Prospects for CP&P violation in $\Lambda_{c}^+$ decay at STCF

CP violation is an excellent tool for probing flavor dynamics as we learnt first with $K_L \to 2 \pi$ and later also with the weak decays of beauty mesons. LHCb 2019 data have shown CP violation for the first time in $D^0\to K^-K^+$ vs. $D^0\to\pi^-\pi^+$. Searching for CP asymmetries is of great interest in charm quark sector in the Standard Model (SM) or even more beyond it. In charm hadron decays, lots of work had focused on two-body final states, and the measurements of CP asymmetries in three- or four-body final states are rare. Dalitz plots have shown an excellent record for three-body final states, and more results are desired for four-body ones. In this work we study CP asymmetries in the decays $\Lambda^+_c \to p K^-\pi^+\pi^0$/$\Lambda \pi^+\pi^+\pi^-$/$pK_S\pi^+\pi^-$, where the SM gives zero values for the first two channels, while $3.3 \times 10^{-3}$ for the last one due to $K^0 - \bar K^0$ oscillation. We performed a fast Monte Carlo simulation study by using electron-positron annihilation data of 1~$\textrm{ab}^{-1}$ at center-of-mass energy $\sqrt{s}\, =\, 4.64$ GeV. The data is expected to be available by the next generation Super Tau Charm Facility proposed by China and Russia with one year (or even less) data taking operation. The results indicate that a sensitivity at the level of 0.2$\sim$0.5% is accessible for these processes, which would be enough to measure nonzero CP-violating asymmetries as large as 1%.


I. INTRODUCTION
Manifestations of CP violation (CPV) predicted by the Cabibbo-Kobayashi-Maskawa (CKM) mechanism [1] in the Standard Model (SM) are in impressive agreement with experimental results, especially for the strange and beauty quark sectors [2][3][4]  1 .CPV in the charm quark sector predicted by the SM is at the order of 10 −3 in singly Cabibbo suppressed decays and much less for doubly Cabibbo suppressed ones [5][6][7].The level of 10 −3 has been near the upper limit of the spread of a substantial range of predictions in the literature, and not really a typical estimate.For the first time CPV has been shown in the weak decays of charm mesons, namely in D 0 → K − K + vs. D 0 → π − π + in the LHCb 2019 data [8].Additionally, CPV has never been observed in the decays of baryons, except for the evidence in the Λ 0 b → pπ − π + π − decay [9].We point out that non-leptonic decays of charmed hadrons mostly occur with by many-body final states (FS) (and even more for beauty ones); crucial information is given there about fundamental dynamics, not as a 'background' for two-body FS.For three-body FS decays, we have a well-known tool, namely Dalitz plots with an excellent record.Yet one has to continue to four-body 1 However, it is not big enough to account for the matterantimatter asymmetry which leaves one reason for searching for New Physics (NP) beyond SM.
ones since we have to learn much more at least.Furthermore, inspired by evidence for CPV in Λ 0 b → pπ − π + π − from LHCb data [9], it is interesting and meaningful to study CPV by the method of triple-product asymmetries in the charmed baryon Λ + c decay.
There is an obvious, but important comment.When discussing CPV in the weak decays of beauty hadrons, one mostly looks at CKM suppressed transition processes.What about CKM favored ones?Indirect CPV has been established in the decay B 0 → J/ψK S ; the CKM favored amplitude of b → ccs gives |V cb V * cs | ∼ O(λ 2 ) ≃ 0.05 << 1.However, the situation is very different for charm hadrons, where the leading source is described by |V cs V * ud | ≃ 1 − λ 2 ≃ 0.95.Furthermore charm baryons can produce direct CPV only.Thus the SM can not explain sizeable CPV asymmetries for V cs V * ud amplitudes in general, and in particular for Λ + c → pK − π + π 0 /Λπ + π + π − .Yet there is a special case, the SM predicts CPV for Λ + c → pK S π + π − at 'around' 3.3 × 10 −3 due to CPV in K 0 − K0 oscillation [6], although it is not due to ∆C = 0.This similar prediction for CPV has been tested for D ± → K S π ± with some success: A CP (D + → K S π + ) = (−0.41± 0.09) %; yet the 'landscape' is more complex for Λ + c .It would be close to a miracle, if new physics (NP) could produce non-zero CPV for Λ + c → pK − π + π 0 /Λπ + π + π − or sizably above 3.3 × 10 −3 for Λ + c → pK S π + π − , but it is possible.Thus experimenters cannot ignore that.With more data and refined analyses in the future one can use much better tools to calibrate favored decays, when one goes for accu-racy.One has to be open-minded about this project.Our community has successful experience with triple-product asymmetries A T−odd and ĀT−odd (see also Sec.II B below).In the weak decays of charm baryons one goes after parity violation(PV) and direct CPV measurements in somewhat different ways.
A T−odd = 0 establish PV by itself and likewise for ĀT−odd : in practice one can test experimental uncertainties by comparing A T−odd vs. ĀT−odd .In the literature, e.g. in [9], PV is also defined as (a P + āP )/2.The SM produces large PV; we will back to that below.As we had said above, the 'landscape' of ∆C = 0 is close to CP invariance; thus one can connect CV (charged conjugation violation) with PV: a P + a C ≃ 0. Using different words to describe the same situations we know that these CP asymmetries are very small at best: Strong final state interactions (FSI) are not the source of CPV.That has to come from new dynamics (ND) with weak phases -yet FSI should show their impact.One has to be realistic: very likely we will not find CPV in these weak decays of Λ + c .Yet it is not a waste of time, and those channels are worth to do in experiment due to the following points: • It is not a miracle to find CPV in Cabibbo suppressed decays of Λ + c ; one can use those mentioned channels to calibrate Singly Cabibbo suppressed (SCS) decays to probe regional CP asymmetries in Λ + c → pπ − π + π 0 /pK − K + π 0 /ΛK + π + π − with accuracy in the future.
• One expects sizable PV in the weak decays of Λ + c .• At least, one can get novel lessons about the impact of strong forces close to thresholds, namely about non-perturbative QCD.
We will consider three decay processes : The current paper is mainly dedicated to the study of physics sensitivities that can be achieved at a future Super Tau Charm Facility (STCF), where the central values for PV and CPV quantities of charmed baryon decays are surely measurable.The new generation STCF is an electron-positron collider to operate at the τ -charm energy region, with peak luminosity above 0.5 × 10 35 cm −2 s −1 at a center-of-mass energy (CME) of √ s ∼4 GeV/c 2 [10][11][12].The facility is discussed strongly and proposed by the Chinese and Russian high energy physics communities in last few years, and is expected to be realized in the coming ten years.With such high luminosity, the proposed STCF can deliver electron-position collisions to accumulate more than 1 ab −1 of integrated luminosity per year, thus providing an excellent opportunity to study charm physics, notably including CPV with charmed meson and baryon decays.
In the electron-positron annihilation process, the Λ c baryon can be produced via the process e + e

