$Z$-boson polarization as a model-discrimination analyzer

Determining the spin of any new particle is critical in identifying the true theory among various extensions of the Standard Model (SM). The degree of $Z$-boson polarization in any two-body decay process $A\to B Z$ is sensitive to the spin assignments of two new particles $A$ and $B$. Considering all possible spin-0, 1/2 and 1 combinations in a renormalizable field theory, we demonstrate that $Z$-boson polarization can indeed play a role of a decisive and universal analyzer in distinguishing the different spin assignments.


Introduction
As the electroweak (EW) scale (v = 246 GeV) and beyond are being explored at the Large Hadron Collider (LHC), it is highly anticipated that the true theory for the origin and stability of the EW scale [1,2,3,4] will be revealed.
One generic prediction in most of new models is the presence of new particles partnered with some or all of the SM particles. For instance, every SM particle in low-energy supersymmetry (SUSY) [5,6,7,8,9,10] has a heavier partner whose spin differs by 1/2. Alternatively, in universal extra dimension (UED) models [11], each SM particle is paired with a tower of Kaluza-Klein (KK) excitations with identical spin. Thus, modelindependent spin measurements are crucial in discriminating among new scenarios.
Since the rest frame of the decaying particle is hardly reconstructible, the direct spin measurement at the LHC are performed through the Lorentz-invariant masses in sufficiently long decay chains [12,13,14]. Such spin-determination methods, however, rely heavily on the final state spins and the chiral structure of couplings [15].
In this paper we analyze the two-body decay of a new heavy state A into a new lighter state B and an on-shell neutral Z-boson as where the Z-boson polarization can be measured through the lepton angular distributions in the decay Z → ℓ + ℓ − with respect to the Z flight direction, reconstructed with great precision. If its branching fraction is sizable, the leptonic decay (1) is highly expected to be a promising tool not only for diagnosing the properties of the new particles [16,17] but also for determining the A and B spins together. If kinematically allowed, several two-body decays like the event topology as the process (1) can occur in each extension of the SM like SUSY, UED and little Higgs (LH) models (See Table 1).

Z Polarization in the Rest Frame
Before describing the two-body decays A → B Z in detail, we briefly summarize how to reconstruct the Z-polarization through the lepton-angle distributions of the leptonic Z-boson decays Z → ℓ + ℓ − , in particular, with ℓ = e and µ. In the rest frame of the decaying Z boson reconstructed with great precision by measuring the lepton momenta, the normalized ℓ − polar-angle distributions with respect to the Z polarization axis defined to be the Z flight direction in the laboratory frame are given by for transversely-polarized (helicity = ±1) and longitudinally-polarized (helicity = 0) Z bosons, respectively, with ξ ℓ = 2v ℓ a ℓ /(v 2 ℓ + a 2 ℓ ). (Although in principle the ℓ − angular distributions for Z bosons with positive and negative helicity can be studied separately, but they are not easy to distinguish practically because of the very small LR asymmetry factor ξ ℓ ≃ −0.147 used for their distinction with v ℓ = s 2 W − 1/4 ≃ −0.02 and a ℓ = 1/4.) Note that the polar-angle distribution can be determined even without knowing the full kinematics of the decay A → B Z.
The general analysis of the Z-boson polarization developed in the two-body decay A → B Z is most transparent if performed in the helicity formalism [18]. In the A rest frame the decay helicity amplitude can be decomposed in terms of the decay polar and azimuthal angles for the momentum direction of the Z boson produced in the A rest frame as where j A , σ A is the spin and helicity of the particle A and σ B , λ are the helicities of the particle B and Z boson, respectively. (For the sake of discussion the Z momentum direction will be referred to as the production axis in the following.) Because of rotational invariance, the reduced matrix elements T λ,σ B , which contains all the dynamical information on the decay process, is independent of the A helicity. After integrating the absolute square of the amplitude over the angles and summing it over the A and B helicities, we can obtain the longitudinal and transverse polarizations, and the LR polarization asymmetry of the produced Z boson in the A rest frame, where the subscript ±1 or 0 denotes the Z helicity.
Combining the Z-boson production process (1) and the lepton angle distribution (3) coherently, ξ ℓ and involving angular distribution linear in cos θ ℓ , we obtain the normalized and correlated lepton angle distribution uniquely determined by the degree of longitudinal Z polarization P A L and the LR asymmetry A A ± . The degree of longitudinal Z-polarization and the LR asymmetry from the two-body decay A → BZ depend crucially on the ABZ vertex structure dynamically, restricted by the spin assignments, and on the masses of the particles A and B kinematically, more specifically the ratios z A,B = m A,B /m Z . Unless we are concerned about any overall numerical factors and higher-dimensional couplings such as those induced from loop corrections, then the ABZ vertex structure for a given spin assignment is uniquely determined up to an overall factor. For each spin assignment of the particles A and B, the general form of the ABZ vertex structure is given by where p, q and k are the incoming A, outgoing B and Z four-momenta, the four-vector indices λ, ν and µ correspond to the states A and B, and the Z boson, and v, a are the vector and axial-vector couplings to the Z-boson in the FF case.
and the LR asymmetry, non-vanishing only for the FF assignment, as where, for notational convenience, we have used the abbreviations with z A,B = m A,B /m Z and the asymmetry A N = (v 2 − a 2 )/(v 2 + a 2 ) defined in terms of the vector and axial-vector couplings v and a. In order for the two-body decay A → BZ to be kinematically allowed, the inequality z A ≥ z B + 1 should be satisfied.
One aspect unique to the SS spin assignment is that the Z-boson is completely longitudinally polarized independently of the masses m A and m B . Another distinctive feature shared by all the spin assignments is that the Z boson becomes longitudinally polarized as z A → ∞, cf. Figure 1.

