Spurion & Symmetry: Meaning, Breaking & Explanations

[princeton118]

What is the meaning of "spurion"? How can it break the symmetry? Which book give some good explanations about this theory?
Thanks!

 

[blechman]

A "spurion" is a code-word for a parameter that breaks a symmetry in some way. The idea is that if you pretend that this parameter is actually a field that transforms in the right way to maintain the symmetry, then you can use it to construct "invariant" operators. Then, at the end of the day, you set this "spurion field" to its actual (constant, non-transforming) value and you would have captured all of the symmetry-breaking operators.

If you know the source of the symmetry breaking, the amazing thing is that this "spurion analysis" will always get you EVERY operator! Of course, if you miss a source of breaking, then all bets are off. That's why Model-Building is as much an art as a science!

The place where this first got its real workout is in the chiral lagrangian (theory of pions) which had an isospin symmetry that exchanged the u and d quarks. People wanted to include the isospin symmetry breaking effects. There are two important ones: electric charge and mass (since quarks have different charges and masses). So what you do is pretend that the charge and mass matrices are actually fields that transform in such a way as to leave the action invariant (we know how to include mass and charge into a theory of fermions, so it is clear how to make M and Q matrices transform to accomplish this. Once we know how these objects transform, we can construct all the operators of a given dimension out of the fields, including these "spurion" fields. Then at the end of the day, setting M and Q equal to their final values gives all the isospin-breaking effects to a given order.

The same kind of analysis can be done in the standard model. There, the Higgs vev is the "spurion" of electroweak symmetry breaking. Therefore we can construct all the operators that break the symmetry by including the right factors of the "spurion field" v. Of course, in the usual standard model, v isn't a spurion but a REAL field that actually does transform, but in other models such as technicolor where there is no actual Higgs field, this is how you do it.

Any good book/lecture review article would mention something about spurions. For an in-depth and beautiful explanation, probably the best place is Georgi's "Weak Interactions" textbook (available on his website), although it is quite technical.

 

[princeton118]

Thanks a lot!

 

[VOLKOV9]

For an in-depth and beautiful explanation, probably the best place is Georgi's "Weak Interactions" textbook (available on his website), although it is quite technical.

Searching the file for "spurion" comes up with no results. Is this the same as what he calls spectators?

 

[blechman]

See chapter 5 when he talks about taking quark masses into account. His source terms "s" and "p" are basically the "spurions".

 

[VOLKOV9]

Thanks!