Low scale quantum gravity in gauge-Higgs unified models

Low scale quantum gravity in gauge-Higgs unified models Jubin Park∗ Department of Physics, Chonnam National University, 300 Yongbong-dong, Buk-gu, Gwangju, 500-757, Republic of Korea arXiv:1501.04381v1 [hep-ph] 19 Jan 2015 Abstract We consider the scale at which gravity becomes strong in linearized General Relativity coupled to the gauge-Higgs unified(GHU) model. We also discuss the unitarity of S-matrix in the same framework. The Kaluza-Klein(KK) gauge bosons, KK scalars and KK fermions in the GHU models can drastically change the strong gravity scale and the unitarity violation scale. In particular we consider two models GHUSM and GHUMSSM which have the zero modes corresponding to the particle content of the Standard Model and the Minimal Supersymmetric Standard Model, respectively. We find that the strong gravity scale could be lowered as much as 1013 (1014 ) GeV in the GHUSM (GHUMSSM ) for one extra dimension taking 1 TeV as the compactification scale. It is also shown that these scales are proportional to the inverse of the number of extra dimensions d. In the d = 10 case, they could be lowered up to 105 GeV for both models. We also find that the maximum compactification scales of extra dimensions quickly converge into one special scale MO near Planck scale or equivalently into one common radius R0 irrespectively of d as the number of zero modes increases. It may mean that all extra dimensions emerge with the same radius near Planck scale. In addition, it is shown that the supersymmetry can help to remove the discordance between the strong gravity scale and the unitarity violation scale. PACS numbers: 11.10.Kk, 11.15.-q, 11.25.Mj, 04.60.Bc Keywords: Linearized General Relativity, gauge-Higgs unified model, Kaluza-Klein, strong gravity scale, perturbative unitarity, compactification, extra dimension ∗ Electronic address: honolov77@gmail.com 1 I. INTRODUCTION The scale at which gravity becomes strong could be lowered as much as TeV scale which is much below the naively expected one (the reduced Planck mass) ∼ 1018 GeV. It is because a large non-minimal coupling of a single scalar field or Kaluza-Klein(KK) gravitons contribute to the renormalization group(RG) running of the reduced Planck mass [1, 2]. Moreover it is also well-known that the strong gravity scale could be different from the unitarity violation scale in linearized General Relativity coupled to matter [3]. One important lesson from these recent studies is that the huge number of KK gravitons becomes a common source that lowers both of the scales, the strong gravity scale and the unitarity violation scale. For instance, in the large extra-dimensional model [4–6]1 there exist 1032 KK gravitons. The low scale quantum gravity is expected and the unitarity violation occurs at a few hundred GeV. Therefore it is an appropriate time to question whether other sources like KK gravitons exist or not, and how they affect both of the scales. Keeping it in mind we focus on the gauge-Higgs unified(GHU) models [7]2 , which naturally provide the KK gauge, KK scalar bosons (and KK fermions if bulk fermions are allowed). We show later that they can really change both of the scales depending on the number of extra dimensions d. On the one hand, the d is a crucial parameter in the GHU models. It constrains the structure of quartic terms of scalar potential.3 In the d = 1, any quartic terms can not be generated at the tree-level in the scalar potential, while