## Abstract

A first search for \(CP\) violation in the Cabibbo-suppressed \({{{\varXi }} ^+_{c}} \rightarrow {p} {{K} ^-} {{\pi } ^+} \) decay is performed using both a binned and an unbinned model-independent technique in the Dalitz plot. The studies are based on a sample of proton-proton collision data, corresponding to an integrated luminosity of \(3.0\,\text {fb} ^{-1} \), and collected by the LHCb experiment at centre-of-mass energies of 7 and \(8\,\text {TeV} \). The data are consistent with the hypothesis of no \(CP\) violation.

## 1 Introduction

The non-invariance of fundamental interactions under the combination of charge conjugation and parity transformation, known as \(CP\) violation (\(CPV\)), is a key requirement for the generation of the baryon–antibaryon asymmetry in the early Universe [1,2,3]. In the Standard Model (SM) of particle physics, \(CPV\) is included through the introduction of a single irreducible complex phase in the Cabibbo–Kobayashi–Maskawa (CKM) quark-mixing matrix [4, 5]. The amount of \(CPV\) predicted by the CKM mechanism is not sufficient to explain a matter-dominated universe [6, 7] and other sources of \(CPV\) are required. The realization of \(CPV\) in nature has been well established in the *K*- and *B*-meson systems by several experiments [8,9,10,11,12,13,14]. The LHCb experiment has observed for the first time \(CPV\) in the charm-meson sector as the difference of the \(CP\) asymmetries between the two-body decays \(D^0\rightarrow K^-K^+\) and \(D^0\rightarrow \pi ^-\pi ^+\) [15]. A similar study using \({{\varLambda }} ^+_{c} \) to \({p} {{K} ^-} {{K} ^+} \) and \({p} {{\pi } ^-} {{\pi } ^+} \) found no evidence for \(CPV\) [16]. Indeed, so far, \(CPV\) has never been observed in any baryon system. Evidence for \(CPV\) in the \(b \) baryon sector reported by the LHCb collaboration in [17] has not been confirmed with more data [18]. Further measurements of processes involving the decay of charm hadrons can shed light on the origin and magnitude of \(CPV\) mechanisms within the SM and beyond.

In two-body decays of charm hadrons, \(CPV\) can manifest itself as an asymmetry between partial decay rates. Multi-body decays offer access to more observables that are sensitive to \(CP\)-violating effects. For a three-body baryon decay the kinematics can be characterised by three Euler angles and two squared invariant masses, which form a Dalitz plot [19]. The Euler angles are redundant if all initial spin states are integrated over. Interference effects in the Dalitz plot probe \(CP\) asymmetries in both the magnitudes and phases of amplitudes. In three-body decays there can be large local \(CP\) asymmetries in the Dalitz plot, even when no significant global \(CPV\) exists. A recent example has been measured in the decay \({{{B} ^+}} \rightarrow {{\pi } ^+} {{\pi } ^-} {{\pi } ^+} \) [20].

In the SM, \(CPV\) asymmetries in the charm sector are expected at the order of \(10^{-3}\) or less [21] for singly Cabibbo-suppressed (SCS) decays. New physics (NP) contributions can enhance \(CP\)-violating effects up to \(10^{-2}\) [22,23,24,25,26,27,28,29,30]. Searches for \(CPV\) in \({{\varXi }} ^+_{c} \) baryon decays^{Footnote 1} provide a test of the SM and place constraints on NP parameters [31,32,33,34,35]. In contrast to SCS decays, in Cabibbo-favoured (CF) charm-quark transitions, such as \({{{\varLambda }} ^+_{c}} \rightarrow {p} {{K} ^-} {{\pi } ^+} \) decays, there is only one dominant amplitude in the SM, resulting in no \(CP\)-violating effects. However this could change with NP, as argued above in the case of SCS decays.

This article describes searches for direct \(CPV\) in the SCS decay \({{{\varXi }} ^+_{c}} \rightarrow {p} {{K} ^-} {{\pi } ^+} \), for \({{{\varXi }} ^+_{c}} \) baryons produced promptly in \({p} {p} \) collisions. The \({{{\varLambda }} ^+_{c}} \rightarrow {p} {{K} ^-} {{\pi } ^+} \) decay is used as a control mode to study in data the level of experimental asymmetries that pollute the measurement. In this paper, the symbol \(H_{c}^+\) is used to refer to both \({{\varXi }} ^+_{c} \) and \({{\varLambda }} ^+_{c} \). It is assumed that the polarisation of charm baryons produced in \({p} {p} \) collisions is sufficiently small, as it is for \(b \)-baryons [36], to justify the integration over the Euler angles. This measurement uses \({p} {p} \) collision data, corresponding to an integrated luminosity of \(3\,\text {fb} ^{-1} \), recorded by the LHCb detector in 2011 (\(1\,\text {fb} ^{-1} \)) at a centre-of-mass energy of \(7\,\text {TeV} \), and in 2012 (\(2\,\text {fb} ^{-1} \)) at a centre-of-mass energy of \(8\,\text {TeV} \). The magnetic field polarity is reversed regularly during the data taking in order to minimise effects of charged particle and antiparticle detection asymmetries. Approximately half of the data are collected with each polarity.

