On electroweak corrections to neutral current Drell–Yan with the POWHEG BOX

Motivated by the requirement of a refined and flexible treatment of electroweak corrections to the neutral current Drell-Yan process, we report on recent developments on various input parameter/renormalization schemes for the calculation of fully differential cross sections, including both on-shell and MS schemes. The latter are particularly interesting for direct determinations of running couplings at the highest LHC energies. The calculations feature next-to-leading order precision with additional higher order contributions from universal corrections such as ∆α and ∆ρ . All the discussed input parameter/renormalization scheme options are implemented in the package of POWHEG-BOX-V2 dedicated do the neutral current Drell-Yan simulation, i.e. Z_ew-BMNNPV , which is used to obtain the presented numerical results. In particular, a comprehensive analysis on physical observables calculated with different input parameter/renormalization schemes is presented, addressing the Z peak invariant mass region as well as the high energy window. We take the opportunity of reporting also on additional improvements and options introduced in the package Z_ew-BMNNPV after svn revision 3376, such as different options for the treatment of the hadronic contribution to the running of the electromagnetic coupling and for the handling of the unstable Z resonance.


Introduction
The neutral current Drell-Yan (NC DY) process plays a particular role in the precision physics programme of the LHC.In fact, considering its large cross section and clean experimental signature, together with the high precision measurement of the Z-boson mass at LEP, this process is a standard candle that can be used for different general purposes such as detector calibration, Parton Distribution Functions (PDFs) constraining and tuning of non-perturbative parameters in the general purpose Monte Carlo event generators.Moreover, in the high tail of the transverse momentum and invariant mass distributions of the produced leptons, the NC DY is one of the main irreducible backgrounds to the searches for New Physics at the LHC.Recently an impressive precision at the sub-percent level has been reached by the experimental analysis in large regions of the dilepton phase space.
In addition to the above general aspects, it has to be stressed that the NC DY process also allows to perform precision tests of the Standard Model electroweak (SM EW) parameters through the direct determination of the W -boson mass [1][2][3][4] and the weak mixing angle in hadronic collisions [5][6][7][8][9].While in the former case the NC DY observables enter only indirectly, the latter can be determined directly, without reference to the charged current DY process.Another interesting possibility is the direct determination of the running of the coupling sin 2 θ , defined in the MS scheme, at the highest available LHC energies, in order to check its consistency with low energy and Z-peak measurements within the SM framework [10].
A fundamental role in simulations for collider phenomenology is played by Monte Carlo event generators capable to consistently match fixed order calculations to parton showers (PS) simulating multiple soft/collinear radiations.The MC@NLO [94] and POWHEG [95,96] algorithms have been developed for the matching of NLO QCD computations to QCD PS and implemented in the public softwares MadGraph5-_aMC@NLO [97] and POWHEG-BOX [98,99].Alternative for-mulations of the above algorithms are used in event generators like Sherpa [100,101] and HERWIG [102,103].More recently, algorithms like NNLOPS [104] (based on reweighting of the MiNLO ′ [105] merging strategy), UNNLOPS [106,107], Geneva [108], and MiNNLO PS [109,110] have been proposed for the matching of NNLO QCD calculations to QCD PS.Though several studies appeared on the approximated inclusion of EW corrections in event generators including higher-order QCD corrections (see, for instance Refs.[111][112][113]), event generators including both NLO QCD and NLO EW corrections consistently matched to both QCD and QED parton showers are only available for a limited number of processes, namely: charged and neutral current Drell-Yan [114][115][116][117][118], HV + 0/1 jet [118], diboson production [119], and electroweak H + 2 jets production [120]  1 .QED correction exclusive exponentiation for DY processes within YFS framework is realized within the event generator KKMC-hh [122][123][124][125][126]. Very recently, the resummation of EW and mixed QCD-EW effects up to next-to-leading logarithmic accuracy has been presented in Ref. [127] for charged and neutral current DY.
The first implementation of QCD and EW NLO corrections and their interplay in a unique simulation framework have been given in Ref. [114] for charged current DY 2 and in Ref. [116] for NC DY.An important improvement in the matching of QED radiation in the presence of a resonance, following the ideas presented in Ref. [99], has been discussed in Ref. [128] for the charged current DY process (W_ew-BMNNP svn revision 3375) and extended to the NC one (Z_ew-BMNNPV svn revision 3376) 3 .After the above mentioned release, additional improvements and options have been introduced, in particular for the NC DY package Z_ew--BMNNPV, motivated by the need of a refined and flexible treatment of EW corrections that allows a consistent internal estimate of the uncertainties affecting the theoretical predictions.They can be schematically enumerated as follows: input parameter/renormalization schemes introduction of known higher order corrections treatment of the hadronic contribution to the running of the electromagnetic coupling ∆ α had scheme for the treatment of the unstable resonance In the following we give a detailed description of the various input parameter schemes and the related higher order corrections.A key ingredient of the latter is given by the running of the electromagnetic coupling α between differ-ent scales.In particular, when low scales are involved in the running, the hadronic contribution is intrinsically nonperturbative and different parametrizations have been developed in the literature relying on low-energy experimental data, which we properly include in our formulation of the electroweak corrections.
The formulae relevant for the various input parameters/ renormalization schemes are presented in a complete and self-contained form, so that they can be implemented in any simulation tool.
Though the Z_ew-BMNNPV package allows to simulate NC DY production at NLO QCD+NLO EW accuracy with consistent matching to the QED and QCD parton showers provided by PYTHIA8 [129,130] and/or Photos [131][132][133], in the present paper we are mainly interested in various aspects of the fixed-order calculation (NLO EW plus universal higher orders).For this reason we show numerical results obtained at fixed-order and including only the weak corrections, since the QED contributions are not affected by the choice of the input parameter scheme and are a gaugeinvariant subset of the EW corrections for the NC DY process.
The layout of the paper is the following: Section 2 provides an introduction to the input parameter schemes available in the code, while general considerations on higher order universal corrections, common to all the schemes, are presented in Section 3. A detailed account of various input/renormalization schemes at NLO accuracy and the related higher order corrections is given in Section 4, while a numerical analysis of the features of the schemes, with reference to cross section and forward-backward asymmetry, as functions of the dilepton invariant mass, is presented in Section 5, together with a discussion on the main parametric uncertainties in Section 6.The treatment of the hadronic contribution to the running of α is discussed in Section 7, while Section 8 is devoted to the description of the improvement of the code with respect to the treatment of the unstable Z resonance.In Section 9 we analyse the effect of the various input/renormalization schemes in the high energy regimes, which will be accessible to the HL-LHC phase and future FCC-hh.A brief summary is given in Section 10.The list of the default parameter values 4 is contained in Appendix A, while the list of the flags activating the available options is given in Appendix B.

Input parameter schemes: general considerations
The input parameter schemes available in the Z_ew-BMNNPV package of POWHEG-BOX-V2 can be divided in three categories: the ones including both the W and the Z boson masses among the independent parameters, the (α 0 , G µ , M Z ) scheme, where we use the notation α 0 for the QED coupling constant at Q 2 = 0, and the schemes with M Z and the sine of the weak mixing angle as free parameters.The latter class includes the schemes that use sin θ l e f f as input parameter, where θ l e f f is the effective weak mixing angle, and an hybrid MS scheme where the independent quantities are α MS (µ 2 ), sin 2 θ MS (µ 2 ) and M Z , with the couplings renormalized in the MS scheme and M Z with the usual on-shell prescription.
The first class of input parameter schemes, namely (α i , M W , M Z ) with α i = α 0 , α(M 2 Z ), G µ , is widely used for the calculation of the EW corrections for processes of interest at the LHC.On the one hand, the fact that the W boson mass is a free parameter is a useful feature, in particular, in view of the experimental determination of M W from charged current Drell-Yan production using template fit methods; on the other hand, the predictions obtained in these schemes can suffer from relatively large parametric uncertainties related to the current experimental precision on M W .This drawback is overcome, for instance, in the (α 0 , G µ , M Z ) scheme used for the calculation of the EW corrections in the context of LEP physics, where all the input parameters are experimentally known with high precision.The third class of input parameter schemes uses the sine of the weak mixing angle as a free parameter.In the Z_ew-BMNNPV package the (α i , sin θ l e f f , M Z ) schemes, with α i = α 0 , α(M 2 Z ), G µ , and the (α MS (µ), sin 2 θ MS (µ 2 ), M Z ) one are implemented.The schemes where sin θ l e f f (sin 2 θ MS (µ 2 )) is a free parameter are particularly useful in the context of the experimental determination of sin θ l e f f (sin 2 θ MS (µ 2 )) from NC DY production at the LHC using template fits [10,118].
The predictions for NC DY production obtained in the schemes that use α(M 2 Z ) or G µ as inputs show a better convergence of the perturbative series compared to the corresponding results from the schemes with α 0 as free parameter 5 .This is a consequence of the fact that, when using α(M 2 Z ) or G µ as independent variables, large parts of the radiative corrections related to the running of α(Q 2 ) from Q 2 = 0 to the electroweak scale are reabsorbed in the LO predictions.On the contrary, the EW corrections in the schemes with α 0 as input tend to be larger, because the running of α(Q 2 ) involves logarithmic corrections of the form log(m 2 /Q 2 ), where m stands for the light-fermion masses (we refer to Sect.7 for the treatment of the light-quark contributions to the running of α(Q 2 )) and Q 2 is the typical large mass scale of the process.
From a technical point of view, the calculation of the one-loop electroweak corrections in the above-mentioned input parameter schemes differs in the renormalization pre-scriptions used for the computation, while the bare part of the Drell-Yan amplitudes remains formally the same and it is just evaluated with different numerical values of the input parameters.For each choice of input scheme, the renormalization is performed as follows: first the electroweak parameters are expressed as a function of the three selected independent quantities, then the counterterms corresponding to these parameters are fixed by imposing some renormalization condition, and finally the counterterms for the derived electroweak parameters are written in terms of the ones corresponding to the input parameters.
The fact that the counterterm part of the Drell-Yan amplitude differs in the considered input parameter schemes implies that also the expression of the universal fermionic corrections changes, as these corrections at NLO can be related to the counterterm amplitude.In fact, they can be computed at O(α 2 ) taking the square of the fermionic universal contributions at NLO (after subtracting the O(α) terms already included in the NLO calculation).We refer to Sect. 3 for details.
The input parameter schemes described in the following sections are formally equivalent for a given order in perturbation theory; however, the numerical results obtained differ because of the truncation of the perturbative expansion.Although there is some arbitrariness in the choice of the input parameter scheme to be used for the calculations, there can be phenomenological motivations to prefer one scheme to the others, depending on the observables under consideration and on the role played by the theory predictions with respect to the interpretation of the experimental measurements 6 .For instance, in the context of cross section or distribution measurements, where the theory predictions are used as a benchmark for the experimental results but do not provide input for parameter determination, those input parameter schemes should be preferred that involve independent quantities known experimentally with high precision in order to minimize the corresponding parametric uncertainties.One should also try to minimize the parametric uncertainties from quantities that enter the calculation only at loop level (such as, for instance, the top quark mass in DY processes).Another aspect that should be taken into account when choosing an input parameter scheme is the convergence of the perturbative expansion in the predictions for the observables of interest, which is mainly related to the possibility of reabsorbing large parts of the radiative corrections in the definition of the coupling at LO.
A different situation is the direct determination of electroweak parameters using template fit methods, as done for example for the W boson mass at Tevatron and LHC.In this case, the theory predictions enter the interpretation of the measurement (with the Monte Carlo templates) and the the-ory uncertainties become part of the total systematic error on the quantity under consideration: it is thus important to use an input parameter scheme where the quantity to be measured is a free parameter that can be varied independently not only at LO, but also at higher orders in perturbation theory.

