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  Generalization of Newton’s inertia principle in flavor phenomenology

   Generalization of Newton’s inertia principle in euclidean 3-space is related to the flavor degree of freedom of the quantum reality. When there are several mass copies, single particle mass must be replaced by the quantum concept of mass matrix; it introduced new additional dimensional quantities besides direction angles of the particle momentum vector – mixing angles and mass ratios of the copies.

   Inertia law states that with no forces (no external parameters in the equations of motion) a particle moves with constant momentum vector. And so, its direction angles in the physically singled out orthogonal coordinate system (Z-axis parallel to momentum vector) are represented by the unit vector [vector (0, 0, 1) for the direction cosine functions] that is an intrinsic geometric solution without external parameters of the well known from euclidean geometry Pythagorean equation.   

   The generalization of Newton’s inertia law in flavor phenomenology is based on the hypothesis that the Standard Model hierarchies of the additional dimensional quantities (mixing angles and mass ratios) also obey the geometric Pythagorean equation at leading approximation without external free parameters. This hypothesis is a first time basic generalization of the Newton’s inertia principle in the framework of flavor phenomenology.  

    The crucial support of this hypothesis comes from the known experimental data for SM Dirac particles (charged leptons and up- and down- quarks) – at leading approximation these data hierarchies are described by cosine-vectors (0, 0, 1) as intrinsic geometric solutions of the Pythagorean equation without external parameters. Another convincing support is from bimaximal approximation for the neutrino mixing angles.  

   The only unknown in the SM at present is neutrino mass spectrum. If neutrinos are Dirac particles, the model points to mass hierarchy with ‘normal ordering’. If neutrino mass hierarchy is of ‘inverted ordering’ or quasi-degenerate, the model points to Majorana neutrino masses.  

Clarification for understanding the relation between Newton’s 1st law inertia-benchmark and SM hierarchies. This relation is based on the facts of independent of time not perturbed by external forces inertial moving particle momentum vector with constant direction angles, on the one hand, and the constant SM mass and mixing hierarchies at leading approximations, on the other hand.  Both the independent of time direction angles of inertial moving particle and SM hierarchies obey the Pythagorean equation (1) and  (2) at the not perturbed approximations.
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                             Comment on Flavor-Inertia Hierarchy Analogy

    Though particle states at rest and inertial movement with constant momentum vector P seem different only by the choice of inertial frame of reference, these states factually are different physical states – the rest-state does not violate the space symmetry, while the P-state singles out one direction in space. This physical difference is related to the additional not symmetric initial condition (external push) that caused P ≠ 0 in reality. The analogy between the SM particle hierarchies and direction angle hierarchies of inertial moving particle relates SM hierarchies only to inertial motion with singled out direction in space P ≠ 0. Particle rest state P = 0 is a special degenerate case. It may be in analogy with a special case of degenerate flavor masses or mixing angles. Flavor-inertia hierarchy analogy relates symmetry violation of SM hierarchies with space symmetry violation of point particle inertial motion. Hence it refers only to violating space symmetry inertial motion with not zero momentum vector P ≠ 0.        

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