Charged Current Drell-Yan Production at N3LO
PPrepared for submission to JHEP
CERN-TH-2020-121, SLAC-PUB-17539
Charged Current Drell-Yan Production at N LO Claude Duhr a Falko Dulat b Bernhard Mistlberger b a Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland. b SLAC National Accelerator Laboratory, 2575 Sand Hill Rd., Menlo Park, CA, 94025 USA
E-mail: [email protected], [email protected],[email protected]
Abstract:
We present the production cross section for a lepton-neutrino pair at theLarge Hadron Collider computed at next-to-next-to-next-to leading order (N LO) in QCDperturbation theory. We compute the partonic coefficient functions of a virtual W ± bosonat this order. We then use these analytic functions to study the progression of the pertur-bative series in different observables. In particular, we investigate the impact of the newlyobtained corrections on the inclusive production cross section of W ± bosons, as well as onthe ratios of the production cross sections for W + , W − and/or a virtual photon. Finally,we present N LO predictions for the charge asymmetry at the LHC.
Keywords: W production, QCD, N LO. a r X i v : . [ h e p - ph ] S e p ontents W -production at N LO 44 Predictions for cross section ratios 105 Conclusion 15
Cross sections for the production of leptons are among the ultimate precision observablesmeasurable at the Large Hadron Collider (LHC) (see for example refs. [1–3]). As a conse-quence, they provide a unique window into the inner workings of collision processes at veryhigh energies. The insights gained by studying them improve our understanding of the col-lider experiment and the fundamental mechanisms of scattering processes alike. To derivemeaningful conclusions from such observations we must put our theoretical prediction forsuch scattering processes at the highest possible level. Here, we take one significant stepin this direction and compute next-to-next-to-next-to leading order (N LO) predictions forthe production cross section of a lepton-neutrino pair in QCD perturbation theory.We focus on the charged current Drell-Yan [4] (CCDY) process, where a W boson isproduced from the annihilation of two quarks with opposite isospin. In nature, the pro-duced W boson decays within the blink of an eye, but the inclusive production probabilityof the bosons is easily related to the probability of producing a pair of fermions consistingof a neutrino and a charged lepton with invariant mass Q . These particles represent astable final state and the charged lepton can be detected by the LHC experiments.To derive theoretical predictions for the inclusive CCDY cross section we use the fac-torisation of hadronic cross sections into parton distribution functions (PDFs) and partoniccross sections. In order to achieve high precision predictions of the hadronic cross section,it is paramount to go beyond the Born approximation for the partonic cross section. Inparticular, we compute the desired partonic cross sections in the framework of perturbativeQCD through N LO in the perturbative expansion in the strong coupling constant. Theanalytic formulæ for the partonic cross sections are some of the main results of this paperand are provided in electronic form together with its arXiv submission.We then move on and study the impact of N LO corrections on various inclusive crosssections involving vector bosons. We focus on the progression of the perturbative seriesin QCD for some of the cleanest hadron collider observables. When combined with the– 1 –vailable N LO results in the literature [5–14], our results are an important input to gaugethe relevance and the impact of N LO corrections on more differential observables, likefiducial cross sections. We also study cross section ratios for vector boson production, andobserve a remarkable perturbative stability for these ratios.This article is structured as follows: In Section 2 we briefly review the computation ofthe N LO corrections to off-shell W production. In Section 3 we present our main result,namely phenomenological predictions for the W cross section at N LO in QCD, and wediscuss the main QCD uncertainties which affect the cross section at this order. In Section 4we extend this analysis to ratios of vector boson cross sections and the charge asymmetryat the LHC. In Section 5 we draw our conclusions.
In this paper we compute higher-order corrections in the strong coupling constant to thecharged-current Drell-Yan (CCDY) cross section, i.e., the inclusive cross section for theproduction of a lepton-neutrino pair of invariant mass Q at a proton-collider with center-of-mass energy √ S . Since we are only interested in QCD corrections, the lepton-neutrinopair can only be produced via the intermediate of an (off-shell) W boson. We can thenfactorise the production of the W boson from its subsequent decay and cast the crosssection in the following form: Q dσdQ ( p p → W ± → (cid:96) ± (–) ν (cid:96) ) = m W π v Q ( Q − m W ) + m W Γ W σ W ± ( τ ) , (2.1)where v is the vacuum expectation value and m W and Γ W are the mass and width of the W boson, and σ W ± denotes the inclusive cross section for the production of an off-shell W ± boson with virtuality Q . Using QCD factorisation, this cross section can be writtenin the form σ W ± ( τ ) = τ σ (cid:88) i,j f i ( τ, µ F ) ⊗ η ± ij ( τ, µ F , µ R ) ⊗ f j ( τ, µ F ) , (2.2)where µ F and µ R denote the factorisation and renormalisation scales, and the f i ( x, µ F )denote the parton density functions (PDFs) to find a parton species i with momentumfraction x inside the proton. Furthermore, τ = Q S and η ± ij is the partonic coefficient for theproduction of an off-shell W boson from the parton species i and j . In the above equationwe made use of convolutions defined by f ( x ) ⊗ g ( x ) = (cid:90) x dx (cid:48) x (cid:48) f ( x (cid:48) ) g (cid:16) xx (cid:48) (cid:17) , (2.3)and we introduced the normalisation factor σ = m W πn c Q v , (2.4)where n c corresponds to the number of colours in QCD.