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Oscillations of B 0 s mesons as a probe of decays with invisible particles

A new experimental technique to study the decays of B0 s mesons into the final states involving unreconstructable particles is proposed. The method uses a characteristic feature of fast flavour oscillations of B0 s mesons. Analysis of the oscillation spectrum of partially reconstructed flavour-tagged B0 s decays provides information about the mass spectrum of invisible states in these decays and removes any background which does not come from the decays of B0 s mesons.


Introduction
Many interesting measurements in heavy flavour physics, both within the Standard Model (SM) framework and beyond the SM, involve decays of heavy-flavoured hadrons into final states with neutral particles that are difficult or impossible to reconstruct with modern collider detectors. The SM decays may involve decays with neutrinos, neutrons or K 0 L mesons. Searches for effects beyond the SM consider such final state particles as heavy neutrinos, axion-like particles or other dark matter candidates.
Various experimental techniques have been developed to deal with such decays. In experiments at lepton colliders, one can use kinematic constraints coming from the welldefined initial state. For instance, at B-factories, where pairs of B mesons are produced at threshold without additional particles, full reconstruction of one of the B mesons provides sufficient kinematic constraints to reconstruct missing particles in the decay of the other B meson [1][2][3]. At the LHCb experiment, which operates in a hadronic environment [4], techniques employing the information about the topology of the decays have been adopted for decays with neutrinos in the final state [5,6]. These techniques take advantage of the large boost of initial-state hadrons in proton interactions at the LHC as well as the excellent vertex resolution of the LHCb detector.
In this paper, we propose a novel technique which uses the pattern of B 0 s oscillations as a tool to study a specific family of B 0 s decays with invisible (unreconstructed) particles in the final state. The method uses a characteristic feature of fast flavour oscillations of B 0 s mesons: they change their flavour many times on the scale of their lifetime. We show that, when combined with the information about the topology of the B 0 s decay, this feature provides a strong kinematic constraint on decays with invisible particles and efficiently cancels any non-B 0 s backgrounds. Contrary to other methods involving topological reconstruction at LHCb, where one has to make an assumption on the nature of the missing particle (e.g. that its mass is equal to zero), this technique can provide the information about the mass (or even the spectrum of invariant masses) of the invisible state(s).
The frequency of B 0 s meson oscillations ∆m s has been precisely measured to be 17.757±0.021 ps −1 , corresponding to the mixing parameter x s = ∆m s /Γ s = 26.81±0.08 [7] (where Γ s is the average B 0 s decay width). The most precise measurement of ∆m s comes from the analysis of B 0 s → D − s π + decays by the LHCb collaboration [8]. The distribution of decay time for mixed (where the flavour of the initial and final state are different) and unmixed B 0 s decays in this analysis is shown in Fig. 1. Observing these oscillations requires knowledge of the initial flavour of the B 0 s (flavour tagging), as well as precise measurement of the decay time. LHCb takes advantage of the large boost of B 0 s mesons in the laboratory frame to transform the measurement of the flight distance of the B 0 s meson to the measurement of its decay time. This transformation requires knowledge of B 0 s momentum. In our proposed approach, we use the fact that the momentum of the partially reconstructed B 0 s meson can be determined from the position of its decay vertex, assuming that the mass of the invisible state is known. Similar topological reconstruction has been used in studies of semileptonic decays at LHCb [9]. With the correctly determined B 0 s momentum, the decay time distribution will exhibit oscillations with a known frequency. One can thus use the amplitude of oscillations at that frequency to constrain the mass (or measure the spectrum of masses) of the invisible particles in partially reconstructed B 0 s decays. Since the method relies on the observation of B 0 s oscillations, the family of decays that could be studied must satisfy two conditions: the decay should be flavour-specific, and the decay vertex should be well-reconstructable. Possible applications of this technique could be studies of decays with heavy invisible (or poorly reconstructed) particles in the final state, both for SM decays, e.g. involving neutrons, and for searches beyond the SM. One particular example of the processes beyond the SM that satisfy the above conditions is the model suggested in [10][11][12], where heavy invisible particle is accompanied by a SM baryon. As a benchmark SM decay mode, one can consider the decay B 0 s → D − s pn, which satisfies both conditions and should have a significant rate since it proceeds via the favoured b → c transition. Reconstruction of decays with SM neutrinos, such as B 0 s → D − s µ + ν µ , can also be considered; in that case, this technique will provide the additional constraint for the reconstruction of kinematic variables related to neutrinos, such as the invariant mass of the µ + ν µ system.

