arXiv subjects
Publications and source records attributed to Eric Braaten.
The Bose-Einstein condensation of atoms can be conveniently formulated as a problem in thermal quantum field theory. There are many properties of the equilibrium system and its collective excitations that can be studied experimentally. The remarkable experimental control over all aspects of the system make it an ideal testing ground for the methods of thermal field theory.
Quantum corrections to the properties of a homogeneous interacting Bose gas at zero temperature can be calculated as a low-density expansion in powers of $\sqrt{ρa^3}$, where $ρ$ is the number density and $a$ is the S-wave scattering length. We calculate the ground state energy density to second order in $\sqrt{ρa^3}$. The coefficient of the $ρa^3$ correction has a logarithmic term that was calculated in 1959. We present the first calculation of the constant under the logarithm. The constant depends not only on $a$, but also on an extra parameter that describes the low energy $3\to 3$ scattering of the bosons. In the case of alkali atoms, we argue that the second order quantum correction is dominated by the logarithmic term, where the argument of the logarithm is $ρa \ell_V^2$, and $\ell_V$ is the length scale set by the van der Waals potential.
The oscillation frequencies of collective excitations of a trapped Bose-Einstein condensate, when calculated in the mean-field approximation and in the Thomas-Fermi limit, are independent of the scattering length $a$. We calculate the leading corrections to the frequencies from quantum fluctuations around the mean field. The semiclassical correction is proportional to $N^{1/5} a^{6/5}$, where $N$ is the number of atoms in the condensate. The correction is positive semidefinite and is zero for surface modes whose eigenfunctions have a vanishing laplacian. The shift in the frequency of the lowest quadrupole mode for an axially symmetric trap is large enough that it should be measurable in future experiments.
I describe the NRQCD factorization approach to the inclusive production of heavy quarkonium, contrasting it with the color-singlet and color evaporation models. These approaches differ dramatically in their assumptions about the roles played by color and spin in the production process. They also differ dramatically in their predictions for the production of charmonium at large transverse momentum.
We propose a new explanation for the longstanding puzzle of the tiny branching fraction of psi' -> rho pi relative to that for J/psi -> rho pi. In the case of J/psi, we argue that this decay is dominated by a higher Fock state in which the ccbar pair is in a color-octet ^3S_1 state and via annihilate process ccbar -> qqbar. In the case of the psi', we argue that the probability for the ccbar pair in this higher Fock state to be close enough to annihilate is suppressed by a dynamical effect related to the small energy gap between the mass of the psi' and the DDbar threshold.
In the mean-field approximation, the number density ρ(r) for the ground state of a Bose-Einstein condensate trapped by an external potential V(r) satisfies a classical field equation called the Gross-Pitaevskii equation. We show that quantum corrections to ρare dominated by quantum fluctuations with wavelengths of order 1/\sqrt{ρa}, where a is the S-wave scattering length. By expanding the equations for the Hartree-Fock approximation to second order in the gradient expansion, we derive local correction terms to the Gross-Pitaevskii equation that take into account the dominant effects of quantum fluctuations. We also show that the gradient expansion for the density breaks down at fourth order.
I present an introduction to the NRQCD factorization method for calculating annihilation rates and inclusive production rates for heavy quarkonium. Using this method, annihilation decay rates and sufficiently inclusive cross sections are factored into long-distance NRQCD matrix elements and perturbative short-distance coefficients. I derive the velocity-scaling rules that are used to estimate the magnitudes of the nonperturbative NRQCD matrix elements and I describe perturbative matching methods for calculating their short-distance coefficients. Some simple applications for which NRQCD factorization methods have dramatic implications are discussed.
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 $α_s(M)$. 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 $v^2$.
Dimensional regularization is incompatible with the standard covariant projection methods that are used to calculate the short-distance coefficients in inclusive heavy quarkonium production and annihilation rates. A new method is developed that allows dimensional regularization to be used consistently to regularize the infrared and ultraviolet divergences that arise in these perturbative calculations. We illustrate the method by calculating the leading color-octet terms and the leading color-singlet terms in the gluon fragmentation functions for arbitrary quarkonium states. We resolve a discrepancy between two previous calculations of the gluon fragmentation functions for the spin-triplet P-wave quarkonium states.
