Time-dependent Dalitz-plot formalism for B_q ->J/\psi\ h+ h-

A formalism for measuring time-dependent CP violation in B_q ->J/\psi h+ h- decays with J/\psi->mu+ mu- is developed for the general case where there can be many h+ h- final states of different angular momentum present. Here h refers to any spinless meson. The decay amplitude is derived using similar considerations as those in a Daltiz like analysis of three-body spinless mesons taking into account the fact that the J/\psi is spin-1, and the various interferences allowed between different final states. Implementation of this procedure can, in principle, lead to the use of a larger number of final states for CP violation studies.


Introduction
Measurement of CP violation in the B 0 and B 0 s systems is important for testing the Standard Model, as new particles can appear in mixing diagrams. Previous measurements have been made in many modes [1]. To measure the phase in B 0 s decays the final states B 0 s → J/ψ K + K − for K + K − masses close to that of the φ meson has been used [2][3][4], as well as B 0 s → J/ψ π + π − [5]. In the latter case the final state is CP odd [6] over most of the π + π − mass range, while in the case of K + K − the final state even in the mass region near the φ meson has both CP odd and even components, that can be resolved using time-dependent angular analysis [7]. In this paper we present a formalism that allows the entire K + K − mass region to be used in CP violation measurements regardless of the final state angular momentum. This formalism can also be applied to B 0 decays, e.g.
The basic concept here is to couple a three-body Dalitz like analysis [8] to the J/ψ h + h − final state, where the J/ψ → µ + µ − and concurrently measure the time-dependent CP violation by splitting the final state into odd and even CP components.

Time-dependent decay rates
The time evolution of the B 0 q -B 0 q system is described by the Schrödinger equation where the M and Γ matrices are Hermitian, and CP T invariance implies that M 11 = M 22 and Γ 11 = Γ 22 . The off-diagonal elements, M 12 and Γ 12 , of these matrices describe the off-shell (dispersive) and on-shell (absorptive) contributions to B 0 q -B 0 q mixing, respectively. The mass eigenstates |B H and |B L of the effective Hamiltonian matrix are given by with |p| 2 + |q| 2 = 1. The decay amplitudes for B 0 q and B 0 q into a self-charge-conjugated final state f , where for this paper f = J/ψ h + h − , are defined as With the additional definitions the time dependent decay rates can be written as [9] where N is a normalization constant, ∆m = m H − m L , ∆Γ = Γ L − Γ H , and Γ = (Γ L + Γ H )/2.

Definition of helicity angles
We express the angular dependence of the decay in terms of "helicity" angles defined as (i) θ , the angle between the µ + direction in the J/ψ rest frame with respect to the J/ψ direction in the B 0 q rest frame; (ii) θ h the angle between the h + direction in the h + h − rest frame with respect to the h + h − direction in the B 0 q rest frame, and (iii) χ the angle between the J/ψ and h + h − decay planes in the B 0 q rest frame. These angles are shown pictorially in 3.2 Time-independent part of the rate for B 0 q decays For the decays of B 0 q → J/ψ h + h − with J/ψ → µ + µ − the decay rate is found by summing over the unobserved lepton polarizations. The time-independent part of the rate is 1 where λ = 0, ±1 is the J/ψ helicity, α = ±1 is the helicity difference between the two muons, J is the spin of the h + h − intermediate state, and H J λ (m hh ) is a helicity amplitude depending on m hh that can be expressed using a formalism similar to that in a Dalitz-plot analyses. We define the term which contains the sum over spin-J as Then Eq. (7) becomes Defining results in Table 1 lists the functions Θ λ λ (θ ). They are invariant under the interchange of λ and λ , i.e. Θ λ λ (θ ) = Θ λλ (θ ), and transform with respect to a change of the sign of both λ and λ as Θ λλ (θ ) = (−1) λ−λ Θ −λ −λ (θ ). Inserting the explicit functional forms in Eq. (11) allows us to express the amplitude as 1 In d J −λ,0 (θ h ), −λ is used instead of λ in order to be consistent with the convention used in [3].
, which contains the helicity amplitudes for B 0 q decays. H λ (m hh , θ h ) and H λ (m hh , θ h ) are related by transversity CP eigenstates [10], that are discussed in Section 4. Using these we find

