# Pion nucleus Drell–Yan process and parton transverse momentum in the pion

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## Abstract

We present a thorough analysis of unpolarized Drell–Yan (DY) pair production in pion–nucleus scattering. On the nucleus side, we use nuclear parton distributions along with parametrisations of the nucleon partonic transverse distribution available in the literature. Partonic longitudinal and transverse distributions of the pion are those obtained in a recent calculation in a Nambu–Jona Lasinio (NJL) framework, with Pauli–Villars regularization. The scale of the NJL model is determined with a minimisation procedure comparing NLO predictions based on NJL evolved pion distributions to rapidity differential DY cross sections data. The resulting distributions are then used to describe, up to next-to-leading logarithmic accuracy, the transverse momentum spectrum of dilepton pairs up to a transverse momentum of 2 GeV. With no additional parameters, fair agreement is found with available pion–nucleus data, confirming the virtues of the NJL description of pion parton structure. We find sizable evolution effects on the shape of the distributions and on the generated average transverse momentum of the dilepton pair. We furthermore discuss the possibility of gaining information about the behavior of the pion unpolarized transverse momentum dependent parton distribution from pion nucleus DY data.

## 1 Introduction

The non perturbative transverse structure of hadrons has attracted recently much attention and the issue of extracting transverse momentum dependendent parton distributions (TMDs) from data taken in different processes in present and forthcoming high-luminosity facilities represent**s** an important goal of nowadays hadronic Physics. In particular, Drell–Yan (DY) pair production [1], discussed in this paper, and semi-inclusive deep inelastic scattering are the main processes under investigation [2].

The cross section for DY pair production, differential in the transverse momentum of the pair, \(q_T\), is a particularly suitable observable for this kind of studies. In particular at small \(q_T\), where the TMD formalism is formulated, fixed order calculation of this process show large logarithmic corrections due to an incomplete cancellation of soft and collinear singularities between real and virtual contributions and need to be resummed to all orders to recover the predictivity of the theory [3, 4, 5].

The description of the \(q_T\) DY spectrum in *pp* collisions has reached a high degree of sophistication [6]. On one side, theoretical improvements have increased the perturbative accuracy of the predictions [7, 8, 9, 10, 11]. On the other side, global fits of DY production at different energies have given access to the non perturbative proton transverse structure [12, 13]. Both aspects have received increasing attention due to the formalisation of new and old concepts in the TMD language [14, 15, 16, 17]. While there are differences between the language used in the modern and the older TMD approaches, physical results should not depend on it. A detailed comparison of the formalisms can be found in Refs. [18, 19].

At high energy colliders, this improved knowledge aims to an increasingly better description of electroweak bosons production, with the Higgs \(q_T\) spectrum being the highlighted case. Measurements of \(q_T\) spectrum of the DY process, at lower centre of mass energies, are instead more sensitive to the hadronic non perturbative transverse structure.

DY pair production in pion–nucleus scattering is a unique probe of pion parton distribution functions (PDFs) and, as such represents a source of information on the pion parton structure. In particular for the \(q_T\) spectrum this was realized long time ago by the authors of Ref. [20]. More recently, phenomenological analyses have appeared [21]. A fit to the \(q_T\) spectrum of DY pairs produced in pion–nucleus collisions has been recently presented in Ref. [22].

Pion TMDs, which could be extracted in principle in a next generation of pion–nucleus DY experiments [23], have received recently considerable theoretical interest [21, 24, 25, 26, 28, 29, 30, 31]. In this paper we study the DY unpolarized pair production in pion–nucleus scattering, to next-to-leading logarithmic (NLL) perturbative accuracy, up to a transverse momentum of the produced lepton pair of 2 GeV. As non perturbative inputs, we use, for the bound nucleons, a longitudinal structure which takes into account nuclear effects and, for the transverse structure, a well established parameterization obtained through a phenomenological fit to proton–proton DY data (called, from now on, KN05 prescription) [12]. For the pion, we use TMDs obtained in a recent calculation [30], within a Nambu–Jona Lasinio (NJL) framework [32], with Pauli–Villars regularization. The corresponding RGE scale of the model is determined in a novel way by comparing the DY unpolarized cross section, integrated over \(q_T\), described by evolved pion PDFs evaluated in the NJL model to the data.