in the d ≥ 2, tree-level quartic terms can be naturally generated from the commutators of zero modes in the field strength [8]. On the other hand, the d significantly changes the total number of KK states. These increased KK states can lower the scale at which unitarity violates in the calculation of tree-level unitarity. More specifically, the partial-wave amplitude for a 2 → 2 elastic scattering [3] via s-channel graviton exchange is given by a2 = − 1 2 3 1 2 GN ECM N , 40 (1) The large extra dimension is introduced in order to solve the hierarchy problem by trading it for geometrical prescriptions such as the AdS geometry with a warping factor. The electroweak scale is protected by a higher dimensional gauge symmetry. The tree-level quadratic terms are also prohibited due to the shift symmetry. See ref. [7] for one explicit example to generate quadratic terms in the monopole background. 2 FIG. 1: Scattering of elementary particles via s-channel graviton exchange. where N ≡ 1 N +NF 3 S
+4NV , and NS , NF , and NV are the number of real scalars, fermions and vector fields in the given model, respectively. Thus, the unitarity bound derived by |Re aJ | ≤ 1/2 shows strong dependence on the total number of the KK states. Generally there are two fundamental energy scales in the GHU models. As easily anticipated, one is the compactification scale(MC ≡ 1/R) of extra dimensions, and the other one is the theory cutoff(Λ CUTOFF) from the effective field theory point of view.4 5 In this letter we introduce one more scale parameter(Λ UNIT ) reflecting the unitarity violation scale. Because of the hierarchy between the Λ CUTOFF and the Λ UNIT there may be some debate. We simply discuss it in the last part of Sec. II. As two interesting benchmark models, we consider GHUSM and GHUMSSM which have the zero modes corresponding to the particle content of the Standard Model(SM) and the Minimal Supersymmetric Standard Model(MSSM), respectively. We find that the strong gravity scale could be lowered as much as a few hundred TeV. We also find that the supersymmetry not only make the maximum compactification scales of the extra dimensions converge into one special scale near Planck scale irrespectively of d, but also help to remove discordance between the strong gravity scale and the unitarity violation scale. This paper is organized as follows. In Sec. II we briefly introduce the model, and show how to obtain these two scales ΛCUTOFF and ΛUNIT . Next we consider aforementioned two models, GHUSM and GHUMSSM in order to show model-dependent results. Finally we analyze their numerical results, and discuss several scenarios depending on hierarchical patterns among three scales ΛCUTOFF, ΛUNIT , and MC . In Sec. III we summarize our paper. 4 5 Various experiments have been performed in order to search for deviations from Newton’s law of gravi  tation, V (r) = −GN m1rm2 1 + αe−r/α . See the ref. [9] for detailed explanation about experiments and current constraints for the compactification radius and scale. From now on, we assume that all extra dimensions have the same radius ∼ R. 3 TABLE I: Scattering amplitudes [3] for (complex) scalars, fermions, and vector bosons via s-channel (n) graviton exchange. They are written in terms of the Wigner dn,m functions [10] in the massless 2 limit. An overall factor −2πGN ECM has been extracted from all amplitudes. The subscripts on the particles indicate their helicities. ss̄ (2) 2 3 f− f¯+ V+ V− V− V+ II. (0) d0,0 − 23 (1 + 12 ξ)2 d0,0 