Electroweak Physics in Inclusive Deep Inelastic Scattering at the LHeC
MMITP/20-038MPP-2020-110
Electroweak Physics in Inclusive Deep InelasticScattering at the LHeC
Daniel Britzger , Max Klein and Hubert Spiesberger Max-Planck-Institut f¨ur Physik, F¨ohringer Ring 6, D-80805 M¨unchen, Germany University of Liverpool, Oxford Street, UK-L69 7ZE Liverpool, United Kingdom PRISMA + Cluster of Excellence, Institut f¨ur Physik, Johannes-Gutenberg-Universit¨at,Staudinger Weg 7, D-55099 Mainz, Germany
Abstract
The proposed electron-proton collider LHeC is a unique facility where electroweakinteractions can be studied with a very high precision in a largely unexplored kine-matic regime of spacelike momentum transfer. We have simulated inclusive neutral-and charged-current deep-inelastic lepton proton scattering cross section data atcenter-of-mass energies of 1.2 and 1.3 TeV including their systematic uncertainties.Based on simultaneous fits of electroweak physics parameters and parton distri-bution functions, we estimate the uncertainties of Standard Model parameters aswell as a number of parameters describing physics beyond the Standard Model, forinstance the oblique parameters S , T , and U . An unprecedented precision at thesub-percent level is expected for the measurement of the weak neutral-current cou-plings of the light-quarks to the Z boson, g u/dA/V , improving their present precision bymore than an order of magnitude. The weak mixing angle can be determined witha precision of about ∆ sin θ W = ± . W -boson mass in the on-shell scheme is possible with an experimental uncertaintydown to ∆ m W = ± Z -pole, and which aspects of the electroweak interac-tion are unique to measurements at the LHeC, for instance electroweak parametersin charged-current interactions. We conclude that the LHeC will determine elec-troweak physics parameters, in the spacelike region, with unprecedented precisionleading to thorough tests of the Standard Model and possibly beyond. a r X i v : . [ h e p - ph ] J u l Introduction
With the discovery of the Standard Model (SM) Higgs boson at the CERN Large HadronCollider (LHC) experiments and subsequent measurements of its parameters, the funda-mental parameters of the SM have been measured directly and with remarkable precision.To further map out the validity of the theory of electroweak interactions, more and higherprecision electroweak measurements have to be performed. Such high-precision measure-ments can also be considered as a portal to new physics, since non-SM contributions maylead to significant deviations for some precisely measurable and calculable observables.The Large Hadron-electron Collider (LHeC) [1–3], planned at the LHC, may complementthe proton ring with an electron beam, allowing to perform deep inelastic scattering (DIS)with electrons and protons at TeV energies. Its electron beam energy may be chosen to be60 or 50 GeV. Considerations in this choice, as for example cost reasons, are discussed inthe forthcoming thorough update of the physics and conceptual accelerator and detectordesign report [3]. In both cases its kinematic reach extends to much higher scales incomparison to HERA, which together with the huge increase of the expected integratedluminosity will allow to perform high-precision electroweak measurements at high scalesin DIS for the first time.Since the discovery of weak neutral currents in 1973 [4, 5] and the formulation ofthe Glashow-Weinberg-Salam model [6–13], deep-inelastic lepton nucleon scattering hasplayed an important role in testing the Standard Model. One of the first measurementsof the electroweak mixing angle, sin θ W , was obtained from polarized electron-deuteronscattering at SLAC [14, 15]. With the advent of HERA, the first electron-proton col-lider, a much larger range of momentum transfers squared, Q , became accessible – animportant prerequisite for probing electroweak interactions in DIS. First measurementsof electroweak effects at HERA were undertaken in Refs. [16, 17], and more thoroughelectroweak analyses have been performed recently, for example with the complete set ofH1 data in Ref. [18].Apart from the LHeC, other options for electron-hadron colliders are presently considered.A DIS option is studied as part of the possible Future Circular Collider (FCC) at CERN,the FCC- eh [19], and will reach center-of-mass energies still higher than at the LHeC.At Brookhaven, the Electron Ion Collider (EIC) [20–22] is under development to performDIS measurements at lower energies but with higher luminosities than were achieved atHERA. For comparison, in Fig. 1 we show single-differential neutral- and charged-current(NC and CC) inclusive DIS cross sections for polarized electron-proton scattering as afunction of Q comparing the future facilities LHeC, FCC- eh , and EIC, with H1 datafrom the past HERA collider [23]. For studies of electroweak effects, data at highervalues of Q will be particularly suitable. However, it is expected, that also the EIC maycontribute to electroweak physics [24]. The LHeC might be realized during the lifetimeof the LHC and could start taking data in the 2030s, and it has recently been describedas a realistic option in the EPPSU deliberation document [25]. The newly proposed1 [GeV Q ] [ pb / G e V / d Q s d -7 -6 -5 -4 -3 -2 -1
10 110
10 p cross section - Polarized e
NC CC LHeCNC CC EICNC CC FCC-ehNC (P=0)
CC HERA-H1
Figure 1: Single differential inclusive DIS cross sections for neutral- and charged-current e − p DIS with longitudinally polarized electrons ( P e = − .
8) at LHeC, EIC, FCC- eh , and HERA.For HERA, unpolarized cross sections are displayed together with data from the H1 experiment. energy-recovery linac (ERL) for a high-quality electron beam, together with the high-luminosity upgrade of the LHC (HL-LHC), are expected to provide more than an order ofmagnitude increase in the reach towards higher Q compared to HERA and furthermorean extraordinary increase of the integrated luminosity compared to what was assumedin all previous studies. This motivates us to perform a novel exploratory study for theLHeC investigating new possibilities for the measurement of electroweak physics effects.Previously, studies of electroweak effects for similar energies have been performed for theLHeC [1] and earlier, to some extent, for the LEP ⊗ LHC proposal [26].We will put the focus on the measurement prospects of inclusive NC and CC cross sec-tions at the LHeC with the aim to determine parameters of the electroweak interaction byanalysing pseudo-data which we simulated with different assumptions on the experimen-tal uncertainties or the center-of-mass energy. Measurements in the regime of space-likemomentum transfer, where the interaction is mediated by gauge boson exchange in the t -channel, are essentially complementary to other experiments, such as proton-proton col-lisions or electron-positron annihilation, or experiments at lower energies, like neutrino ormuon scattering. The potential of experiments at the LHeC with exclusive final states,for example W - or Z -boson production, or production of the Higgs boson, has been stud-ied elsewhere [27–30]; the possible improvement in our knowledge of parton distributionfunctions due to LHeC experiments was described in Refs. [31, 32] (see also Refs. [1, 3, 19]and references therein). 2ur goal is to study tests of the electroweak SM. We therefore start with laying outthe theoretical framework and summarize the SM predictions for NC and CC DIS crosssections, including higher-order electroweak corrections in the following Sec. 2. In subse-quent sections we describe the main features of the cross section predictions (Sec. 3), thesimulated data that we use (Sec. 4), and the methodology for fitting these data to extractelectroweak physics parameters (Sec. 5). Then we present a first group of results in Sec. 6for the determination of mass parameters, i.e. the masses of the W and Z bosons and inSec. 7 for the weak mixing angle. The expected high precision of measurements at theLHeC motivates to also envisage an indirect determination of the top-quark mass throughhigher-order corrections (Sec. 8). These studies will allow one to perform tests of the SMby comparing different determinations of the electroweak physics parameters.A high precision measurement of parameters of the SM is important in order to study thevalidity of the theory of electroweak interactions. In addition, we will study a numberof possible ways to generically parameterize new physics beyond the SM In Sec. 9 westudy the well-known ST U -parameters which describe new physics entering through loopinsertions in the self energy corrections of the gauge bosons. Then we follow the wide-spread convention to generalize the SM gauge-boson fermion couplings by introducing ρ and κ parameters, both for NC (Sec. 10) and for CC (Sec. 11), or, eventually, allowingthe vector and axial-vector coupling constants to be independent free parameters, notobeying any restriction as imposed by the SM (Sec. 12). We will be able to show that inparticular the quark coupling constants, separately for up- and down-type quarks, can bedetermined with a precision at the sub-percent level. The large kinematic reach of theLHeC will also allow us to study the scale-, i.e. Q -dependence of coupling parameters.This opportunity is in fact unique to the LHeC. Finally, we conclude and summarize themost important results in Sec. 13. The impact of the LHeC measurements on possiblefuture global fits of the electroweak SM parameters is discussed in an appendix (A).A summary of our results is also part of the description of the electroweak physics potentialwithin the forthcoming publication of the update [3] of the 2012 Conceptual Design Reporton the LHeC. In this section we lay out the general properties of DIS cross sections, first at leadingorder, taking into account single boson exchange diagrams at tree level.Inclusive NC DIS cross sections are expressed in terms of generalized structure functions˜ F ± , x ˜ F ± and ˜ F ± L at electroweak (EW) leading order (LO) as d σ NC ( e ± p ) dxdQ = 2 πα xQ (cid:104) Y + ˜ F ± ( x, Q ) ∓ Y − x ˜ F ± ( x, Q ) − y ˜ F ± L ( x, Q ) (cid:105) , (1)where α denotes the fine structure constant, x is the Bjorken scaling variable, and y Y ± = 1 ± (1 − y ) encode the helicity dependence of theunderlying lepton quark hard-scattering process. The generalized structure functions canbe separated into contributions from pure γ - and Z -exchange, and their interference [33]:˜ F ± = F − ( g eV ± P e g eA ) κ Z F γZ + [( g eV g eV + g eA g eA ) ± P e g eV g eA ] κ Z F Z , (2)˜ F ± = − ( g eA ± P e g eV ) κ Z F γZ + [2 g eV g eA ± P e ( g eV g eV + g eA g eA )] κ Z F Z , (3)where P e is the degree of longitudinal polarization ( P e = − F L . The naive quark-partonmodel corresponds to the LO approximation of Quantum Chromodynamics (QCD). In thisapproximation the structure functions are calculated from quark and anti-quark partondistribution functions, q ( x ) and ¯ q ( x ): (cid:104) F , F γZ , F Z (cid:105) = x (cid:88) q (cid:2) Q q , Q q g qV , g qV g qV + g qA g qA (cid:3) { q + ¯ q } , (4) x (cid:104) F γZ , F Z (cid:105) = x (cid:88) q [2 Q q g qA , g qV g qA ] { q − ¯ q } . (5)In Eqs. (2) and (3), the coefficient κ Z accounts for the Z -boson propagator and thenormalization of the weak, relative to the electromagnetic, interaction. It is calculated,at LO, as κ Z ( Q ) = Q Q + m Z
