Minimal and non-minimal Universal Extra Dimension models in the light of LHC data at 13 TeV

Universal Extra Dimension (UED) is a well-motivated and well-studied scenario. One of the main motivations is the presence of a dark matter (DM) candidate namely, the lightest level-1 Kaluza-Klein (KK) particle (LKP), in the particle spectrum of UED. The minimal version of UED (mUED) scenario is highly predictive with only two parameters namely, the radius of compactification and cut-off scale, to determine the phenomenology. Therefore, stringent constraint results from the WMAP/PLANCK measurement of DM relic density (RD) of the universe. The production and decays of level-1 quarks and gluons in UED scenarios give rise to multijet final states at the Large Hadron Collider (LHC) experiment. We study the ATLAS search for multijet plus missing transverse energy signatures at the LHC with 13 TeV center of mass energy and 139 inverse femtobarn integrated luminosity. In view of the fact that the DM RD allowed part of mUED parameter-space has already been ruled out by the ATLAS multijet search, we move on to a less restricted version of UED namely, the non-minimal UED (nmUED), with non-vanishing boundary-localized terms (BLTs). The presence of BLTs significantly alters the dark matter as well as the collider phenomenology of nmUED. We obtain stringent bounds on the BLT parameters from the ATLAS multijet plus missing transverse energy search.

, a cosmologically viable dark matter candidate [14,15], a unification scale at a few TeV [16], an explanation for the long life-time of proton [17] and the number of fermion generations to be an integral multiple of three [18], and above all, a chance to probe the model at collider experiments with its promising signatures .
In this work, we study the collider phenomenology of a couple of simple variants of UED scenarios which are characterized by a single flat universal (accessible to all the SM particles) extra dimension (y), compactified on a S 1 /Z 2 orbifold with radius R (oneUED scenarios). In particular, we consider both the minimal (mUED) and non-minimal (nmUED) versions of the oneUED model. The particle spectrum of oneUED contains infinite towers of Kaluza-Klein (KK) modes (identified by an integer n, called the KK-number) for each of the SM fields. The zero modes are identified as the corresponding SM particles. From a 4-dimensional perspective, the conservation of the momentum along fifth direction implies conservation of the KK-number. However, the additional Z 2 symmetry (y ↔ −y), which is required to obtain chiral structure of the SM fermions, breaks the translational invariance along the 5 th dimension. As a result, KK-number conservation breaks down at loop-level, leaving behind only a conserved KK-parity, defined as (−1) n , which is an automatic outcome of the S 1 /Z 2 orbifolding and has several interesting consequences. For example, KK-parity ensures the stability of the lightest KK particle (LKP), allows only the pair productions of level-1 KK particles at the collider, and prohibits KK modes from affecting electroweak (EW) precision observables at tree-level.
OneUED, being a higher dimensional theory, is non-renormalizable and hence, should be treated as an effective theory valid upto a cut-off scale Λ. Apart from the usual the SM kinetic, Yukawa and scalar potential terms for the 5D fields, the oneUED Lagrangian also includes additional SM gauge and Lorentz invariant terms like, the vector-like bulk mass terms [30,31,[50][51][52] for the 5D fermions. Furthermore, one can, in principle, also add kinetic (and Yukawa) terms 1 for all the 5D fields at the orbifold fixed points, i.e., the boundaries of the bulk and the brane [53,54]. The parameters associated with the BLTs are not a priory known quantities (since they are related to ultra-violet (UV) completion for such scenarios) and thus, would serve as extra free parameters of the theory. In the minimal version of oneUED [55], all BLTs are assumed to vanish at the cut-off scale (Λ) and are radiatively generated at the low scale which ultimately appear as corrections to the masses of the KK particles. Therefore, in addition to the SM parameters, the phenomenology of mUED is determined by only two additional parameters namely, the radius of compactification, R and the cut-off scale, Λ. Hence, it's predictions are very specific and easily testable at different high energy physics (HEP) experiments. As a result, verdicts from different non-collider and collider experiments, like the LHC and various DM experiments can easily rule out mUED. It has already been shown in the literature [56][57][58] that the parts of R −1 -Λ plane of mUED which are consistent with the WMAP/PLANCK [59,60] observed relic density (RD) data, are on the verge of being excluded from the direct searches for the KK particles at the LHC. This motivates us to move on to a less stricter version of oneUED with more parameters namely, the BLT parameters. This is where the non-minimal UED comes into the picture. In nmUED, BLT parameters give rise to modifications in the KK particle masses as well as interactions [61][62][63][64][65][66]. The effect of such alterations is rather dramatic at the colliders as well as at the dark matter experiments.
