# Helicity Amplitudes for Generic Multibody Particle Decays Featuring Multiple Decay Chains

Authors: Daniele Marangotto
Journal: Advances in High Energy Physics
 (2020)

---

## Abstract

We present the general expression of helicity amplitudes for generic multibody particle decays characterised by multiple decay chains. This is achieved by addressing for the first time the issue of the matching of the final particle spin states among different decay chains in full generality for generic multibody decays, proposing a method able to match the exact definition of spin states relative to the decaying particle ones. We stress the importance of our result by showing that one of the matching methods used in the literature is incorrect, leading to amplitude models violating rotational invariance. The results presented are therefore relevant for performing numerous amplitude analyses, notably those searching for exotic structures like pentaquarks.


---

## Body

## 1. Introduction

The helicity formalism, proposed by Jacob and Wick [1] in 1959 to treat relativistic processes involving particles with spin, is still one of the most important tools for performing amplitude analyses of particle decays. To date, complex amplitude analyses involving final-state particles with spin and multiple decay chains have been performed, especially for the search of new resonant structures. Pentaquark searches are a typical example: a pentaquark involves at least one baryon in the final state and introduces an additional decay chain. For instance, pentaquark states were discovered by the LHCb collaboration performing an amplitude analysis of the Λb0⟶J/ψpK− baryon decay [2].However, a consistent definition of the final particle spin states for these kinds of decays turned out to be an issue, since the definition of helicity states is different for different decay chains. Various solutions to match the final particle spin states have been proposed [2–5], but none addressed the problem in full generality for generic multibody decays. In this paper, we present a general method for matching spin states, obtained requiring that, for any decay chain, the final particle states are defined by the same Lorentz transformations relatively to the decaying particle spin states.To this end, we first review the definition of spin states in quantum mechanics in Section2, with a particular attention to their phase specifications. The key point we want to stress is that the relative phases among sets of spin states linked by rotations are fully specified by the transformations applied. Therefore, phase differences like those arising when spin states are rotated with respect to their quantisation axis or like the change of sign under 2π angle rotations of fermion states can not be neglected in helicity amplitudes.Then, we revisit the helicity formalism as originally proposed by Jacob and Wick [1], highlighting the different treatments of daughter particle helicity states in two-body processes. We also propose a simpler definition of two-particle helicity states than the standard one, which allows for an easier matching of the final particle spin states.In Section4, we present how to write helicity amplitudes with a consistent definition of the final particle spin states for different decay chains, applicable to any multibody decay topology. We explicitly derive helicity amplitudes for three-body decays.We stress the need for a consistent definition of the final particle spin states in Section5. First, we discuss the consequences of an incorrect phase introduced between amplitudes describing different decay chains on the decay distributions, showing that they produce observable effects on the decay distributions via interference terms. Next, we perform a numerical study on Λc+→pK−π+ helicity amplitude models featuring different methods to match the final particle spin states, checking a general property of the decay distributions following from rotational invariance. We show how the method employed for the amplitude analyses [2, 3] is incorrect, leading to amplitude models violating rotational invariance, while that proposed in this article fully satisfies rotational symmetry.


## 2. On Spin State Definition

In this section, we review the definition of spin states in quantum mechanics underlining the importance of their phase specification, which will be needed for the upcoming discussion of multibody particle decays in the helicity formalism.In quantum mechanics, the spin of a particle is described by a vector of spin operatorsS^=S^x,S^y,S^z, which defines a right-handed spin coordinate system x,y,z. The spin states s,m are defined as the simultaneous eigenstates of the spin squared modulus S∧2 and S^z, with eigenvalues ss+1 and m, respectively. The z-axis is called the quantisation axis, while x- and y-axes will be named the orthogonal axes.The choice of the orthogonal axes specifies the relative phases among spin states, which can be conventionally chosen by defining the action of the “ladder” operatorsS^±=S^x±iS^y transforming s,m into s,m±1 eigenstates (see, e.g., Ref. [6]). The overall phase of the spin states is undefined and can be chosen arbitrarily.Now, let us consider a set of spin statess,m′ defined relative to the original one s,m by applying a rotation R^:
(1)s,m′=R^s,m.The rotationR defines the relative phases among the two sets: for instance, a rotation around the z-axis of angle α introduces a phase difference between original and rotated spin states:
(2)s,m′=R^zαs,m=e−iαS∧zs,m=e−iαms,m.Therefore, once an overall phase for the original set of spin states is conventionally chosen, that of the rotated spin states is defined by the rotation. In other words, the rotated states are completely defined in terms of the original states and the rotation.The fact that the expectation values of the spin operators transform as a vector under rotations (see, e.g., Ref. [6]) can give the deceptive impression that one can represent rotations applied to spin states in the usual Cartesian space, while spin states transform under spin s representations of the SU2 group. This is why graphical descriptions are not used in the present article, though they are widely used in the literature. For instance, it is well known that fermion states change sign for a 2π angle rotation around any axis i:
(3)R^i2πs,m=e−2iπS∧is,m=−12ss,m,even if the spin operator expectation values do not change.When considering sets of spin states relatively defined by rotations, it is important to take into account their relative phase differences, since interference effects can make them observable quantities. In this article, we will show how to properly consider the spin state definition in the case of multibody particle decays with different intermediate states in the helicity formalism (Section4) and the consequences that an incorrect treatment of spin state definitions has on particle decay distributions (Section 5).