– 2 –he partonic coefficients can be expanded into a perturbative series in the renormalisedstrong coupling constant a S = α S ( µ R ) /πη ± ij ( z ) = η ± (0) ij ( z ) + a S η ± (1) ij ( z ) + a S η ± (2) ij ( z ) + a S η ± (3) ij ( z ) + . . . . (2.5)Above we have suppressed arguments of the functions indicating the dependence of thepartonic coefficients on the perturbative scales. At leading order (LO) in α S , it is onlypossible to produce a W boson from the annihilation of two (massless) quarks with oppositeisospin: η +(0) u i ¯ d j ( z ) = | V u i d j | δ (1 − z ) and η − (0) d i ¯ u j ( z ) = | V u j d i | δ (1 − z ) , (2.6)where V u i d j denotes the Cabibbo-Kobayashi-Maskawa quark-mixing-matrix. Beyond LO [4],also other partonic channels open up. Perturbative corrections to the CCDY cross sectionshave been computed at next-to-leading order (NLO) in refs. [15–18] and at next-to-next-to-leading order (NNLO) in refs. [19–22]. The main result of this paper is to present for thefirst time phenomenological results for the production of an off-shell W boson at next-to-next-to-next-to-leading oder (N LO) in perturbative QCD. Before we discuss our resultsin the next sections, we review in this section the main steps of the computation of thethird-order corrections to the partonic coefficients.The partonic coefficients are computable from Feynman diagrams, with z = Q / ˆ s andˆ s the partonic center-of-mass energy. We have followed the same strategy as that for thecomputation of the inclusive cross section for Higgs boson production through gluon fusionand bottom-quark fusion [6, 8, 9, 23] and the inclusive Drell-Yan cross section [10]. Inparticular, the results were obtained using the framework of reverse unitarity [24–28] inorder to compute all required interferences of real and virtual amplitudes contributing tothe N LO cross section. The required phase-space and loop integrals were carried outimplicitly using integration-by-parts (IBP) identities [29–31], together with the method ofdifferential equations [32–36]. This method allows one to represent the required integratedand interfered amplitudes in terms of linear combinations of master integrals . The purelyvirtual corrections are essentially identical to the case of the production of an off-shellphoton, apart from the colour structure involving a cubic Casimir operator, which is absenthere because of the non-diagonal flavour-structure of the charged-current interactions. Thethree-loop corrections for virtual photon production were first computed in refs. [37–43],and recomputed and confirmed in ref. [8]. Contributions with one real parton in the finalstate were considered in refs. [44–49] and the master integrals we used for our calculationwere documented in refs. [44, 48]. Master integrals with two and three real partons wereobtained for the purpose of ref. [6] and are based on results from refs. [23, 50–54].We work in the MS-scheme in conventional dimensional regularisation. The only in-teraction vertex that involves an axial coupling is the vertex involving the W boson. Thenon-diagonal flavour-structure of the charged-current interactions forces the W boson to becoupled twice to same connected fermion line in interference diagrams. As a consequence,vector and axial vector contributions to the hadronic cross section are identical and weonly work with a vector current in the generation of our partonic coefficient functions.The ultraviolet (UV) counterterm for the strong coupling constant has been determined– 3 –hrough five loops in refs. [55–59]. Infrared (IR) divergences are absorbed into the def-inition of the PDFs using mass factorisation at N LO [60–62]. The mass factorisationinvolves convoluting lower-order partonic cross sections with the three-loop splitting func-tions of refs. [63–65]. We have computed all the convolutions analytically in z -space usingthe PolyLogTools package [66]. After combining our interfered matrix elements withthe UV and PDF-IR counterterms we send the dimensional regulator to zero and obtainour final results. Our partonic coefficients are expressed in terms of the iterated integralsin z defined in ref. [6]. While many of these iterated integrals can be expressed in termsof harmonic or multiple polylogarithms [67, 68], some integration kernels involve ellipticintegrals. It is currently unknown how to express them in terms of known functions. Inref. [6] it was shown how to obtain fast converging series representations which allow oneto achieve a relative numerical precision of (at least) 10 − in the whole range z ∈ [0 ,
1] forall partonic coefficients.The analytic results for the partonic coefficients are provided as ancillary materialwith the arXiv submission. Besides the explicit analytic cancellation of the UV and IRpoles, we have performed various checks to establish the correctness of our computation.First, we have checked that all logarithmic terms in the renormalisation and factorisationscales produced from the cancellation of the UV and IR poles satisfy the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equation [69–71]. Second, the limitof soft-gluon emission, which corresponds to z →
1, is independent of the form of thehard interaction process and only depends on the quantum numbers of the initial-statepartons. Hence, the soft-virtual cross sections must be identical to the ones for neutral-current Drell-Yan production (again, apart from the contribution from the cubic Casimiroperator). We have reproduced the soft-virtual N LO cross section of refs. [51, 72–76],and also the physical kernel constraints of refs. [77–79] for the next-to-soft term of thequark-initiated cross section. The high-energy limit of the cross section, which correspondsto z →
0, must also be identical between charged- and neutral-current production, andwe have checked that our partonic cross sections have the expected behaviour in the high-energy limit [80, 81]. Finally, we have checked that we reproduce the numerical results forthe ratios of the inclusive W + and W − cross sections up to NNLO given in ref. [82]. W -production at N LO In this section we discuss our phenomenological results for inclusive (off-shell) W -productionat N LO. All results were obtained in a theory with N f = 5 massless quark flavours (twoup-type and three down-type flavours). The values of the mass and the width of the W boson are m W = 80 .