In this article, we have studied the collider signatures of mUED and nmUED in the context of recent LHC searches for beyond the SM (BSM) scenarios. The level-1 KK particles are expected to be in the mass range of few hundreds of GeV to few TeVs. Being strongly interacting, the level-1 KK quarks (both the singlet, q (1) , and doublet, Q (1) ), and gluons, G (1) , can be copiously produced in pairs at the LHC. These, subsequently, decay into the SM particles and the LKP via cascades involving other level-1 KK particles. Therefore, the pair productions of the level-1 KK particles give rise to generic multijet + multilepton + missing transverse energy 2 (E T / ) signatures at the LHC. Now turning on to the actual ambit of our work, it is worth mentioning that the LHC collaborations have so far performed dedicated analysis in the multijet as well as multilepton channels in the context of supersymmetric and other BSM scenarios. In particular, the ATLAS collaboration have studied the signatures of gluino and/or squark (supersymmetric partner of gluon and quark, respectively) pair productions in multijet plus missing transverse energy channels at the LHC at 13 TeV center of mass energy with 139 fb −1 integrated luminosity. Non-observation of expected signal (over background) results in strong constraints on many sparticle (supersymmetric particle) masses. One can always perform the ditto analysis as done by the ATLAS for any model to constrain the parameter space of that particular model from the LHC data. In this article, we follow this well trodden path and revisit the status of the mUED and nmUED scenarios after the LHC run-II data.
The paper is organized as follows. In the following section 2, we describe the ATLAS multijet analysis strategy and validate our methodology by reproducing the ATLAS results. Next, we first look for the status of the minimal version of Universal Extra Dimension under the lens of LHC data collected at 13 TeV. In Section 4, we describe the non-minimal UED model. Section 4.1 comprises of the LHC phenomenology of the nmUED model followed by the concluding remarks in section 5.

Collider Phenomenology
We have closely followed the latest ATLAS n j + E T / [83] search with 139 fb −1 integrated luminosity data at the 13 TeV LHC. Although, the analysis in Ref. [83] is dedicated for the search of squarks and gluinos in the context of supersymmetry, the model independent 95% CL upper limits on the visible n j + E T / cross-sections ( σ 95 obs ) for different signal regions (SRs), can be used to constrain the parameter space of other BSM scenarios which also give rise to similar final states. A brief description about the reconstruction of various objects (jets, leptons etc.), event selection criteria, definition of different SRs are presented in the following. Object Reconstruction: Jet candidates have been reconstructed using anti-k T [84] algorithm implemented in FastJet [85] with jet radius parameter 0.4. Reconstructed jets with p j T > 20 GeV and |η j | < 2.8 are considered for further analysis. Electron (muon) candidates are required to have p l T > 7(6) GeV and within |η l | < 2.47(2.7). Next, the overlapping between identified leptons and jets in the final state are resolved by discarding any electron/muon candidate lying within a distance ∆R < min(0.4, 0.04 + 10GeV p e/µ T ) of any reconstructed jet candidate. Missing transverse momentum vector p mis T (with magnitude E T / ) is reconstructed using all remaining visible entities, viz. jets, leptons, photons and all calorimeter clusters not associated to such objects. For a signal having n j jets, effective mass m eff is defined as the scalar sum of E T / and the transverse momenta of all the n j jet candidates having p T > 50 GeV. Whereas, H T is calculated as the scalar sum of transverse momentum of all jets with p T > 50 GeV and |η| < 2.8. After reconstructing different physics objects, events are pre-selected for further analysis. As SUSY and other BSM scenarios are expected to reside in the high mass scale region, events are preselected accordingly and thus, in the process, unnecessary events are rejected. The preselection criteria is summarized below. Preselection criteria: Events containing a leading jet with p j 1 T > 200 GeV and atleast a second jet with p j 2 T > 50 GeV are considered for further analysis. Only zero lepton events are considered i.e., events with an isolated electron (muon) with p T > 7(6) GeV are vetoed. Events are required to have sufficiently large missing transverse energy (E T / > 300 GeV) and effective mass (m eff > 800 GeV) in order to be considered for further analysis. Events failing to satisfy ∆φ(j 1,2 , p mis T ) min > 0.4 3 are also rejected. Event Selection and Signal Regions (SRs): To make the search process exhaustive, the ATLAS collaboration [83] have defined various signal regions (SRs). Each signal region is designed to study a particular region of parameter space and hence, the signal regions are made mutually exclusive as far as possible. Number of jets sets a powerful criterion in achieving this. For instance, in the context of supersymmetry, a pair of gluinos typically give more number of jets than squarks in their usual decay modes. Thus, binning different numbers for jets is the first step for segmenting SRs. Moreover, mass splitting of the parent and daughter determines the kinematics of the events. Thus, in addition, specific cuts on different kinematic variables (dubbed as multi-bin search) have also been applied to target specific mass hierarchies among different BSM particles. In Ref. [83], ATLAS collaboration have defined ten signal regions for their model independent study of multijet plus missing energy signatures at the LHC running at 13 TeV center of mass energy with 139 fb −1 integrated luminosity. The signal regions are defined by varying numbers of jet multiplicities (between 2-6) along with the minimum value of the effective mass m eff . In view of the high level of agreement between predicted background and observed yield in all signal regions, a model independent 95% CL upper limit is set on the visible BSM contribution to the multijet cross-section ( σ 95 obs ) for each signal region. In our analysis, we have used the ATLAS derived bounds on σ 95 obs in each signal region to constrain the parameter space of mUED and nmUED. For the sake of completeness, we have summarized the definitions of few relevant signal regions in Table 1 Table 1: Selection criteria which have been used to define model independent search regions with jet multiplicities two and four are shown. The aplanarity variable is defined as A = 3 2 λ 3 , where λ 3 is the smallest eigen value of the normalized momentum tensor of the jets (see Ref. [86] for detail). The model independent 95% CL upper limits derived by the ATLAS [83] on the visible BSM contributions to the multijet cross-sections ( σ 95 obs ) for the above signal regions are also provided. The predictions for mUED and nmUED scenarios for three selected benchmark points (BPs), listed in Table 3 and Table 4, respectively, are also presented.

