CP Asymmetries in Strange Baryon Decays

While indirect&direct CP violations (CPV) had been established in the decays of strange&beauty mesons, none have been done for baryons. There are different"roads"for finding CP asymmetries in the decays of strange baryons; they are highly non-trivial ones. The HyperCP Collaboration had probed CPV in the decays of single $\Xi$&$\Lambda$ [Phys.Rev.Lett 93 (2004) 262001]. We talk about future lessons from $e^+e^-$ collisions at BESIII/BEPCII: probing decays of pairs of strange baryons, namely $\Lambda$, $\Sigma$&$\Xi$. Realistic goals are to learn about non-perturbative QCD. One can hope to find CPV in the decays of strange baryons; one can also dream to find impact of New Dynamics (ND). We point out that a new important era starts with the BESIII/BEPCII data accumulated by the end of 2018.

Actually, it has been argued by Buras et al. for a long time that the SM cannot produce the value given by the data.
We have now obtained the first result from a LQCD group [6]: Obviously we need more lattice data. So far, it is not clear which lesson we can learn here: does it mean that these data are consistent with what the SM gives us, or it is a sign of the impact of ND?
Soon after 1964 it was hoped to probe CPV in the transitions of strange baryons. This is a huge challenge. However, the goal is so important that we should not give up. Present experimental limits are high above the level at which one can even think about the possible impact of ND. In 1998 several situations were proposed, in particular for CP observables in B 0 (t)→ hyperon-antihyperon, where CPV could be found based on predictions at that time [7].
The landscape of fundamental dynamics has changed very much since then. Neutrinos are massless in the SM, but neutrino oscillations have been discovered. The SM produces true large CP asymmetries in ∆B = 0 transitions, and indeed the BABAR/Belle Collaborations have found this with B 0 → J/ψK S and other systems, but no non-zero values of CP asymmetries have yet been established in the decays of baryons in general, beyond the huge asymmetry in matter vs. anti-matter in our Universe (or our 'existence'). However, evidence has been seen by the LHCb experiment in Λ 0 b → pπ − π + π − for regional CPV [8]. Obviously we are talking about direct CPV in ∆B = 1. In two-body final states of beauty mesons the usual scale is ∼ 0.1; for Λ 0 b decays it suggests that the regional scale is sizably enhanced. Can this happen also for strange baryons?
The final states are mostly two-body in the decays of strange baryons. There are two classes of transitions, as given in the PDG2016 data: 1) Re-scattering gives a sizable impact, and it is obvious for Λ and Σ + : CPT invariance is 'usable', telling us about average asymmetry 1) . These widths are basically produced with two hadrons, thus: Γ (Λ → pπ − )+Γ (Λ → nπ 0 ) Γ (Λ→pπ + )+Γ (Λ→nπ 0 ); likewise for Σ + . Therefore, The goal is to establish CP asymmetry in Λ→pπ − and in Σ + → pπ 0 . To find it also in Λ → nπ 0 & Σ + → nπ + would be nice, but not important. The situations are very different for the decays of Λ 0 b , where the final states are mostly many-body.
For the decay Λ → pπ − without the spins of the baryons included, there is only one observable, namely a number. The data depend on production asymmetries in Λ → pπ − vs.Λ →pπ + . That is not a problem for e + e − annihilations for the BESIII experiment or forpp collisions; however, the situation is quite different for pp collisions from LHCb data.
2) The situation is more complex: the impact of rescattering is not obvious, when one cannot use polarized baryons, as one can see in the branching ratios: Successfully probing CPV in strange baryon transitions is a true challenge. Previous predictions have been made based on the SM [10,11]: A later SM prediction was made by combining Λ and Ξ decays [12]: The HyperCP experiment 2) searched for CPV using a 800 GeV proton beam on a Cu target [1]: It is still not clear whether the theoretical uncertainties are included correctly. However, two points might help to reach our goal: 1) The BESIII collaboration can probe pairs of strange baryons. We will discuss that in Section 2.
2) Future BESIII analyses, namely e + e − →J/ψ with the unusual narrow resonance, will be enhanced as a source of strange baryons compared to PDG2016: The final state interactions (FSI) show impact, although our community is so far not able to describe it quantitatively. We will discuss in some detail what we can learn about fundamental dynamics including CP asymmetries.
1) The impact of CPT invariance goes well beyond the same mass and width, as discussed in Ref. [9].
2) The authors of this proposal "HyperCP: Search for CP Violation in Charged-Hyperon Decays" quoted Bigi and Sanda from their recent book CP Violation: "We are willing to stake our reputation on the prediction that dedicated and comprehensive studies of CP violation will reveal the presence of New Physics." At least one of the co-authors of this paper agrees.
It is not trivial at all, and as usual there is a price for the prize.