14 sin θ W cos θ W = Q Q + m Z G F m Z √ πα . (6)Thus, depending on the choice of independent theory parameters, the normalization of κ Z is fixed by an input value for sin θ W , or, alternatively, using the Fermi coupling constant G F . The first option where sin θ W = 1 − cos θ W = 1 − m W /m Z is fixed, is called the on-shell scheme , while the second option with G F as input parameter is known as the modified on-shell scheme .The vector and axial-vector coupling constants of the lepton or quark to the Z -boson, g fV and g fA (with f = e , q and q = u , d ) in Eqs. (2) and (3), are given by the SM electroweaktheory. They depend on the electric charge, Q f , in units of the positron charge, and onthe third component of the weak-isospin of the fermion, I ,f . They are given, at LO, by g fA = I ,f , (7) g fV = I ,f − Q f sin θ W . (8)The CC DIS cross section is written, in the LO approximation, as d σ CC ( e ± p ) dxdQ = 1 ± P e πα θ W x (cid:20) Q + m W (cid:21) × (cid:0) Y + W ± ( x, Q ) ∓ Y − xW ± ( x, Q ) − y W ± L ( x, Q ) (cid:1) . (9)4ere, an incoming electron can scatter only with positively charged quarks. Therefore, inthe naive quark-parton model the structure functions W ± and xW ± are obtained fromparton distribution functions for up-type quarks and down-type anti-quarks as W − = x (cid:0) U + D (cid:1) , xW − = x (cid:0) U − D (cid:1) , (10)where U = u + c and D = ¯ d + ¯ s . For positron scattering, the combinations U = ¯ u + ¯ c and D = d + s are needed and one has W +2 = x (cid:0) U + D (cid:1) , xW +3 = x (cid:0) D − U (cid:1) . (11)At LO of QCD, one has for the longitudinal structure function W ± L = 0.Higher-order perturbative corrections of QCD are included in the MS scheme by using Q -dependent parton distribution functions, q ( x, Q ) and ¯ q ( x, Q ), evolved according to theDokshitzer-Gribov-Lipatov-Altarelli-Parisi equations. In addition, there are correctionsof order O ( α s ) to the relations (4, 5) and (10, 11) between PDFs and structure functions,and the longitudinal structure functions for NC and CC are predictions of perturbativeQCD.We will see below that the precision of LHeC measurements is expected to be at a levelwhich makes the inclusion of higher-order electroweak corrections indispensable. In par-ticular, QED radiative corrections (bremsstrahlung) have to be taken into account. Weassume that these corrections will be removed from the data at the required level of pre-cision. One-loop EW corrections have been calculated in Refs. [34–36] for NC and inRefs. [37, 38] for CC scattering (see also ref. [39] for a study of numerical results). Wehave adapted the implementation in the program EPRC [40] for our present study.The dominating universal higher-order EW corrections can be described by a modificationof the fermion gauge-boson couplings. For NC scattering, vacuum polarization leads tothe running of the fine structure constant. The NC couplings are affected by γZ mixingand Z self energy corrections. These corrections are taken into account by replacingEqs. (7, 8) with corrected couplings g fA = √ ρ NC ,f I ,f , (12) g fV = √ ρ NC ,f (cid:0) I ,f − Q f κ f sin θ W (cid:1) . (13)At LO, the coefficients ρ NC ,f and κ f are unity, but at NLO they are promoted to formfactors which are flavor and scale dependent. Since they depend on Q , they render thecoupling constants ‘effective’ running couplings. The coefficient κ f can be combined withsin θ W to define an effective, flavor and scale-dependent ( µ ) weak mixing angle,sin θ eff W,f ( µ ) = κ f ( µ ) sin θ W . (14)The leptonic weak mixing angle, sin θ effW ,(cid:96) ( m Z ), has been used to describe LEP/SLD ob-servables at the Z -pole (see e.g. [41]). We emphasize that the µ dependence of the5ffective weak mixing angle is not negligible for LHeC physics ( µ = − Q ), while only itsvalue at and close to µ = + m Z was relevant for Z -pole observables.For CC scattering, a corresponding correction factor ρ CC ,eq is introduced for e − q and e + ¯ q scattering, and ρ CC ,e ¯ q for e − ¯ q and e + q scattering, by the replacement of Eqs. (10, 11) with W − = x (cid:0) ρ ,eq U + ρ ,e ¯ q D (cid:1) , xW − = x (cid:0) ρ ,eq U − ρ ,e ¯ q D (cid:1) , (15)and W +2 = x (cid:0) ρ ,e ¯ q U + ρ ,eq D (cid:1) , xW +3 = x (cid:0) ρ ,e ¯ q D − U ρ ,eq (cid:1) . (16)In addition, box graph corrections, which are Q - and energy-dependent, are added asseparate correction terms to the NC and CC cross sections. Higher-order EW correctionsare defined in the on-shell scheme [42, 43], using m Z and m W as independent parameters(see also Refs. [44, 45]).In order to calculate predictions in the SM electroweak theory at LO, only two independentparameters are needed in addition to α . At higher orders, loop corrections involve a non-negligible dependence on the complete set of SM parameters, where the most importantones are the top-quark mass, m t , and the Higgs-boson mass, m H . In addition, hadroniccontributions to the running of the effective couplings have to be provided as independentinput [46, 47], since the corresponding higher-order corrections can not be calculated inperturbation theory.In the on-shell scheme, the masses of all particles are taken as independent input pa-rameters. The weak mixing angle is defined by the masses of the W and Z bosons,sin θ W = 1 − m W /m Z , also at NLO. Since the Fermi constant G F has been measuredwith a very high precision in muon-decay experiments [48] it is often preferred to calculatethe less well-known W -boson mass from the relation G F = πα √ m W θ W − ∆ r , (17)where higher-order corrections enter through the quantity ∆ r = ∆ r ( α, m W , m Z , m t , m H , . . . ) [42], which depends on all mass parameters of the EW SM. The correction∆ r has also to be taken into account when the propagator factor κ Z ( Q ) (see Eq. (6))is calculated, using either α , m W and m Z (the naive on-shell scheme), or α , G F and m Z (the modified on-shell scheme) to fix input parameters. The choice of a scheme forinput parameters is important since it leads to very different sensitivities to parametervariations. The contribution of the weak interaction to inclusive NC and CC DIS cross sectionsbecomes large at high momentum transfers squared and competes with the purely elec-tromagnetic neutral current interaction. This is most clearly illustrated in Fig. 2 where6 [GeV Q ] [ pb / G e V / d Q s d -9 -8 -7 -6 -5 -4 -3 -2 -1
10 110 p cross section - LHeC polarized e p - LHeC eNC -0.8) =60GeV , P= e (E CC NC =60GeV , P=+0.8) e (E CC NC -0.8) =50GeV , P= e (E CC NC =50GeV , P=+0.8) e (E CC NC (P=0%)
CC HERA-H1
Figure 2: Single differential inclusive DIS cross sections for polarized e − p NC and CC DIS atthe LHeC for two different electron beam energies ( E e = 50 and 60 GeV). Cross sections forlongitudinal electron beam polarizations of P e = − . . √ s = 920 GeV withunpolarized ( P = 0) electron beams are displayed. we show predictions for the single-differential cross sections for polarized e − p scatteringas a function of Q . Here, LHeC electron beam energies of E e = 50 GeV and 60 GeV,and a proton beam energy of E p = 7000 GeV are chosen. The LHeC predictions arecompared to data for unpolarized scattering measured at HERA, where the electron andproton beam energies had been E e = 27 . E p = 920 GeV, respectively.At lower values of Q , the NC cross section is dominated by the photon-exchange contri-bution, determined by the structure function F (cf. Eqs. (2, 3)), and much larger thanthe cross section for CC scattering. At values of Q below the mass of the W boson, Q (cid:28) m W , the propagator term in the CC cross section becomes m W / ( m W + Q ) (cid:39) Q .Weak contributions to the NC cross section become important at Q values around theelectroweak scale, Q ≈ m Z . As a consequence, the dependence of the NC cross section onthe longitudinal beam polarization, P e , becomes strong, and the cross sections for positiveand negative helicities differ significantly. Since CC scattering is purely left-handed, thedependence on the longitudinal beam polarization is strongest in this case: the CC crosssection scales linearly with the fraction of left-handed electrons in the beam, i.e. with1 − P e (cf. Eq. (9)). Note that, since DIS is mediated by gauge boson exchange withspacelike momentum transfer, µ = − Q , no resonance of a weak boson is present in the Q -dependent cross section. 7he cross sections increase slowly with the center-of-mass energy, mainly because thereach towards smaller values of the Bjorken variable x gets larger. For an electron beamenergy of E e = 60 GeV, the cross sections for NC or CC scattering in the typical range of Q in 10 000 < Q <
100 000 GeV are larger by about 10 to 15 %, compared to the caseof E e = 50 GeV. The difference of cross sections between E e = 50 and 60 GeV increaseswith Q . In this section, details of the simulation of LHeC pseudo-data used subsequently for anextraction of electroweak parameters are described.In the present analysis simulated double-differential inclusive NC and CC DIS cross sectiondata are exploited. The data have been simulated based on a numerical procedure [49] forthe purpose of the LHeC CDR update [3]. The data are briefly described in the following.The data sets include electron and positron scattering, different lepton beam polarizationsettings, and different proton beam energies. Since a decision about the actual layoutof the LHeC energy-recovery linac for the lepton beam has not yet been taken, we willstudy scenarios for two lepton beam energies, i.e. E e = 50 and 60 GeV. Most of the datawere generated with the nominal LHC proton beam energy of E p = 7000 GeV, but inaddition, a small sample with reduced proton energy of E p = 1000 GeV is also considered.A summary of the data sets is given in Tab. 1. Processes E p Q e P e L Q range No. of data points (NC, CC)[TeV] [fb − ] [GeV ] LHeC-60 LHeC-50NC, CC 7 − − . − . · − E e = 50 GeV and 60 GeV. The majority of the data will be collected with an electron beam ( Q e = −
1) and with alongitudinal beam polarization of P e = − .
8, expected to reach an integrated luminosity ofabout
L (cid:39) − . This will allow us to consider measurements of NC and CC DIS crosssections up to values of Q (cid:39) . A considerably smaller data sample will becollected with a positive electron beam polarization of P e = +0 .
8, i.e. with right-handedelectrons. For this sample, an integrated luminosity of 10 fb − was assumed. Another data In the following, the simulated pseudo-data is simply denoted as data in order to facilitate reading. − is assumed for this sample . Such reduced luminosityvalues still allow to consider measurements with positrons reaching up to Q values of500 000 GeV . Finally, another data sample will be collected with a reduced proton beamenergy. This will be important for a determination of F L and to access higher values of x at fixed medium Q . For this low-energy sample an integrated luminosity of 1 fb − wasassumed.The analysis of all data sets is restricted to Q ≥ in order to avoid regions wherenon-perturbative QCD effects are important, which could deteriorate the determinationsof parton distribution functions. For our purpose, the low- Q region is anyway of lessinterest since it does not contribute much to the sensitivity to EW parameters. CC DISdata are simulated only for Q ≥
100 GeV , since CC scattering events with significantlylower Q may be difficult to measure due to limitations of the trigger system.The data simulation accounts for the acceptance of the LHeC detector, the kinematicreconstruction, and trigger restrictions. The resulting coverage of the kinematic planecan be found, for instance, in Ref. [32]. Source of uncertainty Size of uncertainty Uncertainty on cross section∆ σ NC ∆ σ CC Scattered electron energy scale ∆ E (cid:48) e /E (cid:48) e E h /E h y < .
