Minimal Universal Extra Dimensions in CalcHEP/CompHEP

We present an implementation of the model of minimal universal extra dimensions (MUED) in CalcHEP/CompHEP. We include all level-1 and level-2 Kaluza-Klein (KK) particles outside the Higgs sector. The mass spectrum is automatically calculated at one loop in terms of the two input parameters in MUED: the radius of the extra dimension and the cut-off scale of the model. We implement both the KK number conserving and the KK number violating interactions of the KK particles. We also account for the proper running of the gauge coupling constants above the electroweak scale. The implementation has been extensively cross-checked against known analytical results in the literature and numerical results from other programs. Our files are publicly available and can be used to perform various automated calculations within the MUED model.


Introduction
The Standard Model (SM) of particle physics has been successfully verified by experiment at low energies. Nevertheless, even if the Higgs boson is discovered, the SM will still be considered to be an incomplete theory, as it fails to provide the long-sought missing link between Einstein's General Relativity and Quantum Mechanics. The leading candidate for a quantum theory of gravity, string theory, typically posits the existence of several new ingredients, which are absent in the SM: new spatial dimensions, a symmetry between bosons and fermions (supersymmetry), as well as new gauge interactions. All of these new ingredients are manifestly present at the Planck scale, but it is not at all clear which of them survive down to low energies. Traditionally, supersymmetry and extra gauge interactions have attracted the most attention, and their consequences for collider phenomenology have been extensively studied [1,2]. Within the last 10 years or so, there has been a resurgence of interest in models with extra spatial dimensions, whose presence might be revealed in high energy collider experiments such as the Tevatron at Fermilab, the Large Hadron Collider (LHC) at CERN, or the proposed International Linear Collider (ILC). By now a whole plethora of extra-dimensional models have been described and studied to various extent in the literature. Roughly speaking, they can all be classified according to the following two criteria: • How many and which of the SM particles can access the extra dimensions (the bulk).
The two extremes here are provided by the "large" extra dimension models (also known as ADD, after the initials of their original proponents) [3], in which only gravity can enter into the bulk, and the Universal Extra Dimensions (UED) models [4], in which all SM particles are allowed to propagate in the bulk.
• What is the metric of the bulk. It can be flat (e.g. in UED), or warped [5].
There are several advantages of choosing CalcHEP and CompHEP for this purpose: • CalcHEP and CompHEP can be used for parton-level event generation, preserving the full spin correlations in both production and decay.
• CalcHEP and CompHEP can be easily interfaced [76] to a general purpose event generator such as PYTHIA [77] for the simulation of fragmentation, hadronization and showering.
• CalcHEP and CompHEP can be easily interfaced with a dark matter program such as micrOMEGAs [78] for the calculation of the relic density and detection rates of a generic dark matter candidate.
• The implementation of new models is very straightforward and user-friendly, as we shall demonstrate below with the example of Minimal UED.
The paper is organized as follows. In Section 2 we first review the Minimal UED model (MUED), introducing the relevant new particles, couplings and interactions. In Section 3 we explain how those were incorporated in CalcHEP and CompHEP. Throughout the paper we assume that the readers are already familiar with these programs, so that we only need to explain the additional *.mdl model files related to our UED implementation ‡. In Section 4 we discuss how the implementation can be used to study the collider phenomenology of MUED and show some illustrative results. In the Appendices we list some more technical results which may be useful to some readers. For example, Appendix A contains the fivedimensional UED Lagrangian and Appendix C contains the resulting Feynman rules for the level 1 KK particles after compactification.