Higher-order corrections
At moderate energies, the leading corrections to NC DY production are related to the logarithms of the light fermion masses and to terms proportional to the top quark mass squared.These contributions can be traced back to the running of α(Q 2 ) (i.e. to ∆ α) and to ∆ ρ and are thus related to the counterterm amplitude for the process under consideration.Following Refs.[29,[135][136][137], these effects can be taken into account at O(α 2 ) by taking the square of the part of the counterterm amplitude proportional to ∆ α and ∆ ρ.They can thus be combined to the full NLO calculation after subtracting the part of linear terms in ∆ α and ∆ ρ appearing in the square of the counterterm amplitude that are already present in the NLO computation.
The numerical results for the fermionic higher-order corrections presented in the following are obtained using the one-loop expression for ∆ α (even though the two-loop leptonic corrections are also available in the Z_ew-BMNNPV package and can be activated with the flag dalpha_lep_2loop), while for ∆ ρ we include the leading Yukawa corrections up to O(α 2 S ), O(α S x 2 t ), and O(x 3 t ), with x t = √ 2G µ M 2 top /16π 2 .More precisely, the expression used for ∆ ρ is where ∆ ρ (2) is the two-loop heavy-top corrections to the ρ parameter [138][139][140], δ QCD and δ QCD are the two and threeloop QCD corrections [141][142][143][144], while the three-loop contributions ∆ ρ x 3 t and ∆ ρ x 2 t α S are taken from Ref. [145].The last term in Eq. ( 1) is introduced in order to avoid the double counting of the O(x 2 t α 2 S ) contribution already present in factorized approximation in the product of the QCD corrections and the Yukawa corrections at two loops.The fourloop QCD corrections to the ρ parameter [146] are not included.By inspection of the numerical impact of three-loop QCD corrections (cf.Figs.7 and 8), the phenomenological impact of four-loop QCD corrections to the ρ parameter is expected to be negligible at the LHC.For the numerical studies presented in the following, the scale for the α S factors entering the QCD corrections to ∆ ρ is set to the invariant mass of the dilepton pair.

Input parameter schemes: detailed description
In the following subsections we present a detailed account of the available input parameter schemes at NLO weak accuracy and the related universal higher order corrections (in what follows, the label NLO+HO refers to NLO plus higherorder accuracy).In the last subsection we present a comparison of the radiative corrections obtained with the different parameter schemes for two relevant differential observables (cross section and forward-backward asymmetry as functions of the dilepton invariant mass M ll ) of the NC DY process at the LHC with √ s = 13 TeV, considering µ + µ − final states.In the following, for the sake of simplicity of notation, whenever the complex mass scheme (CMS in the following) is used for the treatment of the unstable gauge bosons, the symbol In these schemes, the input parameters are the W and Z boson masses and The counterterms for the independent parameters are defined as where the subscript b denotes the bare parameter.The expression for δ Z e is fixed by imposing that the NLO EW corrections to the γe + e − vertex vanish in the Thomson limit, while δ M 2 W and δ M 2 Z are obtained by requiring that the gaugeboson masses do not receive radiative corrections.
The analytic expression of the counterterms can be found in Ref. [136] (and in Refs.[147][148][149] if the complex-mass scheme is used).In the following, for the self energies and the counterterms we will use the notation of Ref. [136].
In the schemes with M W and M Z as independent parameters, the sine of the weak-mixing angle is a derived quantity defined as and the corresponding counterterm reads: When α(M 2 Z ) or G µ are used as input parameters, the calculation of the O(α) corrections is formally the same one as in the (α 0 , M W , M Z ) scheme but with the replacements δ Z e → δ Z e − ∆ α(M 2 Z )/2 and δ Z e → δ Z e − ∆ r/2, respectively, that take into account the running of α(Q 2 ) from Q 2 = 0 to the weak scale which is absorbed in the LO coupling (α(M 2 Z ) or G µ ).It is worth noticing that these replacements remove the logarithmically enhanced fermionic corrections coming from ∆ α.The factor ∆ r represents the full one-loop electroweak corrections to the muon decay in the scheme (α 0 , M W , M Z ) after the subtraction of the QED effects in the Fermi theory and reads: with In the Z_ew-BMNNPV package, we implemented a slightly modified version of Eqs.(3.45)-(3.49) of Ref. [29] for the computation of the leading fermionic corrections to neutralcurrent Drell-Yan up to O(α 2 ).More precisely, those equations are modified in such a way to be valid also in the complex-mass scheme.As discussed in Sect.3, in order to combine these higher-order fermionic corrections with the NLO EW results, it is mandatory to subtract those effects that are included in the full one-loop calculation to avoid double counting.In particular, this implies the replacement ∆ ρ → (∆ ρ − ∆ ρ 1−loop ) in the linear terms of the fermionic corrections up to O(α 2 ): if we use, optionally by means of the flag a2a0-for-QED-only 7 , for the overall weakloop factors the same value of α i used in the LO couplings, ∆ ρ 1−loop is computed as a function of α 0 , α(M 2 Z ), or G µ for the (α 0 , M W , M Z ), (α(M 2 Z ), M W , M Z ), and (G µ , M W , M Z ) schemes, respectively.If instead we use α 0 for the overall weak-loop factors, we subtract the quantity ∆ ρ 1−loop | α 0 computed in the α 0 scheme, regardless of the value of α i used as independent parameter.

The
In the (α i , sin 2 θ l e f f , M Z ) schemes (where α i = α 0 , α(M 2 Z ), G µ ) the sine of the effective weak mixing angle is used as input parameter 8 .This quantity is defined from the ratio of the vectorial and axial-vectorial couplings of the Z boson to the leptons g l V and g l A , or, equivalently, in terms of the chiral Zll couplings g l L and g l R , measured at the Z resonance and reads sin 2 θ l e f f ≡ where I l 3 is the third component of the weak isospin for lefthanded leptons 9 .Since sin 2 θ l e f f is used as an independent parameter, this scheme is particularly useful in the context of the direct extraction of sin 2 θ l e f f from NC DY at the LHC using template fit methods at NLO EW accuracy.
The counterterms corresponding to the input parameters are defined as The expressions of δ Z e and δ M 2 Z are determined as in Sect.4.1, while the expression of δ sin 2 θ l e f f is fixed by requiring that the definition in Eq. ( 11) holds to all orders in perturbation theory.More precisely, we write Eq. ( 11) at one loop as sin 2 θ l e f f ≡ where ) represent the Zl L l L and Zl R l R form factors computed at one loop accuracy at the scale M 2 Z and we impose the condition: The δ g l L(R) factors contain both bare vertices and counterterms and since they are functions of δ sin 2 θ l e f f , Eq. ( 17) can be used to compute the counterterm corresponding to the effective weak mixing angle.By inserting the expressions for sin θ l e f f δ Z AZ (18) where δ Z l L(R) are the pure weak parts of the wave function renormalization counterterms for the leptons and δV L(R) are the one-loop weak corrections to the left/right Zll vertices defined as and the vertex functions V a and V − b are given in Eqs.(C.1) and (C.2) of Ref. [136], respectively.No QED correction is included in Eq. ( 18), since the QED contributions to the Zll vertex are the same for left or right-handed fermions and cancel in Eq. (17).When the complex mass scheme is used, the input value for sin 2 θ l e f f remains real: this implies that g LO R/L remain real and the condition (16) still reduces to (17).
As a consequence, the definition in Eq. (18) remains valid in the complex-mass scheme, provided that the CMS expressions for δ Z l L(R) and δ Z AZ are used.Note that the vertex functions V a and V b are computed for a real scale M 2 Z , while the gauge boson masses appearing in the loop diagrams are promoted to complex.If one instead uses a complex-valued s 2 W (see the discussion on fermionic higher-order effects in Sect.4.3), the condition in (17) can still be used but without taking the real part.
As already discussed in Sect.4.1, the counterterms in the (α(M 2 Z ), sin 2 θ l e f f , M Z ) and (G µ , sin 2 θ l e f f , M Z ) schemes can be obtained from the ones in the (α 0 , sin 2 θ l e f f , M Z ) scheme performing the replacements δ Z e → δ Z e −∆ α(M 2 Z )/2 and δ Z e → δ Z e − ∆ r/2, respectively, where ∆ r represents the one-loop electroweak corrections to the muon decay (after subtracting the QED effects in the Fermi theory) in the scheme (α 0 , sin 2 θ l e f f , M Z ) and reads that can be written also as with where we used the short-hand notation s W = sin θ l e f f and c W = cos θ l e f f .From Eqs. ( 18)-( 21) it is clear that the leading fermionic corrections in the schemes with sin 2 θ l e f f as input parameter are only related to δ Z e ∼ ∆ α 2 and ∆ r ∼ ∆ α − ∆ ρ, while the counterterm of sin 2 θ l e f f does not contain terms proportional to the logarithms of the light-fermion masses or to the square of the top quark mass.As a result, the fermionic higher-order corrections in these schemes (after the subtraction of the effects already included in the O(α) calculation) are just overall factors that multiply the LO matrix element squared and read: and for the schemes with α 0 and G µ as input parameters, respectively, while these corrections are zero when α(M 2 Z ) is used as independent parameter.In equation (23), a resummation of the logarithms of the light-fermion masses was performed, while the overall factor in (24) comes from the relation between α and G µ at NLO plus higher orders,