01) 0.0 – 1.1 % included aboveRadiative corrections 0.3 % –Photoproduction background ( y > .
5) 1 % 0.0 or 1.0 % –Uncorrelated uncertainty (efficiency) 0.5 % 0.5 %Luminosity uncertainty (normalization) 1.0 % 1.0 %Table 2: Summary of the assumptions for uncertainties from various sources used in the sim-ulation of the NC and CC cross sections. The first three items are calibration uncertaintiesand affect the event reconstruction. The last four items are uncertainties which can be assigneddirectly to the cross section.
The data include a full set of systematic uncertainties and the individual sources aresummarized in Tab. 2. For the bulk of the phase space, the ‘electron’ reconstructionmethod is used where the kinematic variables x and Q are determined from the energyand polar angle of the scattered electron. Important uncertainties originate from theelectron energy scale and polar angle measurement, and uncertainties of ∆ E (cid:48) e /E (cid:48) e = 0 . This luminosity value may eventually be smaller due to difficulties to generate intense positron beams. θ (cid:48) e = 0 . y the electron methodleads to a deterioration of the measurement resolution ∝ /y . Thus one has to exploitthe hadronic final state in the determination of the inelasticity. The present simulationaccounts for this by using a simple ‘mixed’ (i.e. Q e , y h ) reconstruction method [49] to de-termine x = Q / ( sy ). For the measurement of the hadronic final state, an uncertainty onthe hadronic energy scale of ∆ E h /E h = 0 . . y region is assumed. The statisticaluncertainty of each data point is taken to be at least 0.1 %. A global normalizationuncertainty of 1 % is taken into account, which includes the luminosity uncertainty.Finally, potential additional sources of measurement errors are combined in an uncorre-lated uncertainty component of 0.5 %. These may comprise unfolding and model uncer-tainties, efficiency uncertainties, beam background related uncertainties, possible smallstochastic uncertainties related to the calibration procedure, or uncorrelated componentsof any of the above sources. In fact, the actual size of this uncorrelated uncertainty is verydifficult to estimate for the future LHeC, but we consider the assumption of 0.5 % to berather conservative. In order to address the effect of the unknown size of the uncorrelateduncertainty in some detail, we consider in the following two alternative scenarios, onewith an uncorrelated uncertainty of 0.5 %, as well as one with a more optimistic valueof 0.25 %. These will be denoted in the following as the ‘a’ or ’b’ scenarios, respectively.Our data samples have been simulated for simplicity with an ad hoc and rather coarse x - Q grid (see Tab. 1). Yet, real data may allow a much finer binning, in particular atmedium x values or at higher Q , depending on the actual detector performance and itsresolution. In fact, the effect of a possibly finer binning may be simulated to a very goodapproximation by changing the size of the uncorrelated uncertainty, which would then beequivalent by comparing the ‘a’ and ‘b’ scenarios. The properties of the generated foursets of data samples are summarized in Table 3. Scenario E e Uncorrelated uncertaintyLHeC-50a 50 GeV 0.5 %LHeC-50b 50 GeV 0.25 %LHeC-60a 60 GeV 0.5 %LHeC-60b 60 GeV 0.25 %Table 3: Summary of the LHeC measurement scenarios. The LHeC data scenarios differ by theassumption on the electron beam energy, E e , and the assumptions made for the uncorrelateduncertainty (see text). They will be referred to by the names shown in the first column.
10n previous similar studies (see, e.g. [50,51]) it was often assumed that cross section ratiosare measured. These are for example the ratio of CC over NC cross sections, R CC / NC ,the polarization asymmetry A LR , or the charge asymmetry B ± measuring the differencebetween cross sections for electron and positron scattering. In fact, our inclusive DIS dataimplicitly comprise such a collection of cross section ratios, but while we do not constructthese ratios explicitly we instead leave it to the parameter extraction procedure to exploitthe corresponding information. In the following, however, it is often informative to con-sider these ratios for the purpose of exposing the parameter dependence and estimate thepotential impact of data on the parameter determinations. In such ratios, we can thenexpect that most of the correlated uncertainties, such as normalization errors, becomelargely constrained by the fit while uncorrelated uncertainties are reduced by taking theproperly weighted average of all data. As a consequence, due to the large number of datapoints, in the order of a few hundred, uncertainties at the per mille level can be expectedfor the observables which we are going to study in the following. The two measurementscenarios labeled with ‘a’ and ‘b’ described above will help us to verify this estimate ofexpected parameter uncertainties and its dependence on our assumption for correlatedmeasurement errors. By the time when the LHeC is realized, one should expect that the determination of PDFswill be dominated by NC and CC DIS data obtained with it. The uncertainties of PDFparameterizations will mainly represent the propagated uncertainties of these inclusiveLHeC data. The uncertainties of EW parameters determined from cross section data willtherefore be correlated with PDF uncertainties. In order to account for these correlations,EW parameters have to be determined in a combined fit simultaneously with PDFs. Thisallows the complete set of statistical, as well as correlated and uncorrelated systematicuncertainties to be taken into account. We denote such an approach in the following by‘EW’+PDF fit, while ‘EW’ may be replaced by the parameter of interest.The x -dependence of the PDFs is parameterized at a starting scale of µ = 1 . u and d valence quarkdistributions ( xu and xd ), the up-type and down-type anti-quark distributions ( xU and xD ), and the gluon distribution ( xg ). The choice of the parameterization follows previousLHeC PDF studies [1, 31], which are closely related to HERAPDF-style PDFs [52–54].The following functional form is used: xf = f A x f B (1 − x ) f C (1 + f D x + f E x ) − f A (cid:48) x f B (cid:48) (1 − x ) . , (18)where f denotes any of the five input PDFs, f = u , d , U , D , g . The second term11n this ansatz is taken into account only for the gluon distribution , i.e. u A (cid:48) = d A (cid:48) = U A (cid:48) = D A (cid:48) = 0. The normalization of each PDF is determined by the quark numbersum-rule ( u A , d A ) or the momentum sum-rules ( g A ). For the anti-quark PDF, we fix U A = D A (1 − . D B = U B . Altogether, 13 independent PDFparameters are determined in each fit ( g B , g C , g A (cid:48) , g B (cid:48) , u B , u C , u E , d B , d C , U C , D A , D B , D C ). The values of the PDF parameters used for the generation of pseudo data are notof particular relevance here. They have been obtained from a private fit to HERA data,similar to Refs. [23, 54].QCD higher-order corrections are taken into account at NNLO in the zero-mass variableflavor number scheme. They are implemented for the evolution of the PDFs and thecalculation of the structure functions using QCDNUM [55]. The strong coupling is fixed, α MS ,N f =5 s ( m Z ) = 0 . χ quantity which is subject to the minimization and error propagation is based onnormal-distributed relative uncertainties, χ = (cid:88) ij log ς i σ i V − ij log ς j σ j (19)where the sum runs over all data points, ς i are the measured cross section values and σ i their corresponding theory predictions (cf. Eqs. (1) and (9)), which incorporate the depen-dence on the fit parameters. The covariance matrix V represents the relative uncertaintiesof the data points. The Minuit library is employed, and the resulting uncertainties of thefit parameters are calculated using the HESSE or MINOS algorithm [60]. For our study,we set the data values equal to the predictions, i.e. our data represent an Asimov dataset [61] and resulting uncertainties refer to expected uncertainties. It is important to notethat with the above definition of χ the actual value of the cross section at a given pointdoes not enter the calculation of the uncertainties, but only the relative size of the uncer-tainties is of relevance. A very similar methodology has previously been used by H1 forthe determination of the expected uncertainties in their electroweak analysis of inclusiveDIS data [18]. We have validated that the uncertainties on the PDFs from a pure PDFfit, i.e. with fixed electroweak parameters, are in good agreement with dedicated PDFstudies based on the same LHeC data samples [3, 32]. The second term is commonly considered to be of importance for PDF determinations as it introducesadditional freedom at lower values of x . This may be important to describe LHeC data which probesthe x region down to x (cid:39) · − . However, we find that this has no significant impact on the resultinguncertainties of the electroweak parameters. Weak boson masses
First, we investigate the possibility to determine the fundamental parameters of the EWtheory from LHeC inclusive DIS data. In this initial part of our study we are furtherinterested to understand how the uncertainty estimates depend on the assumptions of thesimulated data.Since our analysis is based on theory predictions derived in the on-shell scheme, the freeparameters at the LO are only the masses of the weak gauge bosons, m W and m Z , andthe fine structure constant α . The latter is fixed in the analysis, i.e. it is considered to beknown with ultimate precision. The weak mixing angle is defined by the ratio of the gaugeboson masses and thus not independent. At higher orders, there is in addition a sensitivityto the top-quark and the Higgs-boson mass, which will be studied in a subsequent section.We determine the expected uncertainties for m W in an m W +PDF fit, where the valueof m Z is considered as an external input, e.g. taken from the LEP+SLD combined mea-surement [62]. For the W -boson mass parameter, we then find expected uncertaintiesof ∆ m W (LHeC-60a) = ± (exp) ± (PDF) MeV = ± (tot) MeV and (20)∆ m W (LHeC-50a) = ± (exp) ± (PDF) MeV = ± (tot) MeVfor the scenarios LHeC-60a and LHeC-50a (cf. section 4), and∆ m W (LHeC-60b) = ± (exp) ± (PDF) MeV = ± (tot) MeV and (21)∆ m W (LHeC-50b) = ± (exp) ± (PDF) MeV = ± (tot) MeVfor LHeC-60b and LHeC-50b, respectively. The breakdown of the uncertainty into contri-butions due to systematic experimental and PDF uncertainties was obtained by repeatingthe fit with PDF parameters kept fixed, which yields the exp uncertainty, while the
PDF uncertainty is then calculated as the quadratic difference from the total uncertainty. Thesize of the uncertainty component associated to the PDFs is found to be of similar sizeas the exp uncertainty.Altogether, we find a relative uncertainty for m W of the order of 10 − , which is compatiblewith our rough initial estimate for cross section ratios . The two scenarios (‘a’ and‘b’) differ by the assumption for the size of the single-bin uncorrelated uncertainty, buthave otherwise the same experimental uncertainties. The dependence of ∆ m W on thisuncertainty component is displayed in the left panel of Fig. 3. Obviously, a good control of In the previous section we have outlined that due to the large number of data points one expectsrelative uncertainties of a per mille for ratios of bin cross sections. Such cross section ratios are determinedby coefficients containing sin θ W (see Eqs. (6) and (9)). Simple error propagation allows one to infer∆ m W /m W = (sin θ W / θ W )(∆ sin θ W / sin θ W ). Therefore a factor of about sin θ W / θ W (cid:39) .
