Measurement of the CP-violating phase $\beta$ in $B^0\rightarrow J/\psi \pi^+\pi^-$ decays and limits on penguin effects

Time-dependent CP violation is measured in the $B^0\rightarrow J/\psi\pi^+\pi^-$ channel for each $\pi^+\pi^-$ resonant final state using data collected with an integrated luminosity of 3.0 fb$^{-1}$ in $pp$ collisions using the LHCb detector. The final state with the largest rate, $J/\psi\rho^0(770)$, is used to measure the CP-violating angle $2\beta^{\rm eff}$ to be $(41.7\pm 9.6_{-6.3}^{+2.8})^{\circ}$. This result can be used to limit the size of penguin amplitude contributions to CP violation measurements in, for example, $B_s^0\rightarrow J/\psi\phi$ decays. Assuming approximate SU(3) flavour symmetry and neglecting higher order diagrams, the shift in the CP-violating phase $\phi_s$ is limited to be within the interval [$-1.05^\circ$, +$1.18^\circ$] at 95% confidence level. Changes to the limit due to SU(3) symmetry breaking effects are also discussed.


Introduction
Measurements of CP violation in neutral B meson decays are used either to search for physics beyond the Standard Model (SM) [1] or set limits on combinations of Cabibbo-Kobayashi-Maskawa couplings (V ij ) [2]. Interpretations of the measurement of the CP-violating phase 2β via the interference of mixing and decays in the (-) B 0 → J /ψ K 0 S channel, and the phase φ s in (-) B 0 s → J /ψφ and J /ψπ + π − decays, 1 are made assuming that the decays are dominated by tree-level processes. However, penguin processes are also possible, and they may have amplitudes large enough to influence the results. Here we use (-) B 0 → J /ψπ + π − decays to set limits on possible changes due to penguin contributions. This mode has both tree and penguin diagrams, as shown in Fig. 1. Theoretical models, to be discussed later, predict that the ratio of penguin to tree amplitudes is greatly enhanced in this decay relative to (-) B 0 → J /ψ K 0 S [3,4]. Thus, the effects of penguin topologies can be investigated by using the J /ψπ + π − decay and comparing differ- 1 CP violation measurements in (-) B 0 → J /ψ K 0 S determine the sum of 2β ≡ 2 arg(−V cd V * cb )/(V td V * tb ) and contributions from higher order diagrams. Similar measurements in the (-) B 0 s system determine φ s which is the sum of −2β s ≡ −2 arg(−V ts V * tb )/(V cs V * cb ) and higher order corrections. ent measurements of the CP-violating phase 2β in J /ψ K 0 S , and individual channels such as (-) B 0 → J /ψρ 0 (770). 2 Next, we discuss the time-dependent decay rate, taking into account that the π + π − system is composed of the resonances previously reported in Ref. [5]. This analysis largely follows the measurement procedure used in the study of CP violation in (-) B 0 s decays into J /ψπ + π − [6]. The total decay amplitude for (-) B 0 at a decay time of zero is taken to be the sum over individual π + π − resonant transversity amplitudes [7], and possibly one non-resonant amplitude, with each component labeled as A i ( A i ). The quantities q and p relate the mass eigenstates to the flavor eigenstates [8]. By introducing the parameter λ i ≡ q p A i A i , relating CP violation in the interference between mixing and decay associated with the state i, the amplitudes A and A can be expressed as the sums of the in- (1)

Penguin and tree amplitudes
The decay B 0 → J /ψ K 0 S can be written as the sum of one tree level amplitude, similar to that shown in Fig. 1(a), but where the virtual W − transforms to a cs pair, and three penguin amplitudes similar to those shown in Fig. 1(b). Here we neglect higher order diagrams. The t-quark mediated penguin amplitude can be expressed in terms of the other two using CKM unitarity. The resulting decay amplitude is [3] A where λ = |V us | = 0.2252 [10] notes the sum of tree and penguin strong amplitudes, and a and θ are the magnitude and phase of the strong parts of the effective penguin amplitude relative to the tree amplitude.
