# Single-diffractive Drell–Yan pair production at the LHC

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## Abstract

We present predictions for single-diffractive low-mass Drell–Yan pair production in *pp* collisions at the LHC at \(\sqrt{s}=13\) TeV. Predictions are obtained adopting a factorised form for the relevant cross sections and are based on a new set of diffractive parton distributions resulting from the QCD analysis of combined HERA leading proton data. We discuss a number of observables useful to characterise the expected factorisation breaking effects.

### Keywords

Fractional Momentum Hadronic Collision Muon Pair Reduced Cross Section Longitudinal Structure Function## 1 Introduction

The diffractive physics program pursued at the HERA *ep* collider in the recent past has substantially improved our knowledge on the dynamics of this class of processes. In the deep inelastic regime, the presence of a hard scale enables the derivation of a dedicated factorisation theorem [1, 2, 3, 4] which allows the investigation of the partonic structure of the colour singlet exchanged in the *t*-channel. From scaling violations of the diffractive Deep Inelastic Scattering (DDIS) structure functions, quite precise diffractive parton distributions functions (dPDFs) have been extracted by performing QCD analysis [5, 6, 7, 8] of available data.

With this tool available, factorisation tests have been conducted in order to investigate the range of validity of this hypothesis in processes other than DDIS. Factorisation has been shown to hold, as expected theoretically, in diffractive dijets production in DIS, where NLO predictions based on dPDFs well describe experimental cross sections [6, 9] both in shape and normalisation. Factorisation breaking effects are expected to appear in diffractive photoproduction of dijets due to the resolved component of the quasi-real photon. In such a case, however, H1 [9] reported a global suppression factor of data over NLO theory around 0.5 while ZEUS [10] found the same ratio compatible with unity. To date, these conflicting results prevent to draw a conclusive statement about factorisation in this case. We note, however, that the measurements of diffractive dijet photoproduction in ultraperipheral collisions in *pp* and *pA* collisions at the LHC [11] may offer an alternative way to settle this issue.

Complementary informations on the nature of diffraction has been provided by hard diffraction measurements in hadronic collisions. As theoretically anticipated in Refs. [1, 2, 3, 12, 13] and experimentally observed in \(p \bar{p}\) collisions at Tevatron [14, 15, 16], factorisation is strongly violated in such a case. In particular, predictions based on a factorised expressions for the relevant cross sections in terms of diffractive parton distributions extracted from HERA data overestimate hard diffraction measurements by a factor \({\mathcal {O}}(10)\) [17]. This conclusion persists even after the inclusion of higher order QCD corrections [18].

A rich program at the LHC is being pursued in diffractive physics by all Collaborations either based on the identification of large rapidity gaps (LRG) [19, 20, 21, 22, 23, 24] or by using dedicated proton spectrometers [25, 26]. Complementing Tevatron (\(\sqrt{s}=1.96\) TeV) results with forthcoming ones from the LHC at higher centre-of-mass energies (\(\sqrt{s}=8,13\) TeV) will give information on the energy dependence, if any, of the suppression factor, the so called rapidity gap survival (RGS) probability. Hopefully, they will allow to study its kinematic dependences, among which the one on the scale characterising the hard process appears to be particularly relevant. In the simplest scenario, it will be possible to clarify whether factorisation may still hold but revisited in a weak form through a global or local rescaling of diffractive PDFs extracted from DDIS and to study their degree of universality among different hard processes in hadronic collisions.

The purpose of the present paper is to present predictions for the single-diffractive Drell–Yan pair production at the LHC at \(\sqrt{s}=13\) TeV, one of the clean and simple measurable process in hadronic collisions. In such a process, the invariant mass of the lepton pair can easily be reconstructed and, depending on experimental capabilities, pushed to rather low values, allowing a detailed characterisation of the hard-scale dependence of the suppression factor. Although estimates of the latter are present in the literature for the specific process at hand [27, 28, 29], we take a conservative approach and avoid to introduce any suppression factor. We further assume factorisation to hold and adopt factorised expressions for the relevant cross sections. A preliminary set of, newly generated, diffractive parton distributions extracted from combined leading proton HERA data will be used for the calculation. In view of the expected factorisation breaking effects in hard, single-diffractive, measurements in hadronic collisions, the obtained values for the cross sections should be considered as upper bounds.