There is presently no successful method for computing decay amplitudes in multi-body charm decays, which could provide reliable predictions on how the \(CP\) asymmetries vary over the phase space of the decay. This situation favours a model-independent approach, which looks for differences between multivariate density distributions for baryons and antibaryons. Therefore, in this article searches for \(CPV\) are performed through a direct comparison between the Dalitz plots of \({{\varXi }} ^+_{c} \) and \({{\varXi }} ^-_{c} \) decays using a binned significance (\(S_{{CP}}\)) method [37] and an unbinned k-nearest neighbour method (kNN) [38,39,40,41], both of which are model independent.

## 2 Detector and simulation

The LHCb detector [42, 43] is a single-arm forward spectrometer covering the pseudorapidity range \(2<\eta <5\). It is designed for the study of particles containing *b* and *c* quarks. The detector includes a high-precision tracking system consisting of a silicon-strip 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 \(4~\mathrm{Tm}\), and three stations of silicon-strip detectors and straw drift tubes placed downstream of the magnet. The tracking system provides a measurement of the momentum, *p*, of charged particles with a relative uncertainty that varies from 0.5% at low momentum to 1.0% at \(200~\mathrm{GeV}/c\). The minimum distance of a track to a primary vertex (PV), the impact parameter (IP), is measured with a resolution of \((15+ 29/p_{{\mathrm {T}}})~ \upmu \mathrm{m}\), where \(p_{{\mathrm {T}}}\) is the component of the momentum transverse to the beam, in \(\mathrm{GeV}/c\). Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors. Photons, electrons and hadrons are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic and a hadron calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers.

Samples of simulated events are used to optimise the signal selection, to derive the angular efficiency and to correct the decay-time efficiency. In the simulation, \({p} {p} \) collisions are generated using PYTHIA [44] with a specific LHCb configuration [45]. Decays of hadronic particles are described by EVTGEN [46], in which final-state radiation is generated using PHOTOS [47]. The interaction of the generated particles with the detector, and its response, are implemented using the GEANT4 toolkit [48] as described in Ref. [49].

## 3 Selection of signal candidates

The online event selection is performed by a trigger consisting of a hardware stage, based on information from the calorimeter and muon systems, followed by two software stages. At the hardware trigger stage, events are required to have either muons with high \(p_{{\mathrm {T}}}\) or hadrons, photons or electrons with a high transverse-energy deposit in the calorimeters. For hadrons, the transverse energy threshold is approximately \(3.5~\mathrm{GeV}/c^2\). In the first software trigger stage at least one good-quality track with \(p_{{\mathrm {T}}} >300~\mathrm{MeV}/c\) is required. In the second software trigger stage an \(H^+_c\) candidate is fully reconstructed from three high-quality tracks not pointing to any PV. The three tracks should form a secondary vertex (SV) which must be well separated from any PV. A momentum \(p>3 \,\text {GeV/}c \) for each track and the scalar sum of \(p_{{\mathrm {T}}}\) for the three tracks \(p_{{\mathrm {T}}} >2 \,\text {GeV/}c \) are required. The combined invariant mass of the three tracks is required to be in the range \(2190{-}2570~\,\text {MeV/}c^2 \). Requirements are also placed on the particle identification criteria of the tracks and on the angle between the vector from the associated PV to the SV and the \(H_c^+\) momentum. The associated PV is the one with smallest difference in vertex fit \(\chi ^2\) when performed with and without the \(H_c^+\) candidate.

