Towards a complete angular analysis of the electroweak penguin decay $\Lambda^0_b\to p h^-\ell^+\ell^-$

TL;DR

This paper develops a comprehensive angular analysis method for the rare baryonic decay $ ext{Lambda}_b^0 o p h^- ext{l}^+ ext{l}^-$ using moments technique, providing a robust approach suitable for low-statistics scenarios.

Contribution

It introduces a complete set of 32 orthonormal moments for analyzing the decay, avoiding likelihood fit ambiguities and enhancing analysis robustness.

Findings

Full set of 32 moments provided for decay analysis

Method is robust under low statistics conditions

Avoids ambiguity issues inherent in likelihood fits

Abstract

We investigate the rare electroweak penguin transition $b \to s \ell^+ \ell^-$ though angular analyses in baryonic $b$-decays. Instead of performing likelihood fits, that might suffer from ambiguity issues, we employ the moments technique, which is a simple counting experiment. The method is known to be robust under low statistics conditions and avoids the ambiguity problem by construction. We provide the full set of 32 orthonormal moments for $\Lambda^0_b\to p h^-\ell^+\ell^-$, where $h\in\{\pi,K\}$, for a polarized $\Lambda^0_b$ and the dihadron system $[ph]$ being in a spin-1/2 state.

Figures

Tables

Table 1. Table 1: The twelve helicity amplitudes for the polarized decay Λ b 0 → p ​ h − ​ ℓ − ​ ℓ + → subscript superscript Λ 0 𝑏 𝑝 superscript ℎ superscript ℓ superscript ℓ {{\mathchar 28931\relax}^{0}_{b}}\rightarrow ph^{-}{\ell^{-}}{\ell^{+}} . The helicities of the Λ b 0 subscript superscript Λ 0 𝑏 {{\mathchar 28931\relax}^{0}_{b}} , spin- J 𝐽 J dihadron and spin-1 dilepton are denotes as λ / 2 𝜆 2 \lambda/2 , λ Λ ∗ / 2 subscript 𝜆 superscript Λ ∗ 2 \lambda_{{\mathchar 28931\relax}^{\ast}}/2 and λ ψ / 2 subscript 𝜆 𝜓 2 \lambda_{\psi}/2 , respectively. All amplitudes carry an additional η = ± 1 𝜂 plus-or-minus 1 \eta=\pm 1 tag for the leptonic-side .city The nomenclature follows Lednicky  Hrivnac  et al. ( 1995 ) .

Amplitude $H_{\lambda }^{J,\eta }$	${\lambda }_{{\Lambda }^{\ast }}/2$	${\lambda }_{\psi }/2$	$\lambda /2\equiv \left({\lambda }_{{\Lambda }^{\ast }}-{\lambda }_{\psi }\right)/2$
$C_{+}^{J,\eta }$	$+3/2$	$+1$	$+1/2$
$A_{+}^{J,\eta }$	$+1/2$	$0$	$+1/2$
$B_{+}^{J,\eta }$	$-1/2$	$-1$	$+1/2$
$C_{-}^{J,\eta }$	$-3/2$	$-1$	$-1/2$
$A_{-}^{J,\eta }$	$-1/2$	$0$	$-1/2$
$B_{-}^{J,\eta }$	$+1/2$	$+1$	$-1/2$

Table 2. Table 2: The moments of the first 20 orthonormal angular functions f i ​ ( Ω ) subscript 𝑓 𝑖 Ω f_{i}(\Omega) in Eq.  III for the spin-1/2 case.

