Possible large $CP$ violation in three body decays of heavy baryon

We propose a new mechanism which can introduce large $CP$ asymmetries in the phase spaces of three-body decays of heavy baryons. In this mechanism, a large $CP$ asymmetry is induced by the interference of two intermediate resonances, which subsequently decay into two different combinations of final particles. We apply this mechanism to the decay channel $\Lambda_b^0 \to p \pi^0\pi^-$, and find that the differential $CP$ asymmetry can reach as large as $50\%$, while the regional $CP$ asymmetry can reach as large as $16\%$ in the interference region of the phase space.


Introduction
CP violation is an important phenomenon in particle physics. Although it has been discovered in the mixing and decay processes of K and B meson systems, including the first discovery of CP violation in K system [1], no CP violation was established in the baryon sector, except an evidence in the decay channel Λ 0 b → pK − [2]. Within the Standard Model, CP violation is originated from the weak phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix [3], along with a strong phase which usually arises from strong interactions. One reason for the smallness of CP violation is that the strong phases are usually small, especially when the strong phases come from a scale that is much larger than the QCD scale. However, non-perturbative effects of the strong interaction at low scales provide possibilities for large strong phases, and hence, large CP violation.
Three-body decays of heavy hadrons can be dominated by intermediate resonances in certain regions of the phase space. When two resonances decay into two different combinations of final particles, it is possible for them to dominate in the same region of the phase space. As a result, the interference effect together with a possible large strong phase can generate a large CP asymmetry.

Differential CP asymmetry
It gets more interesting when one applies the aforementioned interference effect to the decay process of heavy baryons. For the decay process Λ 0 b → pπ 0 π − , there is an overlap region in the phase space for resonances ρ − (770) and N + (1440), which lies right in the corner of the phase space. The decay amplitude for Λ 0 b → pπ 0 π − can be expressed as in the overlap region of the phase space, whereĤ eff is the effective Hamiltonian for the weak decays,Ĥ 1 andĤ 2 are the formal Hamiltonian for the strong decays in which the magnitudes of the coupling constants can be determined from experiments, s and s 0 are the invariant mass squares of the systems π 0 π − and pπ 0 , respectively, m ρ , m N , Γ ρ , and Γ N are the masses and decay widths of ρ 0 (770) and N + (1440), respectively, and the summation over the polarizations of the intermediate particles is understood. The effective HamiltonianĤ eff takes the form [4] where G F is the Fermi constant, V ud , V ub , V cd , V cb , V td , and V tb are the CKM matrix elements, c i (c i = 1, · · · , 10) is the Wilson constant, and O i is the four-Fermion operator, which takes the form with d, b, q, and q ′ being quark fields and α and β being colour indices. Under the factorization hypothesis, the weak decay amplitudes can be expressed as where ǫ ρ − is the polarization vector of ρ − , u N and u Λ b are the spinors for N + (1440) and Λ b , respectively, with f π being the decay constant of the pion, F Λ b →N + and F Λ b →p being the form factors for the transition Λ b → N + (1440) and Λ b → p, respectively, and a i = c i + c i−1 /N c for even i.
Because of the non-purterbative effects of strong interactions, there can be a relative strong phase between the coupling constants ofĤ 1 andĤ 2 . We will denote this relative phase by δ and treat it as a free parameter. The strong decay amplitudes are then expressed as and respectively, where the effective coupling constants g 1 and g 2 can be expressed as with m p being the mass of proton, Γ N + →N π and Γ ρ − →π 0 π − being the partial decay widths for N + (1440) → N (939)π and ρ − (770) → π 0 π − , respectively, and The differential CP asymmetry is then defined as whereM is the decay amplitude of the CP conjugate process, Λ 0 b → pπ + π 0 , and the overlines above |M | 2 and M 2 represent averaging and summing over the spin states of the initial and final particles, respectively. After some algebra, one has where s − is the invariant mass squared of the system pπ − , and In order to obtain the expression for M 2 , all one needs to do is to replace the CKM matrix elements in Eq. (15) with their complex conjugates.
In order to see where the CP asymmetry arises, let's first display the weak and strong phases in λ 1 and λ 2 explicitly. For a fixed point in the phase space, λ 1 and λ 2 can be expressed as where i = 1, 2, λ Tree and , and a, b, c, and d are real quantities. One can see from Eq. (21) that both the strong and weak phases are essential for CP violation. The difference in Eq. (21) is proportional to the sine of the difference of the tree and penguin weak phases. Besides, the four terms in Eq. (21) are also proportional to the sine of the differences of the tree and penguin strong phases, respectively.
In Fig. 1, we present the differential CP asymmetry distribution in the overlap region of the phase space for various values of δ, from which one can see clearly that the interference effect of the two aforementioned resonances does result in a CP asymmetry in the overlap region of the phase space.
Among the input parameters, the Wilson coefficients are taken from Ref. [4], and the rest of the input parameters are from Particle Data Group [5]. In the heavy quark limit and for large recoil final light baryon, both of the from factors for Λ b → p and Λ b → N + (1440) transitions reduce to a single form factor [6]. Since the structure of N + (1440) is still not clear, the decay form factors for the transition Λ b → N + (1440) is not available. Therefore, we set in Fig. 1 the heavy-to-light baryonic tradition form factors equal to each other, i.e.
One interesting behaviour for the differential CP asymmetry in Fig. 1 is that it can be as large as 50% in the overlap region of the phase space. Besides, one can see that the differential CP asymmetry shows large anisotropic behaviour in the overlap region, especially when s is away from the vicinity of ρ − (770) ( √ s > m ρ + Γ ρ ). The reason for this behaviour is that the amplitude of Λ b → pρ − → pπ 0 π − is larger than that of Λ b → N + (1440)π − → pπ 0 π − when s and s 0 are close to m 2 ρ − and m 2 N + , respectively.