Validation
Since we are following the ATLAS multijet analysis, validation of our analysis against the ATLAS results is very important. In Table 17 of Ref. [83], ATLAS collaboration has presented cut-flow table for their simulated gluino pair production events at √ s =13 TeV for gluino mass mg = 2200 GeV and the lightest neutralino (the spin-half supersymmetric partners of the SM EW bosons) mass mχ0 1 = 600 GeV. For validation purpose, we have also generated gluino pairs up to two extra partons in MG5 aMC@NLO [87] with the NNPDF23LO [88] parton distribution functions. Subsequent decays, showering and hadronization are simulated in Pythia 8.2 [89,90]. The CKKW-L merging scheme [91] is employed for matching and merging. Hadronized events are passed into Delphes 3 [92] for object reconstructions and the implementation of cuts. The cut efficiencies supplied by the ATLAS are presented alongside ours in Table 2. The excellent agreement between 3 ∆φ(ji, p mis T )min is the azimuthal angle between the i th jet and missing transverse momentum vector p mis T . Jets are ordered according to their pT hardness (p j 1 T > p j 2 T > .....). 4 For more details, we refer the reader to   [83]. Cut efficiencies resulting from our simulation are presented in the third column for comparison.
the ATLAS analysis and our simulation bolsters our confidence on our method to apply the same for other BSM scenarios like, mUED and nmUED. We must mention that similar exercises have been performed for other signal regions as well and numbers are consistent. However, we do not intend to present them here, but to move on to the actual goal of our study.

mUED after LHC Run-II data
In this section, the mUED model is put to test under the lens of LHC data collected at 13 TeV. As mentioned in the introduction, mUED has one extra spatial dimension (y) compactified on a circle of radius R which signifies the length scale under probe at the LHC. At the tree-level, the masses of KK states for a given KK level are almost degenerate leaving little space for the decay products to get registered at the LHC detector. In mUED, radiative corrections to the masses play a very crucial role to remove the degeneracy. Loop corrections to the KK masses in an orbifolded theory are logarithmically divergent. Since, mUED is an effective theory which remains valid up to certain cut-off scale (Λ), the radiative corrections are proportional to the logarithm of Λ [55,93]. Therefore, the phenomenology of mUED is completely specified by only two parameters: the compactification radius 5 (R) and the cut-off scale 6 (Λ). Low energy observables like muon g − 2 [95,96], flavor changing neutral currents [97][98][99], Z → bb decay [100], the ρ-parameter [101],B → X s γ [102] and 5 The inverse of radius of compactification (R −1 ) determines the overall mass scale of KK particles for a given KK level and hence, determines the production cross-sections of KK particles at the LHC. 6 The cut-off scale (Λ) controls the mass splitting between different KK particles for a given KK level and hence, determines the kinematics of mUED signatures at the colliders. The perturbativity of the U (1) gauge coupling requires that Λ < 40R −1 . It has been argued in Ref. [94] that a much stronger bound arises from the running of the Higgs-boson self-coupling and the stability of the electroweak vacuum. However, the results of Ref. [94] relies on the lowest-order calculations and the inclusion of higher-loops can substantially change these results. Therefore, in our analysis, we varied Λ in the range 3-40R −1 .  on R −1 . Given this upper limit, it is extremely plausible that experiments at the LHC can either discover or rule out mUED which will be the key focus of discussion in the following. In order to discuss the production, decay, and the resulting collider signatures of the KK particles, and to present the numerical results, we have chosen three benchmark points (BPs) listed in Table 3 along with the masses of relevant level-1 KK particles. Being strongly interacting, the level-1 KK gluons G (1) and singlet (q (1) ) as well as doublet (Q (1) ) KK quarks are expected to be copiously pair produced at the LHC at 13 TeV center-of-mass energy. These level-1 KK particles subsequently decay to the SM particles and the LKP via on/off-shell lighter intermediate KK particles. It is important to mention that in the framework of mUED scenario, the mass hierarchies among different level-1 KK particles are determined by the radiative corrections only and hence, are independent of R −1 and Λ. As a result, the decay branching ratios of the level-1 KK particles are also practically independent of R −1 and Λ. As the spectra in Table 3 suggest, G (1) , being the heaviest among the level-1 KK particles, can decay to both singlet (q (1) ) and doublet (Q (1) ) quarks with almost 8 same branching ratios. A singlet level-1 KK quark (q (1) ) decays only to B (1) in association with a SM quark. Similarly, a doublet level-1 KK quark (Q (1) ) decays preferably to W (1)± or Z (1) accompanied by a SM quark. Mass spectra in Table 3 shows that the hadronic decays of the W (1)± are closed kinematically. Therefore, it decays to all three level-1 KK doublet lepton flavors universally (both L (1)± ν and L (0)± ν (1) ). Similarly, Z (1) can decay only to L (1)± l ∓ or ν (1) ν (with branching fractions being determined by the corresponding SM couplings). The level-1 KK leptons finally decay to B (1) and an ordinary (SM) lepton. In all three BPs (in mUED in general), B (1) is the lightest KK particle i.e. the LKP. Therefore, the production and subsequent decays of level-1 KK quarks/gluons at the LHC give rise to a final state consisting of a number of jets and/or leptons and missing transverse momentum. However, the small mass splittings between level-1 KK W ± /Z and leptons as well as level-1 KK leptons and the LKP (see Table 3) would render very soft leptons in the final state. Thus, we concentrate only on the hadronic final states to probe the parameter space of mUED at the LHC at √ s = 13 TeV and 139 fb −1 integrated luminosity data as per the ATLAS strategy [83]. Pair productions of level- m G (1) contour  Table 1) of 13 TeV ATLAS search [83] for multijets +E T / with 139 fb −1 integrated luminosity data. The region left to exclusion lines corresponding to different SRs are ruled out at 95% CL. Level-1 KK gluon mass (m G (1) ) contours are laid over as grey lines along with corresponding masses printed in TeV. The three benchmark points, listed in Table 3, are also shown in filled black dots. The black solid line with green band of 3σ significance surrounding it represents the region that give correct dark matter relic density [58]. The entire region right to the relic density (Ω DM h 2 ) curve is said to be ruled out in view of over-closure of the universe.

BPs
1 KK quarks/gluons are simulated in MG5 aMC@NLO [87] event-generator. The subsequent decays, initial state radiation (ISR), final state radiation (FSR), hadronization etc. are simulated in Pythia 8.2. For the purpose of reconstruction and analysis of the events, we designed our own analysis code with very close proximity to the ATLAS utilized object reconstruction criteria and selection cuts.
The results are summarized in Table 1 and Fig. 1. We present the final exclusion bound in Fig. 1 on the mUED parameter space for each of the SRs listed in Table 1. The region in the R −1 -ΛR plane to the left of a given exclusion curve is ruled out at 95% CL. Fig. 1 also shows level-1 KK gluon mass (in TeV) contours. For large ΛR, the strongest bound comes from 4-jet final state (in particular, SR4j-1.0 signal region) which excludes level-1 KK gluon mass below about 2.37 TeV. Note that the parameter space with lower ΛR 5 seems somewhat less restricted. For small ΛR, the strongest bound of about 2.22 TeV on level-1 KK gluon mass results from SR2j-2.2. The numerical predictions for signal multijet + E T / cross-sections in different SRs are presented in Table 1 for the mUED benchmark points defined in Table 3. BP m 1 represents the part of mUED parameter space characterized by small ΛR ∼ 3 and hence, a highly degenerate mass spectra for level-1 KK particles. As a result, the decays of level-1 quarks/gluons give rise to very soft jets at the LHC. For such a scenario, a mono-jet like final state comprising a single high p T jet, resulting primarily from initial state radiation, accompanied by missing transverse energy is a promising channel. Table 1 shows that BP m 1 is excluded from SR2j-2.2 which is indeed a mono-jet like [109,110] signal region. BP m 2 (BP m 3 ) corresponds to large ΛR ∼ 40(30) and hence, relatively larger mass splittings between level-1 KK particles. At the parton level, the pair (associated) production of level-1 KK gluons (in association with level-1 KK quarks) and their subsequent decays give rise to four (three) hard jets. Additional jets also arise from initial state radiation. Therefore, for large ΛR regions, four-jet channels (in particular, SR4j-1.0) are the most promising ones for estimating the bound as can be seen from Table 1 as well as from Fig. 1.
Although, we do not claim to have performed any dark matter related analysis, for the sake of completeness, we have shown the relic density bound on R −1 -ΛR plane from Ref. [58]. Potential reason for its inclusion is that the bound from dark matter abundance appears to be the most severe one and strips off a large chunk of parameter space. In Fig. 1 the narrow green strip centered around solid black line shows the parameter region with correct dark matter relic density. The band signifies the 3σ tolerance level. The parameter space on the left of the RD line results into relic densities which are smaller than the RD observed by WMAP/PLANCK. Therefore, this region is allowed in the sense that one can always concoct scenarios with multi-component dark matter in order to evade such strict constrain from relic abundance. However, the entire region right to the relic density curve in Fig. 1 corresponds to relic densities larger than the WMAP/PLANCK result and hence, is ruled out. Therefore, we can conclude from Fig. 1 that the region of R −1 -ΛR plane, which is consistent with WMAP/PLANCK RD data, has already been ruled out by the ATLAS multijet + E T / searches at 13 TeV LHC with 139 fb −1 integrated luminosity. Hence, we shift our focus on the non-minimal UED (nmUED) where an enhanced number of parameters offer rich phenomenology. Next section is slotted for discussion on the theoretical set up of the model.