379 GeV and Γ W = 2 .
085 GeV, and the vacuum expectation value is v = 246 .
221 GeV. The elements of the CKM matrix relevant to our computation are [83]: | V ud | = 0 . , | V us | = 0 . , | V ub | = 0 . , | V cd | = 0 . , | V cs | = 0 . , | V cb | = 0 . . (3.1)Note that only the absolute values of the entries of the CKM matrix enter our computation.In our phenomenological results the top quark is absent, which is equivalent to having V td i =– 4 – and only considering N f = 5 massless degrees of freedom in loops. This approximationis motivated because off-diagonal CKM matrix elements are small and diagrams without acoupling of the top quark to the electroweak gauge boson decouple in the limit of infinite topquark mass. Corrections to this approximation, which are expected to be very small, can becomputed separately and are beyond the scope of this article. The strong coupling constantis evolved to the renormalisation scale µ R using the four-loop QCD beta function in theMS-scheme assuming N f = 5 active, massless quark flavours. Unless stated otherwise,all results are obtained for a proton-proton collider with √ S = 13 TeV using the zerothmember of the combined PDF4LHC15 nnlo mc set [84]. ��� ���� ���� ��� ��� ��� ��� �� ��� ��� ��� ��� ��� ��� ��� ��� ��������������������������������� μ � / � σ / σ � � � �� � � � μ � ��������� ��� � + ��� �������������� _ ���� _ ��� = ��� ��� ��� ���� ���� ��� ��� ��� ��� �� ��� ��� ��� ��� ��� ��� ��� ��� ��������������������������������� μ � / � σ / σ � � � �� � � � μ � ��������� ��� � - ��� �������������� _ ���� _ ��� = ��� ��� Figure 1 : The cross sections for producing a W + (left) or W − (right) for µ R = Q =100 GeV as a function of the factorisation scale µ F . The bands are obtained by varying µ R by a factor of 2 up and down. The cross sections are normalised to the leading ordercross section evaluated at µ F = µ R = Q . ��� ���� ���� ��� ��� ��� ��� �� ��� ��� ��� ��� ��� ��� ��� ��� ��������������������������������� μ � / � σ / σ � � � �� � � � μ � ��������� ��� � + ��� �������������� _ ���� _ ��� = ��� ��� ��� ���� ���� ��� ��� ��� ��� �� ��� ��� ��� ��� ��� ��� ��� ��� ��������������������������������� μ � / � σ / σ � � � �� � � � μ � ��������� ��� � - ��� �������������� _ ���� _ ��� = ��� ��� Figure 2 : The cross sections for producing a W + (left) or W − (right) for µ F = Q =100 GeV as a function of the renormalisation scale µ R . The bands are obtained by varying µ F by a factor of 2 up and down. The cross sections are normalised to the leading ordercross section evaluated at µ F = µ R = Q .Figures 1 and 2 show the dependence of the fixed-order cross sections on the factori-sation scale µ F and renormalisation scale µ R , which are introduced by the truncation ofthe perturbative series. We show the variation of the cross section for Q = 100 GeV onone of the two scales with the other held fixed at Q . We observe that the dependence onthe perturbative scales is substantially reduced as we increase the perturbative order. Thedependence on the scales looks very similar to the case of the N LO cross section for theneutral-current process studied in ref. [10]. We notice, that the dependence of the cross– 5 –ection on the renormalisation scale is slightly larger than on the factorisation scale.Next, in order to quantify the size of the N LO corrections, we investigate the K-factors: K N LO W ± ( Q ) = σ (3) W ± ( µ F = µ R = Q ) σ (2) W ± ( µ F = µ R = Q ) ,δ (scale) = δ scale ( σ (3) W ± ) σ (3) W ± ( µ F = µ R = Q ) , (3.2)where σ ( n ) W ± ( µ F = µ R = Q ) is the hadronic cross section including perturbative correctionsup to n th order evaluated for µ F = µ R = Q and δ scale ( σ ( n ) W ± ) is the absolute uncertainty onthe cross section from varying µ F and µ R independently by a factor of two up and downaround the central scale µ cent = Q such that ≤ µ F µ R ≤ Q/ GeV K N LO W + K N LO W − K N LO γ ∗
30 0 . +2 . − . . +2 . − . . +2 . − .
50 0 . +1 . − . . +1 . − . . +1 . − .
70 0 . +1 . − . . +1 . − . . +1 . − . .
379 0 . +1 . − . . +1 . − . . +1 . − .
90 0 . +0 . − . . +0 . − . . +0 . − .
110 0 . +0 . − . . +0 . − . . +0 . − .
130 0 . +0 . − . . +0 . − . . +0 . − .