4.3
The (α 0 , G µ , M Z ) scheme In the (α 0 , G µ , M Z ) scheme, the input parameters are α 0 , G µ , and the mass of the Z boson.The main advantage of using this scheme is that all the independent parameters are experimentally known with high precision and the corresponding parametric uncertainties are small (in particular, compared to the schemes in Sect.4.1, it is independent of the uncertainties related to the experimental knowledge of M W ).
In the scheme under consideration, the sine of the weak mixing angle and the W boson mass are derived quantities.At the lowest order in perturbation theory they can be computed using the relations 10 In the LO matrix element, the value of α used is derived from G µ at In terms of Eqs.(26), it is possible to write the LO amplitude for NC DY as the sum of the photon exchange amplitude proportional to α and the Z exchange amplitude proportional to G µ M 2 Z , namely: where A σ ,τ is the part of the amplitude containing the γ matrices and the external fermions spinors (σ , τ = L, R) and Q q(l) and I σ q(l) 3 being the quark (lepton) charges and third components of the weak isospin.In the complex-mass scheme, the definition of χ Z is s/(s − M 2 Z ).Clearly the Z-boson exchange diagram contains a residual dependence on α from s W in Eq. (28).Two different realizations of the (α 0 , G µ , M Z ) scheme are available in the Z_ew-BMNNPV package: users can select a specific one through the azinscheme4 flag in the powehg.inputfile.If the flag is absent or negative, in Eqs. ( 26) α = α 0 in such a way that the γ f f interaction is evaluated at low scale while the Z f f couplings are computed at the weak scale.If azinscheme4 is positive, α = α 0 /(1 − ∆ α(M 2 Z )): this way also the photon part of the amplitude is evaluated at the weak scale.Note that we compute ∆ α(M 2 Z ) form α 0 rather that taking α(M 2 Z ) as an independent parameter.For dilepton invariant masses in the resonance region or larger, the latter running mode allows to reabsorb in the couplings the mass logarithms originating from the running of α from q 2 = 0 to the weak scale.If not otherwise stated, the numerical results presented for the (α 0 , G µ , M Z ) scheme are obtained with azinscheme4= 1.
The counterterms for the independent quantities are defined as: The expression of the δ Z e and δ M 2 Z counterterms is fixed as in Sect.4.1 (if azinscheme4= 1, there is the additional shift δ Z e → δ Z e − ∆ α(M 2 Z )/2), while δ G µ is determined by requiring that the muon decay computed in the (α 0 , G µ , M Z ) scheme does not receive any correction at NLO (after the subtraction of the QED effects in the Fermi theory), namely: where α in the last term correspond to the loop factor governed by the flag a2a0-for-QED-only.The counterterms for the dependent quantities read where δ opt ,1 is one if the azinscheme4 flag is active and zero otherwise.By looking at the expressions of the counterterms in equations ( 32)- (35), it is clear that at NLO the leading fermionic corrections to the photon exchange amplitude are related to δ Ze , while for the Z exchange amplitude they come from the counterterms of the overall factor G µ M 2 Z and from δ s W /s W , with In order to include these effects beyond O(α), we follow the strategy described, for instance, in [150]: the fermionic higher-order corrections are written as a Born-improved amplitude written in terms of effective couplings α = α 0 /(1 − ∆ α(M 2 Z )) and sin 2 θ l e f f (computed as a function of α 0 , M Z , G µ ) after subtracting those parts of the corrections already present in the NLO result.The sine of effective leptonic weak-mixing angle in the (α 0 , G µ , M Z ) can be computed at NLO using Eq. ( 15): after noticing that the second term in the last line goes like δ s 2 W and it would be zero if the s W counterterm was δ sin 2 θ l e f f (i.e. it had the expression derived according to Eq. ( 16) but with a numerical value of s W fixed by Eq. ( 26)), by adding and subtracting δ sin 2 θ l e f f Eq. ( 15) boils down to where in the second equality the explicit expression of the counterterms δ s 2 W (35) and δ G µ (32) was used and compared to the explicit expression of ∆ r (if the flag azinscheme4 is on, ∆ r must be computed in terms of δ Ze rather than δ Z e ).Equation (37) is the NLO expansion of sin 2, HO where ∆ rHO is obtained from ∆ r by adding to the ∆ ρ term in (21) the higher-order corrections in Eq. ( 1).Note that ∆ rHO depends on ∆ ρ, but not on ∆ α: in fact, if azinscheme4 is equal to one, ∆ r is function of δ Ze , while for azinscheme4 equal to zero the ∆ α factor originally present in ∆ r is subtracted and resummed in the α = α 0 /(1 − ∆ α) factor under the square root in Eq. (38).To summarize, the fermionic higher-order effects are computed in terms of a LO matrix-element squared computed as a function of the effective parameters α 0 /(1 − ∆ α(M 2 Z )) and sin 2, HO θ l e f f and the removal of the double-counting of the O(α) correction is achieved by subtracting its first-order expansion in ∆ r (and ∆ α if the azinscheme4 flag is off).If the complex-mass scheme is used, ∆ r in Eq. ( 37) becomes complex, but we decided to include in Eq. ( 38) − and thus, effectively, resum − only its real part in order to minimize the spurious effects introduced by the CMS prescription.
As a conclusive remark, we recall that α 0 , M Z , and G µ are the input parameters used for the theory predictions/tools [151][152][153][154][155][156][157][158][159][160][161][162][163] developed for the precise determination of the Zboson properties at LEP1 (see for instance [164] for a tuned comparison).The realizations of the (α 0 , G µ , M Z ) scheme described in this section differ from the ones used in the above-mentioned references, even though they are equivalent at the perturbative order under consideration.In fact, a typical strategy in the literature was to perform the calculation in a given scheme, for instance (G µ , M W , M Z ), using the formulae for the NLO (or NLO plus fermionic higherorder corrections) derived in that scheme but computing the numerical value of M W and s W (at the same perturbative accuracy) from (α 0 , G µ , M Z ) through the expression of ∆ r, namely: where the leading fermionic effects related to the running of the parameters α and s W in ∆ r have been resummed [135] and ∆ ρ HO includes also higher-order corrections.In Eq. ( 39) Equation ( 39) is solved iteratively as ∆ r remn is a function of s W .In Ref. [164] a slightly modified version of Eq. ( 39) was used, with ∆ ρ HO promoted to ∆ ρ HO + ∆ ρ X and ∆ r remn changed accordingly to ∆ r remn + where the notation MS just means that, within the brackets, UV poles have been removed and the mass scale µ Dim was replaced with M Z .In the following, the LEP1-like tuned comparison will be performed with the convention from Ref. [164].
A similar strategy could be followed for the (α, sin 2 θ l e f f , M Z ) schemes, where the value of sin 2 θ l e f f can be obtained from the iterative solution of 4.4 The (α MS , s 2 W MS , M Z ) scheme and its decoupling variants In the (α MS , s 2 W MS , M Z ) scheme, the independent parameters are the MS running couplings α MS and s 2 W MS and the Z-boson mass.More precisely, the input parameters are the numerical values of α MS (µ 2 0 ) and s 2 W MS (µ 2 0 ) for a given MS renormalization scale µ 0 selected by the user and the onshell Z mass (internally converted to the corresponding pole value).The values of α MS (µ 2 0 ) and s 2 W MS (µ 2 0 ) are then evolved to α MS (µ 2 ) and s 2 W MS (µ 2 ), where µ is the MS renormalization scale selected for the calculation.Both fixed and dynamical renormalization scale choices are implemented in the code.The numerical results presented in the following are obtained with a dynamical renormalization scales, µ being set to the dilepton invariant mass.
The calculation of the tree-level and bare one-loop amplitudes in the (α MS , s 2 W MS , M Z ) scheme proceeds in the very same way as in the other schemes described above, with the only difference that the electric charge and the sine of the weak-mixing angle are set to e MS (µ 2 ) and s 2 W MS (µ 2 ), respectively.The additional factor of α coming from the virtual and real QED corrections is always set to α 0 .In the numerical studies presented in the next sections, the value of α used in the loop factor in the virtual weak loops corresponds to α MS (µ 2 ), as the flag a2a0-for-QED-only is active, but the code allows the use of α 0 as well.
The renormalization in the weak sector is performed in a hybrid scheme: the Z-boson mass counterterm as well as the external-fermions wave-function counterterms are derived in the on-shell scheme (with the modifications related to the complex-mass scheme choice), while the electric charge and the sine of the weak-mixing angle are renormalized in the MS scheme (possibly supplemented with W -boson and topquark decoupling).
The electric charge counterterm in the (α MS , s 2 W MS , M Z ) scheme reads: where µ Dim is the unphysical dimensional scale introduced with dimensional regularization and cancels in the sum of bare and counterterm amplitudes.The last two terms in Eq. ( 42) implement the top and W decoupling: if µ is greater than M top (M W, thr.), only the part of the top-quark (W ) loop proportional to the combination ∆ UV − log µ  The counterterm corresponding to the sine of the weakmixing angle reads: where δ Z ZA MS and δ Z AZ MS have the usual expression of δ Z ZA and δ Z AZ in the on-shell scheme upon the replacement Note that M 2 W in Eq. ( 44) is computed as Z and does not necessarily coincide with M 2 W, thr. .When the decoupling is active, the O(α) threshold correction for µ = M W, thr. in the running of α MS induces a similar discontinuity in the running of s 2 W MS : the last term in Eq. ( 43) cancels this discontinuity at the W threshold at O(α).
The running of α MS from the scale µ 0 to the scale µ is taken from Eqs. ( 9)-( 13) of Ref. [165], which contain QED and QCD corrections to the fermionic contributions to the β function up to O(α) and O(α 3 s ) [166][167][168][169], respectively.When the calculation is performed in the decoupling scheme, the threshold corrections corresponding to the W and the top-quark thresholds are also implemented: while the former are O(α) effects, the latter are included at O(α 2 ), O(αα S ), and O(αα 2 S ) [165,170].In the code, the running of α MS is only computed between scales µ 0 and µ well within the perturbative regime (µ 2 0 , µ 2 ≫ 4m 2 b ): non-perturbative QCD effects are effectively included through the numerical value of α MS (µ 2 0 ) selected by the user (see Appendix B for the corresponding default value and related discussion).The running of s W MS is taken from Eq. ( 25) of Ref. [171] (see also [172]), which contains O(α 2 ), O(αα S ), O(αα 2 S ), and O(αα 3 S ) corrections to the fermionic part of the β function [166][167][168].As in the case of α MS , when the decoupling is active, the corrections associated with the crossing of the W and the top-quark thresholds at O(α) and at O(α 2 ), O(αα S ), and O(αα 2 S ), respectively, are also computed.For some of the results presented below, the running is only performed at NLO (flag excludeHOrun= 1).
Similarly to the (α 0 , sin 2 θ l e f f , M Z ) scheme, where the fermionic higher-order corrections effectively account for the running of α from the Thomson limit to the weak scale, in the (α MS , s 2 W MS , M Z ) scheme the universal higher-order effects are included through the running of the couplings.
In the Z_ew-BMNNPV package, the choice of leaving α MS (µ 2 0 ) and s 2 W MS (µ 2 0 ) as free parameters is motivated by the possibility of measuring s 2 W MS at the LHC and future hadron colliders from neutral-current Drell-Yan through a template fit approach, as investigated in Ref. [10].Such measurements would require the generation of Monte Carlo templates for different vales of s 2 W MS (µ 2 0 ) (and possibly α MS (µ 2 0 )) to be fitted to the data.While the present study is focused on fixedorder results and in particular on weak corrections, the Z-_ew-BMNNPV can generate the required templates at NLO QCD+NLO EW accuracy with the consistent matching to QCD and QED parton showers.Another possibility could be to use the MS scheme for a precise prediction (rather than determination) of s 2 W at the weak scale as done in Ref. [173][174][175][176][177][178][179] up to full O(α) accuracy (plus higher order corrections to the running of α and ∆ ρ).In this approach α MS (µ 2 0 ) and s 2 W MS (µ 2 0 ) are derived quantities computed as functions of other input parameters: typically the calculation is performed in the (α 0 , G µ , M Z ) scheme given the high accuracy at which these parameters are measured.For what concerns α MS , it can be computed from α 0 via the relation where ) as well as QCD corrections of order α S and α 2 S are also available).At order α, where the (renormalized) sine of the weak-mixing angle and the corresponding counterterm on the right-hand side of the second equality are computed in the (α 0 , G µ , M Z ) scheme.From equation (46) it follows that, at O(α): Fig. 1 Upper panel: cross section distribution as a function of the leptonic invariant mass, at leading order in the (G µ , M W , M Z ) scheme.Lower panel: relative difference between the NLO cross section and the LO one in the three renormalization schemes ∆ r MS being formally identical to ∆ r, but with δ Z e and sin 2 θ l e f f replaced with δ Z e MS (µ 2 ) and δ s 2 W MS .As in the case of Eqs.(37) and (38), we can consider Eq. ( 47) as the NLO expansion of In equation (48) the renormalization scale was identified with M 2 Z , given the input-parameter set used, and ∆ r MS, HO is obtained from ∆ r MS by replacing the O(α) expression of ∆ ρ LO Fig. 2 Upper panel: forward-backward asymmetry distribution as a function of the leptonic invariant mass, at leading order in the (G µ , M W , M Z ) scheme.Lower panel: absolute difference between NLO and LO asymmetry in the three renormalization schemes (G µ , M W , M Z ) (solid blue), (α 0 , G µ , M Z ) (dashed red), and with the one including the higher-order corrections discussed in Sect.3. A last comment is in order concerning the decoupling procedure.We decouple the top-quark and the W boson in the α MS and s 2 W MS running to make contact with Refs.[171,172], mainly motivated by the huge impact of the W decoupling. However we adopt a minimal (and simplified) approach where the top and the W are integrated out only in the renormalization-group equations for α MS and s 2 W MS and in the expression of the NLO counterterms for δ Z e MS and δ s 2 W MS /s 2 W MS , which are closely related to the evolution equations.The heavy degrees of freedom are not integrated out in the calculation of the relevant matrix elements.

