Bootstrapped On-Shell Amplitudes

• The paper introduces bootstrapped on-shell amplitudes built recursively using on-shell data, enforcing unitarity and locality without off-shell Lagrangians.
• It employs analytic properties like factorization, BCFW shifts, and symmetry constraints to reconstruct complex n-point amplitudes from simpler building blocks.
• These methods have practical implications in precision collider computations and effective field theory, streamlining analyses in both QFT and string theory.

Bootstrapped on-shell amplitudes are amplitude representations and calculation methods in quantum field theory (QFT) and string theory that construct scattering amplitudes recursively, using only on-shell data, locality, unitarity, and gauge or global symmetries. Rather than relying on a Feynman diagram expansion from an off-shell Lagrangian, bootstrapped on-shell techniques reconstruct the full n-point amplitudes (tree or loop level, depending on context) from lower-point on-shell amplitudes and certain analytic constraints. These methods, pioneered in field theory via the Britto–Cachazo–Feng–Witten (BCFW) recursion and further generalized to effective theories, strings, and curved backgrounds, exploit and manifest the power of S-matrix consistency, amplitudes’ analytic properties, and symmetry constraints—including high-energy and soft limits, unitarity, and crossing symmetry. Bootstrapped on-shell frameworks have central roles in precision Standard Model computations, effective field theory (EFT) analyses, string duality studies, and the analytic bootstrap of highly symmetric gauge theories.

1. Foundational Principles: Recursion, Locality, and Unitarity

The core of the bootstrapped on-shell program is the construction of scattering amplitudes solely from their analytic properties and symmetry constraints, enforcing:

• Factorization (unitarity): Amplitudes must have simple poles corresponding to physical intermediate states and factorize accordingly. For n-point tree amplitudes, the residue at each physical pole must equal the product of lower-point amplitudes summed/interpolated over the exchanged state(s).
• Locality: Apart from these poles, amplitudes are rational or analytic functions (at tree level), possibly with polynomial “contact” terms.
• Recursion: By performing a complex deformation of momenta (e.g., a BCFW shift), the amplitude $A_n(z)$ as a function of the complex parameter $z$ can be written as a sum over residues at finite $z$ (physical poles) provided the $z \to \infty$ limit falls off suitably:

$$A_n(0) = - \sum_i \text{Res}_{z=z_i} \frac{A_n(z)}{z}$$

• Analyticity and crossing: Amplitudes must satisfy crossing symmetry (equivalence under interchange of channels) and be analytic except at physical singularities or branch points.

In string theory, these principles are extended. The full string amplitudes on the disc (open string) or sphere (closed string) inherit factorization on an infinite spectrum of higher-spin poles, and local monodromy/duality relations interconnect different channels or color orderings (Boels et al., 2010).

2. On-Shell Bootstrap in Field Theory and Effective Theories

In modern QFT, especially gauge theory and EFT contexts, bootstrapped on-shell amplitudes have several concrete incarnations:

• BCFW recursion and extensions: Tree-level amplitudes can be constructed recursively via complex momentum shifts (BCFW or massive analogues), provided good large-$z$ scaling. For example, massless QCD amplitudes and SMEFT amplitudes in physical helicity basis are recursively reducible to three-point on-shell data (Badger, 2016, Herderschee et al., 2019, Ema et al., 22 Mar 2024).
• Massive amplitudes and spinor-helicity: Systems with massive particles, including SM vectors/scalars, can be bootstrapped using massive spinor-helicity variables and little-group covariant superspaces. All three-point and four-point building blocks are classified by symmetry and covariance, and higher-point amplitudes are (up to contact terms) assembled by gluing these blocks, subject to unitarity and leading power counting (Herderschee et al., 2019, Aoude et al., 2019, Durieux et al., 2019, Liu et al., 2022).
• EFT matching and operator expansions: Bootstrapped on-shell amplitudes avoid operator redundancy and gauge-fixing ambiguities by matching directly to the physical amplitude basis. EFT corrections appear as Taylor expansions of analytic (kinematics-dependent) coefficients, with each term’s power reflecting the operator dimension or chiral order (e.g., in SMEFT or HEFT). This leads to manifestly gauge-invariant (and even reparameterization-invariant) matching (Gröber et al., 2 Sep 2025).

3. Analyticity, Constraints, and Higher-Order Bootstrapping

Recent developments exploit further structures—or constraints—on amplitudes’ analytic form and physical behavior:

• Symbol/bootstrap for loop and multileg amplitudes: The analytic S-matrix bootstrap reconstructs, for planar $\mathcal{N}=4$ SYM and related theories, multiloop amplitudes as elements of a finite-dimensional space of multiple polylogarithms (or their symbols), with the symbol letters determined by physical branch points (cluster algebras, Steinmann relations, etc.). Amplitudes are then fixed by symmetry, factorization, and known limits (Papathanasiou, 2022).
• Regge and dual model bootstrap: Four-point (dual resonance) amplitudes (e.g., Veneziano or Virasoro–Shapiro) can be bootstrapped from physical input (spectrum, crossing, finiteness of exchanged spin, boundedness) without reference to a Lagrangian. This yields both well-understood and novel generalizations, including $q$-deformations, by algebraically solving for allowed consistent analytic structures (Cheung et al., 2023, Eckner et al., 16 Jan 2024).
• Numerical linear programming bootstrap: For dual-resonance models (string-like amplitudes), efficient parametrizations in terms of Mandelstam–Regge poles enable the use of linear programming to enforce crossing and unitarity at finite truncation. This reveals, for example, the unique role of the Veneziano amplitude (“super-unitarity” property), and pit structures in parameter space correspond to especially symmetric or physically relevant amplitudes (Eckner et al., 16 Jan 2024).
• Extensions to AdS and cosmological correlators: The Mellin-momentum bootstrap defines on-shell AdS/CFT amplitudes as residues of differential operators in Mellin space, bootstrapped recursively from three-point data and fixed by factorization, soft limits, and flat space correspondence. This provides an S-matrix–like structure to cosmological correlators, with gravitational (and double copy) amplitudes constructed in analogy with flat space (Mei, 2023, Mei et al., 14 Feb 2024, Mei et al., 7 Oct 2024).

4. Applications: Gauge Theories, Strings, and Effective Field Theory

Bootstrapped on-shell amplitudes have widespread applications:

• Precision collider QCD and LHC: On-shell recursion, combined with integrand reduction, underpins next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) calculations of multileg amplitudes (e.g., $pp \to 5$ jets), enabling practical automation and analytic insight (Badger, 2016).
• EFT and Higgs sector: Bootstrapped construction enables unambiguous mapping between SMEFT/HEFT operator expansions and physical n-point amplitudes, clarifying the order at which different kinematic structures appear and allowing precise diagnosis of symmetry realization in the Higgs sector (Gröber et al., 2 Sep 2025).
• String amplitudes and duality: All open string disc amplitudes obey BCFW-type recursion up to possible kinematic reality conditions, and all monodromy (color/ordering) relations are encoded in the CFT origin of vertex operator braiding—establishing a string-specific extension of field-theory recursion (Boels et al., 2010, Cheung et al., 2023).
• Gravity and the double-copy construction: Even in (A)dS, bootstrapped on-shell graviton amplitudes can be built from lower-point building blocks and connected with gauge amplitudes via double copy, making locality and factorization manifest (Mei, 2023, Mei et al., 7 Oct 2024).

5. Mathematical Structures and Algorithmic Implementations

Tables and algebraic structures arising in bootstrapped on-shell amplitudes include:

Context	Building Blocks	Recursion/Structure
Gauge theory/QCD	Three-point amplitudes (spinor-helicity)	BCFW/particle-pair shifts, unitarity cuts, factorization
String theory	Koba–Nielsen integrals/vertex operators	CFT monodromy, BCFW-type recursion, KLT relations
Mellin–momentum/AdS	Three-point Mellin amplitudes	Residues of diff operators, soft limits, double copy
Biadjoint/YM (CHY)	CHY integrals/current factorizations	Off-shell currents in common kinematic variables, on-shell projection

The computational pipeline is often recursive: three-point seed data plus analytic properties plus symmetry yields higher-point amplitudes, with explicit formulas for the connection between kinematics and allowed structures (Gomez, 31 Jul 2025).

6. Generalizations, Exceptional Cases, and Limitations

• Massive amplitudes and spinor basis: Little group covariant spinor-helicity formalism, on-shell superspace, and permutation symmetries generalize the bootstrap to massive multiplets and high-spin fields, with recursive constraints fixing multiparticle couplings (Herderschee et al., 2019, Liu et al., 2022).
• Continuous-spin particles (CSP): The bootstrap can be executed for CSPs, involving infinite towers of helicity, with nontrivial realization of the $ISO(2)$ little group on bi-spinor variables and selection rules arising from analyticity and factorization constraints (Bellazzini et al., 24 Jun 2024).
• Operator redundancies and field redefinitions: On-shell recasting, particularly when using generalized geometric/function space formalisms, guarantees reparameterization (field redefinition) invariance of amplitudes, with explicit transformation rules and manifest “on-shell covariance” (Cohen et al., 2022).
• Deformations and non-Lagrangian systems: Instances exist (e.g. certain dual model bootstraps, or non-analytic deformations) where a unique Lagrangian may not arise or where analytic continuation and mild non-unitarity are required to access all possible amplitude structures (Cheung et al., 2023, Eckner et al., 16 Jan 2024, Bellazzini et al., 24 Jun 2024).

7. Impact and Future Directions

• Bootstrapped on-shell amplitude techniques have reshaped both theoretical and practical quantum field and string theory computations, exposing deep analytic structure and enabling new precision calculations.
• Further development is ongoing in the context of higher-loop and higher-multiplicity computations in realistic theories, the analytic bootstrap for amplitudes and Feynman integrals, and amplitude-based approaches to quantum gravity, celestial holography, and the S-matrix in cosmology.
• Algorithmic recursion and efficient basis representations are expected to further automate and generalize these constructions, moving toward a complete landscape understanding of consistent quantum field and string S-matrices.

This extensive program systematically replaces off-shell, gauge-redundant formulations by transparent, symmetry-, and analyticity-driven constructions, making manifest the immutable properties of physical observables in quantum field and string theory.