New physics in the angular distribution of $B_{c}^{-}\to J/\psi(\to\mu^{+}\mu^{-})\tau^{-}(\to\pi^{-}\nu_{\tau})\bar{\nu}_{\tau}$ decay

Assuming that the NP scale is higher than the electroweak scale, one can integrate out the possible NP particles as well as the SM heavy particles — the $W^{\pm}$, $Z^{0}$, the top quark, and the Higgs boson, thus obtaining the effective Hamiltonian suitable for describing the $b\to c\tau^{-}\bar{\nu}_{\tau}$ transition²²2Neutrinos are assumed to be left-handed in this work. The effective Hamiltonian containing right-handed neutrinos can be found in refs. Dutta:2013qaa ; Ligeti:2016npd ; Mandal:2020htr . It can be derived from the identity $\sigma^{\mu\nu}\gamma_{5}=-\frac{i}{2}\epsilon^{\mu\nu\alpha\beta}\sigma_{\alpha\beta}$ that the operator $({\bar{c}}\sigma^{\mu\nu}(1+\gamma_{5})b)({\bar{\tau}}\sigma_{\mu\nu}\nu_{\tau L})$ is absent. We use the convention $\epsilon_{0123}=-\epsilon^{0123}=1$.

	$\displaystyle{\cal H}_{\rm eff}=\sqrt{2}G_{F}V_{cb}\big{[}$	$\displaystyle g_{V}({\bar{c}}\gamma^{\mu}b)({\bar{\tau}}\gamma_{\mu}\nu_{\tau L})+g_{A}({\bar{c}}\gamma^{\mu}\gamma_{5}b)({\bar{\tau}}\gamma_{\mu}\nu_{\tau L})$
		$\displaystyle+g_{S}({\bar{c}}b)({\bar{\tau}}\nu_{\tau L})+g_{P}({\bar{c}}\gamma_{5}b)({\bar{\tau}}\nu_{\tau L})$
		$\displaystyle+g_{T}({\bar{c}}\sigma^{\mu\nu}(1-\gamma_{5})b)({\bar{\tau}}\sigma_{\mu\nu}\nu_{\tau L})\big{]}+{\rm H.c.},$		(1)

where $G_{F}$ is the Fermi constant, $V_{cb}$ is the CKM matrix element, $\sigma^{\mu\nu}\equiv\frac{i}{2}[\gamma^{\mu},\,\gamma^{\nu}]$, and $\nu_{\tau L}=P_{L}\nu_{\tau}$ denotes the field of left-handed neutrino. The NP effects are encoded in the Wilson coefficients $g_{i}$, which are defined at the typical energy scale $\mu=m_{b}$. In the SM, $g_{V}=-g_{A}=1$ and $g_{S}=g_{P}=g_{T}=0$.

In the calculation, the hadronic matrix elements contain the nonperturbative QCD effects and can be parameterized as the Lorentz invariant form factors. The vector and axial-vector current matrix elements can be written as the following four form factors Beneke:2000wa ; Sakaki:2013bfa ; Harrison:2020gvo

	$\displaystyle\langle J/\psi(k,\varepsilon)|\bar{c}\gamma_{\mu}b|B_{c}(p)\rangle$	$\displaystyle=\frac{2iV(q^{2})}{m_{B_{c}}+m_{J/\psi}}\epsilon_{\mu\nu\alpha\beta}\varepsilon^{*\nu}k^{\alpha}p^{\beta},$		(2)
	$\displaystyle\langle J/\psi(k,\varepsilon)|\bar{c}\gamma_{\mu}\gamma_{5}b|B_{c}(p)\rangle$	$\displaystyle=2m_{J/\psi}A_{0}(q^{2})\frac{\varepsilon^{*}\cdot q}{q^{2}}q_{\mu}$
		$\displaystyle+(m_{B_{c}}+m_{J/\psi})A_{1}(q^{2})\left(\varepsilon^{*}_{\mu}-\frac{\varepsilon^{*}\cdot q}{q^{2}}q_{\mu}\right)$
		$\displaystyle-A_{2}(q^{2})\frac{\varepsilon^{*}\cdot q}{m_{B_{c}}+m_{J/\psi}}\left(p_{\mu}+k_{\mu}-\frac{m_{B_{c}}^{2}-m_{J/\psi}^{2}}{q^{2}}q_{\mu}\right),$		(3)

where $q=p-k$, $\varepsilon^{\mu}$ denotes the polarization vector of $J/\psi$ meson. In our numerical analysis, we will use the vector and axial-vector form factors computed in lattice QCD Harrison:2020gvo ; Harrison:2020nrv .

