A New Method for Indirect Mass Measurements using the Integral Charge Asymmetry at the LHC

Processes producing a charged final state at the LHC have a positive integral charge asymmetry. We propose a novel method for an indirect measurement of the mass of these final states based upon the process integral charge asymmetry. First, the theoretical prediction of the integral charge asymmetry and its related uncertainties are studied through parton level cross sections calculations. Then, the experimental extraction of the integral charge asymmetry of a given signal, in the presence of some background, is performed using particle level simulations. Process dependent templates enable to convert the measured integral charge asymmetry into an estimated mass of the charged final state. Finally, a combination of the experimental and the theoretical uncertainties determines the full uncertainty of the indirect mass measurement. This new method applies to all charged current processes at the LHC. In this article, we demonstrate its effectiveness at extracting the mass of the W boson, as a first step, and the sum of the masses of a chargino and a neutralino in case these supersymmetric particles are produced by pair, as a second step.


Introduction
Contrarily to most of the previous high energy particle colliders, the LHC is a charge asymmetric machine. For charged final states, 1 denoted F S ± , the integral charge asymmetry, denoted A C , is defined by where N (F S + ) and N (F S − ) represent respectively the number of events bearing a positive and a negative charge in the FS. For a F S ± produced at the LHC in p + p collisions, this quantity is positive or null, whilst it is always compatible with zero for a F S ± produced at the TEVATRON in p +p collisions.
To illustrate the A C observable, let's consider the Drell-Yan production of W ± bosons in p + p collisions. It is obvious for this simple 2 → 2 s-channel process that more W + than W − are produced. Indeed, denoting y W the rapidity of the W boson, the corresponding range of the Björken x's: x 1,2 = M W ± √ s ×e ±y W , probes the charge asymmetric valence parton densities within the proton. This results in having more U +D → W + thanŪ + D → W − configurations in the initial state (IS). Here U and D collectively and respectively represent the up and the down quarks.
In the latter case the dominant contribution to A C comes from the difference in rate between the u +d and the d +ū quark currents in the IS. Using the usual notation f (x, Q 2 ) for the parton density functions (PDF) and within the leading order (LO) approximation, this can be expressed as: where the squared four-momentum transfer Q 2 is set to M 2 W . 1 We defined these as event topologies containing an odd number of high pT charged and isolated leptons within the fiducial volume of the detector.

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From equation (1.2), we can see that the Q 2 evolution of the parton density functions (PDFs) governs the Q 2 evolution of A C . The former are known, up-to the NNLO in QCD, as solutions of the DGLAP equations [2]. One could therefore think of using an analytical functional form to relate A C to the squared mass of the s-channel propagator, here M 2 W . However there are additional contributions to the W ± inclusive production. At the Born level, some come from other flavour combinations in the IS of the s-channel, and some come from the u-channel and the t-channel. On top of this, there are higher order corrections. These extra contributions render the analytical expression of the Q 2 dependence of A C much more complicated. Therefore we choose to build process-dependent numerical mass template curves for A C by varying M F S ± . These mass templates constitute inclusive and flexible tools into which all the above-mentioned contributions to A C can be incorporated, they can very easily be built within restricted domain of the signal phase space imposed by kinematic cuts.
The A C for the W ± → ± ν production at the LHC is large enough to be measured and it has relatively small systematic uncertainties since it's a ratio of cross sections. The differential charge asymmetry of this process in p + p collisions have indeed been measured by the ATLAS [3], the CMS [4,5] and the LHCb [6] experiments [7] for the first times in their 2011 datasets.
In this article we exploit the A C to set a new type of constraint on the mass of the charged F S ± as initially proposed in [10,11].
We'll separate the study into two parts. The first one, in section 2, is dedicated to present in full length the method of indirect mass measurement that we propose on a known Standard Model (SM) process. We choose the W ± → ± + / E T inclusive production at the LHC to serve as a test bench.
In the second part, in section 3, we shall repeat the method on a "Beyond the Standard Model" (BSM) process. We choose a SUSY search process of high interest, namelỹ For both the SM and the BSM processes, we obviously tag the sign of the FS by choosing a decay into one (or three) charged lepton(s) for which the sign is experimentally easily accessible.
It's obvious that for these two physics cases other mass reconstruction methods exist. These standard mass reconstruction techniques are all based on the kinematics of the FS. For the W ± → ± + / E T process mass templates based upon the transverse mass allow to extract M W ± with an excellent precision that the new technique proposed here cannot match. In constrast, for theχ ± 1 +χ 0 2 → 3 ± + / E T process, even if astute extensions of the transverse mass enable to acurrately measure some mass differences, no standard techniques is able to measure accurately the mass of the charged FS: M F S ± = Mχ± 1 + Mχ0 2 . Therefore this new mass reconstruction technique should not be viewed as an alternative to the standard techniques but rather as an unmined complement to them. In a few cases, especially where many FS particles escape detection, this new technique can be more accurate than the standard ones. It also has the advantage of being almost model independent.

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For each signal process we sub-divide the method into four steps that are described in four sub-sections. In the first sub-sections 2.1 and 3.1, we start by deriving the theoretical A C template curves at the parton level.
In the second sub-sections 2.2 and 3.2, we place ourselves in the situation of an experimental measurement of the A C of the signal in the presence of some background. For that we generate samples of Monte Carlo (MC) events that we reconstruct using a fast simulation of the response of the ATLAS detector. This enables to account for the bias of the signal A C induced by the event selection. In addition we can quantify the bias of A C due to the residual contribution of some background processes passing this event selection.
Then, in the third sub-sections 2.3 and 3.3, we convert the measured A C into an estimated M F S using fitted experimental A C template curves that account for all the experimental uncertainties.
In the fourth sub-sections 2.4 and 3.4, we combine the theoretical and the experimental uncertainties on the signal A C to derive the full uncertainty of the indirect mass measurement. The conclusions are presented in section 4 and the prospects in section 5.
Note that we'll always express the integral charge asymmetry in % and the mass of the charged final state in GeV throughout this article. The uncertainty on the integral charge asymmetry δA C will also be expressed in % but will always represent an absolute uncertainty as opposed to a relative uncertainty with respect to A C .

