Cubic Scalar Bulk Couplings Overview

• Cubic scalar bulk couplings are effective three-field interactions in higher-dimensional theories, linking bulk dynamics to conformal boundary data.
• They are computed via techniques such as SYK three-point matching, KK reductions in RS models, and minimal coupling in higher-spin formulations, revealing universal symmetry properties.
• These couplings inform theoretical predictions by imposing selection rules, influencing interaction strengths, and clarifying bulk reconstruction in holographic and extra-dimensional contexts.

Cubic scalar bulk couplings denote the effective interactions in the bulk involving three scalar fields, typically arising in higher-dimensional gravitational theories, holographic dualities, and Kaluza-Klein reductions. These couplings are central to understanding the structure of bulk AdS theories reconstructed from conformal data, the phenomenology of extra-dimensional models, and the universality and vanishing theorems in string/M-theory compactifications. The explicit form of such couplings is model-dependent but shares universal structural features across different frameworks.

1. Construction of Cubic Scalar Bulk Couplings in Holographic Dualities

In the context of holography, particularly the AdS/CFT correspondence, cubic scalar bulk couplings are determined by matching bulk interaction terms to CFT three-point functions. For the Sachdev-Ye-Kitaev (SYK) model, the AdS$_2$ bulk dual contains a tower of massive scalar fields $\phi_n$, each dual to a singlet bilinear operator ${\cal O}_n$ of scaling dimension $h_n$ in the boundary theory. The cubic couplings $g_{nmk}$ are fully determined from the six-point function of the boundary Majorana fermions, which, via OPE and conformal limits, reduces in the bulk to an interaction term

$$S_{\rm int} = \frac{1}{\sqrt N} \sum_{m,n,k}g_{mnk} \int d^2x\,\sqrt{g}\; \phi_m \phi_n \phi_k\,,$$

with $g_{mnk}$ expressed in terms of the SYK data, including spectral parameters, OPE coefficients, and universal functions of the scaling dimensions (Gross et al., 2017). The couplings possess full permutation symmetry in $(n,m,k)$ and, in the large-$q$ limit, admit closed-form simplifications involving rational expressions in $1/q$ and triple sums that match those found in generalized free field theories: $$g_{nmk}^{(1)}|_{q\gg1} = -(-1)^{n+m+k} \frac{16}{\sqrt\pi\,q(\epsilon_n+\epsilon_m+\epsilon_k)\alpha_n\alpha_m\alpha_k},$$ with $\epsilon_n$ and $\alpha_n$ explicit functions of $n$ and $q$. The planar and contact contributions scale differently for large mode numbers, leading to distinctive physical regimes.

2. Higher Spin and General Covariant Classification in Constant Curvature Backgrounds

Cubic couplings between scalars and higher-spin gauge fields in constant curvature (A)dS backgrounds are generated via minimal coupling to conserved currents, constructed explicitly for each spin $r$ as

$$J^{(r)}_{\mu_1\cdots\mu_r}(x)=\sum_{m=0}^{\lfloor r/2 \rfloor}\frac{(-1)^m}{R^{2m}\,c_m}\,\nabla_{(\mu_1}\cdots\nabla_{\mu_{r-2m}}\phi^*(x)\; \nabla_{\mu_{r-2m+1}}\cdots\nabla_{\mu_r)}\phi(x),$$

satisfying the divergence-free condition on-shell (Bekaert et al., 2010). The cubic interaction is then

$$S_{\rm int} = \sum_{r=0}^\infty \frac{1}{r!}\int d^dx\,\sqrt{|g|}\;h^{(r)\,\mu_1\cdots\mu_r}(x)\;J^{(r)}_{\mu_1\cdots\mu_r}(x)\,,$$

which can be compactly recast in terms of generating functions or as a Weyl-quantized matrix element in the ambient formalism. The construction remains valid for all $d>2$ provided the scalar mass satisfies the Breitenlohner–Freedman stability bound.

A unique aspect in $d=3$ is the classification of cubic vertices: for triples $(s,0,0)$ (one spin-$s$ field, two scalars), the traceless–transverse basis gives a unique derivative structure: $$\mathcal{L}^{(3)}_{s,0,0}=g_s\;\phi_{\mu_1\cdots\mu_s}\;\partial^{\mu_1}\cdots\partial^{\mu_s}\left(\varphi_2\,\varphi_3\right),$$ which is the only possible Lorentz-invariant current coupling in three dimensions, with $s$ derivatives and a single coupling constant $g_s$ at each spin (Mkrtchyan, 2017). The $s=0$ case yields the canonical cubic self-coupling for scalars.

