NNLO QCD study of polarised W^+ W^- production at the LHC
PPrepared for submission to JHEP
NNLO QCD study of polarised W + W − production atthe LHC Rene Poncelet and Andrei Popescu
Cavendish Laboratory, University of Cambridge,J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
E-mail: [email protected] , [email protected] Abstract:
Longitudinal polarisation of the weak bosons is a direct consequence of Elec-troweak symmetry breaking mechanism providing an insight into its nature, and is instru-mental in searches for physics beyond the Standard Model. We perform a polarisationstudy of the diboson production in the pp → e + ν e µ − ¯ ν µ process at NNLO QCD in the fidu-cial setup inspired by experimental measurements at ATLAS. This is the first polarisationstudy at NNLO. We employ the double pole approximation framework for the polarisedcalculation, and investigate NNLO effects arising in differential distributions. Keywords:
Electroweak bosons, Polarisation, NNLO QCD, Diboson, LHC
CAVENDISH–HEP–21/03 a r X i v : . [ h e p - ph ] F e b ontents Weak boson polarisation is under intense research both on the theoretical and experimentalside. It is a handle to directly probe the Standard Model (SM) electroweak (EW) symmetrybreaking mechanism and is instrumental in constraining the triple and quartic gauge bosoncouplings for beyond SM physics searches.Several processes have been studied theoretically in the context of weak boson polar-isation. Seminal papers covered the V +j process [1, 2]. Later on other processes wereconsidered, such as diboson production [3–6], vector boson scattering (VBS) [7–9]. Top-quark decays are currently under investigation.The amount of statistics of Run 2 at the LHC has enabled polarised measurementin relatively high cross section processes such as V +jet [10–13], boson [14, 15] and top-quark pair production [16, 17]. There are good prospects for polarised VBS signals athigh-luminosity LHC [18] and there already are some results available [19]. It is impossibleto directly select bosons with a specified polarisation, but methods like reweighting proce-dures [10–12, 20] and artificial intelligence techniques [21–24] allow for analysis of polarisedsignals. The main result is the extracted polarisation coefficients which are then comparedto theoretical values. Close attention is paid to differential distributions for longitudinallypolarised bosons, which is a direct probe of the EW symmetry breaking mechanism.The production of weak boson pairs has been extensively studied in the literature.Next-to-leading order (NLO) [25–27], next-to-next-to-leading order (NNLO) [28–30] and– 1 –ombined NLO EW and (N)NLO QCD [31, 32] computations are available for a variety ofsetups and observables. Resummation and parton shower effects have also been studied inthe context of weak boson pair production [33–37]. Recent progress has been made in thecomputation of NLO corrections to cross sections for polarised bosons [5, 6]. There are twomain obstacles that are in the way of direct theoretical calculations with polarised boson inthe experimentally accessible signatures. Firstly, weak bosons are short-lived particles, andthey can be observed only through their leptonic and hadronic decay signatures. They areproduced off-shell and some adjustment is required to make sense of their polarisation state.Secondly, the signatures involve a non-resonant background which cannot be removed in asimple manner, because it is essential for gauge invariance of the whole amplitude. Effectsof the gauge invariance breakdown are severe [8]. The commonly used approach to tackleboth issues is to use on-shell amplitudes which can be obtained either by restricting theintegration phase space in the way of the Narrow Width Approximation [38, 39] or bymeans of an on-shell projection (OSP) [7, 8], also know as pole approximation. Regardlessof particular implementation, a method that uses on-shell amplitudes has an intrinsicuncertainty of O ( Γ M ) but it is still advantageous in comparison with the indirect approachinvolving reweighting which is used in experimental analysis as shown in the case of WZproduction [8].In this paper we address, for the first time, NNLO QCD correction for the polarisedW + W − production. We compute fiducial and differential cross sections at NNLO QCDaccuracy for the LHC at 13 TeV and investigate what are the effects at this precision level.NNLO corrections are particularly important for the differential distributions in dibosonproduction, where NLO scale uncertainty exceeds the intrinsic uncertainty related to thetheoretical definition of boson polarisation. Additionally, we explore how narrow widthapproximation performs in comparison with double pole approximation which is assumedto be more accurate due to its incorporation of off-shell effects.The structure of this paper is as follows. In Section 2.1 we discuss approaches whichwe apply to define boson polarisations in the diboson production process. Then we specifyour setup including the SM parameters, selection cuts, and a list of computational toolsthat we use. Section 3 is dedicated to our results. We present the integrated cross sections,and discuss the pure NNLO QCD corrections in Section 3.2. We add the loop-inducedchannel and discuss its effects in Section 3.3. In Section 3.2 we compare the Narrow-Width-Approximation and the double pole approximation for unpolarised weak bosonsagainst a off-shell computation. In Section 4 we summarise our findings. In this paper we study the resonant production and subsequent decay of (un)polarised W + W − -boson pairs at the LHC in the different flavour di-leptonic decay channel, i.e.pp → W + W − + X → e + ν e µ − ¯ ν µ + X . (2.1) Non-resonant as opposed to double-resonant. More precisely, single-resonant. – 2 – ¯ q W + W − e + ν e µ − ¯ ν µ (a) q ¯ q γ/Z W + W − e + ν e µ − ¯ ν µ (b) q ¯ q l ∓ ( ν l ) ν ( l ∓ ) l (cid:48)± ν (cid:48) l γ/Z W ± (c) gg W + W − e + ν e µ − ¯ ν µ (d) gg γ/Z e + ν e µ − ¯ ν µ (e) Figure 1 . Selected diagrams contributing to the pp → e + ν e µ − ¯ ν µ process. Diagrams (a,b) rep-resent Born double resonant contribution; (c) – general case of Born single resonant contributions(background); (d,e) – loop-induced double resonant contribution; Figure 1(a) and Figure 1(b) show the contributing double resonant