Formalism
Consider the decay B 0 s → AX, where the state A is reconstructed and X is invisible. B 0 s mesons to the final state AX at least do not occur at the same rate or one of them is forbidden. The state A is also required to be short-lived, such that its decay vertex coincides with the decay vertex of the B 0 s , and have a well-reconstructable vertex i.e. it should be reconstructed from at least two charged tracks.
To be specific, let us give an example in the context of searches for Dark Matter (DM) candidates in B meson decays suggested in [10]. In that model, transitions such asb → csψ are expected, where ψ is a heavy neutral DM baryon. This transition gives rise to such B 0 s meson decays as B 0 s → Ξ 0 c ψ, where ψ would act as the invisible particle X in the notation used in our formalism. While the ground state Ξ 0 c is long-lived and does not constrain the B 0 s decay position, the excited Ξ * 0 c state decaying to Λ + c K − forms a well-defined vertex and can serve as the state A in our notation.
Similarly, for the SM decay B 0 s → D − s pn mentioned above, the antineutron serves as the invisible particle X, while the B 0 s decay vertex is provided by the combination D − s p that acts as the state A. In the reconstruction of semileptonic decays such as where the mass of the neutrino is fixed to be zero, one can consider the combination of µ + ν µ as the particle X and treat D − s as A. Although D − s is long-lived, the decay vertex of the B 0 s meson is constrained by the D − s and µ + , thus our technique is still applicable.
Under the conditions formulated above, the momentum of the B 0 s meson can be reconstructed from the momentum of the particle A (p A ) and the angle between the directions of A and B 0 s in the laboratory frame (θ) up to a two-fold ambiguity: where M B , M A and M X are masses of the B 0 s , A and X particles, respectively, ∆ ≡ Figure (natural units = c = 1 are used here). Oscillations result in the modulation of density of flavour-tagged B 0 s decays. Assuming perfect flavour tagging, the decay time distributions for the mixed and unmixed B 0 s decays are where the sign "+" corresponds to unmixed and "−" to mixed decays. Here, ∆Γ s is the width difference between the two B 0 s mass eigenstates, and ∆m s is their mass difference which drives the oscillations.
In the case of partially reconstructed B 0 s decays, one can use the topologically reconstructed momentum p B from Eq. (1) in the calculation of decay time. Since there are two solutions for p B , a sample of partially reconstructed decays will provide two decay time distributions, f (min) (t) and f (max) (t). Only one of the solutions corresponds to the true B 0 s momentum for each decay, thus wrong solutions will not produce a coherent oscillation pattern and will result in a smooth background contribution in both densities.
Once the correct mass M X is taken in the calculation of p B , the distribution of the reconstructed decay time will contain an oscillating term with the frequency corresponding to the mass difference ∆m s . A biased value of mass m X = M X will result in the oscillating term having higher or lower frequency. As a result, by calculating the amplitude of the harmonic corresponding to the mass difference ∆m s in the spectrum of reconstructed decay times as a function of mass m X assigned to the X state, one would observe a maximum in the amplitude at the true mass m X = M X .
Instead of the absolute rates of the B 0 s and B 0 s mesons, it is convenient to study the flavour asymmetry where N mix and N unmix are the mixed and unmixed yields as a function of decay time.
The asymmetry oscillates as a function of decay time. If the correct momentum is used to calculate t from the decay position x, then the oscillations of flavour asymmetry will occur with known frequency ∆m s . The amplitude A B 0 s of the harmonic corresponding to the measured B 0 s oscillation frequency will therefore probe the correctness of the mass hypothesis for the X state: where A B 0 s depends on m X indirectly, through the dependence of reconstructed decay time t on the momentum p B which, in turn, depends on m X . For the correct mass m X = M X , C B 0 s (m X ) should reach maximum, while S B 0 s (m X ) should cross zero 1 . An attractive feature of this approach is that the measurement of the amplitude of high-frequency oscillations does not require detailed knowledge of the low-frequency backgrounds coming from non-B 0 s decays and prompt production. As a result, this method is expected to have low systematic uncertainty due to the background description.
If several invisible states X i contribute to the B 0 s decay, the mass spectrum A B 0 s (m X ) will consist of several peaks at the masses of each X i state. The relative magnitudes of the peaks will give relative branching ratios to the final states with X i states. Finally, if the mass spectrum of X is continuous, the shape of the function A B 0 s (m X ) will constrain the X invariant mass distribution.
Even assuming perfect tagging and vertex resolution, the mass resolution will be limited since it is determined by a finite number of observed oscillation periods. However, with large number of B 0 s decays it should be possible to perform deconvolution of the A B 0 s (m X ) function to reconstruct the true X mass spectrum. Finally, it is possible that the maximum likelihood fit to the two distributions f (min) (t) and f (max) (t) will provide more information than the study of Fourier spectrum of B 0 s flavour asymmetry. This, however, would require a good control of slowly varying components of the decay time distributions. 2 GeV and M X varying between 0 and 3 GeV. The initial B 0 s mesons are generated with transverse momentum p T following an exponential distribution with a mean of 5 GeV and with pseudorapidity η distributed uniformly in the range from 2 to 5. The decay time distribution follows Eq. 3. All the following studies assume perfect flavour tagging; the effect of realistic flavour tagging performance can be taken into account by scaling the size of the data sample.

