Measurement of the $CP$-violating phase $\phi_s$ from $B_{s}^{0}\to J/\psi\pi^+\pi^-$ decays in 13 TeV $pp$ collisions

Decays of $B_{s}^{0}$ and $\overline{B}{}_{s}^{0}$ mesons into $J/\psi\pi^+\pi^-$ final states are studied in a data sample corresponding to 1.9 fb$^{-1}$ of integrated luminosity collected with the LHCb detector in 13 TeV $pp$ collisions. A time-dependent amplitude analysis is used to determine the final-state resonance contributions, the $CP$-violating phase $\phi_s=-0.057\pm 0.060\pm 0.011$ rad, the decay-width difference between the heavier mass $B_{s}^{0}$ eigenstate and the $B^{0}$ meson of $-0.050\pm 0.004\pm 0.004$ ps$^{-1}$, and the $CP$-violating parameter $|\lambda|=1.01_{-0.06}^{+0.08}\pm0.03$, where the first uncertainty is statistical and the second systematic. These results are combined with previous LHCb measurements in the same decay channel using 7 TeV and 8 TeV $pp$ collisions obtaining $\phi_s =0.002\pm0.044\pm0.012$ rad, and $|\lambda| =0.949\pm0.036\pm0.019$.


Introduction
Measurements of CP violation in final states that can be populated both by direct decay and via mixing provide an excellent way of looking for physics beyond the Standard Model (SM) [1]. Heavy, as yet unobserved, bosons corresponding to new forces, or fermions can be present virtually in quantum loops, and thus affect the relative CP phase. Direct decays into non-flavour-specific final states can interfere with those that undergo B 0 s − B 0 s mixing prior to decay. This interference can result in CP violation. In certain B 0 s decays one CP -violating phase that can be measured, called φ s , can be expressed in terms of Cabibbo-Kobayashi-Maskawa matrix elements as −2arg [−V ts V * tb /V cs V * cb ]. It is not predicted in the SM, but can be inferred with high precision from other experimental data giving a value of −36.5 +1.3 −1.2 mrad [2]. This number is consistent with previous measurements, which did not have enough sensitivity to determine a non-zero value [3][4][5][6][7]. In this paper we present the results of a new analysis of the B 0 s → J/ψ π + π − decay using data from 13 TeV pp collisions collected using the LHCb detector in 2015 and 2016. 1 The existence of this decay and its use in CP -violation studies was suggested in Ref. [8].

Detector and simulation
The LHCb detector [9,10] 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 [11], 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 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 GeV. 2 The minimum distance of a track to a primary vertex (PV), the impact parameter (IP), is measured with a resolution of (15 + 29/p T ) µm, where p T is the component of the momentum transverse to the beam, in GeV/c. Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors [12]. Photons, electrons and hadrons are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers [13]. 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.
At the hardware trigger stage, events are required to have a muon with high p T or a hadron, photon or electron with high transverse energy in the calorimeters. The software trigger is composed of two stages, the first of which performs a partial reconstruction and requires either a pair of well-reconstructed, oppositely charged muons having an invariant mass above 2.7 GeV, or a single well-reconstructed muon with high p T and large IP. The second stage applies a full event reconstruction and for this analysis requires two

