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Ivan Gonzalez

Publications and source records attributed to Ivan Gonzalez.

30 records · Page 2Linked to original sources
arXiv2013

Box ladders in non-integer dimension

We construct a family of triangle-ladder diagrams which may be calculated by making use of Belokurov-Usyukina loop reduction technique in d = 4 -2e dimensions. The main idea of the approach proposed in the present paper consists in generalization of this loop reduction technique existing in d = 4 dimensions. The recursive formula relating the result for L-loop triangle ladder diagram of this family and the result for (L-1)-loop triangle ladder diagram of the same family is derived. Since the method proposed in the present paper combines analytic and dimensional regularizations, at the end of the calculation we have to remove the analytic regularization by taking the double uniform limit in which the parameters of the analytic regularization are vanishing. In this limit on the left hand side of the recursive relations we obtain in the position space the diagram in which the indices of the rungs are 1 and all the other indices are 1-e. Fourier transformation of the diagrams of this type gives the momentum space diagrams which have indices of the rungs equal to 1-e and all the other indices 1. Via conformal transformation of the dual space image of this momentum space representation we relate such a family of the triangle ladder momentum diagrams to a family of the box ladder momentum diagrams in which the indices of the rungs are equal to 1-e and all the other indices are 1. Since any diagram from this family can be reduced to one-loop diagram, the proposed generalization of the Belokurov-Usyukina loop reduction technique to non-integer number of dimensions allows us to calculate this family of box-ladder diagrams in the momentum space explicitly in terms of Appell's hypergeometric function F_4 without expanding in powers of parameter e in an arbitrary kinematic region in the momentum space.

hep-th↗
arXiv2012

Belokurov-Usyukina loop reduction in non-integer dimension

Belokurov-Usyukina loop reduction method has been proposed in 1983 to reduce a number of rungs in triangle ladder-like diagram by one. The disadvantage of the method is that it works in d=4 dimensions only and it cannot be used for calculation of amplitudes in field theory in which we are required to put all the incoming and outgoing momenta on shell. We generalize the Belokurov-Usyukina loop reduction technique to non-integer d=4-2e dimensions. In this paper we show how a two-loop triangle diagram with particular values of indices of scalar propagators in the position space can be reduced to a combination of three one-loop scalar diagrams. It is known that any one-loop massless momentum integral can be presented in terms of Appell's function F_4. This means that particular diagram considered in the present paper can be represented in terms of Appell's function F_4 too. Such a generalization of Belokurov-Usyukina loop reduction technique allows us to calculate that diagram by this method exactly without decomposition in terms of the parameter e.

hep-th↗
arXiv2010

The method of brackets. Part 2: examples and applications

A new heuristic method for the evaluation of definite integrals is presented. This method of brackets has its origin in methods developed for the evaluation of Feynman diagrams. The operational rules are described and the method is illustrated with several examples. The method of brackets reduces the evaluation of a large class of definite integrals to the solution of a linear system of equations.

math-ph↗
arXiv2010

Measuring Renyi Entanglement Entropy with Quantum Monte Carlo

We develop a quantum Monte Carlo procedure, in the valence bond basis, to measure the Renyi entanglement entropy of a many-body ground state as the expectation value of a unitary {\it Swap} operator acting on two copies of the system. An improved estimator involving the ratio of {\it Swap} operators for different subregions enables simulations to converge the entropy in a time polynomial in the system size. We demonstrate convergence of the Renyi entropy to exact results for a Heisenberg chain. Finally, we calculate the scaling of the Renyi entropy in the two-dimensional Heisenberg model and confirm that the Néel groundstate obeys the expected area law for systems up to linear size L=28.

cond-mat.str-el↗
arXiv2009

Feynman Diagrams and a Combination of the Integration by Parts (IBP) and the Integration by Fractional Expansion (IBFE) Techniques

In this paper we show how to improve and extend the Integration by Fractional Expansion technique (IBFE) by applying it to certain families of scalar massive Feynman diagrams. The strategy is based on combining this method together with the Integration by Parts technique (IBP). In particular, we want to calculate certain Feynman diagrams which have a triangle loop as a subgraph. The main idea is to use IBP in this subgraph in order to simplify the topology of the original diagram in which it is immersed, using then, in a second step, the IBFE technique. The result we have obtained, after the application of both techniques, represents a simplification in the complexity of the solution, compared with having used only the IBFE technique.

