Analysis of vector boson production within TMD factorization
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Abstract
We present a comprehensive analysis and extraction of the unpolarized transverse momentum dependent (TMD) parton distribution functions, which are fundamental constituents of the TMD factorization theorem. We provide a general review of the theory of TMD distributions, and present a new scheme of scale fixation. This scheme, called the \(\zeta \)prescription, allows to minimize the impact of perturbative logarithms in a large range of scales and does not generate undesired power corrections. Within \(\zeta \)prescription we consistently include the perturbatively calculable parts up to nexttonexttoleading order (NNLO), and perform the global fit of the Drell–Yan and Zboson production, which include the data of E288, Tevatron and LHC experiments. The nonperturbative part of the TMDs are explored checking a variety of models. We support the obtained results by a study of theoretical uncertainties, perturbative convergence, and a dedicated study of the range of applicability of the TMD factorization theorem. The considered nonperturbative models present significant differences in the fitting behavior, which allow us to clearly disfavor most of them. The numerical evaluations are provided by the arTeMiDe code, which is introduced in this work and that can be used for current/future TMD phenomenology.
1 Introduction
The transverse momentum dependent (TMD) distributions are universal functions that describe the interactions of partons in a hadron. The TMD distributions naturally appear within the TMD factorization theorem for the differential cross section of doubleinclusive hard processes. A lot of effort has been made to achieve a comprehensive picture of TMD factorization (for the latest works see [1, 2, 3, 4, 5, 6, 7, 8]). In this work we perform a detailed comparison of the experimental measurements with the theory expectations based on our studies of higherorder perturbative expansions and power corrections for unpolarized TMDPDFs made in Refs. [9, 10, 11, 12].
Among many different spin (in)dependent TMD distributions, the unpolarized TMD parton distribution functions (TMDPDFs) play a central role. From the practical point of view, their precise knowledge is required to extract further TMD distributions and perform other precision measurements. The ideal process to study the unpolarized TMDPDFs is the unpolarized vector boson production. The data on the \(q_T\)dependent crosssection for the Drell–Yan process are collected by many experiments, including the precise measurements done by Tevatron and LHC. The theoretical descriptions of Drell–Yan data were made by many groups using different forms of TMD factorization (see e.g. [8, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]).
This work presents a number of differences with respect to the previous literature. The collection of the improvements forms a completely new point of view in the TMD phenomenology. The main difference of the present work with respect to the more standard ones (here we consider as the most spread out, and de facto standard, analyses those based on the codes ResBos [15, 23] and DYqT/DYRes [17, 18, 21]) are as follows: (i) We extract the parameters related to individual TMDPDFs, which are suitable for phenomenological description of other TMDrelated processes. (ii) We consistently include the perturbative ingredients, such as coefficient functions and anomalous dimensions, at the nexttonexttoleading order (NNLO), introducing and using the \(\zeta \)prescription to solve the problem of perturbative convergence at largeb (where b is the transverse distance). (iii) The TMDPDF parameterization is based on and is consistent with the theory expectation on the TMD behavior with b. To our knowledge this is the first attempt to include in a fit both high and low energy data at NNLO precision. The extraction of TMDs takes into account (for the first time to our knowledge) also LHC data. All this represents for us a clear improvement with respect to the more classical analyses.
In a modern view, a TMD distribution is a cumbersome function of many factors, which mix up perturbative and nonperturbative information. In this context, the issue of the separation of perturbative and nonperturbative physics requires a fine analysis and it is open to different solutions. The \(\zeta \)prescription proposed in this work, is an attempt to consider the perturbative input to a TMD distribution as it is, without artificial regulators. The \(\zeta \)prescription is founded on the fact that the TMD factorization introduces two factorization scales, one for the collinear and one for the soft exchanges. These scales are usually fixed to the same point, while in the \(\zeta \)prescription they are chosen to eliminate the problematic doublelog contributions. In other words, the \(\zeta \)prescription is based on the freedom to select the normalization and factorization scales, which is guaranteed by the structure of the perturbative theory. The \(\zeta \)prescription is essentially different from other used schemes. In particular, it does not strictly solve the problem of the large logarithmic contributions at largeb. It only decreases the power of the logarithmic contributions. However, the xdependence of the remaining logarithmic terms has a form which prevents the blow up of the perturbative series, which is not accidental, but the result of the charge conservation. In this way, the \(\zeta \)prescription postpones the large logarithm problem to the very far domain of bspace, where other factors suppress a TMD distribution. The practical implementation of the \(\zeta \)prescription shows that it is efficacious and it allows a very accurate and sound description of the data.
The description of the nonperturbative parts of TMD distributions is the most interesting task. It is highly nontrivial because the definition of the nonperturbative part is strongly affected by schemes and prescriptions used in the perturbative implementation. In this respect a full NNLO can be clarifying. As an example, we recall that the nonperturbative behavior of the TMDPDFs is often assumed to have a Gaussian shape (see e.g. discussions in [15, 22, 24, 25]). Although the Gaussian ansatz is widely used, it comes into conflict with the usual picture of longdistance strong interaction fueled by lightmeson exchanges. The typically expected behavior at long distances is exponential, which is confirmed also by model calculations [26]. However, the Gaussian shape is often introduced together with the \(b^*\)prescription [27]. Notwithstanding many positive features, the \(b^*\)prescription has a serious issue: it introduces undesired beven power corrections. In turn, these power correction introduced by \(b^*\) can easily simulate the Gaussian behavior (see also discussion in [28]). Once the \(b^*\)prescription is removed the Gaussian ansatz for the TMD shape is no more essential, according to what we find.
An additional remarkable point of the present study is the wide range of energies covered by the data that we have analyzed. The lowest energy measurements included in the fits have \((Q,\sqrt{s})=(4,19.4)\;\text {GeV}\) (E288 experiment [29]), while the most energetic have \((Q,\sqrt{s})=(116150,8\cdot 10^3)\; \text {GeV}\) (ATLAS collaboration [30]). Typically, the low and highenergy data are considered separately. The main reason for a separate scan is the assumed physical picture of strong interactions, which describes different energies. The description of the highenergy data requires a precise perturbative input and it is expected to be less sensitive to the fine nonperturbative dynamics. The situation is the opposite for the lowenergy measurements. Our experience shows that the inclusion of data of different energies is not only possible within the TMD formalism, but it is also desired because it cuts away inappropriate models very sharply. We find also that the precision achieved by LHC is already sensitive enough to the nonperturbative structure of TMDs. We show that low and high energy data are sensitive to different regions of bspace, and consequently to different nonperturbative regimes of the TMDs: high energy data are better described by a Gaussian nonperturbative correction, while low energy data prefer an exponential type of nonperturbative models. The code (arTemiDe) that we have prepared allows to test all these hypotheses, and can be adaptded also to test different nonperturbative inputs for TMDs.
 (i)
Stability The TMD factorization is valid at small\(q_T\) (the dilepton transverse momentum) up to a certain limit. Therefore, an acceptable model should produce a stable and good description within the allowed \(q_T\)range. In other words, the value of \(\chi ^2\) should be sufficiently close to one and the central values of the parameters should be stable independently of the number of included data points (as far as the points belong to the allowed range).
 (ii)
Convergence The agreement with data should improve with the increase of the perturbative order. Given the current state of the art of the theory, we can define four successive perturbative orders, which is enough to test the perturbative convergence. Also, the value of the phenomenological nonperturbative constants that one extracts should converge to some central value.
 (iii)
\(\chi ^2\) minimization Naturally, among the models with similar behavior we select the model with the minimal \(\chi ^2\). We have found that it is difficult to find a model (with one or two parameters), which fulfills the demands (i) and (ii), and that at the same time provides a good \(\chi ^2\) value on the whole set of data points (although it is relatively easy to achieve this, selecting a particular experiment). The models that we test consider a kind of minimal set of parameters which can be enlarged in future studies, refining the fitting hypotheses.
In order to numerically evaluate the theoretical expressions, we have produced the package arTeMiDe. arTeMiDe has a flexible module structure and can be used at any level of TMD theory description, from the evaluation of a single TMDPDF or evolution factor to an evaluation of differential crosssection. The arTeMiDe code is available at [31] and can be used to check our statements or test a possible future/alternative ansatz (for instance [14, 32]). In arTeMiDe we have collected all recent achievements of TMD theory, including NNLO matching coefficient function, and N\(^3\)LO TMD anomalous dimensions. In the current version, arTeMiDe evaluates only unpolarized TMDPDFs and related crosssections, however, we plan to extend it further.
The body of the article is divided as in the following. In Sect. 2 we review the theoretical construction of the DrellYan cross section and summarize the theoretical knowledge on unpolarized TMDPDFs. In this section, we also describe all the theoretical improvements which are original for this work. The main original point, namely \(\zeta \)prescription is presented in Sect. 2.4 and “Appendix B”. The phenomenological studies are presented in Sect. 3. This section includes also a dedicated discussion of the shape of the nonperturbative part of the TMD. The allowed range of validity of the TMD factorization is explored in Sect. 3.4, the presentation of theoretical uncertainties is given in Sect. 3.5. The results of the final fit are presented in Sect. 3.7. A final discussion and conclusions can be found in Sect. 4.
2 Theoretical framework
2.1 Expressions for crosssection for different produced bosons
2.2 TMD parton distributions: evolution
The peculiar feature of the TMD operator is the presence of two types of divergences and, as a consequence, two renormalization factors Z and \({\mathcal {R}}\). Firstly, we have ultraviolet divergences, which have their collinear counterpart in the coefficient function \(C_V\). These divergences are the result of collinear factorization and give rise to the logarithms of the factorization scale \(\mu \). Secondly, we have rapidity divergences, which arise in the factorization of the softgluon exchanges between partons. The singular softgluons exchanges can be collected into the soft factor, which in turn, can be written as a product of rapidity renormalization factors \({\mathcal {R}}\), see e.g. [10, 11, 44]. This procedure introduces the rapidity factorization scale \(\zeta \).
2.3 TMD parton distributions: bspace behavior
The TMDPDF is a genuine nonperturbative function, which is to be fitted by a certain ansatz, which covers the whole domain in bspace. Different intervals of bspace describe different regimes of strong interactions. In Fig. 2 we show schematically the parts of bspace which need a specific treatment for each TMDPDF. In order to construct an optimal and physically meaningful fitting ansatz, the behavior in every part of the bspace should be reproduced. In this section, we collect the main information on the bdependence of TMDPDFs, as it is understood according to the current state of art.
There is a subregion \(b\ll 1/Q\), which should be considered specially. While the TMD distribution is completely perturbative within this region, the contributions of this region to the crosssection strongly overlaps with the Yterm, Eq. (4), which is formally \({\mathcal {O}}(1/(bQ))\). The behavior of TMD distributions within this tiny range together with the Yterm dictates the asymptotics of the crosssection at large \(q_T\). As a consequence, it has a significant influence on the value of the total crosssection. In our current evaluation we restrict ourself to the range of small\(q_T\) (for a dedicated study of the applicability of this approximation in practice, see Sect. 3.4). Therefore, we drop the Yterm and do not need any special treatment of \(b\ll Q^{1}\) region.
 (i)
The OPE contains only even powers of b. Moreover, the coefficient function of n’th order has a prefactor \(x^n\). In other word, the natural scale of OPE is \(x\varvec{b}^2/B^2\) rather then just \(\varvec{b}^2/B^2\).
 (ii)
The higher order OPE contributions induced by renormalons, can be summed together to some effective nonperturbative function under the convolution integral.
 (iii)The \(n=1\) contribution to OPE can be estimated by the leading renormalon contribution of order \(\sim x\varvec{b}^2\) [12]. It has the formwhere \(\varLambda =\varLambda _{QCD}\) is the position of the Landau pole.$$\begin{aligned} C^{\text {ren}}_{q\leftarrow q}(x,\varvec{b};\mu ,\zeta )= & {} 2{\bar{x}}+\frac{2x}{(1x)_+} \nonumber \\&\,\delta ({\bar{x}})\left( {\mathbf {L}}_\varLambda {\mathbf {L}}_{\sqrt{\zeta }}+\frac{2}{3}\right) , \end{aligned}$$(26)
We should mention that the size of the parameter B, as well as, the order of convergence of the smallb OPE, which influences the size of the intermediate region 2, are not known. Our estimations of these characteristic sizes are presented in Sect. 4.
2.4 Definition of scaling parameters
The smallb matching is the starting point for the construction of the majority of phenomenological ansatzes for TMD distributions. It can be considered as an additional collinear factorization, which is performed at some convenient set of scales \((\mu _i,\zeta _i)\). The difference of \((\mu _i,\zeta _i)\) from the initial (defined by process kinematic) scales of TMD distribution is compensated by the evolution factor in Eq. (17). As usual, the allorder expression is independent of \((\mu _i,\zeta _i)\), but in practice, these scales are to be chosen such that the coefficient function \(C_{f\leftarrow f'}\) has good perturbative convergence. This procedure is alike the choice of hardfactorization scale, with one essential difference: the parameter b, which accompanies \(\mu _i\) and \(\zeta _i\) in the logarithms, has no fixed value. It varies from zero to infinity within the Fourier integral.
In this work we use another scheme which we call \(\zeta \)prescription. It is a novel one (to our best knowledge), and it is described in the following.
The \(\zeta \)prescription uses the fact that the TMD operator and hence its smallb OPE depends on two scales \(\mu \) and \(\zeta \), which are entirely independent. This simple fact has been overlooked so far. Indeed, the first typical step is to fix \(\zeta =C_0^2/\varvec{b}^2\), or \(\zeta =\mu ^2\) [1, 12, 52]. It reduces the problem to a single variable problem, which looks simpler, but finally, it does not provide a simple solution for the appearance of large logarithms in the OPE.
The initial point of the \(\zeta \)prescription is the observation that not all logarithms in the coefficient function are dangerous. So, the terms \({\mathbf {L}}^2_\mu \) and \({\mathbf {L}}_\mu {\mathbf {l}}_\zeta \) in Eq. (27) are problematic, while the logarithm in the first term is not. There are several reasons for it. First, the double logarithm contributions violate the normal perturbative counting and at largeb grows faster than the single logarithms. Second, the first term of Eq. (27) comes together with the DGLAP kernel, and thus, it preserves the area (say, the integral over x) of the TMDPDF, due to the conservation of the electromagnetic charge. We remind that logarithms accompanying the DGLAP kernel are related to PDF evolution, while the rest of logarithms are related to the TMD evolution. For this reason, the main problem of convergence is represented by the logarithms that are related to the TMD evolution. The logarithms related to the PDF evolution come with a particular xdependent function. The probabilistic interpretation of PDF ensures their minimal contribution in the very large domain of b. Practically, this fact has been already demonstrated although not entirely realized in the fit [20]. In the realization of Ref. [20], the DGLAP logarithms were left unregulated and they do not influence the convergence of the fit.
Finally, we should also select the value for the parameter \(\mu _0\) that enters in the expression for the evolution factor, Eq. (22). To keep our discussion simple, we set \(\mu _0=\mu _b\).
2.5 Theoretical uncertainties and perturbative ordering
The perturbative orders studied in the fit. For each order we indicate the power of \(a_s\) of each piece that enters in the TMDs. Note, that the order of \(a_s\) and PDF set are related, since the values of \(a_s\) are taken from the PDF set
Name  \(C_V^2\)  \(C_{f\leftarrow f'}\)  \(\varGamma \)  \(\gamma _V\)  \({\mathcal {D}}\)  PDF set  \(a_s\)(run)  \(\zeta _\mu \) 

