Factorization of the forward-backward charge asymmetry and measurements of the weak mixing angle and proton structure at hadron colliders

The forward-backward charge asymmetry (AFB) at hadron colliders is sensitive to both the electroweak (EW) symmetry breaking represented by the effective weak mixing angle, and the proton structure information in the initial state modeled by the parton distribution functions (PDFs). Due to their strong correlation, the precisions of the determination on the weak mixing angle and PDFs using the measured AFB spectrum are limited. In this paper, we define a set of structure parameters which factorize the unique proton information of the relative difference between quarks and antiquarks in the AFB observation. Other than the conventional way of extracting the weak mixing angle fro the convolution of PDF and EW calculations, we propose a new method to simultaneously determine the value of the weak mixing angle and the proton structure terms by fitting to the observed AFB distribution, and point out the necessity of specifying additional observations to further reduce the uncertainties on the proton structure terms respectively, so that the model-independent high precision measurements can be achieved at the future LHC experiments.


INTRODUCTION
The forward-backward asymmetry (A F B ) of the neutral current process f ifi → Z/γ * → f jfj describes the relative difference between cross sections of the forward and backward scattering. It arises from the vector and axial vector couplings of Z to fermions, and is governed by the effective weak mixing angle sin 2 θ ℓ eff , which has been measured first in the electron-positron collisions at the LEP and the SLC [1], then in the proton-antiproton collisions at the Tevatron [2]. At the Large Hadron Collider (LHC), the initial fermions and antifermions of the corresponding Drell-Yan (DY) process pp(qq) → Z/γ * → ℓ + ℓ − are quarks and antiquarks, acting as partons in the proton, with their momentum distributions modeled by the parton distribution functions (PDFs). Since quarks and antiquarks would come from either side of the proton beams, the observed asymmetry at hadron colliders, denoted as A h F B , is diluted from its original A F B value. The reduction in magnitude from A F B to A h F B directly reflects the difference between quark and antiquark momentum fractions in the proton, and can be used to constrain PDFs. Some of the earlier studies of using A h F B to constrain PDFs can be found in Refs. [3][4][5][6].
Though A h F B can be precisely measured at the LHC, it is difficult to use this observable to constrain the PDFs and extract a more precise value of sin 2 θ ℓ eff , due to the strong correlation between them. On the one hand, as noted in Ref. [6], directly using the A h F B observable in a typical global PDF analysis may yield a large bias induced by the limited precision of sin 2 θ ℓ eff , which should be considered as an additional uncertainty in constrain-ing the PDFs. On the other hand, as reported by the ATLAS, CMS and LHCb collaborations [7][8][9], the PDFinduced uncertainty is becoming the most dominant issue in the sin 2 θ ℓ eff measurements, and thus prevents the measurements of sin 2 θ ℓ eff at the LHC to achieve a relative precision at O(10 −4 ). Such precision is recently achieved in the W boson mass measurement by the CDF collaboration [10], which is significantly deviated from the Standard Model (SM) prediction. If such deviation hints new physics beyond the SM, it should also appear as a shift on sin 2 θ ℓ eff . Therefore, it is essential to have a strategy to decouple the parton information from the EW prediction in the A h F B observation, so that sin 2 θ ℓ eff and parton information can be simultaneously determined using the LHC data instead of fitting for one and fixing the other in the conventional methods.
However, such simultaneous determination is challenging in the framework of PDF global analysis, with sin 2 θ ℓ eff as a free parameter. Many of the old experimental data sets included in the PDF analyses were performed at the leading order in EW interaction, and those data are still important for providing PDF information [11]. To achieve a high precision of sin 2 θ ℓ eff determination, all the Wilson coefficients of the scattering processes included in the PDF analysis have to be updated to higher-order in EW interactions. This feature is not yet available in the current PDF global analysis programs.
In this paper, we propose a new method which factorizes the proton information relevant to the A h F B measurement into a set of structure parameters. They are defined to be generally independent of any specific PDF modeling, and thus can be viewed as some new exper-imental observables. These parameters, together with sin 2 θ ℓ eff , can be determined from one single measurement of the A h F B at the LHC, through a carefully-designed fitting procedure, as to be described below. The correlation between the determination of sin 2 θ ℓ eff and proton structure is automatically taken into account in the proposed analysis.
The A F B at hadron colliders is observed in the Collins-Soper (CS) frame [12], which is a center-of-mass frame with theẑ-axis defined as the bisector of the angle formed by the direction of the momentum of one incoming hadron (H A ) and the negative direction of the other incoming hadron (H B ). To decouple the original asymmetry from the proton structure information, we define two scattering angles, i.e. cos θ q and cos θ h , with different choices of H A and H B . For the cos θ q , H A and H B are assigned according to the directions of q andq, respectively. The differential cross section of the DY process can be expressed in term of cos θ q as: where Y , M and Q T are the rapidity, invariant mass and transverse momentum of the Z boson. The index f denote the flavor, and the term α f is proportional to the cross section of the quark-antiquark subprocesses (with flavor f ). The terms A f 0 and A f 4 are the normalized polar angular coefficient functions, while other angular functions are cancelled when integrated over the azimuthal angle. Both A f 0 and A f 4 , as a function of Y , M and Q T , can be calculated in the perturbative expansion of the electroweak and QCD couplings. In this work, they are computed using the ResBos [13] package at approximate next-to-next-to-leading order (NNLO) plus next-to-nextto-leading logarithm (NNLL) in QCD interaction.
