Drell-Yan, ZZ, W+W- production in SM&ADD model to NLO+PS accuracy at the LHC

In this paper, we present the next-to-leading order QCD corrections for di-lepton, di-electroweak boson (ZZ, W+W-) production in both the SM and the ADD model, matched to the HERWIG parton-shower using the aMC@NLO framework. A selection of results at the 8 TeV LHC, which exhibits deviation from the SM as a result of the large extra-dimension scenario are presented.


Introduction
With more accumulated data at the LHC, extra dimension searches at different energies have yielded stringent bounds [1,2] on the model parameters [3,4]. This has also been facilitated by improved theoretical calculations to next-to-leading order (NLO) in QCD that have been available for the large (ADD) [3] and warped (RS) [4] extra dimension models for various processes viz. di-lepton [5], di-boson (γγ [6], ZZ [7], W W [8] (W + W − is denoted as W W )). In extra dimension models, pair production could result from the exchange of virtual Kaluza-Klein (KK) modes. As a result of possible new physics, it is expected that the production rate and potentially certain kinematical distributions may get modified as compared to the SM predictions. Further, it is essential that higher order QCD corrections are included as it leads to reduction in scale uncertainties which in turn improves the theoretical predictions. For extra dimension searches, ATLAS and CMS have investigated the impact of NLO corrections in their analysis by using constant K-factors, which does not necessarily give reliable predictions.
One important recent development has been the implementation of the di-photon production to NLO including Parton Shower (PS) in the AMC@NLO environment for the ADD model [9]. This allows for the generation of fully exclusive events that are NLO accurate for observables inclusive in QCD radiation. If required, these events can be directly passed through a detector simulation. In this paper, we have implemented the rest of the pair production processes (ℓ + ℓ − , ZZ and W W ) that could contribute to the ADD model, to NLO+PS accuracy in the AMC@NLO environment.
To set the notations and the conventions used, we briefly describe the interaction Lagrangian of the massive spin-2 KK modes h ( n) µν with the SM particles, which is through the energy momentum tensor T µν of the SM. The coupling κ is related to the Planck mass in 4-dimension, κ = √ 16π/M P . Using the convention of HLZ [10] the summation of the KK modes in the propagator D(s) is given by The summation over KK modes leads to the integral I(Λ/ √ s), defined in [10], √ s is the center of mass energy, Λ is the UV cutoff of the KK modes which is identified with the fundamental scale M S in 4 + d dimensions [10,11]. Bounds on M S for different extra dimensions d have been obtained by ATLAS and CMS collaborations; for our present analysis we choose the following values M S = 3.7 TeV (d=2), 3.8 TeV (d=3), 3.2 TeV (d=4), 2.9 TeV (d=5), 2.7 TeV (d=6). The rest of the paper is as follows: we briefly describe the framework for matching the NLO results with Parton Shower Monte Carlo in section 2. A selection of the numerical results are presented in section 3 and finally we present our conclusions in section 4.

NLO+PS
In order to provide a more realistic description of a process at the LHC, it is unavoidable to match the NLO QCD results with Parton Shower Monte Carlo. For the present analysis, we adopt the MC@NLO formalism [12] to match the fixed order NLO results with the HERWIG6 [13] parton shower, including the hadronisation contribution by using the automated AMC@NLO framework. The Born and realemission correction for all these processes are computed with MADFKS [14], which uses the FKS subtraction method [15] to compute the real-emission contribution in an automated way, within the MadGraph5 [16] environment. The virtual contributions are implemented separately in this environment for each of these processes, using the analytically calculated results for ℓ + ℓ − [5], ZZ [7] and WW [8] production processes. We have also incorporated an algorithm that takes care of the summation of the KK modes in the ADD model (Eq. 2); this has been made possible by appropriate changes in the spin-2 HELAS routine [9]. The exact numerical cancellations of double and single poles coming from the real and virtual terms in all the subprocesses, for each of the production processes have been checked.
For the Drell-Yan (DY) process, we have generated the events for the process P P → e + e − X, which is phenomenologically same as P P → µ + µ − X, except for the experimental identification of the final state particles. The leading order (LO) partonic contribution comes from the qq → e + e − in both the SM and ADD model, whereas at LO g g → e + e − contributes only to the ADD model. Emission of real gluon and one loop correction due to the virtual gluon, together with the partonic subprocess q(q) g → q(q) e + e − , give all the O(α s ) contributions. The interference between the SM and ADD diagrams also give O(α s ) contribution at the NLO. For the di-boson final states, in addition to similar partonic sub processes, there are contributions due to the interference between the gg initiated box diagrams in SM and the gg initiated Born diagrams in the ADD which is of O(α s ). We have considered all the above contributions in each of these processes of interest for our present analysis.
After generation of events following the above procedure, we let the Z and W ± bosons to decay to leptons at the time of showering. For the ZZ events, we let one Z boson to decay to e + e − and the other one to µ + µ − , while for W W events we let the W + decay to e + ν e and the W − to µ −ν µ . Alternatively, the W ± and Z bosons can be decayed using MadSpin [17] at the time of event generation itself, which retains nearly all spin correlations. We have not chosen to do this, because the inclusion of the sum over the KK modes is non-trivial in this way.

