IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

This is the first of a series of papers dealing with many-particle systems from a unified, nonperturbative point of view. It contains derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers, In a short introduction the general method of approach is summarized, and its relationship to other field-theoretic problems indicated. In the second section the macroscopic properties of the spectra of many-particle systems are described. Asymptotic evaluations are performed which characterize these macroscopic features in terms of intensive parameters, and the relationship of these parameters to thermodynamics is discussed. The special characteristics of the ground state are shown to follow as a limiting case of the asymptotic evaluations. The third section is devoted to the time-dependent field correlation functions, or Green's functions, which describe the microscopic behavior of a multiparticle system. These functions are defined, and related to intensive macroscopic variables when the energy and number of particles are large. Spectral representations and other properties of various one-particle Green's functions are derived. In the fourth section the treatment of nonequilibrium processes is considered. As a particular example, the electromagnetic properties of a system are expressed in ternis of the special two-particle Green's function which describes current correlation. The discussion yields specifically a Quctuation-dissipation theorem, a sum rule for conductivity, and certain dispersion relations. The fifth section deals with the differential equations which determine the Green's functions. The boundary conditions that characterize the Green's function equations are exhibited without reference to adiabatic decoupling. A niethod for solving the equations approximately, by treating the correlations among successively larger numbers of particles, is considered. The first approximation in this sequence is shown to yield a generalized Hartree-like equation. A related, but rigorous, identity for the single-particle Green s function is then derived. A second approximation, which takes certain two-particle correlations into account, is shown to produce various additional e8ects: The interaction between particles is altered in a manner characterized by the intensive macroscopic parameters, and the modification and spread of the energy-momentum relation come into play. In the final section compact formal expressions for the Green*s functions and other physical quantities are derived. Alternative equations and systematic approximations for the Green's functions are obtained.

Theory of the many-particle systems.

We point out that thermal relic abundance of dark matter is strongly altered by a non‐perturbative effect called the Sommerfeld enhancement, when constituent particles of the dark matter are SU(2)L non‐singlet and much heavier than the weak gauge bosons. Typical candidates are the heavy wino and higgsino. We investigate the non‐perturbative effect on the relic abundance of dark matter for the wino as an example. We show that its thermal abundance is reduced by 50% compared to the perturbative result. The wino mass consistent with the observed dark matter abundance turns out to be 2.7 TeV<m<3.0 TeV. This article is based on Ref. [1].

Non‐perturbative Effect on Thermal Relic Abundance of Dark Matter

I disagree with the view Sally Haslett presented (Nursing Standard week ending April 23) of practice nurses in [Illegible word] efforts to promote her app[Illegible word], to save district health authority family planning clinics!

An alternative view.

A weakly interacting massive particle (WIMP) is a leading candidate of the dark matter. The WIMP dark matter abundance is determined by the freeze-out mechanism. Once we know the property of the WIMP particle such as the mass and interaction, we can predict the dark matter abundance. There are, however, several uncertainties in the estimation of the WIMP dark matter abundance. In this work, we focus on the effect from Standard Model thermodynamics. We revisit the estimation of the WIMP dark matter abundance and its uncertainty due to the equation of state (EOS) in the Standard Model. We adopt the up-to-date estimate of the EOS of the Standard Model in the early Universe and find nearly 10% difference in the 1–1000 GeV dark matter abundance, compared to the conventional estimate of the EOS.

Precise WIMP Dark Matter Abundance and Standard Model Thermodynamics

We introduce DRAKE, a numerical precision tool for predicting the dark matter relic abundance also in situations where the standard assumption of kinetic equilibrium during the freeze-out process may not be satisfied. DRAKE comes with a set of three dedicated Boltzmann equation solvers that implement, respectively, the traditionally adopted equation for the dark matter number density, fluid-like equations that couple the evolution of number density and velocity dispersion, and a full numerical evolution of the phase-space distribution. We review the general motivation for these approaches and, for illustration, highlight three concrete classes of models where kinetic and chemical decoupling are intertwined in a way that quantitatively impacts the relic density: i) dark matter annihilation via a narrow resonance, ii) Sommerfeld-enhanced annihilation and iii) `forbidden' annihilation to final states that are kinematically inaccessible at threshold. We discuss all these cases in some detail, demonstrating that the commonly adopted, traditional treatment can result in an estimate of the relic density that is wrong by up to an order of magnitude. The public release of DRAKE, along with several examples of how to calculate the relic density in concrete models, is provided at this http URL

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DRAKE: Dark matter Relic Abundance beyond Kinetic Equilibrium.

The formation of meta-stable dark matter bound states in coannihilating scenarios could efficiently occur through the scattering with a variety of Standard Model bath particles, where light bosons during the electroweak cross over or even massless photons and gluons are exchanged in the t-channel. The amplitudes for those higher-order processes, however, are divergent in the collinear direction of the in- and out-going bath particles if the mediator is massless. To address the issue of collinear divergences, we derive the bound-state formation collision term in the framework of non-equilibrium quantum field theory. The main result is an expression for a more general cross section, which allows to compute higher-order bound-state formation processes inside the primordial plasma background in a comprehensive manner. Based on this result, we show that next-to-leading order contributions, including the bath-particle scattering, are i) collinear finite and ii) generically dominate over the on-shell emission for temperatures larger than the absolute value of the binding energy. Based on a simplified model, we demonstrate that the impact of these new effects on the thermal relic abundance is significant enough to make it worthwhile to study more realistic coannihilation scenarios.

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Dark Matter bound-state formation at higher order: a non-equilibrium quantum field theory approach

In this work, we derive differential equations from path-integral based non-equilibrium quantum field theory, that cover the dynamics and spectrum of non-relativistic two-body fields for any environment. For concreteness of the two-body fields, we choose the full potential non-relativistic Quantum Electrodynamics Lagrangian in this work. After closing the correlation function hierarchy of these equations and performing consistency checks with previous literature under certain limits, we demonstrate the range of physics applications. This includes Cosmology such as Dark Matter in the primordial plasma, Quarkonia inside a quark gluon plasma, and superconductivity and Ferromagnetism in Condensed or strongly Correlated Matter physics. Since we always had to take limits or approximations of our equations in order to recover those known cases, our equations could contain new phenomena. In particular it is based on Greens function that can deal with non-hermite potentials. We propose a scheme for other Lagrangian based theories or higher N-body states such as molecules to derive analog equations.

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Path Integral based Non-equilibrium Quantum Field Theory of Non-relativistic Pairs inside an Environment

Potential non-relativistic Quantum Electrodynamics and the Keldysh-Schwinger formalism is used to derive Kadanoff-Baym-like equations for two-body field correlators. These cover the out-off-equilibrium dynamics and spectrum of heavy particle-antiparticle pairs under ultra-soft transitions inside a plasma background, whose temperature is much smaller compared to the typical relative pair momentum. It is shown that the dynamical equation for the pair phase-space distribution function, containing recombination and dissociation via the electric dipole interactions, is consistent with the previous open quantum system treatment of bound states in the Coulomb limit of the two-body spectral function.

Tobias Binder

Papers

DRAKE: Dark matter Relic Abundance beyond Kinetic Equilibrium.

Dark Matter bound-state formation at higher order: a non-equilibrium quantum field theory approach

Path Integral based Non-equilibrium Quantum Field Theory of Non-relativistic Pairs inside an Environment

A study of potential non-relativistic QED pairs in the Keldysh-Schwinger formalism