Regional CP asymmetry
In order to compare with future experiments, one has to consider the regional CP asymmetry, which can be defined as where Ω is some region of the phase space, Γ Ω andΓ Ω are the regional decay width for Λ 0 b → pπ 0 π − and Λ 0 b → pπ 0 π + , respectively, with the former one taking the form We will focus on an overlap region of the phase space which satisfies m ρ + Γ ρ < √ s < In order to exclude these resonances, we have to keep close to the vicinity of N + (1440). Secondly, it is understandable that the contribution of the interference effect becomes more significant when the two amplitudes in Eq. (1) are comparable. One can easily check that the first term is much larger than the second one when both of the resonances ρ − (770) and N + (1440) are on the mass shell. Consequently, we choose the region Ω OL around the vicinity of N + (1440) but a bit further away from the vicinity of ρ − (770).
In Fig. 2, we present the regional CP asymmetries in Region Ω OL as a function of the strong phase δ. The three curves in Fig. 2 correspond to the CP asymmetries in Region Ω OL for F Λ b →p /F Λ b →N + = 0.5, 1, and 2, respectively, indicating that the regional CP asymmetry is sensitive to the form factors. From Fig. 2, one can see that the interference of the decay amplitudes corresponding to the intermediate resonances ρ ∓ (770) and N ± (1440), together with proper strong phase δ, does result in the regional CP asymmetry in the interference region of phase space. Especially, the regional CP asymmetry in Region Ω OL can reach as large as ±16% when F Λ b →p /F Λ b →N + = 0.5.
To conclude, we want to point out that the interference of ρ(770) with other baryonic resonances such as N (1520) can also result in CP asymmetries, given that the decay amplitudes corresponding to these baryonic resonances are comparable with that corresponding to ρ(770) in the interference regions. Besides, there can also be regional CP asymmetries induced by the interference between the amplitudes corresponding to nearby baryonic resonances, for example, N (1440) and N (1520). Similar behaviour to the latter one has already been proposed in B meson decay processes [7] and observed by the LHCb collaboration [8]. in Hunan province, and the Innovation Team of Nuclear and Particle Physics of USC.