One-loop order effects from one universal extra dimension on λ ϕ 4 theory
M. A. López-Osorio, E. Martínez-Pascual, G. I. Nápoles-Cañedo, J. J. Toscano
aa r X i v : . [ h e p - t h ] F e b One-loop order effects from one universal extradimension on λφ theory M. A. L´opez-Osorio ( a ) , E. Mart´ınez-Pascual ( a ) , G. I.N´apoles-Ca˜nedo ( b ) , J. J. Toscano ( b ) ( a ) Departamento de Ciencias Naturales y Exactas,Centro Universitario de los Valles, Universidad de Guadalajara,Carretera Guadalajara-Ameca Km 45.5, CP 46000, Ameca, Jalisco, M´exico. ( b ) Facultad de Ciencias F´ısico Matem´aticas, Benem´erita Universidad Aut´onomade Puebla, Apartado Postal 1152, Puebla, Puebla, M´exico.E-mail: [email protected]
February 2020
Abstract.
The self-interacting λφ scalar field theory is a warhorse in quantumfield theory. Here we explore the one-loop order impact from one universal extradimension, S / Z , to the self-energy and four point vertex functions associatedto this theory. Such effects come as an infinite number of UV divergencescorresponding to an infinite superposition of excited KK particles around the loop.We show that dimensional regularisation is adequate enough to control them interms of the product of the one dimensional inhomogenous Epstein zeta functiontimes the gamma function. From the analytical properties of these functions, theUV divergences are extracted and the counterterms defined; the latter turn outto be of canonical dimension four at the Lagrangian level. We use both, the MS-scheme and a mass-dependent subtraction scheme to remove divergences. Onlythe latter manifestly satisfy the decoupling theorem. Keywords: universal extra dimensions, Kaluza-Klein excitations, renormalisation,compactification and four dimensional models.
1. Introduction
The idea of extra dimensions in field theories dates back to the 1920s [1, 2], sincethen, many related proposals, extensions and critical judgements about this idea havebeen gravitating in theoretical physics [3, 4, 5, 6, 7]. In particular, phenomenologicalstring theory, naturally formulated in this framework, led the community to revisitthe issue decades later from different perspectives. The use of the braneworld scenarioallows to consistently lower the typical scale of quantum gravity to TeVs by choosingthe number and size of spatial extra dimensions [8, 9] in the so-called large extradimensional models, in contrast with warped [10] or universal [11] extra dimensionalmodels; the latter were particularly inspired by [12], and accessible lectures on allthese models are for instance [13, 14]. Models with universal extra dimensions(UEDs) have recently been a matter of interest since, for example, they are capable ofsuggesting answers to some still valid questions within the Standard Model (SM), asit is the possible explanation of having three generations of fermions as a result of the ne-loop order effects from one... S / Z orbifolds preserve KK parity and provide a dark matter candidate in the form of stableKK partners [17]. Models containing spatial extra dimensions continue to be a topicwith phenomenological importance, and searching of this phenomenon in nature iscontinuously contemplated in the experiments of the Large Hadron Collider [18, 19].The basic idea in a UED model is rather simple [20]: the stage is a factorisablespacetime geometry with compact spatial extra dimensions, commonly an orbifold, onwhich a field theory is defined. The dimensional reduction takes place once the extradimensional content of the fields is harmonically expanded, under certain boundaryconditions, and the extra dimensions themselves are integrated out at the actionlevel; the resulting model, which we referred to it as the effective or dimensionallyreduced theory , comprises an infinite number of fields defined on M , namely, thezero modes (one for each original field) and an infinite number of Kaluza-Klein (KK)mode fields. These fields originally enter as coefficients in the expansions allowedby the corresponding extra spatial geometry. In the dimensionally reduced theory,the KK fields are the only remaining information from the extra dimensions, hencewe attribute any of their contributions on the effective theory as some kind of extradimensional input.In this paper we are interested in the dimensionally reduced theory obtainedfrom the self-interacting λ (5) Φ model, plus all kind of Lorentz invariant operators O r (Φ , ∂ M Φ) of canonical dimension greater than five suppressed by some massscale as in any effective field theory, defined on the five dimensional factorisablespacetime M × S / Z . Here M is the Minkowski spacetime. Reducing thedimension, as in the aforementioned sense, makes the field content Φ( x, ¯ x ), with( x, ¯ x ) = ( x µ , x ) ∈ M × S / Z , to be recast into the zero mode φ ( x ) and the infinitenumber of excited KK fields φ ( k ) ( x ), where k denotes non-zero Fourier labels. Afterintegrating out the extra dimension, the dimensionally reduced theory consists of aninfinite number of fields gathered into four sectors: ( i ) the pure zero mode sector,which corresponds to the well-known real scalar self-interacting λφ Lagrangian, ( ii )the interaction terms between the zero mode and the excited KK fields; ( iii ) the pureKK excited sector, from which the bare propagator of each KK excited field can beread and some quartic interactions among these fields exist; and ( iv ) the sector thatcontains all kind of Lorentz invariant operators O r ( φ, φ ( k ) , ∂ µ φ, ∂ µ φ ( k ) ) of canonicaldimension greater than 4 appropriately suppressed by some mass scale such thatevery sector is of canonical dimension four. Since KK fields actually correspond toFourier coefficients in the expansion of the field Φ( x, ¯ x ), they define the profile ofΦ in the extra dimension and we interpret their presence as an effective input fromthe extra dimension, besides they can be interpreted as heavy fields from an effectivefield theory perspective [21, 22]. Bearing this in mind, with