Using the equation of motion, the scalar and pseudo-scalar matrix elements can be obtained by

	$\displaystyle\langle J/\psi(k,\varepsilon)|\bar{c}b|B_{c}(p)\rangle$	$\displaystyle=0,$		(4)
	$\displaystyle\langle J/\psi(k,\varepsilon)|\bar{c}\gamma_{5}b|B_{c}(p)\rangle$	$\displaystyle=-\varepsilon^{*}\cdot q\frac{2m_{J/\psi}}{m_{b}+m_{c}}A_{0}(q^{2}),$		(5)

Based on the above four form factors $V(q^{2})$ and $A_{0,1,2}(q^{2})$, one can define four independent transversity amplitudes as follows

	$\displaystyle\mathcal{A}_{t}$	$\displaystyle=\mathcal{A}^{SP}_{t}+\frac{m_{\tau}}{\sqrt{q^{2}}}\mathcal{A}^{VA}_{t},$		(6)
	$\displaystyle\mathcal{A}_{0}$	$\displaystyle=g_{A}\frac{m_{B_{c}}+m_{J/\psi}}{2m_{J/\psi}\sqrt{q^{2}}}\left[A_{1}(q^{2})(m_{B_{c}}^{2}-m_{J/\psi}^{2}-q^{2})-A_{2}(q^{2})\frac{Q_{+}Q_{-}}{(m_{B_{c}}+m_{J/\psi})^{2}}\right],$		(7)
	$\displaystyle\mathcal{A}_{\perp}$	$\displaystyle=g_{A}\sqrt{2}A_{1}(q^{2})(m_{B_{c}}+m_{J/\psi}),$		(8)
	$\displaystyle\mathcal{A}_{\parallel}$	$\displaystyle=g_{V}\sqrt{2}V(q^{2})\frac{\sqrt{Q_{+}Q_{-}}}{m_{B_{c}}+m_{J/\psi}},$		(9)

with

$\displaystyle\mathcal{A}^{SP}_{t}=-g_{P}A_{0}(q^{2})\frac{\sqrt{Q_{+}Q_{-}}}{m_{b}+m_{c}},\ \mathcal{A}^{VA}_{t}=g_{A}A_{0}(q^{2})\frac{\sqrt{Q_{+}Q_{-}}}{\sqrt{q^{2}}},$

(10)

where $m_{b}$ and $m_{c}$ are the current quark masses evaluated at the scale $\mu=m_{b}$, and $Q_{\pm}\equiv(m_{B_{c}}\pm m_{J/\psi})^{2}-q^{2}$.

The tensor matrix element can be parameterized as Beneke:2000wa ; Sakaki:2013bfa ; Leljak:2019eyw

	$\displaystyle\langle J/\psi(k,\varepsilon)|\bar{c}\sigma_{\mu\nu}b|B_{c}(p)\rangle=$	$\displaystyle\epsilon_{\mu\nu\alpha\beta}\Bigg{\{}-\varepsilon^{*\alpha}(k+p)^{\beta}T_{1}(q^{2})$
		$\displaystyle+\frac{m_{B_{c}}^{2}-m_{J/\psi}^{2}}{q^{2}}\varepsilon^{*\alpha}q^{\beta}\left[T_{1}(q^{2})-T_{2}(q^{2})\right]$
		$\displaystyle+2\frac{\varepsilon^{*}\cdot q}{q^{2}}p^{\alpha}k^{\beta}\left[T_{1}(q^{2})-T_{2}(q^{2})-\frac{q^{2}}{m_{B_{c}}^{2}-m_{J/\psi}^{2}}T_{3}(q^{2})\right]\Bigg{\}},$		(11)

and $\langle J/\psi|\bar{c}\sigma_{\mu\nu}\gamma_{5}b|B_{c}\rangle=-\frac{i}{2}\epsilon_{\mu\nu\alpha\beta}\langle J/\psi|\bar{c}\sigma^{\alpha\beta}b|B_{c}\rangle$. In the presence of the tensor operators, we find three additional independent transversity amplitudes as follows