## 3. Helicity Formalism Revisited

In this section, we revisit the helicity formalism [1], developed to overcome problems related to the treatment of spin in relativistic processes. In particular, we highlight the different roles played by the daughter particles in two-body processes and the importance of consistently specifying the definition of their helicity states, including phases. This is an aspect almost neglected so far, which becomes essential for a correct treatment of decays characterised by multiple interfering decay chains. For a clearer treatment of such aspects and a simpler matching of the final particle spin definitions among different decay chains (Section 4), we also propose a different way to express two-particle helicity states, which ease the control of their definitions. A review of the description of relativistic processes involving particles with spin is reported in Appendix A; there, the definition of canonical and helicity states used throughout the article is presented.The key point underlying the helicity formalism is the invariance of helicity under rotations, exploited to construct two-particle states which are eigenstates of total angular momentum. Indeed, under rotations, both spin states and the momentum expressing their quantisation axis rotate, so that the projection of the particle spin on the momentum is unchanged.Let us consider a two-body decayA⟶1,2. The particle 1 helicity states p1A,s1,λ1 are defined in the helicity system (see Equation (A.4)):
(4)S1H=L−p1AzR0,−θ1,−ϕ1SA,with p1A as the particle 1 momentum in the A spin reference rest frame SA and θ1,ϕ1 as its spherical coordinates.The particle 2 helicity statesp2A,s2,λ2 are defined in the helicity system:
(5)S2H=L−p2AzR0,−θ2,−ϕ2SA,which is reached by a different rotation with respect to particle 1 helicity states, so that the direct product of particle 1 and 2 states cannot be related to the total angular momentum SA. Exploiting p1A=−p2A, in Equations (13) and (14) of Ref. [1], the direct product of daughter particle helicity states is defined as
(6)p1A,θ1,ϕ1,λ1,λ2≡p1A,s1,λ1⊗−1s2−λ2exp−iπJ^yp2A,s2,λ2.These states can be now related to two-particle states with a definite value of total angular momentum, which are denotedp1A,J,M,λ1,λ2 as (here with the ψ=0 convention described in Appendix A)
(7)p1A,θ1,ϕ1,λ1,λ2=∑J,M2J+14πDM,λ1−λ2Jϕ1,θ1,0p1A,J,M,λ1,λ2.This expression allows writing theA⟶1,2 decay amplitude as
(8)AmA,λ1,λ2θ1,ϕ1=p1A,θ1,ϕ1,λ1,λ2T^sA,mA=Hλ1,λ2DmA,λ1−λ2∗sAϕ1,θ1,0,in which sA,mA are the A spin states defined in the SA system, T^ is the transition operator, and
(9)Hλ1,λ2≡J=sA,M=mA,λ1,λ2T^sA,mAare complex numbers called helicity couplings, describing the decay dynamics. Note that the key point underlying Equation (7) is that the state p1A,0,0,λ1,λ2 (with p1A aligned with the z-axis) is an eigenstate of J^z with eigenvalue λ1−λ2, which is then rotated to
(10)p1A,θ1,ϕ1,λ1,λ2=R^ϕ1,θ1,0p1A,0,0,λ1,λ2.We note en passant that, using the properties of WignerD-matrices (Equation (B.9)), the two-body amplitude can be rewritten as
