The impact of nonminimal Universal Extra Dimensional model on $\Delta B=2$ transitions

We measure the impact of nonvanishing boundary localised terms on $\Delta B=2$ transitions in five-dimensional Universal Extra Dimensional scenario where masses and coupling strengths of several interactions of Kaluza-Klein modes are significantly modified with respect to the minimal counterpart. In such scenario we estimate the Kaluza-Klein contributions of quarks, gauge bosons and charged Higgs by evaluating the one-loop box diagrams that are responsible for the $\Delta B=2$ transitions. Using the loop function (obtained from one-loop box diagrams) we determine several important elements that are involved in Wolfenstein parametrisation. Moreover, with these elements we also study the geometrical shape of unitarity triangle. Besides, we compute the quantity $\Delta M_s$ scaled by the corresponding Standard Model value. Outcomes of our theoretical predictions have been compared to the allowed ranges of the corresponding observables simultaneously. Our current analysis shows that, depending on the parameters in this scenario the lower limit on the inverse of the radius of compactification can reach to an appreciable large value ($\approx 1.48$ TeV or even higher).


I Introduction
The Standard Model (SM) of particle physics has been a tremendously successful theory for explaining the features and interactions of fundamental particles, with many measurements confirming its predictions to extraordinary precision. With the discovery of the Higgs Boson by the ATLAS [1] and CMS [2] at the Large Hadron Collider (LHC) at CERN, all the particles in the SM zoo have been observed experimentally. However, it is known to be incomplete, as there exist several experimental data, such as massive neutrinos, Dark Matter (DM), matter anti-matter asymmetry etc., that cannot be explained in the SM scenario. Therefore, one of the current goals of particle physics is to discover new particles and interactions-generically known as "new physics" (NP) that could provide an explanation for these observations. In principle, there are two ways to search for NP. At the high-energy frontier one tries to produce those new degrees of freedom directly, while at the high-precision frontier one analyses the indirect virtual effects of such new particles. In the second method NP would appear as a discrepancy between SM expectations and experimental measurements.
Considering the latter argument, we would like to mention that, one of the elegant ways to search for new particles is by studying processes known as flavour changing neutral current (FCNC) decays, where a quark changes its flavour without changing its electric charge. One example of such a transition is the decay of a bottom quark (b) into a strange (s) or down (d) quark. In such classification, B q -meson (q = s, d) mixing is a particularly interesting process for indirect NP searches in the quark-flavour sector. Since in the SM such processes are forbidden at treelevel, they are sensitive to new heavy particles appearing as virtual particles in loop diagrams.
Moreover, it is Glashow Iliopoulos Maiani (GIM) suppressed. The physical observables are the mass differences (∆M q ), decay-width differences (∆Γ q ) between the heavy and light neutral B qmeson mass eigenstates, and the flavour-specific CP asymmetries a q fs . Theoretical predictions of B q -mixing observables in both the SM and beyond are governed by ∆B = 2 transitions and the hadronic matrix elements of which are expressed by local four-fermion operators in the effective weak Hamiltonian (given in Eq. 21). These observables are very useful to constrain physics beyond SM (BSM) scenarios. For example, in any BSM scenario if we compute the contribution of new heavy particles (particularly the contribution in box diagram for B q -mixing) then it will contribute to loop-integral function ( Inami-Lim function [3]) of the corresponding operator which controls the ∆B = 2 transitions. Consequently, using the B q -mixing observables we can measure the effects of that BSM scenarios by extracting the elements of the Cabibbo Kobayashi Maskawa (CKM) matrix [4,5] as well as by studying the shape of unitarity triangle (UT) [6]. Generally, four independent parameters are required to define the CKM matrix fully. Out of many parametrisations, Wolfenstein parametrisation [7] is the most famous and has several nice features. In particular it gives very prominent geometrical representation of the structure of the CKM matrix in conjunction with the UT which in turn very helpful to constrain the BSM scenarios. For example the refs. [8][9][10][11][12][13] depicted how the observables are related to ∆B = 2 transition that can constrain different BSM scenarios. In the current article we consider a class of models, namely Universal Extra Dimensional (UED) [14] scenario, with nonvanishing boundary localised terms (BLTs), where the low energy effective Hamiltonians are controlled by local operators that are same as in the SM. In this type of scenario flavour violation and CP violation are entirely governed by the CKM matrix, with Minimal Flavour Violation (MFV) as defined in [6,9,15,16].