p (2) 2/3 d1,0 p (2) 2/3 d−1,0 p (2) 2 2/3 d2,0 p (2) 2 2/3 d−2,0 f+ f¯− ′ ψ′ ψ− + ′ ψ′ ψ+ − s′ s′ −→ p (2) 2/3 d0,1 (2) d1,1 p (2) (2) 2/3 d0,−1 (2) d1,−1 V−′ V+′ V+′ V−′ p p (2) (2) 2 2/3 d0,2 2 2/3 d0,−2 (2) (2) 2d1,2 (2) d−1,1 d−1,−1 (2) 2d2,−1 4d2,2 (2) 4d−2,2 2d2,1 (2) 2d−2,1 (2) 2d−2,−1 2d−1,2 (2) (2) (2) 2d1,−2 (2) 2d−1,−2 (2) 4d2,−2 (2) 4d−2,−2 MODEL AND FUNDAMENTAL ENERGY SCALES The Lagrangian of linearized General Relativity coupled to particle content of the GHU model is given by S = + Z h √ d4 x − g 1 (−2λ + R) 16πGN ! i 1 1 µν g ∂µ φ† ∂ν φ + ξRφ2 + e ψ̄iγ µ Dµ ψ + Fµν F µν , 2 4 (2) where g is the determinant of the metric gµν , λ is the cosmological constant, R is the Ricci scalar, and ξ is a free parameter. The scalar, fermion and vector fields in the Lagrangian stand for the typical fields of the GHU model. In particular, we focus on the non-minimal coupling case, ξ = −1/12, corresponding to the conformal limit of the theory [11]. Now let us start by considering the s-channel scattering of matter particles via exchange of graviton. These all amplitudes in the massless limit are represented in Table I. The partial P (J) wave amplitude aJ is extracted from A = 16π J (2J + 1) aJ dµ,µ′ . In particular, each J = 0 and J = 2 partial wave amplitude can lead to the significant constraints to the ΛUNIT scale and the matter content in the GHU models. Note that the J = 0 partial wave amplitude automatically vanishes due to ξ = −1/12 from a0 ∼ (1 + 12 ξ)2, while the J = 2 partial waves do not change even if massive KK gravitons are involved [2]. As aforementioned, the large number of fields can induce a sizable running of the reduced 4 Plank mass. More specifically, the RG equation for it is given by [12] M P (µ) 2 = M P (0)2 −  1 1 2 N + 2ξN l ξ µ , 16π 2 6 (3)
where Nl ≡ NS +NF −4NV , Nξ is the number of real scalar fields non-minimally coupled to gravity, and µ is the renormalization scale. In general, the strong gravity scale is evaluated when the fluctuations at length scale µ⋆ is close to the reduced Planck scale M P (µ⋆ ). We regard it as the cutoff (ΛCUTOFF ) of the GHU models, µ⋆ = r 1+ M P (0)  ≡ ΛCUTOFF .  1 1 N + 2 ξNξ 16π 2 6 l (4) Before we discuss it in detail, it is worthwhile to mention an interesting relation which is induced by the boundary conditions on compact extra dimensions, (0) 6 (0) GHU : I = NS + NV , (5) where superscripts (0) denote zero modes for scalar and vector fields, and I is the number of generators of the original gauge group GM . For example, with GM = SU(3), if GM is broken into SU(2) × U(1), then we can have “ 8 = 4 + (3 + 1) ” relation, where the 4 represents the (real) degrees of freedom of the Higgs doublet, and 8, 3, and 1 correspond to the number of generators for each gauge generator of SU(3), SU(2) and U(1), respectively. Therefore in (0) general, two parameters Nl (0) Nl (0) (0) and N (0) can be given in terms of NF , NV and I, 1 11 (0) (0) (0) (0) = I + NF − 5NV , N (0) = I + NF + NV . 3 3 (6) Note that they can be used to remove degrees of freedom after fixing the GM and its branching rule to subgroups. Again, let us turn back to the theory cutoff. After compactification, the GHU model becomes the 4-dimensional effective field theory with KK states of scalars and vector fields (and fermions if bulk fermions are allowed). Because they have mass spectra that have the same interval such as 1/R2 , it is natural to assume that the total number of KK states of 6 Here we assume that Aµ (µ = 0, 1, 2, 3) and Ai (i = 5, 6, · · · ) have the opposite boundary conditions of each other. 