Asymptotic freedom: An approach to strong interactions

The discovery of asymptotic freedom has opened up the possibility of extracting new sorts of detailed, dynamical consequences from a strongly interacting quantum field theory. The necessary tools - perturbation theory, the renormalization group, gauge theories, and the operator product expansion - are not new. To anyone familiar with these field theoretic approaches to strong interactions, the novel feature is a simple fact: there is a unique class of theories in which “the origin is an ultraviolet fixed point”. But the consequences are so exciting that it seemed appropriate to review these ideas as they reflect on each other. Many important applications of the renormalization group and the operator product expansion to hadronic physics are omitted; the emphasis here is on recent work based on asymptotically free field theories. No doubt, there are some developments so recent that they are not treated in this article.The discussion of the basic results concerning short distance behavior is informal, but, hopefully, accurate and complete. The specific applications are treated in varying detail.     

Wilson's operator expansion: Can it fail?

Within the framework of some simple models we discuss the status of the operator product expansion (OPE) in the presence of nonperturbative effects. We consider, in particular, the 4d Higgs model, 2d sigma model and the Schwinger model. The general formulation of OPE is presented and it is demonstrated that there exists a consistent procedure allowing one to define unambiguously both coefficient functions and matrix elements of composite operators. One of the key elements of the procedure is the introduction of an auxiliary parameter, the normalization point μ. For the simplest T-products discussed in the literature earlier we construct the corresponding OPE explicitly. Then we check its validity by comparing the results for the two-point functions with independent direct calculations of the same correlators. Although the general procedure is standard and does not vary from one theory to another, numerically the relative role of perturbative and nonperturbative contributions in vacuum condensates is different in different theories. The two extremes considered are the λ?⁴ theory with no spontaneous breaking of the symmetry and the O(N) sigma model in the limit N → ∞. In the former case there is only perturbative contribution to (?²), while in the latter case the perturbative pieces are suppressed by 1/N factors. Numerically QCD is much closer to the O(N) sigma model in the large-N limit. Comments on specific features of QCD are presented.     

Calculation of the Mass of X（3872） by the Mandelstam Generalization of the Gell-Mann-Low Method

On the basis of assuming that the narrow state X（3872） is a molecule state consisting of D0 and D＊0, we apply the Mandelstam generalization of the Ge11-Mann-Low method to calculate the matrix element of quark current between the heavy meson states described by Bether-Salpeter wave function. In calculation of the matrix element of quark current the operator product expansion is used in order to include the nonperturbative contribution of the vacuum condensates. In this scheme we calculate the mass of X（3872）. We believe that this scheme is closer to QCD than the previous work.     

Scalar Hidden-Charm Tetraquark States with QCD Sum Rules

In this article, we study the masses and pole residues of the pseudoscalar-diquark-pseudoscalar-antidiquark type and vector-diquark-vector-antidiquark type scalar hidden-charm cud(cus) tetraquark states with QCD sum rules by taking into account the contributions of the vacuum condensates up to dimension-10 in the operator product expansion.The predicted masses can be confronted with the experimental data in the future. Possible decays of those tetraquark states are also discussed.     

Analysis of the Z(4430) as the First Radial Excitation of the Z_c(3900)

In this article,we take the Zc(3900) and Z(4430) as the ground state and the first radial excited state of the axial-vector tetraquark states with J~(PC) = 1~(+-),respectively,and study their masses and pole residues with the QCD sum rules by calculating the contributions of the vacuum condensates up to dimension-10 in a consistent way in the operator product expansion.The numerical result favors assigning the Z_c(3900) and Z(4430) as the ground state and first radial excited state of the axial-vector tetraquark states,respectively.     

Analysis of the scalar doubly charmed hexaquark state with QCD sum rules

In this article, we study the scalar-diquark–scalar-diquark–scalar-diquark type hexaquark state with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension 16. We obtain a lowest hexaquark mass of \(6.60^{+0.12}_{-0.09}\,\mathrm {GeV}\), which can be confronted with the experimental data in the future.     