15 applies if the relative uncertainty of the cross section ratios is translated into an uncertainty of m W .This results in a relative uncertainty of ∆ m W /m W (cid:39) O (10 − ). ncorrelated uncertainty [%] − − × [ M e V ] W m ∆ Normalization uncertainty [%] − × [ M e V ] W m ∆ Figure 3: Left: The total uncertainty ∆ m W as a function of the size of the uncorrelated uncer-tainty. The horizontal line marks the uncertainty of the present world average. The ‘a’ scenariosLHeC-60a and LHeC-50a (uncorrelated uncertainty of 0.5 %) and the ‘b’ scenarios LHeC-60band LHeC-50b (0.25 %) are indicated by vertical lines. Right: The uncertainty ∆ m W as afunction of the size of the normalization uncertainty of the DIS cross sections. The nominalassumption of 1 % is indicated by a vertical line. All other systematic uncertainties are kept aslisted in Tab. 2. the uncorrelated uncertainty component will help to improve the precision of a potential W -boson mass determination. We re-iterate that a smaller uncorrelated uncertainty canbe achieved through a higher resolution which allows one to choose finer binning of thedata. In the right panel of Fig. 3 we show how the uncertainty of m W depends on thecross section normalization uncertainty for the different LHeC scenarios. Obviously thiscomponent of the uncertainty for the cross section measurement cancels to a large extent,as already discussed in the previous section. Other (correlated) systematic uncertaintycomponents behave similar as the normalization uncertainty.The expected uncertainties ∆ m W are displayed in Fig. 4 and compared with the mea-surements by LEP2 [64], Tevatron [63], ATLAS [65] and the PDG [41]. We concludethat the LHeC can be expected to yield a W -boson mass determination with the small-est experimental uncertainty from a single experiment. It will even be superior to thecurrent world average. Therefore, when real data are available, a detailed assessmentof associated theoretical uncertainties will be needed to determine the accurate centralvalue of the W -boson mass. For example, a theoretical uncertainty due to the top-quarkmass dependence entering through radiative corrections in ∆ r (see Eq. (17)) will haveto be taken into account. Assuming ∆ m t = 0 . m W = 2 . In Fig. 4, the values from LEP2 and Tevatron represent combined results taking into account mea-surements from a number of independent experiments. This procedure benefits from a reduction of thesystematic uncertainties. The same remark applies to the PDG world average. [GeV] W m80.35 80.4 [2020] PDG
LHeC-50aLHeC-50bLHeC-60aLHeC-60bIndirect determinationsATLAS TevatronLEP2Direct measurements
W-boson mass
Figure 4: Determination of the W -boson mass from a combined m W +PDF fit, assumingfixed values for all other EW parameters. Different LHeC scenarios with beam energies of E e = 60 GeV and 50 GeV as described in the text are considered and compared with existingmeasurements [63–65] and with the world average value (PDG2020) [41]. given above, is, however, not sensitive by itself to higher-order corrections beyond NLO,while the actual values would be.The high precision of the W -boson mass parameter requires an in-depth discussion of itsinterpretation and the relation to other, more direct, measurements. We find, that thesensitivity of the DIS cross sections to m W arises mainly from the weak mixing angle in theNC vector couplings g fV , Eqs. (13) (cf. also next section), whereas the contribution fromthe NC and CC normalization, Eqs. (6) and (9), and from the W -boson propagator term inCC DIS, (1 / ( Q + m W )) , cf. Eq. (9), is only small . Therefore, the precise measurementof DIS cross sections yields primarily (only) an indirect determination of the SM massparameters. In fact, the philosophy of this indirect parameter determination is similar tothe one of the so-called ‘global EW fits’ [66–68], where a collection of observables is fittedto SM predictions calculated as a function of properly chosen free theory parameters. The‘measurement’ of m W from inclusive DIS cross sections at the LHeC, therefore, providesa consistency check of the SM and is complementary to direct, true, mass measurementsof the W -boson mass.A determination of the Z -boson mass from an m Z +PDF fit yields expected experimentaluncertainties of ∆ m Z = 11 MeV (13 MeV) for LHeC-60a (LHeC-50a), respectively. Theseuncertainties are of a similar size as those for m W . However, they cannot compete withthe high precision measurements at the Z -pole by LEP+SLD [62]. Moreover, future A determination of m W from the W -boson propagator alone yields an uncertainty of ±
17 or ±
36 MeVfor LHeC-60b or LHeC-50a, respectively. [GeV] W m80.3 80.35 80.4 80.45 [ G e V ] Z m constraint) F (with G LHeC-60aPDG F G W Q sin C.L. Figure 5: Simultaneous determination of the Z -boson and W -boson masses m Z and m W fromLHeC-60a or LHeC-50a data. The additional precision measurement of G F yields a strongconstraint and its combination with the m Z and m W determination leads to a very shallowellipse. e + e − colliders are expected to provide a substantial improvement of the precision of m Z [19, 69, 70].Finally we investigate the possibility to perform a combined determination of m W and m Z . The result is shown in Fig. 5, where the 68 % confidence level contours are displayed.The precision of m W and m Z if taken from the projections of these contours, is onlymoderate. However, the observed strong correlation provides a test of the high-energybehavior of the EW SM theory. Indeed, the 68 % C.L.-contour is aligned along the lineof a constant value of sin θ W (dotted line in Fig. 5). Imposing the additional constraintfor the very precisely known value of G F [48] (dashed line, see Refs. [51, 71]) results ina very shallow ellipse (yellow). Real data would have to lead to a consistent picture ofthe different constraints shown in this figure. Their comparison provides a test for theconsistency of high-energy data from the LHeC with low-energy input from α , G F andsin θ W . θ W In the SM, the weak neutral-current couplings of the fermions are fixed by one singleparameter, i.e. through the weak mixing angle θ W . High-precision measurements of sin θ W in as many as possible different processes are therefore considered as a key to test andto restrict extensions of the SM. Therefore, we study in this section the prospects for a16etermination of sin θ W from DIS data at the LHeC, i.e. we assume the weak mixing anglein the fermion neutral-current couplings as a free fit parameter while all other parametersare fixed. This way we allow the weak neutral-current couplings to deviate from theirSM values, however only in a correlated way, instead of allowing independent, flavor-dependent variations for vector and axial-vector couplings as we will do in a subsequentsection.The highest precision on sin θ W so far has been obtained from interpretations of dedi-cated measurements in e + e − collisions at the Z pole [62]. The results are conventionallyexpressed in terms of a leptonic effective weak mixing angle which is related to the on-shelldefinition of sin θ W by a well-known correction factor,sin θ effW ,(cid:96) = κ (cid:96) ( m Z ) sin θ W . (22)A determination of sin θ W from DIS data can be compared with Z -pole measurements,provided its value is mapped to the definition of the leptonic weak mixing angle. Alsoin DIS one can define an effective, scale- and flavor-dependent weak mixing angle, cf.Eq. (14), sin θ effW ,f ( µ ) = κ f ( µ ) sin θ W . (23)We will now consider sin θ W as a free parameter which is allowed to vary in a sin θ W +PDFfit. Note that we consider in this fit only the sin θ W -dependence in the vector couplings,taken the same for leptons and quarks. SM higher-order corrections are taken into accountas described in Sec. 2 by keeping the Q - and flavor-dependent form factors κ f (seeEq. (13)). Our estimate for the uncertainties in the different LHeC scenarios are∆ sin θ W (LHeC-60a) = ± . (exp) ± . (PDF) = ± . (tot) , (24)∆ sin θ W (LHeC-50a) = ± . (exp) ± . (PDF) = ± . (tot) and ∆ sin θ W (LHeC-60b) = ± . (exp) ± . (PDF) = ± . (tot) , (25)∆ sin θ W (LHeC-50b) = ± . (exp) ± . (PDF) = ± . (tot) . These results are collected in Fig. 6 where we compare with presently available determi-nations of the leptonic weak mixing angle. Here we have neglected additional parametricuncertainties that may enter when the LHeC measurements are mapped to the leptoniceffective weak mixing angle. The determination at the LHeC is superior to any currentsingle measurement and of similar size as the LEP+SLD combination. Even measure-ments in a spacelike region of momentum transfers, i.e. for a non-resonant process, turnsout to be competitive with Z -pole measurements, despite of the fact that the cross sectionreceives large contributions from pure photon exchange at lower Q , which is independentof the weak mixing angle.In the on-shell scheme, sin θ W and m W are related to each other and a measurementof one parameter can be interpreted as a determination of the other. The uncertainty17 ,leff q sin0.231 0.2315 0.232 [PDG20] Standard Model
LHeC-50aLHeC-50bLHeC-60aLHeC-60bLHCTevatronLEP+SLD
W,leff q sin Figure 6: Comparison of determinations of the weak mixing angle. The results fromLEP+SLD [62], Tevatron [72], LHC [73–76] and the SM value refer to the leptonic weak mixingangle, sin θ effW ,(cid:96) , and include the information about the W - and Z -boson masses [76]. They areall obtained from a combination of various separate measurements (not shown individually) (seealso Ref. [68] for additional information). Two scenarios for the simulation of LHeC inclusiveNC/CC DIS data are considered. Here, the estimated uncertainties refer to the fermionic effec-tive weak mixing angle, sin θ effW ,f . With real data, the central values will have to be mapped toeach other by taking into account the proper κ -factors, see Eqs. (22, 23). for sin θ W in scenario LHeC-60b, ∆ sin θ W = ± . W -boson mass of ∆ m W = ± m W = ± m W +PDF fit (Sec. 6) would correspond to ∆ sin θ W = ± . m W is due to the weak NCcouplings; the additional m W -dependence from the CC propagator mass provides littleextra information for the determination of m W .The measurement of sin θ W can be performed in sub-regions of the wide kinematic rangeof Q accessible at the LHeC. The results for twelve Q values obtained from bin-widthand bin-center corrected cross section data are shown in Fig. 7 and Tab. 4. We findthat sin θ W can be determined in the range of about 25 < (cid:112) Q <
700 GeV with aprecision better than 0.1 % and everywhere better than 1 %. We emphasize that DIS iscomplementary to other measurements since the scattering process is mediated by bosonexchange with spacelike momenta, i.e. the scale is given by µ = − Q (cf. Sec. 7). If acalculation of DIS cross sections including higher-order EW corrections in the MS schemeis available, the uncertainty of this Q -dependent sin θ W -determination can be translatedinto a test of the running of the weak mixing angle.18 [GeV] Q100 1000 W q s i n Figure 7: Expected uncertainties of the weak mixing angle determined in sub-regions of Q .Two scenarios for the simulation of LHeC inclusive NC/CC DIS data are displayed, LHeC-50aand LHeC-60b. The SM expectation is displayed as a dotted line. Q i Bin i Expected relative uncertainty of sin θ W [GeV ] LHeC-60b LHeC-60a LHeC-50b LHeC-50a200 1 ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . Table 4: Expected relative uncertainties for the determination of sin θ W as a function of Q forthe four LHeC scenarios. The uncertainties are obtained in a simultaneous fit of 12 parame-ters sin θ W ( µ i ) with µ i = − Q i ( i = 1 , . . . ,
