Boost Asymmetry of the diboson productions in pp collisions

We propose the boost asymmetry of the diboson productions in pp collisions as a new experimental observable, which can provide unique information on the proton structure. The boost asymmetry rises as the difference in the kinematics of the two bosons, that are coupled to the two different quark and antiquark initial states, respectively, and thus reflects different features of the quark and antiquark parton densities. By comparing the kinematics of the two bosons, such as the boson energy or rapidity, the diboson events with antiquark having higher energy than quark can be distinguished from those with quark having higher energy than antiquark. This would provide unique information in some special parton momentum fraction regions, which cannot be directly proved by current W and Z measurements at the Large Hadron Collider or other deep inelastic scattering experients.


I. INTRODUCTION
The vector boson productions at the Large Hadron Collider (LHC) are dominated by the initial state quarkantiquark (q iqj ) scattering, thus are highly sensitive to the corresponding parton densities in a large range of the Bjorken variable x, describing the fraction of the parton momentum to the energy of the proton. In the latest global analysis of the parton distribution functions (PDFs) such as the CT18, MSHT20 and NNPDF4.0, the single Z and W production rates measured at 7 and 8 TeV pp collisions have been used and delivered significant impacts [1][2][3].
Due to the high energy of the proton beam, the vector boson productions at the LHC contain both the q i (x L )q j (x S ) contribution where the initial state quarks carry higher energy than the antiquarks (x L > x S ), and theq j (x L )q i (x S ) contribution where the antiquarks have higher energy. The ratio between the two cross sections, which we call as the quark exchanging fraction, relates to the parton densities of the valence u and d quarks in the small x region, and the sea quarks in the relatively large x region. An example is the forward backward asymmetry (A F B ) of the Drell-Yan pp → q iqj → Z/γ * → ℓ + ℓ − process. Due to the limited knowledge of the dilution effect, which is actually the quark exchanging fraction for the uū and dd cases, the observed A F B is reduced from its original value arising from the electroweak (EW) symmetry breaking [4], and therefore induces large PDF-induced uncertainty on the determination of the weak mixing angle (sin 2 θ ℓ eff ), reported by the ATLAS, CMS and LHCb measurements [5][6][7].
The quark exchanging fraction is unique information that is difficult to acquire from other data such as the Deep-Inelastic Scattering and the fixed-target Drell-Yan experiments. Therefore, the measurements at the LHC are expected to provide important input to expand our knowledge of proton structure. However, the observed Z and W cross sections are always the mixture of the contributions, so they could not distinguish the two initial states and fail to give direct constraint on the quark exchanging fraction. It was proposed to use A F B itself to constrain the dilution effect [8][9][10], but found to have large additional uncertainties due to the correlation with the EW sin 2 θ ℓ eff parameter [4].
Without a direct constraint, the current PDF global analysis has to provide predictions on the quark exchanging fraction by combining the information of the LHC data and other old experimental results [11]. However, the combined fitting highly relies on the assumption that all the data should be consistent, while on the other hand, it is known that, e.g., the ATLAS 5 TeV, 7 TeV and 8 TeV W and Z differential cross section data have tensions with other datasets included in the PDF global analysis [12][13][14]. Therefore, it would be important to have experimental observables which can provide constraint directly on the quark exchanging fraction at the LHC.
In this paper, we propose a set of new experimental observables, the boost asymmetries can be distinguished from each other, so that the quark exchanging fraction can be directly determined. We also perform an impact study of introducing the A V V ′ boost observables to the PDF global analysis. It is demonstrated that the uncertainties of the relevant parton densities can be significantly reduced.