nmUED : Model description
The assumption of vanishing boundary terms in mUED is somewhat unnatural, since they can anyway be generated at the loop-level. Moreover, these boundary-localized terms (BLT) obey all the symmetries of the model [111]. The non-minimal version of the model (nmUED) takes these BLTs into account. Every boundary-localized term is associated with a parameter, which we generally denote by r. The presence of these unknown BLT parameters drastically alters the nmUED mass-spectrum compared to the mUED one. Moreover, the interaction vertices of involving various non-zero KK-modes are non-trivially modified by a multiplicative factor known as, overlap integrals. However, before going into the collider phenomenology of nmUED scenario, we briefly introduce the theoretical set-up of nmUED scenario.
The most general nmUED action is required to be invariant under the gauge symmetry of the SM i.e., invariant under SU (3) C ×SU (2) W ×U (1) Y , as well as the Lorentz symmetry in 4D, and can be written as, where, the individual parts of the full nmUED action, S nmUED , are discussed in the follow-ing. The gauge part of the action is given by, where, G a M N , W i M N , B M N stand for 5D field-strength tensors corresponding to the SU (3) C , SU (2) W and U (1) Y gauge fields, respectively. The symbols M, N run for 0, 1, 2, 3, 5 and the Greek indices run for 0, 1, 2, 3. The actions clearly consist of two parts. The first parts are the usual gauge kinetic term in 5D. The second parts are the brane-(also called boundary) localized kinetic terms (BLKTs). These terms appear only at the boundaries of the brane and the bulk, as can be seen by the presence of delta functions. We consider boundary parameters at the two orbifold-fixed points to be the same which ensures the conservation of KK parity. Now we briefly describe the fermionic parts of the action which can be written as, where, 5D quark (lepton) doublet and singlets are denoted by Q (L) and U/D (E), respectively, j = 1, 2, 3 is the generation index, Γ M = (γ µ , iγ 5 ) denotes γ-matrices in 5D and D M is the gauge covariant derivative in 5D. Finally, the action corresponding to the 5D Higgs field is given by, where, Φ is the 5D Higgs. µ 5 and λ 5 represent the 5D bulk Higgs mass parameter and scalar self-coupling, respectively. The BLKT parameter for scalar field is r Φ ; µ B and λ B are the boundary-localized Higgs mass parameter and the scalar quartic coupling, respectively. We must mention that all the BLT parameters (r i where i stands for G, W , B, Q, L and Φ fields) are dimensionful parameters. However, we express our results in section 4.1 in terms of scaled BLT parameters R i = r i R −1 as is customary.
The nmUED action written in the previous paragraph contains the information of the full theory in 5D. 5D fields can be expanded into x µ and y dependent parts where x µ is the usual 4D space-time coordinates and y is the extra dimension coordinate which is compactified on a S 1 /Z 2 orbifold. Once the mode expansions are fed into the actions and the extra dimensional coordinate, y, is integrated out, we obtain a 4D effective theory involving the SM particles as well as their KK modes. For example, the mode expansions for the 5D gluon can be written as, where, C G = cos (m G (n) πR/2) and S G = sin (m G (n) πR/2). Note that the above expansion together with the boundary conditions give rise to the following transcendental equations: for n odd (4.6) In the framework of nmUED, the mass of the n-th level KK gluon (m G (n) ) is obtained by solving these transcendental equations. The normalization of the wave function (N G (n) ) is given by, Such KK decomposition and transcendental equations are common for all the 5D fields. Therefore, in nmUED scenario, the masses for the KK modes of other SM particles are also given by the solution of transcendental equations similar to those in Eq. 4.6 with appropriate BLT parameters. It is interesting to note that the phenomenology of the level-1 electroweak gauge sector of nmUED is significantly different from that of mUED since the masses and mixings of the level-1 KK EW gauge bosons in nmUED non-trivially depend on the BLT parameters r W , r B and r Φ . In the context of mUED, the masses of the lightest (i.e. the LKP which is the DM candidate in the theory) and next-to-lightest level-1 KK gauge boson are determined by the radiative corrections. In addition, the extent of mixing between the level-1 U (1) Y and SU (2) W KK gauge bosons is minuscule, unless R −1 is very small. Therefore, in mUED, the lightest and next-to-lightest level-1 KK gauge bosons are, for all practical purposes, essentially the level-1 excitations of U (1) Y and SU (2) W gauge bosons, respectively. However, in presence of the various overlap integrals involving the gauge and scalar BLT parameters, the mixing between the level-1 U (1) Y and SU (2) W gauge bosons could be large in the framework of nmUED. Moreover, depending on the choice of r W , r B and r Φ , the LKP in nmUED could be either a level-1 excitation of U (1) Y gauge boson, or a level-1 excitation of SU (2) W gauge boson. These facts, in turn, have profound implications for the dark matter phenomenology. Note that in mUED, due to little freedom available for determining the mass spectrum and mixing, the observed value of dark matter RD provides a stringent upper bound on R −1 which essentially rules out the model at the LHC. However, the additional parameters r W , r B and r Φ in nmUED play a crucial role to lift the RD upper bound on R −1 . It has been shown in Ref. [71] that with proper choice of r W and r B , larger values of R −1 is possible without conflicting with the measured value of dark matter RD. The freedom in setting the mass spectrum of level-1 KK particles at required value also helps specific co-annihilation channels to contribute more and thus, obtain the required RD. We intend to address these issues related to the dark matter phenomenology of nmUED in a future article. In the present article, we focused on the collider bounds on the masses of level-1 KK quarks and gluons in the framework of nmUED.