Numerical Result
In this section, we present some of the kinematical distributions for the production of ℓ + ℓ − , ZZ, W W , both in the SM and ADD to NLO+PS accuracy for the LHC center of mass energy √ S = 8 TeV. Events are generated using the following input parameters: α −1 EW = 132.507, G F = 1.16639 × 10 −5 GeV −2 , m z = 91.188 GeV. Using these electro-weak parameters as inputs, the mass of W boson m w = 80.419 GeV and sin 2 θ w = 0.222 are obtained. The (N)LO events are generated using MSTW(n)lo2008cl68 parton distribution functions (PDF) for the (N)LO and the value of strong coupling constant α s is solely determined by the corresponding MSTW PDF [18] at (N)LO. The factorisation scale µ F and the renormalisation scale µ R are set equal to the invariant mass of the corresponding di-final state. The number of active quark flavor is taken to be five and are treated as massless. We use the following loose cuts at the time of event generation for the DY production:  For showering the DY events, HERWIG6 in MC@NLO formalism is used. Using the following analysis cuts: P l T > 20 GeV (l = e + , e − ), |η l | < 2.5, M e + e − < M S , ∆R ll > 0.4 for showering, the hardest (with maximum P T ) e + and e − are collected. In order to separate leptons from jets, ∆R lj > 0.7 is used. For both ZZ and W W showering, we have identified those final state, stable lepton-pair, whose mother is one of the Z boson (for ZZ showering) or the final state stable lepton-neutrino pair whose mother is one of the W boson (for W W showering) and that is the reason we avoid the cut which is commonly used to reconstruct the Z(W ) boson mass from the invariant mass of the lepton-lepton (lepton-neutrino) pair. For decay products of Z/W , we use the same analysis cuts to plot various differential distributions and they are the following: invariant mass In addition, we have collected only those leptons whose separation from other leptons and jets are greater than 0.4 and 0.7 respectively in the rapidity-azimuthal angle plane.
Here, we describe few selected differential distributions for some of the kinematical observables. To start with, we study the effect of parton shower over the fixed order NLO correction. Fixed order NLO results (dashed brown) along with the NLO+PS results (solid blue) for the log 10 (P T ) distribution of the e + e − (left), ZZ (middle) and W W (right) pair are plotted in fig. 1, using their specific analysis cuts detailed above for extra dimensions d = 2 and its corresponding M S value. In all these plots, the fixed order cross section diverges for P T → 0, while the NLO+PS result shows a converging behavior in the low P T region. The effect of parton shower ensures correct resummation of the Sudakov logarithmic terms which appear in the collinear region leading to a suppression of the cross section in the low P T region.
There is no significant deviation in the high P T region as expected.
In the subsequent plots, we have included fractional scale and PDF uncertainties corresponding to the SM and ADD model distributions. By fractional uncertainty we mean the central value of a particular distribution divided by its extremum value. The scale uncertainties are calculated by considering independent variation of the renormalisation and the factorisation scales in the following way: µ R = ξ R M and µ F = ξ F M. Here, M denotes the invariant mass of the di-final state i.e., M e + e − , M ZZ , M W W as required and ξ R , ξ F can take either of the following values (1, 1/2, 2) independently. The scale uncertainty band is the envelope of the following (ξ F , ξ R ) combinations [9] as described below: (1,1), (1/2,1/2), (1/2,1), (1,1/2), (1,2), (2,1), (2,2). Estimation of the PDF uncertainty is done in the Hessian method as prescribed by the MSTW [18] collaboration. All these uncertainties are determined automatically by following the re-weighting procedure [19] built in AMC@NLO which stores sufficient information in the parton level Les Houches events for this purpose.