NLL/LO  \(a_s^0\)  \(a_s^0\)  \(a_s^2\)  \(a_s^1\)  \(a_s^2\)  nlo  nlo  NLL 
NLL/NLO  \(a_s^1\)  \(a_s^1\)  \(a_s^2\)  \(a_s^1\)  \(a_s^2\)  nlo  nlo  NLO 
NNLL/NLO  \(a_s^1\)  \(a_s^1\)  \(a_s^3\)  \(a_s^2\)  \(a_s^3\)  nlo  nlo  NNLL 
NNLL/NNLO  \(a_s^2\)  \(a_s^2\)  \(a_s^3\)  \(a_s^2\)  \(a_s^3\)  nnlo  nnlo  NNLO 

Uncertainty associated with the perturbative matching of rapidity anomalous dimension This uncertainty arises from the dependence (at the fixed perturbative order) on \(\mu _0\), which should be compensated between the Sudakov factor and the boundary term \({\mathcal {D}}(\mu _0)\) in the TMD evolution factor Eq. (22). This uncertainty can be tested by changing \(\mu _0\rightarrow c_1\mu _0\) and varying \(c_1\in [0.5,2]\).

Uncertainty associated with the hard factorization scale This uncertainty arises from the dependence (at the fixed perturbative order) on the scale \(\mu _f(\sim Q)\) which is to be compensated between the hard coefficient function \(C_V^2\) and the TMD evolution factor. This uncertainty can be tested by changing \(\mu _f\rightarrow c_2\mu _f\) and varying \(c_2\in [0.5,2]\).

Uncertainty associated with the TMD evolution factor This uncertainty arises from the dependence (at the fixed perturbative order) on the initial scale of TMD evolution \(\mu _i\), which is to be compensated between the evolution integral and the \(\mu \)dependence of \(\zeta _i\) in Eq. (22). This uncertainty can be tested by changing \(\mu _i\rightarrow c_3\mu _i\) and varying \(c_3\in [0.5,2]\).

