A spin on same-sign W-boson pair production

We demonstrate that the LHC will be sensitive to quantum correlations between two quarks inside the proton. Same-sign W-boson pair production is the most promising channel for clear measurements of double parton scattering. The left-handed nature of the coupling between quarks and W-bosons makes it a prime probe to measure parton spin correlations. We perform a detailed analysis of double parton scattering, including relevant backgrounds. The analysis reveals that measurements comparing the rate at which two muons from W boson decays are produced in the same compared to opposite hemispheres are especially sensitive to spin correlations between two quarks inside the proton. We provide estimates of the significance of the measurements as a function of the integrated luminosity.


INTRODUCTION
The LHC will allow us to have a precise and deep enough look inside the proton to reveal how the quantum properties of two quarks are interconnected. That correlations between the quantum numbers of quarks and gluons inside the proton can in principle be probed through double parton scattering (DPS) was realized in a series of papers by Mekhfi [1,2]. Although this was an exciting discovery, the papers were somewhat ahead of time and attracted only mild attention. The discussion of quantum correlations, and in particular spin correlations in DPS came back to life in the early days of the LHC [3][4][5]. Although relatively widespread, these were to a large extent still exercises with little impact on experimental searches and observables. Even for the only case so far where it has been shown that spin correlations have a large influence on the cross section [6], no clear observable for their detection was found. The reason was that the spin correlations had a large impact on the (unknown) size of the cross section, but only milder effects on the distributions of final state particles. For a summary of the theoretical status of DPS and correlations between two partons we refer to Ref. [7][8][9][10][11] and references therein. In short, many of the elements required in a proof of factorization of DPS have been established, with a consistent separation between single and double parton scattering and with double parton distributions (DPDs) describing the properties of the two partons inside the proton [12][13][14].
Same-sign W-pair (SSW) production is one of the most studied DPS processes [15][16][17][18][19][20][21][22][23]. In particular Ref. [16] investigated the effect of kinematical correlations between the partons. The contribution to the signal from single parton scattering is suppressed by additional couplings and produces signatures which makes the two relatively easy to separate experimentally. This implies that the parton splitting mixing single and double parton scattering is suppressed [7,13]. LHC is now reaching integrated luminosities large enough to start probing the SSW process and recently first experimental observations or indications of DPS in the SSW final state have been found [24,25].
Since the W-boson only couples to left-handed (right-handed) quarks (antiquarks), the polarized DPDs describing the spin correlations between two partons enter linearly into the cross section [26]. This leads to especially large effects on the cross section from spin correlations. In this letter, we demonstrate the impact which spin correlations can have on the distribution and rate of leptons from the W-boson decays. We identify observables with large sensitivity to the spin correlations, taking relevant backgrounds into consideration. We give estimates of the integrated luminosity necessary for measurements to probe the phase space for spin correlations and possibly make the first determinations of quantum correlations between the proton constituents. Further details, including the effects of other types of interparton correlations, a broader range of observables and with different selection criteria, will be discussed in detail in a forthcoming publication [27].

MEASURING INTERPARTON SPIN-CORRELATIONS
The spin correlations between two partons inside a proton are quantified by polarized DPDs. The SSW process is sensitive to longitudinally polarized quarks (and antiquarks) and the corresponding DPDs describe (at leading order) arXiv:1809.09024v1 [hep-ph] 24 Sep 2018 the difference between the probabilities of finding the two quarks with helicities aligned rather than anti-aligned. These distributions have never been experimentally probed, and are therefore largely unknown. The information that we do have on them, are either thanks to positivity bounds [28], giving upper limits on the polarized distributions, or model calculations [29][30][31]. The correlations decrease with double DGLAP evolution, but the longitudinally polarized distributions relevant in SSW production can remain large up to high scales [32].