The aim of the present paper is to study the performances of the NJL model, widely used to describe the non-perturbative meson structure, against DY differential cross section data for the first time. We also analyze to what extent this process can be used to obtain information on the pion transverse structure in momentum space, as it happens for the proton in the corresponding process.

The paper is structured as follows. In the next section, we present the set-up of the calculation and introduce the ingredients used to describe the proton and pion structure. In the third section, we discuss the results of the calculation of DY cross sections in the kinematics of presently available data for pion-tungsten scattering. Eventually, we draw our conclusions in the last section.

## 2 Setting-up the calculation

### 2.1 Drell–Yan cross section

*n*the order of the perturbative calculation. These large logarithmic corrections can be resummed to all orders by using the Collins-Soper-Sterman (CSS) formalism [6]. In this limit, of interest for the present analysis and neglecting finite corrections in the \(q_T\sim Q\) region, the cross-section can be written as

*a*,

*b*indices run on quark and gluons, \(J_0(b \, q_T)\) is the Bessel function of first kind and \(f_{i/h}\) corresponds to the distribution of a parton

*i*in a hadron

*h*. The cross section in Eq. (2) is differential in \(\tau =Q^2/s\) and

*y*, the rapidity of the DY pair. Momentum fractions appearing in parton distribution functions can be expressed in terms of these variables as

*y*by a suitable transformation. By defining \(A=\sqrt{x_F^2 + 4 \tau }\) one gets

*b*-space in the Sudakov perturbative form factor

*A*,

*B*in Eq. (6) have perturbative expansions in \(\alpha _s\),

### 2.2 Proton structure

*pp*) and proton–nucleus (

*pA*) collisions. Quite recent analyses [15, 16] have appeared which address such an extraction. Since our aim here is to establish the possibility of studying the pion transverse non perturbative structure in pion–nucleus DY experiments, we here intend to minimize the uncertainity coming from the proton structure part of the calculation. We use the well known and widely accepted results of Konychev and Nadolsky (KN05) [12] obtained within the CSS formalism [6] where \(S_{NP}^{pp}(b)\) is extracted from global fit to

*Z*-boson and low mass DY data, updating the results presented in Ref. [13]. The latter is parametrised as

*b*, region both in the Sudakov form factor, Eq. (6), and the parton distributions is given. The authors of Ref. [12] adopt the so-called \(b_\star \)-prescription, substituting

*b*with

*Q*accessible is \(Q_{in}=1.3\) GeV, the \(b_\star \)-prescription entering PDFs calls is used with \(b_{max}=b_0/Q_{in} \simeq 0.86 \, \text{ GeV }^{-1}\) which always guarantees \(\mu _F>Q_{in}\). It is important to remark that the non perturbative form factor is determined not only by fitting the parameters of the chosen functional form, but also by the specific regularisation prescription and its associated parameters adopted to deal with the infrared region. In general all these ingredients have been found to be highly correlated.

In order to present a benchmark of our code and to gauge how theory performs in extrapolation regions, we compare predictions from KN05 to the *pA* data of Ref. [37]. An additional \(\pm 25 \%\) normalisation error is assigned to the data [37]. In the original KN05 analysis, only the data at \(p_{lab}=400\) GeV, \(q_T<1.4\) GeV, \(5<M/\text{ GeV }<9\) were included in the fit. In such a restricted region indeed the theory (solid lines) performs well offering a good benchmark of our code, as shown in the first row of Fig. 1. Since the \(\pi W\) data to be analyzed in the following are at \(p_{lab}\)=252 GeV, it is important to check how well the theory performs in extrapolation regions at lower \(\sqrt{s}\) and higher DY rapidity. Therefore we present in the second and third rows of Fig. 1 the KN05 benchmark (dashed lines) versus data [37] at \(p_{lab}\)= 200 and 300 GeV, which were not included in the KN05 fit. By using Eq. (4) and Eq. (5) and assuming the invariant mass values indicated on the plots, the rapidity coverage of these data can be converted to the range \(0<x_F<0.3\). In both cases we find good agreement between data and theory up to \(q_T \sim 2\) GeV giving us confidence that the KN05 model can be successfully used in this (\(x_F,q_T\)) range at the \(\sqrt{s}\) of interest in this analysis.