Λ−
c , the absolute decay branching fractions of Λ + c → pK − π + as well as other eleven Cabibbo favored (CF) hadronic modes, the branching fractions of SCS decays, the decay with neutron included, semi-leptonic decay and inclusive decays etc [14][15][16][17][18][19].In proton-proton collisions, such as at the LHCb experiment, the Λ + c baryon is abundantly produced directly from proton-proton collision or via beauty baryon decays [20][21][22].Comparing to the Belle II and LHCb experiments, the STCF is of shortage in statistics.However, STCF has several advantages, such as the excellent ratio of signal to background, the perfect detection efficiency, the well controlled systematic uncertainty and the capability of full event reconstruction, etc.By implementing the double tag (DT) method, STCF can perform systematic researches of Λ + c decays, including the absolute measurements of semi-leptonic decays and the decays with a neutron, K L or invisible particles included in final state [23].Besides studying Λ + c physics, STCF will play crucial role in the study of how the Y (4630) state enters e + e − → Λ + c Λ− c production [24], the mixing of axial-vector mesons [25], the proton form factors [26,27], etc.
In what follows, we will perform a careful investigation for the sensitivities on CPV and PV in the decays Λ There is a rich 'landscape' about strong and weak forces; one needs more refined analyses -but we have the tools for that; all we need is more data.

II. OBSERVABLES
The situations between PV & CPV are very different as said above; thus the goals are also different.The first example: with more data one should find non-zero values of PV in these non-leptonic transitions.

