Search for $CP$ violation using $T$-odd correlations in $D^0 \to K^+K^-\pi^+\pi^-$ decays

A search for $CP$ violation using $T$-odd correlations is performed using the four-body $D^0 \to K^+K^-\pi^+\pi^-$ decay, selected from semileptonic $B$ decays. The data sample corresponds to integrated luminosities of $1.0\,\text{fb}^{-1}$ and $2.0\,\text{fb}^{-1}$ recorded at the centre-of-mass energies of 7 TeV and 8 TeV, respectively. The $CP$-violating asymmetry $a_{CP}^{T\text{-odd}}$ is measured to be $(0.18\pm 0.29\text{(stat)}\pm 0.04\text{(syst)})\%$. Searches for $CP$ violation in different regions of phase space of the four-body decay, and as a function of the $D^0$ decay time, are also presented. No significant deviation from the $CP$ conservation hypothesis is found.


Introduction
Violation of the CP symmetry in charm decays is expected to be very small in the Standard Model (SM) [1,2], however, asymmetries at a few times 10 −3 within the SM cannot be excluded according to recent calculations [3][4][5].A significant excess of CP violation (CP V ) with respect to the theoretical predictions would be a signature of physics beyond the SM.The study of CP V in singly Cabibbo-suppressed charm decays is uniquely sensitive to physics beyond the SM, in particular through new contributions in strong penguin and chromomagnetic dipole operators [2].The analysis of singly Cabibbo-suppressed D 0 → K + K − π + π −1 decays allows localised CP V in different regions of phase space to be probed.This approach enhances the sensitivity due to several interfering amplitudes with different relative strong phases contributing to the decay.
The analysis in Ref. [6] quotes a p-value of 9.1% for the compatibility with the CP conservation hypothesis, using D * -tagged promptly-produced D 0 mesons.In the present analysis, a sample of D 0 → K + K − π + π − decays, selected from semileptonic B decays, is used to measure a CP -violating parameter based on T -odd correlations characterised by different sensitivity to CP V [7,8].Using triple products of final state particle momenta in the D 0 centre-of-mass frame, C T ≡ p K + • ( p π + × p π − ) for D 0 and C T ≡ p K − • ( p π − × p π + ) for D 0 decays, two T -odd observables, can be studied [9], where Γ D 0 (Γ D 0 ) is the partial decay width of D 0 (D 0 ) decays to K + K − π + π − in the indicated C T (C T ) range.However, because final state interaction (FSI) effects could introduce fake asymmetries [9,10], these are not theoretically clean CP -violating observables.A well defined CP -violating observable is as FSI effects cancel out in the difference.In contrast to the asymmetry between the phase-space integrated rates in a D 0 → V 1 V 2 decay (where V i indicates a vector meson), a T -odd CP is sensitive to CP violation in interference between even-and odd-partial waves of the V 1 V 2 system [7].
Previous measurements of a T -odd CP are compatible with no CP V : FOCUS measured a T -odd CP = (1.0 ± 5.7 ± 3.7)% [11], and BaBar measured a T -odd CP = (0.10 ± 0.51 ± 0.44)% [12].The physics observables, A T , A T , and a T -odd CP are by construction insensitive to D 0 /D 0 production asymmetries, detector-and reconstruction-induced charge asymmetries.The measurement described in this paper determines the CP -violating observable a T -odd CP with an improved precision.For the first time, a T -odd CP is measured in different regions of phase space and in bins of D 0 decay time, allowing to probe for CP violation both in the decay amplitude and in its interference with the mixing amplitude.