The canonical scales [14,15] are set to the invariant mass of the lepton pair in the DY events. The EW calculation of ResBos is similar to the effective born approximation used in zfitter [16]. The DY events are categorized as forward (F ) for cos θ q > 0, and backward (B) for cos θ q < 0. It is custom to define A F B as where σ F and σ B are the cross sections of F and B events. In this case, the parton densities of the initial state quarks and antiquarks contribute only to α f terms, and A f 0 , A f 4 and A F B are independent of proton structure information. However, such A F B cannot be experimentally observed because the directions of q andq are practically unknown.
At the LHC, the DY process is observed in terms of cos θ h , of which H A is defined as the hadron which points to the same hemisphere aligned to the reconstructed Z boson of the dilepton final state. Since the Z boson is boosted along with the direction of the parton which carries higher energy, cos θ h = cos θ q when q has larger energy thanq, and cos θ h = − cos θ q whenq has the larger energy. We introduce the dilution factor D f (Y, M, Q T ) to represent the probability of having cos θ h = − cos θ q , i.e., the antiquark has higher energy than the quark in the hard scattering subprocess. In this manner, the differential cross section in term of cos θ h can be expressed as: It is clear that only the A f 4 term would be affected by the dilution effect. Accordingly, the experimental observed asymmetry A h F B can be formulated as follow: where A f F B (Y, M, Q T ) denotes the asymmetry calculated using cos θ q for each ff subprocess, which is independent with proton structure.
The derivation of Eq. 4 is straightforward and provided in Sec. S1 of the Supplemental Material [17]. The above equation reveals the nature of the observed A h F B ,which is an average of the asymmetry contributed from each ff subprocess, weighted by its relative cross section, and the associated dilution factor reflects our limited knowledge of the q andq PDFs of the proton.
Here, we assume that the parton distributions of strange quark and antiquark of the proton are the same, s(x) =s(x), though a small violation can be generated at the NNLO via DGLAP parton evolution [18]. Similarly, c(x) =c(x) and b(x) =b(x), with c and b denoting charm and bottom flavor, respectively. Consequently, the dilution factors D s , D c and D b equal to 0.5 in all Z boson kinematic configurations, and their contributions in the numerator of Eq. 4 vanish.
Based on the argument, we propose to factorize the observed A h F B as products of the proton structure parameters and independent EW-dominant parts: where the structure parameters are related to Eq. (4) via In Eq. 5, the sin 2 θ ℓ eff -dependence of the light quark asymmetry A u/d F B (Y, M, Q T ; sin 2 θ ℓ eff ) are explicitly written out, which can be precisely predicted by the SM electroweak calculation. The structure parameters P u/d 0 (Y, Q T ) represent the parton information averaged in a given mass range of M , while the parameters ∆ u/d (Y, M, Q T ) describe the variation around the averaged behavior provided by P u 0 and P d 0 , respectively, in that mass range. Both P f 0 and ∆ f (M ) parameters are structure parameters, representing the parton information. Since A h F B is usually observed in a narrow mass window around the Z pole, the relevant proton structure information and the corresponding uncertainties are dominated by P f 0 parameters.
As an example, we show in the upper panel of Figure 1 the distribution of A h F B (M ), in the mass region of 60 < M < 120 GeV/c 2 , for a pseudo-data sample of the DY process produced at the 13 TeV LHC, generated using the ResBos program with CT18 NNLO PDFs [19]. In the lower panel of Figure 1, the PDF-induced uncertainty of A h F B for each invariant mass bin is decomposed according to Eq. 5, for illustration. As expected, the uncertainty is dominated by those of P u 0 and P d 0 , while the contributions from ∆ u (M ) and ∆ d (M ) are quite small. By its definition in Eq. 6, the P f 0 parameters contain information of the dilution effects of the light quarks and their relative cross sections. Accordingly, the values of P f 0 are expected to be increasing as a function of Y , for the dilution effects should be reduced in the kinematic region where the difference in the quark and anti-quark energy is large. At the LHC energies, both (1 − 2D u ) and (1 − 2D d ) approach to 0 when Y is zero, and toward 1 as the magnitude of Y increases. This trend is demonstrated in Figure 2, where the P f 0 values predicted by CT18 PDFs are shown as a function of Y and Q T . Additionally, in the high Y region where the dilution effects of the u and d quarks are similarly small, the separation of P u 0 and P d 0 could be attributed to the difference in the light quark parton abundance and their corresponding cross sections. Note that if the ratio of P u 0 and P d 0 is employed, the information on s, c and b cross sections cancels out. Therefore, the experimental observation on such ratio can directly probe the ratio of u and d (anti)quark parton densities of the proton as a function of x. These phenomenal conclusions are also checked using predictions of MSHT20 [20] and NNPDF3.1 [21], respectively, as shown in Sec. S2 of the Supplemental Material.