Input parameter schemes: numerical results
In this section we investigate the numerical impact of the radiative corrections to differential observables of the NC DY process at the LHC, according to the above described input parameter schemes.In particular, we focus on the dilepton invariant mass distribution dσ /dM ll and on the forwardbackward asymmetry A FB (M ll ), defined as: where c is the cosine of the lepton scattering angle in the Collins-Soper frame, as a function of the invariant mass M ll .We consider the µ + µ − final state, with √ s = 13 TeV.All results are obtained for an inclusive setup, where no cuts are imposed on the final-state leptons except for an invariant mass cut M ll ≥ 50 GeV.The numerical values of the relevant parameters are specified in Appendix A and the default values for the higher order options, for the hadronic contributions to ∆ α as well as for the W /Z boson width options are adopted.
The upper panels of Figs. 1 and 2 show the LO predictions obtained in the G µ , M W , M Z scheme for the differential cross section and A FB distribution computed as functions of the dilepton invariant mass M ll in the window 50 GeV ≤ M ll ≤ 200 GeV, without additional kinematical cuts on the leptons.While the invariant mass distribution has a Breit-Wigner peak for M ll equal to Z mass, the asymmetry crosses zero and changes sign in the resonance region: because of this behaviour, in the following we quantify the impact of EW corrections or the differences among predictions obtained in different input-parameter schemes in terms of absolute (rather than relative) differences for the A FB distribution.
To analyse the main features of the NLO weak corrections and the higher-order effects discussed in Sects.3 and 4, we consider the (α 0 , G µ , M Z ) scheme together with a representative of the class of schemes using M W as independent parameter and a representative for the class using sin 2 θ l e f f as input.We choose schemes with couplings defined at the weak scale, namely G µ , M W , M Z , G µ , sin 2 θ l e f f , M Z , and (α 0 , G µ , M Z ) with the flag azinscheme4 equal to one (i.e. using α = α 0 /(1 − ∆ α) in the calculation).Other schemes where couplings are defined at low-energy, like the ones involving α 0 , lead to larger corrections with respect to the LO because of the running of the parameters up to the weak scale: such effects tend to reduce the differences w.r.t. to the predictions of schemes relying on α(M 2 Z ) or G µ when moving from LO to NLO and NLO+HO accuracy (see Figs. 9-10).In the MS scheme of Sect.4.4, the running of α MS (µ 2 ) and s 2 W MS (µ 2 ) reabsorbs large part of the corrections in the Born matrix elements: for this reason, the relative corrections with respect to the LO are not shown for the (α MS , s 2 W MS , M Z ) scheme and we only show the MS results at NLO with the best predictions (i.e.NLO+HO) in the other schemes (Figs.9-10).
The lower panel of Fig. 1 shows the NLO relative correction to dσ /dM ll w.r.t. the LO prediction, for three input schemes: (G µ , sin 2 θ l e f f , M Z ) (dotted green line), α 0 , G µ , M Z (dashed red line), and (G µ , M W , M Z ) (solid blue line).The corrections in the first two schemes are very similar, ranging from −1% to about +1%, with the line corresponding to the (G µ , sin 2 θ l e f f , M Z ) scheme slightly above the one for the α 0 , G µ , M Z scheme.When using M W as an independent parameter, the corrections have different shape and are in general larger, ranging form +5% at 40 GeV to −1% around 100 GeV.The picture could be understood as follows.The analytic expression of the one-loop matrix element in the three schemes is identical once the counterterms are expressed in terms of δ Z e (or δ Ze ) and δ s 2 W , the only differences being the actual form of the counterterms (δ Z e and δ s 2 W ) and the ∆ r or ∆ r subtraction terms that factorize on a the tree-level matrix-element in the scheme (G µ , M W , M Z ) or (G µ , sin 2 θ l e f f , M Z ) schemes, respectively 11 .If one replaces the counterterm δ s 2 W with δ s 2 W + δ sin 2 θ l e f f − δ sin 2 θ l e f f , one can split the one-loop matrix-element in a term that corresponds to the one-loop amplitude in the scheme (G µ , sin 2 θ l e f f , M Z ) (up to the above-mentioned subtraction terms which however appear as constant shifts in the relative corrections) plus a reminder that might be written as ∆ s 2 W ∂ M LO /∂ s 2 W which represents the change in the LO matrix-element when the numerical value of s 2 W is shifted by a factor ∆ s 2 W = δ s 2 W − δ sin 2 θ l e f f .In the α 0 , G µ , M Z scheme, ∆ s 2 W is about 2.7 × 10 −4 and the corresponding impact is hardly visible on the scale of the plot, while in the W is much larger, of order 1 × 10 −2 , and it is the main responsible for the shape and the size of the effects shown in Fig. 1.The relative corrections at NLO in the schemes with α 0 or α(M 2 Z ) as input together with M W (sin 2 θ l e f f ) can be obtained from the ones shown in Fig. 1 by removing the constant term −2∆ r (−2∆ r) or replacing it with −2∆ α(M 2 Z ), respectively.The lower panel of Fig. 2 shows the NLO correction to the asymmetry, defined as the absolute difference Similarly to what happens for the cross section, the NLO weak corrections with the schemes (G µ , sin 2 θ l e f f , M Z ) and α 0 , G µ , M Z are very close and in general smaller, falling in the range ±0.002, while the corrections in the (G µ , M W , M Z ) are larger, reaching the value of −0.018 at about 80 GeV.The results for the asymmetry and the ones for the dilepton 11 More precisely, ∆ r − ∆ α or ∆ r − ∆ α, since in all the three schemes there is a term −∆ αM LO .invariant mass basically share the same interpretation detailed above, with the main difference that the effect of the overall subtraction terms like ∆ r and ∆ r largely cancel in A NLO FB .
Figures 3 and 4 show the relative (absolute) NLO weak corrections to the cross section (forward-backward asymmetry) if only the gauge-invariant subset of the bosonic loops is included.At low dilepton invariant masses, large part of these corrections come from the bosonic contribution to ∆ r and ∆ r entering the calculation in the (G µ , M W , M Z ) and Fig. 6 Higher-order correction to the forward-backward asymmetry distribution as a function of the leptonic invariant mass.The three curves correspond to the three different choices of renormalization scheme discussed.
in the (G µ , sin 2 θ l e f f , M Z ) schemes, respectively.For larger M ll values, on the right part of the plot, the bosonic corrections are relatively large (order −5%), while the full NLO weak corrections are at the permille level, pointing out a strong cancellation between bosonic and fermionic corrections.The contribution from ∆ r and ∆ r essentially cancels in A FB and the asymmetry difference in Fig. 4 is dominated by the shift induced in the effective s 2 W by the bosonic part of the O(α) corrections in the (α 0 , G µ , M Z ) and (G µ , M W , M Z ) schemes (δ s 2 W ∼ 3 × 10 −3 and δ s 2 W ∼ 2.5 × 10 −3 , respectively).By comparing Figs. 2 and 4, one notices that a large cancellation between bosonic and fermionic effects is still there in the (α 0 , G µ , M Z ) and (G µ , M W , M Z ) schemes, while in the (G µ , sin 2 θ l e f f , M Z ) bosonic corrections dominate over the fermionic ones and the lines corresponding to this scheme in Figs. 2 and 4 are almost the same in the scale of the plot.
Figs. 5 and 6 show the higher-order universal corrections (i.e.beyond NLO) defined in Sect.3, to the cross section invariant mass distribution (normalized to the LO predictions) and the forward-backward asymmetry, respectively, in the three renormalization schemes.As for the NLO case, the plots display the relative corrections for the cross section distribution and the absolute correction for A FB (M ll ).In the (G µ , sin 2 θ l e f f , M Z ) scheme the corrections are small (order 0.2%) and essentially flat: this is because the corrections in Eq. ( 24) factorize on the LO matrix-element squared and the only dependence on M ll comes from the running of α S in the QCD corrections to ∆ ρ.The corrections in the (α 0 , G µ , M Z ) scheme fall in the range [−0.3%, 0], being basically zero for low dilepton invariant masses and reaching their maximum around the Z peak: the shape of the corrections is determined by the additional shift ∆ s HO W on top of the NLO one (∼ 5 × 10 −5 ) which only affects the Z-boson exchange amplitude, while for small invariant masses the dominant contribution is the γ exchange.The impact of the fermionic higher-order effects in the (G µ , M W , M Z ) scheme is larger than for the other choices of input parameters, ranging from about −0.7% at M ll = 50 GeV to about +0.1% at the Z-peak: in this scheme, the corrections come from the interplay of the shift to s 2 W (−9 × 10 −4 in addition to the NLO shift) and the overall factor 2(∆ ρ − ∆ ρ (α) )c W /s w + ∆ ρ 2 c 2 W /s 2 W coming from the relation between α and G µ .The latter effect enters also the γ-exchange diagram and thus affects also the low-invariant mass region of the plot.When considering the asymmetry (Fig. 6), any overall term common to numerator and denominator of Eq. ( 49) cancels: this is almost the case for the higher-order corrections in the (G µ , sin 2 θ l e f f , M Z ) scheme, where the factorization of the higher-order terms is only approximate, due to the presence of the NLO corrections, leading to a negligible residual effect of order 10 −6 on the asymmetry difference not visible within the resolution of the plot.The impact in the (G µ , M W , M Z ) scheme is larger, with a maximum of 2 × 10 −3 for M ll around 80 GeV: the behaviour is essentially determined by the above-mentioned shift in s 2 W on top of the O(α) one.In the (α 0 , G µ , M Z ) scheme the corrections are negative, reaching the value of about −7 × 10 −4 again at about 80 GeV, and driven by the higher-order shift on s 2 W . Figs. 7 and 8 show the impact of the three-loop QCD correction to ∆ ρ, δ Fig. 8 The same as in Fig. 7 for A FB .The absolute difference between the three-loop QCD contribution to ∆ ρ and the two-loop case is shown.
and the forward-backward asymmetry, respectively, in the three renormalization schemes.The leading δ QCD contribution comes from the replacement of the ∆ ρ (α) terms in the NLO calculation with the expression of ∆ ρ in Eq. (1).In the (G µ , sin 2 θ l e f f , M Z ) scheme, ∆ ρ comes from the relation between α and G µ and simply multiplies the LO matrixelement squared: the corresponding line in Fig. 7 is twice the factor 3x t (1 + x t ∆ ρ (2) ) QCD and the dependence on M ll is the residual scale dependence of the QCD correction.In the (α 0 , G µ , M Z ) scheme, the linear term in ∆ ρ comes from the corrections to the G µ M 2 Z factor in the Zboson exchange diagram: as a consequence, in the low dilepton invariant mass region, where the dominant contribution is from γ exchange, the correction tends to vanish, while for larger values of M ll it does not factorize on the LO matrixelement.In the (G µ , M W , M Z ) scheme, linear terms in ∆ ρ come both from the δ s 2 W counterterm and from the ∆ r relating α and G µ at NLO (∼ c 2 W /s 2 W ∆ ρ).Only the latter contribution factorizes on the Born result and it is the only one affecting the γ-exchange which dominates the cross section in the low-invariant mass limit, where the effect is basically three times (∼ c 2 W /s 2 W ) larger than the one observed in the (G µ , sin 2 θ l e f f , M Z ) scheme.Moving to the asymmetry, the impact of δ QCD in the (G µ , sin 2 θ l e f f , M Z ) scheme is not visible in Fig. 8, since it largely cancels between numerator and denominator in A FB .Also for the other two schemes the effect is tiny, of the order of 10 −5 .The four-loop QCD corrections to ∆ ρ (not included in Z_ew-BMNNPV code) computed at the scale M top should be about five times smaller than the three loop ones [146], but with reduced scale dependence, so that the numerical impact on the M ll and A FB distributions is negligible compared to the other effects discussed in the following.
We close this subsection presenting in Figs. 9 and 10 the predictions for different schemes referred to the ones obtained in the (α(M 2 Z ), sin 2 θ l e f f , M Z ) scheme, for different levels of perturbative accuracy: LO, NLO, and NLO+HO (relative differences for dσ /dM ll and absolute differences for A FB (M ll )).The lower panels, referring to the NLO+HO predictions, contain also the results in the hybrid MS scheme discussed in Section 4.4, (α MS , s 2 W MS , M Z ).The choice of the reference scheme is motivated by the fact that, in the (α(M 2 Z ), sin 2 θ l e f f , M Z ) scheme, the corrections do not involve ∆ α-or ∆ ρ-enhanced terms and thus the higher-order corrections discussed in Sect. 3 are absent.On the other hand, in the other schemes, the corrections can be split in a non-enhanced part − which is formally the same one as in the (α(M 2 Z ), sin 2 θ l e f f , M Z ) scheme with a different numerical value for α and s 2 W − plus a shift in s 2 W from Eq. ( 37) and an overall effect coming from the running of α or from the corrections to the relation between α and G µ when α 0 or G µ are used as input, respectively.When going beyond NLO, the latter effects can have a non-trivial interplay leading, for instance, to mixed contributions of the form ∆ α∆ ρ.As a general comment, the spread of the predictions for the differential cross section based on different input parameter schemes tends to shrink from order 20% at LO to 2% at NLO and few 0.1% with the inclusion of universal additional corrections.The absolute differences for A FB are at the level 0.02 at LO and become of the order of 10 −3 (10 −4 ) when the NLO (NLO plus fermionic higher-order) corrections are included.In the low invariant mass region, dominated by the γ-exchange diagram, the cross section ra-tios computed at LO (upper panel of Fig. 9) reduce to ratios of the values of α used in the numerator and in the denominator (squared).For the schemes employing α(M 2 Z ), including (α 0 , G µ , M Z ) since the azinscheme4 flag is active, the ratios tend to one at low M ll .The same holds for the (G µ , sin 2 θ l e f f , M Z ) scheme, since the value of α computed from G µ and sin 2 θ l e f f at LO is pretty close to α(M 2 Z ).For the schemes based on α 0 the ratios are about 12% smaller, while for the (G µ , M W , M Z ) scheme the corresponding ratio is about 6% smaller than the one for the α(M 2 Z )-based schemes.At the LO, the only difference in the predictions for the cross section computed in schemes using sin 2 θ l e f f as input comes from the value of α used in the LO couplings: this explains the horizontal lines corresponding to the (α 0 , sin 2 θ l e f f , M Z ) and and (G µ , sin 2 θ l e f f , M Z ) schemes.When using the schemes with M W as input or (α 0 , G µ , M Z ), not only the value of α used for the couplings changes with respect to the one used in the denominator, but also s 2 W is different: since a variation of this parameter affects in a different way the Zand γ-exchange amplitudes (the latter only when α is derived from G µ ) which are weighted by the factors Z +iΓ Z M Z and 1/s, respectively, the ratios corresponding to the M W -based schemes in the upper panel of Fig. 9 have a non trivial shape as a function of M ll .For the (α 0 , G µ , M Z ) scheme, the ratio is still close to one since the value of s 2 W computed at LO in this scheme is close to the value of sin 2 θ l e f f used in the denominator (0.2308 versus 0.2315, to be compared with 0.2228 in the M W -related schemes).As any overall constant factor cancels between numerator and denominator in A FB , the LO asymmetry difference with respect to the (α(M 2 Z ), sin 2 θ l e f f , M Z ) scheme is zero when (α 0 , sin 2 θ l e f f , M Z ) or (G µ , sin 2 θ l e f f , M Z ) are used as input parameters (upper panel of Fig. 10).For the other schemes one only sees the impact of the different value of s 2 W used: the three lines for the schemes based on M W overlap, while the one for the (α 0 , G µ , M Z ) is again closer to the one of the sin 2 θ l e f f -based schemes.
At NLO, the spread of the cross section ratios is considerably reduced.In the low M ll region, the ratio stays one for the schemes based on α(M 2 Z ), while for the α 0 -related schemes it is much closer to one compared to the LO results.This is because the NLO corrections in these schemes develop an overall factor 2∆ α that, once added to the LO term, leads to a sort of LO-improved prediction proportional to the effective coupling α 2 0 (1 + 2∆ α) which is just the firstorder expansion of α(M 2 Z ) 2 = α 2 0 /(1 − ∆ α) 2 .Something similar happens for the (G µ , M W , M Z ) scheme including the one-loop corrections to the α-G µ relation contained in ∆ r.Besides changing the effective value of α used in the calculation, the one-loop corrections also change the value of the effective s 2 W used for the M W -based and the (α 0 , G µ , M Z ) schemes: the latter effect is the main responsible for the shape differences in the plots and in particular it explains the change of trend moving from LO to NLO when M W is used as input parameter (s 2 e f f W, LO (M W ) < sin 2 θ l e f f while s 2 e f f W, NLO (M W ) > sin 2 θ l e f f ) 12 .Including the higher-order corrections goes in the direction of further reducing the differences among the predictions in different schemes (lower panel of Fig. 9), as this class of corrections is basically obtained in terms of Bornimproved matrix elements squared written as functions of effective couplings α and s 2 W reabsorbing the leading part of the fermionic corrections up to the scale M Z and numerically close to α(M 2 Z ) and sin 2 θ l e f f .It is worth noticing that this sort of redefinition of the couplings in the LO matrixelement does not affect the part of the one-loop result that is not enhanced by large fermionic corrections and in particular does not apply to the bosonic part of the O(α) result: the different couplings entering this part of the corrections are the main responsible for the residual deviations from one in the lower panel of Fig. 9.As an example, one can take the predictions in the (α 0 , sin 2 θ l e f f , M Z ) scheme: according to Eq. ( 23), the expression for the NLO+HO corrections is identical to the one in the (α(M 2 Z ), sin 2 θ l e f f , M Z ) scheme, when α(M 2 Z ) is obtained from α 0 /(1 − ∆ α), and the only difference is the non-enhanced part of the O(α) result, that is proportional to α 3 0 in the numerator and to α(M 2 Z ) 3 in the denominator of the ratio in Fig. 9.This difference alone leads to an effect of order ±0.2% as shown in the plot.
The lower panel of Fig. 9 also shows the ratio of the MS predictions with respect to the ones in the (α(M 2 Z ), sin 2 θ l e f f , M Z ) scheme.In the Z_ew-BMNNPV package, we implemented the expressions of Refs.[165] and [171,172] for the running of α MS and s 2 W MS from a scale µ 2 0 to a scale µ 213 leaving both µ 2 0 and the actual values of α MS (µ 2 0 ) and s 2 W MS (µ 2 0 ) as free parameters, since we had in mind the determination of s 2 W MS from neutral-current Drell-Yan at the LHC and future hadron-colliders by means of template fits, as in Ref. [10]: the MS results thus depend on α MS (µ 2 0 ) and s 2 W MS (µ 2 0 ) as input parameters.The solid black line shows the ratio of the MS prediction over the one in the ) to the values quoted by the PDG [180] (see also Appendix A).While the numbers fall in the same ballpark as the ones obtained in the other schemes, the discrepancy tends to be a little larger.The source of the differences is twofold: on the one hand, the values in Ref. [180] are computed with a theoretical accuracy that is not matched by the rest of the calculation in Z_ew-BMNNPV and, on the other hand, the param- 12 Since in our simulations the flag a2a0-for-QED-only is switched on, the loop factors for the schemes (α 0 , M W , M Z ), (α(M 2 Z ), M W , M Z ), and (G µ , M W , M Z ) are not the same, leading to slightly different values of s 2 e f f W, NLO (M W ). 13 Under the assumption that both µ 2 0 and µ 2 are in the region above 4m 2  b .
eters used in their computation and the ones employed in the present study are not tuned.The dashed black line corresponds to the MS predictions for a tuned choice of α MS (M 2 Z ) and s 2 W MS (M 2 Z ): α MS (M 2 Z ) is consistently computed from α 0 using the same parameters as in the rest of the calculation, while s 2 W MS (M 2 Z ) is derived from the input parameters (α 0 , G µ , M Z ) as described in Sect.4.4.
The interpretation of Fig. 10 for the asymmetry difference follows closely the one for the dilepton invariant mass cross section distribution, with the main difference that the corrections connected to ∆ α and ∆ r largely cancel between numerator and denominator in A FB (though not exactly, leading for instance to small deviations from zero in the low invariant mass region for the sin 2 θ l e f f -based schemes at NLO), and the spread of the predictions in the considered schemes is mainly due to the different values of s 2 W effectively employed.
The numerical results in Figs.9-10 are obtained under the assumption that the input parameters are actually free parameters to be set to the corresponding experimental values (or to be used as variables in template fit analyses) and no attempt was made to tune the input parameters for the different schemes (with the only exceptions of α(M 2 Z ) − computed from α 0 − and s 2 W MS (µ 2 0 ) in the tuned MS calculation).Another possibility, closer to the strategy used for the numerical predictions for LEP1 studies mentioned in Sect.4.3, would be to take a reference input scheme, say (α 0 , G µ , M Z ), and perform the calculation in other schemes, like the (G µ , M W , M Z ), (G µ , sin 2 θ l e f f , M Z ), or (α MS , s 2 W MS , M Z ) ones, but deriving the numerical values of M W , sin 2 θ l e f f , and s 2 W MS (µ 2 0 ), from the parameters α 0 , G µ , M 2 Z using the quantity ∆ r (∆ r) as in Eqs.(39), (41), and (48).Clearly the tuning procedure reduces all the tuned schemes to the reference one at the considered theoretical accuracy (in our case, NLO plus leading fermionic corrections of order ∆ α 2 , ∆ ρ 2 , ∆ α∆ ρ) and it is expected to reduce the spread of the predictions in the peak region (where the tuning is actually performed) but not necessarily away from the resonance.The effect of tuning is shown in Fig. 11 for the dilepton invariant mass cross section distribution (upper panel) and for the forwardbackward asymmetry (lower panel), which basically correspond to the lower panels of Figs.9-10 but taking as reference the (α 0 , G µ , M Z ) scheme.The maximum spread of the cross section ratios as a function of M ll is of about 0.025%, while the one for the asymmetry difference is of the order of 0.005%.As a technical remark, the plots are obtained in the pole scheme in order to minimize the spurious O(α 2 ) effects induced by the CMS, and the fermionic HO corrections in the (G µ , sin 2 θ l e f f , M Z ) scheme are obtained with a modified version of Eq. ( 24) where ∆ ρ is replaced with ∆ r: the expressions are equivalent at the considered theoretical accuracy, differing by terms at most of order ∆ rremn ∆ ρ, but this way the effective couplings entering the Z f f ver- NLO+HO Fig. 9 Relative difference of the predictions for the dilepton invariant mass cross section distribution at LO (upper panel), NLO (middle panel), NLO+HO (lower panel).The calculation is performed in the CMS.The α values used in the loop factors corresponds to the ones used for the LO couplings.In the tex in the calculation of the fermionic higher-orders in the (G µ , sin 2 θ l e f f , M Z ) and the (α 0 , G µ , M Z ) schemes become identical.
Concerning the (α MS , s 2 W MS , M Z ) scheme, it is interesting to analyze the renormalization-scale dependence of the predictions obtained in this scheme.Fig. 12 shows the ratio of the dilepton invariant mass cross section distribution computed with µ R = 2M ll (µ R = M ll /2) and the one obtained with the default choice µ R = M ll at LO (upper panel) and NLO (lower panel).Regardless of the accuracy in the matrix element calculation, the running of α MS and s 2 W MS is computed at O(α) accuracy (solid and dotted lines) or at O(α) plus the higher-order corrections taken from Refs.[165,171,172] (dashed and dot-dashed lines).In the plots, the finite jumps for M ll = M W (M ll = 2M W ) are a consequence of  the discontinuity in the O(α) running of the MS parameters for µ R = M W at the denominator (at the numerator for the choice µ R = M ll /2).When the HO corrections to the running of α MS and s 2 W MS are included, similar discontinuities appear also per M ll = M top (and At the LO, scale-variation effects are of order ±2% and the size of the jumps related to the W threshold in the running of the couplings is of about a couple of permille, while the jumps originated by the top threshold in the dashed and dotdashed lines are not visible on the scale of the plot.In the NLO calculation, the renormalization-scale dependence of α MS and s 2 W MS cancels against the one of the renormalization counterterms in the one-loop amplitude and the residual scale dependence starts at O(α 2 ).As a consequence, on the one hand scale variation effects are strongly suppressed − 1 Fig. 11 Relative difference (absolute difference) of the predictions for the cross section (forward-backward asymmetry) as a function of the dilepton invariant mass at NLO+HO accuracy.The calculation is performed in the pole scheme.The α values used in the loop factors corresponds to the ones used for the LO couplings.In the (α 0 , G µ , M Z ) the actual value of α is α 0 /(1 − ∆ α).The value of M W used in the (G µ , M W , M Z ) scheme is derived from (α 0 , G µ , M Z ) using Eq. ( 39).Similarly, sin 2 θ l e f f and s 2 W MS are computed by means of Eqs. ( 41) and (48).
(compared to the ones in the upper panel) and enter at the sub-permille level and, on the other hand, the jumps at the W threshold are visibly reduced.This does not happen for the discontinuities at the top threshold, since the matching corrections to the running formulae are beyond O(α).Though the HO corrections to the running of the MS parameters are not matched by the O(α) virtual matrix elements, the size of the renormalization-scale dependence shown by the dashed and dot-dashed lines is close to the one in the solid and dotted plots where only the O(α) running of α MS and s 2 W MS is used.It is thus reasonable to take the numerical impact of the HO contribution to the running of the parameters (some 0.01%, as shown in Fig. 13) as a rough estimate of the missing higher-order corrections to the matrix elements.
As a general remark, while the difference between theoretical predictions obtained with different input parameter and renormalization schemes can be considered as a rough and conservative estimate of the theoretical control over predictions involving weak corrections, there might be motivations to prefer one scheme to the others, like, for instance, the parametric uncertainties connected with the knowledge NLO Fig. 14 Effects of varying the input parameter sin 2 θ l e f f = 0.23154 ± 0.00016 in the (G µ , sin 2 θ l e f f , M Z ) scheme from the central value to the upper and lower ones, at leading and next-to-leading order.Here it is shown the relative difference between the invariant mass distribution obtained with the upper/lower value of sin 2 θ l e f f and the one with the central value sin 2 θ l e f f , c = 0.23154.
of the input parameters, the size of the perturbative corrections, the need of a specific free parameter in the calculation.
In the following we address some of these additional sources of theoretical uncertainty, in particular in Sect.6 we focus on the main parametric uncertainties, in Sect.7 we discuss the treatment of the light quark contributions, and in Sect.8 we consider different available strategies for the treatment of the unstable gauge bosons.