UED [14] scenario is a specific extension of SM with one flat space-like dimension (y) compactified on a circle S 1 of radius R. All the SM fields are exposed to the extra dimension y. The fields appeared on this manifold are generally defined as towers of 4-dimensional (4D) Kaluza-Klein (KK) states while the zero-mode of the KK-towers is recognised as the corresponding 4D SM field. SM chiral fermions are emerged in this scenario by imposing a discrete symmetry Z 2 (y ↔ −y) on the extra spatial dimension. Therefore, the extra dimension is known as an S 1 /Z 2 orbifold and as a consequence physical domain extends from y = 0 to y = πR. Eventually, the y ↔ −y symmetry is appeared as a conserved parity which is designated as KK-parity = (−1) n , where n is known as KK-number and it measures the discretised momentum along the y-direction. Due to the conservation of KK-parity the lightest Kaluza-Klein particle (LKP) with KK-number one (n = 1) cannot decay to a pair of SM particles and becomes absolutely stable. Therefore, the LKP has been treated as a potential DM candidate in this scenario [17][18][19][20][21][22][23][24]. Besides, a few variants of this model can resolve some other demerits of SM, for example, gauge coupling unifications [25][26][27], neutrino mass [28,29] and fermion mass hierarchy [30] etc.
At the n th KK-level the mass of KK-partner of any SM particle can be expressed as , where m is identified as the zero-mode mass (SM particle mass) and it is very small in comparison to R −1 . Eventually, this UED scenario consists of nearly degenerate mass spectrum at each KK-level. Due to this reason, UED scenario suffers from lack of phenomenological importance, particularly, at the colliders. However, radiative corrections [31,32] can cure the problem of degeneracy in the mass spectrum. The radiative corrections can be divided into two categories, e.g., the first one is bulk corrections (which are finite and only nonzero for KK-excitations of gauge bosons) while the other one is considered as boundary localised corrections. The latter is proportional to logarithmically cut-off 1 scale (Λ) dependent terms. One can allow the boundary correction terms as 4D kinetic, mass and other possible interaction terms for the KK-states at the two fixed boundary points (y = 0 and y = πR) of this orbifold. Actually, it is very natural to consider such terms in an extra dimensional theory like UED, because these boundary terms have served as the counterterms for cut-off dependent loop-induced contributions.
There is a special assumption in the minimal version of UED (mUED) model, where the boundary terms are chosen in such a way that the 5D radiative corrections are disappeared at the cut-off scale Λ. Although, this unique assumption can be discarded and without computing the exact radiative corrections one could parametrise these as kinetic, mass as well as other interaction terms localised at the two fixed boundary points. Hence, this typical version is known as nonminimal Universal Extra Dimensional (nmUED) model [33][34][35][36][37][38][39][40][41]. Within this scenario, apart from the radius 1 UED is regarded as an effective theory and it is characterised by a cut-off scale Λ. of compactification (R), coefficients of different BLTs have been considered as free parameters and that can be constrained by various experimental data of several physical observables. In literature there exists number of phenomenological studies in this scenario. For example, bounds on the values of the coefficients of the BLTs have been obtained from the evaluation of electroweak observables [39,41], S, T and U parameters [37,42], DM relic density [43,44], production as well as decay of SM Higgs boson [45], collider study of LHC experiments [46][47][48][49][50][51], Z → bb [52], branching ratios of some rare decay processes of B-meson: e.g., B s → µ + µ − [53], B → X s γ [54] and [56][57][58], flavour changing rare top decay [59,60] and unitarity of scattering amplitudes involving KK-excitations [61].
Within the scope of this article, to the best of our knowledge, for the first time we explore the ∆B = 2 transitions in the nmUED scenario by computing the KK-contributions to the one-loop box diagrams. The function emerging from the box diagrams is not only affected by the radius of compactification but also by the BLT parameters. Now it has already been mentioned that with the help of this function we can extract the CKM elements which in turn gives geometrical representation of the UT. Therefore, if we compare our theoretical prediction with the current allowed ranges of the CKM parameters and elements of the UT then we can easily constrain the parameter space of this nmUED framework. Moreover, from our study we can also measure the lower limit on R −1 and compare the same with the results obtained from our previous analyses [53][54][55]. Similar kind of exercise was executed several years ago in UED framework [9] where the BLT parameters are zero. In this article, considering the present allowed ranges of the observables [62,63] we will also revisit the lower bound on R −1 in UED framework with the BLT parameters as zero.
