Disentangling radiative corrections using high-mass Drell-Yan at the LHC
DDisentangling radiative corrections using high-mass Drell-Yan atthe LHC
Radja Boughezal, ∗ Ye Li, † and Frank Petriello
1, 3, ‡ High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA SLAC National Accelerator Laboratory,Stanford University, Stanford, CA 94309, USA Department of Physics & Astronomy,Northwestern University, Evanston, IL 60208, USA
Abstract
We present a detailed numerical study of lepton-pair production via the Drell-Yan process abovethe Z -peak at the LHC. Our results consistently combine next-to-next-to-leading order QCD cor-rections and next-to-leading order electroweak effects, and include the leading photon-initiatedprocesses using a recent extraction of the photon distribution function. We focus on the effectsof electroweak corrections and of photon-photon scattering contributions, and demonstrate whichkinematic distributions exhibit sensitivity to these corrections. We show that a combination ofmeasurements allows them to be disentangled and separately determined. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - ph ] D ec . INTRODUCTION The ATLAS and CMS experiments are producing measurements of kinematic distribu-tions with cross sections spanning orders of magnitude, and in which all bins have experi-mental errors approaching the percent level. These results allow for unprecedented detail incomparisons of Standard Model (SM) theory with experiment. Perhaps the most strikingexamples of these measurements are those of the Drell-Yan spectra from 7 TeV pp collisionsat both ATLAS and CMS [1, 2]. Both experiments determine the invariant-mass spectrum oflepton pairs from 15 GeV (CMS) or 116 GeV (ATLAS) through 1.5 TeV. The CMS collabo-ration further bins their measurement according to the dilepton rapidity. The extraordinarylever arms provided by these data sets, and the degree to which systematic errors can becontrolled, open unique windows onto radiative corrections in the SM, and into the structureof the proton. Higher-order QCD corrections are crucial in bringing SM into agreement withthese measurements. The Drell-Yan distribution has also been helpful in determining theparton distribution function (PDF) of the photon inside the proton [3].Future measurements of the Drell-Yan spectrum at a 14 TeV LHC will access an evenlarger kinematic range. Along with this larger phase space comes an increased sensitivityto a host of effects. In addition to QCD corrections through next-to-next-to-leading order(NNLO), electroweak Sudakov corrections [4–9] become increasingly more important at high-energy colliders, as has been emphasized recently in the literature [10]. Photon-initiatedcorrections also increase in importance at high energies, as has been recently emphasizedfor both Drell-Yan and W -pair production processes [11, 12]. The ATLAS collaboration hasperformed detailed studies showing the importance of these effects for a host of measurementsin the Drell-Yan channel [13]. One view of these corrections is that they represent additionalsources of theoretical uncertainty which must be sufficiently controlled in order to performinteresting measurements. For example, the measurement of high-mass W W scatteringwill play a central role in Run II of the LHC, as it directly probes whether the discoveredHiggs boson completely unitarizes the SM, or whether additional particles beyond those sofar discovered are needed. Sufficient understanding of electroweak (EW) corrections andphoton-initiated processes, in addition to the usual QCD corrections, will be needed tointerpret these measurements. The experimental control over the Drell-Yan channel allowssuch theoretical effects to be precisely validated before being applied to other processes.Another viewpoint is that the determination of these corrections is interesting in its ownright, and that the experimental control over the Drell-Yan channel at the LHC offers aunique laboratory in which to study in detail the higher-order perturbative structure of theStandard Model. For both of these reasons, the study of high-mass Drell-Yan productionwill be of great interest in Run II of the LHC.A wealth of theoretical information is available for the Drell-Yan process. QCD correc-tions up to the next-to-next-to-leading order (NNLO) in the strong coupling constant havebeen previously calculated, both for the inclusive cross section [14] and for differential quan-tities [15–20]. The NLO EW effects are known [11, 21–25]. Our goal in this manuscript isto use this knowledge to determine how to separately measure the electroweak and photon-initiated corrections affecting the Drell-Yan process. We exhaustively study the availablekinematic distributions which can be measured, at both a 8 TeV and a 14 TeV LHC. We2how that a combination of several differential measurements in high-mass Drell-Yan leptonpair production at a 14 TeV run of the LHC allows one to disentangle the effect of the vari-ous corrections. Although both the electroweak Sudakov logarithms and the photon-initiatedcontributions increase with lepton-pair invariant mass, they affect other distributions in dis-tinct ways. We estimate the observability of these deviations and show the most sensitivephase-space regions by constructing χ distributions that account for both statistical errorsand imprecise knowledge of quark and gluon distribution functions. We also study how dif-ferent choices of basic acceptance cuts on the leptons affect this analysis. For our numericalstudies we use the latest version of FEWZ [20], which consistently combines NNLO QCD cor-rections with both NLO electroweak effects and the leading photon-initiated processes. Forinclusive observables away from phase-space regions in which hadronic radiation is restricted,which form the vast majority of the results presented here, this fixed-order approach repre-sents the appropriate framework in which to perform this study. Near kinematic boundaries,a combination of EW corrections with a QCD parton shower represents a more appropriateframework [26, 27]. We point out where we expect fixed-order perturbation theory to breakdown when presenting our results. We summarize the main conclusions of our study below. • The most sensitive observable to the photon distribution function is the low end ofthe lepton transverse momentum ( p T l ) distribution, due to the underlying t -channelsingularity of the corresponding matrix elements. At lower invariant masses the EWcorrections must be under good control in order to extract the photon PDF, as thetwo effects strongly cancel. The photon-initiated processes increase more quickly withinvariant mass, reducing the effect of this cancellation and making the high-mass, low- p T l region an ideal place from which to determine the photon PDF. This distributionis also expected to be less sensitive