Feynman diagrams.The resonant process is a part of the more general off-shell production of the same leptonicfinal state pp → e + ν e µ − ¯ ν µ + X (2.2)This process has additional contributions from single resonant diagrams, see Figure 1(c).To define the production and decay of polarised intermediate vector-bosons in a gauge-invariant way both W-bosons are required to be on their mass-shell. We consider two com-monly used approximations both resulting in on-shell amplitudes for polarised W-bosons:the so-called pole approximation or, in this case, double pole approximation (DPA), andthe Narrow-Width-Approximation (NWA). Both methods neglect single resonant contri-butions present in the general process pp → e + ν e µ − ¯ ν µ and introduce uncertainties whichare formally of O (Γ W /M W ). While this error estimate holds for inclusive observables, theuncertainty in differential distributions can be significantly larger.NWA considers the limit Γ W /m W → O (Γ W /m W )terms, rendering the W-bosons on-shell, and factorizing the amplitudes and phase spacesof production and decay. The NWA works well with massive short-lived particles suchas weak bosons and top-quarks, and so it is well suited for this study. By constructionthis approximation performs poorly for observables which are sensitive to the off-shellnessof the vector bosons. There exist extensions to the NWA, which attempt to include off-shell effects, such as the Madspin approach [40], which simulates the off-shell effects byusing the Breit-Wigner sampling for the resonant propagator. We do not consider such anextension here. The production and decay are correlated through the boson’s momentaand polarisations, schematically the amplitude factorizes as follows: M pp → e + ν e µ − ¯ ν µ ∼ (cid:88) h,h (cid:48) ∈ Λ M h,h (cid:48) pp → W + W − Γ h W + → e + ν e Γ h (cid:48) W − → µ − ¯ ν µ , (2.3)– 3 –here h, h (cid:48) ∈ Λ = { + , − , L } stand for the W-boson polarisations. By restricting the sumto specific choices for h, h (cid:48) we define the polarised production and decay. We denote thecoherent sum of the transverse polarisations { + , −} as (T) and the longitudinal polarisationas (L).The DPA [25, 41, 42] approach instead considers the off-shell phase space and intro-duces an approximation for the amplitudes alone. In order to guarantee gauge invariance,one defines an on-shell projection to map the off-shell kinematics of the decay productspoint-by-point to the on-shell kinematics. This allows the same factorization as in Eq. (2.3)by neglecting single-resonant diagrams. Boson propagators with the off-shell kinematicsare kept for modelling the Breit-Wigner shape of the off-shell amplitude.DPA has an advantage over NWA in that it generates the off-shell kinematics, howeverit is not uniquely defined. The on-shell mapping has to be specified, whereas NWA approachis unambiguous. We will compare performance of these two methods in Section 3.4.For the polarisation study at NNLO we will use the DPA approach. We follow [7] indefining the on-shell projection, where the following conserved quantities are suggested:1. the total diboson momentum p W + W − ;2. the direction of p W + momentum in diboson centre-of-mass frame (after a direct boostin Lab frame);3. the angles of charged leptons w.r.t to their parent boson momentum (in Lab frame)in their parent W-boson centre-of-mass frame (after a direct boost in Lab frame).The algorithm goes as follows. Consider the diboson mass frame by a direct boostfrom lab frame. In this frame individual boson momenta are equal and back-to-back, butgenerally not on-shell. To correct this, for each boson momenta, we fix the energy to be √ s/ (cid:113) s − M , while the anglesare kept untouched. These modifications do not affect the total momentum of the dibosonsystem. Next, we turn to the decay products. In order to modify momenta of e + , ν e wereconstruct helicity frames in W + -boson rest frame, and calculate angles of the positronusing the original off-shell kinematics. While polar angle definition is unambiguous, therotation of the XY-plane and thus the azimuthal angle is subject to specification. Wediscuss the azimuthal angle definition we use in App. A. With the new on-shell W + -bosonmomentum we construct the new positron momentum in the new parent boson rest framewith the original polar and azimuth angles. The neutrino momentum is trivially inferred.Analogously, we build the new momenta for decay products of W − .The process under consideration allows for unambiguous on-shell mapping, but if thereexists an ambiguity around combining the decay products into parent resonances, such asappearing in ZZ production or NLO EW radiation, the OSP should be revised. A treatmentof non-factorisable corrections and a generic massive particle configuration can be foundin [43].The definition of the boson polarisation vectors entering the polarised production anddecay amplitudes is not unambiguous and needs to be chosen. A particular choice, which– 4 –e employ in this work, based on momenta in the laboratory frame, is ε µ − = 1 √ , cos θ V cos φ V + i sin φ V , cos θ V sin φ V − i cos φ V , − sin θ V ) ,ε µ + = 1 √ , − cos θ V cos φ V + i sin φ V , − cos θ V sin φ V − i cos φ V , sin θ V ) ,ε µ L = 1 M ( p, E sin θ V cos φ V , E sin θ V sin φ, E cos θ V ) , (2.4)for left, right, and longitudinal polarisations respectfully. Here M, p, E are mass, totalmomentum, and energy of the weak boson, and θ V , φ V are its angles in a selected frame.In this study we define