Observation of Time Reversal Violation in the Meson System
Although violation in the meson system has been well established by the B factories, there has been no direct observation of time reversal violation. The decays of entangled neutral mesons into definite flavor states ( or ), and or final states (referred to as or ), allow comparisons between the probabilities of four pairs of -conjugated transitions, for example, and , as a function of the time difference between the two decays. Using 468 million pairs produced in decays collected by the BABAR detector at SLAC, we measure -violating parameters in the time evolution of neutral mesons, yielding and . These nonzero results represent the first direct observation of violation through the exchange of initial and final states in transitions that can only be connected by a -symmetry transformation.
pacs:13.25.Ft, 11.30.Er, 12.15.Ff, 14.40.Lb
The BABAR Collaboration
The observations of -symmetry breaking, first in neutral decays ref:christenson:1964 () and more recently in mesons ref:mixingInducedCP-Bs (); ref:directCP-Bs (), are consistent with the standard model (SM) mechanism of the three-family Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix being the dominant source of violation ref:CKM:1963:1973 (). Local Lorentz invariant quantum field theories imply invariance ref:CPTtheorem (), in accordance with all experimental evidence ref:CPTtests (); ref:TestsConservationLaws (). Hence, it is expected that the -violating weak interaction also violates time reversal invariance.
To date, the only evidence related to violation has been found in the neutral system, where a difference between the probabilities of and transitions for a given elapsed time has been measured ref:Angelopoulos (). This flavor mixing asymmetry is both - and -violating (the two transformations lead to the same observation), independent of time, and requires a nonzero decay width difference between the neutral mass eigenstates to be observed ref:Kabir (); ref:Wolfenstein (); ref:Wolfenstein2 (). The dependence with has aroused controversy in the interpretation of this observable ref:Wolfenstein (); ref:Wolfenstein2 (); ref:Gerber (); ref:TestsConservationLaws (). In the neutral and systems, where and are negligible and significantly smaller, respectively, the flavor mixing asymmetry is much more difficult to detect ref:TviolationBs (). Experiments that could provide direct evidence supporting non-invariance, without using an observation which also violates , involve either nonvanishing expectation values of -odd observables, or the exchange of initial and final states, which are not conjugates to each other, in the time evolution for transition processes. Among the former, there exist upper limits for electric dipole moments of the neutron and the electron ref:edm (). The latter, requiring neutrinos or unstable particles, are particularly difficult to implement.
In this letter, we report the direct observation of violation in the meson system, through the exchange of initial and final states in transitions that can only be connected by a -symmetry transformation. The method is described in Ref. ref:method2012 (), based on the concepts proposed in Ref. ref:bernabeuPLB-NPB () and further discussed in Refs. ref:Wolfenstein2 (); ref:QuinnDiscrete (); ref:BernabeuDiscrete (). We use a data sample of 426 of integrated luminosity at the resonance, corresponding to pairs, and 45 at a center-of-mass (c.m.) energy 40 below the , recorded by the BABAR detector ref:Aubert:2001tu () at the PEP-II asymmetric-energy collider at SLAC. The experimental analysis exploits identical reconstruction algorithms, selection criteria, calibration techniques, and meson samples to our most recent time-dependent asymmetry measurement in decays ref:Aubert:2009yr (), with the exception of and final states. The “flavor tagging” is combined here, for the first time, with the “ tagging” ref:bernabeuPLB-NPB (), as required for the construction of -transformed processes. Whereas the descriptions of the sample composition and time-dependent backgrounds are the same as described in Ref. ref:Aubert:2009yr (), the signal giving access to the -violating parameters needs a different data treatment. This echoes the fundamental differences between observables for and symmetry breaking. The procedure to determine the -violating parameters and their significance is thus novel ref:method2012 ().
In the decay of the , the two mesons are in an entangled, antisymmetric state, as required by angular momentum conservation for a P-wave particle system. This two-body state is usually written in terms of flavor eigenstates, such as and , but can be expressed in terms of any linear combinations of and , such as the and states introduced in Ref. ref:method2012 (). They are defined as the neutral states filtered by the decay to -eigenstates (-even) and , with (-odd), respectively. The and states are orthogonal to each other when there is only one weak phase involved in the decay amplitude, as it occurs in decays to final states ref:CPVreview (), and violation in neutral kaons is neglected.