12) together with the PDF parameters. Absoluteuncertainties ∆ sin θ W can be calculated by multiplication with the value of sin θ W . Mass parameters through higher-order corrections
The sensitivity to the weak boson masses m W and m Z is high since these parameters enterat tree level through the cross section normalization and through the boson propagators.Other SM mass parameters enter only at higher orders. Cross sections are thereforeonly weakly dependent on them. However, measurements with a high precision may stillexhibit some sensitivity. Their investigation is interesting since this provides a test of theSM at the level of quantum corrections which is complementary to direct determinations.The dominant corrections to the gauge boson self energies depend on the top-quark mass m t . In the on-shell scheme they enter in the NC coupling parameters ρ NC and κ andin the CC correction factor ∆ r through the quantity ρ t = (3 α/ π sin θ W )( m t /m W ).Therefore, inclusive DIS cross sections depend quadratically on m t and since ρ t is in theorder of 1 %, one can expect to observe a sizeable effect on the DIS cross sections.We have determined the uncertainties of the top-quark mass m t through DIS cross sectionmeasurements for the four scenarios in an m t +PDF fit. For LHeC-50a and LHeC-50b wefind ∆ m t = ± . ± . E e =60 GeV, the top-quark mass can be determined with an uncertainty of ∆ m t = ± . m t (LHeC-60b) = ± . . (26)The size of the PDF-related uncertainty amounts to about 0.6 GeV and is already includedin the values above. In these studies, the value of m W is considered as an external, i.e.fixed, parameter. However, the dominant theoretical uncertainty for an m t determinationarises in fact from the uncertainty of m W . At present, the W mass is known with anuncertainty of ∆ m W = ±
12 MeV [41]. This corresponds to a theory uncertainty of m t ofabout ± m t = ± . ± . m t at the LHC experiments is dominated by Monte Carlo mod-elling and theoretical uncertainties related to the proper definition of the top-quark mass.These theoretical uncertainties are shared between different LHC measurements and itis expected that they limit the precision of the m t determination also in the future. Incontrast, the definition of the top-quark mass entering in higher-order EW corrections toDIS cross sections is theoretically very clean and free from QCD-related ambiguities. Infact, the definition of m t corresponds to the one used in the calculation of observables inthe SM framework, as it is also done in the global EW fits. It will therefore be justifiedto include possible future data from the LHeC in a determination of a world average of m t . We study this possibility briefly in appendix A. Our results indicate that LHeC data Note that at sufficiently high scales, e.g. Q (cid:38) (2 m t ) , the top-quark contributes also to the QCDevolution of the PDFs. However, these contributions to the inclusive DIS cross sections are very small atthe LHeC. In particular, their sensitivity to the actual value of the top-quark mass can be neglected. [GeV] t m160 170 180 190 [ G e V ] W m C.L. Figure 8: Simultaneous determination of the W -boson and top-quark masses from LHeC-60a orLHeC-50a data. Results from the Z -pole fit using LEP+SLD data [62] and from a global EWfit, where direct measurements of m W and m t have been excluded [66] are also shown. will not improve the present uncertainty of ∆ m t = ± . m t and m W are taken into account.We also consider the possibility to determine the W -boson mass m W simultaneouslytogether with m t . Prospects for such a simultaneous determination of m t , m W , and thePDFs are displayed for selected LHeC scenarios in Fig. 8 and compared with results fromthe LEP+SLD combination of Z -pole measurements [62]. The figure shows also resultsfrom a global EW fit [66], for which the direct m t and m W measurements have beenexcluded. We find that the uncertainties of the LHeC are better than those obtainedfrom the LEP+SLD combined data. For the scenario LHeC-60b, the uncertainty contouris very similar in size as the global EW fit. It is not surprising that both the globalEW fit and the LHeC fit exhibit the same type of correlation since they exploit the same m t /m W -dependent terms of the radiative corrections.One may also attempt to determine the Higgs-boson mass m H from inclusive DIS data. m H also enters through the self-energy corrections in the SM, however, the m H depen-dence is only logarithmic, ∝ log( m H /m W ), i.e. very weak. An m H +PDF fit leads toan uncertainty of ∆ m H = +28 − and +14 − GeV for the scenarios LHeC-50a and LHeC-60b,respectively. This compares well with the precision found for the indirect m H determi-nations from LEP+SLD combined data [62, 66, 67], but is, of course, much less precisethan the direct determination from the LHC experiments nowadays [78, 79]. From Higgsboson production and its decays into fermion pairs, the LHeC has a direct Higgs massmeasurement potential as well, which surely is much better than the indirect one butunlikely competitive to that at the LHC through the 4-lepton and 2-photon decays.21 Oblique parameters S , T , and U Many theories beyond the SM predict additional heavy particles. While these may be tooheavy for a direct detection in present or future experiments, they may contribute througheffective low-energy operators or through higher-order loop corrections to observables.High-precision measurements provide an opportunity to observe in an indirect way theirpresence.Loop insertions with particle-antiparticle pairs in gauge boson self energies, Σ ij ( q ), areparticularly important since they are universal. If the masses of the non-SM particles arelarge, a low- q expansion of the self-energy corrections,Σ ij ( q ) = Σ ij (0) + q F ij ( q ) , ( ij ) = ( γγ ) , ( γZ ) , ( ZZ ) , ( W W ) , (27)and neglecting the q -dependence of F ij ( q ), can provide a sufficiently precise approxima-tion by constant parameters. Taking into account that the electromagnetic U (1) gaugesymmetry has to stay intact and that some of these constants can be absorbed into renor-malization constants, there are three free parameters, usually called S , T and U [80]. Asuitable definition is described in Ref. [41] which we adopt in the following, while theirrelation to alternative definitions [81, 82] is described in Ref. [51].Results for various ST U +PDF fits are presented in Figs. 9 and Tab. 5. These fits areperformed in the on-shell scheme and the SM masses are fixed at their PDG values,in particular the values of m Z and m W . Single-parameter fits of S , T or U can provideuncertainties that are better by a factor of 2 to 5 compared to the present PDG values [41].In 2- and 3-parameter fits we observe a very strong correlation of the parameters. This canbe traced back to the fact that only certain linear combinations of S , T and U contributeto the NC and CC scattering cross sections and the γZ interference contribution. Forinstance, the values of T and U can be disentangled only if their contributions to NC andCC DIS are combined, but not from NC DIS alone. By implication, however, these linearcombinations can be determined with very high precision – a fact which makes the DISmeasurement particularly useful since it is complementary to determinations of S , T and U from Z -pole data (see, for example, Refs. [41, 66, 67]).The ST U +PDF fit to LHeC DIS data in the on-shell scheme can be combined withthe constraint from the G F measurement, cf. Fig. 9. Since G F is known with very highprecision, this constraint amounts essentially to fixing one linear combination of the ST U parameters, the one that enters in ∆ r (see Eq. (17)).New physics parameterized with the help of S , T and U will also affect the G F - m W relation, Eq. (17), through the W -boson self energy correction to the muon decay. In themodified on-shell scheme [83], where m W is calculated from G F , new physics will thereforenot only contribute by corrections to the measured cross sections, but also through amodification of the input parameters. As a consequence, the sensitivity to S , T and U is modified. Results of a ST U +PDF fit in the modified on-shell scheme are collected in22 - T - F GLHeC-60aLHeC-50aC.L.
68 S0.5 - U - F GLHeC-60aLHeC-50aC.L.
68 T0.5 - U - F G Figure 9: Results of 2-parameter fits to pairs of S , T , and U where m Z and m W are fixed SMinput parameters. For each choice of two of the three parameters S , T , or U , the third obliqueparameter is kept equal to zero. 1 σ contours are shown for three LHeC scenarios. The relationhow a direct measurement of G F would constrain the parameters is indicated in addition. Fit parameters Parameter Expected uncertainty Correlation (LHeC-60b)LHeC-60b LHeC-60a LHeC-50b LHeC-50a
S T US +PDF S ± . ± . ± . ± . T +PDF T ± . ± . ± . ± . U +PDF U ± . ± . ± . ± . S + T +PDF S ± . ± . ± . ± .
42 1 .
00 0 . T ± . ± . ± . ± .
23 1 . S + U +PDF S ± . ± . ± . ± .
26 1 .
00 0 . U ± . ± . ± . ± .
13 1 . T + U +PDF T ± . ± . ± . ± .
28 1 . − . U ± . ± . ± . ± .
26 1 . S + T + U +PDF S ± . ± . ± . ± .
58 1.00 0.65 − . T ± . ± . ± . ± .
64 1.00 − . U ± . ± . ± . ± .
36 1.00
Table 5: Results of the
ST U +PDF fits with fixed SM gauge boson masses. From top to bottomwe show the expected uncertainties for 1-, 2- and 3-parameter fits as indicated in the first columnfor all four LHeC scenarios. In the case of 2- and 3-parameter fits, the last columns show thecorrelation matrices.
Figs. 10 and Tab. 6. The uncertainties determined from single-parameter fits are slightlyless favorable in this case. However, the 2- and 3-parameter fits exhibit weaker correlationsleading to smaller uncertainties for their corresponding 1-parameter projections. In themodified on-shell scheme, additional constraints on the
ST U parameters may be obtainedby adding further direct measurements of m W or sin θ effW ,(cid:96) , e.g. from measurements in e + e − or hadron-hadron collisions. The parameter relations of such measurements are alsoindicated in Fig. 10 and in particular external measurements sensitive to sin θ effW ,(cid:96) wouldbe useful for further improvements. 23 - T - W m ) (m effW,l q sinC.L.
68 S0.5 - U - W m ) (m effW,l q sinC.L.
68 T0.5 - U - W m ) (m effW,l q sinC.L. Figure 10: Same as Fig. 9, but the calculations are performed in the modified on-shell scheme,i.e. the value for m Z is fixed, but the W boson mass is calculated from its relation to the Fermiconstant G F . The relations how direct measurements of m W or sin θ effW ,(cid:96) at the Z -pole wouldconstrain the two oblique parameters are additionally indicated by dashed and dotted lines,respectively. In the modified on-shell scheme, the measurement of m W would constrain thesame relation as a measurement of 1 − m W /m Z (dashed line). Fit parameters Parameter Expected uncertainty Correlation (LHeC-60b)LHeC-60b LHeC-60a LHeC-50b LHeC-50a S T U S +PDF S ± . ± . ± . ± . T +PDF T ± . ± . ± . ± . U +PDF U ± . ± . ± . ± . S + T +PDF S ± . ± . ± . ± .