Uncertainty associated with the smallb matching This uncertainty arises from the dependence (at the fixed perturbative order) on the scale of the smallb matching \(\mu _{\text {OPE}}\) which is to be compensated between the smallb coefficient function \(C_{f\leftarrow f'}\) and evolution of PDF. This uncertainty can be tested by changing \(\mu _{\text {OPE}}\rightarrow c_4\mu _{\text {OPE}}\) and varying \(c_4\in [0.5,2]\).
The perturbative orders of each cross section constituent are to be combined consistently. Having at our disposal the NNLO expressions for coefficient function and \(\hbox {N}^3\hbox {LO}\) expressions for anomalous dimensions, we can define four successive selfcontained sets of ordering. This is reported in Table 1. In our definition of orders we use the following logic: (i) The order of the \(a_s\)running should be the same as the order of PDF set, since their extraction are correlated. (ii) The order of \({\mathcal {D}}\) should be the same as the order of \(\varGamma \) since they enter the evolution kernel R with the same counting of logarithms (i.e. \(a_s^n \ln ^{n+1}\mu \)), and oneorder higher then the order of \(\gamma _V\), since it has counting \(a_s^n \ln ^n\mu \). (iii) The order of smallb matching coefficient should be the same as the order of evolution of a PDF, because large logarithms of b are to be compensated by the PDF evolution. (iv) The order of \(\zeta _\mu \) should be such that no logarithms appear in the coefficient function, and the general logarithm counting coincides with the counting of the evolution factor. In Table 1 the order of the \(\zeta _\mu \) is defined as following: NLL is \({\mathbf {l}}_\zeta ={\mathbf {L}}_\mu /2\), NLO has in addition finite part at order \(a_s^0\) (i.e. two first terms of Eq. (30)), NNLL has in addition logarithmic part at order \(a_s^1\) (i.e. the first line of Eq. (30)), and NNLO is given by whole expression Eq. (30). The \({\mathbf {l}}_\zeta \) cases NLL and NNLL are somewhat intermediate cases. In fact, while one achieves a cancellation of logs of the same order in the evolution kernel and the coefficient, one finds that the counting in the coefficient is consistent with the \(a_s L_\mu ^2\sim a_s^0\). A similar counting was introduced in [53]. We postpone a full study of this counting within \(\zeta \)prescription to a future work.
To label the orders we use the nomenclature where the part with ’LO suffix designates the order of coefficient functions, and the part with ’LL suffix designates the order of the evolution factor in the \(a_s \ln \mu \sim 1\) scheme. So, our highest order is NNLL/NNLO, which at the moment the highest available combination of the perturbative series. The order NLL/LO appears to be barely inconsistent, because it requires the LO PDF evolution to match the trivial coefficient function. Therefore, we exclude the NLL/LO from our phenomenological studies.
2.6 Implementation of lepton cuts
The characteristics of the data measured at E288 experiment
E288 200  E288 300  E288 400  

\(\sqrt{s}\)  19.4 GeV  23.8 GeV  27.4 GeV 
Process  \(\hbox {p~+~Cu}\rightarrow \gamma \rightarrow \mu ^+\mu ^\)  \(\hbox {p~+~Cu}\rightarrow \gamma \rightarrow \mu ^+\mu ^\)  \(\hbox {p~+~Cu}\rightarrow \gamma \rightarrow \mu ^+\mu ^\) 
Q range  4–9 GeV  4–9 GeV  5–14 GeV 
\(\varDelta Q\)bin  1 GeV  1 GeV  1 GeV 
y  \(\hbox {y}=0.4\)  \(\hbox {y}=0.21\)  \(\hbox {y}=0.03\) 
Observable  \(E\frac{d^3\sigma }{d^3q}\)  \(E\frac{d^3\sigma }{d^3q}\)  \(E\frac{d^3\sigma }{d^3q}\) 
Ref.  [29]  [29]  [29] 
3 Comparison with experiment
3.1 Review of experimental data
 Lowenergy data set

E288: DrellYan process, at \(4<Q<14\) GeV.

 High energy data set:

CDF/D0: Zboson production at \(\sqrt{s}=1.8,~1.96\) TeV.

ATLAS/CMS/LHCb: Zboson production at \(\sqrt{s}=7,8,13\) TeV.

ATLAS: Vector boson production outside the Zpeak (\(46<Q<66\) and \(116<Q<150\) GeV) at \(\sqrt{s}=8\) TeV.

In the following, we present each included measurement in more detail.

We exclude the data points in the range \(9<Q<11\) GeV, because these data are dominated by the \(\Upsilon \)resonance (\(M_\Upsilon \simeq 9.5\) GeV). The description of \(\Upsilon \)resonance production is beyond the scope of current TMD factorization approach.
 The E288 experiment is made on a copper target. To simulate the effects of copper nuclei we replace the proton PDFs by the following combinationswhere \(Z=29\), \(A=63\) and \(N=AZ=34\), are charge, atomic number and the number of neutrons in copper correspondingly.$$\begin{aligned} u_{Cu}(x)= & {} \frac{Z u(x)+N d(x)}{A}, \nonumber \\ d_{Cu}(x)= & {} \frac{Z d(x)+N u(x)}{A}, \nonumber \\ s_{Cu}(x)= & {} s(x), \end{aligned}$$(43)

The absolute normalization of the E288 \(p_T\)crosssection is unknown. Typically, one includes an additional normalization factor \(N_{E288}\), as a parameter of the fit, see e.g. [13, 15, 19, 20]. There is no agreement on this factor values, it varies from \(\sim 0.8\) [13, 19, 20] to \(\sim 1.2\) [15]. In our analysis we fix \({\mathcal {N}}_{E288}=0.8\).
The theoretical uncertainties for low energy experiments are large, of the order ± 10% at the best (see Sect. 3.5). As a consequence, the value of the crosssection is very sensitive to the choice of the PDF set and the overall normalization factor. For example, we have checked that the E288 data can be fitted also with \(N_{E288}=0.9\) with the same (or better) value of \(\chi ^2\) by an additional variation of \(\mu _b\). However, we consider this as a bad practice and restrict ourself to \({\mathcal {N}}_{E288}=0.8\), as the most conventional solution.
 The data are splitted into different bins with different dilepton invariant mass. For each bin we evaluate the crosssection Eq. (6) aswhere \(Q_{\max ,\min }\) are the boundary of the Qbin.$$\begin{aligned} E\frac{d \sigma }{dq^3}=\int _{Q_{\text {min}}}^{Q_{\text {max}}}dQ\, \frac{2Q}{\pi } \frac{d\sigma }{dQ^2 dy d(q_T)^2}, \end{aligned}$$(44)
The characteristics of the data measured at CDF and D0 collaborations at run 1
CDF run I  D0 run I  

\(\sqrt{s}\)  1.8 TeV  1.8 TeV 
Process  \(p+{\bar{p}}\rightarrow Z\rightarrow e^+e^\)  \(p+{\bar{p}}\rightarrow Z\rightarrow e^+e^\) 
\(M_{ll}\) range  66–116 GeV  75–105 GeV 
y  yintegrated  yintegrated 
Observable  \(\frac{d\sigma }{dq_T}\)  \(\frac{d\sigma }{dq_T}\) 
Exp. \(\sigma _{\text {tot}}\) [pb]  \(248\pm 17\)  \(\sigma =221\pm 11\) 
\(\sigma _{\mathrm{tot}}\hbox {[pb]} \begin{array}{l} {[}17,54]_\text {NLO}\\ {[}17,54]_\text {NNLO}:\end{array}\)  \(\begin{array}{l} 223.8\pm 0.05\\ 237.63 \pm 0.18 \end{array}\)  \(\begin{array}{l}223.8\pm 0.05\\ 237.63 \pm 0.18 \end{array}\) 
Ref.  [55] 
The characteristics of the data measured at CDF and D0 collaborations at run 2
CDF run II  D0 run II  

\(\sqrt{s}\)  1.96 TeV  1.96 GeV 
Process  \(p+{\bar{p}}\rightarrow Z\rightarrow e^+e^\)  \(p+{\bar{p}}\rightarrow Z\rightarrow e^+e^\) 
\(M_{ll}\) range  66–116 GeV  70–110 GeV 
y  yintegrated  yintegrated 
Observable  \(\frac{d\sigma }{dq_T}\)  \(\frac{1}{\sigma }\frac{d\sigma }{dq_T}\) 
Exp. \(\sigma _{\text {tot}}\) [pb]  \(256 \pm 2.91 \)  \(\sigma =255\) 
\(\sigma _{\text {tot}}\hbox {[pb]} \begin{array}{l} {[}17,54]_\text {NLO}\\ {[}17,54]_\text {NNLO}:\end{array}\)  \(\begin{array}{l}245.0\pm 0.06\\ 259.77 \pm 0.22\end{array}\)  \(\begin{array}{l}245.0\pm 0.06\\ 259.77 \pm 0.22\end{array}\) 
Ref.  [58]  [59] 
The characteristics of the Zboson production data measured by ATLAS collaborations
ATLAS  ATLAS  

