Helicity Amplitudes For Generic Multibody Particle Decays .
HindawiAdvances in High Energy PhysicsVolume 2020, Article ID 6674595, 15 pageshttps://doi.org/10.1155/2020/6674595Research ArticleHelicity Amplitudes for Generic Multibody Particle DecaysFeaturing Multiple Decay ChainsDaniele MarangottoUniversità degli Studi di Milano and INFN Milano, Via Celoria 16, 20133 Milano, ItalyCorrespondence should be addressed to Daniele Marangotto; daniele.marangotto@unimi.itReceived 4 October 2020; Accepted 30 November 2020; Published 21 December 2020Academic Editor: Sunny VagnozziCopyright 2020 Daniele Marangotto. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Thepublication of this article was funded by SCOAP3.We present the general expression of helicity amplitudes for generic multibody particle decays characterised by multiple decaychains. This is achieved by addressing for the first time the issue of the matching of the final particle spin states among differentdecay chains in full generality for generic multibody decays, proposing a method able to match the exact definition of spin statesrelative to the decaying particle ones. We stress the importance of our result by showing that one of the matching methods usedin the literature is incorrect, leading to amplitude models violating rotational invariance. The results presented are thereforerelevant for performing numerous amplitude analyses, notably those searching for exotic structures like pentaquarks.1. IntroductionThe helicity formalism, proposed by Jacob and Wick [1] in1959 to treat relativistic processes involving particles withspin, is still one of the most important tools for performingamplitude analyses of particle decays. To date, complexamplitude analyses involving final-state particles with spinand multiple decay chains have been performed, especiallyfor the search of new resonant structures. Pentaquarksearches are a typical example: a pentaquark involves at leastone baryon in the final state and introduces an additionaldecay chain. For instance, pentaquark states were discoveredby the LHCb collaboration performing an amplitude analysisof the Λ0b J/ψpK baryon decay [2].However, a consistent definition of the final particle spinstates for these kinds of decays turned out to be an issue, sincethe definition of helicity states is different for different decaychains. Various solutions to match the final particle spinstates have been proposed [2–5], but none addressed theproblem in full generality for generic multibody decays. Inthis paper, we present a general method for matching spinstates, obtained requiring that, for any decay chain, the finalparticle 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 inquantum mechanics in Section 2, with a particular attentionto their phase specifications. The key point we want to stressis that the relative phases among sets of spin states linked byrotations are fully specified by the transformations applied.Therefore, phase differences like those arising when spinstates are rotated with respect to their quantisation axis orlike the change of sign under 2π angle rotations of fermionstates 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-bodyprocesses. We also propose a simpler definition of twoparticle helicity states than the standard one, which allowsfor an easier matching of the final particle spin states.In Section 4, we present how to write helicity amplitudeswith a consistent definition of the final particle spin states fordifferent decay chains, applicable to any multibody decaytopology. We explicitly derive helicity amplitudes for threebody decays.We stress the need for a consistent definition of the finalparticle spin states in Section 5. First, we discuss the consequences of an incorrect phase introduced between amplitudesdescribing different decay chains on the decay distributions,
2Advances in High Energy Physicsshowing that they produce observable effects on the decaydistributions via interference terms. Next, we perform anumerical study on Λ c pK π helicity amplitude modelsfeaturing different methods to match the final particle spinstates, checking a general property of the decay distributionsfollowing from rotational invariance. We show how themethod employed for the amplitude analyses [2, 3] isincorrect, leading to amplitude models violating rotationalinvariance, while that proposed in this article fully satisfiesrotational symmetry.2. On Spin State DefinitionIn this section, we review the definition of spin states inquantum mechanics underlining the importance of theirphase specification, which