hep-th↗
arXiv2009

Optimized Negative Dimensional Integration Method (NDIM) and multiloop Feynman diagram calculation

We present an improved form of the integration technique known as NDIM (Negative Dimensional Integration Method), which is a powerful tool in the analytical evaluation of Feynman diagrams. Using this technique we study a $% ϕ^{3}\oplus ϕ^{4}$ theory in $D=4-2ε$ dimensions, considering generic topologies of $L$ loops and $E$ independent external momenta, and where the propagator powers are arbitrary. The method transforms the Schwinger parametric integral associated to the diagram into a multiple series expansion, whose main characteristic is that the argument contains several Kronecker deltas which appear naturally in the application of the method, and which we call diagram presolution. The optimization we present here consists in a procedure that minimizes the series multiplicity, through appropriate factorizations in the multinomials that appear in the parametric integral, and which maximizes the number of Kronecker deltas that are generated in the process. The solutions are presented in terms of generalized hypergeometric functions, obtained once the Kronecker deltas have been used in the series. Although the technique is general, we apply it to cases in which there are 2$ $or$ $3 different energy scales (masses or kinematic variables associated to the external momenta), obtaining solutions in terms of a finite sum of generalized hypergeometric series de 1 and 2 variables respectively, each of them expressible as ratios between the different energy scales that characterize the topology. The main result is a method capable of solving Feynman integrals, expressing the solutions as hypergeometric series of multiplicity $(n-1) $, where $n$ is the number of energy scales present in the diagram.

hep-th↗
arXiv2009

Modular application of an Integration by Fractional Expansion (IBFE) method to multiloop Feynman diagrams

We present an alternative technique for evaluating multiloop Feynman diagrams, using the integration by fractional expansion method. Here we consider generic diagrams that contain propagators with radiative corrections which topologically correspond to recursive constructions of bubble type diagrams. The main idea is to reduce these subgraphs, replacing them by their equivalent multiregion expansion. One of the main advantages of this integration technique is that it allows to reduce massive cases with the same degree of difficulty as in the massless case.

hep-th↗
arXiv2009

Valence Bond and von Neumann Entanglement Entropy in Heisenberg Ladders

We present a direct comparison of the recently-proposed valence bond entanglement entropy and the von Neumann entanglement entropy on spin 1/2 Heisenberg systems using quantum Monte Carlo and density-matrix renormalization group simulations. For one-dimensional chains we show that the valence bond entropy can be either less or greater than the von Neumann entropy, hence it cannot provide a bound on the latter. On ladder geometries, simulations with up to seven legs are sufficient to indicate that the von Neumann entropy in two dimensions obeys an area law, even though the valence bond entanglement entropy has a multiplicative logarithmic correction.

cond-mat.str-el↗
arXiv2009

Modular application of an Integration by Fractional Expansion (IBFE) method to multiloop Feynman diagrams II

A modular application of the integration by fractional expansion (IBFE) method for evaluating Feynman diagrams is extended to diagrams that contain loop triangle subdiagrams in their geometry. The technique is based in the replacement of this module or subdiagram by its corresponding multiregion expansion (MRE), which in turn is obtained from Schwinger's parametric representation of the diagram. The result is a topological reduction, transforming the triangular loop into an equivalent vertex, which simplifies the search for the MRE of the complete diagram. This procedure has important advantages with respect to considering the parametric representation of the whole diagram: the obtained MRE is reduced and the resulting hypergeometric series tend to have smaller multiplicity.

hep-th↗
arXiv2008

Definite integrals by the method of brackets. Part 1

A new heuristic method for the evaluation of definite integrals is presented. This method of brackets has its origin in methods developed for theevaluation of Feynman diagrams. We describe the operational rules and illustrate the method with several examples. The method of brackets reduces the evaluation of a large class of definite integrals to the solution of a linear system of equations.

math-ph↗
arXiv2005

Recursive method to obtain the parametric representation of a generic Feynman diagram

A recursive algebraic method which allows to obtain the Feynman or Schwinger parametric representation of a generic L-loops and (E+1) external lines diagram, in a scalar $ϕ^{3}\oplus ϕ^{4}$ theory, is presented. The representation is obtained starting from an Initial Parameters Matrix (IPM), which relates the scalar products between internal and external momenta, and which appears explicitly when this parametrization is applied to the momentum space representation of the graph. The final product is an algorithm that can be easily programmed, either in a computer programming language (C/C++, Fortran,...) or in a symbolic calculation package (Maple, Mathematica,...).

hep-th↗