(11)AmA,λ1,λ2θ1,ϕ1=Hλ1,λ2Dλ1−λ2,mAsA0,−θ1,−ϕ1,so that, compared with Equation (B.5), the Wigner D-matrix is indeed the representation of the helicity rotation R0,−θ1,−ϕ1 aligning the p1A momentum to the z-axis on the A particle spin states sA,mA.Let us now consider the particle 2 state in Equation (6): the rotation exp−iπS^y acts on the helicity state inverting the z-axis direction; therefore, the particle 2 states entering the amplitude (Equation (7)) are actually opposite-helicity states, which represent spin projection eigenstates in the direction opposite to p1A. This different role of particle 1 and 2 states must be properly considered when these particles have a subsequent decay: the amplitude for the particle 2 decay must take into account that it is not referred to p2A,s2,λ2 states but to those obtained applying the inversion exp−iπS^y and the phase factor −1s2−λ2. We stress that these tricky aspects related to the helicity formalism have been neglected or underestimated so far, because for simple processes (like decays via single decay chains or involving spinless particles), they do not have consequences on the decay distributions. However, they matter for the treatment of the more general decays considered in this article.To take into account in a cleaner way the different roles of particles 1 and 2 in the helicity formalism, we propose a simpler definition of the two-particle state (Equation (6)), which allows for an easier matching of the final particle spin definitions among different decay chains (Section 4). We define the two-particle product state as
(12)p1A,θ1,ϕ1λ1,λ¯2=p1A,s1,λ1⊗p2A,s2,λ¯2,in which p2A,s2,λ¯2 represent spin projection eigenstates in the direction opposite to p2A. The operator λ¯^ is the opposite of the helicity:
(13)λ¯^=−S^·pp,and the opposite-helicity reference system of particle 2 is defined by
(14)S2OH=Lp2AzR0,−θ1,−ϕ1SA,that is, the particle 2 rest frame is reached by boosting along its momentum p2A pointing in the direction opposite to the z-axis. Comparing Equations (4) and (14), we see that both particle 1 and particle 2 states are obtained from the same rotation R0,−θ1,−ϕ1, so that their spin is referred to the “same” spin reference system (they only differ by a boost along the z-axis), including the same definition of the orthogonal axes.It is therefore possible to define eigenstates of total angular momentump1A,J,M,λ1,λ¯2 similarly as before, and Equation (7) holds with the substitution −λ2⟶λ¯2:
(15)p1A,θ1,ϕ1,λ1,λ¯2=∑J,M2J+14πDM,λ1+λ¯2Jϕ1,θ1,0p1A,J,M,λ1,λ¯2.The two-body decay amplitude becomes(16)AmA,λ1,λ¯2θ1,ϕ1=p1A,θ1,ϕ1,λ1,λ¯2T^sA,mA=Hλ1,λ¯2DmA,λ1+λ¯2∗sAϕ1,θ1,0,and the helicity values allowed by angular momentum conservation are
(17)∣λ1∣≤s1,∣λ¯2∣≤s2,∣λ1+λ¯2∣≤sA.The amplitudes (Equations (8) and (16)) are the same but for the substitution λ2↔λ¯2, so why bother with a new state definition? The difference is in the definition of particle 2 states: in the standard formulation (Equation (6)), the particle 2 opposite-helicity state is obtained inverting the helicity one, applying two rotations to the initial system SA plus a phase; in our definition, it is just defined by a single rotation from SA. Our choice simplifies both the writing of particle 2 subsequent decay amplitudes and the matching of the final particle spin states among different decay chains.For the purpose of Section4, it is useful to derive the relation between opposite-helicity and canonical states, the analogue of Equation (A.6) for helicity states. It is obtained applying Equation (A.5) along with the relations p2A=−p1A, p2A=p1A, to the definition of canonical states (Equation (A.1)):
(18)S2C=L−p2ASA=Lp1ASA=Rϕ1,θ1,0Lp1AZR0,−θ1,−ϕ1SA=Rϕ1,θ1,0Lp2AZR0,−θ1,−ϕ1SA=Rϕ1,θ1,0S2OH.The rotation is indeed the same as the one from the helicity to the canonical system of particle 1 (see Equation (A.6)):
(19)S1C=Rϕ1,θ1,0S1H.