KK decomposition
The five-dimensional (5D) UED model [4] is simply the Standard Model placed in an extra dimension compactified on an S 1 /Z 2 orbifold, as shown in Fig. 1. Let us label the usual 3 + 1 space-time dimensions with x µ , µ = 0, 1, 2, 3, reserving the coordinate y for the extra dimension. In order to end up with chiral fermions in 4 dimensions and to project out ‡ Our implementation was originally developed for the Second MC4BSM workshop in Princeton, March [24][25][26][27]2007. Since then, the Minimal UED model has been partially implemented in PYTHIA [79], and more fully in CalcHEP, MadGraph, PYTHIA or Sherpa through FeynRules [80,81]. unwanted gauge degrees of freedom, one typically imposes an additional symmetry, thus creating a manifold with boundaries. For example, in the case of the S 1 /Z 2 orbifold shown in Fig. 1, one identifies the opposite points on the circle, which creates two fixed points, denoted with the blue dots. Any 5-dimensional field can now be assigned a definite parity with respect to the orbifold projection P 5 : y → −y. For example, consider a generic scalar field φ(x, y). An even scalar field φ + (x, y) is expanded in Kaluza-Klein (KK) modes as and obeys Neumann boundary conditions at the two fixed points: Here x is the usual 4-dimensional spacetime coordinate x µ , R is the size of the extra dimension and n labels the KK-level. The SM modes correspond to n = 0. In contrast, the KK decomposition of an odd scalar field is missing a zero mode (n = 0) and obeys Dirichlet boundary conditions One can similarly assign a definite P 5 parity to each component of a gauge field A M (x, y), M = 0, 1, 2, 3, 5. The usual 3 + 1 components A µ , µ = 0, 1, 2, 3, are chosen to be even, which ensures the presence of the SM gauge fields A 0 µ (x) at the n = 0 level, while the extradimensional component A 5 is taken to be odd. The corresponding KK expansions of the 5-dimensional gauge fields are given by At the two fixed points y = 0 and y = πR, the components A µ (x, y) (A 5 (x, y)) obey Neumann (Dirichlet) boundary conditions analogous to eq. (2) (eq. (4)).
The KK decomposition of a fermion is rather interesting. Since there is no chirality in 5 dimensions, the KK modes of the SM fermions come in vectorlike pairs, i.e. there is a left-handed and a right-handed KK mode for each SM chiral fermion. For example, the SU(2) W -singlet chiral fermions ψ 0 R (x) of the SM (which happen to be all right-handed) are obtained from the following decomposition where upon compactification, the two KK fermions ψ n R (x) and ψ n L (x) at any given KK level n pair up to give a Dirac fermion of mass n R . Similarly, the SU(2) W -doublet SM fermions Ψ 0 L (x) (which happen to be left-handed) arise from where the massive Dirac fermion at each n is now formed from Ψ n L (x) and Ψ n R (x). From eqs. (7-10) we see that there exist left-handed KK modes ψ n L (x), which are associated with the right-handed SM fermions ψ 0 R (x) and vice versa -there are righthanded KK modes Ψ n R (x), which go along with the left-handed SM fermions Ψ 0 L (x). This often leads to some confusion in the literature when it comes to the labelling of fermion KK partners. It should be understood that the chiral index (L or R) of a KK mode fermion refers to the chirality of its SM partner. Here we shall also utilize an alternative convention, introduced in [82], where the KK fermions are identified by their SU(2) W quantum numbers Table 1. Fermion content of the Minimal UED model. SU (2) W -doublets (SU (2) W -singlets) are denoted with capital (lowercase) letters. KK modes carry a KK index n, and for simplicity we omit the index "0" for the SM zero modes.
This convention was already employed in eqs. (7-10) as well. With those conventions, the fermion content of the Minimal UED model is listed in Table 1.
Finally, notice that the geometry in Fig. 1 is still invariant under the interchange of the two fixed points. The corresponding Z 2 symmetry is the celebrated KK parity and will be a symmetry of the Lagrangian as long as one continues to treat the two boundary points in a symmetric fashion.

KK mass spectrum
At tree level, the mass m n of any KK mode at the n-th KK level is given by where R is the radius of the extra dimension as illustrated in Fig. 1, and m 0 is the mass of the corresponding SM particle (zero mode). The resulting mass spectrum for the first KK level is shown in Fig. 2a for R −1 = 500 GeV, and can be seen to be highly degenerate.
In fact, several of the lightest n = 1 KK modes have no allowed decays and are absolutely stable.
However, this drastic conclusion is completely reversed, once radiative corrections are taken into account [82]. First, the mass spectrum gets renormalized by bulk interactions, which are uniquely fixed in terms of the SM gauge and Yukawa couplings, and thus contain no new parameters beyond those already appearing in the SM. At the same time, the KK masses also receive contributions from terms localized on the boundary points (the two blue dots in where g 1 (g 2 ) is the hypercharge (weak) gauge coupling, v = 246 GeV is the vev of the SM Higgs boson, andδ represents the total one-loop correction, including both bulk (δ) and § Strictly speaking, the true LKP in Fig. 2b is the KK graviton G 1 (not shown). However, due to its extremely weak couplings, G 1 is irrelevant for collider phenomenology. For its astrophysical implications, see [83]. boundary (δ) contributions [82]: Note that for n ≥ 1 the KK mixing angle θ n is in general different from the zero-mode (Weinberg) angle θ 0 ≡ θ W in the SM. For typical values of R −1 and Λ, θ n ≪ θ W , and the neutral gauge boson KK mass eigenstates become approximately aligned with the corresponding interaction eigenstates: γ n ≈ B n and Z n ≈ W 3 n for n ≥ 1. This approximation will be used in our MUED implementation described below in Section 3.

