Search for CP violation using triple product asymmetries in $\Lambda^{0}_{b}\to pK^{-}\pi^{+}\pi^{-}$, $\Lambda^{0}_{b}\to pK^{-}K^{+}K^{-}$ and $\Xi^{0}_{b}\to pK^{-}K^{-}\pi^{+}$ decays

A search for $C$P and $P$ violation using triple-product asymmetries is performed with $\Lambda^{0}_{b}\to pK^{-}\pi^{+}\pi^{-}$, $\Lambda^{0}_{b}\to pK^{-}K^{+}K^{-}$ and $\Xi^{0}_{b}\to pK^{-}K^{-}\pi^{+}$ decays. The data sample corresponds to integrated luminosities of 1.0fb$^{-1}$ and 2.0fb$^{-1}$, recorded with the LHCb detector at centre-of-mass energies of 7TeV and 8TeV, respectively. The $CP$- and $P$-violating asymmetries are measured both integrating over all phase space and in specific phase-space regions. No significant deviation from $CP$ or $P$ symmetry is found. The first observation of $\Lambda^{0}_{b}\to pK^{-}\chi_{c0}(1P)(\to\pi^{+}\pi^{-}, K^{+}K^{-})$ decay is also reported.


, where h 1 =
K − , h 2 = π + for the Λ 0 b → pK − π + π − decay, h 1 = K − fast , h 2 = K + for the Λ 0 b → pK − K + K − decay and h 1 = K − fast , h 2 = π + for the Ξ 0 b → pK − K − π + decay.
The kaon labelled as "fast (slow)" is that with the highest (lowest) momentum among those with the same charge.The triple product C T is defined similarly for X 0 b baryons using the momenta of the charge conjugate particles.

Two T -odd asymmetries are defined based on the operator T that reverses the spin and the momentum of the particles [5][6][7].This operator is different from the time-reversal operator, which reverses also the initial and final state.The asymmetries are defined as
A T = N (C T > 0) − N (C T < 0) N (C > 0) + N (C T < 0) ,(1)A T = N (−C T > 0) − N (−C T < 0) N (−C T > 0) + N (−C T < 0) ,(2)
where N and N are the numbers of X 0 b and X 0 b decays.The P -and CP -violating observables are defined as
a T -odd P = 1 2 A T + A T , a T -odd CP = 1 2 A T − A T ,(3)
and a significant deviation from zero in these observables would indicate P violation and CP violation, respectively.In contrast to the asymmetry between the phase-space integrated rates, a T -odd CP is sensitive to the interference of T -even and T -odd amplitudes and has a different sensitivity to strong phases [8,9].The observables A T , A T , a T -odd and detector-induced charge asymmetries of the final-state particles [10].In the present paper, these quantities are measured integrated over all the phase space and in specific phase-space regions.


Detector and simulation

The LHCb detector [11,12] is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5, designed for the study of particles containing b or c quarks.The detector includes a high-precision tracking system consisting of a siliconstrip vertex detector surrounding the pp interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about 4 Tm, and three stations of silicon-strip detectors and straw drift tubes placed downst eam of the magnet.The magnetic field is reversed periodically in order to cancel detection asymmetries.The tracking system provides a measurement of momentum, p, of charged particles with a relative uncertainty that varies from 0.5% at low momentum to 1.0% at 200 GeV where p T is the component of the momentum transverse to the beam, in GeV/c.Different types of charged hadro herenkov detectors.Photons, electrons and hadrons are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter.Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers.

The online event selection is performed by a trigger, which consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction.Candidates are required

o pass both hardware
and software trigger selections.The hardware trigger identifies the hadron daughters of the X 0 b or events containing candidates generated from hard pp scattering collisions.The software trigger identifies four-body decays that are consistent with a b-hadron decay topology, and which have final-state tracks originating from a secondary vertex detached from the primary pp collision point.

In the simulation, pp collisions are generated using Pythia [13] with a specific LHCb configuration [14].Decays of hadronic particles are described and its response, are implemented using the Geant4 toolkit [17] as described in Ref. [18].


Candidate selection

The analysis is based on data recorded with the LHCb detector at centre-of-mass energies of 7 TeV and 8 TeV, corresponding to integrated luminosities of 1.0 fb −1 and 2.0 fb −1 , respectively.

