Grand Unified Theories without the Desert

We present a Grand Unified Theory (GUT) that has GUT fields with masses of the order of a TeV, but at the same time preserves (at the one-loop level) the success of gauge-coupling unification of the MSSM and the smallness of proton decay operators. This scenario is based on a five-dimensional theory with the extra dimension compactified as in the Randall-Sundrum model. The MSSM gauge sector and its GUT extension live in the 5D bulk, while the matter sector is localized on a 4D boundary.

Our set-up is based on the Randall-Sundrum 5D model [3], where the bulk is a slice of AdS 5 .
This corresponds to a 5D non-factorizable geometry with a the fifth dimension y compactified on an orbifold, S 1 /Z 2 , of radius R with 0 ≤ y ≤ πR. The orbifold fixed points at y * = 0 and y * = πR are 4D boundaries of the five-dimensional space-time. The metric is given by [3] ds 2 = e −2ky η µν dx µ dx ν + dy 2 , where 1/k is the AdS curvature radius and η µν = diag(−1, 1, 1, 1) with µ = 1, ..., 4. The fundamental scale in the 5D theory, M 5 , is related with the 4D Planck mass, M P , by M 3 5 ≃ kM 2 P (for R > 1/k). We assume that all the scales are of roughly the same order of magnitude k ≃ M 5 ≃ M P , with the radius of the extra dimension slightly larger, R ∼ 11/k. The effective scales on the two boundaries are very different. On the y * = 0 boundary the effective scale is M P , while on the y * = πR boundary this is given by ke −kπR ∼ TeV (for the assumed value R ∼ 11/k). We will hence call these two boundaries the M P -boundary and the TeV-boundary respectively.
3 One-loop corrections to the gauge propagator at low energies Let us consider a 5D gauge boson [5,1], A M (x, y), with M = (µ, 5), living in the slice of AdS 5 described above. We want to calculate the one-loop corrections to the gauge propagator at low energies. For simplicity, we will consider a 5D scalar QED theory. We will work in the gauge A 5 (x, y) = 0, so we only have to consider A µ (x, y). At energies below the Kaluza-Klein (KK) masses, only the massless zero-mode of the photon is relevant. This is given by [5,1] This corresponds to a 4D massless state with a y-independent wave-function. Contrary to the graviton case, this gauge mode is not localized by the AdS metric and has the interesting property that it couples to the two boundaries of the orbifold with equal strength.
We want to calculate the one-loop corrections to the propagator of this massless photon generated by a 5D scalar φ with charge 1 and even under the Z 2 . We will regularize this theory with a 5D Pauli-Villars (PV) field Φ of mass Λ. This mass corresponds to the cut-off of the theory that we will take to be Λ < ∼ k. Let us decompose the 5D scalar fields φ and Φ in KK modes. This has been done in Refs. [6,2]. For a 5D scalar particle of mass M, the approximate KK mass spectrum for M < k is given in Table 1. We have defined the n = 0 mode as the mode that becomes massless in the limit M → 0. For M of order k, this mode becomes heavier than some of the KK states since m KK = πke −kπR ≪ k for R > 1/k. This is very different from compactifications in a slice of flat space where the n = 0 mode (defined as explained above) is always the lightest state. This is the effect of the AdS 5 curvature that lower the masses of the KK spectrum but not the mass of the zero mode.
For the scalar φ and the PV field Φ, we can obtain the KK spectrum using Table 1 with the following values for α: From this KK decomposition we can already infer the magnitude of the quantum corrections.
For each KK mode of the field φ there is a KK mode of the PV field whose mass acts as a cut-off scale. Since the masses of the KK modes of φ and Φ are of the same order of magnitude, we do not expect large corrections from them. Nevertheless, the zero mode of the PV field is very heavy, O(Λ), in contrast with the zero mode of φ that is massless. Therefore we expect a large correction coming from this large mass splitting of the zero modes that will reproduce the quantum corrections of an ordinary 4D theory.