KK interactions
The bulk interactions of the KK modes are already fixed by the SM. The 5D MUED Lagrangian is a straightforward generalization of the SM Lagrangian to 5 dimensions, as discussed in Appendix A. Upon compactification, integrating over the extra-dimensional coordinate y, one recovers the bulk interactions among the various KK modes and their SM counterparts (see Appendix C). Since translational invariance holds in the bulk, all these bulk interactions conserve both KK number and KK parity.
However, as already alluded to in the previous subsection, there may also exist "boundary" interactions localized on the fixed points in Fig. 1. They do not respect translational invariance and therefore break KK number by even units. Such interactions may already appear at the scale Λ, being generated by the new physics which is the ultraviolet completion of UED. In the Minimal UED version, one makes the assumption that no such terms are present at the scale Λ. Even so, upon renormalization to lower energy scales, boundary terms are radiatively generated from bulk interactions. This is illustrated in Fig. 3, where we show how an effective coupling between a level 2 KK gauge boson V 2 and two SM Here I 3 is the fermion isospin and Y L (Y R ) is the hypercharge of a left-handed (right-handed) SM fermion. In the case of top quarks, one has to include inδ(m f2 ) the additional corrections proportional to the top Yukawa coupling h t :δ ht m Tn andδ ht m tn , respectively (see [82] for details).
fermions is generated at one loop from a diagram with level 1 KK particles running in the loop. This effective coupling can be expressed in terms of the boundary contributionsδm n (see eq. (13)) to the one-loop mass corrections [82]. The explicit form of this effective coupling is summarized in Table 2 for each different type of level 2 KK gauge boson and for the various possible SM fermion pairs.

Model files
Having reviewed the MUED model, we are now in a position to describe its implementation in CalcHEP and CompHEP. Each one of these programs gives its users an opportunity to incorporate new physics in the already existing framework of the SM, MSSM, etc. To

Particles
New particles are defined in the prtclsN.mdl model file. We incorporate the n = 1 and n = 2 KK modes of the gauge bosons (see Table 3), leptons (see Table 4) and quarks (see Table 5). In Tables 3-5 the KK number is represented by a superscript n = 1 or n = 2, while the subscript is either the Lorentz index (µ) of the vector particles in Table 3

Variables
The input parameters for any given physics scenario are defined in the varsN.mdl model file.
In principle, MUED has only two additional input parameters beyond the SM: the radius R of the extra dimension and the cut-off scale Λ. For convenience, we use the inverse radius R −1 and the number of KK levels ΛR which can fit below the scale Λ. R −1 has dimensions of GeV, while ΛR is dimensionless. Our additions to the varsN.mdl model file are listed in Table 6. As seen from the table, we also include several other variables of interest. RG is used to turn on and off the running of coupling constants, while scaleN is the renormalization scale µ at which the couplings are evaluated. The remaining parameters in Table 6 are some useful numerical constants related to the RGE running of the gauge couplings (see Section 3.5).

Constraints
The funcN.mdl model file is reserved for variables which are not numerical inputs, but are instead computed in terms of the parameters already defined in the varsN.mdl model file. In our case, we use funcN.mdl to supply the masses and two-body decay widths of the KK particles introduced in Section 3.1. Therefore they are automatically computed by CalcHEP/CompHEP at the beginning of each numerical session. The masses for all KK particles are evaluated based on the 1-loop formulas of Ref. [82] and we have also made numerical cross-checks with the results from the private code used in Ref. [82]. Our formulas for the widths have been derived analytically and cross-checked with CalcHEP/CompHEP (see Section 4). A partial list of 2 body decay widths can be found in [14,15,20] and our formulas agree with their expressions. In the older versions of CalcHEP/CompHEP, defining the widths as constraints was very convenient in our implementation, since one did not have to launch a separate numerical session for their calculation, and then enter their numerical values as input parameters. However, the more recent versions of CalcHEP and CompHEP allow for the automatic calculation of the particle widths on the fly, using the interactions defined in the lgrngN.mdl model file. Our implementation thus allows for backward compatibility with older versions of CalcHEP/CompHEP.

Interactions
The new interactions of the KK particles of Section 3.1 are added to the lgrngN.mdl model file. We include the usual bulk interactions, as well as the KK number violating boundary interactions listed in Table 2 [82]. Since the Weinberg angle θ n for any n ≥ 1 is small [82], we ignore the mixing among the neutral KK gauge bosons. Thus the KK-photon γ n is identical to the hypercharge gauge boson B n and the KK Z-boson Z n is identical to the neutral SU(2) W gauge boson W 3 n . We also ignore the mixing between SU(2) W -doublet and SU(2) W -singlet KK fermions.
Our lgrngN.mdl model file includes all interactions of level-1 and level-2 KK particles except for the KK Higgs bosons. The phenomenology of the KK Higgs bosons is very model dependent, depending on the value of the SM Higgs mass m h and the bulk Higgs mass term (see [82] for details). Therefore we omit any interactions involving KK Higgs bosons .
The collider phenomenology of the KK Higgs bosons has been discussed in [23,24,29].
The UED Lagrangian can be easily derived as shown in Appendix C. Here we only point out how to deal with 4-point interactions involving KK gluons, since this case requires special treatment when implemented in CalcHEP/CompHEP.

