Abstract
We present a recipe for constructing the effcient contour which allows one to calculate with high accuracy the Mellin-Barnes integrals, in particular, for the F3 structure function written in terms of its Mellin moments. We have demonstrated that the contour of the stationary phase arising for the F3 structure function tends to the finite limit as Re(z) → –∞. We show that the Q2 evolution of the structure function can be represented as an integral over the contour of the stationary phase within the framework of the Picard-Lefschetz theory. The universality of the asymptotic contour of the stationary phase defined at some fixed value of the momentum transfer square $Q_{0}^{2}$ for calculations with any Q2 is shown.
Highlights
The Mellin–Barnes (MB) integrals are widely used in high-energy physics
The asymptotic behavior of the integrand ΦDIS(z) at large-z has the form euB z A. Based on this expression and following our method of building the asymptotic contour of the stationary phase Cas, we find that after shifting to the saddle point c0 this contour is expressed as zDaIsS = y ctg uBy β+1
This advantage is compensated by using the contour Cas(Q20) if we increase the number of terms in the quadrature formula only by 2–4 units
Summary
The Mellin–Barnes (MB) integrals are widely used in high-energy physics. Their efficient numerical evaluation is an important task. Selecting the imaginary part of the Φ-function and imposing the condition on the contour Im [Φ(z)] = 0, we find the equation for the contour of the stationary phase Cst as x(y) + a sin(uy) − y cos(uy) = 0 The solution of this equation, which provides continuity of the integration contour at the point y = 0, has the form. Let us apply the described method of constructing the asymptotic contour of the stationary phase to calculate the structure function in the Bjorken variable space using the inverse Mellin transform (1). 1 2πi dz ΦDIS(z) , ΦDIS(z) = euBz M3(z), uB = − ln(xB) Based on this expression and following our method of building the asymptotic contour of the stationary phase Cas, we find that after shifting to the saddle point c0 this contour is expressed as zDaIsS = y ctg uBy β+1.
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