The X 0 b candidates are formed from combinations of tracks that originate from a good quality common vertex.The tracks are identified as p, K or π candidates with loose particle identification (PID) requirements.The proton or antiproton identifies the candidate as a X 0 b baryon or X 0 b antibaryon.Reconstructed tracks are required to have
0 b → Λ + c (→ pK − π + )π − 2.23 < m(pK − π + ) < 2.31 GeV/c 2 Λ 0 b → D 0 (→ K − π + )pπ − 1.832 < m(K − π + ) < 1.844 GeV/c 2
p T > 250 MeV/c and p > 1.5 GeV/c, and are required to be displaced from any primary vertex.The latter requirement is imposed by selecting tracks with χ 2 IP > 16, where χ 2 IP is the change of the primary-vertex fit χ 2 when including the considered track.Only X 0 b candidates with a transverse momentum p T > 1.5 GeV/c are retained.To ensure that the X 0 b baryon is produced in the primary interaction, it is required that χ 2 IP (X 0 b ) < 16, and the flight direction of the X 0 b decay, calculated from its associated primary vertex, defined as that with minimum χ 2 IP (X 0 b ), and the decay vertex, must align with the reconstructed particle momentum with an angle that satisfies cos θ > 0.9999.

Decays of X 0 b bary ns to charm hadrons represent a source of background that originates from b → c transitions.Such background is vetoed by rejecting candidates with combinations of two or three final-state particles that have reconstructed invariant masses compatible with weakly decaying charm hadron states or with the J/ψ resonance.Among the vetoed candidates, those listed in Table 1 are used for assessing systematic uncertainties and for selection criteria optimization.Backgrounds from a pion or a kaon misidentified as a proton originating from B 0 and B 0 s decays with a φ or K * (892) 0 resonance are suppressed by vetoing the region within 10 and 70 MeV/c 2 of the φ and K * (892) 0 invariant masses, respectively, after applying the relevant substitution of the particle mass hypotheses.

A boosted decision tree (BDT) classifier [19] is used to suppress combinatorial background.Background from other b hadrons is suppressed by means of PID requirements.The Λ 0 b → pK − π + π − decay, which is the final state of interest with the largest yield, is used to train the classifier, since its kinematics and topology are very similar to those of
Λ 0 b → pK − K + K − and Ξ 0 b → pK − K − π + decays.
The signal training sample is obtained by subtracting the background using the sPlot technique and a fit to the invariant mass distribution [20].The candidates from the sideband region, 5.85 < m(pK − π + π − ) < 6.40 GeV/c 2 , are selected as the background training sample.The discriminating variables included in the BDT are the proton transverse and longitudinal momenta p T and p z ; the impact parameter of the K and π candidate tracks with respect to the X 0 b primary vertex; the χ 2 of the X 0 b decay vertex fit; the angle between the X 0 b momentum and its flight direction; the X 0 b χ 2 IP ; the asymmetry between the transverse momentum of the X 0 b and that of the charged tracks contained in a region defined as ∆η 2 + ∆φ 2 < 1.0, where ∆η (∆φ) is the difference of pseudorapidity (azimuthal angle) between the candidate and the charged tracks.No correlation is found between the discriminating va iables or the BDT output and the reconstructed b-baryon candidate mass.The signal and background training samples are divided into three statistically independent subsamples with selection criteria are optimised by maximising S/ √ e expected yield is estimated using S = S S 0 (B = B B 0 ), where the signal (background) efficiency S ( B ) of each BDT selection requirement is evaluated using Λ 0 b → pK − π + π − (data sideband) control samples; the reference signal (background) yield, S 0 (B 0 ), is obtained from a fit to the reconstructed invariant mass in the range [5.5 − 5.7] GeV/c 2 before applying the BDT selection.

The Λ 0 b → pD 0 (→ K − π + )π − sample is employed to optimise the PID selection since the momentum and pseudorapidity distributions of its final-state particles are similar to those of Λ 0
b → pK − π + π − , Λ 0 b at is maximised is defined as
S PID = ε S (PID) • N S ε S (PID) • N S + ε B (PID) • N B ,(4)
where the signal and background efficiencies of the PID selection criteria, ε S (PID) and ε B (PID), respectively, are determined using the Λ 0 b → pD 0 (→ K − π + )π − sample; N S (N B ) is the number of signal (background) candidates after applying the BDT selectio

Multiple candidates are reconstructed in less
than 1% of the selected events, and in such cases a single candidate is retained at random.

There are three main categories of background considered in the optimization process.Background from partially reconstructed decays is localised in the region at low invariant mass, and originates from
Λ 0 b → pπ + K − ρ − (→ π − π 0 ), Λ 0 b → pπ + π − K * − (→ K − π 0
) and similar decays, where the π 0 meson is not reconstructed.The background from misidentified final-state particles, called cross-feed in the following, consists of four-body Λ 0 b , B 0 and B 0 s decays, where one of them is reconstructed with the wrong mass hypothesis.The combinatorial background results from random combin tions of tracks in the event.