30 1 .
00 0 . T ± . ± . ± . ± .
24 1 . S + U +PDF S ± . ± . ± . ± .
10 1 .
00 0 . U ± . ± . ± . ± .
11 1 . T + U +PDF T ± . ± . ± . ± .
10 1 . − . U ± . ± . ± . ± .
14 1 . S + T + U +PDF S ± . ± . ± . ± .
60 1.00 0.97 − . T ± . ± . ± . ± .
60 1.00 − . U ± . ± . ± . ± .
28 1.00
Table 6: Same as Tab. 5, but in the modified on-shell scheme with m Z and G F as fixed inputparameters, i.e. m W is calculated. ρ NC and κ In the following we consider the option that modifications of the EW interaction by newphysics can be parameterized directly with the help of the NC weak coupling constants.A systematic approach is based on using anomalous parameters ρ (cid:48) NC and κ (cid:48) . The first, ρ (cid:48) NC , affects the SU(2) component of NC couplings, while the second, κ (cid:48) , represents amodification of the weak mixing with the U(1) gauge field. These parameters can bechosen flavor-specific and are introduced by writing [18] g fA = (cid:113) ρ (cid:48) NC ,f ρ NC ,f I ,f , (28) g fV = (cid:113) ρ (cid:48) NC ,f ρ NC ,f (cid:0) I ,f − Q f κ (cid:48) f κ f sin θ W (cid:1) . (29)Here, the un-primed form factors ρ NC and κ f take higher-order SM corrections into ac-count, as described in Sec. 2. In the SM, the anomalous parameters ρ (cid:48) NC and κ (cid:48) areunity. In the presence of physics beyond the SM, they can deviate from unity and be Q -dependent. In particular, a value of ρ (cid:48) NC (cid:54) = 1 corresponds to a modification of theratio of the strengths of NC and CC weak interactions. A similar study of a generalizationof the CC form factor ρ CC will be discussed below in Sec. 11. The parameter κ (cid:48) can alsobe interpreted as a modification of the weak mixing angle sin θ W (see Sec. 7), i.e. thedefinition of the effective weak mixing angle, Eq. (23), is replaced bysin θ effW ,f ( µ ) = κ (cid:48) f ( µ ) κ f ( µ ) sin θ W . (30)We determine the uncertainties of the anomalous form factors ρ (cid:48) NC and κ (cid:48) in a simultane-ous fit together with the PDFs, using the simulated LHeC inclusive NC and CC DIS data.First, we consider universal, i.e. flavor-independent, ρ (cid:48) NC and κ (cid:48) parameters for both thequark and electron couplings. The results are displayed in Fig. 11. In this figure, we com-pare the expected LHeC uncertainties with corresponding results that have been obtainedfrom combined LEP+SLD data for leptonic couplings . At the LHeC, uncertainties areexpected at the level of a few per mille, i.e. of similar size as those of the LEP+SLDcombination. As expected, the scenario LHeC-60b yields the smallest uncertainties, whilefrom the LHeC-50a scenario one should expect the largest ones.The ρ (cid:48) NC - κ (cid:48) fit can be interpreted as a simultaneous determination of sin θ effW ,f and auniversal modification of the normalization of NC weak couplings by ρ (cid:48) NC . We find anuncertainty of ∆ sin θ effW ,f = ± . ± . From the combined measurements of LEP+SLD, the leptonic parameters ρ NC ,(cid:96) and κ (cid:96) have beendetermined [62]. For our comparison, we interpret them as uncertainties of flavor-universal anomalousparameters ρ (cid:48) NC and κ (cid:48) . NC,f ' r f ' k leptons (uncertainty only) LEP+SLD SM
C.L. % Figure 11: Expectation for a determination of ρ (cid:48) NC ,f and κ (cid:48) f at the 68 % confidence level, assum-ing one common anomalous factor for each fermion type. The results for three different LHeCscenarios are compared with the relative uncertainties obtained from an analysis of LEP+SLDcombined data [62] for leptonic couplings. the up- and down-type couplings is particularly intersting since this may help to narrowdown possible explanations of the flavor-structure of the SM. We perform a fit of the fouranomalous parameters ( ρ (cid:48) NC ,u , κ (cid:48) u , ρ (cid:48) NC ,d , and κ (cid:48) d ). The resulting contours at 68 % C.L.for a combination of two of the free parameters is shown in Fig. 12 (left panel for up-type, right panel for down-type quarks). The high-precision data from LEP+SLD didnot allow for a full flavor-separated determination of quark couplings; however there aredeterminations of the couplings of the second- and third-generation quarks, charm andbottom, based on a data analysis using flavor tagging. It is interesting to compare theLHeC analysis, which is dominated by light-quark couplings, with these LEP+SLD resultsfor heavy quarks. This is shown in Fig. 12 and we find that the uncertainties for up-typequarks are superior to those from LEP+SLD and comparable in the case of down-typequarks. The results for different LHeC scenarios are summarized in Tab. 7.The fact that DIS at the LHeC covers a huge range of Q values allows us to perform a testof SM couplings which is not feasible at other experiments: one can determine the scaledependence of the anomalous form factors. Indeed, many models predict flavor-specificand Q -dependent modifications. In order to study such a test, we perform fits of ρ (cid:48) NC and κ (cid:48) to LHeC data split into twelve subsets with different Q ranges. Our findings are shownin Fig. 13 for the scenarios LHeC-60a and LHeC-50a, where we include, for comparison,results obtained from H1 data [18]. At the LHeC we expect highest precision in the regionof about Q ≈
20 000 GeV . In the worst case, for scenario LHeC-50a, we can expect26 C,u ' r u ' k c-quarks (uncertainty only) LEP+SLD SM
C.L. % NC,d ' r d ' k b-quarks (uncertainty only) LEP+SLD SM
C.L. % Figure 12: Expectations at the 68 % confidence level for the simultaneous determination ofanomalous up- and down-type quark couplings, assuming electron couplings fixed at their SMvalue. In the left panel, for up-type quarks, the results are compared with uncertainties fromLEP+SLD for charm-quark anomalous couplings. The right panel for down-type quarks showsa comparison of LHeC results with LEP+SLD [62] determinations of bottom-quark couplings.
Fit parameters Parameter Expected uncertaintyLHeC-60b LHeC-60a LHeC-50b LHeC-50a ρ (cid:48) NC ,u + κ (cid:48) u + ρ (cid:48) NC ,d + κ (cid:48) d +PDF ρ (cid:48) NC ,u ± . ± . ± . ± . κ (cid:48) u ± . ± . ± . ± . ρ (cid:48) NC ,d ± . ± . ± . ± . κ (cid:48) d ± . ± . ± . ± . ρ (cid:48) NC ,u + κ (cid:48) u +PDF ρ (cid:48) NC ,u ± . ± . ± . ± . κ (cid:48) u ± . ± . ± . ± . ρ (cid:48) NC ,d + κ (cid:48) d +PDF ρ (cid:48) NC ,d ± . ± . ± . ± . κ (cid:48) d ± . ± . ± . ± . ρ (cid:48) NC ,f + κ (cid:48) f +PDF ρ (cid:48) NC ,f ± . ± . ± . ± . κ (cid:48) f ± . ± . ± . ± . ρ (cid:48) NC ,f +PDF ρ (cid:48) NC ,f ± . ± . ± . ± . Table 7: Overview of results for the ρ (cid:48) NC and κ (cid:48) fits in different LHeC scenarios. From top tobottom we list results for 4-, 2- and 1-parameter+PDF fits. uncertainties ∆ ρ (cid:48) NC = ± . κ (cid:48) = ± . ρ (cid:48) NC = ± . κ (cid:48) = ± . [GeV] Q100 1000 NC ,f ' r Q100 1000 f ' k Figure 13: Scale dependence of the anomalous form factors ρ (cid:48) NC ,f ( µ ) (left) and κ (cid:48) f ( µ ) (right)with µ = − Q for the scenarios LHeC-50a and LHeC-60b. The highest precision is obtained inthe region of about Q ≈
20 000 GeV for scenario LHeC-60b. The expected uncertainties arecompared to measured values by H1 [18].