For the case of (-) B 0 → J /ψπ + π − decays, the π + π − pairs are in spin states ranging from zero to two. Since they are in a final state with a spin-1 J /ψ resonance, the amplitudes in the different transversity states f need to be distinguished for all spins above zero. For example, the amplitude for each J /ψρ 0 (770) transversity state is where the primed quantities are defined in analogy with the unprimed ones in Eq. (2). For B 0 decays only the sign in front of iγ changes. We are only concerned here with the relative size of the tree and penguin amplitudes. For J /ψ K 0 S the penguin is suppressed relative to the tree by an additional factor of ≡ λ 2 /(1 − λ 2 ) = 0.0534. Thus, comparing even a relatively poor measurement of 2β eff measured in J /ψρ 0 with 2β measured in J /ψ K 0 S allows us to set stringent limits on the penguin contribution. Using approximate SU(3) flavor symmetry the size of the penguin contribution in (-) B 0 → J /ψρ 0 can be related to that in (-) B 0 s → J /ψφ decays as pointed out in Refs. [4,11].
We now turn to the expressions for CP violation in the presence of both tree and penguin amplitudes. The complex-valued CP parameter λ f is given by where β is the phase induced by mixing. Thus the measured phase β eff f is related to β by 3 Note that while q/p and A i /A i are phase convention dependent, λ i is not. 4 We assume Γ d = 0 and |p/q| = 1. The averages of current measurements are Γ d /Γ d = 0.001 ± 0.010 and |p/q| = 1.0005 ± 0.0011 [9].
separating real and imaginary parts gives , and For the J /ψ K 0 S mode we replace a f and θ f in Eq. (6) by − a f and θ f , respectively. In addition, we take a = a and θ = θ . The relationship between the penguin influence on the mixing induced CP violation phase in favored decays and the measurements in We will show that the penguin shift has a weak dependence on |λ f |, resulting in δ P ≈ − 2β f . Since the uncertainty on the current measurement of 2β is ( +1.6 −1.5 ) • , a measurement of 2β f , even with an uncertainty ten times larger, could limit penguin contributions to be well below the current statistical uncertainty, which is the main aim of this analysis.

Detector software and event selection
The LHCb detector [12] 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 [13], a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about 4 Tm, and three stations of siliconstrip detectors and straw drift tubes [14] placed downstream of the magnet. The tracking system provides a measurement of momentum, 5 p, with a relative uncertainty that varies from 0.4% at low momentum to 0.6% at 100 GeV. 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 p transverse to the beam, in GeV. Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors [15]. 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 [16]. The trigger [17] consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage that applies a full event reconstruction [17]. Events selected for this analysis are triggered by a J /ψ → μ + μ − decay, where the J /ψ meson is required at the software level to be consistent with coming from the decay of a (-) B 0 meson by use of either of IP requirements or detachment of the J /ψ meson decay vertex from the primary vertex. In the simulation, pp collisions are generated using Pythia [18] with a specific LHCb configuration [19]. Decays of hadronic particles are described by EvtGen [20], in which final state radiation is generated using Photos [21]. The interaction of the generated particles with the detector and its response are implemented using the Geant4 toolkit [22] as described in Ref. [23].
A (-) B 0 → J /ψπ + π − candidate is reconstructed by combining a J /ψ → μ + μ − candidate with two pions of opposite charge. The like-sign combinations J /ψπ ± π ± are also reconstructed for background studies. The event selection is described in detail in the time-integrated amplitude analysis [5]. The only difference here is that we reject K 0 S → π + π − candidates by excluding the events in the region within ±20 MeV of the K 0 S mass peak.
Only candidates with dimuon invariant mass between −48 MeV and +43 MeV relative to the observed J /ψ mass peak are selected, corresponding a window of about ±3σ . The two muons subsequently are kinematically constrained to the known J /ψ mass.
Other requirements are imposed to isolate B 0 candidates with high signal yield and minimum background. This is accomplished by combining the J /ψ → μ + μ − candidate with a pair of pion candidates of opposite charge, and then testing if all four tracks form a common decay vertex. Pion candidates are each required to have p T greater than 250 MeV, and the scalar sum of the two transverse momenta, p T (π + ) + p T (π − ), must be larger than 900 MeV. To test for inconsistency with production at the PV, the IP χ 2 is computed as the difference between the χ 2 of the PV reconstructed with and without the considered track. Each pion must have an IP χ 2 greater than 9. Pion and kaon candidates are positively identified using the RICH system. The four-track B 0 candidate must have a flight distance of more than 1.5 mm, where the average decay length resolution is 0.17 mm. The angle between the combined momentum vector of the decay products and the vector formed from the positions of the PV and the decay vertex (pointing angle) is required to be less than 2.5 • . Events satisfying this preselection are then further filtered using a multivariate analyzer based on a Boosted Decision Tree (BDT) technique [24]. The BDT uses eight variables that are chosen to provide separation between signal and background. These are the minimum of DLL(μ − π ) of the μ + and μ − , p T (π + ) + p T (π − ), the minimum of IP χ 2 of the π + and π − , and the B 0 properties of vertex χ 2 , pointing angle, flight distance, p T and IP χ 2 , where DLL(μ − π ) is a logarithm of the likelihood ratio between μ and π hypotheses for the muon candidates.
