PHYSICAL REVIEW D 101, 095038 (2020)

Gauge-invariant approach to the beta function in Yang-Mills theories with universal extra dimensions

M. Huerta-Leal, H. Novales-Sánchez, and J. J. Toscano Facultad de Ciencias Físico Matemáticas, Benem´erita Universidad Autónoma de Puebla, Apartado Postal 1152 Puebla, Puebla, Mexico

(Received 20 February 2020; accepted 18 May 2020; published 29 May 2020)

The radiative correction to the beta function is comprehensively studied at the one-loop level in the context of universal extra dimensions. Instead of using cutoffs to regularize one-loop divergences, the dimensional regularization scheme is used. In this approach, large momenta effects are removed from physical amplitudes by adjusting the parameters of the appropriate counterterms. The use of a SUðNÞ- covariant gauge-fixing procedure to quantize the theory is stressed. One-loop contributions of Kaluza- Klein (KK) excitations are characterized by discrete KK sums and continuous momenta sums, which can diverge. Two types of ultraviolet divergences are identified, one arising from poles of the gamma function and associated with short-distance effects in the usual four-dimensional spacetime manifold and the other emerging either from poles of the one-dimensional Epstein function or from the gamma function and corresponding to short-distance effects in the compact manifold. We address the cases of 5 and 4 þ n dimensions (n ≥ 2) separately. In five dimensions, the one-dimensional Epstein function is convergent, so the usual counterterm renormalizes the vacuum-polarization function. For 4 þ n dimensions, the one- dimensional Epstein function is divergent, so renormalization is implemented by interactions of canonical dimension higher than 4, already present in the effective theory. The polarization function is renormalized using both a mass-dependent scheme and a mass-independent scheme, with extra-dimensions effects decoupling in the former case but not in the latter. The beta function is calculated for an arbitrary number of extra dimensions. Our main result is that Yang-Mills theory remains perturbative at the one-loop level, which is in disagreement with the results obtained in the literature by using a cutoff regulator, which suggest that Yang-Mills theory in more than four dimensions ceases to be perturbative. We emphasize the advantages of a mass-dependent scheme in this type of theories, in which decoupling is manifest.

DOI: 10.1103/PhysRevD.101.095038

I. INTRODUCTION discovery of supersymmetry [12,13], and the presence of a massless particle of spin 2 [14], to be identified as the The use of extra dimensions in model building graviton, were two main elements of superstring theory started with the works by Nordström and Kaluza, who that motivated its use to achieve a quantum theory of attempted to unify electromagnetism and gravity by assum- gravity, always with the complicity of extra dimensions. ing the existence of a spatial extra dimension [1,2]. Remarkably, the critical dimension of superstring theory Nevertheless, it was Klein who realized, for the first time, turned out to be just 10, as it was shown by Schwarz [15]. that compactification could be used to explain the lack of The introduction of the Green-Schwarz mechanism [16],to observations of extra dimensions [3]. The ulterior birth of eliminate quantum anomalies arising in string theory, then string theory, as a theory of strong interactions [4–10], triggered the first superstring revolution, during which five would eventually endow great relevance to formulations of consistent superstring formulations were given [17–20]. extra dimensions. The original string-theory formulation Furthermore, a connection, through compactification, already had this ingredient, as 26 spacetime dimensions between superstring theory, featuring a six-dimensional were required to ensure unitarity [11]. The introduction of Calabi-Yau extra-dimensional manifold [21], and four- fermions in string theory [12], which came along with the dimensional supersymmetry was established [22].Asec- ond superstring revolution started with the emergence of the M theory, by Witten [23], who showed that the five Published by the American Physical Society under the terms of superstring formulations known at the time are limits of this the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to single theory, which is a unifying fundamental theory set in the author(s) and the published article’s title, journal citation, 11 spacetime dimensions. The existence of D-branes, and DOI. Funded by SCOAP3. proposed by Polchinski [24] for the sake of string duality,

2470-0010=2020=101(9)=095038(28) 095038-1 Published by the American Physical Society HUERTA-LEAL, NOVALES-SÁNCHEZ, and TOSCANO PHYS. REV. D 101, 095038 (2020) was a major event. It was also shown that supergravity exclusively at loop orders. An appealing characteristic of in 11 dimensions is a low-energy limit of the M theory these models is the small number of added parameters, [25,26]. The AdS=CFT correspondence, which establishes which are a high-energy compactification scale, R−1, and a duality of five-dimensional theories of gravity with gauge the number of extra dimensions, n. Moreover, universal- field theories set in four dimensions [27], is a quite extra-dimensions models include dark matter candidates important result with remarkable practical advantages [42–45], which would be either the first KK excited mode of regarding nonperturbative physics. Among the events the photon or that corresponding to the neutrino. and advances experienced by string theory throughout To write down an effective Lagrangian that extends the the years, and a plethora of papers on the matter, we wish Standard Model in the UED approach, a series of nontrivial to emphasize that its development is the one that got steps must be implemented at the classical level. It is worth modern physics used to extra dimensions. analyzing in some detail how this effective theory is Considerable interest in the phenomenology of extra constructed in order to have a broader understanding of dimensions arose because of the works by Antoniadis et al. its implications at the quantum level. Here, we focus on a and Arkani-Hamed et al. [28–30], who, motivated by the pure (without matter fields) YM theory, whose technical hierarchy problem, proposed the existence of large extra details can be found in Ref. [46]. At some energy scale Λ, dimensions, responsible for the observed weakness of the which is assumed to be far above of the compactification gravitational interaction, at the stunning scale of a milli- scale R−1, one proposes an effective theory governed by the meter. Shortly after, Randall and Sundrum initiated an extended group ISOð1; 3 þ nÞ × SUðN;M4þnÞ, where important branch of extra-dimensional models, the so- ISOð1; 3 þ nÞ is the Poincar´e group defined on the called models of warped extra dimensions, in which the (4 þ n)-dimensional flat manifold M4þn ¼ M4 ×N n, with hierarchy problem was tackled by introducing a spatial M4 the usual spacetime manifold and N n a Euclidean extra dimension and assuming that the associated five- manifold. Note that SUðN;M4þnÞ and SUðN;M4Þ dimensional spacetime is characterized by an anti-de Sitter coincide as Lie groups, but they differ as gauge groups structure [31,32]. The present paper is developed within because they have a different number of connections. another well-known extra-dimensional framework, dubbed The most general effective Lagrangian that respects the universal extra dimensions [33] (UED), proposed by ISOð1; 3 þ nÞ × SUðN;M4þnÞ symmetry can be written as Appelquist et al. [33] and characterized by the assumption YM 4 L ðx; x¯Þ¼L d≤4 ðx; x¯ÞþL d>4 n ðx; x¯Þ, with x ∈ M that every field of a given formulation depends on all the eff ð þnÞ ð þ Þ and x¯ ∈ N n. The first term of this Lagrangian corresponds coordinates of the spacetime with extra dimensions. The to the (4 n)-dimensional version of the usual YM theory, work by these authors included the formulation of a field þ which contains interactions of canonical dimension less than theory with the structure of the four-dimensional Standard or equal to 4 þ n, while the second term includes all Model but defined, rather, on a spacetime with compact interactions of canonical dimension higher than 4 þ n that spatial extra dimensions, where all the dynamic variables are compatible with these symmetries, since the theory is are assumed to propagate, thus leading to an infinite set of nonrenormalizable in the usual sense. From dimensional Kaluza-Klein (KK) modes per each extra-dimensional considerations, the interactions that appear in the field.1 The present investigation takes place around L d 4 ðx; x¯Þ Lagrangian will be multiplied by inverse Yang-Mills (YM) theories defined in a spacetime compris- ð > þnÞ Λ ≫ −1 ing 4 þ n dimensions, with the assumption that the n extra powers of the high-energy scale R , so they will be dimensions are compact and universal. Investigations on naturally suppressed with respect to those that constitute LYM ¯ theoretical and phenomenological aspects of extra- the ðd≤4þnÞðx; xÞ Lagrangian, which do not depend on this dimensional Yang-Mills theories have been explored and scale. This means that this Lagrangian determines the reported in Refs. [36–41]. In models of universal extra dominant contributions to physical observables. This fact dimensions, conservation of extra-dimensional momentum will play a central role in our quantum analysis at the one- yields, after integrating out the extra dimensions, four- loop level. Note that the coupling constant g4þn that appears dimensional KK effective field theories in which KK parity in this Lagrangian is dimensionful and must be rescaled to is preserved, with the consequence that, from the perspective obtain the correct dimensionless YM coupling. of the Feynman-diagrams approach, the very first effects To describe physical phenomenon below the compacti- from the KK modes on low-energy Green’s functions (and fication scale, we need go from the ISOð1; 3 þ nÞ × thus on low-energy observables) occur at one loop [33]. SUðN;M4þnÞ description to a description based in the Such a feature is particularly relevant in the case of physical usual ISOð1; 3Þ × SUðN;M4Þ symmetry, for which it is observables and processes that, within the context of the necessary to implement a procedure of hiding of symmetry. Standard Model in four dimensions, can take place To hide symmetry ISOð1; 3 þ nÞ × SUðN;M4þnÞ in sym- metry ISOð1; 3Þ × SUðN;M4Þ, we need to implement two 1Models of universal extra dimensions have been reviewed in canonical maps, one that transforms covariant objects of Refs. [34,35]. ISOð1; 3 þ nÞ into covariant objects of ISOð1; 3Þ, the other

095038-2 GAUGE-INVARIANT APPROACH TO THE BETA FUNCTION IN … PHYS. REV. D 101, 095038 (2020) that allows us to hide, through a general Fourier series, any continuous sums may diverge, a regularization scheme dynamic role of subgroup ISOðnÞ ⊂ ISOð1; 3 þ nÞ [46]. must be adopted. This is a crucial issue on which we will The zero modes of such series are identified with the gauge focus our attention. In the literature [33,47,48], one-loop fields and gauge parameters of the usual YM theory. To calculations of KK contributions to some observables have match the zero modes with those of the usual theory, the been presented in the cases of one and two extra dimen- dimensionful coupling constant g4þn must be rescaled to sions using as a regulator an explicit cutoff. This approach n 2π 2 identify the four-dimensional coupling g ¼ g4þn=ð RÞ . makes sense, since in an effective theory we must only This procedure, which is qualitatively discussed in the include momenta below a certain energy scale and exclude next section, leads to an effective field theory governed by those effects that are above it. Following this premise, the usual ISOð1; 3Þ × SUðN;M4Þ groups. After compac- integrals are solved using a cutoff, while only a few of the tification and integration on the x¯ coordinates, the first KK modes of infinite sums are kept. This procedure LYM ¯ used in Refs. [33,47,48] seems to be motivated by Wilson’s ðd≤4þnÞðx; xÞ Lagrangian, which does not depend on the Λ scale, unfolds into two terms: one that corresponds approach to renormalization theory, which introduces a 0 cutoff to divide the fundamental path integral into its low- to the usual four-dimensional theory, Lð Þ x , and the other YMð Þ energy and high-energy parts [49], although the authors that contains interactions between usual fields and KK only keep the low-energy effects. This scheme leads, of excitations, LKK x . The important point to be empha- ðd≤4Þð Þ course, to physical observables that are highly dependent Lð0Þ LKK on the regulator. This can invalidate the perturbation sized is that, like YMðxÞ, the ðd≤4ÞðxÞ Lagrangian only contains interactions of canonical dimension less than or expansion even if the four-dimensional coupling constant equal to 4, so they cannot depend on the high-energy scale is small. Based on these results, it is generally assumed in Λ LKK the literature that YM theories in more than four dimen- . This means that the ðd≤4ÞðxÞ Lagrangian only can −1 sions become strongly interacting. This result has such depend on the compactification scale R through masses strong implications that are is worth reviewing more of the KK excitations. This is the reason why this carefully. Using a cutoff as regulator to extract the low- Lagrangian has a dominant role in phenomenological energy effects of the theory leads to the suspicion that predictions, even though its effects first appear at the something could be wrong. The main goal of this work is to one-loop level. Another point worth noting is that, like address this problem from a different perspective of Lð0Þ LKK YMðxÞ, the interactions contained in the ðd≤4ÞðxÞ effective theories. The first thing we must do is introduce Lagrangian are dictated by the SUðN;M4Þ gauge group, a good regularization scheme, bearing in mind that we will so we can expect well-behaved loop amplitudes. As far as face a new series of complications that are not present in L ¯ the ðd>4þnÞðx; xÞ Lagrangian is concerned, it unfolds into conventional effective theories, which is evidenced by the two infinite Lagrangians: one that depends only on zero presence of infinite discrete sums in the one-loop ampli- ð0Þ tudes. Although the cutoff schemes are very intuitive, they modes fields, L ðxÞ, which contains all interactions of ðd>4Þ are not really good regulators because they make it difficult canonical dimension higher than 4 that respect the usual to keep the symmetries of the theory, especially gauge symmetry, and the other one that mixes interactions invariance, which is the essence of YM theories. Definitely, between zero modes and KK excitations. Although the a cutoff is a bad regulator, which should not be used in Lð0Þ ðd>4ÞðxÞ Lagrangian can generate at tree level the same theories with a high content of symmetries, like the YM low-energy observables that can be induced by the theories. Although various regularization schemes are KK available, we know from renormalization theory that it is L d≤4 ðxÞ Lagrangian at the one-loop level, its contribution ð Þ crucial to subtract off the infinities with a procedure that is subdominant due to the suppression factor introduced by the high-energy scale Λ. This means that in this type of preserves all the symmetries of the original theory. In line theories any physically interesting contribution on low- with this observation, we will adopt the dimensional energy observables will necessarily arise from one-loop regularization scheme [50,51], which has proven to be effects induced by the Lagrangians of canonical dimension the best existing regularization scheme, mainly because it preserves the gauge and Poincar´e symmetries. The adop- Lð0Þ LKK less than or equal to 4, namely, YMðxÞ and ðd≤4ÞðxÞ.In tion of this regularization scheme may give the impression this work, we will focus on this type of contributions. that we are going against the genuine spirit of effective At the one-loop level, the amplitudes characterizing KK theories, since dimensional regularization keeps momenta contributions are proportional toP a discreteR sum and a arbitrarily high. However, as has already been shown in the 4 continuousP sum as well, that is, ðkÞ d k, where the literature [52–55], undesirable contributions, that is, the symbol ðkÞ represents a multiple sum running over effects of very high energies, can be removed from physical discrete vectors defined on the compact manifold. The observables by adjusting the parameters of appropriate higher multiplicity of such discrete sums is the dimension counterterms. The required counterterms are already avail- of the compact manifold, n. Since discrete and/or able in the effective Lagrangian, since it contains all

