Observables in Semileptonic Decays

EOS predicts a large number of observables for exclusive semileptonic decays, i.e. decays that proceed through a charged-current transition of the type described by the semileptonic charged-current operators. The available observables depend on the spin of the initial- and final-state hadrons. This section summarises the observables that EOS provides for three common cases: the decay of a pseudoscalar meson to a pseudoscalar meson (\(P\)), the decay of a pseudoscalar meson to a vector meson (\(V\)), and the decay of a spin-\(1/2\) baryon to a spin-\(1/2\) baryon. All three cases are built from the corresponding semileptonic form factors.

Throughout, \(q^2\) denotes the invariant mass squared of the lepton-neutrino pair, and the physical \(q^2\) range extends from \(m_\ell^2\) to \((M_1 - M_2)^2\), where \(M_1\) and \(M_2\) are the masses of the initial- and final-state hadron. For each decay, EOS provides observables that are

fully differential in \(q^2\) and the decay angles;
single differential in \(q^2\), obtained by integrating over the angles; and
integrated over a \(q^2\) interval \([q^2_\text{min}, q^2_\text{max}]\).

In addition, EOS provides a set of normalised observables, in which the dependence on the CKM matrix element \(|V_{UD}|\) is removed (i.e. \(|V_{UD}| = 1\)), and a set of lepton-flavour-universality (LFU) ratios.

Decays to a Pseudoscalar Meson

For a decay \(P_1 \to P_2\, \ell\, \bar\nu\) to a pseudoscalar meson, the kinematics are described by \(q^2\) and a single angle: the helicity angle \(\theta_\ell\) of the charged lepton, defined in the rest frame of the lepton-neutrino pair. The fully differential decay rate is a second-order polynomial in \(\cos\theta_\ell\):

\begin{equation*} \frac{d^2\Gamma}{dq^2\, d\cos\theta_\ell} = a_\ell(q^2) + b_\ell(q^2) \cos\theta_\ell + c_\ell(q^2) \cos^2\theta_\ell\,. \end{equation*}

From this distribution EOS derives, as functions of \(q^2\) and as quantities integrated over a \(q^2\) bin:

the differential decay width \(d\Gamma/dq^2\) and branching ratio \(d\mathcal{B}/dq^2\) (e.g. B->pilnu::dBR/dq2), and the integrated branching ratio (B->pilnu::BR);
the leptonic forward-backward asymmetry \(A_\text{FB}(q^2)\) (B->pilnu::A_FB(q2)), which is sensitive to \(b_\ell(q^2)\);
the flat term \(F_H\) (B->pilnu::F_H), which quantifies the angular-independent contribution to the distribution and vanishes in the massless-lepton limit; and
the longitudinal lepton-polarisation asymmetry \(A_\lambda^\ell\) (B->pilnu::A_l).

LFU ratios compare the rates into different lepton flavours, for example

\begin{equation*} R_{P} = \frac{\mathcal{B}(P_1 \to P_2\, \tau\, \bar\nu)}{\mathcal{B}(P_1 \to P_2\, \ell\, \bar\nu)}\,, \qquad \ell = e, \mu\,, \end{equation*}

which EOS exposes both integrated (B->pilnu::R_pi) and differential in \(q^2\) (B->pilnu::R_pi(q2)).

Decays to a Vector Meson

For a decay \(P \to V(\to P_1 P_2)\, \ell\, \bar\nu\) to a vector meson, the subsequent decay of the vector meson makes the full angular distribution accessible. The kinematics are described by \(q^2\) and three angles: the helicity angle \(\theta_\ell\) of the charged lepton, the helicity angle \(\theta_V\) of the vector meson (defined through its decay products), and the azimuthal angle \(\phi\) between the two decay planes. The fully differential decay rate reads

\begin{align*} \frac{d^4\Gamma}{dq^2\, d\cos\theta_\ell\, d\cos\theta_V\, d\phi} = \frac{9}{32\pi} \Big[ & \left(J_{1s} \sin^2\theta_V + J_{1c} \cos^2\theta_V\right) + \left(J_{2s} \sin^2\theta_V + J_{2c} \cos^2\theta_V\right) \cos 2\theta_\ell \\ & + J_3 \sin^2\theta_V \sin^2\theta_\ell \cos 2\phi + J_4 \sin 2\theta_V \sin 2\theta_\ell \cos\phi + J_5 \sin 2\theta_V \sin\theta_\ell \cos\phi \\ & + \left(J_{6s} \sin^2\theta_V + J_{6c} \cos^2\theta_V\right) \cos\theta_\ell + J_7 \sin 2\theta_V \sin\theta_\ell \sin\phi \\ & + J_8 \sin 2\theta_V \sin 2\theta_\ell \sin\phi + J_9 \sin^2\theta_V \sin^2\theta_\ell \sin 2\phi \Big]\,, \end{align*}

where the twelve angular coefficient functions \(J_i \equiv J_i(q^2)\) encode the full information of the decay. EOS provides each of the \(J_i\) both differentially in \(q^2\) (e.g. B->D^*lnu::J_1c(q2)) and integrated over a \(q^2\) bin (e.g. B->D^*lnu::J_1c).