polarisation vectors in the laboratory frame which is moreaccessible experimentally. However there exist other alternatives, e.g. the diboson centre-of-mass frame, which was also used in the experimental studies [19]. It has been observedthat the frame choices tend to be rather complementary to each other in their discriminationpower to isolate polarisations [6]. To fully specify our computational setup we give a summary of all numerical input param-eters. We use the following set of particle parameters: M osW = 80 . , Γ osW = 2 . ,M osZ = 91 . , Γ osZ = 2 . ,M H = 125 GeV , Γ H = 0 . ,m t = 173 GeV , m b = 4 . M V = M OS V (cid:113) OS V /M OS V ) , Γ V = Γ OS V (cid:113) OS V /M OS V ) ; (2.6)for V = W , Z. All leptons are considered massless, which makes the results insensitiveto the specific lepton flavours as long as they belong to different generations in order forthe diboson reconstruction to remain unique. All other quarks (u , d , c , s) are considered asmassless.We consider the 5-flavour PDF set NNPDF31 [n]nlo as 0118 (IDs: 303400, 303600)approximation for [N]NLO [45] as implemented in LHAPDF [46]. However, we use massivebottom quarks throughout the calculation to avoid contributions from off-shell top-quarkpair production which would enter at NNLO and would be regarded as a separate process.The numerical impact of the mismatch in the number of massless quarks between the PDFand the perturbative part should be small for any observable studied in this work. Wewould like to point out that the scheme we use is — up to NLO — effectively the same asremoving processes with a b -quark in the initial state.– 5 –e use the complex mass scheme framework [47] and the couplings are fixed followingthe G µ scheme with G µ = 1 . · − GeV − . (2.7)Within the frameworks of NWA and DPA we set weak boson (W and Z) widths to zero inthe calculation of couplings and the Weinberg angle, so they remain real.Both factorisation and renormalisation scales are set to W pole mass: µ F = µ R = M W .The cuts we use are presented as fiducial setup in Ref. [5] inspired by ATLAS mea-surements [14]: • minimum transverse momentum of the charged leptons, p T ,(cid:96) >
27 GeV; • maximum rapidity of the charged leptons, | y (cid:96) | < . • minimum missing transverse momentum, p T , miss >
20 GeV; • veto on events containing at least one jet candidate with p T , j >
35 GeV , | η j | < . • minimum invariant mass of the charged lepton-pair system, M e + µ − >
55 GeV.Also, by construction, DPA and NWA contain an implicit cut on diboson invariant mass: M W + W − > · M W .The jet veto is used to reduce giant K-factors [30] otherwise mostly appearing at NLObut also driving up NNLO corrections. CKM matrix is assumed to be diagonal. Finally,the invariant mass cut reduces the Higgs background in the gg -initiated process. Note thatwe did not apply the M W + W − >
130 GeV cut for gg -initiated process cut as suggested in[5] to exclude the Higgs peak region as it has little effect on the results. The computation has been done using
Stripper framework, a
C++ implementation ofthe four-dimensional formulation of the sector-improved residue subtraction scheme [48].
Stripper is a library which supplies a Monte-Carlo generator and automates the subtrac-tion scheme, and it relies on external tools for calculating tree-level, one-loop and two-loopamplitudes. It has been successfully applied to the production of top-quark pairs [49, 50],inclusive jets [51], and three photons [52]. We use
AvH library [53] to provide the Bornamplitudes. The one-loop amplitudes are calculated using O
PEN L OOPS q ¯ q -induced channel were provided by VVamp project [57].Several checks have been performed both on the integrated cross section level and perphase space point. The total cross section for the off-shell setup calculated within Stripperframework was checked against
Matrix at NNLO [58] in the inclusive setup. Our privatebuild of O
PEN L OOPS
Recola used in [5] for various DPA setups. We checked our differential distributions atNLO against the ones provided in [5] as well as the total cross section results for variouspolarised setups. – 6 –
Results
In this section, we present phenomenological results for the polarised signals in W-pairproduction in the fiducial setup on the LHC at a hadronic centre of mass energy of 13 TeV.Diboson production and its further decay into leptons is represented by the diagramsin Figure 1. The loop-induced contribution in Figures 1(d)–1(e) enters the calculationfor the first time at NNLO in α s . It is effectively a LO contribution which introducessubstantial corrections. In what follows we will refer to NNLO corrections to diagramsin Figures 1(a)–1(c) as to ’NNLO (without LI)’, or just ’NNLO’, and to corrections thatinclude the loop-induced channel as to ’NNLO (with LI)’ or ’NNLO+LI’ corrections. Thepolarisation setups are identified by its polarised boson. We will abbreviate polarisationsetups by two letters out of the set { U, T, L } , which correspond to unpolarised, transverse,and longitudinal boson polarisation respectfully. For example, the singly polarised setupwith longitudinal W + boson will read ’LU’.We provide LO, NLO, and NNLO results, the NNLO K-factor which is calculated as σ NNLO /σ NLO at the central scale, and the NNLO+LI which includes the loop-inducedchannel. Scale uncertainties are calculated using the standard independent 7-point varia-tion of µ R , µ F by a factor of 2 around the central scale, with the restriction 1 / ≤ µ R /µ F ≤
2. After discussing the fiducial cross section in the next section, we will turn our focus on’NNLO (without LI)’ corrections to differential distributions Section 3.2. The effects of theloop-induced channel on differential distributions will be explored in Section 3.3. Finally, acomparison on differential level of the DPA and NWA approximations against the off-shellcomputation will be performed in Section 3.4.
