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 Rμναβ, ∇μ∇νϕ, ∇ρFμν and up to quadratic in the vector field strength tensor Fμν; the second is up to linear in ∇μ∇νϕ, contains no second derivatives of vector field and metric, but allows for arbitrary functions/powers of Fμν. 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 G3(ϕ, X)□ϕ in the scalar sector with X = –∇μϕ∇μϕ/2; the Horndeski non-minimal coupling term <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <msub> <mi>w</mi> <mn>0</mn> </msub> <mfenced close=")" open="("> <mi>ϕ</mi> </mfenced> <msub> <mi>R</mi> <mtext mathvariant="italic">βδαγ</mtext> </msub> <msup> <mover accent="true"> <mi>F</mi> <mo stretchy="true">~</mo> </mover> <mi mathvariant="italic">αβ</mi> </msup> <msup> <mover accent="true"> <mi>F</mi> <mo stretchy="true">~</mo> </mover> <mi mathvariant="italic">γδ</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$ {w}_0\left(\phi \right){R}_{\beta \delta \alpha \gamma}{\overset{\sim }{F}}^{\alpha \beta}{\overset{\sim }{F}}^{\gamma \delta} $</EquationSource> </InlineEquation> in the vector field sector, where <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <msup> <mover accent="true"> <mi>F</mi> <mo stretchy="true">~</mo> </mover> <mi mathvariant="italic">μν</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$ {\overset{\sim }{F}}^{\mu \nu} $</EquationSource> </InlineEquation> is the Hodge dual of Fμν; and two interaction terms between the scalar and vector field sectors: [w1(ϕ, X)gρσ + w2(ϕ, X)∇ρϕ∇σϕ]∇β∇αϕ <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math display="inline"> <msup> <mover accent="true"> <mi>F</mi> <mo stretchy="true">~</mo> </mover> <mi mathvariant="italic">αρ</mi> </msup> <msup> <mover accent="true"> <mi>F</mi> <mo stretchy="true">~</mo> </mover> <mi mathvariant="italic">βσ</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$ {\overset{\sim }{F}}^{\alpha \rho}{\overset{\sim }{F}}^{\beta \sigma} $</EquationSource> </InlineEquation>. For the second subclass, which admits 11 possible terms, three of these four, excluding the Horndeski non-minimal coupling term proportional to w0(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.

Linear higher-order Maxwell-Einstein-Scalar theories

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2026年4月10日

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Mohammad Ali Gorji,
Shinji Mukohyama,
Pavel Petrov,
Masahide Yamaguchi

摘要

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∇_νϕ, ∇_ρF_μν and up to quadratic in the vector field strength tensor F_μν; the second is up to linear in ∇_μ∇_νϕ, 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 G₃(ϕ, X)□ϕ in the scalar sector with X = –∇_μϕ∇^μϕ/2; the Horndeski non-minimal coupling term $w_{0} (ϕ) R_{βδαγ} {\tilde{F}}^{αβ} {\tilde{F}}^{γδ}$ $ {w}_0\left(\phi \right){R}_{\beta \delta \alpha \gamma}{\overset{\sim }{F}}^{\alpha \beta}{\overset{\sim }{F}}^{\gamma \delta} $ in the vector field sector, where ${\tilde{F}}^{μν}$ $ {\overset{\sim }{F}}^{\mu \nu} $ is the Hodge dual of F_μν; and two interaction terms between the scalar and vector field sectors: [w₁(ϕ, X)g_ρσ + w₂(ϕ, X)∇_ρϕ∇_σϕ]∇_β∇_αϕ ${\tilde{F}}^{αρ} {\tilde{F}}^{βσ}$ $ {\overset{\sim }{F}}^{\alpha \rho}{\overset{\sim }{F}}^{\beta \sigma} $ . For the second subclass, which admits 11 possible terms, three of these four, excluding the Horndeski non-minimal coupling term proportional to w₀(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.