5 scalar(S), vector(V ) and fermion(F ) fields is all the same, 7 NSKK = NVKK = NFKK ≡ JKK . (7) Note that the small differences among S, V and F modes due to boundary conditions are (0) (0) (0) (0) negligible because NXKK ,Y ≫ ∆NXY , where ∆NXY ≡ |NX − NY | for X, Y = {S, V, F }. Thus, the cutoff scale in the GHU models is mainly dominated by the JKK factor because (0) JKK ≫ Nξ , Nl , ΛCUTOFF ∼ q (0) where the Nl = JKK Nl M̄P (0) 1+ (0) JKK Nl /(96π 2) (0) (0) JKK ∼ Λ CUTOFF 1/R d = Λ (8) (0)
= JKK NS + NF − 4NV states with d extra dimensions is easily calculated by , . In addition, the number of KK CUTOFF MC d . (9) The ΛCUTOFF as a function of MC is obtained with the above two relations (neglecting a constant 1 in a denominator of Eq. (8) ) ΛCUTOFF = (0) Numerically, Nl " MC d M̄P2 (0) (0) Nl /96 π 2 #1/(2+d) . (10) = 1 for the SM which has NS = 4, NF = 45 and NV = 12. For the MSSM (0) which has two Higgs doublets, NS = 98, NF = 61 and NV = 12, the Nl = 111. The ΛCUTOFF for GHUSM and GHUMSSM at the MC = 1 TeV is calculated by ΛSM CUTOFF ΛMSSM CUTOFF 1/(2+d)  (39+3d) ∼ 5.62 × 10 , 1/(2+d)  ∼ 5.06 × 10(37+3d) . (11) We present numerical results of the ΛCUTOFF for both models in Table II. In Table II, the first column d denotes the number of extra dimensions, and the second and the fourth columns show the cutoff scales at MC = 1 TeV. Interestingly, they show that the strong gravity scale could be much lower than the reduced Planck mass ∼ 1018 GeV, and it could appear at 7 For simplicity, we assume that our bulk space is flat. However in the warped (or curved) extra dimension, we should consider the red-shifted (or blue-shifted) energy spectrum. We do not consider it because it is beyond our present interest. 6 TABLE II: The cutoff scale (ΛCUTOFF ) of the GHU model that has the zero modes corresponding to the particle content of the SM(or MSSM) at MC = 1 TeV. The d and Mmax denote the number of extra dimensions and the maximum MC , respectively. Note that the Mmax may be regarded as the upper bound of MC (see the main body). As the d increases, the ΛCUTOFF (MC = 1TeV) drastically decreases. On the contrary, the Mmax slowly increases until the ΛCUTOFF is equal to the reduced Planck mass at µ = 0. GHUSM GHUMSSM d ΛCUTOFF (MC = 1 TeV) [Gev] Mmax [Gev] ΛCUTOFF (MC = 1 TeV) [GeV] Mmax [GeV] 1 1.78 × 1014 2.57 × 1015 3.70 × 1013 2 2.74 × 1011 7.91 × 1016 8.44 × 1010 8.34 × 1017 4 4.21 × 108 4.39 × 1017 1.92 × 108 1.42 × 1018 10 6.49 × 105 1.23 × 1018 4.39 × 105 1.97 × 1018 ∞ 103 2.44 × 1018 103 2.44 × 1018 2.85 × 1017 105 ∼ 1013 or 1014 GeV depending on d = 10 ∼ 1. Additionally, the third and the fifth columns denote the maximum MC (Mmax ) when the cutoff scale (as a function of MC ) is equal to the reduced Planck mass M̄P (µ = 0) by varying the MC from 103 GeV to 1021 GeV (see maximum points around vertical lines in both panels in Fig. 2). We also plot the ΛCUTOFF as a function of MC with a fixed number of d in Fig. 2. The Left(right) panel is corresponding to the case of GHUSM (GHUMSSM ). In each panel, we choose the d = 1 case as a reference case. Its strong gravity region is painted yellow. Additionally the horizontal and vertical lines in both panels are used to denote the reduced Planck mass at µ = 0 and the Mmax when d = 1, respectively. Note that when MC > Mmax , it seems that the ΛCUTOFF can be larger than the horizontal line M̄P (µ = 0). However, it is not consistent because