A model independent spin analysis of fundamental particles using azimuthal asymmetries

# A model independent spin analysis of fundamental particles using azimuthal asymmetries

Fawzi Boudjema
LAPTH, Univ. de Savoie, CNRS,
B.P. 110, F-74941, Annecy-Le-Vieux, FRANCE
Email:
Ritesh K. Singh
Institut für Theoretische Physik und Astrophysik, Universität Würzburg,
D-97074 Würzburg, GERMANY
Email:
###### Abstract:

Exploiting the azimuthal angle dependence of the density matrices we construct observables that directly measure the spin of a heavy unstable particle. A novelty of the approach is that the analysis of the azimuthal angle dependence in a frame other than the usual helicity frame offers an independent cross-check on the extraction of the spin. Moreover, in some instances when the transverse polarisation tensor of highest rank is vanishing, for an accidental or dynamical reason, the standard azimuthal asymmetries vanish and would lead to a measurement with a wrong spin assignment. In a frame such as the one we construct, the correct spin assignment would however still be possible. The method gives direct information about the spin of the particle under consideration and the same event sample can be used to identify the spins of each particle in a decay chain. A drawback of the method is that it is instrumental only when the momenta of the test particle can be reconstructed. However we hope that it might still be of use in situations with only partial reconstruction. We also derive the conditions on the production and decay mechanisms for the spins, and hence the polarisations, to be measured at a collider experiment. As an example for the use of the method we consider the simultaneous reconstruction, at the partonic level, of the spin of both the top and the in top pair production in in the semi-leptonic channel.

Spin, Quantum interference
preprint: LAPTH/1322
arXiv:0903.4705
October 3, 2019

## 1 Introduction

The Standard Model (SM) of particle physics has been successful in explaining all the collider observables till date with a high degree of precision. This is remarkable considering that the particle content of the model is not complete since it requires a scalar spin-less particle, the Higgs. In the SM formulation, this particle is an integral ingredient of the mechanism of electro-weak (EW) symmetry breaking. This mechanism is still not well understood. For example, the fact that in the SM no symmetry protects the mass of the spin-less Higgs poses the hierarchy problem. Any solution to these issues brings in new particles and interactions at TeV scale with varying spin and gauge quantum number assignment. In a collider experiment, where almost all these particles are expected to be produced and decay to the light SM particles, the gauge quantum numbers, in principle, can be re-constructed by adding-up the gauge quantum numbers of all the observed light SM particles. Spin, on the other hand, shows up only in the distributions in various kinematic variables in the production and the decay sub-processes. Since the knowledge of the spin, along with the gauge quantum numbers, can enable us to distinguish amongst various candidate theories of physics beyond the SM (BSM) there has been growing interest in this subject recently in the context of the upcoming Large Hadron Collider (LHC) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] and also in the context of proposed International Linear Collider (ILC) [20, 21, 22, 23, 24, 25].

Most of the BSM models have new particles that are partners of the SM particles based on their gauge quantum numbers assignments. However, the new particles may differ in the spin assignments. In models with supersymmetry (SUSY), the spin of the SM partner differs by owing to the fermionic nature of the SUSY generators. There are however many other models such as UED where the spin of the partner is same as in the SM. In both kind of models a symmetry can be left over, leading to a heavy stable particle in the spectrum which can be the dark matter candidate. In SUSY models, the lightest neutralino, singlino, gravitino or axino can be stable, while in the models with universal extra dimensions (UED) the first Kaluza-Klein (KK) excitation of photon can be stable and are the dark matter candidate. These dark matter candidates can not be detected directly in collider experiments. Thus if these particles appear at the end of a decay chain, the re-construction of spin can be non-trivial, specially at a hadron collider like LHC. Moreover it would be also important to infer the spin, and other properties, of these dark matter candidates since these properties are important for the indirect detection of dark matter in astrophysical experiments.