The BDT is trained on a simulated sample of two million B 0 → J /ψπ + π − signal events generated uniformly in phase space with unpolarized J /ψ → μ + μ − decays, and a background data sample from the sideband 5566 < m( J /ψπ + π − ) < 5616 MeV. Then separate samples are used to train and test the BDT.

The invariant mass of the selected
where the dimuon pair is constrained to have the J /ψ mass, is shown in Fig. 2. There is a large peak at the B 0 s mass and a smaller one at the B 0 mass on top of the background. A double Crystal Ball function with common means models the radiative tails and is used to fit each of the signals [25]. Other components in the fit model take into account background contributions tions, and combinatorial backgrounds. The exponential combinatorial background shape is taken from like-sign combinations, that are the sum of π + π + and π − π − candidates. The shapes of the other components are taken from the simulation with their normalizations allowed to vary. Only the candidates within ±20 MeV of the B 0 mass peak are retained for CP violation measurements;

The signal likelihood
We fit the entire π + π − mass spectrum, by including the resonance contributions found in the amplitude analysis [5], in order to measure the CP-violating parameters of all the states, the most important being B 0 rest frame [26,27]. We perform a simultaneous unbinned maximum likelihood fit to the decay time t, m hh , and the three helicity angles Ω, along with information on the initial flavor of the decaying hadron, i.e. whether it was produced as a B 0 or a B 0 meson. The probability density function (PDF) used in the fit consists of signal and background components that include detector resolution and acceptance effects. The predicted decay time error for each event is used for the decay time resolution model, and similarly the measured per-event misidentification probability is used for determining the initial flavor of the neutral B meson. The π + π − invariant mass distribution is shown in Fig. 3 along with the fitted components of the different resonances using the "Best model" [5] for the π + π − resonance content. Knowledge of the (-) B 0 flavor at production, called "tagging", is necessary to measure CP violation. We use both opposite-side (OS) [28] and same-side pion (SSπ ) tagging information; here we use the same procedure as for same-side kaon tagging used in the (-) B 0 s → J /ψπ + π − and J /ψφ analyses [27], but identify the tag from a pion rather than a kaon. The wrong-tag probability η is estimated based on the output of a neural network trained on simulated data. It is calibrated with data using flavor-specific decay modes in order to predict the true wrong-tag probability of the event (-) ω(η) for an initial flavor (-) B 0 meson, which has a linear dependence on η. The calibration is performed separately for the OS and the SSπ taggers. If events are tagged by both OS and SSπ algorithms, a combined tag decision and wrong-tag probability are given by the algorithm defined in Ref. [28]. This combined algorithm is implemented in the overall fit. The effective tagging power obtained is characterized by ε tag D 2 = (3.26 ± 0.17)%, where D ≡ (1 −2ω avg ) is the dilution, ω avg is the average wrong-tag probability for ω and ω, and ε tag = (42.1 ± 0.6)% is the signal tagging efficiency.
The signal decay time distribution including flavor tagging is where t is the true decay time, Γ is defined in Eq. (1), and A P = −0.0035 ± 0.0081 [29] is the B 0 -B 0 production asymmetry in the LHCb acceptance. The flavor tag parameter q takes values of −1 or +1 if the signal meson is tagged as B 0 , B 0 respectively, or 0 if untagged.
The signal function is convolved with the decay time resolution and multiplied by the acceptance: where ε(m hh , Ω) is the efficiency as a function of the h + h − mass and angles, obtained from the simulation as described in Ref. [5], is the decay time resolution function which depends upon the estimated decay time error for each event δ t , and E t (t) is the decay time acceptance function. The decay time resolution function T (t −t; δ t ) is described by a sum of three Gaussian functions with a common mean. Studies using simulated data show that J /ψπ + π − combinations produced directly in the pp interaction (prompt) have nearly identical resolution to signal events. Specifically, the time resolution is determined using prompt J /ψ decays into a dimuon pair, using a dedicated trigger for calibration purposes, plus two oppositely charged tracks from the primary vertex with the similar selection criteria as for J /ψπ + π − and an invariant mass within ±20 MeV of the B 0 mass. The effective resolution is found to be about 40 fs by using the weighted average widths of the three Gaussians. This is negligibly small compared to the B 0 -B 0 oscillation time.
