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A bstract We define a perturbatively calculable quantity — the on-shell correlator — which furnishes a unified description of particle dynamics in curved spacetime. Specializing to the case of flat and anti-de Sitter space, on-shell correlators coincide precisely with on-shell scattering amplitudes and boundary correlators, respectively. Remarkably, we find that symmetric manifolds admit a generalization of on-shell kinematics in which the corresponding momenta are literally the isometry generators of the spacetime acting on the external kinematic data. These isometric momenta are intrinsically non-commutative but exhibit on-shell conditions that are identical to those of flat space, thus providing a common language for computing and representing on-shell correlators which is agnostic about the underlying geometry. Afterwards, we compute tree-level on-shell correlators for biadjoint scalar (BAS) theory and the nonlinear sigma model (NLSM) and learn that color-kinematics duality is manifested at the level of fields under a mapping of the color algebra to the algebra of gauged isometries on the spacetime manifold. Last but not least, we present a field theoretic derivation of the fundamental BCJ relations for on-shell correlators following from the existence of certain conserved currents in BAS theory and the NLSM.

A bstract We describe on-shell methods for computing one- and two-loop anomalous dimensions in the context of effective field theories containing higher-dimension operators. We also summarize methods for computing one-loop amplitudes, which are used as inputs to the computation of two-loop anomalous dimensions, and we explain how the structure of rational terms and judicious renormalization scheme choices can lead to additional vanishing terms in the anomalous dimension matrix at two loops. We describe the two-loop implications for the Standard Model Effective Field Theory (SMEFT). As a by-product of this analysis we verify a variety of one-loop SMEFT anomalous dimensions computed by Alonso, Jenkins, Manohar and Trott.

A bstract We derive a universal soft theorem for every scattering amplitude with at least one massless particle in an arbitrary theory of scalars. Our results follow from the geometry of field space and are valid for any choice of mass spectrum, potential terms, and higher-derivative interactions. For a vanishing potential, the soft limit of every amplitude is equal to the field-space covariant derivative of an amplitude with one fewer particle. Furthermore, the Adler zero and the dilaton soft theorem are special cases of our results. We also discuss more exotic scenarios in which the soft limit is non-trivial but still universal. Last but not least, we derive new theorems for multiple-soft limits which directly probe the field-space curvature, as well as on-shell recursion relations applicable to two-derivative scalar field theories exhibiting no symmetries whatsoever.

A bstract We describe in detail how a d log representation of Feynman integrals leads to simple differential equations. We derive these differential equations directly in loop momentum or embedding space making use of a localization trick and generalized unitarity. For the examples we study, the alphabet of the differential equation is related to special points in kinematic space, described by certain cut equations which encode the geometry of the Feynman integral. At one loop, we reproduce the motivic formulae described by Goncharov [1] that reappeared in the context of Feynman integrals in [2–4]. The d log representation allows us to generalize the differential equations to higher loops and motivates the study of certain mixed-dimension integrals.

A bstract We study constraints from causality and unitarity on 2 → 2 graviton scattering in four-dimensional weakly-coupled effective field theories. Together, causality and unitarity imply dispersion relations that connect low-energy observables to high-energy data. Using such dispersion relations, we derive two-sided bounds on gravitational Wilson coefficients in terms of the mass M of new higher-spin states. Our bounds imply that gravitational interactions must shut off uniformly in the limit G → 0, and prove the scaling with M expected from dimensional analysis (up to an infrared logarithm). We speculate that causality, together with the non-observation of gravitationally-coupled higher spin states at colliders, severely restricts modifications to Einstein gravity that could be probed by experiments in the near future.

A bstract We revisit the calculation of classical observables from causal response functions, following up on recent work by Caron-Huot et al. [ 1 ]. We derive a formula to compute asymptotic in-in observables from a particular soft limit of five-point amputated response functions. Using such formula, we re-derive the formulas by Kosower, Maybee and O’Connell (KMOC) for the linear impulse and radiated linear momentum of particles undergoing scattering, and we present an unambiguous calculation of the radiated angular momentum at leading order. Then, we explore the consequences of manifestly causal Feynman rules in the calculation of classical observables by employing the causal (Keldysh) basis in the in-in formalism. We compute the linear impulse, radiated waveform and its variance at leading and/or next-to-leading order in the causal basis, and find that all terms singular in the ħ → 0 limit cancel manifestly at the integrand level. We also find that the calculations simplify considerably and classical properties such as factorization of six-point amplitudes are more transparent in the causal basis.

A bstract We derive a soft theorem for a massless scalar in an effective field theory with generic field content using the geometry of field space. This result extends the geometric soft theorem for scalar effective field theories by allowing the massless scalar to couple to other scalars, fermions, and gauge bosons. The soft theorem keeps its geometric form, but where the field-space geometry now involves the full field content of the theory. As a bonus, we also present novel double soft theorems with fermions, which mimic the geometric structure of the double soft theorem for scalars.

A bstract We present the two-body Hamiltonian and associated eikonal phase, to leading post-Minkowskian order, for infinitely many tidal deformations described by operators with arbitrary powers of the curvature tensor. Scattering amplitudes in momentum and position space provide systematic complementary approaches. For the tidal operators quadratic in curvature, which describe the linear response to an external gravitational field, we work out the leading post-Minkowskian contributions using a basis of operators with arbitrary numbers of derivatives which are in one-to-one correspondence with the worldline multipole operators. Explicit examples are used to show that the same techniques apply to both bodies interacting tidally with a spinning particle, for which we find the leading contributions from quadratic in curvature tidal operators with an arbitrary number of derivatives, and to effective field theory extensions of general relativity. We also note that the leading post-Minkowskian order contributions from higher-dimension operators manifest double-copy relations. Finally, we comment on the structure of higher-order corrections.

A bstract We revisit dispersive bounds on Wilson coefficients of scalar effective field theories (EFT) coupled to gravity in various spacetime dimensions, by computing the contributions from graviton loops to the corresponding sum rules at low energies. Fixed-momentum-transfer dispersion relations are often ill-behaved due to forward singularities arising from loop-level graviton exchange, making naive positivity bounds derived from them unreliable. Instead, we perform a careful analysis using crossing-symmetric dispersion relations, and compute the one-loop corrections to the bounds on EFT coefficients. We find that including the graviton loops generically allows for negativity of Wilson coefficients by an amount suppressed by powers of Newton’s constant, G . The exception are the few couplings that dominate over (or are degenerate with) the graviton loops at low energies. In D = 4, we observe that assuming that the eikonal formula captures the correct forward behavior of the amplitude at all orders in G , and for energies of the order of the EFT cutoff, yields bounds free of logarithmic infrared divergences.

A bstract In this work we revisit the “holomorphic modular bootstrap”, i.e. the classification of rational conformal field theories via an analysis of the modular differential equations satisfied by their characters. By making use of the representation theory of PSL(2 , ℤ n ), we describe a method to classify allowed central charges and weights ( c, h i ) for theories with any number of characters d . This allows us to avoid various bottlenecks encountered previously in the literature, and leads to a classification of consistent characters up to d = 5 whose modular differential equations are uniquely fixed in terms of ( c, h i ). In the process, we identify the full set of constraints on the allowed values of the Wronskian index for fixed d ≤ 5.