to potential new physics affecting the high-massDrell-Yan tail, as the decay of a heavy object will generally populate higher p T l bins.Use of p T l will therefore also help disentangle new physics from the Standard Model. • Although the low- p T l region is most sensitive to the photon PDF, there is no benefitto reducing the experimental lepton p T l requirement unless the pseudorapidity ( η l )constraints can also be loosened, due to the kinematics of the underlying process.Relaxing the pseudorapidity cut on the leptons does enhance the observable deviationsfrom photon-initiated processes. • The central region of dilepton rapidity ( Y ll ) is sensitive to the photon PDF, indicatingthat a measurement of the three-dimensional distribution in Y ll , p T l and invariant massshould be a goal for the 14 TeV run. • The electroweak corrections are largest in the high-mass, central η l region of phasespace, due to the underlying angular dependence of the Sudakov logarithms. Onceenough integrated luminosity is collected this distribution offers a window into thestructure of the electroweak radiative corrections.We view our results as an atlas of radiative corrections that can help guide the 14 TeVexperimental study of the Drell-Yan process. Other detailed studies of Drell-Yan productionat the LHC exist [11]. We extend upon this important work in several ways: we more3xhaustively study the invariant-mass dependence of the available kinematic distributions;we control higher-order QCD corrections and uncertainties through the use of the full NNLOQCD corrections, which we show to be crucial in determining the other effects of interest;we estimate the observability of the various corrections using detailed and up-to-date errorestimates; we use the latest results on the photon PDF that are informed by Run I LHCmeasurements; finally, we study the effect of varying the experimental acceptance cuts onthe leptons.Our paper is organized as follows. We present our notation and setup in Section II.Numerical results for an 8 TeV and a 14 TeV LHC are presented in Section III. Finally, weconclude in Section IV. II. SETUP
We describe here the parameters and framework we employ in our study. All numerical re-sults presented are obtained with the program
FEWZ [17, 19, 20], which consistently combinesNNLO QCD corrections with NLO electroweak corrections and the leading photon-initiatedprocesses. The EW corrections and photon-initiated contributions, which are the focus ofour study, have been extensively validated against the literature in Ref. [20], and we donot repeat that comparison here. We use the recent NNPDF 2.3 PDFs [3], which consis-tently include QED corrections and allow for an initial-state photon distribution function,at NNLO in QCD perturbation theory. This is the most recent PDF set which allows foran initial-state photon, and is the only one in which the photon PDF has been constrainedby data within a global fit. The MRST 2004 PDF set [28] also allows for a photon PDF,with its form given primarily by a model parameterization. Another possible approach toincluding the photon PDF would be to use the MRST 2004 set for the photon, while usinga more up-to-date set for the quark and gluon PDFs [29]. We use the G µ scheme as ourelectroweak input scheme, which is known to reduce the size of higher-order electroweakcorrections [11]. In a fixed-order calculation, the same input scheme must be chosen for realand virtual corrections to preserve the cancellation of infrared singularities, meaning that weuse this same scheme for the calculation of the real photonic corrections. Another possibilityis the α (0) scheme. These two choices differ by uncalculated, and most likely small, O ( α )terms. Both are of course completely consistent to the order we are working.We write the cross section in the schematic form σ full = σ NNLO QCD + ∆ σ NLO EW + ∆ σ γ , (1)where σ NNLO QCD contains the leading-order result together with the higher-order QCD cor-rections, ∆ σ NLO EW contains the one-loop electroweak radiative corrections, and ∆ σ γ denotesthe contribution from the leading-order γγ → l + l − process (we will refer to this contributionas ‘photon-induced’ or ‘photon-initiated’ in our work). All three pieces are evaluated usingthe same NNLO PDFs. For further details on the combination of these contributions, werefer the reader to Ref. [20]. We note that we have neglected exclusive and single-dissociativeprocesses, where respectively both or one of the two incoming protons remains intact. Also, qγ scattering processes are known to slightly reduce the size of the photon-initiated contri-butions [11]. We have neglected processes containing real emission of W or Z bosons, which4artially cancel the effects of the EW virtual corrections. Such corrections lead to dibosoncontributions to the cross section. How, and whether or not, they should be included in thetheoretical prediction of the signal depends on the experimental analysis. For example, boththe ATLAS and CMS Drell-Yan measurements explicitly model such diboson backgrounds,where both bosons decay leptonically, separately [1, 2], indicating that this contributionshould not be added to the Drell-Yan signal studied here. The effect of including real-bosonemission with subsequent decay to either jets or neutrinos was studied in Ref. [30]. Thenumerical impact of these additional corrections was to reduce the size of the electroweakcorrections by a few percent in the high-invariant mass tail.We have investigated two different choices of renormalization and factorization scale: adynamical scale set equal to the invariant mass of the produced lepton pair, and a fixed scaleequal to the geometric mean of the upper and lower boundaries of the invariant mass binunder consideration. The two choices give almost identical results, and are indistinguishablein the plots we present. We show the dynamic scale choice. Since the scale uncertaintiesare in general smaller than the PDF and statistical errors, we do not consider them further.In later sections we consider the use of the difference between the NLO and NNLO QCDpredictions as a conservative estimate of the uncertainties coming from uncalculated QCDcorrections. We compute the 1 σ PDF errors on σ full using the procedure suggested by theNNPDF collaboration [31]. When forming our χ function we do not include the uncertaintiesfrom the photon distribution function, as one purpose of our study is to identify distributionsfor which we can control other sources of error reliably enough to extract it. We display thephoton PDF uncertainty separately on our plots.We impose the following basic acceptance cuts on the final-state leptons: p T l >
20 GeV , | η l | < . . (2)In the final section we study the effects of changing these constraints. Photons satisfying (cid:112) ( φ l − φ γ ) + ( η l − η γ ) < . M ll ∈ [0 . , .
