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On the Cut-off Estimate in Lifshitz Five Dimensional Field Theories

We analyze if and to what extent the high energy behaviour of five-dimensional (5D) gauge theories can be improved by adding certain higher dimensional operators of"Lifshitz"type, without breaking the ordinary four-dimensional Lorentz symmetries. We show that the UV behaviour of the transverse gauge field polarizations can be improved by the Lifshitz operators, while the longitudinal polarizations get strongly coupled at energies lower than the ones in ordinary 5D theories, spoiling the usefulness of the construction in non-abelian gauge theories. We conclude that the improved behaviour as effective theories of the ordinary 5D models is not only related to locality and 5D gauge symmetries, but is a special property of the standard theories defined by the lowest dimensional operators.


Introduction
Field theories in more than four space-time dimensions have received a lot of attention in recent years. They have given us a new perspective on various aspects of high energy physics and cosmology. For instance, a fundamental TeV-sized quantum gravity scale might arise from extra dimensions [1] or a TeV scale can naturally emerge from a red-shift effect in a warped extra dimension [2]. Combined with the AdS/CFT idea [3], five-dimensional (5D) theories also give us a new handle to approach strongly coupled QFT [4].
Field theories in extra dimensions are non-renormalizable and, as such, they should be seen as effective theories valid up to a maximum energy Λ, above which they break down. Estimates based on (not too much) Naïve Dimensional Analysis (NDA) and unitarity bounds both give, for a simple 5D gauge theory on a flat segment of length L = πR, where g is the 4D gauge coupling. Based purely on four-dimensional considerations, a 5D gauge theory can be seen as an infinite number of gauge symmetries, non-linearly realized by the pseudo Nambu Goldstone Bosons (pNGB's) coming from the gauge field components A (n) y along the extra dimension y, with m n = n/R. According to this picture, one would naively expect Λ N aive ∼ 4πf (1.2) where f = m 1 /g is the decay constant of the lightest pNGB. We see that Λ = 4/gΛ N aive , and for a sufficiently weak coupling g, the 5D theory remains weakly coupled up to energies parametrically higher than those expected from a generic 4D effective theory [5]. This is also seen in the 4D deconstructed versions of 5D theories [6,7], where the delay in the unitarity breakdown in scattering amplitudes with respect to the naive estimate arises from non-trivial cancellations among different contributions [8].
Aim of this work is to show to what extent the improved high energy behaviour of 5D theories holds and whether it is even possible to improve the situation by modifying the theory by adding higher dimensional operators. We technically address these questions by analyzing a specific class of non-standard 5D theories with anisotropic scaling symmetry (also called, with some abuse of language, Lifshitz field theories, see [9] for a review and references). The reason to consider these theories is twofold. First, a symmetry principle allows to restrict the class of higher dimensional operators to consider in studying generalizations of ordinary 5D theories. Second, Lifshitz field theories are known to possibly have an improved UV behaviour with respect to ordinary theories, exploiting the improved UV behaviour of the particle propagators. In fact, simple UV-completions of 5D theories based on Lifshitz field theories have been shown to be possible [10]. The price to be paid is however high. In these theories the 4D Lorentz invariance is broken at high energies and is generally recovered in the IR only at the price of extreme fine-tunings [11,12]. For these reasons, we consider here 4D Lorentz-invariant theories, where the anisotropy involves the extra dimension only. 1 We focus on pure non-abelian gauge theories compactified on a plain S 1 /Z 2 orbifold, the addition of matter field being straightforward.
We separately study the UV behaviour of the transverse and longitudinal gauge field polarizations. In the former case we estimate the cut-off by a one-loop computation of the gauge coupling corrections induced by the Kaluza-Klein (KK) modes to the zero mode gauge fields. After showing in some detail the form of this correction in the ordinary 5D case, leading to the cut-off estimate (3.8), we show how the Lifshitz operators lead to a parametrically higher cut-off, eq.(3.20). The UV behaviour of longitudinal gauge bosons, on the other hand, is analyzed by looking at their elastic scattering amplitudes, A(W where n is the KK mode of the longitudinal gauge field in the scattering process. Building on previous results [15], we see that, contrary to the ordinary 5D theories, the O( L ) (E being the center of mass energy), no longer cancels. This leads to the breakdown of unitarity at energies lower that those obtained in ordinary theories, with an associated cut-off given by eq.(4.6), spoiling the usefulness of the Lifshitz construction for non-abelian gauge theories. In other words, we get that Λ can be parametrically higher than the estimate (1.1) in abelian gauge theories, while in non-abelian theories the addition of the Lifshitz operators at a sufficiently low scale would result in a decrease of Λ to Λ N aive .