We select events in which one candidate is reconstructed in a or state, and the flavor of the other is identified, referred to as flavor identification (ID). We generically denote reconstructed final states that identify the flavor of the as for and for . The notation is used to indicate the flavor or final states that are reconstructed at corresponding times and , where , i.e., is the first decay in the event and is the second decay. For later use in Eq. (1), we define . Once the state is filtered at time , the living partner is prepared (“tagged”) by entanglement as its orthogonal state. The notation describes the transition of the which decays at , having tagged its state at . For example, an event reconstructed in the time-ordered final states identifies the transition for the second to decay. We compare the rate for this transition to its -reversed (exchange of initial and final states) by reconstructing the final states . Any difference in these two rates is evidence for -symmetry violation. There are three other independent comparisons that can be made between , , and transitions and their -conjugates, , , and , respectively. Similarly, four different () comparisons can be made, e.g., between the transition and its ()-transformed () ref:method2012 ().
Assuming , each of the eight transitions has a general, time-dependent decay rate given by
where indices and stand for and final states, respectively, and the symbol or indicates whether the decay to the flavor final state occurs before or after the decay to the final state . Here, is the average decay width, is the mass difference between the neutral mass eigenstates, and and are model independent coefficients. The sine term, expected to be large in the SM, results from the interference between direct decay of the neutral to the final state and decay after - oscillation, while the cosine term arises from the interference between decay amplitudes with different weak and strong phases, and is expected to be negligible ref:CPVreview (). violation would manifest itself through differences between the or values for -conjugated processes, for example between and .
In addition to , states are reconstructed through the and final states (denoted generically as ), with , , , and (the latter only for ). states are identified through . The candidates are characterized by the difference between the reconstructed energy of the and the beam energy in the c.m. frame, , while for the modes we use the beam-energy substituted invariant mass , where is the momentum in the c.m. frame.
The flavor ID of the other neutral meson in the event, not associated with the reconstructed or , is made on the basis of the charges of prompt leptons, kaons, pions from mesons, and high-momentum charged particles. These flavor ID inputs are combined using a neural network (NN), trained with Monte Carlo (MC) simulated data. The output of the NN is then divided into six hierarchical, mutually exclusive flavor categories of increasing misidentification (misID) probability . Events for which the NN output indicates very low discriminating power are excluded from further analysis. We determine the signed difference of proper time between the two decays from the measured separation of the decay vertices along the collision axis. Events are accepted if the reconstructed and its estimated uncertainty, , are lower than and , respectively. The performances of the flavor ID and reconstruction algorithms are evaluated by using a large sample of flavor-specific neutral decays to and final states (referred to as sample). The resolution function is the same as in Ref. ref:Aubert:2009yr () except that all Gaussian offsets and widths are modeled to be proportional to .
The composition of the final sample is determined through fits to the and distributions, using parametric forms and distributions extracted from MC simulation and dilepton mass sidebands in data to describe the signal and background components. Figure 1 shows the and data distributions for events that satisfy the flavor ID and vertexing requirements, overlaid with the fit projections. The final sample contains 7796 events, with purities in the signal region ( ) ranging between 87% and 96%, and 5813 events, with a purity of 56% in the region.
We perform a simultaneous, unbinned maximum likelihood fit to the distributions for flavor identified and events, split by flavor category. The signal probability density function () is ref:method2012 ()
where is the signed difference of proper time between the two decays in the limit of perfect reconstruction, is the Heaviside step function, with is the resolution function, and are given by Eq. (1). Note that is equivalent to () when a true flavor () tag occurs. Because of the convolution with the resolution function, the distribution for contains predominantly true flavor-tagged events, with contribution from true -tagged events at low , and conversely for . Mistakes in the flavor ID algorithm mix correct and incorrect flavor assignments, and dilute the -violating asymmetries by a factor of approximately . Backgrounds are accounted for by adding terms to Eq. ( Observation of Time Reversal Violation in the Meson System ) ref:Aubert:2009yr (). Events are assigned signal and background probabilities based on the or distributions, for or events, respectively.
A total of 27 parameters are varied in the likelihood fit: eight pairs of coefficients for the signal, and 11 parameters describing possible and violation in the background. All remaining signal and background parameters are fixed to values taken from the sample, -candidate sidebands in , world averages for and ref:pdg2010 (), or MC simulation ref:Aubert:2009yr (). From the 16 signal coefficients ref:epaps (), we construct six pairs of independent asymmetry parameters , , and , as shown in Table 1. The -asymmetry parameters have the advantage that -symmetry breaking would directly manifest itself through any nonzero value of or , or any difference between and , or between and (analogously for - or -symmetry breaking). The measured values for the asymmetry parameters are reported in Table 1. There is another two times three pairs of -, -, and -asymmetry parameters, but they are not independent and can be derived from Table 1 or Ref. ref:epaps ().