$i$	$f_i\left(\Omega \right)\times \left(4\pi \sqrt{2}\right)$	${\Gamma }_i^L$	${\eta }_i^{L\to R}$
0	$1$	$\left({\left|A_{+}^L\right|}^2+{\left|A_{-}^L\right|}^2+{\left|B_{+}^L\right|}^2+{\left|B_{-}^L\right|}^2\right)$	+
1	$\sqrt{3}\cos \theta$	$\frac{1}{\sqrt{3}}\left({\left|A_{+}^L\right|}^2-{\left|A_{-}^L\right|}^2+{\left|B_{+}^L\right|}^2-{\left|B_{-}^L\right|}^2\right)$$P_b$	”
2	$\sqrt{3}\cos {\theta }_h$	$\frac{1}{\sqrt{3}}\left({\left|A_{+}^L\right|}^2-{\left|A_{-}^L\right|}^2-{\left|B_{+}^L\right|}^2+{\left|B_{-}^L\right|}^2\right)\alpha$	”
3	$3\cos \theta \cos {\theta }_h$	$\frac{1}{3}\left({\left|A_{+}^L\right|}^2+{\left|A_{-}^L\right|}^2-{\left|B_{+}^L\right|}^2-{\left|B_{-}^L\right|}^2\right)$$P_b$ $\alpha$	”
4	$\sqrt{5}{d}_{00}^2$	$\frac{1}{\sqrt{5}}\left(-{\left|A_{+}^L\right|}^2-{\left|A_{-}^L\right|}^2+\frac{{\left|B_{+}^L\right|}^2+{\left|B_{-}^L\right|}^2}{2}\right)$	”
5	$\sqrt{15}\cos \theta d_{00}^2$	$\frac{1}{\sqrt{15}}\left(-{\left|A_{+}^L\right|}^2+{\left|A_{-}^L\right|}^2+\frac{{\left|B_{+}^L\right|}^2-{\left|B_{-}^L\right|}^2}{2}\right)$ $P_b$	”
6	$\sqrt{15}\cos {\theta }_h{d}_{00}^2$	$\frac{1}{\sqrt{15}}\left(-{\left|A_{+}^L\right|}^2+{\left|A_{-}^L\right|}^2+\frac{-{\left|B_{+}^L\right|}^2+{\left|B_{-}^L\right|}^2}{2}\right)$$\alpha$	”
7	$\sqrt{45}\cos \theta \cos {\theta }_h{d}_{00}^2$	$-\frac{1}{\sqrt{45}}\left({\left|A_{+}^L\right|}^2+{\left|A_{-}^L\right|}^2+\frac{{\left|B_{+}^L\right|}^2+{\left|B_{-}^L\right|}^2}{2}\right)$ $P_b$ $\alpha$	”
8	$\frac{\sqrt{135}}{4}\sin \theta \sin {\theta }_h{\sin }^2{\theta }_{\ell }\cos {\varphi }_h$	$\frac{4}{\sqrt{15}}Re\left(A_{+}^L{A}_{-}^{L\ast }\right)$ $P_b$ $\alpha$	”
9	$\frac{\sqrt{135}}{4}\sin \theta \sin {\theta }_h{\sin }^2{\theta }_{\ell }\sin {\varphi }_h$	$-\frac{4}{\sqrt{15}}Im\left(A_{+}^L{A}_{-}^{L\ast }\right)$ $P_b$ $\alpha$	”
10	$\frac{\sqrt{135}}{4}\sin \theta \sin {\theta }_h{\sin }^2{\theta }_{\ell }\cos \left({\varphi }_h+2{\varphi }_{\ell }\right)$	$\frac{2}{\sqrt{15}}Re\left(B_{+}^L{B}_{-}^{L\ast }\right)$ $P_b$ $\alpha$	”
11	$\frac{\sqrt{135}}{4}\sin \theta \sin {\theta }_h{\sin }^2{\theta }_{\ell }\sin \left({\varphi }_h+2{\varphi }_{\ell }\right)$	$\frac{2}{\sqrt{15}}Im\left(B_{+}^L{B}_{-}^{L\ast }\right)$ $P_b$ $\alpha$	”
12	$\sqrt{\frac{135}{2}}\sin \theta \cos {\theta }_h\sin {\theta }_{\ell }\cos {\theta }_{\ell }\cos {\varphi }_{\ell }$	$\sqrt{\frac{1}{15}}Re\left(A_{+}^L{B}_{-}^{L\ast }+A_{-}^{L\ast }{B}_{+}^L\right)$ $P_b$ $\alpha$	”
13	$\sqrt{\frac{135}{2}}\sin \theta \cos {\theta }_h\sin {\theta }_{\ell }\cos {\theta }_{\ell }\sin {\varphi }_{\ell }$	$\sqrt{\frac{1}{15}}Im\left(A_{+}^L{B}_{-}^{L\ast }+A_{-}^{L\ast }{B}_{+}^L\right)$ $P_b$ $\alpha$	”
14	$\sqrt{\frac{135}{2}}\cos \theta \sin {\theta }_h\sin {\theta }_{\ell }\cos {\theta }_{\ell }\cos \left({\varphi }_h+{\varphi }_{\ell }\right)$	$-\sqrt{\frac{1}{15}}Re\left(A_{+}^L{B}_{+}^{L\ast }+A_{-}^L{B}_{-}^{L\ast }\right)$ $P_b$ $\alpha$	”
15	$\sqrt{\frac{135}{2}}\cos \theta \sin {\theta }_h\sin {\theta }_{\ell }\cos {\theta }_{\ell }\sin \left({\varphi }_h+{\varphi }_{\ell }\right)$	$\sqrt{\frac{1}{15}}Im\left(A_{+}^L{B}_{+}^{L\ast }-A_{-}^L{B}_{-}^{L\ast }\right)$ $P_b$ $\alpha$	”
16	$\sqrt{\frac{45}{2}}\sin \theta \sin {\theta }_{\ell }\cos {\theta }_{\ell }\cos {\varphi }_{\ell }$	$-\frac{1}{\sqrt{5}}Re\left(-A_{+}^L{B}_{-}^{L\ast }+A_{-}^{L\ast }{B}_{+}^L\right)$ $P_b$	”
17	$\sqrt{\frac{45}{2}}\sin \theta \sin {\theta }_{\ell }\cos {\theta }_{\ell }\sin {\varphi }_{\ell }$	$\frac{1}{\sqrt{5}}Im\left(A_{+}^L{B}_{-}^{L\ast }-A_{-}^{L\ast }{B}_{+}^L\right)$ $P_b$	”
18	$\sqrt{\frac{45}{2}}\sin {\theta }_h\sin {\theta }_{\ell }\cos {\theta }_{\ell }\cos \left({\varphi }_h+{\varphi }_{\ell }\right)$	$-\frac{1}{\sqrt{5}}Re\left(A_{+}^L{B}_{+}^{L\ast }-A_{-}^L{B}_{-}^{L\ast }\right)$ $\alpha$	”
19	$\sqrt{\frac{45}{2}}\sin {\theta }_h\sin {\theta }_{\ell }\cos {\theta }_{\ell }\sin \left({\varphi }_h+{\varphi }_{\ell }\right)$	$\frac{1}{\sqrt{5}}Im\left(A_{+}^L{B}_{+}^{L\ast }+A_{-}^L{B}_{-}^{L\ast }\right)\alpha$	”

