To demonstrate the use of splines in Mellin space we will show the accompanying calculation for a specific cross section and compare the numerical result with a result
2In some configurations one does not see this effect. The reason is a numerical inaccuracy that causes x 6= x1. As pointed out, the slightest shift from the lowest sample point suffices to obtain the expected result.
that uses an additional fit similar to equation (5.4). For this purpose we choose the resummed unpolarized Drell-Yan cross section, which has been calculated in [265]. We have been able to receive the numerical code, which used the GRV parton distribution functions [255] for the numerical calculations in [265]. Because the current state of our numerical spline implementations considers only PDF sets of this decade (see section 5.5), we exchanged the set by an already existing fit to the CT10 set [87] which is valid for scales µ2 in 10 to 640 GeV2 and longitudinal momentum fractions x in 0.01 to 0.9 for valence and 0.01 to 0.6 for sea quarks. This assures a sufficient proximity to the set accessible in our Fortran code implementing the spline interpolation, which is the CT12 set [82].3
For the analytical discussion the exact formular of the cross section is not necessary.
It will therefore not be repeated here, as it is described in detail in [265]. The only important point for our discussion is, that the cross section (and also differential cross sections) take the form incoming protons and h(N) contains all the details we do not need for the discussion.
We also omitted the flavour indices of the quarks and the corresponding sum, that composes the possible initial states with appropriate terms in h(N) (which would also be decorated by the indices), as they also play no role in the following argument.
The only point that matters, is that the PDFs are not necessarily the same, which is indicated by the tilde. Now using splines we can set f(N) → S(N) and ˜f (N ) → ˜S(N ), where n is the number of sample points, which we assume to be equal for S(N) and S(N )˜ for simplicity. The general case with different underlying grids does not add any difficulties, only larger expressions. Now the same argument as in section 5.3 applies: the contour C has to be chosen according to the asymptotic behavior of the integrand. Assuming that h(N) does not change this behavior, the criterion for one single summand is given by
τ−NxNi xNj =exp
3Confusingly the title of [82] states to correspond to CT10. But also the standalone codes of CT12 states to correspond to [82].
100
Figure 5.6: Unpolarized Drell-Yan cross section for a p¯p collider with√
s =30 GeV. The numerical code has been taken from [265]. The upper panel in this plot represents their right panel of figure 4, except for the used PDF set. While in [265] the GRV set [255] is used, we use the CT10 set [87] for the conventional fit and CT12 [82] for the spline interpolation. To distinguish the two plots, we multiplied the spline based cross section with two. The lower panel depicts the quotient of the two approaches.
which implies, that for τ < xixj the contour should approach Re(N) → −∞ while for τ > xixj it should approach Re(N) → ∞. We call these again C− and C+. The sums
In principle the term σ+ should vanish, as the contour of the inverse Mellin transform has to be on the right-hand side of all poles of the integrand. This is not possible for the resummed cross section we want to investigate, because the Landau pole induces a branch cut along the real axis. However, it has been shown that it is valid to integrate
on the left-hand side of the Landau pole. The procedure is known as the minimal prescription [126, 139, 266]. Nevertheless we will ignore σ+, as it is by orders of magnitude smaller than σ− for small boson masses M and consistent with zero for intermediate and large M. In figure 5.6 we show the resummed Drell-Yan cross section for a p¯p collider running at the center of mass energy√
s =30 GeV [265, FIG.4, right panel]. As mentioned before we changed the originally used PDF set [255] to a fit to the CT10 PDFs [87]. For the spline we used CT12 [82], which was the latest set available at the time the Fortran spline code was written. We can confirm that the calculation with the spline agrees with the conventional approach. For M < 16 GeV they agree on sub per mill level. The fluctuations in the lower panel are due to the error of the integration routine. Interestingly, the stated error estimate is lower for the splines as for the fit, which is most likely a numerical artifact. For M > 16 GeV the fluctuations increase, as the integration becomes more unstable (also visible in the upper panel).
Nevertheless we can detect a slight trend of the fitted PDFs to produce a larger cross section than the splines do. Because it is not possible to decide which one gives the
‘correct’ result, we want to emphasize that the fit has a limited validity range for x → 1, which becomes more stressed at the kinematic threshold τ → 1 (which is M →√
s).
An additional feature of the spline interpolation is, that it might be possible to extract more informations about the cross section at threshold. As remnant of the minimal prescription it is known that the differential cross section is not zero at the kinematic threshold τ = 1. It is possible that the spline will not show this remnant, as long as one only considers the part integrated along C−. This would also be in agreement with the observation that the spline produces smaller values for large M (however, this is speculative because we use a slightly different set and we already
know that the PDF fit might not describe the PDFs accurate enough in this region).
Even more interesting it might be possible, that the C+ part reveals the error made by the estimate of the minimal prescription. However, for this purpose it is necessary to perform a very sophisticated integration, as the standard Monte Carlo integration does not sufficiently converge to get any reliable numbers. One main problem is the computational load that is accompanied by using the spline interpolation. We started to address this problem in chapter 2.
Finally we will briefly study the case that the PDFs are the main source of poles of the integrand in Mellin space. The common fit, resulting in beta functions, see equation (5.5), includes an infinite number of poles close to the real axis with Re(N) < 0.
As mentioned before cubic splines exhibit only four poles. If h(N) itself has no poles at all, we are able to calculate the cross section analytically. We split the result for the Drell-Yan example into the diagonal and non-diagonal parts
σ−= X
xixi,jj>τ 3
X
m=0
gm,ig˜m,j
∂Nh(N )exp
Nln xixj
τ
N =−m
+ X
If h(N) contains a finite number of poles, the computation can still be performed analytically with additional terms from the Residuum theorem.