A. Parity asymmetries
It had been realized that it is a crucial test of the SM: charged W ± bosons are left-handed, as we had learnt from π + /K + → µ + ν vs. π + /K + → e + ν; so far, we have not seen right-handed one.2018 PDG data [28] have shown PV in Λ + c → Λl + ν that are consistent with the SM predictions, although with sizable uncertainties: On the other hand, this situation is not well tested in nonleptonic decays.Probing these non-leptonic decays of Λ + c would give new lessons about non-perturbative QCD or even indirect impact of New Dynamics on PV.In these non-leptonic decays of Λ + c T-odd moments should produce sizable PV with different values, see the Eq.( 1).We have added these analyses of PV below.Indeed, one gets a non-trivial test of this experiment.
One should expect large values of PV in those nonleptonic transitions.A small/tiny value of PV would be signal of NP.However, one cannot predict future results of PV even within the SM.It means our community would learn new lessons about the impact of strong forces.So far, no true predictions can be given due to non-perturbative QCD with many resonances in the region of 0.5 -2 GeV, including broad ones like f 0 (500), K * 0 (700) etc.Our main paint is that we describe the travel to use, when our community has the future data to get the information about the underlying dynamics.

B. CP asymmetries
The SM predicts tiny CPV in charm baryon decays; therefore large statistics are required.Obviously one goes after direct CPV.The landscape of data is very 'flat' for CP asymmetries: it is expected to be very unlikely that any evidence for CPV in Cabibbo favored(CF) transitions is found, e.g.
, where CPV was investigated by measuring the decay asymmetry parameters.There are also recent theoretical papers about the decay asymmetry parameters [31][32][33], and especially in Ref. [33] the model calculations are done for the singly Cabibbo-suppressed decays.We also notice that BESIII has measured the absolute branching fraction of Λ + c → Λl + ν with less uncertainties [17].We exploit triple-product asymmetries composed by fourmomenta without recurring to the information of polarization as has been done in Refs.[34,35].
The CF decays of Λ + c baryons with multi-hadrons in final state, such as Λ + c → pK − π + π 0 , Λ + c → Λπ + π + π − and Λ + c → pK S π + π − depict a much 'complex' landscape [19,28], which is believed to give us much more information about the underlying dynamics than that of Λ + c → Λπ + and Λ + c → Λe + ν, but need both more data & refined analyses.To describe the four-body weak decays of Λ + c one has one baryon in the FS, p or Λ, plus three pseudoscalar mesons -kaons or pions.In the rest frame of the charm baryon we have two observables of spin-1/2 s Λc and s p/Λ -and the momenta of the four particles p p/Λ plus the momenta of the three mesons.One can describe T-odd moments in different ways, which give us the same information about the underlying dynamics; however with finite data and lack of perfect control of QCD, some are better than others.We exploit the scalar triple products to construct CPV observables, see the Refs.[35][36][37][38][39][40][41][42][43][44][45][46].These papers came mostly from theorists who had focused on singly Cabibbo transitions.This method has been widely applied in several experiments, see recent ones in Refs.[9,[47][48][49].Some early ones can be found in Refs.[50][51][52].
For these Λ + c decays, the scalar triple products , with pseudo-scalar mesons h i , are defined to study CPV.The momenta p are measured in the rest frame of the Λ + c baryon; When two π + (or two π − ) mesons are that one with the larger momentum is selected.The asymmetries are then defined as: These correspond to A T−odd ĀT−odd moments; CPV observables are δ CP , see Eq. ( 2).Any significant deviation from zero indicates CPV; in particular, one also looks for the number N of events for the direct CPV asymmetries: One can expect sizable values of A T and Ā T due to FSI effects [6,40].It is also possible to find non-zero CPV.In Ref. [36], the authors show that large CPV can indeed happen in NP with the two-Higgs doublet model as an example.The CP violation ∼ 0.18 sin φ with φ denoting the New-Physics CP violating phase.Then it can reach 18% if sin φ is close to 1.
The measurements may vary over the phase space due to resonant contributions or their interference effects, which may be cancelled if integrating over the whole phase space.For the decays Λ + c → pK − π + π 0 , Λ + c → Λπ + π + π − and Λ + c → pK 0 S π + π − , the semi-regional CPV is measured with respect to several bins separated by the dihedral angle, and Monte Carlo (MC) simulation is also exploited to study this case.We stress again that no CP asymmetry has been found yet in the transitions of charm baryons.Therefore, one has to probe CPV with more data and tools, although this is not trivial.