II. THE BOOST ASYMMETRY IN THE V V ′ EVENTS
At the LHC, the diboson events are produced dominantly by the q iqj initial state via the t− and u−channel contributions, of which the Feynman diagrams are shown in Fig. 1. The two bosons are separately coupled to the quarks and antiquarks, and consequently the kinematics of the two different bosons reflect the energy of the corresponding quark or antiquark respectively. The boson kinematic can usually be represented by the rapidity (Y ) of the boson or the lepton from the boson decay. For the W γ, W + W − and W Z processes where the two bosons can be experimentally distinguished, the events can be divided into The boost asymmetry is defined to describe the relative difference between the two categories: where N is the number of observed events. Due to q i andq j have different energy densities, the observation of A boost is expected to have non-zero values. In the following subsections, we will discuss how A V V ′ boost reflect the information of the quark exchanging fractions. The W ± γ production is dominated by the two different ud and dū initial states respectively, where W and γ can couple to either quarks or antiquarks. Since the W + γ and W − γ processes can be measured independently, it is possible to probe the different energy spectrum of ud and dū initial states. For example, according to different boson-quark-couplings (V − q) and parton densities (q(x)), the ud contribution in W + γ events can be further divided into four parts as: γ−u(xL) . When u carries higher energy thand, i.e. in the two u( , and thus partially cancel the asymmetry. However, the cross section of N γ−u(xL) , because the γ − u coupling results in larger contribution than the γ−d coupling by a factor of about 4. Therefore, the cancellation on the asymmetry is not significant. In the same way, whend carries higher energy than u, i.e. in u( γ−u(xS) , and also would be partially cancelled by N Note that the relationship between Y W and Y γ does not exactly reflect the relationship between u andd energy. Apart from the cancellation effect mentioned above, the freedom in the kinematic of internal quark propagators, the contributions from the s−channel processes and higher order effects, and the interference between them would also smear the boost asymmetry. In practice, the W boson kinematic is usually replaced by the rapidity of the leptons from the decay, thus further smears A boost due to the missing neutrino. Nevertheless, these smearing effects of diminishing A boost observation are more related to the EW calculations which can be precisely predicted, and thus independent with the parton densities of the initial state quarks. As a conclusion, A boost in the W + γ events is dominated by the quark exchanging frac- To give numerical results, a sample of pp → W γ → ℓνγ is generated at √ s = 13 TeV using pythia event generator [15], with 700 million events in full phase space corresponding to about 1 ab −1 data produced at the LHC. The boost asymmetry is specifically defined as: in terms of the rapidity of γ and ℓ of the final states. The predicted values of A W γ boost in the W + γ and W − γ events from CT18, MSHT20 and NNPDF4.0 [1][2][3], together with the corresponding PDF uncertainties are summarized in Table I.
±0.006(PDF) ±0.008(PDF) ±0.004(PDF) ±0.004(PDF) The asymmetry A W γ boost in the W + γ event has a positive large value, namely γ is more boosted than the W boson, because of three reasons. Firstly, since the probability of the valence u quark having higher energy is greater than that of the sead quark, N γ−u(xL) is suppressed due to the charge-determined γ − q couplings. Thirdly, the massless γ would be more boosted than the massive W boson. As a result, the boost asymmetry is enhanced.
On the contrary, A W γ boost in the W − γ event has a smaller negative value. In thisūd process, γ can acquire higher energy by coupling to the valence d quark, but the dominating contribution N γ−ū(xS ) , but its boost is limited due to its heavy mass. Consequently, neither W nor γ could significantly lead the boost after the cancellation.
Similarly, A boost can be defined in the W Z events, and is also sensitive to the quark exchanging fraction of ud and dū. Due to the smaller cross section and full lepton decay branching ratios of the W and Z boson, the boost asymmetry in the W Z processes is less significant than that in the W γ ones. However, the A W Z boost observation would be more feasible than A W γ boost at the LHCb, where the precise photon measurement is not practicable. Thus the LHCb A W Z boost observation may provide complementary information in a much forward phase space, which cannot be covered by the acceptance of the ATLAS and CMS detectors.

II-B. A boost in the W + W − process
The W + W − process is dominated by the uū and dd initial states, and has different sensitivities to the quark exchanging fraction from the W γ and W Z events. The W + boson is always coupled to the positive charged u or d quark, while the W − boson is coupled to the negative chargedū or d quark. Therefore, there is no cancellation due to the exchange of the boson-to-quark couplings in the W + W − event. Instead, the cancellation rises between the uū and dd contributions. In the uū subprocess, the large cross section part N W + −u(xL) W − −ū(xS) statistically gives W + the higher energy, while the higher energy W − events are mainly produced by the small cross section N W + −u(xS) W − −ū(xL) contribution, so that W + leads the boost. On the contrary, in the dd subprocess, it is the large cross sectionN W + −d(xL) has W + the higher energy, so that W − leads the boost. Since the overall cross section and boost kinematics of uū differ from those of dd, the observed boost asymmetry in the W + W − is expected to be non-zero, and can be defined in terms of the rapidity of the leptons of the W boson decay in the final state as: The boost asymmetry in the W + W − event was previously discussed in Ref. [16], of which it was expected to have some sensitivity in new physics search. In this work, we will demonstrate that A W W boost is a useful observable in the PDF global analysis. The numerical predictions of pythia with various PDFs are listed in Table II. Finally, we would like to discuss the boost asymmetry in the Zγ production. It is also dominated by uū and dd initial states, thus could have the same sensitivities to the quark exchanging fractions as the W + W − events. However, the sensitivity is almost completely cancelled due to the exchange of the boson-quark-coupling, namely the interchanging possibilities of either higher energy q(x L ) orq(x L ) radiating a photon or Z boson are equal. Even though there is still a sizable asymmetry, as shown in Table III, such asymmetry is purely raised by the mass difference between Z and γ and has very little sensitivity to the quark densities. This is also indicated by the much smaller PDF uncertainties in Table III than those of A W γ boost and A W W boost . Therefore, for the Zγ production at the LHC, even though it has relatively larger cross section than W + W − and is easy to be measured, it would not provide as large PDF-sensitivity boost asymmetry as in the W γ and W W events.