the appropriate termsin the dimensionally reduced theory, in this paper we calculate the one-loop ordercontributions from the extra dimension, or excited KK fields, to the two- and four-point vertex functions associated to the λφ Lagrangian, we regard these functions asstandard vertex functions from which the Green’s functions with only external lightparticles can be reconstructed. One of the main challenges in these calculations relieson the infinite number of interactions between the zero mode and the different KKexcited modes. In turn, they imply at the one-loop order an infinite number of graphs,each of these contributing with a UV divergence and we will consider all of them. To bemore precise, in this scenario, a typical one-order loop amplitude involves a series of UV ne-loop order effects from one... P ∞ k =1 R d k which should be regularised in order to make sense of thesetype of contributions. In the same spirit of a previous work co-authored by us [23] onQED with extra dimensions, we argue that both the series and each UV divergence canbe handled in this purely scalar field theory by means of the well-known dimensionalregularisation, whose ultimate justification is found in the analytical continuation ofdivergent integrals to the complex plane [24]. It turns out that, once the dimensionalregularisation is performed, the infinite number of UV divergences results into the onedimensional inhomogeneous Epstein zeta function [25, 26, 27, 28] times the gammafunction.In this communication we show, using dimensional regularisation and keeping theinfinite number of contributions from excited KK modes, that the extra dimensionaleffect on the two- and four-point vertex functions of the self-interacting λφ to the one-loop order can be encapsulated into the product of the one dimensional inhomogeneousEpstein zeta function times the gamma function. Therefore the whole process ofextracting UV divergences is reduced to analyse the singularities of such product offunctions. Remarkably, the poles of this product can be explicitly obtained in the lowenergy limit m /R − ≪
1, where R is the radius of the extra dimension and m isthe mass associated to the light field φ . At the one-loop order, we show that in both,the two- and four-point standard vertex functions, the extra dimensional contributioncontains divergences of the typical form (1 /ǫ ) with constant factors, where ǫ is theusual parameter in dimensional regularisation ( ǫ = 4 − D ).In order to cancel out the UV divergences already described at the one-loop orderwe use two different types of schemes to define the counterterms at the Lagrangianlevel: the MS-scheme and a mass dependent subtraction scheme with a kinematicalor subtraction scale M . Interestingly, we will see that in the presence of one UED,with the geometry given above, there is no need to go beyond operators of canonicaldimension higher than four to define these counterterms at the one-loop level. Thereason for the examination of two different subtraction schemes is to gain some insightin relation with the decoupling theorem by Appelquist and Carazzone [29]. It isknown that although the MS-scheme removes the UV divergences, the correspondingamplitudes and beta function, do not obey this theorem; in contrast, when using amass dependent subtraction this theorem is manifestly satisfied [30]. This behaviour isfollowed in the context of the dimensionally reduced theories as it was already pointedout in [23] and it will be confirmed in the present communication.As a word of warning, we must say that in this work we define the physical massand coupling constant by handling the infinite number of UV divergences present atrelevant light particle vertex functions to the 1-loop approximation, that is, we do notattempt to give a meaningful complete approach to the renormalization of the wholedimensionally reduced model in this work; nevertheless, our contribution may shedsome light on this goal.In terms of [31] our method is closer to a ‘KK-renormalisation’, in contrast to the‘KK-regularisation’ in which the infinite sums over KK modes are performed beforethe integration over internal loops [32]. A key mathematical resource in our work is theEpstein zeta function [25], which is a generalisation of the well-known Riemann zetafunction [33]; both being very interesting objects in their own right. The systematicuse in physics of zeta-regularisation methods dates back to the 1970s with seminalworks as [34] and [35] in the context of effective theories and path integrals on curvedspacetimes, respectively. Applications of zeta functions can also be spotted in quantumgravity models as well as cosmology [36], string theory [37], and crystallography [38]. ne-loop order effects from one... λφ model. InSec. 3 we study the one-loop order effects of one UED on the two- and four-point vertexfunctions associated to the λφ theory. In Secs. 4 and 5 the UV divergences present inthe two-point and four-point standard vertex function are respectively extracted andthe counterterms to cancel them out are defined in the MS-scheme, the beta function isalso presented in this subtraction scheme. In Sec. 6 we migrate the previous results to amass dependent subtraction scheme and show that the decoupling theorem is satisfied,the beta function calculated in the MS-scheme is recovered when the kinematical massscale is much larger than any mass characterising the model. In Sec.7 the summaryand final remarks are given. By the end, the Appendix A is reserved to gather theFeynman rules of the dimensionally reduced theory. Although in the present paperwe concentrate on the contribution from one UED, in a subsequent paper [42] we willextend our analysis to the case of the contribution from an arbitrary number n ofUEDs, with geometry (cid:0) S / Z (cid:1) n ; such case requires the analysis of higher dimensionalinhomogeneous Epstein zeta functions, and the Lagrangian counterterms to correctlysubtract infinities will exhibit a richer structure.