Decay amplitude
The resonance structure in B 0 s and B 0 s → J/ψ π + π − decays has been previously studied with a time-integrated amplitude analysis using 7 and 8 TeV pp collisions [21]. The final state was found to be compatible with being entirely CP -odd, with the CP -even state fraction below 2.3% at 95% confidence level, which allows the determination of the decay width of the heavy B 0 s mass eigenstate, Γ H . The possible presence of a CP -even component is taken into account when determining φ s [22].
The total decay amplitude for a ( ) B 0 s meson at decay time equal to zero is assumed to be the sum over individual π + π − resonant transversity amplitudes [23], and one nonresonant amplitude, with each component labelled as A i (A i ). Because of the spin-1 J/ψ meson in the final state, the three possible polarizations of the J/ψ generate longitudinal (0), parallel ( ) and perpendicular (⊥) transversity amplitudes. When the π + π − pair forms a spin-0 state the final system only has a longitudinal component, and thus is a pure CP eigenstate. The parameter λ i ≡ q p A i A i , relates CP violation in the interference between mixing and decay associated with the polarization state i for each resonance in the final state. Here the quantities q and p relate the mass and flavour eigenstates, p ≡ B 0 s |B L , and q ≡ B 0 s |B L , where B L is the lighter mass eigenstate [1]. The total amplitudes A and A can be expressed as the sums of the individual with η i being the CP eigenvalue of the state. For each transversity state i there is a CP -violating phase φ i s ≡ − arg(η i λ i ) [24]. Assuming that CP violation in the decay is the same for all amplitudes, then λ ≡ η i λ i and φ s ≡ − arg(λ). Using |p/q| = 1, the decay rates for B 0 s and B 0 s into the J/ψ π + π − final state are 3 ( ) where the -sign before the cos(∆m s t) term and + sign before the sin(∆m s t) term apply to Γ(t), ∆Γ s ≡ Γ L − Γ H is the decay-width difference between the light and the heavy mass eigenstates, ∆m s ≡ m H − m L is the corresponding mass difference, and Γ s ≡ (Γ L + Γ H )/2 is the average B 0 s meson decay width [26]. For J/ψ decays to µ + µ − final states the A i amplitudes are themselves functions of four variables: the π + π − invariant mass m ππ , and three angular variables Ω ≡ (cos θ ππ , cos θ J/ψ , χ), defined in the helicity basis. These angles are defined as θ ππ between the π + direction in the π + π − rest frame with respect to the π + π − direction in the B 0 s rest frame, θ J/ψ between the µ + direction in the J/ψ rest frame with respect to the J/ψ direction in the B 0 s rest frame, and χ between the J/ψ and π + π − decay planes in the B 0 s rest frame [22,24]. (These definitions are the same for B 0 s and B 0 s , namely, using µ + and π + to define the angles for both B 0 s and B 0 s decays.) The explicit forms of the |A(m ππ , Ω)| 2 , |A(m ππ , Ω)| 2 , and A * (m ππ , Ω)A(m ππ , Ω) terms in Eq. (1) are given in Ref. [22].
The analysis proceeds by performing an unbinned maximum-likelihood fit to the π + π − mass distribution, the decay time, and helicity angles of B 0 s candidates identified as B 0 s or B 0 s by a flavour-tagging algorithm [27]. 4 The fit provides the CP -even and CP -odd components, and since we include the initial flavour tag, the fit also determines the CP -violating parameters φ s and |λ|, and the decay width. In order to proceed, we need to select a clean sample of B 0 s decays, determine acceptance corrections, perform a calibration of the decay-time resolution in each event as a function of its uncertainty, and calibrate the flavour-tagging algorithm.

Selection requirements
The selection of J/ψ π + π − right-sign (RS), and wrong-sign (WS) J/ψ π ± π ± final states, proceeds in two phases. Initially we impose loose requirements and subsequently use a multivariate analysis. In the first phase we require that the J/ψ decay tracks be identified as muons, have p T > 500 MeV, and form a good vertex. The identified pions are required to have p T > 250 MeV, not originate from any PV, and form a good vertex with the muons. The resulting B 0 s candidate is assigned to the PV for which it has the smallest χ 2 IP , defined as the difference in the χ 2 of the vertex fit for a given PV reconstructed with and without the considered particle. Furthermore, we require that the smallest χ 2 IP is not greater than 25. The B 0 s candidate is required to have its momentum vector aligned with the vector connecting the PV to the B 0 s decay vertex, and to have a decay time greater than 0.3 ps. Reconstructed tracks sharing the same hits are vetoed.
In addition, background from B + → J/ψ K + decays, 5 where the K + is misidentified as a π + and combined with a random π − , is vetoed by assuming that each detected pion is a kaon, computing the J/ψ K + mass, and removing those candidates that are within ±36 MeV of the known B + mass [14]. Backgrounds from B 0 → J/ψ K + π − or B 0 s → J/ψ K + K − decays with misidentified kaons result in masses lower than the B 0 s peak and thus do not need to be vetoed.
For the multivariate part of the selection, we use a Boosted Decision Tree, BDT [28,29], with the uBoost algorithm [30]. The algorithm is optimized to not further bias acceptance on the variable cos θ ππ . The variables used to train the BDT are the difference between the muon and pion identifications for the muon identified with lower quality, the p T of the B 0 s candidate, the sum of the p T of the two pions, and the natural logarithms of: the χ 2 IP of each of the pions, the χ 2 of the B 0 s vertex and decay tree fits [31], and the χ 2 IP of LHCb RS data and fit