150 0 . +0 . − . . +0 . − . . +0 . − .
500 0 . +0 . − . . +0 . − . . +0 . − .
800 0 . +0 . − . . +0 . − . . +0 . − . Table 1 : The QCD K-factor at N LO for charged-current and neutral-current Drell-Yanproduction. All central values and uncertainties are computed according to eq. (3.2). Theresults for neutral-current Drell-Yan production are taken from ref. [10].Table 1 shows the results for the K-factors for both charged- and neutral-current Drell-Yan production for various values of Q (the results for the neutral-current process are takenfrom ref. [10]). We observe that the K-factor reaches up to 5% for small values of Q , andthe impact of the N LO corrections gets smaller as Q increases. Moreover, we find thatthe K-factors at N LO are identical between the charged- and neutral-current processes,confirming and extending results at lower orders, and hinting towards a universality ofQCD corrections to vector-boson production in proton collisions.Figure 3 shows the values of the cross section normalised to the N LO cross sectionas a function of the virtuality Q . The uncertainty bands are obtained by varying therenormalisation and factorisation scales independently up and down as described abovearound the central scale µ cent = Q . We observe that for Q (cid:38)
50 GeV the scale-variationbands at NNLO and N LO do not overlap. A similar feature was already observed for– 6 – ���������� �� �� �� �� ��� ��� ��� ��� ��� ���������������� � [ ��� ] σ / σ � � � � � - ������ � + ��� �������������� _ ���� _ �� μ ����� = � ����������� �� �� �� �� ��� ��� ��� ��� ��� ���������������� � [ ��� ] σ / σ � � � � � - ������ � - ��� �������������� _ ���� _ �� μ ����� = � Figure 3 : The cross sections for producing a W + (left) or W − (right) as a function of thevirtuality Q normalised to the N LO prediction. The uncertainty bands are obtained byvarying µ F and µ R around the central scale µ cent = Q . The dashed magenta line indicatesthe physical W boson mass, Q = m W .virtual photon production in ref. [10], hinting once more towards a universality of theQCD corrections to these processes. ����������� �� �� �� �� ��� ��� ��� ��� ��� ���������������� � [ ��� ] σ / σ � � � � � - ������ � + ��� �������������� _ ���� _ �� μ ����� = ����� ����������� �� �� �� �� ��� ��� ��� ��� ��� ���������������� � [ ��� ] σ / σ � � � � � - ������ � - ��� �������������� _ ���� _ �� μ ����� = ����� Figure 4 : The cross sections for producing a W + (left) or W − (right) as a function ofthe virtuality Q . The uncertainty bands are obtained by varying µ F and µ R around thecentral scale µ cent = Q/
2. The dashed magenta line indicates the physical W boson mass, Q = m W .Figure 4 shows the scale variation of the cross section with a different choice for thecentral scale, µ cent = Q/
2. It is known that for Higgs production a smaller choice of thefactorisation scale leads to an improved convergence pattern and the bands from scalevariations are strictly contained in one another. We observe here that the two scale choicesshare the same qualitative features.The fact that the scale variation bands do not overlap puts some doubt on whetherit gives a reliable estimate of the missing higher orders in perturbation theory, or whetherother approaches should be explored (cf., e.g., refs. [85, 86]). In ref. [10] it was noted thatfor virtual photon production there is a particularly large cancellation between differentinitial state configurations. We observe here the same in the case of W boson production.This cancellation may contribute to the particularly small NNLO corrections and scalevariation bands, and it may be a consequence of the somewhat arbitrary split of the content– 7 –f the proton into quarks and gluons. If these cancellations play a role in the observedperturbative convergence pattern, then it implies that one cannot decouple the study ofthe perturbative convergence from the structure of the proton encoded in the PDFs. Wewill return to this point below, when we discuss the effect of PDFs on our cross sectionpredictions. ����������� ��� ��� ��� ��� ���� ���� ���� ���� ���������������������� � [ ��� ] σ / σ � � � � � - ������ � + ��� �������������� _ ���� _ �� μ ����� = � ����������� ��� ��� ��� ��� ���� ���� ���� ���� ���������������������� � [ ��� ] σ / σ � � � � � - ������ � - ��� �������������� _ ���� _ �� μ ����� = � Figure 5 : The cross sections for producing a W + (left) or W − (right) as a function ofthe virtuality Q . The uncertainty bands are obtained by varying µ F and µ R around thecentral scale µ cent = Q . The dashed magenta line indicates the physical W boson mass, Q = m W .Figure 5 shows the production cross section for an off-shell W boson normalised to theprediction at N LO for a larger range of virtualities ( Q ≤ Q >
550 GeV) the bands derived from scale variation at NNLOand N LO start to overlap. We also observe a more typical shrinking of the scale variationbands as well as a small correction at N LO. ���� ��� ��� ��� ��� ���� ���� ���� ���� ������ - � �� - � �� - � �� � �� � �� � �� � �� � �� � � [ ��� ] � � � σ � � � � / � � � [ �� ] ����� ������� � + ��� �������������� _ ���� _ �� μ ����� = � ���� ��� ��� ��� ��� ���� ���� ���� ���� ������ - � �� - � �� - � �� � �� � �� � �� � �� � �� � � [ ��� ] � � � σ � � � � / � � � [ �� ] ����� ������� � - ��� �������������� _ ���� _ �� μ ����� = � Figure 6 : The cross sections for producing a lepton-neutrino pair via an off-shell W bosonas a function of the invariant mass of the final state, or equivalently the virtuality of the W boson, cf. eq. (2.1).Figure 6 shows the nominal production cross section of a lepton-neutrino pair at theLHC at 13 TeV centre of mass energy, as defined in