We conclude that the improved behaviour as effective theories of 5D theories is not only related to locality and 5D gauge symmetries, but is a special property of the standard theories defined by the lowest dimensional operator F 2 M N . We also deconstruct the simplest version of our Lifshitz 5D theory. We show that, as expected, the higher dimensional Lifshitz operators are reproduced in the 4D deconstructed model by next-nearest-neighbour terms in field space. The precocious breakdown of unitarity induced by the Lifshitz terms is particularly clear from this perspective using the equivalence theorem [16]. 2 The structure of the paper is as follows. In section 2 we introduce the class of theories we consider. In section 3 we estimate the cut-off Λ of these theories by a one-loop vacuum polarization computation. In section 4 we estimate again the cut-off Λ, but this time by considering the breakdown of unitarity in scattering amplitudes of longitudinal gauge bosons. In Section 5 we deconstruct a simple 5D Lifshitz theory and show the form of the terms corresponding to the higher dimensional Lifshitz operators. In section 6 we conclude. We report in Appendix some details of the one-loop vacuum polarization amplitude. 1 Five-dimensional Lifshitz theories where the anisotropy of the scaling symmetry shows up only in the extra dimension has been considered in [13] and [14] for a λφ 4 theory and a model of Gauge-Higgs unification, respectively. 2 The importance of locality in field space in deconstructed theories has recently been analyzed in [17], where it has been shown that non-local terms always lead to a smaller cut-off Λ. Contrary to the Lifshitz terms considered here, the non-local terms in [17] remain non-local in the 5D limit.

General Set-Up
Lifshitz theories are typically taken to be invariant under anisotropic scale transformations under which the time coordinate scales differently from the spatial coordinates. In this way higher derivative terms in the spatial derivatives and quadratic in the fields can be introduced without violation of unitarity. The improved UV behavior of the propagator turns otherwise non-renormalizable theories in renormalizable ones. Along the lines of [13], we consider here Lifshitz models where time and the ordinary spatial directions scale in the same way, so that this symmetry can be made compatible with the 4D Lorentz symmetry, while the extra dimension scales differently. We focus on pure 5D non-abelian SU (m) gauge theories (the addition of matter being straightforward) compactified on an S 1 /Z 2 orbifold of length L = πR parametrized by the coordinate y, where the terms odd under the parity symmetry y → −y are forbidden and no localized boundary terms are inserted. In this case, the higher derivative Lifshitz terms introduced below do not lead to uncancelled boundary terms in the action variation and the eigenfunctions of a KK field mode n is the usual cos ny/R or sin ny/R, depending on the parity symmetry of the field. 3 For simplicity, we take in the following Neumann (Dirichlet) boundary conditions for the gauge fields A µ (A y ). We assume an anisotropic scale invariance of the form where µ = 0, . . . , 3 parametrizes the ordinary 3+1 space-time directions, φ denotes a generic field and Z is a positive integer. According to eq.(2.1), we can assign to the coordinates and to the fields a "weighted" scaling dimension: Power-counting renormalizability arguments apply, provided one substitutes the standard scaling dimensions of the operators by their "weighted scaling dimensions" [18], i.e. by the dimensions implied by the assignment (2.2). The weighted dimensions of the gluons A µ are fixed by looking at their ordinary 4D kinetic term components. One gets Gauge invariance fixes the weighted dimensions of the 5D gauge coupling g 5 and of the the gluon components A y : For any finite Z the theory remains non-renormalizable, but with a coupling that is less and less irrelevant as Z increases. Notice that the scaling dimensions of A µ and A y are different, 3 We have not systematically studied the effects of the Lifshitz terms for more general interval compactifications.