Physics background
Combinatorial background WS data and fit Figure 1: Results of the simultaneous fit to the J/ψ ππ mass distributions RS (black points) and WS (light-brown points) samples. The solid (blue) curve shows the fit to the RS sample, the long dashed (red) curve shows the signal, the dot-dashed (magenta) curve shows B 0 → J/ψ π + π − decays, the dot-long-dashed (brown) curve shows the combinatorial background, the dotted (black) curve shows the sum of B 0 s → J/ψ η and Λ 0 b background, while the dot-dot-dashed (green) curve shows the fit to WS sample. the B 0 s candidate. In the fit, the B 0 s momentum vector is constrained to point to the PV, the two muons are constrained to the J/ψ mass, and all four tracks are constrained to originate from the same vertex.
Implementing uBoost requires a training procedure. Data background in the J/ψ π + π − mass interval between 200 to 250 MeV above the B 0 s mass and simulated signal are first used. Then, separate samples are used to test the BDT performance. We weight the training simulation samples to match the two-dimensional B 0 s p and p T distributions, and smear the vertex fit χ 2 , to match the background-subtracted preselected data. Finally, the BDT requirement is chosen to maximize signal significance.
To determine the signal and background yields we fit the candidate B 0 s mass distribution. Backgrounds include combinatorics, whose shape is estimated using WS J/ψ π ± π ± candidates modelled by an exponential function, B 0 s → J/ψ η (→ ρ 0 γ) decays with the γ ignored, and Λ 0 b → J/ψ pK − decays with both hadrons misidentified as pions. The latter backgrounds are modelled using simulation. The B 0 s signal shape is parameterized by a Hypatia function [32], where the signal radiative tail parameters are fixed to values obtained from simulation. The same shape is used for the B 0 → J/ψ π + π − decays, with the mean value shifted by the known B 0 s and B 0 meson mass difference [14]. Finally, we fit simultaneously both RS and WS candidates, using the simulated shape for B 0 s → J/ψ η (→ ρ 0 γ) whose yield is allowed to float, and fixing both the size and shape of the Λ 0 b → J/ψ pK − component. The results of the fit are shown in Fig. 1. We find 33 530 ± 220 signal B 0 s within ±20 MeV of the B 0 s mass peak, with a purity of 84%. These decays are used for further analysis. Multiple candidates in the same event have a rate of 0.20% in a ±20 MeV interval around the B 0 s mass peak, and are retained. To subtract the background in the signal region in the amplitude fit we add negatively weighted events from the WS sample to the RS sample, also accounting for