CP asymmetries in J/ψ → pair of strange baryons
In this section, we mostly discuss the decays of J/ψ to final states with only two strange baryons. We also include special cases, in Section 2.6.
Published 2010 data from the BES Collaboration are based on 5.8×10 7 J/ψ events [14], giving 10 % accuracy. To reach the level of O(1%) is not trivial. However, we expect that our community will have data to reach 0.1 % level by the end of 2018. We have to be realistic: this prize cannot be reached easily. On the other hand, it would be very pessimistic to suggest we will follow the path of the Higgs boson, namely searching for 40 years.
The BESIII detector has already collected 1.3×10 9 J/ψ events, and it is expected to have close to 10 10 J/ψ events by the end of 2018 [15]. The situation is more , which we will discuss in Section 2.5.
Further competition will arrive from the LHCb collaboration by probing J/ψ→ΛΛ; the challenge comes in triggering for this channel 1) .

General statements about first steps
First we describe the 'landscape' and the tools one can use. The super narrow vector resonance J/ψ produces a connection of the spins of a pair of hyperons Y of Λ, Σ, and Ξ and final state X of p, Λ: One can measure T -odd moments: where σ and π describe the spins and the momenta of the baryons in the rest frames of Y /Ȳ . It is not surprising to find sizable or even large non-zero values of T -odd moments; below we will give an example from data with large values. It shows the impact of final state interactions (FSI). Non-zero values of T -odd moments do not mean by themselves we have found T violation or CP violation using CPT invariance. However, one can probe direct CP asymmetries without polarized Y andȲ . The crucial point is to use the connection of the spin-1 of the initial state J/ψ with the final state with two spin-1/2. The goal is to find A (X) CP = 0 without directly measuring polarization of Y /Ȳ and X/X baryons and their correlations due to the very narrow resonance J/ψ in their production.
We measure correlations between the pair of final state baryons and pions. In the rest frame of J/ψ one can define C T = ( p X × p π )· pX and conjugate transitions C T = ( pX × pπ)· p X , and compare the event numbers (N ) with positive and negative values Thus are observable CP asymmetries based on CPT invariance. We have taken a different convention forĀ T compared to Ref. [8].
One can compare to its charge-conjugate channel to get rid of a fake CP asymmetry induced by the FSI effect. However, the charge-conjugate of the process J/ψ → ΛΛ → [pπ − ][pπ + ] considered here is itself; or it is an untagged sample that cannot be distinguished by experiment. Such a situation is, to some extent, different from a measurement of D 0 (D 0 )→KKππ/4π.
2) That is not the end of the impact of strong resonances of ∆(1232) with width ∼ 117 MeV and N (1440) with width 250−450 MeV. They will affect the lessons we learn from future data about possible CP violations in the transitions of strange baryons.
3) The "duality" between the worlds of quarks and hadrons is very subtle, in particular close to the thresholds of resonances.
Here we estimate the sensitivities for measuring such observables using the collected J/ψ sample. By the end of 2018, 10 10 J/ψ events will be accumulated by the BE-SIII experiment [15]. The detection efficiency for pions, protons, and kaons with momentum larger than 100 MeV can reach 98%. As for particle identification (PID), pions, kaons and protons can be distinguished with 3σ (three standard deviations) uncertainty below a momentum of 1.0 GeV. Considering the branching fractions of [4] etc., we can estimate the expected numbers of events and the further sensitivities, as shown in Table 1. Probing such a large data sample with refined tools will give us rich information about the underlying dynamics.