\(\sqrt{s}\)  7 TeV  8 TeV 
Process  \(pp\rightarrow Z\rightarrow ee+\mu \mu \)  \(pp\rightarrow Z\rightarrow \mu \mu \) 
\(M_{ll}\) range  66–116 GeV  66–116 GeV 
lepton cuts  \(\begin{array}{l}p_T>20\hbox { GeV}\\ \eta <2.4\end{array}\)  \(\begin{array}{l}p_T>20\hbox { GeV}\\ \eta <2.4\end{array}\) 
y  \(2.4<y<2.4\)  \(2.4<y<2.4\) 
Observable  \(\frac{1}{\sigma }\frac{d\sigma }{dq_T}\)  \(\frac{1}{\sigma }\frac{d\sigma }{dq_T}\) 
Exp.\(\sigma _{\text {fid}}\hbox {[pb]}\)  –  \(537.1\pm 0.63(\pm \, 2.8\%) \) 
\(\hbox {Theor.}\sigma _{\text {fid}}\hbox {[pb]}\)  \(\begin{array}{l} {[}17,54]_\text {NLO}: 448.56\pm 0.19\\ {[}17,54]_\text {NNLO}: 471.53\pm 0.94\end{array}\)  \(\begin{array}{l} {[}17,54]_\text {NLO}: 505.53\pm 0. 21\\ {[}17,54]_\text {NNLO}: 531.39\pm 0.93\\ {[}61]: 507.9^{+2.4}_{0.7}\end{array}\) 
Ref.  [62]  [30] 
 There is a known tension between the values of total crosssection at run I of CDF and D0. Here we restrict ourself to the fit of the shape of the crosssection and normalize the theoretical points on the binbybin integrals in the allowed range of \(q_T\). I.e. we multiply the theoretical crosssection by the factorwhere \(\varDelta q_T\) is the size of \(q_T\)bins. As we show in Sect. 3.6 the obtained normalization factors are very close to one (at NNLO), and the values of partial crosssections are in agreement with the experimental ones within errorbars. In the Tables 3, 4, we also present the values of the total crosssections evaluated by DYNNLO code [17, 54]. In this calculation of the totalcrosssection, we have used the same inputs as in the TMD fits, i.e. the PDF are taken from MMHT2014 set [60].$$\begin{aligned} {\mathcal {N}}=\frac{\sum _{\begin{array}{c} \text {included}\\ {\text {bins}} \end{array}} \varDelta q_T \frac{d\sigma _{\text {exp.}}}{dq_T}}{\sum _{\begin{array}{c} \text {included}\\ {\text {bins}} \end{array}} \varDelta q_T \frac{d\sigma _{\text {th.}}}{dq_T}}, \end{aligned}$$(45)
 The experimental values for crosssection points are obtained by integrating over all values of y, integrating over measure range of Q and averaging in \(q_T\). Consequently, we have used the following expression for the crosssectionwhere \(y_0=\frac{1}{2}\ln (s/Q^2)\), \(q_{T,\text {low}}\) and \(q_{T,\text {high}}\) are boundaries of \(q_T\)bin, and \(\varDelta q_T\) is the size of the \(q_T\)bin.$$\begin{aligned} \frac{d\sigma }{dq_T}= & {} \frac{1}{\varDelta q_T}\int _{q_{T,\text {low}}}^{q_{T,\text {high}}}2 q'_T dq'_T \int _{y_0}^{y_0}dy \nonumber \\&\times \int _{M_{ll,\min }}^{M_{ll,\max }} 2 Q dQ~\frac{d\sigma }{dy d({q'_T}^2)dQ^2}, \end{aligned}$$(46)

The data from the ATLAS detector at 8 TeV run are presented in several sets [30], which corresponds to different treatment of finalstate photon radiation. We have considered the “dressed” set of the data.

The values of crosssection have been calculated using the expression in Eq. (46), where \(y_0=2.4\), as it is presented in the Tables 5, 6.

There is a known tension between the theoretical calculation of the integrated crosssection and the measured one, see e.g. [30, 61]. Moreover the available theoretical crosssection for vector boson production is not precise enough for the present study. Therefore, we normalize the calculated crosssections by a factor, as explained in more detail in the text around Eq. (51). In Sect. 3.6, we compare the obtained values of normalization to the total crosssection. We have found that the values of obtained normalization are practically independent of the nonperturbative input of the TMD model, and at NNLL/NNLO correctly reproduce (within the errorbars) the measured total crosssection.

All data sets from LHC are presented within fiducial crosssections. Therefore, we have implemented the cut leptonic tensor as it is discussed in Sect. 2.6.
The characteristics of the data for the vector boson production off the Zpeak measured by ATLAS collaborations
ATLAS  ATLAS  

\(\sqrt{s}\)  8 TeV  8 TeV 
Process  \(pp\rightarrow Z/\gamma ^*\rightarrow \mu \mu \)  \(pp\rightarrow Z/\gamma ^*\rightarrow \mu \mu \) 
\(M_{ll}\) range  46–66 GeV  116–150 GeV 
Lepton cuts  \(\begin{array}{l} p_T>20\hbox { GeV}\\ \eta <2.4\end{array}\)  \(\begin{array}{l} p_T>20\hbox { GeV}\\ \eta <2.4 \end{array}\) 
y  \(2.4<y<2.4\)  \(2.4<y<2.4\) 
Observable  \(\frac{1}{\sigma }\frac{d\sigma }{dq_T}\)  \(\frac{1}{\sigma }\frac{d\sigma }{dq_T}\) 
\(\hbox {Exp.}\sigma _{\text {fid}}\hbox {[pb]}\)  \(14.96\pm 2.62(\pm 2.8\%) \)  \(5.59\pm 1.52(\pm 2.8\%)\) 
\(\hbox {Theor.}\sigma _{\text {fid}}\hbox {[pb]}\)  –  – 
Ref.  [30]  [30] 
The characteristics of the Zboson production data measured by CMS collaborations
CMS  CMS  

\(\sqrt{s}\)  7 TeV  8 TeV 
Process  \(pp\rightarrow Z\rightarrow ee+\mu \mu \)  \(pp\rightarrow Z\rightarrow \mu \mu \) 
\(M_{ll}\) range  60–120 GeV  60–120 GeV 
Lepton cuts  \(\begin{array}{l}p_T>20\hbox { GeV}\\ \eta <2.1\end{array}\)  \(\begin{array}{l}p_T>15\hbox { GeV}\\ \eta <2.1\end{array}\) 
y  \(y<2.1\)  \(y<2.1\) 
Observable  \(\frac{1}{\sigma }\frac{d\sigma }{dq_T}\)  \(\frac{1}{\sigma }\frac{d\sigma }{dq_T}\) 
Norm. exp.  –  – 
\(\sigma _{\text {fid}}\hbox {[pb]} \begin{array}{l}{[}17,54]_\text {NLO}\\ {[}17,54]_\text {NNLO}\end{array}\)  \(\begin{array}{l}379.43\pm 0.16\\ 398.27\pm 0.71\end{array}\)  \(\begin{array}{l}427.32\pm 0.53\\ 448.04\pm 0.83\end{array}\) 
Ref.  [63]  [64] 
The characteristics of the Zboson production data measured by LHCb collaborations
LHCb  LHCb  LHCb  

\(\sqrt{s}\)  7 TeV  8 TeV  13 TeV 
Process  \(pp\rightarrow Z\rightarrow \mu \mu \)  \(pp\rightarrow Z\rightarrow \mu \mu \)  \(pp\rightarrow Z\rightarrow \mu \mu \) 
\(M_{ll}\) range  60–120 GeV  60–120 GeV  60–120 GeV 
Lepton cuts  \(\begin{array}{l}p_T>20\hbox { GeV}\\ 2<\eta <4.5\end{array}\)  \(\begin{array}{l}p_T>20\hbox { GeV}\\ 2<\eta <4.5\end{array}\)  \(\begin{array}{l} p_T>20\hbox { GeV}\\ 2<\eta <4.5\end{array}\) 
y  \(2<y<4.5\)  \(2<y<4.5\)  \(2<y<4.5\) 
Observable  \(d\sigma (q_T)\)  \(d\sigma (q_T)\)  \(\frac{d\sigma }{dq_T}\) 
Norm. exp.  \(\sigma =76.0\pm 3.1\) pb  \(\sigma =95.0\pm 3.2\) pb  \(\sigma =198.0\pm 13.3\) pb 
\(\sigma _{\text {fid}}\hbox {[pb]}\) \(\begin{array}{l}{[}17,54]_\text {NLO}\\ {[}17,54]_\text {NNLO} \end{array}\)  \(\begin{array}{l}69.85\pm 0.3\\ 74.30\pm 0.21\end{array}\)  \(\begin{array}{l}88.98\pm 0.397\\ 93.50\pm 0.3\end{array}\)  \(\begin{array}{l}185.0\pm 0.09\\ 192.78 \pm 0.82\end{array}\) 
Ref.  [65]  [66]  [67] 

The values of crosssection have been calculated using the expression in Eq. (46), where the limits for yintegration \(y_0\) are taken in accordance to the Tables 7, 8.

Just as in the case of ATLAS data we have normalized the calculated crosssections by the factor provided in Eq. (45) discussed in Sect. 3.6. We have found a good agreement between the theoretical and experimental values for total crosssection for LHCb data.

