Bases

Bases above the EW scale

Derivative basis

• Name: derivative_above
• Lagrangian:

\[\mathcal{L} \subset \frac{a}{f_a}\left(\frac{g_B^2}{16\pi^2} c_{B}B_{\mu\nu}\tilde{B}^{\mu\nu} + \frac{g_W^2}{16\pi^2}c_{W} W^i_{\mu\nu}\tilde{W}^{i,\mu\nu}+\frac{g_s^2}{16\pi} c_{g}G^a_{\mu\nu}\tilde{G}^{a,\mu\nu}\right) + \frac{\partial^\mu a}{f_a} \left(\bar{q}_L c_{q_L} \gamma_\mu q_L + \bar{u}_R c_{u_R} \gamma_\mu u_R + \bar{d}_R c_{d_R} \gamma_\mu d_R + \bar{\ell}_L c_{\ell_L} \gamma_\mu \ell_L + \bar{e}_R c_{e_R} \gamma_\mu e_R \right)\]

• Parameters:
  • cB: Coupling to $B$ bosons $c_B$, float.
  • cW: Coupling to $W$ bosons $c_W$, float.
  • cg: Coupling to gluons $c_g$, float.
  • cqL: Coupling to left-handed quarks $c_{q_L}$, $3\times 3$ numpy.matrix.
  • cuR: Coupling to right-handed up-type quarks $c_{u_R}$, $3\times 3$ numpy.matrix.
  • cdR: Coupling to right-handed down-type quarks $c_{d_R}$, $3\times 3$ numpy.matrix.
  • clL: Coupling to left-handed leptons $c_{\ell_L}$, $3\times 3$ numpy.matrix.
  • ceR: Coupling to right-handed charged leptons $c_{e_R}$, $3\times 3$ numpy.matrix.
• Notes:
  • Fermions are weak eigensates.
• References:
  • Equation (1) of M. Bauer, M. Neubert, S. Renner, M. Schnubel, A. Tham: The Low-Energy Effective Theory of Axions and ALPs. JHEP 04 (2021) 063. DOI, arXiv.

Mass basis

• Name: massbasis_above
• Lagrangian:

\[\mathcal{L} \subset \frac{a}{f_a}\left(\frac{\alpha_\mathrm{em}}{4\pi}c_{\gamma} F_{\mu\nu}\tilde{F}^{\mu\nu} + \frac{\alpha_\mathrm{em}}{2\pi s_w c_w}c_{\gamma Z} F_{\mu\nu}\tilde{Z}^{\mu\nu} + \frac{\alpha_\mathrm{em}}{4\pi s_w^2 c_w^2}c_{Z} Z_{\mu\nu}\tilde{Z}^{\mu\nu} + \frac{\alpha_\mathrm{em}}{2\pi s_w^2}c_{W} W^+_{\mu\nu}\tilde{W}^{-\mu\nu} + \frac{\alpha_s}{4\pi}c_{g} G^a_{\mu\nu}\tilde{G}^{a,\mu\nu}\right) + \frac{\partial^\mu a}{f_a} \left(\bar{u}_L k_U \gamma_\mu u_L + \bar{u}_R k_u \gamma_\mu u_R + \bar{d}_L k_D \gamma_\mu d_L + \bar{d}_R k_d \gamma_\mu d_R + \bar{e}_L k_E \gamma_\mu e_L + \bar{e}_R k_e \gamma_\mu e_R + \bar{\nu}_L k_\nu \gamma_\mu \nu_L \right)\]

• Parameters:
  • cgamma: Coupling to photons $c_\gamma$, float.
  • cgammaZ: Coupling to photon and $Z$ boson $c_{\gamma Z}$, float.
  • cZ: Coupling to $Z$ bosons $c_Z$, float.
  • cW: Coupling to $W$ bosons $c_W$, float.
  • cg: Coupling to gluons $c_g$, float.
  • kU: Coupling to left-handed up-type quarks $k_U$, $3\times 3$ numpy.matrix.
  • ku: Coupling to right-handed up-type quarks $k_u$, $3\times 3$ numpy.matrix.
  • kD: Coupling to left-handed down-type quarks $k_D$, $3\times 3$ numpy.matrix.
  • kd: Coupling to right-handed down-type quarks $k_d$, $3\times 3$ numpy.matrix.
  • kE: Coupling to left-handed charged leptons $k_E$, $3\times 3$ numpy.matrix.
  • ke: Coupling to right-handed charged leptons $k_e$, $3\times 3$ numpy.matrix.
  • kNu: Coupling to left-handed neutrinos $k_\nu$, $3\times 3$ numpy.matrix.
• Notes:
  • Fermions are mass eigensates.
• References:
  • Equations (39) and (43) of M. Bauer, M. Neubert, S. Renner, M. Schnubel, A. Tham: The Low-Energy Effective Theory of Axions and ALPs. JHEP 04 (2021) 063. DOI, arXiv.