Simulation studies
For each of the generated B 0 s decays, the momentum (and subsequently the decay time) is calculated from topological information following Eqs. 1 and 2. Figure 3 shows the distributions of B 0 s decay time as a function of mass m X for the decays generated with M X = 1 GeV. The two plots each correspond to a solution of the B 0 s momentum. The dependence of the oscillation frequency on the assigned mass is clearly seen. Once the distributions of the decay time calculated this way for a certain m X are obtained for the mixed and unmixed decays, the flavour asymmetry as a function of decay time is calculated according to Eq. 4. Only the solution p (min) B is used since it more often gives the right answer for higher B 0 s momenta (and thus higher decay times). An approach using multivariate regression based on event topology information [13] can be used to make the choice of the correct solution more efficient. Figure 4 shows the flavour asymmetry as a function of decay time for a sample of 10 5 events of each initial B 0 s flavour with true M X = 1 GeV and taking different assigned masses m X of 0, 1, and 2 GeV. Perfect vertex resolution is assumed when calculating the decay time. One can see that the amplitude of flavour asymmetry oscillations is maximal for a correct mass hypothesis, while for the wrong masses, the coherence at higher decay times gets worse. In addition, the correct mass hypothesis yields a frequency of oscillations corresponding to the true mass difference ∆m s , while the wrong mass hypotheses yield higher or lower frequencies.
One then obtains the spectrum of X masses by scanning m X in the range between 0 s mesons in a typical analysis involves the requirement of non-zero lifetime and the reconstruction efficiency at low decay times is small (see Fig. 1), only the decays with t > 0.8 ps are used to calculate the amplitude of the B 0 s oscillation harmonic. The X mass spectra (in the form of real and imaginary amplitudes as a function of m X ) for true masses M X of 0, 1, and 2 GeV are shown in Fig. 5. One can see that the mass resolution is better for higher-mass states; this is natural since the method is effectively sensitive to the velocity of the B 0 s mesons. While the spectra presented in the plot contain uncertainties, these are not straightforward to take into account: each value of the amplitude is calculated from the same dataset, so all the values of the amplitude are correlated. The uncertainties are not plotted in Fig. 5. In a real analysis, the experimental uncertainties will need to be represented by the functions σ(m X ) (for the uncertainty) and ρ(m X,1 , m X,2 ) (for the correlation between the two assigned masses) depending on the data sample.
Good vertex resolution is an important factor for the proposed technique. Finite precision in the reconstruction of the position of the B 0 s decay vertex transverse to its flight direction translates to the uncertainty of the angle θ and thus of the momentum p B . The uncertainty in p B results in loss of coherence at higher decay times, which in turn leads to worse X mass resolution. The transverse vertex resolution at LHCb depends on the charge multiplicity and kinematics of the decay, with typical values of a few tens of µm [14]. The effect of finite vertex resolution is illustrated in Figs. 6(a-d), where the transverse component of vertex resolution is taken equal to σ x = 20 and 40 µm.

Conclusion
A novel technique to study B 0 s meson decays with invisible particles at a hadron collider has been presented. It uses the characteristic pattern of fast B 0 s flavour oscillations to obtain information about the B 0 s meson momentum and remove any non-B 0 s backgrounds. The technique has been studied with pseudoexperiments for a particular case of a decay B 0 s → AX, where the state A is reconstructed and constrains the B 0 s vertex, while X is invisible. In this case, the proposed procedure provides sensitivity to the spectrum of X masses which would be impossible with other approaches.
The proposed technique can find its applications in the studies of both SM decays, e.g. involving neutrons, and for searches beyond the SM, such as decays with heavy neutrinos, axion-like particles and other dark matter candidates (e.g. [10][11][12]). The benchmark measurements that should be feasible already with the currently available LHCb dataset could employ Cabibbo-favoured open-charm decays such as B 0 s → D − s pn. A similar approach is probably feasible for the reconstruction of decays with neutrinos (e.g. for the semileptonic decays involving muons or τ leptons) where B 0 s oscillations can provide additional kinematic constraints compared to conventional techniques used at LHCb.