Going beyond first steps
Measuring A T is the first step to probe CP asymmetries. There are various options for intermediate steps like: This is a very promising way to go beyond A T in Eq. (18). By the end of 2018 we can expect that BESIII can probe CPV in the decays of strange baryons to the level of 10 −4 sensitivity, as shown in Table 1 (neglecting systematic errors).
The above method can also be applied to J/ψ → ΛΛππ to probe CPV in Ξ decays. Interestingly, for the case of Ξ baryons, CPV can also be probed by polarized Ξ thanks to the decay chain Ξ 0 →Λπ→(pπ)π, where the CP -violating observable can be related to the helicity amplitudes. A similar proposal was given in Ref. [16] for Λ c decay. Such a CP -violating signal can be extracted by performing an angular analysis, which is again accessible in the BESIII experiment due to the large J/ψ data sample. However, for Λ decays the interference of the helicity amplitude is absent in the angular observable [14], which handicaps accessing CP the same way as in polarized Ξ decay, i.e., measuring the interference of helicity amplitudes.
By fitting Eq. (20) to the data, one can determine α J/ψ and α Λ αΛ. One can make a substitution: where A describes a CP asymmetry observable. As mentioned before, published 2010 data from the BES collaboration show [14] α (p) Λ =−0.755±0.083±0.063, based on previously measured α (p) Λ =0.642±0.013. Their non-zero numbers show the impact of re-scattering; it is large, which is not surprising. We also note that Eq. (20) is derived from considering only spin projection J z =±1 for J/ψ, which is a consequence of the QED process e + e − → γ → cc (J/ψ) [22]. Thus, only the product term α Λ αΛ can be measured 1) .
The present limit on direct CP asymmetry is around a few percent. ND cannot even produce effects close to 1% here. To reach O(0.1%) would be a considerable achievement for A (p) CP based on α (p) Λ ∼α (p) Λ ∼0.64. Measuring semi-local asymmetry would give us new lessons about non-perturbative QCD.
1) In a pp machine, the terms α Λ and αΛ can be separated [24].
It has been said very recently in Ref. [23] that Ref. [14] missed some contributions for on-shell J/ψ → ΛΛ →pπ + pπ − ; so far we are not convinced. Of course, experimental data from BESIII are needed in order to test this.
Even now BESIII has much more data, and by the end of 2018 we will have about 2.6×10 6 events for the J/ψ →ΛΛ → [pπ + ][pπ − ] decay chain, after considering the efficiencies for tracking, particle identification and Λ pair reconstruction [15]. The sensitivity might reach 6×10 −4 by the end of 2018, as listed in Table 1. Now the landscape is different, with some hope to find CP asymmetry here, and also for J/ψ →ΣΣ, which will be discussed below.
Some general comments are as follows. (a) Except n−n (or maybe, even Λ−Λ) oscillations [25], one probes only direct CPV with baryons. (b) It is well-known that the impact (local) penguin operators are crucial for ∆S = 1 transitions for strange mesons, in particular for the non-zero value for . What about decays of strange baryons? The transitions of pairs of strange baryons are not far from the thresholds; thus one has to think about the "duality" between the world of hadrons and that of quarks and gluons. We discuss this further below.
To probe CP asymmetries with accuracy, BESIII can calibrate with transitions where CP asymmetries cannot happen. We have two examples with broad resonances, where the situations are complex: J/ψ → N (1440)N (1440) with Γ N (1440) ∼ (250−450) MeV carrying isospin I = 1/2, and J/ψ →∆(1232)∆(1223) with Γ ∆(1232) ∼ 117 MeV carrying isospin I =3/2. The BESIII experiment has measured the background very well and continues to do so. The total background is very small, which provides good opportunity for a clean probe of the CPV signal.
From knowledge of the previous BES measurement [14] (the main background channels are also listed in this reference) and the ongoing measurement 1) , we can conservatively estimate that the number of combinatorial background events is roughly 10 −3 of the signal events, a very small fraction, such that CPV can be cleanly probed. This is one of the strengths of the BESIII collaboration. We can show this more transparently with an example. By the end of 2018 we can expect 2.6×10 6 signal events, and assuming the CPV is on the level of 10 −4 , one has ∆N = N + −N − = 260. The background can induce ∆N bkg = √ 2.6×10 3 ≈51. This illustrates that a nonzero A T will really indicate the observation of CP asymmetry. However, if the CPV A T is below the level of 10 −5 , the impact of the background is important.
4) Therefore, we use two different diagrams for transitions: "⇒" describes the amplitudes due to QCD(×QED) that conserve P and C symmetries; "→" includes SU (2) L with weak dynamics, including sources of P, C and CP asymmetries. The direct path is while a somewhat indirect path can happen due to offshell intermediate amplitudes: We hypothesize off-line exchanges of kaons or new dynamics.
Once non-zero values for CP asymmetries are found, one can attempt another challenge: do they come from the transition of Ξ→Λπ or Λ→pπ − , or the interferences between them?