## 4. Helicity Amplitudes for Generic Multibody Particle Decays Featuring Multiple Decay Chains

In this section, we present how helicity amplitudes for generic multibody particle decays characterised by multiple decay chains can be written: in particular, we propose an original method to match the final particle spin states among different decay chains able to properly take into account the definition of spin states. For the sake of clarity, we consider a three-body decayA⟶1,2,3, but the method presented to write helicity amplitudes is applicable to any decay topology.Decay amplitudes for multibody particle decays are obtained in the helicity formalism by breaking the decay chain in sequential two-body decays mediated by intermediate states; for instance, a three-body decay is treated by breaking it into two binary decays. Three decay chains, involving three kinds of intermediate states, are possible:A⟶R⟶1,2,3, A⟶S⟶1,3,2, and A⟶U⟶2,3,1.We first consider theA⟶R⟶1,2,3 decay chain: the A⟶R,3 decay can be expressed in the A rest frame by Equation (16):
(20)AmA,λR,λ¯3RA→R,3θR,ϕR=pRA,θR,ϕR,λR,λ¯3RT^sA,mA=HλR,λ¯3RA→R,3DmA,λR+λ¯3R∗sAϕR,θR,0,and the R⟶1,2 decay can be written in the same form, in the R rest frame, by applying Equation (16) to the R state sR,λR as the decaying particle:
(21)AλR,λ1R,λ¯2RR→1,2θ1R,ϕ1R=p1R,θ1R,ϕ1R,λ1R,λ¯2RT^pRA,sR,λR=Hλ1R,λ¯2RR→1,2DmR,λ1R+λ¯2R∗sRϕ1R,θ1R,0.TheR superscript is put on helicity values and angles of particles 1 and 2 to stress that their definition is specific to the A⟶R⟶1,2,3 decay chain.The total amplitude of theA⟶R⟶1,2,3 decay is written introducing R as the intermediate state, and summing the amplitudes over the helicity values λR satisfies the angular momentum conservation requirements (Equation (17)):
(22)AmA,λ1R,λ¯2R,λ¯3RA→R,3→1,2,3Ω=pi,λ1R,λ¯2R,λ¯3RT^sA,mA=∑λRp1R,θ1R,ϕ1R,λ1R,λ¯2RT^pRA,sR,λR×pRA,θR,ϕR,λR,λ¯3RT^sA,mA=∑λRAmA,λR,λ¯3RA→R,3θR,ϕRAλR,λ1R,λ¯2RR→1,2θ1R,ϕ1R=∑λRHλ1R,λ¯2RR→1,2DλR,λ1R+λ¯2R∗sRϕ1R,θ1R,0×HλR,λ¯3RA→R,3DmA,λR+λ¯3R∗sAϕR,θR,0.Note that the angles entering the decay amplitude depend on the phase space variables describing the decay, denoted collectively asΩ.Now, let us consider theA⟶S⟶1,3,2 decay chain. Its associated amplitude is, following Equation (22),
(23)AmA,λ1S,λ¯2S,λ¯3SA⟶S,2⟶1,2,3Ω=∑λSHλ1S,λ¯3SS⟶1,3DλS,λ1S+λ¯3S∗sSϕ1S,θ1S,0×HλS,λ¯2SA⟶S,2DmA,λS+λ¯2S∗sAϕS,θS,0.Helicity values and angles denoted by theS superscripts are defined specifically for the A⟶S⟶1,3,2 decay chain: the definition of the final particle spin states for this decay chain is different from that used for the R intermediate state one.To write the total amplitude of theA⟶1,2,3 decay, amplitudes associated with different intermediate states must be summed coherently to properly include interference effects. The summing can be performed only if the definition of the final particle spin states is the same across different decay chains. Since helicity systems are specific to each decay chain, they must be rotated to a reference set of spin states, for each final particle. Various solutions to match the final particle spin states have been proposed [2–5], but none addressed the problem in full generality for generic multibody decays. In the following, we derive the correct matching of the final particle spin states requiring that, for any decay chain, the final particle states are defined by the same Lorentz transformations relatively to the decaying particle spin states.The definition of the helicity states used to express the helicity amplitudes (Equations (22) and (23)) is given relatively to the A particle spin states (SA reference system) by a sequence of Lorentz transformations. Once a conventional definition of the sA,mA state overall phase is chosen (see Section 2), the helicity states are fully specified by the Lorentz transformation sequence, overall phase included. Therefore, to relate different helicity state definitions, it is mandatory to refer back to the initial SA reference system; i.e., it is not possible to relate the two systems via a direct transformation. To stress this essential point, let us consider the two helicity systems for particle 1 defined by the R and S decay chains, S1HH,R and S1HH,S, respectively. Suppose we find a rotation R^ such that R^S1HH,S=S1HH,R and we rotate the p1S,s1,λ1S states applying the Wigner D-matrix associated with that rotation. However, this does not guarantee that the spin state phase definition is the same between S1HH,R and R^S1HH,S systems, since they are defined with respect to SA by different Lorentz transformation sequences: for a fermion, the two may differ by an overall 2π rotation changing