Sources of theoretical uncertainties on A C
Since these cross sections integration are numerical rather than analytical, they each have an associated statistical uncertainty δσ ± Stat due to the finite sampling of the process phase space. Even though these are relatively small we explicitely include them and we calculate the resulting statistical uncertainty on the process integral charge asymmetry: δ(A C ) Stat for which we treat δσ + Stat and δσ − Stat as uncorrelated uncertainties. Hence: For each cross section calculation we choose the central Parton Density Function (PDF) from a PDF set (or just the single PDF when there's no associated uncertainty set). Whenever we use a PDF set, it contains 2N P DF uncertainty PDFs on top of the central PDF -3 -

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fit, the PDF uncertainty is calculated as proposed in [23]: where A C (0), A C (i) up , and A C (i) down represent the integral charge asymmetries calculated with σ 0 , σ up i , and σ down i , respectively. σ 0 represents the cross section calculated with the central PDF fit. σ up i represent the N P DF upward uncertainty PDFs such that generally σ up i > σ 0 , and σ down i represent the N P DF downward uncertainty PDFs such that generally σ down i < σ 0 . We choose the QCD renormalization and factorization scales: µ R = µ F = µ 0 to be equal, and we choose a process dependent dynamical option to adjust the value of µ 0 to the actual kinematics event by event. The scale uncertainty is evaluated using the usual factors 1/2 and 2 to calculate variations with respect to the central value µ 0 : The total theoretical uncertainty is defined as the sum in quadrature of the 3 sources: (2.5)

Setup and tools for the computation of A C
We calculate the σ + = σ(p + p → W + → + ν) and σ − = σ(p + p → W − → −ν ) cross sections and their uncertainties at √ s =7 TeV using MCFM v5.8 [33][34][35]. We include both the W ± + 0Lp and the W ± + 1Lp matrix elements (ME) in the calculation in order to have a better representation of the W ± inclusive production (the notation "Lp" stands for "light parton", i.e. u/d/s quarks or gluons). We set the QCD scales as µ R = µ F = µ 0 = M 2 (W ± ) + p 2 T (W ± ) and we run the calculation at the QCD leading order (LO) and nextto-leading order (NLO). For both the phase space pre-sampling and the actual cross section integration, we run 10 times 20,000 sweeps of VEGAS [12]. We impose the following parton level cuts: M ( ± ν) > 10 GeV, |η( ± )| < 2.4 and p T ( ± ) > 20 GeV. We artificially vary the input mass of the W ± boson and we repeat the computations for the 3 following couples of respective LO and NLO PDFs: MRST2007lomod [19] -MRST2004nlo [20], CTEQ6L1 [17] -CTEQ6.6 [18], and MSTW2008lo68cl -MSTW2008nlo68cl [22] which are interfaced to MCFM through LHAPDF v5.7.1 [24]. As the LO is sufficient to present the method in detail, we'll restrict ourselves to LO MEs and LO PDFs throughout the article for the sake of simplicity. We shall however provide the theoretical A C mass templates up to the NLO for the W process. And we recommend to establish them using the best theoretical calculations available for any use in a real data analysis, including at the minimum the QCD NLO corrections.
The MRST2007lomod is chosen as the default PDF throughout this article. The two other LO PDFs serve for comparison of the central value and the uncertainty of A C -4 -

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with respect to MRST2007lomod. In that regard, MSTW2008lo68cl is especially useful to estimate the impact of the δ(A C ) P DF .

Modeling of the theoretical A C (W ± → e ± ν e ) template curves
The theoretical MRST2007lomod and MRST2004nlo raw template curves are obtained by sampling A Raw C at different values of M W ± . The corresponding theoretical uncertainties are also calculated: A Raw C ± δA Raw C . This discrete sampling is then transformed into a continuous template curve through a fit using a functional form A Fit C = f (M W ± ) which is constrained by the theoretical uncertainties.
We have considered three different types of functional forms for these fits with f being either a: The types of functional forms that we're considering are not arbitrary, they are all related to parametrizations of solutions of the DGLAP equations for the evolution of the PDFs. The polynomial of logarithms of logarithms is inspired by an expansion of the PDF in series of Log[Log(Q 2 )] as suggested in [2]. The polynomial of logarithms was just the simplest approximation of the aforementioned series that we first considered. And the expansion of the PDF in series of Laguerre polynomials is proposed in [8].
In the appendix A, we give a numerical example of the evolution of the u(x, Q 2 ), u(x, Q 2 ), d(x, Q 2 ),d(x, Q 2 ) proton density functions calculated with QCDNUM [9] and the MSTW2008nlo68cl PDF. We also provide a few toy models to justify the main properties of the functional forms used for A Fit C . Ultimately, the model of the theoretical template curve uses the functional form f for the A Fit C central values and re-calculate their uncertainty δA Fit C by accounting for the correlations between the uncertainties of the fit parameters: The diagonal and off-diagonal elements of the fit uncertainty matrix are denoted V AR(A i ) and COV AR(A i , A j ), they correspond to the usual variances of the parameters and the covariances amongst them, respectively. The number of fit parameters N F P is taken as the minimum integer necessary to get a good χ 2 /N dof for the fit and it is adjustable for each A C template curve.
Comparing the three types of polynomials cited above as functional forms to fit all the A C template curves of sub-sections 2.1 and 3.1, we find that the polynomials of logarithms of logarithms of Q give the best fits. They are henceforth chosen as the default functional form to model the Q evolution of A C throughout this article.  Figure 1 displays the fit to the A C template curve using a polynomial of Log (Log(Q)). In the case of the MRST2007lomod PDF, it is sufficient to limit the polynomial to the degree N F P = 5 to fit the A C template curve in the following (default) range: M W ± ∈ [15, 1500] GeV.
The theoretical CTEQ6L1 and CTEQ6.1 A C template curves are obtained from the signed cross sections used for table 3.
2.1.6 A C (W ± → e ± ν e ) template curves for MSTW2008 The theoretical MSTW2008lo68cl and MSTW2008nlo68cl A C template curves are obtained from the signed cross sections used for table 5. In this case, the PDF uncertainty is provided and it turns out to be the dominant source of theoretical uncertainty on A C .    and NLO with the MRST2004nlo on the right-hand side (r.h.s.). The raw curve with its uncertainty bands, the corresponding fitted curve and the fitted curve with the correlations between the fit parameters uncertainties are displayed on the top, the middle and the bottom rows, respectively.

Comparing the different A C template curves
At this stage, it's interesting to compare the A C template curves produced with different PDFs using MCFM. From figure 4 we can see that the A C of the different PDF used at LO and at NLO are in agreement at the ±2σ level, provided that we switch the reference to a PDF set containing uncertainty PDFs. This figure also displays the ratios for the three families of PDFs used. These ratios are almost flat with respect to M W ± over the largest part of our range of interest. However at the low mass ends they vary rapidly. As we illustrate in the appendix A, these integral charge asymmetry ratios can be fitted by the same functional forms as the A LO C and A N LO

Experimental measurement of
The aim of this sub-section is to study the biases on A C due to two different sources: the event selection and the residual background remaining after the latter cuts are applied.