the enhancement to the M̄P (µ = 0) is not allowed due to the constant 1 in a denominator of Eq. (8). Finally, the red dashed line in Fig. 2 is corresponding to the d = ∞ case. It divides the (MC , ΛCUTOFF) parameter space into the MC < ΛCUTOFF region(allowed region) and the MC > ΛCUTOFF region(forbidden region). We find two interesting facts from the above numerical analysis. Firstly, there exists a tension between d and ΛCUTOFF , that is to say, when the d increases, the ΛCUTOFF dras- 7 1020 1020 Strong Gravity Region 1016 LCUTOFF @GevD LCUTOFF @GevD 1016 d=1 1012 d=2 GHUSM 108 d=4 d=10 10 d=¥ 4 1 1000 Strong Gravity Region 106 109 1012 1015 1018 d=1 1012 d=2 GHUMSSM 108 d=4 d=10 d=¥ 4 10 1021 1 1000 MC @GevD 106 109 1012 1015 1018 1021 MC @GevD FIG. 2: ΛCUTOFF as a function of MC . The left(right) panel is corresponding to the case of GHU model which has the degrees of freedom of the SM(MSSM). The ΛCUTOFF is regarded as the scale at which gravity becomes strong. It also denotes that the perturbativity of the model breaks down. In the d = 1 case the strong gravity region is painted yellow. The horizontal(vertical) line denotes the reduced Planck mass at µ = 0(the maximum MC ≡ Mmax ). Note that a red dashed line is corresponding to the d = ∞ case. It divides the (MC , ΛCUTOFF ) parameter space into two MC < ΛCUTOFF (allowed region) and MC > ΛCUTOFF (forbidden region). tically decreases, and vice-verse. Interestingly, the d = 10 case shows that the ΛCUTOFF could be lowered to a few hundred TeV at MC = 1 TeV.8 Secondly, as the number of zero (0) (0) modes increases (for example, NSM of the SM → NM SSM of the MSSM), it seems that the maximum compactification scales (Mmax ) quickly converge into one special scale (see around the vertical line in the right panel in Fig. 2). It is very intriguing that any GHUMSSM with an arbitrary d finally has one common Mmax near the reduced Planck mass. Actually, we find that all lines meet at one scale near Planck scale (from now on, let us call it “MO ” or equivalently “R0 ” as one common compactification radius). It may mean that all extra dimensions emerge with the same radius near Planck scale, while the extra dimensions which have MC > MO or R < R0 rapidly dissolve in the strong gravity region. In this sense, we may say that all compactification radii of extra dimensions are unified at MO . Note that this situation is analogous to the unification of gauge couplings in the MSSM. Therefore, 8 On the other hand, it implies that the d > 10 case could be excluded from negative experimental data about the low scale quantum gravity below a few hundred TeV in gravitational and collider experiments. 8 the supersymmetry could not only unify the gauge couplings but also unify all radii of extra dimensions into the R0 near Planck scale. Now let us turn out our attention into the J = 2 partial wave amplitude. Because it has additional overall factors due to the degrees of freedom of KK states see Eq. (1) for the
original amplitude , it has this general form of a2 = − 1 2 G GN ECM JKK JKK N (0) , 40 G where JKK is the total number of KK gravitons, and N (0) = (0) (0)