will be needed for the upcomingdiscussion of multibody particle decays in the helicityformalism.In quantum mechanics, the spin of a particle is describedby a vector of spin operators Ŝ ð̂Sx , ̂Sy , ̂Sz Þ, which defines aright-handed spin coordinate system ðx, y, zÞ. The spinstates js, mi are defined as the simultaneous eigenstates ofthe spin squared modulus S 2 and ̂Sz , with eigenvalues sðs 1Þ and m, respectively. The z-axis is called the quantisation axis, while x- and y-axes will be named the orthogonalaxes.The choice of the orthogonal axes specifies the relativephases among spin states, which can be conventionallychosen by defining the action of the “ladder” operatorŝS ̂Sx îSy transforming js, mi into js, m 1i eigenstates(see, e.g., Ref. [6]). The overall phase of the spin states isundefined and can be chosen arbitrarily.Now, let us consider a set of spin states js, mi ′ defined rel̂:ative to the original one js, mi by applying a rotation R̂ js, mi:js, mi ′ Rð1ÞThe rotation R defines the relative phases among the twosets: for instance, a rotation around the z-axis of angle αintroduces a phase difference between original and rotatedspin states:̂ z ðαÞjs, mi e iαS z js, mi e iαm js, mi:js, mi ′ Rð2ÞTherefore, once an overall phase for the original set ofspin states is conventionally chosen, that of the rotated spinstates is defined by the rotation. In other words, the rotatedstates are completely defined in terms of the original statesand the rotation.The fact that the expectation values of the spin operatorstransform as a vector under rotations (see, e.g., Ref. [6]) cangive the deceptive impression that one can represent rotations applied to spin states in the usual Cartesian space, whilespin states transform under spin s representations of the SUð2Þ group. This is why graphical descriptions are not usedin the present article, though they are widely used in the lit-erature. For instance, it is well known that fermion stateschange sign for a 2π angle rotation around any axis i:̂ i ð2πÞjs, mi e 2iπS i js, mi ð 1Þ2s js, mi,Rð3Þeven if the spin operator expectation values do not change.When considering sets of spin states relatively defined byrotations, it is important to take into account their relativephase differences, since interference effects can make themobservable quantities. In this article, we will show how toproperly consider the spin state definition in the case of multibody particle decays with different intermediate states inthe helicity formalism (Section 4) and the consequences thatan incorrect treatment of spin state definitions has on particledecay distributions (Section 5).3. Helicity Formalism RevisitedIn this section, we revisit the helicity formalism [1], developed to overcome problems related to the treatment of spinin relativistic processes. In particular, we highlight the different roles played by the daughter particles in two-body processes and the importance of consistently specifying thedefinition of their helicity states, including phases. This isan aspect almost neglected so far, which becomes essentialfor a correct treatment of decays characterised by multipleinterfering decay chains. For a clearer treatment of suchaspects and a simpler matching of the final particle spin definitions among different decay chains (Section 4), we alsopropose a different way to express two-particle helicity states,which ease the control of their definitions. A review of thedescription of relativistic processes involving particles withspin is reported in Appendix A; there, the definition ofcanonical and helicity states used throughout the article ispresented.The key point underlying the helicity formalism is theinvariance of helicity under rotations, exploited to constructtwo-particle states which are eigenstates of total angularmomentum. Indeed, under rotations, both spin states andthe momentum expressing their quantisation axis rotate, sothat the projection of the particle spin on the momentum isunchanged.Let us consider a two-body decay A 1, 2. The particle1 helicity states jpA1 , s1 , λ1 i are defined in the helicity system(see Equation (A.4)): A SH1 L p1 z Rð0, θ1 , ϕ1 ÞSA ,ð4Þwith pA1 as the particle 1 momentum in the A spin referencerest frame SA and θ1 , ϕ1 as its spherical coordinates.The particle 2 helicity states jpA2 , s2 , λ2 i are defined in thehelicity system: A SH2 L p2 z Rð0, θ2 , ϕ2 ÞSA ,ð5Þwhich is reached by a different rotation with respect to particle 1 helicity states, so that the direct product of particle 1 and2 states cannot be related to the total angular momentum SA .
