Third order fiducial predictions for Drell–Yan at the LHC

The Drell–Yan process at hadron colliders is a fundamental benchmark for the study of strong interactions and the extraction of electro-weak parameters. The outstanding precision of the LHC demands very accurate theoretical predictions with a full account of fiducial experimental cuts. In this letter we present a state-of-the-art calculation of the fiducial cross section and of differential distributions for this process at third order in the strict fixed-order expansion in the strong coupling, as well as including the all-order resummation of logarithmic corrections. Together with these results, we present a detailed study of the subtraction technique used to carry out the calculation for different sets of experimental cuts, as well as of the sensitivity of the fiducial cross section to infrared physics. We find that residual theory uncertainties are reduced to the percent level and that the robustness of the predictions can be improved by a suitable adjustment of fiducial cuts.

Despite this outstanding progress, the accurate description of experimental data is challenged by the presence of fiducial selection cuts in the measurements, whose inclusion in theoretical calculations can potentially compromise the stability of the perturbative expansion [97][98][99][100].An initial estimate of the N 3 LO Drell-Yan cross section with an account of experimental cuts was presented in Refs.[58,101] using the q T -subtraction formalism [102], albeit without a complete assessment of the theoretical and methodological uncertainties.The conclusions of the above study are discussed in detail in Appendix.
In this letter, we present state-of-the-art predictions both for the fiducial Drell-Yan cross section and for dif-ferential distributions of the final-state leptons.We exploit this calculation to carry out, for the first time, a thorough study of the robustness of these theory predictions in the presence of different sets of fiducial cuts.We also present a detailed analysis of the reliability of the computational method adopted, and show that reaching a robust control over the involved systematic uncertainties requires an excellent stability of the numerical calculation in deep infrared kinematic regimes.
Methodology.-The starting point of our calculation for the production cross section dσ DY of a Drell-Yan lepton pair, differential in its phase space and in the pair's transverse momentum p ℓℓ T , is the formula: where dσ N 3 LL DY represents the N 3 LL resummed p ℓℓ T distribution obtained in Ref. [59] with the computer code RadISH [52,103,104], including the analytic constant terms up to O(α 3 s ); the quantity dσ s ) is its expansion up to third order in α s , and dσ NNLO DY+jet is the differential p ℓℓ T distribution at NNLO (i.e.O(α 3 s )), obtained with the NNLOJET code [15,19,20].Eq. ( 1) is finite in the limit p ℓℓ T → 0: by integrating it inclusively over p ℓℓ T one can obtain predictions differential in the leptonic phase space at N 3 LO+N 3 LL perturbative accuracy, allowing for the inclusion of fiducial cuts.An important challenge in the evaluation of the integral of Eq. (1) over p ℓℓ T is given by the fact that both dσ NNLO DY+jet and dσ s ) diverge logarithmically in the limit p ℓℓ T → 0, and only their difference is finite since the large logarithmically divergent terms present in dσ NNLO  DY+jet are exactly matched by those contained in dσ N T /m ℓℓ ) n ).If one integrates inclusively over the leptonic phase space one has n = 2, while the presence of fiducial cuts in general leads to the appearance of linear terms with n = 1 [100,[105][106][107]. Starting from order α 2 s , the corrections are further enhanced by logarithms of p cut T .The presence of these corrections introduces a systematic uncertainty which can be controlled by reducing the value of p cut T to a sufficiently small value.This procedure is computationally demanding especially in the presence of linear corrections, due to the smaller value of p cut T required to achieve the independence of the results of the slicing parameter.Such linear corrections can be resummed at all orders in Eq. ( 1) [56] by applying a simple recoil prescription [108] to dσ N 3 LL DY , and their inclusion would in principle allow for a larger p cut T in the calculation.These effects are accounted for in Eq. ( 1), as discussed in Ref. [59].As a consequence, our Fiducial p ℓℓ T distribution at N 3 LO+N 3 LL (blue, solid) and NNLO+NNLL (red, dotted) compared to ATLAS data from Ref. [112].The binning is linear up to 30 GeV and logarithmic above.
N 3 LO+N 3 LL fiducial predictions obtained by integrating Eq. ( 1) are only affected by a slicing error of order O((p cut T /m ℓℓ ) 2 ).The perturbative expansion of the N 3 LO+N 3 LL fiducial cross section to third order in α s leads to the N 3 LO prediction as obtained according to the q T -subtraction formalism [102].In this case, the outlined procedure to include linear power corrections below p cut T in the N 3 LO computation is analogous to that of Refs.[101,109].Since the fiducial cross section can be computed up to NNLO using the NNLOJET code, which implements a subtraction technique [110,111] that does not require the introduction of a slicing parameter, in the fixed-order results quoted in this letter we apply the above procedure only to the computation of the N 3 LO correction, while retaining the p cut T -independent result up to NNLO.This effectively suppresses the slicing error in our fiducial N 3 LO cross section to O(α 3 s (p cut T /m ℓℓ ) 2 ).In general, the presence of linear fiducial power corrections indicates an arguably undesirable sensitivity of the fiducial cross section to the infrared region in which QCD radiation has small transverse momentum, which compromises the stability of the perturbative series [100].These issues can be avoided by modifying the definition of the fiducial cuts in such a way that the scaling of the power corrections be quadratic across most of the leptonic phase space.In the following we present a calculation of Eq. ( 1) and of the fiducial cross section both for the standard (symmetric) cuts adopted by LHC experiments [112,113], where the same cut is imposed on transverse momentum of the final state leptons, as well as for the modified (product) cuts proposed in Ref. [100], where   2a) and product Eq.(2b) cuts both at fixed perturbative order and including all-order resummation.We report the theoretical uncertainty in percent and, in parentheses, the absolute value of the statistical uncertainty.The latter applies to the last significant figures displayed.At N 3 LO we also separately indicate the slicing error, in absolute value.See the main text for details.
a cut is instead imposed on the product of the transverse momenta of the final state leptons.This state-of-the-art calculation allows us to assess precisely the effect of different types of fiducial cuts on the theoretical prediction for the cross section, as well as on the performance of the computational approach adopted here.
Results.-We consider proton-proton collisions at a centre-of-mass energy √ s = 13 TeV.We adopt the NNPDF4.0parton densities [114] at NNLO with α s (M Z ) = 0.118, whose scale evolution is performed with LHAPDF [115] and Hoppet [116], correctly accounting for heavy quark thresholds.We adopt the G µ scheme with the following EW parameters taken from the PDG [117]: M Z = 91.1876GeV, M W = 80.379 GeV, Γ Z = 2.4952 GeV, Γ W = 2.085 GeV, and G F = 1.1663787 × 10 −5 GeV −2 .We consider two fiducial volumes, in both of which the leptonic invariant-mass window is 66 GeV < m ℓℓ < 116 GeV and the lepton rapidities are confined to |η ℓ ± | < 2.5.The transverse momentum of the two leptons is constrained as Symmetric cuts [112]: Product cuts [100]: The central factorisation and renormalisation scales are chosen to be µ R = µ F = m ℓℓ 2 + p ℓℓ T 2 and the central resummation scale is set to Q = m ℓℓ /2.In the results presented below, the theoretical uncertainty is estimated by varying the µ R and µ F scales by a factor of two about their central value, while keeping 1/2 ≤ µ R /µ F ≤ 2. In addition, for the resummed results, for central µ R = µ F scales we vary Q by a factor of two around its central value.Moreover, a matching-scheme uncertainty is estimated by including the full scale variation of the additive matching scheme of Ref. [59] (27 variations that comprise the one of the central matching scale v 0 introduced in Eq. (5.2) of that article).The final uncertainty is obtained as the envelope of all the above variations, corresponding to 7 and 36 curves for the fixed-order and resummed computations, respectively.We present re-sults for the central member of the NNPDF4.0set.In the fiducial cross sections quoted below at N 3 LO and N 3 LO+N 3 LL, we do not consider the uncertainty related to the missing N 3 LO parton distributions, which are currently unavailable.
