Hard-thermal-loop QED thermodynamics

The weak-coupling expansion for thermodynamic quantities in thermal field theories is poorly convergent unless the coupling constant is tiny.We discuss the calculation of the free energy for a hot gas of electrons and photons to three-loop order using hard-thermal-loop perturbation theory (HTLpt).We show that the hard-thermal-loop perturbation reorganization improves the convergence of the successive approximations to the QED free energy at large coupling,e ～ 2.The reorganization is gauge invariant by construction,and due to the cancellations among various contributions,we obtain a completely analytic result for the resummed thermodynamic potential at three loops.     

Resummation in hot field theories

There has been significant progress in our understanding of finite-temperature field theory over the past decade. In this paper, we review the progress in perturbative thermal field theory focusing on thermodynamic quantities. We first discuss the breakdown of naive perturbation theory at finite temperature and the need for an effective expansion that resums an infinite class of diagrams in the perturbative expansion. This effective expansion which is due to Braaten and Pisarski, can be used to systematically calculate various static and dynamical quantities as a weak-coupling expansion in powers of g. However, it turns out that the weak-coupling expansion for thermodynamic quantities are useless unless the coupling constant is very small. We critically discuss various ways of reorganizing the perturbative series for thermal field theories in order to improve its convergence. These include screened perturbation theory (SPT), hard-thermal-loop perturbation theory, the Φ-derivable approach, dimensionally reduced (DR) SPT, and the DR Φ-derivable approach.     

Heavy quark diffusion in perturbative QCD at next-to-leading order

We compute the momentum diffusion coefficient of a heavy quark in a hot QCD plasma, to next-to-leading order in the weak-coupling expansion. Corrections arise at [see formula]; physically they represent interference between overlapping scatterings, as well as soft, electric scale (p approximately gT) gauge field physics, which we treat using the hard thermal loop effective theory. In 3-color, 3-flavor QCD, the momentum diffusion constant of a fundamental representation heavy quark at next-to-leading order is kappa = 16pi/3alpha(s)(2)T(3)(ln1/g(s)+0.07428+1.9026 g(s)). The convergence of the perturbative expansion is poor.     

Hard-thermal-loop QCD trace anomaly

In this proceedings I summarize results of QCD trace anomaly from recent three-loop hard-thermal-loop perturbation theory (HTLpt) calculations. I focus on the trace anomaly scaled by T ² for pure-glue and N _f = 3 QCD. The comparison to available lattice data suggests that for pure-glue QCD agreement between HTLpt results and lattice data for the trace anomaly begins at temperatures above 8 T _c while when including quarks (N _f = 3) agreement begins already at temperatures above 2 T _c . The results in both cases indicate that at very high temperatures the T ²-scaled trace anomaly increases with temperature in accordance with the predictions of HTLpt.     

Optimization and the ultimate convergence of QCD perturbation theory

I suggest that QCD perturbation theory can be convergent, and that “optimization” of the renormalization scheme choice is essential in achieving this. Arguing that higher orders probe shorter distances, I suggest that the effective expansion parameter (the “optimized” couplant) decreases at high orders, leading to an induced convergence. The mechanism is illustrated by a simple mathematical example. The point is that, even if the perturbation series is divergent in all fixed renormalization schemes, the sequence of “optimized” approximations may still converge. It is emphasized that the limit approached by perturbation theory, if any, will not be the exact result of the full theory. Allegations that QCD series are not Borel-summable are critically re-examined in this light.     

A consistent next-to-leading-order QCD calculation of hadronic diffractive scattering

We calculate the order _s² and order _s³ QCD contributions to colour-singlet exchange in the leading log s approximation. We implement the resulting amplitude at the hadronic level and thus construct the QCD pomeron and odderon to this order of perturbation theory. We show that the structure of the hadronic form factors provides a natural mechanism through which the odderon gets suppressed at t = 0 whereas it dominates the elastic cross section at large t. We also demonstrate that the inclusion of nonperturbative effects through a modification of the gluon propagator accelerates greatly the convergence of the log s expansion, although not enough to provide agreement with the data.     

