Scalar Cubic Lagrangian

• Scalar cubic Lagrangians are theories with cubic interactions of scalar fields and derivatives that maintain second-order equations to prevent Ostrogradsky ghosts.
• They play a pivotal role in modified gravity, Galileon formulations, and cosmological bias modeling through precise algebraic and degeneracy conditions.
• Unified frameworks extend these Lagrangians to canonical, phantom, and tachyonic fields, enabling versatile applications in cosmology and quantum field theory.

A scalar cubic Lagrangian refers to a class of theories in which a scalar field, or fields, appears with interactions that are cubic in fields and/or their derivatives. Such Lagrangians are central in contemporary theoretical physics, notably in the context of Galileon theories, higher-derivative scalar-tensor gravity, large-scale structure bias modeling, and double-copy constructions in gauge/gravity duality. The defining feature is the presence of terms cubic in either the scalar field itself, its derivatives, or in second derivatives, often constructed so as to preserve desirable properties such as the absence of Ostrogradsky ghosts, or to encode non-trivial coupling structures essential for cosmology, gravitational theory, and quantum field theory.

1. Fundamental Structure of Scalar Cubic Lagrangians

Scalar cubic Lagrangians commonly take the form of either interaction terms cubic in the scalar field $\phi$, its derivatives, or combinations thereof. The prototypical example is the cubic Galileon Lagrangian: $\mathcal{L}_3 = -\frac{1}{2} \phi^2 \Box \phi$ where $\Box \phi$ denotes the d'Alembertian acting on $\phi$. This term involves higher derivatives but is arranged such that the equations of motion remain second order in derivatives, thus avoiding the Ostrogradsky instability. The key property in such construction is ensuring that the variation leads to equations of motion of at most second order, which is non-trivial when dealing with cubic (or higher-order) derivative terms.

An important extension appears in scalar-tensor theories where the action includes terms cubic in second derivatives: $$\mathcal{L}_3 = C^{(3)}_{\mu\nu\rho\sigma\alpha\beta}(\phi,X,\nabla\phi,g) \cdot \phi^{\mu\nu} \phi^{\rho\sigma} \phi^{\alpha\beta}$$ with $\phi^{\mu\nu} = \nabla^\mu\nabla^\nu \phi$, $X = g^{\mu\nu}\nabla_\mu\phi\nabla_\nu\phi$, and the tensor $C^{(3)}$ built from allowable contractions that respect coordinate invariance (Achour et al., 2016).

In the context of Horndeski, the most general cubic-order Lagrangian admitting scaling solutions is given by: $\mathcal{L} = X g_2(Y) - g_3(Y) \Box \phi$ with $X = -\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi$, $Y = X e^{\lambda\phi}$, and $g_2$, $g_3$ arbitrary functions (Frusciante et al., 2018).

2. Pathologies and Degeneracy Conditions

The introduction of cubic (or higher) derivatives generally raises the risk of unwanted degree-of-freedom propagation—namely, Ostrogradsky ghosts. The original cubic Galileon structure is carefully designed to prevent this, but when supersymmetry is introduced, multiple fields must be included, resulting in higher-order equations and ghost modes.

For a supersymmetric $N=1$ cubic Galileon, the minimal complex scalar structure $A = \tfrac{1}{\sqrt{2}}(\phi + i\xi)$ yields the SUSY Lagrangian: $$\mathcal{L}_3^{SUSY} = -\frac{1}{2}\left[\phi^2\Box\phi - \xi^2\Box\phi + 2\phi\xi\Box\xi\right]$$ which, upon diagonalization, demonstrates third time derivatives and a ghost mode in the quadratic fluctuation Lagrangian (Koehn et al., 2013). The effective mass $m_g$ of the ghost mode is background-dependent, but an EFT cutoff $\Lambda < m_g$ can render the theory valid below the new scale.

In more general higher-order scalar-tensor theories, degeneracy of the Hessian with respect to all velocities ensures that at most three propagating degrees of freedom persist—two tensors and one scalar. Degeneracy is imposed via algebraic constraints on the coefficients of the cubic contractions, enforced by requiring that the Hessian matrix possesses a nontrivial null eigenvector (Achour et al., 2016).

3. Classification and Construction in Scalar-Tensor Gravity

In scalar-tensor gravity, cubic Lagrangians are classified as either minimally or non-minimally coupled:

• Minimally coupled (f₃=0): Seven classes (3M-I to 3M-VII) are enumerated, including familiar beyond Horndeski examples. For instance, the cubic beyond Horndeski term:

$$L_{BH} = f(\phi,X) X \left[(\Box\phi)^3 - 3\Box\phi\,\phi_{\mu\nu}\phi^{\mu\nu} + 2\phi_{\mu\nu}\phi^{\nu\rho}\phi^{\mu}_{\rho}\right]$$

subject to constraints ensuring Hessian degeneracy.

• Non-minimally coupled (f₃\neq0): Two main classes (3N-I and 3N-II) in which part of the cubic tensor coefficients are free but related by algebraic relations.

Successful theory construction requires matching null eigenvectors of quadratic and cubic sectors if quadratic terms are present, a nontrivial constraint for viable model-building (Achour et al., 2016).