All data sets from LHC are fiducial crosssections. Therefore, we have implemented the cut leptonic tensor as it is discussed in Sect. 2.6.
3.2 arTeMiDe
In order to evaluate the crosssections we have prepared the program package arTeMiDe. The arTeMiDe package is a collection of FORTRAN modules that evaluates individual terms of the TMD factorization formalism, such as TMD evolution factors, TMDPDFs, and combines them into the differential crosssections. arTeMiDe forms a flexible package for TMDPDF phenomenology based on the \(\zeta \)prescription, as described in this article. It is publicly available at the webpage [31].
arTeMiDe version 1.1 evaluates the quark and gluon unpolarized TMDPDFs (although in the discussed fit the gluon TMDPDFs are not necessary) for any given function \(f_{NP}\), at any composition of perturbative orders from LO to NNLO, with or without renormaloninduced power corrections. For the current study, the input PDFs are taken from the MMHT2014 PDF set [60].
The evaluation of the Hankeltype integral over b is one of the main source of numerical errors. Typically, in order to obtain sufficient precision one should include a large number of points into the integral, which is very costly especially at NNLL/NNLO. arTeMiDe evaluates this integral with the Ogata quadratures [69]. The Ogata quadrature is a double exponential quadrature, whose nodes are the zeros of the Bessel function. It provides a fast and precise evaluation of Hankeltype integrals with the minimal number of integrand calls.
The fitting procedure has been performed by minimizing the \(\chi ^2\)function. The minimization of the \(\chi ^2\) distribution has been done using the MINUIT package from the CERN library [70, 71]. The estimation of the statistical uncertainties for nonperturbative parameters is made with the MINOS procedure, performing the variation of parameters in the range \(\chi ^2\pm \varDelta \chi ^2\), with \(\varDelta \chi ^2\) corresponding to the 68% confidence level (i.e. \(\varDelta \chi ^2\simeq \{1.03,2.32,3.55\}\) for 1–3 fitting parameters, correspondingly.) The sources of theoretical uncertainties have been pointed in Sect. 2.5, and parameterized by the constants \(c_{1,2,3,4}\). The variation of these constants in the region (0.5, 2) produces the errorbands. The discussion on the individual contributions of theoretical uncertainties associated with different scales is given in Sect. 3.5.
3.3 Models for nonperturbative part of TMDPDFs
The largeb behavior of TMD distributions is the key point of TMD parametrization and extraction. There is no common agreement on this behavior. Clearly, such an agreement cannot be achieved in general, since the bshape of a TMD distribution is strongly dependent on the largeb prescription. For example, the Gaussian behavior is typically observed in the models based on \(b^*\)prescription. Moreover, the classical fits by ResBos package [15] disfavor other nonperturbative behaviors, such an exponential one (for more recent discussion, see [24]). Also the Gaussian shape is used in DYRes code [21] (together with \(b^*\)prescription) and in DYqT code [18] (together with the minimal prescription). Contrary, the fit made in Ref. [20], which does not employ the \(b^*\)prescription, uses an exponential shape of \(f_{NP}\) and also obtains an agreement with data. We point out that the use of LHC data for TMD extraction is made here for the first time (to our knowledge). Given the precision of LHC data, the consistency and/or goodness of all previous hypotheses has to be rediscussed.
In order to decide the best shape of \(f_{NP}\) within \(\zeta \)prescription, we have considered several subsets of the data. It appears important to include simultaneously both highenergy and lowenergy data because they are sensitive to different parts of the bspace spectrum. We have found that the most optimal data subset is given by the E288 data and the ATLAS Zboson production data, see Tables 2, 5. In this subset, the very small errorbands of ATLAS data are compensated by a large number of points in E288 data, and as a result, we have a certain equilibrium between low and highenergy inputs.
The values of \(\chi ^2/d.o.f\) for different singleparameter nonperturbative functions \(f_{NP}\), minimized on different data sets. The \(\chi ^2/d.o.f\) values correspond to \(\delta _T=0.2\) and NNLL/NNLO
\(\hbox {data/}f_{NP}\)  \(e^{\lambda b}\)  \(e^{\lambda b^2}\)  \(\cosh ^{1}(\lambda b)\) 

ATLAS  4.78  1.43  1.42 
E288  2.70  5.68  3.64 
E288 + ATLAS  8.18  5.77  3.72 
 (i)
The high and low energy data should be considered altogether, because they test different intervals of the bspace spectrum of \(f_{NP}\).
 (ii)
The subset of data points E288 and ATLAS Zboson, is very selective for the \(f_{NP}\). A good fit of this subset guaranties the good fit for the whole set of data points. Nevertheless, in the following sections, we include all experiments, for consistency.
 (iii)
Both theoretically and phenomenologically, we argue that \(f_{NP}\) should be a function of even powers of b with an exponential asymptotic behavior at \(b\rightarrow \infty \). Using a minimal set of two parameters (and the evolution parameter \(g_K\)) we find that one can easily fit the data with a \(\chi ^2/d.o.f\sim 1.2\)–1.3. The addition of more parameters (say for the control of \(b^4\) correction and/or flavor dependence) has the possibility to increase the quality of the fit. However, in this work, we do not consider extra parameters, since the current quality of the fit is already typical and reasonable for the modern TMD extraction (compare e.g. with [22]).
 (iv)
One needs at least two parameters (one to control \(\sim b^2\) behavior at \(b\rightarrow 0\) and another to control the asymptotics) to fit simultaneously low and highenergy data. However, the multiplication by polynomials (e.g. \(f_{NP}\sim (1+\lambda b^2)/\cosh (b)\)) does not work well, which suggests that the asymptotic terms \(\sim b^2 e^{b}\) are disfavored.
 Model 1 This ansatz uses the fact that the simplest evenb function with exponent asymptotics is the hyperbolic cosine. The model readswhere \(\lambda _1[\text {GeV}]>0\) and \(\lambda _2[\text {GeV}^2]>0\) are free parameters. The advantage of this model is its simplicity and independence of the Bjorken variable. The model 1 has a quadratic (Gaussian) behavior at smallb \(f_{NP}\sim e^{\lambda _2 b^2}\) and exponential behavior at largeb \(f_{NP}\sim e^{\lambda _1 b}\).$$\begin{aligned} f_{NP}(b)=\frac{\cosh \left( \left( \frac{\lambda _2}{\lambda _1}\frac{\lambda _1}{2}\right) b\right) }{\cosh \left( \left( \frac{\lambda _2}{\lambda _1}+\frac{\lambda _1}{2}\right) b\right) }, \end{aligned}$$(49)
 Model 2 The model 2 readswhere \(\lambda _1[\text {GeV}]>0\) and \(\lambda _2[\text {GeV}^2]>0\) are free parameters. In this model we attempt to incorporate the theoretical expectations on the zdependence of \(f_{NP}\). So, the model 2 has a \(zb^2\)behavior at smallb \(f_{NP}\sim e^{\lambda _2 zb^2}\) and exponential behavior at largeb \(f_{NP}\sim e^{\lambda _1 b}\).$$\begin{aligned} f_{NP}(z,b)=\exp \left( \frac{\lambda _2 z b^2}{\sqrt{1+z^2 b^2\frac{\lambda _2^2}{\lambda _1^2}}}\right) , \end{aligned}$$(50)
Additionally, to the parameters \(\lambda _{1,2}\) we have studied the parameter \(g_K[\text {GeV}^2]>0\), which parametrizes the nonperturbative contribution to the rapidity evolution kernel \({\mathcal {D}}\) (see Eq. (21)). The importance of this parameter is not clear from the literature. In Ref. [12] we have estimated its size in the large\(\beta _0\) approximation as \(0.01\pm 0.03\;\text {GeV}^2\), i.e. consistent with zero. Also, the fit of [20] shows a negligible influence of this parameter on the final results. Therefore, in the following we consider both possibilities \(g_K=0\) and \(g_K\ne 0\). In Sect. 3.7, we demonstrate that the parameter \(g_K\) is important at lower perturbative order, but its influence is negligible at NNLL/NNLO.
3.4 The domain of TMD factorization
The number of points with \(q_T<\delta _T Q\) for each data set. In the majority of fits we use \(\delta _T=0.2\), see explanation in the text
\(\delta _T\)  0.1  0.125  0.15  0.175  0.2  0.225  0.25  0.275  0.3 