Measurement of the CP -violating asymmetries

For each signal mode, the selected data sample is split into four subsamples according to the X 0 b or X 0 b flavour and the sign of C T or C T .Simulated events and the Λ 0 b → Λ + c (pK − π + )π − control sample indicate that the reconstruction efficiencies for candidates with C T > 0 (−C T > 0) and C T < 0 (−C T < 0) are equal, within statistical uncertainties.For each final state, a simultaneous maximum likelihood fit to the m(pK − h + h − ) distribution of the four subsamples is used to determine the number of signal and background yields and the asymmetries A T and A T .The P -and CP -violating asymmetries, a T -odd P and a T -odd CP , are then obtained according to Eq. ( 3).

The invariant-mass distribution of the X 0 b signal is modelled by the sum of two Crystal Ball functions [22] that share the peak value and width but have tails on opposite sides of the peak.The parameters related to the tails and the relative fraction of the two Crystal Ball functions are determined from fits to simulated samples, and are fixed in fits made to data.The Ξ 0 b signal is also visible in the m(pK − π + π − ) and m(pK − K + K − ) invariant-mass distributions, and its peak value is fitted by imposing a Gaussian constraint using the known value of the mass difference of the Ξ 0 b and Λ 0 b baryons, 174.8 ± 2.5 MeV/c 2 [23].The combinatorial background distribution is modelled by an exponential function with the rate parameter determined from the data.Partially reconstructed Λ 0 b decays are described by a threshold function [24] convolved with a Gaussian function to account for resolution effects, the parameters of which are determined from the fit.The shap ) candidates have an identical final state to the Λ 0 b → pK − π + π − and Λ 0 b → pK − K + K − signal
decays and can potentially contribute to CP violation.These candidates are retained, together with the charmless 4-body decays, for the measurements of the asymmetries described below.Similar decays from Λ 0 b → pK − J/ψ with J/ψ → π + π − are removed due to the significant background from misidentified J/ψ → µ + µ − decays.

Two different approaches have been used to search for P and CP violation: a measurement integrated over the phase space and measurements in specific phase-space regions.The results of the first approach are obtained by fitting th full data sample and found to be compatible with P and CP symmetries, as shown in Table 2.

The CP -violating asymmetries may vary over the phase space due to the interference between resonant contributions.Therefore, measurements in specific phase-space regions may have better sensitivity to CP violation.In order to avoid biases, the binning schemes used to divide up t a.Two binning schemes are used for the Λ 0
b → pK − π + π − (Λ 0 b → pK − K + K − ) decay.
Schemes A and B (C and D) are designed to isolate regions of phase space according to the dominant resonant contributions and to exploit the potential interference of contributions as a function of the angle Φ between the decay planes formed by the pK − (pK − fast ) and the π + π − (K + K − slow ) systems, respectively.Scheme A (C) is defined in Table 4 (6) in Appendix B, while scheme B (D) has twelve (ten) nonoverlapping bins of width π/12 (π/10) in |Φ|.The size of the bins, and the resulting statistical uncertainty, is chosen to have sensitivity at the level of a few percent.The same fit model used for the integrated measurement is employed to fit each phase-space region.The distribu y is shown in Fig. 4 (5), and the results are reported in Table 5 (7) in Appendix B.

The compatibility with the CP -symmetry (P -symmetry) hypothesis is tested for each scheme individually by means of a χ 2 test, where the χ 2 is defined as
R T V −1 R, with R the array of a T -odd CP (a T -odd P
) measurements and V −1 the inverse of the covariance matrix, which is the sum of the statistical and systematic covariance matrices.An average systematic uncertainty, discussed in Section 5, is assumed for all bins.The statistical uncertainties are considered uncorrelated among the bins, while systematic uncertainties are assumed to be fully correlated.The results are consistent with the CP -symmetry hypothesis with a p-value of 0.93 (0.55), based on χ 2 /ndf= 7.2/14 (10.8/12) for scheme A (B) and a p-value of 0.95 (0.99), based on χ 2 /ndf= 2.1/7 (2.2/10) for scheme C (D).A similar χ 2 test is performed on the a T -odd P measurements.The results are consistent with the P -symmetry hypothesis with a p-value of 0.53 (0.80), based on χ 2 /ndf= 13.0/14 (7.8/12) for scheme A (B) and a p-value of 0.18 (0.73), based on χ 2 /ndf= 10.1/7 (6.9/10) for sche