Monte Carlo generation
To quantify these biases we generate Monte Carlo (MC) event samples using the following LO generator: Herwig++ v2.5.0 [41]. We adopt a tune of the underlying event derived by the ATLAS collaboration [27] and we use accordingly the MRST2007lomod [19] PDF. Herwig++ mainly uses 2 → 2 LO ME that we denote in the standard way: 1 + 2 → 3 + 4. For all the non-resonant processes, the production is splitted into bins of M , where M = M (3, 4) is the invariant mass of the two outgoing particles.
For the single vector boson ("V+jets") production, where V stands for W ± and γ * /Z, we mix in the same MC samples the contributions from the pure Drell-Yan process V+0Lp ME and the V+1Lp ME. For all the SM processes a common cut of M > 10 GeV is applied.
All the samples are normalized using the Herwig++ cross section multiplied by a Kfactor that includes at least the NLO QCD corrections. We'll denote NLO (respectively NNLO) K-factor the ratio: σ N LO σ LO (respectively σ N N LO σ LO ). We choose not the apply such higher order corrections to the normalization of the following non-resonant inclusive processes:    The theoretical CTEQ6 A C template curves at LO with CTEQ6L1 (l.h.s.) and NLO with the CTEQ6.6 (r.h.s.). The raw curve with its uncertainty bands, the corresponding fitted curve and the fitted curve with the correlations between the fit parameters uncertainties are displayed on the top, the middle and the bottom rows, respectively.
• light flavour QCD (denoted QCD LF): 2 → 2 MEs involving u/d/s/g partons • heavy flavour QCD (denoted QCD HF): c +c and b +b • prompt photon productions: γ + jets and γ + γ Despite their large cross sections these non-resonant processes will turn out to have very low efficiencies and to represent a small fraction of the remaining background in the event selection used in the analyses we perform. The NNLO K-factors for the γ * /Z(→ ± ∓ ) process are derived from PHOZR [44] with µ R = µ F = M ( ± ∓ ) and using the MSTW2008nnlo68cl PDF for σ N N LO and the MRST2007lomod one for σ LO . The top pairs and single top [45,46] NLO K-factors are obtained by running MCFM v5.8 using the MSTW2008nlo68cl and the MSTW2008lo68cl PDFs for the numerator and the denominator respectively, with the QCD scales set as follows: µ R = µ F =ŝ.

Fast simulation of the detector response
We use the following setup of Delphes v1.9 [29] to get a fast simulation of the ATLAS detector response as well as a crude emulation of its trigger. The generated MC samples are written in the HepMC v2.04.02 format [30] and passed through Delphes.

Analyses of the W ± → ± ν process
We consider only the electron and the muon channels. For these analyses we set the integrated luminosity to Ldt = 1 fb −1 .
Instead of trying to derive unreliable systematic uncertainties for these analyses using Delphes, we choose to use realistic values as quoted in actual LHC data analysis publications. We choose the analyses with the largest data samples so as to reduce as much as possible the statistical uncertainties in their measurements but also to benefit from the largest statistics for the data samples utilized to derive their systematic uncertainties. This choice leads us to quote systematic uncertainties from analyses performed by the CMS -10 - collaboration. Namely we use: The values quoted in equations. (2.7) and (2.8) come from references [4] and [5], respectively. And to get an estimate of the uncertainty on a ratio of number of expected events we use the systematics related to the measurement of the following cross sections ratio which amounts to 1.0% [48].
• |η(e ± )| < 1.37 or 1.53 < |η(e ± )| < 2.4     • Reject events with an additional second track (T rack 2 ) such that: The corresponding selection efficiencies and event yields (expressed in thousanths of events) are reported in table 7. Figure 5 displays the / E T distribution after the event selection in the electron channel (l.h.s.) and in the muon channel (r.h.s.).
The non-resonant background processes represent just ∼ 4% of the total background after the event selection, this justifies the approximation of not to include the NLO QCD corrections to their normalizations.

a.2.
Common procedure for the background subtraction and the propagation of the experimental uncertainty. If we were to apply such an analysis on real collider data, we would get in the end the measured integral charge asymmetry A Meas C of the data sample passing the selection cuts. And obviously we wouldn't know which event come from which sub-process. Since the MC enables to separate the different contributing sub-processes, it's possible to extract the integral charge asymmetry of the signal (S), knowing that of the total background (B). the ratio of the expected number of background events to the expected number of signal events, we can express A Exp C (S +B), the integral charge asymmetry of all remaining events either from signal or from background, with respect to that quantity for signal only events A Exp C (S), and for background only events A Exp C (B). This writes: where the upper script "Exp" stands for "Expected". This formula can easily be inverted to extract A Exp C (S) in what we'll refer to as the "background subtraction equation": Note that these expressions involve only ratios hence their experimental systematic uncertainty remains relatively small.
The uncertainty on A Exp C (S) is calculated by taking account the correlation between the uncertainties of α Exp , A Exp C (B), and A Exp C (S + B).
In order to propagate the experimental uncertainties from equations. (2.7), (2.8), and (2.9) to δA C (S), we perform pseudo-experiments running 10,000,000 trials for each. In these trials all quantities involved in the background subtraction equation (2.11) is allowed to fluctuate according to a gaussian smearing that has its central value as a mean and its total uncertainty as an RMS.  Table 7. Selection efficiencies, event yields and integral charge asymmetries for the W ± → e ± ν e analysis.  Figure 5. / E T distribution after the event selection is applied for the W ± → e ± ν e (l.h.s.) and for the W ± → µ ± ν µ (r.h.s.) analysis.

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the numbers of positively and negatively charged events also fluctuate but in full anticorrelation. This procedure enables to estimate numerically the values of the variances and covariances appearing in equation (2.12).
In a realistic analysis context, A Exp C (S) can be obtained from a full simulation of the signal, A Exp C (B) and α Exp can also be obtained this way or through data-driven techniques. The experimental systematic uncertainties can be propagated as usually done to each of these quantities. And one can extract A Obs C (S) from a data sample using the following form of equation (2.11): provided a good estimate of the number of remaining signal and background events after the event selection as well as the integral charge asymmetries of the signal and of the background are established. The upper script "Obs" stands for observed. (S) using the inputs from the analysis in the electron channel only with their statistical uncertainties: After the background subtraction and the propagation of the experimental systematic uncertainties, we get: 2.2.4. a.4. The A C template curve in the electron channel. In order to establish the experimental A C template curve, we apply a "multitag and probe method". We consider all the W ± → e ± ν e MC samples with a non-nominal W mass as the multitag and the one with the nominal W mass as the probe. We apply equation (2.11) to each of the multitag samples and plot their A Meas C (S) as a function of M W ± . A second degree polynomial of logarithms of logarithms is well suited to fit the template curve as shown in the l.h.s. of figure 6, for the electron channel. The fit to this template curve can expressed by equation (2.15). Note that we do not include the probe sample in the template curve since we want to estimate the accuracy of its indirect mass measurement.      Table 8. Noise to signal ratio, signal statistical significance, and expected and measured integral charge asymmetries for the signal after the event selection in the electron channel.