The spin of a (new) particle determines the Lorentz structure of its couplings with the other SM fermions and bosons. This, in a way, fixes its dominant production and decay mechanisms. In many cases a careful study of the energy dependence of the cross section around threshold can distinguish between spin– and spin– particles for example. Other methods to determine spin involve decay particles correlators. At the heart of these more direct methods is the decay helicity amplitude. For example, the helicity amplitude of a particle with spin– and helicity with decaying into two particles of spins and , with helicity respectively, can be written as [26]

 (1)

Here is the reduced matrix element. This has been written most conveniently in the rest frame of the decaying particle. In fact the helicity here is the projection of the spin on the quantisation axis. The polar angle is measured w.r.t. this quantisation axis and the azimuthal angle is measured around the same quantisation axis with freedom to chose the plane. In most of the examples, it is useful to chose the production plane of the decaying particle as the plane. Boosting along the quantisation axis will leave the value of the helicity unchanged. The angular distribution in these angles encodes the spin information through the rotation matrix which factorises into an overall phase factor carrying the azimuthal angle dependence and the the function carrying the polar angle dependence. The latter can be expressed as [27]

 dsλl(θ) = ∑t(−1)t [(s+λ)! (s−λ)! (s+l)! (s−l)!]1/2t! (s+λ−t)! (s−l−t)! (t+l−λ)! (2) ×(cosθ2)2(s−t)+λ−l(sinθ2)2t+l−λ

with . The sum is taken over all values of which lead to non negative factorials. The differential rates have therefore polynomial dependence on up to degree and the azimuthal modulation coming from the off-diagonal elements of the density matrix ranges up to . One can construct observables to extract the degree of and/or distribution. If the highest mode for, say, the azimuthal dependence can be extracted this would be an unambiguous measure of the spin, of the particle.

Other methods have been used or advocated to determine spin.

1. Exploiting the behaviour of the total cross-section at threshold for pair production [15, 22] or the threshold behaviour in the off-shell decay of the particle [1],

2. distribution [4, 7, 11, 13, 22] in the production (polar) angle relying on a known production mechanism,

3. comparing different spin assignments to intermediate particles in a process for a given collider signature [1, 5, 7, 8, 9, 10, 17, 18, 19],

4. comparing SUSY vs UED for a given collider signature [3, 4, 6, 11, 12, 14, 16, 21, 23, 24, 25],

5. extracting the polar angle dependence [2, 3, 7, 8, 14, 20, 22, 28, 29] or azimuthal angle dependence [23, 24, 25, 29] of the decay distributions.

The first four methods are indirect ways to assess the spin information subject to some assumptions and can only support or falsify a hypothesis. For example, the threshold behaviour depends not only on the spin but the parity of the particle as well [1] and for a particle of given spin it could be used to determine its parity [30]. Further, it has been shown [22] that for pair production in an annihilation, the threshold behaviour alone can not determine the spin of the particle. With the velocity of the produced particle in the laboratory, at threshold the cross section for a scalar scales as behaviour while for a spin– it goes like , except for Majorana fermions which can have behaviour. Note that these characteristics do not take into account Sommerfeld/Coulomb[31] type corrections. The spin–1 particle can also have threshold behaviour and production angle dependence same as that of fermions with only difference coming from the distributions of their decay products. Thus, threshold behaviour and production angle distributions can at best be used only to confirm the spin assignment not to determine it. In the second method, one usually assumes a production topology, like for example –channel pair production through a gauge boson. In this method the production angle dependence will depend upon the spin of the test particle. But still this dependence is not unique and can be obtained for higher spin test particles.

The third and the fourth method uses numerical values of correlators or differences in the distributions, which can be modified by the changes in the couplings or the presence of addition particles in virtual exchange etc. Thus, one can not use this method without having re-constructed the spectrum of the theory experimentally. The last method, which uses decay correlators, gives either the spin of the particle or the absolute lower limit on its spin. We note that the moments of the polar angle distribution discussed in Ref.[28] gives an upper limit on the spin of the particle.