Green functions of vortex operators

We study the euclidean Green functions of the 't Hooft vortex operator, primarily for abelian gauge theories. The operator is written in terms of elementary fields, with emphasis on a form in which it appears as the exponential of a surface integral. We explore the requirement that the Green functions depend only on the boundary of this surface. The Dirac veto problem appears in a new guise. We present a two-dimensional “solvable model” of a Dirac string, which suggests a new solution of the veto problem. The renormalization of the Green functions of the abelian Wilson loop and abelian vortex operator is studied with the aid of the operator product expansion. In each case, an overall multiplication of the operator makes all Green functions finite; a surprising cancellation of divergences occurs with the vortex operator. We present a brief discussion of the relation between the nature of the vacuum and the cluster properties of the Green functions of the Wilson and vortex operators, for a general gauge theory. The surface-like cluster property of the vortex operator in an abelian Higgs theory is explored in more detail.     

Analysis of X(5568) as Scalar Tetraquark State in Diquark-Antidiquark Model with QCD Sum Rules

In this article, we take the X(5568) as the diquark-antidiquark type tetraquark state with the spin-parity J^P=0⁺, construct the scalar-diquark-scalar-antidiquark type current, carry out the operator product expansion up to the vacuum condensates of dimension-10, and study the mass and pole residue in details with the QCD sum rules. We obtain the value M_X=(5.57±0.12) GeV, which is consistent with the experimental data. The present prediction favors assigning the X(5568) to be the scalar tetraquark state.     

Systematic field theory of the RNA glass transition

We prove that the L?ssig-Wiese (LW) field theory for the freezing transition of the secondary structure of random RNA is renormalizable to all orders in perturbation theory. The proof relies on a formulation of the model in terms of random walks and on the use of the multilocal operator product expansion. Renormalizability allows us to work in the simpler scheme of open polymers, and to obtain the critical exponents at 2-loop order. It also allows us to prove some exact exponent identities, conjectured by LW.     

A Linearization Approach for Rational Nonlinear Models in Mathematical Physics

In this paper,a novel method for linearization of rational second order nonlinear models is discussed.In particular,we discuss an application of the δ expansion method(created to deal with problems in Quantum Field Theory) which will enable both the linearization and perturbation expansion of such equations.Such a method allows for one to quickly obtain the order zero perturbation theory in terms of certain special functions which are governed by linear equations.Higher order perturbation theories can then be obtained in terms of such special functions.One benefit to such a method is that it may be applied even to models without small physical parameters,as the perturbation is given in terms of the degree of nonlinearity,rather than any physical parameter.As an application,we discuss a method of linearizing the six Painleve equations by an application of the method.In addition to highlighting the benefits of the method,we discuss certain shortcomings of the method.     

Spontaneous symmetry breaking in massless field theories: Dimensional transmutation and gauge independence

In a reformulation of the Coleman and Weinberg program, we study the parametrization of the spontaneous symmetry breaking solutions of massless field theories. Our analysis is based on a full use of the concept of the dimensional transmutation, namely the parametrization in terms of the independent invariants of the underlying renormalization group. This invariant parametrization provides a convenient setting for the operator interpretation in a given Fock space and is shown to be a powerful tool in studying general aspects of the theory beyond the one-loop approximation. In particular, the gauge independence of the physical matrix elements in the spontaneously broken solution of massless scalar electrodynamics is demonstrated to all orders in perturbation theory.     

