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- TL;DR Summary
- Investigating the force law in particle physics: The Lorentz 4-force and the Higgs force.
To get you started I will derive the Lorentz force law from the QED Lagrangian and then, I let you do the same to a SM-like theory, i.e., an gauge invariant theory of fermionic fields coupled to the Higgs field, both belonging to the fundamental representation of : Here are real parameters of the Higgs potential, and is the Yukawa coupling-matrix. Of course, in the SM one has to specify explicitly the generators , the fermion fields , the Higgs field , and the coupling matrix which determines the fermion mass eignstates after the spontaneous symmetry breaking.
The Lorentz Force Law from the QED Lagrangian:
From (1) we get the field equations for the fermionic matter field :
and the EM gauge field
with the gauge-invariant matter-field current The conserved and gauge-invariant energy-momentum tensor in QED has a gauge-invariant matter-field and electromagnetic-field part where From , one gets, neglecting surface integral at infinity: Notice that on the LHS we have the rate of change of the 4-momentum of the matter field. Now, using the field equation , one can easily show Substituting this in (10), we find The RHS represents the Lorentz 4-force which causes the change of the 4-momentum of the matter field with time.
Your exercise: Apply the above reasoning to the Lagrangian of eq(2) and show that where is the fermionic matter field current. The two integrals on the RHS of (13) represent the Lorentz-like 4-force of the gauge-field and the Higgs-field force respectively. Clearly, the gauge field couples to the matter-field via the coupling constant , whereas the coupling strength of the Higgs-field to the fermions is determined only by the (fermionic) mass matrix . This fact seems to point to a gravitational role for the Higgs field on the microscopic level !
The Lorentz Force Law from the QED Lagrangian:
From (1) we get the field equations for the fermionic matter field
Your exercise: Apply the above reasoning to the
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