A novel strategy for searching for $C\!P$ violations in the baryon sector

Despite the large baryon-anti-baryon asymmetry in the observable Universe, the closely related phenomenon -- the violation of the combined charge and parity symmetry ($C\!P$V) -- has not been observed in the baryon sector in laboratories. In this paper, a new strategy for searching for $C\!P$V in heavy hadron multi-body decays is proposed, in which a set of novel observables measuring $C\!P$V in such decays -- the partial wave $C\!P$ asymmetries (PW$C\!P$As) -- are introduced. This strategy is model-independent and applicable to multi-body decays of heavy hadrons with arbitrary spin configurations in both initial and final states, and with any number of particles in the final state. It is especially applicable for $C\!P$V investigations in multi-body decays of heavy baryons. As applications of this strategy, we suggest to measure the PW$C\!P$As in some decay channels of bottom baryons such as $\Lambda_b^0\to p \pi^-\pi^+\pi^-$, $\Lambda_b\to p K^- \pi^+\pi^-$, $\Lambda^0_b\to p\pi^-K^+K^-$, $\Lambda_b^0\to\Lambda K^+\pi^-$, and $\Lambda_b^0\to p \pi^- K_s$.}


Introduction
The violation of the symmetry of the combined charge-parity (CP ) transformation, as a phenomenon closely related to the matter-antimatter asymmetry in the Universe [1], was first discovered in the neutral kaon system [2], and is accommodated in the Standard Model (SM) of particle physics by the Cabibbo-Kobayashi-Maskawa (CKM) mechanism that describes the transitions between up-and down-type quarks [3,4]. CP violation (CP V) in hadron decays is described by the asymmetry between the particle and antiparticle decay rates, which has been observed in many decay channels of heavy mesons [5][6][7][8][9]. On the other hand, despite the large baryon-anti-baryon asymmetry in the observable Universe, no CP V has ever been observed in the baryon sector in laboratories [10][11][12][13][14][15].
One reason for the negative result in searching for CP Vs in the baryon sector is the substantially lower statistics compared with the meson case. One of the latest examples for the CP V measurements in bottom baryon decays can be found in Ref. [16], from which one can clearly see that the statistics for the bottom baryon case is indeed much lower than that in the bottom meson case (see, for example, Ref. [17]). So although CP asymmetries (CP As) of a few or several tens of percent is expected in some decay channels of heavy baryons [18], no significant evidence of CP V has yet been found experimentally in the baryon sector.
The rich resonance structures in multi-body decays of heavy hadrons provide us opportunities for CP V searching in these decay channels, as the interference between different intermediate resonances may result in large regional CP As (rCP As) distributed in the phase space, although the overall CP A may be small due to cancellations among different parts of the phase space. In fact, it has been observed that for some three-body decay channels of B mesons such as B + → π + π − π + and B + → K + π − π + , very large rCP As are localized in certain small parts of the phase space, thanks to the relatively larger statistics [8]. Some of these rCP As can indeed be explained by the interference between a resonance and another one nearby via the introduction of a non-perturbative relative strong phase between their corresponding amplitudes [19][20][21]. Large rCP As associated with the interference between close resonances are also expected in multi-body decays of heavy baryons. However, the aforementioned low statistics problem leads to the direct measurements of the rCP As for multi-body decays of heavy baryons suffering from large uncertainties and hence no CP V has been established in heavy baryon decays. Even though one can partly fix the low statistic issue by merging small bins in the phase space region into larger ones, the rCP As of the enlarged bins are usually much smaller since there are almost always cancellations among these aforementioned small bins when obtaining the rCP As of the enlarged bins. Therefore, besides the direct measurements of the rCP As or the overall CP As, a more efficient and systematic method for CP V hunting is thus urgent for multi-body decays of heavy baryons.
Motivated by the above discussion, in this paper, we will introduce a set of new CP V observables -the Partial Wave CP Asymmetries -associated with the intermediate resonances in the multi-body decays of heavy hadrons. It will be shown that the Partial Wave CP Asymmetries can make use of the data in a more efficient way, and thus, can potentially overcome the low statistics problem associated with the heavy baryon decays. It is possible that with this new strategy, the establishment of CP V in the baryon sector may be hastened.