the relative sign among spin states by Equation (3).The correct way to proceed is to define a reference spin system for the final particle from the initial systemSA and relate each helicity system to this one by applying a sequence of rotations which turn the Lorentz transformation sequence defining the helicity state into that specifying the reference one. The reference and the helicity systems are different rest frames of the final particle, so no boosts need to be applied. Any spin system can be chosen as the reference one, and we will choose canonical states reached from SA being the simplest possibility.We illustrate the method explicitly deriving the rotation sequences transforming the helicity systems of theA⟶1,2,3 decay final particles, for R and S intermediate state decay chains, to their canonical systems reached from SA. Starting from particle 1, the helicity system S1HH,R is defined from SA by the sequence (see Equation (A.4)):
(24)SRH=L−pRAzR0,−θR,−ϕRSA,(25)S1HH,R=L−p1RzR0,−θ1R,−ϕ1RSRH,in which SRH is the R helicity system reached from SA. The canonical system of particle 1 derived from SRH, following Equation (A.1) by decomposing the Lorentz boost, is
(26)S1HC,R=Rϕ1R,θ1R,0L−p1RzR0,−θ1R,−ϕ1RSRH=Rϕ1R,θ1R,0S1HH,R,so that the rotation Rϕ1R,θ1R,0 from S1HH,R to S1HC,R systems “undoes” the helicity rotation R0,−θ1R,−ϕ1R, acting in the particle 1 rest frame. The spin states p1R,s1,μ1R defined by the S1HC,R system Lorentz transformation are expressed in terms of the helicity states p1R,s1,λ1R as in Equation (A.7):
(27)R^ϕ1R,θ1R,0p1R,s1,λ1R=∑μ1RDμ1R,λ1Rs1ϕ1R,θ1R,0p1R,s1,μ1R.Now, let us consider the canonical system of particle 1S1CC,R derived from the R canonical system SRC, defined following Equation (A.1):
(28)SRC=L−pRASA=RϕR,θR,0L−pRAzR0,−θR,−ϕRSA,(29)S1CC,R=L−p′1RSRC=Rϕ′1R,θ′1R,0L−p1RzR0,−θ′1R,−ϕ′1RSRC,in which primed quantities indicate that the particle 1 momentum used to build the S1CC,R system is different from the one defining the S1HC,R system, being determined from SRC instead of SRH. The momenta p1R and p′1R differ by the additional rotation RϕR,θR,0 used in the definition of SRC:
(30)p′1R=RϕR,θR,0p1R,and the rotation R0,−θ′1R,−ϕ′1R used to align p′1R to the z-axis can be decomposed into R0,−θR,−ϕR, rotating p′1R into p1R, times R0,−θ1R,−ϕ1R, with the helicity rotation aligning p1R with the z-axis:
(31)R0,−θ′1R,−ϕ′1R=R0,−θ1R,−ϕ1RR0,−θR,−ϕR.Applying the above decomposition, along with its inverse,(32)Rϕ′1R,θ′1R,0=RϕR,θR,0Rϕ1R,θ1R,0,to Equation (29) and using the systems defined in Equations (25) and (26), we obtain
(33)S1CC,R=L−p′1RL−pRASA=Rϕ′1R,θ′1R,0L−p1RzR0,−θ′1R,−ϕ′1R×RϕR,θR,0L−pRAzR0,−θR,−ϕRSA=Rϕ′1R,θ′1R,0L−p1RzR0,−θ1R,−ϕ1R×L−pRAzR0,−θR,−ϕRSA=RϕR,θR,0Rϕ1R,θ1R,0S1HH,R=RϕR,θR,0S1HC,R.The rotationRϕR,θR,0 from S1HC,R to S1CC,R systems “undoes” the helicity rotation R0,−θR,−ϕR, still acting in the particle 1 rest frame. The spin states p1R,s1,ν1R defined by the S1CC,R system are expressed in terms of the states p1R,s1,μ1R as
(34)R^ϕR,θR,0p1R,s1,μ1R=∑ν1RDν1R,μ1Rs1ϕR,θR,0p′1R,s1,ν1R.Finally, the systemS1CC,R has to be related to our reference system, the canonical system of particle 1 reached directly from SA:
(35)S1C,A=L−p1ASA,which differs from S1CC,R by
(36)S1C,A=L−p1ALpRALp′1RS1CC,R.The transformationL−p1ALpRALp′1R is equivalent to a rotation, called the Wigner rotation, indicated generically as a Euler rotation Rα1W,R,β1W,R,γ1W,R, in which the associated Euler angles do not have a simple algebraic expression but can be easily computed. The value of the Euler angles can be calculated as follows: the Wigner rotation matrix is found from multiplying the three matrices associated with each Lorentz boost; the unit vectors describing the rotated system are simply the columns of the Wigner rotation matrix; finally, Euler angles are computed using Equation (B.4). Explicit axis-angle expressions for the composition of two Lorentz boosts are reported in Ref. [7]. The canonical spin states p1A,s1,m1 defined by the S1C,A coordinate systems are expressed in terms of the states p′1R,s1,ν1R as
(37)R^α1W,R,β1W,R,γ1W,Rp′1R,s1,ν1R=∑m1Dm1,ν1Rs1α1W,R,β1W,R,γ1W,Rp1A,s1,m1.The different Lorentz transformations applied starting from the initialSA system down to the final S1C,A one, for R intermediate states, are graphically summarised in Figure 1, along with the definition of the many spin states involved in the writing of the helicity amplitude.Figure 1