he total uncertainty are listed in Tabl
3.The main source of systematic uncertainty is due to the experimental reconstruction and analysis technique, which could introduce potential biases in the measured asymmetries.This is tested by measuring the asymmetry a T -odd     The results of the fit are overlaid as described in the legend.The contribution of the cross-feeds to the fit results is barely just visible but is found to be nonnegligible.
CP (Λ + c π − ) forΛ Full fit − K + K − pK → 0 b Λ Comb. bkg. − π + K − pK → 0 b Λ − K + K − pK → 0 b Ξ − π + K + K − K → 0 B − K + K − K + K → 0 s B LHCb ] 2 c ) [GeV/ − K + K − pK ( m 5.Λ Full fit − K + K − pK → 0 b Λ Comb. bkg. − π + K − pK → 0 b Λ − K + K − pK → 0 b Ξ − π + K + K − K → 0 B − K + K − K + K → 0 s B LHCb ] 2 c ) [GeV/ − K + K − pK ( m 5.T C − ( 0 b Λ Full fit − K + K − pK → 0 b Λ Comb. bkg. − π + K − pK → 0 b Λ − K + K − pK → 0 b Ξ − π + K + K − K → 0 B − K + K − K + K → 0 s B LHCb ] 2 c ) [GeV/ − K + K − pK ( m 5.T C − ( 0 b Λ Full fit − K + K − pK → 0 b Λ Comb. bkg. − π + K − pK → 0 b Λ − K + K − pK → 0 b Ξ − π + K + K − K → 0 B − K + K − K + K → 0 s B LHCb
the Cabibbo-favoured Λ 0 b → Λ + c π − decay mode, where negligible CP violation is expected.The measured asymmetry is consistent with zero with a statistical uncertainty of 0 assigned as a systematic uncertainty in each bin of the different binning schemes A, B, C and D.

For the measurements of the triple products C T and C T , the systematic uncertainty from detector-resolution effects, which could introduce a migration of signal decays between The results of the fit are overlaid as described in the legend.The contribution of the B 0 → K − K + K + π − cross-feed to the fit results is barely visible but is found to be nonnegligible.the bins, is estimated from simulated samples of Λ
0 b → pK − π + π − , Λ 0 b → pK − K + K − and Ξ 0 b → pK − K − π + decays
, where neither P -nor CP -violating effects are present.The difference between the reconstructed and generated asymmetry is taken as systematic uncertainty and is less than 0.05% in all cases.

The systematic uncertainties related to the choice o The cross-feed backgrounds are described with one or two Crystal Ball functions with the tail and fraction parameters fixed from fits to simulated samples.Ten thousand pseudoexperiments are generated using the alternative models with the same event yields determined in the fits to data.The nominal model is then fitted to each generated sample and the asymmetry parameters are extracted.As the bias observed is not significantly different from zero, the statistical uncertainty on the mean of the pulls is taken as the systematic uncertainty due to the model.

Further cross-checks are made to test the stability of the results with respect to different periods of data-taking, the different magnet polarities, the choice made in the selection of multiple candidates, and the effect of the trigger and selection criteria.The results of these checks are all statistically compatible with the nominal results, and no systematic uncertainty is assigned.

Table 3: Sources of systematic uncertainty and their relative contributions to the total uncertainty.Where present, the value in brackets shows the systematic uncertainty ssigned to the measurement in specific phase-space regions.
Contribution Λ 0 b → pK − π + π − (%) Λ 0 b → pK − K + K − (%) Ξ 0 b → pK − K − π + (→ pK − π + π − , Λ 0 b → pK − K + K − and Ξ 0 b → pK − K − π + decays
are reconstructed, yielding 19877 ± 195, 5297 ± 83 and 709 ± 45 signal candidates, respectively.Two different measurements are made: one integrated over the phase space, and the other in specific phase-space regions.

No significant asymmetry is observed in the integrated measurements with a sensitivity of 0.8% in
Λ 0 b → pK − π + π − , 1.5% in Λ 0 b → pK − K + K − and 5.2% in Ξ 0 b → pK − K − π + decays,
w

phase space
for Λ 0 b → pK − π + π − and Λ 0 b → pK − K + K − decays are also all found to be consistent with conservation of both P symmetry and CP symmetry.


Appendices

A Observation of the Λ 0 b → χ c0 (1P )pK − decay

The π + π − and K + K − fast invariant-mass distributions, obtained by selecting Λ 0 b candidates within a signal window of ±2σ with respect to the reconstructed Λ 0 b mass peak, are shown in Fig. 6.The invariant mass distributions of the χ c0 (1P ) and χ c2 (1P ) signals are modelled by nonrelativistic Breit-Wigner functions convol ed with a Gaussian function to account for the detector resolution.The mean and width of the signal Breit-Wigner functions are fixed to known values [23], while the detector resolution, identical for the χ c0 (1P ) and χ c2 (1P ) signals, is determined from the data.The background, from random combinations of tracks and from Λ 0 b decays that do not proceed via the χ c0 (1P ) states, is modelled by an exponential function.An unbinned extended maximum