I Introduction
###### Abstract

Non-vanishing boundary localised terms significantly modify the mass spectrum and various interactions among the Kaluza-Klein excited states of 5-Dimensional Universal Extra Dimensional scenario. In this scenario we compute the contributions of Kaluza-Klein excitations of gauge bosons and third generation quarks for the decay process incorporating next-to-leading order QCD corrections. We estimate branching ratio as well as Forward Backward asymmetry associated with this decay process. Considering the constraints from some other observables and electroweak precision data we show that significant amount of parameter space of this scenario has been able to explain the observed experimental data for this decay process. From our analysis we put lower limit on the size of the extra dimension by comparing our theoretical prediction for branching ratio with the corresponding experimental data. Depending on the values of free parameters of the present scenario, lower limit on the inverse of the radius of compactification () can be as high as GeV. Unfortunately, the Forward Backward asymmetry of this decay process would not provide any significant limit on in the present model.

Looking for in non-minimal Universal Extra Dimensional model

Avirup Shaw***email: avirup.cu@gmail.com

Theoretical Physics, Physical Research Laboratory,

## I Introduction

Standard Model (SM) of particle physics has almost been accomplished by the discovery of Higgs boson at the Large Hadron Collider (LHC) [1, 2]. However, the SM scenario is not the ultimate one, because there exists several experimental data in various directions, such as massive neutrinos, Dark Matter (DM) enigma, observed baryon asymmetry etc., that cannot be addressed within the SM. This in turn, ensures that new physics (NP) is indeed a reality of nature. Moreover, experimental data for several flavour (specially physics) physics observables show significant deviation from the corresponding SM expectations. For example, in -physics experiments at LHCb, Belle and Babar have pointed at intriguing lepton flavour universality violating (LFUV) effects for both the charge current ( [3] and  [4]) as well as the flavour changing neutral current (FCNC) ( [5] and [6]) processes. In the latter case, involving processes are described at the quark level by the transition which is highly suppressed in SM. Therefore, even for small deviation between the SM prediction and experimental data, these types of observables have always been very instrumental to probe for the favorable kind among the various NP models that exist in the literature.

Following the same argument, an inclusive decay mode has been considered as one of the harbingers for the detection of NP scenario. The reason is that, this decay mode is one of the most significant and relatively clean decay modes. In most of the cases, experimental data for several observables for this decay mode have been more explored for two regions of di-lepton invariant mass square () spectrum. In these two regions, the experimental data of branching ratio (Br) are given by Babar collaboration111These experimental data have also been used in two recent articles [7, 8] in context of the same decay process. [9]

 Br(B→Xsℓ+ℓ−)expq2∈[1,6]GeV2=(1.60+0.41+0.17−0.39−0.13±0.18)×10−6, Br(B→Xsℓ+ℓ−)expq2∈[14.4,25]GeV2=(0.57+0.16+0.03−0.15−0.02±0.00)×10−6,(ℓ=e,μ). (1)

The SM predictions for the above quantities are [10]

 Br(B→Xsℓ+ℓ−)SMq2∈[1,6]GeV2=(1.62±0.09)×10−6, Br(B→Xsℓ+ℓ−)SMq2∈[14.4,25]GeV2=(2.53±0.70)×10−7,(ℓ=e,μ). (2)

Apart from the branching ratio, there exists an elegant observable for the process which could help for the detection of NP scenario. This is called Forward-Backward asymmetry () and for this decay process for the two distinct regions of the experimental values are given by Belle Collaboration[11]

 AFB(B→Xsℓ+ℓ−)∣∣expq2∈[1,6]GeV2=0.30±0.24±0.04, AFB(B→Xsℓ+ℓ−)∣∣expq2∈[14.4,25]GeV2=0.28±0.15±0.02, (3)

while the corresponding SM expectations are [12, 13, 11]

 AFB(B→Xsℓ+ℓ−)∣∣SMq2∈[1,6]GeV2=−0.07±0.04, AFB(B→Xsℓ+ℓ−)∣∣SMq2∈[14.4,25]GeV2=0.40±0.04. (4)

Therefore, from the above data it is clearly evident that the SM predictions for the respective observables coincide with the experimental data within a few standard deviations. Hence, by investigating these observables one can search any favourable NP scenario and also tightly constrain the parameter space of that scenario. With this spirit, in this article we evaluate the decay amplitude for the process in an extra dimensional scenario. In literature one can find several articles, e.g., [7, 14] which have been dedicated to the exploration of the same decay process in the context of several BSM scenarios.

