Formulae for the Analysis of the Flavor-Tagged Decay B^0_s -->Jpsi phi

Differential rates in the decay B^0_s -->Jpsi phi, with phi -->K^+K^- and Jpsi -->mu^+ mu^- are sensitive to the CP-violation phase beta_s, predicted to be very small in the standard model. The analysis of B^0_s -->Jpsi phi decays is also suitable for measuring the B^0_s lifetime, the decay width difference DeltaGamma_s between the B^0_s mass eigenstates, and the B^0_s oscillation frequency Delta m even if appreciable CP violation does not occur. In this paper we present normalized probability densities useful in maximum likelihood fits, extended to allow for S-wave contributions on one hand and for the effects of direct CP violation on the other. Our treatment of the S-wave contributions includes the strong variation of the S-wave/P-wave amplitude ratio with m(K^+K^-) across the phi resonance, which was not considered in previous work. We include a scheme for re-normalizing the probability densities after detector sculpting of the angular distributions of the final state particles, and conclude with an examination of the symmetries of the rate formulae, with and without an S-wave contribution. All results are obtained with the use of a new compact formalism describing the differential decay rate of B^0_s mesons into Jpsi phi final states.


Introduction
The decay of B 0 s → J/ψφ, a transition of a pseudoscalar into two vector mesons can be thought of as six independent decays. The initial B 0 s system consists of a heavy and a light mass eigenstate, and the J/ψφ system to which it decays is characterized by three distinct orbital angular momentum states. A maximum amount of information about this system can be obtained from analyses which disentangle the two initial states and the three final states. The experimental technique of flavor tagging infers a meson's flavor at production time as B 0 s orB 0 s and is the key to disentangling the two initial states. Flavor-tagged B 0 s → J/ψφ decays are of great interest in particle physics because of their sensitivity to the CKM phases [1] and to anomalous mixing phases from physics beyond the standard model [2]. Recently the CDF and D0 collaborations have constrained the CKM phases in both untagged analyses [3,4], and flavor-tagged analyses [5,6]. These analyses are based on complete differential rates for the decay given in Ref. [7]. They use the angular distributions of the decay products to disentangle the three final states.
In this paper we re-express the differential decay rates in Ref. [7], using a new formalism that makes explicit a number of symmetries that are otherwise hidden. These formulae are then extended to the case in which the final state in the decay B 0 s → J/ψφ includes decays of type B 0 s → J/ψK + K − (kaons in an S-wave state), which has been suggested [8] to be an important effect. After including the S-wave contribution in the theoretical description, we identify the symmetries of the modified formulae.
In addition to S-wave effects, we also investigate other aspects of the differential decay rate formulae. We include the effects of possible direct CP violation. In addition we show how interference between CP odd and CP even J/ψφ final states effectively tags the flavor of the B 0 s meson at decay, allowing for the possibility to observe B 0 s →B 0 s flavor oscillations in a flavor-tagged analysis, even in the absence of CP violation effects. Experimentally, the differential rate formulae are used to construct likelihood functions based on normalized probability density functions (PDFs). In this paper we include normalization constants where appropriate in all expressions for transition amplitudes and PDFs. Detector angular acceptance is an important effect which must be included in these probability densities. However, the inclusion of this effect disturbs the normalization of the PDF. We present a scheme for normalizing the probability density analytically, as required for unbinned maximum likelihood fits.

Phenomenology of the B 0 s → J/ψφ Decay
We first summarize the phenomenology of the B 0 s system and the decay B 0 s → J/ψφ → µ + µ − K + K − . Two flavor eigenstates, |B 0 s and |B 0 s , mix via the weak interaction. The two mass eigenstates are labeled "heavy" and "light". The mass and lifetime differences between the B H s and B L s states can be defined as where m H,L and Γ H,L denote the mass and decay width of B H s and B L s (with this definition both ∆m and ∆Γ are expected to be positive quantities). The heavy state decays with a longer lifetime, τ H = 1/Γ H , while the light state decays with the shorter lifetime τ L = 1/Γ L , in analogy to the neutral kaon system. The mean lifetime is defined to be τ = 1/Γ. Theoretical estimates predict ∆Γ/Γ to be on the order of ∼ 15% [2]. Linear polarization eigenstates of the J/ψ and φ provide a convenient basis for the analysis of the decay [9]. The two vector mesons can have their spins transversely polarized with respect to their momentum and be either parallel or perpendicular to each other. Alternatively, they can both be longitudinally polarized. We denote these states as |P || , |P ⊥ , and |P 0 .
In the standard model, CP violation occurs through complex phases in the CKM matrix [10]. Large phases occur in the matrix elements V ub and V td . While these matrix elements generate large CP violation in the B 0 system, they do not appear in leading order diagrams contributing to either B 0 s ↔B 0 s mixing or to the decay B 0 s → J/ψφ. For this reason the standard model expectation of CP violation in B 0 s → J/ψφ is small. In the limit of vanishing CP violation, the heavy, long-lived mass eigenstate B H s is CP odd and decays to the CP -odd, L=1 orbital angular momentum state |P ⊥ . The light, short-lived mass eigenstate B L s is CP even and decays to both CP -even L=0 and L=2 orbital angular momentum states, which are linear combinations of |P 0 and |P || .
The small CP violation in B 0 s → J/ψφ can be quantified in the following way: we define A i as the decay amplitude B s |H|P i andĀ i as the decay amplitude B s |H|P i where i is one of {||, ⊥, 0}. All CP observables in the system are characterized by three quantities λ i = q pĀ i A i . In the standard model the λ i are given as λ i = ± exp (i2β s ) where the positive and negative sign applies to the CP even and odd final state, and The standard model expectation [11] is 2β s = 0.037 ± 0.002, a very small phase which does not lead to appreciable levels of CP violation. New physics can alter the mixing phase, while leaving λ very nearly unimodular. In this paper we consider, however, also the case in which |λ| = 1.

