Dimensional Recurrence Relations Research Articles

Four-loop HQET propagators from the DRA method

We use dimensional recurrence relations and analyticity to calculate four-loop propagator-type master integrals in the heavy-quark effective theory. Compared to previous applications of the DRA method, we apply a new technique of fixing homogeneous solutions from pole parts of integrals evaluated in different rational space-time dimension points. The latter were calculated from the integration-by-parts reduction of finite integrals in shifted space-time dimension and/or with increased propagators powers. We provide results for epsilon expansions of master integrals near d = 4 and d = 3 using constructed alternative sets of integrals with expansion coefficients having conjectural uniform transcendental weight.

Functional reduction of one-loop Feynman integrals with arbitrary masses

A method of functional reduction for the dimensionally regularized one-loop Feynman integrals with massive propagators is described in detail.The method is based on a repeated application of the functional relations proposed by the author. Explicit formulae are given for reducing one-loop scalar integrals to a simpler ones, the arguments of which are the ratios of polynomials in the masses and kinematic invariants. We show that a general scalar n-point integral, depending on n(n + 1)/2 generic masses and kinematic variables, can be expressed as a linear combination of integrals depending only on n variables. The latter integrals are given explicitly in terms of hypergeometric functions of (n − 1) dimensionless variables. Analytic expressions for the 2-, 3- and 4-point integrals, that depend on the minimal number of variables, were also obtained by solving the dimensional recurrence relations. The resulting expressions for these integrals are given in terms of Gauss’ hypergeometric function 2F1, the Appell function F1 and the hypergeometric Lauricella — Saran function FS. A modification of the functional reduction procedure for some special values of kinematic variables is considered.

Automatic computation of Feynman integrals containing linear propagators via auxiliary mass flow

We proposed a recipe to systematically calculate Feynman integrals containing linear propagators using the auxiliary mass flow method. The key of the recipe is to introduce a quadratic term for each linear propagator and then using differential equations to get rid of their effects. As an application, we calculated all master integrals of vacuum integrals containing a gauge link up to four loops, and we checked the results by nontrivial dimensional recurrence relations.

Massive two-loop heavy particle diagrams

We determine the master integrals for vertex and propagator diagrams that appear in effective field theories containing heavy fields. The integrals involve at least one heavy line, and the standard lines include an arbitrary mass scale. The evaluation is done analytically with modern techniques. We employ the methods of differential equations and dimensional recurrence relations to evaluate said integrals up to two-loop order.

Cutting massless four-loop propagators

Among the unitarity cuts of 4-loop massless propagators two kinds are currently fully known: the 2-particle cuts with 3 loops corresponding to form-factors, and the 5-particle phase-space integrals. In this paper we calculate master integrals for the remaining ones: 3-particle cuts with 2 loops, and 4-particle cuts with 1 loop. The 4-particle cuts are calculated by solving dimensional recurrence relations. The 3-particle cuts are integrated directly using 1→3 amplitudes with 2 loops, which we also re-derive here up to transcendentality weight 7. The results are verified both numerically, and by showing consistency with previously known integrals using Cutkosky rules. We provide the analytic results in the space-time dimension 4 − 2ε as series in ε with coefficients being multiple zeta values up to weight 12. In the supplementary material we also provide dimensional recurrence matrices and SummerTime files suitable for numerical evaluation of the series in arbitrary dimensions with any precision.

Decomposition of Feynman integrals on the maximal cut by intersection numbers

We elaborate on the recent idea of a direct decomposition of Feynman integrals onto a basis of master integrals on maximal cuts using intersection numbers. We begin by showing an application of the method to the derivation of contiguity relations for special functions, such as the Euler beta function, the Gauss 2F1 hypergeometric function, and the Appell F1 function. Then, we apply the new method to decompose Feynman integrals whose maximal cuts admit 1-form integral representations, including examples that have from two to an arbitrary number of loops, and/or from zero to an arbitrary number of legs. Direct constructions of differential equations and dimensional recurrence relations for Feynman integrals are also discussed. We present two novel approaches to decomposition-by-intersections in cases where the maximal cuts admit a 2-form integral representation, with a view towards the extension of the formalism to n-form representations. The decomposition formulae computed through the use of intersection numbers are directly verified to agree with the ones obtained using integration-by-parts identities.

