On the predictivity of the non‐renormalizable quantum field theories

Following a Four Dimensional Renormalization approach to ultraviolet divergences (FDR), we extend the concept of predictivity to non‐renormalizable quantum field theories at arbitrarily large perturbative orders. The idea of topological renormalization is introduced, which keeps a finite value for the parameters of the theory by trading the usual order‐by‐order renormalization procedure for an order‐by‐order redefinition of the perturbative vacuum. One additional measurement is then sufficient to systematically compute quantum corrections at any loop order, with no need of absorbing ultraviolet infinities in the Lagrangian.     

Direct evidence for a Coulombic phase in monopole-suppressed SU(2) lattice gauge theory

Further evidence is presented for the existence of a non-confining phase at weak coupling in SU(2) lattice gauge theory. Using Monte Carlo simulations with the standard Wilson action, gauge-invariant SO(3)–Z2 monopoles, which are strong-coupling lattice artifacts, have been seen to undergo a percolation transition exactly at the phase transition previously seen using Coulomb gauge methods, with an infinite lattice critical point near β=3.2$\beta =3.2$. The theory with both Z2 vortices and monopoles and SO(3)–Z2 monopoles eliminated is simulated in the strong-coupling (β=0$\beta =0$) limit on lattices up to 60⁴. Here, as in the high-β phase of the Wilson-action theory, finite size scaling shows it spontaneously breaks the remnant symmetry left over after Coulomb gauge fixing. Such a symmetry breaking precludes the potential from having a linear term. The monopole restriction appears to prevent the transition to a confining phase at any β  . Direct measurement of the instantaneous Coulomb potential shows a Coulombic form with moderately running coupling possibly approaching an infrared fixed point of α∼1.4$\alpha \sim 1.4$. The Coulomb potential is measured to 50 lattice spacings and 2 fm. A short-distance fit to the 2-loop perturbative potential is used to set the scale. High precision at such long distances is made possible through the use of open boundary conditions, which was previously found to cut random and systematic errors of the Coulomb gauge fixing procedure dramatically. The Coulomb potential agrees with the gauge-invariant interquark potential measured with smeared Wilson loops on periodic lattices as far as the latter can be practically measured with similar statistics data.     

Brief introduction to Wilson loops and large <Emphasis Type="Italic">N</Emphasis>

A brief pedagogical introduction to Wilson loops, lattice gauge theory, and 1/N expansion in QCD is presented.     

Being Discrete about Yang and Mills: Basic Techniques of Euclidean Lattice Gauge Theory

We discuss the formulation of gauge-invariant quantum field theories (without dynamical matter fields) as statistical mechanics systems on four-dimensional Euclidean lattices. Approximation methods including strong- and weak-coupling expansions, mean-field theory and Monte Carlo simulations are reviewed in detail, and Abelian duality transformations are derived. New models are discussed. An action is defined on 2 × 1 rectangular loops of links and its properties are investigated. It is found to result in phase transitions in 2, 3 and 4 dimensions with Z(2) and SU(2) gauge groups. A large class of models with Z(N) symmetry realised on plaquettes is investigated, and several phase diagrams are presented. A mixed model with interactions through both plaquettes and rectangles is found to have a line of phase transitions and a critical point associated with the crossover region in the Wilson SU(2) model.     

Strong-coupling planar diagrams,random surfaces,and the large-N phase transition in U(N) lattice gauge theories

The strong-coupling expansion of U(N) gauge theory on a D-dimensional lattice is reformulated in the limit N → ∞ through a set of diagrammatic rules directly for the free energy and Wilson loops. The strong-coupling planar diagrams are interpreted as surfaces embedded in the lattice. The large-N phase transition is related to the entropy of these surfaces. It is shown that the strong-coupling phase of the U(∞) gauge theory terminates with a phase transition of Gross-Witten type only in 2 and 3 dimensions. When D⩾4 the large-N singularity takes place in a metastable phase because of an earlier first-order transition to the weak-coupling phase of the theory.     

Scaling behaviour of Creutz ratios inSU(2) lattice gauge theory

An analysis of the scaling behaviour of Creutz ratios on large lattices is given forSU(2) gauge theory. The β-interval is 2.5≦β≦2.8. Under a factor 2 scaling test, after multiplicative corrections for lattice artifacts, the Monte Carlo data show deviations from scaling, which are similar for all values of β. The ratios can be fitted successfully by a sum of three perturbative terms and an exponentially decreasing nonperturbative term. For many ratios the latter turns out to be very small, and its size dependence at fixed β is consistent with that of an area term in the Wilson loops. The deviation of the corresponding exponents from the ones expected for an area term gives a coherent cxplanation of the observed departures from scaling. It is well possible that for fixed spatial extension (in lattice units) nonperturbative contributions vanish so fast that they cannot be interpreted as physical effects.     