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The signal significances reported are calculated using a conversion of the confidence level of the signal plus background hypothesis CL S+B into an equivalent number of onesided gaussian standard deviations Z N as proposed in [52] and implemented in RooStats [53]. For these calculations the systematic uncertainty of the background was set to 5%, which completely covers the total uncertainty for the measurement of the inclusive cross section σ(p + p → W ± → ± ν) as reported in [48].
We recalculate the uncertainty on A Meas C (S) accounting for the correlation between the parameters when fitting the A Meas C (S) template curve by applying equation (2.12). This results in a slightly reduced uncertainty as shown in equation (2.16).
The muon channel.
2.2.4. b.1. Event selection in the muon channel. The following cuts are applied: • Tracker Isolation: reject events with additional tracks of p T > 2 GeV within a cone of ∆R = 0.5 around the direction of the µ ± track • Calorimeter Isolation: the ratio of, the scalar sum of E T deposits in the calorimeter within a cone of ∆R = 0.5 around the direction of the µ ± , to the p T (µ ± ) must be less than 0.25 • Reject events with an additional trailing isolated muon: µ ±

2
• Reject events with an additional leading isolated electron: e ±

1
• Reject events with an additional second track (T rack 2 ) such that: The corresponding selection efficiencies and event yields are reported in table 9. The r.h.s. of figure 5 displays the / E T distribution after the event selection. The non-resonant background processes represent ∼ 3% of the total background after the event selection.  Table 9. Event selection efficiencies, event yields and integral charge asymmetries for the W ± → µ ± ν µ analysis.

b.2.
The measured A C in the muon channel. The A Meas C (S) treatment described in paragraph 2.2.4. a.2. is applied to the probe sample in the muon channel, starting from the following inputs: For the nominal W mass, this leads to a measured integral charge asymmetry of: where the uncertainty is also dominated by the value in equation (2.8).
2.2.4. b.3. The template curve in the muon channel. After applying the A Meas C (S) treatment to the tag samples in the muon channel, we get the A Meas C (S) template curve shown in the r.h.s. of figure 6. The fit to this template curve is reported in equation (2.18).  Table 10. Noise to signal ratio, signal statistical significance, and expected and measured integral charge asymmetries for the signal after the event selection in the muon channel.
The values of the noise to signal ratio (α Exp ), the signal statistical significance (Z N ), and the expected (A Exp C ) and the measured (A Meas C ) integral charge asymmetries for the signal after the event selection in the muon channel are reported in table 10.
Again, accounting for the correlation between the parameters when fitting the A Meas C (S) template curve enables to reduce the uncertainty as shown in equation (2.19). ± δM W ± measurements using the experimental A C template curves from the r.h.s. of figure 6 in each of these channels:

Combination of the electron and the muon channels
We combine the electron and muon channels using a weighted mean for the measured W ± mass, the weight is the inverse of the uncertainty on the measured mass. In order to account for the asymmetric uncertainties, we slightly modify the expressions for the weighted mean and the weighted RMS of a quantity x as follows: are respectively the central value, the upward uncertainty and the downward uncertainty of the mass derived in the channel i.
The result of the combination is:

Final result for MRST2007lomod
The next step is to estimate the theoretical uncertainty corresponding to the measured mass and to combine it with the experimental uncertainty. We simply use the central value of the measured W ± mass and we read-off the theoretical template curve the intervals, defined by the intercepts with upper and lower fit curves.
Finally we just sum in quadrature the theoretical and experimental upward and downward uncertainties: Therefore the final result for the MRST2007lomod PDF reads: This constitutes an indirect M W ± mesurement with a relative accuracy of 1.2%, where the experimental uncertainty largely dominates over the (underestimated) theoretical uncertainty.

Final results for the other parton density functions
Since Delphes v1.9 does not store the set of variables (x 1 , x 2 , f lav 1 , f lav 2 , Q 2 ) necessary to access the PDF information from the generator, we slightly modify it so as to retrieve the "HepMC::PdfInfo" object from the HepMC event record and to store it within the Delphes GEN branch as described in [49]. Based upon these variables we can apply PDF re-weightings so as to make experimental A C predictions for the CTEQ6L1 and the MSTW2008lo68cl PDFs. The new event weight is calculated in the standard way: where the "Old PDF" is the default one, MRST2007lomod, and the "New PDF" is either CTEQ6L1 or MSTW2008lo68cl.  Table 11. Number of expected signal events and expected signal A C as a function of M (W ± ) for the electron and muon analyses reweighted to the CTEQ6L1 PDF predictions. We re-run the electron and muon channel analyses and just change the weights of all the selected events. This results in signal event yields, and A Exp C (S), A Exp C (B) as reported in tables 11 and 12 for the CTEQ6L1 PDF and in tables 13 and 14 for the MSTW2008lo68cl one.

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Then we produce the experimental A C template curves for CTEQ6L1 and MSTW2008lo68cl and both analysis channels as displayed in figures 7 and 8.
For the CTEQ6L1 PDF, we find: which leads to the following combined value:     The CTEQ6L1 A C template curves for the W ± → e ± ν e (top) and the W ± → µ ± ν µ (bottom) analyses. The fits to the A Exp C (S) are presented on the l.h.s. These fits with uncertainty bands accounting for the correlation between the uncertainties of the fit parameters are shown on the r.h.s.  Table 13. Number of expected signal events and expected signal A C as a function of M (W ± ) for the electron and muon analyses reweighted to the MSTW2008lo68cl PDF predictions. and it's dominant uncertainty is also experimental, since its theoretical uncertainty is underestimated. This represents an indirect measurement of the W ± mass with a relative accuracy of 0.8%.      Figure 8. The MSTW2008lo68cl A C template curves for the W ± → e ± ν e (top) and the W ± → µ ± ν µ (bottom) analyses. The fits to the A Exp C (S) are presented on the l.h.s. These fits with uncertainty bands accounting for the correlation between the uncertainties of the fit parameters are shown on the r.h.s. For the MSTW2008lo68cl PDF: which leads to the following combined value: The corresponding theoretical uncertainties are:  Table 15. Summary of the indirect mass measurements of M W ± extracted from the integral charge asymmetry of the W ± → ± ν process. Different figures of merit of the accuracy of these measurements are presented.
Therefore the final result for the MSTW2008lo68cl PDF reads: and it's dominant uncertainty comes from δ Theory P DF A C . In this case, this represents an indirect measurement of the W ± mass with a relative accuracy of 2.1%.