3. Kaluza-Klein Reductions and Warped Extra Dimensions

In bulk theories with extra dimensions, scalar cubic couplings emerge from dimensional reduction of higher-dimensional actions. Consider a 5D scalar with bulk cubic self-interaction in a Randall–Sundrum (RS) warped background: $$S_5 = \int d^4x\int_{-\pi R}^{\pi R}dy\,\sqrt{-G}\left\{ \frac12 G^{MN}\partial_M\Phi\partial_N\Phi - \frac12 m^2\Phi^2 - \frac{\lambda_5}{3!}\Phi^3\right\}.$$ Upon KK decomposing $\Phi(x,y)=\sum_n\phi_n(x)f_n(y)$ using profiles orthonormal with respect to the warped measure, the four-dimensional cubic coupling among KK modes is

$$\lambda_{nmk} = \lambda_5 \int_{-\pi R}^{\pi R} dy\,e^{-4k|y|}f_n(y)f_m(y)f_k(y).$$

The resulting effective couplings can be exponentially suppressed or enhanced depending on the localization of the profiles and the warp factor. For the massless zero-mode, the cubic self-coupling receives double suppression by the exponential warp factor, whereas interactions involving excited modes (localized near the TeV brane) can be unsuppressed or even amplified (Chakraborty et al., 2014). For realistic RS parameters, mixings such as $\lambda_{011}$ and $\lambda_{111}$ can be phenomenologically relevant and fall in the TeV$^{-1}$ range.

4. Universal Structure from Exceptional Field Theory and Selection Rules

Exceptional Field Theory (ExFT) organizes the computation of scalar bulk couplings in AdS vacua upliftable from maximal gauged supergravities. Scalar fluctuations $\phi_i(x,y)$ expand into 4D fields and harmonics $Y_\Sigma(y)$ on the internal manifold $M$: $$\phi_i(x, y) = \varphi^{\alpha\Sigma}(x) Y_\Sigma(y).$$ The universal cubic coupling decomposes as

$$g_{i j k} = N_{i j k}\,\mathcal{I}_{i j k},\quad \mathcal{I}_{i j k} = \int_M Y_i Y_j Y_k\,d\text{vol}_M,$$

where $N_{i j k}$ is determined by group-theoretic (gauged SUGRA) data and $\mathcal{I}_{i j k}$ by a triple-overlap invariant of harmonics (Duboeuf et al., 2023). Block-diagonality in KK level and selection rules for nonvanishing $\mathcal{I}_{i j k}$ strictly constrain which cubic couplings survive. In AdS$_5\times S^5$, explicit combinatorial formulas (using Gamma functions or factorials) encode all nonzero cubic couplings of chiral primaries and manifest the vanishing of extremal and near-extremal couplings predicted by earlier conjectures.

Framework	Cubic Coupling Structure	Selection Rule/Constraint
SYK/AdS$_2$ Bulk	$g_{nmk}$ in terms of $h_n$, OPE data	Fully symmetric, holomorphic in $h_i$
RS Warped Models	$\lambda_{nmk} = \lambda_5 \int e^{-4k|y|}f_nf_mf_k$	Profile localization, warp factor
ExFT/AdS$\times S$	$g_{i j k} = N_{i j k}\, \mathcal{I}_{i j k}$	$\mathcal{I}_{i j k}=0$ for forbidden $k$

5. Scalar Couplings Involving Additional Bulk Fields

Couplings involving scalars and bulk antisymmetric tensor fields, such as dilaton–3-form interactions in RS-type models, are induced by terms like $e^{\phi/M^{3/2}}\,Y_{MNPQ}Y^{MNPQ}$. KK reduction and mode expansion yield a four-dimensional effective cubic interaction,

$$\mathcal{L}_{\rm int}=g_3\,\Phi(x)\,A_{\rho\sigma\lambda}(x)\,A^{\rho\sigma\lambda}(x),$$

with $g_3$ computed as a specific overlap integral involving the dilaton and 3-form wavefunctions in the warped background. For the lowest-lying modes, $g_3$ is of order TeV$^{-1}$, thus potentially accessible at the LHC through Drell–Yan type production processes (Alencar et al., 2010).

6. Normalization, Matching, and Boundary Data

Matching the overall cubic coupling constants in the bulk to boundary correlator normalization is crucial for holographic dualities. For AdS$_{d+1}$ scalar self-interactions,

$$S_{\rm cubic} = g_{000} \int_{\rm AdS_{d+1}} d^{d+1}x\sqrt{g}\, \Phi_0(x)^3,$$

where $g_{000}$ is fixed by requiring that the Witten diagram matches the CFT OPE coefficient, yielding

$$g_{000} = \frac{1}{\sqrt N}\; \frac{\pi^{(d-3)/4}2^{(3d-1)/2}\left[\Gamma\left(\frac{d-1}{2}\right)\right]^{3/2}}{\Gamma(d-3)},$$

or, equivalently, as a function of the three-point CFT coefficient (Sleight et al., 2016).

7. Phenomenological and Theoretical Significance

Cubic scalar bulk couplings play a central role in several domains:

• In holographic dualities (e.g., SYK/AdS$_2$), they encode higher-point boundary correlators and clarify bulk reconstruction at finite $N$ or large $q$.
• In extra-dimensional models, the magnitude and structure of these couplings drive the phenomenology of KK excitations and their collider signatures.
• In string/M-theory and ExFT, they organize selection rules, vanishing theorems, and the systematics of couplings, restricting the allowed interactions beyond group-theoretic symmetry constraints.
• Minimal couplings to higher-spin fields and their generating functions provide a comprehensive catalog of interactions in any dimension, with unitarity and geometric constraints arising in curved backgrounds.

The detailed forms of the couplings, their symmetry properties, vanishing loci, and dependence on background parameters (e.g., $q$ in SYK, warp factor in RS, or internal harmonic indices in ExFT) collectively define the rich structure of cubic scalar interactions in modern high-energy theoretical physics.