Before going into the collider phenomenology of nmUED, it is important to mention that the couplings, involving the zero-mode and non-zero mode KK particles, are also modified non-trivially by factors known as, the overlap integrals. These coupling modifications appear once we plug in the KK expansions in 5D Lagrangian and integrate over the extra dimensional coordinate, y. Note that a generic interaction of a level-l gauge boson (V (l) ) with a pair of level-m and k fermion-antifermion (Ψ (m) (x) andΨ (k) (x)) results from the following term in the 5D action after compactifying the extra dimensional coordinate y: where,g is the corresponding gauge coupling in 5D. The connection between 5D gauge couplingg and its 4D counterpart is given by g =g/ √ r V + π R, where r V is the corresponding BLT parameter for the 5D gauge boson V M . Note that the integration over the extra dimensional coordinate, y, is non-zero only for certain combinations of the KK numbers (k, l, m) and hence, acts as a selection rule known as, KK number conservation, for interactions involving different KK level particles. In mUED, the integration over the extra dimensional coordinate in the above equation is either one (for KK number conserving interactions) or zero (for KK number violating interactions) depending on the choice of (k, l, m). However, the presence of BLTs in nmUED result into a y profile of KK excitations which is different from the mUED case and hence, gives rise to non-trivial overlap integral. Depending on the values of the BLT parameters, the overlap integrals can enhance or reduce a particular coupling and thereby, influence the phenomenology of the model. In Fig. 2, the modification factors for the gauge coupling involving a level-1 KK gluon, a level-1 KK quark and a SM quark have been plotted against the gluon and quark scaled 9 BLT parameters. Fig. 2 shows significant deviation from unity in different parts of R G -R Q plane. It can be deduced from Fig. 2 that in certain parts of R G -R Q plane, one could obtain an enhancement (suppression) as large as 13% (36%) in the interaction strength of Q (0) G (1) Q (1) vertex compared to the interaction strength of pure QCD vertex.

Collider Phenomenology
After discussing the nmUED model, mass spectrum and coupling modifications, we are now equipped enough to study its collider phenomenology and impose bounds from the ATLAS search for multijet plus E T / final states. However, before delving into the ATLAS analysis, it is important to discuss the productions of different KK particles and their subsequent decays in the framework of nmUED. The LHC being a proton-proton collider, we only consider   the QCD pair productions of level-1 KK quarks/gluons in our analysis. Unlike mUED 10 , the nmUED QCD pair production cross-sections of level-1 KK particles are determined by radius of compactification as well as the BLT parameters for the quarks and gluons. The inverse of radius of compactification sets the overall mass scale for the level-1 KK particles in nmUED, over which R G and R Q fix the masses of the KK gluons and KK quarks, respectively. In addition, the BLT parameters also govern the strength of interactions involving the SM and level-1 KK particles which are crucial for the productions as well as decays of the level-1 KK particles. For example, the Q (0) G (1) Q (1) coupling (the dependence of which on R G and R Q is shown in Fig. 2) appears in all the relevant QCD production of level-1 KK particles at the LHC namely, To illustrate the dependence of QCD productions of level-1 KK particles on R Q and R G , we have presented the pair/associated production cross-sections of level-1 KK gluon and pair productions of level-1 KK quarks in Fig. 3 and 4, respectively. The left panel of   TeV 11 . The dominant contribution 12 to the G (1) G (1) production (σ(pp → G (1) G (1) )) at the LHC results from the gluon-gluon initiated process with a level-1 KK gluon in the t(u)-channel. The vertices involved in the Feynman diagrams of gg → G (1) G (1) are purely QCD vertices which do not get modified and hence, σ(gg → G (1) G (1) ) depends only on m G (1) . However, some of the Feynman diagrams (in particular, the t(u) channel level-1 KK quark exchange diagrams) for the subdominant qq → G (1) G (1) production channel involve Q (0) G (1) Q (1) vertex which gets modified. Therefore, the variation of σ(pp → G (1) G (1) ) in Fig. 3 (left panel) results from the R G and R Q dependence of quark-antiquark initiated contribution to the total cross-section. An important fact is that the two Feynman diagrams namely, the s-channel gluon exchange diagram and t(u)-channel Q (1) exchange diagram, contributing to the quark-antiquark initiated production of G (1) -pairs, interfere destructively. For a given R G (and hence, fixed R −1 ) in Fig. 2, increasing R Q corresponds to decreasing m Q (1) and hence, stronger destructive interference which tends to decrease the cross-section. On the other hand, the coupling modification factors increase with increasing R Q (see Fig. 2) which tends to increase the cross-section. These two competing 11 In nmUED, the level-1 KK gluon mass is obtained by solving the transcendental equation in Eq. 4.6 and hence, m G (1) depends on both R −1 and RG. For a given value of RG, one can obtain m G (1) = 2 TeV by suitably choosing a value for R −1 . Therefore, the plots in Fig. 3 (as well as the plots in Fig. 4) do not correspond to a particular value of R −1 . To clearly display the dependence of QCD cross-sections on RQ and RG, one needs to minimize the dependence on parton densities and phase space factors and hence, ensures fixed values for the final state particle masses. This motivates us to present the cross-sections in Fig. 3(4) for a fixed m G (1) (m Q (1) ) instead of a fixed R −1 .