In the present article we are particularly focused on an extension of SM with one flat space-like dimension () compactified on a circle of radius . All the SM fields are allowed to propagate along the extra dimension . This model is called as 5-dimensional (5D) Universal Extra Dimensional (UED) [15] scenario. The fields manifested on this manifold are usually defined in terms of towers of 4-Dimensional (4D) Kaluza-Klein (KK) states while the zero-mode of the KK-towers is designated as the corresponding 4D SM field. A discrete symmetry () has been needed to generate chiral SM fermions in this scenario. Consequently, the extra dimension is defined as orbifold and eventually physical domain extends from to . As a result, the symmetry has been translated as a conserved parity which is known as KK-parity , where is called the KK-number. This KK-number () is identified as discretised momentum along the -direction. From the conservation of KK-parity the lightest Kaluza-Klein particle (LKP) with KK-number one () cannot decay to a pair of SM particles and becomes absolutely stable. Hence, the LKP has been considered as a potential DM candidate in this scenario [16, 17, 18, 19, 20, 21, 22, 23]. Furthermore, few variants of this model can address some other shortcomings of SM, for example, gauge coupling unifications [24, 25, 26], neutrino mass [27, 28] and fermion mass hierarchy [29] etc.

At the KK-level all the KK-state particles have the mass . Here, is considered as the zero-mode mass (SM particle mass) which is very small with respect to . Therefore, this UED scenario contains almost degenerate mass spectrum at each KK-level. Consequently, this scenario has lost its phenomenological relevance, specifically, at the colliders. However, this degeneracy in the mass spectrum can be lifted by radiative corrections [30, 31]. There are two different types of radiative corrections. The first one is considered as bulk corrections (which are finite and only non-zero for KK-excitations of gauge bosons) and second one is regarded as boundary localised corrections that proportional to logarithmically cut-off222UED is considered as an effective theory and it is characterised by a cut-off scale . scale () dependent terms. The boundary correction terms can be embedded as 4D kinetic, mass and other possible interaction terms for the KK-states at the two fixed boundary points ( and ) of this orbifold. As a matter of fact, it is very obvious to include such terms in an extra dimensional theory like UED since these boundary terms have played the role of counterterms for cut-off dependent loop-induced contributions. In the minimal version of UED (mUED) models there is an assumption that these boundary terms are tuned in such a way that the 5D radiative corrections exactly vanish at the cut-off scale . However, in general this assumption can be avoided and without calculating the actual radiative corrections one might consider kinetic, mass as well as other interaction terms localised at the two fixed boundary points to parametrise these unknown corrections. Therefore, this specific scenario is called as non-minimal UED (nmUED) [32, 33, 34, 35, 36, 37, 38, 39, 40]. In this scenario, not only the radius of compactification (), but also the coefficients of different boundary localised terms (BLTs) have been considered as free parameters which can be constrained by various experimental data of different physical observables. In literature one can find different such exercise regarding various phenomenological aspects. As for example limits on the values of the strengths of the BLTs have been achieved from the estimation of electroweak observables [38, 40], , and parameters [36, 41], DM relic density [42, 43], production as well as decay of SM Higgs boson [44], collider study of LHC experiments [45, 46, 47, 48, 49, 50], [51], branching ratios of some rare decay processes e.g., [52] and [53], anomalies [54, 55], flavour changing rare top decay [56, 57] and unitarity of scattering amplitudes involving KK-excitations [58].

