Dalitz-plot decomposition for three-body decays

Abstract

We present a general formalism to write the decay amplitude for multibody reactions with explicit separation of the rotational degrees of freedom, which are well controlled by the spin of the decay particle, and dynamic functions on the subchannel invariant masses, which require modeling. Using the three-particle kinematics we demonstrate the proposed factorization, named the Dalitz-plot decomposition. The Wigner rotations, that are subtle factors needed by the isobar modeling in the helicity framework, are simplified with the proposed decomposition. Consequently, we are able to provide them in an explicit form suitable for the general case of arbitrary spins. The only unknown model-dependent factors are the isobar lineshapes that describe the subchannel dynamics. The advantages of the new decomposition are shown through three examples relevant for the recent discovery of the exotic charmonium candidate Zc(4430), the pentaquarks Pc, and the intriguing Λ + c → pK − π + decay. I. INTRODUCTIONPartial-wave decomposition of reaction amplitudes is widely used in the analysis of both fixed target (e.g. COM-PASS, VES, CLAS, GlueX), and collider (e.g. LHCb, BESIII, Belle, BaBar) experiments. It is the most powerful way to account for spin and parity, J P , of various contributions, thus required in quantum number determinations of newly observed resonances. It also provides for the most sensitive way of distinguishing exotic hadrons, including the XY Z states and pentaquark candidates in the heavy quarkonium sector, from usually large contributions by ordinary mesons and baryons. To establish the existence of a resonance in a given partial wave, it is desired to have a representation of the reaction amplitude consistent with the S-matrix principles of unitarity, analyticity and Lorentz invariance. This is nontrivial when dealing with particles with spin, which introduce kinematical singularities and (pseudo)threshold relations between partial waves. Amplitude analysis in the context of the S-matrix constraints has been extensively studied in the past using both covariant [1-3] and noncovariant methods [4][5][6][7]. When several particles with spin are involved, the noncovariant approach is more practical, because spin is universally accounted for through the simple Wigner D-functions. In this paper, we take a step to simplify amplitude construction and discuss a convenient framework which incorporates dynamic subchannel resonances for a multiparticle decay. We present a universal amplitude formula which describes the decay of an arbitrary spin state to three particles, each also with arbitrary spin. Specifically, we write the amplitudes in a factorized form to separate the dependence on the angles that characterize the orientation of the final-state particles (and thus the information about the polarization of the parent particle) from the Dalitz-plot variables that encode the information on the intermediate resonances in the multiparticle final state. We do not focus on details of the two-particle dynamics merely giving ...

Novel approaches in Hadron Spectroscopy

“…There is a recent interest in this, motivated by the fact that the practical implementation of such rotations is highly nontrivial [29,30]. A proposal to write the 1 → 3 reaction as a sum over subchannels in the same reaction plane, and then rotate the whole sum together, was given in [31], and is referred to as "Dalitz plot decomposition". In this way, the dependence of the Wigner rotations on the relevant Mandelstam variables is apparent.…”

Extracting Maximum Information from Polarised Baryon Decays via Amplitude Analysis: The Λc+⟶pK−π+Marangotto1 2020Advances in High Energy Physics5020View full textAdd to dashboardCite

“…First, we introduce the description of the three-body phase space and the Dalitz plot decomposition [26], Section 5.1, which allows to separate invariant mass and decay orientation degrees of freedom in the decay amplitude. In Section 5.2, we perform a general study of the Λ + c ⟶ pK − π + polarised decay rate, while in Section 5.3, we write the complete Λ + c ⟶ pK − π + decay amplitude written in the helicity formalism following the method of Ref.…”

“…To simplify the Λ + c ⟶ pK − π + amplitude model expression, it is useful to separate invariant mass and decay orientation degrees of freedom applying the Dalitz plot decomposition proposed in Ref. [26]. Moreover, the properties following from rotational invariance described in Section 2.2 are enforced by construction.…”

Towards an amplitude analysis of the decay $$ {\varLambda}_b^0\to {pK}^{-}\gamma $$

“…The technical aspects of this transformation and the inclusion of the dynamical part are discussed in appendix B; for completeness, a comprehensive review on Dalitz plot decomposition can be found in ref. [11]. The final amplitude for a decay A → (B → DE)C with defined helicities of all particles is…”