We expect that new consistency constrains should be imposed in this case and more drastic modifications to the spectrum of the theory might arise.
with the latter being smaller than one for Z > 1. This difference will play a crucial role in what follows. The most general Lagrangian involving weighted marginal and relevant operators only (i.e. where the SU (m) generators T a in the fundamental representation are normalized as Tr T a T b = δ ab /2 and Λ L is the energy scale above which the theory effectively behaves as a Lifshitz theory. 4 By properly rescaling the internal dimension and the scale Λ L , without loss of generality, we can set a 0 = a Z−1 = 1. We do not consider here the problem of understanding where the anisotropic symmetry (2.1) comes from but simply assume its presence in the effective theory. The quadratic mixing terms between A µ and A y coming from the second term in eq.(2.5) can be canceled by choosing a generalized R ξ gauge-fixing term of the form The ghost Lagrangian associated to the gauge-fixing (2.6) can easily be derived, though it is not explicitly needed in our analysis. The spectrum of states of the Lagrangian (2.5) is the standard infinite tower of KK modes labelled by an integer n, with the usual wave functions of the form At the quadratic level, the only effect of the higher derivative Lifshitz terms is to modify the masses of the KK modes: The schematic behaviour of the theory is the following. For energies E < 1/R, it is effectively an ordinary Lorentz invariant 4D gauge theory. For 1/R < E < Λ L , the theory behaves as an ordinary 5D gauge theory and for E > Λ L it behaves as a Lifshitz theory where operators are effectively classified by their weighted dimensions. If we take Λ L ∼ 1/R, the theory is never in the "ordinary" 5D regime. We will study in the next two sections the impact of the higher derivative Lifshitz operators on the cut-off of the theory, estimated by using gauge coupling corrections and by unitarity bounds on scattering amplitudes.

Estimate of the Cut-off through Gauge Coupling Corrections
According to NDA, the coefficients of the local operators in a non-renormalizable Lagrangian should be of the same order of the ones induced by radiative corrections at the scale Λ, where Λ is the energy above which the effective theory breaks down. One can also invert the logic and apply NDA to particularly simple operators to estimate the value of Λ itself. The obvious choice of operator in a 5D gauge theory is the kinetic term F 2 µν . The cut-off Λ can then be defined as the scale where the one-loop vacuum polarization correction to the zero mode gauge fields A (0) µ becomes of order one. A naive estimate that just takes into account the phase space of the loop integration and the number of colors would give where 24π 3 is the 5D loop factor, g 4 = g 5 / √ πR is the 4D gauge coupling and m is the quadratic Casimir operator, C 2 (G) = m, for SU (m). A more detailed computation such as the one below (see appendix for further details) shows that this estimate is in fact too naive and optimistic, and a more reliable one is obtained by using the 4D loop factor 16π 2 in eq.(3.1). In light of these possible discrepancies, in what follows we estimate the cut-off for the Lifshitz field theories by computing in detail the one-loop vacuum polarization for A (0) µ . Before considering the Lifshitz case, it is useful to review the ordinary 5D Lorentz invariant computation. The 5D Lorentz invariant model is obtained by taking Z = 1 in eq.(2.5). A useful, though not necessary, way to compute the gauge+ghost contribution to the one-loop gauge coupling correction is to make use of a mass-dependent β-function in 4D. 5 The whole contribution (ghosts included) of the KK resonances of mass M n to the β-function of the 4D gauge coupling is (see the appendix D of [19]) where E is the sliding RG (euclidean) energy scale and M 2 n = n 2 /R 2 is the mass of the KK mode n. We show in the appendix some details on how to obtain eq.(3.3), since we are not aware of any derivation in the literature. The factor −11/3 in eq.(3.3) is the zero mode contribution. When M n → 0, the integral over x is trivial and gives −7/2, which reproduces the contribution of a massless gauge field plus its scalar (longitudinal) component: −7/2 = −11/3 + 1/6. The one-loop gauge contribution can be written as Performing the sum over the KK modes n, we get and, using eq.(3.4), the following RG behaviour for g −2 4 is obtained: (3.6) For R → 0, eq.(3.6) reproduces the usual one-loop logarithmic gauge contribution. We are here interested in using eq.(3.5) to estimate the cut-off of the theory. The latter is defined as the energy Λ where the one-loop factor is comparable to the "tree-level" term g from which one obtains In eq.(3.8), g 0 ≡ g 4 (E 0 ) and we have introduced a superscript (1) to Λ to specify that this is the value of the cut-off for the ordinary theory with Z = 1. In comparing the naive estimate (3.1) with the more refined (3.8) we notice that the former is too optimistic by almost one order of magnitude. If we insist in using naive estimates based on loop factors only, we see that a more reliable estimate is obtained by replacing the 5D loop factor 24π 3 with the 4D loop factor 16π 2 in eq.(3.1).