Differential Rates
The state of an initially pure B 0 s orB 0 s meson after a proper time t has elapsed is denoted as |B 0 s,phys (t) and |B 0 s,phys (t) . Transitions of these states to the detectable µ + µ − K + K − can be written as where H is the weak interaction Hamiltonian. The expression can be written much more simply, by defining time-dependent amplitudes for |B 0 s and |B 0 s to reach the states |P i either with or without mixing: Then: where the time dependence of A i (t) andĀ i (t) is: and where the upper sign indicates a CP even final state, the lower sign indicates a CP odd final state, 4) and the a i are complex amplitude parameters satisfying: The final state µ + µ − K + K − is characterized by three decay angles, described in a coordinate system 1 called the transversity basis [1]. In the J/ψ rest frame, the x-axis is taken to lie along the momentum of the φ and the z-axis perpendicular to the decay plane of the φ. The variables (θ, ϕ) are the polar and azimuthal angles of the µ + momentum in this basis. We also define the angle ψ to be the "helicity" angle in the φ decay, i.e. the angle between the K + direction and the x-axis in the φ rest frame. With these definitions, the muon momentum direction in the J/ψ rest frame is given by the unit vector n = (sin θ cos ϕ, sin θ sin ϕ, cos θ) . (3.6) Let A(t) andĀ(t) be complex vector functions of time defined as where A i (t) have now been normalized. For experimental measurements we are concerned with normalized probability density functions P B and PB for B andB mesons in the variables t, cos ψ, cos θ, and ϕ, which can be obtained by squaring Eq. (3.2). The formulae of Ref. [7] are then equivalent to: which give a picture of a time-dependent polarization analyzed in the decay 2 . The factors of 9/16π are normalization constants, and are present in order that The quantities |a i | 2 give the time-integrated rate to each of the polarization states. The values of A i (t) at t = 0 will be denoted as A i . To translate between the a's and the A's one can use the following two sets of transformations:

Detector Efficiency and Normalization
The detector efficiency ε(ψ, θ, ϕ), when introduced into the above expression, disturbs the normalization of Eq. (3.9). We restore it by dividing by a normalization factor N , Suppose that the efficiency ε(ψ, θ, ϕ) can be parametrized as where c k lm are expansion coefficients, P k (cos ψ) are Legendre polynomials, and Y lm (θ, ϕ) are real harmonics related to the spherical harmonics through the following relations: The products P k (cos ψ)Y lm (θ, ϕ) constitute an orthonormal basis for functions of the three angles. The detector efficiency (obtained, for example, from Monte Carlo simulation) can be fit to the first few of these polynomials. A straight-forward calculation shows that: The numerical factors +π/8 and −π/32, appearing together with c k 2,1 and c k 2,−2 in the infinite series, are the integrals P k (cos ψ) cos(ψ) sin ψd(cos ψ) . (4.5) While this series is infinite, the number of basis functions needed to fit detector efficiencies in a particular analysis is finite and determined chiefly by the size of the data sample. With the factors in Eq. (4.5) the normalizing factor can be adapted to account for all terms used in the expansion of the efficiency. Eq. (4.4) represents an analytic normalization of the fitting function and provides an efficient way to compute the likelihood during a maximum log likelihood fit. The orthonormality of the basis functions has been used to reduce the expression to its final form.

