Search for quark contact interactions and extra spatial dimensions using dijet angular distributions in proton-proton collisions at sqrt(s) = 8 TeV

A search is presented for quark contact interactions and extra spatial dimensions in proton-proton collisions at sqrt(s) = 8 TeV using dijet angular distributions. The search is based on a data set corresponding to an integrated luminosity of 19.7 inverse femtobarns collected by the CMS detector at the CERN LHC. Dijet angular distributions are found to be in agreement with the perturbative QCD predictions that include electroweak corrections. Limits on the contact interaction scale from a variety of models at next-to-leading order in QCD corrections are obtained. A benchmark model in which only left-handed quarks participate is excluded up to a scale of 9.0 (11.7) TeV for destructive (constructive) interference at 95% confidence level. Lower limits between 5.9 and 8.4 TeV on the scale of virtual graviton exchange are extracted for the Arkani-Hamed--Dimopoulos--Dvali model of extra spatial dimensions.


Introduction
High momentum-transfer proton-proton collisions at the CERN LHC probe the dynamics of the underlying interaction at distances below 10 −19 m. Often these collisions produce a pair of jets (dijets) approximately balanced in transverse momentum p T . These dijet events provide an ideal testing ground to probe the validity of perturbative quantum chromodynamics and to search for new phenomena such as quark compositeness or additional, compactified spatial dimensions. A particularly suitable observable for this purpose is the dijet angular distribution [1] expressed in terms of χ dijet = exp(| y 1 − y 2 |), where y 1 and y 2 are the rapidities of the two jets with the highest transverse momenta. Rapidity is defined as y = ln [(E + p z ) / (E − p z )] /2 with E being the jet energy and p z the projection of the jet momentum onto the beam axis.
For the scattering of massless partons, χ dijet is related to the polar scattering angle θ * in the partonic center-of-mass (c.m.) frame by χ dijet = (1 + | cos θ * |)/(1 − | cos θ * |). The choice of the variable χ dijet is motivated by the fact that for Rutherford scattering the angular distribution is approximately independent of χ dijet . In perturbative QCD the dijet angular distribution at small c.m. scattering E-mail address: cms-publication-committee-chair@cern.ch.
angles is approximately independent of the underlying partonic level process and exhibits behavior similar to Rutherford scattering, characteristic of spin-1 particle exchange. Signatures of new physics (NP), such as quark contact interactions (CI) or virtual exchange of Kaluza-Klein [2] excitations of the graviton, that exhibit angular distributions that are more isotropic than those predicted by QCD, could appear as an excess of events at low values of χ dijet .
Models of quark compositeness [3][4][5] postulate interactions between quark constituents at a characteristic scale that is much larger than the quark masses. At energies well below , these interactions can be approximated by a CI characterized by a four-fermion coupling. The effective Lagrangian for flavor-diagonal color-singlet couplings between quarks can be written as [4,5]: where the subscripts L and R refer to the left and right chiral projections of the quark fields respectively and η LL , η R R , and η R L are taken to be 0, +1, or −1. The various combinations of (η LL , η R R , η R L ) correspond to different CI models. The following CI scenarios with color-singlet couplings between quarks are investigated: http (η LL , η R R , η RL ) Note that the models with positive (negative) η LL or η R R lead to destructive (constructive) interference with the QCD terms and a lower (higher) cross section in the limit of high partonic c.m. energies. In all CI models discussed in this Letter, next-to-leading-order (NLO) QCD corrections are employed to calculate the cross sections. In proton-proton collisions the ± LL and ± R R models result in identical tree-level cross sections and NLO corrections, and consequently lead to the same sensitivity. For ± V V and ± A A , as well as for ± (V −A) , the CI predictions are identical at tree-level, but exhibit different NLO corrections and yield different sensitivity.
Measurements of dijet angular distributions at the Fermilab Tevatron have been reported by the CDF [6] and D0 [7,8]  Dijet angular distributions are also sensitive to signatures from the Arkani-Hamed-Dimopoulos-Dvali (ADD) model [15,16] of compactified extra dimensions (EDs) that provides a possible solution to the hierarchy problem of the standard model (SM). In the ADD model, gravity is assumed to propagate in the entire higher-dimensional space, while SM particles are confined to a (3 + 1) dimensional subspace. As a result, the fundamental Planck scale M D in the ADD model is much smaller than the (3 + 1) dimensional Planck energy scale M Pl , which may lead to phenomenological effects that can be tested with protonproton collisions at the LHC. The coupling of the graviton in higher-dimensional space to the SM fields can be described by a (3 + 1)-dimensional tower of Kaluza-Klein (KK) graviton excitations, each coupled to the energy-momentum tensor of the SM field with gravitational strength. The effects of a virtual graviton exchange can therefore be approximated at leading-order (LO) by an effective (3 + 1)-dimensional theory that sums over KK excitations of a virtual graviton. This sum is divergent, and therefore has to be truncated at a certain energy scale of order M D , where the effective theory is expected to break down. Such a theory predicts a non-resonant enhancement of dijet production, whose angular distribution differs from the QCD prediction. Two parameterizations for virtual graviton exchange in the ADD model are considered, namely the Giudice-Rattazzi-Wells (GRW) [17] and the Han-Lykken-Zhang (HLZ) [18] conventions. Though not considered in this paper, another convention by Hewett [19] exists. In the GRW convention the sum over the KK states is regulated by a single cutoff parameter T . The HLZ convention describes the effective theory in terms of two parameters, the cutoff scale M S and the number of extra spatial dimensions n ED . The parameters M S and n ED can be directly related to T [20]. We consider scenarios with 2 to 6 EDs. The case of n ED = 1 is not considered since it would require an ED of the size of the order of the solar system; the gravitational potential at these distances would be noticeably modified and this case is therefore excluded. The case of n ED = 2 is special in the sense that the relation between M S and T also depends on the parton-parton c.m. energy √ŝ .
In this Letter, we extend previous searches for contact interactions to higher CI scales, for a wide range of models that include the exact NLO QCD corrections to dijet production. In addition, we explore various models of compactified extra dimensions. Using a data sample corresponding to an integrated luminosity of 19.7 fb −1 at √ s = 8 TeV, the measured dijet angular distributions, unfolded for detector effects, are compared to QCD predictions at NLO, including for the first time electroweak (EW) corrections.

