# Distinctive ultraviolet structure of extra-dimensional Yang-Mills theories by integration of heavy Kaluza-Klein modes

###### Abstract

One–loop Standard Model observables produced by virtual heavy Kaluza–Klein fields play a prominent role in the minimal model of universal extra dimensions. Motivated by this aspect, we integrate out all the Kaluza–Klein heavy modes coming from the Yang–Mills theory set on a spacetime with an arbitrary number, , of compact extra dimensions. After fixing the gauge with respect to the Kaluza–Klein heavy gauge modes in a covariant manner, we calculate a gauge independent effective Lagrangian expansion containing multiple Kaluza–Klein sums that entail a bad divergent behavior. We use the Epstein–zeta function to regularize and characterize discrete divergences within such multiple sums, and then we discuss the interplay between the number of extra dimensions and the degree of accuracy of effective Lagrangians to generate or not divergent terms of discrete origin. We find that nonrenormalizable terms with mass dimension are finite as long as . Multiple Kaluza–Klein sums of nondecoupling logarithmic terms, not treatable by Epstein–zeta regularization, are produced by four–dimensional momentum integration. On the grounds of standard renormalization, we argue that such effects are unobservable.

###### pacs:

11.10.Kk, 11.15.–q, 14.70.Pw,14.80.Rt## I Introduction

The formulation of a physical theory describing nature at the most fundamental level is one of the main incentives behind investigations framed within high–energy physics. While experiments have set some hints PDG pointing towards this aim, on the theoretical side there are many possibilities available. Among the vast collection of ideas, the conjectural existence of compact extra dimensions Kclss1 ; Kclss2 ; Anto ; ADD ; AADD is the setup for the present work.

The exploration of Standard Model extensions with the ingredient of extra dimensions has been motivated by some of the most intriguing questions nowadays. A world in which dark matter particles are part of the field content of Kaluza–Klein effective theories has been widely investigated ChMS ; ChFM ; SerTaotro ; SerTa ; HooKr ; BHS ; KoMa ; HooPro ; DHKM ; BNP ; BKP ; BMMO . Studies of neutrino physics in extra–dimensional contexts have been carried out as well, including the generation of neutrino masses DDGnu ; MNPL ; MRS ; ADDMR ; ADPY ; BMOZotro ; BMOZ ; OhRi and the physics behind neutrino oscillations DDGnu ; MNPL ; ADPY ; DvSm ; BCS .
The physics of the Higgs boson in extra–dimensional scenarios has been also an object of study AStr ; HaKo ; RizzWe ; FPetr ; AppYee ; CGM ; DLR ; BhKu ; SaKuRa ; BBD ; NThiggs ; BGHHLMSS ; CCGHN ; KNOW ; BBBKP .

Among the different models of extra dimensions, there is the formulation in which the whole Standard Model is defined in the extra–dimensional spacetime, where all the dynamic variables are allowed to propagate, so that those particles included by the Standard Model in four dimensions are the lowest–energy manifestations of such extra–dimensional fields, which describe nature at a higher–energy scale. This framework, commonly known as universal extra dimensions ACD , is the setting of the study performed in the present paper. Investigations centered in the cosmological role of the lightest Kaluza–Klein particle BKP ; CPS , a Kaluza–Klein Higgs boson BBBKP ; DeyRa ; DPR , and experimental data ATLASUED ; ATLASUED2 from the Large Hadron Collider have provided upper bounds on the size of universal extra dimensions, corresponding to energy scales that range around 1 TeV for the case of just one compact dimension.

The recent observation ATLAShiggs ; CMShiggs of a Higgs–like particle at the Large Hadron Collider is the last piece of the realization that nature is governed by gauge symmetry. In this context, the fate of extra–dimensional gauge symmetry at the level of Kaluza–Klein theories becomes a matter of interest. From the four–dimensional viewpoint, extra–dimensional gauge symmetry is split into two disjoint sets of transformations that have been termed the standard gauge transformations and the nonstandard gauge transformations NT5DYM ; CGNT ; LMNT .
The set of standard gauge transformations is a gauge subgroup that is identified with the gauge symmetry characterizing the four–dimensional low–energy formulation, while the rest of the extra–dimensional gauge group, which corresponds to the nonstandard gauge transformations, remains hidden LMNT .
With these ingredients, the quantization of Kaluza–Klein gauge theories can be carried out NT5DYM .
One can take advantage of the mutual independence of the two four–dimensional sets of gauge transformations and execute quantization while leaving four–dimensional gauge invariance untouched NT5DYM . In practice, the presence of this symmetry is convenient, for it introduces simplifications FMNRT ; NTKKint in calculations.

