Classical Yang-Mills Research Articles

Searching for KvBLL calorons in SU(3) lattice gauge field ensembles

We discuss Kraan - van Baal - Lee - Lu (KvBLL) solutions of the classical Yang-Mills equations with temperature in the context of SU(3) lattice gauge theory. We present discretized lattice versions of KvBLL solutions and other dyonic structures, obtained by cooling in order to understand their variety and signature. An analysis of the zero modes of the lattice Dirac operator for different fermionic boundary conditions gives clear evidence for a KvBLL-like background of finite T lattice subensembles with Q = +/-1. Using APE-smearing we are able to study the topological charge density q(x) of the configurations and to corroborate this interpretation.

EFFECTIVE 't HOOFT–POLYAKOV MONOPOLES FROM PURE SU(3) GAUGE THEORY

The well-known topological monopoles originally investigated by 't Hooft and Polyakov are known to arise in classical Yang–Mills–Higgs theory. With a pure gauge theory, it is known that the classical Yang–Mills field equation do not have such finite energy configurations. Here we argue that such configurations may arise in a semi-quantized Yang–Mills theory, where the original gauge group, SU(3), is reduced to a smaller gauge group, SU(2), and with some combination of the coset fields of the SU(3) to SU(2) reduction acting as effective scalar fields. The procedure is called semi-quantized since some of the original gauge fields are treated as quantum degrees of freedom, while others are postulated to be effectively described as classical degrees of freedom. Some speculation is offer on a possible connection between these monopole configurations and the confinement problem, and the nucleon spin puzzle.

Chiral Ring of Sp(N) and SO(N) Supersymmetric Gauge Theory in Four Dimensions

The chiral ring of classical supersymmetric Yang-Mills theory with gauge group Sp(N) or SO(N) is computed, extending previous work (of Cachazo, Douglas, Seiberg, and the author) for SU(N). The result is that, as has been conjectured, the ring is generated by the usual glueball superfield S ~ TrWαWα, with the relation Sh = 0, h being the dual Coxeter number. Though this proposition has important implications for the behavior of the quantum theory, the statement and (for the most part) the proofs amount to assertions about Lie groups with no direct reference to gauge theory.

Production of gluons in the classical field model for heavy ion collisions

The initial stages of relativistic heavy ion collisions are studied numerically in the framework of a $(2+1)$-dimensional classical Yang-Mills theory. We calculate the energy and number densities and momentum spectra of the produced gluons. The model is also applied to noncentral collisions. The numerical results are discussed in the light of RHIC measurements of energy and multiplicity and other theoretical calculations. Some problems of the present approach are pointed out.

Gluons, Chirality and v 2 from the Color Glass Condensate

I review recent applications of the classical Yang-Mills formalism to the dynamics in the central-rapidity region of high-energy heavy-ion collisions.

Calorons, instantons, and constituent monopoles in SU(3) lattice gauge theory

We analyze the zero-modes of the Dirac operator in quenched SU(3) gauge configurations at non-zero temperature and compare periodic and anti-periodic temporal boundary conditions for the fermions. It is demonstrated that for the different boundary conditions often the modes are localized at different space-time points and have different sizes. Our observations are consistent with patterns expected for Kraan - van Baal solutions of the classical Yang-Mills equations. These solutions consist of constituent monopoles and the zero-modes are localized on different constituents for different boundary conditions. Our findings indicate that the excitations of the QCD vacuum are more structured than simple instanton-like lumps.

New findings for topological excitations in SU(3) lattice gauge theory

We probe the SU(3) vacuum using eigenvectors of the Dirac operator with an arbitrary phase for the temporal boundary condition. We consider configurations with topological charge | Q|=1 near the QCD phase transition and at low temperatures on a torus. For all our ensembles we show that the zero mode of the Dirac operator changes its position as one changes the phase of the boundary condition. For ensembles near the QCD phase transition our results closely resemble the behavior of zero modes for Kraan–van Baal solutions of the classical Yang–Mills equations where the individual lumps are interpreted as monopoles. Our findings near T c and on the torus show that for both cases an excitation with topological charge | Q|=1 is built from several separate lumps.