Detector efficiency and resolution
The correlated efficiencies in m ππ and angular variables Ω are determined from simulation. We weight the simulated signal events to reproduce the B 0 s meson p T and η distributions as well as the track multiplicity of the events. The latter may influence the efficiencies of the tracking and particle identification. The calculated efficiencies are shown in Fig. 2 along with the determined efficiency function. The four-dimensional efficiency is parameterized by a combination of Legendre and spherical harmonic moments [33], as where P a are Legendre polynomials, Y bc are spherical harmonics, m min ππ = 2m π + and m max ππ = m B 0 s − m J/ψ , and abcd are efficiency coefficients determined from weighted averages of decays generated uniformly over phase space [6].
The model gives an excellent representation of the simulated data. The efficiency is uniform within about ±4% for cos θ J/ψ and about 10% for χ variables; however the m ππ and cos θ ππ variables show large efficiency variations and correlations (see Fig. 3), due to the χ 2 IP > 4 requirements on the hadrons. The loss of efficiency in the lower m ππ region can be understood as being due to kinematics; at cos θ ππ = ±1 and m ππ 0.6 − 0.8 GeV, which corresponds to the kinematic boundary of m 2 J/ψ π + . One of the pions is almost at rest in B 0 s rest frame, and thus the pion points to the PV, resulting in a very small χ 2 IP for this pion. The χ 2 IP variable is the most useful tool to suppress large pion combinatorial background from the PV.
The reconstruction efficiency is not constant as a function of B 0 s decay time due to displacement requirements applied to the hadrons in the offline selections and on J/ψ candidates in the trigger. It is determined using the control channel B 0 → J/ψ K * (892) 0 , with K * (892) 0 → K + π − , which is known to have a lifetime of τ B 0 = 1.520 ± 0.004 ps [14]. The same weighting procedure is implemented as done for B 0 s → J/ψ π + π − decays. The signal efficiency is calculated as ε is the efficiency of the control channel as measured by comparing data with the known lifetime distribution, and ε The acceptance is checked by measuring the decay width of B + → J/ψ K + decays. The fitted decay-width difference between the B + and B 0 mesons is Γ B + − Γ B 0 = −0.0475 ± 0.0013 ps −1 , where the uncertainty is statistical only, in agreement with the known value of −0.0474 ± 0.0023 ps −1 [14].
From the measured B 0 s candidate momentum and decay distance, the decay time and its uncertainty δ t are calculated. The calculated uncertainty is imbedded into the resolution function, which is modelled by the sum of three Gaussian functions with common means and widths proportional to a quadratic function of δ t . The parameters of the resolution function are determined with a sample of putative prompt J/ψ → µ + µ − decays combined with two pions of opposite charge. Taking into account the decay-time uncertainty distribution of the B 0 s signal, the average effective resolution is found to Table 1: Tagging efficiency, ε tag , and tagging power given as the efficiency times dilution squared, ε tag D 2 , where D = (1 − 2ω) for each category and the total. The uncertainties on ε tag are statistical only, and those for ε tag D 2 contain both statistical and systematic components.

Flavour tagging
Knowledge of the ( ) B 0 s flavour at production is necessary. We use information from decays of the other b hadron in the event (opposite-side, OS) and fragments of the jet that produced the ( ) B 0 s meson that contain a charged kaon, called same-side kaon (SSK) [27]. The OS tagger infers the flavour of the other b hadron in the event from the charges of muons, electrons, kaons, and the net charge of the particles that form reconstructed secondary vertices.
The flavour tag, q, takes values of +1, −1 or 0 if the signal meson is tagged as B 0 s , B 0 s or untagged, respectively. The wrong-tag probability, y, is estimated event-by-event based on the output of a neural network. It is subsequently calibrated with data in order to relate it to the true wrong-tag probability of the event by a linear relation as ( ) where p 0 , p 1 , ∆p 0 and ∆p 1 are calibration parameters, and ω(y) and ω(y) are the calibrated probabilities for a wrong-tag assignment for B 0 s and B 0 s mesons, respectively. The calibration is performed separately for the OS and the SSK taggers using B + → J/ψ K + and B 0 s → D − s π + decays, respectively. When events are tagged by both the OS and the SSK algorithms, a combined tag decision is formed. The resulting efficiency and tagging powers are listed in Table 1. 7 Description of the π + π − mass spectrum We fit the entire π + π − mass spectrum including the resonance contributions listed in Table 2, and a nonresonant (NR) component. We use an isobar model [21]. All resonances are described by Breit-Wigner amplitudes, except for the f 0 (980) state, which is modelled by a Flatté function [34]. The nonresonant amplitude is treated as being constant in m ππ . The previous publication [21] used an unconfirmed f 0 (1790) resonance, reported by the BES collaboration [35], instead of the f 0 (1710) state. We test which one gives a better fit.