### 1.1 Spin through the polar angle

Earlier studies of spin measurements used the average values of or angular asymmetry, with appropriately defined polar angle , in the process of 2-body decay [28] or cascade decay [29]. The numerical values of the angular asymmetries or the moments of angular distribution gave estimates of the spin of the decaying particles in a model independent way. Most of the recent spin studies using decay kinematics focus on a decay chain that can be realised in SUSY or UED models. All the intermediate particles in the decay chains are assumed on-shell such that there is no distortions coming from the shape of the off-shell propagator and that it can be decomposed as a series of two body decays, simplifying the calculations. For example, we look at a –body decay chain of a particle , shown in Fig.1. We look in the rest frame of the intermediate particle whose spin is to be determined. Using crossing symmetry we write the matrix element for the –channel process as [26]

 MλA,λBλD,λE(θBD,ϕ)=(2s+1)  dsλi,λf(θBD)  eiϕ(λi−λf)  Msλi,λf, (3)

where, and . The rotation matrix for spin particle is degree polynomial in and , Eq.(2), which on squaring transforms to a degree polynomial in . This leads to a degree polynomial form of the angular distribution as

 dΓ(A→BDE)dcosθBD=Q0+Q1 cosθBD+ ... +Q2s cos2sθBD. (4)

Thus, we see that the degree of these polynomials is a consequence of the representation of the particle under Lorentz or rotation group, in other words, the spin of the particle, provided is produced on-shell, i.e. constant.

One can also describe the decay in powers of some invariants, the highest power giving a measure of spin. Indeed, with we can write

 dΓ(A→BDE)dm2BD=P0+P1 m2BD+ ... +P2s (m2BD)2s, (5)

obtained from Eq.(4) through by using a transformation of variables. Note that we could have and for some kinematical or dynamical reason, in this case we would set the lower limit on the spin to be . We note that the above method involves two decay products of particle while it measures the spin of the intermediate particle and not the spin of the mother particle . To directly measure the spin of , we need to use the polar angle of every decay products w.r.t. the quantisation axis of . The distribution w.r.t this decay angle looks identical to Eq.(4) with being the spin of .

### 1.2 Spin through azimuthal angle

Another method of direct spin re-construction is to use the azimuthal angle distribution of the decay product about the quantisation axis of the decaying particle . This is the main thrust of the present work. Using the form of the rotation matrices it can be shown, see later, that the azimuthal distribution appearing from the interference of different helicity states, has the general form

 dΓdϕ=a0+2s∑j=1aj cos(jϕ)+2s∑j=1bj sin(jϕ), (6)

with being the even contributions while the being odd contributions. A statistically significant non-zero value of or proves the particle spin to be . The coefficients, and , depend on the dynamics of production and decay processes and we will see that they are proportional to the degree of quantum interference of different helicity states of the particle , or in other words, to the off-diagonal elements of production and decay density matrices. This distribution (but for the odd part) has been proposed in [23] and used in [24] to measure the spin of and bosons at LEP-II and Tevatron, respectively. The azimuthal distribution in the laboratory frame is not simple or , however it is sensitive to the polarisation of the decaying particle as shown in Ref. [32]. In this paper, we study the azimuthal distribution, Eq.(6), in a model independent way to determine the constants s and s in terms of production and decay mechanism and construct collider observables to possibly measure these constants. We construct the observables in two different frames of reference and compare their merits.

This paper is organised as follows. In section 2 we give the angular distribution of decay products for a general process of production and decay with emphasis on the case of spin– and spin– particles. We describe the azimuthal distribution in terms of observables (asymmetries) to be used at colliders or event-generators in section 3. A numerical example of top quark decay chain is given in section 4 for the two different reference frames. We conclude in section 5. Additional expressions are given in the appendices.

