Universality of the Collins-Soper-Sterman nonperturbative function in gauge boson production

We revise the $b_*$ model for the Collins-Soper-Sterman resummed form factor to improve description of the leading-power contribution at nearly nonperturbative impact parameters. This revision leads to excellent agreement of the transverse momentum resummation with the data in a global analysis of Drell-Yan lepton pair and Z boson production. The nonperturbative contributions are found to follow universal quasi-linear dependence on the logarithm of the heavy boson invariant mass, which closely agrees with an estimate from the infrared renormalon analysis.

orders of α s to reliably predict the cross section [1]. The feasibility of all-order resummation is proved by a factorization theorem, first formulated for e + e − hadroproduction [2,3], stated by Collins, Soper, and Sterman (CSS) for the Drell-Yan process [4], and recently proved by detailed investigation of gauge transformations of k T -dependent parton densities [5,6].
The heavy bosons acquire non-zero q T mostly by recoiling against QCD radiation. The CSS formalism accounts for both the short-and long-wavelength  [5]. In the study presented here, we carefully investigate agreement of the universality assumption with the data in a global analysis of fixed-target Drell-Yan pair and Tevatron Z boson production. We revise the nonperturbative model used in the previous studies [8,9] and improve agreement with the data without introducing additional free parameters. Renormalization-group invariance requires F N P (b, Q) to depend linearly on ln Q [3,4]. With our latest revisions put in place, the global q T fit clearly prefers a simple function dependence on the collision energy √ S comparatively to the earlier fits. The slope of the ln Q dependence found in the new fit agrees numerically with its estimate made with methods of infrared renormalon analysis [10,11].
The function F N P (b, Q) primarily parametrizes the "power-suppressed" terms, i.e., terms proportional to positive powers of b. When assessed in a fit, also contains admixture of the leading-power terms (logarithmic in b terms), which were not properly included in the approximate leading-power function In contrast, estimates of F N P (b, Q) made in the infrared renormalon analysis explicitly remove all leading-power contributions from F N P (b, Q) [11]. While the recent studies [9,10,11,12,13] point to an approx- , they disagree on the magnitude of F N P (b, Q) and its Q dependence. The source of these differences can be traced to the varying assumptions about the form of the leading-power The exact behavior of W (b) at b > 2 GeV −1 is of reduced importance, as Our paper follows the notations in Ref. [9]. The form factor W (b) factorizes at all b as [2,3,4] where σ (0) j /S is a constant prefactor [4], and x 1,2 ≡ e ±y Q/ √ S are the Bornlevel momentum fractions, with y being the rapidity of the vector boson. The b-dependent parton densities P j (x, b) and Sudakov function are universal in Drell-Yan-like processes and SIDIS [5]. When the momentum 123 is a dimensionless constant) are much larger than 1 GeV, W (b) reduces to its perturbative part W pert (b), i.e., its leading-power (logarithmic in b) part evaluated at a finite order of α s : Here the finite-order approximations to the leading-power parts of S(b, Q) and is the k T -integrated parton density, computed in our study by using the CTEQ6M parameterization [14].
In Z boson production, the maximum of b W (b) is located at b ≈ 0.25 GeV −1 , and W pert (b) dominates the Fourier-Bessel integral. In the examined low-Q region, the maximum of b W (b) is located at b ≈ 1 GeV −1 , where higher-order corrections in powers of α s and b must be considered. We reorganize Eq. (1) to separate the leading-power (LP) term W LP (b), given by the model-dependent continuation of W pert (b) to b 1 GeV −1 , and the nonperturbative exponent , which absorbs the power-suppressed terms: At b → 0, the perturbative approximation for W (b) is restored: The power-suppressed contributions are proportional to even powers of b [10]. Detailed expressions for some power-suppressed terms are given in Ref. [11]. At impact parameters of order 1 GeV −1 , we keep only the first power-suppressed contribution proportional to b 2 : where a 1 , a 2 , and a 3 are coefficients of magnitude less than 1 GeV 2 , and φ(x) is a dimensionless function. The terms a 2 ln(Q/Q 0 ) and a 3 φ(x j ) arise from , respectively. We neglect the flavor dependence of φ(x) in the analyzed region dominated by scattering of light u and d quarks. F N P is consequently a universal function within this region. The dependence of F N P on ln Q follows from renormalization-group invariance of the soft-gluon radiation [3]. The coefficient a 2 of the ln Q term has been related to the vacuum average of the Wilson loop operator and estimated within lattice QCD as 0.19 +0.12 −0.09 GeV 2 [11].