CDF + D0 run1  27  34  38  41  44  47  49  51  52 
CDF + D0 run2  22  28  32  38  43  49  55  60  63 
ATLAS Zproduction (7 + 8 TeV)  10  12  14  16  18  20  21  23  24 
ATLAS DY (8 TeV)  9  11  12  14  14  16  16  18  18 
CMS (7 + 8 TeV)  8  10  10  12  14  16  16  18  18 
LHCb (7 + 8 + 13 TeV)  18  21  24  27  30  30  33  33  33 
High energy total  94  116  130  148  163  178  190  203  208 
E288 200 GeV  16  20  24  29  35  36  41  44  47 
E288 300 GeV  22  27  33  38  45  46  51  55  59 
E288 400 GeV  33  40  49  57  66  69  76  82  85 
Low energy total  71  87  106  124  146  151  168  181  191 
Total  165  203  236  272  309  329  358  384  399 
The TMD factorization is restricted to the small\(q_T\) range. The size of the allowed \(q_T\)region is a priory unknown. We have not found any phenomenological studies on this point but only some statement on the strong dependence of the fit on the \(q_T\)window. A specific study on TeVatron Zboson production data in Ref. [53] shows that the Yterm contribution is extremely marginal for \(q_T<30\) GeV.
In order to make a qualitative study, we introduce the parameter \(\delta _T\) and we consider all data points with \(q_T<\delta _T Q\). The amount of data points which are allowed by such a restriction are shown in the Table 10. In order to estimate the maximum value of \(\delta _T\) we perform a series of fits with increasing values of \(\delta _T\). Ideally, the \(\chi ^2/d.o.f.\) and the fitting parameters should be stable within and unstable outside of the allowed \(\delta _T\) interval. In this way, considering the dependence on \(\delta _T\) one should find an interval of \(\delta _T\) for which the fit is not sensitive to the Yterm. This point indicates the region of TMDfactorization, and should not depend of the perturbative order.
We have performed such a test for highenergy data set with different oneparameter forms of \(f_{NP}\). We have especially used the one parameter models to guarantee the absence of finetuning of the crosssection. For this reason we also exclude the E288 data, because it is impossible to describe high and lowenergy data with a single nonperturbative parameter. The result of the fits practically agrees for all tested models and orders. In Fig. 3, we present some typical outcome of the fits.
In plots 3 one can see that all models reproduce the data verywell at very small \(\delta _T\), which is expected since the TMD factorization is only valid at \(q_T\ll Q\). Then the value of \(\chi ^2\) slightly grows but keeps less then one until \(\delta _T=0.2\) and after this threshold it jumps to higher values. The next jump is seen at \(\delta _T=0.25\). After \(\delta _T=0.25\) the value of \(\chi ^2\) increases rapidly. We interpret this fact saying that at \(\delta _T=0.2\) we become sensitive to Yterm, and at \(\delta _T=0.25\) the Yterm starts to dominate the crosssection, i.e. we leave the domain of TMD factorization. We have found that the presented plots rather strongly depend on the set of pertubative scales. For some choice of these scales, one can obtain an ideally flat plateau of \(\chi ^2\) for \(\delta _T\leqslant 0.2\). However, the values of the two important thresholds, namely, \(\delta _T=0.2\) (where deviation form TMD factorization appears) and \(\delta _T=0.25\) (where the TMD factorization is completely broken), are stable with perturbative scales.
As a result of these tests, in the following we use the data points with \(q_T\lesssim 0.2\; Q\), or say \(\delta _T=0.2\). The choice of \(\delta _T\) that we make is consistent with [53]. This range includes 163 highenergy and 146 lowenergy data points (in total 309 data points). Comparing this number of points with the literature, we observe that, it is the largest set of points for DrellYan/Zboson production used up to present in a simultaneous fit of TMDPDF (to our knowledge), which also has the largest considered range of energies from \((Q,\sqrt{s})=(4,19.4)\;\text {GeV}\) (from the E288 experiment) to \((Q,\sqrt{s})=(150,8000)\;\text {GeV}\) (from the ATLAS experiment).
3.5 Scale variations and theoretical uncertainties
The theoretical uncertainties of the perturbative inputs are tested by varying the perturbative scales around their central values, as it is discussed in Sect. 2.4. The distribution of uncertainties through orders for a typical high energy experiment is shown in Figs. 4, 5, and for a typical lowenergy experiments in Figs. 6, 7. The complete set of plots for every included experiment can be found in [31].
The uncertainty associated with the TMD evolution factor is parameterized by the \(c_1\)variation. This uncertainty drops down between NLL/NLO and NNLL/NLO orders, that is together with the increase of the perturbative order for \({\mathcal {D}}\) (see Table 1). The size of the band is correlated with the energy of the process, that is, it is less significant for higherenergy experiments.
The uncertainty associated with the hard scale depends on the \(c_2\)variation. This band is independent of \(q_T\). This error is the main one at NLL/LO (which we do not present here), but becomes negligible at higher orders.
The uncertainty associated with the lowenergy behavior of the evolution factor is parameterized by the \(c_3\)variation. We have found that it significantly influences the shape of the crosssection and also it is rather large at small\(q_T\). As expected it is decreases going from NLL/NLO to NNLL/NNLO. At NNLL/NNLO it gives the main contribution to the uncertainty band for the crosssection.
The uncertainty associated with the smallb matching of coefficients and PDFs is represented by the \(c_4\)variation. It is the most interesting error because it checks the convergences of the \(\zeta \)prescription. The corresponding errorband is larger at \(q_T\rightarrow 0\), which corresponds to the contribution of large \({\mathbf {L}}_\mu \) (we remind that in \(\zeta \)prescription, \({\mathbf {L}}_\mu \) grows unrestrictedly). The important observation is that the large uncertainty area significantly shrinks between NLL/NLO and NNLL/NNLO, although the NNLL/NNLO contains a higher power of \({\mathbf {L}}_\mu \). This shows a very good behavior of the \(\zeta \)prescription. In total this error is dominant at NLL/NLO, but becomes smaller (although compatible) to the one coming from the \(c_3\) variation at NNLL/NNLO.
In order to provide a final definition of the theoretical error, we use all scale variations and we take the maximum deviation among them. We have found that our definition of uncertainties is close, as far as one can compare different theoretical expressions, to the common definition used e.g. in [21, 43]. In total, for the highenergy experiments we find that the theoretical uncertainty (at NNLO) is of the order \(2{\text {}}3\)% at the peak. It grows to \(\sim 5{\text {}}6\%\) at maximum allowed \(q_T\), and to \(\sim 10\%\) at \(q_T\rightarrow 0\). This value seems to be smaller (but comparable) to the typical values of uncertainties presented ResBos or DYRes. This is a definite positive point of the \(\zeta \)prescription. Indeed, the main contribution (at high energies) to it comes from the \(c_3\) and \(c_4\)errorbands, which are controlled by \(\zeta \)prescription. The \(c_4\)band would be significantly larger in the presence of doublelogarithms, which are absent due to the \(\zeta \)prescription.
3.6 Normalization
As the TMD factorization approach describes the shape of the differential cross section only in a limited range of \(q_T\), we need some extra input to normalize the curves. In order to compare with the data, we weight the differential crosssection by the total (or fiducial) crosssection. The values of the theory predictions for total crosssections can be obtained from the studies of other groups. For example, one can use the DYNNLO code [17, 54]. Its predictions for the total crosssections are presented in the Tables 3, 4. However, we found that such a strategy is unreliable, because even tiny disagreement in the normalization leads to huge effects in the \(\chi ^2\)minimization. This is especially important for LHC data sets, which have very small errorbands. Additionally, as we demonstrate later, the DYNNLO predictions are worse than that obtained using our normalization factors.
The normalization factors for the crosssection for each experiment. The dimensionalless numbers are ratios of partially integrated cross section over \(q_T\) (51) (theory/experiment, i.e. \({\mathcal {N}}^{1}\)), for the data with the published value of total crosssection. For the data sets with unpublished values of total crosssection, the value of the total crosssection used for normalization is presented. The numbers are given for the model 1. The variation of the scales and models gives the change of numbers in the unrepresented digits. The numbers shown in bold are those which agree with the measured crosssection within the error bars
Order  ATLAS Zboson 7 TeV (pb)  ATLAS Zboson 8 TeV  ATLAS 46–66 8 TeV  ATLAS 116150 8 TeV  CMS 7 TeV (pb)  CMS 8 TeV (pb)  LHCb 7 TeV  LHCb 8 TeV  LHCb 13 TeV 

NLL/NLO  438  0.92  1.01  0.93  369  407  0.92  0.93  0.93 
NNLL/NLO  438  0.92  1.01  0.93  368  407  0.92  0.93  0.93 
NNLL/NNLO  461  0.97  1.08  0.98  387  429  0.97  0.99  0.98 
The results of the \(\chi ^2\)minimization procedure with \(g_K=0\). The values of \(\chi ^2\) are given including the theoretical errorband. The values of extracted parameters are given with statistical errorband (the first pair of numbers) and the theoretical errorband (the second pair of numbers). The visual presentation of this table is given in Fig.9
Order  \(\frac{\chi ^2}{d.o.f.}\)  \(\lambda _1\)  \(\lambda _2\) 

Model 1  
NLL/NLO  \(2.33\,^{+2.76}_{0.68}\)  \(0.321^{+0.008}_{0.007}\,^{+0.095}_{0.100}\)  \(0.271^{+0.014}_{0.013}\,^{+0.155}_{0.063}\) 
NNLL/NLO  \(1.76\,^{+1.25}_{0.48}\)  \(0.289^{+0.004}_{0.004}\,^{+0.007}_{0.121}\)  \(0.424^{+0.051}_{0.045}\,^{+0.673}_{0.139}\) 
NNLL/NNLO  \(1.34\,^{+0.44}_{0.20} \)  \(0.271^{+0.007}_{0.006}\,^{+0.076}_{0.073}\)  \(0.277^{+0.015}_{0.012} \,^{+0.081}_{0.042}\) 
Model 2  
NLL/NLO  \(2.19\,_{0.64}^{+2.34}\)  \(0.329^{+0.008}_{0.008}\,^{+0.047}_{0.101}\)  \(0.289^{+0.019}_{0.017}\,^{+0.276}_{0.008}\) 
NNLL/NLO  \(1.65\,^{+1.32}_{0.39}\)  \(0.236^{+0.005}_{0.004}\,^{+0.070}_{0.064}\)  \(0.440^{+0.049}_{0.044}\,^{+0.573}_{0.126}\) 
NNLL/NNLO  \(1.36\,^{+0.35}_{0.18}\)  \(0.284^{+0.007}_{0.006}\,^{+0.074}_{0.079}\)  \(0.280^{+0.019}_{0.017}\,^{+0.086}_{0.034}\) 
The results of the \(\chi ^2\)minimization procedure with nonzero \(g_K\). The values of \(\chi ^2\) are given with theoretical errorband. The values of extracted parameters are given with statistical errorband (the first pair of numbers) and the theoretical errorband (the second pair of numbers). The visual presentation of this table is given in Fig. 9
Order  \(\frac{\chi ^2}{d.o.f.}\)  \(\lambda _1\)  \(\lambda _2\)  \(g_K\times 10^{2}\) 