Parametric uncertainties
We study in the following the parametric uncertainties induced on dσ /dM ll and A FB (M ll ) by the current experimental errors affecting some of the relevant input parameters for each of the above considered schemes.In particular, we treat the scheme α 0 , G µ , M Z as free from parametric uncertainties due to the input parameters because α 0 , G µ and M Z are known with excellent accuracy in high energy physics.Therefore, for the other two representative schemes, (G µ , M W , M Z ) and (G µ , sin 2 θ l e f f , M Z ), we study the uncertainties induced by the imperfect knowledge of M W and sin 2 θ l e f f , respectively.Figure 14 displays the effect of a variation of sin 2 θ l e f f within the range 0.23154 ± 0.00016 [181] on the dilepton invariant mass distribution computed in the (G µ , Fig. 15 Effects of varying the input parameter sin 2 θ l e f f = 0.23154 ± 0.00016 in the (G µ , sin 2 θ l e f f , M Z ) scheme from the central value to the upper and lower ones, at leading and next-to-leading order.In particular it is shown the absolute difference between the asymmetry distribution obtained with the upper/lower value of sin 2 θ l e f f and the one with the central value sin    sin 2 θ l e f f , M Z ) scheme at LO and NLO accuracy (black and red lines, respectively).In particular, the quantity where s 2 e f f , c stands for the reference sin 2 θ l e f f value (0.23154) and ∆ s 2 e f f = 0.00016, is plotted as a function of M ll .Since the renormalization conditions in the (G µ , sin 2 θ l e f f , M Z ) scheme require that sin 2 θ l e f f is not affected by radiative corrections, the variations of sin 2 θ l e f f have basically the same impact at LO and at NLO.It is worth noticing that the dependence of dσ /dM ll on sin 2 θ l e f f is twofold: on the one hand, it depends on the g V /g A ratio through the Z f f vertices and, on the other hand, there is an overall dependence coming from the relation14 The latter effect is the source of the enhancement in the lowmass region of Fig. 14, as can be understood by comparing the predictions in Fig. 14 with the ones in Fig. 16, obtained in the α 0 , sin 2 θ l e f f , M Z scheme.As a consequence, in order to assess the sensitivity of the dilepton invariant mass distribution on the leptonic effective weak mixing angle (considered as a measure of the g V /g A ratio), one should consider the normalized dσ /dM ll distribution, rather than the absolute one.
In Figure 15 we plot the quantity showing the effects on A FB (M ll ) induced by variations of sin 2 θ l e f f within the same range considered in Fig. 14.Also in this case, the dependence on sin 2 θ l e f f is basically the same at LO and at NLO accuracy.Quantitatively, it amounts to approximately ±3 × 10 −4 in the resonance region and drops quickly away from the Z peak.As A FB is defined through a ratio of differential distributions, the overall spurious dependence on sin 2 θ l e f f related to Eq. ( 53) cancels and the results in Fig. 15 show the sensitivity of the A FB on the effective leptonic weak-mixing angle.
In Figures 17 and 18, we focus on the parametric uncertainty coming from the value of the W -boson mass that affects the predictions obtained in the (G µ , M W , M Z ) scheme.In particular, we plot the quantities in Eqs. ( 52) and ( 54) where we replaced s 2 e f f , c and ∆ s 2 e f f with the reference Wmass value (M c W = 80.385 GeV) and its 1σ error (∆ M W = ±15 MeV15 ).Figures 17 (18) and 14 (15) are very simi-lar.This can be understood for instance at LO, where the variations of M W and sin 2 θ l e f f are related by from which we can see that a shift of 15 MeV in M W corresponds to a shift of −0.0003 in sin 2 θ l e f f , which is approximately twice the shift we are considering in Figs. 14 and 15.The plots show the same pattern also at NLO, though the relation between sin 2 θ l e f f and M W beyond LO is more involved (indeed, at variance with the plots in Figs. 14 and 15, in Figs. 17 and 18 the NLO curves do not overlap with the LO ones).As in the case of Fig. 14, the source of the enhancement in the low invariant mass region of Fig. 17 is the overall dependence on M W originating from the relation where the real part of the masses is taken in the complexmass scheme.Figs.19 and 20 show the sensitivity of dσ /dM ll and A FB (M ll ) to variations of 400 MeV 16 in the top-quark mass value, for the three different input parameter schemes.Since M top enters parametrically only through the loop diagrams, we display only the results obtained with the NLO predictions.The largest part of the top-quark mass dependence at O(α) can be encoded in the ∆ ρ factor defined in Sect.3. In the NLO predictions computed in the (G µ , M W , M Z ) scheme, ∆ ρ enters in two different ways: through the overall factor −2∆ r (∼ 2c 2 W /s 2 W ) and via the counterterm corresponding to s 2 W (δ s 2 W ∼ c 2 W ∆ ρ).The former contribution is responsible for the constant shift of about ±3 × 10 −4 for ∆ M top = ±0.4 GeV clearly visible at low dilepton invariant masses, while the latter is the source of the shape effect in the upper panel of Fig. 19.In the (α 0 , G µ , M Z ) scheme, ∆ ρ enters the O(α) predictions only via the counterterms δ G µ and δ s 2 W : as a consequence, the γ-exchange amplitude is not affected by top-mass variations, as clearly visible in the low dilepton invariant-mass region in the central panel of Fig. 19.In the (G µ , sin 2 θ l e f f , M Z ) scheme, ∆ ρ comes from the overall factor −2∆ r (∼ 2∆ ρ) and induces a constant shift approximately three times smaller than the one coming from ∆ r in the (G µ , M W , M Z ) scheme (given the different coefficients multiplying ∆ ρ in the two calculations, namely c 2 W /s 2 W and 1).The two lines in the lower panel of Fig. 19 are not completely flat because, besides the quadratic terms in M top collected in ∆ ρ, there is a residual subleading dependence on the top-quark mass which leads to a tiny relative effect of order 10 −6 .In the forward-backward asymmetry the overall contributions from ∆ r and ∆ r largely cancel.As a result,  Fig. 20 shows the impact of the ∆ ρ term in δ s 2 W (which, in turn, is a shift of the effective s 2 W entering the calculation) for the (G µ , M W , M Z ) and (α 0 , G µ , M Z ) schemes, while for the (G µ , sin 2 θ l e f f , M Z ) scheme we only see the impact of the non-enhanced M top corrections which is basically two orders of magnitude smaller than the effect observed for the other schemes.

