Gravitational field equations near an arbitrary null surface expressed as a thermodynamic identity

Previous work has demonstrated that the gravitational field equations in all Lanczos-Lovelock models imply a thermodynamic identity TdS=dE+PdV (where the variations are interpreted as changes due to virtual displacement along the affine parameter) in the near-horizon limit in static spacetimes. Here we generalize this result to any arbitrary null surface in an arbitrary spacetime and show that certain components of the Einstein's equations can be expressed in the form of the above thermodynamic identity. We also obtain an explicit expression for the thermodynamic energy associated with the null surface. Under appropriate limits, our expressions reduce to those previously derived in the literature. The components of the field equations used in obtaining the current result are orthogonal to the components used previously to obtain another related result, viz. that some components of the field equations reduce to a Navier-Stokes equation on any null surface, in any spacetime. We also describe the structure of Einstein's equations near a null surface in terms of three well-defined projections and show how the different results complement each other.Comment: v2, 25 pages, no figures, to appear in JHE

Tribimaximal Mixing From Small Groups

Current experimental data on the neutrino parameters is in good agreement with tribimaximal mixing and may indicate the presence of an underlying family symmetry. For 76 flavor groups, we perform a systematic scan for models: The particle content is that of the Standard Model plus up to three flavon fields, and the effective Lagrangian contains all terms of mass dimension <=6. We find that 44 groups can accommodate models that are consistent with experiment at 3 sigma, and 38 groups can have models that are tribimaximal. For one particular group, we look at correlations between the mixing angles and make a prediction for theta13 that will be testable in the near future. We present the details of a model with theta12=33.9, theta23=40.9, theta13=5.1 to show that the recent tentative hints of a non-zero theta13 can easily be accommodated. The smallest group for which we find tribimaximal mixing is T7. We argue that T7 and T13 are as suited to produce tribimaximal mixing as A4 and should therefore be considered on equal footing. In the appendices, we present some new mathematical methods and results that may prove useful for future model building efforts.Comment: 44 pages, 7 figures. Typos corrected, references added, figures update

A Scan for Models of Neutrino Mixing from Non-Abelian Discrete Symmetries

The structure of the neutrino mixing matrix is indicative of an underlying family symmetry that interrelates the three generations of fermions in the Standard Model. We systematically scan the parameter space of 76 discrete non-Abelian family symmetries and construct all models with the Standard Model particle content and up to three flavon fields where we include non-renormalizable interactions of mass dimension five and six. We find that of the 76 groups that we considered, 44 groups can accommodate models that are consistent with experiment at 3sigma, and 38 groups can have models that are tribimaximal. One immediate consequence is that A4 is not "special", but should be considered on equal footing with other groups such as T7 that is the smallest group for which we find tribimaximal mixing, and T13 that has the largest fraction of TBM models. We present the details of a model with theta12=33.9, theta23=49.1, theta13=5.1 to show that a non-zero theta13 can easily be accommodated.Comment: Contribution to the proceedings of "The 2011 Europhysics Conference on High Energy Physics-HEP 2011", July 21-27, 2011, Grenoble, Franc

The Structure of the Gravitational Action and its relation with Horizon Thermodynamics and Emergent Gravity Paradigm

If gravity is an emergent phenomenon, as suggested by several recent results, then the structure of the action principle for gravity should encode this fact. With this motivation we study several features of the Einstein-Hilbert action and establish direct connections with horizon thermodynamics. We begin by introducing the concept of holographically conjugate variables (HCVs) in terms of which the surface term in the action has a specific relationship with the bulk term. In addition to g_{ab} and its conjugate momentum \sqrt{-g} M^{cab}, this procedure allows us to (re)discover and motivate strongly the use of f^{ab}=\sqrt{-g}g^{ab} and its conjugate momentum N^c_{ab}. The gravitational action can then be interpreted as a momentum space action for these variables. We also show that many expressions in classical gravity simplify considerably in this approach. For example, the field equations can be written in a form analogous to Hamilton's equations for a suitable Hamiltonian if we use these variables. More importantly, the variation of the surface term, evaluated on any null surface which acts a local Rindler horizon can be given a direct thermodynamic interpretation. The term involving the variation of the dynamical variable leads to T\delta S while the term involving the variation of the conjugate momentum leads to S\delta T. We have found this correspondence only for the choice of variables (g_{ab}, \sqrt{-g} M^{cab}) or (f^{ab}, N^c_{ab}). We use this result to provide a direct thermodynamical interpretation of the boundary condition in the action principle, when it is formulated in a spacetime region bounded by the null surfaces. We analyse these features from several different perspectives and provide a detailed description, which offers insights about the nature of classical gravity and emergent paradigm.Comment: 31 pages, published version with typos fixe

A Minimal Model of Neutrino Flavor

Models of neutrino mass which attempt to describe the observed lepton mixing pattern are typically based on discrete family symmetries with a non-Abelian and one or more Abelian factors. The latter so-called shaping symmetries are imposed in order to yield a realistic phenomenology by forbidding unwanted operators. Here we propose a supersymmetric model of neutrino flavor which is based on the group T7 and does not require extra Z_N or U(1) factors, which makes it the smallest realistic family symmetry that has been considered so far. At leading order, the model predicts tribimaximal mixing which arises completely accidentally from a combination of the T7 Clebsch-Gordan coefficients and suitable flavon alignments. Next-to-leading order (NLO) operators break the simple tribimaximal structure and render the model compatible with the recent results of the Daya Bay and Reno collaborations which have measured a reactor angle of around 9 degrees. Problematic NLO deviations of the other two mixing angles can be controlled in an ultraviolet completion of the model

Comments on the classification of the finite subgroups of SU(3)

Many finite subgroups of SU(3) are commonly used in particle physics. The classification of the finite subgroups of SU(3) began with the work of H.F. Blichfeldt at the beginning of the 20th century. In Blichfeldt's work the two series (C) and (D) of finite subgroups of SU(3) are defined. While the group series Delta(3n^2) and Delta(6n^2) (which are subseries of (C) and (D), respectively) have been intensively studied, there is not much knowledge about the group series (C) and (D). In this work we will show that (C) and (D) have the structures (C) \cong (Z_m x Z_m') \rtimes Z_3 and (D) \cong (Z_n x Z_n') \rtimes S_3, respectively. Furthermore we will show that, while the (C)-groups can be interpreted as irreducible representations of Delta(3n^2), the (D)-groups can in general not be interpreted as irreducible representations of Delta(6n^2).Comment: 15 pages, no figures, typos corrected, clarifications and references added, proofs revise