Conventions for the Standard Model Effective Field Theory in the Warsaw basis The conventions used in the default Matchete model file for the Standard Model Effective Field Theory in the Warsaw basis ("SMEFT_Warsaw"
The renormalizable part of the Lagrangian ($\mathcal{L}_{\mathrm{SM}}$) agrees with the SM model file definitions.
The dimension-six Warsaw basis Lagrangian is then given by \(\mathcal{L}_{\mathrm{Warsaw}}=\mathcal{L}_{\mathrm{SM}}+\sum_{k}{c_k}{Q_k},\) where the sum runs over all dimension-six operators ($Q_k$) listed below and their Wilson coefficients are denoted $c_k$.
The key differences with respect to the conventions chosen originally for the Warsaw basis [arXiv:1008.4884] are:
We use a different gauge field normalization. Therefore, for every field-strength tensor contained in an operator, we divide this operator by the corresponding gauge coupling. Since we use the minus sign convention for covariant derivatives, we include a minus sign for every gauge coupling in an operator. Our definition of the Levi-Civita symbol ($\epsilon^{\mu\nu\rho\sigma}$) has opposite sign. Thus, we include a minus sign for all operators containing a dual field-strength tensor. This choice ensures that the values of the Wilson coefficients in [arXiv:1008.4884] , with the conventions chosen there, exactly agree with the values of the Wilson coeffiecients determined by Matchete using the conventions of Matchete.
The full list of dimension-five and dimension-six operators defined in the Matchete version of the Warsaw basis is given below.
At mass-dimension five only a single electroweak structure, i.e., the Weinberg operator, is included in the basis:
\[Q_{llHH}^{pr} = \varepsilon_{in} \varepsilon_{jm} H^i H^j [{(\ell_p^n)}^\intercal C \ell_r^m]\] At mass-dimension six there are 63 electroweak structures which we seperate into classes following [arXiv:1312.2014] .
\[\begin{align*} Q_{G} &= -\frac{1}{g_s^3} f^{ABC} G_\mu^{A\nu} G_\nu^{B\rho} G_\rho^{C\mu} \\ Q_{\tilde G} &= \frac{1}{g_s^3} f^{ABC} \tilde G_\mu^{A\nu} G_\nu^{B\rho} G_\rho^{C\mu} \\ Q_{W} &= -\frac{1}{g_L^3} \varepsilon^{IJK} W_\mu^{I\nu} W_\nu^{J\rho} W_\rho^{K\mu} \\ Q_{\tilde W} &= \frac{1}{g_L^3} \varepsilon^{IJK} \tilde W_\mu^{I\nu} W_\nu^{J\rho} W_\rho^{K\mu} \end{align*}\] \[Q_{H}=(H^\dagger H)^3\] \[\begin{align*} Q_{H\Box} &= (H^\dagger H) \Box (H^\dagger H) \\ Q_{HD} &= (H^\dagger D_\mu H)^\ast (H^\dagger D^\mu H) \end{align*}\] \[\begin{align*} Q_{HG} &= \frac{1}{g_s^2} (H^\dagger