Additive Renormalization Research Articles

Resonant tunneling and charge fluctuations in mesoscopic tunnel junctions

Quantum fluctuations in mesoscopic tunnel junctions are studied in the presence of high charging energies. The processes of sequential tunneling, electron cotunneling and resonant tunneling are analyzed within a new nonperturbative resummation technique. The current and the average charge are shown to be related to the spectral density describing the charge excitations of the single electron transistor. An energy-dependent finite lifetime is obtained which leads to the theory of inelastic electron cotunneling and gives important corrections to the classical result for strong tunneling and finite temperatures. An additional energy renormalization leads to a logarithmic temperature dependence of the renormalized system parameters. Several formulas for the line shape of the conductance oscillations are presented which show the possibility to observe experimentally quantum fluctuation effects at realistic temperatures.

Light-front quantization of the sine-Gordon model.

It is shown how to modify the canonical light-front quantization of the (1+1)-dimensional sine-Gordon model such that the zero-mode problem of light-front quantization is avoided. The canonical sine-Gordon Lagrangian is replaced by an effective Lagrangian which does not lead to divergences as [ital k][sup +]=([ital k][sup 0]+[ital k][sup 1])/ [radical]2 [r arrow]0. After canonically quantizing the effective Lagrangian, one obtains the effective light-front Hamiltonian which agrees with the naive light-front (LF) Hamiltonian, up to one additional renormalization. The spectrum of the effective LF Hamiltonian is determined using discrete light-cone quantization and agrees with results from equal-time quantization.

Topological charge on the lattice. The 2D CP N−1 model

We calculate the first two non-zero terms in the perturbative expansion of the additive and multiplicative renormalizations of the topological susceptibility, χ, in the 2D CP N−1 model, when a density of topological charge is defined as a local operator on the lattice. We compare our results with some direct determinations, by Monte Carlo techniques.

Topological susceptibility on the lattice: The two-dimensional O(3) sigma model.

As a test of the procedures used in lattice gauge theories, we study the topological susceptibility $\ensuremath{\chi}$ in a two-dimensional O(3) $\ensuremath{\sigma}$ or ${\mathrm{CP}}^{1}$ model. We determine $\ensuremath{\chi}$ by defining the density of topological charge as a local operator on the lattice. Following the prescriptions of field theory we perform the additive and multiplicative renormalizations needed to extract $\ensuremath{\chi}$ from Monte Carlo data. We also determine $\ensuremath{\chi}$ by the cooling method, finding consistent results. A combined use of cooling and field theory again gives the same result and insight into the renormalization mechanism. Finally we give a direct determination, by Monte Carlo techniques, of both the multiplicative and additive renormalizations, by heating configurations with a definite number of instantons. The results are consistent with perturbation theory.

A simplified formulation of the gamma variate function

The gamma variate function has been often used to describe the dispersion of a bolus as it passes through a series of compartments. For this reason, it is frequently chosen to fit first-pass data in studies quantifying cardiac output and left-to-right cardiac shunts. Although the gamma variate is an appropriate function to model these situations, it has several undesirable mathematical properties. Changes in the alpha and beta parameters affect not only the rise and fall times of the function, but also change the location and magnitude of the function maximum. This makes it difficult to anticipate how the function will be altered by varying the parameters and often requires an additional renormalization step when the gamma variate is being used to fit a curve. A different but entirely equivalent form of the gamma variate is derived in which these problems are eliminated.

Renormalization of polymer networks and stars

We consider polymer networks of arbitrary but fixed topology. We prove the multiplicative renormalizability of these networks in terms of their star-like vertices by relating it to special field theoretic correlation functions. To avoid additive renormalizations these correlation functions have to be differentiated with respect to certain temperature-type variables. For the polymer network this regularization procedure implies a constraint on the molecular weight distribution of its strands. Short strands must be suppressed. This in general excludes “field theoretic” networks which roughly speaking show an exponential chain length distribution. Our proof also holds for networks constructed from strands which differ in chemical constitution. The basic “star-like” exponents previously have been calculated in field theory to order ϵ 3, and we use this expansion in a Padé-Borel analysis, including also exact results established in two dimensions. We find good agreement with Monte Carlo results.