	$\displaystyle\mathcal{A}^{T}_{0}$	$\displaystyle=g_{T}\frac{1}{2m_{J/\psi}}\left[T_{2}(q^{2})(m_{B_{c}}^{2}+3m_{J/\psi}^{2}-q^{2})-T_{3}(q^{2})\frac{Q_{+}Q_{-}}{m_{B_{c}}^{2}-m_{J/\psi}^{2}}\right],$		(12)
	$\displaystyle\mathcal{A}^{T}_{\perp}$	$\displaystyle=g_{T}\sqrt{2}T_{2}(q^{2})\frac{m_{B_{c}}^{2}-m_{J/\psi}^{2}}{\sqrt{q^{2}}},$		(13)
	$\displaystyle\mathcal{A}^{T}_{\parallel}$	$\displaystyle=g_{T}\sqrt{2}T_{1}(q^{2})\frac{\sqrt{Q_{+}Q_{-}}}{\sqrt{q^{2}}},$		(14)

where the superscript $T$ indicates that an amplitude appears only when one considers the tensor operators.

The measurable angular distribution of the five-body $B_{c}^{-}\to J/\psi(\to\mu^{+}\mu^{-})\tau^{-}(\to\pi^{-}\nu_{\tau})\bar{\nu}_{\tau}$ decay can be written as

	$\displaystyle\frac{d^{5}\Gamma}{dq^{2}dE_{\pi}d\cos\theta_{\pi}d\phi_{\pi}d\cos\theta_{J/\psi}}=$	$\displaystyle\frac{3G_{F}^{2}\left|V_{cb}\right|^{2}\left|\bm{p}_{J/\psi}\right|\left(q^{2}\right)^{3/2}m_{\tau}^{2}}{256\pi^{4}m_{B_{c}}^{2}\left(m_{\tau}^{2}-m_{\pi}^{2}\right)^{2}}{\cal B}_{\tau}{\cal B}_{J/\psi}$
		$\displaystyle\times\mathcal{I}\left(q^{2},E_{\pi},\cos\theta_{J/\psi},\cos\theta_{\pi},\phi_{\pi}\right),$		(15)

where $\left|\bm{p}_{J/\psi}\right|=\sqrt{Q_{+}Q_{-}}/(2m_{B_{c}})$ denotes the magnitude of three-momentum of the $J/\psi$ meson in the $B_{c}$ rest frame. ${\cal B}_{\tau}\equiv{\cal B}\left(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\right)$ and ${\cal B}_{J/\psi}\equiv{\cal B}\left(J/\psi\rightarrow\mu^{-}\mu^{+}\right)$ are the branching fractions of $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}$ and $J/\psi\rightarrow\mu^{-}\mu^{+}$ decays, respectively. Here $q^{2}$ is the invariant mass squared of the $\tau^{-}\bar{\nu}_{\tau}$ pair; $\theta_{J/\psi}$ denotes the polar angle of $\mu^{-}$ in the $J/\psi$ rest frame; $E_{\pi}$, $\theta_{\pi}$, and $\phi_{\pi}$ represent the energy, polar angle, and azimuthal angle of $\pi^{-}$ in the $\tau^{-}\bar{\nu}_{\tau}$ center-of-mass frame, respectively. A more intuitive definition of the angles is shown in figure 1. The function $\mathcal{I}\left(q^{2},E_{\pi},\cos\theta_{J/\psi},\cos\theta_{\pi},\phi_{\pi}\right)$ can be decomposed into a set of trigonometric functions as follows