2. The dimensionally reduced scalar model from one UED
In this section we describe the dimensionally reduced scalar model obtained fromthe self-interacting scalar field theory λ (5) Φ , plus compatible operators of canonicaldimension higher than five, with one UED. In the spirit of [43, 44, 45, 46] the startingpoint is the five dimensional field theory S = Z M d x d¯ x
12 ( ∂ M Φ)( ∂ M Φ) − m Φ − λ (5) Φ + X r,s β r,s Λ n + r O (5+ r ) s (Φ , ∂ M Φ) ! , (2.1)where M = M × (cid:0) S / Z (cid:1) , with coordinates ( x ; ¯ x ) = ( x µ ; x ), and Φ is a real valuedscalar field on M . The index M = 0 , , , , λ (5) have canonical dimensions of 3 / −
1, respectively; the last termcontains s different operators O rs of canonical dimension 5 + r , r ≥
1, constructedfrom Φ and ∂ µ Φ, Λ is at an energy scale above which the new physics would beginto manifest itself, and β r,s are dimensionless parameters that depend on the detailsof the underlying physics. Notice that operators of higher canonical dimension aresuppressed by powers of Λ. The field Φ is assumed to fulfil the following periodicity ne-loop order effects from one... x, ¯ x ) = Φ( x, ¯ x + 2 πR ) , (2.2a)Φ( x, ¯ x ) = Φ( x, − ¯ x ) , (2.2b)where R is the radius of the circle S , the latter is equivalent to imposing Neumannboundary condition at the fixed points of the orbifold [14]; relations (2.2) allow theFourier expansion in the extra dimension of the field itselfΦ( x, ¯ x ) = r πR φ (0) ( x ) + r πR ∞ X k =1 φ ( k ) ( x ) cos (cid:18) kx R (cid:19) . (2.3)As we know the coefficients in the Fourier expansion render the precise profile of thefunction that is being expanded, as they modulate the trigonometric functions. Inthis case the field modes define the profile of Φ on points in the extra dimension. Itis in this sense that KK modes contain information about the behaviour of Φ on theextra dimension. Directly from the expansion (2.3), one can see that every KK modehas canonical dimension equal to 1.Using the orthogonality of the trigonometric functions enables an immediateintegration of the compact extra dimension out of the action (2.1) and defines thefollowing dimensionally reduced theory L = L (0) + ∞ X k =1 L (0 k ) + ∞ X k =1 L ( k ) + L d> ; (2.4)where the first term is the purely light field φ (0) ≡ φ self-interacting λφ model, L (0) = 12 (cid:0) ( ∂ µ φ )( ∂ µ φ ) − m φ (cid:1) − λ φ , (2.5a)the myriad interaction terms between light and heavy fields are gathered in the secondterm, where L (0 k ) = − λ φ φφ ( k )2 + 2 φ ( k ) ∞ X l,q =1 φ ( l ) φ ( q ) ∆ ( klq ) , (2.5b)and the third term represents the purely KK excited mode terms, in this case, L ( k ) = 12 (cid:16) ∂ µ φ ( k ) ∂ µ φ ( k ) − m k ) φ ( k ) φ ( k ) (cid:17) − λ φ ( k ) ∞ X l,q,r =1 ∆ ( klqr ) φ ( l ) φ ( q ) φ ( r ) , (2.5c)where m k ) = m + k /R . In the set of Eqs. (2.5), the dimensionless multi-indexedsymbols ∆ ( ··· ) are defined as follows:∆ ( klq ) := √ πR Z d x cos (cid:18) kx R (cid:19) cos (cid:18) lx R (cid:19) cos (cid:18) qx R (cid:19) , (2.6a)∆ ( klqr ) := 2 πR Z d x cos (cid:18) kx R (cid:19) · · · cos (cid:18) rx R (cid:19) . (2.6b) ne-loop order effects from one... ( ··· ) are totally symmetric.The dimensionless universal coupling constant λ is defined as λ (5) / (2 πR ). The term L d> contains all the interactions of canonical dimension higher than 4, such termsmust be included in the 4-dimensionally reduced theory because the 5-dimensionaltheory is nonrenormalisable according to Dyson’s criterion. It is worth mentioningthat in this work we are solely interested on Green’s functions at the one-loop orderthat only contain zero mode particles as external legs, and at most tree level effectscoming from L d> ; remarkably, when the dimensionally reduced theory is coming fromfive dimensions, the counterterms needed to cancel out the UV divergences in this typeof Green’s functions are of canonical dimension four.The Feynman rules for the interactions in (2.5) are gathered in the Appendix A,from them there can be classified essentially two different types of k -point Green’sfunctions depending on the type of external legs, namely: the Standard Green’sfunctions (SGFs), whose external legs only comprise zero mode particles, and the
Non-standard Green’s functions (NSGFs), whose external legs contain at least one excitedKK particle. The latter can either be
Hybrid Green’s functions , with some zero andsome excited particles as external legs, or purely excited KK Green’s functions withonly KK excited particles as external legs. In particular the k − point SGFs, hereafterdenoted by G (0 ... where there are k slots filled with zeroes, will receive contributionsfrom excited KK mode particles within the loops. In fact, the Lagrangian L (0) togetherwith the infinite number of interaction like the first term in (2.5b) provide the necessaryterms to calculate the impact from the fifth dimension to G (0 ... at the one-loop level,in other words, to the four dimensional λφ theory. An interesting observation is thatthe superficial degree of divergence of these one-loop contributions, would indicate thepossibility of removing UV divergences, that is, divergences that arise due to shortdistance effects in M . However, as we will see later, the presence of an infinitenumber of interactions between the zero mode and KK modes will imply an infinitenumber of UV divergences at the one-loop level. This issue is characteristic of thistype of theories and will be tackled on the following section.
3. One-loop structure from one extra dimension
As usual, the renormalised quantities { φ, φ ( k ) , λ } and the bare quantities { φ B , φ ( k ) B , λ B } are connected through the renormalisation factors as follows: φ B = √ Zφ, φ ( k ) B = q Z φ ( k ) φ ( k ) , λ B = Z λ Z φ λ. (3.1)Then, the bare Lagrangian can be written as L B = L (0) + ∞ X k =1 L (0 k ) + ∞ X k =1 L ( k ) + L d> + L (0) c.t. + ∞ X k =1 L ( k ) c.t. + L d> c.t. , (3.2)where L (0) , L (0 k ) and L ( k ) represent the renormalised Lagrangian sectors given inthe set of Eqs. (2.5), while L d> contains interactions of canonical dimension higherthan four written in terms of renormalised quantities. The term L (0) c.t. represents thestandard counterterm of the self-interacting λφ scalar theory, which is L (0) c.t. = 12 δ Z ( ∂ µ φ ) ( ∂ µ φ ) − δm φ − δλ φ , (3.3) ne-loop order effects from one... Figure 1.
