Scattering Amplitudes For All Masses and Spins (1709.04891v2)

Abstract: We introduce a formalism for describing four-dimensional scattering amplitudes for particles of any mass and spin. This naturally extends the familiar spinor-helicity formalism for massless particles to one where these variables carry an extra SU(2) little group index for massive particles, with the amplitudes for spin S particles transforming as symmetric rank 2S tensors. We systematically characterise all possible three particle amplitudes compatible with Poincare symmetry. Unitarity, in the form of consistent factorization, imposes algebraic conditions that can be used to construct all possible four-particle tree amplitudes. This also gives us a convenient basis in which to expand all possible four-particle amplitudes in terms of what can be called "spinning polynomials". Many general results of quantum field theory follow the analysis of four-particle scattering, ranging from the set of all possible consistent theories for massless particles, to spin-statistics, and the Weinberg-Witten theorem. We also find a transparent understanding for why massive particles of sufficiently high spin can not be "elementary". The Higgs and Super-Higgs mechanisms are naturally discovered as an infrared unification of many disparate helicity amplitudes into a smaller number of massive amplitudes, with a simple understanding for why this can't be extended to Higgsing for gravitons. We illustrate a number of applications of the formalism at one-loop, giving few-line computations of the electron (g-2) as well as the beta function and rational terms in QCD. "Off-shell" observables like correlation functions and form-factors can be thought of as scattering amplitudes with external "probe" particles of general mass and spin, so all these objects--amplitudes, form factors and correlators, can be studied from a common on-shell perspective.

• The paper introduces an extended spinor-helicity formalism that uses an SU(2) index to represent massive particles alongside massless ones.
• It characterizes three- and four-particle amplitudes constrained by Poincaré symmetry, unitarity, and locality, influencing gauge and gravity theories.
• The study reframes the Higgs and Super-Higgs mechanisms as infrared unification processes, linking high-energy behavior to UV consistency and quantum corrections.

An Examination of "Scattering Amplitudes For All Masses and Spins"

In the paper "Scattering Amplitudes For All Masses and Spins," Arkani-Hamed, Huang, and Huang present a comprehensive framework for understanding scattering amplitudes in four dimensions for particles with arbitrary mass and spin. This work extends the spinor-helicity formalism previously used for massless particles to encompass massive particles by incorporating an additional $SU(2)$ little group index, which renders the amplitudes as symmetric rank $2S$ tensors for particles of spin $S$. The paper systematically characterizes three-particle and four-particle amplitudes, elucidating how these structures are constrained by Poincaré symmetry and principles of unitarity and locality.

General Framework and Key Results

The authors begin by extending the spinor-helicity formalism to encapsulate massive particles, introducing a representation of spin states as symmetric tensors of $SU(2)$, which is more convenient for maintaining rotational invariance without reference directions. From this groundwork, they characterize all possible three-particle amplitudes by leveraging Poincaré invariance. In the massless regime, distinct amplitudes are completely determined by the helicities of the particles, revealing the stringent constraints imposed by symmetry.

Progressing to four-particle amplitudes, the paper imposes factorization conditions to ensure consistency across all channels, a non-trivial process predominantly for massless theories due to underlying poles in three-particle amplitudes. This leads to recovery of familiar results: interacting massless particles are constrained to spins (0, 1/2, 1, 3/2, and 2), the necessity of gauge symmetries for spin-1 particles via Yang-Mills structures, and spin-2 particles aligning with gravity.

Consistent coupling to gravitons demands universality of gravitational interactions, thereby reinforcing classical physics heuristics like the equivalence principle and charge conservation.

Constraints on Massless Theories and Massive Particle Spin

A significant discussion revolves around why higher-spin massless particles don't exist non-trivially: with consistent four-point amplitudes missing due to incompatibility of factorization conditions across s, t, and u channels. Similar conservation laws dictate charged particles with spins larger than 1 cannot interact consistently within flat spacetime quantum field theories, reminiscent of the Weinberg-Witten theorem that constrains conserved currents’ dimensionality.

Regarding massive particles, the paper explores interactions illustrating that high-spin states (greater than 2) yield unmanageable high-energy behaviors, particularly within Compton scattering. This behavior implies that theories where such spins masquerade as elementary particles lack consistency unless string-theoretic or similar frameworks introduce additional states to regulate these "non-elementary" interactions in the ultraviolet.

The Higgs and Super-Higgs Mechanism Analysis

A novel treatment of the Higgs mechanism is presented, showcasing how massive amplitudes for spin 1 reconcile different helicity components at high energies, effectively uniting massless amplitudes. This reframes the Higgs and Super-Higgs mechanisms as infrared unification processes rather than symmetry breaking—an illuminating conceptual shift that emphasizes coherence with high-energy effective field behaviors, maintaining consistency with gauge symmetries throughout.

One-Loop Extensions and Form Factors

The methodology naturally extends into quantum-level calculations, where generalized unitarity deciphers one-loop amplitudes like electron g-factor $(g-2)$ and running coupling constants (like the QCD beta function). The analytic structure aligns with broad field-theory predictions but offers a refined vantage that circumvents gauge redundancy by viewing these processes through physical degrees of freedom.

Additionally, observables like form factors and correlation functions gain a lucid interpretation through massive representations, correlating well with empirical probing of stress-energy tensors and charge distributions, dovetailing with soft theorem principles.

Outlook and Future Directions

In summary, this paper lays out a coherent scheme for analyzing scattering amplitudes regardless of mass and spin, promulgating an alternative formalism where redundancies inherent in field-theoretic operators are minimized. Spectacularly, it integrates massless and massive amplitude analyses into one continuous picture, accentuating the deep-rooted connections between on-shell consistency, symmetry, and unitarity. As on-shell methods reveal more of their robust landscape, this work opens promising avenues for exploring emergent phenomena in high-energy physics and potential constraints in quantum gravity paradigms. The on-shell formalism endeavors to derive the necessities of string theory or other UV-complete frameworks as natural continuations of these scattering analyses.