Physlib.Particles.StandardModel.Fermions.DownSinglet

The equivalence $\text{DownSinglet} \simeq \text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$ identifies the down-type singlet multiplet with its underlying representation as the tensor product of a right-handed Weyl spinor and a three-dimensional complex Euclidean space representing the color index. It provides the maps to wrap a tensor product into a `DownSinglet` and to extract the underlying tensor value from a `DownSinglet`.

The space $\text{DownSinglet}$, representing the target vector space of a down-type quark singlet multiplet in the Standard Model, is equipped with an additive commutative group structure. This structure is inherited from its underlying representation as the tensor product of right-handed Weyl fermions and the three-dimensional complex color space, $\text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$, via the equivalence `valEquiv`.

The space $\text{DownSinglet}$, representing the target vector space of a down-type quark singlet multiplet in the Standard Model, is equipped with a complex module structure. This defines $\text{DownSinglet}$ as a vector space over the field of complex numbers $\mathbb{C}$. The structure is inherited from its underlying representation as the tensor product of right-handed Weyl spinors and the three-dimensional complex color space, $\text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$, via the equivalence `valEquiv`.

This definition establishes the linear equivalence over the complex numbers $\mathbb{C}$ between the space of down-type quark singlets $\text{DownSinglet}$ and its underlying representation as the tensor product $\text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$. Here, $\text{RightHandedWeyl}$ represents the space of right-handed Weyl spinors (carrying the Lorentz index) and $\mathbb{C}^3$ (represented as `EuclideanSpace ℂ (Fin 3)`) represents the three-dimensional color space.

For any element $d$ in the $\text{DownSinglet}$ space, the linear equivalence $\text{valLinEquiv}$ applied to $d$ is equal to its underlying representation $d.\text{val}$ in the tensor product space $\text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$, where $\text{RightHandedWeyl}$ is the space of right-handed Weyl spinors and $\mathbb{C}^3$ is the three-dimensional color space.

Let $\text{DownSinglet}$ be the complex vector space representing the down-type quark singlet multiplet in the Standard Model. This space is defined as a wrapper around the tensor product $\text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$, where $\text{RightHandedWeyl}$ is the space of right-handed Weyl spinors and $\mathbb{C}^3$ (represented as $\text{EuclideanSpace } \mathbb{C} (\text{Fin } 3)$) is the color space. Given the linear equivalence $\text{valLinEquiv}: \text{DownSinglet} \cong \text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$, for any element $m$ in the tensor product space, the inverse mapping $\text{valLinEquiv}^{-1}$ is given by: $$\text{valLinEquiv}^{-1}(m) = \langle m \rangle$$ where $\langle m \rangle$ denotes the element $m$ as an inhabitant of the $\text{DownSinglet}$ type.

For any two elements $d_1, d_2$ in the $\text{DownSinglet}$ space, the underlying value of their sum is the sum of their individual values: $$(d_1 + d_2).\text{val} = d_1.\text{val} + d_2.\text{val}$$ where $\text{DownSinglet}$ is the vector space representing the down-type quark singlet multiplet, and the `.val` operation extracts the element's representation in the underlying tensor product space $\text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$.

For any complex scalar $r \in \mathbb{C}$ and any element $d$ in the $\text{DownSinglet}$ space, the underlying value of their scalar multiplication is equal to the scalar multiplication applied to the value of $d$: $$(r \cdot d).\text{val} = r \cdot d.\text{val}$$ where $\text{DownSinglet}$ is the vector space representing the down-type quark singlet multiplet, and the `.val` operation extracts the element's representation in the underlying tensor product space $\text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$.

The representation of the special linear group $SL(2, \mathbb{C})$ on the space $\text{DownSinglet}$, which represents the target vector space of a down-type quark singlet multiplet. This action is defined such that for any $\Lambda \in SL(2, \mathbb{C})$, the group acts via the right-handed Weyl representation on the spinor factor and acts trivially on the three-dimensional color space $\mathbb{C}^3$, based on the linear isomorphism $\text{DownSinglet} \cong \text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$.

This definition defines the group representation of the unquotiented Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ on the space of down-type quark singlets $\text{DownSinglet}$. The space $\text{DownSinglet}$ is linearly equivalent to the tensor product $W_R \otimes \mathbb{C}^3$, where $W_R$ is the space of right-handed Weyl spinors and $\mathbb{C}^3$ is the three-dimensional complex color space. For an element $g = (g_3, g_2, z) \in SU(3) \times SU(2) \times U(1)$, the representation $\rho(g)$ acts on a state $\psi \otimes v \in W_R \otimes \mathbb{C}^3$ as: $$\rho(g)(\psi \otimes v) = \bar{z}^2 (\psi \otimes g_3 v)$$ where $g_3 \in SU(3)$ acts on the color index, $\bar{z} = z^{-1}$ is the complex conjugate of the $U(1)$ phase $z$ (representing a hypercharge normalized to $6Y = -2$), and the $SU(2)$ component $g_2$ acts trivially.

