One-Loop order effects from one extra universal dimension on λϕ 4 theory

The self-interacting λ ϕ 4 scalar field theory is the prototype in the study of basic renormalisation issues in quantum field theories. In this paper we explore the one-loop order divergent impact to this theory from one universal extra dimension, S1/Z2 , to the self-energy, four point vertex and the beta function. Such effects come as an infinite number of UV divergences corresponding to an infinite superposition of excited KK particles around the loop. We show that dimensional regularisation is adequate enough to control all of them in terms of the product of the one dimensional inhomogeneous Epstein zeta function times the gamma function. From the analytical properties of these functions, the UV divergences are extracted and the counterterms defined; the latter turn out to be of canonical dimension four at the Lagrangian level. We use both, the MS-scheme and a mass-dependent subtraction scheme to remove divergences. Only the latter manifestly satisfy the decoupling theorem.


Introduction
The idea of extra dimensions in field theories dates back to the 1920s [1,2], since then, many related proposals, extensions and critical judgements about this idea have been gravitating in theoretical physics [3][4][5][6]. In particular, phenomenological string theory, naturally formulated in this framework, led the community to revisit the issue decades later from different perspectives. For instance the first proposal for using large (TeV) extra dimensions in the Standard Model, motivated by the problem of supersymmetry breaking in string theory, was given in [7]. The use of the braneworld scenario allows to consistently lower the typical scale of quantum gravity to TeVs by choosing the number and size of spatial extra dimensions [8,9] in the so-called large extra dimensional models, in contrast with warped [10] or universal [11] extra dimensional models; the latter were particularly inspired by [12], and accessible lectures on all these models are for instance [13,14]. Models with universal extra dimensions (UEDs) have recently been a matter of interest since, for example, they are capable of suggesting answers to some still valid questions within the Standard Model (SM), as it is the possible explanation of having three generations of fermions as a result of the global cancellation of anomalies in the SM extension [15], in addition, they provide a convincing explanation of proton stability [16]. The UED models compactified on  S 1 2 orbifolds preserve KK parity and provide a dark matter candidate in the form of stable KK partners [17]. Models containing spatial extra dimensions continue to be a topic with phenomenological importance, and searching of this phenomenon in nature is continuously contemplated in the experiments of the Large Hadron Collider [18,19].
The basic idea in a UED model is rather simple [20]: the stage is a factorisable spacetime geometry with compact spatial extra dimensions, commonly an orbifold, on which a field theory is defined. The dimensional reduction takes place once the extra dimensional content of the fields is harmonically expanded, under certain boundary conditions, and the extra dimensions themselves are integrated out at the action level; the resulting model, which we referred to it as the effective or dimensionally reduced theory, comprises an infinite number of fields defined on the four dimensional Minkowski spacetime  4 , namely, the zero modes (one for each original field) and an infinite number of Kaluza-Klein (KK) mode fields. These fields originally enter as coefficients in the expansions allowed by the corresponding extra spatial geometry. In the dimensionally reduced theory, the KK fields are the only remaining information from the extra dimensions, hence we regard any of their contributions on the effective theory as some kind of extra dimensional input.
Due to the nature of UED models, in order to obtain meaningful physical information out of the dimensionally reduced theory, one must simultaneously deal with the divergent behaviour at large momenta and the infinite number of interactions. These two ingredients make the renormalisation process not necessarily immediate since an infinite number of UV divergences already appear at the one-loop level; although controlling each UV divergence can be done by traditional means, the infinite sum of them represents a complication and an opportunity to investigate on the generality of analytical continuation when handling such infinities in these models. In recent works co-authored by some of us [21][22][23] it was proven that a consistent removal of infinities in these UED models is possible and hence physical interpretation of the impact of extra dimensions can be given; such works are founded on gauge theories, or gauge theories coupled to fermions where the process of fixing the gauge, the intricate KK mass-generation mechanism after dimensional reduction [24], and other subtleties are present. One aim of the present paper is to use the dimensionally reduced model coming from the well-known scalar self-interacting λ (5) 2 , to emphasise the essence of regularisation and renormalisation process at the one-loop order of this type of models.