## 2 Density matrices, polarisation and azimuthal distributions

To assess the spin of an unstable particle , we look at a general –body production process followed by the decay of as , for example. The other particles ’s produced in association with can be either stable or decay inclusively. The differential rate for such a process is given by (see for example [32]),

 dσ = ∑λ,λ′[(2π)42Iρ(λ,λ′)δ4(kB1+kB2−pA−(n−1∑ipi))d3pA2EA(2π)3 n−1∏id3pi2Ei(2π)3] (7) × [ 1ΓA (2π)42mAΓ′(λ,λ′)δ4(pA−pB−pC)d3pB2EB(2π)3 d3pC2EC(2π)3]

after using the narrow-width approximation for the unstable particle , thereby factoring out the production part from the decay. Here we have , , is the total decay width of , is the mass of and . The production and decay density matrices for are denoted by and , respectively. The terms in square brackets in Eq.(7) are Lorentz invariant combinations. The phase space integration can be performed in any frame of reference without loss of generality.

Since we are interested in the decay distribution of , we perform the phase space integrations in the rest frame . We integrate the first square bracket in Eq.(7) and denote it as

 σ(λ,λ′)=∫(2π)42Iρ(λ,λ′)δ4(kB1+kB2−pA−(n−1∑ipi))d3pA2EA(2π)3 n−1∏id3pi2Ei(2π)3 . (8)

We note that the total integrated production cross-section, without cuts, of the process is given by the sum of diagonal terms , while the off-diagonal terms of denote the production rates for transverse/tensor polarisation states or, in other words, for the quantum interference states. Further, we rewrite , where is the polarisation density matrix for in the corresponding production process. Similarly, we can partially integrate the second term in Eq.(7) and write it as

 ∫1ΓA (2π)42mAΓ′(λ,λ′)δ4(pA−pB−pC)d3pB2EB(2π)3 d3pC2EC(2π)3 =BBC(2s+1)4π ΓA(λ,λ′)dΩB, (9)

where is the branching ratio for the decay , is spin of , is the decay density matrix normalised to unit trace, is the solid angle measure for the decay product .111One can also consider 3–body or higher body decay of in Eq.(7) and write Eq.(9) by integrating all the phase space except . One example of this will be the top-quark decay [32]. Combining Eq.(8) and Eq.(9) in Eq.(7) we get the decay angular distribution as

 1σ dσdΩB=2s+14π∑λ,λ′ PA(λ,λ′)  ΓA(λ,λ′), (10)

where , the total cross-section for production of followed by its decay into state. The polarisation density matrix contains the dynamics of the production process and we will discuss its form for spin– and spin– particle in the following sections.

First we will discuss the general structure of the decay density matrix which can be studied independently of the production mechanism. The decay density matrix for a spin– particle, expressed in terms of helicity amplitudes Eq.(1), is given by

 Γ′s(λ,λ′) = ∑l1,l2Msλl1l2M∗sλ′l1l2 (11) = (2s+14π)ei(λ−λ′)ϕ∑l1,l2dsλl(θ)dsλ′l(θ) |Msl1,l2|2 = ei(λ−λ′)ϕ∑ldsλl(θ)dsλ′l(θ)⎡⎣∑l1(2s+14π)|Msl1,l1−l|2⎤⎦ = ei(λ−λ′)ϕ∑ldsλl(θ)dsλ′l(θ) asl,

where

 asl=(2s+14π)∑l1|Msl1,l1−l|2,|l1|≤s1,  |l1−l|≤s2,  |l|≤s. (12)

For a spin particle there are different ’s that define the decay density matrix with

 Tr(Γ′s(λ,λ′))=∑lasl .

Dividing the by its trace leaves us with independent quantities involving ’s to define the normalised decay density matrix of a spin– particle. Now, the normalised decay density matrix can be written as

 ΓA(λ,λ′)=ei(λ−λ′)ϕ  ∑ldsλl(θ)dsλ′l(θ)asl∑lasl=ei(λ−λ′)ϕ γA(λ,λ′;θ), (13)

where is the reduced normalised decay density matrix with only dependence left. It is important to keep in mind that the dependence is an overall phase and we see clearly that the differential cross section will have a more transparent dependence on the azimuthal angle than the polar angle. Using the relation in Eq.(13) we can re-write Eq.(10) as

