KK-parity non-conservation in UED confronts LHC data

Kaluza-Klein (KK) parity can be violated in five-dimensional universal extra dimensional model with boundary-localized (kinetic or mass) terms (BLTs) at the fixed points of $S^1/Z_2$ orbifold. In this framework we study the resonant production of Kaluza-Klein excitations of the neutral electroweak gauge bosons at the LHC and their decay into an electron-positron pair or a muon-antimuon pair. We use the results (first time in our knowledge) given by the LHC experiment to constrain the mass range of the first KK-excitation of the electroweak gauge bosons ($B^1\ \textrm{and} \ W_3^1$). It is interesting to note that the LHC result puts an upper limit on the masses of the $n=1$ KK-leptons for positive values of BLT parameters and depending upon the mass of $\ell^{+}\ell^{-}$ resonance.


I Introduction
Discovery of the Higgs boson at the Large Hadron Collider (LHC) at CERN is a milestone in the success of Standard Model (SM). However, there are still many unanswered questions and unsolved puzzles, ranging from dark matter to the hierarchy problem to the strong-CP problem. But there is no experimental result that can explain such unsolved problems with standard particle physics. Out of various interesting alternatives, supersymmetry (SUSY) and extra dimensional models are the most popular frameworks for going beyond the SM of particle physics. In this work we consider a typical extra-dimensional model where all SM particles can access an extra space-like dimension y. We use the results [1] presented by the ATLAS Collaborations of search for a high-mass resonances decaying into the ℓ + ℓ − (ℓ ≡ e or µ) pair to constrain the parameter space of such a model where the lowest (n = 1) KK-excitations are unstable due to lack of any specified symmetry.
We are interested in a specific framework, called the Universal Extra Dimension (UED) [2] scenario, characterised by a single flat extra space like dimension y which is compactified on a circle S 1 of radius R and has an imposed Z 2 symmetry (y → −y) to accommodate chiral fermions, hence the compactified space is called S 1 /Z 2 orbifold. From a four-dimensional viewpoint, every field will then have an infinite tower of KK-modes, the zero modes being identified as the SM states. In this orbifold a πR amount of translation in y direction leads to a conserved KK-parity given by (−1) n . The conservation of KK-parity ensures that the lightest n = 1 KK-particle called Lightest Kaluza-Klein (LKP) is absolutely stable and hence is a potential dark matter candidate. As the masses of the SM particles being small compared to 1/R, hence this scenario leads to an almost degenerate particle spectrum at each KK-level. This mass degeneracy could be lifted by radiative corrections. Being an extra dimensional theory and hence being non-renormalizable, this can only be an effective theory characterised by a cut-off scale Λ. So at the two fixed points (y = 0 and y = πR) of S 1 /Z 2 orbifold, one can include four-dimensional kinetic and/or mass terms for the KK-states. These terms are also required as counterterms for cut-off dependent loop-induced contributions [3] of the fivedimensional theory. In the minimal Universal Extra-Dimensional Models (mUED) these terms are fixed by requiring that the five-dimensional loop contributions [4,5] are exactly cancelled at the cut-off scale Λ and the boundary values of the corrections, e.g., logarithmic mass corrections of KK-particles, can be taken to be zero at the scale Λ. There are several existing literature [2], [6]- [19] in which we can find how the experimental results constrain the values of the two basic parameters (R and Λ) of mUED theory.
In this work we generate non-conservation of KK-parity 1 by adding unequal boundary terms at the two fixed boundary points. Consequently n = 1 KK-states are no longer stable. Hence the single production of n = 1 KK-states and its subsequent decay into n = 0 states would be possible via this non-conservation of KK-parity. We will utilise this KK-parity-non-conserving coupling of the B 1 (W 1 3 ) to a pair of SM fermions (n = 0 states) [20] to calculate the (resonance) production cross section of B 1 (W 1 3 ) in pp collisions at the LHC (8 TeV) and its subsequent decays to e + e − /µ + µ − , assuming B 1 and W 1 3 to be the lightest KK-particles. Once B 1 and W 1 3 are produced via KK-parity-non-conserving coupling, the KK-parity-conserving decaying mode being kinematically disallowed, thus the B 1 and W 1 3 decay to a pair of zero-mode fermions via the same KK-parity-non-conserving coupling. A search for high-mass resonances based on 8 TeV LHC pp collision data collected by the ATLAS and CMS have been reported in [1] and [21] respectively. Refs.