### 2.3 Pion structure

A calculation of pion TMDs in a NJL framework, with Pauli–Villars regularisation, has been recently presented in Ref. [30]. Model calculations of meson partonic structure within this approach have a long story of successful predictions [38, 39, 40, 41, 42, 43]. Collinear parton distributions obtained within a model have to be associated to a low momentum scale \(Q_0^2\) and, in order to be used to predict measured quantities, have to be evolved to higher momentum scales according to perturbative QCD (pQCD).

*q*(

*x*) is properly recovered with correct normalisation and the momentum sum rule is exactly satisfied. This is due to the fact that NJL is a field theoretical scheme and the correct support of the PDF, \(0\le x\le 1\), is not imposed but arises naturally. In particular, the momentum sum rule reads \(\int dx~x~q\left( x\right) =0.5\), i.e. the fraction of momentum carried by each quark is one half of the total momentum, since at the scale of the model only valence quarks are present. The dependence on \(\varvec{k_T}\) of the TMD obtained in Ref. [30] is very important for the present study. It is worth stressing that, in this approach, the \(\varvec{k_T}\) dependence is automatically generated by the NJL dynamics and it is not imposed by using any educated guess. This is an important feature of the results of Ref. [30], not found in other approaches [21, 27]. In this paper we will use the pion TMD obtained in Ref. [30] in the chiral limit, the latter allowing for a factorisation of the

*x*and \(k_T\) dependence at the low but undetermined scale \(Q_0^2\) associated to the model:

*T*is given by

**,**behave

**s**as \(k_T^{-6}\) for asymptotic values of \(k_T=|\varvec{k_T}|\) and satisfies the normalisation

*x*-dependence (see, e.g., Ref. [46]). We have therefore used the expression valid in the chiral limit to avoid further irrelevant complications in an evaluation which is already rather involved.

^{1}In the present paper we use a different strategy: we consider \(Q_0^2\) a free parameter of the NJL model which is then fixed with a minimisation procedure, outlined in the following, of the theoretical \(\pi ^- W\) DY cross sections, differential in \(\sqrt{\tau }\) and \(x_F\), against the corresponding experimental ones [44]. Theoretical cross sections are calculated according to

*Z*-boson mass, \(M_Z\). Since the data we are comparing to are obtained on a tungsten target, we take into account nuclear effects by using nuclear PDFs of Ref. [51]. We have carried out a \(\chi ^2\) study to establish the hadronic scale of the model that describes the best the data at NLO in pQCD. Two cases have been considered: an evaluation of the \(\chi ^2\) for the full range of \(x_F\) and another one with a cut \(x_F<0.4\), since the NJL model is expected to better reproduce the pion valence distributions, expected to populate the range of large and positive \(x_F\) . The scales thus determined are

*x*with respect to NLO ones. This

**,**in turn

**,**implies softer pion PDFs at large

*x*. In the present context

**,**this fact would imply a scale \(Q_0^2\) for the NJL model lower than the one already determined by using NLO Wilson coefficients in Eq. (20).

## 3 Predictions for \(\pi W\) collisions data

*M*nor on parton fractional momenta. Such a comparison is meaningful at the typical scale for which the transverse form factors and the longitudinal momentum part factorize. For the pion case this happens at the scale \(Q_0^2\) determined in the previous section. For the proton TMD such a scale is ambiguously defined and, according to KN05 analysis, ranges between \(Q_{in}^2\) and \((b_0/b_{max}^{KN05})^2\). Therefore we choose \(M=Q_{in}=1.3\) GeV in Eq. (12) and fix the product \(x_1 x_2=M^2/s\), see Eq. (4), exploiting the \(\pi ^- W\) kinematics with