In this article we estimate the contributions of KK-excited modes to the decay of in a 5D UED model with non-vanishing BLT parameters. Our calculation includes next-to-leading order (NLO) QCD corrections. To the best of our knowledge, this to be the first article which deals with the decay in the framework of nmUED has not yet been presented in the literature. Considering the present experimental data of the concerned FCNC process we will put constraints on the BLT parameters. Furthermore, we would like to investigate how far the lower limit on to higher values can be extended using non-zero BLT parameters. Consequently, it will be an interesting part of this exercise is to see whether this lower limit of is comparable with the results obtained from our previous analysis [52, 53] or not? Several years ago the same analysis [59] has been performed in the context of minimal version of UED model, however, the present experimental data have been changed with respect to that time. Therefore, it will be a relevant job to revisit the lower bound on in UED model by comparing the current experimental result [9, 11] with theoretical estimation using vanishing BLT parameters.

In the following section II, we will give a brief description of the nmUED model. Then in section III we will show the calculational details of branching ratio and Forward-Backward asymmetry for the present process. In section IV we will present our numerical results. Finally, we conclude the results in section V.

## Ii KK-parity conserving nmUED scenario: A brief overview

Here we present the technicalities of nmUED scenario required for our analysis. For further discussion regarding this scenario one can look into[32, 33, 34, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]. In the present scenario we preserve a symmetry by considering equal strength of boundary terms at both the boundary points ( and ). Consequently, KK-parity has been restored in this scenario which makes the LKP stable. Hence, this present scenario can give a potential DM candidate (such as first excited KK-state of photon). A comprehensive exercise on DM in nmUED can be found in [43].

We begin with the action for 5D fermionic fields associated with their boundary localised kinetic term (BLKT) of strength [37, 43, 52, 53, 54]

 Sfermion=∫d5x[¯ΨLiΓMDMΨL+rf{δ(y)+δ(y−πR)}¯ΨLiγμDμPLΨL +¯ΨRiΓMDMΨR+rf{δ(y)+δ(y−πR)}¯ΨRiγμDμPRΨR], (5)

where and represent the 5D four component Dirac spinors that can be expressed in terms of two component spinors as [37, 43, 52, 53, 54]

 ΨL(x,y)=(ϕL(x,y)χL(x,y))=∑n(ϕ(n)L(x)fnL(y)χ(n)L(x)gnL(y)), (6)
 ΨR(x,y)=(ϕR(x,y)χR(x,y))=∑n⎛⎝ϕ(n)R(x)fnR(y)χ(n)R(x)gnR(y)⎞⎠. (7)

and are the associated KK-wave-functions which can be written as the following [33, 38, 43, 52, 53, 54]

 fnL=gnR=Nfn⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩cos[mf(n)(y−πR2)]cos[mf(n)πR2]for n even,−sin[mf(n)(y−πR2)]sin[mf(n)πR2]for n odd, (8)

and

 gnL=−fnR=Nfn⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩sin[mf(n)(y−πR2)]cos[mf(n)πR2]for n even,cos[mf(n)(y−πR2)]sin[mf(n)πR2]for n odd. (9)

Normalisation constant () for KK-mode can easily be obtained from the following orthonormality conditions [43, 52, 53, 54]

 ∫πR0dy[1+rf{δ(y)+δ(y−πR)}]fmLfnL∫πR0dy[1+rf{δ(y)+δ(y−πR)}]gmRgnR⎫⎬⎭= δnm ; ∫πR0dyfmRfnR∫πR0dygmLgnL⎫⎬⎭= δnm , (10)

and it takes the form as

 Nfn=√2πR[1√1+r2fm2f(n)4+rfπR]. (11)

Here, is the KK-mass of KK-excitation acquired from the given transcendental equations [33, 43, 52, 53, 54]

 rfmf(n)2=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩−tan(mf(n)πR2)for n even,cot(mf(n)πR2)for n odd. (12)

Let us discuss the Yukawa interactions in this scenario as the large top quark mass plays a significant role in amplifying the quantum effects in the present study. The action of Yukawa interaction with BLTs of strength is written as [52, 53, 54]

 SYukawa = −∫d5x[λ5t¯ΨL˜ΦΨR+ry{δ(y)+δ(y−πR)}λ5t¯ϕL˜ΦχR+h.c.]. (13)

The 5D coupling strength of Yukawa interaction for the third generations are represented by . Embedding the KK-wave-functions for fermions (given in Eqs. 6 and 7) in the actions given in Eq. 5 and Eq. 13 one finds the bi-linear terms containing the doublet and singlet states of the quarks. For KK-level the mass matrix can be expressed as the following [52, 53, 54]