Model 1  
NLL/NLO  \(1.17\,^{+1.32}_{0.07} \)  \(0.189^{+0.009}_{0.009}\,^{+0.114}_{0.052}\)  \(0.425^{+0.054}_{0.045}\,^{+0.047}_{0.250}\)  \(2.31^{+0.25}_{0.24}\,^{+1.44}_{1.19}\) 
NNLL/NLO  \(1.21\,^{+1.16}_{0.02}\)  \(0.175^{+0.008}_{0.008}\,^{+0.089}_{0.041}\)  \(0.532^{+0.076}_{0.067}\,^{+0.426}_{0.203}\)  \(1.27^{+0.22}_{0.21}\,^{+1.19}_{1.27}\) 
NNLL/NNLO  \(1.23\,^{+0.30}_{0.13} \)  \(0.228^{+0.016}_{0.013}\,^{+0.034}_{0.060}\)  \(0.306^{+0.031}_{0.026}\,^{+0.265}_{0.063}\)  \(0.73^{+0.24}_{0.23}\,_{0.73}^{+1.09}\) 
Model 2  
NLL/NLO  \(1.18\,^{+1.31}_{0.07}\)  \(0.199^{+0.011}_{0.010}\,^{+0.104}_{0.062}\)  \(0.443^{+0.061}_{0.052}\,^{+0.503}_{0.093}\)  \(2.18^{+0.26}_{0.25}\,^{+1.57}_{1.06}\) 
NNLL/NLO  \(1.22\,^{+1.16}_{0.01}\)  \(0.181^{+0.009}_{0.009}\,^{+0.099}_{0.045}\)  \(0.562^{+0.092}_{0.075}\,^{+0.468}_{0.206}\)  \(1.18^{+0.22}_{0.21}\,^{+1.12}_{1.18}\) 
NNLL/NNLO  \(1.29\,^{+0.26}_{0.18}\)  \(0.244^{+0.016}_{0.015}\,^{+0.035}_{0.069}\)  \(0.306^{+0.034}_{0.029}\,^{+0.216}_{0.050}\)  \(0.59^{+0.24}_{0.27}\,^{+1.01}_{0.59}\) 
3.7 Results of the fits and TMD extraction
We have estimated both statistical and theoretical errors on the fit parameters. The statistical errors are related to the uncertainty of the \(\chi ^2\)minimization and are induced by the experimental errorbands. The statistical errors have been estimated by the MINOS procedure of MINUIT package [71]. The theoretical errors are related to the uncertainty of perturbation series. There is no common procedure for the estimation of the theoretical error. Therefore, we propose the method presented in the following.
The theoretical error is estimated by a set of independent fitting procedures for each variation of the scale constants \(c_{1,2,3,4}\in [0.5,2]\), as discussed in Sect. 2.5. In other words, we set, say, \(c_1=2\) and perform the minimization of \(\chi ^2\). In this way, we obtain a new set of model parameters (and a new value of \(\chi ^2\)). In total, we have 8 independent variations and hence have 8 values of parameters. The final theoretical errorband is given by the maximal positive and minimal deviations from the central value and the results are reported in Table 14. A drawback of this procedure is the variation of a scale can lead to the serious increase in \(\chi ^2\). In other words changing the matching scales affects also the quality of the fit. In general, the size of the band for \(\chi ^2\) value represents the stability of the theoretical model, and they are also reported in Table 14. One can see that the error for \(\chi ^2\) significantly drops with orders.
Example of parameter extraction with the variation of \(c_{1,2,3,4}\) constants, and evaluation of the theoretical error. Bold numbers in brackets represent the deviation of the parameter from its central value
Variation  \(\frac{\chi ^2}{d.o.f.}\)  \(\lambda _1\)  \(\lambda _2\)  \(g_K\times 10^{2}\) 

Model 1 NNLL/NLO  
\(c_{1,2,3,4}=1\)  1.17  0.189  0.425  2.31 
\(c_{1}=2\)  \(1.31\,(\mathbf {+ 0.14}) \)  \(0.201 \,(\mathbf {+ 0.012})\)  \(0.316 \,(\mathbf { 0.109})\)  \(3.00 \,(\mathbf {+ 0.69})\) 
\(c_{1}=0.5\)  \(1.10\,(\mathbf { 0.07}) \)  \(0.184 \,(\mathbf { 0.005})\)  \(0.308 \,(\mathbf { 0.117})\)  \(1.60 \,(\mathbf { 0.71})\) 
\(c_{2}=2\)  \(1.19\,(\mathbf {+ 0.02}) \)  \(0.204 \,(\mathbf {+ 0.015})\)  \(0.223 \,(\mathbf { 0.202})\)  \(2.12 \,(\mathbf { 0.19})\) 
\(c_{2}=0.5\)  \(1.20\,(\mathbf {+ 0.03}) \)  \(0.219 \,(\mathbf {+ 0.030})\)  \(0.226 \,(\mathbf { 0.199})\)  \(1.93 \,(\mathbf { 0.38})\) 
\(c_{3}=2\)  \(1.23\,(\mathbf {+ 0.06}) \)  \(0.251 \,(\mathbf {+ 0.062})\)  \(0.315 \,(\mathbf { 0.110})\)  \(3.75 \,(\mathbf {+ 1.44})\) 
\(c_{3}=0.5\)  \(1.13\,(\mathbf { 0.04}) \)  \(0.160 \,(\mathbf { 0.029})\)  \(0.220 \,(\mathbf { 0.205})\)  \(1.12 \,(\mathbf { 1.19})\) 
\(c_{4}=2\)  \(1.76\,(\mathbf {+ 0.59}) \)  \(0.137 \,(\mathbf { 0.052})\)  \(0.473 \,(\mathbf {+ 0.046})\)  \(2.71 \,(\mathbf {+ 0.40})\) 
\(c_{4}=0.5\)  \(2.49\,(\mathbf {+ 1.32}) \)  \(0.303 \,(\mathbf {+0.114})\)  \(0.175 \,(\mathbf { 0.250})\)  \(1.15 \,(\mathbf {1.16})\) 
Result  \(1.17^{+1.32}_{0.07}\)  \(0.189^{+0.114}_{0.052}\)  \(0.425^{+0.047}_{0.250}\)  \(2.31^{+1.44}_{1.19}\) 
Model 1 NNLL/NNLO  
\(c_{1,2,3,4}=1\)  1.23  0.228  0.306  0.73 
\(c_{1}=2\)  \(1.40\,(\mathbf {+ 0.17}) \)  \(0.242 \,(\mathbf {+ 0.014})\)  \(0.296 \,(\mathbf { 0.010})\)  \(1.21 \,(\mathbf {+ 0.48})\) 
\(c_{1}=0.5\)  \(1.14\,(\mathbf { 0.09}) \)  \(0.221 \,(\mathbf { 0.007})\)  \(0.346 \,(\mathbf {+ 0.020})\)  \(0.12 \,(\mathbf { 0.61})\) 
\(c_{2}=2\)  \(1.22\,(\mathbf { 0.01}) \)  \(0.217 \,(\mathbf { 0.011})\)  \(0.295 \,(\mathbf { 0.011})\)  \(0.86 \,(\mathbf {+ 0.13})\) 
\(c_{2}=0.5\)  \(1.26\,(\mathbf {+ 0.03}) \)  \(0.252 \,(\mathbf {+ 0.024})\)  \(0.326 \,(\mathbf {+ 0.020})\)  \(0.48 \,(\mathbf { 0.25})\) 
\(c_{3}=2\)  \(1.27\,(\mathbf {+0.04}) \)  \(0.260 \,(\mathbf {+ 0.032})\)  \(0.344 \,(\mathbf {+ 0.038})\)  \(1.82 \,(\mathbf {+ 1.09})\) 
\(c_{3}=0.5\)  \(1.31\,(\mathbf {+ 0.08}) \)  \(0.198 \,(\mathbf { 0.030})\)  \(0.358 \,(\mathbf {+ 0.052})\)  \(0.00 \,(\mathbf { 0.73})\) 
\(c_{4}=2\)  \(1.10\,(\mathbf { 0.13}) \)  \(0.168 \,(\mathbf { 0.060})\)  \(0.571 \,(\mathbf {+ 0.265})\)  \(1.27 \,(\mathbf {+ 0.54})\) 
\(c_{4}=0.5\)  \(1.53\,(\mathbf {+ 0.30}) \)  \(0.262 \,(\mathbf {+ 0.034})\)  \(0.243 \,(\mathbf { 0.063})\)  \(0.68 \,(\mathbf { 0.05})\) 
Result  \(1.23^{+0.30}_{0.13}\)  \(0.228^{+0.034}_{0.060}\)  \(0.306^{+0.265}_{0.063}\)  \(0.73^{+1.09}_{0.73}\) 
The values of \(\chi ^2/points\) for individual data sets. The boxes indicate the values of partial \(\chi ^2\) which are responsible for the increment of \(\chi ^2/d.o.f.\) from NLL/NLO to NNLL/NNLO
Data set  Point  Model 1  Model 2  