095038-3 HUERTA-LEAL, NOVALES-SÁNCHEZ, and TOSCANO PHYS. REV. D 101, 095038 (2020) interactions that respect the symmetries of the theory. In the However, for n ≥ 2, short-distance effects in both the usual ð0Þ spacetime and the compact manifold may be present, so the case at hand, the L d 4 ðxÞ Lagrangian can generate all the ð > Þ usual counterterms do not suffice to remove both types of necessary counterterms, since it induces at the tree level all divergences. The reason is that divergences from short- interactions between zero modes fields that are compatible distance effects in the compact manifold appear as coef- 1 3 M4 with the standard symmetry ISOð ; Þ × SUðN; Þ. ficients of polynomials in the external momentum, which A significant complication of field models of extra suggests that these effects are associated with Lagrangian dimensions is related to the types of divergences that can terms of canonical dimension higher than 4. To generate arise. An infinite number of KK-mode fields, circulating in appropriate counterterms, renormalization is implemented the loops of contributing Feynman diagrams, occurs, and, by considering interactions which are nonrenormalizable, moreover, there are two spaces, namely, the usual spacetime according to Dyson’s criterion. As already commented, such manifold and the compact extra-dimensional manifold, so interactions are already available in the effective KK thinking of short-distance effects in both spaces makes Lagrangian, which is constituted by all the interactions sense. A detailed discussion on the matter, in the somewhat governed by the symmetries of the theory. simpler context of extra-dimensional quantum electrody- Asymptotic freedom is a remarkable aspect of quantum namics, has been recently carried in Ref. [56]. As we discuss chromodynamics, taking place in the context of the below, in the same way that there are two spaces, there are 3 SUð ÞC-invariant YM theory description of strong inter- also two types of divergences, so the key to tackle the actions [58–60]. This physical phenomenon, by which problem relies on establishing a connection between the two collisions involving hadrons display a pointlike-particle sorts of divergences and the two types of spaces. Such a behavior of constituent quarks in case of large momentum connection is already suggested by the fact that the one-loop transfer, occurs in the presence of a negative Callan- amplitudes characterizing KK contributions are propor- Symanzik beta function [61,62], β, which is identified from tional, as already mentioned, toP aR discrete sum and a 4 the ultraviolet structure of loop corrections to the gauge- continuous sum as well, that is, ðkÞ d k. Since discrete field propagator. This beta function then provides, at the and/or continuous sums may diverge, we simultaneously desired loop order, an estimation for the running strong regularize them using the dimensional regularization α coupling constant s, in accordance with the renormaliza- scheme. This means that all values of both discrete and tion group equation, that is compatible with perturbation continuous momenta will be kept. As will be seen through- theory at sufficiently large momentum transfer q2. In the out the paper, the dimensional regularization scheme 2 one-loop approximation, α ðq Þ is expressed as becomes central to our work. An important result is that s the discrete sums thus regularized can naturally be expressed α μ2 2 sð Þ in terms of multidimensional Epstein functions [57]. In this α ðq Þ¼ 2 ; ð1:1Þ s 1 α μ2 β q regularization scheme, divergences from continuous sums þ sð Þ log μ2 manifest as poles of the gamma function in the limit as the parameter ϵ ¼ 4 − D goes to zero. We show that the with μ denoting the renormalization scale. In the context 3 divergences associated with discrete sums emerge alter- of SUð ÞC-invariant chromodynamics, the first calculation nately either as poles of the one-dimensional Epstein of a negative beta function, at the one-loop order, was function or as poles of the gamma function. It is in this reported in Ref. [58], whose result established asymptotic sense that we talk about the existence of two types of freedom of quarks and gluons. Ulterior calculations of the divergences. Since these divergences originate in discrete beta function at two, three, and even four loops were sums, we argue that such types of divergences correspond to carried out afterward [63–68]. The presumed occurrence short-distance effects of the KK excitations in the compact of KK excited modes, as a low-energy manifestation of manifold, so they are genuine ultraviolet divergences that extra dimensions, plays a role in the definition of the beta can be removed by renormalization. The KK excitations also function. The efforts of the present investigation are induce divergences that emerge through poles of the gamma aimed at the estimation of effects produced by the function, which have to do with short-distance effects in the presence of universal extra dimensions on the beta usual spacetime manifold. Showing that both types of function, which is done through a calculation, at one divergences can consistently be handled by renormalization loop, of the contributions from the KK excited modes is a main objective of the present work. In general, KK that originate in the (4 þ n)-dimensional gauge fields of excitations can induce both types of divergences, except in the SUðN;M4þnÞ theory. the special case of only one extra dimension, since the one- A major objective of the present work is the calculation dimensional Epstein function is convergent. Thus, the cases and characterization of the impact, at the one-loop level, of n ¼ 1 and n ≥ 2 are addressed separately. For n ¼ 1, we will universal extra dimensions on the beta function of YM show that, at the one-loop level, the counterterms of the theories. For this purpose, we take gauge symmetry, usual theory, describing the zero modes, are enough to the dimensional regularization scheme, and decoupling remove ultraviolet divergences from the KK theory. between physical scales as guiding principles. To this

095038-4 GAUGE-INVARIANT APPROACH TO THE BETA FUNCTION IN … PHYS. REV. D 101, 095038 (2020) aim, we quantize the KK theory by using a SUðN;M4Þ- Lorentz symmetry holds, effective field theories with the covariant gauge-fixing procedure. The Background Field ingredient of Lorentz-invariance nonconservation, inspired Method [69,70] is used to fix the gauge for zero modes, by the spontaneous breaking of Lorentz symmetry in string which are the four-dimensional YM fields, while for theory formulations [72,73] and by the occurrence of KK excited modes, a covariant gauge [41] is considered. Lorentz violation in noncommutative field theory [74], The construction of a SUðN;M4Þ-invariant quantum have been propounded [75–77]. On the other hand, the Lagrangian considerably simplifies the calculation of the choice of the gauge-symmetry group has more often beta function. Regarding the control of divergences, we become the defining trademark of field theory models. already have emphasized the importance of using dimen- For instance, the Lagrangian terms constituting the sional regularization to preserve the original symmetries of Standard Model are determined, in part, by the gauge 3 2 1 the theory and implementing renormalization to remove the group SUð ÞC ×SUð ÞL ×Uð ÞY. This is also the case of effects of large discrete and continuous momenta. In models that involve left-right symmetry [78–83], which are 2 2 1 Ref. [56], some of us showed, for the first time, the based on SUð ÞL ×SUð ÞR ×Uð ÞB−L. Moreover, the advantages of using the dimensional regularization scheme main feature of the so-called 331 models [84,85] is a 3 3 to simultaneously regularize the continuous and discrete particular gauge group, in this case SUð ÞC ×SUð ÞL× 1 sums. On the other hand, we demand extra-dimensions Uð ÞX. Also, theories of grand unification, based on the effects to be of decoupling nature; that is, we require them symmetry group SU(5), were explored and thoroughly to vanish from physical quantities and renormalized ampli- discussed [86,87]. The consideration of spacetime mani- tudes at low energies, in accordance with the Appelquist- folds with extra dimensions has opened alternative paths Carazzone decoupling theorem [71]. The renormalized to define field-theory models.2 In the present study, we vacuum-polarization function, in our gauge-invariant quan- work within the framework set by (4 þ n)-dimensional YM tization procedure, and the beta function shall be quantities theories, by which we refer to a replica of the SUðN;M4Þ- subjected to this requirement. Decoupling of new physics invariant pure-gauge theory, but with all its field content effects means that these quantities must reduce to the usual and symmetries defined on the spacetime with the n extra ones in the limit of a very large compactification scale. dimensions, which we assume to be spatial-like and Fulfillment of this requirement depends crucially on universal. In this section, we briefly discuss this extra- renormalization scheme. We note that a mass-dependent dimensional YM theory, with focus on those aspects that scheme must be followed to achieve this goal. If these goals are relevant for the calculation of beta-function contribu- are achieved, we will have shown that Yang-Mills theory in tions that we are about to execute. We develop our more than four UEDs remains perturbative at the one- discussion in the general context of n extra dimensions, loop level. which, of course, can be straightforwardly particularized to The rest of the paper has been organized as follows. In the case n ¼ 1. We spare the reader from the whole bunch Sec. II, we discuss the main features of YM theories with an of specific details characterizing this formulation and arbitrary number of extra dimensions. In this section, we suggest Refs. [36–41,56] for more detailed discussions will focus more on the conceptual part, avoiding, as much on the matter. as possible, getting into the technical details. Section III is First consider, in general, a spacetime comprising devoted to presenting the calculations needed for our study one timelike dimension and 3 þ n spatial-like dimen- of the phenomenon of asymptotic freedom. The gauge- sions. Assume that, at some high-energy scale (short covariant scheme to quantize the theory is discussed to distances), this spacetime can be characterized by a some extent. Some properties of zeta functions are pre- 4 (4 þ n)-dimensional manifold M þn with metric g ¼ sented, and a number of relevant expressions for our work MN diagð1; −1; …; −1Þ. Here and in what follows, capital are derived. In Sec. IV, the beta function is derived using spacetime indices, like M, N, take the values3 both a mass-independent scheme and a mass-dependent 0; 1; 2; 3; 5; …; 4 þ n. Think of a field-theory formula- scheme. Finally, in Sec. V, we summarize or main results tion defined on this manifold and governed by the extra- and present final remarks. dimensional Poincar´egroupISOð1; 3 þ nÞ. Imagine a process in which we study nature at increasing distances, II. EXTRA-DIMENSIONAL YANG-MILLS starting from the aforementioned high-energy scale. THEORY AND ITS KALUZA-KLEIN While at a certain range of high-energy scales the proper LAGRANGIAN field-theory description is invariant with respect to In general, field theories are defined by symmetries and dynamic variables [55]. While a variety of symmetries 2 relevant to this purpose is available, spacetime and gauge Field-theory models of extra dimensions extend the 4- dimensional standard model (4DSM) in the direction of space- symmetries, in particular, turn out to be essential elements time symmetry groups. for the definition of field-theory descriptions. For instance, 3Note that a convention in which the first extra dimension is even though most models are set under the assumption that labeled by M; N ¼ 5 has been used.

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1 3 ðkÞ ðkÞ ISOð ; þ nÞ, we assume that the afore-described process The specific set ffE ;fO g is determined, in part, by leads us to a lower-energy scale at which n out of the the geometry of the extra dimensions, but an extra- 3 þ n spatial-like dimensions display a compact nature. It dimensional observable is also required to this end. This is said that these n dimensions are compactified.Atthis is the case of the Casimir invariants of ISOðnÞ,among energy scale, also called the compactification scale, an which we choose P¯ 2,withP¯ the momentum operator along appropriate field-theory formulation is not governed by the extra dimensions. Being a Hermitian operator, P¯ 2 has 4 the ( þ n)-dimensional Poincar´e group anymore. Besides an associated orthogonal set of eigenkets fjp¯ ðkÞig,with being a theoretical possibility, the ingredient of compac- 2 2 real eigenvalues ðp¯ ðkÞÞ ¼ p¯ ðkÞ · p¯ ðkÞ.TheP¯ eigenkets tification has the practical use of explaining the absence of ðkÞ ððkÞ measurements of extra dimensions [88–96]. A variety of then define ffE ;fO g from the wave function relations ðkÞ ¯ ¯ ðkÞ geometries, suitable for compactified extra dimensions, is fE;O ¼hxjp i. Using such relations, together with appro- available [97–101]. In general, all the symmetries and ðkÞ ðkÞ priate boundary conditions, f and f are determined to dynamic variables constituting a sensible physical E O be normalized trigonometric functions, so that the field χ is description at this lower-energy scale are different. Fourier expanded, with the following two disjoint cases: From here on, x and x¯ denote, respectively, coordinates even parity: for the four standard spacetime dimensions and the n extra dimensions. In this context, consider any dynamic variable, χ ¯ 1 generically denoted by ðx; xÞ, which we assume to be a χðx; x¯Þ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi χð0ÞðxÞ tensor field with respect to the (4 þ n)-dimensional Lorentz ð2πÞnR 1 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi group. Now, we break ISOð ; þ nÞ invariance by imple- X 2 k menting compactification, for which we assume that each þ χð ÞðxÞ cosfp¯ ðkÞ · x¯g; ð2:1Þ 1 ð2πÞnR E extra dimension is compactified on an orbifold S =Z2 ðkÞ characterized by a radius Rj, with j ¼ 1; 2; …;n. This compactification scheme induces periodicity properties on odd parity: χ, with respect to the extra-dimensional coordinates x¯. Moreover, it allows for the assignment to χ of definite- sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¯ → −¯ X parity properties, with respect to reflections x x. 2 ðkÞ χ χðx; x¯Þ¼ χ ðxÞ sinfp¯ ðkÞ · x¯g: ð2:2Þ The field is then expanded in terms of a complete ð2πÞnR O ðkÞ ¯ ðkÞ ¯ ðkÞ set of orthogonal functions ffE ðxÞ;fO ðxÞg, which exclusively depend on extra-dimensional coordinates R x¯. Such an expansion runs over the multi-index In these equations, we denoted ¼ R1R2 Rn.We ¯ ðkÞ ðkÞ¼ðk1;k2; …;k Þ, where any k is an integer number. have defined discrete extra-dimensional momentaPp ¼ n j … Furthermore, the labels “E” and “O” mean that the Pðk1=RP1;k2=RP2; ;kn=RnÞ as well. The symbol ðkÞ ¼ ¯ → −¯ represents a multiple sum that runs over corresponding function is even or odd under x x. k1 k2 kn Once this expansion has been implemented on every field every discrete vector ðkÞ labeling an independent field χ, the extra-dimensional coordinates x¯ no longer label χðkÞ, with the additional restriction that ðkÞ ≠ ð0Þ¼ degrees of freedom, which are now characterized by the KK ð0; 0; …; 0Þ. That is, this notation summarizes a total ðkÞ ðkÞ χ of 2n − 1 different series as follows: index ðkÞ. Each function fE or fO , in the expansion, lies χðkÞ χðkÞ multiplied by a coefficient E ðxÞ or O ðxÞ, respectively. X X∞ X∞ 0 … 0 0 … These fields, which depend only on the four-dimensional TðkÞ ¼ Tðk1; ; ; Þ þþ Tð ; ;knÞ 1 1 coordinates x of the noncompact spacetime dimensions, are ðkÞ k1¼ kn¼ the new four-dimensional dynamic variables, the KK X∞ X∞ 0 … 0 0 … 0 modes, suitable for the physical description after compac- þ Tðk1;k2; ; ; Þ þþ Tð ; ; ;kn−1;knÞ ð0Þ 1 1 tification. Assume that a constant function, f , belongs to k1;k2¼ kn−1;kn¼ ðkÞ ðkÞ ¯ → −¯ . ffE ;fO g. This function, trivially even under x x, . χð0Þ χð0Þ comes with a four-dimensional field ðxÞ. Fields , X∞ … known as KK zero modes, are identified as the dynamic þ Tðk1; ;knÞ: ð2:3Þ variables that constitute the low-energy description. The k1;…;kn¼1 fields χðkÞ, with ðkÞ ≠ ð0Þ, are known as KK excited modes and are interpreted as degrees of freedom that reflect the Whereas positions of Fourier indices in the entries of ðkÞ presence of extra dimensions from a four-dimensional are not relevant, the number of occupied entries makes effective-theory viewpoint. Thus, only x¯-even extra- a difference. So, in practice, one can use the following dimensional fields χ yield low-energy dynamic variables. definition:

