D-Dimensional 4-Point Higher-Derivative Amplitudes

• The paper establishes a unique polynomial basis using elementary symmetric Mandelstam invariants, facilitating the classification of higher-derivative corrections in effective field theories.
• It utilizes a modular assembly of scalar, color, and tensor blocks, ensuring gauge invariance and enabling the double-copy construction for gravity and gauge amplitudes.
• The analysis demonstrates that locality, Bose/Fermi symmetry, and permutation invariance precisely constrain the amplitude structure, allowing systematic operator counting and anomaly matching.

D-dimensional four-point higher-derivative contact amplitudes are local on-shell scattering amplitudes constructed in arbitrary spacetime dimension $D$ from effective field theories containing operators with increasing numbers of derivatives. Such amplitudes play a central role in classifying the possible higher-dimensional corrections to gauge theory, gravity, and matter interactions both in flat space and curved backgrounds, and underlie the systematic construction of effective operator bases relevant for phenomenology, anomaly matching, and string theory expansions. Their structure at four points is governed by general principles—locality, Bose/Fermi symmetries, and permutation invariance—which severely constrain their form. The resulting amplitudes are characterized by universal polynomial structures in Mandelstam invariants, built from a finite basis of independent tensor and color/kinematic blocks, and are compatible with algebraic frameworks such as color-kinematics duality and the double-copy construction.

1. Polynomial Structure and Symmetry Constraints

The D-dimensional four-point contact amplitude $A_4(s, t, u)$ is a symmetric polynomial, where the on-shell momenta $p_i$ satisfy $p_i^2=0$ and total momentum conservation, and the Mandelstam invariants generate all kinematic dependence

$$s = -(p_1 + p_2)^2, \qquad t = -(p_1 + p_3)^2, \qquad u = -(p_1 + p_4)^2, \qquad s+t+u=0.$$

Locality requires $A_4(s, t, u)$ to be a polynomial in these invariants; Bose symmetry imposes invariance under permutations of external legs, restricting the amplitude to lie in the ring of symmetric polynomials modulo $s + t + u$:

$$A_4(s,t,u)\in \mathbb{C}[s, t, u]^{S_3}/\langle s + t + u\rangle,$$

where $S_3$ is the permutation group on three objects. The minimal basis is provided by the elementary symmetric polynomials (for $D\geq4$)

$\sigma_2 = st + tu + us, \qquad \sigma_3 = stu,$

with the polynomial ring $$\mathbb{C}[s, t, u]^{S_3}/\langle s + t + u\rangle \cong \mathbb{C}[\sigma_2, \sigma_3]$$ (Bonifacio et al., 2021). Thus, for a fixed maximal derivative order $2N$, every allowed four-point contact amplitude can be written uniquely as

$$A_4(s, t, u) = \sum_{a, b \geq 0,\,2a + 3b \leq 2N} c_{a, b}\;\sigma_2^{a}\,\sigma_3^{b}.$$

The number of independent monomials at each degree is counted by integer solutions of $2a+3b=d$; equivalently, by the Hilbert series $H(t) = \frac{1}{(1-t^2)(1-t^3)}$.

For external fields with spin or internal quantum numbers (vectors, gravitons, fermions), the full amplitude further factors into kinematic, color, and polarization structures, each with their own algebraic constraints—see below.

2. Tensor and Kinematic Bases for Gauge and Gravity Amplitudes

For vectors and gravitons, the construction uses gauge-invariant tensor structures built from linearized field strengths and Riemann tensors, projected onto definite permutation parity. For four-point graviton contact amplitudes in higher-derivative gravity,

$$S_\text{eff} = \int d^D x \sqrt{-g}\left\{ \frac{1}{2\kappa_D^2}R + \sum_{p=2}^k \alpha_{2p}I_{2p}[R_{abcd}] \right\},$$

where $I_{2p}$ are independent gauge-invariant contractions of $p$ Riemann tensors, the amplitude for external transverse-traceless gravitons simplifies to (Shawa et al., 2017)

$$A_4^{(2k)}(s, t, u) = \alpha_{2k}\;F_k(s, t, u)\;T_\text{ids},$$

where $F_k(s, t, u) = \sum_{a + b + c = k} s^a t^b u^c$, and $$T_\text{ids} = \epsilon_{xy}^1 \epsilon_{xy}^2 \epsilon_{xy}^3 \epsilon_{xy}^4$$ is the overall polarization contraction.