Let us now consider the Lifshitz theory. Interestingly enough, the expression (3.3) for the β-function still holds, provided we use the modified mass terms (2.8) for the KK gluon mode n. This is best seen in unitary gauge, ξ → ∞, in which A y = 0 and the Lifshitz interactions boil down to higher derivative quadratic terms for the KK gluons. For simplicity, we keep the marginal operators only, setting all couplings a i to zero, except a Z−1 = 1. For further simplicity, let us first take Z = 2. Summing eq. Re π √ a 2 e iπ/4 cot(π √ a 2 e iπ/4 ) , where a 2 is the value for Z = 2 of the variable a Z defined for future use as (3.10) It is straightforward to check that (3.11) Using the above relations, we get where is a numerical factor. The transverse gauge fields A µ enter in a strongly coupled regime for We can also analyze the asymptotic region E ≫ 1/R, Λ L , E 0 for an arbitrary, but finite, Z. We have Re πa ) . Re πa where c is a positive numerical factor of O(1). Using the above relations, we get where .

(3.19)
For generic Z, the would-be cut-off of the theory is estimated to be (3.20) For Z = 1, 2, eq.(3.20) reproduces the previous estimates (3.8) and (3.15). The first numerical values of Λ (Z) are The would-be cut-off of the theory is parametrically high for a sufficiently small 't Hooft coupling g 2 0 m, provided that Λ L ∼ 1/R. 6 Sometimes it is useful to consider how many KK modes N (Z) M ax have a mass below the cut-off of the theory. From an effective field theory point of view, these correspond to the states that we are justified to keep in the theory. 7 Interestingly enough, N and since k Z slightly decreases for increasing values of Z, the actual number of KK states below the cut-off actually decreases with respect to the ordinary Z = 1 theory. When the weighted relevant operators are considered, a i = 0, the sums over the KK modes become rather cumbersome and complicated, but a qualitative physical description can easily be given. The gauge coupling evolution is essentially dictated by the value of Λ L R. For Λ L R ∼ 1, all the terms appearing in eq.(2.8) are of the same order of magnitude and the marginal coupling a Z−1 quickly dominates for n > 1. In this case, the approximation above is justified and the would-be cut-off of the theory is given by eq.(3.20). For Λ L R ≫ 1, up to KK modes of order n ∼ Λ L R, the dominant coupling is the ordinary a 0 term, giving rise to the usual coupling behaviour (3.7). The theory enters in the Lifshitz regime only for E > Λ L . It is then obvious that the Lifshitz operators are significant only in the energy range 1/R < Λ L < Λ (1) .
Similar results also apply in the presence of fermions. The anisotropic scaling (2.1) would demand the presence of higher (covariant) derivative interactions along the internal dimension, that in unitary gauge boil down to a modified KK mass formula for the fermion KK modes, similar to eq.(2.8). The explicit contribution of a fermion to β is reported in appendix, eq.(A.14). The analysis is essentially identical to the one we did for the gauge case. In particular, the gauge coupling correction still scales as (Λ L R)(E/Λ L ) 1/Z , as in eq.(3.18). The results shown here apply then for abelian theories as well, where at one-loop level the matter contribution is the only one.