Detector
The LHCb detector [13] 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 silicon-strip vertex detector surrounding the pp interaction region [14], 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 [15] placed downstream of the magnet.The tracking system provides a measurement of momentum, p, with a relative uncertainty that varies from 0.4% at low momentum to 0.6% at 100 GeV/c.The minimum distance of a track to a primary vertex, the impact parameter, is measured with a resolution of (15 + (29 GeV/c)/p T ) µm, where p T is the component of p transverse to the beam.Different types of charged hadrons are distinguished using information from two ringimaging Cherenkov (RICH) detectors [16].Photon, electron and hadron candidates 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 [17].The trigger [18] 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.
Events are required to pass both hardware and software trigger selections.The software trigger identifies decays, where X indicates any system composed of charged and neutral particles, by requiring a four-track secondary vertex with a scalar sum of p T of the tracks greater than 1.8 GeV/c.The D 0 daughter tracks are required to have p T > 0.3 GeV/c and momentum p > 2 GeV/c.The muon track is selected with p T > 1.2 GeV/c and p > 2 GeV/c.Tracks have to be compatible with detached decay vertices of B and D 0 decays.Therefore, a requirement is imposed for all the tracks in the signal candidate on the χ 2 IP , i.e. the difference in χ 2 of a given primary vertex reconstructed with and without the considered particle, to be greater than 9.The invariant mass of the D 0 µ system is required to be less than 6.2 GeV/c 2 .
In the simulation, pp collisions are generated using Pythia [19,20] with a specific LHCb configuration [21].Decays of hadronic particles are described by EvtGen [22].The interaction of the generated particles with the detector and its response are implemented using the Geant4 toolkit [23,24] as described in Ref. [25].

Selection
The analysis is based on data recorded by the LHCb experiment, at center-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 D 0 candidates are formed from combinations of kaon and pion candidate tracks and then combined with a muon candidate track to reconstruct the semileptonic B decay.The flavour of the D 0 is identified by the charge of the muon, i.e. a negative charge identifies a D 0 meson and a positive charge identifies a D 0 meson.The information from the RICH system is used to distinguish between kaons and pions, while the information from the muon system is used to identify muon candidates.The D 0 meson decay vertex is required to be downstream of the B decay vertex.The invariant mass of the D 0 µ system is required to be in the range [2.6, 5.2] GeV/c 2 .
Two main sources of peaking background in m(K + K − π + π − ), the reconstructed invariant mass of D 0 candidates, are present and consist of D 0 → K 0 S K + K − decays, and D 0 → K + K − π + π − decays from D 0 mesons that originate at the interaction point, referred to as "prompt" charm decays in the following.The small component of D 0 → K 0 S K + K − events is vetoed by requiring the invariant mass of the π + π − system to be more than 3σ away from the known K 0 S mass [26], where σ = 4.5 MeV/c 2 is the resolution determined from the fit to data.The contribution of prompt charm decays is estimated by fitting the distribution of the logarithm of the χ 2 IP of the D 0 meson.The prompt component and the signal component from semileptonic B decays accumulate at 0 and at 5, respectively.The fraction of prompt D 0 decays in our sample is measured to be f prompt = (1.20 ± 0.08)% and the effect of its presence is accounted for as a systematic uncertainty.The distributions of D 0 decays where pions and kaons have not been correctly identified have been studied and do not peak in m(K + K − π + π − ).When multiple candidates are reconstructed, one candidate per event is retained, by random choice.This happens in 0.7% of the events.The signal yield for events with a sample purity of about 75%.
By using identical kinematic selection criteria as for the signal, Cabibbo-favoured D 0 → K − π + π − π + decays are also reconstructed with a signal yield of about of 6.211 × 10 6 and a purity of about 95%.These decays are used for control checks and for assessing systematic uncertainties.

Asymmetry measurements
The selected data sample is split into four subsamples according to the charge of the muon candidate, which determines the flavour of the D 0 , and the sign of C T (C T ).The reconstruction efficiencies are equal, within their uncertainties, for C T > 0 (−C T > 0) and for C T < 0 (−C T < 0) according to studies based on simulated events and on the D 0 → K − π + π − π + control sample.A simultaneous maximum likelihood fit to the m(K + K − π + π − ) distribution of the four subsamples is used to determine the number of signal and background events, and the asymmetries A T and A T .The fit model consists of two Gaussian functions with common mean for the signal and an exponential function for Table 1: Number of signal decays obtained from the fit to data for each of the four samples defined by the D 0 /D 0 flavour and the sign of C T or C T .