095038-6 GAUGE-INVARIANT APPROACH TO THE BETA FUNCTION IN … PHYS. REV. D 101, 095038 (2020)   X Xn n X∞ X∞ afore-alluded splitting is a canonical transformation that ¼ : ð2:4Þ maps covariant objects of SOð1; 3 þ nÞ into covariant 1 l 1 1 ðkÞ l¼ k1¼ kl¼ objects of SOð1; 3Þ [39,40]. To land on a KK effective Lagrangian consistently comprising the low-energy theory, Note that such an effective theory, often referred to as KK namely, the YM theory in four dimensions, we assume that theory, is defined only in four dimensions of spacetime; a Aμ is even with respect to x¯ → −x¯, but the definite parity of once the extra dimensions have been compactified and the the scalar fields Aa under such a transformation is taken to dynamic variables Fourier expanded, the whole dependence μ¯ be odd. Equations (2.1) and (2.2) embody a second on extra dimensional coordinates in the action SYM ¼ R 4þn canonical transformation [39,40], which, after implemen- 4þn LYM ¯ d x 4þnðx; xÞ lies within trigonometric functions, tation, yields sets of KK modes, recognized as dynamic which can be straightforwardly integrated out, leading to variables of the KK Lagrangian. Furthermore, as a by- a Lagrangian, product of the last canonicalpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi transformation, the dimen- Z 2πR n sionless quantity g ¼ g4þn= ð Þ is identified as the LYM n ¯LYM ¯ 4 KK ðxÞ¼ d x 4þnðx; xÞ; ð2:5Þ SUðN;M Þ coupling constant. The whole set of KK modes, together with the two canonical maps generating defined in four spacetime dimensions. them, is illustrated in Eq. (2.8): We start by considering a YM theory set on a (4 þ n)- 8 < a ð0Þa ðkÞa dimensional spacetime with those features previously Aμðx; x¯Þ ↦ Aμ ðxÞ;Aμ ðxÞ Aa ¯ ↦ Mðx; xÞ ð2:8Þ described and which is governed by the extended : a ðkÞa 4 Aμ¯ ðx; x¯Þ ↦ Aμ¯ ðxÞ: groups ISOð1;3þnÞ×SUðN;M þnÞ. The gauge-symmetry group introduces N2 − 1 connections, denoted as Aa ðx; x¯Þ, M After usage of the canonical maps, and subsequent where a ¼ 1; 2; …;N2 − 1 is the gauge index. The theory to straightforward integration of the extra dimensions in the be addressed is then given by the Lagrangian term LYM action,R the four-dimensional KK Lagrangian term KK ¼ n ¯LYM 1 d x 4þn arises. The effective-theory description pro- YM a aMN L4 ¼ − F F þ L d 4 ðF; DFÞ; ð2:6Þ LYM þn 4 MN ð > þnÞ vided by KK is characterized by low-energy symmetries, among which four-dimensional Poincar´e invariance is L F D 1 3 where ðd>4þnÞð ; FÞ includes all interactions which central. With respect to the Lorentz group SOð ; Þ, the 4 ð0Þa ðkÞa ðkÞa have canonical dimension higher than þ n and which KK fields Aμ and Aμ are 4-vectors, whereas the Aμ¯ 4 are compatible with the ISOð1; 3 þ nÞ ×SUðN;M þnÞ are scalars. About gauge symmetry, the effectuation of symmetry. This must be so because the theory is non- compactification entails the occurrence of hidden sym- renormalizable in the usual sense. This effective metries [39]. Originally characterized by the gauge group 4 Lagrangian is given in terms of the SUðN;M þnÞ YM SUðN;M4þnÞ, set on 4 þ n spacetime dimensions, the F a curvature components MN and its derivatives, defined as (4 þ n)-dimensional Yang-Mills theory has been mapped usual [102], into a KK theory that manifests gauge invariance corre- sponding to the low-energy group SUðN;M4Þ, defined on F a ∂ Aa − ∂ Aa abcAb Ac MN ¼ M N N M þ g4þnf M N; ð2:7aÞ four spacetime dimensions. Collaterally, the gauge trans- 4 Aa formations of the ( þ n)-dimensional connections M Dab δab∂ − abcAc M ¼ M g4þnf M; ð2:7bÞ split into two disjoint sets of four-dimensional gauge transformations [36–41]: standard gauge transformations, with fabc the structure constants of the group. The which constitute the gauge group SUðN;M4Þ and with 4þn SU N;M coupling constant, g4 , is dimensionful, ð0Þa ð Þ þn respect to which KK zero mode Aμ behaves as a gauge −n=2 with units ðmassÞ . field, and nonstandard gauge transformations, under YM Having defined the Lagrangian term L4 , we imple- ðkÞa þn which KK excited modes Aμ are sort of like gauge ment compactification through a couple of canonical fields, in the sense that they follow a transformation that is transformations to go from the (4 þ n)-dimensional per- reminiscent of a gauge transformation, but which does not spective to the KK effective theory, set in four spacetime M4 dimensions. Because of compactification, Aa , which at correspond to SUðN; Þ. Furthermore, let us remark that M ðkÞa first was a (4 þ n)-vector of SOð1; 3 þ nÞ, is split into the KK excited modes Aμ are not connections of a 4 SOð1; 3Þ 4-vector Aμ and the set of SOð1; 3Þ scalar fields SUðN;M Þ, but, rather, they transform as matter fields, Aa Aa … Aa in the adjoint representation of this group [36–41]. Hence, f 5; 6; ; 4þng. From now on, we utilize greek indices like μ, ν ¼ 0, 1, 2, 3 to denote four-dimensional Lorentz gauge symmetry governing the KK effective theory does indices and use indices μ¯; ν¯ ¼ 5; 6; …; 4 þ n to label not forbid the presence of mass terms for vector KK- ðkÞa extra-dimensions coordinates. The implementation of the excited-mode fields Aμ . This is to be contrasted with the

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ð0Þa situation of KK zero modes Aμ , which, being four- where dimensional gauge fields, are restricted to be massless. All ð0Þa ð0Þa ð0Þa abc ð0Þb ð0Þc scalar KK modes, on the other hand, transform as matter Fμν ¼ ∂μAν − ∂νAμ þ gf Aμ Aν ð2:13Þ fields with respect to both sets of gauge transformations; ðkÞa are the curvature components associated with the standard the scalar fields Aμ¯ are, in spite of their gauge origin, gauge group SUðN;M4Þ. Note that the curvature compo- matter fields under SUðN;M4Þ, which opens the possibil- F ð0Þa F ðkÞa ity for them to become massive. nents f μν ðxÞ; μν ðxÞg transform in the adjoint repre- A remarkable outcome of compactification is the occur- sentation of this group. Then, the Lagrangian (2.11a) contains the Yang-Mills term associated with the standard rence of mass terms for the whole set of KK excited modes, 4 which we refer to as the KK mass-generating mechanism, SUðN;M Þ group plus terms involving interactions among or KK mechanism for short. To illustrate more explicitly the connection components and matter fields. Note that ðkÞa consequences of this mechanism, we present the effective F μν ðxÞ contains the kinetic terms for the matter fields. Lagrangian that results after compactification of the ðkÞa ðkÞa ð0Þa As far as the curvatures fF μν¯ ; F μ¯ ν¯ ; F μ¯ ν¯ g are con- (4 n)-dimensional version of the pure Yang-Mills theory. þ cerned, which transform in the adjoint representation of the So, after integrating over the extra coordinates in (2.5),we gauge group SUðN;M4Þ, they are given by have an effective Lagrangian given by ðkÞa ð0Þab ðkÞb ðkÞ ðkÞa LYM LYM L F μν¯ ¼ Dμ Aν¯ þ pν¯ Aμ KK ¼ KKðd≤4Þ þ ðd>4Þ; ð2:9Þ X ðrÞb ðsÞc abc Δ0 μ þ gf ðrskÞA ðxÞAν¯ ; ð2:14aÞ LYM where KKðd≤4Þ contains only interactions of canonical ðrsÞ dimension less than or equal to 4, which emerge from ðkÞa ðkÞ ðkÞa ðkÞ ðkÞa compactification of the (4 þ n)-dimensional version of the F μ¯ ν¯ ¼ pμ¯ Aν¯ − pν¯ Aμ¯ YM theory. This Lagrangian can conveniently be written as X abc Δ0 ðrÞb ðsÞc þ gf ðrskÞAμ¯ Aν¯ ; ð2:14bÞ LYM LYM LYM LYM ðrsÞ KKðd≤4Þ ¼ v-v þ v-s þ s-s ; ð2:10Þ X F ð0Þa abc ðkÞb ðkÞc where μ¯ ν¯ ¼ gf Aμ¯ ðxÞAν¯ : ð2:14cÞ ðkÞ X 1 0 0 μν 1 μν YM ð Þa ð Þ ðkÞa ðkÞ L − ¼ − F μν F − F μν F ; ð2:11aÞ ð0Þab v v 4 a 4 a In the above expressions, Dμ is the covariant derivative ðkÞ in the adjoint representation of SUðN;M4Þ. In addition, the symbols Δ and Δ0 are given by 1 X ðrksÞ ðrksÞ LYM F ðkÞa F ðkÞa μ v-s ¼ 2 μν¯ ðxÞ ν¯ ðxÞ; ð2:11bÞ Z Z 1 2πR 2πR1 ðkÞ Δ n … n ¯ ðkÞ ¯ ðsÞ ¯ ðrÞ ¯ ðrksÞ ¼ 0 d xfE ðxÞfE ðxÞfE ðxÞ; ð Þ 0 0 X fE 1 0 0 μ¯ ν¯ 1 μ¯ ν¯ LYM − F ð ÞaF ð Þ − F ðkÞaF ðkÞ s-s ¼ 4 μ¯ ν¯ a 4 μ¯ ν¯ a : ð2:11cÞ ð2:15aÞ ðkÞ Z Z 1 2πRn 2πR1 Δ0 … dnxf¯ ðkÞ x¯ fðsÞx¯ fðrÞ x¯ : The Lagrangian L d 4 in (2.9) arises from compactifica- ðrksÞ ¼ ð0Þ O ð Þ O Þ E ð Þ ð > Þ f 0 0 L E tion of ðd>4þnÞ in (2.6), which contains all interactions of canonical dimension higher than 4, which are compatible ð2:15bÞ with the standard symmetry ISOð1; 3Þ ×SUð1; M4Þ. Note that the Lagrangians (2.11b) and (2.11c) correspond In the above expressions, the curvature components to a scalar kinetic sector and to a scalar potential, F ð0Þa F ðkÞa f μν ðxÞ; μν ðxÞg are given by respectively. This means that masses for the gauge, X ðkÞa ðkÞa Aμ ðxÞ, and scalar, Aμ¯ ðxÞ, fields can arise from the F ð0Þa ð0Þa abc ðkÞb ðkÞc μν ¼ Fμν þ gf Aμ Aν ; ð2:12aÞ Lagrangians (2.11b) and (2.11c), respectively. ðkÞ Any KK excited mode χðkÞ, labeled by a specific multi- index ðkÞ, acquires a KK mass, ðkÞa ð0Þab ðkÞb ð0Þab ðkÞb F μν ¼ Dμ Aν − Dν Aμ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     X 2 2 2 abc ðrÞb ðsÞc k1 k2 k þ gf Δ Aμ Aν ; ð2:12bÞ n ðkrsÞ mðkÞ ¼ þ þþ : ð2:16Þ ðrsÞ R1 R2 Rn

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ðkÞa Mass terms for vector KK fields Aμ emerge in a same number of pseudo-Goldstone bosons. On the other straightforward manner from Eqs. (2.11b) and (2.14a), hand, the dH gauge fields pointing along directions which contrasts with the case of scalar KK excited modes corresponding to unbroken generators remain massless ðkÞa ðkÞa … ðkÞa and are the connections of the gauge subgroup H, which belonging to the set fA5 ;A6 ; ;A4 ng, with fixed KK þ governs the resultant theory. So, the remaining dH unbro- index ðkÞ, since mixings among all the fields of such a set ken generators define the Lie algebra of H. Regarding the take place, as it follows from Eqs. (2.11c) and (2.14b). KK mechanism, note that the complete set of orthogonal The mixing for this set of scalar KK modes is given by the k k functions ffð Þ;fðð Þg is not unique. To pick a particular set, real and symmetric mixing matrix MðkÞ, with entries E O an extra-dimensional observable, namely, the ISOðnÞ MðkÞ 2 δ − ¯ ðkÞ ¯ ðkÞ ¯ 2 μ¯ ν¯ ¼ mðkÞ μ¯ ν¯ pμ¯ pν¯ . Things can be conveniently Casimir invariant P , was utilized, though other options, k k arranged so that, denoting the orthogonal-diagonalization yielding different sets ffð Þ;fðð Þg, are available. The 4 E O matrix of MðkÞ by RðkÞ, the diagonalization definition of a such a set determines a canonical trans- ðkÞ TMðkÞ ðkÞ 2 δ 1 − δ ðR R Þμ¯ ν¯ ¼ mðkÞ μ¯ ν¯ ð μ¯;4þnÞ can be executed. formation that maps the extra-dimensional fields into the Note that the eigenvalues of RðkÞ TMðkÞRðkÞ are m2 , except four-dimensional KK modes, thus defining a theory gov- ðkÞ erned by four-dimensional Poincar´e invariance. In other for that corresponding to μ¯ ν¯ 4 n, which is 0. The ¼ ¼ þ words, the map ISOð1; 3 þ nÞ → ISOð1; 3Þ takes place. null eigenvalue implies the presence of massless scalar KK Furthermore, while the extra-dimensional theory is invari- ðkÞa excited modes, which we denote as AG and which turn ant with respect to some gauge group defined on the out to be kind of pseudo-Goldstone bosons, in the sense spacetime with extra dimensions, after this map, the that a nonstandard gauge transformation that eliminates resulting theory is manifestly governed by a gauge group them from the theory exists, indicating that such fields characterized by the same generators, though defined in represent unphysical degrees of freedom. After the change four dimensions. Consider a connection of the gauge group of basis, induced by diagonalization, the resulting mass- in extra dimensions and assume that it has been mapped eigenfields basis involves, for any fixed KK index ðkÞ, the into its set of KK modes. The corresponding KK zero mode 0ðkÞa 0ðkÞa … 0ðkÞa points along the direction of the constant function set of scalar fields fA 1 ;A2 ; ;An−1 g, all of them fð0Þ ¼hx¯jp¯ ð0Þi, determined by the P¯ 2 eigenket jp¯ ð0Þi. with mass mðkÞ, and the aforementioned pseudo-Goldstone ðkÞa The zero mode remains massless and transforms as a bosons AG . Such a diagonalization, with the associated gauge field with respect to the four-dimensional gauge set of resulting fields, is illustrated in Eq. (2.17): group, which resembles what happens with the gauge fields pointing toward the directions associated to unbroken ðkÞa 4þn ↦ ðkÞa 0ðkÞa n−1 generators in the EHM. Moreover, the remaining 2n − 1 fAμ¯ gμ¯¼5 AG ; fA n¯ gn¯¼1: ð2:17Þ eigenkets jp¯ ðkÞi, with ðkÞ ≠ ð0Þ, are analogues of the The KK-mechanism procedure bears features that evoke broken gauge-group generators from the EHM, in the – ðkÞ the Englert-Higgs mechanism (EHM) [103 105], respon- sense that they define independent directions fO and sible for mass generation in the Standard Model. A gauge- ðkÞ fE along which vector fields with masses acquired by invariant scalar potential with degenerate minima, which the KK mechanism are directed, with the presence of the can be characterized by the set of points constituting a same number of associated pseudo-Goldstone bosons. It is hypersphere with radius determined by some vacuum worth emphasizing that, in contrast with the case of the expectation value, is the starting point of the EHM. The EHM, the KK mechanism does not involve broken gauge hypersphere points are connected to each other by gauge generators, since the extra-dimensional and the four-dimen- symmetry associated to some group G, of dimension dG,so sional gauge groups share the same generators. they represent physically equivalent vacuum states. To pick one of such minima, a specific constant vector, associated III. ONE-LOOP CALCULATION to a particular point on the hypersphere, is taken. Such a choice induces a map G ↦ H that breaks the gauge group The purpose of this section is to present the calculation of the one-loop contribution of the KK modes to the G down into one of its subgroups H ⊂ G, of dimension dH. tensor polarization associated with the zero-mode gauge This procedure breaks dG − dH generators of G, thus ð0Þa leaving dH unbroken generators. Any gauge field pointing field Aμ . toward the direction defined by a broken generator becomes massive, which yields the emergence of an A. Gauge-fixed renormalized action associated pseudo-Goldstone boson. Hence, the resulting Our goal is the calculation, at one loop, of the beta − set of fields involves dG dH massive gauge fields and the function associated to the gauge group SUðN;M4Þ, in the context of the KK theory discussed in the previous section. 4In this equation, the repeated index μ¯ is not summed. To this aim, we are interested in those couplings that

