Physlib.Particles.StandardModel.Fermions.LeptonSinglet

The equivalence $\text{LeptonSinglet} \simeq \text{RightHandedWeyl}$ identifies the vector space of charged-lepton singlets in the Standard Model with the underlying space of right-handed Weyl spinors.

The type `LeptonSinglet`, representing the space of charged-lepton singlets in the Standard Model, is equipped with an additive commutative group structure. This structure is induced by transporting the additive group properties of right-handed Weyl spinors via the equivalence $\text{LeptonSinglet} \simeq \text{RightHandedWeyl}$.

The space of charged-lepton singlets, $\text{LeptonSinglet}$, is equipped with the structure of a module over the field of complex numbers $\mathbb{C}$. This defines $\text{LeptonSinglet}$ as a complex vector space, where the scalar multiplication and addition are inherited from the space of right-handed Weyl spinors through the identification $\text{LeptonSinglet} \simeq \text{RightHandedWeyl}$.

The linear isomorphism between the space of charged-lepton singlets $\text{LeptonSinglet}$ and the space of right-handed Weyl spinors $\text{RightHandedWeyl}$ over the complex numbers $\mathbb{C}$. This identification allows a charged-lepton singlet, which carries only a Lorentz index and no gauge indices, to be treated as its underlying Weyl spinor.

For any charged-lepton singlet $l$ in the space $\text{LeptonSinglet}$, the image of $l$ under the linear isomorphism $\text{valLinEquiv} : \text{LeptonSinglet} \to \text{RightHandedWeyl}$ is equal to its underlying right-handed Weyl spinor $l.\text{val}$. That is, $$\text{valLinEquiv}(l) = l.\text{val}$$ where $\text{RightHandedWeyl}$ is the space of right-handed Weyl spinors.

For any right-handed Weyl spinor $m \in \text{RightHandedWeyl}$, the inverse of the linear isomorphism $\text{valLinEquiv} : \text{LeptonSinglet} \simeq_{\mathbb{C}} \text{RightHandedWeyl}$ maps $m$ to the charged-lepton singlet formed by that spinor, denoted $\langle m \rangle$: $$\text{valLinEquiv}^{-1}(m) = \langle m \rangle$$ where $\text{LeptonSinglet}$ is the vector space of the $(1, 1)_{-6}$ Standard Model representation.

For any two charged-lepton singlets $l_1$ and $l_2$, the underlying right-handed Weyl spinor of their sum is equal to the sum of their individual underlying spinors, expressed as: $$(l_1 + l_2).\text{val} = l_1.\text{val} + l_2.\text{val}$$ where $\text{val}$ denotes the projection from the `LeptonSinglet` space to its representation as a right-handed Weyl spinor.

For any complex scalar $r \in \mathbb{C}$ and any charged-lepton singlet $l$, the underlying right-handed Weyl spinor of their scalar product is equal to the scalar product of $r$ and the underlying spinor of $l$, expressed as: $$(r \cdot l).\text{val} = r \cdot l.\text{val}$$ where $\text{val}$ denotes the projection from the $\text{LeptonSinglet}$ space to its representation as a right-handed Weyl spinor.

The representation of the special linear group $SL(2, \mathbb{C})$—the double cover of the restricted Lorentz group—on the complex vector space of charged-lepton singlets $\text{LeptonSinglet}$. This representation is defined by transporting the right-handed Weyl representation of $SL(2, \mathbb{C})$ through the linear isomorphism $\text{valLinEquiv} : \text{LeptonSinglet} \cong \text{RightHandedWeyl}$. Specifically, for an element $\Lambda \in SL(2, \mathbb{C})$, the action is given by the composition: $$\text{valLinEquiv}^{-1} \circ \rho_R(\Lambda) \circ \text{valLinEquiv}$$ where $\rho_R(\Lambda)$ is the right-handed Weyl representation of $\Lambda$ acting on the underlying spinor space.

