1. What is the significance of the smallness of the Higgs mass in particle physics?
The smallness of the observed Higgs mass compared to the expected scale of the Standard Model (SM) is a deep mystery in particle physics. The Higgs mass is expected to have a power-law dependence on high scales where physics beyond the SM manifests itself. This smallness, known as the 'Higgs hierarchy' or 'naturalness' problem, is difficult to understand. One solution to this problem is Higgs compositeness, where new interactions produce light composites including the Higgs boson, providing dynamical stabilization of the hierarchy. Composite Higgs models can also be studied through holographic implementations, where the large hierarchy is manifested as a warped extra dimension. The appearance of the IR brane can be interpreted as a spontaneous breaking of conformal symmetry, with the radion excitation corresponding to fluctuations of the IR brane. The stabilization of the dilaton/radion field at large values is crucial for the hierarchy problem. An alternative stabilization mechanism, relevant stabilization, involves a relevant operator with a small coefficient in the UV, altering the nature of the RS phase transition and the shape of the potential for the dilaton. This has implications for the spectrum of gravitational waves emitted during the phase transition and could be detected by next-generation gravitational wave observatories.
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2. How does the operator dL affect the CFT hierarchy?
The operator dL, defined as g d O, plays a crucial role in the CFT hierarchy. When the coupling O(1) is present, it leads to a large explicit breaking of the CFT, preventing the generation of a hierarchy. However, if the coupling g d is small, an approximate CFT in the UV is maintained, allowing for the creation of a large hierarchy due to the running of the coupling. The running of g d as a function of the renormalization scale u is given by g d (u) = g d (u UV ) (u UV /u)^4-d. Assuming a small coupling in the UV, the breaking of scale invariance occurs when the running coupling becomes significant at a scale f far below the UV scale. This results in a UV/IR hierarchy, which can be utilized in composite Higgs models.
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3. How is the dilaton potential calculated in the 5D picture?
In the 5D picture, the effective dilaton potential is calculated by considering the stabilization mechanism. The minimum of this potential corresponds to a hierarchically small dilaton VEV, which is smaller than the UV cutoff scale given by the AdS curvature u UV = k. This hierarchy is essential for solving the naturalness problem. The calculation is performed in the Randall-Sundrum (RS) background, with the metric ds^2 = e^-2ky dx^2 - dy^2. The UV and IR branes are the two orbifold fixed points at y = 0 and y = yc, respectively. This metric is a solution to the Einstein equations.
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4. What triggers the phase transition in CFT?
The phase transition in CFT is triggered as the universe cools down. It occurs from the hot deconfined phase to the cold confined phase. In the dual 5D picture, the hot phase is described by a black brane solution to the Einstein equations, which becomes the AdS-Schwarzschild metric when the UV brane approaches the AdS boundary. The cold phase is dual to the usual RS picture with an IR brane. The phase transition proceeds via nucleation of IR brane bubbles within the black brane background. The critical temperature of the phase transition, Tc, is determined by matching the free energies of the confined and deconfined phases. The former is given by the dilaton effective potential in eq.(3.13), F conf (kh) V (kh) (for T M KK (kh)), while the latter is F deconf (T) = -p^2 N^2 T^4 /8 + V0. The constant term V0 can be found by identifying a common limit to the two phases, leading to V0 = 3N^2 l(2-n)^2 p^2 n kh^4. Solving for F conf (kh) = F deconf (Tc), the critical temperature is thus Tc = kh p^12l^2 - n n^1/4.
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