In this communication we show that dimensional regularisation, whose ultimate justification is found in the analytical continuation of divergent integrals to the complex plane [25], is adequate enough to control the infinite number of contributions from excited KK modes to the two-and four-point vertex functions of the selfinteracting λf 4 to the one-loop order. They can be encapsulated into the product of the one dimensional inhomogeneous Epstein zeta function [26][27][28][29] times the gamma function. Therefore the whole process of extracting UV divergences is reduced to analyse the singularities of such product of functions. Remarkably, the poles of this product can be explicitly obtained in the low energy limit m 2 /R −2 =1, where R is the radius of the extra dimension and m is the mass associated to the zero mode field f. At the one-loop order, we show that in both, the two-and four-point standard vertex functions, the extra dimensional contribution contains divergences of the typical form (1/ò) with constant factors, where ò is the usual parameter in dimensional regularisation (ò=4−D).
In order to cancel out the UV divergences already described at the one-loop order we use two different types of schemes to define the counterterms at the Lagrangian level: the MS-scheme and a mass dependent subtraction scheme with a kinematical or subtraction scale M 2 . Interestingly, we will see that in the presence of one UED, with the geometry  S 1 2 , there is no need to go beyond operators of canonical dimension higher than four to define these counterterms at the one-loop level. The reason for the examination of two different subtraction schemes is to gain some insight in relation with the decoupling theorem by Appelquist and Carazzone [30]. It is known that although the MS-scheme removes the UV divergences, the corresponding amplitudes and beta function, do not obey this theorem; in contrast, when using a mass dependent subtraction this theorem is manifestly satisfied [31]. This behaviour is followed in the context of the dimensionally reduced theories as it was already pointed out in [21] and it will be confirmed in the present communication.
As a word of warning, we must say that in this work we define the physical mass of the zeroth mode and the effective coupling constant by handling the infinite number of UV divergences present at relevant light particle vertex functions to the 1-loop approximation, that is, we do not attempt to give a meaningful complete approach to the renormalization of the whole dimensionally reduced model in this work; nevertheless, our contribution may shed some light on this goal.
In terms of [32] our method is closer to a 'KK-renormalisation', in contrast to the 'KK-regularisation' in which the infinite sums over KK modes are performed before the integration over internal loops [33]. A key mathematical resource in our work is the Epstein zeta function [26], which is a generalisation of the well-known Riemann zeta function [34]; both being very interesting objects in their own right. The systematic use in physics of zeta-regularisation methods dates back to the 1970s with seminal works as [35] and [36] in the context of effective theories and path integrals on curved spacetimes, respectively. Applications of zeta functions can also be spotted in quantum gravity models as well as cosmology [37], string theory [38], and crystallography [39]. Complete modern treatises on this topic are [28,29]. Probably the most famous application of the zetaregularisation is found in the Casimir effect [40] where the calculation of the physical vacuum energy of a quantised field in the presence of external boundary configurations is performed. More recently, the computation of the vacuum energy for scalar fields at different temperature limits has been solved by the use of the Epstein zeta function [41,42]. We mention in passing that the emergence of the Epstein zeta function in our regularisation proposal may be not as surprising as it looks since the boundaries of the compact extra dimensions may resemble the external boundary configurations in the Casimir effect.
The rest of the paper is organised as follows. In section 2 we describe the dimensionally reduced model for the case of one UED, we point which terms are necessary to calculate the contributions from extra dimensions to the λf 4 model. In section 3 we study the one-loop order effects of one UED on the two-and four-point vertex functions associated to the λf 4 theory. In sections 4 and 5 the UV divergences present in the two-point and fourpoint standard vertex function are respectively extracted and the counterterms to cancel them out are defined in the MS-scheme, the beta function is also presented in this subtraction scheme. In section 6 we migrate the previous results to a mass dependent subtraction scheme and show that the decoupling theorem is satisfied, the beta function calculated in the MS-scheme is recovered when the kinematical mass scale is much larger than any mass characterising the model. In section 7 the summary and final remarks are given. By the end, the appendix is reserved to gather the Feynman rules of the dimensionally reduced theory. Although in the present paper we concentrate on the contribution from one UED, in a subsequent paper we will extend our analysis to the case of the contribution from an arbitrary number n of UEDs, with geometry ( )  S ; such case requires the analysis of higher dimensional inhomogeneous Epstein zeta functions, and the Lagrangian counterterms to correctly subtract infinities will exhibit a richer structure.