factors explain the pattern of σ(pp → G (1) G (1) ) variation on R Q and R G as displayed in Fig. 3 (left panel).
The associated production cross-section (σ(pp → G (1) U (1) )) of a level-1 KK gluon in association with a level-1 up-type KK quark 13 as a function of R G and R Q is presented in Fig. 3 (right panel) for fixed G (1) mass of 2 TeV. Here, the mass of the level-1 up-type KK quark, however, is not constant over the R G -R Q plane. At the LHC, the G (1) U (1) associated production is a quark-gluon initiated process which proceeds via the exchange of a level-1 KK quark or KK gluon in the t(u)-channel. Although, both the Feynman diagrams contributing to σ(pp → G (1) U (1) ) contain coupling which depends on BLT parameters, the large variation of σ(pp → G (1) U (1) ) in Fig. 3 (right panel) mainly occurs due to the variation of the level-1 KK quark mass in the final state. The R G -R Q dependence of the production cross-sections of level-1 KK quark-quark pair (σ(pp → U (1) U (1) )) and KK quark-antiquark pair (σ(pp → U (1)Ū (1) )) are presented in the left panel and right panel, respectively, of Fig. 4 for m U (1) = 2 TeV. In order to generate a fixed level-1 KK quark mass, R −1 needs to be varied with R Q . The variation of R −1 is also depicted as the y 2 -axis in Fig. 4. The dominant contribution to σ(pp → U (1) U (1) ) comes from the quark-quark fusion process at the LHC through a level-1 KK gluon in the t(u)-channel. The resulting variation of σ(pp → U (1) U (1) ) with respect to R G -R Q is shown in Fig. 4 (left panel). On the other hand, σ(pp → U (1)Ū (1) ) receives contributions from quark-antiquark and gluon-gluon initiated processes. While the gluon-gluon initiated channel depends only on m U (1) and hence, independent of R G and R Q for a fixed m U (1) = 2 TeV; the mild variation of σ(pp → U (1)Ū (1) ) over R G -R Q plane (see the right panel of Fig. 4) results from the sub-dominant quark-antiquark initiated process.
After discussing the productions of level-1 KK quarks/gluons, we will now discuss the decays of various level-1 KK particles and the resulting signatures at the LHC. Mass hierarchy among various KK particles plays a crucial role in determining the the decay cascades of level-1 KK quarks/gluons and hence, the topology and kinematics of the final states at the LHC. While, for mUED, the mass hierarchy among KK particles of a given  Table 4: nmUED benchmark points and mass spectra of relevant level-1 KK particles.
level is completely determined by the radiative corrections, the nmUED mass spectrum is determined by the BLT parameters which are free parameters of the theory. For example, R G < R Q would render a mass hierarchy similar to mUED with KK gluon being more massive than KK quarks while R G > R Q results into KK quarks being heavier than the   KK gluons. In order to discuss the decays, and the resulting collider signatures nmUED as well as present the numerical results, we have chosen three benchmark points (BPs) listed in Table 4 along with the masses of relevant level-1 KK particles. The BPs in Table 4 are characterized by R −1 and (R Q , R G ). We have assumed fixed values for the BLT parameters 14 in the EW sector namely, R W , R Φ , R B and R L . We consider R W = −0.02 = R Φ and set R B to zero. This particular choice of the EW BLT parameters gives rise to a LKP which is dominantly the level-1 KK excitation of the U (1) Y gauge boson (B (1) ) with significant mixing with the level-1 KK excitation of the neutral SU (2) L gauge boson (W 3 (1) ). Note that the EW level-1 KK gauge sector is highly degenerate (see Table 4) with the dominantly SU (2) L level-1 KK gauge bosons namely, W ±(1) and Z (1) , being slightly heavier than the LKP. It has been shown in Ref. [71] that such an EW level-1 sector of nmUED enhances the dark matter annihilation/co-annihilation cross-sections and hence, allows larger values of R −1 without conflicting with the WMAP/PLANCK measured value of the RD. We have also fixed the BLT parameters for leptons at R L = −0.01 and scanned over negative 15 values of R Q and R G .