eq. (2.1).Figure 7 shows the variation of K-factors as a function of the energy of the hadroncollider for Q = 100 GeV. The orange, blue and red bands correspond to predictionswith the perturbative cross section truncated at NLO, NNLO and N LO, and the sizeof the band is obtained by performing a 7-point variation of ( µ F , µ R ) around the centralscale µ cent = Q . We observe that the NLO, NNLO and N LO K-factors are relativelyindependent of the centre of mass energy. Furthermore, we see that the bands due to scale– 8 – � �� �� �� �� �� �� �� ������������ ��� [ ��� ] σ / σ � � � � ����������� ��������������������� σ �� → � + � [ �� ] � � → � + + ���������� _ ���� _ �� μ ����� = � = ��� ��� �� �� �� �� �� �� �� �� ������������ ��� [ ��� ] σ / σ � � � � ����������� ��������������������� σ �� → � + � [ �� ] � � → � - + ���������� _ ���� _ �� μ ����� = � = ��� ��� Figure 7 : The cross sections for producing a W + (left) or W − (right) as a function ofthe hadronic centre of mass energy for Q = 100 GeV. The uncertainty bands are obtainedby varying µ F and µ R around the central scale µ cent = Q (see text for details).variation at NNLO and N LO do not overlap for a large range of center of mass energies.However, the gap is narrowed at the extreme end of the range of energies considered here.Parton distribution functions are extracted from a large set of measurements and areconsequently subject to an uncertainty related to the input as well as to the methodologyused to extract the PDFs. Here, we follow the prescription of ref. [84] for the compu-tation of PDF uncertainties δ (PDF) using the Monte Carlo method. Furthermore, alsothe strong coupling constant is an input parameter for our computation. The PDF set PDF4LHC15 nnlo mc uses α S = 0 .
118 as a central value and two additional PDF sets areavailable that allow for the correlated variation of the strong coupling constant in thepartonic cross section and the PDF sets to α up S = 0 . α down S = 0 . δ ( α S ) on our cross section following the prescription ofref. [84]. We combine the PDF and strong coupling constant uncertainties in quadratureto give δ (PDF + α S ) = (cid:112) δ (PDF) + δ ( α S ) . (3.3)In our computation we use NNLO-PDFs, because currently there is no available PDFset extracted from data with N LO accuracy. It is tantalising to speculate if the observedconvergence pattern is related to the mismatch in perturbative order used for the PDFs andthe partonic cross section. We estimate the potential impact of this mismatch on our crosssection predictions using a prescription introduced in ref. [5] that studies the variation ofthe NNLO cross section as NNLO- or NLO-PDFs are used. This defines the PDF theoryuncertainty δ (PDF-TH) = 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ (2), NNLO-PDFs W ± − σ (2), NLO-PDFs W ± σ (2), NNLO-PDFs W ± (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.4)Here, the factor is introduced as it is expected that this effect becomes smaller at N LOcompared to NNLO. – 9 – ( ��� ) δ ( ��� + α � ) δ ( ��� + α � )+ δ ( ��� - �� ) ��� ��� ��� ��� ���� ���� ���� ���� ���� - �� - � - � - � - � - � - � - � - � - ������������� � [ ��� ] � � ��� � � � � � � [ % ] ��� � α � ���������������� �� ������������ _ ���� _ ��� � → � + + � δ ( ��� ) δ ( ��� + α � ) δ ( ��� + α � )+ δ ( ��� - �� ) ��� ��� ��� ��� ���� ���� ���� ���� ���� - �� - � - � - � - � - � - � - � - � - ������������� � [ ��� ] � � ��� � � � � � � [ % ] ��� � α � ���������������� �� ������������ _ ���� _ ��� � → � - + � Figure 8 : Sources of uncertainty as a function of Q for the W + (left) and W − (right)K-factors. δ (PDF), δ (PDF+ α S ) and the sum of δ (PDF+ α S ) and δ (PDF-TH) are shownin orange, red and green respectively. The dashed magenta line indicates the physical Wboson mass, Q = m W .Figure 8 displays the uncertainties δ (PDF), δ (PDF+ α S ) and δ (PDF-TH) as a functionof Q in orange, red and green respectively. In particular, the green band indicates the sum δ (PDF + α S ) + δ (PDF-TH). Our findings for δ (PDF) are compatible with the results of forexample refs. [84, 87] where PDF effects on W boson cross sections were discussed in moredetail. We observe that the estimate for δ (PDF-TH) plays a significant role especially forlow values of Q . The traditional PDF uncertainty has a stronger impact for larger valuesof Q . Overall, we observe that the relative size of δ (PDF) and δ (PDF-TH) is large incomparison to the effect of varying the scales. We conclude that future improvements inthe precision of the prediction of this observable will have to tackle the problem of theuncertainties discussed here. In particular, we emphasize that the relatively large size of δ (PDF-TH) can potentially have a substantial impact on the central value of the N LOcorrection, especially for smaller values of Q . As discussed above, there are large intricatecancellations between different initial state channels at N LO. This implies that a smallrelative change of quark vs. gluon parton densities at N LO may have an enhanced effecton the perturbative cross section as a result. We can only wonder if the usage of trueN LO parton densities could lead to N LO predictions that are fully contained in the scalevariation band of the previous order. However, in the absence of N LO PDFs, we canonly stress the importance estimating an uncertainty due to the missing N LO PDFs andsuggest δ (PDF-TH) as a possible estimator. In the previous section, we have seen that the conventional variation of the perturbativescales by a factor of 2 does not give a reliable estimate of the size of the missing higherorders. This motivates us to study the ratios of cross sections for the production of gaugebosons with virtuality Q : R XY ( Q ) = σ X ( Q ) σ Y ( Q ) , X, Y ∈ { W ± , γ ∗ } . (4.1)– 10 –ndeed, since the charged- and neutral-current Drell-Yan processes show very similar K-factors and dependences on the perturbative scales, it is conceivable that some of theuncertainties (e.g., PDF effects) cancel in the ratio, and the ratios may exhibit an enhancedperturbative stability. In the remainder of this section we analyse and compare differentprescriptions to estimate the missing higher orders in the perturbative expansion of thecross section ratios for √ S = 13 TeV. We focus here on the following prescriptions: • Prescription A:
We take the ratio of the perturbative expansion of the numeratorand the denominator computed at a given order in perturbation theory. We choosethe renormalisation and factorisation scales in the numerator and denominator ina correlated way, i.e., we always choose the same values for the scales in both thenumerator and the denominator. • Prescription B:
Similar to Prescription A, but we do not correlate the renormalisa-tion and factorisation scales between the numerator and the denominator. In otherwords, we perform independently a 7-point variation of the scales in the numeratorand the denominator, and we take the envelope of the values obtained. • Prescription A (cid:48) : We choose the scales in a correlated way, but we expand the ratioin perturbation theory and only retain terms through a given order in the strongcoupling. • Prescription B (cid:48) : Similar to Prescription A (cid:48) , but we choose the scales in an uncor-related way. • Prescription C:
We take the relative size of the last considered order compared tothe previous one as an estimator of the perturbative uncertainty: δ (pert.) = ± (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − R ( n ) XY ( Q ) R ( n − XY ( Q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × . (4.2)The superscript n on R ( n ) XY ( Q ) indicates the order at which we truncate the pertur-bative expansion. The values of µ F = µ R = µ cent are chosen in a correlated way inthe numerator and the denominator of R XY . This estimator is based on the expec-tation that in a well-behaved perturbative expansion the subleading terms should besmaller than the last known correction. By construction, it leads to an estimate ofthe missing higher orders that is symmetric around the central value.Note that Prescriptions A, A (cid:48) , B and B (cid:48) are fully equivalent to all orders in perturbationtheory. The truncation of the perturbative series can, however, introduce differences inthe results obtained from these four prescriptions, especially at low orders in perturbationtheory. For example, the bands obtained by varying the scales will always be larger ifthe scales are varied in an uncorrelated way, because in that case the size of the band isobtained by taking the envelope of a strictly larger set of values.Table 2 shows the prediction for R W + W − for Q = m W computed at the first feworders in perturbation theory. First, we see that the central value is extremely stable in– 11 –LO NNLO N LO µ cent m W m W / m W m W / m W m W /
2A 1 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . A (cid:48) . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . B 1 . +8 . − . . +12 . − . . +2 . − . . +2 . − . . +2 . − . . +2 . − . B (cid:48) . +5 . − . . +8 . − . . +1 . − . . +2 . − . . +2 . − . . +4 . − . C 1 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . Table 2 : The ratio R W + W − for Q = m W computed for different values of µ cent and withthe different prescriptions mentioned in the text.perturbation theory, changing only at the permille level as we go from NNLO to N LO. Thecentral value is pretty much independent of the choice of the central scale µ cent and whetherthe ratio is expanded in perturbation theory or not (primed vs. unprimed prescriptions).While in general the different prescriptions lead to vastly different estimates of the missinghigher orders, the predictions are similar between the primed vs. unprimed prescriptions,especially as we increase the perturbative order. This is to be expected: If the perturbativeorder is increased, the differences stemming from expanding or not the denominator ofthe ratio should decrease, which is indeed what we observe. We therefore only discussthe unprimed prescriptions from now on. Second, we observe that Prescription B leadsto an estimate that is more than an order of magnitude larger than for PrescriptionsA and C. In particular, this makes one wonder if correlated scales (Prescription A &C) tend to underestimate the size of the missing higher-order terms beyond N LO. Webelieve that results obtained from uncorrelated scales (Prescription B) lead to estimatesthat are too conservative. Indeed, since the central value of the ratio only receives permille-level corrections from NNLO to N LO and exhibits extremely good perturbative stability,one expects higher-order corrections to be at the sub-permille level, which is indeed thesize of the band obtained by varying the scales in a correlated way (Prescription A &C). It would be unreasonable to expect that the missing higher-order corrections shiftthe central value by 1% or even more, which is the size of the bands obtained from theuncorrelated prescription (Prescription B). A correlated prescription is also motivated bythe fact that the neutral- and charged-current processes are expected to receive very similarQCD corrections, a fact which is corroborated by the results from the previous section.Finally, we observe that Prescription C leads to an estimate that is always slightly larger(by a factor ∼ Q = m W ) than the one obtained