Event selection
A detailed description of the CMS detector, together with a definition of the coordinate systems used and the relevant kinematic variables, can be found in Ref. [27]. The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diameter, providing an axial field of 3.8 T. Within the solenoid are the silicon pixel and strip trackers, which cover the region of pseudorapidity |η| < 2.5, and the lead tungstate crystal electromagnetic and the brass and scintillator hadronic calorimeters, which surround the tracking volume and cover |η| < 3. Muons are measured in gasionization detectors embedded in the steel flux-return yoke of the solenoid with a coverage of |η| < 2.4.
Events are reconstructed using a particle-flow technique [28,29] which combines information from all CMS subdetectors to identify and reconstruct in an optimal way the individual particle candidates (charged hadrons, neutral hadrons, electrons, muons, and photons) in each event. These particle candidates are clustered into jets using the anti-k T algorithm [30] as implemented in the FastJet package [31] with a size parameter R = 0.5. Jet energy scale corrections [32] derived from data and Monte Carlo (MC) simulation are applied to account for the response function of the calorimeters for hadronic showers. The CMS trigger system uses a two-tiered system comprising a level-1 trigger (L1) and a high-level trigger (HLT) to select physics events of interest for further analysis. The selection criteria used in this analysis are the inclusive single-jet triggers, which require one L1 jet and one HLT jet with various thresholds on the jet p T , as well as trigger paths with thresholds on the dijet mass and scalar sum of the jet p T . The p T of jets is corrected for the response of the detector at both L1 and the HLT. The efficiency of each singlejet trigger is measured as a function of dijet mass M jj using events selected by a lower-threshold trigger.
Events with at least two reconstructed jets are selected from an inclusive jet sample and the two highest-p T jets are used to measure the dijet angular distributions for different ranges in M jj . In units of TeV the M jj ranges are (1.9, 2.4), (2.4, 3.0), (3.0, 3.6), (3.6, 4.2), and >4.2. The lowest M jj range is chosen such that the trigger efficiency exceeds 99% in all bins of χ dijet considered in this analysis. The two highest M jj ranges were chosen to maximize the expected sensitivity to the new physics signals considered. Events with spurious jets from noise and noncollision backgrounds are rejected by applying loose quality criteria [33] to jet properties and requiring a reconstructed primary vertex within ±24 cm of the detector center along the beam line and within 2 cm of the detector center in the plane transverse to the beam. The main primary vertex is defined as the one with the largest summed p 2 T of its associated tracks. The phase space for this analysis is defined by selecting events with χ dijet < 16 and y boost < 1.11, where y boost = 1 2 |y 1 + y 2 |. This choice of values restricts the two jets within |y| < 2.5. The highest value of M jj observed in this data sample is 5.2 TeV.