The Becchi–Rouet–Stora–Tyutin quantization BRS1 ; BRS2 ; Tyutin ; GPS of gauge theories in five dimensions was detailed in a paper by some of us NT5DYM . Among the main points of that work, a four–dimensional set of SU()–covariant gauge–fixing functions was proposed in order to maintain four–dimensional gauge invariance in the quantum Kaluza–Klein theory. This was then utilized to integrate out NTKKint all the Kaluza–Klein excited modes and derive an effective Lagrangian featuring four–dimensional gauge invariance. In the present paper, we go further in this direction and perform such calculation for the case of extra dimensions. We resort to the results reported in Ref. LMNTotro , in which a full analysis of the Kaluza–Klein Lagrangian generated by the –dimensional Yang–Mills theory was performed.
We find that a gauge independent result can be obtained by the delicate interplay of the contributions from the pure–gauge, pseudo–Goldstone and ghost–antighost Kaluza–Klein sectors.

Nonrenormalizability of extra–dimensional formulations manifests in our results, which include multiple infinite Kaluza–Klein sums.
Using a regularization scheme GLMNNT , which is based on the Epstein–zeta function Eps ; PoTi ; NaPa ; Zuck ; Hard ; Sieg ; Glass ; Glassotro , we identify and isolate divergences that are inherent in such multiple sums.
Our effective Lagrangian expansion involves nonrenormalizable terms whose mass–dimension is as large as 6. We determine that Kaluza–Klein sums in these terms produce a divergence if the number of extra dimensions is greater than 1.
Similarly to the general discussion of Ref. GLMNNT , we find that improvements in the accuracy of this effective Lagrangian would modify the ultraviolet behavior, by Kaluza–Klein sums, as long as the number of extra dimensions is large enough. Besides such discrete divergences, there are terms in our effective Lagrangian expansion that include standard divergences (short distance effects on the standard four–dimensional spacetime manifold) and nondecoupling logarithms that are affected by multiple Kaluza–Klein sums. Nevertheless, following previous results GLMNNT , we argue that these nondecoupling effects are unobservable, since they can be absorbed by the standard renormalization procedure, used to eliminate standard divergences.

A complete and detailed study about the quantization of gauge theories comprising extra dimensions is underway and will be presented elsewhere inprog . Anyway, in the present paper we provide some advances on that matter, including the tree--level couplings from the ghost--antighost sector that contribute to standard Green’s functions^{1}^{1}1Throughout the paper we use the term standard Green’s function to refer to any Green’s function generated by loop diagrams in which all external legs are Kaluza–Klein zero modes. at the one–loop level and a generalization to extra dimensions of the Kaluza–Klein covariant gauge–fixing functions given in Ref. NT5DYM . A simple relation among one–loop contributions from Kaluza–Klein pseudo–Goldstone bosons and those from the ghost–antighost sector is observed, which also occurs in five dimensions NTKKint ; FMNRT .

This document has been organized in the following way: we develop a brief discussion on the Kaluza–Klein model in Section II, which includes the mass–spectrum of the Kaluza–Klein scalars and the Kaluza–Klein couplings that are necessary to carry out the main calculation; then, in Section III, we provide some results on the quantization of the Kaluza–Klein theory; Section IV is dedicated to the integration of the Kaluza–Klein excited modes, covering the proof of gauge independence and the Epstein–zeta regularization of divergences from Kaluza–Klein sums; finally, in Section V, we give a summary of the paper.

## Ii Tree–level Kaluza–Klein couplings contributing at one loop

Theoretical aspects of field theories in extra dimensions have been considered in diverse works ADD ; ACD ; DDGed1 ; RS1 ; RS2 ; DDGed2 ; MPR ; Hol ; DoPo ; BDP ; Uek ; NT5DYM ; LMNT ; LMNTotro . In this section, we provide some results that are necessary ingredients to perform the main calculation of the paper. In what follows, we use the notation of Refs. LMNTotro ; GLMNNT , where fully detailed discussions on all these results can be found.

