Understanding the large-distance behavior of transverse-momentum-dependent parton densities and the Collins-Soper evolution kernel

There is considerable controversy about the size and importance of nonperturbative contributions to the evolution of transverse-momentum-dependent (TMD) parton distribution functions. Standard fits to relatively high-energy Drell–Yan data give evolution that when taken to lower $Q$ is too rapid to be consistent with recent data in semi-inclusive deeply inelastic scattering. Some authors provide very different forms for TMD evolution, even arguing that nonperturbative contributions at large transverse distance $b_{\mathrm{T}}$ are not needed or are irrelevant. Here, we systematically analyze the issues, both perturbative and nonperturbative. We make a motivated proposal for the parametrization of the nonperturbative part of the TMD evolution kernel that could give consistency: with the variety of apparently conflicting data, with theoretical perturbative calculations where they are applicable, and with general theoretical nonperturbative constraints on correlation functions at large distances. We propose and use a scheme- and scale-independent function $A(b_{\mathrm{T}})$ that gives a tool to compare and diagnose different proposals for TMD evolution. We also advocate for phenomenological studies of $A(b_{\mathrm{T}})$ as a probe of TMD evolution. The results are important generally for applications of TMD factorization. In particular, they are important to making predictions for proposed polarized Drell–Yan experiments to measure the Sivers function.

Figure 1

$b_{\mathrm{T}}$-space integrand for the Drell–Yan process. These plots are from Fig. 4 of Ref. [2] and show the results of using different values of $b_{\max }$: (a) for $Z$ production $\sqrt{s}=1.96\mathrm{TeV}$; (b) for $\sqrt{s}=38.8\mathrm{GeV}$ and $Q=11\mathrm{GeV}$. The two fits with $b_{\max }=1.5{\mathrm{GeV}}^{-1}$ correspond to two different choices for the ratio $C_3={\mu }_{b_{*}}{b}_{*}$, which gives a measure of sensitivity to truncation of perturbative expansions. The curve labeled “Qiu–Zhang,” with $b_{\text{max}}=0.3{\mathrm{GeV}}^{-1}$, uses the Qiu–Zhang [3, 92] parametrization. The normalization of $\tilde{W}$ differs from that defined in Eq. (2).

Figure 2

The $A(b_{\mathrm{T}})$ function defined in Eq. (30) for the BLNY and KN fits, together with some of its components. The thick, solid, red curve shows a purely perturbative, but RG-improved, calculation, using the leading-order and next-to-leading-order terms in Eq. (37). The thin, blue, dashed curve shows the same quantity with the CS taming procedure, i.e., the second term in (50), with the value $b_{\max }=0.5{\mathrm{GeV}}^{-1}$ for the BLNY fit. The thin, blue, dot-dashed curve shows the result after also adding in the $g_K(b_{\mathrm{T}};b_{\max }=0.5{\mathrm{GeV}}^{-1})$ function from BLNY [1]; i.e., it is the actual $A(b_{\mathrm{T}})$ for the BLNY fit. The thick, black, dashed, and dot-dashed curves similarly show the cutoff perturbative component and the full $A$ for the KN fit [2] with $b_{\max }=1.5{\mathrm{GeV}}^{-1}$. In (a), we plot the results with a linear scale out to $5{\mathrm{GeV}}^{-1}=1\mathrm{fm}$. In (b), we show the same functions, but with a logarithmic horizontal scale extending to $1{\mathrm{GeV}}^{-1}=0.2\mathrm{fm}$, to show better the region of small $b_{\mathrm{T}}$, to which large-$Q$ interactions are most sensitive.

Figure 3

Like Fig. 2, but showing a comparison between the BSY approximation, the KN fit, and the purely perturbative, RG-improved, calculation. The thick, red, solid curve is the next-to-leading-order perturbative calculation. The thick, black, dot-dashed, and thin, blue dot-dashed curves are the same KN and BLNY curves as in Fig. 2. The solid brown curve is for $A$ at $Q=1.0\mathrm{GeV}$ in the BSY [13] method, from Eq. (51). The dashed blue curve is the BSY calculation for $Q=10.0\mathrm{GeV}$. In (a), we plot the results with a linear scale out to $\text{5\hspace{0.17em}\hspace{0.17em}}{\mathrm{GeV}}^{-1}=1\mathrm{fm}$. In (b), we show the same functions, but with a logarithmic horizontal scale extending to $1{\mathrm{GeV}}^{-1}=0.2\mathrm{fm}$, to show better the region of small $b_{\mathrm{T}}$, to which large $Q$ interactions are most sensitive.

Figure 4

Like Fig. 2, but now we show the comparison between $A(b_{\mathrm{T}})$ for the Qiu–Zhang model of Refs. [3, 92] (brown line with step), the purely perturbative RG-improved calculation (red), and the full $A$ functions from the BLNY fit (thin blue dot-dashed) and the KN fit (thick black dot-dashed). Since by construction the Qiu–Zhang method agrees with the RG-improved perturbative calculation below the $b_{\max }$ cutoff, we do not bother with the logarithmic plot for low $b_{\mathrm{T}}$.

Figure 5

Typical two-loop graph with a heavy quark loop that contributes to $\tilde{K}$. The notation is as in Ref. [9]: The cross denotes a derivative of a spacelike Wilson line with respect to its rapidity.

Figure 6

(a) Comparison of different treatments of $g_K(b_{\mathrm{T}};b_{\max })$ in calculation of $\tilde{K}(b_{\mathrm{T}};Q)$ for $Q=2.0\mathrm{GeV}$. The solid blue and dot-dashed red curves are the calculation using the $g_K(b_{\mathrm{T}};b_{\max })$ parametrization from Eq. (79) with, respectively, $b_{\max }=1.5$ and $0.5{\mathrm{GeV}}^{-1}$. The value at $b_{\mathrm{T}}=\infty$ is set by $g(b_{\max }=1.5{\mathrm{GeV}}^{-1})=0.3$. The thin dotted blue and dashed red curves are the leading-order RG improved calculations of $\tilde{K}(b_{*};Q)$ using $b_{\max }=1.5$ and $0.5{\mathrm{GeV}}^{-1}$ and no $g_K(b_{\mathrm{T}};b_{\max })$. The black dotted and dashed curves are using the KN and BLNY fits for $g_K(b_{\mathrm{T}};b_{\max })$ with $b_{\max }=1.5$ and $0.5{\mathrm{GeV}}^{-1}$ respectively. (b) Calculation of $\tilde{K}(b_{\mathrm{T}};Q)$ using Eq. (79) with $b_{\max }=1.5$ and $0.5{\mathrm{GeV}}^{-1}$ and several values of $Q$: $Q=2$, 10, and 100 GeV. Note the change in the meaning of the line types between graphs (a) and (b).

Figure 7

(a) The values of the function $A(b_{\mathrm{T}})$ that corresponds to $\tilde{K}(b_{\mathrm{T}})$ for the curves in Fig. 6. (b) The function $A(b_{\mathrm{T}})$ for a selection of $b_{\max }$ values in the range $0.7\mathrm{GeV}\lesssim C_1/b_{\max }\lesssim 1.25\mathrm{GeV}$. They use (79) and (80), with our parameter values. For $b\lesssim 2.5{\mathrm{GeV}}^{-1}$, we have checked that $A(b_{\mathrm{T}})$ varies by less than 20% with the changes in the cutoff scale corresponding to the range plot (b), so long as $0.2\lesssim g_0(1.5{\mathrm{GeV}}^{-1})\lesssim 0.6$.