Summary of the M W ± measurements and their accuracy
We sum up the indirect mass measurements of M W ± extracted from the integral charge asymmetry of the W ± → ± ν inclusive process within table 15. Therein we also present a few figures of merit of the accuracy of these measurements: In this notation, M Fit W ± and δM Fit W ± represent the indirectly measured M W ± and its uncertainty, and M T rue W ± stands for the nominal W ± boson mass. The first figure of merit (1.) reflects the intrinsic resolution power of the indirect mass measurement, irrespective of its possible biases, it's expressed in %. The second and the third ones measure the accuracy with respect to the nominal W ± boson mass: firstly as a relative uncertainty in % irrespective of the precision of the method (2.) and secondly as a compatibility between the nominal and the predicted masses given the precision of the method (3.), expressed in number of standard deviations (σ). 3 Inclusive production ofχ ± 1 +χ 0 In this section we repeat the types of calculations done in section 2.1 but now for a process of interest in R-parity conserving SUSY searches, namely the p + p →χ ± 1 +χ 0 We use Resummino v1.0.0 [14] to calculate the p + p →χ ± 1 +χ 0 2 cross sections at different levels of theoretical accuracy. At fixed order in QCD we run these calculations at the LO and the NLO. In addition, we also run them starting from the NLO MEs and including the "Next-to-Leading Log" (NLL) analytically resummed corrections. The latter, sometimes refered to as "NLO+NLL" will simply be denoted "NLL" in the following.
We calculate these cross sections at √ s = 8 TeV using "Simplified Models" [13] for the following masses: Regarding the phase space sampling, a statistical precision of 0.1% is requested for the numerical integration of the cross sections.
3.1.1 A C (χ ± 1 +χ 0 2 ) template curves for MRST The theoretical MRST A C template curves are obtained by computing the A C based upon the cross sections of the signed processes used for table 17. They are displayed in figure 9.
3.1.2 A C (χ ± 1 +χ 0 2 ) template curves for CTEQ6 The theoretical CTEQ6 A C template curves are obtained by computing the A C based upon the cross sections of the signed processes used for table 19. They are displayed in figure 10.
template curves for MSTW2008 The theoretical MSTW2008lo68cl A C template curves are obtained by computing the A C based upon the cross sections of the signed processes used for table 21. They are displayed in figure 11.

Comparing the different A C template curves
Here again we compare the A C template curves produced with different PDFs using Resummino this time. From figure 12 we can see that the A C of the different PDF used at LO and at NLO are in agreement only at the ±3σ level. This figure also displays the ratios for the three families of PDFs used.

Experimental measurement of
The aim of this sub-section is to repeat, in the context of the considered SUSY signal, a study similar to that of section 2.2.    We use Simplified Models to generate our signal in the two configurations shown in figure 13.
The second signal configuration, denoted S2, supposes that the lightest part of the SUSY mass spectrum is made ofχ ± 1 ,χ 0 2 , andχ 0 1 , in order of decreasing mass. The charged    sleptons are supposed to be much heavier. In addition, the following SUSY decays are all supposed to have a braching ratio of 100%:χ ± 1 → W ± (→ ± ν)+χ 0 1 ,χ 0 2 → Z 0 (→ ± ∓ )+χ 0 1 . In practice, within the MSSM, these braching ratios not only depend on the envisaged mass hierarchy, but also on the fields composition of theχ 0 2 , theχ ± 1 , and theχ 0 1 . Regarding the SM leptonic decays of the W ± and the Z 0 gauge bosons, we used their actual SM branching ratios. This will have the obvious consequence of a much smaller event yield for the S2 signals compared to the S1 signals of same mass.
The hypotheses common to configurations S1 and S2 are that the lightest SUSY particle (LSP) is theχ 0 1 , and that theχ 0 2 and theχ ± 1 are mass degenerate.

Monte Carlo generation
We generate a new set of MC samples. We report here only the MC parameters that are different from those used in sub-section 2.2.1. We use the following LO generator: Herwig++ v2.5.2 for the SUSY signal and for most of the background processes.
For the W + HF processes, the renormalization scale is set to where the i index runs over the number of FS partons N F S p , and where M 2 T = M 2 + p 2 T . In particular for the signal samples, we test distinct mass hypotheses in different configurations.
For the S1 signal, we vary Mχ0 2 in the range [100,700] GeV by steps of 100 GeV, and we set Mχ0 1 = Mχ0 2 /2 and M˜ ± = [Mχ0 2 + Mχ± 1 ]/2. For the S2 signal, we produce a single "S2a" sample, i.e. with Mχ0 2 − Mχ0 1 < M Z , for which we set Mχ0 2 = 100 GeV, Mχ0 1 = 50 GeV. This enables to explore the case where thẽ χ ± 1 and theχ 0 2 decay through a W ± and through a Z that are both off-shell. For the other S2 samples, denoted "S2b" and described in the following paragraph, both the W ± and the Z bosons are on-shell. In addition, we vary

Analysis of theχ
We considered only the electron and the muon channels. For these analyses we set the integrated luminosity to Ldt = 20 fb −1 . The latter cut is applied on the so-called "stransverse mass": M T 2 . We used a boostcorrected calculation of this variable as described in [56] and implemented in MCTLib [57]. The event selection efficiencies, event yields, signal significances and the expected integral charge asymmetries are reported in table 23. Figure 14 displays the / E T distribution after the event selection.
In this simple version of the analysis, we keep the same event selection for both teh S1 and the S2 signals. Therefore these signals samples share the same residual background as well as the same bias from the event selection. In these conditions, we could use a common A C template curve for both of them. However, because we choose many overlapping masses between these two signal samples, we split them into two seperate sets of experimental A C template curves. The S1 A C template curve, that include the propagation of the realistic experimental uncertainties into each term of equation (2.11), are displayed in figure 15, the S2 ones are displayed in figure 16. And the final signal template curves for which the uncertainties account for the correlations between the parameters used to fit the A Meas C template curves are shown in figure 17, on the l.h.s. for S1 and on the r.h.s. for S2. 75,50] 0.45 ± 0.01 1097. 43  ν e /µ ± ν µ /τ ± ν τ /qq ) + LF 0.00 ± 0.00 0.00 --W ± (→ e ± ν e /µ ± ν µ /τ ± ν τ ) + HF 0.082 ± 0.004 0.96 -(36.93 ± 1.76) tt 0.00 ± 0.00 0.00 -t + b, t + q(+b) 0.00 ± 0.00 0.00 --W + W, W + γ * /Z, γ * /Z + γ * /Z 0.283 ± 0.002 106.78 -(26.95 ± 0.25) W + + W − + W ± , W + + W − + γ * /Z, 0.576 ± 0.004 1.77 -(29.84 ± 0.34) W ± + γ * /Z + γ * /Z, γ * /Z + γ * /Z + γ * /Z γ + γ, γ + jets, γ + W ± , γ + Z 0.00 ± 0.00 0.00 -γ * /Z + LF 0.00 ± 0.00 0.00 -γ * /Z + HF 0.00 ± 0.00 0.00 --QCD HF 0.00 ± 0.00 0.00 --QCD LF 0.00 ± 0.00 0.00 --  Table 24. Noise to signal ratio, signal statistical significance, and expected and measured integral charge asymmetries for the S1 and S2 signal samples for the p + p →χ ±  Figure 14. Distribution of the / E T after the event selection. The background, the S1, and the S2 signals are the filled yellow, the hollow brown, and the hollow red histograms, respectively. This enables us to perform a closure test of our method on the S1 signal sample as displayed at the top of figure 18, where we can fit of the input versus the measured Mχ±