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 ∆B = 2 transition in nmUED scenario. Consequently, we present the mechanism of the extraction CKM parameters and the elements of UT. In section IV we will present our numerical results. Finally, we will summarise the results in section V.
II A concise overview of KK-parity conserving nmUED scenario In this section we overview the salient features of the nmUED scenario necessary for our current analysis. One can find detailed description of this scenario in [33][34][35][36][37][38][39][40][46][47][48][49][50][51][52][53][54][55][56]. In order to conserve the KK-parity, coefficients of boundary terms at both the boundary points (y = 0 and y = πR) are kept equal. Therefore, one has a stable LKP in this scenario and hence the present scenario can provide a potential DM candidate (such as first excited KK-state of the photon). In this type of scenario, one can find an extensive study on DM in [44].
In the above expressions, N f n represents the normalisation constant for n th KK-mode wave-function and can readily be derived from the following orthonormality conditions [44,[53][54][55][56] πR 0 and the compact form of this normalisation constant is given by [44,[53][54][55][56] N f n = 2 πR KK-mass of n th KK-excitation is represented by m f (n) and it can be obtained from the following transcendental equations [34,44,[53][54][55][56] r To this end, we would like to discuss the Yukawa interactions in this scenario, as the large top quark mass plays a pivotal role in enhancing the quantum effects in the present work. The action of Yukawa interaction including BLTs with coefficient r y is given by [53][54][55][56] In the above action λ 5 t represents the 5D coupling strength of Yukawa interaction for the third is the 5D Higgs doublet field and Φ = iσ 2 Φ * . Inserting the KK-wavefunctions for fermions (given in Eqs. 2 and 3) in the actions given in Eq. 1 and Eq. 9, one obtains the bi-linear terms containing the doublet and singlet states of the quarks. The resulting mass matrix for the n th KK-level for third generation of quark can be written as the following [53][54][55][56] − φ L The integral I nm 1 is nonvanishing for both the cases of n = m and n = m. However, in the limit r y = r f , this integral becomes unity (when n = m) or zero (n = m). Besides, the integral I nm 2 is nonvanishing only for n = m and becomes unity for r y = r f . At this point we would like to mention that, in our analysis in order to evade the complicacy of mode mixing and construct a simpler form of fermion mixing matrix we choose the condition of equality (r y =r f ) [52][53][54][55][56]. This equality condition 2 (r y = r f ) has been maintained in the rest of our analysis.
With the above mentioned equality condition (r y = r f ), the mass matrix (given in Eq.
where, α tn = 1 2 tan −1 is identified as the mixing angle. The gauge eigen states Ψ L (x, y) and Ψ R (x, y) can be expressed in terms of mass eigen states T 1 t and T 2 t by the following relations [53][54][55][56] φ (n) Both the mass eigen states where, r W , r B and r φ are designated as the coefficients of the BLKTs for the respective fields. 5D field strength tensors are expressed as are considered as the 5D gauge fields corresponding to the gauge groups SU (2) L and U (1) Y respectively. 5D covariant derivative can be where,g 2 andg 1 are represented as the 5D gauge coupling constants. Generators of SU (2) L and U (1) Y gauge groups are represented by σ a 2 (a ≡ 1 . . . 3) and Y 2 respectively. Each of the gauge and scalar fields which are involved in the above actions (Eqs. 13 and 14) can be manifested using appropriate KK-wave-functions as [52-56, 59, 64] and where both the 5D SU (2) L and U (1) Y gauge bosons are generically illustrated by (V µ , V 4 ).
To this end, we would like to discuss some important points by which one could understand the following gauge and scalar field structure as well as the corresponding KK-wave-functions. The Therefore, under this circumstances, it would be very difficult to diagonalise the bulk and boundary actions simultaneously by the same 5D field redefinition 3 except the condition r W = r B . Hence, in the following, we will keep the equality condition r W = r B [52-56, 59, 64] and as a consequence we obtain the same structure (like mUED scenario) of mixing between KK-excitations of the neutral component of the gauge fields (i.e., the mixing between W 3(n) and B (n) ) in nmUED scenario.
Thereafter, the mixing between W 3(1) and B (1) (i.e., the mixing at the first KK-level) provides the Z (1) and γ (1) . This γ (1) (first excited KK-state of the photon) is completely stable due to the conservation of KK-parity and it possesses the lowest mass among the first excited KK-states in the nmUED particle spectrum. Furthermore, it could not decay to pair of SM particles. Therefore, this γ (1) can be considered as a viable DM candidate in this scenario [44].