This paper is organized as follows. In Sec. 2, we introduce the set of observables, the Partial Wave CP Asymmetries, and discuss its measurement method and its advantages comparing with other observables or techniques for CP V measurements. In Sec. 3, we focus mainly on the situation of the interference of nearby resonances, in which the nonperturbative effect of strong interaction could amplify the CP V signal. We briefly discuss the correlation of the underline dynamics for CP V with this newly introduced observables for this situation. An set of important Selection Rules which tell weather the interference effect are present or absent in certain Partial Wave CP Asymmetries are also presented. In Sec. 4, we apply the newly introduced observables to the four-body decay process, Λ 0 b → pπ − π + π − . In the last section, we briefly give the conclusion.

Partial Wave CP Asymmetries
Without lose of generality, consider a multi-body decay, H → h 1 h 2 h 3 · · · h n , where H is a heavy hadron, while h k (k = 1, 2, 3, · · · , n) are light ones. We will focus on the region of the phase space where an intermediate resonance R a dominates, i.e., the invariant mass squared of the h 1 h 2 system, s 12 , lies between (m Ra − α Ra ) 2 and (m Ra + β Ra ) 2 , with m Ra the mass of R a , and α Ra and β Ra of the same other as the decay width of R a . Then, this multi-body decay will be dominated by the cascade decay H → R a h 3 · · · h n , R a → h 1 h 2 . Assuming that the initial particle H is produced unpolarized, the differential decay width can then be expressed as dΓ ∝ |M| 2 dc θ * 1 , (2.1) after summing over the helicities (or the z-components of the spins) of both the initial and the final particles, and integrating over the phase space variables -s 12 is integrated from (m Ra − α Ra ) 2 to (m Ra + β Ra ) 2 for the phase space integration-except c θ * 1 , where θ * 1 is the relative angle between the momentum of h 1 and that of H in the rest frame of the h 1 h 2 system, and c θ * 1 is its cosine, |M| 2 is the spin-averaged and phase space-integrated (except c θ * 1 ) square of the decay amplitude, which can be further expanded with the aid of the Legendre's polynomials as where the coefficients w (j) can be expressed as according to the orthogonality properties of P j s. Since this decay process is a weak one, CP V may show up as the difference between w (j) andw (j) , wherew (j) corresponds to w (j) for the CP conjugate processH →h 1h2h3 · · ·h n . One can then introduce a set of observables, which will be called as the Partial Wave CP Asymmetries (PWCP As), and for the jth-wave: The PWCP As are experiment-friendly observables, which can be extracted easily by fitting the data with Eq. (2.2). Another equivalent but more straightforward method is described as follows. According to the orthogonality property of P j one can see that different events collected in the data do not contribute to w (j) equally. Instead, they contribute with a weight which is proportional to P j (c θ * 1 ). For an event labeled as k, one can easily determine its corresponding c θ * 1 , which will be denoted as c θ * 1,k . Accordingly, one can introduce the so called P j -weighted event yield, which is proportional to w (j) . 1 The P j -weighted event yield for the CP conjugate process N j−weighted can be obtained in the same way. With the above P j -weighted event yields N j−weighted andN j−weighted , one can easily obtain the experimental values of the PWCP As through From the measurement methods proposed above, one can see that with the newly introduced observables PWCP As one can make use of the data in a more efficient way, comparing with the rCP As which are widely measured in multi-body decays of B mesons 1 Equivalently, to obtain N j−weighted , one can also firstly divide the range of c θ * 1 , from −1 to 1, into M small intervals, with each interval corresponding to an average value of c θ * 1 , which will be denoted as c θ * 1,m , (m = 1, 2, · · · , M ). Secondly, count the event number in each interval, which will be denoted as Nm for the interval m. Then, one can easily see that the Pj-weighted event yields N j−weighted can be expressed as . This method will be used to obtain the PWCP A in the simulation in Sec.