2] TeV , M ll ∈ [0 . , .
5] TeV , M ll ∈ [0 . ,
1] TeV . (3)For 14 TeV collisions, we add on another high-mass bin: M ll ∈ [0 . , .
2] TeV , M ll ∈ [0 . , .
5] TeV , M ll ∈ [0 . ,
1] TeV , M ll ∈ [1 ,
3] TeV . (4)These choices are meant to illustrate the behavior of the various corrections as the invariantmass is changed. We note that we have also studied the region M ll ∈ [3 ,
14] TeV at a 14 TeVmachine, and have found that the event rates are too small to allow discrimination betweendifferent effects until at least 3000 fb − is reached.In each invariant mass region we study the following three distributions: the dileptonrapidity Y ll , the lepton pseudorapidity η l , and the lepton transverse momentum p T l . Wenote that the lepton means the negatively-charged lepton; to avoid too great a proliferation5f plots we do not show the anti-lepton distributions. In the final section we also study dis-tributions for the harder and softer leptons, ordered in p T . We have also studied the dileptontransverse momentum distribution p T ll , but have found that this distribution does not helpin distinguishing between the various radiative corrections we are considering. The reasonfor this is clear: the photon-initiated contributions contain only an l + l − pair in the finalstate, and therefore populate only the p T ll = 0 bin in our calculation. Higher-order correc-tions with an additional photon radiated will not significantly change this conclusion, sincethese effects are suppressed by α , unlike QCD radiation which contributes as α s . Similarly,QED radiation effects only minimally shift the electroweak corrections away from p T ll = 0.Therefore, all deviations in p T ll are concentrated near the origin, and this distribution doesnot help disentangle the various higher-order effects. The impact of multiple photon radi-ation on several distributions was studied in Ref. [11], and found to be much smaller thanthe effects we are considered here.To denote a cross section restricted to a given invariant mass bin, and to a bin in anotherkinematic variable, we will generically use the notation σ x ( i ), which is shorthand for thefollowing expression where the bin boundaries are explicitly written as arguments of thebin-integrated result: σ x ( i ) ≡ σ x (cid:0) [ M downll , M upll ] , [ v down , v up ] (cid:1) , (5)where x = full , NNLO QCD , NLO EW , γ . v denotes either the absolute value of the dileptonrapidity or the lepton rapidity, or the lepton transverse momentum. We only study restric-tions in a single variable v at a time in this manuscript, as it makes our results simpler tovisualize. III. NUMERICAL RESULTS
We will show numerical results for distributions for both 8 TeV and 14 TeV pp collisions,for the invariant mass regions defined in Eqs. (3) and (4). In order to determine the observ-ability of the various deviations induced by photon and electroweak effects, we will comparethem to the estimated statistical and PDF errors. The scale variation coming from miss-ing QCD corrections has been studied previously, including in the course of the Drell-Yanmeasurements [1, 2], and has been found to be smaller than the considered error sources.Later in this manuscript we will consider the difference between NLO QCD and NNLO QCDas a measure of the theoretical uncertainty. The experimental systematics require detailedinvestigation by the experimental collaborations, and we therefore do not attempt to includethem. Assuming L inverse femtobarns of integrated luminosity, the relative error for a givenbin i is δ L ( i ) = (cid:115) σ full ( i ) L + (cid:18) ∆ PDF ( i ) σ full ( i ) (cid:19) , (6)where σ full ( i ) is the cross section in the i -th bin (subject to acceptance cuts) expressed infemtobarns. From this we can form a χ function to quantitatively determine the significance6 IG. 1: Shown are the deviations induced by photon-initiated contributions and electroweak cor-rections to the dilepton invariant mass distribution at a 14 TeV LHC. of the neglect of a particular radiative correction x = NLO EW , γ, NLO EW + γ : χ x, L ( i ) = L × [ σ full ( i ) − σ x ( i )] σ full ( i ) + L ∆ ( i ) . (7)We caution that these χ distributions are only meant to illustrate the most promising regionsof phase space in which to pursue the measurements of interest. In addition to the lack ofexperimental systematic errors, we have neglected possible bin-to-bin correlations that mayappear. These χ distributions are not meant to serve as a numerical fit of the photon PDFor of other effects.To begin, we orient the reader by showing in Fig. 1 the relative deviations induced byEW corrections and photon-initiated processes to the invariant mass distribution at a 14TeV LHC. Specifically, these are the deviations induced by the ∆ σ NLO EW and ∆ σ γ contri-butions in Eq. (1), relative to the full cross section σ full . Both corrections grow in magnitudewith invariant mass, with the photon corrections reaching +30% at 3 TeV and the EWcontributions reaching − . Results for an 8 TeV LHC We begin by studying the | Y ll | , | η l | , and p T l distributions at an 8 TeV LHC. In order toestimate statistical errors we assume 20 fb − of integrated luminosity, consistent with theamount of data collected separately by the ATLAS and CMS experiments. Our results showthe deviations induced by the ∆ σ NLO EW and ∆ σ γ contributions in Eq. (1), relative to thefull cross section σ full (with the inclusion of acceptance cuts). We begin with the dileptonrapidity distribution in the invariant mass range M ll ∈ [120 , χ functions ofEq. (7) obtained by turning off the photon-initiated processes, the EW corrections, or both,are shown in the right panel. The dashed line in the left panel, indicating the estimatederrors from statistics and imperfect quark and gluons PDFs, is dominated by the PDF er-ror component for this invariant-mass bin. Both deviations from photon-induced processesand electroweak corrections are larger than the estimated error, and peak near central ra-pidity. The importance of simultaneously controlling both corrections is clear; they almostcompletely cancel. Any attempt to extract the photon PDF without accounting for EWcorrections, or vice versa, would lead to incorrect results. The error coming from imperfectknowledge of the photon distribution function is large, reaching ± χ values reach five per bin if the EW corrections are neglected, and three per binif the photon PDF is set to zero. If other sources of error can be controlled, then measure-ment of this distribution should provide a handle on the photon content of the proton. Ifboth corrections are set to zero simultaneously, the χ value remains under two for most ofthe kinematic range, quantitatively emphasizing the need to control both effects in order tomeasure either one.In Fig. 3 we show results for the | Y ll | distribution in two higher invariant mass bins, M ll ∈ [200 , M ll ∈ [500 , p T l , and the photon PDF has a smaller downward slope with increasingBjorken- x compared to the sea-quark distributions, as can be checked using Refs. [3] and [31].This makes the high-mass, central-rapidity phase space region a good place to extract thephoton PDF with relatively fewer complications from EW corrections. The estimated erroron the photon PDF also grows rapidly with invariant mass, indicating that any experimentalmeasurement in this region will improve upon the current determination of this quantity. Theestimated error from statistics and uncertainty in the quark and gluon PDFs increases from ± .