The total amplitude, aggregating all derivative orders up to $2k$, is

$$A_4(s, t, u) = T_{\text{ids}} \left[ \frac{1}{2\kappa_D^2}(s^2 + t^2 + u^2) + \alpha_6(s - t - u)(s^2 + t^2 + u^2) + \alpha_8(s^4 + t^4 + u^4 + s^2 t^2 + s^2 u^2 + t^2 u^2) + \cdots \right],$$

which holds for general $D$.

In gauge theory, the amplitude is organized in terms of color factors ($c_s, c_t, c_u$), kinematic building blocks (often Jacobi-satisfying numerators), and fully permutation-invariant scalar polynomials. Contact terms must be local: kinematic numerators contain channel invariants to cancel propagator poles,

$A_4^\text{gauge}(s, t, u) = \sum_{\text{channels}} c_g\,n_g(s, t, u)/g, \quad\text{with $n_g \sim g\, (\text{or higher})$},$

where $n_g$ are algebraic combinations of $\sigma_2, \sigma_3$ and structure tensors ($J^{(0)}, J^{(1)}, J^{(2)}$ as described in (Carrasco et al., 2019)).

For amplitudes with external matter, all building blocks are classified using a modular approach—scalar polynomials, explicit color tensors, and a finite table of spinor or field-strength “spin blocks”—for a complete D-dimensional basis (Carrasco et al., 7 Nov 2025).

3. Modular Assembly: Color, Spin, and Scalar Building Blocks

A rigorous and scalable framework for assembling general D-dimensional four-point higher-derivative contact amplitudes is provided by the modular “LEGO-like” construction (Carrasco et al., 7 Nov 2025). This framework consists of:

• Scalar Mandelstam polynomials: $s$, $t-u$, and masses, organized into spaces $\mathcal{P}_D^{(h_1|h_2)}$ labeled by definite permutation parities of external legs.
• Gauge-invariant field-strength/kinematic blocks: All vector and tensor amplitudes are built from linearized field strengths $F_i^{\mu\nu}$ and their traces, with gauge invariance manifest by construction.
• Spinor multilinears: Fermionic blocks spanning the full $D$-dimensional Clifford basis, with Fierz and EOM reductions applied to reach a minimal independent set.
• Color-weight structures: All possible four-point color tensors, including adjoint and symmetric-trace $d^{a_1 a_2 a_3 a_4}$ factors, are classified and combined with scalar blocks.
• Permutation-invariance basis: Transition to elementary symmetric polynomials $\sigma_2$, $\sigma_3$ renders permutation symmetry manifest.

The full amplitude is then spanned by the direct sum

$$\mathcal{A}_D^{(h_1|h_2)}(1234) \in \bigoplus_{d_1 + d_2 = D,\,p_i, q_i = \pm 1} \left[\, \mathcal{C}_0^{(p_1|p_2)} \otimes \mathcal{P}_{d_1}^{(p_1 q_1 | p_2 q_2)} \otimes n_{X, d_2}^{(q_1 | q_2)} \,\right],$$

after applying Bose/Fermi projectors to enforce (anti-)symmetry as required by statistics. The completeness of the basis is guaranteed even for evanescent operators—contact structures vanishing only in $D=4$ due to special Clifford or duality identities are retained for proper dimensional regularization and loop matching.