BES [35]
The amplitude A R (m ππ ) is used to describe the mass line shape of resonance R. It is combined with the B 0 s and resonance decay properties to form the m ππ dependent expression Here P B is the J/ψ momentum in the B 0 s rest frame, P R is the momentum of either of the two hadrons in the dihadron rest frame, m B is the B 0 s mass, m 0 is the mass of resonance R, 6 J R is the spin of the resonance R, L B is the orbital angular momentum between the J/ψ meson and π + π − system, and L R the orbital angular momentum in the π + π − system, and thus is the same as the spin of the π + π − resonance. The terms F are the Blatt-Weisskopf barrier factors for the B 0 s meson and R resonance, respectively [37]. The shape parameters for the f 0 (980) and f 0 (1500) resonances are allowed to vary.

Likelihood definition
The decay-time distribution including flavour tagging is R(t, m ππ , Ω, q|y) = 1 1 + |q| [1 + q (1 − 2ω(y))] Γ(t, m ππ , Ω) wheret is the true decay time, ( -) Γ is defined in Eq. (1), and A P is the production asymmetry of B 0 s mesons. The fit function for the signal is modified to take into account the decay-time resolution and acceptance effects resulting in where ε(m ππ , Ω) is the efficiency as a function of m ππ and angular variables, T (t −t|δ t ) is the decay-time resolution function, and ε The free parameters in the fit are φ s , |λ|, Γ H − Γ B 0 , the amplitudes and phases of the resonances, and the shape parameters of some resonances. The other parameters, including ∆m s , and Γ L , are fixed to the known values [14] or other measurements mentioned below. The signal function is normalized by summing over q values and integrating over decay time t, the mass m ππ , and the angular variables, Ω, giving We assume no asymmetries in the tagging efficiencies, which are accounted for in the systematic uncertainties. The resulting signal PDF is The fitter uses a technique similar to sPlot [38] to subtract background from the log-likelihood sum. Each candidate is assigned a weight, W i = +1 for the RS events and negative values for the WS events. The likelihood function is defined as where s W ≡ i W i / i W 2 i is a constant factor accounting for the effect of the background subtraction on the statistical uncertainty.
The decay-time acceptance is assumed to be factorized from other variables, but due to the χ 2 IP cut on the two pions, the decay time is correlated with the angular variables. To avoid bias on the determination of Γ H from the decay-time acceptance, the simulated B 0 s signal is weighted in order to reproduce the m ππ resonant structure observed in data by using the preferred amplitude model that is determined by the overall fit. An iterative procedure is performed to finalize the decay-time acceptance. This procedure converges in three steps beyond which Γ H does not vary. When we apply this method to pseudoexperiments that include the correlation mentioned before, the fitter reproduces the input values of φ s , Γ H and |λ|.

Fit results
We first choose the resonances that best fit the m ππ distribution. Table 3 lists the different fit components and the value of −2 ln L. In these comparisons, the mass and width of most resonances are fixed to the central values listed in Table 2, except for the f 0 (980) and f 0 (1500) resonances, whose parameters are allowed to vary. We find two types of fit results, one with positive interference among the contributing resonances and the other with negative interference, defined by the sum of the resulting fit fractions. The first listed Solution I is better than Solution II by four standard deviations, calculated by taking the square root of the −2 ln L difference. We take Solution I for our measurement and II for systematic uncertainty evaluation. The models corresponding to Solutions I and II are very similar to those found in our previous analysis of the same final state [21].
For the fit we assume that the CP -violation quantities (φ s i , |λ i |) are the same for all the resonances. We also fix ∆m s to the central value of the world average

# Resonance content
Int −2 ln L I f 0 (980) + f 0 (1500) + f 0 (1790) + f 2 (1270) + f 2 (1525)+NR − −4850 17.757 ± 0.021 ps −1 [14], and fix Γ L to the central value of 0.6995 ± 0.0047 ps −1 from the LHCb B 0 s → J/ψ K + K − results [6]. The fit values and correlations of the CP -violating parameters are shown in Table 4 for Solution I. The shape parameters of f 0 (980) and f 0 (1500) resonances are found to be consistent with our previous results [21]. The angular and decay-time fit projections are shown in Fig. 4. The m ππ fit projection is shown in Fig. 5, where the contributions of the individual resonances are also displayed. The first two solutions give very similar fit values for φ s and Γ H . We also find that the CP -odd fraction is greater than 97% at 95% confidence level. The resonant content for Solutions I and II are listed in Table 5.