III. MEASUREMENT PROCEDURE
The STCF project is in research and development stage.To maximize the physics potential, a BESIIIlike detector but with much improved performance for each sub system is proposed.From inside to outside, the STCF detector consists of a tracking system, a particle identification (PID) system, a high granularity electromagnetic calorimeter (EMC) and a muon detector with high µ/π separation capability.To be competitive on high precision measurements, and to cope with high event rate and radiation dose, several advanced technologies are proposed to be the STCF sub-detectors, such as a thin silicon detector or a micro pattern gas detector for the inner tracking system, a Cherenkov based PID system, crystal LYSO or pure CsI based electromagnetic calorimeter etc.To investigate the physics potential capability and optimize the detector design, a fast simulation tool dedicated to the STCF detector has been developed, where the detection efficiency and measurement resolution of each sub-detector are parameterized according to an empirical formula and the BESIII detector performance, and the parameters are adjustable flexibly.The event generators for both signal and background processes are migrated from the BESIII experiment.The tool has been validated by the BESIII full simulation package [53] using Geant4, and provides a perfect platform to perform physics studies with huge statistics.A note dedicated to this tool is under preparation.
To study the sensitivities of CPV and PV in the decays Λ + c → Λπ + π + π − , Λ + c → pK − π + π 0 and Λ + c → pK S π + π − , both signals and inclusive MC samples are generated based on the STCF fast simulation tool, where the parameters for each sub-detector are from BESIII.In this study, the Λ + c signal is originated from the process e + e − → Λ + c Λ − c at the CME of √ s=4.64 GeV, where the peak of the production cross section lies.The study is performed based on the integrated luminosity of 1 ab −1 , which is expected to be achieved at STCF within one year (or even less) of data taking.In the simulation, e + e − collisions are simulated by the KKMC generator [54], which takes into account the beam energy spread and the ISR correction, where the beam energy spread is assigned to be same value as that of BEPCII.To study the poten-tial background and optimize event selection, an inclusive MC sample, which includes Λ + c Λ− c pair production, l + l − (l = e, µ, τ ) events, open charm processes, ISR-produced low-mass ψ states, and the continuum process e + e − → q q with q = u, d, s quarks [55] are generated with the integrated luminosity of 1 ab −1 , where the decays of intermediate states, such as Λ + c baryons, charmed mesons, charmonium state, and light hadrons, is performed according to the branching fractions quoted from PDG.To study the signal shape and detection efficiency, the signal MC samples of Λ + c → pK − π + π 0 , Λ + c → Λπ + π + π − and Λ + c → pK S π + π − are generated with uniform distribution in phase space; no intermediate state in the two or three bodies is considered.The real data will show the impact of intermediate states, such as ρ, K * , ∆ etc.
In this analysis, the single tag method is implemented to improve the statistics.Candidate events are selected with the similar criteria (including charged tracks, π 0 and K S candidates selection, PID, etc.) as in Ref. [18] according to the final state of signal.The signal yields are dertermined by performing a binned maximum likelihood fit to the distribution of the beam constrained mass M BC , which is defined as Λc /c 2 , with E beam denoting the energy of the electron/positron beam and p Λc the three-momentum of the Λ + c candidate calculated from the momenta of the final-state particles in the initial e + e − center-of-mass system.Figure 1 shows the M BC distributions for Λ + c → pK − π + π 0 , Λ + c → Λπ + π + π − and Λ + c → pK s π + π − decays corresponding to 1 ab −1 of an inclusive MC sample, where ∆E, defined as ∆E = E beam − E Λc with E Λc denoting the energy of Λ c candidate summing over the energy of the corresponding final state particles, is required to be within three times of its resolution.Clear Λ + c signals with low background are observed.Detailed studies by the inclusive MC sample indicate that there is no peaking background in the M BC distributions.Thus, in the fit to determine the signal yields, the shape of background is described by an ARGUS function [56] with fixed highend truncation, and those of signal are obtained from the signal MC samples.
For semi-regional CPV, one may discretize the dihedral angle and/or the invariant mass into different bins, as in Ref. [9].In the intermediate state regions, strong phases are enhanced and thus can provide opportunity for large CP asymmetries due to large interference.Since the components of intermediate states are unknown due to the lack of experimental data, in this study, we split the phase space into different bins along the dihedral angle Φ distribution only, and the binning along the invariant mass distribution is not considered.Here, Φ is the angle between the decay planes formed by the pπ 0 and K − π + (pπ − and K 0 S π + , Λπ + fast and π + slow π − ) for the process Λ In the future, once collecting huge data at STCF, we can have a better understanding of the underlying dynamics of the Λ + c decay, including the impact of broad intermediate states, such as K * 0 (700)/κ and f 0 (500)/σ etc , and )  the analyses of semi-regional CPV can be refined3 .

