Editing Yukawa interaction (section)

==Yukawa coupling to the Higgs in the Standard Model==
A Yukawa coupling term to the [[Higgs field]] affecting [[spontaneous symmetry breaking]] in the Standard Model is responsible for fermion masses in a symmetric manner.

Suppose that the potential <math>~V(\phi)~</math> has its minimum, not at <math>~\phi = 0~,</math> but at some non-zero value <math>~\phi_0~.</math> This can happen, for example, with a potential form such as <math>~V(\phi) = \lambda\,\phi^4~ - \mu^2\,\phi^2 </math>. In this case, the Lagrangian exhibits [[spontaneous symmetry breaking]]. This is because the non-zero value of the <math>~\phi~</math> field, when operating on the vacuum, has a non-zero [[vacuum expectation value]] of <math>~\phi~.</math>

In the [[Standard Model]], this non-zero expectation is responsible for the fermion masses despite the chiral symmetry of the model apparently excluding them.
To exhibit the mass term, the action can be re-expressed in terms of the derived field <math> \phi' = \phi - \phi_0~,</math> where <math>~\phi_0~</math> is constructed to be independent of position (a constant). This means that the Yukawa term includes a component
<math display="block">~g \, \phi_0 \, \bar\psi \, \psi~</math>
and, since both {{mvar|g}} and <math>\phi_0</math> are constants, the term presents as a mass term for the fermion with equivalent mass <math>~g\,\phi_0~.</math> This mechanism is the means by which spontaneous symmetry breaking gives mass to fermions. The scalar field <math>\phi'~</math> is known as the [[Higgs field]].

The Yukawa coupling for any given fermion in the Standard Model is an input to the theory. The ultimate reason for these couplings is not known: it would be something that a better, deeper theory should explain.