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Applying variational principles to fermion masses and mixings


I have recently completed a URSS summer project with Jonathan Davis and under the supervision of Prof. Paul Harrison, of the Particle Physics Group. I produced a poster summarising the project. My research partner and I also wrote wrote a primer as a general introduction to the field, as well as a technical report to suggest further research.

The project was to make work towards finding a flavour-symmetric potential (or action) which, when extremised covariantly, will predict the fermion masses and mixing angles in both the leptonic and quark sectors.


The Standard Model of Particle Physics has been incredibly successful at predicting the phenonema observed in accelerators, particle showers and many other events. It does not, however, incorperate non-zero neutrino masses, nor predict the absolute values for the quark masses. The mixing angles in the electroweak interaction are also not predicted by the model, and have to be 'put in by hand' from experimental data.

Since from a flavour perspective, quarks and leptons are essentially identical in construction, a theory predicting the masses and mixings for quarks should also work for leptons.

Solar and atmospheric neutrino data lead Harrison, Perkins and Scott to propose the tri-bimaximal mixing (TBM) ansatz (below) in their 2002 paper.


A distinctive feature of the above mixing is the tri-maximally mixed ν2 eigenvector, suggesting an underlying maximisation process. Another feature is the |Ue3|2 element equalling zero. Since the mixing is a zeroth order approximation, the size of |Ue3|2 will determine whether CP violation in the leptonic sector is small or zero. This hints at a minimisation process.

Consequently, there is the implication that TBM should be reproducable from an extremisation process, demonstrated by Harrison and Scott in their 2005 paper.

The action in this paper was constructed to be basis invariant, i.e. the action does not depend on the choice of basis in the weak Lagrangian. This is a consistent approach, since it does not make sense for the laws of physics to depend on your point of view. A theory requiring a particular point of view to work is clearly unphysical.

The extremisation was carried out with respect to the Yukawa couplings, providing a method of extremising covariantly; this is desirable for the same reason as the basis invariant choice of action.


The project deals with the extension of Prof. Harrison's work in extremising actions in order to produce the TBM ansatz. The extremization of general mixing invariants carried out in the 2005 paper use Lagrange multipliers to remove some constraints. We found that, when the Lagrange multipliers are removed, extremisation conditions were satisfied trivially by zero-mixing.

We tried actions similar in construction to the ones in this paper, and found that TBM was obtained using Lagrange multipliers, but the removal of Lagrange multipliers again lead to zero-mixing.

As the data suggests a mixing slightly deviated from S3 symmetry, we tried parameterising deviations of the neutrino mass matrix from S3 symmetry. This did remove some of the trivialities, although the solution was still zero-mixing.

All the actions have been constructed out sums of traces of quadratics of commutators of powers of mass matrices e.g.

Commutator Action,

where q is a parameter to be predicted by the theory, and L and N are the lepton and neutrino mass matrices respectively. These elements can be formed from a transform of K, the CP conserving analogue of J, the Jarlskog invariant. Similar elements can be constructed from transforms of J, and are cubics of commutators. Extremising a combination of elements constructed from K and J should provide a solution where CP is not entirely conserved nor violated, in line with the data. This was demonstrated in Harrison and Scott's 2009 paper, although TBM was not produced.

We have tried some more general combinations of these quantities, and have come close to producing self-consistent, non-trivial solutions.


Paul Harrison

p dot f dot harrison at warwick dot ac dot uk

Paul Harrison's research profile


Jonathan Davis

j dot h dot davis at warwick dot ac dot uk

Jonathan's ePortfolio