The total cross-section results for various polarisation setups are presented in Table 1. Italso includes unpolarised calculations performed in the frameworks of DPA, NWA, and theoff-shell calculation. Scale uncertainties are presented in percentage values with respectto the central scale result as sub- and superscripts. Monte-Carlo numerical errors on thecentral scale values are indicated in parentheses and correspond to the last significant digitof the result.In the unpolarised setups we see that DPA undershoots the off-shell calculation by2.5%. DPA is not supposed to fully match the off-shell calculation as it only includesdouble-resonant contributions relevant for the diboson production. This fraction persistsafter inclusion of NNLO corrections both with and without the loop-induced channel.In contrast to DPA, NWA result overshoots the off-shell result by 1%. In both casesthe differences to the complete off-shell are well within their expectation of O (Γ W /M W ),even though this estimate is only exact for inclusive phase space integration. The scaleuncertainty decreases by a factor of 3 with NNLO corrections, however after introductionof the loop-induced channel bounces back to 80% and 50% of the NLO level for the higherand lower band correspondingly. Higher order corrections to the loop-induced channel,– 7 – O NLO NNLO K NNLO
NNLO+LIoff-shell 202 . +4 . − . . +1 . − . . +0 . − . . +1 . − . unpol. (nwa) 202 . +4 . − . . +1 . − . . +0 . − . . +1 . − . unpol. (dpa) 196 . +4 . − . . +1 . − . . +0 . − . . +1 . − . W + L (dpa) 50 . +5 . − . . +1 . − . . +0 . − . . +1 . − . W − L (dpa) 57 . +5 . − . . +1 . − . . +0 . − . . +0 . − . W + T (dpa) 141 . +4 . − . . +1 . − . . +0 . − . . +1 . − . W − T (dpa) 144 . +4 . − . . +1 . − . . +0 . − . . +1 . − . W + L W − L (dpa) 6 . +4 . − . . +3 . − . . +1 . − . . +2 . − . W + L W − T (dpa) 44 . +5 . − . . +1 . − . . +0 . − . . +0 . − . W + T W − L (dpa) 50 . +5 . − . . +1 . − . . +0 . − . . +0 . − . W + T W − T (dpa) 99 . +3 . − . . +1 . − . . +0 . − . . +2 . − . Table 1 . Total cross-sections (in fb) for the unpolarised, singly-polarised and doubly-polarisedW + W − production at the LHC. Unpolarised calculation is performed in three ways: full off-shell,and using approximations: DPA, NWA. Polarised setups are calculated using DPA. Uncertaintiesare computed with 7-point scale variations by a factor of 2 around the central scale. K -factors arepresented as ratios of NNLO QCD (without LI) over NLO integrated cross-sections. which are formally of O (cid:0) α (cid:1) (part of N LO), are expected to cure this behaviour but areleft for future work.Next we consider singly-polarised setups. Missing interferences between longitudinaland transverse polarisations of W + (W − ) and restrictions on the leptonic phase space dueto cuts, give rise to differences between the sum of the single polarised setups and the fullyunpolarised DPA setup. The NNLO K-factors are of the same magnitude across setups,with a slightly larger value for the longitudinal polarisations. The scale uncertainty featuresthe same behaviour as for the unpolarised setups, except in the longitudinal setups it isnot amplified by the loop-induced channel.NNLO corrections (without LI) to the doubly-polarised setups can be estimated tobe roughly 25% of NLO corrections, except for TL setup where NNLO corrections are abit larger. It was shown that at NLO the double-longitudinal polarisation of the dibosonsystem is, among other polarisation setups, particularly affected by QCD corrections [5].This is also true at NNLO as represented by the corresponding K-factor. As will be shownfurther, the profile of its corrections is also distinctly different on the differential level. Thescale variation band goes down at NNLO, however the loop-induced channel brings it backto NLO level at both LL and TT setups, whereas for LT and TL setups it remains on thesame level.Of interest are the polarisation fractions, i.e. the fractions of the cross section for vari-ous polarised boson configurations. Although NNLO corrections differ among the differentpolarisations, there is no significant difference in the polarisation fractions with respect to– 8 – − − d σ / d M e + , µ − [ f b ] LHC 13 TeV, fiducial cuts NNPDF31 as 0118 7-scale scheme ( µ = M W ) NLO off-shellNLO unpol. (dpa)Summed W + ,W − pol. NLO W + L W − L (dpa)NLO W + L W − T (dpa)NLO W + T W − L (dpa)NLO W + T W − T (dpa)0 . . . R a t i o t oo ff - s h e ll M e + ,µ − . . . K N L O (a) at NLO − − d σ / d M e + , µ − [ f b ] LHC 13 TeV, fiducial cuts NNPDF31 as 0118 7-scale scheme ( µ = M W ) NNLO off-shellNNLO unpol. (dpa)Summed W + ,W − pol. NNLO W + L W − L (dpa)NNLO W + L W − T (dpa)NNLO W + T W − L (dpa)NNLO W + T W − T (dpa)0 . . . R a t i o t oo ff - s h e ll M e + ,µ − . . . . K NN L O (b) at NNLO (without LI) Figure 2 . Distribution of charged lepton pair invariant mass at different orders of QCD. Doublepolarised setups are shown. From top-down: absolute value differential distribution, ratio to off-shell result, K-factor of the corresponding order. Monte-Carlo errors are shown in the middle andlower panes as grey bands. Scale variation bands are shown as coloured bands in the upper pane.
NLO, indicating that the fractions are rather independent of higher order QCD correc-tions. In particular, the fraction of doubly longitudinal polarised W, which gets the largestcorrections, is still small.
In this section we will explore NNLO QCD effects on the differential distributions as theyappear without the loop-induced channel. Observables which allow discrimination betweendifferent boson polarisations are of particular interest, theoretically and experimentally.The key quantity here is again the (differential) polarisation fractions. A general featureof differential polarisation fractions is that at high energies the longitudinal componentvanishes as the weak bosons get effectively massless. Naturally, regions of large invariantmass or transverse momentum are populated by transversely polarised W-bosons. Closeto the W-pair production threshold, where the diboson system is produced with smallmomentum, the contribution of the longitudinal component is largest.This characteristic can be seen in the invariant mass distribution of the charged lep-ton pair, shown in Figure 2. We show the NLO (left) and NNLO (right) predictions forthe absolute cross section, the differential polarisation fraction, and the NLO and NNLO– 9 – d σ / d c o s Θ e + , µ − [ f b ] LHC 13 TeV, fiducial cuts NNPDF31 as 0118 7-scale scheme ( µ = M W ) NLO off-shellNLO unpol. (dpa)Summed W + ,W − pol. NLO W + L W − L (dpa)NLO W + L W − T (dpa)NLO W + T W − L (dpa)NLO W + T W − T (dpa)0 . . . R a t i o t oo ff - s h e ll − . − . − . − .