Sample
Signal Decays the background.The two asymmetries A T and A T are included in the fit model as The CP -violating asymmetry a T -odd CP is then calculated from A T and A T .Negligible correlation is found between these two asymmetries.The results of the fit are shown in Fig. 1.The number of signal decays for each subsample are listed in Table 1.
Three different approaches have been followed to search for CP V : a measurement integrated over the phase space, measurements in different regions of phase space, and measurements as a function of the D 0 decay time.The results of the first approach are obtained by fitting the full data sample and are A T = (−7.18± 0.41)%, and A T = (−7.55± 0.41)%, where the uncertainties are statistical only.The CP -violating asymmetry calculated from the two partial asymmetries is a T -odd CP = (0.18 ± 0.29)%.The relatively large asymmetries observed in A T and A T are due to FSI effects [9,10], which are known to be relevant in charm mesons decays.These effects are difficult to predict, since they involve non-perturbative strong interactions [27].However, experimental measurements provide solid anchor points for future calculations.
The measurement in different regions of the phase space is performed by dividing the sample using a binning scheme based on the Cabibbo-Maksimowicz [28] variables , Φ, defined as the K + K − and π + π − invariant masses, the cosine of the angle of the K + (π + ) with respect to the opposite direction to the D 0 momentum in the K + K − (π + π − ) rest frame, and the angle between the planes described by the two kaons and pions in the D 0 rest frame, respectively.
The background-subtracted distributions for D 0 (D 0 ) events with C T > 0 and C T < 0 (−C T > 0 and −C T < 0) in m π + π − and m K + K − are shown in Fig. 2. The background subtraction is performed using m(K + K − π + π − ) sidebands.Clear indications of ρ 0 → π + π − and φ → K + K − resonances are seen in the data.The distributions of cos(θ K + ), cos(θ π + ), and Φ variables are shown in Fig. 3, where FSI-induced differences are clearly evident [29].Effects of CP V would lead to different distributions for D 0 and D 0 events.
The phase space is divided in 32 regions such that the number of signal events is similar in each region; the definition of the 32 regions is reported in Table 3 in Appendix A.
The same fit model used for the integrated measurement is separately fitted to data in each bin.The signal shapes are consistent among different bins, while significant variations are found in the distribution of the combinatorial background.The distributions of the measured asymmetries in the 32-region binning scheme are shown in Fig. 4 and the results are reported in  test, where the χ 2 is defined as R T V −1 R, where R is the array of a T -odd CP measurements, and V −1 is the inverse of the covariance matrix V , defined as the sum of the statistical and systematic covariance matrices.An average systematic uncertainty, whose evaluation is discussed in Sec. 5, is assumed for the different bins.The statistical uncertainties are considered uncorrelated among the bins, while systematic uncertainties are assumed to be fully correlated.The contribution of systematic uncertainties is small compared to the statistical ones, as shown in Table 2.The results are consistent with the CP conservation hypothesis with a p-value of 74%, based on χ 2 /ndf = 26.4/32,where ndf is the number of degrees of freedom.Four alternative binning schemes, one with 8 regions and three with 16 regions, are also tested.These are described in Appendix A. Results are compatible with the CP conservation hypothesis with a p-value of 24% for the case of 8 regions and 28%, 62%, 82% for the three different phase space divisions in 16 regions.The A T and A T asymmetries are significantly different among the different regions.This effect can be explained by the rich resonant structure of the hadronic four-body decay [30] that produces different FSI effects over the phase space.(c, d) cos(θ π + ), and (e, f) Φ for different values of C T (C T ).The asymmetric distributions with respect to 0 for cos(θ K + ) and cos(θ π + ) variables, and with respect to π/2 for the Φ variable, are due to the dynamics of the four-body decay.
The a T -odd CP distribution in D 0 decay time is shown in Fig. 5 and the results for the different decay time bins are reported in Table 5 in Appendix B. The compatibility with the CP conservation hypothesis is verified by means of a χ 2 test considering statistical and systematic uncertainties as in the previous case; a value of χ 2 /ndf = 1.3/4 is obtained, corresponding to a p-value of 86%.Consistent results are obtained when using different    divisions of the decay time in 3 and 5 intervals compatible with the CP conservation hypothesis with p-values of 92% and 83%, respectively.This result is consistent with no time-dependent CP V .The A T and A T asymmetry parameters do not show any significant dependence as a function of the decay time, and the results are compatible with constant functions with p-values of 80% and 38%, respectively.

