Operator basis

New physics contributions are always defined as contributions to dimension-6 operators in an effective field theory (EFT) below the $W$ mass in the $\overline{\text{MS}}$ renormalization scheme with naive dimensional regularization. The operators are grouped into “sectors” relevant for a particular class of processes. Operators in one sector close under renormalization, but individual operators can appear in several “sectors”.

Meson-antimeson mixing (bsbs, bdbd, sdsd)

The following lists the operators for the case of $B_s$ mixing.

Operator Wilson coefficient
$O_V^{LL} = (\bar s_L \gamma^\mu b_L)(\bar s_L \gamma_\mu b_L)$ CVLL_bsbs
$O_V^{RR} = (\bar s_R \gamma^\mu b_R)(\bar s_R \gamma_\mu b_R)$ CVRR_bsbs
$O_S^{LL} = (\bar s_R b_L)(\bar s_R b_L)$ CSLL_bsbs
$O_S^{RR} = (\bar s_L b_R)(\bar s_L b_R)$ CSRR_bsbs
$O_T^{LL} = (\bar s_R \sigma^{\mu\nu} b_L)(\bar s_R \sigma_{\mu\nu} b_L)$ CTLL_bsbs
$O_T^{RR} = (\bar s_L \sigma^{\mu\nu} b_R)(\bar s_L \sigma_{\mu\nu} b_R)$ CTRR_bsbs
$O_V^{LR} = (\bar s_L \gamma^\mu b_L)(\bar s_R \gamma_\mu b_R)$ CVLR_bsbs
$O_S^{LR} = (\bar s_R b_L)(\bar s_L b_R)$ CSLR_bsbs

Here and in the following, $\sigma^{\mu\nu}=\frac{i}{2}[\gamma_\mu, \gamma_\nu]$.

Semi-leptonic FCNCs (bsee, bdee, sdee, bsmumu, …, bstautau, …)

The following lists the operators for the case of $b\to se^+e^-$.

Operator Wilson coefficient
$O_1 = (\bar s_L \gamma^\mu T^a c_L)(\bar c_L \gamma_\mu T^a b_L)$ C1_bs
$O_1^\prime = (\bar s_R \gamma^\mu T^a c_R)(\bar c_R \gamma_\mu T^a b_R)$ C1p_bs
$O_2 = (\bar s_L \gamma^\mu c_L)(\bar c_L \gamma_\mu b_L)$ C2_bs
$O_2^\prime = (\bar s_R \gamma^\mu c_R)(\bar c_R \gamma_\mu b_R)$ C2p_bs
$O_3 = (\bar s_L \gamma^\mu b_L)\sum_q(\bar q \gamma_\mu q)$ C3_bs
$O_3^\prime = (\bar s_R \gamma^\mu b_R)\sum_q(\bar q \gamma_\mu q)$ C3p_bs
$O_4 = (\bar s_L \gamma^\mu T^a b_L)\sum_q(\bar q \gamma_\mu T^a q)$ C4_bs
$O_4^\prime = (\bar s_R \gamma^\mu T^a b_R)\sum_q(\bar q \gamma_\mu T^a q)$ C4p_bs
$O_5 = (\bar s_L \gamma^{\mu_1}\gamma^{\mu_2}\gamma^{\mu_3} b_L)\sum_q(\bar q \gamma_{\mu_1}\gamma_{\mu_2}\gamma_{\mu_3} q)$ C5_bs
$O_5^\prime = (\bar s_R \gamma^{\mu_1}\gamma^{\mu_2}\gamma^{\mu_3} b_R)\sum_q(\bar q \gamma_{\mu_1}\gamma_{\mu_2}\gamma_{\mu_3} q)$ C5p_bs
$O_6 = (\bar s_L \gamma^{\mu_1}\gamma^{\mu_2}\gamma^{\mu_3} T^a b_L)\sum_q(\bar q \gamma_{\mu_1}\gamma_{\mu_2}\gamma_{\mu_3} T^a q)$ C6_bs
$O_6^\prime = (\bar s_R \gamma^{\mu_1}\gamma^{\mu_2}\gamma^{\mu_3} T^a b_R)\sum_q(\bar q \gamma_{\mu_1}\gamma_{\mu_2}\gamma_{\mu_3} T^a q)$ C6p_bs
$O_{3Q} = (\bar s_L \gamma^\mu b_L)\sum_q Q_q (\bar q \gamma_\mu q)$ C3Q_bs
$O_{3Q}^\prime = (\bar s_R \gamma^\mu b_R)\sum_q Q_q (\bar q \gamma_\mu q)$ C3Qp_bs
$O_{4Q} = (\bar s_L \gamma^\mu T^a b_L)\sum_q Q_q (\bar q \gamma_\mu T^a q)$ C4Q_bs
$O_{4Q}^\prime = (\bar s_R \gamma^\mu T^a b_R)\sum_q Q_q (\bar q \gamma_\mu T^a q)$ C4Qp_bs
$O_{5Q} = (\bar s_L \gamma^{\mu_1}\gamma^{\mu_2}\gamma^{\mu_3} b_L)\sum_q Q_q (\bar q \gamma_{\mu_1}\gamma_{\mu_2}\gamma_{\mu_3} q)$ C5Q_bs
$O_{5Q}^\prime = (\bar s_R \gamma^{\mu_1}\gamma^{\mu_2}\gamma^{\mu_3} b_R)\sum_q Q_q (\bar q \gamma_{\mu_1}\gamma_{\mu_2}\gamma_{\mu_3} q)$ C5Qp_bs
$O_{6Q} = (\bar s_L \gamma^{\mu_1}\gamma^{\mu_2}\gamma^{\mu_3} T^a b_L)\sum_q Q_q (\bar q \gamma_{\mu_1}\gamma_{\mu_2}\gamma_{\mu_3} T^a q)$ C6Q_bs
$O_{6Q}^\prime = (\bar s_R \gamma^{\mu_1}\gamma^{\mu_2}\gamma^{\mu_3} T^a b_R)\sum_q Q_q (\bar q \gamma_{\mu_1}\gamma_{\mu_2}\gamma_{\mu_3} T^a q)$ C6Qp_bs
$O_9 = \frac{e^2}{16\pi^2}(\bar s_L \gamma^\mu b_L)(\bar e \gamma_\mu e)$ C9_bsee
$O_9^\prime = \frac{e^2}{16\pi^2}(\bar s_R \gamma^\mu b_R)(\bar e \gamma_\mu e)$ C9p_bsee
$O_{10} = \frac{e^2}{16\pi^2}(\bar s_L \gamma^\mu b_L)(\bar e \gamma_\mu\gamma_5 e)$ C10_bsee
$O_{10}^\prime = \frac{e^2}{16\pi^2}(\bar s_R \gamma^\mu b_R)(\bar e \gamma_\mu\gamma_5 e)$ C10p_bsee
$O_S = \frac{e^2}{16\pi^2}m_b(\bar s_L b_R)(\bar e e)$ CS_bsee
$O_S^\prime = \frac{e^2}{16\pi^2}m_b(\bar s_R b_L)(\bar e e)$ CSp_bsee
$O_P = \frac{e^2}{16\pi^2}m_b(\bar s_L b_R)(\bar e \gamma_5 e)$ CP_bsee
$O_P^\prime = \frac{e^2}{16\pi^2}m_b(\bar s_R b_L)(\bar e \gamma_5 e)$ CPp_bsee