Let us discuss on the gauge fixing action (considering a generic BLKT parameter r V for gauge bosons) for nmUED scenario [52-56, 59, 64] 3 However, in general one can proceed with r W = r B , but in this case the mixing between B and W 3 in the bulk and on the boundary points generate off-diagonal terms in the neutral gauge boson mass matrix.
where M W (M Z ) is identified as the mass of the SM W ± (Z) boson. For an extensive study on gauge fixing action/mechanism in nmUED we refer to [64]. The above action (given in Eq. 18) is very intricate and at the same time very important for this nmUED scenario where we will compute one-loop diagrams (required for present calculation) in Feynman gauge. Due to the impact of the BLKTs, the Lagrangian leads to a non-homogeneous weight function for the fields with respect to the extra dimension. This inhomogeneity enforces us to define a y-dependent gauge fixing parameter ξ y as [52-56, 59, 64] where ξ is independent of y. The above relation behaves as renormalisation of the gauge fixing parameter since the BLKTs are in some sense contributed as the counterterms taking into account the unknown ultraviolet correction in loop calculations. With this in mind, we treat ξ y as the bare gauge fixing parameter while ξ can be viewed as the renormalised gauge fixing parameter taking the values 0 (Landau gauge), 1 (Feynman gauge) or ∞ (Unitary gauge) [64].
In this nmUED scenario appropriate gauge fixing procedure enforces the equality condition of coefficient of the BLKTs of gauge and scalar fields, i.e., r V = r φ [52-56, 59, 64]. In this limit, KK-masses for the gauge and the scalar fields are equal (m V (n) (= m φ (n) )) and can be derived from the same transcendental equation (Eq. 8). At the n th KK-level the physical gauge fields (W µ(n)± ) and charged Higgs (H (n)± ) have the same 4 mass eigen value and is given by [52-56, 59, 64] Besides, in the t-Hooft Feynman gauge, the mass of Goldstone bosons (G (n)± ) corresponding to the gauge fields W µ(n)± have the same value M W (n) [52-56, 59, 64].
To this end, we would like to focus on the interactions that will involve in our calculation. We can derive these interaction by integrating out the 5D action over the extra space-like dimension (y) using the specific y-dependent KK-wave-function for the respective fields in 5D action. Consequently, some of the interactions are modified by so called overlap integrals with respect to their mUED counterparts. The actual form of the overlap integrals have been given in Appendix A.
Detailed discussions on these overlap integrals have been given in [53]. 4 In the same way one can find the mass eigen values for the KK-excited Z boson and pseudo scalar A. Also, their mass eigen values are identical to each other at any KK-level. For example at n th KK-level it takes the form

III ∆B = 2 transitions in nmUED scenario
The effective Hamiltonian which governs the ∆B = 2 transitions in the SM [65,66] can easily be modified for the nmUED scenario as follows represents the short distance Quantum Chromo Dynamics (QCD) corrections [65,66]. The function S(x t , r f , r V , R −1 ) represents the total contribution in nmUED scenario as given below . m V (n) and m f (n) can be obtained from transcendental equation given in Eq. 8. Moreover, depicts the SM contribution [9,67,68] and is obtained from the box diagrams with (W ± , t) and (G ± , t) exchanges with the m t independent terms eliminated by the GIM mechanism.
The function S n (x t , x f (n) , x V (n) ) indicate the KK-contributions which is obtained by evaluating the box diagrams shown in Fig. 1 with where the function F (x i (n) , x j (n) ) is representing the sum of the contribution of the diagrams corresponding to a given pair ( Finally the compact form of this function obtained from one-loop box diagrams (see Fig. 1) is given by With the increasing value of KK-modes (i.e., with higher value n) the masses of the fields and T 2(n) u become degenerate in nature and consequently the function S n (x t , x f (n) , x V (n) ) diminishes with larger values of n. Therefore, only a few terms in the sum given in Eq. 23 are relevant. Here, I n 1 and I n 2 are two overlap integrals and their expressions are given in the Appendix A.
It has already been addressed that, due to the presence of different BLTs in the nmUED action, the KK-masses and couplings (modified by I n 1 and I n 2 ) involving KK-excitations are nontrivially modified with respect to their UED counterparts. Therefore, it would not be possible to evaluate the function S n in nmUED scenario simply by rescaling the same of the UED model [9]. Hence, we have computed the function S n independently using the box diagram (Fig. 1) for the nmUED scenario. Moreover, it is quite cleare from Eq. 27 that the function S n is drastically different from that of the UED expression. However, if we set the boundary terms to zero; i.e., r V = 0 and r f = 0, then we can easily reproduce the result of the UED version from our expression.
