Search for $CP$ violation in $D_s^+\to K_S^0 \pi^+$, $D^+\to K_S^0 K^+$ and $D^+\to \phi \pi^+$ decays

A search for charge-parity ($CP$) violation in Cabibbo-suppressed $D_s^+\to K_S^0 \pi^+$, $D^+\to K_S^0 K^+$ and $D^+\to \phi \pi^+$ decays is reported using proton-proton collision data, corresponding to an integrated luminosity of 3.8 fb$^{-1}$, collected at a center-of-mass energy of 13 TeV with the LHCb detector. High-yield samples of kinematically and topologically similar Cabibbo-favored $D_{(s)}^+$ decays are analyzed to subtract nuisance asymmetries due to production and detection effects, including those induced by $CP$ violation in the neutral kaon system. The results are \begin{align*} \mathcal{A}_{CP}(D_s^+\to K_S^0 \pi^+)&=\left(\phantom{-}1.3\phantom{0}\pm1.9\phantom{0}\pm0.5\phantom{0}\right)\times10^{-3},\\ \mathcal{A}_{CP}(D^+\to K_S^0 K^+)&=\left(-0.09\pm0.65\pm0.48\right)\times10^{-3},\\ \mathcal{A}_{CP}(D^+\to \phi \pi^+)&=\left(\phantom{-}0.05\pm0.42\pm0.29\right)\times10^{-3}, \end{align*} where the first uncertainties are statistical and the second systematic. They are the most precise measurements of these quantities to date, and are consistent with $CP$ symmetry.

, where Γ is the partial decay rate. If CP symmetry is violated in the decay, A CP = 0. An experimentally convenient quantity to measure is the "raw" asymmetry of the observed yields N , . ( The raw asymmetry can be approximated as where A P (D + (s) ) is the asymmetry of the D + (s) -meson production cross-section [32,33] and A D (f + ) is the asymmetry of the reconstruction efficiency for the final state f + . When f + = K 0 S h + (with h = K, π), the detection asymmetry receives contributions from the h + hadron (indicated as companion hadron in the following), A D (h + ), and from the neutral kaon, A D (K 0 ). Relevant instrumental effects contributing to A D (h + ) may include differences in interaction cross-sections with matter between positive and negative hadrons and slightly charge-asymmetric performance of the reconstruction algorithms. The contribution to A D (K 0 ) arises from K 0 and K 0 mesons having different interaction cross-sections with matter and from their propagation in the detector being affected by the presence of CP violation in the K 0 -K 0 system. When f + = φ(→ K + K − )π + , the detection asymmetry is mostly due to the charged pion, as the contributions from the oppositely charged kaons cancel to a good precision.
The detection and production asymmetries are canceled by using the decays D + → K 0 S π + , D + s → K 0 S K + and D + s → φπ + , which proceed through the Cabibbo-favored c → sdu transition. In the SM, these decays are expected to have CP asymmetries that are negligibly small compared to the Cabibbo-suppressed modes, when effects induced by the neutral kaons are excluded [30,34]. Hence, their raw asymmetries can be approximated as in Eq. (3), but with A CP = 0. The CP asymmetries of the decay modes of interest are determined by combining the raw asymmetries as follows: where the contribution from A D (K 0 ) is omitted and should be subtracted from any of the measured asymmetries where it is present. The LHCb detector [35,36] is a single-arm forward spectrometer designed for the study of particles containing b or c quarks. The detector elements that are particularly relevant to this analysis are: a silicon-strip vertex detector that allows for a precise measurement of the impact parameter, i.e., the minimum distance of a charged-particle trajectory to a pp interaction point (primary vertex); a tracking system that provides a measurement of the momentum of charged particles; two ring-imaging Cherenkov detectors that are able to discriminate between different species of charged hadrons; and a calorimeter system that is used for the identification of photons, electrons and hadrons. The polarity of the magnetic field is periodically reversed during data-taking to mitigate the differences between reconstruction efficiencies of oppositely charged particles.
The online event selection is performed by a trigger, which consists of a hardware stage followed by a two-level software stage. In between the two software stages, an alignment and calibration of the detector is performed in near real-time and their results are used in the trigger [37]. Events with candidate D + (s) decays are selected by the hardware trigger by imposing either that one or more D + (s) decay products are associated with large transverse energy deposits in the calorimeter or that the accept decision is independent of the D + (s) decay products. In the first level of the software trigger, one or more D + (s) decay products must have large transverse momentum and be inconsistent with originating from any primary vertex. In the second level, the candidate decays are fully reconstructed using kinematic, topological and particle-identification criteria. The D + (s) → K 0 S h + candidates are made by combining charged hadrons with K 0 S → π + π − candidates that decay early enough for the final-state pions to be reconstructed in the vertex detector. This requirement suppresses to a negligible level possible CP -violation effects due to interference between Cabibbo-favored and doubly Cabibbo-suppressed amplitudes with neutral-kaon mixing in the control-sample decays D + → K 0 S π + and D + s → K 0 S K + [34].