Given the exploratory nature of the analysis, more intended as a feasibility study, theoretical predictions are calculated to leading order accuracy. We take into account, however, the virtual photon decay into leptons so that cross sections can be studied as a function of, measurable, final state leptons kinematics. This allows us to explore the phase space available for the process and to estimate the impact of typical experimental cuts on the transverse momenta and rapidities of the leptons.

From QCD analyses performed in DDIS and anticipating the results of the next section, we know that the colour singlet exchanged in the *t*-channel is a gluon-enriched state. Since gluonic contributions to Drell–Yan production starts to \({\mathcal {O}}(\alpha _s)\) in perturbation theory, an accurate estimation of the suppression factor will require the inclusion of higher order corrections. The impact of the latter and a detailed report on the extraction of diffractive parton distributions to NLO accuracy will be presented in a companion publication.

The paper is organised as follows. In Sect. 2 we report in some details the extraction of diffractive PDFs from combined HERA leading proton data. In Sect. 3, making use of such distributions, we present results for single-diffractive Drell–Yan production in *pp* collisions at the LHC at \(\sqrt{s}=13\) TeV. In Sect. 4 we summarise our results.

## 2 FIT overview

*p*is detected in the final state. In Eq. (1)

*X*stands for the unobserved part of the hadronic final state and we indicate particles four-momenta in parentheses. In the

*lp*centre-of-mass system, diffractive DIS events are then characterised by outgoing protons with a large momentum fraction of the incident proton and quite small values of the transverse momentum measured with respect to the collision axis, i.e. in the target fragmentation region of the incident proton. The kinematic variables used to describe the DIS process are the conventional Lorentz invariants

*t*at the proton vertex:

*lp*cross section, \(\sigma _r^{D(4)}\), which depends on the diffractive transverse and longitudinal structure functions \(F_2^{D(4)}\) and \(F_L^{D(4)}\), respectively. In the one-photon exchange approximation, it reads

*i*runs on the flavour of the interacting parton. The hard-scattering coefficients \(C_ {ki}\) (\(k=2,L\)) are perturbatively calculable as a power expansion in the strong coupling \(\alpha _s\) and depend upon \(\mu _F^2\) and \(\mu _R^2\), the factorisation and renormalisation scales, respectively. The \(C_{ki}\) coefficient functions are the same as in fully inclusive DIS. Diffractive PDFs \(f_{i/p}^D(\beta ,\mu _F^2,{x}_{I\!\!P},t)\) appearing in Eq. (5) are proton-to-proton fracture functions [30] in the very forward kinematical limit and can be interpreted as the number density of interacting partons at a scale \(\mu _F^2\) and fractional momentum \(\beta \) conditional to the detection of a final state proton with fractional momentum \(1-{x}_{I\!\!P}\) and invariant momentum transfer

*t*. The

*t*-unintegrated diffractive PDFs appearing in Eq. (5) obey standard DGLAP [31, 32, 33] evolution equations [34]. The same statement holds when they are integrated over

*t*in a limited range [35]:

*t*, is integrated in the restricted range \(0.09< |t| < 0.55\) Ge\(\hbox {V}^2\) in order to minimise systematic uncertainties originating from

*t*-extrapolation of the various measurements outside their respective measured ranges. The reduced cross sections in Eq. (4) are integrated over

