Determination of the sign of the decay width difference in the B_s system

The interference between the K+K- S-wave and P-wave amplitudes in B_s ->J/psi K+K- decays with the K+K- pairs in the region around the phi(1020) resonance is used to determine the variation of the difference of the strong phase between these amplitudes as a function of K+K- invariant mass. Combined with the results from our CP asymmetry measurements in B_s ->J/psi phi decays, we conclude that the B_s mass eigenstate that is almost CP =+1 is lighter and decays faster than the mass eigenstate that is almost CP =-1. This determines the sign of the decay width difference DeltaGamma_s == Gamma_L -Gamma_H to be positive. Our result also resolves the ambiguity in the past measurements of the CP violating phase phi_s to be close to zero rather than pi. These conclusions are in agreement with the Standard Model expectations.

The interference between the K + K − S-wave and P-wave amplitudes in B 0 s → J/ψK + K − decays with the K + K − pairs in the region around the φ(1020) resonance is used to determine the variation of the difference of the strong phase between these amplitudes as a function of K + K − invariant mass. Combined with the results from our CP asymmetry measurement in B 0 s → J/ψφ decays, we conclude that the B 0 s mass eigenstate that is almost CP = +1 is lighter and decays faster than the mass eigenstate that is almost CP = −1. This determines the sign of the decay width difference ∆Γs ≡ ΓL − ΓH to be positive. Our result also resolves the ambiguity in the past measurements of the CP violating phase φs to be close to zero rather than π. These conclusions are in agreement with the Standard Model expectations.
To be submitted to Physical Review Letters The decay time distributions of B 0 s mesons decaying into the J/ψφ final state have been used to measure the parameters φ s and ∆Γ s ≡ Γ L − Γ H of the B 0 s system [1][2][3]. Here φ s is the CP violating phase equal to the phase difference between the amplitude for the direct decay and the amplitude for the decay after oscillation. Γ L and Γ H are the decay widths of the light and heavy B 0 s mass eigenstates, respectively. The most precise results, presented recently by the LHCb experiment [3], show no evidence of CP violation yet, indicating that CP violation is rather small in the B 0 s system. There is clear evidence for the decay width difference ∆Γ s being nonzero. It must be noted that there exists another solution φ s = 2.99 ± 0.18 (stat) ± 0.06 (syst) rad ∆Γ s = −0.123 ± 0.029 (stat) ± 0.011 (syst) ps −1 (II) arising from the fact that the time dependent differential decay rates are invariant under the transformation (φ s , ∆Γ s ) ↔ (π −φ s , −∆Γ s ) together with an appropriate transformation for the strong phases. In the absence of CP violation, sin φ s = 0, i.e. φ s = 0 or φ s = π, the two mass eigenstates become also CP eigenstates with CP = +1 and CP = −1. They can be identified by the decays into final states which are CP eigenstates. In B 0 s → J/ψK + K − decays, the final state is a superposition of CP = +1 and CP = −1 for the K + K − pair in the P-wave configuration, and CP = −1 for the K + K − pair in the S-wave configuration. Higher order partial waves are neglected. These decays have different angular distributions of the final state particles and are distinguishable.
Solution I is close to the case φ s = 0 and leads to the light (heavy) mass eigenstate being almost aligned with the CP = +1 (CP = −1) state. Similarly, solution II is close to the case φ s = π and leads to the heavy (light) mass eigenstate being almost aligned with the CP = +1 (CP = −1) state. A fit to the observed decay time distribution shows that it can be well described by a superposition of two exponential functions corresponding to CP = +1 and CP = −1, compatible with no CP violation [3]. In this fit the decay width to the CP = +1 final state is found to be larger than that to CP = −1.
Thus the mass eigenstate that is predominantly CP even decays faster than the CP odd state. For solution I, we find ∆Γ s > 0, i.e. Γ L > Γ H , and for solution II, ∆Γ s < 0, i.e. Γ L < Γ H . In order to determine if the decay width difference ∆Γ s is positive or negative it is necessary to resolve the ambiguity between the two solutions.