	$\displaystyle\mathcal{I}\left(q^{2},E_{\pi},\cos\theta_{J/\psi},\cos\theta_{\pi},\phi_{\pi}\right)=$	$\displaystyle\sum_{i=1}^{12}\mathcal{I}_{i}\left(q^{2},E_{\pi}\right)\Omega_{i}\left(\cos\theta_{J/\psi},\cos\theta_{\pi},\phi_{\pi}\right)$
	$\displaystyle\equiv$	$\displaystyle\mathcal{I}_{1c}\cos^{2}\theta_{J/\psi}+\mathcal{I}_{1s}\sin^{2}\theta_{J/\psi}$
		$\displaystyle+\left(\mathcal{I}_{2c}\cos^{2}\theta_{J/\psi}+\mathcal{I}_{2s}\sin^{2}\theta_{J/\psi}\right)\cos 2\theta_{\pi}$
		$\displaystyle+\left(\mathcal{I}_{6c}\cos^{2}\theta_{J/\psi}+\mathcal{I}_{6s}\sin^{2}\theta_{J/\psi}\right)\cos\theta_{\pi}$
		$\displaystyle+\left(\mathcal{I}_{3}\cos 2\phi_{\pi}+\mathcal{I}_{9}\sin 2\phi_{\pi}\right)\sin^{2}\theta_{\pi}\sin^{2}\theta_{J/\psi}$
		$\displaystyle+\left(\mathcal{I}_{4}\cos\phi_{\pi}+\mathcal{I}_{8}\sin\phi_{\pi}\right)\sin 2\theta_{\pi}\sin 2\theta_{J/\psi}$
		$\displaystyle+\left(\mathcal{I}_{5}\cos\phi_{\pi}+\mathcal{I}_{7}\sin\phi_{\pi}\right)\sin\theta_{\pi}\sin 2\theta_{J/\psi}.$		(16)

The twelve angular observables $\mathcal{I}_{i}(q^{2},E_{\pi})$ can be completely expressed in terms of the seven transversity amplitudes defined in subsection 2.2 and the dimensionless factors listed in appendix A.3. Explicitly, we have

