Hamiltonian reduction of SU(2) gluodynamics

The Hamiltonian reduction of the Yang-Mills theory with the structure group SU(2) to a nonlocal model of a self-interacting 3 × 3 positive semidefinite matrix field is presented. Analysis of the field transformation properties under the action of the Poincaré group is carried out. It is shown that, in the strong coupling limit, the classical dynamics of a reduced system can be described by the local theory of interacting nonrelativistic spin-0 and spin-2 fields. A perturbation theory in powers of the inverse coupling constant g ^−2/3 that allows calculating the corrections to a leading long-wave approximation is suggested.     

Dual field theory of strong interactions

A dual field theory of strong interactions is derived from a Lagrangian of the Yang-Mills and Higgs fields. The existence of a magnetic monopole of mass 2397 MeV and Dirac chargeg=(137/2)e is incorporated into the theory. Unification of the strong, weak, and electromagnetic forces is shown to converge at the mass of the intermediate vector bosonW ^±. The coupling constants of the strong and weak interactions are derived in terms of the fine-structure constant=1/137.     

Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops

We prove conjecture due to Erickson-Semenoff-Zarembo and Drukker-Gross which relates supersymmetric circular Wilson loop operators in the N=4{\mathcal N=4} supersymmetric Yang-Mills theory with a Gaussian matrix model. We also compute the partition function and give a new matrix model formula for the expectation value of a supersymmetric circular Wilson loop operator for the pure N=2{\mathcal N=2} and the N=2^*{\mathcal N=2^*} supersymmetric Yang-Mills theory on a four-sphere. A four-dimensional N=2{\mathcal N=2} superconformal gauge theory is treated similarly.     

Stability and isolation phenomena for Yang-Mills fields

In this article a series of results concerning Yang-Mills fields over the euclidean sphere and other locally homogeneous spaces are proved using differential geometric methods. One of our main results is to prove that any weakly stable Yang-Mills field overS ⁴ with groupG=SU₂, SU₃ orU ₂ is either self-dual or anti-self-dual. The analogous statement for SO₄-bundles is also proved. The other main series of results concerns gap-phenomena for Yang-Mills fields. As a consequence of the non-linearity of the Yang-Mills equations, we can give explicitC ⁰-neighbourhoods of the minimal Yang-Mills fields which contain no other Yang-Mills fields. In this part of the study the nature of the groupG does not matter, neither is the dimension of the base manifold constrained to be four.Laboratoire Associé au C.N.R.S. No. 169Research partially supported by Volkswagen Grant and NSF Grant MCS-77-23579     

Geometroelectrodynamics

We present an elementary particle model that can be thought of as a unification of certain topological ideas abstracted from the string model and the standard Yang-Mills theory. The basic dynamical entity of the model is a spacelike 3-surfaceX ³ in some metric spaceH and is interpreted as a particle. The dynamics of the model is based on two ideas. First the model is formally a Yang-Mills theory on the surfaceX ⁴ representing the orbit(s) of the particle(s) inH. Secondly the Yang-Mills structure onX ⁴ is constructed using only the natural geometric structures of the space H by a process which we call induction. It is found that some rather general requirements highly fix the choice of the space H. In fact the minimal model, for which the space H is the product of Minkowski space and the 2-sphere, is obtained by requiring that the symmetry group of the theory is the product of the Poincaré group and the color groupSO(3). The unique feature of the minimal model is that it affords a purely topological mechanism for quark confinement.     

Removable singularities in Yang-Mills fields

We show that a field satisfying the Yang-Mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite. We obtain the result that every Yang-Mills field overR ⁴ with bounded functional (L ² norm) may be obtained from a field onS ⁴=R ⁴{}. Hodge (or Coulomb) gauges are constructed for general small fields in arbitrary dimensions including 4.     

On the possibility to measure the π0→γγ decay width and the γ?γ→π0 transition form factor with the KLOE-2 experiment

A possibility of KLOE-2 experiment to measure the width \varGamma_{p⁰ ?gg}\varGamma_{\pi^{0} \to\gamma\gamma} and the π ⁰ γγ ^∗ form factor F(Q ²) at low invariant masses of the virtual photon in the space-like region is considered. This measurement is an important test of the strong interaction dynamics at low energies. The feasibility is estimated on the basis of a Monte-Carlo simulation. The expected accuracy for \varGamma_{p⁰ ?gg}\varGamma_{\pi^{0} \to\gamma\gamma} is at a per cent level, which is better than the current experimental world average and theory. The form factor will be measured for the first time at Q ²≤0.1 GeV² in the space-like region. The impact of these measurements on the accuracy of the pion-exchange contribution to the hadronic light-by-light scattering part of the anomalous magnetic moment of the muon is also discussed.     