Further, we would like to mention that in our computation of one-loop box diagrams we consider only those interactions in which zero-mode field couples to a pair of KK-excitations with equal KK-numbers. Moreover, in the KK-parity conserving nmUED scenario one can also have nonzero interactions involving KK-excitations with KK-numbers n, m and p, where n + m + p is an even integer. However, we have explicitly verified that the final results would not change significantly even if one considers the contributions of all the possible off-diagonal interactions [52][53][54][55].
To this end, applying the same procedure as in the SM, we can calculate the mass differences ∆M q for nmUED scenario. For this purpose, we can readily adopt the technique of UED scenario as given in [9], because the basic structures (also the structure of operator responsible for B q -mixing) of both UED and nmUED scenarios are similar. Therefore, all other physical aspects (apart from the function S n obtained from the one-loop box diagrams) of both the UED and nmUED scenarios are same. Hence, one can easily write the expression of mass difference in nmUED scenario as Here, F Bq represents the B q -meson decay constant andB q the renormalisation group invariant parameter related to the hadronic matrix element of the operator ∆B = 2 [69].
At this moment, we would like to make a few comments on the QCD factor η B (given in Eq. 22) which has been evaluated within the SM including next-to-leading oder (NLO) QCD corrections.  Fig. 1 can, in principle, differ from the known QCD corrections to the SM box diagrams [65,66] that have been included in η B . However, as the QCD coupling constant α s (µ t ) is small and the QCD corrections to the SM box diagrams are of order of a few percent, therefore one can expect that the difference between the QCD corrections to the diagrams in Fig. 1 and to the SM box diagrams is insignificant [9]. Therefore, in the following, we will use the same QCD factor and hadronic matrix element in SM as well as in the concerned nmUED scenario. With this we will study the ∆B = 2 transitions in nmUED scenario and try to estimate the impact of BLT parameters on the CKM elements and UT.

III.1 Effects of BLTs on CKM parameters and Unitarity Triangle in the nmUED scenario
In nmUED scenario, we evaluate the KK-contributions to the function S n which is significantly affected by BLT parameters (r V , r f ). Therefore, using this function S n we can study the effects of BLT parameters on the elements of the CKM matrix and in particular on the shape of the UT.
However, in order to execute this strategy we need to recall some features of CKM matrix and the UT as depicted in Fig. 2. Using the Wolfenstein parametrisation [7] as generalised to higher orders 5 in λ, the CKM matrix can be written as [67] Here λ, A, ρ and η are the Wolfenstein parameters [7]. The most commonly used UT arises from 5 One should note that here, terms greater than O(λ 4 ) have been neglected. However, this may give wrong results. In order to improve the accuracy of the UT a correction term ≈ O(λ 5 ) has been included to V td element. by dividing each side by the best known one, V cd V * cb . Its vertices are exactly C = (0, 0), B = (1, 0), and A = (ρ,η), whereρ andη of the UT is given by [67]  The lengths R b and R t can be expressed as [9,67] while the angles γ and β of the UT are connected directly to the complex phases of the CKM elements V td and V ub , respectively, via The value of R b (given in 32), i.e., the length of the side AC is determined from |V ub /V cb |. |V cb | and Moreover, we would like to mention that in the nmUED scenario as there are no new complex phases beyond the KM phase, the angle β as extracted by means of a ψK S in B → ψK s is common to both nmUED scenario and SM. The world average value of the angle β which we will use in our following analysis is given below [70] β = (22.2 ± 0.7) • .
Before proceeding further, we would like to mention that, in view of the previous discussions we can easily categorise the present nmUED scenario as an MFV model [15,16]. The reason is that this nmUED scenario (like UED scenario [9]) is a class of extensions of the SM in which only the SM operators in the effective weak Hamiltonian are relevant and flavour violation is entirely governed by the CKM matrix. Moreover, CP violation is governed solely by the KM phase.
Besides, one of the fascinating features of the MFV models is the existence of the universal UT (UUT) [6] that can be constructed from quantities in which all the dependence on NP cancels out or is negligible like in tree level decays. Now, in spite of the existence of common UUT for both the nmUED and SM scenarios, a crucial difference is S(x t , r f , r V , R −1 ) = S 0 (x t ). Therefore, only one from the following observables: ε K 6 , ∆M d and ∆M s will agree with the experimental data for only one of the two scenarios (SM and nmUED).
