Conventions

EOS implicitly uses the following conventions in the predictions of flavour physics observables.

Units

EOS uses natural units \((\hbar = c =1)\). Most quantities are expressed in powers of GeV. Exceptions include lifetimes, which are expressed in seconds. Units are specified in the definition of observables and parameters (which can be found in this documentation (here and here, respectively), or directly from python, as described in the basic examples). EOS currently implements the following units:

  • GeV^3: \(\textrm{GeV}^3\)

  • GeV^-4: \(\textrm{GeV}^{-4}\)

  • GeV^-1: \(\textrm{GeV}^{-1}\)

  • undefined: \(\textrm{undefined}\)

  • GeV^2: \(\textrm{GeV}^2\)

  • s^-1: \(\textrm{s}^{-1}\)

  • GeV s: \(\textrm{GeV}\,\textrm{s}\)

  • ps^-1: \(\textrm{ps}^{-1}\)

  • 1: \(1\)

  • fm^2: \(\textrm{fm}^2\)

  • GeV: \(\textrm{GeV}\)

  • s: \(\textrm{s}\)

  • GeV^-2: \(\textrm{GeV}^{-2}\)

Weak Effective Theory

We follow the conventions of the WCxf format for the Wilson coefficients of the effective Hamiltonian at mass dimension six. Each sector of the effective Hamiltonian is defined by a unique name, corresponding to the flavour quantum numbers of the fermion fields.

Semileptonic Charged-Current Operators

We use the Bern class-III notation for the semileptonic charged-current operators. For down-type (\(D\)) to up-type (\(U\)) semileptonic decays, the sector of the effective theory is described by the Lagrangian:

\begin{align*} \mathcal{L}^{UD\ell\nu} = -\frac{4 G_\text{F}}{\sqrt{2}} V_{UD} \sum_{i} \mathcal{C}_i^{UD\ell\nu}(\mu) \, \mathcal{O}_i + \text{h.c.}\,, \end{align*}

where \(V_{UD}\) is the CKM matrix element, \(G_\text{F}\) is the Fermi constant. The operator basis \(\mathcal{O}_i\) reads:

\begin{align*} \mathcal{O}_{VL}^{UD\ell\nu} & = [\bar{U} \gamma_\mu P_L D] [\bar{\ell} \gamma^\mu P_L \nu]\,, & \mathcal{O}_{VR}^{UD\ell\nu} & = [\bar{U} \gamma_\mu P_R D] [\bar{\ell} \gamma^\mu P_L \nu]\,, \\ \mathcal{O}_{SL}^{UD\ell\nu} & = [\bar{U} P_L D] [\bar{\ell} P_L \nu]\,, & \mathcal{O}_{SR}^{UD\ell\nu} & = [\bar{U} P_R D] [\bar{\ell} P_L \nu]\,, \\ \mathcal{O}_{T}^{UD\ell\nu} & = [\bar{U} \sigma_{\mu\nu} D] [\bar{\ell} \sigma^{\mu\nu} P_L \nu]\,. \end{align*}

The Wilson coefficients \(\mathcal{C}_i^{UD\ell\nu}(\mu)\) are defined at the sector-specific scale \(\mu^{UD\ell\nu}\) and are dimensionless. Their parameters use the prefix UDlnul, where U, D, and l are understood as metavariables.