SIMULTANEOUS FIT OF sin 2 θ ℓ eff AND P f 0 Some progresses on reducing the PDF and EW correlation and potential bias have been made in the past. In Ref. [6], it was found that excluding the A h F B data around the Z pole region in a PDF global analysis could reduce the bias. To further reduce the bias, at the expense of sensitivity, a new observable was proposed later in Ref. [22], which uses the gradient information of A h F B (M ) spectrum to perform the PDF global fitting, while using its average value around the Z-pole to determine sin 2 θ ℓ eff . However, the residual bias given by the above two methods is still sizable. Alternatively, Ref. [23] explored the impact of the imperfect knowledge of the proton structure on the determination of EW parameters (e.g. M W ) via nuisance parameter formalism, by including the bin-by-bin correlation of the kinematic distributions with respect to PDF variations. Though the analysis would effectively reduce the PDF uncertainty, it relies on the information of specific PDF sets, which are not automatically updated in the process of measur- ing the EW parameter. In short summary, none of these analyses provided a strategy to simultaneously fit sin 2 θ ℓ eff and parton information.
Based on the factorization of Eq. 5, we propose a new method to simultaneously determine the EW and proton structure parameters by fitting to the observed A h F B (Y, M, Q T ) data. One can employ Eq. 5 to build theoretical templates, with sin 2 θ ℓ eff , P u 0 (Y, Q T ) and P d 0 (Y, Q T ) as fitting parameters. Their values are therefore determined by requiring the best agreement between theory templates and the observed data. Due to lack of sufficient constraints by using the A h F B distribution only, the ∆ f (Y, M, Q T ) terms would have to be fixed to the prediction of current PDFs, and thus induce extra theoretical uncertainties to the fitted parameters.
For numerical test, a ResBos+CT18 pseudo-data sample of 2.5 billion DY events is used, corresponding to 1 ab −1 integrated luminosity at the LHC 13 TeV pp collisions. Similar tests are also made using MSHT20 and NNPDF3.1, of which the results are given in Sec. S3 of the Supplemental Material. The fitted results of P u 0 , P d 0 and sin 2 θ ℓ eff parameters, as a function of Y , are depicted in Figure 3. The statistical fluctuations of the fitted P u 0 , P d 0 and sin 2 θ ℓ eff values (labelled as "Fitted unc."), are comparable to the difference between the fitted values and their input or predicted values in the pseudo-data (labelled as "δ"), which manifests the closure of the factorization and the fitting method.
The theoretical uncertainties associated with the input ∆ f terms in the fit (labelled as "∆ f -induced unc.") are estimated by repeating the fitting procedure with different ∆ f (Y, M, Q T ) predictions given by the CT18 PDF error sets, instead of the central set, with P f 0 and sin 2 θ ℓ eff as free fitting parameters, and quoting the variations of their fitted values, respectively. For comparison, the variations of P f 0 values predicted by different CT18 PDF error sets, and the PDF-induced uncertainty on the sin 2 θ ℓ eff extracted from the convnentional method of setting sin 2 θ ℓ eff as the only fitting parameter, are also depicted (labelled as "Original PDF unc."), respectively. The factorization method provides a new perspective for the issue of the correlation between PDFs and sin 2 θ ℓ eff . Firstly, the ∆ f -induced uncertainties arise from the variation of proton structure information in a small region of the Bjorken variable x, while the original PDF ones are dominated by the average magnitude of the structure parameters. Secondly, we note that the ∆ f terms do not contribute large uncertainty on A h F B distribution itself, cf. Figure 1. However, when A h F B is the sole data included in the fit, the P f 0 terms and sin 2 θ ℓ eff are higly (negatively) correlated. To improve the resolution power of the proposed factorization form of A h F B to the determination of the P f 0 terms and sin 2 θ ℓ eff , we could either introduce new ∆ f -sensitive observables in the PDF global fitting to reduce the uncertainty on ∆ f itself, or add additional P f 0 -sensitive data in the simultaneous fit, so that the correlation between sin 2 θ ℓ eff and P f 0 can be reduced. Of course, this kind of new ∆ f or P f 0 -sensitive observables, preferably at the LHC, ought to be sin 2 θ ℓ effindependent.

CONCLUSION
The A h F B measurement at hadron colliders not only is an ideal observable for the determination of sin 2 θ ℓ eff , but also provides unique proton structure information on the relative difference between quarks and antiquarks. It is essential to factorize the proton structure information with well defined observables, so that both sin 2 θ ℓ eff and the parton information can be simultaneously determined. In this article, we proposed a novel method to do just that, which is based on the factorization property of Eq. (5), and demonstrated how the method works. The A h F B factorization provides a novel method fo handle the proton distribution uncertainties in the electroweak measurement at the LHC, in contrast to the conventional PDF error estimation. It should also be pointed out that by introducing other observables in the same LHC data, which are sensitive to either of the two types of factorized ∆ f or P f 0 terms, the precision of the simultaneous fit based on the A h F B factorization could be further improved, so that the proton structure information and the EW sin 2 θ ℓ eff parameter could be determined modelindependently in the future LHC experiment.