Four-particle problem using Feynman diagrams

We present an exact diagrammatic approach for the problem of dimer-dimer scattering in 3D for dimers being a resonant bound state of two fermions in a spin singlet state, with corresponding scattering length a. We recover exactly the previously known result a _B = 0.60a, where a _B is the dimer-dimer scattering length. A detailed discussion of how one can “sum all the diagrams” in this case is presented. Applications to the study of 4-particle bound states of various complexes in 2D are briefly presented.     

Vertex models on Feynman diagrams

We discuss the statistical mechanics of vertex models on both generic (“thin”) and planar (“fat”) random graphs. Such models can be formulated as the N → 1 and N → ∞ limits of N × N complex matrix models, respectively. From the graph theoretic perspective one is using matrix model and field theory inspired methods to count various classes of directed graphs. For the thin random graphs we use saddle point methods to solve the models in the thermodynamic, large number of vertices limit and note that, as in the case of the eight-vertex model on the square lattice, various other models such as the Ising model appear as particular limits. The generic solution of the fat graph model is rather more elusive, but we show that for several choices of the couplings the models can be reduced to eigenvalue integrals and their critical behaviour deduced.     

A technique for generating Feynman diagrams

We present a simple technique that allows to generate Feynman diagrams for vector models with interactions of order2n and similar models (Gross-Neveu, Thirring model) using a bootstrap equation that uses only the free field value of the energy as an input. The method allows to find the diagrams to, in principle, arbitrarily high order and applies to both energy and correlation functions. It automatically generates the correct symmetry factor (as a function of the number of components of the field) and the correct sign for any diagram in the case of fermion loops. We briefly discuss the possibility of treating QED as a Thirring model with non-local interaction.     

Feynman diagrams to three loops in three-dimensional field theory

The two-point integrals contributing to the self-energy of a particle in a three-dimensional quantum field theory are calculated to two-loop order in perturbation theory as well as the vacuum ones contributing to the effective potential to three-loop order. For almost every integral an expression in terms of elementary and dilogarithm functions is obtained. For two integrals, the master integral and the Mercedes integral, a one-dimensional integral representation is obtained with an integrand consisting only of elementary functions. The results are applied to a scalar λφ⁴ theory.     

Simple method to make asymptotic series of Feynman diagrams converge

We show that, for two nontrivial lambda phi(4) problems (the anharmonic oscillator and the Landau-Ginzburg hierarchical model), improved perturbative series can be obtained by cutting off the large field contributions. The modified series converge to values exponentially close to the exact ones. For lambda larger than some critical value, the method outperforms Padé's approximants and Borel summations. The method can also be used for series which are not Borel summable such as the double-well potential series. We show that semiclassical methods can be used to calculate the modified Feynman rules, estimate the error, and optimize the field cutoff.     

Planar matrices and arrays of Feynman diagrams

Recently, planar collections of Feynman diagrams were proposed by Borges and one of the authors as the natural generalization of Feynman diagrams for the computation of k = 3 biadjoint amplitudes. Planar collections are one-dimensional arrays of metric trees satisfying an induced planarity and compatibility condition. In this work, we introduce planar matrices of Feynman diagrams as the objects that compute k = 4 biadjoint amplitudes. These are symmetric matrices of metric trees satisfying compa...     

A duality property of planar Feynman diagrams

It is found that the Fourier-transform of the amplitude of a planar Feynman diagram G can be written as the amplitude of the Feynman diagram G?, where G? is the dual of G in the sense of graph theory of graph theory, the propagators of G? being the Fourier-transformed of the ordinary ones.     

Feynman diagram approach to atomic collisions

A quantum field theoretic formulation of atomic collision phenomena involving non-relativistic free and bound systems is developed and a calculational procedure in terms of Feynman diagrams is prescribed. Matrix elements of several atomic collision processes have been calculated. In most cases standard quantum mechanical results are reproduced. But in some cases new terms appear in the scattering matrix whose contribution though negligibly small in the low energy region, become important at higher energies. A large portion of this work formed material for an invited talk delivered by one of us (T.P.) at the Second National Workshop on Atomic and Molecular Physics at Visva-Bharati, Santiniketan (India) held from 18–23 November 1979.     

Calculation of infrared-divergent Feynman diagrams with zero mass threshold

Two-loop vertex Feynman diagrams with infrared and collinear divergences are investigated by two independent methods. On the one hand, a method of calculating Feynman diagrams from their small momentum expansion [1] extended to diagrams with zero mass thresholds [2] is applied. On the other hand, a numerical method based on a two-fold integral representation is used [3], [4]. The application of the latter method is possible by using lightcone coordinates in the parallel space. The numerical data obtained with the two methods are in impressive agreement. Received: 22 April 1997 / Published online: 20 February 1998     

Calculation of Feynman diagrams from their small momentum expansion

A new powerful method to calculate Feynman diagrams is proposed. It consists in setting up a Taylor series expansion in the external momenta squared (in general multivariable). The Taylor coefficients are obtained from the original diagram by differentiation and putting the external momenta equal to zero, which means a great simplification. It is demonstrated that it is possible to obtain by analytic continuation of the original series high precision numerical values of the Feynman integrals in the whole cut plane. For this purpose conformal mapping and subsequent resummation by means of Padé approximants or Levin transformation are applied.Supported by Bundesministerium für Forschung und Technologie     

Resistance of Feynman diagrams and the percolation backbone dimension

We present an alternative view of Feynman diagrams for the field theory of random resistor networks, in which the diagrams are interpreted as being resistor networks themselves. This simplifies the field theory considerably as we demonstrate by calculating the fractal dimension D(B) of the percolation backbone to three loop order. Using renormalization group methods we obtain D(B)=2+epsilon/21-172epsilon(2)/9261+2epsilon(3)[-74 639+22 680zeta(3)]/4 084 101, where epsilon=6-d with d being the spatial dimension and zeta(3)=1.202 057... .     

Loops on surfaces, Feynman diagrams, and trees

We relate the author’s Lie cobracket in the module additively generated by loops on a surface with the Connes–Kreimer Lie bracket in the module additively generated by trees.     

Geometric approach to asymptotic expansion of Feynman integrals

We present an algorithm that reveals relevant contributions in non-threshold-type asymptotic expansion of Feynman integrals about a small parameter. It is shown that the problem reduces to finding a convex hull of a set of points in a multidimensional vector space.     

A new technique for the evaluation of Feynman diagrams in the high energy,large momentum transfer limit

The problem of finding scattering amplitudes in the high energy, large momentum transfer limit is reconsidered. We propose a novel technique of evaluating Feynman diagrams in this regime with higher order perturbative corrections in mind. The relation to other methods is discussed and the ladder diagrams in φ³ theory, recalculated up to sixth order, serve as examples.     

An algorithm to isolate poles in Feynman diagrams preserving background gauge invariance and supersymmetry

We regularize the heat-kernel expansion of propagators in the presence of background fields. This leads to a renormalized perturbation series which is gauge-invariant and supersymmetric with respect to background variations. This procedure permits isolation of singularities in the computation of the effective action, obeying the above symmetries.