In the offline analysis, tighter selection requirements are placed on the track-reconstruction quality, the *p* and \(p_{{\mathrm {T}}}\) of the final-state particles. For protons \(10<p < 100~\,\text {GeV/}c \) is required, while kaons and pions momentum satisfies \(3<p< 150 \,\text {GeV/}c \). Only \(H_c^+\) candidates with \(p_{{\mathrm {T}}}\) in the range \(4< p_{{\mathrm {T}}} < 16~\,\text {GeV} \) are retained. Additional requirements are also made on the SV fit quality, and the minimum significance of the displacement from the SV to any PV in the event. This reduces the contribution of charm baryons from \(b \)-hadron decays to less than 5% of the prompt signal. Reconstructed particles are accepted if their momenta are within a region defined by \(|p_x|<0.2p_z\) and \(|p_x|>0.01p_z\), where \(p_x\) and \(p_z\) are the momentum components along the *x* and *z* axes.^{Footnote 2} This requirement has a signal loss of 25%, and is imposed to avoid large detection asymmetries that are present in the excluded kinematic regions. Differences between particles and antiparticles in reconstruction efficiencies are also observed for \(H^+_c\) candidates where \(p<20~\mathrm{GeV}/c\) for all charged tracks. These differences do not cancel by simply averaging the data acquired with opposite magnet polarities. To minimise the reconstruction asymmetry, the momentum of all tracks is required to be greater than \(20~\mathrm{GeV}/c\). This requirement rejects about 20% of the selected charm-baryon candidates.

The distributions of the invariant-mass, \(M(pK^-\pi ^+)\), of selected \({{\varLambda }} ^+_{c} \) and \({{\varXi }} ^+_{c} \) candidates are presented in Figs. 1 and 2, respectively, with fit curves overlaid. The fit model comprises a sum of two Gaussian functions describing the signal and a second-order Chebyshev polynomial function describing the combinatorial background. No additional source of background is found to contribute significantly, according to studies in data reconstructed with different mass hypotheses.

The final samples used for the \(CPV\) search comprise all candidates with \(M(pK^-\pi ^+)\) within \(\pm 3\sigma \) around \(m({{{\varLambda }} ^+_{c}})\) or \(m({{{\varXi }} ^+_{c}})\), where \(\sigma \) is the weighted average of the two fitted Gaussian widths and \(m({{{\varLambda }} ^+_{c}})\) and \(m({{{\varXi }} ^+_{c}})\) are the masses of the \({{\varLambda }} ^+_{c} \) and \({{\varXi }} ^+_{c} \) baryons [50]. There are approximately 2.0 million \({{\varLambda }} ^+_{c} \) candidates (0.4 million in the 2011 and 1.6 million in the 2012 data sample) and 0.25 million \({{\varXi }} ^+_{c} \) candidates (0.05 million in the 2011 and 0.2 million in the 2012 data sample). The purity for \({{\varLambda }} ^+_{c} \) decays is 94% for 2011 and 98% for 2012 and that for \({{\varXi }} ^+_{c} \) decays is 77% for 2011 and 78% for 2012, where purity is defined as the number of signal candidates obtained from the fit to the invariant-mass distribution divided by the total number of candidates.

## 4 Methods

The Dalitz plot for \(H_{c}^+\rightarrow pK^-\pi ^+\) is formed by the squares of the invariant masses of two pairs of the decay products: \(M^2(K^-\pi ^+)\) and \(M^2(pK^-)\). Comparisons of the Dalitz plots of \(H_{c}^+\) and \(H_{c}^-\) candidates are performed using the binned \(S_{CP} \) and the unbinned kNN methods, described in the following. For both the binned \(S_{{CP}}\) and unbinned kNN methods, a signal of \(CPV\) is established if a *p*-value lower then \(3\times 10^{-7}\) is found, corresponding to an exclusion of \(CP\) symmetry with a significance of five standard deviations. However, in case that no \(CPV\) is found, there is no model-independent mechanism for setting an upper limit on the amount of \(CPV\) in the Dalitz plot.

### 4.1 Binned \(S_{CP} \) method

The \(S_{CP} \) method [37] has been used before for searches of \(CPV\) testing in charm and beauty decays [41, 51,52,53,54]. This method is used to search for localised asymmetries in the phase space of the decay \(H_{c}^+\rightarrow pK^-\pi ^+\) and is based on a bin-by-bin comparison between the Dalitz plots of baryons, \(H_{c}^+\), and antibaryons, \(H_{c}^-\). The Dalitz plots of \(H_{c}^+\) and \(H_{c}^-\) are divided using an identical binning. For each bin *i* of the Dalitz plot, the significance of the difference between the number of \(H_{c}^+\) (\(n^i_+\)) and \(H_{c}^-\) (\(n^i_-\)) candidates, is computed as

where the factor \(\alpha \) is defined as \(\alpha =\frac{n_+}{n_-}\) and \(n_+\), \(n_-\) are the total number of \(H_{c}^+\), \(H_{c}^-\) candidates. This factor accounts for asymmetries arising in the production of \(H_{c}^+\) baryons, as well as in the detection of the final-state particles. The production and global detection asymmetries do not to depend on the Dalitz plot position.