The dimensionally reduced scalar model from one UED
In this section we describe the dimensionally reduced scalar model obtained from the self-interacting scalar field theory λ (5) Φ 4 , plus compatible operators of canonical dimension higher than five, with one UED. This dimensionally reduced model will be the starting point in order to investigate on the impact of the fifth dimension to the λf 4 theory. In the spirit of [24,[43][44][45] we begin with where ; 5 , and Φ is a real valued scalar field on . The index M=0, 1, 2, 3, 5 and Einstein sum convention for repeated indices is understood. In the scalar theory (2.1), the scalar field Φ and the coupling constant λ (5) have canonical dimensions of 3/2 and −1, respectively; the last term contains ¢ s different operators ¢ +  s r 5 of canonical dimension 5+r, r1, constructed from Φ and ∂ μ Φ, Λ is at an energy scale above which the new physics would begin to manifest itself, and b ¢ r s , are dimensionless parameters that depend on the details of the underlying physics. Notice that operators of higher canonical dimension are suppressed by powers of Λ. The field Φ is assumed to fulfil the following periodicity and parity conditions where R is the radius of the circle S 1 , the latter is equivalent to imposing Neumann boundary condition at the fixed points of the orbifold [14]; relations (2.2) allow the Fourier expansion in the extra dimension of the field itself As we know the coefficients in the Fourier expansion render the precise profile of the function that is being expanded, as they modulate the trigonometric functions. In this case the field modes define the profile of Φ on points in the extra dimension. It is in this sense that KK modes contain information about the behaviour of Φ on the extra dimension. Directly from the expansion(2.3), one can see that every KK mode has canonical dimension equal to 1.
Using the orthogonality of the trigonometric functions enables an immediate integration of the compact extra dimension out of the action(2.1) and defines the following dimensionally reduced theory the myriad interaction terms between light and heavy fields are gathered in the second term, where and the third term represents the purely KK excited mode terms, in this case, In the set of equations (2.5), the dimensionless multi-indexed symbols Δ (L) are defined as follows: Directly from their definition, the multi-indexed objects Δ (L) are totally symmetric. The dimensionless effective universal coupling constant λ is defined as λ (5) /(2πR). The term >  d 4 contains all the interactions of canonical dimension higher than 4, such terms must be included in the 4-dimensionally reduced theory because the 5-dimensional theory is nonrenormalisable according to Dyson's criterion. It is worth mentioning that in this work we are solely interested on Green's functions at the one-loop order that only contain zero mode particles as external legs, and at most tree level effects coming from >  ; d 4 remarkably, we will show that when the dimensionally reduced theory is coming from five dimensions, the counterterms needed to cancel out the UV divergences in this type of Green's functions are of canonical dimension four.
The Feynman rules for the interactions in (2.5) are gathered in the appendix, from them there can be classified essentially two different types of k-point Green's functions depending on the type of external legs, namely: the Standard Green's functions (SGFs), whose external legs only comprise zero mode particles, and the Non-standard Green's functions (NSGFs), whose external legs contain at least one excited KK particle. The latter can either be Hybrid Green's functions, with some zero and some excited particles as external legs, or purely excited KK Green's functions with only KK excited particles as external legs. In particular the k− point SGFs, hereafter denoted by G (0K0) where there are k slots filled with zeroes, will receive contributions from excited KK mode particles within the loops. In fact, the Lagrangian ( )  0 together with the infinite number of interaction like the first term in (2.5b) provide the necessary terms to calculate the impact from the fifth dimension to G (0K0) at the one-loop level, in other words, to the four dimensional λf 4 theory. An interesting observation is that the superficial degree of divergence of these one-loop contributions, would indicate the possibility of removing UV divergences, that is, divergences that arise due to short distance effects in  4 . However, as we will see later, the presence of an infinite number of interactions between the zero mode and KK modes will imply an infinite number of UV divergences at the one-loop level. This issue is characteristic of this type of theories and will be tackled on the following section.

One-loop structure from one extra dimension
Then, the bare Lagrangian can be written as . .
The contributions ( ) 2) contain interactions between the light and heavy fields, and among pure heavy fields, whose specific structure will not be needed here. The where the term within parenthesis is nothing but the light scalar field self-energy which consists of three contributions: the first one, M (0) (p 2 ), comes from the well-known zero mode self-interacting term λf 4 in ( )  0 , the second one, ( ) , is the result of the infinite number of interactions between f and excited KK modes present in ( )  k 0 , henceforth we have an infinite sum of excited modes around the loop (see figure 1), and finally the third one, M c.t. , is the usual counterterm for the self-energy, that is, which comes from (3.3). This term is sufficient to cancel out all the UV divergences present in the self-energy as it will be seen below. The first two contributions to the self-energy in equation (3.5) are written in short as follows: where the symbol k stands for {0,k}.