In Fig. 1, we start by showing the transversemomentum distribution of the Drell-Yan lepton pair in the fiducial volume Eq. (2a), obtained with Eq. ( 1), compared to experimental data [112].In the figure we label the distributions by the perturbative accuracy of their inclusive integral over p ℓℓ T .Our state-of-the-art N 3 LO+N 3 LL prediction provides an excellent description of the data across the spectrum, with the exception of the first bin at small p ℓℓ T which is susceptible to nonperturbative corrections not included in our calculation.We point out that the term dσ NNLO DY+jet − dσ in Eq. ( 1) gives a non-negligible contribution even for p ℓℓ T ≤ 15 GeV.The residual theoretical uncertainty in the intermediate p ℓℓ T region is at the few-percent level, and it increases to about 5% for p ℓℓ T ≳ 50 GeV.A more accurate description of the large-p ℓℓ T region requires the inclusion of EW corrections, which we neglect in our calculation.
We now consider the fiducial cross section with symmetric cuts.In order to gain control over the slicing systematic error, we choose p cut T as low as 0.81 GeV.In the first column of Tab.I, denoted as N k LO, we show the fixed-order results to O(α k s ).The second column of Tab.I displays the result obtained including resummation effects.In the fixed-order case, the theoretical uncertainty at N 3 LO, estimated as discussed above, is supplemented with an estimate of the slicing uncertainty obtained by varying p cut T in the range [0.45, 1.48] GeV and taking the average difference from the result with p cut T = 0.81 GeV.In the resummed case, we quote the total theoretical uncertainty including also the matching scheme variation.In both cases the statistical uncertainty is reported in parentheses.
We observe that the new N 3 LO corrections decrease the fiducial cross section by about 2.5%, and the final prediction at N 3 LO has larger theoretical errors than the NNLO counterpart, whose uncertainty band does not capture the N 3 LO central value.This indicates a poor convergence of the fixed-order perturbative series for this process, which is consistent with what has been observed in the inclusive case in Refs.[10][11][12].In the resummed case, the theoretical uncertainty is more reliable and within errors the convergence of the perturbative series is improved.The presence of linear power corrections is also responsible for the moderate difference between the fixed-order and the resummed prediction for the symmetric cuts, which as previously discussed indicates a sensitivity of the cross section to the infrared region of small p ℓℓ T .This ultimately worsens further the perturbative convergence of the fixed-order series thereby challenging the perspectives to reach percent-accurate theoretical predictions within symmetric cuts.
A possible solution to this problem [100] is to slightly modify the definition of the fiducial cuts as in Eq. (2b) in order to reduce such a sensitivity to infrared physics.We present for the first time theoretical predictions up to N 3 LO and N 3 LO+N 3 LL for this set of cuts, reported in the third and fourth column of Tab.I.The relative difference between the fixed-order and resummed calculations for the fiducial cross section never exceeds 0.04%, which indicates that the predictions with product cuts can be computed accurately with fixed-order perturbation theory.Nevertheless, we still observe a more reliable estimate of the theoretical uncertainties when resummation is included.
In order to study the stability of our predictions against variations of the infrared parameter p cut T , in Fig. 2 we show the dependence of the N k LO correction (i.e. the O(α k s ) term in the expansion of the fiducial cross section) on p cut T down to p cut T ≃ 0.4 GeV.In the case of symmetric cuts Eq. (2a), we observe that the inclusion of the linear power corrections is essential to reach a plateau at small p cut T , achieving the necessary independence of the result on the slicing parameter.We thus obtain an excellent control over the estimate of the slicing error quoted in Tab.I. Furthermore, Fig. 2 clearly shows that the omission of such linear corrections leads to an incorrect result for the fiducial cross section computed with the q T -subtraction method, unless dσ NNLO DY+jet can be computed precisely down to p cut T ≪ 1 GeV.Conversely, in the case of the product cuts, we observe a much milder dependence of the N k LO correction on p cut T , and the further inclusion of power corrections does not lead to any visible difference, consistent with the fact that such corrections are quadratic in most of the phase space [100].As an additional sanity check, we have repeated the test of Fig. 2 for each individual flavour channel contributing to the N 3 LO Drell-Yan cross section.The results are collected in the Appendix, together with a discussion on alternative approaches to q T subtraction employing a fitting procedure [118], and a comparison to the literature [58,101].
Finally, the computation presented in this letter al- lows us to obtain, for the first time, N 3 LO+N 3 LL predictions for the kinematical distributions of the final-state leptons.A particularly relevant distribution is the leptonic transverse momentum, which plays a central role in the precise extraction of the W -boson mass at the LHC [2,6].Fig. 3 shows the differential distribution of the negatively charged lepton at three different orders, for our default value p cut T = 0.81 GeV.Unlike for the fiducial cross section, the inclusion of p ℓℓ T resummation in this observable is crucial to cure local (integrable) divergences in the spectrum due to the presence of a Sudakov shoulder [119] at p ℓ − T ∼ m ℓℓ /2.The figure shows an excellent convergence of the perturbative prediction, with residual uncertainties at N 3 LO+N 3 LL of the order of a few percent across the entire range.
Conclusions.-In this letter, we have presented state-of-the-art predictions for the fiducial cross section and differential distributions in the Drell-Yan process at the LHC, through both N 3 LO and N 3 LO+N 3 LL in QCD.These new predictions are obtained through the combination of an accurate NNLO calculation for the production of a Drell-Yan pair in association with one jet, and Lepton transverse momentum distribution up to N 3 LO+N 3 LL order in the fiducial phase space Eq.(2a).The labels indicate the order in the fiducial cross section.
the N 3 LL resummation of logarithmic corrections arising at small p ℓℓ T .The high quality of these results allowed us to carry out a thorough study of the performance of the computational method adopted, reaching an excellent control over all systematic uncertainties involved.We presented predictions for two different definitions of the fiducial volumes, relying either on symmetric cuts Eq. (2a) on the transverse momentum of the leptons, or on a recently proposed product cuts Eq. (2b) which is shown to improve the stability of the perturbative series.Our results display residual theoretical uncertainties at the O(1%) level in the fiducial cross section, and at the few-percent level in differential distributions.These predictions will play an important role in the comparison of experimental data with an accurate theoretical description of the Drell-Yan process at the LHC.

