Observation of Time Reversal Violation in the $B^0$ Meson System

The BABAR Collaboration

The observations of $CP$-symmetry breaking, first in neutral $K$ decays ref:christenson:1964 () and more recently in $B$ 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 $CP$ violation ref:CKM:1963:1973 (). Local Lorentz invariant quantum field theories imply $CPT$ invariance ref:CPTtheorem (), in accordance with all experimental evidence ref:CPTtests (); ref:TestsConservationLaws (). Hence, it is expected that the $CP$-violating weak interaction also violates time reversal invariance.

To date, the only evidence related to $T$ violation has been found in the neutral $K$ system, where a difference between the probabilities of $K^0\to \overline{K}{}^0$ and $\overline{K}{}^0\to K^0$ transitions for a given elapsed time has been measured ref:Angelopoulos (). This flavor mixing asymmetry is both $CP$- and $T$-violating (the two transformations lead to the same observation), independent of time, and requires a nonzero decay width difference $\Delta {\Gamma }_K$ between the neutral $K$ mass eigenstates to be observed ref:Kabir (); ref:Wolfenstein (); ref:Wolfenstein2 (). The dependence with $\Delta {\Gamma }_K$ has aroused controversy in the interpretation of this observable ref:Wolfenstein (); ref:Wolfenstein2 (); ref:Gerber (); ref:TestsConservationLaws (). In the neutral $B$ and $B_s$ systems, where $\Delta {\Gamma }_d$ and $\Delta {\Gamma }_s$ 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 $T$ non-invariance, without using an observation which also violates $CP$, involve either nonvanishing expectation values of $T$-odd observables, or the exchange of initial and final states, which are not $CP$ 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 $T$ violation in the $B$ meson system, through the exchange of initial and final states in transitions that can only be connected by a $T$-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 ${\text{,fb}}^{-1}$ of integrated luminosity at the $\text{mathchar}28935\text{relax}\left(4S\right)$ resonance, corresponding to $468\times {10}^6$ $B\overline{B}$ pairs, and 45 ${\text{,fb}}^{-1}$ at a center-of-mass (c.m.) energy 40 $MeV$ below the $\text{mathchar}28935\text{relax}\left(4S\right)$, recorded by the BABAR detector ref:Aubert:2001tu () at the PEP-II asymmetric-energy $e^{+}{e}^{-}$ collider at SLAC. The experimental analysis exploits identical reconstruction algorithms, selection criteria, calibration techniques, and $B$ meson samples to our most recent time-dependent $CP$ asymmetry measurement in $B\to c\overline{c}{K}^{(\ast )0}$ decays ref:Aubert:2009yr (), with the exception of ${\eta }_c{K}_S^0$ and $J/\psi K^{\ast 0}(\to K_S^0{\pi }^0)$ final states. The “flavor tagging” is combined here, for the first time, with the “$CP$ tagging” ref:bernabeuPLB-NPB (), as required for the construction of $T$-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 $T$-violating parameters needs a different data treatment. This echoes the fundamental differences between observables for $T$ and $CP$ symmetry breaking. The procedure to determine the $T$-violating parameters and their significance is thus novel ref:method2012 ().

In the decay of the $\text{mathchar}28935\text{relax}\left(4S\right)$, the two $B$ 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 $B^0$ and $\overline{B}{}^0$, but can be expressed in terms of any linear combinations of $B^0$ and $\overline{B}{}^0$, such as the $B_{+}$ and $B_{-}$ states introduced in Ref. ref:method2012 (). They are defined as the neutral $B$ states filtered by the decay to $CP$-eigenstates $J/\psi K_L^0$ ($CP$-even) and $J/\psi K_S^0$, with $K_S^0\to \pi \pi$ ($CP$-odd), respectively. The $B_{+}$ and $B_{-}$ states are orthogonal to each other when there is only one weak phase involved in the $B$ decay amplitude, as it occurs in $B$ decays to $J/\psi K^0$ final states ref:CPVreview (), and $CP$ violation in neutral kaons is neglected.

We select events in which one $B$ candidate is reconstructed in a $B_{+}$ or $B_{-}$ state, and the flavor of the other $B$ is identified, referred to as flavor identification (ID). We generically denote reconstructed final states that identify the flavor of the $B$ as ${\ell }^{-}X$ for $\overline{B}{}^0$ and ${\ell }^{+}X$ for $B^0$. The notation $(f_1,f_2)$ is used to indicate the flavor or $CP$ final states that are reconstructed at corresponding times $t_1$ and $t_2$, where $t_2>t_1$, i.e., $B_1\to f_1$ is the first decay in the event and $B_2\to f_2$ is the second decay. For later use in Eq. (1), we define $\Delta \tau =t_2-t_1>0$. Once the $B_1$ state is filtered at time $t_1$, the living partner $B_2$ is prepared (“tagged”) by entanglement as its orthogonal state. The notation $B_2\left(t_1\right)\to B_2\left(t_2\right)$ describes the transition of the $B$ which decays at $t_2$, having tagged its state at $t_1$. For example, an event reconstructed in the time-ordered final states $({\ell }^{+}X,J/\psi K_S^0)$ identifies the transition $\overline{B}{}^0\to B_{-}$ for the second $B$ to decay. We compare the rate for this transition to its $T$-reversed $B_{-}\to \overline{B}{}^0$ (exchange of initial and final states) by reconstructing the final states $(J/\psi K_L^0,{\ell }^{-}X)$. Any difference in these two rates is evidence for $T$-symmetry violation. There are three other independent comparisons that can be made between $B_{+}\to B^0$ $(J/\psi K_S^0,{\ell }^{+}X)$, $\overline{B}{}^0\to B_{+}$ $({\ell }^{+}X,J/\psi K_L^0)$, and $B_{-}\to B^0$ $(J/\psi K_L^0,{\ell }^{+}X)$ transitions and their $T$-conjugates, $B^0\to B_{+}$ $({\ell }^{-}X,J/\psi K_L^0)$, $B_{+}\to \overline{B}{}^0$ $(J/\psi K_S^0,{\ell }^{-}X)$, and $B^0\to B_{-}$ $({\ell }^{-}X,J/\psi K_S^0)$, respectively. Similarly, four different $CP$ ($CPT$) comparisons can be made, e.g., between the $\overline{B}{}^0\to B_{-}$ transition and its $CP$ ($CPT$)-transformed $B^0\to B_{-}$ ($B_{-}\to B^0$) ref:method2012 ().

