Charm mixing and CP violation at LHCb

The LHCb experiment has collected the world's largest sample of charmed hadrons. Using data corresponding to an integrated luminosity of 3.0 fb−1 recorded in 2011 and 2012, measurements of direct and indirect CP violation in the charm sector and of D0 mixing parameters were performed. Results from several decay modes are presented with complementary time-dependent and time-integrated analyses.


Introduction
The charm sector is a promising place to probe for the presence of physics beyond the Standard Model (SM) because CP violation (CPV) in the charm sector is expected to be very small in the SM [1]. Since evidence of D 0 −D 0 oscillations was first reported [2], [3], [4] there is growing interest in this subject. Details of the measurement of D 0 −D 0 oscillations achieved recently at the LHCb are discussed in Section 2. CPV arises when two or more amplitudes with different weak and strong phases contribute to the same final state. This is possible in the singly Cabibbosuppressed (SCS) D decays, where significant tree and penguin contributions can be expected. The time-integrated CP asymmetry measurements in two-body SCS D 0 decays are discussed in Section 3. The searches for CP asymmetry in three-body SCS D + decays are discussed in Section 4. Conclusions are presented in Section 5.

Search for CP in D 0 −D 0 oscillations
To measure D 0 −D 0 oscillations, the D 0 flavour has to be identified at both the production and decay. The flavour of the D 0 meson at the production time is identified by the charge of the slow pion (π s ) from D * + → D 0 π + s and D * − →D 0 π − s decays. The flavour at the time of decay is identified using the almost flavour-specific K ± π ∓ final state. The D 0 → K + π − decays are called wrong-sign (WS) decays and the D 0 → K − π + decays are called right-sign (RS) decays. The RS decays are dominated by the Cabibbo-favoured (CF) decay amplitude, whereas the WS amplitude includes contributions from both the doubly-Cabibbo-suppressed (DCS) D 0 → K + π − decay and D 0 −D 0 mixing followed by the favouredD 0 → K + π − decay. Assuming negligible CPV, the time-dependent ratio, R, of WS to RS decay rates is approximated by [1] where t/τ is the decay time expressed in units of the average D 0 lifetime (τ ), R D is the ratio of DCS to CF decay rates, x = x cos δ + y sin δ, y = y cos δ − x sin δ, where δ is the strong phase difference between DCS and CF amplitudes. The parameters x = ∆m/Γ and y = ∆Γ/2Γ, where ∆m and ∆Γ are the mass and the decay width differences between the two mass eigenstates and Γ is the average D 0 decay width.
The measurement of the time-dependent WS/RS ratio is performed with 3 fb −1 in thirteen bins of D 0 decay time, chosen to have a similar number of candidates in each bin [5]. The number of RS and WS decays are determined using fits to the M (D 0 π + s ) distributions in each bin. The measured WS/RS ratios as a function of t/τ are fit separately for D * + and D * − decays to the time-dependence of Equation 1. Three fits are performed to the data; the first allows direct and indirect CPV; the second allows only indirect CPV; and the third is a CPconserving fit. Figure 1 shows the central values and confidence regions in the (x 2 , y ) plane. Assuming no CPV, the resulting of the probabilities of observing CP asymmetry as large or larger than those in data, p-value, for the fit with direct and indirect (indirect only) CPV allowed, is 91% (81%), showing that the data are compatible with CP symmetry. This allows to limit the amount of CPV in D 0 −D 0 mixing, using the relations x ± = |q/p| 1 (x cos φ ± y sin φ) and y = |q/p| 1 (y cos φ ∓ x sin φ), to 0.75 < |q/p| < 1.24 at 68.3% CL and the amount of direct CPV in the DCS decay to is the DCS/CF rate ration for D 0 (D 0 ) decays. Assuming CP conservation, the mixing parameters are measured to be x 2 = (5.5 ± 4.9) × 10 −5 , y = (4.8 ± 1.0) × 10 −3 and R D = (3.568 ± 0.066) × 10 −3 . in the (x 02 , y 0 ) plane. For each fit, 104 WS-to-RS ratio data points are used, corresponding to 13 ranges of decay time, distinguishing D ⇤+ from D ⇤ decays, TOS from TOS decays, and 2011 data from 2012 data. The consistency with the hypothesis of CP symmetry is determined from the change in 2 between the fit without and with CP violation, taking into account the di↵erence in number of degrees of freedom. The resulting p value, for the fit with direct and indirect (indirect only) CP violation allowed, is 91% (81%), showing that the data are compatible with CP symmetry.
The uncertainties incorporate both statistical and systematic contributions, since all relevant systematic e↵ects depend on the true values of the mixing parameters, and are thus incorporated into the fit 2 . These include the uncertainty in the fraction of charm mesons from b-hadron decays, and their bias on the observed decay time; the uncertainty in the fraction of peaking background; and the uncertainty in the determination of the instrumental asymmetry. The statistical uncertainty is determined in a separate fit and used to calculate the systematic component by subtraction in quadrature.

