Search Results for « Infrared finite scattering theory »

Any non-trivial scattering with massless fields in four spacetime dimensions will generically produce an out-state with memory. Scattering with any massless fields violates the standard assumption of asymptotic completeness -- that all "in" and "out" states lie in the standard (zero memory) Fock space -- and therefore leads to infrared (IR) divergences in the standard $S$-matrix amplitudes. In this paper we define an infrared finite scattering theory which assumes only (1) the existence of in/out algebras and (2) that Heisenberg evolution is an automorphism of these algebras. The resulting "superscattering" map $\$$ allows for transitions between different in/out memory states and agrees with the standard $S$-matrix when it is defined. We construct $\$$-amplitudes by defining (3) a "generalized asymptotic completeness" which accommodates states with memory in the space of asymptotic states and (4) a complete basis of improper states which generalize the usual $n$-particle momentum basis to account for states with memory. Using only general properties of $\$$, we prove an analog of the Weinberg soft theorems in quantum gravity and QED which imply that all $\$$-amplitudes are well-defined in the infrared. We comment on how one must generalize this framework to consider $\$$-amplitudes for theories with collinear divergences (e.g., massless QED and Yang-Mills theories).

It has been known since the earliest days of quantum field theory (QFT) that infrared divergences arise in scattering theory with massless fields. These infrared divergences are manifestations of the memory effect: At order $1/r$ a massless field generically will not return to the same value at late retarded times ($u\ensuremath{\rightarrow}+\ensuremath{\infty}$) as it had at early retarded times ($u\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$). There is nothing singular about states with memory, but they do not lie in the standard Fock space. Infrared divergences are merely artifacts of trying to represent states with memory in the standard Fock space. If one is interested only in quantities directly relevant to collider physics, infrared divergences can be successfully dealt with by imposing an infrared cutoff, calculating inclusive quantities, and then removing the cutoff. However, this approach does not allow one to treat memory as a quantum observable and is highly unsatisfactory if one wishes to view the $S$-matrix as a fundamental quantity in QFT and quantum gravity, since the $S$-matrix itself is undefined. In order to have a well-defined $S$-matrix, it is necessary to define ``in'' and ``out'' Hilbert spaces that incorporate memory in a satisfactory way. Such a construction was given by Faddeev and Kulish for quantum electrodynamics (QED) with a massive charged field. Their construction can be understood as pairing momentum eigenstates of the charged particles with corresponding memory representations of the electromagnetic field to produce states of vanishing large gauge charges at spatial infinity. (This procedure is usually referred to as ``dressing'' the charged particles.) We investigate this procedure for QED with massless charged particles and show that, as a consequence of collinear divergences, the required dressing in this case has an infinite total energy flux, so that the states obtained in the Faddeev-Kulish construction are unphysical. An additional difficulty arises in Yang-Mills theory, due to the fact that the ``soft Yang-Mills particles'' used for the dressing contribute to the Yang-Mills charge-current flux, thereby invalidating the procedure used to construct eigenstates of large gauge charges at spatial infinity. We show that there are insufficiently many charge eigenstates to accommodate scattering theory. In quantum gravity, the analog of the Faddeev-Kulish construction would attempt to produce a Hilbert space of eigenstates of supertranslation charges at spatial infinity. Again, the Faddeev-Kulish dressing procedure does not produce the desired eigenstates because the dressing contributes to the null memory flux. We prove that there are no eigenstates of supertranslation charges at spatial infinity apart from the vacuum. Thus, analogs of the Faddeev-Kulish construction fail catastrophically in quantum gravity. We investigate some alternatives to Faddeev-Kulish constructions but find that these also do not work. We believe that if one wishes to treat scattering at a fundamental level in quantum gravity---as well as in massless QED and Yang-Mills theory---it is necessary to approach it from an algebraic viewpoint on the ``in'' and ``out'' states, wherein one does not attempt to ``shoehorn'' these states into some prechosen ``in'' and ``out'' Hilbert spaces. We outline the framework of such a scattering theory, which would be manifestly infrared finite.

A scattering zipper is a system obtained by concatenation of scattering events with equal even number of incoming and out going channels. The associated scattering zipper operator is the unitary equivalent of Jacobi matrices with matrix entries and generalizes Blatter-Browne and Chalker-Coddington models and CMV matrices. Weyl discs are analyzed and used to prove a bijection between the set of semi-infinite scattering zipper operators and matrix valued probability measures on the unit circle. Sturm-Liouville oscillation theory is developed as a tool to calculate the spectra of finite and periodic scattering zipper operators.

The functional Schrodinger picture formulation of quantum field theory and the variational Gaussian approximation method based on the formulation are briefly reviewed. After presenting recent attempts to improve the variational approximation, we introduce a new systematic method based on the background field method, which enables one to compute the order-by-order correction terms to the Gaussian approximation of the effective action.

