Extraction of polarization sensitivity in charm-baryon three-body decays in LHCb

The polarization P of a decaying particle can be measured via the following equation: dσ d cos β = N(1 + αP cos β) (1) This equation is a particular case when the decaying particle is only longitudinally polarized, i.e. has spin projection along or against the direction of motion. In practice, what can be measured is αP by measuring the distribution of the angle β. The asymmetry parameter α plays a crucial role and it is assumed to be known in order to obtain the polarization P. The α parameter is truly a universal property of the particle and is experiment independent, whereas the polarization P of a decaying particle depends on the production mechanism, i.e. the beam or the parent decay chain, if any. Thus, P is dependent on the conditions of the specific experiment where it is measured. We begin with the case of a two-body decay, such as Λ → pπ, where the decaying Λ baryon has spin- 1 2 and the proton and pion have spins 1 2 and 0, respectively. In this case, the asymmetry parameter α in Eq. (1) can be expressed as follows: α = |H1 2 | 2 − |H− 1 2 | 2 |H1 2 | 2 + |H− 1 2 | 2 , (2) where |Hi | 2 are helicity couplings and the ± 1 2 indices refer to the helicity of the proton. The complete derivation is given in Appendix A. Equation (2) shows α written in terms of helicity couplings. In order to get α different than 0, parity conserving and parity violating couplings are both needed. To see this and identify parity conserving and parity violating parts of the decay we change the helicity basis to the parity, or LS, basis. The expression for the α in terms of LS couplings clearly shows the importance of both parity conserving and parity violating couplings. The change of basis takes place linearly through Eq. (3), where the coefficients are given by the Clebsch-Gordan coefficients Hλ = X L s 2L + 1 2 · 1/2 + 1 1/2, λ; 0, 0|1/2, λ L, 0; 1/2, λ|1/2, λ HL, (3) H1 2 = r 3 2 HS − r 1 3 HP ! (4) H− 1 2 = r 3 2 HS + r 1 3 HP ! (5) In Eqs. (4) and (5), H1 2 and H− 1 2 (Hλ) are couplings in the helicity basis, while HS and HP (HL) are couplings in LS (parity) basis. The labels S and P refer to S- and P-wave couplings (see Section 3), both of which the spin-1/2 Λ particle can have, one being parity conserving while the other one parity violating. It is evident from Eq. (6) that both parity violating and parity conserving couplings are needed in order to obtain a non-zero α. α = − √ 3Re (HSH∗ P ) 3|HS| 2 + |HP | 2 . (6) This quantity is important since it is really a fundamental property of a baryonic decay and once measured, its value is used to measure the polarization. A recent example where an updated measurement of α(Λ → pπ) impacted many previous polarization measurements is given by [3, 4]. There are also LHCb measurements of polarization in certain decays such as Λb → ΛJ/ψ [1]. In this project, we explore polarization in charmed baryon (Ξc/Λc) decays. There is little knowledge on the asymmetry parameter α for these type of decays. Knowledge on the asymmetry parameter opens 3 up the possibility of using Λ + c decay angle to improve sensitivity of the angular analysis in searches for exotic hadrons in system with charm baryon in the final state. Also, measurements of the polarization of the Λ + c /Ξc are needed in searches for new physics using electromagnetic dipole moment (EDM) measurements as proposed by the SELDOM project [6]. The report is structured as follows: in Sec. 2 we present the general formalism of three-body decays which leads to the expression of the asymmetry parameter α in the polarized case. In Sec. 3 we construct general decay amplitude using Dalitz-plot decomposition (DPD) as well as provide the details of the toy model. In Sec. 4 the α observable is discussed. Sec. 5 discusses the model ambiguities and Sec. 6 discusses the fitting strategy to unpolarized data. Finally. Sec. 7 consists of conclusions and outlook.