*s*calculated according to a beam energy of \(p_{lab}=252\) GeV. This comparison is presented in Fig. 3, where our result for the pion non-perturbative form factor, \(S_{NP}^{\pi }(b)\), is also compared to the parametrisation of the non-perturbative pion form factor of Ref. [22] (called hereafter WLS). The approach of Ref. [22] is rather different from ours, both in the spirit and in the physical ingredients used. As a matter of fact, in that paper the proton non-perturbative form factor, \(S_{NP}^{pp}(b)\), has a structure similar to that of our Eq. (12) and for the pion the same form has been assumed, with the corresponding parameters obtained from a fit of the same cross section data used in the present paper. As a result, the pion non-perturbative form factor depends on both the hard scale and the parton momenta. A fit is then performed up to \(q_T\simeq 3\) GeV. We reiterate that our goal here is not to fit but rather to assume a well known structure for the proton non-perturbative form factor and to test the pure NJL predictions for the pion against the data. The purpose of the comparison with the WLS parametrisation is therefore mainly illustrative and quantitative conclusions can be hardly reached. All the distributions presented in Fig. 3 reduce to unity in the \(b \rightarrow 0\) limit, since they are all normalised to unity in transverse momentum space. In the top panel of Fig. 3 we compare the NJL transverse distribution to the pion parametrisation of Ref. [22] (called hereafter WLS) obtained from a fit of the same cross section data used in the present paper. One may notice that, for this model, the width of the distribution is smaller with respect to the NJL one, implying a larger average transverse momentum. In the bottom panel of Fig. 3 one may notice that the NJL pion transverse distribution develops a larger tail with respect to the gaussian drop of the proton distributions. Moreover the

*b*-space width of the KN05 proton with \(M=1.3\) GeV is larger with respect to the pion one. When transformed back in \(k_T\) space, this implies that the intrinsic transverse momentum in the pion is larger than the one in the proton, in agreement with the general expectations, since the pion is a much smaller system with respect to the proton. It is worth mentioning that both the KN05 and WLS non perturbative form factors have an explicit, althought slightly different, dependence upon the hard scale

*M*, in both cases set equal to the invariant mass of the dilepton pair. Therefore we plot in each panels, as a representative case, the curves corresponding to both form factors evaluated with the scale set to \(M=4\) GeV. Comparing the latter curves to the ones with \(M \sim 1\) GeV, we conclude that the

*M*-dependence generates a sizable non perturbative evolution of the form factor which is more pronounced for KN05 proton model than for the WLS pion model.

*M*. The comparison is performed up to a \(q_T \sim 2\) GeV, where we have checked that the KN05 gives an adequate description of

*pp*data. All three different predictions, to be discussed in the following, capture the normalisation of the data and share a tendency to slightly overestimate the data at very small \(q_T\) and to underestimate them at larger \(q_T\). This effect progressively disappears increasing the mass of the lepton pair. Comparing the two curves corresponding to \(b_{max}^{NJL}= 2.44\) Ge\(\text{ V }^{-1}\) and \(b_{max}^{NJL}= 1.5\) Ge\(\text{ V }^{-1}\), one may notice a substantial stability upon variation of the regulators on the pion side. On the same plot, in order to investigate the sensitivity to the pion transverse distribution, we additionally show the predictions obtained by substituting the pion transverse factor, Eq. (19), with \(\sqrt{S_{NP}^{pp}(b)}\). As shown in Fig. 3, the non perturbative transverse distributions for the proton and pion differ at low scales. The corresponding curve, indicated with

*pp*on the plot, is barely distinguishable from the other two. Such a comparison supports the hypothesis that the effect of the perturbative evolution, driven by Eq. (23), is to wash away differences in the non perturbative structure found at the hadronic scale. This result implies a reduced sensitivity to non perturbative structure. A quantitative analysis of the results yields that the \(\chi ^2/d.o.f.\) of the shown distributions slowly decreases from values of order 4 to values of order 1 with increasing

*M*from 4 GeV to 13 GeV.

^{2}These numbers are remarkably good if one considers that we are presenting pure model predictions without a fitting of parameters

*a posteriori*. Besides, we observe that the agreement with data of the NJL distributions is slightly better than those obtained with a proton-like non-perturbative form factor for the pion, in any

*M*bin, and, more importantly, that the difference in \(\chi ^2/d.o.f.\) reaches 30 % for the lowest values of

*M*. The region of low (but still perturbative)

*M*is therefore selected as the most promising to access non-perturbative details of the pion transverse structure.