Table 3. Table 3: The 12 additional terms for the spin-1/2 dihadron, for the electroweak penguin case in Eq.  III .

$i$	$f_i\left(\Omega \right)\times \left(4\pi \sqrt{2}\right)$	${\Gamma }_i^L$	${\eta }_i^{L\to R}$
20	$\frac{3}{\sqrt{2}}\sin \theta \sin {\theta }_{\ell }\cos {\varphi }_{\ell }$	$Re\left(A_{+}^L{B}_{-}^{L\ast }+A_{-}^{L\ast }{B}_{+}^L\right)P_b$	-
21	$\frac{3}{\sqrt{2}}\sin \theta \sin {\theta }_{\ell }\sin {\varphi }_{\ell }$	$Im\left(A_{+}^L{B}_{-}^{L\ast }+A_{-}^{L\ast }{B}_{+}^L\right)P_b$	”
22	$3\sqrt{\frac{3}{2}}\sin \theta \sin {\theta }_{\ell }\cos {\theta }_h\cos {\varphi }_{\ell }$	$\frac{1}{\sqrt{3}}Re\left(A_{+}^L{B}_{-}^{L\ast }-A_{-}^{L\ast }{B}_{+}^L\right)P_b\alpha$	”
23	$3\sqrt{\frac{3}{2}}\sin \theta \sin {\theta }_{\ell }\cos {\theta }_h\sin {\varphi }_{\ell }$	$\frac{1}{\sqrt{3}}Im\left(A_{+}^L{B}_{-}^{L\ast }-A_{-}^{L\ast }{B}_{+}^L\right)P_b\alpha$	”
24	$\sqrt{3}\cos {\theta }_{\ell }$	$-\frac{\sqrt{3}}{2}\left({\left|B_{+}^L\right|}^2-{\left|B_{-}^L\right|}^2\right)$	”
25	$3\cos {\theta }_h\cos {\theta }_{\ell }$	$\frac{1}{2}\left({\left|B_{+}^L\right|}^2+{\left|B_{-}^L\right|}^2\right)\alpha$	”
26	$3\cos \theta \cos {\theta }_{\ell }$	$-\frac{1}{2}\left({\left|B_{+}^L\right|}^2+{\left|B_{-}^L\right|}^2\right)P_b$	”
27	$3\sqrt{3}\cos {\theta }_h\cos {\theta }_{\ell }\cos \theta$	$\frac{1}{2\sqrt{3}}\left({\left|B_{+}^L\right|}^2-{\left|B_{-}^L\right|}^2\right)P_b\alpha$	”
28	$\frac{3}{\sqrt{2}}\sin {\theta }_{\ell }\sin {\theta }_h\cos \left({\varphi }_h+{\varphi }_{\ell }\right)$	$Re\left(A_{+}^L{B}_{+}^{L\ast }+A_{-}^L{B}_{-}^{L\ast }\right)\alpha$	”
29	$\frac{3}{\sqrt{2}}\sin {\theta }_{\ell }\sin {\theta }_h\sin \left({\varphi }_h+{\varphi }_{\ell }\right)$	$-Im\left(A_{+}^L{B}_{+}^{L\ast }-A_{-}^L{B}_{-}^{L\ast }\right)\alpha$	”
30	$3\sqrt{\frac{3}{2}}\sin {\theta }_{\ell }\sin {\theta }_h\cos \left({\varphi }_h+{\varphi }_{\ell }\right)\cos \theta$	$\frac{1}{\sqrt{3}}Re\left(A_{+}^L{B}_{+}^{L\ast }-A_{-}^L{B}_{-}^{L\ast }\right)P_b\alpha$	”
31	$3\sqrt{\frac{3}{2}}\sin {\theta }_{\ell }\sin {\theta }_h\sin \left({\varphi }_h+{\varphi }_{\ell }\right)\cos \theta$	$-\frac{1}{\sqrt{3}}Im\left(A_{+}^L{B}_{+}^{L\ast }+A_{-}^L{B}_{-}^{L\ast }\right)P_b\alpha$	”