At this stage, considering the above mentioned facts, we can present a picture which shows how the quantities |V td |,ρ,η and γ are affected by the BLT parameters (r V , r f ) and R −1 . In order to obtain these results in nmUED scenario we adopt the following procedure which is quite similar as given in [9] albeit for UED scenario. At first we use the B 0 d −B 0 d mixing constraint ∆M d . Using the central value of experimental data of ∆M d = (0.5065±0.0019)ps −1 [62,70] and utilising the function given in Eq. 23 we can determine 7 the value of |V td | in nmUED scenario from Eq. 28. After that, using this |V td | and with the allowed value of λ = 0.224747 +0.000254 −0.000059 [70] and V cb = (41.0 ± 1.4) × 10 −3 [62] we will find R t via Eq. 33. Finally, using the following relations [15] η = R t sin(β),ρ = 1 − R t cos(β) and cot γ = we will compute the values ofρ,η and γ respectively, as functions of R −1 and BLT parameters (r V , r f ) in nmUED scenario. With this procedure we can obtain the parameter space which satisfies 6 In view of the discussions given in ref. [9] the effect of the KK-modes on the charm-and mixed charm-top contributions is totally insignificant. Considerable effect for these modes are only top contributions which has been indicated by the same function S(x t , r f , r V , R −1 ) as in the case of ∆M q . Therefore, the K L − K S mass difference ∆M K , is practically dominated by internal charm contributions and essentially unaffected by the KKmodes. Consequently, in the present nmUED scenario we have also found the same phenomenon. We hence refrain from providing the details of ε K in the present article. 7 Here, we use Additionally, with the above mentioned conditions we would also like to impose another condition simultaneously by studying the effects of BLT parameters (r V , r f ) and R −1 on ∆M s . For this purpose, we would like to mention that |V ts | is very close to |V cb | due to CKM unitarity. As |V cb | is common for both the SM and nmUED scenario, therefore, |V ts | is common with an excellent accuracy to both models and consequently we can have the following relation 8   [53][54][55]. 8 In view of the discussions given just before the subsection III.1, we assume that all other quantities like QCD factor, renormalisation group invariant parameter are same for both SM and nmUED scenarios. 9 Here, we would like to mention that, in order to measure the effect of Georgi-Machacek model on ∆M s , the same procedure has been adopted in ref. [72]. 10 In earlier studies, usually 20-30 KK-levels have been taken while adding up the contributions from KK-modes. 11 We have taken M W = 80.379 GeV [62] for SM W ± boson mass and m t (m t ) = 165.25 GeV [63] for SM top quark mass. 12 The summation of KK-contribution is convergent in UED type models with one extra space-like dimension, as far as one-loop calculation is concerned [74].

IV.1 Constraints and choice of range of BLT parameters
Here we highlight the following constraints that have been considered in our analysis.
• In this nmUED scenario, comprehensive analyses on different rare B-decay processes (FCNC type), for example B s → µ + µ − [53], B → X s γ [54] and B → X s + − [55] have been performed. Moreover, these processes have always been played significant role for searching any favourable kind of NP scenario. In all these cases the expressions of these observables are the functions of the same set of parameters i.e., r V , r f and R −1 which are also involved in the expressions of the observables of the current article. Therefore, in this article it is very necessary that we should deal with such parameter space which is satisfied by the experimental data of these observables. Using the expressions of Br(B s → µ + µ − ), Br(B → X s γ) and Br(B → X s + − ) given in [53], [54] and [55] we have considered the branching ratios of these rare decay processes as constraints in our present study. In the following we present the latest experimental data for branching ratios of these processes Observables Experimental value Br(B → X s γ)  Table 1: Experimental data for branching ratios of B s → µ + µ − , B → X s γ and B → X s + − .
• Electroweak precision test (EWPT) is a crucial and significant tool for constraining any kind of BSM physics. Using the technique of the correction to Fermi constant G F at tree level one can perform the corrections to Peskin-Takeuchi parameters S, T, and U in the nmUED model. This is a distinctive feature of this nmUED scenario with respect to the minimal version of the UED model where these corrections emerge at one-loop processes. A detailed exercise on EWPT in this nmUED model has been given in [53,56]. S, T, and U in the nmUED model are the functions of r V , r f and R −1 . Following the similar approach provided in refs. [53,56] we have imposed EWPT as one of the constraints in the analysis.