4.
currentle. This is because instead of dividing the value range of c θ * 1 into small bins and making use of only the events in each bins when measuring the rCP As, each of the PWCP As makes use of all the events distributed in the whole range of c θ * 1 from −1 to 1. This can be seen clearly when the integration over c θ * 1 from −1 to 1 is performed based on the orthogonality properties of P j , indicating the usage of the events over the whole range of c θ * 1 . This can also be seen in more detail from the measurement method proposed above. Making use of all the events distributed in the whole range of c θ * 1 is very important when statistics is not large enough, as in the case of heavy baryon decays. Another important feature of the PWCP As which can be seen from the above discussion is that, they provide a more efficient model-independent approach for hunting for CP V than the currently used amplitude analysis technique [22] in multi-body decays of heavy hadrons, in which model dependence could potentially arise [23].
3 Contribution of the interference of nearby resonances to PWCP A: underlining dynamics and Selection Rules CP is nothing but the CP A parameter for the decay H → R a h 3 · · · h n . 2 However, CP V is usually relatively small in the single-resonance-dominance situation. Hence, interference with a second resonance is required to generate larger CP V. In this situation, the decay amplitude can then be approximated by the sum of the two cascade decays, In this situation, the PWCP As are no longer just CP V observables for H → R a h 3 · · · h n . Instead, they will also contain contribution from the resonance R b , and moreover, that from the interference between the two resonances R a and R b . The interference of nearby resonances has great impact on CP V in multi-body decays of bottom hadrons, either through rCP As, as has been observed in some three-body decays of B meson, and/or through PWCP As, which will be seen in more details in what follows.
It is crucially important to find out the correlation between the origin of CP V in the PWCP As, A (j) CP , and the underlining dynamics. To achieve this, one needs to substitute the decay amplitudes into |M| 2 . After some algebra, one arrive at the expression of w (j) as where s Ra/R b are the denominators of the Breit-Wigner propagators and take the form the notation " · · · " represents the integral over the phase space except c θ * 1 , and CP is also the rCP A for the whole Ra-dominated phase space region, (mR a − αR a ) 2 < s12 < (mR a + βR a ) 2 .
where X, Y = a, b, j a/b are the spins of R a/b , M Jmz,a/b σλ 3 ··· are the covariant decay amplitudes for the weak decays H → R a/b h 3 · · · h n with σ, λ 3 , · · · being the helicities of R a/b , h 3 , · · · defined in the rest frame of H , respectively, G j a/b λ 1 λ 2 are the helicity decay amplitudes [24] for the strong decay processes R a/b → h 1 h 2 in the rest frame of the h 1 h 2 system with λ 1(2) being the helicities of h 1 (2) in the same frame, and the notation · · · | · · · 's are the Clebsch-Gordan coefficients. Among the three terms in Eq. (3.1), the first two represent the contributions from the resonances R a and R b alone, respectively, while the last one represents the effect of the interference between R a and R b .
As of aforementioned, the PWCP As get their contributions from the difference between w j andw j . This difference is originated from the interference of amplitudes with different weak and strong phases. . All the three terms in Eq. (3.1) could contribute to the PWCP As. The contributions to the PWCP As of the first two terms in Eq. (3.1) come from the weak decay process H → R a/b h 3 · · · h n alone, i.e., the interference of the tree and penguin amplitudes for the decay via the same resonances R a or R b . These two terms are proportional to the sine of the strong phase difference between the amplitudes of the tree and penguin operators for the same resonances R a or R b , respectively, which is usually small unless some particular mechanism enters to generate a large phase difference. 3 On the other hand, the last term in Eq. (3.1) represents the interference between the two resonances R a and R b . Its contribution to the PWCP As is proportional to the sine of the strong phase difference between the amplitude through the resonance R a and that through R b , i.e., tree and penguin amplitudes from different resonances. The strong phase difference between different resonances could be large because of the non-perturbative strong interaction effects, permitting the existence of large rCP As and/or PWCP As. In fact, large rCP As which are correlated with the aforementioned interference of different resonances have been observed in some three-body decays of B meson. Similarly, the presence of the last term in w j in Eq. (3.1) indicates that it can contribute to the PWCP As, resulting in the PWCP As which are large enough to be potentially detectable.