We begin by assuming that, at some high–energy scale, spacetime looks like a plane manifold, , comprising dimensions and being characterized by a Minkowski–like metric, . Uppercase indices run over all the spacetime coordinates, so that . Any field formulation nested in this spacetime will be governed by the extra–dimensional Poincaré group ISO(). We also assume that all fields propagate in the whole spacetime, so that they are functions of –vector coordinates . We consider the SU()–invariant Lagrangian

(1) |

given in terms of extra–dimensional gauge fields, which are denoted by and which define the Yang–Mills curvatures as . Here, the is the SU() coupling constant, with dimensions , and represents the structure constants of the gauge group. Lowercase indices correspond to the gauge group, which means that .

So far, experiments have not found any evidence PDG ; ATLASUED ; ATLASUED2 pointing to the actual existence of extra dimensions, which can be explained as long as these extra dimensions are small enough.
The transition of the Lagrangian from the extra–dimensional manifold to the four–dimensional perspective is implemented GLMNNT by two canonical transformations.
The first of such transformations maps covariant objects of ISO() into ISO()–covariant objects:
extra–dimensional vector gauge fields are split into ISO() 4–vectors , with , and a set of ISO() scalars , in which .
This transformation maintains invariance under the extra–dimensional Poincaré group, though keeping it hidden and only showing manifest invariance with respect to ISO(1,3). As we consider lower energies, the compact structure of extra dimensions becomes apparent and the extra–dimensional Poincaré group ISO is explicitly broken by compactification. We assume that the resulting spacetime is , where represents the standard four–dimensional spacetime and the submanifold stands for all compact dimensions, whose radii are . With this structure of the compact extra dimensions, the gauge fields acquire periodicity properties, , and parity properties, , as well.
The implementation in the Lagrangian of the explicit breaking of the extra–dimensional Poincaré group ISO(1,) is carried out by LMNT ; GLMNNT a second canonical transformation, which turns out to be Fourier expansions of the fields that are consistent with the periodicity and parity properties adopted by them. These expansions are commonly known as Kaluza–Klein towers.

After using the aforementioned Fourier series, all dependence on the extra–dimensional coordinates of the Yang–Mills Lagrangian , Eq. (1), can be integrated out, which generates a four–dimensional effective theory, , whose dynamic variables are the Kaluza–Klein modes. To make sure that the low–energy limit of is just the Yang–Mills theory in four dimensions the correct parity conditions for the extra–dimensional fields are , and . With this choice, the 4–vector unfolds into a set of gauge Kaluza–Klein modes, of which () are zero modes, , and are excited modes, .
For excited modes, the Kaluza–Klein indices are nonnegative integer numbers, but the case in which all of these indices are simultaneously zero is excluded, for it corresponds to zero modes.
The zero modes are recognized as the dynamic variables of the low–energy theory, and consistently they behave as gauge fields with respect to the standard gauge transformations.
The ISO(1,3) scalars , on the other hand, are decomposed into scalar Kaluza–Klein modes, . Again, the only combination of Kaluza–Klein indices that is excluded is .

The general case of extra dimensions brings intricate expressions, which are difficult to read. For this reason, a convenient notation, which evokes intuition and directly generalices the results of the five–dimensional case NT5DYM is desirable. To this aim, we define , which we use to express zero modes more briefly as . We represent all other possible arrangements of Kaluza–Klein indices generically by , so that Kaluza–Klein excited modes are compactly denoted by and . Then, we use the sum LMNTotro ; GLMNNT

(2) | |||||

which comprises all possible arrangements of Kaluza–Klein indices for a given number of extra dimensions. In terms of the sum given in Eq. (2), all results have the same structure that is found in the case of just one extra dimension NT5DYM . Notice that this definition involves multiple infinite sums.

