Linear higher-order Maxwell-Einstein-Scalar theories

A bstract In the context of the Higher-Order Maxwell-Einstein-Scalar (HOMES) theories, which are invariant under spacetime diffeomorphisms and U(1) gauge symmetry, we study two broad subclasses: the first is up to linear in R μναβ , ∇ μ ∇ ν ϕ , ∇ ρ F μν and up to quadratic in the vector field strength tensor F μν ; the second is up to linear in ∇ μ ∇ ν ϕ , contains no second derivatives of vector field and metric, but allows for arbitrary functions/powers of F μν . Under these assumptions, we systematically derive the most general form of the action that leads to second-order (or lower) equations of motion. We prove that, among 41 possible terms in the first subclass, only four independent higher-derivative terms are allowed: the kinetic gravity braiding term G 3 ( ϕ , X )□ ϕ in the scalar sector with X = –∇ μ ϕ ∇ μ ϕ /2; the Horndeski non-minimal coupling term $$ {w}_0\left(\phi \right){R}_{\beta \delta \alpha \gamma}{\overset{\sim }{F}}^{\alpha \beta}{\overset{\sim }{F}}^{\gamma \delta} $$ w 0 ϕ R βδαγ F ~ αβ F ~ γδ in the vector field sector, where $$ {\overset{\sim }{F}}^{\mu \nu} $$ F ~ μν is the Hodge dual of F μν ; and two interaction terms between the scalar and vector field sectors: [ w 1 ( ϕ , X ) g ρσ + w 2 ( ϕ , X )∇ ρ ϕ ∇ σ ϕ ]∇ β ∇ α ϕ $$ {\overset{\sim }{F}}^{\alpha \rho}{\overset{\sim }{F}}^{\beta \sigma} $$ F ~ αρ F ~ βσ . For the second subclass, which admits 11 possible terms, three of these four, excluding the Horndeski non-minimal coupling term proportional to w 0 ( ϕ ), are allowed. These independent terms serve as the building blocks of each subclass of HOMES. Remarkably, there is no higher-derivative parity-violating term in either subclass. Finally, we propose a new generalization of higher-derivative interaction terms for the case of a charged complex scalar field.