Time Development
The short oscillation length of the B 0 s meson [12], 2πc/∆m ∼ 106 µm, requires us to account for resolution effects when fitting the rates of flavor-tagged decays, even using the best silicon vertex detectors, which have proper decay length resolutions on the order of 25 µm. Certain time-dependent functions arising from particle-antiparticle oscillations, particularly those expressed as the product of exponential decays and harmonic functions with frequency ∆m, must be convolved with one or more Gaussian components describing detector resolution. This convolution can be carried out analytically, using the method described in Ref. [13] for the evaluation of certain integrals which are equivalent to complex error functions. In this step one requires that various components of the time dependence first be separated from Eq. (3.8). The time development of A 0 (t) and A (t) amplitudes are identical, but differs from that of A ⊥ (t). We begin by decomposing and and we define

Sensitivity to ∆m
It can be noticed that the time development of the interference term, expressions 5.9 and 5.10, contain undiluted mixing asymmetries even in the case of no CP violation, i.e., when β s = 0. Let us try to better understand the mechanism by which the flavor of the B 0 s meson is tagged at decay time, by first rewriting Eq. (3.1) using the B H s and B L S states in the expansion rather than the B 0 s andB 0 s states: Now, we take the limit of zero CP violation in the B 0 s system, such that P || |H|B H s = P 0 |H|B H s = P ⊥ |H|B L s = 0, and only three of the six terms in Eq. (6.1) remain: When the expression is squared, the interference terms are the cross terms involving both the product of a CP -even and a CP -odd amplitudes. The time dependence of these terms is contained in the factor:

Incorporating Direct CP Violation
An asymmetry either in the decay rate (|Ā i /A i | = 1) or in the mixing (|q/p| = 1) such that |λ| = 1 is direct CP violation. In the case of direct CP violation λ does not lie on the unit circle in the complex plane, and we need two parameters to describe it which we will take to be C ≡ Re(λ) and S ≡ Im(λ). Experimentally, even if one sets out to extract β s assuming the constraint |λ| = 1, it is nonetheless of interest to test that constraint, since sensitivity to C and S arise from very different features of the detector. In that case we must revisit not only the functional form of the differential decay rates, but also the normalization. The amplitudes in Eq. (3.3) must now be written as: These amplitudes can readily be seen to reduce to those of Eq. (3.3) in the limit of C 2 +S 2 ≡ |λ| 2 → 1. The normalization of detector efficiency, Eq. (4.4), becomes: Finally, the explicit time development, Eqs. (5.7), (5.8), (5.9) and (5.10), must be replaced with the more general forms: which can be seen to reduce to expression 5.7, 5.8 and 5.9, 5.10 as |λ| 2 → 1.

Incorporating a Contribution from B 0 s → J/ψK + K − (Kaons in an S-Wave State)
It has been suggested [8] that a contribution from S-wave K + K − under the φ peak in B 0 s → J/ψφ decay may contribute up to 5-10% of the total rate. A normalized probability density for the decay B 0 s → J/ψK + K − (kaons in an S-wave state) can be worked out by considering the polarization vector of the J/ψ in the decay and proceeding as in [9]. The resulting expressions do not depend at all on the angle ψ (which is the helicity angle in the φ decay). In the previous expression where the time-dependent amplitudes, reflect the CP -odd nature of the J/ψKK final state. When both P -wave and S-wave are present, the amplitudes must be summed and then squared. The P wave has a resonant structure due to the φ-propagator, while the S-wave amplitude is flat (but can have any phase with respect the P -wave). Suppose that in our experiment we accept events for which the reconstructed mass m(K + K − ) ≡ µ lies within a window µ lo < µ < µ hi . The normalized probability in this case is where we use a nonrelativistic Breit-Wigner to model the φ resonance 3 3 We shall have more to say about that, later. a flat model for the S-wave mass distribution and define and In these equations, F s is the S-wave fraction; µ φ is the φ mass (1019 MeV/c 2 ); Γ φ is the φ width (4.26 MeV/c 2 ), and δ s is the phase of the S-wave component relative to the P -wave component.
In the presence of an S-wave contribution, the normalization of Eq. (4.4) must be generalized; in order to do this we first define the quantities and Then the normalizing factor appropriate for Eq. (8.4) is where N is given in Eq. (4.4), and 8.12) and The numerical factors +π/2 and −π/8 appearing together with c k 2,1 and c k 2,−2 in the infinite series are the integrals P k (cos ψ) sin ψd(cos ψ) . (8.14) We now work out the explicit time and mass dependence of the differential rates. We will use Eq. (5.5) together with the analogous equation for the pure S-wave differential rate: and where the vector B =x = (1, 0, 0). The full probability densities, which can be used in a time-, angle-, and φ mass-dependent fit, are obtained by expanding Eq. (8.4). We get Re F(µ) (A − ×n) · (B ×n) · |f − (t)| 2 and ρB (θ, ψ, ϕ, t, µ) = In case one does not want to observe the φ-mass variable µ, one can integrate it out. Then one obtains

