PhyslibAlpha.Particles.BeyondTheStandardModel.TwoHDM.EffectivePotential

The space of effective potentials for the two-Higgs-doublet model, consisting of real-valued functions $V: \text{TwoHiggsDoublet} \to \mathbb{R}$ that map a Higgs field configuration to a real number.

An effective potential $V: \text{TwoHiggsDoublet} \to \mathbb{R}$ is invariant under the global gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ if, for every gauge transformation $g \in \mathcal{G}$ and every Higgs field configuration $\phi \in \text{TwoHiggsDoublet}$, the potential satisfies $V(g \cdot \phi) = V(\phi)$.

An effective potential $V: \text{TwoHiggsDoublet} \to \mathbb{R}$ has a maximum mass dimension less than or equal to $n$ if it can be represented as a multivariate polynomial in the real-linear components of the field configurations with a total degree of at most $n$. Specifically, there exists a multivariate polynomial $p$ over $\mathbb{R}$ whose variables are elements of the dual space $(\text{TwoHiggsDoublet})^*$ (the space of real-linear functionals on field configurations), such that for every configuration $\phi$, $V(\phi)$ is the evaluation of $p$ at the values of these functionals at $\phi$, and the total degree of $p$ is less than or equal to $n$.

Let $V: \text{TwoHiggsDoublet} \to \mathbb{R}$ be an effective potential of the two-Higgs-doublet model with a maximum mass dimension less than or equal to $n$. This means $V$ can be represented as a multivariate polynomial of degree at most $n$ in the real-linear functionals of the field configurations. For any finite index set $\iota$ and any real-linear map $L: \mathbb{R}^\iota \to \text{TwoHiggsDoublet}$ (representing a linear parametrization of the field), there exists a multivariate polynomial $P$ over $\mathbb{R}$ with variables indexed by $\iota$ such that for all $a \in \mathbb{R}^\iota$, the potential evaluated at the parameterized configuration $L(a)$ is equal to the evaluation of the polynomial $P$ at $a$, i.e., $V(L(a)) = P(a)$.