1 (0) . N + N + 4N F V 3 S (12) For one instructive example, let us consider the large extra dimensions scenario where the gravitons propagate in the bulk, while all matter and gauge fields are confined to the 3-dimensional G membrane. In this case we have the JKK = 10 32 and the JKK = 1 [1, 2]. By applying the unitarity condition |a2 | ≤ 1/2, the energy scale at which tree-level unitarity violates is given by (0) 2 2 ECM (0) 2 E 1 20 = G CM = G (0) GN N JKK JKK JKK JKK (0) (13) (0) where ECM ≡ 20(GN N (0) )−1 . Numerically, ECM ≈ 6 × 1018 GeV for the SM, and ECM ≈ 4 × 1018 GeV for the MSSM. The unitarity violation in the large extra dimensions scenario thus occurs at the ECM , SM ΛUNIT ΛMSSM UNIT r (6 × 1018 )2 = 600 GeV , 1032 × 1 ∼ 400 GeV , ≡ ECM ∼ (14) Note that they are approximate estimates due to the massless limit of KK gravitons (See Ref. [2] for more exact numbers). Similarly, many KK states of scalar, vector and fermion fields in the context of GHU models behave like KK gravitons when considering the theory cutoff and the unitarity. As aforementioned, we introduce another parameter Λ UNIT reflecting the unitarity violation scale. Because the NKK is in inverse proportion to MC (see Eq. (13)), the ΛUNIT ≡ ECM is proportional to MC . Namely, if the MC increases, the number of KK states decreases and it can raise the scale of unitarity violation, while if the MC decreases, then the NKK increases and the ΛUNIT decreases. Numerically, if we take MC = 1 TeV with GHUSM d = 1, then ΛCUTOFF = 1.78 × 1014 GeV (see Table II) and the number of KK states is !1 1.78 × 1014 = 1.78 × 1011 . (15) JKK ∼ 3 10 9 1032 1030 Number of KK states 10 Number of KK states GHUMSSM GHUSM 27 d=1 1022 d=2 1017 d=4 d=10 1012 d=1 d=2 18 10 d=4 d=10 12 10 106 107 100 1000 1024 106 109 1012 1015 1018 MC @GevD 1 1000 106 109 1012 1015 1018 MC @GevD FIG. 3: The number of KK states JKK as a function of compactification scale MC . The number d denotes the number of extra dimensions. The left(right) panel corresponds to the case of the GHUSM (GHUMSSM ). These JKK numbers drastically decrease as the MC scale increases because
d
d JKK ∼ ΛCUTOFF /(1/R) = ΛCUTOFF /MC . We plot the JKK as a function of MC in Fig. 3 for both GHUSM (left panel) and GHUMSSM (right panel). They show that the JKK drastically decreases as the MC increases. SM With this JKK , the ΛGHU is easily calculated by UNIT r (6 × 1018 )2 GHUSM ΛUNIT ∼ ∼ 1.42 × 1013 GeV. 1.78 × 1011 (16) It is interesting that the theory cutoff and the unitarity violation scale do not coincide in GHUSM SM the GHUSM (ΛGHU CUTOFF 6= ΛUNIT ). In the same way, we calculate the JKK and the ΛUNIT by varying d from 0 to ∞. These numerical results are presented in Table III. As the d increases, the JKK rapidly increase and the ΛUNIT drastically decreases. In particular, the d = 10 case shows that the unitarity violation scale could be lowered as much as ∼ 10 TeV GHUSM GHUSM similarly to the previous case of the theory cutoff. It is also found that ΛCUTOFF > ΛUNIT GHUMSSM MSSM in the GHUSM , while ΛGHU in the GHUMSSM . It is thus expected that there CUTOFF ≈ ΛUNIT is different physics at around ΛUNIT in each model. In the following subsections, we discuss several scenarios depending on the hierarchical patterns among ΛUNIT , ΛCUTOFF, and MC . 10 TABLE III: The d , JKK , ΛUNIT and the radius of extra dimensions for both GHUSM and GHUMSSM are presented in sequence. They are evaluated by taking MC = 1 TeV and the ΛCUTOFF values in Table II. Interestingly, numerical results show that ΛCUTOFF > ΛUNIT in the GHUSM , while ΛCUTOFF ≈ ΛUNIT in the GHUMSSM . In addition the d = 10 case shows very low unitarity violation  scales ∼ 104 (GHUSM ), 105 (GHUMSSM ) GeV. For one reference, we present the experimental constraint of gravitation, R ≤ 44 µm [9] or equivalently 1/R ≥ 4.5 × 10−3 eV, and the collider  constraint MC > 