The topological susceptibility of the 2D O(3) σ model

We study the topological susceptibility, χ, of the 2D O(3) σ model by defining a density of topological charge as a local polynomial of the spin variables on the lattice. Following the field theoretical prescriptions we perform the additive and multiplicative renormalizations needed to extract χ from Monte Carlo data. By numerical simulations we show that the field theoretical definition and the cooling method give a consistent determination of χ.

Renormalization and topological susceptibility on the lattice

We study the renormalization of the topological susceptibility, χ, when the density of topological charge is defined as a local operator on the lattice. In the 2D O(3) δ model we determine both multiplicative and additive renormalizations by heating configurations with a definite number of instantons. We find results that are consistent with perturbation theory. The method is also applicable to the physically interesting case of 4D non-abelian gauge theories.

PARITY ANOMALY IN CHERN-SIMONS THEORY

Ultraviolet (uv) divergences are canceled in the effective action of the D=3 Chern-Simons (CS) gauge theory but regularization is needed. It is impossible to introduce gauge invariant regularization and conserve the parity of the classical action. As a result, in the limit when regularization is removed the finite contribution to the effective action induced by parity-violating regulators remains which being added to the classical action leads to additive integer-valued renormalization of the coupling constant.

Topological charge, renormalization and cooling on the lattice

We show that a definition of the topological susceptibility χ on the lattice in terms of local fields, with the proper additive and multiplicative renormalizations, is well defined, self-consistent and compatible with results we obtain by the cooling method. Moreover, we can follow the behaviour of the multiplicative renormalization during cooling, and check its dependence on the coupling constant.

The trilinear gluon condensate on the lattice

We study the operator g 3ƒ abcF μν aF νϱ bF ϱμ c on the lattice. We perform the additive and multiplicative renormalizations needed to define its condensate G 3 to one loop and calculate the mixing with lower dimensional operators. We also perform a Monte Carlo calculation and use our analytical results to extract the numerical value of G 3. In all this, we pay special attention to problems particular to the extraction from Monte Carlo data of matrix elements of higher dimensional operators in general.

Topological charge, renormalization and cooling on the lattice

Abstract We show that a definition of the topological susceptibility χ on the lattice in terms of local fields, with the proper additive and multiplicative renormalizations, is well defined, self-consistent and compatible with results we obtain by the cooling method. Moreover, we can follow the behaviour of the multiplicative renormalization during cooling, and check its dependence on the coupling constant.

A new theoretical constraint on the static critical behaviour of the specific heat

We show from a field-theoretical approach that, if we admit that the additive renormalization function of the specific heat C is singular at the fixed point, we obtain a coherent formulation of the critical behaviour of C. Especially we show that the α<0 case, which corresponds to a cusp for C, is dominated by a critical constant B cr generated by the long-range correlations of the fluctuations. We derive a universal combination between the leading and first correction amplitudes and B cr, which will have a great importance in the analysis of experimental data.

Anomalous dimensions and the Adler-Bardeen theorem in supersymmetric Yang-Mills theories

We analyze the Adler-Bardeen anomaly in a way which does not depend on a particular renormalization scheme and find that the validity of the Adler-Bardeen theorem is related to the fact that g 2 F ∗ F does not need any multiplicative renormalization in a gauge invariant regularization scheme. In supersymmetric theories the Adler-Bardeen theorem is still valid provided F ∗F is minimally renormalized. Only in a particular subtraction scheme, however, does this choice of renormalization of F ∗F agree with that obtained by rearranging the ∂ μ J 5 μ component of the supercurrent anomaly equation. In general schemes, F ∗ F acquires an anomalous dimension through an additional finite renormalization.