 (14)

Here, is identified as the mass of SM top quark while is obtained from the solution of the transcendental equations given in Eq. 12. and are the overlap integrals which are given in the following[52, 53, 54]

 Inm1=⎛⎜⎝1+rfπR1+ryπR⎞⎟⎠×∫πR0dy[1+ry{δ(y)+δ(y−πR)}]gmRfnL,

and

 Inm2=⎛⎜⎝1+rfπR1+ryπR⎞⎟⎠×∫πR0dygmLfnR.

The integral is non vanishing for both the conditions of and . However, for , this integral becomes unity (when ) or zero (). On the other hand, the integral is non vanishing only when and becomes unity in the limit . At this stage we would like to point out that, in our analysis we choose a condition of equality (=) to elude the complicacy of mode mixing and develop a simpler form of fermion mixing matrix [51, 52, 53, 54]. Following this motivation, in the rest of our analysis we will maintain the equality condition333However, in general, one can choose unequal strengths of boundary terms for kinetic and Yukawa interaction for fermions. .

Implying the alluded equality condition () the resulting mass matrix (given in Eq. 14) can readily be diagonalised by following bi-unitary transformations for the left- and right-handed fields respectively [52, 53, 54]

 (15)

with the mixing angle . The gauge eigen states and are related with the mass eigen states and by the given relations [52, 53, 54] (16)

Both the physical eigen states and share the same mass eigen value at each KK-level. For KK-level it takes the form as .

In the following we present the kinetic action (governed by gauge group) of 5D gauge and scalar fields with their respective BLKTs [38, 60, 51, 52, 53, 54, 56]

 Sgauge = −14∫d5x[WaMNWaMN+rW{δ(y)+δ(y−πR)}WaμνWaμν (17) + BMNBMN+rB{δ(y)+δ(y−πR)}BμνBμν],
 Sscalar = (18)

where, , and are identified as the strength of the BLKTs for the respective fields. 5D field strength tensors are written as

 WaMN ≡ (∂MWaN−∂NWaM−~g2ϵabcWbMWcN), (19) BMN ≡ (∂MBN−∂NBM).

and () are represented as the 5D gauge fields corresponding to the gauge groups and respectively. 5D covariant derivative is given as , where, and represent the 5D gauge coupling constants. Here, and are the generators of and gauge groups respectively. 5D Higgs doublet is represented by . Each of the gauge and scalar fields which are involved in the above actions (Eqs. 17 and 18) can be expressed in terms of appropriate KK-wave-functions as [60, 51, 52, 53, 54, 56]

 Vμ(x,y)=∑nV(n)μ(x)an(y), V4(x,y)=∑nV(n)4(x)bn(y) (20)

and

 Φ(x,y)=∑nΦ(n)(x)hn(y), (21)

where generically represents both the 5D and gauge bosons.

Before proceeding further, we would like to mention a few important remarks which would aid the reader to understand the following field and the corresponding KK-wave function structure of the gauge as well as the scalar fields. We know that physical neutral gauge bosons generate due to the mixing of and fields and hence the KK-decomposition of neutral gauge bosons become very intricate in the present extra dimensional scenario because of the existence of two types of mixings both at the bulk as well as on the boundary. Therefore, in this situation without the condition , it would be very difficult to diagonalise the bulk and boundary actions simultaneously by the same 5D field redefinition444However, in general one can proceed with , but in this situation the mixing between and in the bulk and on the boundary points produce off-diagonal terms in the neutral gauge boson mass matrix.. Hence, in the following we will sustain the equality condition [60, 51, 52, 53, 54, 56]. Consequently, similar to the mUED scenario, one obtains the same structure of mixing between KK-excitations of the neutral component of the gauge fields (i.e., the mixing between and ) in nmUED scenario. Therefore, the mixing between and (i.e., the mixing at the first KK-level) gives the and . This (first excited KK-state of photon) is absolutely stable by the conservation KK-parity and it possesses the lowest mass among the first excited KK-states in the nmUED particle spectrum. Moreover, it can not decay to pair of SM particles. Therefore, this has been played the role of a viable DM candidate in this scenario [43].