5% in the first invariant-mass bin, to ±
10% in the second bin. This is caused primarilyby decreased statistics, and not by a change in the PDF errors. In both bins it is smaller thanthe expected photon-induced contribution at central rapidities. The electroweak correctionsinduce an approximately −
4% correction that is flat over the entire kinematic range.We next study the lepton p T l distribution, beginning with the invariant mass region M ll ∈ [120 , IG. 2: Shown are the deviations induced by photon-initiated contributions and electroweak cor-rections to the dilepton rapidity distribution (left panel) at an 8 TeV LHC, for the invariant massrange M ll ∈ [120 , χ deviation for each binassuming 20 fb − of integrated luminosity. corrections are flat over the entire distribution. The reason for this peak is clear, and hasbeen pointed out previously in the literature [11]. The photon processes proceed via t -channel lepton exchange, which have a collinear singularity regulated by the cuts on p T l and η l , leading to an enhancement at low p T l (the lepton mass would regulate this singularityin the absence of cuts). The size of the peak is larger than the estimated errors. The χ distribution is shown in the right panel of Fig. 4. The low- p T l region is very sensitive to thephoton PDF, with a χ value reaching ten per bin. The importance of controlling the EWcorrections is again clear; when both the EW and photon-initiated corrections are turnedoff, the χ value drops to two. We note that the region near p T l ∼
60 GeV has been shadedout. This is the Jacobian peak coming from the kinematic boundary of the leading-orderprocess. Fixed-order perturbation theory breaks down near this boundary, and the shadingis meant to remove it from consideration.The same results for the region M ll ∈ [200 , p T l increases in this bin, primarily because of low statistics, the size ofthe photon deviation increases, so that the χ function indicates as significant a deviation asin the lower invariant mass bin. The photon-initiated corrections increase quickly with mass,and the relative importance of the electroweak corrections decreases in this higher invariantmass range, making this region a cleaner place from which to extract the photon PDF. Wesee from the left panel in Fig. 5 that the uncertainty from the photon distribution function isfar larger than the estimated errors from other sources, indicating that measurement of thisdistribution would very significantly improve our knowledge of this quantity. The χ valuesreach over ten for low p T l , and do not decrease much upon simultaneously neglecting EWcorrections, quantitatively demonstrating their reduced importance in this region. We note9
IG. 3: Shown are the deviations induced by photon-initiated contributions and electroweak cor-rections to the dilepton rapidity distribution at an 8 TeV LHC, for the invariant mass ranges M ll ∈ [200 , M ll ∈ [500 , M ll ∈ [120 , χ deviationfor each bin assuming 20 fb − of integrated luminosity. The region near the Jacobian peak, wherefixed-order perturbation theory breaks down, has been shaded out. IG. 5: Shown are the deviations induced by photon-initiated contributions and electroweak cor-rections to the lepton transverse momentum distribution (left panel) at an 8 TeV LHC, for theinvariant mass range M ll ∈ [200 , χ deviationfor each bin assuming 20 fb − of integrated luminosity. The region near the Jacobian peak, wherefixed-order perturbation theory breaks down, has been shaded out. that the maximum χ value, indicating the phase-space region with the most sensitivity tothe photon PDF, occurs somewhat above the lower bound of p T l >
20 GeV. This is becausethe leading-order kinematics of the γγ → l + l − process implies that p T l = √ ˆ sx e η + x e − η , (8)where ˆ s = x x s s is the standard partonic Mandelstan invariant. The constraint | η l | < . p T l cut to enhance the effect of photon-induced processes does not help unless the η l cut can simultaneously be relaxed. We willstudy the effect of relaxing this cut in a later section.Finally, we show in Fig. 6 the invariant mass bin M ll ∈ [500 , p T l region. However, the χ function indicatesthat this mass range can still discriminate between different photon distribution functions.The electroweak corrections are smaller, and constant over the entire kinematics range. The χ function in the lower p T l bins does not significantly change if the EW corrections areturned off in addition to the photon contributions.We next study the lepton pseudorapidity distribution. Results for the electroweak andphoton-induced deviations are shown in Fig. 7 for all three invariant mass bins. Both cor-rections are relatively flat over the entire kinematic range, and tend to cancel. Structurein the distributions only appears in the highest invariant mass bin, but is too small to be11