Equations

$$H_{\lambda }^{{\lambda }_p\eta }$$

$$+d_{0\eta }^1({\theta }_{\ell })A_{\lambda }^{J,\eta }{d}_{\frac{\lambda }{2}\frac{{\lambda }_p}{2}}^J({\theta }_h)e^{i\lambda \frac{{\varphi }_h}{2}}$$

$$\displaystyle\hskip 28.45274pt\;\;+d^{1}_{-\lambda\eta}(\theta_{\ell})B^{J,\eta}_{\lambda}\,d^{J}_{-\frac{\lambda}{2}\frac{\lambda_{p}}{2}}(\theta_{h})e^{\mp i\lambda\left(\frac{\phi_{h}}{2}+\phi_{\ell}\right)}\Bigg{]}$$

$$\times h_{\frac{{\lambda }_p}{2}}^J,$$

$$\alpha =|h_{+}^J{|}^2-|h_{-}^J{|}^2$$

$$1=|h_{+}^J{|}^2+|h_{-}^J{|}^2$$

$$h_{-\frac{{\lambda }_p}{2}}^J=(-1{)}^{J+1/2}P{h}_{\frac{{\lambda }_p}{2}}^J,$$

$$\frac{d\Gamma }{d{q}^{2}d\Omega }$$

$$=(4\pi \sqrt{2})\sum _i{\Gamma }_i(q^2)f_i(\Omega ),$$

$$d\Omega \equiv d\cos \theta d\cos {\theta }_{h}d\cos {\theta }_{\ell }d{\varphi }_{h}d{\varphi }_{\ell },$$

$$\rho$$

$${\Gamma }_i(q^2)\equiv {\Gamma }_i^L(q^2)+{\eta }_i^{L\to R}{\Gamma }_i^R(q^2),$$

$$H_{\lambda }^{{\lambda }_p,\eta }({\delta }_W,{\delta }_S)={\overline{H}}_{-\lambda }^{-{\lambda }_p,-\eta }(-{\delta }_W,{\delta }_S),$$

$$\frac{dN}{d\Omega }\sim$$

$$+2Re(H_{+}^{+\eta }{H}_{-}^{\ast +\eta }){\rho }_{+-}$$

$$+H_{+}^{-\eta }{H}_{+}^{\ast -\eta }{\rho }_{++}+H_{-}^{-\eta }{H}_{-}^{\ast -\eta }{\rho }_{--}$$

