PhyslibAlpha.Particles.BeyondTheStandardModel.TwoHDM.SwapDoublet

In the context of the Two-Higgs-Doublet Model (2HDM), given a configuration $H = (\Phi_1, \Phi_2)$ where $\Phi_1, \Phi_2 \in \mathbb{C}^2$, the function `swapDoublet` is the $\mathbb{R}$-linear map that exchanges the two doublets: $$(\Phi_1, \Phi_2) \mapsto (\Phi_2, \Phi_1)$$ This map commutes with the gauge action and transforms gauge-invariant polynomial potentials into gauge-invariant polynomial potentials, effectively swapping the physical alignments of $\Phi_1$ and $\Phi_2$.

In the Two-Higgs-Doublet Model, for any configuration $\phi = (\Phi_1, \Phi_2)$, the first doublet component of the swapped configuration $\text{swapDoublet}(\phi)$ is equal to the second doublet $\Phi_2$ of the original configuration.

In the Two-Higgs-Doublet Model, for any Higgs field configuration $\phi$ consisting of two doublets $(\Phi_1, \Phi_2)$, the second doublet of the swapped configuration $\text{swapDoublet}(\phi)$ is equal to the first doublet of the original configuration $\phi$. That is, $(\text{swapDoublet } \phi).\Phi_2 = \phi.\Phi_1$.

For any configuration $\phi = (\Phi_1, \Phi_2)$ in the Two-Higgs-Doublet Model (2HDM), where $\Phi_1, \Phi_2 \in \mathbb{C}^2$ are the two complex scalar doublets, applying the doublet swap map $\text{swapDoublet}$ twice returns the original configuration: $$\text{swapDoublet}(\text{swapDoublet}(\phi)) = \phi$$

For any element $g$ of the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ and any configuration $\Phi = (\Phi_1, \Phi_2)$ in the Two-Higgs-Doublet Model, the `swapDoublet` map, which exchanges the two doublets, commutes with the gauge action: $$\text{swapDoublet}(g \cdot \Phi) = g \cdot \text{swapDoublet}(\Phi)$$

For any configuration $\phi$ of the two-Higgs-doublet model, the $0$-th component of its Gram vector (indexed by `Sum.inl 0`) remains unchanged under the $\mathbb{R}$-linear map `swapDoublet` that exchanges the two Higgs doublets $\Phi_1$ and $\Phi_2$. That is, the $0$-th component of the Gram vector of the swapped configuration is equal to that of the original configuration.

For any configuration $\phi = (\Phi_1, \Phi_2)$ of the two Higgs doublet model, let $r \in \mathbb{R}^4$ be its Gram vector. The components $r_\mu$ (for $\mu \in \{0, 1, 2, 3\}$) are defined by the expansion of the Gram matrix $G$ in the Pauli basis $\{\sigma_0, \sigma_1, \sigma_2, \sigma_3\}$ such that $G = \frac{1}{2} \sum_{\mu=0}^3 r_\mu \sigma_\mu$. Let $\text{swapDoublet}$ be the $\mathbb{R}$-linear map that exchanges the two doublets $(\Phi_1, \Phi_2) \mapsto (\Phi_2, \Phi_1)$. Then the $r_1$ component of the Gram vector (indexed by $\text{Sum.inr } 0$) is invariant under this transformation: $$(\text{swapDoublet } \phi).r_1 = \phi.r_1$$

For any configuration $\phi = (\Phi_1, \Phi_2)$ of the two Higgs doublet model, let $\text{gramVector}(\phi) \in \mathbb{R}^4$ be its Gram vector with components $r_\mu$. Under the $\mathbb{R}$-linear map `swapDoublet` that exchanges the two doublets $(\Phi_1, \Phi_2) \mapsto (\Phi_2, \Phi_1)$, the $r_2$ component of the Gram vector (indexed by `Sum.inr 1`) flips its sign: $$\text{gramVector}(\text{swapDoublet}(\phi))_2 = - \text{gramVector}(\phi)_2$$

In the two-Higgs-doublet model, for any configuration $\phi = (\Phi_1, \Phi_2)$, let $r = (r_0, r_1, r_2, r_3)$ be its associated Gram vector in $\mathbb{R}^4$. The $\mathbb{R}$-linear map `swapDoublet` exchanges the two doublets, $(\Phi_1, \Phi_2) \mapsto (\Phi_2, \Phi_1)$. This theorem states that under this exchange, the third spatial component of the Gram vector $r_3$ (corresponding to the index `Sum.inr 2`) changes sign: $$(\text{swapDoublet } \phi).\text{gramVector}_3 = - \phi.\text{gramVector}_3$$

Let $V: \text{TwoHiggsDoublet} \to \mathbb{R}$ be an effective potential. If $V$ has a maximum mass dimension less than or equal to $n$, then the potential obtained by swapping the doublets, $\phi \mapsto V(\text{swapDoublet}(\phi))$, also has a maximum mass dimension less than or equal to $n$. An effective potential is defined to have a maximum mass dimension $\le n$ if it can be represented as a multivariate polynomial of total degree at most $n$ in the real-linear components of the field configurations, and $\text{swapDoublet}$ is the $\mathbb{R}$-linear map that exchanges the two Higgs doublets $(\Phi_1, \Phi_2) \mapsto (\Phi_2, \Phi_1)$.

Let $V$ be an effective potential in the two-Higgs-doublet model (2HDM). If $V$ is gauge-invariant, then the potential formed by pre-composing $V$ with the doublet-swapping map, defined by $\phi \mapsto V(\text{swapDoublet}(\phi))$, is also gauge-invariant. Here, $\text{swapDoublet}$ is the $\mathbb{R}$-linear map that exchanges the two Higgs doublets $(\Phi_1, \Phi_2) \mapsto (\Phi_2, \Phi_1)$.