In all the plots ADD represents the full contribution of the SM and ADD model contributions including interference. We use a consistent graphical representation for the rest of the kinematic distributions. In each case, the upper inset gives the distribution in SM (solid blue) as well as in ADD model (dashed brown) to NLO+PS accuracy. For the same distribution, the middle (ADD) and lower (SM) insets provide fractional scale (solid brown) and PDF (dashed black) uncertainties. Various kinematical observable in the DY process are given in fig. 2, 3, 4 and 5. In fig. 2, we have shown the invariant mass distribution (left) and transverse momentum distribution (right) of the e + e − pair for d = 2 with its associated M S value. The effect of large extra dimension is dominant in the high invariant mass region and hence we focus in the region M e + e − > 600 GeV to study the other distribution viz. P T , rapidity, angular distribution of the e + e − pair and also look at some of the distributions of the individual leptons. In fig. 2, note that there is an increase in the scale and PDF uncertainties with increase in P T as is well known, see for example [20]. In fig. 3, the rapidity distribution of e + e − pair (left) and the angular distribution (right) are given for d = 2. For the rapidity distribution the deviation from the SM is only prominent in the central region. The angle made by the lepton pair in its center of mass frame with respect to one of the incoming hadron is denoted by θ * . The angular distribution is a good discriminator for the full range to distinguish the ADD from the SM. fig. 4   For the W W production process, the relevant plots are presented in fig. 9 and fig. 10, wherein the decays of W ± bosons to leptons and neutrinos are included at the stage of showering. For the choice of M S values associated with specific number of extra dimensions, we do not find any significant deviation from the SM. In the very high invariant mass region of the four-body final state for d = 5, 6 there is some deviation form the SM. In fig. 9, we have given the invariant mass (M e + µ − ✟ ✟ E T ) distribution (left) of the final state decay products of W ± and the total missing transverse energy distribution (right) which comes from the final state neutrinos for d = 5. For completeness in fig. 10, we also provide the transverse momentum distribution of the final state positron (left) along with its rapidity distribution (right) for d = 6. Only mild difference between the SM and ADD in the high invariant mass region is observed. We zoom into this very high invariant mass region to look for deviations from the SM for these exclusive observable. We have studied dσ/d ✚ ✚ E T , dσ/dP e + T and dσ/dη e + in the region when the invariant mass lies between 2 TeV and M S .  Table 1: Lower bounds on M S for various extra dimensions d at the 14 TeV LHC with integrated luminosity of 10 fb −1 at 3-sigma and 5-sigma signal significance.
Using the dilepton process, we present the search sensitivity for the extra dimensions d = 2 − 6, for 14 TeV LHC. The total cross section σ is calculated using the invariant mass distribution of the di-lepton pair for signal plus background and the background only. For a particular choice of extra dimension d, we find the minimum luminosity by varying the scale M S at 3-sigma (3σ) and 5-sigma (5σ) signal significance. We define the required minimum luminosity as L = max{L 3σ(5σ) , L 3N S (5N S ) }, where L 3σ(5σ) is the integrated luminosity at 3-sigma (5-sigma) signal significance and L 3N S (5N S ) describes the integrated luminosity to get at least 3(5) signal events. Now we can get the corresponding M S value for 10 fb −1 luminosity by inversion which is tabulated in table 1. Of course, a full analysis including the effects of detector simulation, non-reducible backgrounds etc. can be better performed by the experimental collaborations.

Conclusion
The main objective of this work has been to make available, the ℓ + ℓ − , ZZ, W + W − production results to NLO+PS accuracy for the large extra dimension model which is implemented in the AMC@NLO framework. All the subprocesses that contribute to NLO in QCD have been included for each of these processes. A selection of results for 8 TeV LHC has been presented for various distributions in an attempt to identify region of interest for extra dimension searches. Scale and PDF uncertainties for each of these distributions have also been studied. In addition, we have presented the search sensitivity for the extra dimensions d = 2 − 6, for 14 TeV LHC at 10 fb −1 .
With the earlier implementation of the di-photon final state to the same accuracy [9], this work completes the rest of the di-final state process (but for di-jet) in large extra dimension searches. In the ADD model, these codes can be used to generate events of the di-final states discussed in this paper to NLO+PS accuracy and are available on the website http://amcatnlo.cern.ch.