Contributions to the decay rates of P-wave charmonium states that are proportional to $n_f α_s^3$, where $n_f$ is the number of flavors of light quarks, are calculated in the framework of nonrelativistic QCD using the threshold expansion method. Dimensional regularization is used to regularize the infrared divergences that arise from the emission of a soft gluon. Our results are consistent with the original calculations of Barbieri et al.
Sufficiently inclusive observables in the decay of the tau lepton can be calculated using the methods of perturbative QCD. These include the asymmetry parameter $A_τ$ that determines that angular distribution of the total hadron momentum in the decay of a polarized tau. It should be possible to measure $A_τ$ accurately using existing data from LEP. Reliable estimates of theoretical errors are essential in order to determine whether a given observable is sufficiently inclusive to be calculated using perturbative methods. The theoretical uncertainties due to higher orders in $α_s$ can be estimated using recent calculations to all orders in the large-$(33-2N_f)$ limit. These estimates indicate that tau decay data can be used to determine $α_s(M_Z)$ to a precision of 2% or better.
The NRQCD factorization approach for calculating inclusive production of heavy quarkonium gives unambiguous predictions for the polarization of quarkonium states. The factorization formula for polarized states can be obtained by using the threshold expansion method to calculate the short-distance coefficients and then using symmetries of NRQD to reduce the NRQCD matrix elements. A particularly dramatic prediction of the NRQCD factorization framework is that prompt psi's and psi-primes's produced at the Tevatron should be predominantly transversely polarized at large transverse momentum.
The low-density expansion for a homogeneous interacting Bose gas at zero temperature can be formulated as an expansion in powers of $\sqrt{ρa^3}$, where $ρ$ is the number density and $a$ is the S-wave scattering length. Logarithms of $ρa^3$ appear in the coefficients of the expansion. We show that these logarithms are determined by the renormalization properties of the effective field theory that describes the scattering of atoms at zero density. The leading logarithm is determined by the renormalization of the pointlike $3 \to 3$ scattering amplitude.
In recent years, there have been significant developments in heavy quarkonium production, both in theory and in experiment. These developments may have important implications for the use of charmonium as a probe in heavy-ion collisions. A summary of these developments is presented below. Those interested in more details are referred to a recent review article by Braaten, Fleming, and Yuan.
We present a general method for calculating inclusive cross sections for the production of heavy quarkonium states with definite polarization within the NRQCD factorization approach. Cross sections for polarized production can involve additional matrix elements that do not contribute to cross sections for unpolarized production. They can also include interference terms between parton processes that produce $Q \bar Q$ pairs with different total angular momentum. The interference terms cancel upon summing over polarizations. Our method can be generalized to $N$ dimensions and is therefore compatible with the use of dimensional regularization to calculate radiative corrections. We illustrate the method by applying it to the production of $J/ψ$ via the parton processes $q \bar q \to c \bar c$ and $g g \to c \bar c$.
The inclusive cross section for the production of $J/ψ$ in $e^+ e^-$ annihilation is studied using a recently developed factorization formalism. In addition to the conventional color-singlet contribution, we include contributions from $c \bar c$ pairs that are produced at short distances in color-octet states. The color-octet contributions are suppressed by the smaller probability for the color-octet $c \bar c$ pair to evolve into a state that includes a $ψ$, but they may dominate near the upper endpoint of the $ψ$ energy spectrum. The signature for the color-octet contribution is a dramatic change in the angular distribution of the $ψ$ near the endpoint.
Effective-field-theory methods are used to separate the free energy for a nonabelian gauge theory at high temperature $T$ into the contributions from the momentum scales $T$, $gT$, and $g^2T$, where $g$ is the coupling constant at the scale $2 πT$. The effects of the scale $T$ enter through the coefficients in the effective lagrangian for the 3-dimensional effective theory obtained by dimensional reduction. These coefficients can be calculated as power series in $g^2$. The contribution to the free energy from the scale $gT$ can be calculated using perturbative methods in the effective theory. It can be expressed as an expansion in $g$ starting at order $g^3$. The contribution from the scale $g^2T$ must be calculated using nonperturbative methods, but nevertheless it can be expanded in powers of $g$ beginning at order $g^6$. We calculate the free energy explicitly to order $g^5$. We also outline the calculations necessary to obtain the free energy to order $g^6$.
Recent data from the Tevatron have revealed that the production rate of prompt charmonium is more than an order of magnitude larger than the best theoretical predictions of a few years ago. This surprising result can be understood by taking into account new production mechanisms that include fragmentation and the formation of charmonium from color-octet $c \bar c$ pairs.