$$+2Re(H_{+}^{-\eta }{H}_{-}^{\ast -\eta }){\rho }_{-+}.$$

$$C_{\lambda }^{J,\eta }$$

$$A_{\lambda }^{J,\eta }$$

$$B_{\lambda }^{J,\eta }$$

$$h_{{\lambda }_p}^J$$

$${\Gamma }_i=\sum _{k\in N_{\mathrm{data}}}{f}_i({\Omega }_k)/(4\pi \sqrt{2}).$$

$$a_{\pm }^2$$

$$b_{\pm }^2$$

$$a_{\pm }^2$$

$$b_{\pm }^2$$

$$P_b{\alpha }_{\Lambda }$$

$$\frac{{\alpha }_{\Lambda }}{P_b}$$

$$P_b{\alpha }_{\Lambda }$$

$$\frac{{\alpha }_{\Lambda }}{P_b}$$

Full text

I Introduction

Among the dedicated $B$-factories, LHCb is unique in its ability to have access to all species of $b$-hadrons. Within the detector acceptance, the production rates of $B:{{\mathchar 28931\relax}^{0}_{b}}:B_{s}$ are approximately in the ratio $4:2:1$ Aaij et al. (2012). The large number of ${{\mathchar 28931\relax}^{0}_{b}}$ baryons available in a clean environment, thanks to the excellent vertexing and particle identification of the LHCb detector, has led to discoveries such as the Pentaquark states Aaij et al. (2015a), the first evidence of $C\!P$ violation in $b$-hadron decays Aaij et al. (2016a), among others.

In this work, we employ decays of the ${{\mathchar 28931\relax}^{0}_{b}}$ baryon to study the rare electroweak penguin transition $b\rightarrow s{\ell^{+}}{\ell^{-}}$. In the Standard Model, this is a suppressed flavor-changing neutral current, occuring only via loop and box diagrams, and is therefore sensitive to new heavy particles entering the loop. In the mesonic sector, there has been some very interesting tensions with the Standard Model recently Descotes-Genon et al. (2016), so it is important to probe this in as many modes as possible. LHCb has already made observations of the ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{\mathchar 28931\relax}{\mu^{+}}{\mu^{-}}$ Aaij et al. (2015b), ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow p{{K}^{-}}{\mu^{+}}{\mu^{-}}$ Aaij et al. (2017a) and ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow p{{\pi}^{-}}{\mu^{+}}{\mu^{-}}$ Aaij et al. (2017b) decays, using Run 1 data. Updates to these results, including Run 2 data, that will increase the statistical precision by a factor of four, are expected soon. The valence quark content of the ${{\mathchar 28931\relax}^{0}_{b}}$ is $|bud\rangle$, where the $[ud]$ system often acts as a spectator diquark in a spin and isospin singlet. Therefore the properties of the ${{\mathchar 28931\relax}^{0}_{b}}$ are expected to reflect those of the underlying $b$-quark. This is different from the $B$-meson states, where spin information of the $b$-quark is lost during the hadronization with the lighter quark to form a spin-0 meson. Therefore, the ${{\mathchar 28931\relax}^{0}_{b}}$ mode can provide complementary information relative to the $B_{(s)}$ meson modes.

The LHCb tracking system Alves et al. (2008) consists of an innermost tracker (VELO) surrounding the interaction point, followed by two large-area trackers, one upstream (TT) and another downstream (tracking stations) of the LHCb magnet. Since the ${\mathchar 28931\relax}$ flies considerably before decaying weakly, most of the ${\mathchar 28931\relax}$ candidates decay outside the VELO (downstream tracks). In the low dimuon mass squared (${q^{2}}$) region, both the dimuon and dihadron system have sufficient break-up momenta and are more likely to be swept out of acceptance. Therefore, the ${\mathchar 28931\relax}$ acceptance tends to be poor in the low ${q^{2}}$ region. However, it is this low ${q^{2}}$ (large-recoil) region that is most suited for theoretical predictions of observables with reduced uncertainties from hadronic effects. The high ${q^{2}}$ (low-recoil) region receives contribution from broad $c\bar{c}$ resonances and is not theoretically clean.