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75 1 . cos Θ e + ,µ − . . . . K N L O (a) at NLO d σ / d c o s Θ e + , µ − [ f b ] LHC 13 TeV, fiducial cuts NNPDF31 as 0118 7-scale scheme ( µ = M W ) NNLO off-shellNNLO unpol. (dpa)Summed W + ,W − pol. NNLO W + L W − L (dpa)NNLO W + L W − T (dpa)NNLO W + T W − L (dpa)NNLO W + T W − T (dpa)0 . . . R a t i o t oo ff - s h e ll − . − . − . − .
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75 1 . cos Θ e + ,µ − . . K NN L O (b) at NNLO (without LI) Figure 3 . Distribution of cosine of angle between charged leptons. Double polarised setups areshown. Individual plot substructure is the same as in Figure 2.
K-factors, respectively. The tail features strong positive corrections for the setups withlongitudinally polarised bosons, reaching 50%. At NNLO the scale variation bands getnotably reduced across polarisation setups, particularly in the tail of the distribution, how-ever there the off-shell calculation still shows a substantial scale uncertainty. Finally, thedouble-longitudinal setup shows a larger correction in the low invariant mass region.It is worth pointing out that this is the region where the most of the production crosssection is coming from. This implies that observables which are sensitive to the thresholdregion, or bulk region, are especially well suited to study boson polarisations. In particular,this includes angular observables of the final state charged leptons which both have a strongsensitivity to polarisations and are shaped by the bulk region.For example, consider the angular separation between two charged leptons in Figure 3.Back-to-back configurations, i.e. where cos Θ e + ,µ − ≈ −
1, are dominated by the doubly-transverse setup, while the regions where the two leptons are aligned, have large contri-butions from setups containing a longitudinal boson. NNLO corrections reduce the scaledependence to the sub-percent level and show a rather small and flat K-factor. A notableexception is the LL setup as it receives strong corrections up to 10 −
15% in magnitude andshape. However, due to its overall small contribution it does not affect the polarisationfractions.Similar effects can be observed in the rapidity distributions. Figure 4 features the– 10 – d σ / d y e + [ f b ] LHC 13 TeV, fiducial cuts NNPDF31 as 0118 7-scale scheme ( µ = M W ) NLO off-shellNLO unpol. (dpa)Summed W + ,W − pol. NLO W + L W − L (dpa)NLO W + L W − T (dpa)NLO W + T W − L (dpa)NLO W + T W − T (dpa)0 . . . R a t i o t oo ff - s h e ll . . . . . . y e + . . . . K N L O (a) at NLO d σ / d y e + [ f b ] LHC 13 TeV, fiducial cuts NNPDF31 as 0118 7-scale scheme ( µ = M W ) NNLO off-shellNNLO unpol. (dpa)Summed W + ,W − pol. NNLO W + L W − L (dpa)NNLO W + L W − T (dpa)NNLO W + T W − L (dpa)NNLO W + T W − T (dpa)0 . . . R a t i o t oo ff - s h e ll . . . . . . y e + . . . . K NN L O (b) at NNLO (without LI) Figure 4 . Symmetrised distribution of e + rapidity at different orders of QCD. Double polarisedsetups are shown. Individual plot substructure is the same as in Figure 2. symmetrised version of e + rapidity distribution. Here, LL polarisation receives a significantcorrection for higher rapidities.To point out another feature of the longitudinally polarised signals, it is instructive toinvestigate transverse momentum distributions for leptons and W-bosons. First, considerthe transverse momentum distributions of the charged leptons p T (e + ) and p T ( µ − ). InFigure 5 we show p T (e + ) (left) and the ratio of the differential cross sections (right)d σ p T ( µ − ) d σ p T (e + ) ≡ d σ/ d p T ( µ − )d σ/ d p T (e + ) . (3.1)There is a striking difference in the ratio for the LT polarised setup, which can be explainedthrough asymmetries in the decay of W + and W − . The charged leptons have a largerprobability to get emitted forward (in the flight direction of the parent W boson) fortransversely polarised bosons than for longitudinal ones. This can be seen in Figure 5(c)and Figure 5(d) demonstrating the distribution in the opening angle between the chargedleptons and their parent W-boson (for precise definition of the angles, see App. A). Thuswe expect the transverse boson to produce a harder p T spectrum for its decay productsthan the longitudinal one, i.e. the ratio d σ p T ( µ − ) / d σ p T (e + ) is expected to be smaller than1 for TL setup and larger than 1 for LT setup. The magnitude of this effect is affected byanother asymmetry between the e + and µ − introduced with the phase space cuts, which– 11 – − − d σ / d p T ( e + ) [ f b ] LHC 13 TeV, fiducial cuts NNPDF31 as 0118 7-scale scheme ( µ = M W ) NNLO off-shellNNLO unpol. (dpa)Summed W + ,W − pol. NNLO W + L W − L (dpa)NNLO W + L W − T (dpa)NNLO W + T W − L (dpa)NNLO W + T W − T (dpa)0 . . . R a t i o t oo ff - s h e ll p T ( e + ) . . . K NN L O (a) Positron transverse momentum . . . . . . . . d σ p T ( µ − ) / d σ p T ( e + ) LHC 13 TeV, fiducial cuts NNPDF31 as 0118 7-scale scheme ( µ = M W ) NNLO off-shellNNLO unpol. (dpa)Summed W + ,W − pol. NNLO W + L W − L (dpa)NNLO W + L W − T (dpa)NNLO W + T W − L (dpa)NNLO W + T W − T (dpa)0 50 100 150 200 250 300 350 400 p T − . − . − . − . − . − . − . L og ∆ σ p T (b) Ratio of the muon and positron transverse mo-mentum distributions − . − . − . − .