A numerical comparison between the Dalitz plots of the \(H_c^+\) and \(H_c^-\) candidates is made using a \(\chi ^2\) test defined as

A *p*-value for the hypothesis of no \(CPV\) is obtained from the \(\chi ^2\) distribution considering that the number of degrees of freedom is equal to the total number of bins minus one, due to the constraint on the factor \(\alpha \) of the overall \(H_c^+\) and \(H_c^-\) normalisation.

In the hypothesis of no \(CPV\), the \(S_{CP} \) values are expected to be distributed according to the normal distribution with a mean of zero and a standard deviation of unity. The test is performed using only bins with a minimum of 10 \(H_c^+\) and 10 \(H_c^-\) candidates. In case of \(CPV\), a deviation from the normal distribution is expected, generating a *p*-value close to zero.

### 4.2 Unbinned kNN method

The kNN method is based on the concept of a set of nearest neighbour candidates (\(n_k\)) in a combined sample of two data sets: baryons and antibaryons. As an unbinned method, the kNN approach is more sensitive to a \(CPV\) search in a sample with limited data, compared to that of the binned \(S_{{CP}}\) method. The kNN method is used here to test whether baryons and antibaryons share the same parent distribution function [38,39,40]. To find the \(n_k\) nearest neighbour events of each \(H_c^+\) or \(H_c^-\) candidate, an Euclidean distance between closest points in the Dalitz plot is used. A test statistic *T* for the null hypothesis is defined as

where \(I(i,k)=1\) if the *i*th candidate and its *k*th nearest neighbour have the same charge and \(I(i,k)=0\) otherwise.

The test statistic *T* is the mean fraction of like-charged neighbour pairs in the sample of \(H_c^+\) and \(H_c^-\) decays. The advantage of the kNN method, in comparison with other proposed methods for unbinned analyses [38], is that the calculation of *T* is simple and fast and the expected distribution of *T* is well known. Under the hypothesis of no \(CPV\), *T* follows a normal distribution with a mean, \(\mu _T\), and a variance, \(\sigma _T\), where

with \(n=n_+ + n_-\) and \(D=2\) is the dimensionality of the tested distribution. A good approximation of \(\sigma _T\) is obtained even for \(D = 2\) for the current values of \(n_+\), \(n_-\) and \(n_k\) [38].

For \(n_+ = n_-\) the mean \(\mu _T\) can be expressed as

and is called the reference value, \(\mu _{TR}\). For large *n*, \(\mu _{TR}\) asymptotically tends to 0.5.

To increase the power of the kNN method, the Dalitz plot is divided into regions defined around the expected resonances. It can provide one of the necessary conditions for observation of \(CPV\): large relative strong phases in the final states of interfering amplitudes of the intermediate resonance states. The Dalitz plot is partitioned into six regions for the decays of the \({{\varLambda }} ^+_{c} \) control mode and eleven regions for signal \({{\varXi }} ^+_{c} \) decays according to the present of resonances of the phase space, as shown in Fig. 3. The definitions of the regions are also given in Tables 1 and 2 for \({{\varLambda }} ^+_{c} \) and \({{\varXi }} ^+_{c} \) baryons, respectively. For \({{\varLambda }} ^+_{c} \) decays the \(K^*(892)\), \(K^* (1430)\), \({{\varDelta }} (1232)\), \({{\varLambda }} (1520)\), \({{\varLambda }} (1670)\), \({{\varLambda }} (1690)\) resonances are seen in data, whilst for \({{\varXi }} ^+_{c} \) decays additional resonances are seen, namely \({{\varLambda }} (1520)\), \({{\varLambda }} (1600)\), \({{\varLambda }} (1710)\), \({{\varLambda }} (1800)\), \({{\varLambda }} (1810)\), \({{\varLambda }} (1820)\), \({{\varLambda }} (1830)\), \({{\varLambda }} (1890)\), \({{\varDelta }} (1600)\), \({{\varDelta }} (1620)\) and \({{\varDelta }} (1700)\). For \({{\varLambda }} ^+_{c} \) decays there are four independent regions (R1–R4), whilst the region R2 is further split into the high \(M^2(pK^-)\) region (R6) and the low \(M^2(pK^-)\) region (R5). For \({{\varXi }} ^+_{c} \) there are seven independent regions (R1–R7), whilst the region R2 is split in mass \(M^2(pK^-)\) in two regions at larger mass (R9) and smaller mass (R8), \(\hbox {R2}= \hbox {R8} \cup \hbox {R9}\), similarly for R10 and R11, where \(\hbox {R10} = \hbox {R4} \cup \hbox {R5}\), and \(\hbox {R11} = \hbox {R4} \cup \hbox {R5} \cup \hbox {R6} \cup \hbox {R7}\). Region R0 is the full Dalitz plot.

