Dilatonic Dark Matter: Experimental Detection of Dilaton and Extra Dimension Young Min Cho Konkuk University, Seoul 143-701, Republic of Korea DOI: http://dx.doi.org/10.3204/DESY-PROC-2016-03/Cho Youngmin The Kaluza-Klein dilaton in higher-dimensional unified theory has important implications. First, as the gravitationally interacting massive particle (GIMP) it becomes an ideal can- didate of the dark matter of the universe. Second, as the massive scalar graviton (MSG) it generates the fifth force which modifies the Einstein’s gravity. Most importantly, it repre- sents the extra space itself, so that detecting the dilaton we can confirm the existence of the extra space. We present a simple way to detect the dilaton experimentally, detecting the two photon decay of the remnant dilaton. We show how the fifth force constraint and the dark matter constraint restrict the dilaton mass and the size of the extra space. Moreover, based on the fact that the dilaton mass comes from the curvature of the extra space, we propose the geometric mass generation mechanism that the curvature creates the mass as an alternative to the Higgs mechanism. All existing unified theories are based on the higher-dimensional space-time, which makes the experimental confirmation of the extra space an important issue. In this talk we discuss the physical consequence of the higher-dimensional unification, and present a simple way to detect the extra space experimentally. An unavoidable consequence of higher-dimensional unification is the appearance of gravi- tationally interacting massive particles (GIMPs) or massive scalar gravitons (MSGs) known as the Kaluza-Klein (KK) dilaton and internal gravitons, in 4-dimensional physics [1, 2]. They come in as the gravitons of the extra space, so that they actually represent the very extra space itself. This means that by detecting them we could confirm the existence of the extra space and justify the higher-dimensional unification. The dilaton plays very important roles in physics. First, as the GIMP it becomes an ideal candidate of the dark matter. Second, as the MSG it generates the fifth force which could violate the equivalence principle. Third, it realizes the Dirac’s large number hypothesis making the cosmological constant space-time dependent. Moreover, it could play the role of the quintessence in cosmology [4, 3]. Finally, it appears in all higher-dimensional unified theories, which strongly implies that the dilaton must exist. This makes the experimental detection of the dilaton very important. Unfortunately the dilaton (and the internal gravitons) so far have not been received the due attention, because the extra space has traditionally been assumed to be extremely small (of the Planck scale) [1, 5, 6]. A small extra space could make them very heavy and decay quickly, in which case they leave almost no trace in the present universe. And they cannot easily be produced because it will cost too much energy. In this case they become almost irrelevant. Patras 2016 1 PATRAS 2016 23 YOUNG MIN CHO But this is not entirely true. First, even a small extra space can make the dilaton massless [1, 2]. Moreover, there is the possibility that the extra space may not be so small, which could make the dilaton very light [2]. Indeed this is exactly the proposal made by Arkani-Hamed, Dimopoulos, and Dvali (ADD) to resolve the hierarchy problem [7]. The mass of the dilaton is very important for the following reasons. First, it determines the range of the fifth force [8, 9]. Moreover, it determines the energy density of the dark matter [4, 10]. And obviously we have to know the mass to detect the dilaton. So it is crucial for us to understand how the dilaton acquires the mass and to figure out what is the mass range. We show that the cosmological constraint on the dark matter rules out the dilaton mass between 160 MeV and 276 MeV, and argue that the size of the extra space may not be smaller 9 than 10− m. Moreover, we suggest to detect the dilaton detecting the two photon decay of the remnant dilatons in the universe. Finally, from the fact that the curvature of the extra space makes the dilaton massive, we propose a geometric mass generation mechanism that mass can be generated by curvatureas an alternative to the Higgs mechanism. To demonstrate these points, we consider the Kaluza-Klein theory, the protopype of all higher-dimensional unified theories. To