With the aid of the properties of the Clebsch-Gordan coefficients and the parityconservation requirement for the processes R a/b → h 1 h 2 , it is easy to show that the interference and non-interference terms satisfy the following Selection Rules (SRs): the interference terms show up only when 1) j = |j a − j b |, · · · , j a + j b and 2) (−) j π a π b is positive, where π a and π b are the parities of R a and R b , respectively; while the non-interference terms show 3 Note that if there were only one resonance Ra which dominates, w j would only contain the first term.
In this situation, the PWCP As are reduced to from which one can see that the PWCP As are indeed observables for the decay H → Rah3 · · · hn and they contain only the weak decay amplitudes M Jmz ,a σλ 3 ··· .
up only when 1) j is even, and 2) j = 0, · · · , 2j a/b . In Table 1, we list the values of j for the interference and non-interference terms appear in w (j) andw (j) according to the SRs for some spin-parity configurations of the resonances R a and R b .

Applications to baryon decays
The previously introduced PWCP As provide a systematic way to investigate the underlining dynamics of CP V in multi-body decays of heavy hadrons. We propose to search for CP V first through the measurements of the PWCP As in multi-body decay channels via the transitions b → uūd or b → uūs, for which large weak phases are expected. Candidates 4 Take the decay channel Λ 0 b → pπ − π + π − as an example. The rich resonance structure of this decay channel makes it also a perfect channel for the measurement of PWCP As. The dominance of the resonances such as ∆ ++ (1232) and ρ 0 (770) has been observed by LHCb in this decay channel [13,14]. Based on a simple isospin symmetry analysis one can deduce that the resonance ∆ 0 (1232) should also dominate. Meanwhile, since the nearby resonance N 0 (1440) has a width as large as 350 MeV, the interference effect between the two resonances, ∆ 0 (1232) and N 0 (1440) (referred as ∆ 0 and N 0 respectively hereinafter), around the vicinity of ∆ 0 , could be large. As a consequence, the interference between the two cascade decays, CP , as is conventionally done, would miss the interferenceinduced potentially large CP V. It could only be possible to find CP V corresponding to the interference effect between N 0 and ∆ 0 through the measurement of A (2) CP . To simplify the numerical estimation, while confining the invariant mass square of the pπ − system, s pπ − , to be around m 2 ∆ , we further constrain the invariant mass of the remaining π + π − pair to be around the ρ 0 (770) (referred as ρ 0 hereinafter), which will not reduce the statistics significantly due to the dominance of ρ 0 . In this situation, the decay process is dominated by two coherent cascade decays . Via the generalized factorization approach, the amplitudes of Λ 0 b → ∆ 0 ρ 0 and Λ 0 b → N 0 ρ 0 can both be parameterized as: where X = ∆ 0 or N 0 , X|u/ ε * (1 − γ 5 )b|Λ 0 b is the matrix element for the transition Λ 0 b → X with ε the polarization vector of ρ 0 , and with V ub , V ud , V tb , and V td the CKM matrix elements, a i,X = c eff i + c eff i−1 /N eff,X C for i = 2, 4 (c eff i and N eff,X C are the effective Wilson coefficients and color number, respectively). The can be respectively expressed in the following helicity forms: where we have also written the weak decay amplitudes in the helicity form, F ∆/N σ , the parameters c ∆,N P −conserve/violate contain the form factors for the transitions Λ 0 b → ∆ 0 /N 0 , the decay amplitudes for the strong decay processes ∆/N → pπ − , and some other common factors corresponding to the ρ 0 resonance which also depend on the phase space variables.