Gluon production in the Color Glass Condensate model of collisions of ultrarelativistic finite nuclei

We extend previous work on high energy nuclear collisions in the Color Glass Condensate model to study collisions of finite ultrarelativistic nuclei. The changes implemented include (a) imposition of color neutrality at the nucleon level and (b) realistic nuclear matter distributions of finite nuclei. The saturation scale characterizing the fields of color charge is explicitly position-dependent, Λ s = Λ s ( x T ). We compute gluon distributions both before and after the collisions. The gluon distribution in the nuclear wavefunction before the collision is significantly suppressed below the saturation scale when compared to the simple McLerran–Venugopalan model prediction, while the behavior at large momentum p T ⪢ Λ s remains unchanged. We study the centrality dependence of produced gluons and compare it to the centrality dependence of charged hadrons exhibited by the RHIC data. We demonstrate the geometrical scaling property of the initial gluon transverse momentum distributions for different centralities. Classical Yang–Mills results for p T < Λ s are simply matched to perturbative QCD computations for p T > Λ s —the resulting energy per particle is significantly lower than the purely classical estimates. Our results for nuclear collisions can be used as initial conditions for quantitative studies of the further evolution and possible equilibration of hot and dense gluonic matter produced in heavy ion collisions. Finally, we study pA collisions within the classical framework. Our results agree well with previously derived analytical results in the appropriate kinematical regions.

Forced tunneling and turning state explosion in pure Yang-Mills theory

We consider forced tunneling in QCD, described semiclassically by instanton--anti-instanton field configurations. By separating topologically different minima we obtain details of the effective potential and study the turning states, which are similar to the sphaleron solution in electroweak theory. These states are alternatively derived as minima of the energy under the constraints of fixed size and Chern-Simons number. We study, both analytically and numerically, the subsequent evolution of such states by solving the classical Yang-Mills equations in real time, and find that the gauge field strength is quickly localized into an expanding shell of radiating gluons. The relevance to high-energy collisions of hadrons and nuclei is briefly discussed.

Functional approach to classical Yang-Mills theories

Sometime ago it was shown that the operatorial approach to classical mechanics, pioneered in the 30's by Koopman and von Neumann, can have a functional version. In this talk we will extend this functional approach to the case of classical field theories and in particular to the Yang-Mills ones. We shall show that the issues of gauge-fixing and Faddeev-Popov determinant arise also in this classical formalism.

A local Langevin equation for slow long-distance modes of hot non-Abelian gauge fields

The effective theory for the dynamics of hot non-Abelian gauge fields with spatial momenta of order of the magnetic screening scale g^2 T is described by a Boltzmann equation. The dynamical content of this theory is explored. There are three relevant frequency scales, g T, g^2 T and g^4 T, associated with plasmon oscillations, multipole fluctuations of color charge, and with the non-perturbative gauge field dynamics, respectively. The frequency scale g T is integrated out. The result is a local Langevin-type equation. It is valid to leading order in g and to all orders in log(1/g), and it does not suffer from the hard thermal loop divergences of classical thermal Yang-Mills theory. We then derive the corresponding Fokker-Planck equation, which is shown to generate an equilibrium distribution corresponding to 3-dimensional Yang-Mills theory plus a Gaussian free field.

Selfdual solution of classical yang—mills fields through a q-analog of ADHM construction

The ADHMN construction of selfdual Yang—Mills gauge fields is reformulated by introducing a functional space defined on a discrete point set. Using this formalism, a q-analogous object of the BPS monopole is obtained.

YANG–MILLS DUALITY AND THE GENERATION PUZZLE

The fermion generation puzzle has survived into this century as one of the great mysteries in particle physics. We consider here a possible solution within the Standard Model framework based on a non-Abelian generalization of electric–magnetic duality. First, by constructing in loop space a non-Abelian generalization of the Abelian dual transform (Hodge *), one finds that a "magnetic" symmetry exists also in classical Yang–Mills theory dual to the original ("electric") gauge symmetry. Secondly, from a result of 't Hooft's, one obtains that for confined color SU(3), the dual symmetry [Formula: see text] is spontaneously broken and can play the role of the "horizontal symmetry" for generations. Thirdly, such an identification not only offers an explanation why there should be three and apparently only three generations of fermions with the remarkable mass and mixing patterns seen in experiment, but allows even a calculation of the relevant parameters giving very sensible results. Other testible predictions follow ranging from rare hadron decays to cosmic ray air showers.