H) G_{\mu\nu}^A G^{A\mu\nu} \\ Q_{H\tilde G} &= -\frac{1}{g_s^2} (H^\dagger H) \tilde G_{\mu\nu}^A G^{A\mu\nu} \\ Q_{HW} &= \frac{1}{g_L^2} (H^\dagger H) W_{\mu\nu}^I W^{I\mu\nu} \\ Q_{H\tilde{W}} &= -\frac{1}{g_L^2} (H^\dagger H) \tilde W_{\mu\nu}^I W^{I\mu\nu} \\ Q_{HB} &= \frac{1}{g_Y^2} (H^\dagger H) B_{\mu\nu} B^{\mu\nu} \\ Q_{H \tilde B} &= -\frac{1}{g_Y^2} (H^\dagger H) \tilde B_{\mu\nu} B^{\mu\nu} \\ Q_{H W B} &= \frac{1}{g_L g_Y} (H^\dagger \sigma^I H) W_{\mu\nu}^I B^{\mu\nu} \\ Q_{H \tilde W B} &= -\frac{1}{g_L g_Y} (H^\dagger \sigma^I H) \tilde W_{\mu\nu}^I B^{\mu\nu} \end{align*}\] \[\begin{align*} Q_{uH}^{pr} &= (H^\dagger H)(\bar q_p u_r H^c) \\ Q_{dH}^{pr} &= (H^\dagger H)(\bar q_p d_r H) \\ Q_{eH}^{pr} &= (H^\dagger H)(\bar\ell_p e_r H) \end{align*}\] \[\begin{align*} Q_{uG}^{pr} &= -\frac{1}{g_s} (\bar q_p \sigma^{\mu\nu}T^A u_r) H^c G_{\mu\nu}^A \\ Q_{uW}^{pr} &= -\frac{1}{g_L} (\bar q_p \sigma^{\mu\nu}u_r)\sigma^I H^c W_{\mu\nu}^I \\ Q_{uB}^{pr} &= -\frac{1}{g_Y} (\bar q_p \sigma^{\mu\nu}u_r) H^c B_{\mu\nu} \\ Q_{dG}^{pr} &= -\frac{1}{g_s} (\bar q_p \sigma^{\mu\nu}T^A d_r)H G_{\mu\nu}^A \\ Q_{dW}^{pr} &= -\frac{1}{g_L} (\bar q_p \sigma^{\mu\nu}d_r)\sigma^I H W_{\mu\nu}^I \\ Q_{dB}^{pr} &= -\frac{1}{g_Y} (\bar q_p \sigma^{\mu\nu}d_r) H B_{\mu\nu} \\ Q_{eW}^{pr} &= -\frac{1}{g_L} (\bar\ell_p \sigma^{\mu\nu}e_r)\sigma^I H W_{\mu\nu}^I \\ Q_{eB}^{pr} &= -\frac{1}{g_Y} (\bar\ell_p \sigma^{\mu\nu}e_r) H B_{\mu\nu} \end{align*}\] \[\begin{align*} Q_{Hq}^{(1)\,pr} &= (H^\dagger i \overleftrightarrow{D}_\mu H)(\bar q_p \gamma^\mu q_r) \\ Q_{Hq}^{(3)\,pr} &= (H^\dagger i \overleftrightarrow{D}_\mu^I H)(\bar q_p \sigma^I\gamma^\mu q_r) \\ Q_{H u}^{pr} &= (H^\dagger i \overleftrightarrow{D}_\mu H)(\bar u_p \gamma^\mu u_r) \\ Q_{H d}^{pr} &= (H^\dagger i \overleftrightarrow{D}_\mu H)(\bar d_p \gamma^\mu d_r) \\ Q_{Hl}^{(1)\,pr} &= (H^\dagger i \overleftrightarrow{D}_\mu H)(\bar\ell_p \gamma^\mu \ell_r) \\ Q_{Hl}^{(3)\,pr} &= (H^\dagger i \overleftrightarrow{D}_\mu^I H)(\bar\ell_p \sigma^I\gamma^\mu \ell_r) \\ Q_{H e}^{pr} &= (H^\dagger i \overleftrightarrow{D}_\mu H)(\bar e_p \gamma^\mu e_r) \\ Q_{Hud}^{pr} &= i(H^{c\,\dagger} D_\mu H)(\bar u_p \gamma^\mu d_r) \ \ [+\mathrm{H.c.