Treatment of ∆ had
The contributions to the running of α coming from the charged leptons and the top quark can be computed perturbatively in  terms of the corresponding contributions to the photon selfenergy and its derivative, namely: For the light-quark contributions, on the contrary, Eq. ( 57) cannot be used because of the ambiguities related to the definition of the light-quark masses arising from non-perturbative QCD effects.In the literature, a common strategy to compute the light-quark contribution to ∆ α is the introduction of light fermion masses as effective parameters which are used to calculate the analogous of Eq. ( 57) for the quark sector: The light-quark masses are chosen is such a way that the resulting hadronic running of α from 0 to M 2 Z corresponds to the one obtained from the experimental results for inclusive hadron production in e + e − collisions using dispersion relations (∆ α had fit ), namely: This approach is implemented in Z_ew-BMNNPV and it is used as a default.We stress that the light-quark masses are only used for the self-energy corrections but they do not enter the vertex and box diagrams.In particular, they are not used for the QED corrections, where the light-quarks mass singularities are regularized by means of dimensional regularization.
Starting from revision 4048, a more accurate treatment of the hadronic vacuum polarization is available in Z_ew--BMNNPV.The code contains an interface to the routines of Refs.[185][186][187][188][189][190][191][192] and [193][194][195][196] (HADR5X19.F and KNT v3.0.1, respectively) for the calculation of the hadronic running of α based on the experimental data for inclusive e + e − → hadron production at low energies in terms of dispersion relations.This interface can be activated using the input flag da_had_-from_fit=1 and the flag fit=1,2 can be used to switch between the two routines for ∆ α had fit . is worth noticing that these routines only provide results in the range [0, q 2 max ]: for larger values of q 2 , we define ∆ α had fit (q 2 ) = ∆ α had fit (q 2 max ) + ∆ α had pert.(q 2 ) − ∆ α had pert.(q 2 max ).
The starting point for the calculation for da_had_from_fit=1 is the relation While Eq. ( 58) is a definition of ∆ α had pert., Eq. ( 61) can be considered as a definition of . On the one hand, Eq. ( 61) is used in the one-loop corrections to the photon propagator to replace the combination Σ had AA (s) − sδ Z had A with −s∆ α had fit (q 2 )+isImΣ had AA (s) (where the factor δ is the light-quark contribution to the photon wave function renormalization counterterm) and, on the other hand, is used for the counterterms related to the electric charge and the photon wave function.More precisely, since the self-energy in Eq. ( 61) can be computed perturbatively for q 2 much larger than Λ 2 QCD , we take q 2 = M 2 Z and tune the quark masses using Eq. ( 59).This way, the formal expression of the counterterms is the same as the one used in the default computation (da_had_from_fit=0).The calculations for da_had_from_fit set to 0 and 1 are not equivalent: in fact, even though both of them rely on the tuning of the light-quark masses from Eq. ( 59), the corrections to the photon propagator are different since ∆ α had fit (s) ̸ = ∆ α had pert.
The input scheme used here is (α 0 , M W , M Z ).
We notice that the electric-charge and wave function counterterms could also be defined from Eq. ( 61) setting the lightquark masses to zero in the photon self-energy diagrams: on the one hand, this would lead to differences of order and, on the other hand, setting the lightquark masses to zero would require several modifications to the routines used for the evaluation of the virtual one-loop corrections.
The impact of the improved treatment of the hadronic running of α is only visible for the input parameter schemes that use α 0 as an independent parameter, since for the other schemes the terms that depend logarithmically on the lightquark masses cancel.Fig. 21 shows the dependence of dσ /dM ll on the uncertainty δ ∆ α had fit for each of the two adopted parameterizations, with the (α 0 , M W , M Z ) scheme.The effect of changing ∆ α had from its central value by a shift of ±δ ∆ α had is at the level of ±0.022% for HADR5X19.F and ±0.027% for KNT v3.0.1, respectively.Variations of ∆ α had mainly affect the δ Z e counterterm, as the light-quark mass logarithms in δ Z e have been traded for ∆ α had , leading to an almost constant shift in Fig. 21.Changing ∆ α had also affects the NLO corrected γ propagator, but the numerical impact is tiny as the bare self-energy diagrams do not involve logarithmically enhanced light-quark mass terms: this effect, being only present for the γ-mediated amplitude, induces a small shape effect in Fig. 21.In A FB the contribution from δ Z e largely cancels, leading to an absolute change in A FB at the 10 −6 level as shown in Fig. 22.
Fig. 22 Change in the asymmetry distribution if one takes the central value for ∆ α had fit plus its uncertainty δ ∆ α had fit , as in Fig. 21.

Treatment of the Z width
The unstable nature of the Z vector boson is considered by default through the complex-mass scheme [147][148][149], according to which the squared vector boson masses are taken as complex quantities ), in the LO and NLO calculation.The input values for M V and Γ V are assumed to be the on-shell ones, M OS V and Γ OS V , and are converted internally in the initialization phase to the corresponding pole values using the relations [197,198] The pole parameters M V and Γ V are used throughout the code for the matrix element calculations.In the CMS, the couplings that are functions of the gauge-boson masses become necessarily complex quantities.In particular, in the schemes with M W and M Z as input parameters, the quantities s 2 W and δ s W /s W of Eq. ( 5) and Eq. ( 6) are calculated in terms of µ W and µ Z .Since sin 2 θ l e f f is defined through the real part of the g V /g A ratio, it is considered as a real quantity when used as a free parameter.Similarly, the input parameters α 0 , α(M 2 Z ), and G µ are real.As a consequence, in the input parameter schemes with only one vector boson mass, M Z , the input couplings are taken as real quantities.
When performing calculations in the input parameter/ renormalization schemes having G µ among the free parameters (with the only exception of the (α 0 , G µ , M Z ) one), G µ is usually traded for α G µ by means of Eqs.(53) or (56).Since the gauge-boson masses enter in this relations, α G µ might in principle acquire an imaginary part if the CMS scheme is employed.In the code, we follow the standard procedure of taking a real-valued α G µ to minimize the spurious higherorder terms associated with the overall factor [Im(α G µ )] 2 .More precisely, in Eqs. ( 53) and ( 56), we always use the real part of the gauge-boson masses 17 .
The CMS preserves gauge invariance order by order in perturbation theory and this feature guarantees also that the higher-order unitarity violations are not artificially enhanced.For this reason it is the scheme commonly adopted for multiparticle NLO calculations.However, for neutral-current Drell-Yan it is possible to adopt other strategies for the treatment of the Z resonance.In particular, in the Z_ew-BMNNPV package, we implemented the so-called pole and factorization schemes following Secs.3.3.iiand 3.3.iii of Ref. [29], respectively 18 .These schemes can be switched on by means of the flags PS_scheme 1 and FS_scheme 1, respectively.
In Figs.23 we show the relative difference between the pole (factorization) scheme, blue (red) line, w.r.t. the CMS, for dσ /dM ll , considering different input parameter schemes.A feature common to all schemes is the oscillation of few 0.01% of amplitude around the Z resonance, as already shown separately for u ū and d d partonic initial states in Ref. [29].Over the whole range 50 GeV < M ll < 200 GeV, the shapes of the differences are similar for the three considered input parameter schemes, with larger differences appearing with the (G µ , M W , M Z ) and (α 0 , G µ , M Z ) schemes.The structure at the WW threshold present in Fig. 6 of Ref. [29] is not visible, within the available statistical error, because of a partial cancellation between the contributions of up-and down-type quark channels.
The same comparison between pole (factorization) scheme and CMS is shown for A FB as a function of M ll in Fig. 24.In this case we plot the absolute difference instead of the relative difference with respect to the CMS.For A FB the difference is smooth around the Z resonance, being of the order of few 10 −5 for the pole scheme and of the order of 10 −4 for the factorization scheme.Contrary to the dσ /dM ll case, in Fig 24 the WW threshold enhancement, of the order of 5 × 10 −4 , is clearly visible.