Fit-based implementation of qT subtraction
As an alternative to choosing a specific p cut T , a fitting procedure can be employed either by extrapolating Fig. 2 of the main letter down to p cut T → 0 or by analytically integrating a fit function.In the first method we directly fit the N 3 LO curve in Fig. 2 in the range p cut T ∈ [p cut T, min , p off T ] using a parametrisation that follows the known analytic structure of power corrections with the coefficients b n being extracted from the fit.We perform Markov chain Monte Carlo fits based on generative probabilistic models for the data that facilitate a straighforward marginalisation over nuisance parameters and the propagation of uncertainties and their correlations.The latter are relevant for the extrapolation at the cumulant level, as the integral of dσ NNLO DY+jet features large offdiagonal entries in the covariance matrix.In both fitting methods we find that too small values of p off T do not allow for the inclusion of sufficiently many data points to constrain the fit parameters, yielding uncertainties as large as O(100%) for the N 3 LO correction.On the other hand, choosing larger values of p off T ≳ 50 GeV leads to a more stable determination of the N 3 LO correction, which in both cases is in agreement with the result quoted in Tab.I of the letter within uncertainties.The nominal uncertainty is slightly smaller than the slicing uncertainty obtained with the procedure described in the letter.However, increasing the value of p off T induces a higher sensitivity to yet subleadingpower terms in Eqs. ( 3)-( 4), which constitute an additional source of systematic uncertainty that must be reliably assessed.A naive extension of the fitting functional forms to account for an additional tower of next-to-subleading terms increases the number of fitting parameters, ultimately leading to uncertainties which are significantly larger than those quoted in Tab.I.For this reason, we choose to adopt the slicing procedure described in the main letter, which allows for a reliable assessment of all the sources of uncertainty involved.
Guaranteeing the cancellation of such divergences requires high numerical precision in the NNLO distribution dσ NNLO DY+jet down to very small values of p ℓℓ T .Setting dσ NNLO DY+jet − dσ N 3 LL DY O(α 3 s ) = 0 for p ℓℓ T ≤ p cut T introduces a slicing error of order O((p cut FIG. 1.Fiducial p ℓℓ T distribution at N 3 LO+N 3 LL (blue, solid) and NNLO+NNLL (red, dotted) compared to ATLAS data from Ref.[112].The binning is linear up to 30 GeV and logarithmic above.