Fluctuations and the energy-optimal control of chaos

We compute the pressure of a finite-density quark-gluon plasma at zero temperature to leading order in hard-thermal-loop perturbation theory, which includes the fermionic excitations and Landau damping. The result is compared with the weak-coupling expansion for finite positive chemical potential &mgr; through order alpha(2)(s) and with a quasiparticle model with a mass depending on &mgr;.     

A variational principle for bound states in quantum field theory

We develop a Rayleigh-Ritz variational method for estimating relativistic, multi-particle bound state energies in any (weak-coupling) quantum field theory. A comparison is made with bound state energies derived from the Bethe-Salpeter equation in the Wick-Cutkosky model. Possible applications to QCD are discussed.     

Nonperturbative weak-coupling analysis of the quantum Liouville field theory

Two independent weak-coupling expansions are developed for the Liouville quantum field theory on a circle. In the first, the coupling of the nonzero modes is treated as a perturbation on the exact solution to the zero-mode problem (quantum mechanics with an exponential potential). The second approach is a weak-coupling approximation to an explicit operator solution which expresses various Liouville operators as functions of a free massless field using a Bäcklund transformation. It is shown that the free state space associated with the latter solution must be restricted to the sector which is odd with respect to a type of “parity.” Various matrix elements are computed to order g¹⁰ using both approaches, yielding identical results.     

KAM Theory in Configuration Space and Cancellations in the Lindstedt Series

The KAM theorem for analytic quasi-integrable anisochronous Hamiltonian systems yields that the perturbation expansion (Lindstedt series) for any quasi-periodic solution with Diophantine frequency vector converges. If one studies the Lindstedt series by following a perturbation theory approach, one finds that convergence is ultimately related to the presence of cancellations between contributions of the same perturbation order. In turn, this is due to symmetries in the problem. Such symmetries are easily visualised in action-angle coordinates, where the KAM theorem is usually formulated by exploiting the analogy between Lindstedt series and perturbation expansions in quantum field theory and, in particular, the possibility of expressing the solutions in terms of tree graphs, which are the analogue of Feynman diagrams. If the unperturbed system is isochronous, Moser’s modifying terms theorem ensures that an analytic quasi-periodic solution with the same Diophantine frequency vector as the unperturbed Hamiltonian exists for the system obtained by adding a suitable constant (counterterm) to the vector field. Also in this case, one can follow the alternative approach of studying the perturbation expansion for both the solution and the counterterm, and again convergence of the two series is obtained as a consequence of deep cancellations between contributions of the same order. In this paper, we revisit Moser’s theorem, by studying the perturbation expansion one obtains by working in Cartesian coordinates. We investigate the symmetries giving rise to the cancellations which makes possible the convergence of the series. We find that the cancellation mechanism works in a completely different way in Cartesian coordinates, and the interpretation of the underlying symmetries in terms of tree graphs is much more subtle than in the case of action-angle coordinates.     

Borel convergence of the variationally improved mass expansion and dynamical symmetry breaking

A modification of perturbation theory, known as the delta expansion (variationally improved perturbation), gave rigorously convergent series in some D=1 models (oscillator energy levels) with factorially divergent ordinary perturbative expansions. In a generalization of the variationally improved perturbation technique appropriate to renormalizable asymptotically free theories, we show that the large expansion orders of certain physical quantities are similarly improved, and prove the Borel convergence of the corresponding series for , with the new (arbitrary) mass perturbation parameter. We argue that non-ambiguous estimates of quantities relevant to dynamical (chiral) symmetry breaking in QCD are possible in this resummation framework. Received: 25 February 2002 / Published online: 8 May 2002     

Perturbative calculation of critical exponents for the Bose–Hubbard model

We develop a strategy for calculating critical exponents for the Mott insulator-to-superfluid transition shown by the Bose–Hubbard model. Our approach is based on the field-theoretic concept of the effective potential, which provides a natural extension of the Landau theory of phase transitions to quantum critical phenomena. The coefficients of the Landau expansion of that effective potential are obtained by high-order perturbation theory. We counteract the divergency of the weak-coupling perturbation series by including the seldom considered Landau coefficient a ₆ into our analysis. Our preliminary results indicate that the critical exponents for both the condensate density and the superfluid density, as derived from the two-dimensional Bose–Hubbard model, deviate by less than 1 % from the best known estimates computed so far for the three-dimensional XY universality class.     