Kähler metrics and gauge kinetic functions for intersecting D6‐branes on toroidal orbifolds – The complete perturbative story

We systematically derive the perturbatively exact holomorphic gauge kinetic function, the open string Kähler metrics and closed string Kähler potential on intersecting D6‐branes by matching open string one‐loop computations of gauge thresholds with field theoretical gauge couplings in 𝒩 = 1 supergravity. We consider all cases of bulk, fractional and rigid D6‐branes on T⁶/Ω ℛ and the orbifolds T⁶/(ℤ_N × Ω ℛ) and T⁶/(ℤ₂ × ℤ_2M × Ω ℛ) without and with discrete torsion, which differ in the number of bulk complex structures and in the bulk Kähler potential. Our analysis includes all supersymmetric configurations of vanishing and non‐vanishing angles among D6‐branes and O6‐planes, and all possible Wilson line and displacement moduli are taken into account. The shape of the Kähler moduli turns out to be orbifold independent but angle dependent, whereas the holomorphic gauge kinetic functions obtain three different kinds of one‐loop corrections: a Kähler moduli dependent one for some vanishing angle independently of the orbifold background, another one depending on complex structure moduli only for fractional and rigid D6‐branes, and finally a constant term from intersections with O6‐planes. These results are of essential importance for the construction of the related effective field theory of phenomenologically appealing D‐brane models. As first examples, we compute the complete perturbative gauge kinetic functions and Kähler metrics for some T⁶/ℤ₂ × ℤ₂ examples with rigid D‐branes of [1]. As a second class of examples, the Kähler metrics and gauge kinetic functions for the fractional QCD and leptonic D6‐brane stacks of the Standard Model on T⁶/ℤ₆^′T⁶/ℤ₆^′ from [2] are given.     

Breaking E6 via trinification in F‐theory

We consider E₆ GUT model in F‐theory approach where E₆ is broken via trinification to the Standard Model (SM) gauge group using non‐abelian fluxes. Including the gauge singlet wave function we found hierarchically small values for both the μ term as well as Dirac neutrino masses.     

Supersymmetric Wilson loops in super-Chern–Simons-matter theory

We study supersymmetric Wilson loop operators in ABJM theory from both sides of the AdS₄/CFT₃ correspondence. We first construct some supersymmetric Wilson loops. The perturbative computations are performed in the field theory side at the first two orders. A fundamental string solution ending on a circular loop is also studied.     

On the quark-antiquark potential at short distances

We investigate the static quark-antiquark potential up to distances of 8 lattice units for pure SU(2) gauge theory on lattices with anisotropic couplings. The action is the Wilson action with a coupling for time-like plaquettes which differs from those for space-like ones. Numerical simulations are performed in a large range of β The potential is obtained by fitting “cooled” Wilson loops with up to four exponential terms. An interpolation of the potentials by a sum of a perturbative term, a linear term and by lattice artifacts shows poor scaling in comparison with he isotropic case. If the coupling in the time-like region is reduced, the linear term is much smaller than in the isotropic case, and vice versa. Consequences for the bag picture for hadrons are discussed.     

A Hierarchical Model of Quantum Anharmonic Oscillators: Critical Point Convergence

We study the two-dimensional gauge theory of the symmetric group S_n describing the statistics of branched n-coverings of Riemann surfaces. We consider the theory defined on the disc and on the sphere in the large-n limit. A non trivial phase structure emerges, with various phases corresponding to different connectivity properties of the covering surface. We show that any gauge theory on a two-dimensional surface of genus zero is equivalent to a random walk on the gauge group manifold: in the case of S_n, one of the phase transitions we find can be interpreted as a cutoff phenomenon in the corresponding random walk. A connection with the theory of phase transitions in random graphs is also pointed out. Finally we discuss how our results may be related to the known phase transitions in Yang-Mills theory. We discover that a cutoff transition occurs also in two dimensional Yang-Mills theory on a sphere, in a large N limit where the coupling constant is scaled with N with an extra logN compared to the standard t Hooft scaling.     

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.     