To this end, we would like to discuss the range of values of BLT parameters used in our present exercise. In general values of the BLT parameters may be positive or negative. However, it is clearly seen from Eq. 7 that, for r f /R = −π the zero-mode solution becomes divergent and beyond r f /R = −π the zero-mode fields become ghost-like. Therefore, any values of BLT parameters lower than −πR should not be considered, but for the purpose of completeness we have taken some negative values of BLT parameters in our numerical analysis. However, from the study of electroweak precision data [53,56] large portion of negative values of BLT parameters have been disfavoured.

IV.2 Results
At this stage, considering the above mentioned constraints, we would like to find the parameter space which satisfy the 2σ allowed ranges of the CKM elements and (∆Ms)exp (∆Ms) SM simultaneously. In the present version of nmUED scenario we have three independent free parameters e.g., inverse of radius of compactification R −1 , dimensionless scaled BLT parameters for boson R V (= r V /R) and fermion R f (= r f /R). In order to find the allowed parameter space we have chosen the following ranges for the free parameters: Using the above ranges of free parameters we have obtained a region in R V − R f plane shown in Fig. 3 parameters [53,56]. Now, in the present analysis we have found a distinguishable observation with respect to our previous analyses on rare B-decay process [53][54][55]. For example, in contrast to the previous analyses [53][54][55], here we obtain lower limit of R −1 with larger values (> 800 GeV) for several combination of positive values of scaled BLT parameters (R V , R f ). In the following we have picked up two sets of nonvanishing scaled BLT parameters (R V , R f ) from the allowed region shown in the Fig. 3 and consequently using the Fig. 4 we will discuss the characteristic dependence of the observables which have been considered in this article with respect to R −1 .
Moreover, depending on the BLT parameters we can estimate the lower bound on the R −1 (using the Fig. 4) that are obtained from the present analysis. Let us focus on the BP1. Speciality of this benchmark point is that, when the curves of each observable intersect the 2σ allowed range of the corresponding observables, then the value of R −1 in each case is around 865 GeV. This shows that the lower limit on R −1 for this combination of scaled BLT parameters is 865 GeV. Let us now discuss the nature of these curves. First of all, using the Eq. 28 we have derived the value of |V td | and it is governed by the function S which is solely dependent on the model parameters R V , R f and R −1 . Now for a fixed value of R V and R f the KK-masses are controlled by R −1 . With the increasing values of R −1 the KK-masses are increased, hence the loop function S is decreased. Therefore, naturally, the value of |V td | is increased with R −1 . However, after a certain value of R −1 , when the KK-masses are very high then decoupling behavior of the KK-mode contribution arises and consequently there is no variation of |V td | with R −1 . After determination of |V td | we have derived R t using Eq. 33. Thereafter, with this R t we have derivedρ,η and γ from the Eq. 36. It is evident from Eq. 33 as for the given values of λ and V cb , R t is proportional to |V td |, therefore R t will follow the same characteristics as |V td | with R −1 .
Moreover, it is also evident from the Eq. 36 thatη is proportional to R t (for a given value of β), therefore,η will be increased with the increasing values of R −1 and obviouslyρ will be decreased.
Further, the increasing behaviour of γ with respect to R −1 is a resultant factor of increment ofη (with R −1 ) and decrement ofρ (with R −1 ). At the KK-mode decoupling limit, the values of all observables are saturated and do not show any variation with R −1 . Now the behaviour of the curve given in the last panel (4c) of the Fig. 4 can be explained in the following way. As we have already mentioned that the nmUED scenario is a class of MFV models, therefore, the NP contribution to one-loop box diagram is always positive. Hence, (∆Ms) (∆Ms) SM (where (∆M s ) = (∆M s ) SM + (∆M s ) NP ) is greater than one. Further, with the increasing values of R −1 (basically KK-mass) the one-loop function S is decreased, therefore, the quantity (∆Ms) (∆Ms) SM is decreased and after a certain large value of R −1 it will become one.
All the above mentioned explanations for all the observables also hold good for BP2. However, for BP2 the lower limit on R −1 is changed. From each panel of the Fig. 4 it is clear that the lower limit of R −1 is around 1.48 TeV for BP2. Now the enhancement of lower limit with respect to the BP1 can easily be explained in the following way. Since in this case of BP2, the values of the BLT parameter are higher than that of the BP1, therefore the KK-masses are decreased. Hence, in order to compensate the KK-mass decrement larger value of R −1 is required. Therefore, for BP2 we obtain larger value of lower limit on R −1 .