Theory of strings with boundaries: Fluctuations, topology and quantum geometry

We discuss Polyakov's quantization of the string in the presence of a boundary allowing for an arbitrary topology for the world sheet. In addition to the dynamical conformal factor discovered by Polyakov, there are a finite number of new degrees of freedom if the surface is more complicated than a sphere or a disc. The quantization of the Liouville theory in an arbitrary topology is discussed. A one-loop calculation shows that the model is renormalizable if one performs a mass renormalization and an additive field renormalization. The renormalization group equations have a perturbative infrared unstable fixed point in all topologies.

Aspects of an Unitary Classical Molecular Theory of the Crystalline and the Fluid State at Thermal Equilibrium

AbstractIt is shown that, in spite of the existence of localized particles in crystals, an unitary partition function and grand partition function, resp., exists for the fluid and the crystalline state if an auxiliary external potential is introduced and, following BOGOLYUBOV'S idea, its zero limit is taken after carrying out the thermodynamic limit. Therefore, an unitary description of both the states (including the phase boundary between them) based on molecular distribution functions is possible. A new class of exact correlation functions characterizing localized particles is considered. The excess part of the free energy functional which results from a functional integration of the direct correlation function shows finite self‐interaction contributions of localized particles which must be removed by an additional renormalization procedure.

Dimensional renormalization of scalar field theory in curved space-time

The regularization and renormalization of an interacting scalar field φ in a curved spacetime background is performed by the method of continuation to n dimensions. In addition to the familiar counter terms of the flat-space theory, c-number, “vacuum” counter terms must also be introduced. These involve zero, first, and second powers of the Reimann curvature tensor Rαβψδ. Moreover, the renormalizability of the theory requires that the Lagrange function couple φ2 to the curvature scalar R with a coupling constant η. The coupling η must obey an inhomogeneous renormalization group equation, but otherwise it is an arbitrary, free parameter. All the counter terms obey renormalization group equations which determine the complete structure of these quantities in terms of the residues of their simple poles in n − 4. The coefficient functions of the counter terms determine the construction of φ2 and φ4 in terms of renormalized composite operators 1, [φ2], and [φ4]. Two of the counter terms vanish in conformally flat space-time. The others may be computed from the theory in purely flat space-time. They are determined, in a rather intricate fashion, by the additive renormalizations for two-point functions of [φ2] and [φ4] in Minkowski space-time. In particular, using this method, we compute the leading divergence of the R2 interaction which is of fifth order in the coupling constant λ.

The effective potential in a rapidly oscillating irregular field

The averaged behavior of physical systems under the action of fields which are inhomogeneous in space and rapidly and irregularly oscillating in time is considered. The problem is studied analytically for random stationary fields, the temporal spectrum of which has a “hump” shape; the relation between the width of the hump and its central frequency is arbitrary involving the well-known limiting cases: monochromatic field and “white noise”. It is shown that a finite spectrum width leads to an additional renormalization of the well-known effective potential and the appearance (contrary to the monochromatic case) of some effective damping for averaged motions. It is essential that both damping and potential vary not monotonously and change sign when the spectrum width increases.

Local operator products in gauge theories, II

We study the operator product expansion of two gauge-invariant currents. We discuss the operators that may appear in the expansion. We study the hadronic matrix elements of gauge-variant operators that may appear in the operator product expansion and show that in the physical applications contemplated so far their hadronic physical matrix elements either vanish or cannot be isolated from strong interactions. We also study the gauge dependence of the Green's functions of the product of two (weak or e.m.) currents and show that is corresponds to the gauge dependence of the additional renormalization counterterms due to weak and electromagnetic interactions.

Equivalence of the sine-Gordon and Thirring models and cumulative mass effects

We investigate the equivalence of the sine-Gordon and Thirring models on the basis of the short-distance behavior in the massive Thirring model. We find that for $\mathrm{dim}\overline{\ensuremath{\psi}}\ensuremath{\psi}\ensuremath{\ge}1$ there is a new additive renormalization effect originating from the occurrence of nonleading mass singularities in the spinor vacuum expectation values. In the sine-Gordon language this effect makes its appearance as a "cumulative" mass effect. It leads to a breakdown of the naive variational method.