}] \end{align*}\] \[\begin{align*} Q_{qq}^{(1)\,prst} &= (\bar q_p \gamma_\mu q_r)(\bar q_s \gamma^\mu q_t) \\ Q_{qq}^{(3)\,prst} &= (\bar q_p \gamma_\mu \sigma^I q_r)(\bar q_s \gamma^\mu \sigma^I q_t) \\ Q_{ll}^{prst} &= (\bar\ell_p \gamma_\mu \ell_r)(\bar\ell_s \gamma^\mu \ell_t) \\ Q_{l q}^{(1)\,prst} &= (\bar\ell_p \gamma_\mu \ell_r)(\bar q_s \gamma^\mu q_t) \\ Q_{l q}^{(3)\,prst} &= (\bar\ell_p \gamma_\mu \sigma^I \ell_r)(\bar q_s \gamma^\mu \sigma^I q_t) \end{align*}\] \[\begin{align*} Q_{uu}^{prst} &= (\bar u_p \gamma_\mu u_r)(\bar u_s \gamma^\mu u_t) \\ Q_{dd}^{prst} &= (\bar d_p \gamma_\mu d_r)(\bar d_s \gamma^\mu d_t) \\ Q_{ee}^{prst} &= (\bar e_p \gamma_\mu e_r)(\bar e_s \gamma^\mu e_t) \\ Q_{ud}^{(1)\,prst} &= (\bar u_p \gamma_\mu u_r)(\bar d_s \gamma^\mu d_t) \\ Q_{ud}^{(8)\,prst} &= (\bar u_p \gamma_\mu T^A u_r)(\bar d_s \gamma^\mu T^A d_t) \\ Q_{eu}^{prst} &= (\bar e_p \gamma_\mu e_r)(\bar u_s \gamma^\mu u_t) \\ Q_{ed}^{prst} &= (\bar e_p \gamma_\mu e_r)(\bar d_s \gamma^\mu d_t) \end{align*}\] \[\begin{align*} Q_{qu}^{(1)\,prst} &= (\bar q_p \gamma_\mu q_r)(\bar u_s \gamma^\mu u_t) \\ Q_{qu}^{(8)\,prst} &= (\bar q_p \gamma_\mu T^A q_r)(\bar u_s \gamma^\mu T^A u_t) \\ Q_{qd}^{(1)\,prst} &= (\bar q_p \gamma_\mu q_r)(\bar d_s \gamma^\mu d_t) \\ Q_{qd}^{(8)\,prst} &= (\bar q_p \gamma_\mu T^A q_r)(\bar d_s \gamma^\mu T^A d_t) \\ Q_{qe}^{prst} &= (\bar q_p \gamma_\mu q_r)(\bar e_s \gamma^\mu e_t) \\ Q_{lu}^{prst} &= (\bar\ell_p \gamma_\mu \ell_r)(\bar u_s \gamma^\mu u_t) \\ Q_{ld}^{prst} &= (\bar\ell_p \gamma_\mu \ell_r)(\bar d_s \gamma^\mu d_t) \\ Q_{le}^{prst} &= (\bar\ell_p \gamma_\mu \ell_r)(\bar e_s \gamma^\mu e_t) \end{align*}\] \[\begin{align*} Q_{quqd}^{(1)\,prst} &= (\bar q_{pi} u_r) \varepsilon^{ij} (\bar q_{sj} d_t) \\ Q_{quqd}^{(8)\,prst} &= (\bar q_{pi} T^A u_r) \varepsilon^{ij} (\bar q_{sj} T^A d_t) \\ Q_{lequ}^{(1)\,prst} &= (\bar \ell_{pi} e_r) \varepsilon^{ij} (\bar q_{sj} u_t) \\ Q_{l equ}^{(3)\,prst} &= (\bar \ell_{pi} \sigma_{\mu\nu} e_r) \varepsilon^{ij} (\bar q_{sj} \sigma^{\mu\nu} u_t) \end{align*}\] \[Q_{ledq}^{prst}=(\bar \ell_{pi} e_r)(\bar d_s q_t^i)\] \[\begin{align*} Q_{duq}^{prst} &= \varepsilon_{\alpha\beta\gamma} [{({d^\alpha_p})}^\intercal C u^\beta_r] [{(q_s^{\gamma i})}^\intercal C \varepsilon_{ij} \ell_t^j] \\ Q_{qqu}^{prst} &= \varepsilon_{\alpha\beta\gamma} [{({q^{\alpha i}_p})}^\intercal C \varepsilon_{ij} q^{\beta j}_r] [{(u_s^{\gamma})}^\intercal C e_t] \\ Q_{qqq}^{prst} &= \varepsilon_{\alpha\beta\gamma} [{({q^{\alpha i}_p})}^\intercal C q^{\beta j}_r] \varepsilon_{im} \varepsilon_{jn} [{(q_s^{\gamma n})}^\intercal C \ell_t^m] \\ Q_{duu}^{prst} &= \varepsilon_{\alpha\beta\gamma} [{({d^\alpha_p})}^\intercal C u^\beta_r] [{(u_s^{\gamma})}^\intercal C e_t] \end{align*}\]