1.59 TeV with CL=95% from p p̄ → dijet, angular distrib. [10] GHUSM GHUMSSM d JKK ΛUNIT [GeV] Radius[m] JKK ΛUNIT [eV] Radius[m] 1 1.78 × 1011 1.42 × 1013 1.38 × 10−25 3.70 × 1010 2.08 × 1013 9.49 × 10−26 2 7.51 × 1016 2.19 × 1010 9.01 × 10−23 7.12 × 1015 4.74 × 1010 4.16 × 10−23 4 3.14 × 1022 3.39 × 107 5.83 × 10−20 1.36 × 1021 1.09 × 108 1.82 × 10−20 10 1.33 × 1028 5.21 × 104 3.79 × 10−17 2.66 × 1026 2.45 × 105 8.04 × 10−18 ∞ 1 103 1.97 × 10−15 1 103 1.97 × 10−15 A. ΛUNIT > ΛCUTOFF The theory enters into the strong interaction region above ΛUNIT scale because the perturbativity of the model breaks down. The (perturbative) effective field theory remains valid below this scale. However it is not consistent because the ΛCUTOFF is already smaller than the ΛUNIT . B. ΛUNIT < ΛCUTOFF In this case, there exists an intermediate energy gap between the strong gravity scale
and the unitarity violation scale see Fig. 4 (a) . In order to make the theory consistent we should assume some mechanism or new physics that can restore the unitarity. Actually, this scenario happens in the GHUSM . Here the ΛCUTOFF is about ten times larger than ΛUNIT (see Table II and Table III). As one candidate of new physics, the stringy effects may help to remedy the unitarity violation. If it really happens, they may leave some new physics signals at that scale. 11 E Quantum Gravity Quantum Gravity Quantum Gravity LCUTOFF LUNIT LUNIT » LCUTOFF MC MC LUNIT » LCUTOFF » MC HaL HbL HcL
FIG. 4: Schematic illustration of energy scales Λ CUTOFF , Λ UNIT , MC in linearized General Rel- ativity coupled to the gauge-Higgs unified(GHU) model. The yellow regions show the quantum gravity region. The red regions correspond to the weak gravity region in which KK states can live. The left panel (a) shows that there exists the discordance between Λ CUTOFF and Λ UNIT . The central panel (b) shows that the Λ UNIT has the same order of magnitude as the Λ CUTOFF scale. The right panel (c) shows that three scales coincide. It is thus expected that there is different physics at around the ΛUNIT scale in each case. C. ΛUNIT ≈ ΛCUTOFF
In the GHUMSSM case, this scenario is realized see Fig. 4 (b) . There is no unnatural discordance between the ΛUNIT and the ΛCUTOFF scales. Any new physics is not needed in order to remedy the unitarity violation. It is worthwhile to recall that the zero modes increased by supersymmetry can reduce the gap between the ΛUNIT and the ΛCUTOFF scales (0) due to the reduced ECM and increased KK numbers in the Eq. (13). D. ΛUNIT ≈ ΛCUTOFF ≈ MC In this scenario, there exists only one new physics scale. It is thus impossible to have any KK states except zero modes because there is no room for them. Whole spectrum consists of all zero modes. Consequently the effective GHU models may not be distinguishable from the SM if there are no additional zero modes. 12 III. CONCLUSION In summary, we have studied the strong gravity scale and the unitarity violation scale in linearized General Relativity coupled to particle content of the GHU model. The KK gauge bosons, KK scalars and KK fermions in the GHU models drastically change both of the scales. In particular we have considered the two interesting benchmark models, GHUSM and GHUMSSM in order to show model-dependent difference. We have found that the strong gravity scale could be lowered as much as 1013 (1014 ) GeV in the GHUSM (GHUMSSM ) by taking MC = 1 TeV and d = 1. It is also shown that these scales are proportional to the inverse of d. In