[1] and [21] present the expected and observed exclusion upper limits on cross section times branching ratio at 95% C.L. for the combined dielectron and dimuon channels for resonance search. In this article we have used the above results to constrain the masses of the n = 1 level KK-fermions and B 1 ( W 1 3 ) of the model. In an earlier article [22] we have reported the production of n = 1 KK-excitation of gluon and its subsequent decay to tt pair at the LHC. Both the production and decay are governed by KK-parity-non-conserving interaction. Constraints have been also derived by comparing the tt cross section with LHC data from CMS [23] and ATLAS [24] Collaborations.
The plan of this article is as follows. At first we present the relevant couplings and masses in the framework of UED with asymmetric boundary localized kinetic terms. We then review the expected ℓ + ℓ − signal from the combined production of the B 1 and W 1 3 at the LHC and their subsequent decay. This is compared with the ATLAS [1] 8 TeV results and the restrictions on the couplings and KK-excitation masses are exhibited. Finally we will summarise the results in section V.

II KK-parity-non-conserving UED in a nutshell
In non-minimal version of five-dimensional UED theory, we put boundary-localized kinetic terms (BLKTs) [20], [25] - [30] at the orbifold fixed points (y = 0 and y = πR). If Ψ L,R are the free fermion fields, zero modes of which are the chiral projections of the SM fermions. In presence of BLKTs, the five-dimensional action can be written as [20], [31]: With σ µ ≡ (I, σ) andσ µ ≡ (I, − σ), σ being the (2 × 2) Pauli matrices. r a f , r b f are the free BLKT parameters which are equal for Ψ L and Ψ R for the purpose of illustration.
The KK-decomposition of five-dimensional fermion fields using two component chiral spinors are introduced as 2 [20], [31]: Using appropriate boundary conditions [20], we can have the solutions for f n L and g n R which are simply denoted by f and g for illustrative purposes.: f n (y) = N n cos(m n y) − r a f m n 2 sin(m n y) , 0 ≤ y < πR, f n (y) = N n cos(m n y) + r a f m n 2 sin(m n y) , −πR ≤ y < 0.
Where the KK-masses m n for n = 0, 1, . . . are solutions of the transcendental equation [26]: The non-trivial wave-functions are combinations of a sine and a cosine function which are different from case of mUED where they are either only sine or cosine function. The departure of wave functions from mUED theory and the fact that the KK-masses are solutions of Eq. (5) rather than just n/R are the key features of this non-minimal Universal Extra Dimensional (nmUED) model.
In our analysis we study the KK-parity-non-conserving UED in two ways. In the first case, we take equal strength of BLKTs (at two fixed point y = 0 and y = πR) for fermion, i.e., r a f = r b f ≡ r f , while the other case has the BLKT at one of the fixed points only: r a f = 0, r b f = 0. In the later situation Eq. (5) becomes: In both cases the mass eigenvalues can be solved from the transcendental equations (Eqs. 5 and 6) using numerical technique.
For small values of r a f R (<< 1) the approximate KK-mass formula becomes (using Eq. 6): It is clear from the above expression that for r a f > 0, the KK-mass diminishes with r a f . This result also holds good when the BLKTs are present at both the boundary points.
N n being the normalisation constant and determined from orthonormality condition [20]: and is given by: for equal strength of boundary terms (r b f = r a f ≡ r f ). And for the other situation when r b f = 0 and we use r a f ≡ r f one has: Now it is evident from the Eqs. 9 and 10 that, for Our concern here only with the zero modes and the n = 1 KK-wave-functions of the fivedimensional fermion fields.
Masses and y-dependent wave functions for the electroweak gauge bosons are very similar to the fermions and can be obtained 3 in a similar manner. We do not repeat these in this article. This can be readily available in [20].
As KK-masses obtained from transcendental equations are similar for fermions and gauge bosons, so we use r a α , r b α to pameterised the strengths of BLKT with α = f (fermions) or V (gauge bosons) for the purpose of discussions. It has been assumed in the following that the KK-quarks are either mass degenerate or heavier than the KK-leptons. However they do not enter in our analysis. For simplicity we have assumed that BLKTs for U(1) and SU(2) gauge bosons are same, so that B 1 and W 1 3 are degenerate in mass. We are only interested in the n = 1 state.
