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Roman N. Lee

Publications and source records attributed to Roman N. Lee.

At least 55 records · Page 3Linked to original sources
arXiv2018

Evaluating `elliptic' master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points

This is a sequel of our previous paper where we described an algorithm to find a solution of differential equations for master integrals in the form of an $ε$-expansion series with numerical coefficients. The algorithm is based on using generalized power series expansions near singular points of the differential system, solving difference equations for the corresponding coefficients in these expansions and using matching to connect series expansions at two neighboring points. Here we use our algorithm and the corresponding code for our example of four-loop generalized sunset diagrams with three massive and two massless propagators, in order to obtain new analytical results. We analytically evaluate the master integrals at threshold, $p^2=9 m^2$, in an expansion in $ε$ up to $ε^1$. With the help of our code, we obtain numerical results for the threshold master integrals in an $ε$-expansion with the accuracy of 6000 digits and then use the PSLQ algorithm to arrive at analytical values. Our basis of constants is build from bases of multiple polylogarithm values at sixth roots of unity.

hep-ph↗
arXiv2018

Three-loop massive form factors: complete light-fermion corrections for the vector current

We compute the three-loop QCD corrections to the massive quark-anti-quark-photon form factors $F_1$ and $F_2$ involving a closed loop of massless fermions. This subset is gauge invariant and contains both planar and non-planar contributions. We perform the reduction using FIRE and compute the master integrals with the help of differential equations. Our analytic results can be expressed in terms of Goncharov polylogarithms. We provide analytic results for all master integrals which are not present in the large-$N_c$ calculation considered in Refs. [1,2].

hep-ph↗
arXiv2018

DREAM, a program for arbitrary-precision computation of dimensional recurrence relations solutions, and its applications

We present the Mathematica package DREAM for arbitrarily high precision computation of multiloop integrals within the DRA (Dimensional Recurrence & Analyticity) method as solutions of dimensional recurrence relations. Starting from these relations, the package automatically constructs the inhomogeneous solutions and reduces the manual efforts to setting proper homogeneous solutions. DREAM also provides means to define the homogeneous solutions of the higher-order recurrence relations (and can construct those of the first-order recurrence relations automatically). Therefore, this package can be used to apply the DRA method to the topologies with sectors having more than one master integral. Two nontrivial examples are presented: four-loop fully massive tadpole diagrams of cat-eye topology and three-loop cut diagrams which are necessary for computation of the width of the para-positronium decay into four photons. The analytical form of this width is obtained here for the first time to the best of our knowledge.

hep-ph↗
arXiv2018

Meromorphic solutions of recurrence relations and DRA method for multicomponent master integrals

We formulate a method to find the meromorphic solutions of higher-order recurrence relations in the form of the sum over poles with coefficients defined recursively. Several explicit examples of the application of this technique are given. The main advantage of the described approach is that the analytical properties of the solutions are very clear (the position of poles is explicit, the behavior at infinity can be easily determined). These are exactly the properties that are required for the application of the multiloop calculation method based on dimensional recurrence relations and analyticity (the DRA method).

hep-ph↗
arXiv2017

Solving differential equations for Feynman integrals by expansions near singular points

We describe a strategy to solve differential equations for Feynman integrals by powers series expansions near singular points and to obtain high precision results for the corresponding master integrals. We consider Feynman integrals with two scales, i.e. nontrivially depending on one variable. The corresponding algorithm is oriented at situations where canonical form of the differential equations is impossible. We provide a computer implementation of our algorithm in a simple example of four-loop generalized sun-set integrals with three equal non-zero masses. Our code provides values of the master integrals at any given point on the real axis with a required accuracy and a given order of expansion in the regularization parameter $ε$.

hep-ph↗
arXiv2017

Normalized Fuchsian form on Riemann sphere and differential equations for multiloop integrals

We consider the question of reducibility of the differential system to normalized Fuchsian form on the Riemann sphere. The differential equations for the multiloop integrals in $ε$-form constitute a particular example of the normalized Fuchsian form. We formulate the algorithmic criterion of reducibility. We also consider the question of the proper choice of variable in the differential system suitable for its reduction to $ε$-form.