Since each solution corresponds to a different set of strong phases, one may attempt to resolve the ambiguity by using the strong phases either as predicted by factorisation, or as measured in B 0 → J/ψK * 0 decays. Unfortunately these two possibilities lead to opposite answers [4]. A direct experimental resolution of the ambiguity is therefore desirable.
In this Letter, we resolve this ambiguity using the de- The total decay amplitude is a coherent sum of S-wave and P-wave contributions. The phase of the P-wave amplitude, which can be described by a spin-1 Breit-Wigner function of the invariant mass of the K + K − pair, denoted by m KK , rises rapidly through the φ(1020) mass region. On the other hand the phase of the S-wave amplitude should vary relatively slowly for either an f 0 (980) contribution or a non-resonant contribution. As a result, the phase difference between the S-wave and P-wave amplitudes falls rapidly with increasing m KK . By measuring this phase difference as a function of m KK , and taking the solution with a decreasing trend around the φ(1020) mass as the physical solution, the sign of ∆Γ s is determined and the ambiguity in φ s is resolved [5]. This is similar to the way the BaBar collaboration measured the sign of cos 2β using the decay , where 2β is the weak phase characterizing mixing-induced CP asymmetry in this decay.
The analysis is based on the same data sample as used in Ref. [3], which corresponds to an integrated luminosity of 0.37 fb −1 of pp collisions collected by the LHCb experiment at the Large Hadron Collider at the centre of mass energy of √ s = 7 TeV. The LHCb detector is a forward spectrometer and is described in detail in Ref. [7]. The trigger, event selection criteria and analysis method are very similar to those in Ref. [3], and here we discuss only the differences. The fraction of K + K − S-wave contribution measured within ±12 MeV of the nominal φ(1020) mass is 0.042 ± 0.015 ± 0.018 [3]. (We adopt units such that c = 1 and = 1.) The S-wave fraction depends on the mass range taken around the φ(1020). The result of Ref. [3] is consistent with the CDF limit on the S-wave fraction of less than 6% at 95% CL (in the range 1009-1028 MeV) [2], smaller than the DØ result of (12 ± 3)% (in 1010-1030 MeV) [8], and consistent with phenomenological expectations [9]. In order to apply the ambiguity resolution method described above, the range of m KK is extended to 988-1050 MeV. Figure 1 shows the µ + µ − K + K − mass distribution where the mass of the µ + µ − pair is constrained to the nominal J/ψ mass. We perform an unbinned maximum likelihood fit to the invariant mass distribution of the selected B 0 s candidates. The probability density function (PDF) for the signal B 0 s invariant mass m J/ψKK is modelled by two Gaussian functions with a common mean. The fraction of the wide Gaussian and its width relative to that of the narrow Gaussian are fixed to values obtained from simulated events. A linear function describes the m J/ψKK distribution of the background, which is dominated by combinatorial background.
This analysis uses the sWeight technique [10] for background subtraction. The signal weight, denoted by W s (m J/ψKK ), is obtained using m J/ψKK as the discriminating variable. The correlations between m J/ψKK and other variables used in the analysis, including m KK , decay time t and the angular variables Ω defined in Ref. [3], are found to be negligible for both the signal and background components in the data. Figure 2 shows the m KK distribution where the background is subtracted statistically using the sWeight technique. The range of m KK is divided into four intervals: 988-1008 MeV, 1008-1020 MeV, 1020-1032 MeV and 1032-1050 MeV. Table I gives the number of B 0 s signal and background candidates in each interval. In this analysis we perform an unbinned maximum likelihood fit to the data using the sFit method [11], an extension of the sWeight technique, that simplifies fitting in the presence of background. In this method it is only necessary to model the signal PDF, as background is cancelled statistically using the signal weights.