Summary from the experimental side
We are studying non-perturbative QCD in a novel situation; it is not trivial to know how much to apply it and where. Just below we give comments about available theoretical tools. The hope is to find signs of CP asymmetries in the decays of strange baryons; therefore the goal is not to go for accuracy. We 'paint' the landscape to find CP asymmetries. Once our community has found a non-zero value somewhere, one can discuss the correlations with other situations.

Comments about tools for CP asymmetries
The realistic goal is to get new lessons about nonperturbative QCD. One can hope to find CP asymmetries in the decays of strange baryons; one can also dream to find the impact of ND [9]. When one goes after accuracy, one needs truly consistent parametrization of the CKM matrix [26]. However, that is not the goal now; therefore one can use the well-known Wolfenstein parametrization.
We have local operators for ∆S =1 amplitudes. One describes the scattering of left-handed quarks s L +u L → d L +u l with refined tree amplitudes and local penguin operators. Challenges come from true strong dynamics in different ways. In particular µ ∼ 1.0 GeV describe the "fuzzy" boundaries between perturbative and nonperturbative QCD. On the other side the landscape is also complex, since the baryons carry spin-1/2; therefore there are more observables. In other words, the amplitudes can be described with S-or P -waves.
There is another challenge, namely to connect quark and hadronic amplitudes. This issue of "duality" is well known, and it is not just another assumption based on true quantum field theory. However, it does not work well when one has to deal with thresholds that are important in the on-shell transitions J/ψ →ΛΛ and J/ψ →Σ − Σ + ; in these cases we have broad resonances like ∆(1232) and N (1440).

Future outlook
As mentioned above, a realistic goal is to measure non-leptonic decays of Λ, Σ + and Ξ with more data to learn new lessons about non-perturbative QCD. We can 'hope' to find CP asymmetries in BESIII data by the end of 2018 and 'dream' of finding the impact of ND. When it is not enough to work on data and their analyses, 'hoping' or even 'dreaming' to learn from that, then -in our view -one is in the wrong business.
Of course, we need much more data, but also powerful analyses to reach even the realistic goal. Here we have listed the directions where more data should improve our understanding of fundamental dynamics (see Table 1). In a future super tau-charm factory [27][28][29], there will be an unprecedentedly high peak luminosity of 10 35 cm −2 ·s −1 , with 10 12 J/ψ events, a data sample 100 times as large as the ongoing BEPCII/BESIII, which will result in a decreasing of the (statistical) uncertainty by 10 times.
We do not give predictions here for CP asymmetries. Our goal here is to point out the paths where our community can learn more in the future. We need more thinking in general, and analysis of correlations in different transitions.
Analyzing e + e − →Λ − c Λ + c can also give us novel lessons about non-perturbative QCD and even allow us to compare e + e − →Λ − c Λ + c →Λ+X vs. e + e − →Λ − c Λ + c →Λ+X. We add a comment that one first sees this as a technical challenge: applying "dispersion relations" has a long history in nuclear physics, hadrodynamics, high-energy physics (HEP), and also other branches of physics. Dispersion relations are based on central statements of quantum field theory, while their results depend on low en-ergy data, with some 'judgment' (see e.g., Ref. [30]). If the 2018 data we have discussed above are produced as planned, there is a good chance of convincing members of our community to work on that. There are two tasks in applying dispersion relations: (i) as an input for them, one needs more data from N π collisions at low energies; and (ii) non-trivial thinking is needed about where and how to apply them for good reasons.
Finally, in the future our community could have a novel competition between BESIII, LHCb and theorists -actually, different parts of the same team.
XWK thanks Hai-Yang Cheng for the careful reading, encouragement and interesting discussion, and Xinxin Ma for useful discussion.