Bases below the EW scale

$k_F$ basis

• Name: kF_below
• Lagrangian:

\[\mathcal{L} \subset \frac{a}{f_a}\left(\frac{\alpha_\mathrm{em}}{4\pi}c_{\gamma} F_{\mu\nu}\tilde{F}^{\mu\nu} + \frac{\alpha_s}{4\pi}c_{g} G^a_{\mu\nu}\tilde{G}^{a,\mu\nu}\right) + \frac{\partial^\mu a}{f_a} \left(\bar{u}_L k_U \gamma_\mu u_L + \bar{u}_R k_u \gamma_\mu u_R + \bar{d}_L k_D \gamma_\mu d_L + \bar{d}_R k_d \gamma_\mu d_R + \bar{e}_L k_E \gamma_\mu e_L + \bar{e}_R k_e \gamma_\mu e_R + \bar{\nu}_L k_\nu \gamma_\mu \nu_L \right)\]

• Parameters:
  • cgamma: Coupling to photons $c_\gamma$, float.
  • cg: Coupling to gluons $c_g$, float.
  • kU: Coupling to left-handed up-type quarks $k_U$, $2\times 2$ numpy.matrix.
  • ku: Coupling to right-handed up-type quarks $k_u$, $2\times 2$ numpy.matrix.
  • kD: Coupling to left-handed down-type quarks $k_D$, $3\times 3$ numpy.matrix.
  • kd: Coupling to right-handed down-type quarks $k_d$, $3\times 3$ numpy.matrix.
  • kE: Coupling to left-handed charged leptons $k_E$, $3\times 3$ numpy.matrix.
  • ke: Coupling to right-handed charged leptons $k_e$, $3\times 3$ numpy.matrix.
  • kNu: Coupling to left-handed neutrinos $k_\nu$, $3\times 3$ numpy.matrix.
• References:
  • Equations (54) and (43) of M. Bauer, M. Neubert, S. Renner, M. Schnubel, A. Tham: The Low-Energy Effective Theory of Axions and ALPs. JHEP 04 (2021) 063. DOI, arXiv.

Vector-axial basis

• Name: VA_below
• Lagrangian:

\[\mathcal{L} \subset \frac{a}{f_a}\left(\frac{\alpha_\mathrm{em}}{4\pi}c_{\gamma} F_{\mu\nu}\tilde{F}^{\mu\nu} + \frac{\alpha_s}{4\pi}c_{g} G^a_{\mu\nu}\tilde{G}^{a,\mu\nu}\right) + \frac{\partial^\mu a}{2f_a} \left(\bar{u} \gamma_\mu (c_u^V + c_u^A \gamma_5) u + \bar{d} \gamma_\mu (c_d^V + c_d^A \gamma_5) d + \bar{e} \gamma_\mu (c_e^V + c_e^A \gamma_5) e + 2\bar{\nu}_L c_\nu \gamma_\mu \nu_L \right)\]

• Parameters:
  • cgamma: Coupling to photons $c_\gamma$, float.
  • cg: Coupling to gluons $c_g$, float.
  • cuV: Vectorial coupling to up-type quarks $c_u^V$, $2\times 2$ numpy.matrix.
  • cuA: Axial coupling to up-type quarks $c_u^A$, $2\times 2$ numpy.matrix.
  • cdV: Vectorial coupling to down-type quarks $c_d^V$, $3\times 3$ numpy.matrix.
  • cdA: Axial coupling to down-type quarks $c_d^A$, $3\times 3$ numpy.matrix.
  • ceV: Vectorial coupling to charged leptons $c_e^V$, $3\times 3$ numpy.matrix.
  • ceA: Axial coupling to charged leptons $c_e^A$, $3\times 3$ numpy.matrix.
  • cnu: Coupling to left-handed neutrinos $c_\nu$, $3\times 3$ numpy.matrix.