Yang-Mills duality and the generation puzzle

The fermion generation puzzle has survived into this century as one of the great mysteries in particle physics. We consider here a possible solution within the Standard Model framework based on a non-Abelian generalization of electric–magnetic duality. First, by constructing in loop space a non-Abelian generalization of the Abelian dual transform (Hodge *), one finds that a magnetic exists also in classical Yang–Mills theory dual to the original (electric) gauge symmetry. Secondly, from a result of 't Hooft's, one obtains that for confined color SU(3), the dual is spontaneously broken and can play the role of the horizontal symmetry for generations. Thirdly, such an identification not only offers an explanation why there should be three and apparently only three generations of fermions with the remarkable mass and mixing patterns seen in experiment, but allows even a calculation of the relevant parameters giving very sensible results. Other testible predictions follow ranging from rare hadron decays to cosmic ray air showers.

Classical sphaleron rate on fine lattices

We measure the sphaleron rate for hot, classical Yang-Mills theory on the lattice, in order to study its dependence on lattice spacing. By using a topological definition of Chern-Simons number and going to extremely fine lattices (up to beta=32, or lattice spacing a = 1 / (8 g^2 T)) we demonstrate nontrivial scaling. The topological susceptibility, converted to physical units, falls with lattice spacing on fine lattices in a way which is consistent with linear dependence on $a$ (the Arnold-Son-Yaffe scaling relation) and strongly disfavors a nonzero continuum limit. We also explain some unusual behavior of the rate in small volumes, reported by Ambjorn and Krasnitz.

Classical Yang-Mills equations invariant under subgroups of the conformal group without coupling to matter

Classical Yang-Mills field equations on Minkowski space with gaugegroup SU(2) are determined. Solutions invariant up to a gauge transformation under a four-dimensional subgroup of the conformal group are evaluated for the Yang-Mills field in the absence of matter. Some properties of the solutions of the equations such as their asymptotic behaviour are obtained.PACS No.: 03.50-z

Classical Yang-Mills field equations coupled to a scalar triplet invariant under subgroups of the conformal group

Classical Yang-Mills-Scalar field equations on Minkowski space with gauge group SU (2) are determined. Solutions invariant up to a gauge transformation under a four-dimensional subgroup of the conformal group are determined for the case of a scalar matter triplet field.

One-parameter family of selfdual solutions in classical Yang-Mills theory

The ADHM construction, which yields (anti-)selfdual configurations in classical Yang-Mills theories, is applied to an infinite dimensional l 2 vector space, and as a consequence, a family of (anti-)selfdual configurations with a parameter q is obtained for SU(2) Yang-Mills theory. This l 2 formulation can be seen as a q-analog of Nahm's monopole construction, so that the configuration approaches the BPS monopole at q→1 limit.

Gluon minijet production in nuclear collisions at high energies

We show explicitly that the leading soft gluon $p_T$ distribution, predicted by Kovner, McLerran, and Weigert after solving classical Yang-Mills equations, can be understood in terms of conventional QCD perturbation theory. We also demonstrate that the key logarithm in their result represents the logarithm in DGLAP evolution equations.

Lax equations in 10-dimensional supersymmetric classical Yang-Mills theories

In a recent paper (hep-th/9811108), Saveliev and the author showed that there exits an on-shell light cone gauge where the non-linear part of the field equations reduces to a (super) version of Yang's equations which may be solved by methods inspired by the ones previously developed for self-dual Yang-Mills equations in four dimensions. Here, the analogy between these latter theories and the present ones is pushed further by writing down a set of super partial linear differential equations whose consistency conditions may be derived from the SUSY Y-M equations in ten dimensions, and which are the analogues of the Lax pair of Belavin and Zakharov. On the simplest example of the two pole ansatz, it is shown that the same solution-generating techniques are at work, as for the derivation of the celebrated multi-instanton solutions carried out in the late seventies. The present Lax representation, however, is only a consequence of (instead of being equivalent to) the field equations, in contrast with the Belavin Zakharov Lax pair.