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Summer Project: Lepton Masses and Mixing Angles


I have recently completed a URSS summer project with Daniel Busbridge and under the supervision of Prof. Paul Harrison, of the Particle Physics Group; the poster produced for this project is available here. You can also read the report I and my partner wrote for the project, or a primer introducing the field.

The project essentially involves finding an action principle which can be minimized to produce the mixing angles and masses for the leptons; the action must be fully basis invariant, following the symmetries of the weak Lagrangian. The action is constructed using traces of commutators of the charged lepton and neutrino mass matrices; minimization gives several constraints which fix the elements of the mixing matrix. The ultimate goal is to obtain the correct lepton masses and the tribimaximal form of the PMNS matrix shown below:

Tribimaxmial Matrix


Modern Particle Physics, the study of the fundamental elements of matter, including electrons and protons, is based upon the Standard Model, a theory which has been rigorously proven by experiment to be an extremely accurate model of nature. There is a problem however: many of the properties of the particles, specifically their masses and mixing angles, are not predicted by the Standard Model and must be added in. This stands in stark contrast to the fundamental aim of physics: to reduce the laws of physics to more general ones with, very importantly, a sense of inevitability about them i.e. that once the law has been discovered it should seem as if there could have been no other way of describing nature. The purpose of this project was to solve this problem, to find a way to predict these masses and mixing angles, and the relations between them, which would otherwise have been added in, using a device known as an Action principle. This is essentially a mathematical device, which when minimized (i.e. made as small as possible) gives the correct laws of nature. An analogue can be made with light, where the action is dependant upon the time the light takes to travel between two points. This means that when the action is minimzed the light takes the path for which it gets between two points the quickest, which is a straight line in free space.

The action considered in this paper has been chosen to be basis invariant. This means that the action should not depend on the choice of basis in the weak lagrangian; essentially this amounts to saying that the action itself must reflect the symmetries of nature. It must work in the same way regardless of the frame of reference chosen; this can be taken in analogue to the translational invariance of space i.e. that all physical laws in the vacuum of space must be the same regardless of where you are. Chosing a law which did not reflect this symmetry would clearly be foolish, and this should therefore extend to the action used here also.


The project dealt with extending Prof. Harrison's work from his 2005 paper. We started by studying the situation where extremisation is carried out without Lagrange multipliers. It was found that the method employed in the aformentioned paper did not work in this case, and so work was done in extending the theory such that small deviations of the neutrino mass matrix from S3 symmetry can be parametrised. This removed certain trivialities from the method but still resulted in incorrect predictions for the masses and mixing angles.

Currently we are working on finding new actions to extremsise. The current idea extends the action considered in Prof. Harrison's 2009 paper involving the traces of the square and cube of the commutator. Traces of the cube of the commutator can be generated using the CP-violating parameter J (the Jarlskog invariant), while traces of the square can be constructed using its CP-conserving analogue K. The idea is to extremise both of these at once and so reach a point where CP is neither violated nor conserved exactly, in line with data.

Skills Gained

The project has exposed me to the far more creative method of thinking required for theoretical physics, which is something not usually experienced in great deal during my degree. This has been highly enjoyable and put me in excellent condition for a future PhD, for which creative thinking will be highly important.

Jonathan Davis

j dot h dot davis at warwick dot ac dot uk

Project Supervisor

Prof. Paul Harrison

Project Partner

Daniel Busbridge