from Prescription A at N LO. Wehave observed that the size of the higher-order terms estimated from Prescription A alwaysencompasses the next order in perturbation theory. In order words, Prescription A appearsto account for the effect of missing higher orders even though it estimates relatively smallresidual uncertainties for the point Q = m W . Below we study ratios of cross sections as afunction of a range of different values of Q allowing us to comment on Prescription A inmore detail. – 12 – ���������� ������������������������� � � + � - ����� � � + � - ����������������� _ ���� _ �� μ ����� = �������������� ��� ��� ��� ��� ���� ���� ���� ���� ������������������������������������ � [ ��� ] � � + � - / � � + � - ( � ) ����������� ��������� � � + � - ����� � � + � - ����������������� _ ���� _ �� μ ����� = �������������� ��� ��� ��� ��� ���� ���� ���� ���� �������������������� � [ ��� ] � � + � - / � � + � - ( � ) ����������� ������������������������� � � + � - ����� � � + � - ����������������� _ ���� _ �� μ ����� = �������������� ��� ��� ��� ��� ���� ���� ���� ���� ������������������������������������ � [ ��� ] � � + � - / � � + � - ( � ) Figure 9 : The upper panels show the ratio R W + W − (left) with bands computed withPrescription A (left), Prescription B (middle) and Prescription C (right). The lower panelsshow the same normalised to the value of the ratio computed at N LO. The dashed magentaline indicates the physical W boson mass, Q = m W .Figure 9 shows the ratio R W + W − as a function of the virtuality Q . The bands werecreated using Prescription A (left), B (middle) and C (right). Just like for on-shell produc-tion, we observe that correlated and uncorrelated scales lead to vastly different estimatesfor the size of the bands. In particular, Prescription B gives an extremely conservative es-timate of the bands over the whole range of Q considered. Unlike for on-shell production,the bands obtained from Prescription A do not overlap for Q >
200 GeV, which indicatesthat Prescription A does not correctly capture the size of higher-order corrections in thisrange of virtualities. In general, this calls into question whether correlated scale variationsshould be used as an uncertainty estimator for ratios of cross sections with very similarK-factors. Instead, Prescription C seems to give the most reliable estimator of the size ofthe residual perturbative corrections for ratios of cross sections over the whole range ofvirtualities considered. ����������� �� �� �� �� ��� ��� ��� ��� ��� ��������������������� � [ ��� ] � � + � - ����� � � + � - ��� �������������� _ ���� _ �� μ ����� = ������������� � ����������� �� �� �� �� ��� ��� ��� ��� ��� �������������� � [ ��� ] � � + γ * ����� � � + γ * ����������������� _ ���� _ �� μ ����� = �������������� ����������� �� �� �� �� ��� ��� ��� ��� ��� ������������ � [ ��� ] � � - γ * ����� � � - γ * ����������������� _ ���� _ �� μ ����� = �������������� Figure 10 : The ratios R W + W − (left), R W + γ ∗ (middle) and R W + γ ∗ (right) as a function ofthe virtuablity Q . The uncertainty bands are obtained with Prescription C for the centralscale µ cent = Q . The dashed magenta line indicates the physical W boson mass, Q = m W .In fig. 10 we extend our analysis from R W + W − (left) to R W + γ ∗ (middle) and R W + γ ∗ (right). In all cases the bands are estimated using Prescription C. Similar to our discussionin the previous paragraph, we find that Prescription C delivers reliable estimates also forthe latter two ratios. In all cases we find that the residual perturbative uncertainty is verysmall, making ratios of production cross sections of electroweak gauge bosons very stableunder perturbative corrections, and therefore ideal precision observables.While Prescription C seems to give reliable estimates for all cross section ratios con-sidered over a wide range of kinematics, we have to point out that Prescription C has theobvious shortcoming that it gives a vanishing result whenever two consecutive perturbative– 13 –rders give identical numerical predictions. The same fact leads to rather unconventionalshapes of the uncertainty band as a function of Q . Consequently, prescription C on its ownwould not serve as a good estimator of perturbative uncertainties. For example, it pre-dicts vanishing perturbative uncertainty of R W + W − at NLO around Q = 700 GeV, whichis clearly unreliable. In order to estimate perturbative uncertainties we therefore suggestthat multiple different perspectives and estimators should be considered, in order to testthe quality of the estimates obtained. For the future, it would be interesting to investi-gate other prescriptions to estimate the impact of missing perturbative orders, includingprescriptions based on statistical methods [85, 86]. A more detailed study of these effects,however, would go beyond the scope of this paper.We finish by presenting results for an observable which is closely related to the ratio R W + W − . The lepton-charge asymmetry is defined as: A W ( Q ) = σ W + ( Q ) − σ W − ( Q ) σ W + ( Q ) + σ W − ( Q ) = R W + W − ( Q ) − R W + W − ( Q ) + 1 . (4.3) ��� ���� ���� ������������������������ � � ������ ������������ �������������� _ ���� _ �� μ ����� = ������������� � ��� ��� ��� ��� ���� ���� ���� ���� �������������������������� � [ ��� ] � � / � � ( � ) Figure 11 : The lepton-charge asymmetry A W as a function of the virtuality Q . Theuncertainty bands are obtained with Prescription C. The dashed magenta line indicatesthe physical W boson mass, Q = m W .Figure 11 shows our predictions for the lepton-charge asymmetry as a function of Q atdifferent orders in perturbation theory. All uncertainty bands are obtained from