Functional reduction of Feynman integrals

A method for reducing Feynman integrals, depending on several kinematic variables and masses, to a combination of integrals with fewer variables is proposed. The method is based on iterative application of functional equations proposed by the author. The reduction of the one-loop scalar triangle and box integrals with massless internal propagators to simpler integrals is described in detail. The triangle integral depending on three variables is represented as a sum over three integrals depending on two variables. By solving the dimensional recurrence relations for these integrals, an analytic expression in terms of the 2F1 Gauss hypergeometric function and the logarithmic function was derived.By using the functional equations, the one-loop box integral with massless internal propagators, which depends on six kinematic variables, was expressed as a sum of 12 terms. These terms are proportional to the same integral depending only on three variables different for each term. For this integral with three variables, an analytic result in terms of the F1 Appell and 2F1 Gauss hypergeometric functions was derived by solving the recurrence relation with respect to the spacetime dimension d. The reduction equations for the box integral with some kinematic variables equal to zero are considered.

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).

On Higher-Dimensional Fibonacci Numbers, Chebyshev Polynomials and Sequences of Vector Convergents

We study higher-dimensional interlacing Fibonacci sequen\-ces, generated via both Chebyshev type functions and $m$-dimensional recurrence relations. For each integer $m$, there exist both rational and integer versions of these sequences, where the underlying prime congruence structures of the rational sequence denominators enables the integer sequence to be recovered. From either the rational or the integer sequences we construct sequences of vectors in $\mathbb{Q}^m$, which converge to irrational algebraic points in $\mathbb{R}^m$. The rational sequence terms can be expressed as simple recurrences, trigonometric sums, binomial polynomials, sums of squares, and as sums over ratios of powers of the signed diagonals of the regular unit $n$-gon. These sequences also exhibit a ``rainbow type'' quality, and correspond to the Fleck numbers at negative indices, leading to some combinatorial identities involving binomial coefficients. It is shown that the families of orthogonal generating polynomials defining the recurrence relations employed, are divisible by the minimal polynomials of certain algebraic numbers, and the three-term recurrences and differential equations for these polynomials are derived. Further results relating to the Christoffel-Darboux formula, Rodrigues' formula and raising and lowering operators are also discussed. Moreover, it is shown that the Mellin transforms of these polynomials satisfy a functional equation of the form $p_n(s)=\pm p_n(1-s)$, and have zeros only on the critical line $\Re (s)=1/2$.

Complete algebraic reduction of one-loop tensor Feynman integrals

We set up a new, flexible approach for the tensor reduction of one-loop Feynman integrals. The 5-point tensor integrals up to rank R=5 are expressed by 4-point tensor integrals of rank R-1, such that the appearance of the inverse 5-point Gram determinant is avoided. The 4-point tensor coefficients are represented in terms of 4-point integrals, defined in $d$ dimensions, $4-2\epsilon \le d \le 4-2\epsilon+2(R-1)$, with higher powers of the propagators. They can be further reduced to expressions which stay free of the inverse 4-point Gram determinants but contain higher-dimensional 4-point integrals with only the first power of scalar propagators, plus 3-point tensor coefficients. A direct evaluation of the higher dimensional 4-point functions would avoid the appearance of inverse powers of the Gram determinants completely. The simplest approach, however, is to apply here dimensional recurrence relations in order to reduce them to the familiar 2- to 4-point functions in generic dimension $d = 4-2\eps$, introducing thereby coefficients with inverse 4-point Gram determinants up to power $R$ for tensors of rank $R$. For small or vanishing Gram determinants - where this reduction is not applicable - we use analytic expansions in positive powers of the Gram determinants. Improving the convergence of the expansions substantially with Pad\'e approximants we close up to the evaluation of the 4-point tensor coefficients for larger Gram determinants. Finally, some relations are discussed which may be useful for analytic simplifications of Feynman diagrams.

Dimensional recurrence relations: an easy way to evaluate higher orders of expansion in ε

Applications of a method recently suggested by one of the authors (R.L.) are presented. This method is based on the use of dimensional recurrence relations and analytic properties of Feynman integrals as functions of the parameter of dimensional regularization, $d$. The method was used to obtain analytical expressions for two missing constants in the $\epsilon$-expansion of the most complicated master integrals contributing to the three-loop massless quark and gluon form factors and thereby present the form factors in a completely analytic form. To illustrate its power we present, at transcendentality weight seven, the next order of the $\epsilon$-expansion of one of the corresponding most complicated master integrals. As a further application, we present three previously unknown terms of the expansion in $\epsilon$ of the three-loop non-planar massless propagator diagram. Only multiple $\zeta$ values at integer points are present in our result.