Contributions to the 2-point SVF Γ ( p ) for oneextra dimension without the counterterm. where δ Z = Z − , δm = m B Z − m , δλ = λ B Z − λ. (3.4)The contributions L ( k ) c.t. and L d> c.t. in Eq. (3.2) contain interactions between the lightand heavy fields, and among pure heavy fields, whose specific structure will not beneeded here.The inverse of the 2-point connected SGF is the 2-point standard vertex function(SVF), G (00) − c = i Γ (00)2 . At the one-loop order, in general, the extradimensionaleffects will impact the SVFs by the insertion of excited KK mode particles circulatingaround the loop. In the particular case of the 2-point SVF, to the 1-loopapproximation, we haveΓ R ( p ) = p − m − M (0) ( p ) + ∞ X k =1 M ( k ) ( p ) + M c.t. ( p ) ! , (3.5)where the term within parenthesis is nothing but the light scalar field self-energy whichconsists of three contributions: the first one, M (0) ( p ), comes from the well-known zeromode self-interacting term λφ in L (0) , the second one, P ∞ k =1 M ( k ) ( p ), is the result ofthe infinite number of interactions between φ and excited KK modes present in L (0 k ) ,henceforth we have an infinite sum of excited modes around the loop (see Fig. 1), andfinally the third one, M c.t. , is the usual counterterm for the self-energy, that is, M c.t. ( p ) = δm − p δ Z , (3.6)which comes from (3.3). This term is sufficient to cancel out all the UV divergencespresent in the self-energy as it will be seen below. The first two contributions to theself-energy in Eq. (3.5) are written in short as follows: M ( k ) = λ Z d k (2 π ) ik − m k ) , (3.7)where the symbol k stands for { , k } .If we want to analyse the corrections to the coupling constant of the φ -theorydue to the extra dimension at the one-loop, we consider the four point SVF Γ R ( p i )Γ R ( p i ) = − iλ + Γ (0)1-loop ( p i ) + ∞ X k =1 Γ ( k )1-loop ( p i ) + Γ c.t. ( p i ) ! , (3.8)with Γ (0)1-loop ( p i ) and Γ ( k )1-loop ( p i ) the loop contributions which modify the couplingconstant at low energy and Γ c.t. ( p i ) the counterterm that will remove the UV infinities. ne-loop order effects from one... φ aroundthe loop, whose presence is due to the common term λφ in L (0) , whereas the secondone corresponds to an infinite sum of excited KK mode particles circulating aroundthe loop (see Fig. 2), whose source is the infinite number of interactions between thelight and heavy fields (see the first term in (2.5b)). Each of these terms contains asum over the Mandelstam variables, that is,Γ ( k )1-loop ( p i ) = X { p } ∆Γ ( k ) ( p ) (3.9)where the symbol P { p } indicates a sum over the three Mandelstam variables, and∆Γ ( k ) ( p ) = − λ Z d l (2 π ) il − m k ) i ( p − l ) − m k ) . (3.10) Figure 2.
Contributions to the 4-point SVF Γ ( p i ) withoutthe counterterm. The third term in the parenthesis in Eq. (3.8) isΓ c.t. ( p i ) = − iδλ. (3.11)which comes from the counterterm (3.3). As we will see, this is the only countertermneeded to cancel out the divergences in (3.8). This result is remarkable, and can berephrased by saying that the effects from one extra dimension to the self-interacting λφ scalar theory become finite with the aid of counterterms of canonical dimensionfour given by (3.3); however, as we will see in a subsequent paper [42] when more extradimensions are involved this is no longer true, as operators of canonical dimensionhigher than four are needed.Notice that (3.7) and (3.10) have identical structures to those that appear inthe pure λφ theory to the one-loop, with the difference that these expressionsalso represent excited KK mode contributions. The above suggests that applyingdimensional regularisation will reveal an infinite number of UV divergences inEqs. (3.5) and (3.8) which will define the corresponding counterterms. We will show inthe following subsection that this infinite number of UV divergences can be controlledby using the so-called one dimensional Epstein zeta function [25, 26, 27, 28, 36]. In order to deal with the vast number of UV divergences that at the one-loop levelimpact the two- and four-point SVFs, Eqs. (3.5) and (3.8), respectively, we nowintroduce dimensional regularisation, which in turn permits to spot the emergenceof the one dimensional inhomogeneous Epstein zeta function (see [25, 27, 28]), E c ( s ) := ∞ X k =1 k + c ) s (3.12) ne-loop order effects from one... c vanishes.In a scalar field theory a general vertex function contains N − point scalar integrals, i.e. integrals over loop momentum F N ∝ Z d l Q Nj =1 (cid:2) ( l − p j ) − m j + iε (cid:3) , (3.13)where m j are internal masses of particles within the loop that appear in any genericvertex function to one-loop order in a given context, and p j are related to the externalmomenta with p ≡ e.g. (3.7) and (3.10). For our divergent integrals we willapply dimensional regularisation by promoting the four dimensional spacetime to a D dimensional spacetime and define s := N − D/