*t*in such a range and the diffractive PDFs in Eq. (6) are defined accordingly. For \({x}_{I\!\!P}<0.03\) the data set overlaps with high-statistics LRG data set and for \(0.03<{x}_{I\!\!P}<0.09\) it provides the best experimental information available on diffractive DIS cross sections. The combination procedure, in general, allows a reduction of the systematic uncertainties via cross-calibration of the various measurements. The direct detection of the forward proton allows one to avoid any systematics associated with the large rapidity gap selection. By definition, these data are free from the proton dissociative background which has been found to contribute around 23% of the diffractive DIS cross sections based on LRG selection [5]. Therefore this set of data provides the most precise knowledge about the absolute normalisation of diffractive DIS cross sections. These advantages, however, come at the price of increased uncertainties relative to LRG data given the reduced statistics of the sample. Diffractive parton distributions extracted form this data set will be used in the context of single hard diffraction in hadronic collisions in conjunction with ordinary parton distributions. In order to avoid any mismatch between inclusive and diffractive PDFs we adopt leading order CTEQL1 parton distribution set [37] evolved in the zero-mass variable-flavour-number scheme (ZM-VFNS). The evolution of diffractive PDFs is performed within the same scheme and to the same accuracy by using QCDNUM17 [38] program. The QCD parameters are the ones quoted in Ref. [37]. In particular we set the charm and bottom masses to \(m_c=1.3\) GeV and \(m_b=4.5\) GeV, respectively, and the strong coupling is evaluated at one loop setting \(\alpha _s^{n_F=5}(M_Z^2)=0.130\). In general factorisation theorem [1, 2, 3, 4] for diffractive DIS in the form of Eq. (5) holds at fixed values of \({x}_{I\!\!P}\) and

*t*so that the parton content of the colour-singlet exchange described by \(f_i^{D}\) is uniquely controlled by the kinematics of the outgoing proton. Therefore, at least in principle, dPDFs may differ at different values of \({x}_{I\!\!P}\) and

*t*. This idea has been successfully tested [7] in the analysis of LRG data from Ref. [5]. In the present context, given the limited number and accuracy of the data points in each \({x}_{I\!\!P}\) bin, we use a simpler approach, namely a fully factorised \(\beta -{x}_{I\!\!P}\) ansatz for the flavour-symmetric singlet and gluon diffractive parton (momentum) distributions defined at the initial scale \(Q_0^2\):

Left: Best-fit parameters. Right: Breakdown of \(\chi ^2\) contributions in each \({x}_{I\!\!P}\) bin

Parameter | \(p_i \pm \delta p_i\) | \({x}_{I\!\!P}\) | \(\chi ^2\) | Fitted points |
---|---|---|---|---|

\(f_0\) | \(-\)1.208 ± 0.022 | 0.00035 | 4.44 | 4 |

\(f_1\) | 48.2 ± 11.9 | 0.0009 | 6.78 | 10 |

\(f_2\) | 1.42 ± 0.13 | 0.0025 | 21.36 | 16 |

\(A_q\) | 0.0039 ± 0.0007 | 0.0085 | 20.34 | 24 |

\(B_q\) | \(-\)0.237 ± 0.026 | 0.0160 | 20.70 | 26 |

\(C_q\) | 0.5 | 0.0250 | 27.24 | 25 |

\(D_q\) | 22.6 ± 2.8 | 0.0350 | 13.85 | 24 |

\(E_q\) | 2.28 ± 0.20 | 0.0500 | 28.69 | 27 |

\(A_g\) | 0.057 ± 0.011 | 0.0750 | 13.10 | 26 |

\(B_g\) | 0.41 ± 0.13 | 0.0900 | 10.51 | 10 |

\(C_g\) | 0.5 | Total | 167.0 | 192 |

We have further performed two consistency checks detailed below. The first one concerns the diffractive longitudinal structure function which contributes starting from \({\mathcal {O}}(\alpha _s)\) and it is absent to the accuracy of the present calculation. Since its dominant contributions appear in the large-*y* region, the fit has been repeated with the cut \(y<0.5\) imposed. The second one addresses the issue, reported in previous analyses [5, 6, 7, 8], of the inclusion in the fit of the lowest \(Q^2\) points. For such a reason, the minimisation has been repeated by including only data points for which \(Q^2>6\) Ge\(\hbox {V}^2\). In both cases we observe a modest decrease in the \(\chi ^2/d.o.f\). However, as shown in Fig. 3, the resulting parametrisations are compatible, within uncertainties, with the ones obtained without imposing the cuts. Given the substantial stability of the results against variation of the phase space boundary of data included in the fit, we consider the “no cut” scenario as our default choice and use the corresponding best-fit parametrisations in the next section.