In the inset is shown the variation of M (1) on ∆R α ≡ (r b α − r a α )/R for two different R α . Right panel: M (1) with respect to R α when the BLKT is present only at the y = 0 fixed point. The inset shows a magnified portion of the variation which gives the actual range of R α that is considered later. The insets of both panels shows small variation of M (1) with respect to BLKTs. Here α = f (fermions) and V (gauge bosons).
In Fig. 1 we have shown in plots a dimensionless quantity M (1) ≡ m α (1) R, in the two different cases. The left panel reflects the mass profile for n = 1 KK-excitation when the BLKTs are presents at the two fixed points (y = 0 and y = πR), while the right panel shows the case when the BLKTs are present only at y = 0. In both cases the KK-mass decreases with the increasing values of BLKT parameter. The detailed illustrations of this non-trivial KK-mass dependence has been discussed in a previous article and can be found in [22].
III Interacting coupling of V 1 (B 1 or W 1 3 ) with zeromode fermions Coupling of the states B 1 or W 1 3 to two zero-mode fermions f 0 is given by, Further, the five-dimensional gauge couplings g 5 (G) which appears above is related to the usual coupling g through We denote the zero-mode fermion wave-functions by f 0 L and g 0 R while the KK (n = 1)-gauge boson wave functions are denoted by a 1 depending on the values of chosen BLKTs.
Let us first discuss the case in which BLKTs are presented at both the fixed points. Here we assume for the fermions : r a f = r b f = r f , but for the gauge bosons : r a V = r b V . Using y-dependent wave-functions and proper normalisation [20] we get: which vanishes 4 when ∆R V = 0.
Where M (1) ≡ m V (1) R is the scaled KK-mass, and R f ≡ r f /R, R a V ≡ r a V /R, and R b V ≡ r b V /R are the scaled dimensionless variables defined earlier.
Now we turn to the case which could be considered the most asymmetric one, namely, the BLKT for the fermion and the gauge boson are present only at the y = 0 fixed point. We obtain for this case [20]: and this becomes zero if we put R V = R f . Here 5 R V ≡ r V /R and R f ≡ r f /R.
The Fig. 2 depicts the KK-parity-non-conserving coupling strength in the two different cases.
In the left panel (BLKTs are present at two fixed points y = 0 and y = πR) we plot the square of the coupling for a fixed value 6 of R a V = 10 as a function of R f for several choices of ∆R V . The right panel shows the same thing with respect to R f (BLKTs are present only at the y = 0) for different values of R V . In both cases the KK-parity-non-conserving coupling decreases with the increasing values of fermion BLKT parameters. The detailed analysis of this coupling strength with respect to BLKT parameters can be found in a previous article [22].

IV Production and decay of B 1 (W 1 3 ) via KK-parity-nonconservation
We are now in a position to discuss the main result of this paper. From now onwards for the SM particles we will not explicitly write the KK-number (n = 0) as a superscript. At the LHC 5 As the BLKTs are present at only one fixed point so we use R f and R V with no superscript for fermions and gauge bosons respectively. 6 We have checked that the results are quite similar for the other value of R a V that we consider later.
we study the resonant production of B 1 (W 1 3 ), via the process pp (qq) → B 1 (W 1 3 ) followed by B 1 (W 1 3 ) → ℓ + ℓ − . This results into an ℓ + ℓ − resonance at the B 1 (W 1 3 ) mass. The final state leads to two leptons (e or µ), with invariant mass peaked at m V (1) which is the KK-mass (n = 1) of gauge bosons. It should be noted that both the production and the decay of n = 1 KK-excitations of electroweak gauge bosons are driven by KK-parity-non-conserving couplings which depend on R f , R a V and ∆R V (R f , R V when BLKTs are present at only one fixed point). If in future such a signature is observed at the LHC, then it would be a good channel to measure such KK-parity-non-conserving couplings. Production of B 1 (W 1 3 ) (which we generically denote by V 1 ) in pp collisions is driven qq fusion. A compact form of the production cross section in proton proton collisions can be written as [20]: Here, q i andq i denote a generic quark and the corresponding antiquark of the i-th flavour respectively. f q i p (fq i p ) is the parton distribution function for quark (antiquark) within a proton.