 1σ dσdΩB = 2s+14π[  ∑λ PA(λ,λ)  γA(λ,λ) (14) + ∑λ≠λ′ R[PA(λ,λ′)]  γA(λ,λ′) cos((λ−λ′)ϕ) − ∑λ≠λ′ I[PA(λ,λ′)]  γA(λ,λ′) sin((λ−λ′)ϕ)⎤⎦,

which is similar to Eq.(6) after integrating out . Thus we have a simple looking distribution of the decay product and also the coefficients of the different harmonics in the distribution. We emphasise again that the dependence enters only through terms with , in other words the off-diagonal elements of the production and decay density matrices. When integrating over the full space without any cuts, the information contained in these terms will be lost. Another point to stress is that the form of the distribution Eq.(14) remains same in any other frame as long as is measured around the momentum axis of the particle with some suitable reference for . The measurement of the (with ) modulation that stems from the part describing the decay depends on the size of the corresponding factor which is controlled by the interactions of particle . The factors describe the different polarisations with which the particle is produced and the factor depends on the dynamics controlling the decay. One of the aims of this paper is to analyse how one can use this understanding to maximise these modulations and especially the modulation with which is the most unambiguous measure of the spin- of the decaying particle. For illustration and as a guide, in the following, we take a close look at the production and decay density matrices for spin– and spin– particles to identify the conditions on the production and decay dynamics for the spin to be measured.

### 2.1 Spin–12 particle

For the decay of spin– particle, , the normalised decay density matrix, in the rest frame or rather the helicity rest frame [33], can be written as

 (15)

using Eq.(11). Here and are defined in terms of reduced matrix elements in Eq.(12) and for spin– particles given as

 a1/21/2 = (12π)∑l1|M1/2l1,l1−1/2|2|l1|≤s1,  |l1−1/2|≤s2 a1/2−1/2 = (12π)∑l1|M1/2l1,l1+1/2|2|l1|≤s1,  |l1+1/2|≤s2. (16)

The explicit calculation of for the spin– particle decaying into two body final state is given in the Appendices C.1 and C.2 for decays into a lighter spin– and either a scalar or spin– considering general effective operators. We have restricted ourselves to operators of dimension 4. It can be seen that is zero for a pure vector or pure axial-vector coupling in the case of decay to a spin-1. It can also be small depending on the masses of the daughter particles.

The polarisation density matrix for a spin– particle can be parameterised as

 (17)

where is the transverse polarisation of in the production plane, is the transverse polarisation of normal to the production plane and is the average helicity or polarisation along the momentum of or polarisation along the quantisation axis. Combining the expression of and in Eq.(14) we get the angular distribution of spin– particle as [32]

 1σ1dσ1dΩB=14π[1+αη3cosθ+αη1sinθcosϕ+αη2sinθsinϕ]. (18)

The averaged azimuthal distribution is given by

 1σ1dσ1dϕ=12π[1+αη1π4cosϕ+αη2π4sinϕ]. (19)

Here we note that the or the modulation of the azimuthal distribution is proportional to the transverse polarisation of the spin– particles and also to the analysing power . Thus, it is important that the production process yields a non-zero value of either or . A non-zero indicates –violation or the presence final state interaction (absorptive parts). A non-zero can be obtained either with parity violation, which is present in the electro-weak sector of the SM or with appropriate initial beam polarisations. Further, we also need to know the analysing power of the particle. For spin– particle, it is given in Eqs.(57) and (59). We see that the decay vertex has to be at least partially chiral, i.e. parity violating for . That is, we need effectively chiral production and at least partially chiral decay for the fermions for their spin to be measured.

It is educative to realise that Eq.(18) can be cast into

 1σ1dσ1dΩB=14π[1+α→pB|→pB|.→η].with→η=(η1,η2,η3) (20)

with the polarisation vector. Performing a general rotation will leave unchanged. In the new frame, after rotation, we can define a new averaged azimuthal distribution as

 1σ1dσ1dϕ′=12π[1+αη′1π4cosϕ′+αη′2π4sinϕ′]. (21)

If the rotation is done along the direction (normal to the production plane), then but will pick up a contribution from , the average helicity. If the azimuthal distribution in this new frame is more conducive to a spin measurement, in the sense of catching the dependence.