High energy regime
The analysis presented so far has been focused on the physics at the Z peak.To complete this study, we examine now the behaviour of the corrections and the interplay among different renormalization and input schemes in the high-energy regime, that can be relevant also in view of the upcoming programme of the LHC and at future high-energy machines.In this section, we present the results for a specific initialstate quark flavour focusing, for brevity, on d-quarks: in this way PDFs contributions exactly cancel when studying the relative effect of weak corrections with respect to the LO, as well as in the ratio of predictions obtained for different schemes.The main motivation of our choice is the fact that, at high dilepton invariant masses, PDFs are poorly constrained and typically affected by large errors.If one considers the contribution of all quark flavors at the same time, the above-mentioned relative corrections and ratios have a residual dependence on PDFs which tends to induce, for high M ll values, quite large unphysical distortions.The actual size of this effect clearly depends on the specific PDF set used.
Fig. 24 Difference of the pole/factorization scheme with respect to the default complex-mass scheme in the forward-backward asymmetry distribution at NLO.The difference PS-CMS is shown by the solid blue curve, while the FS-CMS one by the dashed red one.Upper panel: (G µ , M W , M Z ); middle panel: (α 0 , G µ , M Z ); lower panel: In Fig. 25 we repeat the study in Fig. 9 for dilepton invariant masses in the range between 1 and 12 TeV.With respect to the peak region, we find a quite different behaviour.First of all, the dilepton invariant mass ratios at LO are flat.In the upper panel of Fig. 9, the shapes came from the variations in the values of s 2 W entering the g Z f f couplings in the Z-boson exchange diagram and the M ll dependence was originated by the different propagators of the γ and the Z as well as by PDFs weighting the different quark flavours: in Fig. 25 not only we consider only d-quarks, but also the Zboson propagator is effectively 1/s (since s ≫ M 2 Z ) so that the only s dependence in the differential cross section is the Fig. 25 Relative difference of the predictions for dilepton invariant mass cross section distribution at LO (upper panel), NLO (middle panel), NLO+HO (lower panel), in the range 1 − 12 TeV.The calculation is performed with the same inputs of Fig. 9.
overall flux factor (see for instance Eq. (2.12) of [29] with χ Z = 1.)Moving to the NLO results, the level of agreement between the different input parameter/renormalization schemes is of the order of 1% at the left edge of the plot, but it gets worse as the partonic center of mass energy increases, with a 10% spread at 12 TeV.The inclusion of the fermionic higher-order effects discussed in the previous sections improves the picture only around 1 TeV, but it does not reabsorb the differences among the predictions in the considered schemes at large M ll .
The behaviour shown in Fig. 25 can be understood as follows.In the considered dilepton invariant mass range, the bosonic part of the weak NLO corrections is dominated by the so-called Sudakov logarithms, which are double and sin-gle logarithms of kinematic invariants over the gauge-boson masses.These logs correspond to the infrared limit of the weak corrections, where the gauge-boson masses are small compared to the energy scales involved and act as (physical) cutoff for the soft and/or collinear virtual weak corrections .Besides the Sudakov corrections, there is another class of logarithmic corrections coming from parameter renormalization.When using dimensional regularization, counterterms contain logarithms of the unphysical mass-dimension scale µ Dim in the combination where ε = (4 − D)/2, D being the number of space-time dimensions, and r ct is related to particle masses in on-shell based schemes or directly to the renormalization scale µ R in the MS scheme.This contribution cancels against similar terms appearing in the bare loop diagrams (where r ct will be some other scale, say r bare ) leaving contribution of the form log(r 2 bare /r 2 ct ).In the Drell-Yan parameter-renormalization counterterms only vertex diagrams enter, so the only possible scale (in particular in the limit of vanishing gauge-boson masses) is M ll .The functional form of both the Sudakov and the parameter-renormalization logarithms is the same in any scheme, but the coefficients multiplying the logarithms (including the LO-like amplitude where they appear and the LO amplitude in interference with it) differ numerically as they are function of the actual α and s 2 W values used.As a result, the logarithms appearing in the numerator and denominator in the NLO ratios of Fig. 25 have different coefficients and they do not cancel, leaving logarithmically enhanced remnants.It is worth emphasising that the inclusion of the fermionic higher-order corrections, by definition has no impact on the Sudakov corrections (as they are bosonic), but they are also irrelevant for the parameter-renormalization logs, as the corrections in Sect. 3 do include an effective running of the parameters, but only up to the weak scale.
In order to prove the argument above, we implemented a private version of the Z_ew-BMMNPV including the routines used in [225,226] for the evaluation of the Sudakov corrections in ALPGEN [227]  19 .Though it is true that the Sudakov corrections alone are not a good approximation for the full NLO weak corrections to neutral-current Drell-Yan (as pointed-out, for instance, in [232]), this is mainly due to the large cancellations between fermionic and bosonic corrections (as shown in Fig. 26) and to a large UV contribution from parameter-renormalization logarithms.For the onshell renormalization based schemes, like the (α 0 , G µ , M Z ), (G µ , sin 2 θ l e f f , M Z ), and (G µ , M W , M Z ) shown in the plot, the fermionic O(α) corrections are of the order of 10 − 20% and mainly come from the fermionic loops entering param- 19 For more recent implementations of Sudakov logarithms in other frameworks, see Refs.[228][229][230][231].
Fig. 26 Comparison among the predictions for the invariant cross section distribution, obtained by including the full NLO weak corrections, the fermionic-only and bosonic-only corrections, or the approximation which includes the Sudakov logarithms, the parameterrenormalization logs and the leading fermionic corrections stemming from ∆ α and ∆ ρ.Four schemes are shown: the MS in both its runningand fixed-scale realizations, the latter one with µ eter renormalization.The sum of the Sudakov corrections and the logarithms from parameter renormalization is a reasonable approximation of the NLO corrections, reproducing their shape with an essentially constant shift of about 5 − 6%.Similar consideration apply to the MS scheme, if fixed renormalization scale is used (µ R = M Z in Fig. 26).The picture changes considerably if the MS scheme is used with the renormalization scale set to the dilepton invariant mass: in this way, the corrections related to parameter renor-malization are reabsorbed in the running couplings and the remaining corrections are smaller than in the other schemes (the fermionic ones, in particular, boil-down to an almost flat −4% effect).The figure also singles-out the part of the fermionic contributions coming from the universal enhanced terms ∆ α and ∆ ρ.
To conclude, in Fig. 27 we repeat the same study of the lower panel of Fig. 25 Fig. 27 Relative difference of the predictions for the dilepton invariant mass cross section distribution at NLO+HO, after the subtraction of the leading logarithmic corrections (Sudakov plus parameterrenormalization logs).
been subtracted from the NLO+HO predictions.Despite the approximations in the calculation of the logarithmic corrections, it is still possible to read a clear trend in Fig. 27: the ratios fall in the few per mille range (as in the case of Fig. 9 for the near-resonance region) and they tend to be flat 20 .
As thoroughly discussed in the literature, when electroweak Sudakov logarithms start to become dominant a resummation algorithm needs to be adopted, in order to obtain reliable predictions.A very recent discussion on automated resummation algorithms of Sudakov logarithms in simulation tools can be found in Ref. [233].This issue is left to future investigations for the case of DY processes with POWHEG-BOX.

Conclusions
The precision physics program of the LHC requires flexible and precise simulation tools to be used for different purposes.The NC DY process, thanks to its large cross section and clean signature, plays a particular role in this context and its NLO electroweak corrections are a mandatory ingredient for every kind of analysis.In the present paper we have addressed the issue of the input parameter/renormalization schemes in the gauge sector for the electroweak corrections to NC DY.In particular, we have provided the relevant expressions for the counterterms at NLO precision in various realizations of on-shell renormalization schemes as well as of an hybrid MS/on-shell renormalization scheme.Among the on-shell schemes, we considered explicitly the following combinations: The hybrid scheme we considered, containing the input quantities (α MS (µ 2 ), s 2 W MS (µ 2 ), M Z ), is of interest for possible future direct determinations of the running of the electroweak couplings at high energies.In addition to the NLO expressions of the counterterms, we provided, for each considered scheme, the expressions for the higher-order universal corrections due to ∆ α and ∆ ρ.All the relevant expressions are presented in a self-contained way, so that they could be easily adopted in any simulation tool.For the present phenomenlogical study, all the discussed input parameter schemes have been implemented in the code Z_ew-BMNNPV of the POWHEG-BOX-V2 framework, which has been used to obtain illustrative phenomenological results on the differential distributions dσ /dM µ + µ − and A FB (M µ + µ − ), with inclusive acceptance of the leptons.The main features of the various input parameter schemes have been quantitatively analysed for the two considered observables, with focus on the invariant mass window which includes the Z peak and on the high energy region.The latter is characterized by the presence of Sudakov logarithms, whose impact has been analysed in detail with a comparison among different schemes.In addition to the effects of the electroweak corrections, we illustrated the parametric uncertainties on the two considered observables, associated with the different schemes.For the schemes with α 0 as input parameter, we included the possibility to calculate ∆ α by means of two different parameterizations based on dispersion relations using e + e − collider data.A section is devoted to the discussion of the improved treatment of the unstable Z boson with respect to the original version of the code.While the full complex-mass scheme is the new default, also the pole and the factorization schemes are available as options in the code.The numerical impact of the different width options on the considered observables are shown for three representative input parameter schemes.

Appendix A: Input parameters
We list below the values of the EW parameters used for the phenomenological results presented in the paper: The numerical values of s 2 W MS and sin 2 θ l e f f correspond to the estimates of Ref. [184].The numerical values of the fermionic masses are given below The light-quark masses are used for the calculation of ∆ α had , as detailed in section 7. Depending on the input parameter scheme adopted, we use the values in Eq. (A.1)only for the independent parameters, while the other ones are either derived or not used at all.The only exception is α 0 , which is always used as input parameter for the calculation of the one-loop QED corrections (not discussed in the present paper) and enters the loop factor α 0 /(4π).While the choice of α 0 for the photon-fermion coupling is motivated by the physical scale of the γ f f splitting and by the required cancellation of the infrared divergences between virtual and real contributions, the natural choice of α entering the weak loop factor α/(4π) is given by α of the input parameter scheme at hand.However, we leave to the user the freedom of using the loop factor α 0 /(4π) also in the pure weak corrections by setting to 0 the flag a2a0-for-QED-only.We stress that the different choices of α for the weak loop factor introduce differences at O(α 2 ).
For the parton distribution functions (PDFs), we use the NNPDF31_nlo_as_0118_luxqed set [234][235][236] provided by the LHAPDF-6.2framework [237] and set the factorization scale to the invariant mass of the dilepton system.

Appendix B: Input flags
In the following, we briefly describe the input-parameter flags for the Z_ew-BMNNPV package that have been used to produce the results shown in the main text.These are only a subset of the available input flags and we refer to the user manual for the complete list of process-specific input options.The non process-specific input flags can be found in the POWHEG-BOX-V2 documentation.