A bstract Any non-trivial scattering with massless fields in four spacetime dimensions will generically produce an “out” state with memory which gives rise to infrared divergences in the standard S -matrix. To obtain an infrared-finite scattering theory, one must suitably include states with memory. However, except in the case of QED with massive charged particles, asymptotic states with memory that have finite energy and angular momentum have not been constructed for more general theories (e.g. massless QED, Yang-Mills and quantum gravity). To this end, we construct direct-integral representations over the “Lorentz orbit” of a given memory and classify all “orbit space representations” that have well-defined energy and angular momentum. We thereby provide an explicit construction of a large supply of physical states with memory as well as the explicit action of the BMS charges all states. The construction of such states is a key step toward the formulation of an infrared-finite scattering theory. While we primarily focus on the quantum gravitational case, we outline how the methods presented in this paper can be applied to obtain representations of the Poincaré group with memory for more general quantum field theories.

Models incorporating moderately heavy dark matter (DM) typically need charged (scalar) fields to establish admissible relic densities. Since the DM freezes out at an early epoch, thermal corrections to the cross sections can be important. In a companion paper [arXiv:1812.04247v2] we established that the infrared (IR) divergences accruing from scalar-photon interactions cancel to all orders in perturbation theory. The corresponding infrared finiteness of thermal fermionic QED has already been established. Here, we study the IR behaviour at finite temperatures, of a theory of dark matter interacting with charged scalars and fermions, which potentially contains both both linear and sub-leading logarithmic divergences. We prove that the theory is IR-finite to all orders with the divergences cancelling when both absorption and emission of photons from and into the heat bath are taken into account. While 4-point interaction terms are known to be IR finite, their inclusion leads to a neat exponentiation. The calculation follows closely the technique used for the scalar finite temperature theory.

The Jupiter-Saturn 2:5 near-commensurability is analyzed in a fully analytic Hamiltonian planetary theory. Computations for the Sun-Jupiter-Saturn system, extending to the third order of the masses and to the 8th degree in the eccentricities and inclinations, reveal an unexpectedly sensitive dependence of the solution on initial data and its likely nonconvergence. The source of the sensitivity and apparent lack of convergence is this near-commensurability, the so-called great inequality. This indicates that simple averaging, still common in current semi-analytic planetary theories, may not be an adequate technique to obtain information on the long-term dynamics of the Solar System. Preliminary results suggest that these difficulties can be overcome by using resonant normal forms.

Infrared divergences have long been heralded to cancel in sufficiently inclusive cross sections, according to the famous Kinoshita-Lee-Nauenberg theorem which mandates an initial and final state sum. While well motivated, this theorem is much weaker than necessary: for finiteness, one need only sum over initial or final states. Moreover, the cancellation generically requires the inclusion of the forward scattering process. We provide a number of examples showing the importance of this revised understanding: in ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}Z$ at next-to-leading order, one can sum over certain initial and final states with an arbitrary number of extra photons, or only over final states with a finite number of photons, if forward scattering is included. For Compton scattering, infrared finiteness requires the indistinguishability of hard forward-scattered electrons and photons. This implies that in addition to experimental limits on the energy and angular resolution, there must also be an experimental limit on the momentum at which electric charge can be observed. Similar considerations are required to explain why the rate for $\ensuremath{\gamma}\ensuremath{\gamma}$ to scatter into photons alone is infrared divergent, but the rate for $\ensuremath{\gamma}\ensuremath{\gamma}$ to scatter into photons or charged particles is finite. This new understanding sheds light on the importance of including degenerate initial states in physical predictions, the relevance of disconnected Feynman diagrams, the importance of dressing initial or final-state charged particles, and the quest to properly define the $S$ matrix.

New arguments are presented to emphasize the interest of the infrared finite coupling approach to power corrections in the context of Sudakov resummation. The more regular infrared behavior of some peculiar combinations of Sudakov anomalous dimensions, free of Landau singularities at large Nf, is pointed out. A general conflict between the infrared finite coupling and infrared renormalon approaches to power corrections is explained, and a possible resolution is proposed, which makes use of the arbitrariness of the choice of exponentiated constant terms. A simple ansatz for a 'universal' non-perturbative Sudakov effective coupling at large Nf follows naturally from these considerations. In this last version, a new result is presented: the striking emergence of an infrared finite perturbative effective coupling in the Drell-Yan process at large Nf (at odds with the infrared renormalon argument) within the framework of Sudakov resummation for eikonal cross sections of Laenen, Sterman and Vogelsang. Some suggestions for phenomenology at finite Nf, alternative to the shape function approach, are given.