*t*-range, \(|t| < 1\) Ge\(\hbox {V}^2\), used in Ref. [5] to define the DDIS cross sections. A further upward normalisation shift is generated by the proton dissociative contribution which amounts to 23% of the DDIS cross section measured with the large rapidity gap method [5] and it is absent in the present analysis. In the left panel of Fig. 4, the comparison is performed at \({x}_{I\!\!P}=0.0025\) where both data sets overlap. In this case, the larger data sample in this \({x}_{I\!\!P}\)-bin used in the fit of Ref. [5] induce smaller experimental uncertainties (yellow band) on dPDFs with respect to the ones presented in this work. The virtue of the present parametrisations can be better appreciated for \({x}_{I\!\!P}>0.03\), being the latter the maximal \({x}_{I\!\!P}\)-value included in the analysis of Ref. [5]. In the right panel of Fig. 4, the comparison is therefore performed at \({x}_{I\!\!P}=0.05\). Diffractive PDFs from Ref. [5] are in a full extrapolation range and we observe a substantial increase of their uncertainties (green error band) driven by the theoretical errors associated with the parametrisations of the flux factor at large \({x}_{I\!\!P}\). On the contrary, our parametrisations, despite the lower statistical power of the data set used in this analysis, show a much reduced error. To conclude, we note that the singlet distribution from Ref. [5] is stable in shape and normalisation between the LO and NLO versions and shows the presence of a large-\(\beta \) bump in nearly the same position with respect to the one obtained in the present analysis. On the contrary we observe a steeper behaviour of the LO FitB gluon with respect to the NLO FitB one and to the one from our fits. This is probably due to the fixed flavour number scheme used in Ref. [5] which, by construction, induces a larger gluon distribution with respect to the one obtained with a variable-flavour-number scheme.

## 3 Single-diffractive Drell–Yan production

*z*is used to characterise final state hadrons and is defined by

*z*is just the observed proton energy, \(E_p^*\), scaled down by the beam energy, \(\sqrt{s}/2\). Hard diffractive events are then characterised by low values of the invariant \({x}_{I\!\!P}\) and

*t*, both in the same range of values as the one measured in DDIS.

In Eq. (8), we assume that the proton with momentum \(P_1\) is moving in the \(+z\) direction and the leading proton with momentum *P* is produced quasi-collinearly to \(P_1\) at large and positive rapidities. At the cross section level, diffractive parton distributions for the proton with momentum \(P_1\) will be used. The same process, of course, may occur also in the opposite emisphere and, since the hadronic initial state is symmetric, will be not considered here.

Outline of the muon-pair and proton phase space regions and the corresponding fiducial cross section for single-diffractive Drell–Yan pair production, \(\sigma ^{SD,DY}\), with associated experimental and theoretical errors

\(pp \rightarrow \mu ^+ \mu ^- \; p \; X\) | \(\sqrt{s}=13\) TeV |
---|---|

Muon-pair kinematics | \(|y^{\mu }|<2.45\) |

\(2<\hbox {M}_{\mu \mu }<20\) GeV | |

No cuts on muon \(p_t\) or \({\varvec{p}}\) | |

Proton kinematics | \(0.09<|t|<0.55\) Ge\(\hbox {V}^2\) |

\(10^{-4}<{x}_{I\!\!P}<10^{-1}\) | |

\(\sigma ^{SD,DY}\) | 1635 ± 60 (exp) \(\phantom {}^{+670}_{-460}\) (scale) pb |

*Y*, and difference \(\bar{y}\):

In the present analysis we focus on diffractive processes tagged with dedicated instrumentation [25, 26]. We choose the proton fractional momentum loss to be in the range \(10^{-4}<{x}_{I\!\!P}<10^{-1}\), with maximal overlap with the range measured at HERA [36]. Predictions presented in the following are integrated over the *t*-range of the data [36] out of which dPDFs are estracted, i.e. \(0.09<|t|<0.55\) Ge\(\hbox {V}^2\). We set the centre-of-mass energy of the *pp* collisions to \(\sqrt{s}=13\) TeV. The invariant mass of the muon pair is required to be in the range \(2<\hbox {M}_{\mu \mu }<20\) GeV, a range of virtualities in line with those measured at HERA. We assume that the \(J/\Psi \) and \(\Upsilon \) contributions, which both lie within this mass range, can be properly subtracted from the data sample. We require both muons to have rapidity \(|y^{\mu }|<2.45\) but we do not apply cuts either on the muons transverse or three momenta.