In the following, we have given the gauge fixing action (contains a generic BLKT parameter for gauge bosons) appropriate for nmUED model[60, 51, 52, 53, 54, 56]

 Sgaugefixing = −1ξy∫d5x∣∣∂μWμ++ξy(∂yW4++iMWϕ+{1+rV(δ(y)+δ(y−πR))})∣∣2 (22) −12ξy∫d5x[∂μZμ+ξy(∂yZ4−MZχ{1+rV(δ(y)+δ(y−πR))})]2 − 12ξy∫d5x[∂μAμ+ξy∂yA4]2,

where is the mass of SM boson. For a detailed study on gauge fixing action/mechanism in nmUED we refer to [60]. The above action (given in Eq. 22) is somewhat intricate and at the same time very crucial for this nmUED scenario where we will calculate one loop diagrams (required for present calculation) in Feynman gauge. In the presence of the BLKTs the Lagrangian leads to a non-homogeneous weight function for the fields with respect to the extra dimension. This inhomogeneity compels us to define a -dependent gauge fixing parameter as [60, 51, 52, 53, 54, 56]

 ξ=ξy(1+rV{δ(y)+δ(y−πR)}), (23)

where is not dependent on . This relation can be treated as renormalisation of the gauge fixing parameter since the BLKTs are in some sense played the role of counterterms taking into account the unknown ultraviolet contribution in loop calculations. In this sense, is the bare gauge fixing parameter while can be seen as the renormalized gauge fixing parameter taking the values (Landau gauge), (Feynman gauge) or (Unitary gauge) [60].

In the present scenario appropriate gauge fixing procedure enforces the condition  [60, 51, 52, 53, 54, 56]. Consequently KK-masses for the gauge and the scalar field are equal () and satisfy the same transcendental equation (Eq. 12). At the KK-level the physical gauge fields () and charged Higgs () share the same555Similarly one can find the mass eigen values for the KK-excited boson and pseudo scalar . Moreover, their mass eigen values are also identical to each other at any KK-level. For example at KK-level it takes the form as . mass eigen value and is given by[60, 51, 52, 53, 54, 56]

 MW(n)=√M2W+m2V(n). (24)

Moreover, in the ât-Hooft Feynman gauge, the mass of Goldstone bosons () corresponding to the gauge fields has the same value [60, 51, 52, 53, 54, 56].

Additionally, we would like to mention that, as in the present article we are dealing with a process that involves off-shell amplitude, hence we need to use the method of background fields [61, 59]. We have already mentioned that the same decay process has already been calculated in the article [59] in the context of 5D UED and further the authors have also used the same background fields. For this reason, in the Appendix A of that article [59] the authors have discussed the background field method and also given the corresponding prescription for the 5D UED scenario. We can readily adopt this prescription in the present nmUED scenario because the basic structure of both these models are similar. We hence refrain from providing the details of this method in the present scenario. However, using that prescription (given in [59]) we can easily evaluate the Feynman rules necessary for our present calculation. In Appendix B we give the necessary Feynman rules derived for the 5D background field method in the 5D nmUED scenario in Feynman gauge.

Up to this we provide the relevant information of the present scenario. At this stage it is important to mention that the interactions for our calculation can be evaluated by integrating out the 5D action over the extra space-like dimension () after plugging the appropriate -dependent KK-wave-function for the respective fields in 5D action. As a consequence some of the interactions are modified by so called overlap integrals with respect to their mUED counterparts. The expressions of the overlap integrals have been given in Appendix B. For further information of these overlap integrals we refer the reader to check [52].