Cubic Lagrangians in Horndeski theories are further generalized by explicit scaling analysis, showing that Lagrangians of the form $Xg_2(Y)-g_3(Y)\Box\phi$ together with specific field-matter couplings $Q(\phi)$ admit a rich set of cosmological sequences, including scaling solutions and $\phi$-matter-dominated epochs ($\phi$MDE), with physical stability checked via ghost and Laplacian conditions (Frusciante et al., 2018).

4. Applications in Cosmological Bias and Large-Scale Structure

Cubic operators and their Lagrangian realizations are crucial in modeling halo bias and matter clustering in cosmology. Halo overdensity fields are expanded up to cubic order, involving bias parameters associated with both local and non-local operators (e.g. tidal terms and Galileons).

The extraction of bias parameters up to cubic order from simulations or data employs cross-correlations of basis sets of operators—such as $\delta^3$, $\mathcal{G}_3$, $\mathcal{G}_2\delta$, and $\Gamma_3$—with halo fields and expresses the results in terms of auto- and cross-spectra. This methodology rigorously quantifies non-local cubic contributions and is used for detection in Eulerian space; in Lagrangian space, non-local quadratic tidal terms are always present, while cubic non-local bias terms are only weakly detected at high halo mass (Abidi et al., 2018).

The underlying structure of these bias models directly links to the formalism of scalar cubic Lagrangians, showing that relevant physical operators in large-scale structure correspond to non-trivial cubic interaction terms in effective field theory language.

5. Roles in Double-Copy Gravity and Higher-Spin Interactions

In the BRST and double-copy approaches, scalar cubic Lagrangians have pivotal significance in the gravity/gauge correspondence. For instance, the BCJ procedure enables the construction of a cubic gravity Lagrangian by squaring two BRST-complete Yang-Mills theories. Off-shell dictionary definitions bundle the graviton, Kalb–Ramond two-form, and dilaton scalar into a rank-two field expressive of cubic interactions of the form: $$\mathcal{L}_{grav}^{(3)} \sim h^{\mu\nu}[h^{\rho\sigma}\partial_\rho\partial_\sigma h_{\mu\nu} - \partial_\mu h^{\rho\sigma}\partial_\nu h_{\rho\sigma} - ...]$$ with precise terms inherited from the cubic vertices in the gauge sector (Borsten et al., 2020, Ferrero et al., 2020).

In higher-spin theories, the scalar cubic vertex connects two massless scalars and one massive/spin-$s$ field. The BRST approach constructs the interaction vertex as a sum over trace operator insertions: $$|V^{(3)}\rangle_{m,(0,0),s} = \sum_{j=0}^{[s/2]} t_j U^{(s)}_j \mathcal{L}^{(3)}_{s-2j}$$ ensuring Lorentz covariance, gauge invariance, and correct irreducibility constraints, with explicit projection operators removing contradictory Lagrangian dynamics and maintaining degree-of-freedom counting (Buchbinder et al., 2022).

6. Unified Lagrangians for Canonical and Non-Canonical Scalar Fields

Unified frameworks for scalar cubic Lagrangians have been developed to encapsulate various scalar field models—canonical, phantom, and tachyonic—within a single parameterized Lagrangian. The general form involves a parameter $\alpha$ that governs the kinetic structure: $$\mathcal{L}_{unified} = f(\alpha)\, \partial_\mu\phi\,\partial^\mu\phi - V(\phi) \sqrt{1 - \alpha\, \partial_\mu\phi\,\partial^\mu\phi}$$ with $f(\alpha) = \frac{1}{2}(1 + \alpha)$, allowing for smooth transition between tachyon ($\alpha=-1$), quintessence ($\alpha=0$), and phantom ($\alpha=1$) forms (Joshi, 2023, Joshi et al., 2023).

For example, setting $\alpha=1$ yields the standard tachyonic Lagrangian, while $\alpha=-1$ or $0$ reproduces the canonical and phantom cases. Mathematical consistency of the equations of motion under this generalization is shown, providing a practical foundation for cosmological modeling of dark energy and early universe dynamics.

7. Summary and Perspectives

Scalar cubic Lagrangians occupy a central role across multiple research domains—higher-derivative gravity, supersymmetry, cosmological bias, quantum field theory, and gauge/gravity duality. Their construction is governed by the competing demands of avoiding ghost degrees of freedom (via second-order equations or degeneracy conditions), modeling non-trivial physical interactions (as in dark energy and inflation), capturing observational phenomena (halo bias and clustering), and maintaining consistency in extended theories (BRST invariance, irreducibility for higher-spin, double-copy procedures).

Recent advances have revealed deep connections between scalar cubic structures in effective field theory, cosmological modeling, and fundamental quantum gravity frameworks, with precise algebraic conditions dictating their stability and physical viability. Unified approaches integrating canonical and non-canonical scalar sectors have further enhanced the versatility of these models in theoretical and observational cosmology. The continued development of scalar cubic Lagrangian theory remains integral to progress in fundamental physics, with ongoing refinement in classification, application, and unification.