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75 1 . cos Θ e + d σ / d c o s Θ e + [ f b ] LHC 13 TeV, fiducial cuts NNPDF31 as 0118 7-scale scheme ( µ = M W ) NNLO off-shellNNLO unpol. (dpa)Summed W + pol. NNLO W + L (dpa)NNLO W + T (dpa) (c) cos ∠ (e + , W + ) − . − . − . − .
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25 0 .
50 0 .
75 1 . cos Θ µ − d σ / d c o s Θ µ − [ f b ] LHC 13 TeV, fiducial cuts NNPDF31 as 0118 7-scale scheme ( µ = M W ) NNLO off-shellNNLO unpol. (dpa)Summed W − pol. NNLO W − L (dpa)NNLO W − T (dpa) (d) cos ∠ ( µ − , W − ) Figure 5 . (Upper row) Distribution of lepton transverse momentum at NNLO (without LI)for e + (left) and its comparison with distribution for µ − (right). Individual plot substructure in(a) is the same as in Figure 2. The plot (b) structure is the following: the upper plot featuresthe ratio of muon over positron transverse momenta distributions; the lower plot in (b) featuresLog (cid:12)(cid:12)(cid:12) σ pT (e + ) − σ pT ( µ − ) σ pT (e + )+ σ pT ( µ − ) (cid:12)(cid:12)(cid:12) , where σ p T is the differential transverse momentum distribution. Doublepolarised setups are shown. (Lower row) Distributions of charged lepton scattering angle cosinecalculated in the parent W-boson CM frame at NNLO (without LI) for e + (left) and µ − (right).Parent boson single polarised setups are are shown. Individual plots show the absolute valuedistributions. Single-polarised setups are shown. – 12 –akes the µ − spectrum from transverse W − harder than the e + spectrum from transverseW + . Unfortunately, as in the case of the invariant mass distribution, the longitudinalcontributions vanish at high transverse momentum, which makes it more difficult to exploitthis observation in experimental measurements. The loop-induced gg -channel only appears at α order and has quite a substantial effecton the cross-section. Its effects on various differential distributions have been alreadyinvestigated in [5]. In this section we will briefly comment on it again in the context of theNNLO calculation. . . . . . . U np o l a r i s e d W + Unpolarised W − Longitudinal W − Transverse W − . . . . . . L o n g i t ud i n a l W + . . . . . . T r a n s v e r s e W + p T ( e + ) Ratioto NLO
LO NLO NNLO NNLO+LI
Figure 6 . Ratio to NLO of e + transverse momentum distribution at various orders of perturbativeQCD. Each of 9 panes represents a selected polarisation setup calculated within DPA framework.W + and W − polarisation setups { U, L, T } are cycled across vertical and horizontal plots respectfully.Coloured regions represent scale variation bands, and grey bands – Monte-Carlo uncertainties. Usually, the loop-induced channel provides a glimpse into the NNLO effects in general.However, due to the simple kinematics structure of the diboson production, especially itsdouble-resonant contribution, the Born kinematics configuration is not enough to get theright distribution shape for many observables. Using Table 1 it can be pointed out that– 13 –oop-induced channel increases scale variation bands up to almost NLO QCD level in allsetups except the ones that contain exactly one longitudinally polarised weak boson.In Figure 6 we present relative corrections to the differential distribution of positrontransverse momentum, with respect to the NLO calculation. Here we observe that NNLOcalculation has brought the scale variation down, and that it is within a reasonable distancefrom NLO given the scale uncertainty. However, the loop-induced contribution drasticallychanges the picture, and its effect depends on the polarisation setup.Setups that include a longitudinal W-boson receive mild corrections from the loop-induced channel at low p T but are hugely affected in the tail ( p T >
120 GeV). Also, thescale variation band in the tail becomes larger than for any other approximation. Thisfact begs for introduction of the higher order correction to the loop-induced channel, whichwould be of order N LO in α s . Note that since the tail is not a part of the cross sectionbulk, the effect of sharp increase in the K-factor and in the scale band is not visible on theintegrated level in Table 1. . . . . U np o l a r i s e d W + Unpolarised W − Longitudinal W − Transverse W − . . . . L o n g i t ud i n a l W + . . . . T r a n s v e r s e W + y e + Ratioto NLO
LO NLO NNLO NNLO+LI
Figure 7 . Ratio to NLO of symmetrised e + rapidity distribution at various orders of perturbativeQCD. Same plot structure as in Figure 6. Setups containing a transversely polarised W-boson represent a different correction inthe loop-induced channel at NNLO. Here we observe an overall positive shift of order 10%– 14 –hich diminishes by the end of the distribution. This behaviour is expectedly replicatedby the unpolarised setup as the transverse contribution dominates the cross section.Similar effects can be observed in the charged leptons invariant mass distribution.Next, we show the rapidity of e + distribution in Figure 7. The setups containingtransverse polarisations as well as the unpolarised setup feature about a 10% positivecorrection which is compatible with what we observed at the bulk of the positron p T distribution. The case is different however for setups containing a longitudinal W. Inparticular, a strong correction receives the LL polarisation, which is expected due to itsoverall large K-factor in Table 1. The gg loop-induced channel follows the shape of NNLOcorrections except in the TT setup, where the loop induced corrections appear to be largerthan NLO. . . . . U np o l a r i s e d W + Unpolarised W − Longitudinal W − Transverse W − . . . . L o n g i t ud i n a l W + . . . . T r a n s v e r s e W + ϕ e + ,µ − Ratioto NLO
LO NLO NNLO NNLO+LI
Figure 8 . Ratio to NLO of azimuthal separation between charged leptons at various orders ofperturbative QCD. Same plot structure as in Figure 6.