Advances in High Energy Physics3Exploiting pA1 pA2 , in Equations (13) and (14) of Ref. [1],the direct product of daughter particle helicity states isdefined as A p1 , θ1 , ϕ1 , λ1 , λ2 pA1 , s1 , λ1 ð 1Þs2 λ2 exp iπ̂J y pA2 , s2 , λ2 :ð6ÞThese states can be now related to two-particle states witha definite value of total angular momentum, which aredenoted jpA1 , J, M, λ1 , λ2 i as (here with the ψ 0 conventiondescribed in Appendix A)rffiffiffiffiffiffiffiffiffiffiffiffi 2J 1 JDM ,λ1 λ2 ðϕ1 , θ1 , 0Þ pA1 , J, M, λ1 , λ2 :4πJ ,M A p1 , θ1 , ϕ1 , λ1 , λ2 ð7ÞThis expression allows writing the A 1, 2 decayamplitude as T jsA , mA iA mA ,λ1 ,λ2 ðθ1 , ϕ1 Þ pA1 , θ1 , ϕ1 , λ1 , λ2 ̂ s H λ1 ,λ2 DmAA,λ1 λ2 ðϕ1 , θ1 , 0Þ,ð9Þare complex numbers called helicity couplings, describing thedecay dynamics. Note that the key point underlying Equation(7) is that the state jpA1 , 0, 0, λ1 , λ2 i (with pA1 aligned with thez-axis) is an eigenstate of ̂J z with eigenvalue λ1 λ2 , which isthen rotated to A p1 , θ1 , ϕ1 , λ1 , λ2 R̂ ðϕ1 , θ1 , 0Þ pA1 , 0, 0, λ1 , λ2 :ð10ÞWe note en passant that, using the properties of WignerD-matrices (Equation (B.9)), the two-body amplitude canbe rewritten assA mA ,λ1 ,λ2 ðθ1 , ϕ1 Þ H λ1 ,λ2 DλA1 λ2 ,mA ð0, θ1 , ϕ1 Þ, A p A , s , λ p A , s , λ p1 , θ1 , ϕ1 λ1 , λ21 1 12 2 2 ,ð11Þso that, compared with Equation (B.5), the Wigner D-matrixis indeed the representation of the helicity rotation Rð0, θ1 , ϕ1 Þ aligning the pA1 momentum to the z-axis on the A particle spin states jsA , mA i.Let us now consider the particle 2 state in Equation (6):the rotation exp ð iπ̂Sy Þ acts on the helicity state invertingthe z-axis direction; therefore, the particle 2 states enteringthe amplitude (Equation (7)) are actually opposite-helicitystates, which represent spin projection eigenstates in thedirection opposite to pA1 . This different role of particle 1and 2 states must be properly considered when these particleshave a subsequent decay: the amplitude for the particle 2decay must take into account that it is not referred to jpA2 ,s2 , λ2 i states but to those obtained applying the inversionð12Þ i represent spin projection eigenstates inin which jpA2 , s2 , λ2 λ is the oppositethe direction opposite to pA2 . The operator bof the helicity:b λ Ŝ · p ,pð8Þin which jsA , mA i are the A spin states defined in the SA system, T̂ is the transition operator, andH λ1 ,λ2 h J sA , M mA , λ1 , λ2 jT̂ jsA , mA iexp ð iπ̂Sy Þ and the phase factor ð 1Þs2 λ2 . We stress thatthese tricky aspects related to the helicity formalism havebeen neglected or underestimated so far, because for simpleprocesses (like decays via single decay chains or involvingspinless particles), they do not have consequences on thedecay distributions. However, they matter for the treatmentof the more general decays considered in this article.To take into account in a cleaner way the different roles ofparticles 1 and 2 in the helicity formalism, we propose a simpler definition of the two-particle state (Equation (6)), whichallows for an easier matching of the final particle spin definitions among different decay chains (Section 4). We define thetwo-particle product state asð13Þand the opposite-helicity reference system of particle 2 isdefined by A SOH2 L p2 z Rð0, θ1 , ϕ1 ÞSA ,ð14Þthat is, the particle 2 rest frame is reached by boosting along itsmomentum pA2 pointing