We proceed our discussion presenting in Fig. 5 the comparison between theory predictions against the same data, now integrated in the mass range \(4<M<8.55\) GeV in a number of \(x_F\) bins. We remind the reader that we have verified that the KN05 model gives a satisfactory description of *pp* data up to \(x_F \sim 0.3\). Up to this \(x_F\) value, as shown in the first row of Fig. 5, the description \(\pi ^-W \) data is fair, as already observed in Fig. 4. Beyond that range, however, the width of the theoretical curves decreases more rapidly than observed in the data, with data substantially undershooted beyond \(q_T \sim 1\) GeV. This effect is more pronounced as \(x_F\) increases. In this region of relatively large pion fractional momenta it would be tempting to invoke, in order to describe the data, an *x*-dependent non perturbative structure. Such an interesting hypothesis, however, cannot be tested unless fixed order contributions at finite \(q_T\) are included in the calculation. Moreover, given our working assumptions, the failure to agree with data at large values of \(q_T\) was somehow expected. Whether it is due to the breakdown of the \(x-k_T\) factorization, the inclusion of a more complex NP Sudakov form factor or the matching with the so called Y-term, further studies shall be pursued to answer that question.

**s**are minimal, implying a reduced sensitivity to details of the non perturbative transverse factor. Therefore if one aims to better appreciate the strictly non perturbative form factor, one has to confine in corners where TMD evolution is minimised, but still in a perturbative range. These phase space regions can be identified by extrapolation from the right plot as the one at the lowest, but still perturbative, values of the invariant masses of the pair, as already noticed above while discussing Fig. 4.

## 4 Conclusions

A thorough analysis of DY pair production in pion–nucleus scattering has been presented. The main goal of our work has been the test of model predictions, obtained within the Nambu–Jona–Lasinio model for the transverse pion structure. In particular we have focused on the study of differential transverse momentum spectra of DY pairs produced in *pA* collisions calculated in the CSS framework at NLL accuracy borrowing from the literature the longitudinal and transverse proton structure. The pion is treated in the Nambu–Jona–Lasinio model. No further assumption has been made: even the momentum scale associated to the model is obtained via a minimization procedure of NLO theory to DY experimental longitudinal spectra. The latter turns out to be a low one, in line with that normally used, which could be predicted within the spirit of the model without fitting “a posteriori”. The agreement found between our pion–nucleus theoretical cross sections and experimental data is rather successful, confirming the predictive power of the NJL model, for both the longitudinal pion parton distributions and its transverse structure. We notice that the theory tends to systematically undershoot the data on the higher end of the considered \(q_T\) interval. All interpretations of this effect, however, are not conclusive without the inclusion of the finite, fixed order, contributions which populate the \(q_T \sim Q\) region and are neglected in our calculation.

The possibility to distinguish between different non perturbative transverse momentum distributions in DY data appears instead more questionable. In this complicated scenario, a possible strategy would be the measurement of DY pion–nucleus \(q_T\)-spectra, in bins of \(x_F\), at low values of the mass of the pair, as the present study suggests to look into this kinematical window to emphasize the non-perturbative content of the pion. Further analyses of the pion non-perturbative form factor, as a function of the hard scale, should be pursued so we could progress on that point. In the very same window, new data could allow a deeper investigation of the dependence of the non perturbative form factor upon the hard scale of the process.

## Footnotes

## Notes

### Acknowledgements

We thank P. Nadolsky for a useful mail exchange about details of the fit presented in Ref. [12]. This work was supported in part by the Mineco under contract FPA2016-77177-C2-1-P, by GVA-Prometeo/II/2014/066, by the Centro de Excelencia Severo Ochoa Programme grant SEV-2014-0398 and by UNAM through the PIIF project Perspectivas en Física de Partículas y Astropartículas. F.A.C. and S.S. thank the Department of Theoretical Physics of the University of Valencia for warm hospitality and support; F.A.C, A.C. and S.N. thank the INFN, sezione di Perugia, and the Department of Physics and Geology of the University of Perugia for warm hospitality and support.

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