A search for time-integrated CPV in D
The CP asymmetry of a decay of a D 0 meson to a CP eigenstate, A CP (f ), can be expressed in terms of two contributions, a direct component associated with CPV in the decay amplitudes and an indirect component associated with CPV in the mixing or in the interference between mixing and decay. It can be written to the first order as [6] A where a dir CP (f ) is a direct CPV for the decay, t is the average proper time of the reconstructed sample, and a ind CP is the indirect CPV. The indirect component is universal for CP eigenstates in the SM whilst the direct component depends in general on the final state.
The time-integrated CPV asymmetry difference between D 0 → K − K + and D 0 → π − π + , ∆A CP , is performed with 3 fb −1 in which the D 0 mesons are produced in inclusive semileptonic B meson decays to the D 0 µ −ν µ X final state, where X means other possible particles [7]. The charge of the accompanying muon is used to identify the flavour of the D 0 meson (B → D 0 µ −ν µ X or B →D 0 µ + ν µ X). The measured raw time-integrated asymmetry is a sum of physics and detector asymmetries, where A D (µ + ) is the detection asymmetry for selecting a muon from semileptonic B decay chain and A P (B) is the production asymmetry of the B mesons. The detector and production asymmetries are removed by the difference of A RAW (K − K + ) and A RAW (π − π + ). A binned maximum likelihood fit to the mass distributions is performed to determine the numbers of signal candidates as 2.1×10 3 for D 0 → K − K + decays and 0.77×10 3 for D 0 → π − π + decays. The fractional difference in an average decay time of the D 0 candidates for the two samples is ∆ t /τ = 0.014 ± 0.004. The small value of ∆ t /τ implies that the measured ∆A CP is equal to the difference in direct CPV with negligible indirect CPV correction. For the first time in LHCb, also the individual CP asymmetry in D 0 → K − K + decays is measured. In this case the determination of the detector and production asymmetries uses control samples of CFB → D 0 (→ Kπ)µνX, D + → Kππ and D + →K 0 (→ ππ)π decays where CPV effects are negligible.
4. Search for CPV in the decay D + → π − π + π + Many-body decays benefit from rich resonance structures with interfering amplitudes modulated by strong-phase variations across the phase space. Searches for localised asymmetries can bring complementary information on the nature of the CPV to the time-integrated decay rate measurements. The investigation of the Cabibbo-suppressed decay D + → π − π + π + with 1 fb −1 is performed across the Dalitz plot using two model-independent techniques, one binned and the other unbinned searches [8].
The binned method, as employed in previous LHCb analyses [9], [10], is based on a bin-by-bin comparison between the D + and D − Dalitz plots. For each bin of the Dalitz plot, the significance of the difference between the number of D + and D − candidates, S i CP , is computed as where N + i (N − i ) is the number of D + (D − ) candidates in the ith bin and N + (N − ) is the sum of N + i (N − i ) over all bins. The parameter α removes the contribution of global asymmetries which may arise due to production and detection asymmetries, as well as from CPV. In the absence of localised asymmetries, the S i CP values follow a standard normal distribution. Therefore, CPV can be detected as a deviation from this behaviour.
Two binning schemes are used, a uniform grid with bins of equal size and an adaptive binning schemes where the bins have the same population ( Figure 2). No single bin in any of the binning schemes presents an absolute S i CP value larger than 3. Assuming no CPV, the probabilities of observing local asymmetries across the phase-space of the D + meson decay as large or larger than those in data are above 50% in all the tested binned schemes. All results are consistent with no CPV .
The unbinned method is based on the concept of nearest-neighbour events (kNN) in a combined D + and D − sample to test whether they share the same parent distribution function [11], [12], [13]. A test statistic, T , is defined as the average fraction of like-charged neighbour pairs. For the null hypothesis of no CPV, this follows a Gaussian distribution the control channel (see Fig. 6). It is accounted for in the kNN method as a deviation of the measured value of µ T from the reference value µ TR shown in Fig. 12. In the signal sample the values µ T 0.5 = (98 ± 15) ⇥ 10 7 and (µ T µ TR )/ (µ T µ TR ) = 6.5 in the full Dalitz plot is a consequence of the 0.4% global asymmetry similar to that observed in the control mode and consistent with the previous measurement from LHCb [20].
The pull values of T and the corresponding p-values for the hypothesis of no CP V are shown in Fig. 13 for the same regions. To check for any systematic e↵ects, the test is repeated for samples separated according to magnet polarity. Since the sensitivity of the method increases with n k , the analysis is repeated with n k = 500 for all the regions. All p-values are above 20%, consistent with no CP asymmetry in the signal mode.
12 Figure 2. Distributions of S i CP across the D + Dalitz plot, with the adaptive binning scheme of uniform population for the total D + → π − π + π + decays with (a) 49 and (c) 100 bins. The corresponding one-dimensional S i CP distributions (b) and (d) are shown with a standard normal function superimposed (solid line).
with predictable mean µ T and variance σ 2 T [8]. CPV can be detected as a deviation from this behaviour.
The kNN method is applied with the two region definitions (Figure 3). The pull values of T and the corresponding p-values for the hypothesis of no CPV are shown in Figure 4. All p-values are above 20%, consistent with no CP asymmetry.