In addition to $J/\psi K_S^0$, $B_{-}$ states are reconstructed through the $\psi \left(2S\right)K_S^0$ and ${\chi }_{c1}{K}_S^0$ final states (denoted generically as $c\overline{c}{K}_S^0$), with $J/\psi ,\psi \left(2S\right)\to e^{+}{e}^{-},{\mu }^{+}{\mu }^{-}$, $\psi \left(2S\right)\to J/\psi {\pi }^{+}{\pi }^{-}$, ${\chi }_{c1}\to J/\psi \gamma$, and $K_S^0\to {\pi }^{+}{\pi }^{-},{\pi }^0{\pi }^0$ (the latter only for $J/\psi K_S^0$). $B_{+}$ states are identified through $J/\psi K_L^0$. The $J/\psi K_L^0$ candidates are characterized by the difference $\Delta E$ between the reconstructed energy of the $B$ and the beam energy in the $e^{+}{e}^{-}$ c.m. frame, $E_{beam}^{\ast }$, while for the $c\overline{c}{K}_S^0$ modes we use the beam-energy substituted invariant mass $m_{ES}=\sqrt{(E_{beam}^{\ast }{)}^2-(p_B^{\ast }{)}^2}$, where $p_B^{\ast }$ is the $B$ momentum in the c.m. frame.

The flavor ID of the other neutral $B$ meson in the event, not associated with the reconstructed $B_{+}$ or $B_{-}$, is made on the basis of the charges of prompt leptons, kaons, pions from $D^{\ast }$ 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 $w$. Events for which the NN output indicates very low discriminating power are excluded from further analysis. We determine the signed difference of proper time $\Delta t=t_{\beta }-t_{\alpha }$ between the two $B$ decays from the measured separation of the decay vertices along the collision axis. Events are accepted if the reconstructed $|\Delta t|$ and its estimated uncertainty, ${\sigma }_{\Delta t}$, are lower than $20$ $ps$ and $2.5$ $ps$, respectively. The performances of the flavor ID and $\Delta t$ reconstruction algorithms are evaluated by using a large sample of flavor-specific neutral $B$ decays to $D^{(\ast )-}[{\pi }^{+},\rho (770{)}^{+},a_1\left(1260{)}^{+}\right]$ and $J/\psi K^{\ast 0}(\to K^{+}{\pi }^{-})$ final states (referred to as $B_{flav}$ sample). The $\Delta t$ resolution function is the same as in Ref. ref:Aubert:2009yr () except that all Gaussian offsets and widths are modeled to be proportional to ${\sigma }_{\Delta t}$.

The composition of the final sample is determined through fits to the $m_{ES}$ and $\Delta E$ 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 $m_{ES}$ and $\Delta E$ data distributions for events that satisfy the flavor ID and vertexing requirements, overlaid with the fit projections. The final sample contains 7796 $c\overline{c}{K}_S^0$ events, with purities in the signal region ($5.27<m_{ES}<5.29$ $GeV/c^2$) ranging between 87% and 96%, and 5813 $J/\psi K_L^0$ events, with a purity of 56% in the $|\Delta E|<10$ $MeV$ region.

A total of 27 parameters are varied in the likelihood fit: eight pairs of $(S_{\alpha ,\beta }^{\pm },C_{\alpha ,\beta }^{\pm })$ coefficients for the signal, and 11 parameters describing possible $CP$ and $T$ violation in the background. All remaining signal and background parameters are fixed to values taken from the $B_{flav}$ sample, $J/\psi$-candidate sidebands in $J/\psi K_L^0$, world averages for ${\Gamma }_d$ and $\Delta m_d$ ref:pdg2010 (), or MC simulation ref:Aubert:2009yr (). From the 16 signal coefficients ref:epaps (), we construct six pairs of independent asymmetry parameters $(\Delta S_T^{\pm },\Delta C_T^{\pm })$, $(\Delta S_{CP}^{\pm },\Delta C_{CP}^{\pm })$, and $(\Delta S_{CPT}^{\pm },\Delta C_{CPT}^{\pm })$, as shown in Table 1. The $T$-asymmetry parameters have the advantage that $T$-symmetry breaking would directly manifest itself through any nonzero value of $\Delta S_T^{\pm }$ or $\Delta C_T^{\pm }$, or any difference between $\Delta S_{CP}^{\pm }$ and $\Delta S_{CPT}^{\pm }$, or between $\Delta C_{CP}^{\pm }$ and $\Delta C_{CPT}^{\pm }$ (analogously for $CP$- or $CPT$-symmetry breaking). The measured values for the asymmetry parameters are reported in Table 1. There is another two times three pairs of $T$-, $CP$-, and $CPT$-asymmetry parameters, but they are not independent and can be derived from Table 1 or Ref. ref:epaps ().