the d = 10 case, they could be lowered up to 105 GeV for both of the models. We have also found that the maximum compactification scales (Mmax ) of extra dimensions quickly converge into one special scale “MO ” near Planck scale or equivalently into one common radius “R0 ” irrespectively of d, when the number of zero modes increases (for (0) (0) example, NSM → NMSSM ). It may mean that there is the unification of compactification radii near Planck scale analogously to the unification of gauge couplings in the MSSM. Moreover, it is also interesting that the supersymmetry helps to remove the discordance between the ΛUNIT and the ΛCUTOFF scales. Consequently, it may reveal that the supersymmetry can play another important role in extra dimensions. Finally, our method can be easily applied to the other extra dimensional models that have these KK states. Acknowledgments J. P was supported by the National Research Foundation of Korea (NRF) grant (No. 2013R1A2A2A01015406). J. P thanks J.S. Lee for his valuable comments. [1] M. Atkins and X. Calmet, Phys. Lett. B 695, 298 (2011) [arXiv:1002.0003 [hep-th]]. [2] M. Atkins and X. Calmet, Eur. Phys. J. C 70, 381 (2010) [arXiv:1005.1075 [hep-ph]]. [3] T. Han and S. Willenbrock, Phys. Lett. B 616, 215 (2005) [hep-ph/0404182]. [4] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 429, 263 (1998) [hep-ph/9803315]. 13 [5] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 436, 257 (1998) [hep-ph/9804398]. [6] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999) [hep-ph/9905221]. [7] N. S. Manton, Nucl. Phys. B 158, 141 (1979). D. B. Fairlie, Phys. Lett. B 82, 97 (1979). P. Forgacs and N. S. Manton, Commun. Math. Phys. 72, 15 (1980). Y. Hosotani, Phys. Lett. B 126, 309 (1983). ; ibid. 129 (1983) 193. Y. Hosotani, Annals Phys. 190, 233 (1989). I. Antoniadis and K. Benakli, Phys. Lett. B 326, 69 (1994). I. Antoniadis, K. Benakli and M. Quiros, New J. Phys. 3, 20 (2001) [arXiv:hep-th/0108005]. C. Csaki, C. Grojean and H. Murayama, Phys. Rev. D 67, 085012 (2003) [hep-ph/0210133]. G. Burdman and Y. Nomura, Nucl. Phys. B 656, 3 (2003) [arXiv:hep-ph/0210257]. C. A. Scrucca, M. Serone and L. Silvestrini, Nucl. Phys. B 669, 128 (2003) [arXiv:hep-ph/0304220]. C. A. Scrucca, M. Sernoe, A. Wulzer and L. Silvestrini, JHEP 0402, 049 (2004). G. Cacciapaglia, C. Csaki and S. C. Park, JHEP 0603, 099 (2006) [arXiv:hep-ph/0510366]. A. Aranda and J. L. Diaz-Cruz, Phys. Lett. B 633, 591 (2006) [arXiv:hep-ph/0510138]. B. Grzadkowski and J. Wudka, Phys. Rev. Lett. 97, 211602 (2006) [arXiv:hep-ph/0604225]. A. Aranda and J. Wudka, Phys. Rev. D 82, 096005 (2010) [arXiv:1008.3945 [hep-ph]]. J. Park and S. K. Kang, JHEP 1204, 101 (2012) [arXiv:1111.5422 [hep-ph]]. G. Panico, M. Safari and M. Serone, JHEP 1102, 103 (2011) [arXiv:1012.2875 [hep-ph]]. [8] W. F. Chang, S. K. Kang and J. Park, Phys. Rev. D 87, no. 9, 095005 (2013) [arXiv:1206.3366 [hep-ph]]. [9] E. G. Adelberger, J. H. Gundlach, B. R. Heckel, S. Hoedl and S. Schlamminger, Prog. Part. Nucl. Phys. 62, 102 (2009). [10] J. Beringer et al. [Particle Data Group Collaboration], Phys. Rev. D 86, 010001 (2012). [11] C. G. Callan, Jr., S. R. Coleman and R. Jackiw, Annals Phys. 59, 42 (1970). [12] F. Larsen and F. Wilczek, Nucl. Phys. B 458, 249 (1996) [hep-th/9506066]. D. N. Kabat, Nucl. Phys. B 453, 281 (1995) [hep-th/9503016]. D. V. Vassilevich, Phys. Rev. D 52, 999 (1995) [gr-qc/9411036]. X. Calmet, S. D. H. Hsu and D. Reeb, Phys. Rev. D 77, 125015 (2008) [arXiv:0803.1836 [hep-th]]. 14