Although the strong coupling constants can be extracted from the corresponding branching ratios, the weak transition form factors are not available. Moreover, the relative strong phase δ ≡ δ N − δ ∆ is not available either due to its non-perturbative nature. These prevent us from an accurate prediction for the PWCP As in Λ 0 b → pπ − π + π − . Nonetheless, for the purpose of illustrating the behaviours of the PWCP As, we simply set all the c ∆,N P −conserve/violate to be the same. Then, the PWCP As are simply functions of the strong phase δ, which are shown in Fig. 1. 5 The dependence of A CP on the strong phase δ can be clearly seen from Fig. 1, which is an indication of the presence of the interference term in A For comparison, the dependences of the differential CP As on c θ * 1 (θ * 1 is the angle between the proton and ρ 0 for now) for δ = 4 are shown in Fig. 2. The whole range of c θ * 1 is divided into three parts according to the sign of the Legendre polynomial P 2 (c θ * 1 ), which are denoted as I, II, and III, respectively, in Fig. 2 The rCP As of these three parts A Reg I, II, III CP are also shown in this figure, from which one can see that the rCP As tend to change signs in Region I and III comparing with the negative sign in Region II. This is a bad news for A (0) CP , because there will be cancellation between Region II and I + III. On the contrary, this is a good news for A (2) CP , as the extra P 2 in A (2) CP results in constructive contributions from all the three parts. Indeed, the numerical values for the PWCP As A (0) CP 5 In drawing this figure, we have used the effective Wilson's coefficients calculated according to Ref. [27] with q 2 /m 2 b = 0.3. Besides, the effective color number is set to be N eff,∆ C = 2 and N eff,N C = 2.5, and the CKM matrix elements are taken from Ref. [28]. and A (2) CP are calculated to be A (0) CP = 6.8% and A (2) CP = 21.8% for δ = 4, respectively. In order to see more clearly, the δ-dependence of the rCP As of Region I, II, and III are also presented in Fig. 1, from which is can be seen that for quite a large range of the strong phase δ, the PWCP A A (2) CP is much lager than the rCP As. In order to illustrate the advantage of the newly introduced PWCP As with respect to the significance and statistics, a simulation for the aforementioned decay process, Λ 0 b → pπ − π + π − , for δ = 4, is performed. The range of c θ * 1 is uniformly divided into fourteen bins, so that the aforementioned three regions will approximately accommodate three (for Region I and III) or eight (for Region II) bins. The distributions of the simulated event yields are presented in Fig. 3. In total, the event yields for the baryon and the anti-baryon decay are simulated to be 2202 and 1946, respectively. 6 The values and uncertainties of various CP V observables calculated from the simulated data are list in Table 2, in which the uncertainties are assumed to be originated only from those of the event yields in each bin, which is estimated to be the square root of these event yields. From this table, one can see clearly that the relative uncertainty of A (2) CP is evidently reduced comparing with those of A (0) CP and other rCP As. In fact, according the simulation, the significance of A (2) CP is 6.7σ, comparing with the significance of other observables, which are 3.1σ, 4.3σ, 1.8σ, and 3.2σ, respectively.
Another interesting observation from Fig. 1 is that for certain values of δ, the A CP ≈ 0. This is mainly because the strong phase originated from the Breit-Wigner factor in the propagatror of ∆ 0 , 1/(s pπ − − m 2 ∆ + im ∆ Γ ∆ ), varies rapidly when s pπ − is around the mass squared of ∆ 0 . As a consequence, when combined with certain values of δ, large cancellation may come up, resulting in almost zero A CP 's defined in the aforementioned three intervals are illustrated in Fig. 4. On the experimental side, even if the search of CP V through the measurement of PWCPAs gained nothing for the interval from (m ∆ −Γ ∆ ) 2 to (m ∆ +Γ ∆ ) 2 , one can change this interval for another try to avoid the potentially cancellation.

Conclusion
CP V is now able to be investigated through multi-body decays of heavy hadrons such as B meson, D meson, bottom and charmed baryons. In view of the yet negative outcome in searching for CP Vs in the baryon sector, we introduce a set of new observables, the PWCP As, which can be used to measure CP Vs in multi-body decays of heavy hadrons. As explained in this work, the PWCP As provide a systematic and model-independent way in the investigation of CP As in multi-body decays of heavy hadrons. We propose to search for CP Vs through the measurements of the PWCP As in multi-body decay channels of bottom and charmed baryons, such as Λ 0 b → pπ − π + π − , Λ b → pK − π + π − , Λ 0 b → pπ − K + K − , Λ 0 b → ΛK + π − , and Λ 0 b → pK s π − . It is possible that CP V in some multi-body decay channels of bottom and charmed baryons can be established with the strategy proposed in this paper. 7 Table 1. The values of j for which the non-interference and interference terms appear in w (j) and w (j) according to SRs in the cases of integer and half-integer spins of R a and R b .