11 Electroweak effects in charged-current scattering
The LHeC provides a unique opportunity to investigate charged-current scattering pro-cesses over many orders of magnitude in the momentum transfer Q in a single experiment.This is a consequence not only of the excellent detector performance like precise tracking,highly granular calorimetry and high-bandwidth triggers; particularly important is thefact that in CC DIS the event kinematics can be fully reconstructed from the measure-ment of the hadronic final state and the incoming electron beam four-momentum.Higher-order EW corrections to the CC DIS cross sections are collected in the effectivecouplings of the fermions to the W boson as shown in Eqs. (15, 16). To allow for physicsbeyond the SM, we introduce new anomalous, primed parameters, ρ (cid:48) CC , eq and ρ (cid:48) CC , e ¯ q [18],in a similar way as for the case of NC scattering. The modified CC structure functionsthen become W − = x (cid:0) ( ρ CC ,eq ρ (cid:48) CC ,eq ) U + ( ρ CC ,e ¯ q ρ (cid:48) CC ,e ¯ q ) D (cid:1) , (31) xW − = x (cid:0) ( ρ CC ,eq ρ (cid:48) CC ,eq ) U − ( ρ CC ,e ¯ q ρ (cid:48) CC ,e ¯ q ) D (cid:1) , (32) W +2 = x (cid:0) ( ρ CC ,eq ρ (cid:48) CC ,eq ) U + ( ρ CC ,e ¯ q ρ (cid:48) CC ,e ¯ q ) D (cid:1) , (33) xW +3 = x (cid:0) ( ρ CC ,e ¯ q ρ (cid:48) CC ,e ¯ q ) D − ( ρ CC ,eq ρ (cid:48) CC ,eq ) U (cid:1) . (34)The prospects for a determination of these anomalous couplings with LHeC data areobtained by performing a fit of the two parameters ρ (cid:48) CC ,eq and ρ (cid:48) CC ,e ¯ q together with the28 C,eq ' r q CC , e ' r C.L. % Figure 14: Expected uncertainties of anomalous CC coupling parameters ρ (cid:48) CC ,eq and ρ (cid:48) CC ,e ¯ q forthree different LHeC scenarios compared with results from the H1 measurement [18]. Fit parameters Parameter Expected uncertaintyLHeC-60b LHeC-60a LHeC-50b LHeC-50a ρ (cid:48) CC ,eq + ρ (cid:48) CC ,e ¯ q +PDF ρ (cid:48) CC ,eq ± . ± . ± . ± . ρ (cid:48) CC ,eq + ρ (cid:48) CC ,e ¯ q +PDF ρ (cid:48) CC ,e ¯ q ± . ± . ± . ± . ρ (cid:48) CC ,eq +PDF ρ (cid:48) CC ,eq ± . ± . ± . ± . ρ (cid:48) CC ,e ¯ q +PDF ρ (cid:48) CC ,e ¯ q ± . ± . ± . ± . ρ (cid:48) CC ,f +PDF ρ (cid:48) CC ,f ± . ± . ± . ± . ρ (cid:48) CC ,f + κ (cid:48) f +PDF ρ (cid:48) CC ,f ± . ± . ± . ± . κ (cid:48) f ± . ± . ± . ± . Table 8: Expected uncertainties of anomalous CC coupling parameters ρ (cid:48) CC ,eq and ρ (cid:48) CC ,e ¯ q from2-parameter (upper part) and 1-parameter fits (lower part). The last two lines show the resultsfrom a fit combining the CC parameter ρ (cid:48) CC with the NC κ (cid:48) parameter (see Eq. (30)). PDFs. The expected uncertainties for the LHeC-50a and LHeC-60a scenarios are dis-played in Fig. 14 and collected in Tab. 8. We find that these parameters can be deter-mined with a relative uncertainty of better than 0.3 %. For the LHeC-60b scenario, evensmaller uncertainties can be achieved and we find in 1-parameter+PDF fits relative un-certainties below one per mille. We can also consider a fit combining the CC parameters ρ (cid:48) CC ,eq = ρ (cid:48) CC ,e ¯ q =: ρ (cid:48) CC ,f with the anomalous NC parameter κ (cid:48) (see Eq. (30)). Resultsfor this case are also shown in Tab. 8. Since the determination of the ρ (cid:48) CC parameters29 [GeV] Q10 100 CC ,f ' r Figure 15: Scale dependent determination of the anomalous CC coupling parameters, assuming ρ (cid:48) CC ,eq = ρ (cid:48) CC ,e ¯ q = ρ (cid:48) CC ,f . For comparison, values from H1 [18] are also displayed. are strongly correlated with the normalization uncertainty of the data, the study benefitsfrom the simultaneous analysis of NC and CC DIS data. By doing so, not only the PDFsare constrained, but also systematic uncertainties that are common to NC and CC DISdata, mainly the luminosity uncertainty, are reduced by the NC DIS data, and thereforesmaller uncertainties are obtained in this analysis than in a fit with CC DIS data alone.Event rates at the LHeC are expected to be large and will cover a large Q range between10 GeV and almost 1000 GeV . It is therefore possible to determine the anomalousCC couplings in different Q ranges. Our results are shown in Fig. 15, assuming oneflavor-independent parameter ρ (cid:48) CC ,f = ρ (cid:48) CC ,eq = ρ (cid:48) CC ,e ¯ q . The Q range is split into twelvebins, and for each bin the coupling parameter ρ (cid:48) CC ,f was allowed to vary independently ina EW+PDF fit. We find uncertainties below 0.3 % in the Q bins up to about 500 GeV .They are dominated by the normalization uncertainties of the data. Higher center-of-massenergies, i.e. with E e = 60 GeV instead of 50 GeV, has therefore only a moderate impacton the size of the uncertainties in the central Q region. However, a larger beam energyallows one to extend the measurement to higher Q values.
12 SM weak neutral-current couplings
The NC DIS cross sections are determined by products of the weak neutral-current cou-pling constants of the quarks. They enter through the γZ interference and Z exchangeterms in the generalized structure functions defined in Eqs. (4, 5). Here we focus onthe inclusive measurement of DIS cross sections and do not discuss the possibility that30ndividual quark flavors might be identified in the final state (e.g. for charm and bottom).Therefore a full flavor separation of quark couplings will not be possible. However, mainlytwo effects allow us to separate the up-type and the down-type quark contributions tothe cross section: first, they carry different electric charge and contribute with differentweights to the γZ interference terms; second, they affect, through ˜ F ± , the dependence onthe polarization and the lepton charge. In fact, these effects due to the weak interactionare important predominantly at higher values of Q , corresponding to large x , where theup- and down-valence quark PDFs dominate. A determination of vector and axial-vectorcouplings of up-type and down-type quarks can therefore be interpreted, with high pre-cision, as a determination of the coupling constants of the up- and down-quarks, i.e. oflight quarks only. Only for the down-type couplings, a contribution from strange quarkshas some relevant size.We perform an EW+PDF fit where the vector and axial-vector couplings of up-type ( u and c ) and down-type quarks ( d , s and b ), denoted as g uV , g dV and g uA , g dA , are free parametersin one single fit. Other parameters, in particular the lepton couplings, are fixed . The fitparameters are identified with the coupling constants defined in the Born approximation,and Q dependent higher-order corrections are calculated in the SM formalism in the1-loop approximation. We have verified that the results of this analysis is consistent withthose of the anomalous form factors described above in Sec. 10 (cf. Fig. 12). The resultinguncertainties of the fits are summarized in Tab. 9 for different LHeC scenarios. Fig. 16shows the results for the LHeC-50a scenario, compared with the current most precisemeasurements. All other LHeC scenarios result in even smaller uncertainties.Current determinations of light-quark couplings from e + e − , ep or p ¯ p collisions all appearwith a similar precision. Future measurements at the LHeC, however, will greatly improvethe measurement of these EW parameters. The scenario with E e = 60 GeV and theoptimistic assumptions for systematic uncertainties is particularly promising, see Tab. 9.In this table we also show results from fits where only two couplings are free fit parameterswhile the other couplings are fixed at their SM value. We find that the uncertainties oflight-quark couplings can be improved by more than an order of magnitude through LHeCdata.At the LHeC, the vector and axial-vector couplings can be disentangled without anysign-ambiguity, since the DIS cross sections receive important contributions from theinterference of photon and Z -boson exchange diagrams. This is in contrast with Z -poleobservables where only squares of the couplings are accessible. The determination ofquark NC couplings will provide a unique possibility for testing the EW SM theory.Such a measurement cannot be performed with a comparably high precision in otherexperiments. A fit to determine the electron couplings will not be competitive with corresponding determinationsfrom Z -pole observables (see last two lines of Tab. 9). A g0.4 0.5 0.6 u V g C.L. dA g0.6 - - - d V g - - - C.L. Figure 16: Weak-neutral-current vector and axial-vector couplings of up-type quarks to the Z -boson (left), and those of the down-type quarks (right) at 68 % confidence level for simulatedLHeC data with E e = 50 GeV (scenario LHeC-50a). The LHeC expectations are compared withresults from the combined LEP+SLD experiments [62] and single measurements by D0 [84] andH1 [18]. The standard model expectations are at the crossing of the horizontal and vertical lines. Coupling PDG Expected uncertaintiesparameter LHeC-60b LHeC-60a LHeC-50b LHeC-50a g uA . +0 . − . ± . ± . ± . ± . g dA − . +0 . − . ± . ± . ± . ± . g uV . ± . ± . ± . ± . ± . g dV − . +0 . − . ± . ± . ± . ± . g uA ± . ± . ± . ± . g uV ± . ± . ± . ± . g dA ± . ± . ± . ± . g dV ± . ± . ± . ± . g eA − . ± . ± . ± . ± . ± . g eV . ± . ± . ± . ± . ± . Table 9: Present values and uncertainties of the light-quark ( g uA , g dA , g uV , g dV ) and electron( g eA , g eV ) weak neutral-current couplings from the PDG [41] and the uncertainties expected forinclusive DIS measurements in different LHeC scenarios, obtained in a simultaneous EW+PDFfit. The upper section shows results of a fit where all 4 couplings are free fit parameters, thelower three sections are results from two-parameter+PDF fits. The proposed LHeC experiment at CERN’s HL-LHC will provide a unique opportunityfor high precision electroweak physics in neutral- and charged-current interactions in ayet completely unexplored kinematic regime of spacelike momentum transfer.In this article we have simulated inclusive NC and CC deep-inelastic scattering crosssection data at electron-proton center-of-mass energies of 1.2 and 1.3 TeV, and assessedtheir sensitivity to a number of parameters of the electroweak theory and the sensitivity topossible generic extensions beyond the SM. Our theoretical calculations include next-to-next-to-leading order QCD corrections to the DGLAP evolution of PDFs and their relationto the structure functions and the full set of one-loop electroweak corrections to electron–(anti-)quark t -channel scattering in the on-shell renormalization scheme. Our simulatedpseudo-data comprise a full set of experimental systematic uncertainties and have beenused also elsewhere to study the prospects for a determination of parton distributionfunctions. The latter are implicitly also included in our analysis framework, while weextend the phenomenological analysis towards electroweak parameters.The sensitivity of inclusive NC/CC DIS data at the LHeC to important independent para-meters of the electroweak Standard Model are summarized in Tab. 10. At the LHeC, thehigh experimental precision for SM parameters is due to the fact that a large kinematicrange of space-like momentum transfer can be used to obtain a large amount of DIS data Expected uncertaintyFit parameters Parameter Unit LHeC-60b LHeC-60a LHeC-50b LHeC-50a m W +PDF m W MeV ± ± ± ± m W + m Z +PDF m W MeV ± ± ± ± m Z MeV ± ± ± ± θ W +PDF sin θ W − ± ± ± ± m t +PDF m t GeV ± . ± . ± . ± . m W + m t +PDF m W MeV ± ± ± ± m t GeV ± . ± . ± . ± . m H +PDF ln m H GeV ± . ± . ± . ± . m H +PDF m H GeV +14 −
13 +24 −
20 +17 −
15 +28 − Table 10: Prospects for the determination of Standard Model parameters from simulated inclu-sive NC and CC DIS data at the LHeC. Scenarios for electron beam energies of E e = 50 and60 GeV, and with two assumptions for experimental uncertainties, denoted scenarios ‘a’ and ‘b’,are studied. Q bins and including both NC and CC scattering, as wellas scattering with positrons and with polarized electrons, is equivalent to a large numberratios of bin cross sections. In particular, the ratios of NC cross sections at large andsmall Q carry information about electroweak parameters. Therefore, correlated (normal-ization) uncertainties cancel to a large extent and play a minor role, while uncorrelateduncertainties are reduced by the implicit averaging over a set of several hundreds of in-dependent measurements. This explains why we observe that the uncertainties for mostof the electroweak parameters are found at the per mille level while the uncertaintiesfor individual cross section data points are in the order of percent. For instance theweak mixing angle can be determined with a high experimental uncertainty of down to ± . · − which corresponds to 0.65 per mille, as shown in Tab. 10. The large number ofdata points at different Q will also allow a determination of the scale dependence of theweak mixing angle in the spacelike regime of about 25 < (cid:112) Q <