NLL/NLO  NNLL/NLO  NNLL/NNLO  NLL/NLO  NNLL/NLO  NNLL/NNLO  
CDF run1  30  0.67  0.68  0.64  0.67  0.67  0.64 
D0 run1  14  0.50  0.52  0.60  0.49  0.51  0.62 
CDF run2  36  1.22  1.36  1.30  1.17  1.29  1.33 
D0 run2  7  2.52  2.69  2.75  2.45  2.64  2.79 
ATLAS (7TeV) Zboson  9  1.54  1.55  2.01  1.60  1.59  2.27 
ATLAS (8TeV) Zboson  9  2.32  2.48  2.69  2.46  2.70  2.79 
ATLAS (8TeV) 4666 GeV  5  0.04  0.05  0.16  0.05  0.04  0.20 
ATLAS (8TeV) 116150 GeV  9  0.30  0.35  0.31  0.30  0.36  0.30 
CMS (7 TeV)  7  1.38  1.39  1.36  1.38  1.38  1.36 
CMS (8 TeV)  7  1.38  1.38  1.54  1.38  1.37  1.58 
LHCb (7 TeV)  10  0.26  0.26  0.31  0.25  0.26  0.33 
LHCb (8 TeV)  10  0.11  0.12  0.27  0.11  0.12  0.32 
LHCb (13 TeV)  10  0.50  0.50  0.28  0.50  0.50  0.27 
High energy data  163  0.95  1.00  0.94  0.94  1.00  1.04 
E288(200) 45 GeV  5  3.86  4.28  3.86  4.25  4.59  4.30 
E288(200) 56 GeV  6  3.00  3.03  1.92  3.05  3.07  1.92 
E288(200) 67 GeV  7  1.68  1.68  0.84  1.66  1.67  0.79 
E288(200) 78 GeV  8  1.10  1.10  0.93  1.13  1.11  1.00 
E288(200) 89 GeV  9  1.83  1.84  0.78  1.89  1.87  1.87 
E288(300) 45 GeV  5  1.93  2.20  4.09  2.24  2.44  4.90 
E288(300) 56 GeV  6  1.15  1.18  1.15  1.19  1.21  1.21 
E288(300) 67 GeV  7  0.84  0.83  0.66  0.85  0.83  0.69 
E288(300) 78 GeV  8  1.18  1.17  0.90  1.16  1.17  0.86 
E288(300) 89 GeV  9  1.13  1.14  1.13  1.11  1.36  1.10 
E288(300) 1112 GeV  12  1.08  1.08  1.00  1.11  1.10  1.04 
E288(400) 56 GeV  6  2.11  2.04  1.12  1.94  1.92  1.01 
E288(400) 67 GeV  7  2.59  2.68  2.55  2.59  2.64  2.55 
E288(400) 78 GeV  8  0.83  0.97  2.02  0.99  1.07  2.44 
E288(400) 89 GeV  9  1.36  1.31  1.37  1.37  1.32  1.54 
E288(400) 1112 GeV  12  1.08  1.06  1.25  1.05  1.05  1.17 
E288(400) 1213 GeV  12  0.88  0.88  1.10  0.87  0.88  1.14 
E288(400) 1314 GeV  12  0.39  0.38  0.72  0.39  0.39  0.71 
Low energy data  146  1.38  1.41  1.35  1.50  1.48  1.49 
Total  309  1.17  1.21  1.23  1.18  1.22  1.29 
As expected the theoretical error is reduced with the increase of the perturbative order. In particular, the band on the value of \(\chi ^2\) is significantly smaller at NNLL/NNLO. The distribution of parameter values over perturbative orders presented in Table 14 is typical. The variation of \(c_1\) does not represent the main contribution to the errorband. It implies that the lowenergy matching for the rapidity anomalous dimension is not so important (in comparison to other matchings), as typically expected.
The variation of \(c_2\) is almost negligible. Here, however, we recall that \(c_2\) influences only the common normalization factor, and thus the effect of its variation could be underestimated due to our fitting procedure. The variation of \(c_3\) and \(c_4\) produces the most part of the errorband and the strongest variation of \(\chi ^2\). At \(g_K=0\) these variation are moreorless equivalent. At \(g_K\ne 0\) there is a clear pattern. In this case, the variation of \(c_3\) gives the main errorband on \(g_K\), while the variation of \(c_4\) gives the main errorband on parameters \(\lambda _{1,2}\). It is very natural since the variation of \(c_3\) tests the lowenergy normalization point of the evolution factor, and \(c_4\) tests the uncertainties of perturbation determination of the TMDPDF.
4 Conclusion
The unpolarized DrellYan process at small\(q_T\) offers the simplest application of the TMD factorization formalism, and as such it has been studied by many groups. In this work, we have revised the main points of the practical implementation of TMD factorization, and reveal some new aspects of the TMD phenomenology. Altogether it allows us to critically reanalyze the available DrellYan data and to extract consistently the unpolarized TMDPDFs, within some approximation. The primary aim of our analysis is to answer some general questions for the TMD approach such as: Up to which \(q_T\) the TMD factorization works? What is the best asymptotical behavior of a TMD distributions? How convergent is TMD formalism at higher orders of perturbative expansion? The answers to these questions are naturally affected by the used prescriptions for the practical implementation of the TMD formalism. Even so, these important issues of TMD phenomenology are undiscussed in the literature or discussed very superficially. Implementing consistently the TMD factorization formalism, we are able to fit a large set of DrellYan data points which ranges from low (\(Q=4\;\text {GeV}\)) to high (\(Q=116\)–\(150\;\text {GeV}\)) dilepton invariant masses on a wide interval of center of mass energies and using a limited set of parameters (two for the nonperturbative part of TMDPDF and one for the nonperturbative part of the TMD evolution).
In this work, we have formulated and used the \(\zeta \)prescription, which is one of the main new theoretical contributions of this article. The \(\zeta \)prescription consists of a particular choice of the rapidity evolution scale \(\zeta =\zeta _\mu \), which depends on \(\mu \), b and the parton flavor (quark or gluon). This choice corresponds to the equievolution line in the space of TMD scales, and thus a TMD distribution is \(\mu \)independent along this line. As a consequence, all logarithms related to the TMD evolution, which are essentially double logarithms, are eliminated from the smallb OPE. It significantly improves the perturbative convergence and the radius of convergences for the smallb OPE. The value of \(\zeta _\mu \) is dictated by the differential equation (29), which can be solved orderbyorder in perturbation theory. We stress that the \(\zeta \)prescription does not strictly solve the problem of largeb logarithms, which are still present in the matching coefficients. However, these logarithms are not related anymore to the TMD scales. Moreover, these logarithms are accompanied by the xdependent coefficients which preserve the integral over x in accordance with the probability interpretation of PDFs. Note, that the \(\zeta \)prescription is universal for all TMDs of the leading dynamical twist, due to the universality of TMD ultraviolet and rapidity renormalization factors. There are multiple possibilities to apply \(\zeta \)prescription, see some discussion in “Appendix B.1”. In this work, we have used the simplest one, which can be certainly improved. A further study of the \(\zeta \)prescription will be done elsewhere.
Within our implementation of TMD factorization and TMD distributions, which has a generic form, we have three independent perturbative series: one for hard matching, one for rapidity evolution, and one for smallb matching. To defend the approach we provide the estimation of the perturbative uncertainty by variation of associated scales by factor 2, see Sect. 3.5. We have considered several successive perturbative orders (see Table 1), and demonstrate that the theory uncertainties and the agreement with the data improve with the increase of the perturbative order. The agreement of the theory with the experiment resulting in our fit is a consequence of the \(\zeta \)prescription to a large extent. The lowest possible combination of perturbative order, namely NLL/LO, produces very large theoretical errorbands and thus has been excluded from the present study.
The theoretical uncertainty on the extracted parameters is shown in Figs. 8, 9. Several improvements of the current approach can certainly help the reduction of errors and allow us to understand them better. In fact one can cite the inclusion of factorization breaking corrections (Yterms), and also QED/isospin breaking corrections for LHC data. These issues will be a matter of discussion of future works.
Another aspect that we point out, is the practical limitation of TMD factorization. To make the discussion quantitative we introduce the parameter \(\delta _T\), which is the highest allowed ratio \(q_T/Q\) accounted in the fit. Clearly, at very low \(\delta _T\) the TMD formalism should perfectly work, e.g. provide small values of \(\chi ^2\)distribution. Our expectation is that within the domain of the TMDfactorization the value of \(\chi ^2/d.o.f.\) is largely constant and starts to grow outside of this domain. Indeed, for the best models, the observed picture agrees with the expectation. In this way, we have shown that TMD factorization as it is, i.e. in the absence of Yterm, is capable of describing the data with \(q_T\lesssim 0.2\ Q\), i.e. \(\delta _T=0.2\). With some risk, one can prolong it to \(\delta _T=0.25\). After \(\delta _T=0.25\) the TMD factorization loses any agreement with the experiment. This analysis is unique, or at least we do not know any analog in the literature.
The fit and the plots of the data has been done with the help of arTeMiDe, version 1.1, available at [31]. This is a code package for the numerical evaluation of TMD distributions and related crosssections. It has a flexible structure and allows to consider an arbitrary combination of perturbative orders up to NNLO for coefficient functions and \(\hbox {N}^3\hbox {LO}\) for evolution factors. In the current version, it evaluates only unpolarized TMDPDFs, but we expect to update it for polarized cases and TMD fragmentation functions, as well as, to include the Yterm, in the future.
Notes
Acknowledgements
We thank Miguel G. Echevarria for the help on the initial stages of this work, and V. Bertone for useful comments on arTeMiDe code. I.S. also thanks Stefano Melis for discussions about some technical points. I.S. is supported by the Spanish MECD Grant FPA201453375C22P and FPA201675654C22P and the group UPARCOS.
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