Conclusions
Using data collected with the LHCb experiment in 2011 and 2012 D 0 −D 0 oscillations are observed by evaluating the time-dependence of the ratio of D 0 → K + π − and D 0 → K − π + decays. Assuming CP conservation, the mixing parameters are measured to be x 2 = (5.5 ± 4.9) × 10 −5 , y = (4.8 ± 1.0) × 10 −3 and R D = (3.568 ± 0.066) × 10 −3 . Study of D 0 andD 0 decays separately shows no evidence for CPV. The measurement of time-integrated difference in CP asymmetry between D 0 → K − K + and D 0 → π − π + decays also does not show evidence for direct CP violation. No evidence for CP violation is found in searches for CP asymmetries in three-body SCS decays D + → π − π + π + .

Conclusion
A search for CP V in the decay D + ! ⇡ ⇡ + ⇡ + is performed using pp collision data corresponding to an integrated luminosity of 1.0 fb 1 collected by the LHCb experiment at a centre-of-mass energy of 7 TeV. Two model-independent methods are applied to a sample of 3.1 million D + ! ⇡ ⇡ + ⇡ + decay candidates with 82% signal purity.
The binned method is based on the study of the local significances S i CP in bins of the Dalitz plot, while the unbinned method uses the concept of nearest neighbour events in the pooled D + and D sample. Both methods are also applied to the Cabibbo-favoured D + s ! ⇡ ⇡ + ⇡ + decay and to the mass sidebands to control possible asymmetries not 13 Figure 3. Dalitz plot for D + → π − π + π + candidates divided into (a) seven regions R1-R7 and (b) three regions P1-P3. originating from CP V .

Regions
No single bin in any of the binning schemes presents an absolute S i CP value larger than 3. Assuming no CP V , the probabilities of observing local asymmetries across the phase-space of the D + meson decay as large or larger than those in data are above 50% in all the tested binned schemes. In the unbinned method, the p-values are above 30%. All results are consistent with no CP V .
14 Figure 4. (a) Pull values of T and (b) the corresponding p-values for D + → π − π + π + candidates restricted to each region obtained the kNN method. The horizontal blue lines in (a) represents pull values −3 and +3. The region R0 corresponds to the full Dalitz plot.