IG. 6: Shown are the deviations induced by photon-initiated contributions and electroweak cor-rections to the lepton transverse momentum distribution (left panel) at an 8 TeV LHC, for theinvariant mass range M ll ∈ [500 , χ deviationfor each bin assuming 20 fb − of integrated luminosity. The region near the Jacobian peak, wherefixed-order perturbation theory breaks down, has been shaded out. observed over the estimated errors. We will see this structure again when we consider the | η l | distribution at a 14 TeV LHC. We do not show the χ distributions for this variable,since it is not particularly sensitive to either effects we are interested in extracting.We conclude this section by considering the impact of higher-order QCD corrections onthe distributions studied. As emphasized in Ref. [11], without sufficient control over QCD,other effects are swamped by its uncertainty. A crucial aspect of our analysis is the inclusionof the NNLO QCD corrections. We show in Fig. 8 the corrections induced by both NLOQCD and NNLO QCD on the dilepton rapidity distribution for the lowest two invariant massbins. While the change from LO to NLO is large, the additional shift in going from NLO toNNLO QCD is small, typically less than or equal to the estimated uncertainty from othersources. Even adding the difference between NLO and NNLO as an estimate of uncertaintyfrom higher-order corrections to the χ function of Eq. (7), which we feel is an overestimateof this error, does not significantly reduce the observability of the photon-initiated or EWterms. To avoid too large a proliferation of plots we do not show the QCD deviations forother observables and other mass bins, but simply note that the above comments remain truewith two exceptions: the p T l distribution near the Jacobian peak, where we anyway expectfixed-order perturbation theory to break down, and the p T l distribution right at the lowercut, where the NNLO-NLO QCD corrections are comparable to or slightly larger than theuncertainty from other sources. Since the most sensitive region to the photon PDF is abovethe lower p T l cut as discussed above (due to the η l cut), this does not have a large effect onthe presented results, but a careful accounting of QCD effects in this region is important.12 IG. 7: Shown are the deviations induced by photon-initiated contributions and electroweak cor-rections to the lepton pseudorapidity distribution at an 8 TeV LHC, for the invariant mass ranges M ll ∈ [120 , M ll ∈ [200 , M ll ∈ [500 , Since the η l cut also has a significant effect near this boundary, relaxing it slightly may reducethe impact of higher-order QCD. We show that this is indeed the case in a later section. B. Results for a 14 TeV LHC
We next proceed to study the | Y ll | , | η l | , and p T l distributions at a 14 TeV LHC. Wenow additionally consider the invariant mass bin M ll ∈ [1000 , − for the lower three bins, an amount expected afterroughly one year of LHC operation. For the new bin we assume 100 fb − , consistent withroughly two years of LHC run time, since the event rate for this bin is lower than the others.13 IG. 8: Shown are the deviations induced by QCD corrections to the dilepton rapidity distributionat an 8 TeV LHC, for the invariant mass ranges M ll ∈ [120 , M ll ∈ [200 , Since the errors for this bin are dominated by statistics and not PDF errors, uncertaintyestimates for different values of integrated luminosity can be approximated by a simplerescaling of the presented results.We start with the dilepton rapidity distribution. The deviations coming from electroweakcorrections and photon-induced processes for all four invariant mass bins are shown in Fig. 9.Several trends are apparent from the plot. Just like in 8 TeV collisions, the photon contri-butions are peaked toward central rapidity, while the electroweak corrections are flat. Theelectroweak corrections grow slightly as the invariant mass is increased, changing from − −
7% at central rapidity when going from the lowest bin to the highest bin. The pho-ton terms grow more quickly, increasing from +5% to +20% when going from the lowestbin to the highest bin. They are larger than the estimated statistical+PDF uncertainty forcentral rapidities in all bins. The error on the photon PDF is much larger than the otherconsidered sources of uncertainty, especially in the higher-invariant mass bins. This has al-ready been noted in the literature [3]. Improved control over this quantity will be criticalto enable high-mass searches at Run II of the LHC. The corresponding χ distributions areshown in Fig. 10. The most sensitive bin is the M ll ∈ [200 , p T l , as discussed in the presentation of 8 TeV results. They grow from8% in the M ll ∈ [120 , M ll ∈ [1 ,
3] TeV; photon-initiated14
IG. 9: Shown are the deviations induced by photon-initiated contributions and electroweak cor-rections to the dilepton rapidity distribution at a 14 TeV LHC. Clockwise from the top left, theplots show the following invariant mass ranges: M ll ∈ [120 , M ll ∈ [200 , M ll ∈ [500 , M ll ∈ [1000 , − are assumed, while 100 fb − are assumed for M ll ∈ [1000 , scattering is expected to be an extremely large component of the total Drell-Yan crosssection at Run II of the LHC. The electroweak corrections grow much more mildly withinvariant mass, reaching only −
7% in the highest invariant mass bin. They are flat as afunction of lepton p T l . The χ distributions for p T l are shown in Fig. 12 for each invariantmass bin. The χ values are in general larger than those for the dilepton rapidity onesfor a given invariant mass bin, indicating that the p T l distribution is particularly sensitiveto the photon PDF. In the M ll ∈ [120 , IG. 10: Shown are the χ distributions for the dilepton rapidity distribution. Clockwise from thetop left, the plots show the following invariant mass ranges: M ll ∈ [120 , M ll ∈ [200 , M ll ∈ [500 , M ll ∈ [1000 , − are assumed, while 100 fb − are assumed for M ll ∈ [1000 , Above this first bin the photon-initiated corrections increase quickly enough in size that theelectroweak contributions become relatively less important. The p T l distribution is a goodplace to extract the photon PDF while lessening the effect of EW corrections. As mentionedin the introduction, another good reason to use the p T l distribution to extract the photonPDF is that potential