The parameters of the B 0 s → J/ψ K + K − decay time distribution are estimated from a simultaneous fit to the  four intervals of m KK by maximizing the log-likelihood function where N k = N sig;k + N bkg;k . Θ P represents the physics parameters independent of m KK , including φ s , ∆Γ s and the magnitudes and phases of the P-wave amplitudes. Note that the P-wave amplitudes for different polarizations share the same dependence on m KK . Θ S denotes the values of the m KK -dependent parameters averaged over each interval, namely the average fraction of S-wave contribution for the k-th interval, F S;k , and the average phase difference between the S-wave amplitude and  3. Distribution of (a) K + K − S-wave signal events, and (b) K + K − P-wave signal events, both in four invariant mass intervals. In (b) the distribution of simulated B 0 s → J/ψ φ events in the four intervals assuming the same total number of P-wave events is also shown (dashed). Note the interference between the K + K − S-wave and P-wave amplitudes integrated over the angular variables has vanishing contribution in these distributions.
the perpendicular P-wave amplitude for the k-th interval, δ S⊥;k . P sig is the signal PDF of the decay time t, angular variables Ω, initial flavour tag q and the mistag probability ω. It is based on the theoretical differential decay rates [5] and includes experimental effects such as decay time resolution and acceptance, angular acceptance and imperfect identification of the initial flavour of the B 0 s particle, as described in Ref. [3]. The factors W p;k account for loss of statistical precision in parameter estimation due to background dilution. Their values are given in Table I.
The fit results for φ s , ∆Γ s and the fractions of S-wave contribution and phase differences between the S-wave and P-wave amplitudes for the four m KK intervals are given in Table II. Figure 3 shows the estimated K + K − S-wave and P-wave contributions in the four m KK intervals. The shape of the measured P-wave m KK dis-  tribution is in good agreement with that of B 0 s → J/ψ φ events simulated using a spin-1 relativistic Breit-Wigner function for the φ(1020) amplitude. In Fig. 4, the phase difference between the S-wave and the perpendicular Pwave amplitude is plotted in four m KK intervals for solution I and solution II. Figure 4 shows a clear decreasing trend of the phase difference between the S-wave and P-wave amplitudes in the φ(1020) mass region for solution I, as expected for the physical solution. To estimate the significance of the result we perform an unbinned maximum likelihood fit to the data by parameterizing the phase difference δ S⊥;k as a linear function of the average m KK value in the k-th interval. This leads to a slope of −0.050 +0.013 −0.020 rad/MeV for solution I and +0.050 +0.020 −0.013 rad/MeV for solution II, where the uncertainties are statistical only.
The difference of the ln L value between this fit and a fit in which the slope is fixed to be zero is 11.0. Hence the negative trend of solution I has a significance of 4.7 standard deviations. Therefore, we conclude that solution I, which has ∆Γ s > 0, is the physical solution. The trend of solution I is also qualitatively consistent with that of the phase difference between the K + K − S-wave and P-wave amplitudes versus m KK measured in the decay D + s → K + K − π + by the BaBar collaboration [12]. Several possible sources of systematic uncertainty on the phase variation versus m KK have been considered. A possible background from decays with similar final states such as B 0 → J/ψ K * 0 could have a small effect. From simulation, the contamination to the signal from such decays is estimated to be 1.1% in the m KK range of 988-1050 MeV. We add a 2.2% contribution of simulated B 0 → J/ψ K * 0 events to the data and repeat the analysis. The largest observed change is a shift of δ S⊥;4 by 0.06 rad, which is only 20% of its statistical uncertainty and has negligible effect on the slope of δ S⊥ versus m KK . The effect of neglecting the variation of the values of F S and δ S⊥ in each m KK interval is determined to change the significance of the negative trend of solution I by less than 0.1 standard deviations. We also repeat the analysis for different m KK ranges, different ways of dividing the m KK range or different shapes of the signal and background m J/ψKK distributions. The significance of the negative trend of solution I is not affected. To measure precisely the S-wave lineshape and determine its resonance structure more data are needed. However, the results presented here do not depend on such detailed knowledge.
In conclusion the analysis of the strong interaction phase shift resolves the ambiguity between solution I and solution II. Values of φ s close to zero and positive ∆Γ s are preferred. It follows that in the B 0 s system, the mass eigenstate that is almost CP even is lighter and decays faster than the state that is almost CP odd. This is in agreement with the Standard Model expectations [13]. It is also interesting to note that this situation is similar to that in the neutral kaon system.