IV. RESULTS AND DISCUSSIONS
Following the approaches described in Sec.III, we report in Table I the physics sensitivities for direct CPV, as defined in Eq. ( 6), as well as for PV and CPV observables constructed from the T -odd moments elaborated in Eqs. ( 1) and (2).The physics sensitivities include the statistical uncertainties only; systematic uncertainties are expected to be well under control 4 .By error propagation, according to Eqs. ( 6), ( 1) and ( 2), if we ignore the impact of the statistical uncertainty from background contamination, and assume As discussed previously, the sensitivity on CPV may be enlarged in some regions of phase space due to the enhancement of the strong phase and interference.This kind of CPV is called semi-regional CPV or localized CPV, and is of great interest for both theorists and experimentalists.In this study, we also perform a sensitivity study for semi-regional CPV for the three Λ + c decay models, individually.Due to the lack of information on the intermediate states, the studies are performed only by binning the dihedral angle Φ, as defined in Sec.III, based on MC samples generated with a phase-space model.The measurements with real data are expected to be of better sensitivity due to the contribution from intermediate states.In this study, we discretize the dihedral angle Φ into ten bins with equal steps from 0 to π, and measure the T-odd moments CPV in each bin.As shown in Table II, the sensitivities for Λ + c → pK − π + π 0 , Λπ + π + π − and pK 0 S π + π − in each bin are around 0.0080, 0.016, and 0.013, respectively, which are smaller by a factor 1/ √ 10 relative to global CPV values, since the statistics is reduced by a factor 10 in each bin.Searching for CPV and PV in charmed baryon decays certainly provide complementary and comprehensive information to understand the underlaying dynamics of charmed hadrons and test the SM, and is of great interest both for theorists and experimentalists.The future Super Tau Charm Facility (STCF) proposed by Chinese and Russian scientists may provide great platform for these kinds of studies due to its characters of high luminosity, broad center-of-mass energy acceptance, abundant production, clean environment, etc.In this work, we propose to study direct CPV by measuring the asymmetries of decay branching fractions between charge conjugate modes as well as PV and CPV by constructing T −odd moments in Λ + c decays to multi-hadron final states.We study the physics sensitivities for CPV and PV in the decays Λ + c → pK − π + π 0 , Λ + c → Λπ + π + π − and Λ + → pK S π + π − by performing a fast simulation, where the Λ + c is assumed to be from the e + e − annihilation to Λ + c Λ− c pair at center-of-mass energy of √ s = 4.64 GeV with 1 ab −1 e + e − integrated luminosity, i.e. expected to be available in one year (or even less) operating at future STCF.The results indicate that the physics sensitivities are around 0.25∼0.5% for the three decay modes, individually, which is at the level of potential CPV in charm hadron sector or for an unambiguous PV observation.We also discuss how semi-regional CPV may be enlarged due to the enhancement of the strong phase and interference, and perform the sensitivity study for the same decay modes by binning the dihedral angle distribution.Simulations cannot give predictions, in particular for many-body final states.In the future, with huge real data collected at STCF, we can also study the intermediate states and their impact.Many exciting results are expected at STCF, providing excellent information for non-perturbative QCD studies.

2 ,
the statistical uncertainties for A CP , (a P +ā P )/2 and δ CP are 1/ √ 2N , where N Λ + c and N Λ− c are the numbers of Λ + c and Λ− c candidate events, and N (C T > 0) ( N ( C T > 0)) and N (C T < 0) ( N ( C T < 0)) are the numbers of candidate events with C T > 0 and C T < 0 for the Λ + c ( Λ− c ) candidates, respectively.Thus, as shown in Table I, the three measured variables have the same sensitivities, mostly due to the small impact from the background, and provide complementary and more comprehensive information to search for PV and CPV in Λ + c hadronic decays.With an e + e − → Λ + c Λ − c data sample of 1 ab −1 integrated luminosity at √ s = 4.64 GeV collected by STCF, the physics sensitivities to search for PV and CPV are at the few permille level for three interesting decay modes, individually, which are at the level of potential CPV in charm sector and unambiguous PV if observed.
− → Λ + , 8.54, and 8.16 pb −1 , respectively).With these data sets, BESIII is very productive, and has published several interesting results, such as the production cross section of e + e − → Λ +