095038-9 HUERTA-LEAL, NOVALES-SÁNCHEZ, and TOSCANO PHYS. REV. D 101, 095038 (2020) contribute at one loop to low-energy Green’s functions, their approach, gauge fixing is parametrized by a gauge- characterized by Feynman diagrams in which the external fixing parameter, usually denoted as ξ, whose different ð0Þa fields are exclusively zero-mode gauge fields Aμ . Such values correspond to different gauges. In such an approach, ξ 0 ’ Lagrangian terms include KK excited-mode vector fields the Landau gauge, ¼ , and the Feynman- t Hooft gauge, ðkÞa ðkÞa ξ ¼ 1, are commonly utilized. When massive gauge fields Aμ and scalar fields A0 , but note that contributions n¯ unitary gauge ð0Þa are present, a customary choice is the , which, from the zero mode Aμ must be considered as well [102]. in this scheme, is obtained by taking the limit as ξ → ∞. So, an effective action Γ½Að0Þ, which results from integrat- The field-antifield formalism and the BRST symmetry ð0Þa ing out quantum fluctuations of Aμ and KK excited-mode constitute an efficacious mean through which the quanti- fields, is defined and calculated. zation of gauge systems can be achieved [106,111–118].In We address gauge fixing in the KK theory within the this framework, the field content defining some gauge framework of the Becchi-Rouet-Stora-Tyutin (BRST) for- theory gets systematically extended. First, a set of ghost malism. The extended-action proper solution for the four- and antighost fields is added to the theory; more precisely, dimensional gauge group SUðNÞ has been discussed in per each gauge parameter participating in the theory, a detail in Ref. [106], while a generalization to extra ghost-antighost pair is introduced. Also, a set of auxiliary dimensions and the corresponding KK theory are found fields is included. Then, a further enlargement of the field in Refs. [36,38,41]. In this approach, gauge fixing in the content takes place by the incorporation of antifields, one Standard Model in extra dimensions has been discussed in per each field already defined. Moreover, a symplectic Refs. [107,108]. The implementation of these techniques to structure, known as the antibracket is defined, with each the (4 þ n)-dimensional YM theory and its KK effective field-antifield pair being canonical conjugate variables. description yields the quantum Lagrangian The resultant increased set of fields is then understood to define an extended action, which is assumed to satisfy the 0 LYM LYM LGF LG Lð ÞYM LKK Batalin-Vilkovisky master equation. BRST transforma- ¼ þ þ þ c:t: þ c:t: ; ð3:1Þ QKK KK KK KK tions, which include gauge transformations, are generated LYM LGF by the extended action, governed by BRST symmetry. where KK is the classical Lagrangian, Eq. (2.9); KK is the gauge-fixing term; and LG is the corresponding Having established the master equation, the main objective KK is the determination of a proper solution, which is Lð0ÞYM ghost-antighost sector. In addition, c:t: is the usual distinguished from other extended actions by suitable LKK YM counterterm, whereas c:t: contains the remaining boundary conditions connecting it with the original action, counterterms of the effective theory. previous to incrementation of the field content. The next goal is gauge fixing, which is nontrivially performed B. Gauge fixing in the Kaluza-Klein theory through the definition of a fermionic functional aimed at Field formulations aimed at furnishing sensible quantum the elimination of the whole set of antifields. The idea is to descriptions of nature are usually built on the grounds of kill two birds with one stone by getting rid of antifields and, gauge symmetry. The essence of gauge symmetry resides in collaterally, fixing the gauge. This process ends with the the presence of more degrees of freedom than those strictly emergence of a quantum action, which depends on general required by some given system for its description [109]. gauge-fixing functions. At this point, gauge invariance has Gauge transformations link a whole family of different been completely removed in a general framework in which mathematical configurations which, in order for gauge sets of ad hoc gauge-fixing functions, with minimal symmetry to make physical sense, must lead to the exact restrictions, can be defined to establish a particular gauge same physical results. In other words, any observable configuration. intended to be genuinely physical must be gauge indepen- To put the gauge-fixing procedure that will be introduced dent. Even though gauge symmetry is a main element for below in perspective, and also for clarity purposes, it is the definition of field theories, it turns out that quantization convenient to present a brief discussion about the one-loop requires gauge fixing to be carried out, which means renormalization of pure YM theories in the linear gauge choosing a specific gauge, thus resulting in a formulation [110]. Bare quantities are related with renormalized 0 pffiffiffiffiffiffi 0 0 pffiffiffiffiffiffi that is not gauge invariant anymore. ð Þa ð Þa ð Þa ð0Þa ones as follows: ABμ ¼ ZAAμ , CB ¼ ZCC , Being associated to local symmetry groups, gauge 0 pffiffiffiffiffiffi pffiffiffiffiffi C¯ ð Þa Z C¯ ð0Þa, and g Z g, with Cð0Þa and C¯ ð0Þa transformations are defined by functions, known as gauge B ¼ C B ¼ g parameters, which depend on spacetime coordinates. The the ghost and antighost fields, respectively. The bare selection of a set of specific spacetime-dependent functions Lagrangian is given by to play the role of gauge parameters fixes the gauge, establishing a particular gauge configuration. A systematic path to pick a gauge, among the so-called linear gauges, 0 0 0 Lð Þ Lð Þ Lð ÞYM was developed long ago by the authors of Ref. [110].In QYM;B ¼ QYM þ c:t: ; ð3:2Þ

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Lð0Þ where QYM is the renormalized quantum Lagrangian, given by

0 1 0 0 μν 1 0 μ 2 ð Þ ð Þa ð Þ ð Þ ¯ ð0Þa μ ab ð0Þb L − Fμν F − ∂μA C −∂ Dμ C ; : QYM ¼ 4 a 2ξ ð a Þ þ ð Þ ð3 3Þ

Lð0ÞYM while c:t: is the counterterm Lagrangian,

0 δ 0 0 g pffiffiffiffiffiffiffiffiffiffiffi 0 0 Lð ÞYM A ∂ ð Þa − ∂ ð Þa ∂μ ð0Þaν − ∂ν ð0Þaν δ − 1 1 ∂ ð Þa − ∂ ð Þa abc ð0Þbμ ð0Þcν c:t: ¼ 4 ð μAν νAν Þð A A Þþ2 ½ ZgZAð A Þþ ð μAν νAν Þf A A 2 g 0 0 δ − 1 1 abc ð Þb ð Þb ade ð0Þdμ ð0Þeν þ 4 ½ZgZAð A Þþ f Aμ Aν f A A ; pffiffiffiffiffiffiffiffiffiffiffi 0 ¯ ð0Þa 2 ð0Þa abc ¯ ð0Þa μ ð Þb ð0Þc − δCC ∂ C − ½ ZgZAðδ þ 1Þ − 1gf C ∂ ðAμ C Þð3:4Þ

ð0Þa with δA ¼ ZA − 1 and δC ¼ ZC − 1. The determination of As displayed in Eq. (3.5), gauge-fixing functions f 0 ZA, at a given order, is achieved through calculation of LGFð Þ define the gauge-fixing Lagrangian term YM , exclu- vacuum polarization, whereas Zg, and hence the renormal- sively constituted by KK zero modes, thus being meant for ized coupling constant, requires the calculation of the the specification of a gauge configuration among those 3- and 4-gauge-boson vertex functions. Note that this defined by the symmetry group SUðN;M4Þ. On the other Lagrangian has five counterterms, which depend on three hand, LGFðkÞ, shown in Eq. (3.6), is a gauge-fixing renormalization parameters, namely, ZA, ZC, and Zg. Thus, KK there must be two relations among the counterterms. Such Lagrangian made of both zero- and excited-mode KK ðkÞa relations emerge as a consequence of BRST symmetry fields and is defined by gauge-fixing functions f . LGFðkÞ governing the quantum Lagrangian. In Abelian theories, The purpose of KK is to pick a gauge configuration such as QED, the Ward identity ensures the equality, to all allowed by invariance associated to nonstandard gauge orders of perturbation theory, between the fermion self- transformations. energy counterterm and the vertex-function counterterm. pffiffiffiffiffi Symmetry with respect to nonstandard gauge trans- This in turn implies that Z Z3 1, where e Z e and pffiffiffiffiffi e ¼ B ¼ e formations can be removed from the KK effective LYM ABμ ¼ Z3Aμ, with e and Aμ the renormalized electric Lagrangian QKK without touching the gauge group charge and electromagnetic field, respectively. An impor- SUðN;M4Þ. The trick lies in noticing that only standard tant goal of the present work is the introduction of gauge- gauge transformations are associated to this four- ð0Þa ðmÞa fixing procedures for the gauge fields Aμ and Aμ , such dimensional gauge group. In this context, a set of gauge- ðkÞa that the relation Z Z ¼ 1, analogous to the aforemen- fixing functions f , suitably defined to transform g A 4 tioned Abelian property, holds, since this will considerably covariantly under SUðN;M Þ, shall get the job done. ð0Þa So, we use the gauge-fixing functions simplify calculations. This relation implies that FBμν ¼ pffiffiffiffiffiffi 0 a ð0Þab μ ðkÞa ð Þ ðkÞa Dμ ðkÞb − ξ ZAFμν , so the counterterm of this sector becomes f ¼ A mðkÞAG ; ð3:7Þ 0 0 μν δA ð Þa ð Þ − Fμν Fa , which is gauge invariant. With this in ð0Þab 4 given for the first time in Ref. [41].Here,Dμ is the mind, we turn to discuss gauge-fixing procedures for the 4 covariant derivative of SU N;M , in the adjoint repre- ð0Þa ðkÞa ð Þ gauge zero modes Aμ and the KK excited modes Aμ .In ðkÞa 4 sentation. Our choice of functions f , given in Eq. (3.7), both cases, SUðN;M Þ gauge symmetry is maintained at thus leaves the issue of zero-mode gauge fixing to the 0 the level of the effective action Γ½Að Þ, thus simplifying the LGFð0Þ Lagrangian term YM . determination of the beta function and the renormalized Implementation of the background field method [69,70] coupling constant. ð0Þa LGF on zero-mode gauge fields Aμ is now carried out. To We write the gauge-fixing sector as KK ¼ LGFð0Þ LGFðkÞ this aim, gauge fields are conveniently rescaled as YM þ KK , with ð0Þa ð0Þa gAμ → Aμ , so the YM Lagrangian becomes 0 1 LGFð Þ − ð0Þa ð0Þa YM ¼ f f ; ð3:5Þ 1 ð0Þa 0 μν 1 ð0Þa 0 μν 2ξ − Fμν Fð Þa → − Fμν Fð Þa : ð3:8Þ 4 4g2 1 X LGFðkÞ − ðkÞa ðkÞa ð0Þa KK ¼ 2ξ f f : ð3:6Þ Moreover, this redefinition of gauge fields Aμ removes ðkÞ coupling-constant factors from both the YM field strength