If we want to analyse the corrections to the coupling constant of the f 4 -theory due to the extra dimension at the one-loop, we consider the four point SVF Γ 4R (p i ) the loop contributions which modify the coupling constant at low energy and Γ c.t. (p i ) the counterterm that will remove the UV infinities. The first term in the parenthesis of equation (3.8) corresponds to the light field f around the loop, whose presence is due to the common term λf 4 in ( )  0 , whereas the second one corresponds to an infinite sum of excited KK mode particles circulating around the loop (see figure 2), whose source is the infinite number of interactions between the light and heavy fields (see the first term in (2.5b)). Each of these terms contains a sum over the Mandelstam variables, that is, The third term in the parenthesis in equation (3.8) is which comes from the counterterm (3.3). As we will see, this is the only counterterm needed to cancel out the divergences in (3.8). This result is remarkable, and can be rephrased by saying that the effects from one extra  dimension to the self-interacting λf 4 scalar theory become finite with the aid of counterterms of canonical dimension four given by (3.3); however, as we will see in a subsequent paper when more extra dimensions are involved this is no longer true, as operators of canonical dimension higher than four are needed.
Notice that (3.7) and (3.10) have identical structures to those that appear in the pure λf 4 theory to the oneloop, with the difference that these expressions also represent excited KK mode contributions. The above suggests that applying dimensional regularisation will reveal an infinite number of UV divergences in equations (3.5) and (3.8) which will define the corresponding counterterms We will show in the following subsection that this infinite number of UV divergences can be controlled by using the so-called one dimensional Epstein zeta function [26][27][28][29]37].

Regularisation
In order to deal with the vast number of UV divergences that at the one-loop level impact the two-and fourpoint SVFs, equations (3.5) and (3.8), respectively, we now introduce dimensional regularisation, which in turn permits to spot the emergence of the one dimensional inhomogeneous Epstein zeta function (see [26,28,29] where m j are internal masses of particles within the loop that appear in any generic vertex function to one-loop order in a given context, and p j are related to the external momenta with p 1 ≡0, e.g.(3.7) and (3.10). For our divergent integrals we will apply dimensional regularisation by promoting the four dimensional spacetime to a D dimensional spacetime and define s≔N−D/2. Comparing equation (3.13) with (3.7) we have N=1 and s→−1 as the four dimensional limit, whereas from (3.13) and (3.10) we have N=2 and s→0 as the four dimensional limit 3 . It is particularly interesting the emergence of the one dimensional inhomogeneous Epstein zeta function ( ) E s c 1 2 subsequent to both dimensional regularisation and the non-cutoff of the series involving excited KK modes. Before going into that analysis, let us formally consider the contribution from all excited KK modes to the self-energy (see (3.5) and 2 when the Lorentzian metric is used. From this viewpoint of the excited KK modes contribution, there are basically two types of potential sources of UV divergences or divergences due to small distances: one is coming from small distances in  4 or very large value of continuum momentum, that obviously makes the integral over p diverge, and the other is coming when we allow the integer quantised momentum to increase indefinitely, in such case p 5 increases itself and the series over k might be not well defined. Therefore one requires an analytical continuation to regularise this behaviour. In fact, when dimensional regularisation is applied such analytical continuation is actually behind the curtains justifying all integrals in D dimensions as we learned from 't Hooft and Veltman [25]. Similar conclusion can be drawn for the excited KK modes contribution to the 4-point SVF.