We build time-dependent asymmetries to visually demonstrate the -violating effect. For transition ,
where . With this construction, is defined only for positive values. Neglecting reconstruction effects, . We introduce the other three -violating asymmetries similarly. Figure 2 shows the four observed asymmetries, overlaid with the projection of the best fit results to the distributions with and without the eight -invariance restrictions: , , and ref:epaps ().
Using large samples of MC simulated data, we determine that the asymmetry parameters are unbiased and have Gaussian errors. Splitting the data by flavor category or data-taking period give consistent results. Fitting a single pair of coefficients, reversing the sign of under , or or exchanges, and the sign of under exchange, we obtain identical results to those obtained in Ref. ref:Aubert:2009yr (). Performing the analysis with decays to and final states instead of the signal and , respectively, we find that all the asymmetry parameters are consistent with zero.
In evaluating systematic uncertainties in the asymmetry parameters, we follow the same procedure as in Ref. ref:Aubert:2009yr (), with small changes ref:epaps (). We considered the statistical uncertainties on the flavor misID probabilities, resolution function, and parameters. Differences in the misID probabilities and resolution function between and final states, uncertainties due to assumptions in the resolution for signal and background components, compositions of the signal and backgrounds, the and , and the branching fractions for the backgrounds and their properties, have also been accounted for. We also assign a systematic uncertainty corresponding to any deviation of the fit for MC simulated asymmetry parameters from their generated MC values, taking the largest between the deviation and its statistical uncertainty. Other sources of uncertainty such as our limited knowledge of , , and other fixed parameters, the interaction region, the detector alignment, and effects due to a nonzero value in the time dependence and the normalization of the , are also considered. Treating and as orthogonal states and neglecting violation for flavor categories without leptons, has an impact well below the statistical uncertainty. The total systematic uncertainties are shown in Table 1 ref:epaps ().
The significance of the -violation signal is evaluated based on the change in log-likelihood with respect to the maximum (). We reduce by a factor to account for systematic errors in the evaluation of the significance. Here, , where is the maximum log-likelihood, is the log-likelihood with asymmetry parameter fixed to its total systematic variation and maximized over all other parameters, and is the change in at confidence level () for one degree of freedom (d.o.f). Figure 3 shows contours calculated from the change in two dimensions for the -asymmetry parameters and . The difference in the value of at the best fit solution with and without violation is with eight d.o.f., including systematic uncertainties. Assuming Gaussian errors, this corresponds to a significance equivalent to standard deviations (), and thus constitutes direct observation of violation. The significance of and violation is determined analogously, obtaining and , respectively, equivalent to and , consistent with violation and invariance.
In summary, we have measured -violating parameters in the time evolution of neutral mesons, by comparing the probabilities of , , , and transitions, to their conjugate. We determine for the main -violating parameters and , and observe directly for the first time a departure from invariance in the meson system, with a significance equivalent to . Our results are consistent with current -violating measurements obtained invoking invariance. They constitute the first observation of violation in any system through the exchange of initial and final states in transitions that can only be connected by a -symmetry transformation.
We are grateful for the excellent luminosity and machine conditions provided by our PEP-II colleagues, and for the substantial dedicated effort from the computing organizations that support BABAR. The collaborating institutions wish to thank SLAC for its support and kind hospitality. This work is supported by DOE and NSF (USA), NSERC (Canada), CEA and CNRS-IN2P3 (France), BMBF and DFG (Germany), INFN (Italy), FOM (The Netherlands), NFR (Norway), MES (Russia), MINECO (Spain), STFC (United Kingdom). Individuals have received support from the Marie Curie EIF (European Union), the A. P. Sloan Foundation (USA) and the Binational Science Foundation (USA-Israel).
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Observation of Time Reversal Violation in the Meson System
The BABAR Collaboration
The following includes supplementary material for the Electronic Physics Auxiliary Publication Service.
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|Background fractions and content||0.024||0.020||0.024||0.016||0.012||0.004||0.007||0.007|
|violation for flavor ID categories||0.026||0.010||0.007||0.005||0.014||0.005||0.003||0.002|