095038-11 HUERTA-LEAL, NOVALES-SÁNCHEZ, and TOSCANO PHYS. REV. D 101, 095038 (2020) and the YM covariant derivative, which are thus given as Lagrangian into the renormalized Lagrangian and the ð0Þa ð0Þa ð0Þa abc ð0Þb ð0Þb ð0Þ Fμν ¼∂μAν −∂νAν þf Aμ Aν and Dμ ¼ ∂μ− counterterm, ð0Þa a a 4 Aμ t , respectively, with t the SUðN;M Þ generators ð0ÞYM LYM ¼ LYM þ L ; ð3:10Þ in some group representation. Next, we split the gauge field BKKðd≤4Þ QKKðd≤4Þ c:t: ð0Þa ð0Þa Aμ into a classical background field, Aμ and a where LYM is the renormalized quantum Lagrangian, ð0Þa ð0Þa ð0Þa ð0Þa QKKðd≤4Þ fluctuating quantum field, Qμ ,asAμ → Aμ þ Qμ . which is given by ð0Þa The classical field Aμ is a fixed field configuration, YM 0 YM k ð0Þa LYM L ð Þ L ð Þ LGF LG whereas the fluctuating field Qμ is taken as an integration QKKðd≤4Þ ¼ KK þ KK þ KK þ KK: ð3:11Þ ð0Þa μ variable in the functional integral. Furthermore, while A YMð0Þ ð0Þa The Lagrangian term L , in the right-hand side of this is found to transform as a gauge field, Qμ does it as a KK equation, is interpreted as the four-dimensional YM theory, matter field in the adjoint representation of SUðN;M4Þ. defined, as usual [102], in four spacetime dimensions, but Such transformation laws are consistent with covariance ð0Þa modified by the implementation of the background- of the YM curvature, expressed after splitting as Fμν → field splitting previously discussed. This Lagrangian term 0 0 0 0 0 0 0 ð Þa Dð Þab ð Þb − Dð Þab ð Þb abc ð Þb ð Þc ð0Þa Fμν þ μ Qν ν Qμ þ f Qμ Qν . In this is thus given only in terms of background fields Aμ and context, we implement a gauge-fixing procedure for ð0Þa μ ð0Þa fluctuation fields Q . The second Lagrangian term, fluctuating quantum fields Qμ . We find it convenient YMðkÞ which has been denoted as L , is constituted by to fix the gauge covariantly with respect to the back- KK couplings in which either both KK zero and excited modes ground gauge field, for which the zero-mode gauge-fixing participate or only KK excited-mode fields are involved functions [see Eqs. (2.11a)–(2.11c), (2.12a), (2.12b), (2.13), and (2.14a)–(2.14c)]. An analogous separation is implemented 0 ð0Þab 0 μ 0 fð Þa ¼ Dμ Qð Þb ; ð3:9Þ LG LGð Þ LGðkÞ in the ghost-antighost sector, that is, KK ¼ KK þ KK . Then, we consider KK couplings comprising the sum of LYMð0Þ LGFð0Þ LGð0Þ to be inserted in Eq. (3.5), are introduced. zero-mode Lagrangian terms KK þ KK þ KK . ðkÞa ð0Þa Besides fields Aμ and Qμ , this equation includes C. New physics effects at one loop zero-mode ghost and antighost fields. Moreover, this Since interactions of canonical dimension greater sum is a gauge-fixed Lagrangian with respect to the ð0Þa than 4 will not be considered in the one-loop calculation, fluctuating fields Qμ ; nonetheless, it is invariant under for the moment, we leave this type of interactions background-field gauge transformations. Note that gauge aside and rather focus on the terms of the quantum LGFð0Þ LGð0Þ invariance of KK implies gauge invariance of KK . Lagrangian defined by (3.1) with canonical dimension less Taking the Feynman-’t Hooft gauge, these Lagrangian than or equal to 4. As usual, we decompose the bare terms are written as

1 YMð0Þ GFð0Þ Gð0Þ ð0Þa ð0Þab 0 ρ μν 0 μν ð0Þc ¯ 0 ð0Þab 0 μ 0 L þ L þ L ¼ Qμ ðDρ Dð Þbc g þ 2fabcFð Þb ÞQν þ Cð ÞaðDμ Dð Þbc ÞCð Þc þ ð3:12Þ KK KK KK 2g2

In this expression, only couplings which contribute to vacuum-polarization Feynman diagrams, at one loop, have been written explicitly, whereas the presence of other couplings is indicated by the ellipsis. Note that in the above expression explicit gauge invariance is preserved. From Eq. (3.11), we also take the sum of Lagrangian terms

X     1 0 1 0 0 0 YMðkÞ GFðkÞ GðkÞ ðkÞaμ ð Þab ð0Þbcρ bc 2 ð Þab ð Þbc abc ð Þm ðkÞcν L L L A gμν Dρ D δ m − 1 − Dμ Dν 2g f Fμν A KK þ KK þ KK ¼ 2 ð þ ðkÞÞ ξ þ s ðkÞ X X 1 ðkÞa ð0Þab 0 ρ 2 ðkÞc 1 ðkÞa ð0Þab 0 ρ 2 ðkÞc − A0 Dρ Dð Þbc δbcm A0 − A Dρ Dð Þbc δbcξm A 2 n¯ ½ þ ðkÞ n¯ 2 G ½ þ ðkÞ G ðkÞ ðkÞ X ð0Þab 0 ρ 2 ¯ ðkÞa Dρ Dð Þbc δbcξ ðkÞc þ C ½ þ mðkÞC þ; ð3:13Þ ðkÞ

095038-12 GAUGE-INVARIANT APPROACH TO THE BETA FUNCTION IN … PHYS. REV. D 101, 095038 (2020) which, as it occurs with the zero-mode Lagrangian (3.12), Lagrangian can be written as the sum of a gauge sector is manifestly invariant under standard gauge transforma- and a ghost sector. However, in this gauge-invariant tions. In this expression, C¯ ðkÞa and CðkÞa are KK-excited- approach to quantization, the ghost fields (and also the ð0Þa mode ghost fields, which arise as part of the quantization Qμ fields) do not have to be renormalized because they ’ procedure [36,38,41]. We work in the Feynman- t Hooft only appear inside loops. So, Eq. (3.4) reduces to gauge, which was chosen in the case of the standard theory as well. A simplification introduced by this gauge consists YMðkÞ ð0ÞYM δA ð0Þa ð0Þμν L L − Fμν F : 3:14 in the elimination, from KK , of the term involving the c:t: ¼ 4 2 a ð Þ ð0Þ ð0Þ g covariant-derivatives factor Dμ Dν . Moreover, in this gauge, the unphysical masses of pseudo-Goldstone bosons L and ghost fields are the same as those of the KK gauge and We define the Lagrangian term β as the sum of scalar fields. The ellipsis in this equation represents other Eqs. (3.12), (3.13), and (3.14), but with all the terms couplings, which occur either with the involvement of both indicated by ellipses, in such expressions, as well as those KK zero and excited modes or among KK-excited-mode of canonical dimension higher than 4, removed. We are L fields only. interested in integrating out, from β, all the zero-mode ð0Þa 0 Since explicit gauge invariance is preserved in fluctuating quantum fields Qμ , ghost fields Cð Þa, and the Lagrangians given by Eqs. (3.12) and (3.13), the antighost fields C¯ ð0Þa, together with all the KK excited 0 Lð ÞYM Γ ð0Þ counterterm Lagrangian c:t: must be gauge invariant modes. With this in mind, the effective action ½A is as well, which implies the relation ZAZg ¼ 1. This defined, by Eq. (3.15),as

Y Y Z  Z  0 0 iΓ½Að Þ D ð ÞaD ð0ÞaD ¯ ð0ÞaD ðkÞaD 0ðkÞaD ðkÞaD ðkÞaD ¯ ðkÞa 4 L e ¼ Qμ C C Aμ A n¯ AG C C exp i d x β ðkÞ x;a;μ;n¯  Z   −1 ð0Þa ð0Þμν ð0ÞYM 1 4 −2 þ1 ¼ exp i d x Fμν F þ L ðDetΔ ð0Þ Þ ðDetΔ ð0Þ Þ 4g2 a c:t: Q C Y −1 1 −n¯ −1 Δ 2 Δ þ Δ 2 Δ 2 × ðDet AðmÞ Þ ðDet CðmÞ Þ ðDet 0ðmÞ Þ ðDet ðmÞ Þ : ð3:15Þ A n¯ AG ðmÞ R eff ð0Þ 4 eff Defining the effective Lagrangian Lβ by Γ½A ¼expfi d xLβ g and performing Gaussian integrals, we write down the equation Z Z   −1 0 0 μν 0 0 4 eff 4 ð Þa ð Þ ð ÞYM i ð0Þ 2 ð Þa a d xL ¼ d x Fμν F þ L þ Tr logfgμν ⊗ ð−ðD Þ Þþ2iFμν ⊗ T g β 4g2 a c:t: 2 a X 0 ð0Þ 2 i ð0Þ 2 2 ð Þa a − iTr log − D Tr log gμν ⊗ − D − m · 1 2iFμν ⊗ T f ð Þ gþ2 f ð ð Þ ðkÞ aÞþ a g ðkÞ   n X i − 1 Tr log − Dð0Þ 2 − m2 · 1 : 3:16 þ 2 f ð Þ ðkÞ ag ð Þ ðkÞ

In this expression, the symbol “Tr” denotes a trace over not be understood as a number, but rather as a 4 × 4 matrix spacetime coordinates and over internal degrees of associated to the four-dimensional Lorentz group. In the ð0Þa freedom as well. We have used Kronecker-product symbols same sense, Fμν is also a 4 × 4 matrix. The second to emphasize that the arguments of the logarithms are trace in Eq. (3.16) has been generated by the zero-mode different from each other, living in different spaces. The ghost-antighost terms explicitly shown in Eq. (3.12). 1 n notation a has been used for the identity matrix of size The last trace, involving the factor 2 − 1, is the total 2 2 ðN − 1Þ × ðN − 1Þ that corresponds to the adjoint rep- ðkÞa contribution from KK scalars A¯ , from pseudo- resentation of the gauge group SUðN;M4Þ. Note that the n AðkÞa first and third traces, respectively, coming from contribut- Goldstone bosons G , and from ghost and antighost ðkÞa ¯ ðkÞa ð0Þa KK-excited-mode fields C and C , all summed ing terms with zero-mode background fields Qμ in together. Note that, in this covariant gauge-fixing pro- ðkÞa Eq. (3.12) and with KK-excited-mode vector fields Aμ cedure, the ghost-antighost contribution is minus twice the in Eq. (3.13), include the symbol gμν, which should scalar contribution.

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˜ ð0Þa ð0Þa Consider the Fourier transform, Aμ ðpÞ, of the vector field Aμ ðxÞ, defined by

Z 4 ð0Þa d p − ˜ ð0Þa Aμ ðxÞ¼ e ip·xAμ ðpÞ: ð3:17Þ ð2πÞ4

In terms of this Fourier transform, the one-loop correction given by Z   Z 1 1 4 4 loop 2 ð0Þa ð0Þμν d p ˜ ð0Þa ˜ ð0Þb 2 μν μ ν ab loop 2 d x − Π ðp Þ Fμν Fa ¼ − Aμ ð−pÞAν ðpÞðp g − p p Þδ Π ðp Þþ ð3:18Þ 4g2 KK 2g2 ð2πÞ4 KK is established. The idea is to calculate the traces in the effective Lagrangian given in Eq. (3.16), with the purpose of Πloop 2 expressing their sum in the form of Eq. (3.18), aiming at the identification of the KK ðp Þ loop polarization function. The Πμνab one-loop contribution to the polarization tensor KK 1L is given by

Πμνab 2 μν − μ ν δabΠloop 2 KK1LðpÞ¼iðp g p p Þ KK ðp Þ: ð3:19Þ R 4 eff Equations (3.20)–(3.23), displayed below, provide the traces defining d xLβ , in Eq. (3.16): KK-zero-modes gauge trace:

Z 4 0 0 0 i ð0Þ 2 ð Þa a i d p ˜ ð Þa ˜ ð Þa Tr logfgμν ⊗ ð−ðD Þ Þþ2iFμν ⊗ T g¼ Aμ ð−pÞAν ðpÞ 2 a 2 ð2πÞ4  Z Z Z  1 d4q ðpμ þ 2qμÞðpν þ 2qνÞ d4q ðp2gμν − pμpνÞ d4q 1 × 4N − − þ gμν þ; ð3:20Þ 2 ð2πÞ4 q2ðq þ pÞ2 ð2πÞ4 q2ðq þ pÞ2 ð2πÞ4 q2

KK-excited-modes gauge trace:

Z 4 0 0 0 i ð0Þ 2 2 ð Þa a i d p ˜ ð Þa ˜ ð Þa Tr logfgμν ⊗ ð−ðD Þ − m · 1 Þþ2iFμν ⊗ T g¼ Aμ ð−pÞAν ðpÞ 2 ðkÞ a a 2 ð2πÞ4  Z Z 1 d4q ðpμ þ 2qμÞðpν þ 2qνÞ d4q ðp2gμν − pμpνÞ × 4N − − 2 ð2πÞ4 ½q2 − m2 ½ðq þ pÞ2 − m2 ð2πÞ4 ½q2 − m2 ½ðq þ pÞ2 − m2  ðkÞ ðkÞ ðkÞ ðkÞ Z 4 μν d q 1 þ g 2π 4 2 − 2 þ; ð3:21Þ ð Þ q mðkÞ

KK-zero-modes ghost-antighost trace:

Z 4 0 2 d p ˜ ð0Þa ˜ ð0Þa −iTr logf−ðDð ÞÞ g¼−i Aμ ð−pÞAν ð−pÞ ð2πÞ4  Z Z  1 d4q ðpμ þ 2qμÞðpν þ 2qνÞ d4q 1 × N − þ gμν þ; ð3:22Þ 2 ð2πÞ4 q2ðq þ pÞ2 ð2πÞ4 q2

KK-excited-modes ghost-antighost trace:

Z 4 0 2 2 d p ˜ ð0Þa ˜ ð0Þa −iTr logf−ðDð ÞÞ − m · 1 g¼−i Aμ ð−pÞAν ð−pÞ ðkÞ a ð2πÞ4  Z Z  1 d4q ðpμ þ 2qμÞðpν þ 2qνÞ d4q 1 N − gμν ; : × 2 2π 4 2 − 2 2 − 2 þ 2π 4 2 − 2 þ ð3 23Þ ð Þ ½q mðkÞ½ðq þ pÞ mðkÞ ð Þ q mðkÞ

All the terms explicitly shown in Eqs. (3.20)–(3.23) are given in terms of loop integrals, which, after being regularized in a gauge-invariant manner and then being solved through standard methods [50,52,102,119,120], fit into the structure of Eq. (3.19). Ellipses in Eqs. (3.20)–(3.23), on the other hand, stand for terms not contributing to the polarization function Πloop 2 KK ðp Þ, in accordance with Eq. (3.19).