To see this explicitly, let us now calculate the one-loop contribution from the scalar φ to the propagator of the photon A (0) µ . Defining the photon self-energy by Π µν (q) = [q 2 η µν − q µ q ν ]Π(q), we find, at zero momentum, that the one-loop contribution is given by where M φ 0 and M φ n are respectively the masses of the zero mode and n-KK mode of the field φ, and similarly for Φ. We denote by b 0 and b KK the beta-function coefficients of the zero mode and KK modes respectively. In the example here we have b 0 = b KK = 1/3. Using Table 1 with where as usual we have introduced an infra-red cut-off µ, and we have replaced the sum over n by an integral. Evaluating the integral and considering that Λ < ∼ k, we obtain For µ ≪ Λ < k the KK contribution is small and can be neglected. We then obtain that the contribution to the gauge boson propagator is dominated by the zero mode and gives exactly the same contribution as in 4 dimensions: This is the main result of this paper. It shows that in these 5D theories the contribution to the massless-mode gauge-boson propagator depends logarithmically on the high-energy cut-off Λ, It is important to see if the result of Eq. (7) can be understood as a running of the gauge coupling similar to the 4D case. Of course, this cannot be the case if matter is localized on the TeV-boundary, since on that boundary our effective scale (cut-off scale) is TeV, and above this energy effects must be considered from the fundamental (string) theory. Nevertheless, if matter is localized on the M P -boundary, where the effective scale is M P , we will show in the next section that the effective gauge couplings can be considered to run logarithmically with the energy similarly to the 4D case. 4 In order to understand what is the behavior of the theory at energies above the TeV, we will derive here the 5D propagator of the gauge boson in the AdS 5 slice. Since we are interested in the propagator at high energies from the point of view of the M P -boundary, we will consider the limit R → ∞. In this limit we do not need to do KK decomposition and we can work directly in 5D.
Let us take the gauge A 5 (x, y) = 0 and consider only the transverse part of A µ (x, y), i.e.
we impose ∂ µ A µ (x, y) = 0. It is shown in Ref. [9] that the transverse part decouples from the non-transverse part in the equations of motion. Moreover, only the propagator of the transverse part is relevant to sources localized on the M P -boundary since the current there is transverse, Following similar steps to those in the graviton case [7,8], we want to calculate the Green function for the gauge boson defined as with ∂ µ J µ = 0. It can be shown that e k(y+y ′ ) G(x, y; x ′ , y ′ ) is also the Green function of a scalar with mass −3k 2 + 2kδ(y) [2]. Let us change the extra dimensional coordinate to z = e ky /k.
Taking the 4D Fourier transform of the Green function we have that G p (z, z ′ ) must satisfy the equation Solving Eq. (10) with the Neumann boundary condition on the M P -boundary, we find where H (1) ν = J ν + iY ν is the Hankel function of order ν, J ν and Y ν are Bessel functions, and we have defined z > (z < ) as the greater (lesser) of z and z ′ . In the case where the coordinate z ′ is on the M P -boundary, z ′ = 1/k, the Green function simplifies to We are interested in the limit r = |x − x ′ | ≫ 1/k where the Green function is dominated by the small-momentum values of the Fourier transform (p ≪ k) and the Hankel function H Let us now study the Green function of Eq. (12) in two different limits. First let us consider the propagator at large z, z ≫ r. In this case we find a falloff of the Green function that is similar to the graviton case [7,8]. Eq. (13) implies that at large momentum p > ∼ 1/z = ke −ky the M P -boundary decouples from the TeV-boundary. This is, in fact, a check of consistency of the theory that shows that the two different effective scales on the boundaries can coexist.