The extra–dimensional curvature inherits, from the gauge fields defining it, specific periodicity and parity transformation properties with respect to the extra coordinates, which means that it can also be expanded in Kaluza–Klein towers. The implementation of the first canonical transformation in the extra–dimensional Lagrangian separates the extra–dimensional curvature into three components that possess definite transformation properties under ISO(), that is, , with being a 2–tensor, transforming as a vector, and behaving as an ISO(1,3) scalar. The second canonical transformation, corresponding to the Kaluza–Klein towers, produces a set of Kaluza–Klein excitations for each of these components, which are given in terms of the Kaluza–Klein modes of the extra–dimensional gauge fields as LMNTotro

(7) |

where is the four–dimensional Yang–Mills curvature and is the SU covariant derivative. Both of these objects include the SU coupling constant, , which is related to its extra–dimensional counterpart by . In addition, we have defined

(8) |

with the underlining of indicating that can be either zero or a natural number, which depends on the zero and nonzero Kaluza–Klein indices in any concrete combination that we take in .
Some of these Kaluza–Klein modes include the objects and . While their precise definitions, given in terms of Kronecker deltas, can be found in Ref. LMNTotro , it is worth commenting that in what follows we do not consider any term in which they appear. The reason is that couplings that incorporate these objects do not contribute at the one–loop level to standard Green’s functions, but they do it since higher orders.

It turns out that the Kaluza–Klein Lagrangian is expressed in terms of the Kaluza–Klein modes of the curvature in a relatively simple manner. The precise expression reads LMNTotro

(9) |

In the next two subsections we extract from Eq. (9) all those Kaluza–Klein couplings that contribute to standard Green’s functions at one loop.

### ii.1 Kaluza–Klein scalars and mass spectrum

The emergence of Kaluza–Klein scalar modes, after compactification, is an interesting characteristic of the SU() theory. In the case of just one extra dimension, the number of Kaluza–Klein scalars is , which exactly matches the number of Kaluza–Klein gauge modes . Remarkably, the gauge excited modes are massive, even though they originate from massless five–dimensional gauge fields. The scalar Kaluza–Klein modes behave like pseudo–Goldstone bosons in the sense that they are massless and can be eliminated from the theory by a specific gauge transformation NT5DYM , just like if they had given their physical degrees of freedom to the excited Kaluza--Klein gauge fields. With the assumption of more extra dimensions the analysis grows in difficulty, although the same mechanism for generation of gauge masses^{2}^{2}2By gauge masses we mean that the corresponding mass terms are invariant under the standard gauge group . takes place LMNTotro ; GLMNNT . Some complications in the scalar sector arise because for extra dimensions the number of Kaluza–Klein scalars is greater than the number of Kaluza–Klein gauge modes. Another difficulty is introduced by the presence of mixings among some of the scalar Kaluza–Klein modes. The whole set of Kaluza–Klein scalars can be split into two types of fields, according to whether or not they participate in such scalar mixings: Kaluza–Klein scalars mix, whereas do not take part in mixings.
All this information is enclosed by the fifth term of the right–hand side of Eq. (9).

The fifth term of Eq. (9) can be written as

(10) |

The only term explicitly shown in the right–hand side of the last equation comprises a set of mixing matrices, , each of which corresponds to a fixed combination of Kaluza–Klein indices in the sum over . All the information concerning Kaluza–Klein scalar mixings is contained in these matrices, with components concisely expressed as

(11) |

This is the structure of the inertia tensor associated to a single massive particle located at . The shape of any mixing matrix is determined by the number of nonzero Kaluza–Klein indices in the combination that distinguishes it: the number of mixed Kaluza–Klein scalars matches the number of nonzero Kaluza–Klein indices, whereas all the remaining, and unmixed, scalars have definite mass,

(12) |

from the onset. Indeed, by performing appropriate interchanges of columns and rows in any mixing matrix , with nonzero Kaluza–Klein indices, it can be rearranged as a block matrix that looks like

(13) |

where , in the block , is the identity matrix and is an nondiagonal matrix that mixes Kaluza–Klein scalars. It is worth emphasizing that, while any scalar mixing matrix can be manipulated to fit this generic shape, the sizes of the and matrices depend on the number of nonzero Kaluza–Klein indices, so that the precise structure of each clearly depends on its corresponding combination . A detailed description of the specific mixing pattern followed by the set of Kaluza–Klein scalars has been carried out in Ref. LMNTotro .