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D. Divergence structure From Eqs. (3.20)–(3.23), the one-loop contribution to the tensor polarization can be written as Z   4 μν μν X μν μν μνab ab 2 d q TV þ TS TV ðnÞþTS ðnÞ Π 1 ðpÞ¼δ ig N þ ; ð3:24Þ KK L ð2πÞ4 q2ðq þ pÞ2 ½q2 − m2 ½ðq þ pÞ2 − m2 ðkÞ ðkÞ ðkÞ

μν μν where the TV and TS Lorentz tensors represent the zero-mode gauge-field and ghost-antighost contributions, respectively, which are given by

μν −2 2 μ 2 ν 4 2 μν − 4 2 μν − μ ν TV ¼ ð q þ pÞ ð q þ pÞ þ ðp þ qÞ g ðp g p p Þ; ð3:25aÞ μν 2 μ 2 ν − 2 2 μν TS ¼ð q þ pÞ ð q þ pÞ ðp þ qÞ g : ð3:25bÞ μν ðkÞa μν On the other hand, TV ðnÞ represents the gauge KK-modes Aμ contribution, while TS ðnÞ brings together the ðkÞa contributions of the corresponding ghost-antighost a sort of pseudo-Goldstone boson associated with Aμ , as well as those contributions from physical scalars. They are given by

μν −2 2 μ 2 ν 4 2 − 2 μν − 4 2 μν − μ ν TV ðnÞ¼ ð q þ pÞ ð q þ pÞ þ ½ðp þ qÞ mðkÞg ðp g p p Þ; ð3:26aÞ   μν n T n − 1 − 2q p μ 2q p ν 2 p q 2 − m2 gμν : 3:26b S ð Þ¼ 2 f ð þ Þ ð þ Þ þ½ ð þ Þ ðkÞ g ð Þ

So far, we have been talking about a quantum field as genuine ultraviolet divergences, which allow us to theory that is unusual in the sense that it comprises an handle them by renormalization in the context of an infinite number of fields, while no mention of conse- effective theory [52–55]. quences that this could have on radiative corrections has To shed light on what the nature of the new kind of been made. As already commented in the Introduction, the divergencesP that can arise from the discrete sums nested in presence of such an infinite number of fields can become a the symbol ðkÞ is, we address the problem from a different difficult problem to handle because eventually divergences perspective. The investigation of the present paper is different from the usual ones mayP ariseR from the presence of motivated by string theory, which is the spirit of KK 4 discrete and continuous sums ðkÞ d k that emerge from theories; that is, we assume that it represents the contri- excited-modes contributions. If the number of KK excita- bution of an infinite number of KK particles. However, tions were finite, no matter how large, we would be in a technically speaking, the loop amplitudes, in which an conventional scenario of calculating the one-loop contri- infinite number of excited KK modes circulate, may be bution of a large, but finite, number of particles to a given subjected to another interpretation. Before compactifica- loop amplitude. However, in our case in which the number tion, consider some particle propagating in the 4 þ n Pof KK fields is infinite, the discrete sums nested in the spacetime dimensions. From this point of view, the discrete ðkÞ symbol [see Eqs. (2.3) and (2.4)] may or may not momenta qμ¯ can be interpreted as the components of the converge. The presence of both discrete and continuous total momentum of such an extra-dimensional particle infinite sums is strongly linked to the existence of two along the compact manifold, that is, qM ¼ qμ þ qμ¯ . different spaces, namely, the usual four-dimensional Thus, this new type of divergences, if present, can be spacetime manifold and the compact n-dimensional mani- considered as genuine ultraviolet divergences in the sense fold, the presence of the infinite number of KK fields that they correspond to very-high-energy effects or, equiv- having to do precisely with this new manifold. As it is alently, to very-short-distance effects. To show this, let us to emphasized in the Introduction and commented for the first rewrite the expression given in (3.24) in the equivalent time in Ref. [56], these types of divergences can be treated manner

Z   4 μν μν X μν μν μνab ab 2 d q TV þ TS TV ðnÞþTS ðnÞ Π 1 ðpÞ¼δ ig N þ ; ð3:27Þ KK L ð2πÞ4 q2ðq þ pÞ2 ½q2 − q¯ 2½ðq þ pÞ2 − q¯ 2 ðqÞ

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2 → 2 where notation has been changed as m k m q ¼ regularizing simultaneously both types of divergent quan- 2 ð Þ ð Þ q¯ μ¯ q¯ μ¯ ≡ q¯ . This expression suggests that sums over tities, because, as previously argued, they have the same discrete squared momenta q¯ 2 comprise very-high-energy origin in the sense that the sums in consideration involve effects for KK indices ðqÞ with very large components, just the magnitudes qμ and q¯ μ¯ , which are linked to the usual as it occurs for continuous sums over very large momenta and compact manifolds through Fourier transform (qμ) and ¯ q. In other words, while very large continuous momenta qμ Fourier series (qμ¯ ). TheP main idea behind this is to characterize very-short-distance effects in the infinite mani- express the regularized ðkÞ sums in terms of Epstein 4 fold M , very-large-discrete momenta qμ¯ can be associated zeta functions [57,121,122] defined in the complex plane through analytic continuation. The inhomogeneous with very-short-distancepffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi effects in the compact manifold, 2 2 2 l-dimensional Epstein function is defined as since 1= q¯ ¼ R =q → 0 as q → ∞. So, when the discrete sums diverge, we have a genuine ultraviolet X∞ 1 divergence. Later on, we will discuss how to distinguish c2 El ðsÞ¼ 2 2 2 s : ð3:28Þ a type of ultraviolet divergence from the other. ðk1 þþk þ c Þ ðk1;…;k Þ¼1 l We now proceed to regularize the expression given by l Eq. (3.27). As emphasized in the Introduction, we will address the one-loop impact of the effective theory not from A special case corresponds to c ¼ 0, which leads to the the cutoff approach but using the dimensional regulariza- homogeneous Epstein function. tion scheme [50,51]. In the case under consideration, we As usual, we promote the ordinary four-dimensional must also deal with the new type of divergences that can spacetime to D dimensions. After that and once Feynman arise from discrete sums. Our approach consists in parametrization has been implemented, Eq. (3.27) becomes

Z Z  μν μν  2 1 2 2−D X μν g N 4πμˆ 2 T 0 T Π ab δab ð Þ D ð Þ ðkÞ KK 1LðpÞ¼ 2 dx D d q 2 2 2 þ 2 2 2 ; ð3:29Þ ð4π Þ 0 π 2 ðq − Δ Þ ðq − Δ Þ i ð0Þ ðkÞ ðkÞ

μˆ ϵ 4 − Δ2 − 1 − 2 Δ2 2 Δ2 where is the scale associated with dimensional regularization, ¼ D, ð0Þ ¼ xð xÞp , and ðkÞ ¼ mðkÞ þ ð0Þ. In addition,     μν 2 T ¼ − 2 1 − q2 þ 2ð1 − xÞ2p2 gμν þð1 − 2xÞ2pμpν þ 4ðp2gμν − pμpνÞð3:30aÞ ð0Þ D       μν D 2 n T ¼ 2 1 þ − 1 q2 − ð1 − xÞ2p2 þ m2 gμν þ 1 þ ð1 − 2xÞ2pμpν þ 4ðp2gμν − pμpνÞ: ð3:30bÞ ð0Þ 2 D ðkÞ 2

Once the integrals on the momentum space have been performed, the one-loop polarization function can be expressed as

Z    2 −ϵ     2 −ϵ  2 1 ϵ Δ 2 X ϵ Δ 2 Πloop 2 g N Γ ð0Þ − n Γ ðkÞ KK ðp Þ¼ 2 dx fðxÞ 2 þ fðxÞ gðxÞ 2 ; ð3:31Þ ð4πÞ 0 2 4πμˆ 2 2 4πμˆ ðkÞ where fðxÞ¼4 − gðxÞ, with gðxÞ¼ð1 − 2xÞ2. Note that the KK-modes contribution contains, besides the gauge contribution characterized by the fðxÞ function, the contribution of n scalar matter fields, which correspond − 1 ðkÞa ðkÞa ðkÞa to the n physical scalars An¯ and the pseudo-Goldstone boson AG associated with the Aμ gauge boson. Note now that

ϵ  2 −   ϵ   ϵ     X Δ 2 −2 − X −2 − Xn ðkÞ R 2 2 2 −ϵ R 2 n 2 ϵ k c 2 Ec ; 3:32 4πμˆ 2 ¼ 4πμˆ 2 ð þ Þ ¼ 4πμˆ 2 l 2 ð Þ ðkÞ ðkÞ l¼1 l

Δ2 2 ð0Þ where in the last step Eqs. (2.4) and (3.28) have been used. In addition, c ¼ R−2 , where, for the sake of simplicity, the same 1 radii for the n orbifolds S =Z2, R ≡ R1 ¼¼Rn, have been assumed. Using this expression, the polarization function (3.31) becomes

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Z    2 −ϵ    ϵ       2 1 Δ 2 −2 − Xn g N ϵ 0 n R 2 n ϵ 2 ϵ Πloop 2 Γ ð Þ − Γ c KK ðp Þ¼ 2 dx fðxÞ 2 þ fðxÞ gðxÞ 2 El : ð3:33Þ 4π 0 2 4πμˆ 2 4πμˆ 2 2 ð Þ l¼1 l c2 l l−1 … − 1 − 3 Since the l-dimensional Epstein function El ðsÞ has poles at s ¼ 2 ; 2 ; ; 2 ; 2 ; , except zero [123], it is clear that c2 ϵ ϵ → 0 El ð2Þ converges for . However, the way this happens is subtle, which can be appreciated more clearly by expressing the Epstein function in terms of the Riemann zeta function, whose properties are well known in the literature. An important result [122] consists in expressing the l-dimensional Epstein function in terms of the one-dimensional Epstein function as     l−1 Xl−1 − 1 p 2 ð−1Þ l p Γðs − 2Þ 2 p Ec ðsÞ¼ ð−1Þpπ 2 Ec s − : ð3:34Þ l 2l−1 Γ s 1 2 p¼0 p ð Þ The reduction to the Riemann zeta function is achieved through the following power series in c2 [56,124],

X∞ k 2 ð−1Þ Γðk þ sÞ Ec ðsÞ¼ ζð2k þ 2sÞc2k; ð3:35Þ 1 k! Γ s k¼0 ð Þ where the Riemman function is defined by

X∞ 1 ζðsÞ¼ ; ð3:36Þ ns n¼1 which has a simple pole at s ¼ 1. We can see, from Eqs. (3.34) and (3.35), that finite terms which are the ratio of ζð1þsÞ 1 two quantities that diverge when s tends to zero, as ΓðsÞ , arise for l> . This fact has nontrivial consequences when a Γ c2 product of the form ðsÞEl ðsÞ, as the one appearing in the polarization function (3.33), is considered. Using the above results, we write           Xn ϵ ϵ 1 Xn Xr X∞ −1 k 2 − ϵ n 2 n n−r ð Þ k þ r n þ 2 Γ Ec ¼ π 2 Γ ζð2k þ r − n þ ϵÞc k: ð3:37Þ 2 l 2 2n−1 − 1 k! 2 l¼1 l r¼1 l¼1 l k¼0

Note that this power series in c2 is actually a power series in 2k þ r − n ¼ 1, which correspond to poles, when ϵ → 0, the external momentum p2. A careful analysis of this of the gamma and Riemann functions, respectively. Note 2kþr−nþ2 expression leads us to conclude that there are two types of that the products of the form Γð 2 Þζð2k þ r − n þ 2Þ divergences for ϵ → 0. One type of divergences, whose converge when 2k þ r − n þ 2 ¼ −2; −4; − , since the main feature is to be independent of both the external negative even integers correspond to the so-called trivial momenta and the compactification scale, can easily be zeros of the Riemann zeta function. These types of identified from the k ¼ 0 term in (3.37). This type of divergences, which arise from nonconvergence of products divergences, which do not depend on the external mo- Γ ϵ c2 ϵ ð2ÞEl ð2Þ, can also be considered as genuine ultraviolet mentum, are usual ultraviolet divergences that emerge from divergences because, as already commented, they corre- M4 short-distance effects in the standard manifold .An- spond to effects of large discrete momentum or, equiv- other type of ultraviolet divergences, different to the usual alently, to short distances effects in the compact manifold. ones in the sense that they emerge as coefficients of the For the analysis that follows, it is convenient to rewrite p2 external momentum or, more precisely, from the ratio R−2, the above expression so that it explicitly shows those terms occurs for terms with k ≠ 0 in the series (3.37). This that are divergent. Once this rearrangement is implemented, can happen for an even integer 2k þ r − n ≤ 0 or for we get

        n    Xn X½2 n ϵ 2 ϵ ϵ 1 1 ϵ c þ Γ E ¼ g 0 ðnÞΓ ζðϵÞþF 0 ðnÞþ f ðnÞ pffiffiffi Γ ζð1 þ ϵÞ l 2 l 2 ð Þ 2 ð Þ ðkÞ π 2 l¼1   k¼1 ϵ Γ ζ ϵ 2k 2 þ gðkÞðnÞ 2 ð ÞþFðkÞðnÞ c þ Fðn; c Þ; ð3:38Þ

095038-17 HUERTA-LEAL, NOVALES-SÁNCHEZ, and TOSCANO PHYS. REV. D 101, 095038 (2020) where   1 g 0 n 2 1 − ; 3:39a ð Þð Þ¼ 2n ð Þ     Xn−1 Xr 1 n n−r r − n F 0 ðnÞ¼ π 2 Γ ζðr − nÞ;n>1 ð3:39bÞ ð Þ 2n−1 − 1 2 r¼1 l¼1 l       Xn Xr 1 n r−n 2k þ r − n F ðnÞ¼ π 2 Γ ζð2k þ r − nÞ;n>1 ð3:39cÞ ðkÞ 2n−1 − 1 2 r≠n−ð2k−1Þ;n−2k l¼1 l     1 Xn Xr X∞ −1 k 2 − 2 n n−r ð Þ k þ r n 2 Fðn; c Þ¼ π 2 Γ ζð2k þ r − nÞc k: ð3:39dÞ 2n−1 − 1 ! 2 1 1 l n k r¼ l¼ k¼½2þ1

      n ϵ ϵ ϵ In the above expressions, the symbol ½2 means the floor of c2 2 n n Γ E1 ¼ Γ ζðϵÞþFð1;c Þ; ð3:42Þ 2, that is, the largest integer less than or equal to 2. On the 2 2 2 other hand, the functions fðkÞðnÞ and gðkÞðnÞ are given by   where the Fð1;c2Þ function is free of ultraviolet divergen- n−X2kþ1 k 1 n ð−1Þ 2k−1 f ðnÞ¼ π 2 ; ð3:40aÞ cies and is given by ðkÞ 2n−1 − 1 k! l¼1 l X∞   ð−1Þs nX−2k 1 2 ζ 2 2s 1 n ð−1Þk Fð ;c Þ¼ ð sÞc g ðnÞ¼ πk: ð3:40bÞ 1 s ðkÞ 2n−1 − 1 ! s¼ 1 l k ∞   l¼ X c2 ¼ − log 1 þ k2 These functions have the following properties: k¼1 log Γ 1 ic 2 ; 3:43 1 2 − 1 0 ¼ ðj ð þ Þj Þ ð Þ fðkÞð Þ¼¼fðkÞð k Þ¼ ; ð3:41aÞ

where in the last step we used the software Mathematica, g k ð1Þ¼¼g k ð2kÞ¼0: ð3:41bÞ ð Þ ð Þ by Wolfram [125], to perform the infinite sum. These relations imply that in the case of only one extra dimension no divergences associated with the power series IV. VACUUM POLARIZATION 2 1 0 1 0 in p emerge, since in this case fðkÞð Þ¼ and gðkÞð Þ¼ AND BETA FUNCTION for all k 1; 2; …. In the case n 2, g 2 0 for all k, ¼ ¼ ðkÞð Þ¼ Asymptotic freedom is a physical phenomenon in which 2 ≠ 0 2 0 2 3 … 3 but fð1Þð Þ and fðkÞð Þ¼ for k ¼ ; ; .Ifn ¼ , an interacting theory becomes asymptotically noninteract- 3 ≠ 0 3 ≠ 0 3 0 besides fð1Þð Þ ,wehavegð1Þð Þ ,butfðkÞð Þ¼ ing because of some coupling constant that decreases as 3 0 2 3 … 2 and gðkÞð Þ¼ for all k ¼ ; ; . Thus, for n ¼ and shorter distances are explored. This aspect, known to n ¼ 3, divergences arise only in the first term (k ¼ 1) of the characterize non-Abelian gauge theories [58–60] and power series in p2. However, for the cases n ¼ 4 and which is central for understanding the strong interaction, n ¼ 5, divergences arise in the first (k ¼ 1) and second is studied through renormalization-group techniques. It (k ¼ 2) terms of the power series in p2; the case n ¼ 6 and turns out that beta functions corresponding to non- n ¼ 7 lead to divergences in the first (k ¼ 1), second Abelian gauge theories that include a sufficiently small (k ¼ 2), and third (k ¼ 3) terms of the power series in p2; number of fermions are negative, which thus yield the and so on. As we will see below, the (3.38) decomposition asymptotic-freedom behavior. The purpose of this section of the multidimensional Epstein functions will play and is to investigate one-loop impact of universal extra dimen- central role in our analysis. sions on the beta function. As already commented, the case n ¼ 1 is the only one The case n ¼ 1 involves features that make it worthy of which does not have ultraviolet divergences arising from special attention, so we will divide our analysis into two short-distance effects in the compact manifold. In this case, scenarios, one with only one extra dimension and the other the expression (3.38) becomes with an arbitrary number n ≥ 2 of extra dimensions.

