Eric Braaten

Universality in few-body systems with large scattering length

Abstract Particles with short-range interactions and a large scattering length have universal low-energy properties that do not depend on the details of their structure or their interactions at short distances. In the 2-body sector, the universal properties are familiar and depend only on the scattering length a. In the 3-body sector for identical bosons, the universal properties include the existence of a sequence of shallow 3-body bound states called “Efimov states” and log-periodic dependence of scattering observables on the energy and the scattering length. The spectrum of Efimov states in the limit a → ± ∞ is characterized by an asymptotic discrete scaling symmetry that is the signature of renormalization group flow to a limit cycle. In this review, we present a thorough treatment of universality for the system of three identical bosons and we summarize the universal information that is currently available for other 3-body systems. Our basic tools are the hyperspherical formalism to provide qualitative insights, Efimovs radial laws for deriving the constraints from unitarity, and effective field theory for quantitative calculations. We also discuss topics on the frontiers of universality, including its extension to systems with four or more particles and the systematic calculation of deviations from universality.

Free energy of QCD at high temperature

Effective-field-theory methods are used to separate the free energy for a non-Abelian gauge theory at high temperature {ital T} into the contributions from the momentum scales {ital T}, {ital gT}, and {ital g}{sup 2}{ital T}, where {ital g} is the coupling constant at the scale 2{pi}{ital T}. The effects of the scale {ital T} enter through the coefficients in the effective Lagrangian for the three-dimensional effective theory obtained by dimensional reduction. These coefficients can be calculated as power series in {ital g}{sup 2}. The contribution to the free energy from the scale {ital gT} can be calculated using perturbative methods in the effective theory. It can be expressed as an expansion in {ital g} starting at order {ital g}{sup 3}. The contribution from the scale {ital g}{sup 2}{ital T} must be calculated using nonperturbative methods, but nevertheless it can be expanded in powers of {ital g} beginning at order {ital g}{sup 6}. We calculate the free energy explicitly to order {ital g}{sup 5}. We also outline the calculations necessary to obtain the free energy to order {ital g}{sup 6}. {copyright} {ital 1996 The American Physical Society.}

Gluon fragmentation into heavy Quarkonium

The dominant production mechanism for heavy-quark--antiquark bound states with large transverse momentum is fragmentation, the splitting of a high energy parton into a quarkonium state and other partons. We show that the fragmentation functions [ital D]([ital z],[mu]) describing these processes can be calculated using perturbative QCD. We calculate the fragmentation functions for a gluon to split into [ital S]-wave quarkonium states to leading order in the QCD coupling constant.

Hard-Thermal-Loop Resummation of the Free Energy of a Hot Gluon Plasma

We calculate the free energy of a hot gluon plasma to leading order in hard-thermal-loop perturbation theory. Effects associated with screening, gluon quasiparticles, and Landau damping are resummed to all orders. The ultraviolet divergences generated by the hard-thermal-loop propagator corrections can be canceled by a counterterm which depends on the thermal gluon mass. The deviation of the hard-thermal-loop free energy from lattice QCD results for T{gt}2T{sub c} has the correct sign and roughly the correct magnitude to be accounted for by next-to-leading order corrections. {copyright} {ital 1999} {ital The American Physical Society}

Three-body recombination in bose gases with large scattering length

An effective field theory for the three-body system with large scattering length is applied to three-body recombination to a weakly bound s-wave state in a Bose gas. Our model independent analysis demonstrates that the three-body recombination constant alpha is not universal, but can take any value between zero and 67.9Plancks over 2pia(4)/m, where a is the scattering length. Other low-energy three-body observables can be predicted in terms of a and alpha. Near a Feshbach resonance, alpha should oscillate between those limits as the magnetic field B approaches the point where a-->infinity. In any interval of B over which a increases by a factor of 22.7, alpha should have a zero.

Exact relations for a strongly interacting fermi gas from the operator product expansion

The momentum distribution in a Fermi gas with two spin states and a large scattering length has a tail that falls off like 1/k4 at large momentum k, as pointed out by Tan. He used novel methods to derive exact relations between the coefficient of the tail in the momentum distribution and various other properties of the system. We present simple derivations of these relations using the operator product expansion for quantum fields. We identify the coefficient as the integral over space of the expectation value of a local operator that measures the density of pairs.

Z0 decay into charmonium via charm quark fragmentation

In decays of the Z 0 , the dominant mechanism for the direct production of charmonium states is the decay of the Z 0 into a charm quark or antiquark followed by its fragmentation into the charmonium state. We calculate the fragmentation functions describing the splitting of charm quarks into S-wave charmonium states to leading order in the QCD coupling constant. Leading logarithms of M Z /m c are summed up using Altarelli-Parisi evolution equations. Our analytic result agrees with the complete leading order calculation of the rate for Z 0 → ψcc. We also use our fragmentation functions to calculate the production rate of heavy quarkonium states in W ± , top quark, and Higgs boson decays

Low-energy universality and the new charmonium resonance at 3870 MeV

The recently-discovered narrow charmonium resonance near 3870 MeV is interpreted as a hadronic molecule whose constituents are the charm mesons D^0 and \bar D^{*0} or \bar D^0 and D^{*0}. Because of an accidental fine-tuning of the molecule to very near the D^0 \bar D^{*0} threshold, it has some universal properties that are completely determined by the unnaturally large D^0 \bar D^{*0} scattering length a. Its narrow width can be explained by the suppression by a factor of 1/a of decay modes other than the decay of a constituent \bar D^{*0} or D^{*0}. Its production rates are also suppressed by a factor of 1/a. A particularly predictive mechanism for generating the large scattering length is the accidental fine-tuning of a P-wave charmonium state to the D^0 \bar D^{*0} threshold.

Three-Body Recombination into Deep Bound States in a Bose Gas with Large Scattering Length

An effective field theory for the three-body system with large two-body scattering length a is applied to three-body recombination into deep bound states in a Bose gas. The recombination constant alpha is calculated to first order in the short-distance interactions that allow the recombination. For a < 0, the dimensionless combination m alpha/(Plancks constant a(4)) is a periodic function of ln (absolute value a) that exhibits resonances at values of a that differ by multiplicative factors of 22.7. This dramatic behavior should be observable near a Feshbach resonance when a becomes large and negative.

Perturbative QCD fragmentation functions as a model for heavy-quark fragmentation

The perturbative QCD fragmentation functions for a heavy quark to fragment into heavy-light mesons are studied in the heavy-quark limit. The fragmentation functions for {ital S}-wave pseudoscalar and vector mesons are calculated to next-to-leading order in the heavy-quark mass expansion using the methods of heavy-quark effective theory. The results agree with the {ital m}{sub {ital b}}{r_arrow}{infinity} limit of the perturbative QCD fragmentation functions for {ital {bar b}} into {ital B}{sub {ital c}} and {ital B}{sub {ital c}}{sup *}. We discuss the application of the perturbative QCD fragmentation functions as a model for the fragmentation of heavy quarks into heavy-light mesons. Using this model we predict the fraction {ital P}{sub {ital V}} of heavy-light mesons that are produced in the vector meson state as functions of the longitudinal momentum fraction {ital z} and the transverse momentum relative to the jet axis. The fraction {ital P}{sub {ital V}} is predicted to vary from about 1/2 at small {ital z} to almost 3/4 near {ital z}=1.