1 KK quarks into a SM quark and G (1) are kinematically forbidden for BP nm 3 , m Q (1) > m W (1) /m B (1) , usually result into a level-1 doublet KK quark decaying to a SM quark in association with a W (1)± / Z (1) / B (1) . Note that for the level-1 singlet KK quarks, the decay into W (1)± is highly suppressed. In the scenario with R G > R Q (see BP nm 1 in Table 4), the level-1 KK quarks, being heavier than G (1) , dominantly decay into a SM quark in association with a level-1 KK gluon. On the contrary, G (1) undergoes a tree-level 3-body decay via an off-shell Q (1) into a SM quark-antiquark pair in association with a level-1 EW boson (W (1)± /Z (1) /B (1) ). The level-1 KK EW bosons subsequently decay into a pair of SM leptons and the LKP. The leptons arising from the decay of W (1)± /Z (1) are usually very soft and hence, often remains invisible at the LHC detectors. Therefore, the pair/associated productions of level-1 KK quarks and KK gluons give rise to multijet in association with large missing transverse energy final states which will be discussed in the following in the context of recent ATLAS search [83] for multijet + missing transverse energy final states at the LHC at 13 TeV center of mass energy and 139 fb −1 integrated luminosity.
The pair and associated productions of level-1 KK quarks and KK gluons are generated in MadGraph 16 . The MadGraph generated events are fed into Pythia 8.2 for simulating decays, ISR, FSR and hadronization. We use our own analysis code for the object reconstructions and computation of nmUED contributions to the multijet + E T / cross-sections in different SRs defined by the ATLAS collaboration [83] (see section 2 and Table 1). The nmUED predictions for the visible cross-sections for the benchmark points (listed in table 4) are presented in Table 1. Fig. 5 and 6 show the visible cross-sections in signal regions SR2j-1.6 (top left panel), SR2j-2.2 (top right panel) and SR4j-1.0 (bottom panel) as a function of R Q and R G for R −1 = 1.8 TeV (Fig. 5) and 1.9 TeV (Fig. 6). Clearly, the reddish cells of Fig. 5 and 6 correspond to nmUED cross-sections which are larger than the ATLAS observed 95% CL upper bound on σ 95 obs (see Table 1) in the respective signal regions and hence, the corresponding parameter points are ruled out.
The complementarity of SRs is clearly visible in Fig. 5. While, SR2j-2.2 and SR2j-1.6 signal regions are more effective to probe the low (represented by BP nm 1 ) and high (represented by BP nm 2 ) R Q regions, respectively; the intermediate (represented by BP nm 3 ) R Q region is susceptible to SR4j-1.0 (see Figs. 5, 6 and Table 1). This particular pattern of effectiveness of different SRs in different parts of R Q -R G plane can be understood from the characteristics of multijet + E T / signatures in different parts of R Q -R G plane. For example, the region characterized by higher values for both R Q and R G (top-right corner of R Q -R G plane and represented by BP nm 2 ) gives rise to a nmUED scenario with nearlydegenerate masses for the level-1 quarks, gluons and the LKP (see Table 4). Although, the pair and associated production cross-sections of G (1) and Q (1) in this region are large, the final state jets are usually too soft to pass the preselection criteria. The production of G (1) and Q (1) in association with a hard ISR jet gives rise to a monojet + E T / signature. It has already been discussed in the context of mUED phenomenology that the selection criteria of SR2j-2.2 is essentially a monojet like selection criteria and hence, effective to probe this part of R Q -R G plane. On the other hand, in the region characterized by low R Q and/or low R G (represented by BP nm   While the pair production of G (1) leads to four hard jets at parton level, two hard jets arise from the pair production of Q (1) . Therefore, this part of R Q -R G plane is susceptible to both SR2j-1.6 and SR4j-1.0. However, our analysis shows that SR2j-1.6 is more efficient to probe this region. While, all parts of R Q -R G plane are ruled out from complementary signal regions for R −1 = 1.8 TeV (see Fig. 5), the ATLAS multijet search can probe only some part of R Q -R G plane for R −1 = 1.9 TeV (see Fig. 6). Our final results are summarized in Fig. 7

Summary and Conclusion
To summarize, we have studied one Universal Extra Dimension scenarios against the dataset recorded by the ATLAS collaboration in proton-proton collisions at a center of mass energy √ s = 13 TeV, corresponding to an integrated luminosity of 139 fb −1 . The phenomenology of the minimal version of UED (mUED) is completely determined by the two new parameters: namely the compactification radius R and the cut-off scale Λ. Our  [GeV] Figure 7: The lower bounds on R −1 are presented as function of R Q and R G . The signal region which leads to a particular lower bound on R −1 for a given R Q and R G is also specified in the figure.
study clearly shows that mUED parameter space is completely ruled out by the ATLAS multijet + E T / analysis together with the dark matter relic density data. Next, we bring in boundary-localized terms (with R G and R Q as BLT parameters for gluon and quark fields, respectively) as an extension of mUED, called non-minimal UED. Introduction of such terms alters the phenomenology substantially. Mass spectrum in nmUED is determined by the transcendental equations coming from the boundary terms. Some interaction vertices are also altered as a result of integration of extra dimensional mode functions of the concerned particles. We have discussed strong production cross sections for the variations of the gluon and quark BLT parameters. We have performed a detailed cut-based analysis emulating the ATLAS multijet + E T / channel. Excluded regions of nmUED parameter space are shown in terms of R −1 , R G and R Q .