## 5 Control mode, background and sensitivity studies

The \(S_{{CP}}\) and kNN methods are tested using the \({{{\varLambda }} ^+_{c}} \rightarrow {p} {{K} ^-} {{\pi } ^+} \) control mode where the \(CP\) asymmetry is expected to be null [22,23,24,25,26,27,28,29,30]. The sidebands of \({{{\varXi }} ^+_{c}} \rightarrow {p} {{K} ^-} {{\pi } ^+} \) candidates in the mass regions \(2320<M({p} {{K} ^-} {{\pi } ^+})<2445~\mathrm{MeV}/c^2\) and \(2490<M({p} {{K} ^-} {{\pi } ^+})<2650~\mathrm{MeV}/c^2\) are used to check that the background does not introduce spurious asymmetries.

The measured total raw asymmetry is defined as

and it depends on the production asymmetry of \(H_c^+\) baryons and on the detection asymmetries that arise through charge-dependent selection efficiencies due to track reconstruction, trigger selection and particle identification. The measured value of \(A_{\mathrm{Raw}}\) in each region of the Dalitz plot of \({{{\varLambda }} ^+_{c}} \rightarrow {p} {{K} ^-} {{\pi } ^+} \) decays is presented in Fig. 4. The measured \(A_{\mathrm{Raw}}\) value integrated over the Dalitz plot equals \(-0.0230\,\pm \, 0.0016\) and \(-0.0188\pm 0.0008\) in the 2011 and 2012 data samples, where the uncertainties are statistical only. Within uncertainties, \(A_{\mathrm{Raw}}\) in all regions amounts to about \(-2\)%. There is no significant difference in the measurement of \(A_{\mathrm{Raw}}\) between the 2011 and 2012 data samples. Since the production and detection asymmetries of \({{\varLambda }} ^+_{c} \) baryons can depend on the baryon pseudorapidity, \(\eta \), and \(p_{{\mathrm {T}}}\), the dependence of \(A_{\mathrm{Raw}}\) in regions of the Dalitz plot is checked in bins of \(\eta \) and \(p_{{\mathrm {T}}}\) of the \({{{\varLambda }} ^+_{c}} \) baryon. It is observed that the value of \(A_{\mathrm{Raw}}\) globally changes from bin to bin of \(\eta \) and \(p_{{\mathrm {T}}}\) of the \({{\varLambda }} _{c} \) candidates, but for a given bin of \(\eta \) and \(p_{{\mathrm {T}}}\) a constant behaviour of \(A_{\mathrm{Raw}}\) in regions of the Dalitz plot is maintained.

In the \(S_{CP} \) method the production asymmetry and all global effects are considered by introducing the \(\alpha \) factor, following the strategy described in Sect. 4.1. The *p*-values obtained are larger than 58%, consistent with the absence of localised asymmetries. As an example, Fig. 5 shows the distribution of \(S^i_{CP} \) for \({{{\varLambda }} ^+_{c}} \rightarrow {p} {{K} ^-} {{\pi } ^+} \) decays considering uniform binning, and for two granularities of the Dalitz plot: 28 and 106 bins in the 2012 sample. Alternatively the Dalitz plot is divided into different size bins with the same number of events in each bin. The *p*-values obtained are larger than 34%, consistent with the hypothesis of absence of localised asymmetries.

Following the strategy described in Sect. 4.2, the results of the kNN method in regions of the Dalitz plot for the \({{{\varLambda }} ^+_{c}} \rightarrow {p} {{K} ^-} {{\pi } ^+} \) control mode are presented in Fig. 6, for \(n_k = 50\). The pulls, \((\mu _T - \mu _{TR} )/{\varDelta }(\mu _T - \mu _{TR} )\), where \({\varDelta }(\mu _T - \mu _{TR} )\) is the statistical uncertainty on the difference \((\mu _T - \mu _{TR} )\), are different from zero in all regions. The largest pull value is observed when integrated over the full Dalitz plot. This asymmetry is the result of the nonzero production asymmetry that is presented in Fig. 4 and discussed above. Pulls of the test statistic *T*, \(((T - \mu _T )/\sigma _T )\), vary within \(-3\) and \(+3\), consistent with the hypothesis of absence of localised asymmetries in any region. The difference among data-taking years are consistent with statistical fluctuations. Figure 6 illustrates how the larger 2012 data sample improves the power of the kNN method. In Run 2 (years of data taking 2016, 2017 and 2018) the yield is expected to be about three times larger than that from Run 1.