## Iii B→Xsℓ+ℓ− in nmUED

The semileptonic inclusive decay is quite suppressed in the SM, however it is very compelling for finding NP signature. Therefore, several -physics experimental collaborations (Belle, Babar) have been involved to measure several observables (mainly decay branching ratio, Forward-Backward asymmetry) associated with this decay process. In the context of SM, the dominant perturbative contribution has been evaluated in [62] and later two loop QCD corrections666Research regarding higher order perturbative contributions has been studied extensively and has already reached at a high level accuracy. For example, one can find NNLO QCD corrections in [63] and including QED corrections in [64, 65]. Moreover, updated analysis of all angular observables in the decay has been given in [10]. It also contains all available perturbative NNLO QCD, NLO QED corrections and includes subleading power corrections. have been described in the refs. [66, 67]. Since in this particular decay mode, a lepton-antilepton pair is present, therefore, more structures contribute to the decay rate and some subtleties arise in theoretical description for this process. For the decay to be dominated by perturbative contributions then one has to eliminate resonances that show up as large peaks in the di-lepton invariant mass spectrum by judicious choice of kinematic cuts. Consequently this leads to “perturbative di-lepton invariant mass windows”, namely the low di-lepton mass region , and also the high di-lepton mass region with .

In this section we will describe the details of the calculation of the branching ratio and Forward-Backward asymmetry of in nmUED model. Since the basic gauge structure of the present nmUED model is similar to SM, therefore, leading order (LO) contributions to electroweak dipole operators are one loop suppressed as in SM. However, in the present model due to the presence of large number of KK-particles we encounter more one loop diagrams in comparison to SM. Hence, we will evaluate the total contributions of these KK-particles to the electroweak dipole operators and just simply add them to SM contribution. With this spirit following the same technique of the ref.[59] we will evaluate the Wilson Coefficients (WCs) of the electroweak dipole operators at the LO level. Then following the prescription of as given in [66, 67] we will include NLO QCD correction to the concerned decay process.

### iii.1 Effective Hamiltonian for B→Xsℓ+ℓ−

The effective Hamiltonian for the decay at hadronic scales can be written as [59]

 Heff(b→sℓ+ℓ−)=Heff(b→sγ)−GF√2V∗tsVtb[C9V(μ)Q9V+C10A(MW)Q10A], (25)

where represents the Fermi constant and are the elements of Cabibbo-Kobayashi-Maskawa (CKM) matrix. In the above expression (Eq. 25) apart from the relevant operators777The explicit form of the effective Hamiltonian for is given in [59, 53]. for there are two new operators [59]

 Q9V=(¯sb)V−A(¯ℓℓ)V,Q10A=(¯sb)V−A(¯ℓℓ)A, (26)

where and refer to vector and axial-vector current respectively. They are produced via the electroweak penguin diagrams shown in Fig. 1 and the other relevant Feynman diagrams needed to maintain gauge invariance (for nmUED scenario) has been given in [52].

For the purpose of convenience the above WCs (given in Eq. 25) can be defined in terms of two new coefficients and as [59, 67]

 C9V(μ) = α2π~C9(μ), (27) C10A(μ) = α2π~C10(μ), (28)

where,

 ~C10(μ)=−Y(xt,rf,rV,R−1)sin2θw. (29)

The function in the context of nmUED scenario has been calculated in [52]. is the Weinberg angle and represents the fine structure constant. The operator does not evolve under QCD renormalisation and its coefficient is independent of . On the other hand using the results of NLO QCD corrections to in the SM given in [66, 67] we can readily obtain this coefficient in the present nmUED model under the naive dimensional regularisation (NDR) renormalisation scheme as

 ~Ceff9(q2) = ~CNDR9~η(q2m2b)+h(z,q2m2b)(3C(0)1+C(0)2+3C(0)3+C(0)4+3C(0)5+C(0)6) −12h(1,q2m2b)(4C(0)3+4C(0)4+3C(0)5+C(0)6)−12h(0,q2m2b)(C(0)3+4C(0)4) +29(3C(0)3+C(0)4+3C(0)5+C(0)6),

where,

 ~CNDR9(μ)=PNDR0+Y(xt,rf,rV,R−1)sin2θw−4Z(xt,rf,rV,R−1)+PEE(xt,rf,rV,R−1). (31)

The value888The analytic formula for has been given in [67]. of is [59] and is [67]. Using the relation given in [67, 59] we can express the function in nmUED scenario as

 Z(xt,rf,rV,R−1)=C(xt,rf,rV,R−1)+14D(xt,rf,rV,R−1), (32)

while the function for nmUED scenario has been calculated in [52]. The function given in the Eq. III.1 represents single gluon corrections to the matrix element and it takes the form as [67]