From the angular coefficients EOS derives further observables, as functions of \(q^2\) and integrated over a \(q^2\) bin, including:

the differential decay width \(d\Gamma/dq^2\) and branching ratio \(d\mathcal{B}/dq^2\) (e.g. B->D^*lnu::dBR/dq2), and the integrated branching ratio (B->D^*lnu::BR);
the leptonic forward-backward asymmetry \(A_\text{FB}(q^2)\) (B->D^*lnu::A_FB(q2));
the longitudinal polarisation fraction \(F_\text{L}\) of the vector meson (B->D^*lnu::F_L); and
a set of angular and CP asymmetries \(A_\text{C}^{1,2,3}\) and \(A_\text{T}^{1,2,3}\) (e.g. B->D^*lnu::A_C^1, B->D^*lnu::A_T^1).

As in the pseudoscalar case, EOS provides LFU ratios, for example

\begin{equation*} R_{V} = \frac{\mathcal{B}(P \to V\, \tau\, \bar\nu)}{\mathcal{B}(P \to V\, \ell\, \bar\nu)}\,, \qquad \ell = e, \mu\,, \end{equation*}

available both integrated (B->D^*lnu::R_D^*) and differential in \(q^2\) (B->D^*lnu::R_{D^*}^{tau/mu}(q2)).

Decays to a Spin-1/2 Baryon

For the decay of a spin-parity \(J^P = 1/2^+\) baryon to another \(1/2^+\) baryon, \(\mathcal{B}_1 \to \mathcal{B}_2\, \ell\, \bar\nu\) (e.g. \(\Lambda_b \to \Lambda_c\, \ell\, \bar\nu\)), the final-state baryon is itself unstable. Its subsequent decay \(\mathcal{B}_2 \to \mathcal{B}_3\, \pi\) (e.g. \(\Lambda_c \to \Lambda\, \pi\)) acts as a polarisation analyser and thereby makes the full angular distribution accessible. The kinematics are described by \(q^2\) and three angles: the helicity angle \(\theta_\ell\) of the charged lepton (in the lepton-neutrino rest frame), the helicity angle \(\theta_\mathcal{B}\) of the daughter baryon (defined through \(\mathcal{B}_2 \to \mathcal{B}_3\, \pi\)), and the azimuthal angle \(\phi\) between the two decay planes. Assuming the initial-state baryon \(\mathcal{B}_1\) to be unpolarised, the fully differential decay rate reads

\begin{align*} \frac{d^4\Gamma}{dq^2\, d\cos\theta_\ell\, d\cos\theta_\mathcal{B}\, d\phi} = \frac{3}{8\pi} \Big[ & \left(K_{1ss} \sin^2\theta_\ell + K_{1cc} \cos^2\theta_\ell + K_{1c} \cos\theta_\ell\right) \\ & + \left(K_{2ss} \sin^2\theta_\ell + K_{2cc} \cos^2\theta_\ell + K_{2c} \cos\theta_\ell\right) \cos\theta_\mathcal{B} \\ & + \left(K_{3sc} \sin\theta_\ell \cos\theta_\ell + K_{3s} \sin\theta_\ell\right) \sin\theta_\mathcal{B} \sin\phi \\ & + \left(K_{4sc} \sin\theta_\ell \cos\theta_\ell + K_{4s} \sin\theta_\ell\right) \sin\theta_\mathcal{B} \cos\phi \Big]\,, \end{align*}

where the ten angular coefficient functions \(K_i \equiv K_i(q^2)\) encode the full information of the decay. A non-zero polarisation of the initial-state baryon would introduce additional angular structures, and hence further angular coefficients, beyond these ten. EOS provides the \(K_i\) integrated over a \(q^2\) bin and normalised to the decay width (e.g. Lambda_b->Lambda_clnu::K_1ss).

From the angular coefficients EOS derives further observables, as functions of \(q^2\) and integrated over a \(q^2\) bin, including:

the differential decay width and branching ratio \(d\mathcal{B}/dq^2\) (Lambda_b->Lambda_clnu::dBR/dq2), and the integrated branching ratio (Lambda_b->Lambda_clnu::BR);
the leptonic forward-backward asymmetry \(A_\text{FB}^\ell\) (Lambda_b->Lambda_clnu::A_FB^l(q2));
the hadronic forward-backward asymmetry \(A_\text{FB}^h\) (Lambda_b->Lambda_clnu::A_FB^h(q2)), which arises from the polarisation of the daughter baryon;
the combined lepton-hadron forward-backward asymmetry \(A_\text{FB}^{h\ell}\) (Lambda_b->Lambda_clnu::A_FB^c(q2)); and
the fraction \(F_0\) (Lambda_b->Lambda_clnu::F_0(q2)).

As in the meson cases, EOS provides LFU ratios, for example

\begin{equation*} R_{\mathcal{B}_2} = \frac{\mathcal{B}(\mathcal{B}_1 \to \mathcal{B}_2\, \tau\, \bar\nu)}{\mathcal{B}(\mathcal{B}_1 \to \mathcal{B}_2\, \ell\, \bar\nu)}\,, \qquad \ell = e, \mu\,, \end{equation*}

available as an integrated observable (Lambda_b->Lambda_clnu::R(Lambda_c)).