Narrow Results

This study investigates the vertex corrections of the optical conductivity in high-$T_c$ superconductors for hole-spin wave scattering in the superconducting state. The Feynman diagram technique is used to determine the mathematical expressions of the ladder diagrams in the calculation of the current–current correlation function. The sum of these vertex terms leads to a self-consistent form of the vertex function. By evaluating the frequency summation through contour integration, a general expression for the current–current correlation function is obtained, which allows us to determine the zeroth-order term and the first-order correction of the optical conductivity explicitly. Our findings for the dc conductivity in a clean superconductor indicate that the zeroth-order term is zero, a result attributed to the conservation of momentum and energy. Meanwhile, the first-vertex correction reveals the Drude behavior in the superconducting state at zero frequency.

We give a detailed introduction to the classical Chern–Simons gauge theory, including the mathematical preliminaries. We then explain the perturbative quantization of gauge theories via the Batalin–Vilkovisky (BV) formalism. We then define the perturbative Chern–Simons partition function at any (possibly non-acylic) reference flat connection using the BV formalism, using a Riemannian metric for gauge fixing. We show that it exhibits an anomaly known as the “framing anomaly” when the Riemannian metric is changed, that is, it fails to be gauge invariant. We explain how one can deal with this anomaly to obtain a topological invariant of framed manifolds.

The Feynman diagram generator FeynArts and the computer algebra program FormCalc allow for an automatic computation of 2→2 and 2→3 scattering processes in High Energy Physics. We have extended this package by four new kinematical routines and adapted one existing routine in order to accomodate also two- and three-body decays of massive particles. This makes it possible to compute automatically two- and three-body particle decay widths and decay energy distributions as well as resonant particle production within the Standard Model and the Minimal Supersymmetric Standard Model at the tree- and loop-level. The use of the program is illustrated with three standard examples: , , and .

We describe a new application of an existing perfect sampling technique of Corcoran and Tweedie to estimate the self energy of an interacting Fermion model via Monte Carlo summation. Simulations suggest that the algorithm in this context converges extremely rapidly and results compare favorably to true values obtained by brute force computations for low dimensional toy problems. A variant of the perfect sampling scheme which improves the accuracy of the Monte Carlo sum for small samples is also given.

Corrections to scaling in the 3D Ising model are studied based on nonperturbative analytical arguments and Monte Carlo (MC) simulation data for different lattice sizes $L$. Analytical arguments show the existence of corrections with the exponent $(\gamma -1)∕\nu \approx 0.38$, the leading correction-to-scaling exponent being $\omega \le (\gamma -1)∕\nu$. A numerical estimation of $\omega$ from the susceptibility data within $40\le L\le 2560$ yields $\omega =0.21(29)$, in agreement with this statement. We reconsider the MC estimation of $\omega$ from smaller lattice sizes, $L\le 384$, using different finite-size scaling methods, and show that these sizes are still too small, since no convergence to the same result is observed. In particular, estimates ranging from $\omega =0.866(21)$ to $\omega =1.247(73)$ are obtained, using MC data for thermodynamic average quantities, as well as for partition function zeros. However, a trend toward smaller $\omega$ values is observed in one of these cases in a refined estimation from extended data up to $L=1536$. We discuss the influence of $\omega$ on the estimation of critical exponents $\eta$ and $\nu$.

A new program created in C/C$++$ language generates automatically the analytic expression of grand potential and prints it in Latex2e format and in textual data. Another code created in Mathematica language can translate the textual data into a mathematical expression and help any interested to evaluate the thermodynamic quantities in analytic or numeric forms.

We present some methods which aim to express multi-leg and multi-loop Feynman diagrams in a form suited for a fast and reliable numerical evaluation, without any restriction on masses and phase-space region of the process.

The multi-particle productions in neutrino–nucleon collisions at high energy are investigated through the analysis of the data of the experiment CERN-WA-025 at neutrino energy less than 260 GeV and the experiments FNAL-616 and FNAL-701 at energy range 120–250 GeV. The general features of these experiments are used as base to build a hypothetical model that views the reaction through a Feynman diagram of two vertices. The first of which concerns the weak interaction between the neutrino and the quark constituents of the nucleon. At the second vertex, a strong color field is assumed to play the role of particle production, which depend on the momentum transferred from the first vertex. The wave functions of the nucleon constituent quarks are determined using the variation method and relevant boundary conditions are applied to calculate the deep inelastic cross sections of the virtual diagram.