4. Algebraic Organization, Bijection to Effective Operators, and Counting

The unique expression of $A_4$ in terms of $\sigma_2, \sigma_3$ and polarization/color blocks facilitates the classification of all independent local four-point operators at fixed mass dimension. The algebraic structure (commutative algebra, Cohen–Macaulay property) ensures that each amplitude monomial corresponds bijectively to a local operator modulo equations of motion, integration by parts, and symmetry redundancies (Bonifacio et al., 2021, Carrasco et al., 2019). Counting is governed by the plethystic or Molien series:

$H'(t) = \frac{1}{(1-t^2)(1-t^3)},$

and in lower $D$ Gram determinant constraints can further reduce the number of independent monomials, but for $D\geq4$ the classification is exhaustive up to any desired order.

The double-copy construction, manifest in the block-wise assembly of color and kinematics, enables immediate generation of gravitational amplitudes from gauge-theoretic ones (Carrasco et al., 2019, Carrasco et al., 7 Nov 2025). The double copy replaces color factors with a second copy of kinematic building blocks, subject to permutation and statistics constraints. All four-point higher-derivative gravitational contact terms of the $R^4$, $D^4R^4$, $D^6R^4$, ... type, and their map to string effective actions, are thereby reproduced.

5. Explicit Examples and Special Sectors

• Gravity: For the six-derivative (e.g., $R^3$) sector, the amplitude is $$A_4^{(6)}(s, t, u) = \alpha_6 (s-t-u) (s^2+t^2+u^2) T_{\text{ids}}$$, and for eight-derivative (e.g. $R^4$) it is $$A_4^{(8)}(s, t, u) = \alpha_8 (s^4 + t^4 + u^4 + s^2 t^2 + s^2 u^2 + t^2 u^2) T_{\text{ids}}$$ (Shawa et al., 2017).
• Gauge theory: All four-point single-trace color-dual amplitudes with higher-derivative insertions ($F^4$, $F^3$, etc.) can be encoded via three scalar block families: $\sigma_2^x \sigma_3^y \cdot J^{(0)}$, $J^{(1)}$, $J^{(2)}$, with color factors $c_g$ and $d^{a_1a_2a_3a_4}$; see (Carrasco et al., 2019, Chen et al., 2023).
• Four-fermion and mixed amplitudes: All known dimension-6/7/8 contact operators in SMEFT, LEFT, and maximal SYM are contained in this modular basis (Carrasco et al., 7 Nov 2025).
• Vanishing and special structures: In $D=4$, certain amplitudes vanish (e.g., the Gauss-Bonnet $R^2$ term), or split into self-dual/anti-self-dual helicity sectors (He et al., 2016).

6. Dimension-Dependence, Evanescent Structures, and Generalizations

The modular classification naturally accommodates $D$-dependent effects, such as evanescent operators critical for loop-level matching and anomalies. Amplitudes that are non-trivial in $D$ but vanish in $D=4$ must be retained for consistency in dimensional regularization frameworks (Carrasco et al., 7 Nov 2025). Color and spin block degeneracies in low dimensions, as well as Gram determinant constraints on invariants, are correctly accounted for in the basis counting and structure.

The framework interfaces directly with string theory amplitudes, effective actions beyond four-point (via higher $n$-point generalizations), and the full double-copy web, enabling generation of operator towers for both gauge and gravity sectors. Algebraic Hopf structure (quasi-shuffle) can be used to organize BCJ numerators universally, even in higher-derivative extensions (Chen et al., 2023).

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In summary, D-dimensional four-point higher-derivative contact amplitudes are spanned by a finite set of modular building blocks—scalar Mandelstam polynomials, color/kinematic structures, and spin/polarization tensors—identified and enumerated using the tools of symmetric function theory, commutative and Clifford algebra, and algebraic symmetrization. This provides a universal, dimension-agnostic basis for all local four-point contact interactions in effective field theories, string-inspired models, and amplitude-bootstrap programs, with explicit algebraic handles for systematic classification, double-copy construction, and loop-level computation.