Cross section unfolding and uncertainties
The measured χ dijet distributions, defined as (1/σ dijet )(dσ dijet / dχ dijet ), are corrected for migration effects due to the finite jet energy and position resolutions of the detector. Fluctuations in the jet response cause event migrations in χ dijet as well as in dijet mass. Therefore, a two-dimensional unfolding in these variables is performed using the D'Agostini method [34] as implemented in the RooUnfold package [35]. The unfolding corrections are determined from a response matrix that maps the true M jj and χ dijet distributions onto the measured ones. This matrix is derived using particle-level jets from herwig++ version 2.5.0 [36,37] with the tune of version 2.4. The jets are smeared in p T with a double-sided Crystal-Ball parameterization [38] of the response, which takes into account the full jet energy response including non-Gaussian tails.
The unfolding correction factors as a function of χ dijet vary from less than 3% in the lowest M jj range to less than 20% in the highest M jj range.
The main experimental systematic uncertainties in this analysis are caused by the jet energy scale, the jet energy resolution, and the unfolding modeling and detector simulation. The overall jet energy scale uncertainty varies between 1% and 2% and has a dependence on pseudorapidity of less than 1% per unit of η [32].
The jet energy scale uncertainty is divided into 21 uncorrelated sources [39]. The effect of each source is propagated to the dijet angular distributions and then summed in quadrature to take into account uncorrelated p T -and η-dependent sources that could cancel if varied simultaneously. The resulting uncertainty in the χ dijet distributions due to the jet energy scale uncertainties is found to be less than 2.0% (2.6%) at low (high) M jj over all χ dijet bins, and the maximum uncertainty in a given M jj bin is typically found to be in the lowest χ dijet bin.
The jet energy resolution is known to within 10% of its value in the phase space considered in this analysis [32]. The systematic uncertainty in the χ dijet distributions due to this effect was evaluated by varying the width of the Gaussian core of the Crystal-Ball parameterization of the response by ±10% and comparing the resultant unfolding corrections before and after these changes. The resulting uncertainty in the χ dijet distributions is 0.5% (1.5%) in the lowest (highest) M jj range. In addition, a systematic uncertainty in the tails of the jet response function is evaluated by determining a correction factor using a Gaussian ansatz [32] rather than the double-sided Crystal-Ball (Gaussian with tails) function to parameterize the response. Since the Gaussian assumption corresponds to the extreme case of the complete absence of a tail, the associated uncertainty has been taken to be 50% of the difference between this correction and the nominal correction based on the Crystal-Ball function. This covers the uncertainty in the understanding of the tails from jet resolution tail measurements. The size of this uncertainty varies from less than 1% in the lowest M jj range to less than 13% in the highest M jj range.
A systematic uncertainty in the unfolding due to the use of a parameterized model of the jet p T and position resolutions to determine the unfolding correction factors is estimated by comparing the smeared χ dijet distributions to the ones from a detailed simulation of the CMS detector using Geant4 [40]. This uncertainty is found to be less than 0.4% (5%) in the lowest (highest) M jj range. A further systematic uncertainty in the unfolding for the modeling of the dijet spectra with herwig++ [0.1% (1.2%) in the lowest (highest) M jj range], is estimated from a comparison of the unfolding corrections from herwig++ with those obtained from pythia 8 version 8.165 [41] with tune 4C [42].
The uncertainty from additional interactions in the same proton bunch crossing as the interaction of interest, called pileup, is determined in simulation by varying the minimum bias cross section within its measured uncertainty of 6% [43]. No significant effect is observed. Though in the statistical analysis of the data the uncertainties are treated separately, for display in tables and figures, the total experimental systematic uncertainty in the χ dijet distributions is calculated as the quadratic sum of the contributions due to the uncertainties in the jet energy calibration, jet p T resolution, and unfolding correction. The total uncertainty including statistical uncertainties is less than 2.5% (49%) for the lowest (highest) M jj range. Experimental uncertainties are evaluated for both the QCD background and signal predictions, however, the resulting uncertainties do not differ significantly.