Let us now consider the opposite limit, r ≫ z, that also corresponds to the case when z = 1/k, i.e. the gauge propagator on the M P -boundary. In this case we have, From Eq. (14) we can derive the static potential on the M P -boundary: We see that it differs from the Coulomb potential in 4D by a logarithmic term. It means that the gauge coupling on the M P -boundary grows, at the tree-level, logarithmically with the energy. This is in contrast with 5D theories in flat space where at the classical level the coupling grows linearly with the energy. In a theory with finite R this "running" will be present at energies above the mass of the first KK mode ∼ m KK = πke −kπR (below m KK we have a single massless gauge boson as in 4D). We then have entering in the strong coupling regime. We must stress that this "running" of the gauge coupling is a tree-level effect, not a quantum one. As a consequence, it will be universal for the different groups of the SM and will not affect gauge-coupling unification.
From the tree-level behaviour of the propagator in Eq. (11), we learn that the theory remains weakly coupled for p < ∼ 1/z. This suggests that the theory can be renormalized as long as we keep our cut-off scale below 1/z, i.e. Λ < ∼ ke −ky . Notice that this cut-off scale depends on the position in the extra dimension. This should be expected in a theory with the metric (1), since the effective scale of the 4D space-time at the position y is given by ∼ ke −ky . Using the cut-off Λ = ke −ky , we can calculate quantum corrections in a very simple way. We just need the 5D propagators for r > z. For the case of the 5D massless scalar discussed in the previous section, the propagator behaves as in 4D [7,8], k d 4 p/(2π) 4 (1/p 2 ), giving then the same one-loop correction to the gauge coupling as in 4D.

GUTs in a slice of AdS 5
Let us now proceed to show that theories with gauge bosons in a slice of AdS 5 can have gauge-coupling unification. We will take a top-down approach. We will assume that we have a supersymmetric GUT in the slice of AdS 5 and show that this theory, when broken to the MSSM group, leads to a successful prediction for the gauge couplings at low energies.
As a toy example, let us consider an SU(5) theory. Due to the Z 2 orbifold symmetry, the massless gauge sector of this theory consists of a N = 1 vectormultiplet [2]. They contain the SM gauge bosons plus the GUT gauge bosons, X and Y , that complete the SU(5) representation.
The KK spectrum consists of N = 2 vectormultiplets with masses ∼ (n − 1 4 ) m KK . Let us now consider that on the M P -boundary we have a chiral supermultiplet, in the 24 representation of SU(5), whose scalar gets a vacuum expectation value (VEV) equal to M GUT ∼ 10 16 GeV (slightly below k ≃ M P ) breaking the SU(5) group down to the MSSM. This can be achieved in the same way as in ordinary 4D SU(5) theories, since our theory on the boundary is 4D N = 1 supersymmetric. It is easy to calculate the KK spectrum of the resulting theory. The n = 0 MSSM gauge bosons remain massless, while the GUT gauge bosons, X and Y , have masses M X,Y ≃ M GUT 1 . The KK mass spectrum (n ≥ 1), however, is not modified by the VEV of the 24 (up to corrections of O(M 2 n /k 2 )) and therefore the KK modes approximately respect the SU(5) symmetry. Consequently, only the zero modes (as we claimed before) will give a relative one-loop contribution to the SM gauge couplings which, at energies µ, is given by where b i is the contribution of the massless modes to the beta-function coefficient of the MSSM gauge group i. Therefore, in order to have the same predictions for the gauge coupling as in 4D supersymmetric GUTs, we must just demand that the massless states of the theory be those of the MSSM. This will be the case of the gauge sector, as we already explained. For the Higgs sector we can, as usual, assume that they arise from a 5 and5 of SU(5). Since we need to have only the SU(2) L -doublet light, we will need a mechanism that provides a doublet-triplet mass splitting inside the 5 and5. Several mechanisms exist in the literature for 4D. It is not clear if these mechanisms can also work in 5D. Nevertheless, we can just rely on these mechanisms by assuming that the Higgs live on the M P -boundary.