Asymptotic behaviour of supersymmetric Yang-Mills theories in the two-loop approximation

A calculation of the renormalisation group function β(g) to O(g⁵) for a supersymmetric Yang-Mills theory is presented. It is found that in no case do both the leading terms in the perturbation expansion of β(g) vanish.     

Light Quark Hadron Mass Scale

With a symmetry procedure based on Noether's theorem, the field equation of motion is obtained from the Dirac Hamiltonian H(D_μ) of a massless quark interacting with a gluon. The equation of motion is the Yang-Mills equation with external current which is spin-dependent and follows from the group algebra. In addition to the pure gauge solution we find a gauge covariant solution which follows from current conservation and sets the mass scale m₀/M = g². This gluon field is due to the density of dipole moments squared and represents four harmonic oscillators with quadratic constraints; the gluon can be written as a string potential or as a 1/x potential with a sharp cutoff. The chiral symmetry group G^spin × G^D gives the light quark hadron degenerate multiplet mass spectrum in terms of m₀[SU(2) × SU(2)] with the spinorial decomposition and the multipole breaks into dipoles. Scaling from atomic lengths it is found that g = em₀/nM for light quarks is the quark charge e/3 renormalized by m₀/M and g is magnetic. Thus quarks occur at the ends of spinning magnetic strings with dipole lengths ∼m₀⁻¹. The mass scale is that of a degenerate magnetic multipole with charge n = 3, 4… .     

The infrared structure of the quark-antiquark scattering kernel in quantumchromodynamics: A clue to the nature of the quark binding potential?

The equal rate, near-mass-shell behavior of the two particle irreducible, spin-independent, color singlet channel quark-antiquark (qq?) scattering kernel is studied to all orders in perturbation theory. A general functional form in terms of the long-distance Yang-Mills invariant charge g(t),t→0, t being the momentum transfer, is determined. Under certain conditions, the behavior of the Bethe-Salpeter iteration of this kernel in the qq? threshold regime suggests the existence of a universal qq? effective potential of the form g²_eff[g(t)]/t.     

Asymptotic behavior of the β function in the ϕ<Superscript>4</Superscript> theory: A scheme without complex parameters

The previously-obtained analytical asymptotic expressions for the Gell-Mann-Low function β(g) and anomalous dimensions in the ϕ⁴ theory in the limit g → ∞ are based on the parametric representation of the form g = f(t), β(g) = f ₁(t) (where t ∝ g ₀^−1/2 is the running parameter related to the bare charge g ₀), which is simplified in the complex t plane near a zero of one of the functional integrals. In this work, it has been shown that the parametric representation has a singularity at t → 0; for this reason, similar results can be obtained for real g ₀ values. The problem of the correct transition to the strong-coupling regime is simultaneously solved; in particular, the constancy of the bare or renormalized mass is not a correct condition of this transition. A partial proof has been given for the theorem of the renormalizability in the strong-coupling region.     

Signals of a Weibel instability in the melting color glass condensate

Based on our work hep-ph/0510121, we discuss further the numerical study of classical SU(2) 3+1-D Yang-Mills equations for matter produced in a high-energy heavy-ion collision. The growth of the amplitude of fluctuations as exp(Γ ) (where g ²μ is a scale arising from the saturation of gluons in the nuclear wave function) is shown to be robust over a wide range of initial amplitudes that violate boost invariance. We argue that this growth is due to a non-Abelian Weibel instability, the scale of which is set by a dynamically generated plasmon mass. We discuss the relation of Γ to the prediction from kinetic theory.     

Perturbation theory for the radial distribution function

Perturbation theory is used to consider expansions for the radial distribution function, g ₂(r), of a fluid with a soft core. We consider the Lennard-Jones (12, 6) potential and divide it into repulsive and attractive regions. In the repulsive region we expand the function exp (β u(r))g ₂(r) about a hard sphere value. For the first-order contribution of the attractive region we consider a simple approximation to the exact analytical expression. The resulting g ₂(r) is accurate at densities below about ρσ ³=0·5.     