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A. Case n =1 [126–130]. This means that if only Green’sfunctions As already discussed, in the case of one extra with zero-mode fields as external legs are considered dimension, no ultraviolet divergences from short-dis- and no interactions of canonical dimension higher than tance effects in the compact manifold arise. In other 4 are inserted in loop diagrams, the theory given by the words, the coefficients of p2k (or c2k)in(3.38) are all quantum Lagrangian (3.11) is self-contained, with a finite. As we will see, this theory has much in common counterterm given by Eq. (3.14) if a gauge-invariant with conventional field theories. An important aspect is quantization procedure has been implemented, as in the that any vertex function of canonical dimension higher case at hand. than 4 is free of ultraviolet divergences, which has been From Eqs. (3.14) and (3.33), the renormalized polari- explicitly verified in many phenomenological studies zation function is given by

ϵ Z    2 −    ϵ     2 1 Δ 2 −2 − g N ϵ 0 1 R 2 ϵ 2 ϵ Π5D 2 Γ ð Þ − Γ c − δ KKðp Þ¼ 2 dx fðxÞ 2 þ fðxÞ gðxÞ 2 E1 A; ð4:1Þ ð4πÞ 0 2 4πμˆ 2 4πμˆ 2 2

ϵ c2 ϵ ðkÞa with Γð2ÞE1 ð2Þ given by Eqs. (3.42) and (3.43). In this where the contributions from the gauge field Aμ and its ðkÞa expression, δA is the contribution of the gauge-invariant pseudo-Goldstone boson AG have been explicitly dis- counterterm (3.14). Note that, in this case, the only con- played. This aims at emphasizing that the longitudinal tribution from matter fields comes from pseudo-Goldstone components of the gauge fields behave as matter fields, ðkÞa ðkÞa bosons AG , associated to the gauge fields Aμ . since their contributions have signs opposite to the con- In usual YM theories, the beta function is calculated by tributions from the transverseP polarization states. It is ζ 0 ∞ −1 2 using a mass-independent renormalization scheme, such as important to note that ð Þ¼ k¼1 ¼ = quantifies MS or MS. In pure YM theories, without matter fields, the the usual ultraviolet divergences induced by the infinite beta function, besides being gauge independent, must be number of KK excited modes. scale independent. However, in our case, the mass spectrum Using that, in this scheme, the one-loop beta function of the theory comprises a wide range of energies, thus can be calculated from the renormalization factor ZA in the 1 ∂ 1 ∂δ suggesting that usage of a mass-dependent scheme that 2 ZA 2 A form, βðgÞ¼− 2 g ∂ ¼ − 2 g ∂ , we then find that allows us to study the sensitivity of beta function to new g g     physics effects is suitable. To contrast differences, we will 21 β β 1 ζ 0 calculate the beta function in both types of schemes. 5DðgÞ¼ ðgÞ þ 22 ð Þ   23 1. Mass-independent scheme β ¼ 44 ðgÞ; ð4:3Þ We use the MS scheme, in which the counterterm is determined by the pole of the divergence. From Eqs. (4.1) 3 g 11N β g − 2 and (3.42),wehave where ð Þ¼ ð4πÞ ð 3 Þ is the usual one-loop beta      function. From this result, we can appreciate that KK 2 g N 11 11 1 2 excited modes globally contribute to the beta function as δ ¼ þ − ζð0Þ ; ð4:2Þ A ð4πÞ2 3 3 6 ϵ matter fields. Operatively, this is caused by the negative value of the Riemann function ζð0Þ. In this scheme, the counterterm is given by

Z         2 Δ2 −2 g N 1 0 1 1 R Π5D 2 − ð Þ − Γ 1 2 KKðp Þ¼ 2 dx fðxÞ log −γ 2 þ fðxÞ gðxÞ log 3 −γ 2 þ logðj ð þ icÞj Þ : ð4:4Þ ð4πÞ 0 4πe μˆ 2 2 16π e μˆ

Note that new physics effects in both the beta function 2. Mass-dependent scheme and the renormalized polarization function are nondecou- The result obtained above for the beta function in pling, which is a disconcerting result, since effects of a mass-independent scheme, which does not depend on heavy physics must decouple. This suggests that a mass- the compactification scale R−1, is surprising because, on dependent renormalization scheme must be used. physical grounds, one would expect new physics effects to

095038-19 HUERTA-LEAL, NOVALES-SÁNCHEZ, and TOSCANO PHYS. REV. D 101, 095038 (2020) decouple in the limit of a very large compactification scale. scheme. To determine the counterterm δA, we use the While simple to implement, mass-independent schemes renormalization condition have the serious disadvantage that heavy particles do not Π5D ðp2 ¼ −μ2Þ¼0; ð4:5Þ decouple at energies lower their masses, in accordance with KK the Appelquist-Carazzone decoupling theorem [71]. Here, where μ is the scale of the kinematical point. This condition we recalculate the beta function by using a mass-dependent leads to the counterterm

Z    2 −ϵ    ϵ     2 1 Δ¯ 2 −2 − g N ϵ 0 1 R 2 ϵ 2 ϵ δ Γ ð Þ − Γ c¯ A ¼ 2 dx fðxÞ 2 þ fðxÞ gðxÞ 2 E1 ; ð4:6Þ ð4πÞ 0 2 4πμˆ 2 4πμˆ 2 2 ¯ 2 Δ ∂δ Δ¯ 2 1 − μ2 ¯2 ð0Þ β μ2 A where ð0Þ ¼ xð xÞ and c ¼ R−2 . The one-loop beta function is given by 5DðgÞ¼g ∂μ2, so taking into account that ¯ 2 ϵ 2 Δ −2 ∂ c¯ ϵ 2 μ2 ∂ ð0Þ − ϵ μ2 E1 ð2Þ − ϵ ¯2 c¯ 1 ϵ ∂μ2 ð4πμˆ 2Þ ¼ 2 and that ∂μ2 ¼ 2 c E1 ð þ 2Þ, we then find

Z   3 1 g N 1 2 β β − − ¯2 c¯ 1 5DðgÞ¼ ðgÞ 2 dx fðxÞ gðxÞ c E1 ð Þ ð4πÞ 0 2 Z   3 1 1 X∞ 1 − μ2 β − g N − xð xÞ ¼ ðgÞ 2 dx fðxÞ gðxÞ 2 2 : ð4:7Þ 4π 0 2 m x 1 − x μ ð Þ k¼1 ðkÞ þ ð Þ

First all, note that the usual one-loop beta function of a pure compactification scale R−1. Since c¯2 ≪ 1, this function YM theory coincides in both mass-independent and mass- can be expressed as a power series in c¯2, which behaves as dependent schemes, which implies that it is scale indepen- 2 c¯2 2 4 2 c¯ E1 ð1Þ¼ζð2Þc¯ − ζð4Þc¯ þ. At first order in c¯ , the dent. As far as the KK excited-modes contribution is beta function can be written as concerned, it is given by a growing positive function on 2 −2 μ =R , which can be explicitly calculated [125]:       37 μ2 ∞ β β 1 ζ 2 X ¯2 1 5DðgÞ¼ ðgÞ þ ð Þ −2 þ : ð4:9Þ c π¯ π¯ − 1 220 R 2 ¯2 ¼ 2 ½ c cothð cÞ : ð4:8Þ k¼1 k þ c

Note that this function vanishes in the limit as c¯ → 0, which It is important to note that both the zero- and excited-modes contributions have the same sign. Furthermore, the correc- is expected from the decoupling theorem [71]. Also, note μ2 ¯ 0 276648 −2 that it diverges for large c, though keep in mind that such a tion is of the order of ð . ÞðR Þ. scenario is beyond the range of validity of our effective From Eqs. (4.1) and (4.6), the renormalized polarization theory, which is valid for energies μ lower than the function can be written in the μ-scheme as follows:

Z        2 Δ¯ 2 X∞ 2 Δ¯ 2 g N 1 0 1 m þ 0 Π5D 2 ð Þ − ðkÞ ð Þ KKðp Þ¼ 2 dx fðxÞ log 2 þ fðxÞ gðxÞ log 2 2 4π 0 Δ 2 Δ ð Þ ð0Þ k¼1 mðkÞ þ ð0Þ Z   ¯ 2     2 2 2 1 Δ 1 Γð1 þ iΔ Þ g N ð0Þ − ð0Þ ¼ 2 dx fðxÞ log 2 þ fðxÞ gðxÞ log ¯ 2 : ð4:10Þ 4π 0 Δ 2 Γ 1 Δ ð Þ ð0Þ ð þ i ð0ÞÞ

The last term of this expression clearly shows the decou- B. Case n ≥ 2 pling nature of new physics effects, since this term vanishes Previously, we have shown that in a YM theory with only −1 in the limit of a very large compctification scale R . This one extra dimension the usual counterterm δA is enough to result must be compared with that obtained from the MS remove divergences if no loop insertions of canonical scheme. dimension higher than 4 are introduced. Also, we have

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0 emphasized that for a number of extra dimensions equal LYM LYMð Þ where d≤4 and c:t: are given by Eqs. (3.11) and to or higher than 2, additional divergences arise as QKKð Þ LYMð0Þ coefficients of powers of external momentum. We have (3.14), respectively. Moreover, KKðd>4Þ represents inter- argued that, since this type of divergences are associated actions of canonical dimension higher than 4, which with discrete sums, they must be attributed to short- induce the counterterms needed to remove the divergen- distance effects in the compact manifold. However, these LYM ces generated at one loop by the Lagrangian QKKðd≤4Þ. divergences cannot be removed by the usual counterterm Such a Lagrangian can be written as δA, which means that additional interactions introducing n the required counterterms must be considered. Because X½2 0 λ YMð Þ ðkÞ a α1 α μν a such divergences multiply powers of external momen- L Dα Dα Fμν D D k F : KKðd>4Þ ¼ −2 k ð 1 k Þ ð Þ tum, the new type of interactions must be of canonical k¼1 ðR Þ dimension higher than 4. Of course, this type of inter- ð4:12Þ actions is not renormalizable in the power-counting sense, but it is renormalizable in a wider sense, since Note that our gauge-invariant renormalization scheme our effective Lagrangian includes all the interactions implies that D α ¼ Dα ,sinceZ Z ¼ 1. So, the corre- allowed by symmetries, so there is a counterterm B i i A g sponding counterterm is gauge invariant as well and is available to cancel any divergence [52–55].Becauseof given by gauge invariance, the building blocks of such interac- tions must be the YM curvature and its covariant n X½2 ðkÞ 0 δ derivatives. With this in mind and taking into account YMð Þ A a α1 α μν a L ¼ ðDα Dα FμνÞ ðD D k F Þ ; the result given by Eq. (3.38), the required bare c:t:ðd>4Þ −2 k 1 k k¼1 ðR Þ Lagrangian is of the form ð4:13Þ 4 0 0 0 LYMð þnÞ ¼ LYM þ LYMð Þ þ LYMð Þ þ LYMð Þ ; BKK QKKðd≤4Þ KKðd>4Þ c:t: c:t:ðd>4Þ δðkÞ ≡ λ − λ where A ZA BðkÞ ðkÞ. ð4:11Þ Then, up to one-loop order, the contribution from the Lagrangian (4.12) to the polarization function is given by

n   n   X½2 2 X½2 2 4 p k p k Πð þnÞDðp2Þ¼ λ þ Πloopðp2Þ − δ þ δðkÞ ; ð4:14Þ KK ðkÞ R−2 KK A A R−2 k¼1 k¼1 where the first term of this expression is the tree- counterterms, we proceed as in the case n ¼ 1,dis- level contribution of the Lagrangian (4.12); the second cussed above, by using both the MS scheme and the μ term represents the one-loop contribution induced scheme. by the Lagrangian (3.11), which is given by Eq. (3.33); the third term is the contribution of the 1. Mass-independent scheme usual counterterm; and the last term represents the contribution of the counterterms induced by interactions The counterterms are determined by the coefficients of n of dimension higher than 4. To determine the ½2þ1 the poles of the divergences:

     g2N 11 11 n 2 δ ¼ þ − ζð0Þ ; ð4:15aÞ A ð4πÞ2 3 3 6 ϵ

n Z   g2N X½2 1 n δðkÞ − −1 k − − 1 − k A ¼ 2 ð Þ ½fðkÞðnÞ gðkÞðnÞ dx fðxÞ gðxÞ ½xð xÞ : ð4:15bÞ 4π 0 2 ð Þ k¼1

Note that the counterterm δA in this expression physics effects do not decouple from the is identical to that of Eq. (4.2), except for the value beta function nor from the polarization function 4 of n. Consequently, the beta function is given by Πð þnÞDðp2Þ, as it occurs in the case of one extra β β 22þn KK ð4þnÞD ¼ ð 44 Þ.Itisevidentthatthenew dimension.