In this respect, the excited ${\mathchar 28931\relax}^{\ast}\rightarrow pK$ and $N^{\ast}\rightarrow p\pi$ states can be particularly advantageous. These decays occur promptly inside the VELO and is relatively impervious to the low acceptance problem at high recoil. Given than angular observables have often proven to be sensitive to New Physics (NP) contributions Descotes-Genon et al. (2013), it is desirable to measure as many angular observables for the ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow ph^{-}{\ell^{-}}{\ell^{+}}$ ($h\in\{K,\pi\}$) cases as well. The goal of this paper is to set up the formalism to measure these observables employing the so-called moments technique, as outlined in Refs. Dey (2015, 2017) for the $B$-meson case.

II The angle conventions

We follow the conventions in Korner Koerner (2014) and Singleton Singleton (1991) for ${{\mathchar 28931\relax}^{0}_{b}}$ angular analyses. A pictorial representation of the angles for the specific case of the ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{\mathchar 28931\relax}(\rightarrow p{{\pi}^{-}}){{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}(\rightarrow{\mu^{-}}{\mu^{+}})$ decay is given in Fig. 1. In the laboratory frame, the ${{\mathchar 28931\relax}^{0}_{b}}$ transverse polarization is along the normal to the plane formed by the $pp$ beam direction and the ${{\mathchar 28931\relax}^{0}_{b}}$ flight direction. This normal direction is shown as $\hat{n}_{\rm pol}$ in Fig. 1a. We then boost to the ${{\mathchar 28931\relax}^{0}_{b}}$ rest frame and measure the polar angle $\theta$, of the ${\mathchar 28931\relax}$ with $\hat{n}_{\rm pol}$ in the ${{\mathchar 28931\relax}^{0}_{b}}$ rest frame.

Next, we rotate the system such that the ${\mathchar 28931\relax}$ flight direction points along the $+\hat{z}_{h}$ direction and $\hat{y}_{h}$ is along $\hat{z}_{h}\times\hat{n}_{\rm pol}$. That is, $\hat{n}_{\rm pol}$ lies in the $\hat{x}_{h}$-$\hat{z}_{h}$ plane with a positive intercept along the $\hat{x}$ direction. This fixes the quadrant of the azimuthal angles. The hadronic-side axes are labelled by the subscript $h$ here, and also in Fig. 1b.

The leptonic-side coordinate axes are defined in Fig. 1b and are labeled by the subscript $\ell$. The leptonic and hadronic systems have a common $\hat{y}$ axis and back-to-back $\hat{x}$ and $\hat{z}$ axes, conforming with the conventions in Korner Koerner (2014) and Singleton Singleton (1991). Figure 1c illustrates the signs of the azimuthal angles $\phi_{h,\ell}$.

We note here that compared to the ATLAS definitions Aad et al. (2014); Hrivnac et al. (1995), our azimuthal angles are shifted as $\phi_{h,\ell}\rightarrow\phi_{h,\ell}+\pi$.

II.1 The decay amplitudes for ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow ph^{-}{\ell^{-}}{\ell^{+}}$

Following Lednicky Hrivnac et al. (1995), for the decay chain ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{\mathchar 28931\relax}(\rightarrow p{{\pi}^{-}}){{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}(\rightarrow{\mu^{-}}{\mu^{+}})$, we choose the spin-quantization axis to be along the $+\hat{z}_{h}$ direction, that is, the ${\mathchar 28931\relax}$ flight direction in the ${{\mathchar 28931\relax}^{0}_{b}}$ rest frame. This is shown in Fig. 1b. The topology is easily generalized to ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow ph^{-}{\ell^{-}}{\ell^{+}}$, with $h\in\{\pi,K\}$ and the dihadron system (${\mathchar 28931\relax}^{(\ast)}$, or $N^{\ast}$) being in a spin-$J$ state. For spin $J>1/2$, there are twelve amplitudes $\{A^{J,\eta}_{\pm},B^{J,\eta}_{\pm},C^{J,\eta}_{\pm}\}$ corresponding to the different helicity configurations of the dihadron and the dilepton systems as listed in Table 1. For spin $J=1/2$ states, only the $A^{\eta}_{\pm}$ and $B^{\eta}_{\pm}$ amplitudes contribute, as in the case of the ground state ${\mathchar 28931\relax}$.