2. Comparing Eq. (3.13) with (3.7) wehave N = 1 and s → − N = 2 and s → ‡ .It is particularly interesting the emergence of the one dimensional inhomogeneousEpstein zeta function E c ( s ) subsequent to both dimensional regularisation and thenon-cutoff of the series involving excited KK modes. Before going into that analysis,let us formally consider the contribution from all excited KK modes to the self-energy(see (3.5) and (3.7)), ∞ X k =1 M ( k ) = λ ∞ X k =1 Z d p (2 π ) ip − m k ) = λ ∞ X k =1 Z d p (2 π ) i (cid:18) p − k R − m (cid:19) ≡ λ ∞ X k =1 Z d p (2 π ) i ( p µ p µ + p ¯ µ p ¯ µ − m ) (3.14)where p ¯ µ ≡ p = k/R is essentially the quantized momentum of a particle enforcedby periodic conditions on the extra dimension, and p ¯ µ p ¯ µ = p p = − ( k/R ) when theLorentzian metric is used. From this viewpoint of the excited KK modes contribution,there are basically two types of potential sources of UV divergences or divergences dueto small distances: one is coming from small distances in M or very large value ofcontinuum momentum, that obviously makes the integral over p diverge, and the otheris coming when we allow the integer quantised momentum to increase indefinitely, insuch case p increases itself and the series over k might be not well defined. Thereforeone requires an analytical continuation to regularise this behaviour. In fact, whendimensional regularisation is applied such analytical continuation is actually behindthe curtains justifying all integrals in D dimensions as we learned from ‘t Hooft andVeltman [24]. Similar conclusion can be drawn for the excited KK modes contributionto the 4-point SVF.Applying dimensional regularisation to the divergent integrals in the 2-pointSVF (3.5), with s = 1 − D/
2, we obtainΓ R ( p ; s ) = p − m − λ π m (cid:18) πµ m (cid:19) s h Γ ( s ) + c s E c ( s )Γ( s ) i − δm + p δ Z , (3.15) ‡ At this point we decide to use the variable s which is conventional in the literature of ‘zeta functions’(Riemann, Hurwitz, Epstein, . . . ), in the following section we will insert ǫ := 4 − D which is commonlyused in dimensional regularisation. ne-loop order effects from one... c := m /R − , µ is the common unit of mass introduced in dimensionalregularisation and Eq. (3.12) has been used. The parameter c controls the scaleenergy of the heavy modes, | c | ≪ M (0) , this is the well-known divergent term presentin the φ self-interacting scalar theory, and the second term comes from P ∞ k =1 M ( k ) which is the consequence of an infinite number of interactions of φ with the excitedKK modes. Since the term within the brackets in Eq. (3.15) is independent of p wecan remove p δ Z by setting δ Z ≡
0; therefore the first term in the L (0) c.t vanishes at theone-loop order level. Needless to say that the possible UV divergences can be readas poles of either Γ( s ) or the product of the one-dimensional inhomogeneous Epsteinzeta function E c ( s ) times the gamma function Γ( s ), hence we must be careful inhandling the limit s → −
1, or equivalently, D = 4. In particular, within the product E c ( s )Γ( s ) one should avoid the direct application of the ‘limit of a product’ rule. Infact, we must exhaust algebraic manipulations in this product before taking the limit.This is the main technical difference with respect to [22], whose analysis is performedfor the case of one spatial extra dimension S . If UV divergences are expected to beremoved from SVFs in the low energy limit, we need to isolate divergent sources outof asymptotic formulae and judiciously define the necessary counterterms.The same procedure of dimensional regularisation can be applied to the 4-pointSVF, leading to the expressionΓ R ( p i ; s ) = − iλµ s + iλ π µ s X { p } Z d z "(cid:18) m + p z ( z − πµ (cid:19) − s Γ( s )+ (cid:18) πµ R − (cid:19) s E c ( p,z ) ( s )Γ( s ) (cid:21) − iδλ, (3.16)being c ( p, z ) := ( m + p z ( z − /R − . The first term in the squared bracketsrepresents the correction to the coupling constant coming from the light fieldinteracting term λφ , this is, Γ (0)1-loop ( p i ); while the second term comes from the extra-dimensional contribution P ∞ k =1 Γ ( k )1-loop ( p i ) and δλ , as previously stated, defines thecorresponding counterterm.
4. One-loop mass corrections in the presence of one extra dimension
The counterterms defined in these subsections in order to deal with the one-loop effectsfrom the fifth dimension at the level of the mass and coupling constant of the self-interacting scalar field theory λφ will be inspired by the MS-scheme [47, 48], i.e. amass independent scheme at which the UV divergences of Feynman graphs will becancelled out by terms defined as poles at D = 4. Interestingly, as we have mentionedsuch terms define the countertemrs of canonical dimension four, see (3.3).In the presence of a single extra dimension, we analyse the UV divergences in Γ R and Γ R . The non-trivial source of these poles is found in E c Γ in the correspondingfour-dimensional limit (see Eqs. (3.15) and (3.16)). Our problem then comes down toinvestigate the structure of the poles of E c Γ and this will be done in the limit | c | ≪ ne-loop order effects from one... § , namely E c ( s )Γ( s ) = ∞ X k =0 ( − k k ! Γ( k + s ) ζ (2 k + 2 s ) c k . (4.1)As it can be seen from (4.1), the 2-point SVF can be written down as follows inthe low energy limit,Γ R ( p , ǫ ) = p − m − λm π (cid:18) πµ m (cid:19) ǫ/ (cid:20) Γ (cid:16) ǫ − (cid:17) + ∞ X k =0 ( − k k ! Γ (cid:16) ǫ − k (cid:17) ζ (2 k + ǫ − c k + ǫ − (cid:21) − δm , (4.2)where the four dimensional limit is obtained when ǫ →
0, with ǫ/ ≡ s + 1 = 2 − D/ s → −
1. The first term in the squared brackets contributes with thewell-known UV divergence, namely the one resulting from the following asymptoticexpansion, k (cid:18) πµ m (cid:19) ǫ/ Γ (cid:16) ǫ − (cid:17) ∼ − ǫ + (finite) + O ( ǫ ) , (4.3)here, the ‘(finite)’ part contains typical constants like the Euler-Mascheroni γ E anda logarithm that includes in its argument the ratio 4 πµ /m . The analysis on thesecond term in (4.2) is subtler, the possible sources of divergences may come eitherfrom the arguments of the gamma or the Riemann zeta function ζ . Therefore, thepoles will be found when the argument of the gamma function is either a negativeinteger or zero, and/or when the argument of the Riemann zeta function becomesone. These conditions can be expressed as follows: − k + 2 = 2 j , j ∈ N (4.4a) − k + 2 = − . (4.4b)The Eq. (4.4a) is only satisfied for k = 0 and k = 1, and there is no positive integer k that fullfils the identity (4.4b). Therefore the only source of UV divergences withinΓ R for one extra dimension is encoded in the gamma function, this will not be thecase for greater number of extra dimensions [42]. At k = 0 the relevant asymptoticexpression as ǫ → (cid:18) πµ c m (cid:19) ǫ/ Γ( ǫ/ − ζ ( ǫ − ∼ ζ (3)2 π + (finite) + O ( ǫ ) . (4.5)At k = 1 the relevant asymptotic expression for small ǫ is (cid:18) πµ c m (cid:19) ǫ/ Γ( ǫ ζ ( ǫ ) ∼ ζ (0) ǫ + (finite) + O ( ǫ ) . (4.6) § We must say that in the sense of [26] this expression is nothing but a non -regularised version ofthe product E c ( s )Γ( s ). k We use the twiddle sign ∼ to mean asymptotic equalities. ne-loop order effects from one... ǫ goes tozero, the product Γ( ǫ/ − ζ ( ǫ −