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Mass-dependent scheme.—The structure of the new counterterms suggests to use the following renormalization conditions:

Πð4þnÞD 2 −μ2 0 KK ðp ¼ Þ¼ ; ð4:16aÞ

d Πð4þnÞD 2 0 2 KK ðp Þ ¼ ; ð4:16bÞ dp p2¼−μ2 . .

n ½2 d Πð4þnÞD 2 0 n ðp Þ ¼ : ð4:16cÞ 2 ½2 KK dðp Þ p2¼−μ2

From these renormalization conditions, the derivation of the counterterms for relatively large values of n does not seem to be a simple task. Fortunately, the symmetry of the theory allows one to write closed expressions for any value of n. The counterterms induced by interactions of canonical dimension higher than 4 can be determined, for any value of n, from the expression

n − δð½2 mÞ −λ − ϵ A ¼ ð½n−mÞ Að½n−mÞðn; Þ 2 2   m 2 X −1 k n − ! n μ ð Þ ð½2 m þ kÞ ð½2−mþkÞ − ½λ n − þ δ þ A n − ðn; ϵÞ ! n − ! ð½2 mþkÞ A ð½2 mþkÞ −2 k 1 k ð½2 mÞ R ¼ Z     2 −2 n − n − 2 1 ½2 m ½2 m ¯ g N R n − d Fðn; c Þ − − 1 − ½2 m 2 dx fðxÞ 2 þ hðxÞ½ xð xÞ 2 n − ; ð4:17Þ ð4πÞ 0 μ dðc¯ Þ½2 m where the definitions − n hðxÞ¼fðxÞ 2 gðxÞ; ð4:18aÞ Z       g2N 1 1 1 ϵ ϵ ϵ ffiffiffi Γ þ ζ 1 ϵ Γ ζ ϵ − 1 − k AðkÞðn; Þ¼ 2 dxhðxÞ fðkÞðnÞ p ð þ ÞþgðkÞðnÞ ð ÞþFðkÞðnÞ ½ xð xÞ ð4:18bÞ ð4πÞ 0 π 2 2

λ n n ϵ have been introduced. Notice that the counterterms in terms of ½2 and A½2ðn; Þ. This result allows us to n n n ϵ ½2−1 ½2 ½2−1 (4.17) are divergent through the quantity AðkÞðn; Þ, which δ δ δ determine A . In turn, A and A allow us to determine n −3 contains divergences arising from short-distance effects in δ½2 A , and so on. the compact manifold. Also note that, in expression (4.17), As far as the usual counterterm is concerned, it can be n n − 1 0 δ½2 m goes from 0 to ½2 . So for m ¼ , A is determined in derived, for any value of n, from the expression

Z         n  2 ¯ 2 −ϵ X½2 2 g N 1 ϵ Δ 2 n dkF n; c¯ δ Γ ϵ −1 k ¯2 k ð Þ A ¼ 2 fðxÞ 2 þ þ hðxÞ Að0Þðn; Þþ ð Þ ðc Þ 2 k ; ð4:19Þ 4π 0 2 4πμˆ 2 ¯ ð Þ k¼0 dðc Þ where     −2 −ϵ R 2 ϵ A 0 n; ϵ g 0 n Γ ζ ϵ F 0 n : : ð Þð Þ¼ ð Þð Þ 4πμˆ 2 2 ð Þþ ð Þð Þ ð4 20Þ

This counterterm contains only divergences that arise from short-distance effects in the usual spacetime manifold. The ultraviolet divergence of the zero mode appears in the first term of (4.19), proportional to fðxÞ, while the ultraviolet ϵ divergences induced by the KK excited modes are contained in the factor Að0Þðn; Þ.

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On the other hand, the renormalized polarization function is given by

Z    2    2 1 Δ¯ Πð4þnÞD 2 g N ð0Þ − n KK ðp Þ¼ 2 dx fðxÞ log 2 4π 0 Δ 2 ð Þ ð0Þ  n  n    X½2 k ¯2 X½2 1 kδðkÞ 2 k 2 2 2 2 k d Fðn; c Þ d p þ hðxÞ Fðn; c Þ − Fðn; c¯ Þ − ðc − c¯ Þ þ −2 : ð4:21Þ ¯2 k k! R k μ2 k¼1 dðc Þ k¼1 dð μ2 Þ We now proceed to derive the beta function. Of course, there is a beta function for each coupling constant, but we will focus on the usual beta function, which emerges from the standard countererm (4.19). The usual beta function does not depend on the scale, so the (4 þ n)-dimensional beta function can be written as

Z  n  g2N 1 d X½2 dpF n; c¯2 β β ¯2 −1 p ¯2 p ð Þ ð4þnÞDðgÞ¼ ðgÞþ 2 dxhðxÞc 2 ð Þ ðc Þ 2 p ; ð4:22Þ 4π 0 ¯ ¯ ð Þ dc p¼0 dðc Þ where the term p ¼ 0, within square brackets, corresponds to Fðn; c¯2Þ. To investigate the impact of extra dimensions on the beta function, we need to determine the sign of the new physics contribution. The calculation of the derivatives on the Fðn; c¯2Þ function is straightforward. In addition, both the finite sum on the index p and the parametric integral on the variable x can be performed. Once this is done, we get       2 Xn Xr X∞ 2 k g N 1 n n−r 2k þ r − n μ β 4 ðgÞ¼βðgÞþ π 2 × Iðn;kÞSðn;kÞΓ ζð2k þ r − nÞ : ð4:23Þ ð þnÞD 4π 2 2n−1 − 1 2 −2 ð Þ 1 1 l n R r¼ l¼ k¼½2þ1

In this expression, Sðn; kÞ is the finite sum over the p index, which is given by   1 Γ 1 −1 Γ k − n ; −1 ðk þ ; Þ − ð ½2 Þ Sðn; kÞ¼ n ; ð4:24Þ e Γðk þ 1Þ Γðk − ½2Þ where Γða; zÞ is the incomplete gamma function and e is the Euler-Napier constant. On the other hand, Iðn; kÞ is the parametric integral over the x variable, which can be expressed as Z pffiffiffi 1 π Γ k 1 16k 22 n 1 − k ð þ Þð þ þ Þ Iðn; kÞ¼ dxhðxÞ½xð xÞ ¼ 2kþ1 1 : ð4:25Þ 0 2 ð2k þ 1Þð2k þ 3ÞΓðk þ 2Þ

Note that Iðn; kÞ > 0 for any value of n and k.In value, for increasing values of the k index. Then, the n fact, all the quantities that appear in Eq. (4.23) are contribution k ¼½2þ1 dominates the contributions com- positive, with the exception of the sum Sðn; kÞ,which ing from the other terms, which alternate positive and n n is negative for k ¼½2þ1, positive for k ¼½2þ2, neg- negative signs but are insignificant in absolute value. n ative for k ¼½2þ3, and so on. However, the contribution Keeping only the dominant contribution, Eq. (4.23) to the beta function decreases considerably, in absolute becomes

          2 Xn Xr g N 1 n n−r n n β 4 g β g π 2 I n; 1 S n; 1 ð þnÞDð Þ¼ ð Þþ 4π 2 2n−1 2 þ 2 þ ð Þ 1 1 l − 1  r¼ l¼     n 2 n 1 2½ þr − n þ 2 n μ ½2þ × Γ 2 ζ 2 þ r − n þ 2 þ; ð4:26Þ 2 2 R−2 where the ellipsis denote subdominant terms. In this case,     1 Γ n 2; −1 n 1 −1 ð½2þ Þ 0 S n; þ ¼ þ n < ; ð4:27Þ 2 e Γð½2þ2Þ β 0 which means that ð4þnÞDðgÞ < .

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The above results show that the radiative correction of fields, generates loopP amplitudesR that involve discrete 4 induced by the KK excitations has the same sign as that of and continuous sums ðkÞ d k, each of which may the usual theory, which means that the KK excitations diverge for large discrete momentum kμ¯ or for large behave like genuine Yang-Mills fields. The new physics continuous momentum kμ. We regularized both types of β → β −1 effect decouples, since ð4þnÞDðgÞ ðgÞ when R goes potentially divergent sums by using the dimensional- to infinity. This is the main result of the present work. regularization approach, which allowed us to parametrize 2 the one-loop amplitudes as products Γ ϵ Ec ϵ , with V. CONCLUSIONS ð2Þ l ð2Þ c2 El ðsÞ the l-dimensional Epstein zeta function representing In the present paper, we have shown that Yang-Mills the regularized discrete sums. In our case, the multidimen- theories in more than four universal extra dimensions 2 p2 2 sional Epstein functions depend on c ∝ −2, with p the remain perturbative at the one-loop level. We have focused R the problem not from the very intuitive approach of using a external momentum. It was argued that divergences arising cutoff regulator but from the perspective of the dimensional as poles of the Epstein function correspond to genuine regularization scheme. In this approach, large contributions ultraviolet divergences, since they are associated with large of continuous and discrete momenta are kept, but they are values of discrete momenta kμ¯ or, equivalently, with short- removed by adjusting parameters of appropriate counter- distance effects in the compact manifold. Thus, this type of terms, which are already available, since the effective divergences can be removed through renormalization, just Lagrangian contains all interactions that respect the sym- as it is done in the case of ultraviolet divergences emerging metries of the theory. from short-distance effects in the usual four-dimensional A comprehensive study of the beta has been spacetime manifold. Results available in the literature were c2 presented. The effective KK theory was quantized using used to reduce El ðsÞ into a finite sum of one-dimensional aSUðN;M4Þ-covariant gauge-fixing procedure, which Epstein functions. This result plays a relevant role because considerably simplifies the calculations. This class of it allows us to distinguish the two types of ultraviolet theories, which are characterized by an infinite number divergences already commented. To see this, note that

            l−1 Xl−1 − 1 ϵ p ϵ 2 ϵ ϵ 2 ϵ ð−1Þ l p Γð2 − 2Þ 2 ϵ p Γ Ec ¼ Γ Ec þ ð−1Þpπ 2 Ec − : ð5:1Þ 2 l 2 2 1 2 2l−1 Γ ϵ 1 2 2 p¼1 p ð2Þ

The first term of this expression represents the usual generate the required counterterms, which are already ultraviolet divergences induced by the KK excited modes, available since the effective theory contains all the inter- c2 ϵ since the Epstein function E1 ð2Þ converges for ϵ → 0 and actions that respect the symmetries of the theory. To carry does not depend on c2. The divergences arise exclusively out practical calculations, we have used another important from the pole of the gamma function. The other type of result of the literature that allows us to express the one- divergences arises from the second term of the above dimensional Epstein function in terms of products of expression. Since the factor Γ ϵ is canceled by the gamma and Riemman functions through a power series ð2Þ 2 2 denominator of the sum over index p, divergences arise in c . This expansion in powers of c is valid in our case ϵ 2 ϵ because this effective theory aims at the description of from products Γð − pÞEc ð − pÞ of this sum when ϵ → 0. 2 2 1 2 2 physics at energies lower than the compactification scale It can be appreciated from this expression that for even 2 R−1. At this level, the poles of the Γ ϵ − p Ec ϵ − p values of p divergences arise as poles of the gamma ð2 2Þ 1 ð2 2Þ function when ϵ → 0, while for odd values of p, diver- products translate into poles of both the Riemann and gences emerge as poles of the one-dimensional Epstein gamma functions. This decomposition of the Epstein c2 function has a high practical value, since both the gamma function in this limit, since E1 ðsÞ has poles at function and the Riemann zeta function already appear 1 − 1 − 3 s ¼ 2 ; 2 ; 2 ; . This is the type of ultraviolet diver- in libraries of modern technical computing systems, such gences which we have identified as short-distance effects in as Mathematica. To be confident about our results, we the compact manifold. However, note that this type of have systematically used this program for the present divergences does not exist in the five-dimensional theory, investigation. since in this case l ¼ 1. Because this divergent Epstein The case of one extra dimension was studied separately. function depends on power of the external momentum, it Our motivation to do this is that this theory has some only can be associated with interactions of canonical peculiarities that make it substantially different from the dimension higher than 4. Using gauge invariance as a more general case of an arbitrary number of extra dimen- guide, we have introduced all interactions needed to sions. An interesting property of the five-dimensional

095038-24 GAUGE-INVARIANT APPROACH TO THE BETA FUNCTION IN … PHYS. REV. D 101, 095038 (2020) formulation is that no divergences associated with the one- The systematic use of multidimensional zeta functions dimensional Epstein function emerge, so the only diver- that arise from infinite discrete sums regularized in the gences are the usual ultraviolet divergences generated by dimensional-regularization scheme played a central role in the KK zero and excited modes. This means that the usual our study. Our conclusion is that short-distance effects of δ ð0Þa ð0Þμν counterterm − A Fμν F (in our gauge-invariant quan- zero mode and excited modes in the spacetime manifold 4 a ϵ 4 − tization scheme) is enough to remove divergences. Another arise as poles of the gamma function when ¼ D important property of this case is that all Green’s functions approaches zero, while short-distance effects of excited of canonical dimension higher than 4 in which external legs modes in the compact manifold emerge as poles of the one- are all zero modes are free of ultraviolet divergences as long dimensional Epstein function. as no interactions of dimension higher than 4 are inserted in We emphasize that our identification of two types of the loops. This has been verified in phenomenological ultraviolet divergences turns out to be natural from the applications. perspective that we have two different spaces, one infinite The beta function was studied in both the five- and the other compact. We think that this fact, along with our approach to this type of effective theories, constitutes dimensional and the (4 þ n)-dimensional cases. The analy- sis was performed in a mass-independent scheme and in a an important contribution of this work. mass-dependent scheme as well. In the former scheme, it To finish, we would like to comment that our method for n ≥ 2 was found that neither the beta function nor the renormal- handling divergences that arise (for ) from poles of the Epstein function could be very useful in phenomeno- ized polarization function exhibits decoupling of extra logical studies of universal extra dimensions, especially in dimensions effects, while in the latter scheme, the decou- some physical processes that are naturally suppressed in the pling is manifest. In the (4 þ n)-dimensional case, explicit Standard Model, so they can be very sensitive to new expressions for the counterterms, renormalized polarization physics effects. This is the case of physical processes that function, and beta function, valid for any value of n, were first arise at one loop, as rare Higgs boson decays or flavor presented. It was shown that the excited KK modes induce violation, which are free of short-distance effects in the a radiative correction that reinforces the value of the usual usual manifold, but not in the compact manifold for n ≥ 2. beta function, since it has the same sign. On physical grounds, this result was expected, since the KK excitations ACKNOWLEDGMENTS of the YM zero mode are also gauge fields governed by the non-Abelian symmetry. However, we have seen that We acknowledge financial support from Consejo proving this fact is not a trivial task, mainly because Nacional de Ciencia y Tecnología (CONACYT). H. N. S. dimensional regularization must be used along with an and J. J. T. also acknowledge financial support from appropriate renormalization scheme. CONACYT through Sistema Nacional de Investigadores.

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