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.     

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.     

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.     

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.     

Operator approach to analytical evaluation of Feynman diagrams

The operator approach to analytical evaluation of multiloop Feynman diagrams is proposed. We show that the known analytical methods of evaluation of massless Feynman integrals, such as the integration-by-parts method and the method of “uniqueness” (which is based on the star-triangle relation), can be drastically simplified by using this operator approach. To demonstrate the advantages of the operator method of analytical evaluation of multiloop Feynman diagrams, we calculate ladder diagrams for the massless ϕ ³ theory (analytical results for these diagrams are expressed in terms of multiple polylogarithms). It is shown how operator formalism can be applied to calculation of certain massive Feynman diagrams and investigation of the Lipatov integrable chain model. The text was submitted by the authors in English.     

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     

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.     

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... .     

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.     

Differential reduction of generalized hypergeometric functions from Feynman diagrams: One-variable case

The differential-reduction algorithm, which allows one to express generalized hypergeometric functions with parameters of arbitrary values in terms of the same functions with parameters whose values differ from the original ones by integers, is discussed in the context of evaluating Feynman diagrams. Where this is possible, we compare our results with those obtained using standard techniques. It is shown that the criterion of reducibility of multiloop Feynman integrals can be reformulated in terms of the criterion of reducibility of hypergeometric functions. The relation between the numbers of master integrals obtained by differential reduction and integration by parts is discussed.     

Calculation of Feynman diagrams with zero mass threshold from their small momentum expansion

A method of calculating Feynman diagrams from their small momentum expansion [1] is extended to diagrams with zero mass thresholds. We start from the asymptotic expansion in large masses [2] (applied to the case when all $M_i^2$ are large compared to all momenta squared). Using dimensional regularization, a finite result is obtained in terms of powers of logarithms (describing the zero-threshold singularity) times power series in the momentum squared. Surprisingly, these latter ones represent functions, which not only have the expected physical “second threshold” but have a branchcut singularity as well below threshold at a mirror position. These can be understood as pseudothresholds corresponding to solutions of the Landau equations. In the spacelike region the imaginary parts from the various contributions cancel. For the two-loop examples with one mass M, in the timelike region for q² ≈ M² we obtain approximations of high precision. This will be of relevance in particular for the calculation of the decay Z → bb?in the m _b = 0 approximation.     

The symmetry group of Feynman diagrams and consistency of the BPHZ renormalization scheme

We study the relation between the symmetry group of a Feynman diagram and its reduced diagrams.We then prove that the counterterms in the BPHZ renormalization scheme are consistent with adding counterterms to the interaction Hamiltonian in all cases,including that of Feynman diagrams with symmetry factors.