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# On-shell heavy particle effective theories

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## Abstract

We introduce on-shell variables for Heavy Particle Effective Theories (HPETs) with the goal of extending Heavy Black Hole Effective Theory to higher spins and of facilitating its application to higher post-Minkowskian orders. These variables inherit the separation of spinless and spin-inclusive effects from the HPET fields, resulting in an explicit spin-multipole expansion of the three-point amplitude for any spin. By matching amplitudes expressed using the on-shell HPET variables to those derived from the one-particle effective action, we find that the spin-multipole expansion of a heavy spin-$s$ particle corresponds exactly to the multipole expansion (up to order $2s$) of a Kerr black hole, that is, without needing to take the infinite spin limit. Finally, we show that tree-level radiative processes with same-helicity bosons emitted from a heavy spin-$s$ particle exhibit a spin-multipole universality.

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Massive gravity has seen a resurgence of interest due to recent progress which has overcome its traditional problems, yielding an avenue for addressing important open questions such as the cosmological constant naturalness problem. The possibility of a massive graviton has been studied on and off for the past 70 years. During this time, curiosities such as the vDVZ discontinuity and the Boulware-Deser ghost were uncovered. We re-derive these results in a pedagogical manner, and develop the St\"ukelberg formalism to discuss them from the modern effective field theory viewpoint. We review recent progress of the last decade, including the dissolution of the vDVZ discontinuity via the Vainshtein screening mechanism, the existence of a consistent effective field theory with a stable hierarchy between the graviton mass and the cutoff, and the existence of particular interactions which raise the maximal effective field theory cutoff and remove the ghosts. In addition, we review some peculiarities of massive gravitons on curved space, novel theories in three dimensions, and examples of the emergence of a massive graviton from extra-dimensions and brane worlds.
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Recent work involving virtual excitation of a spin (3/2) baryon resonance is seen to contain two distinct problems. The Feynman propagator for spin (3/2) from the Rarita-Schwinger formalism has often been mistaken for its on-mass-shell limit. In nonrelativistic work the direct channel exchange of a Delta resonance is normally included, but with the exclusion of a term for the intermediate anti-Delta. It is shown that both of these terms are of equal importance in the resonance region for the case of nucleon Compton scattering.
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We implement at the Lagrangian level a natural'' electromagnetic coupling prescription, different from the minimal one, and proposed a long time ago by Weinberg. This prescription yields, for elementary particles of [ital arbitrary][minus][ital spin], a gyromagnetic ratio [ital g]=2 at the tree level. This value is already known to arise in renormalizable theories for spin 1/2 and spin 1, is suggested by the classical, relativistic equations of the spin polarization, and is also found for arbitrary-spin, charged excitations of the open string.
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Recently, by using the known structure of one-loop scattering amplitudes for gluons in Yang-Mills theory, a recursion relation for tree-level scattering amplitudes has been deduced. Here, we give a short and direct proof of this recursion relation based on properties of tree-level amplitudes only.
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We investigate all the four-body graviton interaction processes: $gX\rightarrow \gamma X$, $gX\rightarrow gX$, and $gg\rightarrow gg$ with $X$ as an elementary particle of spin less than two in the context of linearized gravity except the spin-3/2 case. We show explicitly that gravitational gauge invariance and Lorentz invariance cause every four-body graviton scattering amplitude to be factorized. We explore the implications of this factorization property by investigating polarization effects through the covariant density matrix formalism in each four-body graviton scattering process. Comment: 45 pages, figures are included (uses pictex), RevTex
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We present an Effective Field Theory (EFT) formalism which describes the dynamics of non-relativistic extended objects coupled to gravity. The formalism is relevant to understanding the gravitational radiation power spectra emitted by binary star systems, an important class of candidate signals for gravitational wave observatories such as LIGO or VIRGO. The EFT allows for a clean separation of the three relevant scales: r_s, the size of the compact objects, r the orbital radius and r/v, the wavelength of the physical radiation (where the velocity v is the expansion parameter). In the EFT radiation is systematically included in the v expansion without need to separate integrals into near zones and radiation zones. We show that the renormalization of ultraviolet divergences which arise at v^6 in post-Newtonian (PN) calculations requires the presence of two non-minimal worldline gravitational couplings linear in the Ricci curvature. However, these operators can be removed by a redefinition of the metric tensor, so that the divergences at arising at v^6 have no physically observable effect. Because in the EFT finite size features are encoded in the coefficients of non-minimal couplings, this implies a simple proof of the decoupling of internal structure for spinless objects to at least order v^6. Neglecting absorptive effects, we find that the power counting rules of the EFT indicate that the next set of short distance operators, which are quadratic in the curvature and are associated with tidal deformations, do not play a role until order v^10. These operators, which encapsulate finite size properties of the sources, have coefficients that can be fixed by a matching calculation. By including the most general set of such operators, the EFT allows one to work within a point particle theory to arbitrary orders in v.
Article
A rigorous QCD analysis of the inclusive annihilation decay rates of heavy quarkonium states is presented. The effective-field-theory framework of nonrelativistic QCD is used to separate the short-distance scale of annihilation, which is set by the heavy quark mass $M$, from the longer-distance scales associated with quarkonium structure. The annihilation decay rates are expressed in terms of nonperturbative matrix elements of 4-fermion operators in nonrelativistic QCD, with coefficients that can be computed using perturbation theory in the coupling constant $\alpha_s(M)$. The matrix elements are organized into a hierarchy according to their scaling with $v$, the typical velocity of the heavy quark. An analogous factorization formalism is developed for the production cross sections of heavy quarkonium in processes involving momentum transfers of order $M$ or larger. The factorization formulas are applied to the annihilation decay rates and production cross sections of S-wave states, up to corrections of relative order $v^3$, and of P-wave states, up to corrections of relative order $v^2$. Comment: Revised to clarify the velocity-scaling rules for spin-flip transitions, to correct error estimates, and to emphasize probabilities of Fock states, rather than amplitudes, 117 pages in REVTEX plus 11 Postscript figures. Erratum to Phys. Rev. D article included as a separate file, 4 pages in REVTEX