Cut-off from Unitarity Bounds in Scattering Amplitudes
In the last section we have shown that the cut-off of Lifshitz field theories, as obtained by a detailed computation of the vacuum polarization correction of the transverse polarizations of the 6 Notice that eq.(3.20) does not hold for parametrically large Z -in which case it would predict that Λ (Z) → 0 for Z → ∞ (kZ ∝ Z for large Z) -because the limit of taking large energies does not commute with the large Z limit. 7 Naive truncations of this kind should be considered with care, because they can lead to a breakdown of the 5D non-linearly realized gauge symmetries.
zero mode field A (0) µ , can be parametrically higher than the one in ordinary theories. We show here that the estimate (3.20) does not apply in non-abelian gauge theories, since the scattering amplitudes of longitudinal components of the gauge fields break unitarity well before the energy (3.20) is reached and even before the ordinary 4D value (3.8). This result could also be obtained by analyzing the gauge coupling corrections to the longitudinal components of A (n) µ , but this computation is rather cumbersome, while we will see how it is straightforward, building on previous works, to get the bounds coming from scattering amplitudes.
Let us briefly review the behaviour of the scattering amplitudes of longitudinal KK gauge bosons in 5D theories, focusing for simplicity to elastic processes [15,5]. This amplitude could grow as fast as E 4 , where E is the center of mass energy of the incoming fields. In [15] it has been shown that the O(E 4 ) and O(E 2 ) terms in the amplitude of longitudinal 5D gauge boson scattering vanish, whenever the following relations hold: where k and n are KK levels, g nnnn is the quartic gauge coupling of KK gauge fields at level n, g nnk is the trilinear coupling of two KK n and one KK k gauge fields, and M 2 n is the mass of the KK n gauge field. In ordinary 5D theories, eqs.(4.1) are satisfied and the amplitude does not grow with the energy, as already claimed in [5,8]. Unitary violation is detected from the O(E 0 ) terms in the amplitude and arises from the multiplicity of states in a coupled channel analysis [5]. For a SU (m) theory compactified on a segment, the maximum number of KK states N M ax that can enter in a scattering process without leading to a violation of unitarity is given by [5] leading to a cut-off estimate roughly in agreement with eq.(3.8).
As we already mentioned, in the unitary gauge A y = 0, no new interactions arise from the higher derivative Lifshitz terms, and the couplings g nnnn and g nnk are the same as in the ordinary 5D theories. The first constraint in eq.(4.1) is then automatically satisfied. The only effect of the Lifshitz interactions is to modify the gauge boson KK masses as given in eq.(2.8). It is straightforward to check that, due to the modification in the mass formula, the second relation in eq.(4.1) is no longer satisfied in the Lifshitz case. For illustration, let us consider Z = 2. For plain S 1 /Z 2 compactifications, the sum over k in eq.(4.1) reduces to two terms, k = 0 and k = 2n, which are the only two states that can be exchanged in the scattering process, due to the conservation of the 5D momentum, mod Z 2 . A simple computation gives While the transverse components of the gauge fields remain weakly coupled for energies above the ordinary bound (4.3), the longitudinal components show a breakdown of unitarity at energies below eq.(4.3). We get, from eq.(4.5): 8 When Λ L ≃ 1/R, eq.(4.6) is the energy one would expect from 4D considerations for a pNGB with mass M 1 ≃ 1/R and "pion" decay constant f = M 1 /g 0 , which would give Λ ≃ 4πf , equal to the naive estimate (1.2). As expected, the ordinary 5D result (4.3) is recovered for Λ L → ∞, in which case one has to look at the O(E 0 ) terms. A similar result is obtained, by the equivalence theorem, by studying the scattering of the pNGB's in a different gauge, such as Landau or Feynman gauge. In these gauges, the higher derivative Lifshitz terms give rise to derivative quartic interactions among the pNGB's that reproduce the behaviour (4.5). We will briefly come back to this point in the next section, when the 4D deconstructed version of the theory is considered.