The mechanism of alkaline hydrolysis of amides: a comparative computational and experimental study of the hydrolysis of N‐methylacetamide,N‐methylbenzamide,and acetanilide

Theoretical computations and experimental kinetic measurements were applied in studying the mechanistic pathways for the alkaline hydrolysis of three secondary amides: N‐methylbenzamide, N‐methylacetamide, and acetanilide. Electronic structure methods at the HF/6‐31+G(d,p) and B3LYP/6‐31+G(d,p) levels of theory are employed. The energies of the stationary points along the reaction coordinate were further refined via single point computations at the MP2/6‐31+G(d,p) and MP2/6‐311++G(2d,2p) levels of theory. The role of water in the reaction mechanisms is examined. The theoretical results show that in the cases of N‐methylbenzamide and N‐methylacetamide the process is catalyzed by an ancillary water molecule. The influence of water is further assessed by predicting its role as bulk solvent. The alkaline hydrolysis process in aqueous solution is characterized by two distinct free energy barriers: the formation of a tetrahedral adduct and its breaking to products. The results show that the rate‐determining stage of the process is associated with the second transition state. The entropy terms evaluated from theoretical computations referring to gas‐phase processes are significantly overestimated. The activation barriers for the alkaline hydrolysis of N‐methylbenzamide and acetanilide were experimentally determined. Quite satisfactory agreement between experimental values and computed activation enthalpies was obtained. Copyright © 2008 John Wiley & Sons, Ltd.     

Monte Carlo studies of wilson loops in four-dimensional U(2) gauge theory

Wilson loops are calculated using Monte Carlo simulations for pure U(2) gauge theory on a 6⁴ lattice. The loops appear to contain an area law piece in both the high and low temperature regions. The string tension is discontinuous at β = β_c, where β_c is the critical inverse temperature. This suggests that the first-order phase transition in U(2) gauge theory is not a deconfining phase transition. The determinant of the Wilson loop, however, extracts the U(1) part of the theory and appears to lose the area law at low temperature.     

From asymptotic freedom to quark confinement

Asymptotic freedom is the perturbative expression of paramagnetic screening by the QCD vacuum. When extended to large external fields the perturbative analysis leads to a catastrophe owing to the appearance of negative eigenvalues of the matrix describing the fluctuations of the gluonic field. This is met by a self-consistently determined magnetic-moment density. There results almost complete screening of the external field and the action is dominated by fluctuations of the magnetic moment. Such behaviour is more typical of very strong coupling e ⪢ 1 rather than that expected for e = O(1), the latter being the effective coupling at the QCD length scale. This kind of situation arises in the response to large Wilson loops, so that the derivation of the area law from strong-coupling lattice gauge theory can be rationalized.     

Critical exponents predicted by grouping of Feynman diagrams in φ4 model

Different perturbation theory treatments of the Ginzburg‐Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ⁴ model with O(n) symmetry. As a result, equations for calculation of the two‐point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.     

Effect of non‐extensivity parameter q on the damping rate of dust ion acoustic waves in non‐extensive dusty plasma

Kinetic theory has been applied to study the damping characteristics of dust ion acoustic waves (DIAWs) in a dusty plasma comprising q‐non‐extensive distributed electrons and ions, while the dust particles are considered extensive following the Maxwellian velocity distribution function. It is found that the results of the three‐dimensional velocity distribution function are more accurate compared to the results of the one‐dimensional velocity distribution function. The numerical solution of the dispersion relation is carried out to study the effect of the non‐extensivity parameter q on the dispersion, the damping rate, and the range of the values of the normalized wavenumber ( k λ_D) for which the DIAWs are weakly damped. It is found that the change in the value of the electron non‐extensivity parameter q_e has a minor effect on the dispersion, the damping rate, and the range of the values of the normalized wavenumber ( k λ_D) for which the DIAWs are weakly damped, while on the other hand, ion non‐extensivity parameter q_i has a strong effect on these arguments. The effect of other parameters, such as the ratio of electron to ion number density and ratio of electron to ion temperature, on the damping characteristics of DIAWs is also highlighted.     

M‐strings,elliptic genera and N = 4 string amplitudes

We study mass‐deformed N = 2 gauge theories from various points of view. Their partition functions can be computed via three dual approaches: firstly, (p,q)‐brane webs in type II string theory using Nekrasov's instanton calculus, secondly, the (refined) topological string using the topological vertex formalism and thirdly, M theory via the elliptic genus of certain M‐strings configurations. We argue for a large class of theories that these approaches yield the same gauge theory partition function which we study in detail. To make their modular properties more tangible, we consider a fourth approach by connecting the partition function to the equivariant elliptic genus of ℂ² through a (singular) theta‐transform. This form appears naturally as a specific class of one‐loop scattering amplitudes in type II string theory on T², which we calculate explicitly.