Symmetries
In this section we examine the symmetries of our differential rate formulae, starting from the simplest case, K + K − in a P -wave, Eq. (3.8), but considering also the case where both P and S waves are included, Eq. (8.4). In the case of pure P -wave, one can readily spot that the probability densities in Eq. (3.8) are invariant to the following transformations: • A simultaneous rotation of the vectors A(t) andn • An inversion of the vector A(t) • Complex-conjugation of the vector A(t) The symmetry to simultaneous rotation of the vectors A(t) andn corresponds to the well-known freedom to choose a convenient basis in which to work. An example of an alternative basis is the helicity basis, which derives from the transversity basis by a cyclic permutation of the coordinate axis:x T =ẑ H , etc. One can take the angles in Eq. (3.6) to be the polar and azimuthal angles in the helicity basis, but then one must transform A(t) accordingly, i.e, by permuting the elements of A(t) in the defining equation, Eq. (3.7). Then, Eq. (3.8) remains valid in the helicity basis. This rotational invariance implies that the choice of basis is irrelevant to the final result since the likelihood is invariant to the choice (though we do not rule out the possibility that the quality of the efficiency expansion, Eq. (4.2), may depend on the choice of basis, as pointed out in [14]).
A more interesting symmetry is the symmetry that results from transforming A(t) to its complex conjugate. If we take, by convention, a 0 to be real and let δ = arg(a ), and δ ⊥ = arg(a ⊥ ), then as we will demonstrate below, this conjugation transformation is equivalent to the simultaneous transformation: However we can find a symmetry transformation that carries one set of physically meaningful parameters into another. Such a symmetry is the transformation of the terms in Eq. (9.2) into their negative complex conjugate. This transformation is equivalent to the combined transformation already described, in addition to: The latter transformation carries us from a point on one side of the φ mass peak to another point located symmetrically on the other side. This symmetry is useful when considering likelihood functions in which the dependence on µ is integrated out. If we integrate symmetrically about the φ mass peak, we can consider the contribution to the integral coming from a slice in φ mass on one hand and the a symmetrically-located slice in φ mass on the other hand. While the contribution of either slice is not invariant to the transformation above, the contribution of both slices certainly is, and the combined transformation: is again a symmetry of the integrated likelihood. We note, however, that this symmetry requires the symmetry of the nonrelativistic φ-propagator, Eq. (8.5), and applies to the likelihood integrated over a finite symmetric interval of integration. Symmetries of the likelihood function for B 0 s → J/ψφ, in the presence of S-wave contribution for a fixed value of µ = m(K + K − ) were discussed in a recent publication [15]. These formula can also used to fit for data falling within a narrow window in µ. Under those assumptions we can drop the φ propagator from the expression in Eq. (9.2), absorb the Breit-Wigner terms into the amplitudes A(t), and write our model for the rates as Then one can see that the transformation in which δ s → −δ s replaces δ s → π − δ s in Eq. 9.4 accomplishes a complex conjugation of the terms in Eq. 9.5 and is a symmetry of the likelihood at fixed µ.
In the more general case one can notice from Eqs. 8.19 and 8.20 that the probability densities integrated over µ are invariant to complex conjugation of both A(t) and the overlap integral I mu of Eq. 8.10. This can be accomplished with a more complicated adjustment of δ s . With a nonrelativistic Breit Wigner the required transformation is δ s → 2δ BW − δ s where δ BW ≡ arg (log ((µ hi − µ φ + iΓ φ /2)/(µ lo − µ φ + iΓ φ /2))). The phase δ BW reduces to δ BW = 0 in the limit of an infinitesimally thin interval in µ, and to δ BW = −π/2 in the limit of a finite symmetric interval. This demonstrates real differences in the two formulations, and underscores the need for caution when applying the formulae of Ref. [15] to a finite interval in µ = m(K + K − ).

Conclusion
In this paper we have presented a compact formalism to easily access physical observables in the analysis of the decay B 0 s → J/ψφ. This formalism has practical applications, since complex vectors and their vector-algebraic operations can be easily implemented in highlevel computer languages in order to model and generate such decays, but also because the symmetries of the formulae under operations such as rotation and complex conjugation are apparent and provide better physical insight into this complicated decay mode. The normalized probability densities can be used for the experimental extraction of physical parameters, in scenarios with no CP violation, with mixing-induced CP violation, or even with direct CP violation. In case of mixing induced CP violation, the effect of the S-wave contribution has also been included in the decay rate formulae.