*Y*is shown in four different ranges of the pair invariant mass and integrated over \({x}_{I\!\!P}\). In all mass bins, the distributions show a maximum in the negative rapidity range, a signal that the interacting parton from the dissociated proton carries, on average, slightly more momentum with respect to the one originating from the scattered proton. In Fig. 8 we present single-differential distributions as a function of \({x}_{I\!\!P}\) in four different invariant-mass ranges. As the invariant mass increases, we observe a progressive flattening of the distributions at small \({x}_{I\!\!P}\). This effect is due to the phase space reduction induced by the constraint \(M_{\mu \mu }^2=\beta {x}_{I\!\!P}x_2 s\), which at low \({x}_{I\!\!P}\) disfavours the production of increasingly massive pair. In Fig. 9 we present single-differential cross section as a function of \(\beta \), the fractional momentum of the interacting parton with respect to the one of the colour singlet exchanged in the

*t*-channel, integrated in various bins of \(M_{\mu \mu }\) and \({x}_{I\!\!P}\). Such distributions offer insight in the sensitivity of the cross section to diffractive parton distributions, modulo kinematics effects. In the lowest \({x}_{I\!\!P}\) bin the distributions span all the allowed \(\beta \) range and progressively shrinks at large \(\beta \) as \({x}_{I\!\!P}\) increases, a natural consequence of momentum conservation. As already shown in Figs. 6 and 9, the distributions in the pair rapidity

*Y*are asymmetric around \(Y=0\). The asymmetry decreases both as the mass of the pair increases and as \({x}_{I\!\!P}\) increases. Such an effect is absent in the inclusive Drell–Yan case initiated by a symmetric initial state. This effect is better appreciated considering the absolute and relative asymmetries, \(A_a\) and \(A_r\), respectively, defined by

*R*of diffractive to inclusive cross sections

*Y*stands for the selected hard-scattering process (Drell–Yan in this case) and

*X*for the unobserved part of the final state. At Tevatron the ratio

*R*has been measured in a variety of final state [14, 15, 16] and it shows a quite stable behaviour with a value close to 1%. For the single-diffractive Drell–Yan production in

*pp*collisions at \(\sqrt{s}=13\) TeV, the ratio

*R*is presented in the right panel of Fig. 10. Given our leading order estimate of the inclusive Drell–Yan cross section,

*R*varies between 6 and 8% and decreases mildly as a function of the invariant mass of the pair, \(M_{\mu \mu }\). This prediction, however, does not take into account the RGS suppression factor. In this respect it would be interesting to check whether the data follow at least the shape of the ratio as a function of \(M_{\mu \mu }\).

## 4 Conclusions

In this paper we have considered the single-diffractive production of low-mass Drell–Yan pair in *pp* collisions at the LHC at \(\sqrt{s}=13\) TeV. Predictions are based on a fully factorised approach for the cross section which makes use of a set of diffractive parton distributions obtained from a QCD fit to combined leading proton DIS data from HERA. A number of distributions are presented both in terms of Drell–Yan pair and scattered proton variables. Examples of asymmetries and ratio are constructed in order to minimise theoretical and experimental uncertainties. In view of the foreseen measurements of this type of process at the LHC Run-II, these results constitute a baseline for the characterisation of the expected factorisation breaking effects.

## Notes

### Acknowledgements

This work is supported in part through the project “Hadron Physics at the LHC: looking for signatures of multiple parton interactions and quark gluon plasma formation (Gossip project)”, funded by the “Fondo ricerca di base di Ateneo” of the Perugia University. We warmly thank Marta Ruspa for numerous discussions on the LHC measurements and for providing us details of the analysis presented in Ref. [36]. We also thank Sergio Scopetta for reading the manuscript before submission.

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