	$\displaystyle\mathcal{I}_{1c}=$	$\displaystyle S_{1}\left|\mathcal{A}_{\perp}\right|^{2}-2R_{1}{\rm Re}\left[\mathcal{A}_{\perp}\mathcal{A}_{\perp}^{T*}\right]+4S_{1}^{T}\left|\mathcal{A}_{\perp}^{T}\right|^{2}+(\perp\leftrightarrow\parallel),$		(17)
	$\displaystyle\mathcal{I}_{1s}=$	$\displaystyle S_{t}\left|\mathcal{A}_{t}\right|^{2}+(S_{1}-S_{3})\left|\mathcal{A}_{0}\right|^{2}-2(R_{1}-R_{3}){\rm Re}\left[\mathcal{A}_{0}\mathcal{A}_{0}^{T*}\right]+4(S^{T}_{1}-S^{T}_{3})\left|\mathcal{A}_{0}^{T}\right|^{2}$
		$\displaystyle+\frac{1}{2}\left\{S_{1}\left|\mathcal{A}_{\perp}\right|^{2}-2R_{1}{\rm Re}\left[\mathcal{A}_{\perp}\mathcal{A}_{\perp}^{T*}\right]+4S_{1}^{T}\left|\mathcal{A}_{\perp}^{T}\right|^{2}+(\perp\leftrightarrow\parallel)\right\},$		(18)
	$\displaystyle\mathcal{I}_{2c}=$	$\displaystyle S_{3}\left|\mathcal{A}_{\perp}\right|^{2}-2R_{3}{\rm Re}\left[\mathcal{A}_{\perp}\mathcal{A}_{\perp}^{T*}\right]+4S^{T}_{3}\left|\mathcal{A}_{\perp}^{T}\right|^{2}+(\perp\leftrightarrow\parallel),$		(19)
	$\displaystyle\mathcal{I}_{2s}=$	$\displaystyle-2S_{3}\left|\mathcal{A}_{0}\right|^{2}+4R_{3}{\rm Re}\left[\mathcal{A}_{0}\mathcal{A}_{0}^{T*}\right]-8S^{T}_{3}\left|\mathcal{A}_{0}^{T}\right|^{2}$
		$\displaystyle+\frac{1}{2}\left\{S_{3}\left|\mathcal{A}_{\perp}\right|^{2}-2R_{3}{\rm Re}\left[\mathcal{A}_{\perp}\mathcal{A}_{\perp}^{T*}\right]+4S^{T}_{3}\left|\mathcal{A}_{\perp}^{T}\right|^{2}+(\perp\leftrightarrow\parallel)\right\},$		(20)
	$\displaystyle\mathcal{I}_{6c}=$	$\displaystyle 2\mathrm{Re}\left[S_{2}\mathcal{A}_{\parallel}\mathcal{A}_{\perp}^{*}-R_{2}\left(\mathcal{A}_{\perp}\mathcal{A}_{\parallel}^{T*}+\mathcal{A}_{\parallel}\mathcal{A}_{\perp}^{T*}\right)+4S_{2}^{T}\mathcal{A}_{\parallel}^{T}\mathcal{A}_{\perp}^{T*}\right],$		(21)
	$\displaystyle\mathcal{I}_{6s}=$	$\displaystyle\mathrm{Re}\Big{[}S_{2}\mathcal{A}_{\parallel}\mathcal{A}_{\perp}^{*}-R_{2}\left(\mathcal{A}_{\perp}\mathcal{A}_{\parallel}^{T*}+\mathcal{A}_{\parallel}\mathcal{A}_{\perp}^{T*}\right)+4S_{2}^{T}\mathcal{A}_{\parallel}^{T}\mathcal{A}_{\perp}^{T*}$
		$\displaystyle-\sqrt{2}R_{t}\mathcal{A}_{t}\mathcal{A}_{0}^{*}+2\sqrt{2}R_{t}^{T}\mathcal{A}_{t}\mathcal{A}_{0}^{T*}\Big{]},$		(22)
	$\displaystyle\mathcal{I}_{3}=$	$\displaystyle S_{3}\left|\mathcal{A}_{\perp}\right|^{2}-2R_{3}\mathrm{Re}\left[\mathcal{A}_{\perp}\mathcal{A}_{\perp}^{T*}\right]+4S^{T}_{3}\left|\mathcal{A}_{\perp}^{T}\right|^{2}-(\perp\leftrightarrow\parallel),$		(23)
	$\displaystyle\mathcal{I}_{9}=$	$\displaystyle 2\mathrm{Im}\left[S_{3}\mathcal{A}_{\parallel}\mathcal{A}_{\perp}^{*}+R_{3}\left(\mathcal{A}_{\perp}\mathcal{A}_{\parallel}^{T*}-\mathcal{A}_{\parallel}\mathcal{A}_{\perp}^{T*}\right)+4S^{T}_{3}\mathcal{A}_{\parallel}^{T}\mathcal{A}_{\perp}^{T*}\right],$		(24)
	$\displaystyle\mathcal{I}_{4}=$	$\displaystyle\sqrt{2}\mathrm{Re}\left[S_{3}\mathcal{A}_{\perp}\mathcal{A}_{0}^{*}-R_{3}\left(\mathcal{A}_{0}\mathcal{A}_{\perp}^{T*}+\mathcal{A}_{\perp}\mathcal{A}_{0}^{T*}\right)+4S^{T}_{3}\mathcal{A}_{0}^{T}\mathcal{A}_{\perp}^{T*}\right],$		(25)
	$\displaystyle\mathcal{I}_{8}=$	$\displaystyle\sqrt{2}\mathrm{Im}\left[S_{3}\mathcal{A}_{\parallel}\mathcal{A}_{0}^{*}+R_{3}\left(\mathcal{A}_{0}\mathcal{A}_{\parallel}^{T*}-\mathcal{A}_{\parallel}\mathcal{A}_{0}^{T*}\right)+4S^{T}_{3}\mathcal{A}_{\parallel}^{T}\mathcal{A}_{0}^{T*}\right],$		(26)
	$\displaystyle\mathcal{I}_{5}=$	$\displaystyle\frac{1}{2\sqrt{2}}\mathrm{Re}\Big{[}2S_{2}\mathcal{A}_{\parallel}\mathcal{A}_{0}^{*}-2R_{2}\left(\mathcal{A}_{0}\mathcal{A}_{\parallel}^{T*}+\mathcal{A}_{\parallel}\mathcal{A}_{0}^{T*}\right)+8S_{2}^{T}\mathcal{A}_{\parallel}^{T}\mathcal{A}_{0}^{T*}$
		$\displaystyle+\sqrt{2}R_{t}\mathcal{A}_{t}\mathcal{A}_{\perp}^{*}-2\sqrt{2}R_{t}^{T}\mathcal{A}_{t}\mathcal{A}_{\perp}^{T*}\Big{]},$		(27)
	$\displaystyle\mathcal{I}_{7}=$	$\displaystyle\frac{1}{2\sqrt{2}}\mathrm{Im}\Big{[}2S_{2}\mathcal{A}_{\perp}\mathcal{A}_{0}^{*}+2R_{2}\left(\mathcal{A}_{0}\mathcal{A}_{\perp}^{T*}-\mathcal{A}_{\perp}\mathcal{A}_{0}^{T*}\right)-8S_{2}^{T}\mathcal{A}_{0}^{T}\mathcal{A}_{\perp}^{T*}$
		$\displaystyle-\sqrt{2}R_{t}\mathcal{A}_{t}\mathcal{A}_{\parallel}^{*}+2\sqrt{2}R_{t}^{T}\mathcal{A}_{t}\mathcal{A}_{\parallel}^{T*}\Big{]}.$		(28)