Experimental result for the S2 signal
As in the previous sub-section, using the S2 signal A C template curves 16, we can get the results reported in table 27. The closure test on the S2 signal samples is displayed at the bottom of figure 18.    Table 25. Measured A C (S) of the S1 signal samples with their full experimental uncertainty. Indirect mass measurement and their full experimental uncertainty as a function of the signal sample.  Table 26. Closure tests with a forced fit parameter for the S1 signal samples.    Here again the fit indicates, within the uncertainties, that the indirect mass measurement is linear and unbiased.
The checks, forcing the parameters of the fit functions, tend to confirm these indications, as presented in table 28.

Final result for the S1 signal
For the S1 sub-samples with a signal significance in excess of 5σ, the indirect measurements of Mχ± 1 + Mχ0 2 are performed with an overall accuracy better than 6% for input masses    Table 29. Final results for the S1 samples with experimental and theoretical uncertainties.

Final result for the S2 signal
For the S2 sub-samples with a signal significance in excess of 3σ, the indirect measurements of Mχ± 1 +Mχ0 2 are performed with an overall accuracy better than 4.5% for respective input masses We sum up the indirect mass measurements of Mχ± 1 + Mχ0 2 extracted from the integral charge asymmetry of theχ ± 1 +χ 0 2 → 3 ± + / E T inclusive process within tables 31 (S1 signal) and 32 (S2 signal).
For the S1 signal at LO, this new method enables to get an accuracy better than 6% for the range with 5σ sensitivity to the signal and better than 10% elsewhere. Whereas for the S2 signal at LO, we get an accuracy better than 4.5% for the range with 3σ sensitivity to the signal and better than 11.2% elsewhere. All these indirect measurements are statistically compatible with the total uncertainty of the method.
One should bear in mind however that these results do not account for the dominant theoretical uncertainty (δ(A C ) P DF ). (GeV) δM Fit (GeV) δM Fit

Dilepton mass edge
In this sub-section, we'll compare the ICA (Integral Charge Asymmetry) indirect mass measurement technique with two other direct mass measurement techniques.
But before entering this topic, let us mention the issue of the combinatorics within the trilepton search topology we've chosen. For our signal, resolving this combinatorics consists in matching the correct dilepton to its parentχ 0 2 whilst associating the third lepton to its parentχ ± 1 . Theχ 0 2 leptonic decay yields two leptons with opposite-signs (OS) and same flavours (SF). In events with mixed flavours (e + e − µ ± or µ + µ − e ± ), the correct assignment is obvious: the dilepton of SF comes from theχ 0 2 and the single lepton with the other flavour comes from theχ ± 1 . However in order to exploit the full signal statistics, one also needs to resolve this combinatorics in tri-electron and tri-muon events. For each of these event topology involving a single flavour, there are always two combinations of OS dileptons and one combination of same-sign (SS) dilepton. Therefore we adopt a statistical solution to lift the combinatorics. In the calculation of any physical observable, for each 3e ± or 3µ ± event, we fill the corresponding histogram with two entries from the two OS dileptons -50 -

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with a weight of +1 and with one entry from the single SS dilepton with a weight of -1. This systematically subtracts from the observable histogram the wrong combination which associates a lepton from theχ ± 1 decay with one of theχ 0 2 decay.
3.6.1. a. Experimental observable. The fact that the OS-SF dilepton coming from the second neutralino decay has an edge in its invariant mass was noted long ago in [58]. It has been used extensively in the litterature [69][70][71][72], including in a few reviews like [75] and in references therein. For the S1 signal, we have the following mass hierarchy Mχ0 2 = Mχ± 1 > M˜ ± > Mχ0 1 and we considerχ 0 2 andχ ± 1 two-body decays proceeding through an intermediate slepton. In this case, the edge is given by: For the S2 signal, we have the following mass hierarchy Mχ0 2 = Mχ± 1 > Mχ0 1 and we considerχ 0 2 andχ ± 1 decays proceeding through W ± and Z bosons. In these cases, the edge is given by: for aχ 0 2 three-body decay proceeding through an off-shell Z * (S2a), and by for aχ 0 2 two-body decay proceeding through an on-shell Z (S2b). In light of these formulae, we see that the mass reconstruction capabilities of this method that we'll call DileME, for "Dilepton Mass Edge", regard exclusively the reconsctruction of mass differences.
The main systematic uncertainties of the DileME method come from the lepton energy scales. These are known to a 0.05% accuracy in the ATLAS experiment at the LHC Run1, both for the electrons [73] and the muons [74]. Since the dilepton invariant mass is: The index with values 1 or 2 refers to either of the two OS-SF leptons from theχ 0 2 decay, and α 1,2 is the angle in space between their flight directions. Neglecting the uncertainty on the angle, the relative uncertainty on M ± ∓ writes: 3.6.1. b. Theoretical shape. For unpolarizedχ 0 2 and for their two-body decays, the theoretical shape of the dilepton invariant mass is known [66] Table 33. Dilepton mass edge measurements for the S1 samples.
As seen in subsection 3.2.2, the main background process in theχ ± 1 +χ 0 2 → 3 ± + / E T analysis is the W ± + γ * /Z 0 → 3 ± + / E T process, which constitutes an irreducible background. The OS-SF dilepton coming from the γ * /Z 0 decay forms a peak centered around M Z . Therefore, we model the invariant mass distribution of events surviving our selection using the following 6-parameters functional form: In order to account for the detector finite resolution, we convoluted the previous functional form with a gaussian distribution centered on zero and with an RMS set to C 2 . The other parameters represent: • C 0 : M Max ± ∓ , i.e. the position of the dilepton edge; • C 1 : N Exp S , i.e. the number of expected signal events under the triangle; • C 3 : N Exp B , i.e. the number of expected background events under the Z peak; • C 4 : M Z , i.e. the position of the Z peak; and, • C 5 : Γ Z , i.e. the width of the Z peak.
For the S2b signal samples, we expect N Exp events under the Z peak. After a few trials we find it is sufficient to use the same triangle distribution to describe both the two-body and the three-bodyχ 0 2 decay in these fits. The results of these fits are presented in tables 33 and 34. The plots 21 and 22 illustrate a few of these fits. Obviously the highest Mχ0 2 mass hypotheses unable any measurement of the dilepton invariant mass edge because of their unsufficient signal-to-noise ratio. This situation is met for Mχ0 2 ≥ 700 GeV for the S1 samples and Mχ0 2 ≥ 400 GeV for the S2 samples.   First of all we notice, that ICA and DileME methods do not give access to the same informations:

Meas. M Edge
, respectively. We notice that the DileME method is very accurate: better than 3.5% (and most often better than 1%) for the S1 samples, and better than 0.5% for the S2a sample. However, for the S2b signal samples, it fails to extract any sensible informations about the mass difference because of the resonant mode of theχ 0 2 decay. For the sample (105, 13.8) S2b sample, the correct mass difference is found by chance, whereas for the other S2b samples, the DileME method systematically provides a wrong answer: In regard of these observations, we conclude that the ICA and DileME methods complement very well each other. and The stranverse mass has two important properties. On the one hand, it's very effective to discriminitate R-parity conserved SUSY signals from their SM background processes. On the other hand it enables to measure the mass of the parent particles (X) and (Y) and of children particle (χ) and for this second purpose, we'll denote this method MT2 in the rest of this article.
Regarding the signal and background discrimination described in section 3.2, we arbitrarily chose the following assignment: • ± 2 ↔ visible energy (B), where the index i = 1, 2, 3 refers to the decreasing p T of the leptons, and we set M trial χ = 0 GeV. This choice does not accurately reflect the actual kinematics of our signal samples, but it is sufficient to provide a good and simple signal to background discrimination applicable to all of them.
On the contrary, in the current section, in order to assess the mass measurement capability of the MT2 method we have to properly assign the OS-SF dilepton to theχ 0 2 decay, say into the visible energy (A), and the additional lepton to theχ ± 1 decay into the visible energy (B). This precise assignment is done via the solution we adopted to solve the trilepton combinatorics which is presented in the preamble of the current section.
The main systematic uncertainties for the MT2 method come from the reconstruction of the different objects in our search topology. As inferred from [54], we consider as sources of uncertainty: the trigger, the reconstruction, the identification, the energy resolution and the isolation for both the electrons and the muons. The resulting uncertainties are 4.6% (e ± ) and 1.1% (µ ± ), respectively. These changes in the electrons and muons kinematics are propagated onto a corrected missing transverse energy / E Corr T . Then, the impact of the uncertainties of the calorimeter cluster energy scale, of the jet energy scale and the jet energy resolution, and of the pile-up on the / E T , are also summed in quadrature, amounting to an uncertainty of 0.8% with which the / E : This procedure is re-iterated for each value of M trial χ , as reported in table 35. and, -56 -

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Whereas, for three-body decays, the f down and f up functions are: and, It's important to note, that for f 2−body down , small values of M trial χ are not always permitted. In the particular for our simplified models, we have the following relations: Mχ0 2 = 2Mχ0 1 , and for the S1 samples, M˜ ± = 3 2 Mχ0 1 . Therefore we need to keep M trial χ ≥ 135 256 × Mχ0 1 in order for f 2−body down to be defined. For the MT2 method, we need to perform two series of fits. We start with primary fits to the M T 2 distributions for each signal sample so as to measure their M Max T 2 . Then we proceed with the secondary fits for each signal sample. The latter use as inputs the different M Max T 2 values obtained for each M trial χ hypothesis and they enable simultaneoulsy to measure the mass of the parent particle, here Mχ0 2 = Mχ± 1 , of the end daughter particle Mχ0 1 and, for the S1 samples, the mass of the intermediate particle, M˜ ± . The 2-body functional forms are utilized to fit the S1 samples and the 3-body ones are utilized to fit the S2 samples. Note that these functional forms also provide the prior knowledge of the M Max T 2 for each signal hypothesis which serve as starting points in the minimization process of the primary fits.
Here are a few important observations that justify our strategy for the primary fits: • the M T 2 distribution of the remaining background events cluster into a Z peak which is located at M Z + M trial χ , • the M T 2 distribution of the S2b samples also cluster into a Z peak which is located at M Z + M trial χ and which may either be truncated or exhibit an asymmetric shoulder, • S1 samples: without an analytical description of the full M T 2 distribution, we just fit the M T 2 falling edge.
This leads us to use similar functional forms as for the dilepton mass distributions for the primary fits, but with 8 parameters: In order to account for the detector finite resolution, we convoluted the previous functional form with a gaussian distribution centered on zero and with an RMS set to C 4 . The other parameters represent: Signal S1 [Mχ0 2 , M˜ ± , Mχ0 1 ] GeV [100, 75,50] Undef. measurements of the S1 samples for M trial χ = 0 GeV.
• C 0 : M Max T 2 , i.e. the position of the M T 2 end-point; • C 1 : slope of the first line; • C 2 : height of the kink between the two lines; • C 3 : slope of the second line;                     respectively. The precision of the MT2 mass measurements are summarized hereafter: • S1 signal: Even though the MT2 method, appears to be slightly less accurate than ICA (itself being much less accurate than DileME ), it provides much more informations on different individual particles mass than ICA, or DileME, or even a combination of ICA and DileME.  However M T 2 end-points are known to be sometimes difficult to measure [76], especially for small signals in the presence of some background.
The last remark, is that ICA appears to have a higher mass reach than DileME and MT2. This is mostly due to the ICA reduced systematic uncertainty in its background subtraction.
So, we see that the three methods have quite different advantages and drawbacks, they also have different systematic uncertainties. They are therefore complementary and the best SUSY mass informations can be extracted by combining them.