Quaternionic-valued Gravitation in 8D, Grand Unification and Finsler Geometry

A unification model of 4D gravity and SU(3)×SU(2)×U(1) Yang-Mills theory is presented. It is obtained from a Kaluza-Klein compactification of 8D quaternionic gravity on an internal CP ²=SU(3)/U(2) symmetric space. We proceed to explore the nonlinear connection formalism used in Finsler geometry to show how ordinary gravity in D=4+2 dimensions has enough degrees of freedom to encode a 4D gravitational and SU(5) Yang-Mills theory. This occurs when the internal two-dim space is a sphere S ². This is an appealing result because SU(5) is one of the candidate GUT groups. We conclude by discussing how the nonlinear connection formalism of Finsler geometry provides an infinite hierarchical extension of the Standard Model within a six dimensional gravitational theory due to the embedding of SU(3)×SU(2)×U(1)⊂SU(5)⊂SU(∞).     

Electric-magnetic duality in non-abelian gauge theories

The duality transformation of the vacuum expectation value of the operator which creates magnetic vortices (the 't Hooft loop operator in the Higgs phase), is performed in the radial gauge (x_uA_u^a(x) = 0). It is found that in the weak coupling region (small g) of a pure Yang-Mills theory the dual operator creates electric vortices whose strength is $\text{1}\text{g}$. The theory is self-dual in this region, and the effective coupling of the dual Lagrangian is $\text{1}\text{g}$. (It is self-dual also in the extreme strong coupling region.) Thus the above duality transformation reduces to electric-magnetic duality where the electric field in the 't Hooft loop operators transforms into a magnetic field in the dual operator. In a spontaneously broken gauge theory these results are valid only within the region where the vortices (or the monopoles) are concentrated, or in directions of the algebra space of unbroken symmetry, as self-duality holds only for this subset of fields. Noting that the 't Hooft loop operator project into the subspace of these field configurations we find that it is an electric-magnetic duality for the spontaneously broken theory as well. In the strong coupling region a strong coupling expansion in powers $\text{1}\text{g}$ is suggested.     

Conformal properties of the BPST instantons of the generalised Yang-Mills system

A manifestly O(4p+1) invariant formulation of generalised Yang-Mills (GYM) theory on S ^4p {r|r·r=1} R ^4p+1 is given, and the corresponding BPST instantons and anti-instantons are shown to be solutions of the equations of motion.     

Massive gauge field in source theory. I

The vacuum polarization is calculated for a massive gauge field according to the methods of source theory. The spectral integral turns out to be convergent and the numerical coefficient, which is (1/3)(e²/4π²) in spinor electrodynamics, is here (3f/16)(g²/4π²) where f depends on the group and f = 1 for SU(2).     

Anharmonic Oscillator and Double-Well Potential: Approximating Eigenfunctions

A simple uniform approximation of the logarithmic derivative of the ground state eigenfunction for both the quantum-mechanical anharmonic oscillator and the double-well potential given by V=m ² x ²+g x ⁴ at arbitrary g ≥ 0 for m ²>0 and m ²<0, respectively, is presented. It is shown that if this approximation is taken as unperturbed problem it leads to an extremely fast convergent perturbation theory Mathematics Subject Classifications(2000) 34L40, 34B08, 41A99     

A connection between ν-dimensional Yang-Mills theory and (ν−1)-dimensional,non-linear σ-models

We study non-linear σ-models and Yang-Mills theory. Yang-Mills theory on the ν-dimensional lattice ? ^v can be obtained as an integral of a product over all values of one coordinate of non-linear σ-models on ? ^v?1 in random external gauge fields. This exhibits two possible mechanisms for confinement of static quarks one of which is that clustering of certain two-point functions of those σ-models implies confinement of static quarks in the corresponding Yang-Mills theory. Clustering is proven for all one-dimensional σ-models, for theU(n) ×U(n) σ-models,n=1, 2, 3, ..., in two dimensions, and for the SU(2) × SU(2) σ-models for a large range of couplingsg ² ? O(ν). Arguments pertinent to the construction of the continuum limit are discussed. A representation of the expectation of Wilson loops in terms of expectations of random surfaces bounded by the loops is derived when the gauge group is SU(2),U(n) or O(n),n=1, 2, 3, ..., and connections to the theory of dual strings are sketched.