Systematic uncertainties
The systematic uncertainties for the CP -violating parameters, λ and φ s , are smaller than the statistical ones. They are summarized in Table 6 along with the uncertainty on Γ H − Γ B 0 . The uncertainty on the decay-time acceptance is found by varying the parameters of the acceptance function within their uncertainties and repeating the fit. The same procedure is followed for the uncertainties on the B 0 lifetime, ∆m s , Γ L , m ππ and angular efficiencies, resonance masses and widths, flavour-tagging calibration, and allowing for a 2% production asymmetry [39]; this uncertainty also includes any possible difference in flavour tagging between B 0 s and B 0 s . Simulation is used to validate the method for the time-resolution calibration. The uncertainties of the parameters of the time-resolution model are estimated using the difference between the signal simulation and prompt J/ψ simulation. These uncertainties are varied to obtain the effects on the physics parameters. Resonance modelling uncertainty includes varying the Breit-Wigner barrier factors, changing the default values of L B = 1 for the D-wave resonances to one or    two, the differences between the two best solutions, and replacing the NR component by the f 0 (500) resonance. Furthermore, including an isospin-violating ρ(770) 0 component in the fit, results in a negligible contribution of (1.1 ± 0.3)%. The process B + c → π + B 0 s can affect the measurement of Γ H − Γ B 0 . An estimate of the fraction of these decays in our sample is 0.8% [5]. Neglecting the B + c contribution leads to a bias of 0.0005 ps −1 , which is added as a systematic uncertainty. Other parameters are unchanged.
Corrections from penguin amplitudes are ignored because their effects are known to be small [40][41][42] compared to the current experimental precision.

Conclusions
Using B 0 s and B 0 s → J/ψ π + π − decays, we measure the CP -violating phase, φ s = −0.057 ± 0.060 ± 0.011 rad, the decay-width difference Γ H − Γ B 0 = −0.050 ± 0.004 ± 0.004 ps −1 , and the parameter |λ| = 1.01 +0.08 −0.06 ± 0.03, where the quoted uncertainties are statistical and systematic. These results are more precise than those obtained from the previous study of this mode using 7 TeV and 8 TeV pp collisions (Run 1) [4]. To combine the Run-1 results with these, we reanalyze them by fixing ∆m s = 17.757 ± 0.021 ps −1 from Ref. [14], and Γ L = 0.6995 ± 0.0047 ps −1 from the LHCb B 0 s → J/ψ K + K − results [6]. We remove the Gaussian constraint on ∆Γ s and let Γ H vary. Instead of taking the uncertainties of flavour tagging and decay-time resolution into the statistical uncertainty, we place these sources in the systematic uncertainty and assume 100% correlation with our new results. The updated results are: φ s = 0.075 ± 0.065 ± 0.014 rad and |λ| = 0.898 ± 0.051 ± 0.013 with a correlation of 0.025. We then use the updated φ s and |λ| Run-1 results as a constraint into our new φ s fit. 7 The combined results are Γ H − Γ B 0 = −0.050 ± 0.004 ± 0.004 ps −1 , |λ| = 0.949 ± 0.036 ± 0.019, and φ s = 0.002 ± 0.044 ± 0.012 rad. The correlation coefficients among the fit parameters are 0.025 (ρ 12 ), -0.001 (ρ 13 ), and 0.026 (ρ 23 ). Our results still have uncertainties greater than the SM prediction and are slightly more precise than the measurement using B 0 s → J/ψ K + K − decays, based only on Run-1 data, which has a precision of 0.049 rad [5]. Hence this is the most precise determination of φ s to date.