$T^a$ is a generator of $SU(3)_c$ in the fundamental representation.

In addition, there are the dipole operators

For these operators, the effective Wilson coefficients C7eff_bs, C7effp_bs, C8eff_bs, C8effp_bs, are used which are defined as

where $y=(0,0,-\frac{1}{3},-\frac{4}{9},-\frac{20}{3},-\frac{80}{9})$, $z=(0,0,1,-\frac{1}{6},20,-\frac{10}{3})$, and analogously for the primed operators.

Semi-leptonic LFV FCNCs (bsemu, bsmue, …, sdtaumu, …)

The following lists the operators for the case of $b\to se^+\mu^-$. Note that these Wilson coefficients are distinct from the case $b\to s\mu^+e^-$!

Operator Wilson coefficient
$O_9 = \frac{e^2}{16\pi^2}(\bar s_L \gamma^\mu b_L)(\bar \mu \gamma_\mu e)$ C9_bsemu
$O_9^\prime = \frac{e^2}{16\pi^2}(\bar s_R \gamma^\mu b_R)(\bar \mu \gamma_\mu e)$ C9p_bsemu
$O_{10} = \frac{e^2}{16\pi^2}(\bar s_L \gamma^\mu b_L)(\bar \mu \gamma_\mu\gamma_5 e)$ C10_bsemu
$O_{10}^\prime = \frac{e^2}{16\pi^2}(\bar s_R \gamma^\mu b_R)(\bar \mu \gamma_\mu\gamma_5 e)$ C10p_bsemu
$O_S = \frac{e^2}{16\pi^2}m_b(\bar s_L b_R)(\bar \mu e)$ CS_bsemu
$O_S^\prime = \frac{e^2}{16\pi^2}m_b(\bar s_R b_L)(\bar \mu e)$ CSp_bsemu
$O_P = \frac{e^2}{16\pi^2}m_b(\bar s_L b_R)(\bar \mu \gamma_5 e)$ CP_bsemu
$O_P^\prime = \frac{e^2}{16\pi^2}m_b(\bar s_R b_L)(\bar \mu \gamma_5 e)$ CPp_bsemu

Decays with neutrinos in the final state (bsnuenue, bdnuenue, …, sdnuenutau, …)

This also includes decays with two differently flavoured neutrinos in the final state, which cannot be distinguished experimentally. Note that bsnuenumu and bsnumunue are distinct cases, etc. The following lists the operators for the case of $b\to s\nu_e\bar\nu_e$. Note that neutrinos are always assumed to be massless and left-handed.

Operator Wilson coefficient
$O_L = \frac{e^2}{16\pi^2}(\bar s_L \gamma^\mu b_L)(\bar \nu_e \gamma_\mu(1-\gamma_5) \nu_e)$ CL_bsnuenue
$O_R = \frac{e^2}{16\pi^2}(\bar s_R \gamma^\mu b_R)(\bar \nu_e \gamma_\mu(1-\gamma_5) \nu_e)$ CR_bsnuenue

Charged-current decays (buenu, …, bctaunu, …,)

The following lists the operators for the case of $b\to c\tau^-\bar\nu$.

Operator Wilson coefficient
$O_V = (\bar c_L \gamma^\mu b_L)(\bar \tau_L \gamma_\mu \nu_L)$ CV_bctaunu
$O_V^\prime = (\bar c_R \gamma^\mu b_R)(\bar \tau_L \gamma_\mu \nu_L)$ CVp_bctaunu
$O_S = m_b(\bar c_L b_R)(\bar \tau_R \nu_L)$ CS_bctaunu
$O_S^\prime = m_b(\bar c_R b_L)(\bar \tau_R \nu_L)$ CSp_bctaunu
$O_T = (\bar c_R \sigma^{\mu\nu} b_L)(\bar \tau_R \sigma_{\mu\nu}\nu_L)$ CT_bctaunu