Theoretical predictions
The normalized dijet angular distributions are compared to the predictions of perturbative QCD. The NLO calculation is provided by NLOJet++ version 4.1.3 [44,45] within the fastNLO framework version 2 [46,47]. The factorization (μ F ) and renormalization (μ R ) scales are defined to be the average p T of the two jets, p T1,2 . Electroweak corrections for dijet production have been derived in Ref. [48], the authors of which provided us with the corresponding corrections for the χ dijet distributions. These corrections change the predictions of the normalized χ dijet distributions by up to 4% (14%) at low (high) M jj . Since fast re-evaluation techniques for different choices of PDFs or scales are not yet available for the electroweak correction part of the theory, the factors have been applied here without additional uncertainties. A figure showing these corrections can be found in Appendix A. The impact of non-perturbative effects such as hadronization and multiple parton interactions is estimated using pythia 8 and herwig++. These effects are found to be negligible.
The is determined from the 22 uncertainty eigenvectors of CT10 [50] using the procedure described in Ref. [50], and is found to be less than 0.6% (1.0%) at low (high) M jj . A summary of the systematic uncertainties in the theoretical predictions is given in Table 1 together with the experimental ones. In the highest M jj range, the dominant experimental contribution is the statistical uncertainty while the dominant theoretical contribution is the QCD scale uncertainty.
For calculating the CI terms as well as the interference between the CI terms and QCD terms at LO and NLO in QCD the cijet program version 1.0 [51] has been employed. The CI models at LO are cross-checked with the implementation in pythia 8 and found to be consistent. The ADD predictions are calculated with pythia 8.