Deconstructed 4D Model
It is interesting to analyze the deconstructed version of our set-up. Let us briefly recall the deconstruction of an ordinary 5D SU (m) pure gauge theory on an interval [6,7]. The Lagrangian of a linear moose with N sites and N − 1 link variables U i is given by where the U i 's transform as U i → g i+1 U i g † i under gauge transformations and have only "nearestneighbour" interactions with the gauge fields A µ,i+1 and A µ,i . The covariant derivative is For simplicity we have taken in eqs.(5.1) and (5.2) a universal decay constant f and a universal coupling constant g. The gauge group SU (m) N is non-linearly realized, because the link fields are "σ-model" fields that can be written as in terms of would-be Goldstone bosons π i (x) = π a i T a . In the unitary gauge U i = 1, the Lagrangian (5.1) contains an N × N mass matrix for the gauge fields of the form that has eigenvalues M 2 n = 4g 2 f 2 sin 2 πn 2N , n = 0, . . . , N − 1 .
We can define L = N a, a = 1 gf , p 5 = πn L , (5.6) so that a can be interpreted as the lattice spacing of the interval and L its length. The 5D gauge coupling g 5 is given by g 2 5 = ag 2 . For n ≪ N , we have where R = L/π is the radius of the 5D covering circle of S 1 /Z 2 . In the unitary gauge, π 1 = π 2 = . . . = π N −1 = 0 and SU (m) N is spontaneously broken to the diagonal subgroup SU (m).
In the canonical basis, the gauge coupling g 4 of the unbroken gauge fields is g 2 4 = g 2 /N = g 2 5 /L, in agreement with what expected from a 5D theory.
Let us generalize the deconstruction above and include the higher derivative operators appearing in eq.(2.5). For simplicity, we consider only the Z = 2 case. The higher derivative terms in the extra dimension suggest that in the deconstructed theory "next-nearest-neighbour" interactions should be present. The deconstructed Lagrangian can be written as 8) where U N +1 = U 1 and U N = 0 in the last term, andf is the "Lifshitz" pion decay constant. As we will shortly see, the last term in eq.(5.8) corresponds to the Tr (D y F µy ) 2 term in eq.(2.5). In the unitary gauge U i = 1, all π i 's vanish and SU (m) N is spontaneously broken to the diagonal subgroup SU (m), like in the ordinary case reviewed above. The quadratic terms for the gauge fields coming from the last term in eq.(5.8) give rise to the following N × N mass matrix: The total mass matrix for the gauge fields is given by with M 2 as in (5.4). By explicit computation, we find that the mass matrix (5.9) has eigenvalues M 2 n = 16g 2f 2 sin 4 πn 2N , n = 0, . . . , N − 1 . The Lifshitz scale Λ L is defined as For n ≪ N , we have which reproduces eq.(2.8) for Z = 2.
The leading behaviour of the amplitude (4.5) can be reproduced in the deconstructed model. It is actually easier, by using the equivalence theorem, to look at the ππ → ππ scattering. In the ordinary linear moose (5.1), the O(E 2 ) term schematically reads where we have used eq.(5.6) in the second equality. For simplicity we have omitted in eq.(5.15) gauge and site indices and have written only the structure of the amplitude. In the limit N → ∞, the O(E 2 ) term in the amplitude vanishes, in agreement with the 5D expectation. In the linear "Lifshitz" moose (5.8), additional derivative quartic couplings arise and we get an extra term ∆A contributing to the amplitude: 9 where in the last equality we have used eqs.(5.6) and (5.13). The factors of N now cancel and the O(E 2 ) term no longer vanishes in the 5D limit. Instead, using eq.(5.16), we get a cut-off Λ ∼ 4πΛ L /g, in agremeent with eq.(4.6). One might ask if the Lifshitz terms, even if not introduced at tree-level, are radiatively generated. The answer is clearly negative in the 5D limit, because of SO(5) symmetry. A similar conclusion is reached in the 4D deconstructed model by using spurions [7]. When the gauge coupling g is switched off, the Lagrangian (5.1) has an SU (m) 2N −2 global symmetry under which U i → L i+1 U i R † i , where L i and R i are independent SU (m) matrices (i = 1, . . . , N −1). This global symmetry is explicitly broken by the gauge fields A i , but it can formally be restored by introducing spurion fields q i . The gauge fields and the spurions transform as and q i → R i q i L † i+1 , respectively. By writing the covariant derivative (5.2) as the SU (m) 2N −2 global symmetry is formally restored. The original model is eventually recovered by setting q i = 1. It is not difficult to see that no SU (m) 2N −2 invariant operators can be constructed by using the spurions, leading to Tr (DU † i+1 DU i ) when q i = 1. We conclude that the Lifshitz operators are not generated by quantum corrections.