We define τ ≡ m 2 V (1) /S P P , where √ S P P is the proton-proton centre of momentum energy. Γ(V 1 → q iqi ) represents the decay width of V 1 into the quark-antiquark pair and is given by Γ = and Y q R being the weak-hypercharges for the left-and right-chiral quarks) for B 1 and Γ = for the W 1 3 . Here g V 1 qq is the KK-parity-non-conserving coupling of the V 1 with the SM quarks -see Eqs. (13) and (14). We use a parton-level Monte Carlo code with parton distribution functions as parametrised in CTEQ6L [32] for determination of the numerical values of the cross sections. In our analysis we set the pp centre of momentum energy at 8 TeV and the factorisation scales (in the parton distributions) at m V (1) . To obtain the event rate one must multiply the cross sections with appropriate branching ratio 8 of B 1 or W 1 3 into e + e − /µ + µ − . Here we have assumed without any loss of generality that B 1 and W 1 3 are lighter than the n = 1 KK-excitation of the fermions. 7 ATLAS results have been used in this paper as it used a higher accumulated data set. However we have checked that the limits derived from CMS [21] data are almost the same. 8 The branching ratio of B 1 (W 1 3 ) to e + e − and µ + µ − is approximately 30 103 ( 2 21 ).
Which implies that they are the lightest KK-particle and they can decay only to a pair of SM particles via KK-parity-non-conserving coupling-see Eqs. (13) and (14).
At this end, let us comment about the values of the BLKT parameters used in our analysis. The BLKTs imposed in Eq. 1 are not five-dimensional operators in four-dimensional effective theory but some sort of boundary conditions on the respective fields at the orbifold fixed points. The masses (solutions of transcendental equations) and profiles in the y-directions for the fields are consequences of these boundary conditions. In fact, four-dimensional effective theory only contains the canonical kinetic terms for the fields and their KK-excitations along with their mutual interactions. The effect of BLKTs only shows up in modifications of some of these couplings via an overlap integral ( see Eq. 11) and also in deviations of the masses from UED values of n/R (in the n-th KK-level). So as long as these overlap integrals are not very large ( 1) we do not have any problem with the convergence of perturbation series. In Fig. 2, we have shown that the values of the overlap integrals (involves in the couplings determined by the five-dimensional wave functions) are 1 for the entire range of the strengths of the BLKTs which have used in this article and never grow with these strengths. Furthermore it has been shown in [33] that theories with large strength of the BLKTs (relative to their natural cutoff scale, Λ) were found to be perturbatively consistent and are thus favoured. Such results in Ref. [33] is in agreement with our observation that the numerical values of the overlap integral diminishes with increasing magnitude of BLKT coefficients (r i 's).
We now present the main numerical results for two distinct cases, either BLKTs are present at both fixed points or only at one of the two, in following subsections.
IV.1 BLKTs are present at y = 0 and y = πR In Fig. 3 we present the results for the case when the fermion BLKTs are symmetric at the two fixed points but unequal values of the gauge BLKTs break the KK-parity. Here we show the region of parameter space excluded by the ATLAS 8 TeV data [1] for two different choices of R a V . Each panel depicts that the region to the left of a curve in the m V (1) − R f plane is excluded by the ATLAS data.
For a chosen R a V there is an one-to-one correspondence of m V (1) with 1/R which is shown on the upper axis of the panels, as the KK-mass is rather insensitive to ∆R V . Also, for any displayed value of 1/R we can estimate the first excitation of fermion KK-mass M f (1) = m f (1) R ( plotted on right-side axis) which is determined by R f .
The exclusion plots can be understood easily in conjunction with Fig. 1 and Fig. 2. For a given ∆R V and R a V the KK-parity-non-conserving couplings are almost insensitive to R f . Thus R f has no steering on the production of e + e − /µ + µ − . The signal rate thus solely depend on R a V and ∆R V . Coupling (and inturn signal strength) increases with ∆R V . Thus with higher and higher strength of KK-parity non-conservation one can exclude higher and higher masses (and higher values of 1/R) as revealed in Fig. 3.