The purpose of this letter, is to test if and when experimental measurements can provide valuable information about these polarized distributions and thereby access non-trivial information on quantum correlations between proton constituents. To answer this question, we will examine the situations where the bounds are saturated at a low input scale Q 0 in such a way that they produce maximal effects of polarization on the cross section. This gives us the relation between the longitudinally polarized and unpolarized distributions for quarks and antiquarks as The subscripts ∆a (a) denote longitudinally polarized (unpolarized) partons, x i is the momentum fraction of parton i and y is the transverse distance between the two partons. The exponent reads n = 1 if a and b are both quarks or both anti-quarks and otherwise n = 2. The scale Q 0 should be a low scale, chosen around the scales where perturbative calculations start to be valid. The reason is that once the bounds are saturated at a low scale Q 0 , they will be satisfied at all larger scales, but typically be violated if perturbation theory is used to evolve them down to even lower scales. We will make this ansatz at the low scale of Q 0 = 1 GeV. Even the unpolarized DPDs are (at best) poorly known. There are a larger number of model calculations compared to the polarized distributions [29][30][31][33][34][35][36][37]. Additional constraints on the unpolarized distributions come from sum rules valid after integrating a single DPD over the transverse distance y [11,[38][39][40]. A common base-line ansatz is to use the approximation that the two partons are uncorrelated, and that there is no significant interdependence between the parton types or momentum fractions and the transverse distance y. We will adopt this scenario throughout this letter, as it allows us to focus on the spin-correlations. Our conclusions are fairly stable under changes to these assumptions. More details on other types of parton correlations, and other extensions of this simple ansatz, will be the subject of a forthcoming publication [27]. The ansatz gives us unpolarized DPDs at a low scale equal to The separation of the y dependence allows us to express the y-dependence at the cross section level in terms of σ eff = 1/ d 2 yG(y; m W )G(y; m W ) which we will set equal to 15 mb. This value is in the range extracted by the CMS collaboration in SSW production [24]. However, one should be careful not to over-interpret this quantity. The reason for this is, among other things, that the extraction makes assumptions on the shape of the DPS cross section in variables which can be changed by, for example, the spin correlations. Both unpolarized and polarized DPDs are evolved from the initial scale Q 0 to the scale of the W-boson mass with (unpolarized and polarized) double DGLAP evolution (without the 1 → 2 splitting), see e.g. [28].
The total cross section, as well as the distributions of the final state leptons, strongly depend on the polarized distribution. The total DPS cross section varies by up to around 30 percent depending on the model of the polarized distributions. The cross section for double W, including spin correlations, is given in equation (3.5) of reference [26]. The polarized contributions, due to the change in angular momentum when the partons have their helicities aligned rather than anti-aligned, have a direct impact on the rapidity distributions of the muons.
The SPS contribution to the SSW final state produces additional particles, i.e. pp → W + W + jj with two forward jets. Cuts on these jets effectively remove this background. SSW production is therefore often praised for its background free nature. However, this is a truth with modification, and significant work is necessary to produce a background free DPS sample for the process. Main backgrounds are coming from the WZ final state (where one of the muons from the Z boson escapes detection) and QCD, in particular tt production. The top quark decaying into a b quark will produce a µ + , and a second same sign muon can be produced by subsequent B meson decay. The tt cross section is, in the absence of cuts etc., many orders of magnitude larger than the DPS signal, but there are a number of effective ways to get this background under control.
We aim to produce a DPS signal which is as pure as possible, and therefore impose restrictive cuts. We analyze particle level final states obtained using a combination of Monte Carlo generators in order to include the effects of the Underlying Event modeling. The DPS signal events come from Herwig 7.1.2. [41,42], where we apply a robust reweighing procedure to mimic our spin correlations at parton level. Herwig is also used for the tt production. Diboson samples were obtained via a combination of MadGraph5 aMC@NLO at LO [43] Table I: Signal and background cross sections in fb for the production of two positively charged muons, with the selection in (3).
defined using the default anti-k t algorithm of FastJet [46] with a pseudo-radius R = 0.4. Our kinematical cuts are where η i is the pseudo-rapidity of the muon from hard interaction i, k lead T (k subl T ) is the transverse momentum of the muon with the largest (second-largest) transverse momentum. k µ3 T is the transverse momentum of a third muon, / E T the missing transverse momenta and k jet1 T and k jet2 T are the transverse momenta of the two hardest jets. dR = (φ 1 − φ 2 ) 2 + (η 1 − η 2 ) 2 , where φ i is the azimuthal angle of µ i , measure the distance between the two muons. On top of this we apply b-tagging with efficiencies 75% for k jet T ∈ {25 − 30} GeV, 80% for k jet T ∈ {30 − 40} GeV and 85% for k jet T ∈ {40 − 50} GeV [47,48]. With these cuts, the cross sections are given in Tab. I. The two remaining relevant backgrounds are the W Z and tt, which we will now discuss separately. W Z is a relatively clean process theoretically and we will make the assumption that this process can be efficiently subtracted by precise theoretical calculations supported by data driven methods. Already today the theoretical calculations for the total cross section have been made with high precision [49,50]. We will assume a 1% remaining uncertainty on this subtraction. The tt is substantially more difficult to calculate, however, there are a number of additional ways to significantly suppress this background further. In particular, demanding tight isolation of the produced muons is a very strong discriminant to separate prompt muons from muons produced by meson decays. Using vertex localization to further discriminate between these two cases can additionally aid the separation and improvements on b-tagging can also help [51][52][53]. A detailed examination of all these possibilities should be done by the experimental collaborations performing the actual measurement. Based on our investigations we will assume that this type of discrimination in combination with data driven subtractions, can reduce the top background to 1% of the above cross section and reducing the signal by only 10%. For example, we could reduce the tt background by more than 95%, keeping more than 90% of the signal through crude muon isolation requirements. We will investigate how sensitive the measurement is to these assumptions by including an additional uncertainty to the measurements.
