Localization of scalar fields on Branes with an Asymmetric geometries in the bulk
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1 Localization of scalar fields on Branes with an Asymmetric geometries in the bulk Vladimir A. Andrianov in collaboration with Alexandr A. Andrianov V.A.Fock Department of Theoretical Physics Sankt-Petersburg State University Dual Year Russia-Spain 2011 Particle Physics, Nuclear Physics and Astroparticle Physics Theory, Experiment, Accelerator Technology Barcelona 8-11 November 2011
2 Plan of the talk 1. Description of the model of minimal interaction of a gravity with scalar matter fields. 2. Full action up to quadratic order in fluctuations in the vicinity of a background metric. 3. Separation of equations for the physical degrees of freedom (a specially chosen gauge). 4. Scalar sector in gauge. 5. Branon mass spectrum in the theory with a potential. 6. Asymmetric background solutions and related defects of the cosmological constant. 7. Conclusion and remarks.
3 Formulation of the model Conformally flat metric
4 EOM:
5 --- + metric:, infinitesimal gauge transformations:
6 The full action to the quadratic order represents the sum
7 Separation of equations for the physical degrees of freedom The scalar components: using the parametrization we can calculate the components of the quadratic action,
8
9 Scalar sector in gauge and integration by parts This equation allows to calculate the mass spectrum of scalar branons
10 The spectrum of the branon:
11 These formulas allow to calculate the spectrum of quadratic fluctuations of the boson field minimally interacting to gravity!
12 Branon mass spectrum in the theory with potential induced by five-dimensional fermions we assume that is a small parameter, which characterize the interaction of gravity and matter fields use the warped metric in gaussian normal coordinates: system of three equations five-dimensional cosmological constant!
13 the metric is completely determined by matter! and the conformal factor solutions to the asymmetric brane are possible, that corresponds to In the case of a symmetric a centrifugal barrier! Numerical calculations show that at the leading order in the gravitational constant there are neither zero-modes, no resonances at And localized scalar states don't exist near a symmetric brane with potential
14 In the main approximation in the potential with asymmetric brane : Numerical calculations show that at zero mass normalizable localized states don't arise, but localized states with nonzero mass arise when t> t min, t min = 0.21!!! They are resonances, since exponentially decreases at infinity and the barrier is penetrable, although the probability of its penetration is very small.
15 Asymmetric background solutions and defect of cosmological constant the exact asymptotics of the metric and the scalar field with
16 For different asymptotics one must introduce an asymmetry in the cosmological constant or break the symmetry under the reflection : dimensionless function EOM with defect : the cosmological "constant" should depend on y so that the relation were satisfied on the solutions of the equations of motion This is possible only if its (fixed) functional dependence of y coincides exactly with the solution
17 easy to obtain: The equation has three solutions, one of them realizes an unstable state, it is the maximum. To calculate two other solutions, we assume and then obtain : It should be compared with: The relations between the asymmetry parameter t, the asymptotics of the defect and the cosmological function : The asymptotics of the defect of scalar matter and of the cosmological constant completely determine the asymmetry of the conformal factor and of the cosmological function
18 Conclusions In the model with a minimal interaction of gravity and scalar fields for the symmetric anti-de Sitter geometry there are no localized states in the vicinity of the brane at least for potential; In the case of anti-de Sitter geometries asymmetric against reflection of the fifth coordinate such states occur;
19 THANK YOU! Dual Year Russian-Spain 2011 Particle Physics, Nuclear Physics and Astroparticle Physics Theory, Experiment, Accelerator thechnology Barcelona 8-11 November 2011
20 Multicomponent scalar field model with spontaneously broken translational symmetry (V.A.,A.A. and O.O.Novikov) Integrability for multicomponent kink solution!
21 N = 2 (A.&V.Andrianovs, P.Giacconi, R.Soldati)
22 Localization of massive fermions on a brane Consider two types of bispinors in order to localize light massive fermions on an asymmetric thick brane Ψ 1 = Ψ Ψ 2 Nontrivial configurations of Φ and Φ 2 4 eventually lead to CP breaking in the Yukawa vertices ΨΦ (x) Ψ on a brane
23 Mass spectrum equations Two sets of solutions corresponding ± m For upper signs Zero approximation m = 0, massless Dirac fermion No CP breaking!
24 Next approximation To provide a normalizable right-handed component one has to impose Φ, Φ 0 This gives the equation for complex mass spectrum if 2 4 CP breaking!
25 Generation of asymmetric brane Two scalar doublets Mass spectrum equation Solution even function odd function F + F (0) (1) L L provides an asymmetric brane localization for fermions
26 Conclusions 1) A model of domain wall ("thick brane") in the noncompact five-dimensional space-time with asymmetric geometries on both sides of the brane is generated by self-interacting fermions in the presence of gravity; 2)The asymmetric geometry in the bulk is provided by the asymmetry of scalar field potential and a corresponding defect of the cosmological constant; 3)The defect of matter fields is accompanied by a defect of the cosmological "constant" in order to ensure consistency of the equations of motion; 4) In the model with a minimal interaction of gravity and scalar fields for the symmetric anti-de Sitter geometry there are no localized states in the vicinity of the brane; 5) In the case of anti-de Sitter geometries asymmetric against reflection of the fifth coordinate such states occur; 6) There is only slowly decaying resonance when a conformal factor for anti-de Sitter spaces on both sides of the brane have different signs. This case is of a physical interest because the lifetime of the resonance is longer than the expected lifetime of the proton. It is expected that the tunneling probability is of order, However, if the conformal factor starts to grow on one side of the brane, bound states may appear as zero-modes, but in this case there is a problem with localization of a massless graviton on the brane.
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