Prescrip-tion C. Just like the cross section ratios studied earlier, we observe a good perturbativestability and a very small residual dependence on the perturbative scales at N LO. Inparticular, for Q = m W , we find A (2) W ( m W ) = 0 . +0 . − . ,A (3) W ( m W ) = 0 . +0 . − . . (4.4)– 14 –o far we have only discussed the uncertainties on cross section ratios from the trun-cation of perturbative orders. We therefore conclude by commenting on uncertainties onratios related to PDFs, similar to those considered in Section 3. Just like in the case of theperturbative uncertainty, a choice has to be made whether or not to treat PDF and strongcoupling constant variation as correlated in numerator and denominator of the ratio. Thefact that PDFs and α S are universal quantities suggests indeed a correlated treatment. Inthis case, ratios of cross sections could provide a remarkable tool to reduce some of thelargest theoretical uncertainties afflicting the observables in question. However, in neutralcurrent and charged current DY production cross section different combinations of partondistribution functions play a dominant role. This could indeed spoil a correlated treatmentof PDF uncertainties and should be studied in more detail. In this paper we have computed for the first time the N LO corrections to the inclusiveproduction cross section of a lepton-neutrino pair at a proton-proton collider in QCDperturbation theory. One of the main results of this article are analytic formulæ for thepartonic coefficient functions for this cross section, which we make available as ancillarymaterial with the arXiv submission of this paper.We have studied the phenomenological impact of our results by providing numericalresults for (off-shell) W -production at the LHC. All our results are differential in thevirtuality of the W boson, or equivalently in the invariant mass of the lepton-neutrino pair.We find that the N LO corrections are at the level of a few percent and they stabilise theperturbative progression of the series. We have also studied ratios of cross sections for W + , W − and γ ∗ , as well as the charge asymmetry at the LHC, i.e., the amount of produced W + bosons relative to the amount W − bosons. We find that these ratios feature a remarkableperturbative stability and that they can be predicted with very high precision. For thefuture, it would be interesting to study, in addition to the QCD corrections discussed here,how the inclusive production probability of weak bosons is impacted by electroweak andQED corrections, for example the electroweak and mixed QCD-electroweak corrections toDrell-Yan processes of refs. [88–98]. While such a study is beyond the scope of this article,we would like to stress their importance here.Combined with the results for virtual photon production of ref. [10], our results havealso allowed us to investigate the progression of the perturbative series through N LO inQCD for one of the simplest classes of hadron collider observables, namely fully inclusivevector boson production cross sections. Understanding the perturbative convergence ofthis class of processes is an important proxy to understand the precision that can bereached more generally for LHC observables at this order in perturbation theory. Herewe summarise our main observations and conclusions, and the possible implications fora precision physics program at the LHC. First, we observe in all cases that the N LOcorrections shift the central value of the predictions by a few percent. This implies that,very likely, also for more differential observables percent-level precision can only be achievedafter the inclusion of N LO corrections. It would be interesting to develop and extend– 15 –echniques to perform differential calculations at N LO (see, e.g., refs. [12–14, 99–101] forfirst steps in this direction). Second, we observe that the uncertainties related to PDFsgenerically dominate over the residual perturbative uncertainty at N LO. This motivates topush for determining PDFs at this order in perturbation theory, by extracting them fromexperimental measurements confronted to theory calculations performed at the same order,and to evolve them using the (yet unknown) DGLAP evolution kernels at N LO. Finally, weobserve that the conventional method to estimate the missing higher-order terms by varyingthe factorisation and renormalisation scales by a factor of two around a hard scale does notgive reliable results at N LO. This calls for an improved method to estimate the missinghigher-order terms, e.g., by studying the progression of the perturbative series as done here,or by considering statistically-motivated techniques such as those of refs. [85, 86]. However,we have also observed that the determination of the residual perturbative uncertainty maynot be completely decoupled from the study of PDF effects: Indeed, we observe substantialcancellations between different partonic channels. It is tantalising to speculate if thesecancellations are responsible for the non-overlapping scale variation bands at N LO, andif they persist once a complete set of N LO PDFs is available. A detailed study of theseeffects, however, goes beyond the scope of this paper and is left for future work.
Acknowledgements
We would like to thank L. Harland-Lang and R. Thorne for useful comments and forpointing out a mistake in fig. 8 of the first version of our article on the arXiv. This work issupported in part by the European Research Council grant No 637019 “MathAm” (CD).FD and BM were supported by the Department of Energy, Contract DE-AC02-76SF00515.BM was also supported by the Pappalardo Fellowship.
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