From the current analysis we have obtained significantly better result with respect to our previous analyses [53][54][55] on rare decays of B-meson in the same version of nmUED scenario. As for example, we can push the lower limit on R −1 to appreciable higher range (≈ 1.48 TeV or even higher) for favourable choice BLT parameters. These lower limits are so far the most stringent one in the current version of nmUED scenario. However, in our previous studies [53][54][55] have shown that, after a certain values (e.g., R V = R f = 6) the lower limit on R −1 cannot be extended beyond ∼ 800 GeV.
Before we conclude, we would like to remark on the lower limit on R −1 which is achieved in the UED scenario considering the current analysis on ∆B = 2 transitions. We can achieve the UED results from our analysis when BLT parameters vanish, i.e., for R V = R f = 0. In this set up KKmass for n th KK-level simply emerges as nR −1 . On the other hand, the overlap integrals I n 1 and I n 2 become unity. Hence, with this limit, the function S n (x t , x f (n) , x V (n) ) is converted into its UED form. We have independently checked that when the BLT parameters are zero the expressions of the function S n (x t , x f (n) , x V (n) ) is exactly matched with that of the given in ref. [9] 13 . It is clear from Fig. 4, that for R V = R f = 0, we obtain the lower limit on R −1 is around 470 GeV. Definitely, this limit is not very striking one, but close to those values that have been achieved from previous analyses in UED scenario. For example (g − 2) µ [76], ρ-parameter [77], FCNC process [8,9,78,79], Zbb [52,80] and electroweak observables [81][82][83] put a lower bound of about 300-600 GeV on R −1 .
Besides, from the analysis of projected tri-lepton signal at 8 TeV LHC one can obtain lower limit on R −1 up to 1.2 TeV [84][85][86]. At this stage it is needed to mention that the value of lower limit  13 In the article [9], the authors have not considered any radiative corrections to the KK-masses in their analysis.
Therefore, the KK-mass at the n th KK-mode is nR −1 . It has already been alluded that for the vanishing BLT parameters (i.e., R f = R V = 0) we can reproduce the results of the minimal version of Universal Extra Dimensional scenario. Therefore, using our analysis, we have revisited the lower limit on R −1 in the framework of minimal Universal Extra Dimensional scenario in the vanishing BLT limit. In that case the value of the lower limit on R −1 becomes 470 GeV and it is indeed comparable to those values that are determined from the earlier analyses exist in the literature. However, this value is excluded from recent collider analysis at the LHC.
Nevertheless, in the presence of different nonvanishing BLT parameters we can enrich the results of lower limit on R −1 in the current version of nonminimal Universal Extra Dimensional scenario. As for example if we choose a set of BLT parameter (R V = 6.72 and R f = 6.58 : BP1 of Fig. 4) from the allowed parameter space given in Fig. 3 then using our analysis we can obtain the lower limit of R −1 ≈ 865 GeV. Moreover, this limit can be even higher (e.g., ≈ 1.48 TeV) if we choose another set of BLT parameter (R V = 8.76 and R f = 9.06: BP2 of Fig. 4) from the allowed parameter space shown in Fig. 3. Definitely, these results (lower limit on R −1 ) in the current version of nonminimal Universal Extra Dimensional scenario are very promising in comparison to the limits obtained from our earlier analyses on rare decays of B-meson [53][54][55] where, after a certain values of BLT parameters (e.g., R V = R f = 6) the lower limit on R −1 cannot be extended beyond ∼ 800 GeV.
Acknowledgements The author would like to give thanks to Anirban Biswas for computational support.

Appendices
A Feynman rules required for the study of ∆B = 2 tran-

sitions in nmUED
This Appendix contains the Feynman rules required for our calculations with the assumption that all momenta and fields are incoming: 1) G µ f 1 f 2 : ig s T a αβ γ µ C, where C takes the following form:  2) S ± f 1 f 2 = g 2 √ 2M W (n) (P L C L + P R C R ), where C L and C R are expressed in the following way: (A-2) 3) W µ± f 1 f 2 : ig 2 √ 2 γ µ P L C L , where C L takes the following form [53]: where the fermion fields f ≡ u, d, T 1 t , T 2 t .
The mass parameters m (i) x are expressed in the following way [53]: where m i is identified as the mass of the zero-mode up-type fermion and c in = cos(α in ) and s in = sin(α in ) with α in as defined earlier.