We show how, modulo the distinction between the antipode and the "twisted" or "renormalized" antipode, Connes and Kreimer's algebraic paradigm trivializes the proofs of equivalence of the (corrected) Dyson–Salam, Bogoliubov–Parasiuk–Hepp and Zimmermann procedures for renormalizing Feynman amplitudes. We discuss the outlook for a parallel simplification of computations in quantum field theory, stemming from the same algebraic approach.

When performing a full calculation within the standard model (SM) or its extensions, it is crucial that one utilizes a consistent set of signs for the gauge couplings and gauge fields. Unfortunately, the literature is plagued with differing signs and notations. We present all SM Feynman rules, including ghosts, in a convention-independent notation, and we table the conventions in close to 40 books and reviews.

An overview is presented on the current status of main mathematical computation methods for the multiloop corrections to single-scale observables in quantum field theory and the associated mathematical number and function spaces and algebras. At present, massless single-scale quantities can be calculated analytically in QCD to 4-loop order and single mass and double mass quantities to 3-loop order, while zero-scale quantities have been calculated to 5-loop order. The precision requirements of the planned measurements, particularly at the FCC-ee, form important challenges to theory, and will need important extensions of the presently known methods.

In this paper, a short review of the operator/Feynman diagram/dessin d’enfants correspondence in the rank 3 tensor model is presented, and the cut and join operation is given in the language of dessin d’enfants as a straightforward development. We classify operators of the rank 3 tensor model up to level 5 with dessin d’enfants. (Based on the talk given by R. Y. at the international workshop “Randomness, Integrability and Representation Theory in Quantum Field Theory” at the Osaka City University Media Center on March 25, 2021.)

Quantum electrodynamics (QED) with self-conjugated equations with spinor wave functions for fermion fields is considered. In the low order of the perturbation theory, matrix elements of some of QED physical processes are calculated. The final results coincide with cross-sections calculated in the standard QED. The self-energy of an electron and amplitudes of processes associated with determination of the anomalous magnetic moment of an electron and Lamb shift are calculated. These results agree with the results in the standard QED. Distinctive feature of the developed theory is the fact that only states with positive energies are present in the intermediate virtual states in the calculations of the electron self-energy, anomalous magnetic moment of an electron and Lamb shift. Besides, in equations, masses of particles and antiparticles have the opposite signs.

An attempt to directly use the synchronous gauge ($g_{0\lambda }=-{\delta }_{0\lambda }$) in perturbative gravity leads to a singularity at $p_0=0$ in the graviton propagator. This is similar to the singularity in the propagator for Yang–Mills fields $A_{\lambda }^a$ in the temporal gauge ($A_0^a=0$). There the singularity was softened, obtaining this gauge as the limit at $\epsilon \to 0$ of the gauge $n^{\lambda }{A}_{\lambda }^a=0$, $n^{\lambda }=(1,-\epsilon {({\partial }^j{\partial }_j)}^{-1}{\partial }^k)$. Then the singularities at $p_0=0$ are replaced by negative powers of $p_0\pm i\epsilon$, and thus we bypass these poles in a certain way. Now consider a similar condition on $n^{\lambda }{g}_{\lambda \mu }$ in perturbative gravity, which becomes the synchronous gauge at $\epsilon \to 0$. Unlike the Yang–Mills case, the contribution of the Faddeev–Popov ghosts to the effective action is nonzero, and we calculate it. In this calculation, an intermediate regularization is needed, and we assume the discrete structure of the theory at short distances for that. The effect of this contribution is to change the functional integral measure or, for example, to add nonpole terms to the propagator. This contribution vanishes at $\epsilon \to 0$. Thus, we effectively have the synchronous gauge with the resolved singularities at $p_0=0$, where only the physical components $g_{jk}$ are active and there is no need to calculate the ghost contribution.