Conclusions
We propose a new method to measure the mass of charged final states using the integral charge asymmetry A C at the LHC.
At first we detail and test this method on the p + p → W ± → ± ν inclusive process. Then we apply it on a SUSY search of interest, namely the p + p →χ ± 1 +χ 0 2 → 3 ± + / E T inclusive process.
For each process, we start by calculating the central values of A C using cross section integrators with LO MEs and with three different LO PDFs. MCFM is used for the SM process and Resummino is used for the SUSY process. The same tools are also used to estimate the theoretical unceratinties on A C . These calculations are repeated varying -70 -  the mass of the charged final state. Over the studied mass ranges we find that A C is a monotically increasing function of M (F S ± ). This function is well described by a polynomial of logarithms of logarithms of M (F S ± ). The PDF uncertainty turns out to be the dominant source of the theoretical uncertainty. The experimental extraction of A C requires a quantitative estimate of the biases caused by the event selection and by the residual background. To this end MC samples are generated for the considered signal and its related background processes. These samples are passed through a fast simulation of the ATLAS detector response. Realistic values for the systematic uncertainties are taken from publications of LHC data analyses. The full experimetal uncertainties as well as the effect of the residual background are consistently propagated through a central value and uncertainties of the measured A C . This way the measured A C of each signal sample can be translated into a central value and uncertainties of an indirect measurement of the corresponding M (F S ± ). The theoretical uncertainties of each measured M (F S ± ) is summed in quadrature with the experimental uncertainties so as to provide the full uncertainty for this new method.
For the p + p → W ± → ± ν inclusive process, M W ± can be indirectly measured with an overall accuracy better than 1.2%. We note that the dispersion of the central values of M W ± indirectly measured with the three PDFs are compatible with the total uncertainty of the MSTW2008lo68cl prediction.

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For the p + p →χ ± 1 +χ 0 2 → 3 ± + / E T inclusive process, without accounting for δ(A C ) P DF , Mχ± 1 +Mχ0 2 can be measured with an overall accuracy better than 6% for a sensitivity to the signal in excess of 5σ and with an accuracy better than 4.5% for a sensitivity to the signal in excess of 3σ. These indirect mass measurements are independent of the details of the decay chains of the signal samples. For the considered SUSY process, basic closure tests indicate the indirect mass estimate does not need any linearity nor offset corrections.
We recommend to apply this method using at least NLO A C templates both for the theoretical and the experimental parts. Indeed, the most precise cross sections and event generations constitute more reliable theory predictions and are in better agreement with the data than LO predictions. NLO or NLL theoretical templates reduce the theoretical uncertainty, as shown in table 21 for example. Besides, the measurements of dA C (W ± → ± ν) dη( ± ) by the LHC experiments [3][4][5][6][7] were found to agree well with NLO theory predictions. Even if our asymmetry ratios of the A C theoretical templates: figure 4 and figure 12, reveal important shape difference of the higher orders with respect to LO, the size of the corrections remain nevertheless quite modest.
Finally, the comparison of the ICA (Integral Charge Asymmetry) method for SUSY mass measurements, to the DileME (Dilepton Mass Edge) and to the MT2 (stransverse mass), shows that these three methods are quite complementary.
• the DileME method is the most precise one, but it can only access a mass difference and it has a strong bias in certain situations (S2b signal); • the MT2 method is the least precise one, it may be difficult to exploit in certain cases, but it provides constraints on individual mass (parent, possibly intermediate and end daughter particle); • the MT2 method is slightly more precise than MT2, it has the largest mass reach, but it can only access a mass sum.

Prospects
In this article we have envisaged two production processes for which the mass measurement from the integral charge asymmetry is applicable. One SM inclusive process p+p → W ± → 1 ± + / E T and one SUSY inclusive process p + p →χ ± 1 +χ 0 2 → 3 ± + / E T . Here are the typical physics cases where we think the indirect mass measurement is applicable and complementary with respect to usual mass reconstruction techniques: • Initial state (IS): processes induced by q +q, or q + g • Final state (FS): situations where the clasiscal reconstruction techniques are degraded because of bad energy resolution for some objects (τ ± had , jets, b-jets,. . . ) combined with a limited statistical significance (i.e. channels with τ ± had compared to channels with e ± or µ ± ) and especially where many particles are undetected -73 -
Other physics cases could be searches for W ± → µ ± ν and for W ± → tb.

1,2,3
Note, that with the increasing center-of-mass energies and the increasing integrated luminosities of the LHC runs in the years to come, all the vector boson fusion production modes of the above cited processes could also become testable. This new method only applies after a given event selection and it is indicative of the mass of the final state produced by a charged current process, only when the event selection provides a good statistical significance for that process. Further studies should determine wether a differential charge asymmetry can be used to improve the separation between a given signal and its related background processes and therefore improve the sensitivity to some of this signal properties.
Differential charge asymmetries have been extensively used in other search contexts. For example, in attempts to explain the large forward-backward asymmetries of the tt production measured at the TEVATRON by both the CDF [59] and the D0 [60] experiments, some studies were carried out at the LHC to constrain possible contributions from an extra W ± boson. See for example [61,62], using a differential charge asymmetry with respect to a three-body invariant mass, and also [63], using an integral charge asymmetry, and the references therein. Such analyses, using charge asymmetries with respect to the tt system rapidity, invariant mass and transverse momentum, have also been performed by the ATLAS and CMS collaborations, see [64] and [65], respectively. We should also mention the differential charge asymmetry with respect to a two-body invariant mass which served as a discriminant between some BSM underlying models [66,67], namely SUSY versus Universal Extra Dimension [68] models, in the study of some specific decay chains.
For what concerns the current article, a first look at the differential charge asymmetry versus the pseudo-rapidity of the charged lepton coming from the chargino decay, reveals promising shape differences between the SM background and the p + p →χ ± 1 +χ 0 2 SUSY signals. However detailed results are awaiting further studies.   Figure 27. Evolutions of the quark PDFs (top), of the quark currents in the IS and of A C (bottom) calculated with QCDNUM using the MSTW2008nlo68cl parametrization.
A.2 Toy models for the main properties of A Fit C Hereafter, we make the hypothesis that quark currents and A C can be fitted by the different polynomials of functions of Q evoked above. We want to figure out how the coefficients of such polynomials arrange so as to give the A C template curves presented in sub-section 2.1, i.e. monotonically increasing functions of Q with a monotonically decreasing slope. Note that this is well suited for x 1,2 which are not too large (below the maximum of the valence peaks for x·u(x, Q 2 ) and x·d(x, Q 2 )). For large x 1,2 (beyond these peaks), A C monotonically decreases with a monotonically decreasing slope.
Again, let's consider the simplest case where the first degree polynomials are sufficient. If we denote x = Q, and f (x) the fit function, we can write the charged cross sections: Provided that lim x→+∞ |f (x)| = +∞ (which holds for all the fit functions we considered), it appears that A C has an asymptote given by:  The derivative of A C (x) can be expressed as: Hence the condition to get a monotonically increasing A C (x) writes: And finally, that fact that A C can be fitted with the same functional form as σ + (x) and σ − (x) relies on the (approximate) fullfilment of the following second degree functional equation: ( The fits of σ + (x), σ − (x) and A C with the 3 considered functional forms are performed and the corresponding values of the fit parameters are presented in table 58.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.