physics beyond the Standard Model is expected to populate the high p T l region, and the chance of confusing these two effects is therefore reduced. We note thatas in 8 TeV collisions, we have shaded out the region near the Jacobian peak in order toindicate the breakdown of fixed-order perturbation theory.Finally, results for the lepton | η l | distributions are shown in Fig. 13. For the lower two in-variant mass bins, the deviations from both photon-initiated processes and from electroweakcorrections are relatively flat over the entire kinematic range. The size of the deviationsis generally at or below the expected uncertainties arising from statistics and imperfect16 IG. 11: Shown are the deviations induced by photon-initiated contributions and electroweakcorrections to the p T l distribution at a 14 TeV LHC. Clockwise from the top left, the plots showthe following invariant mass ranges: M ll ∈ [120 , M ll ∈ [200 , M ll ∈ [500 , M ll ∈ [1000 , − areassumed, while 100 fb − are assumed for M ll ∈ [1000 , knowledge of quark and gluon PDFs. Structure begins to appear in the electroweak cor-rections in the M ll ∈ [500 , M ll ∈ [1000 , −
15% and are peaked to-ward central pseudorapidity. This arises because of the strong angular dependence of theSudakov logarithms which dominate the electroweak corrections at high invariant masses.A detailed discussion of the angular dependence of Sudakov logarithms is given in Ref. [32].For example, it is shown there that in the partonic center-of-mass frame, the EW Sudakovlogarithms are largest for the scattering angle cos( θ CM ) ≈ d ¯ d → µ + µ − channel,17 IG. 12: Shown are the χ distributions for the lepton transverse momentum distribution. Clock-wise from the top left, the plots show the following invariant mass ranges: M ll ∈ [120 , M ll ∈ [200 , M ll ∈ [500 , M ll ∈ [1000 , − are assumed, while 100 fb − are assumed for M ll ∈ [1000 , while they are peaked toward forward scattering for the u ¯ u → µ + µ − channel. A combi-nation of these underlying processes leads to the effect seen in Fig. 13. Distributions suchas the dilepton rapidity are relatively insensitive to this dependence. When forming thisobservable the four-momenta of the two leptons are added, removing the dependence on theCM-frame scattering angle and therefore removing the angular structure of these correc-tions. The p T l distribution is more sensitive to the t -channel enhancement of the underlyingmatrix elements. The lepton | η l | distribution offers a window onto the underlying structureof the Sudakov logarithms. We see from Fig. 13 that while large, the deviations caused bythe electroweak corrections are small compared to the estimated errors, at least with therelatively fine binning chosen. More luminosity or a coarser binning is needed to uncover18 IG. 13: Shown are the deviations induced by photon-initiated contributions and electroweakcorrections to the | η l | distribution at a 14 TeV LHC. Clockwise from the top left, the plots showthe following invariant mass ranges: M ll ∈ [120 , M ll ∈ [200 , M ll ∈ [500 , M ll ∈ [1000 , − areassumed, while 100 fb − are assumed for M ll ∈ [1000 , the underlying structure of the Sudakov corrections. The need to control the photon PDFusing other distributions is clear; in agreement with the trend observed so far, the two typesof corrections tend to cancel.We again conclude this section by considering the impact of higher-order QCD correc-tions on the distributions studied in Fig. 14. We show the dilepton rapidity and lepton p T l distribution in the invariant mass region M ll ∈ [500 , p T l distribution, but the corrections do become large19
IG. 14: Shown are the deviations induced by QCD corrections to the dilepton rapidity distribution(left panel) and lepton p T l distribution (right panel) at a 14 TeV LHC, for the invariant mass range M ll ∈ [500 , near the lower kinematic boundary. Relaxing the η l cut reduces the size of these corrections,as we now show. C. Effects of modified acceptance cuts
An additional issue that we wish to address in this study is what can be gained bymodifying the acceptance cuts away from the values in Eq. (2), which were used so farin our study. These might be tightened by the experimental collaborations due to triggerrequirements during higher luminosity running, or they may be loosened because of improvedanalysis techniques that allow for additional kinematic regions to be accessed. To probe suchpossibilities, we will consider in this section two potential changes in the acceptance cuts onthe leptons. • We will study the effect of loosening the pseudorapidity cut on the leptons to | η l | < .
0. Although this is an aggressive relaxation of this bound, a similar loosening maybe possible in the electron channel by using calorimetric information in the forwardregion . • We consider the effect of a staggered transverse momentum cut on the leptons; wedemand that the harder lepton satisfy p T >
40 GeV, while the softer one satisfies p T >
20 GeV. We thank S. Stoynev for discussions on this topic. IG. 15: Shown are the deviations induced by photon-initiated contributions and electroweak cor-rections to the dilepton rapidity distribution (left panel) and the lepton pseudorapidity distribution(right panel), for the invariant mass range M ll ∈ [200 , | η l | >
4. The bands show the errors coming from the photon distributionfunction. The dashed lines show the estimated errors coming from statistics and from uncertaintiesin the quark and gluon distribution functions.
For simplicity, we restrict this study to the invariant mass bin M ll ∈ [200 , | η l | < .