Of course, scalar mixing can be eradicated from the Kaluza–Klein theory by diagonalizing the mixing matrices . All such matrices have the same eigenvalue spectrum: 1 zero eigenvalue and nonzero eigenvalues, all of them being equal to . This corresponds to a mass–eigenstates basis characterized by 1 massless scalar, , and massive scalars, , with . For extra dimensions, the total number of scalar mixings in the Kaluza–Klein theory is , each one providing one massless scalar . Hence, the total number of massless scalars is , which consistently coincides with the number of gauge Kaluza–Klein excited modes . To each mixing matrix there corresponds a rotation matrix such that

(14) |

In our notation, any matrix with primed indices is a diagonal matrix, contrastingly to matrices with unprimed indices , which are nondiagonal. Things can always be accommodated in such a manner that, after diagonalization, the last entry of any resulting diagonal matrix is the zero eigenvalue, which means that . This allows a straightforward extraction of the massless scalars from

(15) |

to finally express the scalar–mass terms in Eq. (10) as

(16) |

### ii.2 Kaluza–Klein couplings

Now that we have discussed the pure–scalar sector of the Kaluza–Klein theory, we proceed to set apart from all those couplings that contribute to standard Green’s functions through one–loop diagrams and which are thus necessary for the integration of the Kaluza–Klein excited modes. An exhaustive catalog of Kaluza–Klein couplings, in the general context of the full extra–dimensional Standard Model, can be found in Ref GLMNNT .

From the expression of the zero mode , exhibited in Eq. (II), it is clear that the first term of Eq. (9) includes the four–dimensional Yang–Mills Lagrangian, defined in terms of the four–dimensional Yang–Mills curvature and which we denote by . We express this term as

(17) |

Besides the low–energy theory, this equation shows explicitly a term contributing at one loop to standard Green’s functions. Ellipsis, on the other hand, represents couplings whose lowest–order contributions to such Green’s functions enter at the two–loop level. Another term generating pure–gauge interactions within is the third one, which we write as

(18) | |||||

The second term of Eq. (9) produces only quartic interactions of Kaluza–Klein excited modes, so we omit it and pass to the Kaluza–Klein gauge–scalar interactions that are situated in the fourth term of this equation. This term can be expressed as

(19) | |||||

As the third term of this equation shows, the Kaluza–Klein gauge–scalar sector includes mass terms for the whole set of Kaluza–Klein gauge excited modes . The scalar fields in the first term of Eq. (19) can be directly rotated into the mass–eigenstate fields and , just by using orthogonality of . An interesting feature of Eq. (19) is the presence, in its second term, of Kaluza–Klein gauge–scalar couplings involving the SU() covariant derivative. It turns out that the relation

(20) |

holds for any combination . This yields an exact cancellation of most gauge–scalar couplings in this term, in which the only Kaluza–Klein scalars that survive the rotation , and thus participate in such gauge–scalar couplings, are the pseudo–Goldstone bosons . Before compactification, the interactions among components of the extra–dimensional gauge vector field are explicitly governed by the extra–dimensional SU() gauge symmetry group. Once the compactness of extra dimensions is implemented in the Lagrangian by the aforementioned canonical transformations, and extra–dimensional gauge invariance is hidden, there emerge the Kaluza–Klein excited gauge modes and the scalar modes as well. After the explicit breaking of the ISO() group takes place, the couplings of extra–dimensional gauge fields evolve into the couplings and mass terms characterizing the four–dimensional formulation. In particular, a link between gauge excited modes and a subset of the Kaluza–Klein scalar spectrum is developed. Such link, which manifests through the generation of gauge masses for the gauge excited modes and the emergence of nonphysical scalars , also selectively allows the presence of bilinear gauge–scalar couplings: the only Kaluza–Klein scalars that bilinearly couple to Kaluza–Klein gauge modes are the pseudo–Goldstone bosons , while such interactions are exactly eliminated for the rest of the scalar spectrum. As we show later, a convenient set of gauge–fixing functions allows us to trade the only existing gauge–scalar bilinear couplings by gauge–dependent mass terms for the pseudo–Goldstone bosons .