Options for EW corrections
no_ew: it is possible to switch off electroweak correction by setting no_ew 1 in the powheg.inputfile.By default no_ew= 0. no_strong: allows to switch off the QCD corrections when set to 1.By default no_strong= 0. ew_ho: the fermionic higher-order corrections discussed in section 3 are included by using the flag ew_ho 1.By default ew_ho=0.includer3qcd : if set to 1, the expression for ∆ r used in the higher-order corrections includes the three-loop QCD corrections.Default 0. includer3qcdew: when equal to 1, the three-loop mixed EW-QCD effects are included in the formula for ∆ r used for the fermionic higher-order corrections.Default 0. includer3ew : same as the previous flag, but for the threeloop EW corrections.Default 0. dalpha_lep_2loop : if set to 1, ∆ α includes the two-loop leptonic corrections from Ref. [238] when computing higherorder corrections.Default 0. QED-only: for the NC DY, the EW corrections can be split in QED and pure weak corrections in a gauge invariant way.By setting QED-only 1 in the input card, only pure QED corrections are computed.weak-only: when set to 1, only the virtual weak part of the EW corrections is computed.
Note that events can be generated without QCD corrections only at LO accuracy or at LO plus weak (potentially higher-order) corrections: the generation of events including NLO QED corrections but not the NLO QCD ones is not allowed.
Options for EW input parameter schemes scheme: this is the main flag for the choice of the EW input scheme.All the available schemes use the on-shell Z mass M OS Z (Zmass) as independent parameter (internally converted to the corresponding pole value), but differ in the choice of the remaining two independent parameters.Default 0.
scheme 0: the second EW input parameter is α 0 .
scheme 1: α(M 2 Z ) is taken as free parameter.scheme 2: G µ is used as input parameter.
scheme 3: in this scheme, the actual input parameter is α 0 .However, for each phase-space point, the matrix elements are evaluated using the on-shell running of α from q 2 = 0 to the partonic center of mass of the event (computed in terms of the Born-like momenta kn_pborn).
The additional factor of α coming from real and virtual QED corrections is always set to α 0 .The loop factor from the virtual weak corrections is set by default to α 0 , but α(q 2 ) is employed when the flag a2a0-for-QED-only is equal to 1 in the input card.The running is performed by default at NLO accuracy, and contains the two-loop leptonic corrections when the flag dalpha_lep_2loop is set to 1. scheme 4: the (α 0 , G µ , M Z ) scheme of Sect.4.3 is used.
scheme 5: the calculation is performed in the MS scheme of Sect.4.4 (see below for input flags specific to this scheme choice).
When scheme is equal to 0, 1, 2, or 3, the third EW input parameter is by default the on-shell W mass M OS W (Wmass) (internally converted to the corresponding pole value).If the flag use-s2effin is different from zero, the third independent parameter is the effective weak-mixing angle as described in Sect.4.2.use-s2effin: should be set to the desired value of the effective weak-mixing angle sin 2 θ l e f f .If this flag is present and scheme is equal to 0, 1, 2, and 3, the calculation is performed in the (α 0 , sin 2 θ l e f f , M Z ), (α(M 2 Z ), sin 2 θ l e f f , M Z ), (G µ , sin 2 θ l e f f , M Z ), and (α(q 2 ), sin 2 θ l e f f , M Z ) schemes, respectively.This flag is not compatible with the options scheme= 4, 5. a2a0-for-QED-only: regardless of the scheme used, the additional loop factor coming from the virtual weak corrections is set by default to α 0 .If the flag is set to 1, the purely weak loop factor is set to the same value used for the LO matrix element in the selected scheme.Note that the additional factor of α from real and virtual QED corrections is always equal to α 0 .
Besides M OS Z , also the on-shell Z width (Zwidth) is a free parameter of the calculation (internally converted to the corresponding pole value).The same holds for the on-shell W width (Wwidth), when M OS W is taken as an EW input parameter, though this is only relevant when the complex-mass scheme is used.
Options for the hadronic running of α and light-quark masses da_had_from_fit: if set to 1, the calculation of the hadronic corrections to the photon propagator and its derivative is based on the experimental data for inclusive e + e − → hadron production at low energies in terms of dispersion relations.Default 0. fit: if da_had_from_fit=1 and fit= 1(2), the HADR5X19.F (KNT v3.0.1) routine is used for the calculation of the hadronic vacuum polarization.The option fit=0 is left for cross checks, as with this option the quark loops under consideration are computed as in the case da_had_from_fit=0.mq_only_phot: if set to 1, the light-quark masses are set to 0 in the W , Z, and mixed γZ self-energy corrections and in their derivatives, since their light-quark mass dependence is regular and tiny in the massless quark limit.Default 0.
The treatment of the hadronic vacuum polarization is critical for the derivative of the photon propagator and thus for the electric-charge and photon wave-function counterterms in the on-shell scheme.As a consequence, it is critical for those schemes that use α 0 as input parameter, while the impact is minor when α(M 2 Z ), G µ , or α(q 2 ) are used.In the context of the MS calculation (scheme= 5), the electric-charge and photon wave-function counterterms do not depend on the light-quark masses, while in the hadronic corrections to the bare photon propagator the light-quark mass dependence is regular and tiny in the limit m q → 0: as a result, in this scheme the da_had_from_fit flag is not needed.The non perturbative effects in the MS running of α MS (and, indirectly, of s 2 W MS ) are included in the starting values of the evolution α MS (µ 2 0 ) and s 2 W MS (µ 2 0 ), that should correspond to a scale µ 2 0 sufficiently larger than 4m 2 b .
Options for the MS scheme The flags below are only effective when scheme= 5. running_muR_sw: if set to 1, the calculation is performed in the MS scheme with dynamical renormalization scale.For each phase-space point, the MS scale is set to the dilepton invariant mass in the Born-like kinematic (kn_pborn momenta) and the couplings α MS and s 2 W MS are evolved accordingly [165,171,172].Default 0. MSbarmu02: if running_muR_sw= 1, this parameter is the starting scale of the evolution of α MS and s 2 W MS .Otherwise, this is the (constant) value of the MS renormalization scale.It should be sufficiently larger than 4m 2  b .By default it is the pole Z mass computed internally from the input parameter M OS Z .MSbar_alpha_mu02: is the value of α MS (µ 2 0 ) for MSbarmu02 = µ 2 0 .MSbar_alpha_mu02 has a default value only if MSbar-mu02 is not present in the input card and corresponds to α MS (M 2 Z ) computed as a function of α 0 according to eq. (10.10) of Ref. [239] which includes effects up to O(αα 2 S ) (see also Ref. [170]).If excludeHOrun= 1, the NLO relation α 0 and α MS (M 2 Z ) is used.MSbar_sw2_mu02: same as MSbar_alpha_mu02, but for s 2 W MS .The default value is s 2 W MS (M 2 Z ) = 0.23122.decouplemtOFF: if set to 1, switches off the top-quark decoupling.Default 0. decouplemwOFF: same as decouplemtOFF, but for the W decoupling. MW_insw2_thr: allows to tune the position of the W threshold in the MS running of α end s excludeHOrun: if set to 1, the MS running of α end s 2 W is performed at NLO accuracy.Default 0 (i.e. the higher-order effects in Refs.[165,171,172] are included).OFFas_aMS: when set to 1, the running of α MS and s 2 W MS does not include the corrections O(αα S ) and O(αα 2 S ).Default 0. OFFas2_aMS: if equal to 1, the running of α MS and s 2 W MS does not include the corrections O(αα 2 S ) .Default 0. ewmur_fact: this entry sets the factor by which the renormalization scale is multiplied.It can be used for studying scale variations, e.g. by setting it to the standard values 2 or 1/2.Default 1.
Remaining EW parameters alphaem : α 0 .Default value: 1/137.0359909956391.When scheme= 0, it is used for both the couplings in the LO matrix element and in the NLO corrections.For other schemes, it is used for the extra power if α in the real and virtual QED corrections (and in the loop factor for the virtual weak corrections if a2a0-for-QED-only is not set to 1).alphaem_z: α(M 2 Z ).Default value: 1/128.95072067974729.It is only used if scheme= 1. See above (alphaem) for the additional power of α in the NLO corrections.gmu : G µ .Default value: 1.1663787 × 10 −5 GeV −2 .It is only used if scheme= 2. See above (alphaem) for the additional power of α in the NLO corrections.azinscheme4 : if it is positive, the electromagnetic coupling of the (α 0 , G µ , M Z ) scheme is set to α = α 0 /(1 − ∆ α)) in the evaluation of the matrix elements (and in the α/4π loop factor if a2a0-for-QED-only= 1).Default 0. It is only active when scheme=4.Zmass : on-shell Z-boson mass M OS Z .Default value: 91.1876 GeV.Internally converted to the corresponding pole value.Zwidth : on-shell Z-boson width Γ OS Z .Default value: 2.4952 GeV.Internally converted to the corresponding pole value.Wmass : on-shell W -boson mass M OS W . Default value: 80.385 GeV.Internally converted to the corresponding pole value.This parameter is only used if the flag use-s2effin is absent when if scheme= 0, 1, 2, 3. Otherwise, it is computed from the independent EW parameters.Wwidth : on-shell W -boson width Γ OS W . Default value: 2.085 GeV.Internally converted to the corresponding pole value.This parameter is only relevant when complexmasses= 1 if scheme is set 0, 1, 2, 3 while use-s2effin is absent.Hmass : Higgs-boson mass, only entering weak corrections.Default value: 125 GeV.Tmass : top-quark mass.Default value: 173 GeV.Elmass : electron mass.Default value: 0.51099907 MeV.Since the calculation is performed for massive final-state leptons, this is the parameter used in the phase-space generator when running the code for the process pp → e − e + .
For a description of the role of light-quark masses in the calculation, we refer to Sect.7 and to the flags for the hadronic running of α and light-quark masses.

2
-charge counterterm, while for µ < M top (M W, thr. ) the full top-quark (W ) loop enters the counterterm expression.Note that the discontinuity at O(α) on the W threshold cancels the corresponding discontinuity in the running of α MS (µ 2 ).In equation 42, δ D, W (δ D, top ) is equal to one if the W (top) decoupling is enabled together with the threshold corrections and zero otherwise (flags decouplemtOFF, decouplemwOFF, OFFthreshcorrs).

Fig. 3
Fig.3Relative corrections to the invariant mass cross section distribution obtained by considering only NLO bosonic contributions.

Fig. 4
Fig. 4 Absolute corrections to the forward-backward asymmetry obtained by considering only NLO bosonic contributions.

Fig. 5
Fig.5Higher-order correction to the cross section distribution as a function of the leptonic invariant mass.The three curves correspond to the three different choices of renormalization scheme discussed.

( 3 )
QCD in Eq. (1), normalized to the LO predictions, to the invariant mass cross section distribution

Fig. 10
Fig.10Absolute difference of the predictions for the forwardbackward asymmetry as a function of the dilepton invariant mass at LO (upper panel), NLO (middle panel), NLO+HO (lower panel).The calculation is performed in the CMS.The α values used in the loop factors corresponds to the ones used for the LO couplings.In the (α 0 , G µ , M Z ) the actual value of α is α 0 /(1 − ∆ α).

Fig. 12 1 M
Fig.12Relative difference of the dilepton invariant mass distribution obtained for µ R = 2M ll (red and green lines) or µ R = M ll /2 (black and blue curves) with respect to the predictions obtained with the de-fault choice µ R = M ll .The MS running of α MS (µ 2 R ) and s 2 W MS (µ 2 R) is computed at O(α) in the solid and dotted lines, while in the dashed and dot-dashed lines the running includes the higher-order effects described in the text.The distributions are computed at LO in the upper panel and at NLO in the lower one.

Fig. 13 2 R ) and s 2 W MS (µ 2 R
Fig.13 Relative difference between the dilepton invariant mass distributions computed in the MS scheme with and without including the HO corrections to the running of α MS (µ 2 R ) and s 2 W MS (µ 2 R ) described in the main text.
Fig.15Effects of varying the input parameter sin 2 θ l e f f = 0.23154 ± 0.00016 in the (G µ , sin 2 θ l e f f , M Z ) scheme from the central value to the upper and lower ones, at leading and next-to-leading order.In particular it is shown the absolute difference between the asymmetry distribution obtained with the upper/lower value of sin 2 θ l e f f and the one with the central value sin 2 θ l e f f , c = 0.23154.

Fig. 16
Fig.16 Effects induced on the dilepton invariant mass distribution by a variation of the input parameter sin 2 θ l e f f = 0.23154 ± 0.00016 in the (α 0 , sin 2 θ l e f f , M Z ) scheme from the central value to the upper and lower ones, at LO and NLO.Same notation and conventions of Fig.14.

Fig. 17
Fig. 17Effects of varying the input parameter M W = 80.385 ± 0.015 GeV in the (G µ , M W , M Z ) scheme from the central value to the upper and lower ones, at LO and NLO.Here it is shown the relative difference between the invariant mass distribution obtained with the upper/lower value of M W and the one with the central value M c W = 80.385 GeV.

Fig. 18
Fig. 18 Effects of varying the input parameter M W = 80.385 ± 0.015 GeV in the (G µ , M W , M Z ) scheme from the central value to the upper and lower ones, at LO and NLO.In particular it is shown the absolute difference between the asymmetry distribution obtained with the upper/lower value of M W and the one with the central value M c W = 80.385 GeV.

Fig. 19
Fig. 19 Effects of varying the top-quark mass M top = 173.0± 0.4 GeV in the three considered schemes, from the central value to the upper and lower ones.The top-quark mass enters only the next-to-leading order corrections.Here it is shown the relative difference between the invariant mass distribution obtained with the upper/lower value of M top and the one with the central value M c top = 173.0GeV.

Fig. 20
Fig. 20 Effects of the top-quark mass M top = 173.0± 0.4 GeV in the three considered schemes, from the central value to the upper and lower ones.The top-quark mass enters only the NLO corrections.Here it is shown the absolute difference between the asymmetry distribution obtained with the upper/lower value of M top and the one with the central value M c top = 173.0GeV.

( 1 MFig. 23
Fig.23 Difference of the pole/factorization scheme with respect to the default complex-mass scheme in the dilepton invariant mass cross section distribution at NLO.The difference PS-CMS is shown by the solid blue curve, the FS-CMS one by the dashed red one.Upper panel: (G µ , M W , M Z ); middle panel: (α 0 , G µ , M Z ); lower panel: (G µ , sin 2 θ l e f f , M Z ).

2 W
and the corresponding argument of the decoupling logarithms.If absent, this parameter is computed as M 2, th W = Re[M 2 Z (1 − s 2 W MS (µ 2 0 ))].OFFthreshcorrs: when set to 1, the threshold corrections in the MS running of α end s 2 W are switched off.Default 0.