Molecular g-tensors from analytical response theory and quasi-degenerate perturbation theory in the framework of complete active space self-consistent field method

The molecular g-tensor is an important spectroscopic parameter provided by electron para magnetic resonance (EPR) measurement and often needs to be interpreted using computational methods. Here, we present two new implementations based on the first-order and second-order perturbation theories to calculate the g-tensors within the complete-active space self-consistent field (CASSCF) wave function model. In the first-order method, the quasi-degenerate perturbation theory (QDPT) is employed for constructing relativistic CASSCF states perturbed with the spin–orbit coupling operator, which is described effectively in one-electron form with the flexible nuclear screening spin–orbit approximation introduced recently by us. The second-order method is a newly reported approach built upon the linear response theory which accounts for the perturbation with respect to external magnetic field. It is implemented with the coupled–perturbed CASSCF (CP-CASSCF) approach, which provides an equivalent of untruncated sum-over-states expansion. The comparison of the performances between the first-order and second-order methods is shown for various molecules containing light to heavy elements, highlighting their relative strength and weakness. The formulations of QDPT and CP-CASSCF approaches as well as the derivation of the second-order Douglas–Kroll–Hess picture change of Zeeman operators are given in detail.     

Perturbative quantum chromodynamics

A large variety of modern perturbative aspects of QCD is critically reviewed from a theoretical as well as phenomenological point of view. The first part of this review is devoted to the classical more formal approach of summing leading logs: After a brief discussion of the basic concepts of renormalization theory, we review the renormalization group and its predictions for the effective (running) coupling constant in any field theory (asymptotic freedom as well as ‘fixed point’ theories). Using, in addition, the operator product expansion for deep inelastic scattering we calculate scaling violations of structure functions and show how to compare these results with experiment. Furthermore, dynamical calculations of parton distributions are discussed, as well as σ_L/σ_T, jets in leptoproduction and subleading corrections. We then proceed to show how these renormalization group improved results can be also derived using a simple perturbative language (Kogut-Susskind; Altarelli-Parisi) or by summing parton (Bethe-Salpeter) ladders. The universal validity (process independence) of the resulting Q² dependencies of parton distributions is emphasized and their factorization from the uncalculable non-perturbative piece (infrared divergences) is discussed. These latter results enable us to make rather unambiguous predictions for processes other than deep inelastic scattering, to which the remainder of this review is devoted. The hard scattering processes discussed indetail include hadronic (Drell-Yan) production of lepton pairs as well as their transverse momenta, the hadronic production of heavy quark flavors, semi-inclusive processes and fragmentation functions, high-p_T reactions and some recent topics and problems of jet production in e⁺e^? annihilation.     

Gauge-invariant sigma models in N=2 superspace

We construct a new broad class of gauge-invariant nonlinear sigma models in N=2 superspace. The dimensional reduction of the theories obtained to d=2 leads to N=4 nonlinear sigma models which are devoid of ultraviolet divergence in all orders of perturbation theory.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 49–53, February, 1989.     

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.     

Operator product expansion in two-dimensional quantum gravity

We consider correlation functions of operators and the operator product expansion in two-dimensional quantum gravity. First we introduce correlation functions with geodesic distances between operators kept fixed. Next, by making two of the operators closer, we examine whether there exists an analog of the operator product expansion in ordinary field theories. Our results suggest that the operator product expansion holds in quantum gravity as well, though special care should be taken regarding the physical meaning of fixing geodesic distances on a fluctuating geometry.     

Spontaneous symmetry breaking of Slavnov symmetry A restriction on the gauge condition

It is shown that the condition of vanishing vacuum expectation value of the gauge operator, using a gauge-fixing term quadratic in this operator, does not necessarily follow from the Slavnov identity, due to a possible spontaneous breakdown of the Slavnov symmetry. For a consistent renormalization such a condition may have to be imposed order by order in perturbation theory, depending on the choice of the gauge. This restriction on non-singular gauges is particularly relevant for the discussion of the spontaneously broken realizations of the gauge symmetry.     

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     

Finite element methods for low-dimensional fermion field theories

We examine lattice fermion models based on operator difference equations derived using the method of finite elements, motivated by the absence of species doubling in such formulations. Interacting fermion theories in (0+1) and (1+1) spacetime dimensions are constructed, and shown to be consistent in the sense that suitably defined equal time commutation relations are preserved from timeslice to timeslice. Using weak coupling perturbation theory to calculate the two-point function, we show that the non-locality inherent in the differencing scheme results in a violation of microcasuality in the continuum limit.