Results
In Fig. 1 the measured χ dijet distributions, corrected for instrumental effects and normalized by their respective event counts, for all M jj ranges, are compared to theoretical predictions. The data are well described by NLO calculations that incorporate EW corrections. No significant deviation from the SM predictions is observed. The distributions are also compared to predictions for  The measured χ dijet distributions are used to determine exclusion limits on CI models that include full NLO QCD corrections to dijet production induced by contact interactions calculated with cijet. Limits are also extracted for CI models calculated at LO with cijet and ADD models implemented in pythia 8. To take into account the NLO QCD and EW corrections in these LO models, the prediction is obtained where the QCD terms are corrected to NLO with EW corrections while the CI (ADD) terms are calculated at LO. The variations due to theoretical uncertainties associated with scales and PDFs are applied only to the QCD terms of the prediction, thereby treating the effective new physics terms as fixed benchmark terms.
In Fig. 2, the χ dijet distributions for the two highest M jj ranges are compared to various CI and ADD models. Only the two highest M jj ranges are used to determine limits of CI and ADD model parameters since the added sensitivity from the lower M jj ranges is negligible. We quantify the significance of an NP signal with respect to the SM-only hypothesis by means of the likelihood for the SMonly, L SM , and the likelihood for the SM with new physics, L SM+NP .
8 . 6 ± 1.2 Table 3 Observed and expected exclusion limits at 95% CL for various ADD models in LO. The uncertainties in the expected limits considering statistical and systematic effects for the SM-only hypothesis are also given. We note that there is an observed difference between the NLO QCD calculations with EW corrections and the NLO QCD-only hypothesis in the above defined likelihood ratio, which corresponds to a significance of 1.1 standard deviation.

Model
The agreement of the data with the SM-only hypothesis is estimated by calculating p SM (q ≤ q obs ) for each M jj bin separately.
The largest difference is found in the M jj range 3.0-3.6 TeV with a probability of 17% to obtain a deviation from the SM-only hypothesis larger than the observed, corresponding to a significance of 1.4 standard deviations. Including the two highest M jj bins in the likelihood reduces this significance to 0.9 standard deviations, corresponding to a probability of 39%. A modified-frequentist approach [53,54,52] is used to set exclusion limits on the scale . Limits on the SM + NP models are set based on the quantity CL s = p SM+NP (q ≥ q obs )/(1 − p SM (q ≤ q obs )), which is required to be 0.05 for a 95% confidence level (CL) exclusion. The observed and expected exclusion limits on different CI and ADD models obtained in this analysis at 95% CL are listed in Tables 2 and 3 respectively. Note that the CI predictions with exact NLO QCD corrections show a smaller enhancement at low χ dijet relative to QCD than do the corresponding LO CI predictions, as described in detail in Ref. [55], and therefore result in less stringent limits. These results are also summarized in Fig. 3. The limits on M S for the different n ED (n ED ≥ 2) directly follow from the limit for T . As a cross check, the limits for the CI scale + LL/R R are also determined for the case in which the data are not corrected for detector effects and instead the simulation predictions are convoluted with the detector resolutions. The extracted limits are found to agree with the quoted ones within 1%. We also quantify the effect of the inclusion of EW corrections in the QCD prediction on the + LL/R R (LO) observed limit, which would be reduced from 10.3 TeV to 9.8 TeV if EW corrections were neglected.

Summary
Normalized dijet angular distributions have been measured with the CMS detector over a wide range of dijet invariant masses. No significant deviation from the standard model predictions is observed. Lower limits are set on the contact interaction scale for a variety of quark compositeness models that include NLO QCD corrections and on the cutoff scale for the ADD models with extra dimensions. The 95% confidence level lower limits on the contact interaction scale are in the range 8.8-15.2 TeV. The improved description of the data resulting from the inclusion of the electroweak corrections yields approximately 5% higher limits. The lower limits on the cutoff scales in the ADD models, T (GRW) and M S (HLZ), are in the range 5.9-8.4 TeV. These results represent the most stringent set of limits on contact interaction scale, modelled at NLO, and the best limits on the benchmark ADD model to date.

Acknowledgements
We would like to thank S. Dittmaier and A. Huss for providing us with the electroweak correction factors. We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centers and personnel of the Worldwide LHC Computing Grid for delivering so effectively the comput-  Fig. A.4 shows the EW corrections to the dijet angular distributions. The corrections are based on the same calculations and tools used to derive the EW corrections to inclusive jet and dijet production cross sections published in Ref. [48]. The authors of Ref. [48] have provided the exact numbers to be applied to the dijet angular distribution as presented in this paper. The EW corrections change the predictions of the normalized χ dijet distributions by up to 4% (14%) at low (high) M jj .