Applying dimensional regularisation to the divergent integrals in the 2-point SVF (3.5), with s=1−D/2, we obtain where c 2 ≔m 2 /R −2 , μ is the common unit of mass introduced in dimensional regularisation and equation (3.12) has been used. The parameter c controls the scale energy of the heavy modes, |c|=1 is the low energy limit. The first term in the squared brackets is derived from M (0) , this is the well-known divergent term present in the f 4 self-interacting scalar theory, and the second term comes from which is the consequence of an infinite number of interactions of f with the excited KK modes. Since the term within the brackets in equation (3.15) is independent of p 2 we can remove p 2 δ Z by setting δ Z ≡0; therefore the first term in the ( )  c t . 0 vanishes at the one-loop order level. Needless to say that the possible UV divergences can be read as poles of either Γ(s) or the product of the one-dimensional inhomogeneous Epstein zeta function ( ) E s c 1 2 times the gamma function Γ(s), hence we must be careful in handling the limit s→−1, or equivalently, D=4. In particular, within the product ( ) ( ) G E s s c 1 2 one should avoid the direct application of the 'limit of a product' rule. In fact, we must exhaust algebraic manipulations in this product before taking the limit. This is the main technical difference with respect to [46], whose analysis is performed for the case of one spatial extra dimension S 1 . If UV divergences are expected to be removed from SVFs in the low energy limit, we need to isolate divergent sources out of asymptotic formulae and judiciously define the necessary counterterms.
The same procedure of dimensional regularisation can be applied to the 4-point SVF, leading to the expression

One-loop mass corrections in the presence of one extra dimension
The counterterms defined in these subsections in order to deal with the one-loop effects from the fifth dimension at the level of the mass and coupling constant of the self-interacting scalar field theory λf 4 will be inspired by the MS-scheme [47,48], i.e. a mass independent scheme at which the UV divergences of Feynman graphs will be cancelled out by terms defined as poles at D=4. Interestingly, as we have mentioned such terms define the countertemrs of canonical dimension four, see (3.3).
In the presence of a single extra dimension, we analyse the UV divergences in Γ 2R and Γ 4R . The non-trivial source of these poles is found in G E c 1 2 in the corresponding four-dimensional limit (see equations (3.15) and (3.16)). Our problem then comes down to investigate the structure of the poles of G E c 1 2 and this will be done in the limit |c|=1, where the formula for this product is known, see [27] 4 , namely As it can be seen from (4.1), the 2-point SVF can be written down as follows in the low energy limit, where the four dimensional limit is obtained when ò→0, with ò/2≡s+1=2−D/2, that is, s→−1. The first term in the squared brackets contributes with the well-known UV divergence, namely the one resulting from the following asymptotic expansion 5 , here, the '(finite)' part contains typical constants like the Euler-Mascheroni γ E and a logarithm that includes in its argument the ratio 4πμ 2 /m 2 . The analysis on the second term in (4.2) is subtler, the possible sources of divergences may come either from the arguments of the gamma or the Riemann zeta function ζ. Therefore, the poles will be found when the argument of the gamma function is either a negative integer or zero, and/or when the argument of the Riemann zeta function becomes one. These conditions can be expressed as follows: 4 We must say that in the sense of [27] this expression is nothing but a non-regularised version of the product ( ) ( ) G E s s c 1 2 , see also [21]. 5 We use the twiddle sign ∼ to mean asymptotic equalities.
The equation (4.4a) is only satisfied for k=0 and k=1, and there is no positive integer k that fullfils the identity (4.4b). Therefore the only source of UV divergences within Γ 2R for one extra dimension is encoded in the gamma function, this will not be the case for greater number of extra dimensions as we will report elsewhere. At k=0 the relevant asymptotic expression as ò→0 is the following At k=1 the relevant asymptotic expression for small ò is From the relation (4.5) notice that although the gamma function diverges as ò goes to zero, the product Γ(ò/2−1)ζ(ò−2) is finite. In other words, the term corresponding to k=0 does not contribute to the pole at all. This is not the case for the term k=1, as we can see from the asymptotic equality (4.6) which shows the typical ultraviolet divergence 1/ò as ò→0.
Introducing the asymptotic relations (4.3), (4.5) and (4.6) into the 2-point SVF (4.2) we have for small ò Since the series left after isolating the divergent terms (at k = 0 and k = 1) converges in the limit |c|=1, we conclude that the 2-point SVF (3.15) will become UV finite at the one-loop level if we define the proper counterterm to avoid the ultraviolet divergence that we just found. In the spirit of the MS-scheme prescription [47,48], where the UV divergences of a Feynman graph must be cancelled out by counterterms that can be read from the poles at D=4, we define When the heavy fields become infinitely massive, that is, in the limit c→0 or equivalently R −1 →∞, we expect in equation (4.7) the decoupling of the KK contribution, however this does not happen, notice the term proportional to 1/c 2 which diverges in the limit c→0; even more, the '(finite)' part in this expression contains a logarithmic divergent term proportional to ( ) pc log 2 . Therefore, although the MS-scheme removes the UV divergences, it is not concerned about the decoupling theorem; nevertheless a mass dependent scheme to cancel out divergences ensures the decoupling as we will show in section 6. This is also true in the dimensionally reduced version of QED as it was shown in [21].