700 GeV. Experimentaluncertainties below ± · − will be possible in the range 60 (cid:46) (cid:112) Q (cid:46)
450 GeV. Thisanalysis will be equivalent to a determination of the running of the weak mixing angle, anopportunity which is unique to the LHeC. Only small and well-known correction factorsfrom theory will be needed to turn the result of such measurements into a determinationof the effective weak mixing angle, or the weak mixing angle in the MS scheme, whichthen can be compared with other measurements.In the on-shell scheme, the weak mixing angle is defined by the ratio of the weak bosonmasses, m W and m Z . The measurement of inclusive DIS cross sections can therefore beinterpreted as an indirect determination of the W -boson mass. We find from our combined m W +PDF fit an experimental uncertainty down to ± − . This high precision will improve all present measurements and providea highly valuable validation of future improved direct m W measurements with O (MeV)accuracy.The high precision of the Born-level parameters of the EW theory also suggests a deter-mination of the dominant parameters of the higher-order EW corrections, most notablythe top-quark mass m t . The value of m t can be determined with an uncertainty downto ± . m H , is logarithmic only, i.e. sub-dominant. Therefore, its uncertainty is fairly largeand compares in size with the indirect determinations from the LEP+SLD data. Non-Standard Model contributions to one-loop weak boson self energy corrections, usuallydescribed by the so-called oblique parameters S , T and U , can also be determined inde-pendently when NC and CC DIS data are considered together, and uncertainties of a fewpercent are expected. 34 simultaneous determination of the weak boson masses, m W and m Z , or a simultaneousdetermination of m W together with m t , yield moderate uncertainties. These, however,compare well with the uncertainties that can be achieved nowadays in global EW fits. Wehave studied in a simplified formalism the potential impact of LHeC NC/CC DIS datato such a global EW fit and found only small improvements; the correlation between m W and m t or m W and m Z are not very different in DIS than in other observables.The SU(2) × U(1) gauge symmetry predicts the weak NC couplings of fermions, g fV and g fA ,in a unique way. Modifications of the SM can be studied in a rather model-independentmanner by considering these coupling constants as free parameters, or alternatively, byintroducing multiplicative anomalous factors to the ρ NC ,f and κ f form factors, which canbe considered to be flavor and Q dependent. While the NC coupling parameters havebeen measured at the Z -pole with high precision, in particular for leptons and b quarks,the couplings of the light quarks, u and d , are experimentally measured only with a ratherpoor precision. The LHeC, in contrast, has a high sensitivity to the NC couplings of lightquarks, and DIS data will provide the unique opportunity to determine the light-quarkcouplings with per mille accuracy, either for the vector and axial-vector coupling constants( g uV , g uA , g dV and g dA ), or for the anomalous form factors, ρ (cid:48) NC f and κ (cid:48) f . In addition, their Q dependence can be determined with percent precision in the range 60 (cid:46) (cid:112) Q (cid:46)
600 GeV.Many precise data for parameters of the NC sector of the EW theory can be found inthe literature, based on measurements extending to highest energies. In contrast, high-precision measurements of CC interactions are often restricted to low energies. Here, theLHeC will offer once more unique opportunities. Anomalous electron–quark and electron–anti-quark form factors can be determined with an accuracy in the order of O (0 . Q dependence can be measured with per mille uncertainties in a Q rangeup to about 400 GeV .We have compared LHeC inclusive DIS pseudo data at two different center-of-mass ener-gies, √ s = 1 . √ s , also the range of Q dependent studies can be extended to some extent.Further improvements will be obtained by higher resolution of the detector, which wouldallow us to obtain measurements for a larger set of independent data points. Altogether,the expected uncertainties from the four LHeC scenarios studied in this article differ byabout a factor of 2 to 3. Such an improvement is indeed interesting in view of the fact thatmany parameters are measured with comparable uncertainties in e + e − or hadron-hadroninteractions.In this study, we employed calculations in the on-shell renormalization scheme includingthe full set of one-loop EW corrections. For our main purpose, i.e. to investigate prospectsfor the uncertainties of EW parameters, this provides a valid framework. However, oncereal LHeC data are available, a more careful study of higher-order corrections will benecessary. On the one hand, one has to match the high experimental precision of the35ata with correspondingly accurate theoretical predictions. On the other hand, a con-clusive test of the SM can only be achieved by comparing as many as possible differentmeasurements, i.e. also observables in processes other than DIS. Therefore, a consistentframework for the calculation of higher-order corrections to all observables in questionwill be needed. For example, different definitions of the effective weak mixing angle haveto be matched to each other. We believe that our study provides a motivation for theinvestigation and calculation of higher-order corrections to DIS observables beyond the1-loop level, and in different renormalization schemes.In conclusion, the high center-of-mass energy, the large kinematic range, and the largeintegrated luminosity at the LHeC will allow for the first time to perform precision elec-troweak measurement at high momentum transfer in NC and CC DIS. Rich aspects ofthe electroweak theory can be probed with highest precision without being limited byuncertainties from parton distribution functions. Instead, electroweak parameters andPDFs can be determined in simultaneous fits taking into account their mutual correla-tions. In many cases, the results are complementary to direct measurements in e + e − orhadron-hadron collisions, such as the indirect determination of the W -boson mass or thetop-quark mass. A unique measurement can be expected for the scale-dependence of theweak mixing angle, as well as of non-SM coupling parameters. A particularly impor-tant and outstanding precision is expected for the determination of weak neutral-currentcouplings of the light quarks. Acknowledgements
We very gratefully acknowledge the numerous interactions with members of the LHeCStudy Group of which we had been a part also for the here presented study. We wouldlike to thank S. Schmitt, A. Sch¨oning and Z. Zhang for many related discussions duringthe earlier stages of this work. We thank J. Erler and R. Kogler for valuable discussionson the interpretation of Z -pole data, and S. Kluth for continuous support and valuablecomments to the manuscript. 36 A simplified global EW fit including LHeC data
The validity of the Standard Model is frequently studied in so-called ‘global EW fits’ [41,66,67]. A large number of precision measurements are simultaneously fitted and comparedwithin a consistent theoretical framework. On the one hand, such a general approachallows one to perform a comprehensive consistency test of the SM; on the other hand onecan expect to obtain the most precise determination of SM parameters. In this appendix,we are interested in estimating the potential impact of LHeC inclusive NC/CC DIS dataon the parameter determination within such a global fit. We describe in the following asimplified approach which is suited for our purpose.It is common to consider a set of data as input for a global EW fit comprising α and G F ,the mass parameters m Z , m W , m H , and m t , partial decay widths of the gauge bosons, Γ W and Γ Z , the effective weak mixing angle sin θ effW ,(cid:96) , partial decay widths and asymmetryobservables in e + e − collisions [62] R (cid:96) , R b , R c , A (cid:96) , A c , A b , A ,(cid:96) FB , A ,c FB , A ,b FB , QCD contri-butions to the QED vacuum polarization ∆ α , and sometimes also measurements of α s , m b , and m c . With the assumption that α and G F are known with ultimate precision, thefit parameters are chosen as m Z , m H , m t , ∆ α and α s . Sometimes also m b and m c aretaken as parameters to be determined by the fit.Since we are interested only in estimating uncertainties and how these are affected byLHeC data, rather than in the actual fit values, some simplifications can be made in thefollowing. One can observe that some of the fit parameters are determined with a highprecision directly from measurements. This is the case for m Z , m H , m b , m c , ∆ α and α s . If these direct data were not taken into account in the global fit, these parameterscould be determined only with a much larger uncertainty. We do not expect that thiswill change if LHeC data are taken into account. Many other observables, like the Z -polecross section ratios and asymmetries have much larger experimental uncertainties thanpredicted by the global fit. However, in the case of m W and m t , the uncertainties fromdirect measurements is of a similar size as those from a global fit. These parameters aretherefore of special interest since for them one might expect a significant impact fromLHeC data.Therefore, we perform a simplified global EW fit, where we consider only the determina-tion of the SM parameters m Z , m W , m H and m t . All other parameters do not provideadditional constraints or are known with good precision. The calculations are performedin the on-shell scheme. The fit parameters are chosen as m W , m t and m H . When usinginput data for α , G F and the SM masses from PDG16 [85] we obtain uncertainties of∆ m W = ± . ± . m t = ± .
87 GeV ( ± . m W and m t are included (excluded) in the fit. This result is very similar to the one ofthe general global EW fit [66–68] and confirms that the simplifications of our approachare justified for our purpose.We can therefore proceed and include LHeC-60b inclusive NC/CC DIS data into the fit.37 [GeV] t m170 172 174 [ G e V ] W m t and m W m C.L. (95%) [GeV] t m173 173.5 174 174.5 [ G e V ] W m t and m W m C.L. (95%)
Figure 17: Left: Expected uncertainties from a simplified global EW fit including LHeC-60binclusive NC/CC DIS data. The results are compared with expected uncertainties from thesimplified global EW fit without LHeC data and to the direct measurements of m W and m t .Right: A magnified view of the left figure. Input data Fit parameters ∆ m W [MeV] ∆ m t [GeV] Correlation ρ m W m t m W , m t , m Z , m H (PDG16) m W , m t ± . ± .
87 +0 . m W , m t (PDG20) m W , m t ± ± .
67 (0) m W , m t , m Z , m H (PDG20) m W , m t ± . ± .
63 +0 . m W , m t , m Z , m H , NC/CC DIS m W , m t , PDF ± . ± .
63 +0 . Table 11: Results for expected uncertainties of m W and m t from a simplified global fit, withand without NC/CC DIS data using the LHeC-60b scenario. Moreover, we include the PDF parameters as well, and we use recent values for m W , m t ,and m H from PDG20 [41]. When including LHeC-60b NC/CC DIS data, we obtain avery moderate improvement for m W : ∆ m W = ± . m W = ± . m W - m t –plane are displayed in Fig. 17. Numerical valuesof the validation fits, and the study including LHeC-60b data are collected in Tab. 11. Nosignificant improvement of ∆ m t is seen. This was to be expected, since m t contributesto NC/CC DIS only through higher-order corrections. Obviously, already other than theLHeC NC/CC DIS data are sufficient to provide good constraints on m W and m t . LHeCDIS data probe essentially the same relation of m W and m t and they cannot competewith direct measurements of these mass parameters.Therefore, the strength of the LHeC EW analysis is its complementary with measurementsin e + e − or pp collisions, in particular the fact that the dominant uncertainty in the W -boson mass determination is due to its correlation with the PDF determination. The38mpact of the latter may be considerably reduced if measurements at the LHeC willbecome available [86]. Commonly, the global fit is repeated with direct measurements ofthe parameters of interest excluded, i.e. without m W and m t . This was studied in Sec. 8.39 eferences [1] LHeC Study Group, J. Abelleira Fernandez et al. , “A Large Hadron ElectronCollider at CERN: Report on the Physics and Design Concepts for Machine andDetector,” J. Phys. G
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