In the SM, the angular observables $\mathcal{I}_{7}$, $\mathcal{I}_{8}$, and $\mathcal{I}_{9}$ are vanishing. Therefore, in future measurements, a non-vanishing $\mathcal{I}_{7}$, $\mathcal{I}_{8}$, or $\mathcal{I}_{9}$ would be a solid signal of NP, which induces a complex contribution to the amplitude.

The differential decay rate (2.3) depends on five parameters $q^{2}$, $E_{\pi}$, $\theta_{J/\psi}$, $\theta_{\pi}$ and $\phi_{\pi}$, and a complete experimental analysis may be limited by statistics. Integrating over the $E_{\pi}$ and after a proper normalization, we can get the following angular function

	$\displaystyle\widehat{\mathcal{I}}\left(q^{2},\cos\theta_{J/\psi},\cos\theta_{\pi},\phi_{\pi}\right)\equiv$	$\displaystyle\frac{\int{\frac{d^{5}\Gamma}{dq^{2}dE_{\pi}d\cos\theta_{\pi}d\phi_{\pi}d\cos\theta_{J/\psi}}}dE_{\pi}}{\frac{d\Gamma}{dq^{2}}}$
	$\displaystyle=$	$\displaystyle\frac{9}{8\pi}\sum_{i=1}^{12}\widehat{\mathcal{I}}_{i}\left(q^{2}\right)\Omega_{i}\left(\cos\theta_{J/\psi},\cos\theta_{\pi},\phi_{\pi}\right),$		(29)

with the twelve normalized angular observables $\widehat{\mathcal{I}_{i}}(q^{2})$ defined as

$\displaystyle\widehat{\mathcal{I}}_{i}\left(q^{2}\right)\equiv\frac{\int{\mathcal{I}_{i}\left(q^{2},E_{\pi}\right)}dE_{\pi}}{\int{\left(3\mathcal{I}_{1c}-\mathcal{I}_{2c}+6\mathcal{I}_{1s}-2\mathcal{I}_{2s}\right)}dE_{\pi}}.$

(30)

Our choice of the normalization in eq. (3.1) results the relationship $3\widehat{\mathcal{I}}_{1c}\left(q^{2}\right)-\widehat{\mathcal{I}}_{2c}\left(q^{2}\right)+6\widehat{\mathcal{I}}_{1s}\left(q^{2}\right)-2\widehat{\mathcal{I}}_{2s}\left(q^{2}\right)=1$. The cancellations through normalization to the decay rate lead to the observation that the observables $\widehat{\mathcal{I}_{i}}(q^{2})$ have less theoretical uncertainty to facilitate the discussion of the NP effects. In section 4, we will analyze numerically the entire set of observables $\widehat{\mathcal{I}_{i}}(q^{2})$ within the SM and in some NP benchmark points.