The interaction cross-section of charged hadrons with matter depends on the charged hadron momentum. As such, the detection asymmetries of the proton and kaon-pion systems are momentum dependent. Pseudoexperiments are performed to check whether the detection asymmetries related to particles reconstructed in the final state can generate a spurious \(CP\) asymmetry. The proton detection asymmetry varies from about 5% at low momentum to 1% at \(100~\mathrm{GeV}/c\) and is estimated using simulations. The kaon-pion detection asymmetry is measured to vary from \(-1.4\)% at low momentum to \(-0.7\)% at \(60~\mathrm{GeV}/c\) [55]. The combined effect of the two asymmetries is found to cancel approximately and does not generate a spurious \(CP\) asymmetry in the Dalitz plot.

These studies are repeated using the candidates in the sideband of the \({{{\varXi }} ^+_{c}} \rightarrow {p} {{K} ^-} {{\pi } ^+} \) mass distribution. No spurious \(CP\) asymmetry is found for both methods. For further cross-checks, the control samples are divided according to the polarity of the magnetic field. The *p*-values are found to be distributed uniformly.

The expected statistical powers of both methods are obtained by performing pseudoexperiments. One hundred samples of \({{{\varXi }} ^+_{c}} \rightarrow {p} {{K} ^-} {{\pi } ^+} \) decays are generated, each with a yield and purity equivalent to that observed in the combined 2011 and 2012 data samples, resulting in 200 000 \({{\varXi }} ^+_{c} \) decays generated in each pseudoexperiment. In this model, the two-dimensional Dalitz plots are generated assuming that the \({{\varXi }} ^+_{c} \) baryons are produced unpolarised. This model is built by including the resonances observed in the data, using the same software as in Ref. [56]. The same resonances as described in Sect. 4.2 are included. The statistical powers of the two methods are found to be comparable. Both methods are sensitive to a 5% \(CP\) asymmetry in the \(K^*(892)\) and \({{\varDelta }} (1232)\) resonance regions with 3 and 5 sigma significances that would be observed in 69% and 10% of the cases for the kNN method and 17% and 10% of the cases for the \(S_{CP} \) method, respectively.

## 6 Results

### 6.1 Binned \(S_{CP} \) method

The binned \(S_{CP} \) method is applied to look for local \(CP\) asymmetries in \({{{\varXi }} ^+_{c}} \rightarrow {p} {{K} ^-} {{\pi } ^+} \) decays following the strategy described in Sect. 4.1. The distribution of \(S_{CP} ^i\) for \({{{\varXi }} ^+_{c}} \rightarrow {p} {{K} ^-} {{\pi } ^+} \) decays considering uniform binning, and for two granularities of the Dalitz plot: 29 and 111 bins are shown in Fig. 7 for the combined 2011 and 2012 data samples. The normalization factor \(\alpha \), defined in Eq. 1, is determined to be \(1.029\pm 0.004\). The measured *p*-values using a \(\chi ^2\) test are larger than 32%, consistent with no evidence for \(CPV\). The obtained \(S_{CP} \) distributions agree with a normal distribution. It is also checked that the results in the 2011 and 2012 data samples are consistent with each other.

### 6.2 Unbinned kNN method

The unbinned kNN method is applied to look for \(CP\) asymmetry in \({{{\varXi }} ^+_{c}} \rightarrow {p} {{K} ^-} {{\pi } ^+} \) decays, following the strategy described in Sect. 4.2. The results are presented in Fig. 8 for \(n_k = 50\) for the merged 2011 and 2012 data samples. The measured pull values, \(((\mu _T - \mu _{TR} )/{\varDelta }(\mu _T - \mu _{TR} ))\), are different from zero. The largest value of pull is observed integrated over the full Dalitz plot. This is due to the expected nonzero production and detector asymmetries, that is presented in Fig. 9. The measured \(A_{\mathrm{Raw}}\) is constant within uncertainties in all regions.

The pulls of the test statistic *T*, \(((T - \mu _T )/\sigma _T )\), shown in Fig. 8 vary within \(-3\) and \(+3\), consistent with the hypothesis of absence of localised asymmetries. To check for any systematic effects the kNN test is repeated for the individual 2011 and 2012 data samples as well as for samples separated according to the polarity of the magnetic field. All obtained results are compatible within uncertainties and no systematic effects are observed.

Since the sensitivity of the method can depend on the \(n_k\) parameter, the analysis is repeated with different values of \(n_k\) from 10 up to 3000. Only *T* and \(\sigma _T\) depend on \(n_k\). Pulls of the statistic *T* for the entire Dalitz plot are shown in Fig. 10. All results show no significant deviation from the hypothesis of \(CP\) symmetry.