Finally, in angular distributions related to the lepton emission angles we see an overallshift which does not affect the distribution shapes. However, a notable difference can beobserved in azimuthal separation between the charged leptons which is a distribution highlysusceptible to interference effects even at inclusive level. In Figure 8 we note that LI channelhas a large overall shift in TT and unpolarised setups and features a rather interesting– 15 –ehaviour in LL setup. LI channel barely has any effect on LL setup at φ e + ,µ − < α in the gg loop-induced channel,to bring the scale variation down. This is left for future work. d σ / d y e + [ f b ] LHC 13 TeV, fiducial cuts NNPDF31 as 0118 7-scale scheme ( µ = M W ) NLO off-shellNLO unpol. (dpa) NLO unpol. (nwa)0 . . . R a t i o t oo ff - s h e ll . . K N L O . . . . . . y e + . . N o r m a li s e d R a t i o t oo ff - s h e ll (a) Symmetrised rapidity of positron (NLO) . . . . . . . . d σ / d ϕ e + [ f b ] LHC 13 TeV, fiducial cuts NNPDF31 as 0118 7-scale scheme ( µ = M W ) NLO off-shellNLO unpol. (dpa) NLO unpol. (nwa)0 . . R a t i o t oo ff - s h e ll . . . K N L O . . . . . . . ϕ e + . . . N o r m a li s e d R a t i o t oo ff - s h e ll (b) Symmetrised azimuthal angle of positron emis-sion in its parent boson CM (NLO) Figure 9 . Comparison between off-shell calculation, DPA, and NWA for selected distributions.Top three panes have the same structure as in Figure 2; the bottom plot shows the ratio distributionsto off-shell calculation normalised according to integrated cross-section value.
As we discussed in Section 2.1 double pole and narrow width approximations considerthe same set of double-resonant diagrams, however they are different in how they generatethe phase space. DPA is able to incorporate off-shell effects via generating off-shell kine-matics and subsequently projecting it on-shell to ensure gauge invariance of the amplitude.NWA is thus considered to be a less precise approach [5]. It is therefore instructive tocompare NWA and DPA in the diboson production setting to inspect differences in theirperformance. – 16 – d σ / d c o s Θ e + [ f b ] LHC 13 TeV, fiducial cuts NNPDF31 as 0118 7-scale scheme ( µ = M W ) NLO off-shellNLO unpol. (dpa) NLO unpol. (nwa)0 . . . R a t i o t oo ff - s h e ll . . K N L O − . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . cos Θ e + . . . N o r m a li s e d R a t i o t oo ff - s h e ll (a) Cosine of positron emission polar angle d σ / d c o s Θ e + , µ − [ f b ] LHC 13 TeV, fiducial cuts NNPDF31 as 0118 7-scale scheme ( µ = M W ) NLO off-shellNLO unpol. (dpa) NLO unpol. (nwa)0 . . . R a t i o t oo ff - s h e ll . . . . K N L O − . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . cos Θ e + ,µ − . . N o r m a li s e d R a t i o t oo ff - s h e ll (b) Cosine of angle between charged leptons (NLO) Figure 10 . Comparison between off-shell calculation, DPA, and NWA for selected distributions.Same plot substructure as in Figure 9.
The approximations differ on the integrated level which we discussed in Section 3.1.Moving on to the differential distributions, we note that by construction, NWA approach, isunable to describe the weak boson invariant masses, so we will not discuss this observable.In fact, due to the absence of single-resonant diagrams, DPA also does not describe thefull off-shell amplitude, and produces a rather symmetrical shape that is different from theoff-shell result [5].There are distributions where NWA and DPA show the same shape but feature anoverall shift that we see on the integrated level. The symmetrised rapidity and azimuthalangle of emission are such examples and are shown at NLO for positron in Figure 9. Thereader can also find a discussion on the definition of azimuthal angle of emission in App. A.For these distributions, it makes sense to compare them normalised by their integratedvalues. The bottom panes of Figure 9 show that the normalised shapes between DPA andNWA agree well within their Monte-Carlo errors at NLO. This behaviour is replicated atNNLO also with the inclusion of the loop-induced channel.Another distribution that showcases similarities in DPA and NWA performance is the– 17 –osine of angle between the charged leptons featured on Figure 10(b). Here the approxi-mations agree with each other across the entire range except for the first and the last bins.Fiducial cuts affect the last bin and are responsible for its large MC errors, but in the caseof the first one there is a true difference. This is independent of QCD order and can beobserved already at LO. Perhaps, the DPA mapping underperformes in the point whereleptons are emitted in the opposite directions.In Figure 10(a) we show a comparison between the charged lepton emission angle in theDPA and NWA frameworks. DPA features a distribution that is slightly further from theoff-shell result. Near the first bin the distribution is affected by the fiducial cuts and so theapproximations become further away from the off-shell calculation. The same conclusionscan be reached for angular distributions of the muon and are replicated at NNLO QCDincluding the loop-induced channel. d σ / d ϕ e + , µ − [ f b ] LHC 13 TeV, fiducial cuts NNPDF31 as 0118 7-scale scheme ( µ = M W ) NLO off-shellNLO unpol. (dpa) NLO unpol. (nwa)0 . . . . R a t i o t oo ff - s h e ll . . . . K N L O . . . . . . . ϕ e + ,µ − . . . . N o r m a li s e d R a t i o t oo ff - s h e ll (a) Azimuthal separation between charged leptons(NLO) − − − d σ / d p T , l ± ( l e a d i n g ) [ f b ] LHC 13 TeV, fiducial cuts NNPDF31 as 0118 7-scale scheme ( µ = M W ) NLO off-shellNLO unpol. (dpa) NLO unpol. (nwa)0 . . . R a t i o t oo ff - s h e ll . . . . K N L O p T,l ± (leading) . . . N o r m a li s e d R a t i o t oo ff - s h e ll (b) Leading lepton transverse momentum (NLO) Figure 11 . Comparison between off-shell calculation, DPA, and NWA for selected distributions.Same plot substructure as in Figure 9.