Graphical summary of the spin state definitions associated with the decay amplitude for final particle 1 and intermediate stateR. The solid lines indicate the Lorentz transformation sequence involved in the writing of the helicity amplitude. Once a conventional choice for sA,mA initial A particle spin states is set, all the other spin states are completely specified by their Lorentz transformation sequence. Momentum labels of spin states have been suppressed to avoid cluttering the notation.Combining Equations (27), (34), and (37), the relation between particle 1 canonical spin states defined by the S1C,A system and particle 1 helicity states defined by the S1HH,R system, those expressing the decay amplitude (Equation (22)), is
(38)R^α1W,R,β1W,R,γ1W,RR^ϕR,θR,0R^ϕ1R,θ1R,0p1R,s1,λ1R=∑μ1RDμ1R,λ1Rs1ϕ1R,θ1R,0×∑ν1RDν1R,μ1Rs1ϕR,θR0∑m1Dm1,ν1Rs1α1W,R,β1W,R,γ1W,Rp1A,s1,m1.The transformation sequence for particle 2 is almost identical to that for particle 1, but for the Lorentz boost involving its momentum, thanks to the use of the opposite-helicity states we introduced in Section3. The sequence of helicity rotations is indeed the same as for particle 1, since particle 2 opposite-helicity states are defined by the same helicity rotation specifying particle 1 helicity states (see Equation (18)). The relation between particle 2 canonical spin states (S2C,A system) and particle 2 opposite-helicity states (S2OHH,S system) is thus the same as Equation (38) apart from the Wigner rotation:
(39)Rα2W,R,β2W,R,γ2W,R=L−p2ALpRALp′2R.For particle 2, the advantage of using our proposed definition for two-particle product helicity states (Equation (12)) instead of the usual one (Equation (6)) is evident: in the latter case, two rotations must be performed, one corresponding to the helicity rotation and one for the inversion.For particle 3, the spin rotation to its canonical system is much simpler since only one helicity rotation has to be “undone,” with no Wigner rotation needed since a single direct boost from particleA to particle 3 rest frames is involved, leading to
(40)p3A,s3,λ3R=∑m3Dm3,λ3Rs3ϕR,θR,0p3As3,m3.The final particle spin rotations are introduced in the decay amplitudes as follows. The amplitude forR intermediate states we are interested in is
(41)AmA,m1,m2,m3A→R,3→1,2,3Ω=pim1,m2,m3T^sA,mA,in which the final particles state
(42)pi,m1,m2,m3=s1,m1⊗s2,m2⊗s3,m3is the product of the canonical spin states reached from the SA system for each final particle.For the sake of clarity, we consider a much simpler example: the case of a single rotation on a single particle state. Suppose we know the transition amplitudeAλ=s,λT^i between a given initial state i and a final state s,λ, but we would like to express the amplitude with respect to some rotated spin states:
(43)R^α,β,γs,λ=∑mDm,λsα,β,γs,m.To this end, we write the transition amplitude for the rotated states introducing thes,λ states as a set of intermediate states:
(44)Am=s,mT^i=∑λs,mR^α,β,γs,λs,λT^i=∑λDm,λsα,β,γAλ,in which the rotation operator R^α,β,γ acts as a transition operator between the final particle spin states. Note the parallel with Equation (11): the Wigner D-matrix is indeed the representation of the rotation applied.Generalising the case of a single rotation for a single particle, the amplitude forR intermediate states (Equation (41)), expressed for the final particle canonical state (Equation (42)) in terms of the amplitude for helicity states (Equation (22)), becomes
(45)AmA,m1,m2,m3A⟶R,3⟶1,2,3Ω=∑λ1R,μ1R,ν1RDm1,ν1Rs1α1W,R,β1W,R,γ1W,RDν1R,μ1Rs1ϕR,θR,0Dμ1R,λ1Rs1ϕ1R,θ1R,0×∑λ2R,μ2R,ν2RDm2,ν2Rs2α2W,R,β2W,R,γ2W,RDν2R,μ2Rs2ϕR,θR,0Dμ2R,λ2Rs2ϕ1R,θ1R,0×∑λ3RDm3,λ3Rs3ϕR,θR,0×AmA,λ1R,λ¯2R,λ¯3RA⟶R,3⟶1,2,3Ω.The amplitude forS intermediate states can be written analogously, taking into account the different decay topologies (interchanging particle 2 and 3 roles):
(46)AmA,m1,m2,m3A⟶S,2⟶1,2,3Ω=∑λ1S,μ1S,ν1SDm1,ν1Ss1α1W,S,β1W,S,γ1W,SDν1S,μ1Ss1ϕS,θS,0Dμ1S,λ1Ss1ϕ1S,θ1S,0×∑λ2SDm2,λ2Ss2ϕS,θS,0×∑λ3S,μ3S,ν3SDm3,ν3Ss3α3W,S,β3W,S,γ3W,SDν3S,μ3Ss3ϕS,θS,0Dμ3S,λ3Ss3ϕ1S,θ1S,0×AmA,λ1S,λ¯2S,λ¯3SA⟶S,2⟶1,2,3Ω.Finally, the total decay amplitude is obtained summing the amplitudes associated with each intermediate state:(47)AmA,m1,m2,m3A⟶1,2,3Ω=∑iAmA,m1,m2,m3A⟶Ri,3⟶1,2,3Ω+∑jAmA,m1,m2,m3A⟶Sj,2⟶1,2,3Ω+∑kAmA,m1,m2,m3A⟶Uk,1⟶1,2,3Ω,in which the amplitude for U intermediate states can be written similarly as for R and S states.For a generic multibody decay, the generalisation of the method for the matching of the final particle spin states is straightforward. Each final particle helicity state must be rotated “undoing” all the helicity rotations applied along its decay chain, in reversed order. One additional Wigner rotation is needed, describing the difference between the direct boost defining the particle canonical state and the boost sequence applied along its decay chain.