It is important to observe that the picture we have described so far in terms of azimuthal dependence through and (or higher for higher spins) may be very much impacted if cuts are applied to the cross section. If the cuts are -dependent, the azimuthal distributions may no longer have the simple form of Eq.(19) but would carry “spurious” dependence that would prevent the spin reconstruction as advocated here. Indeed, we could have a much more complicated dependence of the form

 1σ1dσ1dϕ=12π[Fc(ϕ)+αη1π4Gc(ϕ)cosϕ+Hc(ϕ)αη2π4sinϕ]. (22)

unless only independent cuts are applied as suggested in Ref. [25].

### 2.2 Spin–1 particle

For the decay of spin– particle, , the normalised decay density matrix is given by

 (23)

where,

 (24)

and

 a11 = (34π)∑l1|M1l1,l1−1|2|l1|≤s1,  |l1−1|≤s2 a10 = (34π)∑l1|M1l1,l1|2|l1|≤min(s1,s2) a1−1 = (34π)∑l1|M1l1,l1+1|2|l1|≤s1,  |l1+1|≤s2

The explicit calculation of the analysing power parameter (the vector part) and (the rank– tensor) for a spin– particle decaying in two body final state is given in the Appendices C.3, C.4 and C.5. In particular for decays into massless fermions assuming dimension–operators. For the decay in the SM we have for massless and .

The polarisation density matrix of spin– particle has two parts: the vector polarisation which we define here as and is identical to that for a spin– and the tensor polarisation described through a symmetric traceless rank– tensor , =0. The density matrix is parameterised as [33]

 (26)

Again using Eq.(14) we can write the angular distribution for spin– particle as

 1σ2 dσ2dΩB = 38π[(23−(1−3δ) Tzz√6)+α pzcosθ+√32(1−3δ) Tzzcos2θ (27) + (α px+2√23(1−3δ) Txzcosθ)sinθ cosϕ + (α py+2√23(1−3δ) Tyzcosθ)sinθ sinϕ + (1−3δ)(Txx−Tyy√6)sin2θcos(2ϕ) + √23(1−3δ) Txy sin2θ sin(2ϕ)].

The averaged distribution is

 1σ2dσ2dϕ = 34π[23+αpxπ4cosϕ+αpyπ4sinϕ+23(1−3δ)(Txx−Tyy√6)cos(2ϕ) (28)

We note that the and modulation of the azimuthal distribution is proportional to the transverse polarisations, and , and the transverse components of the tensor polarisations, and . Again, in this frame all the modulations are –even and the modulations are –odd. To determine the spin we need from the production part and from the decay part in the –even production process. Since there is no symmetry that sets to be one-third, the dynamics in the decay part is not constrained. In other words, the decay mechanism does not require any parity violation.

As we have done for spin– it is instructive to rewrite Eq.(27) in terms of invariants under rotations. If one defines a rank– tensor out of the tensor product of the unit vector describing the momentum of the decay product , 222. The scalar product is ., using the fact that is traceless Eq.(27) writes in terms of (rotation) invariants as

 1σ2 dσ2dΩB = 14π[1+α 32 →pB|→pB|.→p+(1−3δ)√32 T.PB]. (29)

We can then rewrite Eq.(29) in another frame, in particular one where we make a rotation around the axis, transverse to the production plane. This will not mix the CP-odd and CP-even tensors but may make some asymmetries in the new frame larger.

### 2.3 spin–32 and spin–2

For spin– and spin– particles we give the decay density matrix in Appendix B. Since we need the coefficient of the highest harmonics to be non-zero for the spin to be determined, we note that for spin– we need parity violating interaction in the decay process, i.e., and/or whose combination defines the analysing power of highest rank. For spin– particles we need . is the analysing power of rank–, the highest rank for spin– to be non-vanishing (see Eq.(52)). This can be achieved without parity violating interactions in the decay process. We note that parity violating interactions are required in the decay of fermions for its spin to be measured along with its (transverse) polarisation being non-zero . For the bosons, on the other hand, we only need its transverse polarisation being non-zero either due to parity violation in the production process or due to polarisation of the initial beams.