The forward-backward asymmetry of $\pi^{-}$ meson as a function of $q^{2}$ can be defined as

	$\displaystyle A_{FB}(q^{2})\equiv$	$\displaystyle\frac{\int_{0}^{1}\frac{d^{2}\Gamma}{dq^{2}d\cos\theta_{\pi}}d\cos\theta_{\pi}-\int_{-1}^{0}\frac{d^{2}\Gamma}{dq^{2}d\cos\theta_{\pi}}d\cos\theta_{\pi}}{\frac{d\Gamma}{dq^{2}}}$
	$\displaystyle=$	$\displaystyle\frac{3}{2}\left(\widehat{\mathcal{I}}_{6c}+2\widehat{\mathcal{I}}_{6s}\right).$		(31)

This asymmetry observable only exists in $\tau$ channel, and specifically for the $\tau^{-}\to\pi^{-}\nu_{\tau}$ decay. Obviously, it can be expressed linearly in terms of angular observables $\widehat{\mathcal{I}_{i}}(q^{2})$.

By integrating over the lepton-side parameters $E_{\pi}$, $\theta_{\pi}$, $\phi_{\pi}$, one can obtain the two-fold differential decay rate as follows

$\displaystyle\frac{d^{2}\Gamma}{dq^{2}d\cos\theta_{J/\psi}}=\frac{3}{8}\frac{d\Gamma}{dq^{2}}\left[2P_{L}^{J/\psi}\left(q^{2}\right)\sin^{2}\theta_{J/\psi}+P_{T}^{J/\psi}\left(q^{2}\right)\left(1+\cos^{2}\theta_{J/\psi}\right)\right],$

(32)

where

$\displaystyle P_{L}^{J/\psi}\left(q^{2}\right)\equiv\frac{d\Gamma_{L}/dq^{2}}{d\Gamma_{L}/dq^{2}+d\Gamma_{T}/dq^{2}},\quad P_{T}^{J/\psi}\left(q^{2}\right)\equiv\frac{d\Gamma_{T}/dq^{2}}{d\Gamma_{L}/dq^{2}+d\Gamma_{T}/dq^{2}},$

(33)

are the longitudinal and transverse polarization fractions of the $J/\psi$ meson, respectively. The differential decay rates for the longitudinally and transversely polarized intermediate state $J/\psi$ are given, respectively, by

	$\displaystyle\frac{d\Gamma_{L}}{dq^{2}}\equiv$	$\displaystyle\frac{d\Gamma^{\lambda_{J/\psi}=0}}{dq^{2}}$
	$\displaystyle=$	$\displaystyle\mathcal{N}\bigg{\{}3q^{2}\left|\mathcal{A}_{t}\right|^{2}+\left(m_{\tau}^{2}+2q^{2}\right)\left|\mathcal{A}_{0}\right|^{2}$
		$\displaystyle+24m_{\tau}\sqrt{q^{2}}\mathrm{Re}\left[\mathcal{A}_{0}\mathcal{A}_{0}^{T*}\right]+16\left(2m_{\tau}^{2}+q^{2}\right)\left|\mathcal{A}_{0}^{T}\right|^{2}\bigg{\}},$		(34)
	$\displaystyle\frac{d\Gamma_{T}}{dq^{2}}\equiv$	$\displaystyle\frac{d\Gamma^{\lambda_{J/\psi}=+1}}{dq^{2}}+\frac{d\Gamma^{\lambda_{J/\psi}=-1}}{dq^{2}}$
	$\displaystyle=$	$\displaystyle\mathcal{N}\bigg{\{}\left(m_{\tau}^{2}+2q^{2}\right)\left|\mathcal{A}_{\perp}\right|^{2}+24m_{\tau}\sqrt{q^{2}}\mathrm{Re}\left[\mathcal{A}_{\perp}\mathcal{A}_{\perp}^{T*}\right]$
		$\displaystyle+16\left(2m_{\tau}^{2}+q^{2}\right)\left|\mathcal{A}_{\perp}^{T}\right|^{2}+(\perp\leftrightarrow\parallel)\bigg{\}},$		(35)