## 5. Effects of an Incorrect Matching of the Final Particle Spin States on Decay Distributions

We stress the need for a correct matching of the final particle spin states by considering the consequences an incorrect matching can have. We first present a general discussion of the observable effects one can introduce in the decay distributions, and even in the simple case, the incorrect matching just introduces a relative phase among spin states belonging to different decay chains. Then, we present a numerical study in which we compare the method for the matching of the final particle spin states presented in this article to the one employed in Refs. [2, 3]: we show that the latter is not correct since it breaks rotational invariance.We first discuss the general case of a particleA decay to i=1,..,n final state particles, passing through two intermediate states R and S. The associated amplitudes are denoted as AmA,miR and AmA,miS. Let us suppose that, due to an incorrect matching of the final particle spin state definition, a phase difference expiψΩ is introduced between the two amplitudes. Since the definition of the spin systems depends on the phase space variables, the incorrect phase will be in general function of the decay phase space. The effects of such a phase are in a certain sense analogous to the phase effects seen in interferometry experiments: in that case, the phase difference between particles following different paths is a physical effect produced by a difference in energy potential felt by the particle; in our case, it is an unnatural effect caused by an incorrect definition of spin states among different decay chains (the analogue of paths). Also, the mere change of sign of fermion states under 2π angle rotations is observable, being measured in neutron interferometry [8, 9].The polarised decay rate for definite initialmA and final spin projections mi (see Equation (B.5) of Appendix C) is
(48)pmA,miΩ=AmA,miR+expiψΩAmA,miS2=AmA,miR2+AmA,miS2+2Reexp−iψΩAmA,miRAmA,miS∗.The incorrect phase affects the decay rate modifying the interference terms between different decay chains, changing their functional form in terms of phase space variables. In an amplitude fit to experimental data, it means that fit parameters depending on interference effects between different decay chains can be biased due to the incorrect phase space dependence introduced. Note that the incorrect phase also affects the unpolarised decay rate: there is no guarantee that the sum over the spin states mitigates its effect.A numerical study illustrating these kinds of unnatural effects is performed on the decay distributions of the three-bodyΛc+→pK−π+ decay obtained from its helicity amplitudes. The three-body decay phase space is described by 5 degrees of freedom: two “Dalitz” two-body invariant masses and three angles describing the orientation of the decay with respect to the A reference system SA, called orientation angles in the following. For the Λc+→pK−π+ decay, they can be chosen to be mpK−2,mK−π+2, the cosine of the proton polar angle in the SΛc+ system, cosθp, the proton azimuthal angle, ϕp, and the signed angle between the plane formed by the proton and the Λc+ quantisation axis and the plane formed by the kaon and the pion, named χ.We test a property of the decay distributions following from rotational invariance, which must be satisfied irrespective of the amplitude model considered: for zeroΛc+ polarisation, the orientation angle distributions cosθp,ϕp,χ must be uniform. Indeed, in the absence of a polarisation vector, nothing specifies a direction in the SΛc+ system. This property provides a necessary test for the correctness of an amplitude model: if the model produces anisotropic orientation angle distributions for zero polarisation, it is wrong, violating rotational invariance.We considerΛc+→pK−π+ helicity amplitudes written closely following the three-body ones described in Section 4 (details specific to the Λc+→pK−π+ decay are reported in Appendix D) applying two different methods for matching proton spin states among different decay chains: the one presented in this article and the one employed for the amplitude analyses [2, 3]. For the Λc+→pK−π+ case, the latter method consists in applying a single rotation to the proton states aligning their quantisation axes to a reference one, which are defined as the direction opposite to the momentum of the particle recoiling against the proton in the proton rest frame. Taking the K∗⟶K−π+ decay chain as the reference, the rotation applied to the proton helicity states is
(49)R^0,βpR,0ppR,1/2,λp=dλp′,λp1/2βpRppR,1/2,λp′,with angles