## 3 The azimuthal distribution at event-generators/colliders

The azimuthal distributions Eqs.(19), (28) etc. are given in the rest frame of the decaying particle. To be able to measure the spin we need to construct the above mentioned azimuthal angle in terms of quantities defined in the lab frame of a collider experiment. Before considering other frames let us first define some asymmetries in the rest frame.

### 3.1 Asymmetries in the rest frame

We start with re-writing the rest frame azimuthal distribution in terms of some simple asymmetries that we define below. Let us first define

 I2s(ϕ1,ϕ2)=ϕ2∫ϕ1dϕ dσ2sdϕ (30)

For we define the following asymmetries and calculate them using Eq.(19):

 A11 = I1(−π/2,π/2)−I1(π/2,3π/2)I1(0,2π)=αη12 B11 = I1(0,π)−I1(π,2π)I1(0,2π)=αη22. (31)

These asymmetries have been used in Ref. [32] as a probe of the polarisation of the top-quark. Eq.(19) can be re-written in terms of these asymmetries as

 1σ1dσ1dϕ=12π[1+πA112cosϕ+πB112sinϕ]. (32)

Similarly for we further define similar asymmetries and calculate them in terms of vector and tensor polarisations as follows,

 A12 = I2(−π/2,π/2)−I2(π/2,3π/2)I1(0,2π)=3αpx4 B12 = I2(0,π)−I2(π,2π)I2(0,2π)=3αpy4 A22 = I2(−π/4,π/4)−I2(π/4,3π/4)+I2(3π/4,5π/4)−I2(5π/4,7π/4)I2(0,2π) =2π (1−3δ) (Txx−Tyy√6) B22 = I2(0,π/2)−I2(π/2,π)+I2(π,3π/2)−I2(3π/2,2π)I2(0,2π) (33) =2π (1−3δ)(√23Txy).

Thus Eq.(28) can be re-written as

 1σ2dσ2dϕ=12π[1+πA122cosϕ+πB122sinϕ+πA222cos(2ϕ)+πB222sin(2ϕ)]. (34)

We see that the distribution in the rest frame of the decaying particle has very simple form in terms of the above mentioned asymmetries. It is clear that for higher spins we need to cut the in more and more parts. The important observation to make is that the coefficient of the , for spin– and for spin– are determined exactly in the same way in terms of the asymmetries, this generalises also in the same to higher spin particles and similarly for other coefficients of .

Next we write the asymmetries in terms of spin-momentum correlators. To this effect, we first define the following spin vectors in the helicity rest frame

 sx=(0,1,0,0),  sy=(0,0,1,0),  sz=(0,0,0,1) (35)

which are orthogonal to the 4–momenta of the particle , . These spin vectors satisfy the conditions and . The asymmetries for the spin– case can then be written as

 A11=σ(sx.pB<0)−σ(sx.pB>0)σ(sx.pB<0)+σ(sx.pB>0),     B11=σ(sy.pB<0)−σ(sy.pB>0)σ(sy.pB<0)+σ(sy.pB>0). (36)

The asymmetries for the spin– case can be written as

 A12 = σ(sx.pB<0)−σ(sx.pB>0)σ(sx.pB<0)+σ(sx.pB>0),     B12=σ(sy.pB<0)−σ(sy.pB>0)σ(sy.pB<0)+σ(sy.pB>0), A22 = σ([sx.pB]2−[sy.pB]2>0)−σ([sx.pB]2−[sy.pB]2<0)σ([sx.pB]2−[sy.pB]2>0)+σ([sx.pB]2−[sy.pB]2<0), B22 = σ([sx.pB][sy.pB]>0)−σ([sx.pB][sy.pB]<0)σ([sx.pB][sy.pB]>0)+σ([sx.pB][sy.pB]<0)  . (37)