2) is finite. In other words, the term correspondingto k = 0 does not contribute to the pole at all. This is not the case for the term k = 1,as we can see from the asymptotic equality (4.6) which shows the typical ultravioletdivergence 1 /ǫ as ǫ → ǫ Γ R ( p , ǫ ) = p − m − λm π ∞ X k =2 ( − k k ! Γ ( k − ζ (2 k − c k − + (cid:20) λm π (cid:18) ǫ (cid:19) + λm ζ (0)16 π (cid:18) ǫ (cid:19)(cid:21) + λm ζ (3)2 π c + (finite) + O ( ǫ ) − δm . (4.7)Since the series left after isolating the divergent terms (at k = 0 and k = 1) convergesin the limit | c | ≪
1, we conclude that the 2-point SVF (3.15) will become UV finiteat the one-loop level if we define the proper counterterm to avoid the ultravioletdivergence that we just found. In the spirit of the MS-scheme prescription [47, 48],where the UV divergences of a Feynman graph must be cancelled out by countertermsthat can be read from the poles at D = 4, we define δm := λm π (cid:18) ǫ (cid:19) + λm ζ (0)16 π (cid:18) ǫ (cid:19) . (4.8)and hence the second term in L (0) c.t. (3.3) becomes totally defined. On the right handside of Eq. (4.8) the first term is the typical pole in the self-energy coming from L (0) ,and the second term is the extra dimensional or KK contribution. The presence of thefactor ζ (0) = − / ¶ When the heavy fields become infinitely massive, that is, in the limit c → R − → ∞ , we expect in Eq. (4.7) the decoupling of the KK contribution,however this does not happen, notice the term proportional to 1 /c which diverges inthe limit c →
0; even more, the ‘(finite)’ part in this expression contains a logarithmicdivergent term proportional to log(2 πc ). Therefore, although the MS-scheme removesthe UV divergences, it is not concerned about the decoupling theorem; neverthelessa mass dependent scheme to cancel out divergences ensures the decoupling as we willshow in Sec. 6. This is also true in the dimensionally reduced version of QED as itwas shown in [23].
5. One-loop corrections to the coupling constant in the presence of oneextra dimension
We now apply a similar treatment to Γ R ( p ; s ) given in Eq. (3.16). This expressiondiverges as s →
0, or equivalently as ǫ → ǫ/ ≡ s . Since our interest is toisolate UV divergences, the idea is then to examine the pole structure of the Γ R ( p i ; s )and for that we will take advantage of the expression in the low energy limit, | c | ≪ ¶ Remember that the Euler-zeta function ζ E ( s ) = P ∞ k =1 1 n s diverges in the Cauchy sense at s = 0,however once this function has been analytically continued into the Riemann zeta function ζ ( s ) ithas well defined value. ne-loop order effects from one... R ( p i ; ǫ ) = − iλµ ǫ + iλ µ ǫ π X { p } " Γ (cid:16) ǫ (cid:17) Z d z (cid:18) m + p z ( z − πµ (cid:19) − ǫ/ + ∞ X k =0 ( − ) k k ! (cid:18) πµ m (cid:19) ǫ/ Γ (cid:16) ǫ k (cid:17) ζ ( ǫ + 2 k ) F k ( p ) c ǫ +2 k − iδλ (5.1)where F k ( p ) is defined as F k ( p ) := (cid:18) πµ m (cid:19) k Z d z (cid:18) m + p z ( z − πµ (cid:19) k = k X l =0 (cid:18) kl (cid:19) ( − l ( l !) (2 l + 1)! (cid:18) p m (cid:19) l . (5.2)The contribution to Γ R ( p i ; s ) from the light particle around the loop is encodedin the first term in the squared brackets of (5.1). This is the well-known contributionfrom L (0) , the corresponding input to the pole can be calculated usingΓ (cid:16) ǫ (cid:17) (cid:18) m + p z ( z − πµ (cid:19) − ǫ/ ∼ ǫ + (finite) + O ( ǫ ) , (5.3)where the ‘(finite)’ part in (5.3) contains the Euler-Mascheroni constant and alogarithm which depends on momentum as ( m + p z ( z − / πµ is integrated in the z variable. Regarding the second term in the squared brackets of (5.1), the divergentterms can be known from the analysis of the arguments of the gamma and Riemannzeta functions, in fact, these functions diverge whenever − k = 2 j, j ∈ N (5.4a) − k = − , (5.4b)respectively. In this case, there is only one solution for (5.4a) that is when k = 0, andthere is no solution for (5.4b) since there is no positive integer k that fullfils it. Hence,the only source of UV divergence at Γ R comes also from the gamma function only.When k = 0 the asymptotic of such term can be obtained from (4.6). Introducing theasymptotic behaviour relations (5.3) and (4.6) into (4.2), we get for small ǫ Γ R ( p i ; ǫ ) = − iλµ ǫ + iλ µ ǫ π X { p } ∞ X k =1 ( − ) k k ! Γ ( k ) ζ (2 k ) F k ( p ) c k + " iλ π (cid:18) µ ǫ ǫ (cid:19) + 3 iλ ζ (0)16 π (cid:18) µ ǫ ǫ (cid:19) + (finite) + O ( ǫ ) − iδλ (5.5)from which the counterterm δλ can be established, so that the finite version of the4-point SVF, at the one-loop level, is obtained by defining iδλ := 3 iλ π (cid:18) ǫ (cid:19) + 3 iλ ζ (0)16 π (cid:18) ǫ (cid:19) . (5.6) ne-loop order effects from one... δλ the last counterterm in L c.t. becomes completelydefined. On the right hand side of (5.6), the first term is the typical UV divergence inthe φ theory L (0) , and the second term is the UV divergence coming from the extradimension at the one-loop level in the low energy limit, which is again characterizedby the regularised value ζ (0); the factor 3 comes from the sum P { p } . In connectionwith the extension of this result to the presence of higher number of UEDs, it isessential to stress that (5.6) is momentum independent, the reason being that the UVdivergence comes from the term k = 0 in (5.1) which contains the factor F ( p ) = 1;when more extra dimensions are compactified, the resulting Γ R of theory containsUV divergences within terms that involve genuine functions of momentum F k ( p ), thisfact implies a richer structure in the related counterterms [42].Notice that the needed counterterms to cancel out UV divergences present in theSVFs Γ R and Γ R are of canonical dimension four, that is, we do not appeal to anyinteraction included in L d> c.t. ; however, this is a very special case since in the presenceof more extra dimensions there will be needed counterterms of canonical dimensionhigher than four and interactions in L d> c.t. will become relevant [42].Directly from the 4-point SVF, the beta function β ( λ ) can be calculated at theone-loop order, in this case, β ( λ ) = lim ǫ → µ ∂λ∂µ = lim ǫ → µ ∂∂µ (cid:18) λµ ǫ + 3 iλ π (cid:18) µ ǫ ǫ (cid:19) + 3 iλ ζ (0)16 π (cid:18) µ ǫ ǫ (cid:19)(cid:19) = 3 λ π + 3 iλ ζ (0)16 π (5.7)= 3 λ π . (5.8)The beta function measures the strength of the coupling constant with energy. Onthe right hand side of Eq. (5.7) the first term is the well-known value of the betafunction for the self-interacting φ theory in the MS-scheme, whereas the second termis the contribution due to the fifth dimension. The value of β ( λ ) is positive, whichmeans that the strength of the coupling increases as the energy increases. Notice thatthe extra dimensional contribution reduces by a factor 1 / R − → ∞ ( c →
0) the decouplingtheorem is not fulfilled either for the Γ R or the beta function. Indeed, regarding Γ R ,see Eq. (5.5), one must say that inside the ‘(finite)’ part there is a term proportionalto log(2 πc ) which makes the decoupling theorem not evident. In addition, the betafunction is simply an R -independent constant, therefore the decoupling theorem is notmanifest at all. This is not a surprise since as it is well known: the major differencebetween dimensional regularisation together with the MS-scheme and other schemesis that the β function is independent of the scale [23, 30]. In order to restore scaledependence we should migrate to a mass dependent subtraction scheme, this will bethe main idea in the following section.
6. Decoupling of the KK contribution in the presence of one extradimension
In this section we will migrate to a mass dependent subtraction scheme by the choice p = − M , with M the kinematical or subtraction scale. We begin by analysing the ne-loop order effects from one... s + 1 = ǫ/ M D ( p ) := λ π m (cid:18) πµ m (cid:19) ǫ h Γ (cid:16) ǫ − (cid:17) + E c ( ǫ − ǫ − c ǫ − i − δm + p δ Z , (6.1)where none of the terms in the squared brackets depend on p , hence δ Z = 0. Wedetermine the counterterm δm using the typical kinematical condition M D ( p ) (cid:12)(cid:12) p − M = 0 , (6.2)therefore δm = − λm π (cid:18) πµ m (cid:19) ǫ/ Γ (cid:16) ǫ − (cid:17) h E c (cid:16) ǫ − (cid:17) c ǫ − i . (6.3)In this scheme the self-energy M D ( p ) = 0 for all p . At the one-loop level, therenormalised mass is not impacted by extra dimensions once the UV divergences arecancelled out, this is because M D ( p ) does not depend on the external momentum p unlike in the dimensionally reduced QED [23]. At this level, not much can be saidabout the decoupling, and we might have to extend our calculations to two loops if wewant to explicitly see the decoupling of the KK contribution. Still, the 4-point SVFis more enlightening in this aspect.For the 4-point SVF we have an explicit dependence on the external momenta atthe one-loop order. The 4-point SVF (3.16) when ǫ/ s is introduced becomes,Γ R D ( p i , ǫ ) = − iλµ ǫ + iλ π µ ǫ X { p } Z dz "(cid:18) m + p z ( z − πµ (cid:19) − ǫ/ Γ( ǫ/ ∞ X k =1 m k ) + p z ( z − πµ ! − ǫ/ Γ( ǫ/ − iδλ. (6.4)In order to determine δλ , we appeal to the following condition:Γ R D ( s, t, u = − M , ǫ ) = − iλµ ǫ . (6.5)then, iδλ = 3 iλ π µ ǫ Z dz Γ( ǫ/ (cid:18) m − M z ( z − πµ (cid:19) − ǫ/ + ∞ X k =1 m k ) − M z ( z − πµ ! − ǫ/ , (6.6)using the asymptotic expression (5.3), we have for small ǫiδλ = 3 iλ π Z dz ( ǫ − γ E + log(4 π ) − log (cid:18) m − z ( z − M µ (cid:19) + ∞ X k =1 " ǫ − γ E + log(4 π ) − log m k ) − z ( z − M µ ! + O ( ǫ ) , (6.7) ne-loop order effects from one... D ( p i ) = − iλ − iλ π Z dz X { p } (cid:20) log (cid:18) m + z ( z − p m − z ( z − M (cid:19) + ∞ X k =1 log m k ) + z ( z − p m k ) − z ( z − M ! . (6.8)The decoupling theorem is now manifest in Eq. (6.8), when the mass of the heavymodes is much larger than our kinematical mass, m ( k ) ≫ M , we haveΓ R D ( p i ) = − iλ − iλ π Z dz X { p } log (cid:18) m + z ( z − p m − z ( z − M (cid:19) , (6.9)then only the contribution of the light field is present. Even more, the β function canbe calculated in this mass dependent scheme using [30] β ( λ ) = M ∂δλ ∂M = 3 λ π Z dz " z (1 − z ) M m + z (1 − z ) M + ∞ X k =1 z (1 − z ) M m k ) + z (1 − z ) M , (6.10)in particular, when m ≪ M , the β function approaches to β ( λ ) = 3 λ π " Z d z ∞ X k =1 z (1 − z ) M m k ) + z (1 − z ) M . (6.11)In addition, in the limit M ≪ m ( k ) , we recover the well-known value for the self-interacting scalar theory, since all KK contributions decouple, leading to the value β ( λ ) = 3 λ π . (6.12)If on the other hand, we permit M to be extremely large, i.e. M ≫ m ( k ) and M ≫ m ,simultaneously, the value for the β function formally becomes the one obtained in theMS-scheme (5.7), that is, β D ( λ ) = 3 λ π " Z d z ∞ X k =1 z (1 − z ) M m k ) + z (1 − z ) M = 3 λ π (cid:20) Z d z ζ (0) (cid:21) = 3 λ π + 3 λ ζ (0)16 π . (6.13)