IV.2 BLKTs are present only at y = 0
Now let us concentrate on the case of fermion and gauge BLKTs at only one fixed point. In this case we display the exclusion curves in the m V (1) − R f plane for several choices of R V in Fig. 4. The region below a curve has been disfavoured by the ATLAS data. One can explain the Fig. 4 on the basis of Fig. 1 and Fig. 2. It is revealed in the left panel of Fig. 1, M (1) ≡ m V (1) R has mild variation with R V . So we can take the mass of V 1 to be approximately proportional to 1/R (the relevant values of 1/R are displayed in the upper axis of the panel in Fig. 4). In our model we estimate the cross section times branching ratio corresponding to any m V (1) , and comparing this with the ATLAS data we can have a specific value of (R V , R f ) pair on each curve via KK-parity-non-conserving coupling. Alternatively, it is evident from the Fig. 4, as m V (1) increases, the production of the B 1 (W 1 3 ) decreases. As a compensation, the KK-parity-non-conserving coupling must increase as we increase the m V (1) . So it is clear from the right panel of Fig. 2, increasing value of KK-parity-non-conserving coupling is achieved by the higher values of R V for a fixed value of R f . In this case also, the KK-fermion mass of first excitation can be obtained in a correlated way from the right-side axis of this plot.
Let us pay some attention to Fig. 4. For a given curve (specified by a R V ), the allowed area in m V (1) − R f plane is bounded by the curve itself and a line parallel to m V (1) axis corresponding to the value of R f determined by the specific value of R V of that curve. The choice of the R f < R V ensures mass hierarchy among KK-electroweak gauge boson and KK-leptons. So the bounded region implies that for a given value of m V (1) , R f is bounded from below which in turn imposes an upper limit on the mass of the n = 1 KK-lepton.

V Conclusions
In summary, we have investigated the phenomenology of KK-parity non-conservation in the UED model where all the SM fields propagate in 4+1 dimensional space time. We have achieved this non-conservation due to inclusion of asymmetric 9 BLTs at the two fixed points of this orbifold. These boundary (kinetic in our case) terms can be thought of as a cut-off (Λ) dependent log divergent radiative corrections [4] which remove the degeneracy in the KK-mass spectrum of the effective 3+1 dimensional theory.
With positive values of BLKTs, we have studied electroweak interaction, in two alternative ways. In the first case we put equal strengths of fermion BLKTs at the two fixed points and parametrised by r f , while for electroweak gauge boson we have considered unequal strengths of BLKTs (r a V = r b V ). Equal strengths of electroweak gauge boson BLKTs would preserve the Z 2 -parity. In the other situation we have considered the fermion and electroweak gauge boson BLKTs are present only at the y = 0 fixed point. These BLKTs modify the field equations and the boundary conditions of the solutions lead to the non-trivial KK-mass excitations and wave-functions of fermions and the electroweak gauge bosons in the y-direction in both cases. In this platform we have calculated KK-parity-non-conserving coupling between the n = 1 KK-excitation of the electroweak gauge bosons and pair of SM fermions (n = 0) in terms of r f , r a V , r b V and 1/R when BLKTs are present at both fixed points and r f , r V , and 1/R for the other case. This driving coupling vanishes in the ∆R V = 0 limit in the first case and for R f = R V in the second.
Finally we estimate the single production of V 1 at the LHC and its subsequent decay to ℓ + ℓ − , both the production and decay are controlled by the KK-parity-non-conserving coupling. We compare our results with the ℓ + ℓ − resonance production signature at the LHC running at 8 TeV pp centre of momentum energy [1,21]. The lack of observation of this signal with 20 f b −1 accumulated luminosity by ATLAS collaboration [1] at the LHC already excludes a large part of the parameter space (spanned by r f , r a V , r b V and 1/R in one case and r f , r V and 1/R in the other). Here we consider the B 1 (W 1 3 ) is lighter than the corresponding fermion and the bounds on the mass of the former are the same as that on the ℓ + ℓ − resonance from the data.
At the end, we also like to point out another important observation regarding the excluded parameter space of this model that the ℓ + ℓ − resonance search disfavoured more parameter space in comparison to the tt resonance search which we performed in our previous article [22].