Let $\text{DownSinglet}$ be the representation space for down-type quark singlets, which is isomorphic to the tensor product $W_R \otimes_{\mathbb{C}} \mathbb{C}^3$ of right-handed Weyl spinors $W_R$ and the three-dimensional complex color space $\mathbb{C}^3$. For any element $g$ in the unquotiented Standard Model gauge group $SU(3) \times SU(2) \times U(1)$, let $z \in U(1)$ be its $U(1)$ component and $g_3 \in SU(3)$ be its $SU(3)$ component. For any spinor $\psi \in W_R$ and color vector $v \in \mathbb{C}^3$, the gauge representation $\rho(g)$ acts on the pure tensor $\psi \otimes v$ as: $$\rho(g)(\psi \otimes v) = (\bar{z}^2 \psi) \otimes (g_3 v)$$ where $\bar{z}$ denotes the complex conjugate of the $U(1)$ phase $z$.

Let $\rho$ be the representation of the unquotiented Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ on the space of down-type quark singlets $\text{DownSinglet} \cong W_R \otimes \mathbb{C}^3$. Let $\{\psi_k\}_{k \in \{0, 1\}}$ be the basis for right-handed Weyl fermions and $\{e_i\}_{i \in \{0, 1, 2\}}$ be the standard basis for the complex color space. For any gauge group element $g \in \mathcal{G}$, the action of $\rho(g)$ on the tensor basis element $\psi_k \otimes e_i$ is given by: $$\rho(g)(\psi_k \otimes e_i) = \sum_{i' \in \{0, 1, 2\}} (\bar{z}^2 (g_3)_{i'i}) (\psi_k \otimes e_{i'})$$ where $g_3 \in SU(3)$ and $z \in U(1)$ are the respective components of $g$, $(g_3)_{i'i}$ denotes the matrix entry of $g_3$ at row $i'$ and column $i$, and $\bar{z}$ denotes the complex conjugate of the phase $z$.

Let $g_1, g_2$ be elements of the unquotiented Standard Model gauge group $SU(3) \times SU(2) \times U(1)$. Let $z_1, z_2 \in U(1) \subset \mathbb{C}$ be their respective $U(1)$ components and $M_1, M_2$ be their respective $SU(3)$ matrices. The gauge representation $\rho$ on the down-type quark singlet space $\text{DownSinglet}$ satisfies $\rho(g_1) = \rho(g_2)$ if and only if for all indices $i, i' \in \{1, 2, 3\}$, the hypercharge-weighted color matrix elements are equal: $$\bar{z}_1^2 (M_1)_{i'i} = \bar{z}_2^2 (M_2)_{i'i}$$ where $\bar{z}$ denotes the complex conjugate of the $U(1)$ phase $z$ (which represents the hypercharge contribution).

Let $g = (g_3, g_2, z)$ be an element of the unquotiented Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$. The element $g$ belongs to the kernel of the gauge representation $\rho$ on the down-type singlet space if and only if there exists a complex scalar $a \in \mathbb{C}$ such that the $SU(3)$ component is a scalar matrix $g_3 = a \mathbb{I}$ and the $U(1)$ component $z$ satisfies $a \bar{z}^2 = 1$, where $\bar{z}$ denotes the complex conjugate of $z$.

The central subgroup $\mathbb{Z}_6$ of the unquotiented Standard Model gauge group $SU(3) \times SU(2) \times U(1)$ is contained in the kernel of the gauge representation $\rho$ acting on the space of down-type quark singlets $\text{DownSinglet}$. This means that the $\mathbb{Z}_6$ subgroup acts trivially on the $(3, 1)_{-2}$ representation.

For any valid choice of a discrete quotient $Q \in \text{GaugeGroupQuot}$, the associated central subgroup $\Gamma_Q \subseteq SU(3) \times SU(2) \times U(1)$ is contained within the kernel of the representation $\rho$ of the unquotiented gauge group on the down-type quark singlet space $\text{DownSinglet}$. That is, $\Gamma_Q \subseteq \ker(\rho)$.

For a given quotient parameter $Q \in \text{GaugeGroupQuot}$, this definition provides the group representation of the global Standard Model gauge group $\text{GaugeGroup}(Q)$ on the space of down-type quark singlets $\text{DownSinglet}$. The space $\text{DownSinglet}$ carries the $(3, 1)_{-2}$ representation, where the indices denote the triplet representation of $SU(3)$, the singlet representation of $SU(2)$, and the $U(1)$ hypercharge $Y$ (normalized as $6Y = -2$). This representation is obtained by descending (lifting) the representation of the un-quotiented group $G_I = SU(3) \times SU(2) \times U(1)$ to the quotient group $\text{GaugeGroup}(Q) = G_I / \Gamma_Q$, utilizing the fact that the discrete central subgroup $\Gamma_Q$ is contained in the kernel of the action on $\text{DownSinglet}$.