Changes between Version 6 and Version 7 of ZeeBabu

Timestamp:

Jun 30, 2022, 3:51:21 PM (2 years ago)

Author:

Richard Ruiz

Comment:

updated Lagrangian

ZeeBabu

@@ -19 +19 @@
 == Model Description ==
 
-This effective/simplified model extends the Standard Model (SM) field content by introducing three right-handed (RH) neutrinos, which are singlets under the SM gauge symmetry (no color, weak isospin, or weak hypercharge charges). Each RH neutrino possesses one RH Majorana mass. After electroweak symmetry breaking, the Lagrangian with three heavy Majorana neutrinos ''N''i (for i=1,2,3) is given by [ [#Atre 6] ]
+The Zee-Babu model extends the Standard Model (SM) by two complex scalars, $k$ and $h$.
+Neither carries color or weak isospin but both are charged under weak hypercharge.
+$k$ and $h$ are assigned lepton number $L=+2$, which is normalized such that SM leptons carry $L=+1$. 
+In terms of the SM Lagrangian $(\mathcal{L}_{\rm SM})$, the Lagrangian of the Zee-Babu model  $(\mathcal{L}_{\rm ZB})$ is
 {{{ 
 #!latex 
-\begin{equation}
-\mathcal{L} = \mathcal{L}_{\rm SM} + \mathcal{L}_{N} + \mathcal{L}_{N~\text{Int.}}
-\end{equation}
+\begin{align}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mathcal{L}_{\rm ZB} = \mathcal{L}_{\rm SM} + \mathcal{L}_{\rm Kin.} + \mathcal{L}_{\rm Yuk.} + \mathcal{L}_{\rm ZB\ scalar}
+ + \delta\mathcal{L}_{\nu}
+ \ .
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{align}
 }}}
-The first term is the Standard Model Lagrangian. In the mass basis, i.e., after mixing with active neutrinos, the heavy Majorana neutrinos' kinetic and mass terms are
+Here, the weak hypercharge operator is normalized such that the electromagnetic charge operator is $\hat{Q}=\hat{T}_L^3+\hat{Y}$, and $Y_k = -2\ (Y_h = -1)$. 
+The weak hypercharge coupling is denoted by $g_Y\approx 0.36$.
+As neither $k$ nor $h$ mix with SM states, the mass eigenstates, denoted by $k^{--}$ and $h^-$, are aligned with their gauge states and carry the electric charges $Q_k=-2$ and $Q_h=-1$, respectively.
+
+
+The kinetic part of the Lagrangian for $k$ and $h$ is given by the following covariant derivatives
+{{{
+#!latex 
+\begin{align}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mathcal{L}_{\rm Kin.} = (D_\mu k)^\dagger (D^\mu k) + (D_\mu h)^\dagger (D^\mu h), 
+  \quad\text{with}\quad
+  D_\mu = \partial_\mu +i g_Y \hat{Y} B_\mu\ .
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{align}
+}}}
+
+The Yukawa part of $\mathcal{L}_{\rm ZB}$ describes the coupling of SM leptons to $k$ and $h$. It is given by
+{{{
+#!latex 
+\begin{align}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mathcal{L}_{\rm Yuk.} & \ ~  = 
+ f_{ij}\ \overline{\tilde{L}^i} L^j h^\dagger
+ +
+ g_{ij}\ \overline{(e_R^c)^i} e_R^j k^\dagger + \text{H.c.}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{align}
+}}}
+
+The scalar potential of $k$ and $h$, including couplings to the SM Higgs doublet $\Phi$, is given by
+{{{
+#!latex 
+\begin{align}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+- \mathcal{L}_{\rm ZB\ scalar} &=\ 
+\tilde{m}_k^2 k^\dagger k  +\ \tilde{m}_h^2 h^\dagger h\
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
++\ \lambda_k (k^\dagger k)^2\ +\ \lambda_{h} (h^\dagger h)^2\
++\ \lambda_{hk} (k^\dagger k)(h^\dagger h)\
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\nonumber\\
+&
++\ \left(\mu_{\not L}\ h h k^\dagger + \text{H.c.}\right)\
++\ \lambda_{kH} (k^\dagger k) \Phi^\dagger \Phi\ 
++\ \lambda_{hH} (h^\dagger h) \Phi^\dagger \Phi 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\ .
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{align}
+}}}
+After EWSB, the physical masses of $k$ and $h$ are, respectively,
 {{{ 
 #!latex 
-\begin{equation}
-\mathcal{L}_{N} = \frac{1}{2}\overline{N_k} i\!\not\!\partial N_k - \frac{1}{2}m_{N_k} \overline{N_k}N_k, \quad k=1,\dots,3,
-\end{equation}
-}}}
+\begin{align}
+m_k^2 = \tilde{m}^2_k + \frac{\lambda_{kH}}{2}v^2
+\quad\text{and}\quad 
+m_h^2 = \tilde{m}^2_h + \frac{\lambda_{hH}}{2}v^2
+\ .
+\end{align}
+}}} 
 
-and its interactions with the Weak gauge and Higgs bosons are given by
-{{{ 
-#!latex 
-\begin{eqnarray}
-\mathcal{L}_{N~\text{Int}} = 
-&-&\frac{g}{\sqrt{2}} W_{\mu}^{+}\sum_{k=1}^{3}\sum_{\ell=e}^{\tau} \overline{N_k}V_{\ell k}^{*}\gamma^{\mu}P_{L}\ell^{-}
-+{\rm H.c.}
-\\
-&-&\frac{g}{2\cos\theta_W}Z_{\mu}\sum_{k=1}^{3}\sum_{\ell=e}^{\tau} \overline{N_k}V_{\ell k}^{*}\gamma^{\mu}P_{L}\nu_\ell
-+{\rm H.c.}
-\\
-&-&\frac{g m_N}{2 M_W}         h \sum_{k=1}^{3}\sum_{\ell=e}^{\tau} \overline{N_k}V_{\ell k}^{*}P_{L}\nu_\ell
-+{\rm H.c.}
-\end{eqnarray}
-}}} 
+
 Neutrino masses (mNk) and mixing parameters (Vlk) between heavy mass eigenstate and (active) flavor eigenstates are taken to be independent, phenomenological parameters. This allows for maximum flexibility and model independence when calculating rates. Therefore, some care is required by the user. 
 The lepton number- and flavor-violating interactions of the Lagrangian allow for modeling of the Type I, Inverse, and Linear seesaw mechanisms at both lepton, hadron, and lepto-hadron colliders.