obtain the 4-dimensional physics, we have to make the dimensional reduction. There are two contending dimensional reduction schemes, the popular zero mode approximation and the dimensional reduction by isometry. In the zero mode ap- proximation one treats the (4+n)-dimensional space as real and views the 4-dimensional physics as a low energy approximation [5, 6]. In the dimensional reduction by isometry, however, the isometry in the unified space makes the extra space invisible and determines the 4-dimensional physics [1, 11]. A sensitive issue in the zero mode approximation is how one can actually make the approx- imation. A simple way to do this is to impose a proper isometry as an approximate symmetry. In this case the zero mode approximation becomes very much like the dimensional reduction by isometry. In general, however, the two methods have totally different 4-dimensional physics. This is because the zero mode approximation produces an infinite tower of the KK excited modes in 4-dimension. To show how the dimensional reduction by isometry works, we suppose that the (4+n)- dimensional unified space has an n-dimensional isometry G and assume the unified space to be a principal fiber bundle P(M,G) made of the 4-dimensional space-time M and the n-dimensional group manifold G. Let γµν and φab be the metrics on M and G. In this setting the (4+n)- dimensional Einstein-Hilbert action becomes [1, 2] 1 IEH = √γ√φ RM (γµν ) −16πG0 n 1 (∂ φ)(∂ φ) Re2κ2 h − γµν µ ν + √n φ ρ γµαγνβF a F b − 4n φ2 4 ab µν αβ γµν 1 + (D ρab)(D ρ ) + Rˆ (ρ ) 4 µ ν ab κ2 √n φ G ab +Λ + λ( detρ 1) d4xdnθ, P | ab| − γ = det γ , φ = deti φ , | µν | | ab| 1/n ρ = φ− φ , ( det ρ = 1), ab ab | ab| 1 1 Rˆ (ρ ) = f df bρac + f mf nρacρbdρ . (1) G ab 2 ab cd 4 ab cd mn 2 Patras 2016 24 PATRAS 2016 DILATONIC DARK MATTER:EXPERIMENTAL DETECTION OF DILATON AND EXTRA ... Here G0 is the (4 + n)-dimensional gravitational constant, RM is the scalar curvature of M, e a is the coupling constant of the KK gauge field Fµν , κ is the scale of the extra space, Λ0 is the (4+n)-dimensional cosmological constant, and λ is a Lagrange multiplier. Notice that RˆG(ρab) is the normalized curvature of the fiber space G. To proceed, we introduce the Pauli metric (also called the Einstein metric in string theory) gµν and the Kaluza-Klein dilaton σ by [1, 4] n g = √φ γ , φ = φ exp 2 σ . (2) µν µν h i n + 2 r But, since κ is left arbitrary, we can always normalize φ = 1, (3) h i without loss of generality. With this understanding we have the following Lagrangian in the Pauli frame [1, 4] ˆ VG 1 2 1 ab µ = √g R + 2 (∂µσ) + (Dµρ )(D ρab) L −16πG0 4 h Rˆ n + 2 n + G exp σ + Λ exp σ κ2 − n P − n + 2 r r e2κ2 n + 2 + exp σ ρ F a F bµν , 4 n ab µν r i Vˆ = dnθ = κn. ( φ = 1) (4) G h i R This describes the well-known unification of gravitation with the gauge field, provided that [1, 2] κn 1 e2κ2 = , = 1, (5) 16πG0 16πG 16πG where G is Newton’s constant. To understand the meaning of (5), notice that by the first equality κ sets the strength of the higher-dimensional gravity l0, 1/(n+2) κ n/2(n+2) l0 = G0 = lp, (6) lp where lp = √G is the Planck scale. This is precisely what one has in the ADD proposal, which confirms that a large extra space can indeed make the higher-dimensional gravity strong [8, 7]. However, from the second equality κ is determined by the fine structure constant of the KK gauge boson αKK [1, 2], 2 2 4 2 e κ = lp. (αKK = ) . (7) αKK 4π This means that a large extra space makes αKK extremely small. In fact this tells that, for the 5 42 ADD proposal which favors κ 10− nm, we need αKK 10− for n = 6. (For n = we 32 ' ' ∞ have α 4 10− , which sets the upper limit of α .) KK ' × KK Patras 2016 3 PATRAS 2016 25 YOUNG MIN CHO To find the mass of the KK dilaton, we assume that (4) has the unique vacuum gµν = η , σ = 0, ρ = δ , A a = 0, and find the dilaton potential h i µν h i h abi ab h µ i 1 Rˆ n + 2 V (σ) = h Gi exp( σ) 16πG κ2 n −r h n +Λ exp( σ) + V , (8) 0 − n + 2 0 r i where RˆG = RˆG(δab) is the (dimensionless) vacuum curvature of the extra space and V0 is a constanth i introduced to assure that (8) has no non-vanishing 4-dimensional vacuum energy. Notice that Λ0 and V0 are completely fixed by the vacuum condition dV (0)/dσ = 0 and V (0) = 0. From (8) we find the mass of the KK dilaton µ (identifyingσ ˆ = σ/√16πG as the physical dilaton), 2 Rˆ 1 Rˆ µ2 = h Gi = h Gi α m2, (9) n κ2 2n KK p where mp is the Planck mass.