0. Resultsfor the dilepton rapidity and lepton pseudorapidity distributions are shown in Fig. 15. Bothof these observables have an increased phase space upon changing the | η l | cut. However,the expected errors are large in the new regions of phase space, and not much is gainedfrom these distributions for this change in the cut. Something more interesting occurs in thelepton transverse momentum distribution, shown in Fig. 16. This plot should be contrastedwith the result shown with | η l | < . p T , instead of decreasing near the lower p T boundary. The reason for thiswas explained below Eq. (8). The previous pseudorapidity cut was restricting the low p T region due to the connection between these variables shown in Eq. (8). This constraint isnow lifted. This has an added benefit. With the previous pseudorapidity cut, additionallow p T phase space regions were opened beyond LO in QCD, increasing the size of the QCDcorrections there and potentially reducing the power of this region in constraining the photonPDF. With the relaxed cuts, all of the available phase space is open already at LO, reducingthe impact of QCD corrections. This can be seen in Fig. 17, where the impact of QCDcorrections on p T l at NLO and NNLO is shown. When | η l | < .
5, the shift from NLO QCDto NNLO QCD is larger than the estimated errors from statistics and PDFs when p T l < | η l | < .
0. Only right at the lower boundary of p T l = 20 GeV areQCD corrections large. The exact numerical values for the p T l and η l cuts used in this studyare meant to be illustrative only, but the importance of carefully considering the impact ofphase-space restrictions on QCD radiation in future measurements should be clear from this21 IG. 16: Shown are the deviations induced by photon-initiated contributions and electroweakcorrections to the lepton transverse momentum distribution for the invariant mass range M ll ∈ [200 , | η l | <
4. The bands show theerrors coming from the photon distribution function. The dashed lines show the estimated errorscoming from statistics and from uncertainties in the quark and gluon distribution functions. example.We now demand that the harder of the two leptons satisfies p T >
40 GeV while the softerhas p T >
20 GeV, while keeping | η l | <
4. The major effect of this staggered cut arises fromthe fact that at Born level, the two leptons are both forced to have p T >
40 GeV, since theyare back-to-back in the transverse plane. The region where the softer lepton is in the range p T ∈ [20 ,
40] GeV only opens up at NLO in QCD when there is additional radiation for theleptons to recoil against. This enhances the impact of QCD corrections in exactly the phase-space region where there is the most sensitivity to the photon PDF. The deviations inducedby higher-order QCD corrections to the softer and harder lepton transverse momentumdistributions are shown in Fig. 18. The shift when going from NLO to NNLO in QCDreaches over 20% for the softer lepton in the region below 40 GeV. This shows that higher-order QCD can potentially mask other effects appearing in the low p T l region if the cutsenhance their effect. One should appreciate this effect in future experimental analyses.
IV. SUMMARY AND CONCLUSIONS
In this note we have mapped the structure of radiative corrections affecting high-massDrell-Yan production in both 8 TeV and 14 TeV LHC collisions. We have carefully studiedthe effect of photon-induced processes and electroweak corrections in all relevant kinematicvariables, for a host of invariant mass regions. We have estimated the observability of theseeffects given expected statistical errors, and uncertainties coming from quark and gluonPDFs. The considered effects are larger than the expected errors over a large kinematic22
IG. 17: Shown are the deviations induced by QCD corrections to the lepton p T l distribution forthe invariant mass range M ll ∈ [200 , | η l | < | η l | < .
5. The two lines indicatethe deviation of NLO QCD minus LO relative to the full result, and NNLO minus NLO relative tothe full result. range, making high-mass Drell-Yan production an ideal place to understand perturbativecorrections in the Standard Model, and to better determine the proton structure. Ouranalysis helps guide future measurements by showing what corrections must be accountedfor in which distributions, and also indicates how to individually extract each effect. Bydoing this one can determine whether such effects as electroweak Sudakov logarithms areunder theoretical control, and can therefore be applied in other processes. Our study alsohelps inform future studies of proton structure. We have performed this study using themost up-to-date theoretical tools: NNLO QCD corrections combined with NLO electroweakeffects, together with NNPDF photon PDFs for the leading photon-initiated processes.The main physics conclusions of our study have already been presented in the Introduc-tion, so we conclude with several general comments on our results. • Not surprisingly, our results show the importance of simultaneously controlling allsources of radiative corrections. While carefully chosen observables can reduce the sizeof EW corrections or photon-initiated effects, in general attempting to determine onewithout accounting for the other will lead to incorrect results. • In our view the experimental collaborations should attempt to measure all possiblekinematic distributions. They are all interesting for different reasons. The dileptonrapidity shows sensitivity to the photon PDF at central values. The lepton transversemomentum distribution is especially sensitive to the photon PDF. The lepton η l dis-tribution allows the angular structure of EW Sudakov logarithms to be probed. Theyall provide a different window into the structure of the Standard Model. • The interplay between QCD corrections and experimental cuts, and particularly the23
IG. 18: Shown are the deviations induced by QCD corrections to the harder (right panel) andsofter (left panel) lepton p T l distributions for the invariant mass range M ll ∈ [200 , | η l | < opening up of new phase space regions at higher orders, should be carefully studied.If not, large QCD uncertainties may mask the other effects one wishes to measure.We look forward to the continued precision study of the Drell-Yan process during Run II ofthe LHC. V. ACKNOWLEDGMENTS
We are grateful to U. Klein, A. Kubik, M. Schmitt, and S. Stoynev for many helpfuldiscussions.The work of R. B. was supported by the U.S. Department of Energy, Division of High En-ergy Physics, under contract DE-AC02-06CH11357. The work of Y. L. was supported by theU.S. Department of Energy under contract DEAC0276SF00515. The work of F. P. was sup-ported by the U.S. Department of Energy, Division of High Energy Physics, under contractDE-AC02-06CH11357 and the grants DE-FG02-95ER40896 and DE-FG02-08ER4153.This research used resources of the National Energy Research Scientific Computing Cen-ter, which is supported by the Office of Science of the U.S. Department of Energy underContract No. DE-AC02-05CH11231. [1] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B , 223 (2013) [arXiv:1305.4192 [hep-ex]].[2] S. Chatrchyan et al. [CMS Collaboration], arXiv:1310.7291 [hep-ex].