We evaluate the mutual information between the input and the output of a two layer network in the case of a noisy and nonlinear analogue channel. In the case where the nonlinearity is small with respect to the variability in the noise, we derive an exact expression for the contribution to the mutual information given by the nonlinear term in first order of perturbation theory. Finally we show how the calculation can be simplified by means of a diagrammatic expansion. Our results suggest that the use of perturbation theories applied to neural systems might give an insight on the contribution of nonlinearities to the information transmission and in general to the neuronal dynamics.

In a recent work we have introduced a novel approach to study the effect of weak nonlinearity in the transfer function on the information transmitted by an analogue channel, by means of a perturbative diagrammatic expansion. We extend here the analysis to all orders in perturbation theory, which allows us to release any constraint concerning the magnitude of the expansion parameter and to establish the rules to calculate easily the contribution at any order. As an example we explicitly compute the information up to the second order in nonlinearity, in presence of random Gaussian connectivity and in the limit when the output noise is not small. We analyze the first and second order contributions to the mutual information as a function of the nonlinearity and as a function of the number of output units. We believe that an extensive application of our method via the analysis of the different contributions at distinct orders might be able to fill a gap between well-known analytical results obtained for linear channels and nontrivial treatments which are required to study highly nonlinear channels.

In the present work the renormalized field theory for the Lagrangian formalism in terms of Hubbard operators is given. It is shown that starting from our path-integral representation found recently, it is possible to contruct the perturbative formalism and the standard Feynman diagram approach for operators verifying the Hubbard algebra. We show that by means of the introduction of proper ghost variables, propagators and vertices can be renormalized to each order. Our Lagrangian approach is checked using the Heisenberg ferromagnet and the antiferromagnet simpler models. In particular, the renormalized ferromagnetic magnon propagator coming from our formalism is studied in details, and it is shown how the softening of the magnon frequency is predicted by the model.

Double permutation (DP) method is developed here for designing Feynman diagrams for mass operator (MO) of interacting electrons in high orders of perturbation theory (PT). The derived expression allows finding the Young diagrams for the class of permutations corresponding to disconnect Feynman diagrams. The classification of DPs, carried out before, allows to identify the permutations corresponding to disconnected, singly connected (improper) diagrams and to derive expressions for intolerant cycles of permutations. Ordering the nonprimed digits in natural order in the cycles of DP, we avoid the permutations, corresponding to the Feynman diagrams of the same topology because of other numbering of nodes. Thus, the numbers of considered permutations is sufficiently reduced: (from 24 to 6 and from 720 to 42) in the second and the third orders of PT. All 414 expressions (diagrams) for MO in the fourth order of PT were designed using this method. The use of group theory allows us to conclude that no more Feynman diagrams can be designed. The developed method can be used as algorithm for Feynman diagrams designing for MO of interacting electrons (one sort fermions) in higher orders of PT.

We study the magnetic field dependences of the conductivity in heavily doped, strongly disordered 2D quantum well structures within wide conductivity and temperature ranges. We show that the exact analytical expression derived in our previous paper [S. A. Alavi and S. Rouhani, Phys. Lett. A320, 327 (2004)], is in better agreement with the existing equation, i.e., Hikami et al. expression [ Prog. Theor. Phys.63, 707 (1980)] and Littman and Schmid expression [J. Low Temp. Phys.69, 131 (1987)], with the experimental data even in low magnetic field for which the diffusion approximation is valid. On the other hand from theoretical point of view we observe that our equation is also rich because it establishes a strong relationship between quantum corrections to the conductivity and the quantum symmetry Su_q(2). It is shown that the quantum corrections to the conductivity is the trace of Green function made by a generator of Su_q(2) algebra. Using this fact we show that the quantum corrections to the conductivity can be expressed as a sum of an infinite number of Feynman diagrams.

We refine a Le and Murakami uniqueness theorem for the Kontsevich Integral in order to specify the relationship between the two (possibly equal) main universal Vassiliev link invariants: the Kontsevich Integral and the perturbative expression of the Chern-Simons theory. As a corollary, we prove that the Altschuler and Freidel anomaly -that groups the Bott and Taubes anomalous terms- is a combination of diagrams with two univalent vertices and we explicitly define the isomorphism of which transforms the Kontsevich integral into the Poirier limit of the perturbative expression of the Chern-Simons theory for framed links, as a function of α