Z Polarization in the Laboratory Frame
Although the Z boson itself is fully reconstructed, the rest of the event, in particular, the rest frame of the decaying particle A, may not be reconstructed. In this situation, a natural axis for the Z-polarization is the Z-boson's momentum direction in the laboratory frame, which will be called the detection axis. The most natural experimental observable for Z decays is then the polar-angle θ ′ distribution of the charged lepton ℓ − with respect to the Z-boson direction of motion in the Z-boson rest frame.
The degree of longitudinal Z polarization P D L and the LR asymmetry A D ± determined along the detection axis is related to the degree of longitudinal Z polarization P A L and the LR asymmetry A A ± computed in the rest frame of the decaying particle A by a Wigner rotation connecting the two spin bases [19]. The detected polarization P D L and the LR asymmetry A D ± are given by in terms of the computed polarization P A L and LR asymmetry A A ± , respectively, and the Wigner angle ω between the production and detection axes in the Z rest frame. This angle is determined by the composition of boosts Λ ZL from the Z rest frame to the laboratory frame, followed by Λ LP from the laboratory frame to the A rest frame, and finally Λ P Z from the A rest frame to the Z rest frame, where M(Λ) and R(θ) are the representation matrices for the Lorentz transformations. Explicitly, the Wigner angle ω can be extracted from the expression where E Z = γ Z m Z and p Z = γ Z β Z m Z are the energy and absolute momentum of the Z in the A rest frame, Θ is the polar angle of the Z boson in the A rest frame and β A is the speed of the particle A in the laboratory frame.
The energy and absolute momentum of the Z boson are fixed with the masses of the particles in the decay A → BZ as The phase space integration over cos Θ can be carried out as the matrix elements are independent of cos Θ after averaging over the A and B spins. The only nontrivial complication arises from the boost spectrum D(β A ) between the A rest frame and the laboratory frame which depends on the A production mechanism and the parton distribution functions. In principle the (normalized) distribution D can be computed for a given model. If so, the averages of the degree of observed longitudinal polarization and the observed LR asymmetry are given by with the double-integration expressions defined by that allow fully detailed quantitative predictions for the spin assignments of the particles A and B.
There are two extreme kinematic limits for which we do not have to rely on any detailed information on the boost distributions in practice. Firstly, if the particle A is produced near threshold with β A → 0, then cos ω → 1 rendering the difference between the production and detection axes negligible. Secondly, if the mass splitting, m A − m B , of the particles A and B is much larger than m Z . In this case the Z boson is highly boosted with E Z and p Z much larger than m Z even in the A rest frame except for the far backward region with Θ very close to π.
In order to analyze the boost dependence quantitatively, we consider the averages of sin 2 ω and cos ω over the angle Θ, sin 2 ω/ cos ω = 1 2 d cos Θ sin 2 ω/ cos ω, of which the analytic forms are as a function of β A , respectively, satisfying the boundary condition that sin 2 ω (0) = 0 and cos ω (0) = 1. Here, the Z-boson boost factors, γ Z = E Z /m Z and β Z = p Z /E Z , with E Z and p Z in Eqs. (26) and (27) fixed with the masses m A and m B in the decay. As β A → 1, the two average functions approach asymptotically to cos As a numerical illustration, the dependence of the averages sin 2 ω and cos ω on the boost β A is shown in Figure 2 for three characteristic sets

Conclusions
We have computed the expected Z polarization for various spin assignments from a wellmotivated class of models beyond the SM. In addition, we have provided a detailed analytic description about how to determine the degree of Z-boson polarization even if the decaying particle is not at rest. As demonstrated numerically, Z bosons produced in any two-body decay A → BZ involving two new particles A and B can provide us with an important and powerful handle for determining the spins of the new particles, unless Z is not at rest nor boosted with extremely high energies.