The preferred F N P is correlated in the fit with the assumed large-b behavior of W LP . We examine this correlation in a modified version of the b * model [3,4].
The shape of W LP is varied in the b * model by adjusting a single parameter b max . Continuity of W and its derivatives, needed for the numerical stability of the Fourier transform, is always preserved. We set relatively low b max = 0.5 GeV −1 was a choice of the previous q T fits [8,9].
However, it is natural to consider b max above 1 GeV −1 in order to avoid ad hoc modifications of W pert (b) in the region where perturbation theory is still we must choose the factorization scale µ F such that it stays, at any b and b max , above the initial We keep the usual choice We perform a series of fits for several choices of b max by closely following the previous global q T analysis [9]. We consider a total of 98 data points from production of Drell-Yan pairs in E288, E605, and R209 fixed-target experi- To test the universality of F N P , we individually examine each bin of Q. We choose F N P = a(Q)b 2 and independently fit it to each of the 5 experimental data sets to determine the most plausible normalization in each experiment.
We then set the normalizations equal to their best-fit values and examine χ 2 at each Q as a function of a(Q). For b max = 1 − 2 GeV −1 , the best-fit values of a(Q) follow a nearly linear dependence on ln Q [cf. Fig. 1]. The slope a 2 ≡ da(Q)/d(ln Q) is close to the renormalon analysis expectation of 0.19 GeV 2 [11]. The agreement with the universal linear ln Q dependence worsens if b max is chosen outside the region 1-2 GeV −1 . Since the best-fit a(Q) are found independently in each Q bin, we conclude that the data support the universality of F N P , when b max lies in the range 1 − 2 GeV −1 . In another test, we find that each experimental data set individually prefers a nearly quadratic dependence on b, F N P = a(Q)b 2−β , with |β| < 0.5 in all five experiments. To further explore the issue, we simultaneously fit our model to all the data.
We parametrize a(Q) as a(Q) ≡ a 1 +a 2 ln [Q/(3.   If a very large b max comparable to 1/Λ QCD is taken, W LP (b) essentially coincides with W pert (b), extrapolated to large b by using the known, although not always reliable, dependence of W pert (b) on ln b. Similar, but not identical, extrapolations of W pert (b) to large b are realized in the models [12,13], which describe the Z data well, in accord with our own findings. In Z boson production, our best-fit a(M Z ) = 0.85 ± 0.10 GeV 2 agrees with 0.8 GeV 2 found in the extrapolation-based models, and it is about a third of 2.7 GeV 2 predicted by the BLNY parametrization. Our results support the conjecture in [12] that a 3 is small if the exact form of W pert (b) is maximally preserved. To describe the low-Q data, the model [12] allowed a large discontinuity in the first deriva- where switching from the exact W pert (b) to its extrapolated form occurs [cf. Fig. 4(b)]. In the revised b * model, such discontinuity does not happen, and W LP (b) is closer to the exact W pert (b) in a wider b range at low Q than in the model [12]. The two models differ substantially at b ≈ 1 GeV −1 , as seen in Fig. 4(b).
To summarize, the extrapolation of W pert (b) to b > 1.5 GeV for C 3 = b 0 , and {0.247 ± 0.016, 0.158 ± 0.023, −0.049 ± 0.012} GeV 2 for C 3 = 2b 0 . In Ref. [15], the experimental errors are propagated into various theory predictions with the help of the Lagrange multiplier and Hessian matrix methods, discussed, e.g., in Ref. [14]. We find that the global fit places stricter constraints on F N P at Q = M Z than the Tevatron Run-1 Z data alone.
Theoretical uncertainties from a variety of sources are harder to quantify, and they may be substantial in the low-Q Drell-Yan process. In particular, χ 2 for the low-Q data improves by 14 units when the scale parameter C 3 in µ F is increased from b 0 to 2b 0 , reducing the size of the finite-order W pert (b) at low Q. The best-fit normalizations N f it also vary with C 3 . The dependence of the quality of the fit on the arbitrary factorization scale µ F indicates importance of O(α 2 s ) corrections at low Q, but does not substantially increase uncertainties at the electroweak scale. Indeed, the O(α 2 s ) corrections and scale dependence are smaller in W and Z production. In addition, the term a 2 ln Q, which arises from the soft factor S(b, Q) and dominates F N P at Q = M Z , shows little variation with C 3 [cf. Fig. 2c]. Consequently, the revised b * model with b max ≈ 1.5 GeV −1 reinforces our confidence in transverse momentum resummation at electroweak scales by exposing the soft-gluon origin and universality of the dominant nonperturbative terms at collider energies.