The Lagrangian for the quartic interactions with KK gluons is the following
The color structure of these 4-point interactions cannot be directly written down in CalcHEP/CompHEP format. Hence, to implement this vertex in CalcHEP/CompHEP, we use the following trick. We introduce three auxiliary tensor fields t a µν , s a µν and u a µν in the same way as the original CalcHEP/CompHEP approach for SM gluons. Then one can rewrite the Lagrangian as It is easy to show that the functional integration over the three auxiliary tensor fields reproduces the 4-gluon interactions (14).

Running of the coupling constants
Due to the additional contributions from the KK modes to the beta functions, the gauge couplings run faster in theories with extra dimensions. The RGE for α i ≡ g 2 i 4π is given by [84] where δ is the number of extra dimensions, µ 0 is some reference energy scale, X δ = 2π δ/2 δΓ(δ/2) , are the SM beta function coefficients, while correspond to the contributions of the Kaluza-Klein states at each massive KK excitation level [85,86]. The solution to (16) becomes The effect of the RGE running (19) can be accounted for by setting the RG parameter in Table 6 to 1 and choosing the appropriate renormalization scale via scaleN.

Code validation
In general, the availability of CalcHEP/CompHEP model files opens the door to a number of applications related to collider phenomenology and dark matter searches. Each such individual study contributes to the validation of the code. Further consistency checks are provided by comparing to existing analytical and/or numerical results in the literature.
• For starters, we have compared the KK mass spectrum calculated with our implementation to the results shown in Fig. 2, which were obtained independently in Ref. [82]. Using identical inputs, and neglecting the running of the gauge couplings (as was done in [82]), we found perfect agreement.
• The interaction vertices of Appendix C can be independently derived with the automated tool LanHEP [87]. We checked some of the more technically challenging cases (especially the self-interactions of gauge bosons) and also found agreement.
• To minimize the possibility of typing mistakes, we computed analytically the crosssections for a selected number of simple scattering processes, and compared to the analytical expressions derived by CalcHEP/CompHEP.
• We have similarly checked that the KK particle widths calculated from our analytical expressions agree with those computed with CalcHEP/CompHEP by means of our MUED implementation. • Our analytic formulas for decay widths agree with the expressions given in [14,15,20].
• Our implementation was used for the analytic calculation of all (co)annihilation crosssections of level 1 KK particles [59] and the results were in complete agreement with [44,58].
• Our model files have already been used for various collider studies [17, 19-22, 26, 59, 88-90]. One example is shown in Fig. 4, which shows the strong production cross-section of level 1 KK particles at the imminent LHC energy of 7 TeV.
• We have compared results for various production cross-sections in MUED to those in published papers [12,13] and find agreement.
• Our model files were also cross-checked against the known analytical expressions for various invariant mass distributions [18,92,93].
• Our model files have also been tested by other groups, for example in creating Pythia UED [79,94,95], which implemented the matrix elements for certain processes in PYTHIA [77]. Another extensive comparison to an independent MUED implementation via FeynRules was done in [81].

Outlook
Moving forward, it is important to be mindful of the limitations of our implementation.
First of all, it is still Minimal UED, and the spectrum is quite constrained, given in terms of only 2 parameters: R −1 and Λ. If a signal consistent with UED is discovered at the LHC or the Tevatron, one would like to start testing the data with a more general UED framework, which allows for the presence of arbitrary boundary terms at the scale Λ. Work along these lines has already started and a beta version of the corresponding UED model files is available from the authors upon request.

Acknowledgments
We The Lagrangian for the 5-dimensional UED model is written as in terms of 5-dimensional fields decomposed as discussed in Section 2.1: Here H(x, y) is the 5D Higgs scalar field and (B µ (x, y), B 5 (x, y)), (W µ (x, y), W 5 (x, y)) and where τ a , a = 1, 2, 3, are the usual Pauli matrices and λ A , A = 1, 2, ..., 8, are the usual Gell-Mann matrices. The 5D field strength tensors for U(1) Y , SU(2) W and SU(3) c are defined as follows where ǫ abc and f ABC are the structure constants for SU(2) W and SU(3) c , respectively. The parameter ξ in (A.2) is the gauge fixing parameter in the generalized R ξ gauge.
The 5-dimensional (4-dimensional) gauge couplings are denoted by g Finally, Q(x, y) and L(x, y) are the SU(2) W -doublet fermions from Table 1, while u(x, y), d(x, y) and e(x, y) are the corresponding SU(2) W -singlet fermions from Table 1 where g M N is the 5D metric and g µν = (+ − −−) is the usual 4D metric.
The covariant derivatives act on 5D fields as follows 1 B M u(x, y) , where the fermion hypercharges are