## Iii Aspects of quantization

Gauge symmetry is a profound concept Dir ; Sunder ; GiTy ; HeTe that characterizes successful and accurate physical formulations, realized within field theory, that are aimed to the quantum description of nature. The essence of gauge invariance is the incorporation of more degrees of freedom than those which are strictly necessary to describe a given physical system. While this symmetry manifests as the invariance of Lagrangians under gauge transformations, the Dirac’s algorithm Dir dives into the depths of this concept and even provides tools to determine Castel the corresponding gauge transformations. This instrument was used in Refs. NT5DYM ; LMNT to develop a careful and complete study of gauge symmetry in the context of extra–dimensional gauge theories. Kaluza–Klein effective descriptions that originate in gauge extra–dimensional theories are invariant under two disjoint sets of gauge transformations: the standard gauge transformations, with respect to which the zero modes behave as gauge fields; and the nonstandard gauge transformations, which transform the Kaluza–Klein excited vector modes as gauge fields.
While there are two types of transformations that are independent of each other, it is indeed the full extra–dimensional gauge group the one which governs the interactions of the Kaluza–Klein Lagrangian. Nevertheless, the sets of canonical transformations that take the extra–dimensional Lagrangian into the Kaluza–Klein theory hide LMNT gauge symmetry living in extra dimensions,
in such a way that the ordinary four–dimensional world displays explicit invariance only under the SU() group.

Though gauge symmetry is usually evoked to construct models, the quantization process by path integral requires HeTe this overdescription to be removed. The framework to execute the quantization of gauge systems is provided by the field–antifield formalism GPS ; FA1 ; FA2 ; FA3 ; FA4 ; FA5 and the Becchi–Rouet–Stora–Tyutin symmetry BRS1 ; BRS2 ; Tyutin , better known as the BRST symmetry. The main point is the determination of a proper solution to the master equation, which arises after a series of extensions of the field spectrum are carried out. In particular, the ghost and antighost fields are introduced, and a set of auxiliary fields enter the game as well. The resulting set of fields is then doubled by introducing an antifield per each field, and a symplectic structure, known as the antibracket, is defined. The proper solution turns out to be the generator of the BRST transformations, which include, as a particular case, the gauge transformations.
In this context, the fixation of the gauge, intended to remove all degeneracy associated to gauge symmetry, is carried out in a nontrivial manner.
This is accomplished by defining a fermionic functional that is used to eliminate all the aforementioned antifields and collaterally fix the gauge.
The main outcome of this procedure is the derivation of a quantum action that depends on general gauge–fixing functions. At this level, gauge symmetry is no longer present and the system is properly quantized.

The quantization of Yang–Mills theories in five spacetime dimensions has been carried out NT5DYM in this approach. The simplest strategy NT5DYM ; inprog consists in generalizing the well–known proper solution that corresponds GPS to the four–dimensional version of this formulation to the case in which there exist extra dimensions.
The transition to the four–dimensional perspective gives rise to a richer Kaluza–Klein theory that now includes, besides the Lagrangian, gauge–fixing and ghost–antighost sectors.
Since the two coexisting sets of four–dimensional gauge transformations are independent of each other, it is possible to remove only NT5DYM ; inprog invariance under the nonstandard gauge transformations. In such manner, the four–dimensional SU() symmetry is still valid and the zero modes are still gauge fields, similarly to what happens, for instance, with the background field method DeWitt ; DeWittbook ; IPS ; tH ; GNW ; KluZu ; Boul ; Hart ; Abbott1 ; Abbott2 ; AGS . In practice, this is achieved by introducing NT5DYM ; inprog an ad hoc set of gauge–fixing functions that are SU()–covariant. This modus operandi to fix the gauge has been of benefit in phenomenological calculations MTTR framed within other formulations, such as the 331 model 331PP ; 331F .
In a forthcoming paper inprog , this picture will be discussed in full detail. However, since the present paper requires some results concerning the quantum version of the Kaluza–Klein theory from Yang–Mills in extra dimensions, we provide here all indispensable expressions for the main calculation.

In the –dimensional case, the quantization procedure sketched above generates a quantum Kaluza–Klein Lagrangian, , that can be split into a sum of three parts as , where is the gauge–fixing term, defined completely by the gauge–fixing functions, here denoted by . The term represents the sector of ghost and antighost fields, and is determined in part by the election of the gauge–fixing functions. Different sets of these functions have been propounded DDGed2 ; MPR ; Uek for the case of just one extra dimension. We generalize the proposal of Ref. NT5DYM , given for the framework of one extra dimension, and provide the following set of SU()–covariant gauge–fixing functions, which is suitable for extra dimensions:

(21) |

where is the gauge–fixing parameter, whose different values correspond to different choices of the gauge. Using these functions, the gauge–fixing term can be written as