Analytic perturbation theory in analyzing some QCD observables

This paper is devoted to the application of the recently devised ghost-free analytic perturbation theory (APT) for the analysis of some QCD observables. We start with a discussion of the main problem of the perturbative QCD, ghost singularities, and with a resume of its resolving within the APT. By a few examples in various energy and momentum transfer regions (with the flavor number f=3,4 and 5) we demonstrate the effect of the improved convergence of the APT modified perturbative QCD expansion. Our first observation is that in the APT analysis the three-loop contribution () is as a rule numerically inessential. This gives hope for a practical solution of the well-known problem of the asymptotic nature of the common QFT perturbation series. The second result is that the usual perturbative analysis of time-like events with the large term in the coefficient is not adequate at . In particular, this relates to decay. Then for the “high” () region it is shown that the common two-loop (NLO, NLLA) perturbation approximation widely used there (at ) for the analysis of shape/events data contains a systematic negative error at the 1–2 per cent level for the extracted values. Our physical conclusion is that the value averaged over the data appreciably differs, , from the currently accepted “world average” (=0.118). Received: 30 July 2001 / Published online: 5 November 2001     

Screened perturbation theory at four loops

We study the thermodynamics of massless ⁴-theory using screened perturbation theory, which is a way to systematically reorganise the perturbative series. The free energy and pressure are calculated through four loops in a double expansion in powers of g² and m/T, where m is a thermal mass of order gT. The result is truncated at order g⁷. We find that the convergence properties are significantly improved compared to the weak-coupling expansion.     

Is asymptotic freedom enough?

Quantum field theories with strong interactions are usually required to be not only renormalizable but also asymptotically free, in order to avoid diseases such as the Landau ghost. In this paper we suggest an even more restrictive requirement: “asymptotic convergence”, which means that at high energies it must be possible to formulate a convergent resummation procedure for the perturbation expansion. Such a convergent resummation technique exists in QCD in the infinite color limit (N → ∞). We give an outline of a proof of this statement, and a brief account of possible consequences of our asymptotic convergence condition on model building.     

Quasilinearization method: Nonperturbative approach to physical problems

The general properties of the quasilinearization method (QLM), particularly its fast quadratic convergence, monotonicity, and numerical stability, are analyzed and illustrated on different physical problems. The method approaches the solution of a nonlinear differential equation by approximating the nonlinear terms by a sequence of linear ones and is not based on the existence of a small parameter. It is shown that QLM gives excellent results when applied to different nonlinear differential equations in physics, such as Blasius, Lane-Emden, and Thomas-Fermi equations, as well as in computation of ground and excited bound-state energies and wave functions in quantum mechanics (where it can be applied by casting the Schrödinger equation in the nonlinear Riccati form) for a variety of potentials most of which are not treatable with the help of perturbation theory. The convergence of the QLM expansion of both energies and wave functions for all states is very fast and the first few iterations already yield extremely precise results. The QLM approximations, unlike the asymptotic series in perturbation theory and 1/N expansions, are not divergent at higher orders. The method sums many orders of perturbation theory as well as of the WKB expansion. It provides final and accurate answers for large and infinite values of the coupling constants and is able to handle even supersingular potentials for which each term of the perturbation series is infinite and the perturbation expansion does not exist.     

广义非简谐振子的多尺度微扰理论

应用多尺度微扰理论到广义非简谐振子, 得到了一阶经典和量子微扰解. 特别是 我们的量子解在极限条件下能方便地转变为经典解, 并且坐标和动量算符的对易 关系的简化十分自然. 与Taylor级数解相比较, 无论是在经典还是在量子解 中频率移动都出现在各阶振动表达式中, 所以多尺度微扰解是弱耦合非简谐振动的较好解法.