wave CPVs in bottom baryon decays can be large enough, and probed experimentally to search for baryon CPVs. This work opens up avenues for deeply understanding the dynamics involved in baryon decays and for unveiling CPV in these processes.



Acknowledgement.—The authors would like to express their gratitude to Pei-Rong Li for generously providing an access to computing resources. Special thanks are extended to Ding-Yu Shao, Yan-Qing Ma, Jian Wang and Jun Hua for their valuable comments. This work was supported in part by Natural Science Foundation of China under
grant No. 12335003, and by the Fundamental Research Funds for the Central Universities under No. lzujbky-2024-oy02. 







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