 ~η(q2m2b)=1+αsπω(q2m2b), (33)

where is the QCD fine structure constant. The explicit form of the functions , and other WCs (e.g., given in Eq. III.1) required for the present decay process have been given in Appendix A. The functions and which we evaluate in this article are given in the following

 D(xt,rf,rV,R−1)=D0(xt)+∞∑n=1Dn(xt,xf(n),xV(n)), (34)

and

 E(xt,rf,rV,R−1)=E0(xt)+∞∑n=1En(xt,xf(n),xV(n)), (35)

with , and . and can be obtained from transcendental equation given in Eq. 12. The functions and are the corresponding SM contributions at the electroweak scale [59, 66, 67, 68, 69]

 D0(xt)=−49lnxt+−19x3t+25x2t36(xt−1)3+x2t(5x2t−2xt−6)18(xt−1)4lnxt , (36)
 E0(xt)=−23lnxt+x2t(15−16xt+4x2t)6(1−xt)4lnxt+xt(18−11xt−x2t)12(1−xt)3 . (37)

Now we will depict the nmUED contribution to the electroweak penguin diagrams. We have already mentioned that the KK-masses and couplings involving KK-excitations are non-trivially modified with respect to their UED counterparts due to the presence of different BLTs in the nmUED action. Therefore, it would not be possible to obtain the expressions of and functions in nmUED simply by rescaling the results of UED model [59]. Consequently, we have evaluated the functions and independently for the nmUED scenario. These functions ( and ) represent the KK-contributions for KK-mode which are computed from the electroweak penguin diagrams (given in Fig. 1) in nmUED model for photon and gluon respectively. Furthermore, it is quite evident from Eqs. III.1 and III.1 that they are remarkably different from that of the UED expression (given in Eqs. 3.31 and 3.32 of ref. [59]). However, from our expressions (given in Eqs. III.1 and III.1) we can reconstruct the results of UED version (given in the Eqs. 3.31 and 3.32 of the ref. [59]) if we set the boundary terms to zero i.e., .

To this end, we would like to mention that in our calculation of one loop penguin diagrams (in order to measure the contributions of KK-excitation to the decay of ) we consider only those interactions which couple a zero-mode field to a pair of KK-excitations carrying equal KK-number. Although, in nmUED scenario due to the KK-parity conservation one can also have non-zero interactions involving KK-excitations with KK-numbers where is an even integer. However, we have explicitly checked that the final results would not change remarkably even if one considers the contributions of all the possible off-diagonal interactions [51, 52, 53].

For the KK-level the electroweak photon penguin function (which is obtained from penguin diagrams given in Fig. 1) takes the form as

 Dn(xt,xf(n),xV(n)) = 23En(xt,xf(n),xV(n))−136(−1+xf(n)−xV(n))4 [(−1+xf(n)−xV(n)){−2(In1)2(43x2f(n)−65xf(n)(1+xV(n)) +16(1+xV(n))2)+(In2)2(11x2f(n)−7xf(n)(1+xV(n)) +2(1+xV(n))2)}−6x2f(n){(In2)2xf(n)+2(In1)2 (6−5xf(n)+6xV(n))}ln(xf(n)1+xV(n))] +136(−1+xt+xf(n)−xV(n))4[(−1+xt+xf(n)−xV(n)){ (In1)2(11x3t+x2f(n)(−86+11xt)−x2t(93+7xV(n))+32(1+xV(n))2 +2xt(1+xV(n))(66+xV(n))+xf(n)(xt(−179+22xt−7xV(n)) 130(1+xV(n))))+(In2)2(11x2f(n)+11x2t−7xt(1+xV(n)) +2(1+xV(n))2+xf(n)(22xt−7(1+xV(n))))}−6(xt+xf(n))2 +12(1+xV(n)))}ln(xt+xf(n)1+xV(n))],

while the function is regarded as the corresponding contribution for penguins given by the first two diagrams of Fig. 1. The expression of the function in nmUED is given as the following

 En(xt,xf(n),xV(