Although we have not explicitly worked out the deconstructed version of eq.(2.5) for general Z, we expect that the introduction of "next-next-. . .-nearest-neighbour" interactions should reproduce the corresponding 5D higher-derivative terms for any Z. No new results are expected to arise by considering higher values of Z.

Conclusions
We have studied the cut-off estimate in 5D field theories, where certain higher derivative (Lifshitz) operators are added to the action. By a detail one-loop vacuum polarization computation, we have argued that the transverse polarizations of the gauge fields have a softer UV behaviour with respect to the ones in ordinary 5D theories. On the other hand, the same higher derivative terms negatively affect the longitudinal polarizations of the gauge fields. Because of these operators, the O(E 2 ) terms in the scattering amplitude of longitudinal gauge bosons no longer vanish, in contrast to the usual 5D case, and lead to an earlier breakdown of unitarity with respect to the standard 5D situation. Of course, this problem does not occur for abelian gauge theories, in which the Lifshitz operators do improve the UV behaviour of the theory. We have then considered (for the special case Z = 2) the deconstructed version of the 5D Lifshitz models and shown how similar conclusions are reached from this perspective. As expected, the Lifshitz terms correspond to next-nearest-neighbour interactions in field space.
Our analysis explicitly shows that the relatively good UV behaviour of standard 5D theories, for which Λ > Λ N aive , as defined in eqs.(1.1) and (1.2), do not only come from 5D locality and 5D gauge symmetries, both preserved in our Lifshitz construction, but are peculiar of the standard 5D action. The 4D deconstructed models are useful in this respect, since they show how the Lifshitz terms break the global symmetries responsible for the good UV behaviour of ordinary 5D theories. From the Lifshitz field theory point of view, our results explicitly show that care has to be used in determining the UV behaviour (e.g. renormalizability) of Lifshitz theories based only on the effective UV dimension of the couplings. It is crucial to also pay attention to the effective dimensions of the fields, even when they can be gauged away (like the fields A y in our case), since they can lead to a precocious strong coupling behaviour.

Acknowledgments
I thank Roberto Iengo for discussions at the early stages of this project.

A Derivation of the 5D β-function
The β-function (3.3) is conveniently computed using background field methods and a background field gauge fixing. We write the gauge field as A tot =Ā + A, whereĀ is the classical background value and A is the quantum fluctuation. The 5D Lagrangian reads, including gauge fixing and ghosts, the covariant derivative with respect to the background field only. In eq.(A.1), we have explicitly included the counterterm δZ for the 4D background field strengthF µν (x), omitting all the others that do not play any role in the computation. We choose in the following ξ = 1 so that all quadratic gauge mixing terms vanish. As well known, the gauge symmetries of the classical background allow to compute the β-function directly from the two-point function Ā µ (−p)Ā ν (p) . The effective one-loop Lagrangian for the zero mode backgroundĀ µ (x) reads We choose a standard momentum subtraction renormalization scheme, by demanding that The mass dependent β-function is given by In the above graphs M n = n/R is the mass of the KK mode running into the loop, the single wiggly lines represent the gluon fluctuations A µ and A y , dashed lines represent the ghost fields and the double wiggly lines represent the background fieldĀ µ . Notice the presence of a quartic interaction among ghost and gauge fields in this gauge, leading to the graph (f ). Summing all the contributions, we get 16π 2 (η µν p 2 − p µ p ν ) where C is an irrelevant divergent constant. The finite wave function correction Z is determined by the renormalization condition (A.4), that fixes the counter-term δZ. Using eq.(A.5), we can finally get eqs. that gives rise to the following contribution to the β-function: . (A.14)