The interference term
Next we calculate the complex term Replacing the explicit terms leads to 4 Time-dependent Dalitz-plot formalism Here we discuss the general formalism which includes S, P, D or higher waves of the h + h − intermediate states.
Apart from the proper decay-time t, the decay of B 0 q → J/ψ h + h − , J/ψ → µ + µ − can be described by four variables, we choose to use m hh and three helicity angles (θ , θ h , χ), where (m hh , cos θ h ) space is used instead of the usual variables in a Dalitz-plot analysis: m 2 hh , m 2 J/ψ h + ; the advantage is the former has an rectangle phase space which is easier for calculating the normalization.
Assuming |p/q| = 1, the differential decay rates in Eqs. (5) and (6) can be written in terms of the five variables t, m hh , θ , θ h , χ as The functions |A f | 2 , |A f | 2 and A * f A f are defined in Eqs. (12), (13) and (15) respectively. We now substitute in Eq. (8) explicit variables for H J λ (m hh ) in terms of our chosen Dalitzplot variables m hh and θ h , resulting in where the function A R (m hh ) describes the mass squared shape of the resonance R, that in most cases is a Breit-Wigner function, P B is the J/ψ momentum in the B 0 q rest frame, P R is the momentum of either of the two hadrons in the dihadron rest frame, m B is the Spin q mass, L B is the orbital angular momentum between the J/ψ and h + h − system, and L R the orbital angular momentum in the h + h − decay, and thus is the same as the spin of the h + h − resonance. F  18) is based on previous Dalitz plot analyses [6,11], but here all allowed values of L B and L R are included.
In order to use CP relations, it is convenient to replace the helicity complex coefficients ( -) h R λ by the transversity complex coefficients Here ( -) a R 0 corresponds to longitudinal polarization of the J/ψ meson, and the other two coefficients correspond to polarizations of the J/ψ meson and h + h − system transverse to the decay axis: ( -) a R for parallel polarization of the J/ψ and h + h − and ( -) a R ⊥ for their perpendicular polarization.
In the SM, if we assume that only one diagram contributes to the decay and there is no direct CP violation, then the CKM weak phase only appears as q p = e −iφs for the B 0 s decays and q p = e −2iβ for the B 0 decays. The ( -) a i amplitudes only contain strong phases, Note that for the h + h − system both C and P are given by (−1) L R , so the CP of the h + h − system is always even. The total CP of the final state is (−1) L B , since the CP of the J/ψ is also even. The final state CP parities for S, P, and D-waves are shown in Table 2.
Direct CP violation can also be considered, i.e.ā R i = η R i a R i . The complex coefficients can be parameterized as where c R i , b R i , δ R i and φ R i are real numbers that can be determined in the experiment. Note that b R i and φ R i are CP violating, while c R i and δ R i are CP conserving. The direct CP asymmetry for a particular intermediate state R with the transversity i component is In the case that direct CP violation is present, the experiment measures an "effective" phase that is the sum of the CP violation due to the interference between mixing the decay and direct CP violation, given by To implement this procedure data need to be fit with the probability density functions (PDFs) given in Eqs. (16) and (17). The normalization can be computed by first integrating over t, θ and χ analytically, then by using numerical integration for the remaining variables; the terms containing variable χ in Eqs. (12), (13) and (15) are zero when integrating over χ ∈ [−π, π]. The data can be either flavour tagged or not [12]. In the latter case the two PDFs are averaged.
Without considering m hh dependence, time-dependent angular analysis for φ s determination in B 0 s → J/ψ φ decay [2][3][4] cannot distinguish between two ambiguous solutions, one that is (φ s , ∆Γ) and the other being (π − φ s ,−∆Γ), because the time-dependent differential decay rates are invariant under this transformation together with a similar transformation for the strong phases. This ambiguity has been resolved by the LHCb collaboration [13] using the P-wave φ interference with the K + K − S-wave [14] as a function of dikaon invariant mass as suggested in [15]. Our Dalitz-plot formalism automatically takes the strong phases as a function of m hh into account in the complex function A R (m hh ) in Eq. (18), and thus provides only one solution for (φ s , Γ s ) without any ambiguity.

Conclusions
We have presented a method that can be used to extract the CP violating phase for neutral B meson decays into a spin-1 resonance that decays to a dilepton pair and a π + π − or K + K − pair, using the full set of mass and angular variables. Thus CP violation can be measured using a much larger set of final states. For example, the K + K − mass range in B 0 s → J/ψ K + K − can be used including higher mass states such as the f 2 (1525) [16].
The amplitudes A i are related to A i as q p where η i is CP eigenvalue of the i component; η S and η ⊥ = −1, and η 0 and η = 1. We express the amplitutes as functions of either the helicity distributions or transversity distributions as the sums From Eqs. (15) and (27), we compute the interference terms as Each term is listed in Table 4. In Ref. [3] the time-dependent and angular-dependent rate for B 0 s → J/ψ φ is written as where the time-dependent function h k (t) = N k e −Γt [a k cosh ∆Γt 2 + c k cos(∆mt) + b k sinh ∆Γt 2 + d k sin(∆mt)], and the f k (Ω tr ) represent angular-dependent functions. Comparing with Eq. (5) it can be seen that a k corresponds to |A f | 2 + |A f | 2 2 , c k to |A f | 2 − |A f | 2 2 , b k to −Re( q p A * f A f ) and d k to −Im( q p A * f A f ). Using Table 4, we find the same equations as shown in Ref. [3].  (29). When two signs appear, the upper one corresponds to q k and the lower toq k . ( -) q k r k g k (Ω hel ) g k (Ω tr ) |A 0 | 2 |A 0 | 2 sin 2 θ cos 2 θ h cos ψ tr (1 − sin 2 θ tr cos 2 φ tr ) |A | 2 |A | 2