## 7 Conclusions

Model-independent searches for \(CP\) violation in \({{{\varXi }} ^+_{c}} \rightarrow {p} {{K} ^-} {{\pi } ^+} \) decays are presented using the binned \(S_{CP} \) and the unbinned kNN methods. The \({{{\varLambda }} ^+_{c}} \rightarrow {p} {{K} ^-} {{\pi } ^+} \) candidates and the sideband regions of \({{{\varXi }} ^+_{c}} \rightarrow {p} {{K} ^-} {{\pi } ^+} \) candidates are used to ensure that no spurious charge asymmetries affect the methods. Both methods are sensitive to \(CP\) asymmetry larger than a 5% in the regions around the \(K^*(892)\) and the \({{\varDelta }} \)(1232). The obtained results are consistent with the absence of \(CP\) violation in \({{{\varXi }} ^+_{c}} \rightarrow {p} {{K} ^-} {{\pi } ^+} \) decays.

## Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors' comment: The datasets analysed during the current study are available from the corresponding author on reasonable request.]

## Notes

Unless stated explicitly, the inclusion of charge-conjugate states is implied throughout.

The LHCb coordinate system is right-handed, with the

*z*axis pointing along the beam axis,*y*the vertical direction, and*x*the horizontal direction. The (*x*,*z*) plane is the bending plane of the dipole magnet.

## References

A.D. Sakharov, Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe. Pisma Zh. Eksp. Teor. Fiz.

**5**, 32 (1967). https://doi.org/10.1070/PU1991v034n05ABEH002497A.D. Sakharov, Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe. Usp. Fiz. Nauk

**161**, 61 (1991)M. Dine, A. Kusenko, The origin of the matter–antimatter asymmetry. Rev. Mod. Phys.

**76**, 1 (2003). https://doi.org/10.1103/RevModPhys.76.1. arXiv:hep-ph/0303065N. Cabibbo, Unitary symmetry and leptonic decays. Phys. Rev. Lett.

**10**, 531 (1963). https://doi.org/10.1103/PhysRevLett.10.531M. Kobayashi, T. Maskawa, CP-violation in the renormalizable theory of weak interaction. Prog. Theor. Phys.

**49**, 652 (1973). https://doi.org/10.1143/PTP.49.652M.B. Gavela, P. Hernandez, J. Orloff, O. Pène, Standard model CP violation and baryon asymmetry. Mod. Phys. Lett. A

**9**, 795 (1994). https://doi.org/10.1142/S0217732394000629. arXiv:hep-ph/9312215T. Vieu, A.P. Morais, R. Pasechnik, Electroweak phase transitions in multi-Higgs models: the case of trinification-inspired THDSM. JCAP

**1807**, 014 (2018). https://doi.org/10.1088/1475-7516/2018/07/014. arXiv:1801.02670J.H. Christenson, J.W. Cronin, V.L. Fitch, R. Turlay, Evidence for the \(2\pi \) decay of the \(K_2^0\) meson. Phys. Rev. Lett.

**13**, 138 (1964). https://doi.org/10.1103/PhysRevLett.13.138BaBar Collaboration, B. Aubert et al., Observation of CP violation in the \(B^0\) meson system. Phys. Rev. Lett.

**87**, 091801 (2001). https://doi.org/10.1103/PhysRevLett.87.091801. arXiv:hep-ex/0107013Belle Collaboration, K. Abe et al., Observation of large CP violation in the neutral \(B\) meson system. Phys. Rev. Lett.

**87**, 091802 (2001). https://doi.org/10.1103/PhysRevLett.87.091802. arXiv:hep-ex/0107061BaBar Collaboration, B. Aubert et al., Observation of direct CP violation in \(B^0~\rightarrow ~K^+\pi ^-\) decays. Phys. Rev. Lett.

**93**, 131801 (2004). https://doi.org/10.1103/PhysRevLett.93.131801. arXiv:hep-ex/0407057Belle Collaboration, Y. Chao et al., Evidence for direct CP violation in \(B^0 \rightarrow K^+ \pi ^-\) decays. Phys. Rev. Lett.

**93**, 191802 (2004). https://doi.org/10.1103/PhysRevLett.93.191802. arXiv:hep-ex/0408100LHCb Collaboration, R. Aaij et al., First observation of CP violation in the decays of \({B}^0_{s}\) mesons. Phys. Rev. Lett.