A notable deviation can be observed between the setups at the beginning of vari-ous transverse momentum and invariant mass distributions, which represent the bulk ofthe cross section. In Figure 11(b) we present the leading lepton p T which shows that at– 18 – T <
50 GeV NWA overshoots the off-shell calculation, whereas DPA undershoots. Thisregion is the origin of the total cross section results, as at higher p T the distribution fallsnearly exponentially. DPA and NWA differ by 2% in the tail of the distribution and bothdeviate significantly from the full off-shell calculation due to single-resonant effects [26].Similar effects of the two approximations can be observed in the diboson invariant massdistribution. W-boson transverse momentum distribution features the same interplay be-tween DPA and NWA, however here the approximations are closer to the off-shell result.NNLO K-factor shapes are the same across three setups, however the loop-induced chan-nel appears to make a difference in the distribution tail where DPA and NWA receive 20%larger corrections. This effect also observed in the W-boson transverse momentum profile.Finally, quite an interesting difference in the behaviour between DPA and NWA canbe observed in the azimuthal angular separation between charged leptons. In Figure 11(a)one can see that the ratio to full off-shell calculation is distinctly different between NWAand DPA for φ e + ,µ − > .
75. They have matching values up until this threshold and divergeuntil the end of the distribution at φ = π . In this region of disagreement, NWA overshootsthe off-shell computation. Expectedly, it appears to be the peak of the distribution, thuscontributing to the overall large NWA integrated cross section value. This behaviourpersists with introduction of higher orders. K-factor both for NNLO corrections and forthe inclusion of loop-induced contribution has the same shape across setups and thus doesnot introduce any differences to the shapes of the ratio plots. In this paper we compute, for the first time, polarised diboson production through NNLO inQCD within the framework of double pole approximation. In the calculations we considereda fiducial phase space that emulates experimental setting which was already used in recenttheoretical and experimental studies of the polarised processes.NNLO corrections effects are twofold. With the exclusion of the loop-induced channel,the corrections show a controlled and predictable behaviour, particularly in the regionsthat represent the bulk of the cross section. Among notable effects we would point outsignificant corrections to the tail of transverse momentum and invariant mass distributions,especially in the case of longitudinally polarised setups. The scale uncertainty is broughtdown by a factor of 3 across all polarisation setups.However, the loop-induced channel changes the picture at NNLO significantly. Thetails of transverse momentum distributions are affected to the point that they reach theirlevel at LO. The effect is particularly strong for the doubly longitudinally polarised setup.This behaviour prompts for further calculation at order N LO to bring the scale uncertaintydown.Finally, we compared the narrow width and double pole approximations in the unpo-larised setup. NWA overshoots the off-shell result by 1%, while DPA is lower by 2.5%,which falls within their expected approximation error. Distributions look similar in thecase of distributions for rapidity and charged lepton emission angles. We observed a sig-nificant deviation between approaches at low transverse momenta and pointed out a local– 19 –ifference between the charged lepton azimuthal separation, where NWA features a morevolatile behaviour in comparison with DPA, undershooting and overshooting the off-shellresult. Generally, we observe similar results between the methods with only slight variationin particular observables and generally the same behaviour with respect to their ability todescribe the full off-shell result.
Acknowledgements
We are grateful to Jean-Nicolas Lang, Giovanni Pelliccioli, and Ansgar Denner for provid-ing their polarisation-capable private version of
Recola and giving us explanations. Weacknowledge helpful discussions with Alexander Mitov and useful comments from Heriber-tus Bayu Hartanto and the Cambridge Pheno Group. This research has received fundingfrom the European Research Council (ERC) under the European Union’s Horizon 2020 Re-search and Innovation Programme (grant agreement no. 683211). A.P. is also supportedby the Cambridge Trust, and Trinity College Cambridge.
A Azimuthal angle of emission
In this appendix we briefly discuss ways to define charged lepton emission angles.There are two reference frames commonly used in the literature: the helicity (HE)frame and Collins-Soper (CS) frame. The helicity coordinate system is defined in [1]. Aswe observed in the literature, it is common, however, to simplify its construction by usinga fixed reference momentum. In order to be able to compare the distributions, we followsuit. Here is the full algorithm we use to construct the X (cid:48) Y (cid:48) Z (cid:48) helicity frame.Denote the first proton momentum as reference, and define Z (cid:48) axis by the direction ofW-boson in the Lab frame. Then proton momenta P , P are boosted into the W-bosonrest frame where they become P (cid:48) , P (cid:48) . We build (cid:126)Y (cid:48) axis in the direction of [ (cid:126)P (cid:48) × (cid:126)P (cid:48) ] vectorwhich is a perpendicular to the plane based on boosted proton momenta. Finally, X (cid:48) axisis defined such that X (cid:48) Y (cid:48) Z (cid:48) coordinate system is right-handed, i.e. (cid:126)X (cid:48) ∼ [ (cid:126)Y (cid:48) × (cid:126)Z (cid:48) ].Another choice is the Collins-Soper frame which originates from [59]. In short, theconstruction goes by boosting proton momenta into the boson rest frame where they aredenoted by P (cid:48) , P (cid:48) . Then Z (cid:48) axis is defined as a bisection between vectors { (cid:126)P (cid:48) , − (cid:126)P (cid:48) } . The X (cid:48) axis is chosen as a bisection between {− (cid:126)P (cid:48) , − (cid:126)P (cid:48) } . Finally Y (cid:48) axis is uniquely definedto complete the right-handed system.We find useful the discussion of these frames in [3], where the authors also present thedistributions corresponding to different frame choices. References [1] Z. Bern et al.,
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