The decay time distribution is influenced by acceptance effects that are introduced by track reconstruction, trigger and event selection. The decay time acceptance is obtained using control samples of (-) B 0 → J /ψ (-) K * 0 (→ K ∓ π ± ) decays, corrected by the acceptance ratio between J /ψ K ∓ π ± and J /ψπ + π − derived from simulation.
The acceptance function for the control sample is defined as (10) where a, n, t 0 , β 1 , β 2 are parameters determined by the fit. The decay time distribution of (-) where f 0 is the signal fraction, and N B 0 and N 0 bkg are normalizations necessary to construct PDFs of signal and background, respectively. The background acceptance function in Eq. (11) uses the same form as the signal and its parameters a 0 bkg , n 0 bkg and τ 0 bkg are obtained from mass sideband regions of 5180-5205 MeV and 5400-5425 MeV. The lifetime is constrained to τ B 0 = 1.519 ± 0.007 ps [10].
We use the product of the acceptance A(a, n, t 0 , β 1 , β 2 ) determined from (-) B 0 → J /ψ (-) K * 0 and the correction ratio found from simulation as the time acceptance function for (12) with parameter values and correlations given in Table 1.

Measurements of 2β eff
The CP-violating parameters are determined from a fit that uses the amplitude model with six final state π + π − resonances. In our previous amplitude analysis [5] we used two parameterizations of the f 0 (500) resonance, "default" and "alternate". The default used a Breit-Wigner resonance shape, with relatively poorly measured parameters, while the alternate used a function suggested by Bugg [30], with more theoretically motivated shape parameters. In this analysis we choose to switch to the shape suggested by Bugg, while the Breit-Wigner shape of the previous default parameterization is used to assess systematic uncertainties. A Gaussian Table 1 Parameter values and correlations for the acceptance function ε t (t) in Eq. (12  As J /ψρ is the final state with the largest contribution, we treat it specially and perform two fits. In both cases all resonances other than the ρ share a common CP violation parameter λ . For Fit 1 the three ρ transversity states share the same CP violation parameter λ, while for Fit 2 each ρ transversity state has its own CP violation parameter λ i . The results are shown in Table 2. The statistical uncertainties are within ±15% of the precision estimated using toy Monte Carlo simulation. To determine 2β f we use the measured value in b → ccs transitions of (42.8 +1. 6 −1.5 ) • found in (-) B 0 decays [9]. Our measurement of 2β eff is consistent with this value for both Fit 1 and Fit 2. The correlation between α ρ CP and 2β eff ρ is −0.01 in Fit 1. Table 3 shows the correlation matrix for the CP-violating parameters in Fit 2. Table 4 lists the fit fractions and three transversity fractions of contributing resonances from Fit 1, consistent with the results shown in the amplitude analysis [5]. For a P -or D-wave resonance, we report its total fit fraction by summing all three transversity components. This time-dependent analysis determines the phase difference between the CP-odd component of ρ(770) ⊥ and the CP-even component of ρ(770) 0 to be (167 ± 11) • in Fit 1.  This quantity is not accessible in the time-integrated amplitude analysis. Fig. 4 shows the decay time distribution superimposed with the fit projection. The statistical significances of the CP measurements are ascertained by fitting the data requiring that CP-violating components Table 3 The correlation matrix for the CP-violating parameters determined using Fit 2, where 2β eff 1.00 Table 5 Results allowing for different CP-violating effects for resonances other than the ρ in an extension of Fit 1.  Table 5.
We find that these fits all give consistent values of the CP-violating parameters.
The systematic uncertainties evaluated for both fit configurations are summarized for the CP-violating phases in Table 6 and for the magnitudes of the asymmetries in Table 7. They are small compared to the statistical ones. The two largest contributions result from the resonance fit model and the resonance parameters. Fit model uncertainties are determined by adding an additional resonance to the default six-resonance model, either the f 0 (980), the f 0 (1500), the f 0 (1700), or non-resonant π + π − , replacing the f 0 (500) model by a Breit-Wigner function, and using the alternative Gounaris-Sakurai model shapes [31] for the various ρ mesons.
The largest variation among those changes is assigned as the systematic uncertainty for modeling. Including a non-resonant component gives the largest negative change on 2β eff for the ρ and ρ 0 categories.