Following Korner Koerner (2014), the full helicity amplitude can be concisely written in terms of the Wigner $d$-matrices as:

[TABLE]

where $\lambda_{p}/2$ is the helicity of the outgoing proton and $h^{J}_{\frac{\lambda_{p}}{2}}$ is the amplitude for the ${\mathchar 28931\relax}^{\ast}\rightarrow ph$ decay. Here, $\eta=\lambda_{{\mu^{-}}}-\lambda_{{\mu^{+}}}=\pm 1$ and correspond to the handedness of the leptonic current. For left(right)-handed amplitudes, $\eta=-1$($+1$), while for the parity conserving electromagnetic transition ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow{\mu^{-}}{\mu^{+}}$, the left- and right-handed amplitudes are the equal.

If only a single spin-$J$ component contributes, it is useful to define the asymmetry parameter

[TABLE]

with the normalization set as

[TABLE]

If the ${\mathchar 28931\relax}^{\ast}$ state has a definite parity $P$, and decays strongly, then

[TABLE]

resulting in $\alpha=0$. The weakly decaying ${\mathchar 28931\relax}\rightarrow p{{\pi}^{-}}$ effectively includes both odd and even parity components, resulting in a non-zero $\alpha_{\Lambda}$.

III The differential rate

The differential rate expression is given by

[TABLE]

where ${q^{2}}$ is the invariant dilepton mass squared, the phase-space element is

[TABLE]

and the spin summation is over $\{\lambda_{p},\eta,\lambda,\lambda^{\prime}\}\in\pm 1$. The density matrix $\rho$ is given by

[TABLE]

where $\vec{\sigma}$ are the spin-1/2 Pauli matrices. The $f_{i}$’s in Eq. III are a set of orthonormal basis functions and the $\Gamma_{i}$’s are the corresponding moments observables

[TABLE]

where the sign $\eta_{i}^{L\rightarrow R}=\pm 1$ depends on the signature of $f_{i}(\Omega)$ under $\theta_{\ell}\rightarrow\pi+\theta_{\ell}$.

III.1 Spin-1/2 case

The expansion of Eq. III for dihadron spin-1/2-only case comprises 32 angular moments, as listed in Tables 2 and 3, including the two sets of $\eta_{i}^{L\rightarrow R}$ signs. Here $d^{2}_{00}\equiv\frac{1}{2}(3\cos^{2}\theta_{\ell}-1)$. For the parity conserving ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow{\ell^{-}}{\ell^{+}}$ decay in ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{\mathchar 28931\relax}{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, only the first 20 terms with $\eta_{i}^{L\rightarrow R}=+1$ in Table 2 contribute and our results are consistent with those listed in Ref. Hrivnac et al. (1995). For the electroweak penguin case, 12 additional terms occur, with $\eta_{i}^{L\rightarrow R}=-1$. These are listed in Table 3.

III.2 $C\!P$ conjugation

The charge conjugation is performed “explicitly”. With reference to Fig. 1, for the $C\!P$ conjugated ${{\kern 1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}}{}^{0}_{b}}$ decay, we “follow” the ${\mu^{+}}$ and the $\bar{p}$. In the absence of direct $C\!P$ violation, this leads to the same form of the angular expression for ${{\mathchar 28931\relax}^{0}_{b}}$ and ${{\kern 1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}}{}^{0}_{b}}$ in terms of the corresponding helicity amplitudes. The helicity amplitudes between $C\!P$ conjugates are however related by

[TABLE]

where $\delta_{W}$ and $\delta_{S}$ are the weak and strong phases. The polarization $P_{b}$ and the asymmetry $\alpha$ also flip sign. As can be checked from Tables 2 and 3, this effectively means that the signs of the measured moments odd in $\phi_{\ell,h}$ are flipped between $C\!P$ conjugates. To keep the signs of the measured moments the same between the $C\!P$ conjugates, we further flip $\phi_{\ell,h}\rightarrow-\phi_{\ell,h}$ for the ${{\kern 1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}}{}^{0}_{b}}$. This is convenient to experimentalists since $C\!P$ symmtries can be measured by combining the ${{\mathchar 28931\relax}^{0}_{b}}$ and ${{\kern 1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}}{}^{0}_{b}}$ data samples.

It is important to note that the moments and amplitudes that we provide are given for the ${\mathchar 28931\relax}^{0}_{b}$ (containing a $b$-quark), since this is consistent with the conventions in several semileptonic and electroweak penguin theory papers.