The D + (s) candidates reconstructed in the trigger are used directly in the offline analysis [38,39]. The candidates with a K 0 S meson in the final state are further selected offline using an artificial neural network (NN), based on the multilayer perceptron algorithm [40], to suppress background due to random combinations of K 0 S mesons and hadrons not originating from a D + (s) → K 0 S h + decay. The quantities used in the NN to discriminate signal from combinatorial background are: the K 0 S candidate momentum; the transverse momenta of the D + (s) candidate and of the companion hadron; the angle between the D + (s) candidate momentum and the vector connecting the primary and secondary vertices; the quality of the secondary vertex; and the track quality of the companion hadron. The NN is trained using signal and background data samples, obtained with the sPlot method [41], from a O(1%) fraction of candidates randomly sampled. In the D + s → K 0 S π + case, thanks to similar kinematics, background-subtracted D + → K 0 S π + decays are exploited as a signal proxy to profit from larger yields. The thresholds on the NN response are optimized for the D + s → K 0 S π + and D + → K 0 S K + decays by maximizing the value of S/ √ S + B, where S and B stands for the signal and background yield observed in the mass ranges 1.93 < m(K 0 S π + ) < 2.10 GeV/c 2 and 1.83 The contribution of D + (s) mesons produced through decays of b hadrons, referred to as secondaries throughout, is suppressed by requiring that the D + (s) impact parameter in the plane transverse to the beam (TIP) is smaller than 40 µm. The remaining percent-level contribution is evaluated by means of a fit to the TIP distribution when such requirement is released, as shown in Fig. 1 for the D + s → K 0 S π + decay. The impact of the secondary background on the results is accounted for in the systematic uncertainties.
Typical sources of background from D + (s) meson and Λ + c baryon decays are: the D + s → K 0 S K + and Λ + c → K 0 S p decays, where the kaon and the proton are misidentified as a pion, when the signal is the D + s → K 0 S π + decay; the D + → K 0 S π + and Λ + c → K 0 S p decays, where the pion and the proton are misidentified as a kaon, in the D + → K 0 S K + case; and the Λ + c → φp decay, where the proton is misidentified as a pion, when the signal is the D + → φπ + decay. These are all reduced to a negligible level using particle-identification requirements and kinematic vetos.
Fiducial requirements are imposed to exclude kinematic regions that induce a large asymmetry in the companion-hadron reconstruction efficiency. These regions occur because low momentum particles of one charge at large (small) angles in the bending plane may be deflected out of the detector acceptance (into the noninstrumented beam pipe region), whereas particles with the other charge are more likely to remain within the acceptance. About 78%, 93% and 94% of the selected candidates are retained by these fiducial requirements for D + (s) → K 0 S π + , D + (s) → K 0 S K + and D + (s) → φπ + decays, respectively. Detection and production asymmetries may depend on the kinematics of the involved particles. Therefore, the cancellation provided by the control decays is accurate only if the kinematic distributions agree between any pair of signal and control modes, or pair of control modes entering Eqs. (4)- (6). Differences are observed, and the ratio between background-subtracted [41] signal and control sample distributions of transverse momentum, azimuthal angle and pseudorapidity are used to define candidate-by-candidate weights. The background-subtracted candidates of the control decays are weighted such that their distributions agree with those of the signal using an iterative procedure. The process consists of calculating the weights in each one-dimensional distribution of the weighting variables and repeating the procedure until good agreement is achieved among all the distributions. For the measurements of the D + s → K 0 S π + and D + → φπ + CP asymmetries, the D + s → φπ + and D + → K 0 S π + control samples are weighted so that the D + (s) meson and companion-pion kinematic distributions agree with their respective signal samples to cancel the D + (s) production and companion-pion detection asymmetries. In the case of the A CP (D + → K 0 S K + ) measurement, the D + kinematic distributions of the D + → K 0 S π + sample are weighted to those of the D + → K 0 S K + signal to cancel the D + production asymmetry, and the K + distributions of the D + s → K 0 S K + decays are weighted to those of the D + → K 0 S K + signal to cancel the kaon detection asymmetry. The D + → K 0 S π + and D + s → K 0 S K + control decays then introduce their own additional nuisance asymmetries, which need to be corrected for using the D + s → φπ + control decay. Hence, the D + s and companion-pion kinematic distributions of the D + s → φπ + sample are made to agree with those of the D + s → K 0 S K + and D + → K 0 S π + samples, respectively, to cancel the D + s production and companion-pion detection asymmetries. Simultaneous