(22) |

The second term of the right–hand side of this equation cancels the only gauge–scalar couplings allowed by the theory, that is, those involving the pseudo–Goldstone bosons and which arise from Eq. (19).
We will profit from this cancellation later, when we integrate out the Kaluza–Klein excited modes. In general, simplifications are introduced by the elimination of these couplings because it reduces the number of Feynman diagrams in any calculation aimed to derive contributions to some standard Green’s function. Since this cancellation was produced by the introduction of covariant gauge–fixing functions, it is clear that the resulting simplifications are a direct consequence of the preservation of gauge symmetry. In other words, the presence of symmetries come along with simplifications in phenomenological calculations.
However, as the third term of Eq. (22) shows, the removal of such unphysical couplings leaves, as a remnant, an unphysical mass term for these spurious scalar degrees of freedom. In the Feynman–’t–Hooft gauge, defined by the condition , these masses coincide with those of the Kaluza–Klein gauge excited modes.
The first and third terms in Eq. (22) contribute to light Green’s functions since the one–loop level and thus are relevant for the present calculation.

The ghost–antighost–fields term, , involves several couplings of Kaluza–Klein ghost fields, , and antighost fields, , with gauge and scalar Kaluza–Klein modes. After inserting the covariant gauge–fixing functions, only two types of these couplings contribute to standard Green’s functions at one loop. One of them is a gauge–dependent mass term, whereas the other one is a kinetic term.

We end this section by showing the whole set of couplings that we shall consider in the integration of heavy Kaluza–Klein modes:

(23) | |||||

## Iv Gauge–independent integration of Kaluza–Klein excitations

In this section, we carry out the functional integration of all the Kaluza–Klein excited modes and derive the first nonrenormalizable terms LLR ; BuWy of an effective Lagrangian expansion governed by the low–energy dynamic variables and symmetries. To this end, we follow the procedure devised by the authors of Ref. BiSa , which was adjusted and implemented, in Ref. NTKKint , to the integration of heavy Kaluza–Klein modes from Yang–Mills theories with one extra dimension.

We begin by defining the effective action, , by

(24) |

where , so that this expression involves solely the functional integration of Kaluza–Klein excited modes. Gaussian integration of all the terms in Eq. (23), according to the definition given in Eq. (24), yields

(25) | |||||

Before following through, some explanation on this equation is opportune. The symbol Tr stands for a trace over all degrees of freedom, including spacetime points. Each trace Tr acts also on matrices that live in different spaces and coexist within the arguments of the logarithms in the different terms, where they multiply each other. For example, look at the second term of the right–hand side of Eq. (25). One way of making sense of the argument of its logarithm is by imagining its terms as block matrices, which come from the gauge group and whose entries are blocks of size , because of the spacetime indices. This means that, for instance, is the SU() covariant derivative in the adjoint representation of the gauge group and in matrix form, and is the identity matrix in this gauge–group space, whereas is the matrix corresponding to the inverse of the metric tensor. This term of Eq. (25) contains all the contributions from the Kaluza–Klein gauge excited modes and, as it can be appreciated, is gauge dependent, since it carries the gauge–fixing parameter . The third and fourth terms of this equation are scalar contributions from physical scalars and pseudo–Goldstone bosons , respectively, and the fifth term comprehends all contributions from ghost and antighost fields, and . Note that the global negative sign within the logarithm of the ghost–sector contribution can be relegated to the zero–point energy. The resulting expression is proportional to the fourth term, which comes from the Kaluza–Klein pseudo–Goldstone modes . Clearly, the ghost–antighost contributions are minus twice times those produced by the pseudo–Goldstone bosons. This relation, which had already been noticed in the 331 model MTTR and in the five–dimensional Yang–Mills theory NTKKint , is a direct consequence of our set of gauge–fixing functions and illustrates the simplifications supported by the presence of gauge symmetry. This result is general, within this gauge–fixing prescription, and remains the same in any one–loop calculation. Of course, both of these unphysical contributions are gauge dependent. Finally, we point out that the argument of the logarithm of the third term includes the object , which is an identity matrix of size that appears because of the presence of physical scalars per each combination and per each value of the gauge index .

### iv.1 The gauge trace

In this subsection, we derive a result that is not exclusive of Kaluza–Klein theories, but also applies in other gauge formulations. For that reason, only for now we change our notation to avoid any reference to the particular case of extra–dimensional models. Inspired by the method of Ref. BiSa , and following the Appendix of Ref. NTKKint , we consider a generic trace of the form