(50)cosβpΛ∗=p^K∗p⋅p^K−p,cosβpΔ∗=p^K∗p⋅p^π+p,for the Λ∗→pK− and Δ∗++→pπ+ decay chains, respectively.For the numerical study, we consider aΛc+→pK−π+ amplitude model consisting of three resonance states, one per decay channel. The detailed specification of the helicity couplings and resonance descriptions employed is reported in Appendix D. The code reproducing the amplitude model is based on a version of the TensorFlowAnalysis package [10] adapted to five-dimensional phase space three-body amplitude fits [11]; this package depends on the machine learning framework TensorFlow [12] and the ROOT package [13].The phase space distributions are described by a set of five millions Monte Carlo pseudodata generated according to the two amplitude models for zero polarisation. We stress that the two amplitude models only differ by the proton spin matching method.The distributions associated with the method presented in this article are shown in Figure2: the orientation angle distributions are precisely isotropic, given the accuracy allowed by the large pseudodata sample.Figure 2
Phase space decay distributions for theΛc+→pK−π+ amplitude models written using the proton spin matching method presented in this article. “Model” refers to the full Λc+→pK−π+ amplitude model distributions, while “D,” “L,” and “K” labels indicate the distributions associated with single Δ∗++,Λ∗,K∗ resonance contributions.The distributions associated with the spin matching method used in Refs. [2, 3] are shown in Figure 3: the orientation angle distributions are evidently anisotropic, also featuring a step in the ϕp distribution. This method is thus incorrect, leading to an amplitude model violating rotational invariance. Moreover, it is important to note that the invariant mass distributions are different for the two methods: therefore, an incorrect spin matching can also affect Dalitz plot analyses, in which orientation angles are integrated over the decay rate, as in spin-zero meson decays or when the polarisation of the decaying particle is not considered.Figure 3
Phase space decay distributions for theΛc+→pK−π+ amplitude models written using the proton spin matching method used in Refs. [2, 3]. “Model” refers to the full Λc+→pK−π+ amplitude model distributions, while “D,” “L,” and “K” labels indicate the distributions associated with single Δ∗++,Λ∗,K∗ resonance contributions.To conclude, we discuss why the matching method of Refs. [2, 3] is not correct. The proton spin rotations (Equation (49)) ensure that the definition of the proton quantisation axis is the same among different decay chains. Therefore, the proton states can differ among different decay chains by a rotation of the orthogonal axes (Equation (2)) or by the sign of the proton state (Equation (3)), which introduce an incorrect phase difference among decay chains. This is exactly the case considered at the beginning of this section. Indeed, the decay distributions associated with single resonances are the same for the two matching methods (up to an irrelevant normalisation factor) (Figure 3), since the phase difference expiψΩ cancels for single resonance decay rates. However, the incorrect phase affects the decay distributions of the full amplitude model, which includes interference effects among different decay chains, as in Equation (48).


## 6. Conclusions

We presented the general expression of helicity amplitudes for generic multibody particle decays characterised by multiple decay chains. We demonstrated the importance of a precise specification of spin states for correct writing of helicity amplitudes, proposing an original method to match the final particle spin states among different decay chains, applicable to any multibody decay topology.To this end, we reviewed the definition of spin states in quantum mechanics, with a particular attention to their phase specifications, and the helicity formalism, considering a simpler definition of two-particle helicity states than the standard one. The proposed method to match the final particle spin states was obtained requiring that, for any decay chain, the final particle states are defined by the same Lorentz transformations relatively to the decaying particle spin states, by applying a sequence of rotations. Helicity amplitudes were explicitly written for three-body decays.We discussed the consequences of an incorrect phase introduced between amplitudes describing different intermediate states on the decay distributions, showing that they produce observable effects on the decay distributions via interference terms. We also tested numerically our spin matching method against the one used for the amplitude analyses [2, 3], in the case of Λc+→pK−π+ decay amplitudes. We showed how the latter leads to amplitude models violating rotational invariance while the first does not. This incorrect behaviour follows from a wrong phase introduced between spin states among different decay chains.



---

*Source: 6674595-2020-12-21.xml*