INTO THE PDF GLOBAL FITTING
In this section, we present an impact study of introducing A V V ′ boost into the PDF global fitting, by using pseudo-data samples of W γ and W W events corresponding to 1 ab −1 data at the 13 TeV LHC as new experimental input. The error PDF Updating Method Package (ePump) [17] is used which can efficiently update the PDF with a new data input in the way equivalent to the PDF global fitting based on the Hessian approach. The central and error set predictions of CT18 PDFs on A W γ boost and A W W boost are used as inputs to ePump. The results of using A W γ boost to do the PDF updating are shown in Fig. 2. With information of the quark exchanging fractions introduced, the uncertainties on the valence-u, valence-d,ū andd PDFs are largely reduced. The proposed boost asymmetry contains unique information of the valence quarks in the small x S region and the sea quarks in the large x L region. As depicted in the figures, the uncertainties of the seaū andd quark PDFs significantly reduced in the large x L region, while the valence u and d ones have better precision in the small x S region. Although A W γ boost can also offer constraints on q i (x L )q j (x S ), the single Z and W observations at the LHC would certainly cover such information with much larger statistics. For example, it was concluded in Ref. [18] that the up and down valence quark PDFs can be better constrained in the large x L region by the high mass Drell Yan data. Therefore, the improvements on the small x S region antiquarks and the large x L region valence quarks are not significant by using the boost asymmetry. PDFs, before and after the PDF updating using A W γ boost . The blue band corresponds to the uncertainty before updating and the red band is after updating.
The impact of using A W W boost is less significant. Firstly, the cross section of the W + W − process is much smaller than the W γ process. Secondly, unlike the W γ measurement which can distinguish the ud and dū initial states, the W + W − process cannot separate the uū from the dd ones. Thirdly, the sensitivity is of A W W boost is further reduced due to the double neutrinos in both the W boson decay. Its leading effect, based on the CT18 updating, is on the valence u and d quark PDFs as shown in Fig. 3. However, a potential impact of measuring A W W boost is expected beyond its own sensitivity. As discussed in the introduction, the A F B observation in the single Z Drell-Yan process at the LHC is believed to have high sensitivity on the quark exchanging fraction of uū and dd, but not available in practice due to the strong correlation with sin 2 θ ℓ eff . Reported in Ref. [19], the A F B spectrum at the LHC has been analytically factorized with proton structure parameters representing the relevant parton information, so that the structure parameters can be determined together with sin 2 θ ℓ eff by simultaneous fit, with the correlation automatically taken into account. It is pointed out that the precision of the simultaneous fit is expected to be significantly improved if other data could be introduced in the fit, which ought to be sin 2 θ ℓ effindependent and providing the information on the quark exchanging fraction exactly same as A F B provides. The observable A W W boost is an ideal input satisfying the requirement proposed in Ref. [19], thus can be used to improve the precision both on the PDF and sin 2 θ ℓ eff by further reducing the correlation between them in the A F B measurement. The detailed numbers given in the discussion could be different when higher order calculations and other PDFs are used in this test. However, the phenomenal conclusion that A V V ′ boost can provide important information to the PDFs should be independent with the choice of event generators and PDFs. In this work, A V V ′ boost is defined in terms of the boson and lepton rapidity. It could be defined with other kinematic variables such as boson and lepton energy under a specific experimental apparatus, if needed. CONCLUSION We propose the boost asymmetry in the W γ, W Z, and W + W − events at the LHC as new experimental observables to constrain the PDFs. The kinematics of the two different bosons separately reflect the parton information of quarks and antiquarks in the initial state, respectively, and result in an asymmetry on the boosted boson. Such boost asymmetry can be used to constrain the quark exchanging fractions directly, which cannot be observed in the current single W and Z measurements. The observation on A boost in the W γ and W Z events is sensitive to the quark exchanging fraction of ud and dū, while in the W W events it is sensitive to that of uū and dd initial states. The W ± γ events are particularly useful because the different strength of photon couplings to up and down quarks can further enhance the asymmetry. Impact study shows a reduction on the PDF uncertainties of relevant quarks when introducing A boost in to the PDF global analysis. The asymmetries would be helpful not only for PDF improvement, but also to other related topics such the measurement of A F B and the EW sin 2 θ ℓ eff parameter.