7. Summary and final remarks
In this paper we have presented the one-loop order contributions from one extradimension to the self-energy and 4-point vertex functions associated to the λφ self-interacting scalar theory in the context of a UED. We explore such contributionsemploying the dimensionally reduced Lagrangian obtained from a field theory definedon the five dimensional spacetime M × S /Z ; this theory corresponds to the effective ne-loop order effects from one... λ (5) Φ model plus all of the Lorentz invariantoperators of canonical dimension higher that five Eq. (2.1). The structure of thedimensionally reduced theory, defined on the Minkowski spacetime M , is given bythe following sectors: ( i ) the pure λφ theory, with φ the lightest field in the Kaluza-Klein tower that comes naturally after compactification, ( ii ) a sector that has aninfinite number of interactions between φ and the excited KK fields, ( iii ) a sectorwith a kinematical term and interactions of purely KK excited fields φ ( k ) , and ( iv ) asector that comes from the dimensional reduction of the Lorentz invariant operators ofcanonical dimension higher than five. We suggested that dealing with sectors ( i ) and( ii ), together with the necessary counterterms, allows to investigate on the impact ofthe fifth dimension to the self-energy and 4-point vertex functions of the λφ model.Regarding the self-energy, we show that at the one-loop order, besides the usualscalar particle around the loop, there appears a contribution given by an infinitesuperposition of KK excited modes. This contribution, which before regularisationseems to contain an infinite number of UV divergences, can analytically be recognisedwith the one dimensional inhomogeneous Epstein zeta function times the gammafunction once dimensional regularisation is performed. In fact, this product can alsobe written as a series of products between the gamma and the Riemann zeta functionsfrom which one is able to extract the term responsible for the UV divergence comingfrom the extra-dimensional contribution. The emergent UV divergence is proportionalto ζ (0) /ǫ , where ǫ → − δm φ .In other words, there was no need to rely on operators of canonical dimension higherthat four to cancel out all the UV divergences present at Γ R to the one-loop order.The UV divergences were removed using two different subtraction schemes: the MS-scheme and a mass dependent subtraction scheme. In none of them the decouplingtheorem is manifest, although the latter scheme may show this property if we go to atwo-loop order in the expansion.We also investigate the impact of the fifth dimension to the 4-point vertex functionof the λφ model. Again, it was proved that at the one-loop order, besides theusual light particle around the loop there exists a contribution formed by an infinitenumber of excited KK particles. Therefore one has to deal with a seemingly infinitenumber of UV divergences at a finite order in the perturbative expansion. Usingdimensional regularisation we show that the infinite number of these divergences can beencapsulated, again, into the product of the one dimensional inhomogeneous Epsteinzeta function times the gamma function. As we previously mention, this productfinds a more tractable form in the low energy limit, where it can be expressed asa series of products between the gamma and the Riemann zeta functions, so thatextracting the UV divergences comes down to localise poles of these products. Asin the case of the self-energy, it was proved that besides the UV divergence comingfrom the light particle around the loop, the extra-dimensional contribution containsone UV divergence which behaves as ζ (0) /ǫ . Hence in order to cancel them out onlyLagrangian counterterms of canonical dimension four are needed, in fact they can begathered into − δλ φ . Again we use the MS-scheme and a mass dependent subtractionscheme with a kinematical mass denoted by M to cancel out divergences. Althoughthe former is not concerned about the decoupling theorem, the latter explicitly shows ne-loop order effects from one... R and, more importantly,at the level of the beta function. In this regard, the beta function associated to the λφ pure model is recovered in the limit of excited KK modes much heavier than M . Moreover, the beta function calculated in the MS-scheme is recovered in the casewhere the kinematical mass is much heavier than the heavy and light fields. All theaforementioned results will be extended in a forthcoming paper [42] to the case wheremore than one extra dimension is present.It becomes stimulating to realize that the whole dimensionally reduced modelstill contains ingredients which could be the source of some theoretically interestinginvestigations. Besides the problem to systematize the study of the complete zoo ofNSGFs at the one-loop level, the term proportional to − λφ P kln φ ( k ) φ ( l ) φ ( n ) ∆ ( kln ) in the sector (2.5b) contains, once the multi-indexed ∆ ( kln ) is explicitly written, theterm proportional to − λφ P k φ (2 k ) φ ( k ) φ ( k ) . Such term would give rise to the presenceof an infinite number of φ − φ (2 k ) transitions at the one-loop level with a φ ( k ) particlewithin the loop (see the Feynman rule (A.3)), in addition there will also be an infinitenumber of mixings among excited KK modes; these effects resemble the interesting γ - Z mixing in the Weinberg-Salam model which has been studied in linear and non-linear R ξ gauges [49, 50] and whose resolution relies on a particular gauge. Acknowledgements
We acknowledge financial support from CONACyT (Mexico). M.A.L.-O, E.M.-P. andJ.J.T. acknowledge SNI (Mexico).
Appendix A. Feynman rules for the dimensionally reduced theory
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