with the factor

$\displaystyle\mathcal{N}\equiv\frac{G_{F}^{2}\left|V_{cb}\right|^{2}\left|\bm{p}_{J/\psi}\right|}{192\pi^{3}m_{B_{c}}^{2}}\left(1-\frac{m_{\tau}^{2}}{q^{2}}\right)^{2}\mathcal{B}_{\tau}\mathcal{B}_{J/\psi}.$

(36)

The polarization observables $P_{L,T}^{J/\psi}(q^{2})$ constructed above are not affected by $\tau$ decay dynamics since we have integrated over all the lepton-side kinematic parameters, so they are also applicable to light leptons $\mu$ and $e$. We denote $\left[P_{L,T}^{J/\psi}\right]_{\tau}$ and $\left[P_{L,T}^{J/\psi}\right]_{\mu}$ as extraction from $B_{c}\to J/\psi\tau\nu$ and $B_{c}\to J/\psi\mu\nu$ decays respectively, and define the following ratios to probe the universality of lepton flavor

$$R\left(P_{L,T}^{J/\psi}\right)\equiv\frac{\left[P_{L,T}^{J/\psi}\right]_{\tau}}{\left[P_{L,T}^{J/\psi}\right]_{\mu}}.$$

(37)

The $q^{2}$ distribution of the decay rate can be obtained by adding up eqs. (34) and (35) as follows

$\displaystyle\frac{d\Gamma}{dq^{2}}=\frac{d\Gamma_{L}}{dq^{2}}+\frac{d\Gamma_{T}}{dq^{2}}.$

(38)

Our $d\Gamma/dq^{2}$ (apart from ${\cal B}_{\tau}\mathcal{B}_{J/\psi}$) is consistent with that in refs. Harrison:2020nrv ; Sakaki:2013bfa .

After integrating over the variables $\theta_{J/\psi}$ and $\phi_{\pi}$, one has

	$\displaystyle\frac{d^{3}\Gamma}{dq^{2}dE_{\pi}d\cos\theta_{\pi}}=$	$\displaystyle\frac{G_{F}^{2}\left|V_{cb}\right|^{2}\left|\bm{p}_{J/\psi}\right|\left(q^{2}\right)^{3/2}m_{\tau}^{2}}{64\pi^{3}m_{B_{c}}^{2}\left(m_{\tau}^{2}-m_{\pi}^{2}\right)^{2}}{\cal B}_{\tau}{\cal B}_{J/\psi}$
		$\displaystyle\times\left[\mathcal{I}_{1c}+2\mathcal{I}_{1s}+\left(\mathcal{I}_{6c}+2\mathcal{I}_{6s}\right)\cos\theta_{\pi}+\left(\mathcal{I}_{2c}+2\mathcal{I}_{2s}\right)\cos 2\theta_{\pi}\right].$		(39)

This three-fold differential decay rate can be used to indirectly reveal the information of $\tau$ asymmetries in $B_{c}^{-}\to J/\psi\tau^{-}\bar{\nu}_{\tau}$ decay Kiers:1997zt ; Nierste:2008qe ; Tanaka:2010se ; Sakaki:2012ft ; Ivanov:2017mrj ; Alonso:2017ktd ; Asadi:2020fdo . The following discussion of this subsection will follow closely refs. Alonso:2017ktd ; Asadi:2020fdo , which give a detailed analysis of the $\tau$ properties in $B\to D^{(*)}\tau^{-}\bar{\nu}_{\tau}$ decays using $\tau\to d\nu(d=\pi,\,\rho)$.