Systematic uncertainties
The sources of systematic uncertainty and their relative contributions to the total uncertainty are listed in Table 2.
The contamination from prompt D 0 decays affects the asymmetry A according to A → A(1 − f ) + f A d , where f is the fraction of the contamination in the selected sample and A d is its own asymmetry.The uncertainties are evaluated by using as input the fraction f prompt and the asymmetries of the prompt charm sample.These events correspond to random combinations of muons with D 0 originating at the primary vertex and show no significant evidence of correlation between the flavour of the D 0 and the charge of the muon.Assuming that the flavour mistag rate for the prompt D 0 and D 0 samples is 0.5, these are related to the signal asymmetries as:

CP
, and The detector bias is tested by measuring a T -odd In this case, a kinematic selection is required between the pair of pions with identical charge to define the C T and C T triple products.The pion with higher momentum is used to calculate the triple product.Since this is a Cabibbo-favoured decay, the CP -violating effects are assumed to be negligible, and any significant deviation from zero is considered as a bias introduced by the experimental technique and the detector reconstruction.The asymmetry obtained on the control sample is compatible with no bias, a T -odd CP (D 0 → K − π + π − π + ) = (0.05 ± 0.04)%.A systematic uncertainty equal to the statistical uncertainty of this measurement is assigned.The test was repeated for different regions of phase space with consistent results.
The systematic uncertainty from detector resolution on C T is estimated from a simulated sample of D 0 → K + K − π + π − decays where neither FSI nor CP -violating effects are present.The difference between the reconstructed and generated asymmetry is considered as a systematic uncertainty due to this effect.
The fit models for signal and background are modified, ensuring good fit quality, to account for model-dependent uncertainties.The signal shape is described with a Gaussian function plus a second Gaussian function with a low-mass power-law tail.The background is described with a third-order polynomial function.Alternative models are fitted to the data and for each model 1000 simulated samples are generated according to fit results.The nominal model is then fitted to the simulated samples and the asymmetry parameters are extracted.Since the bias is not significantly different from zero, its statistical uncertainty is taken as the systematic uncertainty due to this source.
Wrongly identified muon candidates could affect the CP -violating asymmetry as where ∆ω ≡ ω + − ω − is the difference between the probability of assigning a wrong D 0 (ω + ) and D 0 (ω − ) flavour.Similar considerations enable the estimation of the uncertainty on A T (A T ) as (A T + A T )ω + ((A T + A T )ω − ).The mistag probabilities are measured by reconstructing the double-tagged decay channel B → D * + µ − X, with D * + → D 0 π + and D 0 → K + K − π + π − , and calculating the fraction of events for which the charge of the muon is identical to the charge of the soft pion from D * + decay.Mistag probabilities of ω + = (5.2± 1.0) × 10 −3 and ω − = (4.7 ± 1.0) × 10 −3 are measured for D 0 and D 0 flavour, respectively.
Since the various contributions to the systematic uncertainty are independent, the total uncertainty is obtained by summing them in quadrature, and it is very small.In particular, the a T -odd CP observable is insensitive to the production asymmetry of D 0 and D 0 and to reconstruction-induced charge asymmetries.Further cross-checks are made for establishing the stability of the results with respect to the different periods of data taking, different magnet polarities, the choice made in the selection of multiple candidates, and the effect of selection through particle identification criteria.All these tests reported effects compatible to statistical fluctuations, and therefore are not included in the systematic uncertainty.
The total systematic uncertainties reported in Table 2 are used for the measurement of the asymmetries in all regions of phase space and in all bins of D 0 decay time.

Conclusions
In conclusion, a search for CP V in D 0 → K + K − π + π − decays produced in B → D 0 µ − X transitions has been performed.The data sample consists of about 171 300 signal decays.Three different approaches have been followed to exploit the full potential of the data: a measurement integrated over the phase space, measurements in different regions of the phase space, and measurements as a function of the D 0 decay time.
The results from the phase space integrated measurement, A T = (−7.18± 0.41(stat) ± 0.13(syst))%, A T = (−7.55± 0.41(stat) ± 0.12(syst))%, a T -odd CP = ( 0.18 ± 0.29(stat) ± 0.04(syst))%, are consistent with those measured in Ref. [12], with significantly improved statistical and systematic uncertainties.The evaluation of the systematic uncertainties is based mostly on high statistics control samples.An analysis of the asymmetries in different regions of the phase space is made for the first time and the results are consistent with CP conservation.Relatively large variations of A T and A T over the phase space are measured, which are due to FSI effects produced in the rich resonant structure of the decay [29,30].For the first time the a T -odd CP asymmetry is measured as a function of the D 0 decay time and does not show any significant structure at the observed sensitivity.These results further constrain extensions of the SM [2].