IV Ambiguities in the amplitude solutions

Expanding out Eq. III, the differential rate is

[TABLE]

The above form is invariant under the the following set of simultaneous transformations: $P_{b}\rightarrow-P_{b}$ and $H^{\lambda_{p}\eta}_{\lambda}\rightarrow\lambda H^{-\lambda_{p}\eta}_{-\lambda}$. The $H^{\lambda_{p}\eta}_{\lambda}\rightarrow\lambda H^{-\lambda_{p}\eta}_{-\lambda}$ transformation can be affected by:

[TABLE]

In particular, the flip in Eq. 11d means that the sign of $\alpha$ will also flip. The simultaneous flip of $\alpha$ and $P_{b}$ is discussed in more detail in the subsequent section. In the case of $P_{b}=0$, further ambiguities are also possible.

The underlying reason for these ambiguities is that, as in the mesonic cases Hofer and Matias (2015), the spins of the final state leptons are not measured, but averaged over. Further, if the spin of the proton is also not measured (as for the ${\mathchar 28931\relax}^{\ast}$ and $N^{\ast}$ cases), this leads to further ambiguities. The ambiguities imply that the full set of $\Gamma_{i}$ elements are not independent, but relations exist among them Hofer and Matias (2015). For an observables fit, it is necessary to identify these relations and the minimal set of independent observables that can be the variable parameters in the fit. The novelty of the moments technique is that ambiguity issue is rendered irrelevant since no fit is performed. Once the full set of moments (which might not be independent) and the covariance matrix is extracted from the data, these can be compared against model predictions.

V Solving for the ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{\mathchar 28931\relax}(\rightarrow p{{\pi}^{-}}){{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}(\rightarrow{\mu^{-}}{\mu^{+}})$ amplitudes

Since the basis in Eq. III is constructed out of an orthonormal set of angular functions, the moments observables can be extracted out by weighting the data events by the angular functions themselves:

[TABLE]

The full description of the formalism to perform background subtraction and detector acceptance correction was provided in Ref. Dey (2015) and was successfully used to perform an angular analysis of the ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}{\mu^{-}}{\mu^{+}}$ system using the full Run I LHCb dataset in Ref. Aaij et al. (2016b).

For the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow{\mu^{-}}{\mu^{+}}$ mode, only the twenty moments in Table 2 contribute. Further, the overall normalization is set to unity by switching to the set of normalized moments $\overline{\Gamma}_{i}\equiv\Gamma_{i}/\Gamma_{0}$. Denoting $A_{\pm}\equiv a_{\pm}e^{i\alpha_{\pm}}$ and $B_{\pm}\equiv b_{\pm}e^{i\beta_{\pm}}$, and $\alpha_{\Lambda}$ as the weak-decay asymmetry, the magnitudes are solved as:

[TABLE]

and also

[TABLE]

These two sets of solutions yield the relations:

[TABLE]

from which both $P_{b}^{2}$ and $\alpha_{\Lambda}^{2}$ can be solved out. This is the essence of the moments technique, where no fit is required. The full set of the complex amplitudes $A_{\pm}$ and $B_{\pm}$ can be solved out from the moments, up to an overall sign ambiguity in $P_{b}$ and $\alpha_{\Lambda}$. Of course, if the sign of $\alpha_{\Lambda}$ is known from elsewhere, the ambiguity is broken and the sign of $P_{b}$ is also known.

VI Outlook and further work for higher spins

In the present work, we gave the prescription for performaing an angular analysis of ${{\mathchar 28931\relax}^{0}_{b}}$ decays in a manner that bypasses any relevant ambiguity issue. The moments calculations presented hold for generic spin-1/2 $\Lambda^{\ast}$ or $N^{\ast}$ hadronic final states. The next step would comprise extending the calculations (using the same formalism as in Sec. II.1) to spin-3/2 and 5/2 cases. The procedure is straightforward, but the explicit expansion and rearrangement of terms has to be performed; this is currently a work in progress and will be covered in a forthcoming paper. Existing quark model predictions Mott and Roberts (2015) for ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow\Lambda^{\ast}{\ell^{+}}{\ell^{-}}$ indicate that $J\leq 5/2$ would suffice for $m(pK)\leq 1900$ MeV. Given that most of the $\Lambda^{\ast}$ statistics lie in the low $m(pK)$ region, the calculation would enable an angular analysis for ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow p{{K}^{-}}{\mu^{+}}{\mu^{-}}$ in the high recoil region at LHCb.