Finally, we must implement the matter sector. Since they form complete SU(5) multiplets, 5 and 10, they are irrelevant to gauge-coupling unification (they will not contribute at the one-loop level to the relative corrections to the gauge couplings). The matter sector, however, must satisfy important constraints from proton decay. Since there are very light KK modes of the X and Y bosons (m KK ∼ TeV), we must worry about proton-decay operators induced by these modes. If we analyze the y-dependent wave-function of these modes, however, we find that they are peaked on the TeV-boundary [1]. Therefore, proton-decay constraints can be satisfied by just placing the matter sector on the M P -boundary. In this case, even if we sum over the full KK tower of the X and Y bosons, we obtain that the strength of the dimension-six proton-decay operator is given by where g n and m n are respectively the coupling to the M P -boundary and the mass of the KK of the X, Y bosons that can be derived following Ref. [2]. We see that the result is similar to that in a 4D theory where one finds g 2 /M 2 GUT . The operator (18) is, however, slightly larger in our theory than in 4D theories because of the factor πkR in Eq. (18). This enhancement is due to the fact that the gauge coupling grows (at tree-level) with the energy according to Eq. (16).
Notice that, at the scale M GUT , the theory is close to the strong coupling regime. This is why we expect in these GUTs a proton decay rate for p → πe closer to the experimental limit than in 4D GUTs.
Up to now we have just assumed that m KK (approximately the mass of the lightest KK state) is an independent parameter of the theory that we have taken to be close to the weak scale by choosing R ≃ 11/k. Nevertheless, it would be interesting to relate m KK with the weak scale. One way to do this is by associating the supersymmetry-breaking scale with m KK . A realization of this is given in Ref. [2]. By assuming different boundary conditions for bosons and fermions on the TeV-boundary, we can get a fermion-boson mass splitting of O(m KK ).
This breaks supersymmetry and induces a Higgs mass of O(m KK ) [10]. If this mass is negative, this will trigger electroweak symmetry breaking. This scenario therefore links the scale m KK with the weak scale.

GUT physics at TeV energies
Although this theory resembles the ordinary 4D supersymmetric GUT, it has very different implications at TeV-energies. While 4D supersymmetric GUTs predict that only the MSSM fields have masses of the order of the weak scale, with a big "desert" up to the GUT scale, our theory has plenty of new physics at the TeV. There are the KK states not only of the SM but also of the GUT fields and graviton. It has been shown in Ref. [1] that the KK modes of the SM gauge bosons have sizeable couplings to the SM fermions living on the M P -boundary (≃ 0.2g) and therefore they could be seen as resonances in TeV colliders. On the other hand, the KK modes of the GUT fields couple very weakly to the M P -boundary (this is why the proton decay rate is small). These modes, however, can be produced at TeV energies by processes mediated by virtual SM gauge bosons (that live in the 5D bulk and propagate between the two boundaries). At these energies, also graviton KK modes can be produced. In fact, since the effective scale on the TeV-boundary is ∼ ke −kπR ∼ TeV, quantum gravity or string effects can be important and possible to test.

Conclusion
We have shown that, in theories with gauge bosons propagating in the 5D bulk of the Randall-Sundrum model (a slice of AdS 5 ), the gauge coupling gets logarithmic corrections similar to those in 4D. These theories contain light (TeV) KK excitations, but only the (massless) zero modes contributes to the renormalization of the gauge coupling.
We have proposed a GUT where the gauge bosons live in the 5D bulk, while matter is localized on the M P -boundary. The theory has the massless spectrum of the MSSM and predicts the right value for the gauge couplings at low energies. On the other hand, we find, at the TeV scale, KK modes of the GUT fields. These modes are very weakly coupled to the fermions of the SM and consequently proton decay rates are suppressed. Nevertheless, they couple to the SM gauge bosons with sizeable couplings, providing the possibility to test GUTs at TeV colliders.