least-squares fits to the mass distributions of weighted D + (s) and D − (s) candidates determine the raw asymmetries for each decay mode considered. To avoid experimenter bias, the raw asymmetries of the Cabibbo-suppressed signals were shifted by unknown offsets sampled uniformly between −1% and 1%, such that the results remained blind until the analysis procedure was finalized. In the fits, the signal and control decays are modeled as the sum of a Gaussian function to describe the core of the peaks, and a Johnson S U distribution [42], which accounts for the asymmetric tails. The combinatorial background is described by the sum of two exponential functions. All shape parameters are determined from the data. In each fit, signal and control decays share the same shape parameters apart from a mass shift, which accounts for the known difference between the D + s and D + masses [31], and a relative scale factor between the peak widths, which is also determined from the data. The means and widths of the peaks, as well as all background  shape parameters, are allowed to differ between D + (s) and D − (s) decays. The projections of the fits to the combined D + (s) and D − (s) data are shown in Fig. 2. The samples contain approximately 600 thousand D + s → K 0 S π + , 5.1 million D + → K 0 S K + , and 53.3 million D + → φπ + signal candidates, together with approximately 30.5 million D + → K 0 S π + , 6.5 million D + s → K 0 S K + , and 107 million D + s → φπ + control decays. The raw asymmetries are, where relevant, corrected for the neutral-kaon detection asymmetry. The net correction is estimated following Ref. [43] to be (+0.084 ± 0.005)% for Table 1: Summary of the systematic uncertainties (in units of 10 −3 ) on the measured quantities. The total is the sum in quadrature of the different contributions.

Source
A , and (−0.068 ± 0.004)% for A CP (D + → φπ + ), where the uncertainty is dominated by the accuracy of the detector modeling in the simulation. The asymmetries are combined following Eqs. (4)-(6) to obtain where the uncertainties are only statistical.
Several sources of systematic uncertainty affecting the measurement are considered as reported in Table 1. The dominant contribution is due to the assumed shapes in the mass fits. This is evaluated by fitting with the default model large sets of pseudoexperiments where alternative models that describe data equally well are used in generation. For A CP (D + s → K 0 S π + ) and A CP (D + → K 0 S K + ), the second leading contribution is due to the residual contamination from secondary D + (s) decays, which introduces a small difference between the asymmetry of D + (s) -meson production cross-sections of the signal and control modes. For A CP (D + → φπ + ), instead, the second leading systematic uncertainty arises from neglected kinematic differences between the φ-meson decay products. These differences, mainly caused by the interference between the S-wave and φπ + decay amplitudes in the K + K − -mass region under study, result in an imperfect cancelation of the charged-kaon detection asymmetry. Other subleading contributions are due to the inaccuracy in the equalization of the kinematic distributions between signal and control samples, and to the uncertainty in the neutral-kaon detection asymmetry.
In addition, several consistency checks are performed to investigate possible unexpected biases by comparing results obtained in subsamples of the data defined according to the data-taking year and magnetic-field polarity, the per-event track multiplicity, the configurations of the hardware-and software-level triggers, and the D + (s) momentum. A χ 2 test has been performed for each cross-check and the corresponding p values are consistent with being uniformly distributed; the lowest (largest) p value is 4% (86%). Therefore, the observed variations in results are consistent with statistical fluctuations and no additional sources of systematic uncertainties are considered.
In summary, using proton-proton collision data collected with the LHCb detector at a center-of-mass energy of 13 TeV, and corresponding to 3.8 fb −1 of integrated luminosity, the following CP asymmetries are measured: where the first uncertainties are statistical and the second systematic. Effects induced by CP violation in the neutral kaon system are subtracted from the measured asymmetries. The results represent the most precise determination of these quantities to date and are consistent with CP symmetry. They are in agreement with previous LHCb determinations based on independent data samples collected at center-of-mass energies of 7 and 8 TeV [28,29], as well as with measurements from other experiments [22][23][24][25][26][27]. The results are combined with previous LHCb measurements using the BLUE method [44]. The systematic uncertainties are considered uncorrelated, apart from those due to the neutral-and chargedkaon detection asymmetries that are fully correlated. The combination yields where the first uncertainties are statistical and the second systematic. No evidence for CP violation in these decays is found.