3] R. D. Ball et al. [ The NNPDF Collaboration], arXiv:1308.0598 [hep-ph].[4] V. V. Sudakov, Sov. Phys. JETP , 65 (1956) [Zh. Eksp. Teor. Fiz. , 87 (1956)].[5] V. S. Fadin, L. N. Lipatov, A. D. Martin and M. Melles, Phys. Rev. D , 094002 (2000)[hep-ph/9910338].[6] J. H. Kuhn, A. A. Penin and V. A. Smirnov, Eur. Phys. J. C , 97 (2000) [hep-ph/9912503].[7] A. Denner and S. Pozzorini, Eur. Phys. J. C , 461 (2001) [hep-ph/0010201].[8] J. H. Kuhn, S. Moch, A. A. Penin and V. A. Smirnov, Nucl. Phys. B , 286 (2001) [Erratum-ibid. B , 455 (2003)] [hep-ph/0106298].[9] A. Denner, M. Melles and S. Pozzorini, Nucl. Phys. B , 299 (2003) [hep-ph/0301241].[10] For recent summaries of this issue, see K. Mishra, T. Becher, L. Barze, M. Chiesa, S. Dittmaier,X. Garcia i Tormo, A. Huss and T. Kasprzik et al. , arXiv:1308.1430 [hep-ph]; J. M. Campbell,K. Hatakeyama, J. Huston, F. Petriello, J. Andersen, L. Barze, H. Beauchemin and T. Becher et al. , arXiv:1310.5189 [hep-ph].[11] S. Dittmaier and M. Huber, JHEP , 060 (2010) [arXiv:0911.2329 [hep-ph]].[12] A. Bierweiler, T. Kasprzik, J. H. K¨uhn and S. Uccirati, JHEP , 093 (2012)[arXiv:1208.3147 [hep-ph]].[13] See the talk by U. Klein at the LPCC electroweak working group meeting, https://indico.cern.ch/getFile.py/access?contribId=14&sessionId=2&resId=0&materialId=slides&confId=203748 .[14] R. Hamberg, W. L. van Neerven and T. Matsuura, Nucl. Phys. B , 343 (1991) [Erratum-ibid. B , 403 (2002)].[15] C. Anastasiou, L. J. Dixon, K. Melnikov and F. Petriello, Phys. Rev. Lett. , 182002 (2003)[hep-ph/0306192].[16] C. Anastasiou, L. J. Dixon, K. Melnikov and F. Petriello, Phys. Rev. D , 094008 (2004)[hep-ph/0312266].[17] K. Melnikov and F. Petriello, Phys. Rev. D , 114017 (2006) [hep-ph/0609070].[18] S. Catani, L. Cieri, G. Ferrera, D. de Florian and M. Grazzini, Phys. Rev. Lett. , 082001(2009) [arXiv:0903.2120 [hep-ph]].[19] R. Gavin, Y. Li, F. Petriello and S. Quackenbush, Comput. Phys. Commun. , 2388 (2011)[arXiv:1011.3540 [hep-ph]].[20] Y. Li and F. Petriello, Phys. Rev. D , 094034 (2012) [arXiv:1208.5967 [hep-ph]].[21] C. M. Carloni Calame, G. Montagna, O. Nicrosini and A. Vicini, JHEP , 109 (2007)[arXiv:0710.1722 [hep-ph]].[22] F. A. Berends, R. Kleiss, J. P. Revol and J. P. Vialle, Z. Phys. C , 155 (1985).[23] U. Baur, S. Keller and W. K. Sakumoto, Phys. Rev. D , 199 (1998) [hep-ph/9707301].[24] U. Baur, O. Brein, W. Hollik, C. Schappacher and D. Wackeroth, Phys. Rev. D , 033007(2002) [hep-ph/0108274].[25] A. Arbuzov, D. Bardin, S. Bondarenko, P. Christova, L. Kalinovskaya, G. Nanava andR. Sadykov, Eur. Phys. J. C , 451 (2008) [arXiv:0711.0625 [hep-ph]].[26] C. Balazs and C. P. Yuan, Phys. Rev. D , 5558 (1997) [hep-ph/9704258].[27] L. Barze, G. Montagna, P. Nason, O. Nicrosini, F. Piccinini and A. Vicini, Eur. Phys. J. C , 2474 (2013) [arXiv:1302.4606 [hep-ph]].[28] A. D. Martin, R. G. Roberts, W. J. Stirling and R. S. Thorne, Eur. Phys. J. C , 155 (2005) hep-ph/0411040].[29] See, for example, the talk by S. Dittmaier at the 2013 Les Houches workshop: https://phystev.in2p3.fr/wiki/_media/2013:groups:lh13_ew.pdf .[30] U. Baur, Phys. Rev. D , 013005 (2007) [hep-ph/0611241].[31] R. D. Ball et al. [NNPDF Collaboration], Nucl. Phys. B , 153 (2012) [arXiv:1107.2652[hep-ph]].[32] J. -y. Chiu, R. Kelley and A. V. Manohar, Phys. Rev. D , 073006 (2008) [arXiv:0806.1240[hep-ph]]., 073006 (2008) [arXiv:0806.1240[hep-ph]].