To evaluate the uncertainties due to the fixed parameters of resonances, we repeat the amplitude fit by varying the mass and width of all the resonances used in the six-resonance model within their errors one at a time, and add the changes in quadrature. To evaluate the systematic uncertainties due to the other fixed parameters including those in the decay time acceptance, the background decay time PDF, the m(π + π − ) distribution, the angular acceptance, and background mass PDF, the data fit is repeated by varying the fixed parameters from their nominal values according to the error matrix, one hundred times for each source. The matrix elements are determined using simulation, data, and like-sign J /ψπ ± π ± data. The r.m.s. of the fitted physics parameter of interest is taken as its uncertainty for each source.
The acceptance model for each of the three angles as a function of m hh is determined independently. To evaluate the reliability of this method we parameterize the mass and angle efficiencies as a combination of Legendre polynomials and spherical harmonics that takes into account all correlations. The amplitude fit is repeated using the new acceptance parameterizations; changes are found to be small and taken as the systematic uncertainty.
In the nominal fit the background is divided into three sources: background to the ρ 0 component from (-) K * 0 when the kaon is misidentified as a pion, and the remaining background. The latter includes the reflections from (-) p decays, where both the kaon and proton are misidentified, and combinatorial background. The dependence on m hh of the decay time distribution for this remaining background is modeled by using different decay time PDFs in different m hh regions. We also change the background modeling The systematic uncertainty due to the tagging parameter calibration is given by the difference in quadrature of the statistical uncertainties for each physics parameter between the nominal fit and an alternative fit where the tagging parameters are Gaussian constrained by their total uncertainties. The systematic uncertainty due to the asymmetry of B 0 − B 0 meson production is estimated by varying the central value A P = −0.0035 ± 0.0081 [29] by its uncertainty.

Discussion of results and conclusions
We compare the ρ-only Fit 1 result of 2β J /ψρ = 2β eff = (41.7 ± 9.6 +2.8 −6.3 ) • with the Cabibbo-favored B to charmonium result, denoted J /ψ K 0 S . The measured difference is Since the result is consistent with zero we determine limits on the magnitude of the CP-violating phase shift due to a possible penguin component in b → ccs decays, δ P . The limit is evalu- B 0 s → J /ψφ decays, and neglecting higher order diagrams [3], we find δ P = (0.05 ± 0.56) • = 0.9 ± 9.8 mrad. At  Table 7 Systematic uncertainties for the magnitude of the asymmetries α i CP (×10 −3 ). Statistical uncertainties are also shown.  shows how δ P varies as a function of θ − θ , indicating that the 95% CL limit on penguin pollution can increase to at most ±1.2 • . The variation in δ P is proportional to a/a . Thus, when changing a/a over the interval 0.5 to 1.5, the limit on the penguin shift at 95% CL varies between ±0.9 • to ±1.8 • , even allowing for maximal breaking between θ and θ . It may be expected that the effect of penguin contributions in other decays, such as (-) B 0 → J /ψ K 0 S , should be limited to similar values, even if there is no strict flavor symmetry relating the mode to (-) B 0 → J /ψρ 0 . Our limit is consistent with theoretical predictions [32].
We also set limits on the strong decay amplitude. Fig. 7 shows the 68% and 95% confidence levels contours for the penguin amplitude parameters of a and θ with a −2 ln L change of 2. The decay (-) B 0 → J /ψπ 0 proceeds through a similar diagram to that shown in Fig. 1, and thus the CP-violating parameters S and C should be similar to those we find in (-) B 0 → J /ψρ 0 . These parameters are related to the parameter λ f via the relationships where we set the CP eigenvalue η f = 1 to compare with the CP-even mode  Table 8 shows the comparison of S f and C f  . 7. Contours corresponding to 68% (dashed) and 95% (solid) confidence levels for ndf of two, respectively, for the penguin amplitude parameters a and θ .
from this measurement with that obtained from the Belle [33] and BaBar [34] Collaborations. Our measurements are in good agreement with the Belle results.
In conclusion, the measured value of the penguin contribution is δ P = (0.05 ± 0.56) • = 0.9 ± 9.8 mrad. Taking the maximum breaking in phase and a range of breaking 0.5 < a/a < 1.5 the uncertainty on δ P becomes ±18 mrad. The measured value of φ s currently has an uncertainty of about 35 mrad, and the value of 2β of 1.5 • or 26 mrad [9]. Thus our limit is smaller than the current uncertainties, but will need to become more precise as the CP-phase measurements improve.