This paper describes applications of extrapolation for the computation of coefficients in an expansion of infrared divergent integrals. An extrapolation procedure is performed with respect to a parameter introduced by dimensional regularization. While this treats typical IR singularities at the boundaries of the integration domain, special care needs to be taken in cases where the integrand is singular in the interior of the domain as well as on the boundaries. A double extrapolation is devised for a class of massless vertex integrals. Quadruple precision results are presented, demonstrating high accuracy. The computations are supported by the use of general adaptive integration programs from the QUADPACK package, in iterated integrations with highly singular integrand functions.
For infrared (IR) divergent loop integrals, the integrand functions have non-integrable singularities through vanishing denominators. Based on dimensional regularization, the integral is expanded as a function of a parameter that approaches zero [9].
In this paper we report numerical results for the leading coefficients obtained by convergence acceleration (extrapolation) of a sequence of integral approximations. The methods are explained in detail in [4], where also results are given using double precision arithmetic. In the present paper we give extensive results using quadruple precision and show that in many cases the accuracy can be improved to near the relative machine accuracy.
According to the asymptotic behavior of the integral, we can explore linear or nonlinear extrapolation techniques. The asymptotic expansion gives rise to linear systems of the form
where S ℓ is generally a scaled version of the integral I(ε), approximated numerically. A linear system solver or a linear extrapolation method can be used if the ϕ k are known functions of ε ℓ .
Otherwise, a nonlinear extrapolation may be suitable, depending on the nature of the ϕ k functions. For (small) ε > 0, the numerical integral approximation may be affected by singular integrand behavior which occurs at the boundaries and/or in the interior of the integration domain. The sample integrals in this paper pertain to classes of one-loop vertex integrals which are two-dimensional, over the unit triangle { (x, y) | 0 ≤ y ≤ 1x ≤ 1 }. An iterated or repeated numerical integration can be performed efficiently with the general adaptive integration programs DQAGS/DQAG from QUADPACK [14].
The expansions derived symbolically in [9] generally involve hypergeometric functions. We calculate the hypergeometric function numerically in Section 3, using an extrapolation to handle the singularity in the integration interval. In Section 4 we present results for the case of one offshell (p 2 3 = 0) and two on-shell (p 2 1 = p 2 2 = 0) particles. The coefficients of the divergent terms in the integral expansion are calculated with an extrapolation as the parameter ε introduced by dimensional regularization goes to zero. The integrals in the sequence have integrand singularities on the boundaries of the integration region.
Section 5 addresses IR divergent integrals with one on-shell and two off-shell particles, where integrand singularities may occur in the interior as well as on the boundaries of the integration domain. In this case, the integrals in the extrapolation sequence with respect to ε involves an extrapolation to deal with the interior singularity.
For a sequence {S(ε ℓ )}, which converges to the limit S = lim ε ℓ →0 S(ε ℓ ), an extrapolation may be performed with the goal of creating sequences which convergence faster than the given sequence, based on its asymptotic expansion as ε → 0. In the context of series convergence we consider the limit of its partial sums. Some extrapolation methods allow summing divergent series to a value referred to as anti-limit.
A linear extrapolation yields solutions to linear systems of the form
of order (ν + 1) × (ν + 1) for increasing values of ν [12,2]. The sequence of ε ℓ may be geometric or another type of sequence that decreases to 0. As an example, Romberg integration relies on the Euler-Maclaurin expansion of the integral as a function of the step size ε = h. Then (2.1) is assumed to be an expansion in even powers of h, for the composite trapezoidal rule values S(h) with h = 2 -ℓ , ℓ ≥ 0. Values for a 0 ≈ S are obtained for successive ν by solving the (ν + 1) × (ν + 1) systems of (2.2) implicitly using the Neville algorithm.
More general sequences of ε include the sequence by Bulirsch, of the form 1/b ℓ with b ℓ = 2, 3, 4, 6, 8, 12, . . . . (consisting of powers of 2, alternating with 1.5× the preceding power of 2). The type of sequence selected influences the stability of the process, which was found more stable with the geometric sequence than with the harmonic sequence (with the Bulirsch seuence in between) [12]. On the other hand there is a trade-off with the computational expense of S(ε), which may become prohibitive for fast decreasing ε. For the computations in subsequent sections we use scaled versions of b ℓ , e.g., b ℓ /16.
If the functions of ε in the asymptotic expansion (2.1) are not known, a nonlinear extrapolation or convergence acceleration may be suitable [17,16,11,8]. As an example of a nonlinear extrapolation method, the ε-algorithm [17] implements the sequence-to-sequence transformation by [15] recursively; and can be applied when the ϕ functions are of the form ϕ k (ε) = ε β k log ν k (ε), under some conditions on ν k and β k and if a geometric sequence is used for ε. The actual form of the underlying ε-dependency does not need to be specified.
for relative integration error tolerance of 10 -25 and the Bulirsch sequence for extrapolation
A representation of the hypergeometric funct
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