One-loop corrections to the coupling constant in the presence of one extra dimension
We now apply a similar treatment to Γ 4R (p;s) given in equation (3.16). This expression diverges as s→0, or equivalently as ò→0 with ò/2≡s. Since our interest is to isolate UV divergences, the idea is then to examine the pole structure of the Γ 4R (p i ;s) and for that we will take advantage of the expression in the low energy limit, |c|=1, equation (4.1). Then the 4-point SVF can be expressed as 6 Remember that the Euler-zeta function ( ) z = å = ¥ s E k n 1 1 s diverges in the Cauchy sense at s=0, however once this function has been analytically continued into the Riemann zeta function ζ(s) it has well defined value. where F k (p 2 ) is defined as The contribution to Γ 4R (p i ;s) from the light particle around the loop is encoded in the first term in the squared brackets of (5.1). This is the well-known contribution from ( )  0 , the corresponding input to the pole can be calculated using where the '(finite)' part in (5.3) contains the Euler-Mascheroni constant and a logarithm which depends on momentum as (m 2 +p 2 z(z−1))/4πμ 2 is integrated in the z variable. Regarding the second term in the squared brackets of (5.1), the divergent terms can be known from the analysis of the arguments of the gamma and Riemann zeta functions, in fact, these functions diverge whenever respectively. In this case, there is only one solution for (5.4a) that is when k=0, and there is no solution for (5.4b) since there is no positive integer k that fullfils it. Hence, the only source of UV divergence at Γ 4R comes also from the gamma function only. When k=0 the asymptotic of such term can be obtained from (4.6).
Introducing the asymptotic behaviour relations (5.3) and (4.6) into (4.2), we get for small ò which means that the strength of the coupling increases as the energy increases. Notice that the extra dimensional contribution reduces by a factor 1/2 the beta function in the absence of the fifth dimension. It is important to emphasize the fact that when R −1 →∞ (c→0) the decoupling theorem is not fulfilled either for the Γ 4R or the beta function. Indeed, regarding Γ 4R , see equation (5.5), one must say that inside the '(finite)' part there is a term proportional to ( ) pc log 2 which makes the decoupling theorem not evident. In addition, the beta function is simply an R-independent constant, therefore the decoupling theorem is not manifest at all. This is not a surprise since as it is well known: the major difference between dimensional regularisation together with the MS-scheme and other schemes is that the β function is independent of the scale [21,31], even in several types of massive fields in two-and four-dimensional curved spacetimes [49]. In order to restore scale dependence we should migrate to a mass dependent subtraction scheme, this will be the main idea in the following section.

Decoupling of the KK contribution in the presence of one extra dimension
In this section we will migrate to a mass dependent subtraction scheme by the choice p 2 =−M 2 , with M the kinematical or subtraction scale. We begin by analysing the one-loop contribution of the self-energy given by (see (3.15) with s+1=ò/2) where none of the terms in the squared brackets depend on p 2 , hence δ Z =0. We determine the counterterm δm 2 using the typical kinematical condition In this scheme the self-energy M 5D (p 2 )=0 for all p 2 . At the one-loop level, the renormalised mass is not impacted by extra dimensions once the UV divergences are cancelled out, this is because M 5D (p 2 ) does not depend on the external momentum p 2 unlike in the dimensionally reduced QED [21]. At this level, not much can be said about the decoupling, and we might have to extend our calculations to two loops if we want to explicitly see the decoupling of the KK contribution. Still, the 4-point SVF is more enlightening in this aspect. For the 4-point SVF we have an explicit dependence on the external momenta at the one-loop order. The 4-point SVF (3.16) when ò/2=s is introduced becomes, In order to determine δλ, we appeal to the following condition: where we have explicitly written all the '(finite)' terms Then the UV-divergent free 4-point SVF is obtained by the substitution of (6.7) into (6.4) giving

ðA:3Þ
Finally, from the purely excited KK sector ( )  k we read the following interaction rule: ðA:4Þ Notice that in fact, from the rule (A.3) we have a mixing phenomenon between f and f (2 k) that mimics the well-known γ-Z mixing present in the Weinberg-Salam model [50,51].