Hadronic Matrix Elements of Local Operators

EOS uses the following conventions for the hadronic matrix elements of local operators, i.e. the decay constants and the semileptonic form factors.

Definition of Decay Constants

EOS uses meson to vacuum elements for the decay of both pseudoscalar (\(P\)) and vector (\(V\)) mesons. The decay constants are defined as:

\begin{align*} \braket{0 | \bar{q}_1 \gamma^\mu \gamma_5 q_2 | P(p)} & = i f_P p^\mu, \\ \braket{0 | \bar{q}_1 \gamma^\mu q_2 | V(p, \epsilon)} & = f_V m_V \epsilon^\mu, & \braket{0 | \bar{q}_1 \sigma^{\mu\nu} q_2 | V(p, \epsilon)} & = i f_V^T \left( \epsilon^\mu p^\nu - \epsilon^\nu p^\mu \right), \end{align*}

Here \(q_1\) and \(q_2\) are the quark fields, \(p\) is the momentum of the initial-state meson, and \(\epsilon\) is the polarization vector of the vector meson.

The parameters describing these decay constants use the prefix decay-constant.

Definition of Semileptonic Form Factors

EOS uses the following definitions for semileptonic form factors.

In the case of the decay of a pseudoscalar meson (\(P_1\)) to another pseudoscalar meson (\(P_2\)), the form factors are defined as:

\begin{align*} \braket{P_2(k) | \bar{q}_1 \gamma^\mu q_2 | P_1(p)} & = f_+^{P_1 \to P_2}(q^2) \left[ \left(p + k\right)^\mu - \frac{M_{P_1}^2 - M_{P_2}^2}{q^2} q^\mu\right] + f_0^{P_1 \to P_2}(q^2) \frac{M_{P_1}^2 - M_{P_2}^2}{q^2} q^\mu\,, \\ \braket{P_2(k) | \bar{q}_1 \sigma^{\mu\nu} q_2 | P_1(p)} & = \frac{i f_T^{P_1 \to P_2}(q^2)}{M_{P_1} + M_{P_2}} \left[ (p + k)^\mu q^\nu - q^\mu (p + k)^\nu\right]\,. \end{align*}

Here, \(q = p - k\) is the momentum transfer.

In the case of the decay of a pseudoscalar meson (\(P\)) to a vector meson (\(V\)), the form factors are defined as:

\begin{align*} \braket{V(k, \eta) | \bar{q}_1 \gamma^\mu q_2 | P(p)} & = \frac{2 V^{P \to V}(q^2)}{M_P + M_V} \varepsilon^{\mu\nu\alpha\beta} \eta^*_\nu p_\alpha k_\beta\,, \\ \braket{V(k, \eta) | \bar{q}_1 \gamma^\mu \gamma_5 q_2 | P(p)} & = i \eta_\nu^* \left[ A_1^{P \to V}(q^2) (M_P + M_V) g^{\mu\nu} - A_2^{P \to V}(q^2) \frac{(p + k)^\mu q_\nu}{M_P + M_V} - (A_3 - A_0) \frac{2 M_V q^\mu q^\nu}{q^2}\right]\,, \\ \braket{V(k, \eta) | \bar{q}_1 \sigma^{\mu\nu} q_2 | P(p)} & = 2 T_1^{P \to V} \varepsilon^{\mu\nu\alpha\beta} \eta^*_\nu p_\alpha k_\beta\,, \\ \braket{V(k, \eta) | \bar{q}_1 \sigma^{\mu\nu} \gamma_5 q_2 | P(p)} & = i T_2^{P \to V} \left[ \eta^{\mu*} (p + k)^\nu - \eta^{\nu*} (p + k)^\mu \right] + \frac{i T_3^{P \to V}}{M_P^2 - M_V^2} (\eta^* \cdot q) \left[q^\mu (p+k)^\nu - q^\nu (p+k)^\mu\right]\,, \end{align*}

where we abbreviate:

\begin{equation*} A_3^{P \to V}(q^2) = A_1^{P \to V}(q^2) \frac{M_P + M_V}{2 M_V} - A_2^{P \to V}(q^2) \frac{M_P - M_V}{2 M_V}\,. \end{equation*}

Here, \(\eta\) is the polarization vector of the vector meson, and again \(q = p - k\) is the momentum transfer.

In the case of the decay of a spin-parity \(J^P = 1/2^+\) baryon (\(\mathcal{B}_1\)) to another \(1/2^+\) baryon (\(\mathcal{B}_2\)), EOS uses the following helicity-based definitions. Abbreviating \(s_\pm \equiv (M_{\mathcal{B}_1} \pm M_{\mathcal{B}_2})^2 - q^2\), the vector and axialvector matrix elements are defined as:

\begin{align*} \braket{\mathcal{B}_2(k, s') | \bar{q}_1 \gamma^\mu q_2 | \mathcal{B}_1(p, s)} & = \bar{u}(k, s') \bigg[ f_t^V(q^2)\, (M_{\mathcal{B}_1} - M_{\mathcal{B}_2}) \frac{q^\mu}{q^2} + f_0^V(q^2)\, \frac{M_{\mathcal{B}_1} + M_{\mathcal{B}_2}}{s_+} \left( (p + k)^\mu - \frac{M_{\mathcal{B}_1}^2 - M_{\mathcal{B}_2}^2}{q^2} q^\mu \right) \\ & \qquad\qquad + f_\perp^V(q^2) \left( \gamma^\mu - \frac{2 M_{\mathcal{B}_2}}{s_+} p^\mu - \frac{2 M_{\mathcal{B}_1}}{s_+} k^\mu \right) \bigg] u(p, s)\,, \\ \braket{\mathcal{B}_2(k, s') | \bar{q}_1 \gamma^\mu \gamma_5 q_2 | \mathcal{B}_1(p, s)} & = -\bar{u}(k, s')\, \gamma_5 \bigg[ f_t^A(q^2)\, (M_{\mathcal{B}_1} + M_{\mathcal{B}_2}) \frac{q^\mu}{q^2} + f_0^A(q^2)\, \frac{M_{\mathcal{B}_1} - M_{\mathcal{B}_2}}{s_-} \left( (p + k)^\mu - \frac{M_{\mathcal{B}_1}^2 - M_{\mathcal{B}_2}^2}{q^2} q^\mu \right) \\ & \qquad\qquad + f_\perp^A(q^2) \left( \gamma^\mu + \frac{2 M_{\mathcal{B}_2}}{s_-} p^\mu - \frac{2 M_{\mathcal{B}_1}}{s_-} k^\mu \right) \bigg] u(p, s)\,, \end{align*}

while the tensor and axialtensor matrix elements are defined as:

\begin{align*} \braket{\mathcal{B}_2(k, s') | \bar{q}_1\, i \sigma^{\mu\nu} q_\nu q_2 | \mathcal{B}_1(p, s)} & = -\bar{u}(k, s') \bigg[ f_0^T(q^2)\, \frac{q^2}{s_+} \left( (p + k)^\mu - \frac{M_{\mathcal{B}_1}^2 - M_{\mathcal{B}_2}^2}{q^2} q^\mu \right) \\ & \qquad\qquad + f_\perp^T(q^2)\, (M_{\mathcal{B}_1} + M_{\mathcal{B}_2}) \left( \gamma^\mu - \frac{2 M_{\mathcal{B}_2}}{s_+} p^\mu - \frac{2 M_{\mathcal{B}_1}}{s_+} k^\mu \right) \bigg] u(p, s)\,, \\ \braket{\mathcal{B}_2(k, s') | \bar{q}_1\, i \sigma^{\mu\nu} q_\nu \gamma_5 q_2 | \mathcal{B}_1(p, s)} & = -\bar{u}(k, s')\, \gamma_5 \bigg[ f_0^{T5}(q^2)\, \frac{q^2}{s_-} \left( (p + k)^\mu - \frac{M_{\mathcal{B}_1}^2 - M_{\mathcal{B}_2}^2}{q^2} q^\mu \right) \\ & \qquad\qquad + f_\perp^{T5}(q^2)\, (M_{\mathcal{B}_1} - M_{\mathcal{B}_2}) \left( \gamma^\mu + \frac{2 M_{\mathcal{B}_2}}{s_-} p^\mu - \frac{2 M_{\mathcal{B}_1}}{s_-} k^\mu \right) \bigg] u(p, s)\,. \end{align*}

Here, \(u(p, s)\) and \(u(k, s')\) are the Dirac spinors of the initial- and final-state baryon, and again \(q = p - k\) is the momentum transfer. The ten form factors are labelled by the current (\(V\), \(A\), \(T\), \(T5\)) and by their helicity: \(t\) (timelike), \(0\) (longitudinal), and \(\perp\) (transverse); the tensor currents have no timelike component. Within EOS the corresponding parameters carry the labels time, long, and perp.

Parametrisation of Semileptonic Form Factors

EOS provides one general-purpose parametrisation of all semileptonic form factors. It is referred to as the BSZ2015 parametrisation. For a generic form factor \(F(q^2)\), it reads:

\begin{equation*} F(q^2) = \frac{1}{1 - q^2 / M_R^2} \left[\sum_{i=0}^N \alpha^{(F)}_{i} \left(z(q^2) - z(0)\right)^i \right]\,. \end{equation*}

Here \(M_R\) correspond to the mass of the first resonance seen by that form factor and \(\alpha_i^{(F)}\) are free parameters. The coefficients \(\alpha_i^{(F)}\) are treated as unconstrained free parameters: there is no expectation for the magnitude of any individual coefficient, nor is there an expectation that the series of coefficients converges. We use the conformal mapping \(q^2 \mapsto z(q^2) = z(q^2; t_+, t_0)\), which reads

\begin{equation*} z(q^2; t_+, t_0) = \frac{\sqrt{t_+ - q^2} - \sqrt{t_+ - t_0}}{\sqrt{t_+ - q^2} + \sqrt{t_+ - t_0}}\,. \end{equation*}

In the above, \(t_+ \equiv (M_1 + M_2)^2\) represents the two-body threshold for the respective form factor for a reaction with hadron masses \(M_{1,2}\), and \(t_0 < (M_1 - M_2)^2\) is a free parameter. Examples for accessing the parameters \(\alpha_i^{(F)}\) are:

Transition	Form Factor	Parameter
\(B\to K\)	\(f_+\)	B->K::alpha^f+_0@BSZ2015
\(D\to K\)	\(f_0\)	D->K::alpha^f0_2@BSZ2015
\(B\to K^*\)	\(V\)	B->K^*::alpha^V_1@BSZ2015