Appendices A Measured asymmetries in regions of phase space
The definitions of the 32 regions of phase space of the four-body D 0 → K + K − π + π − decay are reported in Table 3.The measurements in each region of phase space for the a T -odd CP , A T , and A T are reported in Table 4.
Table 3: Definition of the 32 regions of the five-dimensional phase space of the four-body

Figure 1 :
Figure 1: Distributions of the K + K − π + π − invariant mass in the four samples defined by D 0 (D 0 ) flavour and the sign of C T (C T ).The results of the fit are overlaid as a solid line, and a dashed line is used for representing the background.The normalised residuals (pulls) of the difference between the fit results and the data points, divided by their uncertainties, are shown on top of each distribution.

Figure 2 :
Figure 2: Sideband-subtracted distributions of D 0 (D 0 ) candidates in variables of (a, b) m π + π − and (c, d) m K + K − for different values of C T (C T ).The veto for D 0 → K 0 S K + K − decays is visible in the m π + π − distribution.

Figure 3 :
Figure 3: Sideband-subtracted distributions of D 0 (D 0 ) candidates in variables of (a, b) cos(θ K + ),(c, d) cos(θ π + ), and (e, f) Φ for different values of C T (C T ).The asymmetric distributions with respect to 0 for cos(θ K + ) and cos(θ π + ) variables, and with respect to π/2 for the Φ variable, are due to the dynamics of the four-body decay.

Figure 4 :
Figure 4: Distributions of the asymmetry parameters (a) A T , (b) A T and (c) a T -odd CP in 32 regions of the phase space.

Figure 5 :
Figure 5: Distributions of the asymmetry parameters (a) A T , (b) A T and (c) a T -odd CP as a function of the D 0 decay time.For a T -oddCP, the value of the χ 2 /ndf for the CP conservation hypothesis, represented by a dashed line, is also quoted.The scale is broken for the first and last bin.
place of the angular variables (16 bins).B Measured asymmetries in intervals of D 0 decay time The values measured in 4 different intervals of the D 0 decay time for a T -odd CP , A T , and A T are reported in Table

Table 4 in
Appendix A. The compatibility with the CP conservation hypothesis is tested by means of a χ 2

Table 2 :
Considered sources of systematic uncertainty and their relative contributions to the total uncertainty.

Table 5 :
Measurements of a T -oddCP, A T and A T in different intervals of D 0 decay time, t, expressed in ps.The uncertainties are statistical only.A common systematic uncertainty of 0.13%, 0.12% and 0.04% should be added to the asymmetries A T , A T and a T -odd CP , respectively.This uncertainty is considered fully correlated among the bins.−7.49 ± 0.88 −6.67 ± 0.81 −7.64 ± 0.80 −6.84 ± 0.79 A T (%) −8.51 ± 0.87 −6.44 ± 0.82 −7.60 ± 0.79 −7.68 ± 0.79 f Università di Ferrara, Ferrara, Italy g Università di Firenze, Firenze, Italy h Università di Urbino, Urbino, Italy i Università di Modena e Reggio Emilia, Modena, Italy j Università di Genova, Genova, Italy k Università di Milano Bicocca, Milano, Italy l Università di Roma Tor Vergata, Roma, Italy m Università di Roma La Sapienza, Roma, Italy n Università della Basilicata, Potenza, Italy o AGH -University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunications, Kraków, Poland p LIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain q Hanoi University of Science, Hanoi, Viet Nam r Università di Padova, Padova, Italy s Università di Pisa, Pisa, Italy t Scuola Normale Superiore, Pisa, Italy u Università degli Studi di Milano, Milano, Italy v Politecnico di Milano, Milano, Italy