The representation of the unquotiented Standard Model gauge group $\mathcal{G}_I = SU(3) \times SU(2) \times U(1)$ on the complex vector space of charged-lepton singlets $\text{LeptonSinglet}$. For an element $g = (g_3, g_2, z) \in \mathcal{G}_I$, where $z \in U(1)$ is the hypercharge component, the action on a lepton singlet $\psi$ is given by scalar multiplication: $$g \cdot \psi = \bar{z}^6 \psi$$ where $\bar{z}$ denotes the complex conjugate of $z$. This defines the $(1, 1)_{-6}$ representation, indicating that the $SU(3)$ and $SU(2)$ components act trivially and the hypercharge $Y$ is normalized such that $6Y = -6$.

For any element $g$ of the unquotiented Standard Model gauge group $\mathcal{G}_I = SU(3) \times SU(2) \times U(1)$ and any right-handed Weyl spinor $\psi$ belonging to the space of charged-lepton singlets, the action of $g$ on $\psi$ is given by: $$g \cdot \psi = (z^*)^6 \psi$$ where $z \in U(1)$ is the projection of $g$ onto its $U(1)$ factor and $z^*$ denotes its complex conjugate.

Let $\mathcal{G}_I = SU(3) \times SU(2) \times U(1)$ be the unquotiented Standard Model gauge group. For any element $g \in \mathcal{G}_I$, let $z \in U(1)$ be its hypercharge component. Let $\{\mathbf{b}_0, \mathbf{b}_1\}$ be the standard basis for the space of charged-lepton singlets (which is identified with the space of right-handed Weyl spinors). The gauge action of $g$ on the basis vector $\mathbf{b}_k$ for $k \in \{0, 1\}$ is given by scalar multiplication: $$g \cdot \mathbf{b}_k = \bar{z}^6 \mathbf{b}_k$$ where $\bar{z}$ denotes the complex conjugate of $z$. This demonstrates that the gauge action is diagonal in the standard Weyl basis.

Let $\mathcal{G}_I = SU(3) \times SU(2) \times U(1)$ be the unquotiented Standard Model gauge group. For any two gauge group elements $g_1, g_2 \in \mathcal{G}_I$, let $z_1, z_2 \in U(1) \subset \mathbb{C}$ be their respective projections onto the $U(1)$ factor. The representation of $g_1$ on the space of charged-lepton singlets is equal to the representation of $g_2$ if and only if $\bar{z}_1^6 = \bar{z}_2^6$, where $\bar{z}$ denotes the complex conjugate.

Let $g$ be an element of the unquotiented Standard Model gauge group $\mathcal{G}_I = SU(3) \times SU(2) \times U(1)$, and let $z \in U(1)$ be its projection onto the $U(1)$ factor. The element $g$ lies in the kernel of the representation on the charged-lepton singlet space (i.e., it acts as the identity) if and only if $\bar{z}^6 = 1$, where $\bar{z}$ denotes the complex conjugate of $z$.

Consider the representation $\rho$ of the unquotiented Standard Model gauge group $\mathcal{G}_I = SU(3) \times SU(2) \times U(1)$ on the complex vector space of charged-lepton singlets $\text{LeptonSinglet}$. Let $Z_6$ be the central subgroup of $\mathcal{G}_I$ corresponding to the $\mathbb{Z}_6$ discrete quotient. Then $Z_6$ is a subgroup of the kernel of $\rho$: $$Z_6 \subseteq \ker(\rho)$$ This means that every element of the central subgroup $Z_6$ acts trivially on the charged-lepton singlet space.

For any valid discrete quotient $Q$ of the Standard Model gauge group, the associated central subgroup $N_Q \subseteq SU(3) \times SU(2) \times U(1)$ is contained within the kernel of the representation of the unquotiented gauge group on the space of charged-lepton singlets. This implies that every element in the subgroup $N_Q$ acts trivially on the charged-lepton singlet space.

For any supported global form of the Standard Model gauge group $\text{GaugeGroup}(Q)$ indexed by the quotient parameter $Q$, this definition provides the corresponding $(1, 1)_{-6}$ representation on the complex vector space of charged-lepton singlets $\text{LeptonSinglet}$. The representation is constructed by descending the unquotiented representation of $SU(3) \times SU(2) \times U(1)$ to the quotient group $\text{GaugeGroup}(Q)$, which is possible because the relevant central subgroups (such as $\mathbb{Z}_2$, $\mathbb{Z}_3$, or $\mathbb{Z}_6$) are contained within the kernel of the representation.