FIG. 2 .
FIG. 2. Dependence of the extracted N k LO corrections to the fiducial cross sections shown in Tab.I on the p cut T infrared parameter, both for the symmetric and product cuts.In the latter case, the NLO correction has been rescaled by a factor 1/4.The dashed vertical line indicates our default value p cut T = 0.81 GeV.The blue band is obtained by combining linearly the statistical and slicing errors.
FIG. 3.Lepton transverse momentum distribution up to N 3 LO+N 3 LL order in the fiducial phase space Eq.(2a).The labels indicate the order in the fiducial cross section.

FIG. 5 .
FIG. 5. Dependence on p cutT of the extracted N k LO fiducial cross sections using the data from Ref.[55].The blue band is obtained by combining linearly the statistical and slicing errors.The dashed line indicates p cut T = 4 GeV, as used in[58,101].

and extract ∆σ N 3 LO 0 .
The lower edge of the fitting range p cut T, min is set to 0.45 GeV.In the second method, similarly to what has been done in Ref.[118], we consider a fit up to relatively large p ℓℓ T ≤ p off T values of the residual quadratic power corrections d[∆σ non-sing DY ] α 3 s , which denotes the α 3 s coefficient of the difference dσ NNLO DY+jet − dσ N 3 LL DY O(α 3 s ) .The fiducial cross section is then obtained by performing a slicing calculation with p cut T = p off T and adding the analytic integral of the above fit over p ℓℓ T ∈ [0, p off T ].In this case, the functional form is d[∆σ non-sing DY ]

TABLE I .
Fiducial cross sections for the symmetric Eq. (