in the direction opposite to the z-axis.Comparing Equations (4) and (14), we see that both particle 1and particle 2 states are obtained from the same rotation Rð0, θ1 , ϕ1 Þ, 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 i similarly as before,angular momentum jpA1 , J, M, λ1 , λ2 :and Equation (7) holds with the substitution λ2 λ2rffiffiffiffiffiffiffiffiffiffiffiffi 2J 1 J :DM,λ λ ðϕ1 , θ1 , 0Þ pA1 , J, M, λ1 , λ2124πJ ,M A p1 , θ1 , ϕ1 , λ1 , λ2ð15ÞThe two-body decay amplitude becomes ̂A mA ,λ1 ,λ 2 ðθ1 , ϕ1 Þ pA1 , θ1 , ϕ1 , λ1 , λ2 T j sA , mA i s H λ1 ,λ 2 Dm A,λ λ ðϕ1 , θ1 , 0Þ,A1ð16Þ2and the helicity values allowed by angular momentum conservation are s , λ λ s : λ1 s1 , λ2212Að17ÞThe amplitudes (Equations (8) and (16)) are the same , so why bother with abut for the substitution λ2 λ2new state definition? The difference is in the definition of
4Advances in High Energy Physicsparticle 2 states: in the standard formulation (Equation (6)),the particle 2 opposite-helicity state is obtained inverting thehelicity one, applying two rotations to the initial system SAplus a phase; in our definition, it is just defined by a single rotation from SA . Our choice simplifies both the writing of particle2 subsequent decay amplitudes and the matching of the finalparticle spin states among different decay chains.For the purpose of Section 4, it is useful to derive the relation between opposite-helicity and canonical states, the analogue of Equation (A.6) for helicity states. It is obtainedapplying Equation (A.5) along with the relations pA2 pA1 ,pA2 pA1 , to the definition of canonical states (Equation (A.1)):The R superscript is put on helicity values and angles ofparticles 1 and 2 to stress that their definition is specific tothe A Rð 1, 2Þ, 3 decay chain.The total amplitude of the A Rð 1, 2Þ, 3 decay iswritten introducing R as the intermediate state, and summingthe amplitudes over the helicity values λR satisfies the angularmomentum conservation requirements (Equation (17)): DR R R ̂,2,3A A RR,3 1,λΩ p,λ,λðÞfgRRi123 T j sA , mA i ,λ m A ,λ 1 , λ2 3 D R T̂ pA , s , λ pR1 , θR1 , ϕR1 , λR1 , λ2R R RλR D R T̂ js , m i pAR , θR , ϕR , λR , λAA3 A A R,3 R ðθR , ϕR ÞA R 1R,2 R θR1 , ϕR1 SC2 L pA2 SA L pA1 SA Rðϕ1 , θ1 , 0ÞL pA1 Z Rð0, θ1 , ϕ1 ÞSA Rðϕ1 , θ1 , 0ÞL pA2 Z Rð0, θ1 , ϕ1 ÞSA Rðϕ1 , θ1 , 0ÞSOH2 :,2 H R 1R R DThe rotation is indeed the same as the one from the helicity to the canonical system of particle 1 (see Equation (A.6)):SC1 Rðϕ1 , θ1 , 0ÞSH1:In this section, we present how helicity amplitudes forgeneric multibody particle decays characterised by multipledecay chains can be written: in particular, we propose anoriginal method to match the final particle spin states amongdifferent decay chains able to properly take into account thedefinition of spin states. For the sake of clarity, we consider athree-body decay A 1, 2, 3, but the method presented towrite helicity amplitudes is applicable to any decay topology.Decay amplitudes for multibody particle decays areobtained in the helicity formalism by breaking the decaychain 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, involvingthree kinds of intermediate states, are possible: A Rð 1, 2Þ, 3, A Sð 1, 3Þ, 2, and A Uð 2, 3Þ, 1.We first consider the A Rð 1, 2Þ, 3 decay chain:the A R, 3 decay can be expressed in the A rest frameby Equation (16):mA ,λR ,λ3,3D