The most promising variable for detection of spin correlations is the asymmetry between the rate at which the two muons are created in opposite (σ − = σ(η 1 η 2 < 0)) compared to same (σ + = σ(η 1 η 2 > 0) hemispheres of the detector In the absence of inter-parton correlations, the two muons are produced in two independent hard scatterings and the asymmetry from DPS will be zero. Therefore a non-zero asymmetry in DPS would be a clear indication of parton correlations. In particular polarization, with its direct influence on the muon rapidities, can produce large values of this asymmetry. It can be generated also by correlations between the longitudinal momentum fractions of the two partons [16,22], but this typically give lower values of the asymmetry [27]. A less inclusive version of this asymmetry is the distribution in the product of the muon rapidities, shown in Fig. 1a. This distribution is very sensitive to the spin correlations, and without correlations between the partons, it has to be symmetric around η 1 η 2 = 0. In contrast, a non-symmetric η 1 η 2 distribution can only be generated by parton correlations. The distribution as a function of the sum of the rapidities shows a clear dependence on the spin correlations and is an interesting observable to study DPS, as shown by Fig. 1b. However, the slope of the distribution in the absolute sum of the rapidities is affected by the details of the properties of the single parton distributions used in the modeling of the unpolarized DPDs, and therefore it is a less clear indicator of partonic correlations.
The asymmetry calculated from the η 1 η 2 distribution for our selection, which is a pure DPS sample, is shown in Tab. II. An additional cut around central pseudorapidities |η i | > η min increases the asymmetry, but naturally  Table II: Asymmetry and DPS cross section for different cuts on the pseudo-rapidities of the two muons. decreases the cross section.
Next, we will examine how sensitive ATLAS and CMS will be to the asymmetry. In order to do so, we estimate how many standard deviations away from zero a measurement of the asymmetry will be. Since it provides the best balance between the value of the asymmetry and the size of the cross section, the cut |η i | > 0.6 will be chosen, where the asymmetry is 0.11. We assume a Poissonian uncertainty on the number of DPS events with the two muons in the same/opposite hemispheres. We then create Gaussian distributions of the signal cross sections in the two hemispheres and use these to test how many standard deviations a measured asymmetry deviates from zero. In order to estimate the effects on our assumptions on the background subtractions, we multiply the uncertainty of the cross-section per hemisphere by 1 + bσ BG /σ DPS , where σ BG is the remaining 1% of the tt and W Z cross sections (given in Tab. I). b is a parameter we set to unity for our central predictions and vary from 1/2 to 2. This gives an uncertainty band, indicating the sensitivity of our predictions on the background assumptions. Fig. 2 shows the significance of a measurement as a function of the integrated luminosity. The central curve corresponds to b = 1 while the band is obtained by the variation of b. With the µ + µ + channel alone, a 3-sigma observation could be approached with the full high-luminosity LHC (HL-LHC) integrated luminosity of 3000 fb −1 . A detailed study of the backgrounds also for final states with electrons is beyond the scope of this article. If we assume that a similar sensitivity can be achieved for electrons as for muons, the DPS signals are equal and therefore effectively enhanced by a factor of 4 including all combinations of positively charged muons and electrons (µ + µ + , e + µ + , and e + e + ). This decreases the integrated luminosity necessary to measure the asymmetry, as shown for |η i | > 0.6 in Fig. 2. A 2-sigma hint is possible with below 400 fb −1 , a 3-sigma observation can be achieved with less than 1000 fb −1 and more than 5-sigma observation of a non-zero asymmetry is reachable with a full 3000 fb −1 [54]. On top of this, combinations of measurements by CMS and ATLAS as well as additional possibilities with negatively charged leptons can further increase the sensitivity of the measurements. This means that there might be possible to see first indications of spin correlations before the start of the high-luminosity LHC.
(a) Figure 2: Estimate of the significance of a measured non-zero asymmetry with |η i | > 0.6, as the distance in standard deviations of a measured asymmetry as in Tab. II from zero. Blue band for µ + µ + final state only and red band including all positively charged combinations of two light leptons (e and µ).

CONCLUSIONS
Quantum number correlations between two partons inside a hadron belong to the experimentally unexplored area of high energy physics. The measurement of same-sign muon pairs in W-boson pair production is one of the best candidates to detect these correlations, improving our knowledge on how closely the proton constituents are bound to one another. The left-handed nature of the coupling between quarks and W-bosons gives rise to a large dependence of the cross section on the polarized double parton distributions. We have presented a model of the polarized double parton distributions describing the spin correlations, with a large impact on the rate and distribution of the measured muons. We have demonstrated that the LHC has the potential to extract non-trivial information on spin-correlations between two quarks inside the proton through the measurement of the rate at which the leptons are produced in the same rather than opposite hemispheres. Due to their direct relation to the distribution of the final state leptons, the spin correlations are a prime suspect for the generation of a large asymmetry. This asymmetry could be the first of a series of measurements at the LHC (HL-LHC), pinpointing how the quantum properties of quarks and gluons are connected to each other inside the proton.