**110**, 221601 (2013). https://doi.org/10.1103/PhysRevLett.110.221601. arXiv:1304.6173LHCb Collaboration, R. Aaij et al., Observation of CP violation in \({{B}}^\pm \rightarrow {{D}} {{K}}^\pm \) decays. Phys. Lett. B

**712**, 203 (2012). https://doi.org/10.1016/j.physletb.2012.04.060. arXiv:1203.3662 [Erratum Phys. Lett. B**713**, 351 (2012). https://doi.org/10.1016/j.physletb.2012.05.060]LHCb Collaboration, R. Aaij et al., Observation of CP violation in charm decays. Phys. Rev. Lett.

**122**, 211803 (2019). https://doi.org/10.1103/PhysRevLett.117.211803. arXiv:1903.08726LHCb Collaboration, R. Aaij et al., Search for CP violation in \({\Lambda }_{{c}} \rightarrow p {{K}}^- {{K}}^+\) and \({\Lambda }_{{c}} \rightarrow p\pi ^-\pi ^+\) decays. JHEP

**03**, 182 (2018). https://doi.org/10.1007/JHEP03(2018)182. arXiv:1712.07051LHCb Collaboration, R. Aaij et al., Measurement of matter–antimatter differences in beauty baryon decays. Nat. Phys.

**13**, 391 (2017). https://doi.org/10.1038/nphys4021. arXiv:1609.05216LHCb Collaboration, R. Aaij et al., Search for CP violation in \({\Lambda }^0_{{b}} \rightarrow {p\pi ^-\pi ^+\pi ^-}\) decays. arXiv:1912.10741

C. Zemach, Three pion decays of unstable particles. Phys. Rev.

**133**, B1201 (1964). https://doi.org/10.1103/PhysRev.133.B1201LHCb Collaboration, R. Aaij et al., Observation of several sources of CP violation in \({{B}}^+ \rightarrow \pi ^+\pi ^+\pi ^-\) decays. Phys. Rev. Lett.

**124**, 031801 (2020). https://doi.org/10.1103/PhysRevLett.124.031801. arXiv:1909.05211S. Bianco, F.L. Fabbri, D. Benson, I. Bigi, A Cicerone for the physics of charm. Riv. Nuovo Cim.

**26N7**, 1 (2003). https://doi.org/10.1393/ncr/i2003-10003-1. arXiv:hep-ex/0309021I. Shipsey, Status of charm flavor physics. Int. J. Mod. Phys. A

**21**, 5381 (2006). https://doi.org/10.1142/S0217751X06034525. arXiv:hep-ex/0607070M. Artuso, B. Meadows, A. Petrov, Charm meson decays. Ann. Rev. Nucl. Part. Sci

**58**, 249 (2008). https://doi.org/10.1146/annurev.nucl.58.110707.171131S. Bianco, I.I. Bigi, 2019 Lessons from \(\tau (\Omega _c^0)\) and CP asymmetry in charm decays. arXiv:2001.06908

Y. Grossman, S. Schacht, The emergence of the \(\Delta U=0\) rule in charm physics. JHEP

**07**, 020 (2019). https://doi.org/10.1007/JHEP07(2019)020. arXiv:1903.10952H.-N. Li, C.-D. Lü, F.-S. Yu, Implications on the first observation of charm CPV at LHCb. arXiv:1903.10638

H.-Y. Cheng, C.-W. Chiang, Revisiting CP violation in \(D\rightarrow PP\) and \(VP\) decays. Phys. Rev. D

**100**, 093002 (2019). https://doi.org/10.1103/PhysRevD.100.093002. arXiv:1909.03063L. Calibbi, T. Li, Y. Li, B. Zhu, Simple model for large CP violation in charm decays, B-physics anomalies, muon g-2, and dark matter. arXiv:1912.02676

M. Chala, A. Lenz, A.V. Rusov, J. Scholtz, \(\Delta A_{CP}\) within the standard model and beyond. JHEP

**07**, 161 (2019). https://doi.org/10.1007/JHEP07(2019)161. arXiv:1903.10490A. Dery, Y. Nir, Implications of the LHCb discovery of CP violation in charm decays. JHEP

**12**, 104 (2019). https://doi.org/10.1007/JHEP12(2019)104. arXiv:1909.11242Y. Grossman, A.L. Kagan, Y. Nir, New physics and CP violation in singly Cabibbo suppressed D decays. Phys. Rev. D

**75**, 036008 (2007). https://doi.org/10.1103/PhysRevD.75.036008. arXiv:hep-ph/0609178I.I. Bigi, Probing CP asymmetries in charm baryons decays. arXiv:1206.4554

Y. Grossman, S. Schacht, U-spin sum rules for CP asymmetries of three-body charmed baryon decays. Phys. Rev. D

**99**, 033005 (2019). https://doi.org/10.1103/PhysRevD.99.03