hep-th↗
arXiv2017

The $n_f^2$ contributions to fermionic four-loop form factors

We compute the four-loop contributions to the photon quark and Higgs quark form factors involving two closed fermion loops. We present analytical results for all non-planar master integrals of the two non-planar integral families which enter our calculation.

hep-ph↗
arXiv2016

Analytic three-loop static potential

We present analytic results for the three-loop static potential of two heavy quarks. The analytic calculation of the missing ingredients is outlined and results for the singlet and octet potential are provided.

hep-ph↗
arXiv2016

Evaluating the last missing ingredient for the three-loop quark static potential by differential equations

We analytically evaluate the three-loop Feynman integral which was the last missing ingredient for the analytical evaluation of the three-loop quark static potential. To evaluate the integral we introduce an auxiliary parameter $y$, which corresponds to the residual energy in some of the HQET propagators. We construct a differential system for 109 master integrals depending on $y$ and fix boundary conditions from the asymptotic behaviour in the limit $y\to \infty$. The original integral is recovered from the limit $y\to 0$. To solve these linear differential equations we try to find an $ε$-form of the differential system. Though this step appears to be, strictly speaking, not possible, we succeed to find an $ε$-form of all irreducible diagonal blocks, which is sufficient for solving the differential system in terms of an $ε$ expansion. We find a solution up to weight six in terms of multiple polylogarithms and obtain an analytical result for the required three-loop Feynman integral by taking the limit $y\to 0$. As a by-product, we obtain analytical results for some Feynman integrals typical for HQET.

hep-ph↗
arXiv2015

One-loop pentagon integral in $d$ dimensions from differential equations in $ε$-form

We apply the differential equation technique to the calculation of the one-loop massless diagram with five onshell legs. Using the reduction to $ε$-form, we manage to obtain a simple one-fold integral representation exact in space-time dimensionality. The expansion of the obtained result in $ε$ and the analytical continuation to physical regions are discussed.

hep-ph↗
arXiv2015

Introducing SummerTime: a package for high-precision computation of sums appearing in DRA method

We introduce the Mathematica package SummerTime for arbitrary-precision computation of sums appearing in the results of DRA method. So far these results include the following families of the integrals: 3-loop onshell massless vertices, 3-loop onshell mass operator type integrals, 4-loop QED-type tadpoles, 4-loop massless propagators. The package can be used for high-precision numerical computation of the expansion coefficients of the integrals from the above families around arbitrary space-time dimension. In addition, this package can also be used for calculation of multiple zeta values, harmonic polylogarithms and other transcendental numbers expressed in terms of nested sums with factorized summand.

hep-ph↗
arXiv2015

Reducing differential equations for multiloop master integrals

We present an algorithm of the reduction of the differential equations for master integrals the Fuchsian form with the right-hand side matrix linearly depending on dimensional regularization parameter $ε$. We consider linear transformations of the functions column which are rational in the variable and in $ε$. Apart from some degenerate cases described below, the algorithm allows one to obtain the required transformation or to ascertain irreducibility to the form required. Degenerate cases are quite anticipated and likely to correspond to irreducible systems.

hep-ph↗
arXiv2013

Critical points and number of master integrals

We consider the question about the number of master integrals for a multiloop Feynman diagram. We show that, for a given set of denominators, this number is totally determined by the critical points of the polynomials entering either of the two representations: the parametric representation and the Baikov representation. In particular, for the parametric representation the corresponding polynomial is just the sum of Symanzik polynomials. The relevant topological invariant is the sum of the Milnor numbers of the proper critical points. We present a Mathematica package Mint to automatize the counting of the master integrals.

hep-ph↗
arXiv2012

The Dimensional Recurrence and Analyticity Method for Multicomponent Master Integrals: Using Unitarity Cuts to Construct Homogeneous Solutions

We consider the application of the DRA method to the case of several master integrals in a given sector. We establish a connection between the homogeneous part of dimensional recurrence and maximal unitarity cuts of the corresponding integrals: a maximally cut master integral appears to be a solution of the homogeneous part of the dimensional recurrence relation. This observation allows us to make a necessary step of the DRA method, the construction of the general solution of the homogeneous equation, which, in this case, is a coupled system of difference equations.

hep-ph↗