H A R Rλ R ,λ 3 sAϕ ,θ ,0 , R ð R R ÞmA ,λR λ3ð20Þand the R 1, 2 decay can be written in the same form, inthe R rest frame, by applying Equation (16) to the R state jsR , λR i as the decaying particle: D R T̂ pA , s , λ A R 1R,2 R θR1 , ϕR1 pR1 , θR1 , ϕR1 , λR1 , λ2R R Rλ R ,λ 1 , λ 2ð21Þ ,2 sRR R H R 1Dϕ,θ,0:R RR R1 1λ 1 ,λ 2mR ,λ1 λ2λ 1 ,λ 2λRλ R ,λ 1 , λ 2 sR RλR ,λR1 λ2,3 H A RD Rλ R ,λ 3ð19Þ4. Helicity Amplitudes for Generic MultibodyParticle Decays Featuring MultipleDecay Chains D R T̂ js , m iA A R,3 R ðθR , ϕR Þ pAR , θR , ϕR , λR , λAA3mA ,λR ,λ3λRð18Þ sA ϕR1 , θR1 , 0 m A ,λ R λ3RðϕR , θR , 0Þ:ð22ÞNote that the angles entering the decay amplitude dependon the phase space variables describing the decay, denoted collectively as Ω.Now, let us consider the A Sð 1, 3Þ, 2 decay chain.Its associated amplitude is, following Equation (22),,2 ,3,3A A SS , 2 1DðΩÞ H S 1S SS SmA ,λ1 ,λ2 ,λ3λSλ 1 ,λ 3 sS,2D H A S Sλ S ,λ 2 SλS ,λS1 λ3 sA SmA ,λS λ2ϕS1 , θS1 , 0ð23Þðϕ S , θ S , 0 Þ:Helicity values and angles denoted by the S superscriptsare defined specifically for the A Sð 1, 3Þ, 2 decaychain: the definition of the final particle spin states for thisdecay chain is different from that used for the R intermediatestate one.To write the total amplitude of the A 1, 2, 3 decay,amplitudes associated with different intermediate states mustbe summed coherently to properly include interferenceeffects. The summing can be performed only if the definitionof the final particle spin states is the same across differentdecay chains. Since helicity systems are specific to each decaychain, they must be rotated to a reference set of spin states,for each final particle. Various solutions to match the finalparticle spin states have been proposed [2–5], but noneaddressed the problem in full generality for generic multibody decays. In the following, we derive the correct matchingof the final particle spin states requiring that, for any decaychain, the final particle states are defined by the same Lorentztransformations relatively to the decaying particle spin states.The definition of the helicity states used to express thehelicity amplitudes (Equations (22) and (23)) is given relatively to the A particle spin states (SA reference system) bya sequence of Lorentz transformations. Once a conventionaldefinition of the jsA , mA i state overall phase is chosen (see
Advances in High Energy Physics5Section 2), the helicity states are fully specified by the Lorentztransformation sequence, overall phase included. Therefore,to relate different helicity state definitions, it is mandatoryto refer back to the initial SA reference system; i.e., it is notpossible to relate the two systems via a direct transformation.To stress this essential point, let us consider the two helicitysystems for particle 1 defined by the R and S decay chains,,ŜS1HH ,R and SHH, respectively. Suppose we find a rotation R1SHH,RŜ S1 ,S SHHandwerotatethejp,s,λistatessuch that R11 1 1applying the Wigner D-matrix associated with that rotation.However, this does not guarantee that the spin state phasê S1HH ,S systems,definition is the same between S1HH ,R and Rsince they are defined with respect to SA by different Lorentztransformation sequences: for a fermion, the two may
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 value
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