Flavour physics and you, part I: Origins
This is the first of a few posts on flavour physics aimed at the level of a physics student.
Quarks and leptons come in [at least, but probably only] three “flavors.” The up and down quarks make up just about all visible matter in the universe. The electron and neutrino mediate just about all of chemistry. But nature seems to have provided two heavier copies of these particles for us, and nobody is sure why there should be three copies (or generations/families). The study of this apparent redundancy and the interplay between the different copies is referred to as flavour physics.
Fig. 1. The Standard Model: three generations… why? Image from Symmetry Magazine.
Naively, these heavy generations shouldn’t affect the physics of the light generation that dominates our visible universe. But then I wouldn’t be interested in this field.🙂 At the heart of flavour physics is the Cabibbo–Kobayashi–Maskawa (CKM) matrix. 
Where the CKM matrix comes from
Let’s focus on the quark sector. Before spontaneous symmetry breaking, the Standard Model is a theory of massless fermions. The masses come from the Yukawa couplings to the Higgs boson in the Lagrangian:
Here the s are the Yukawa couplings, is the Higgs doublet, is the left-handed quark doublet, is the right-handed up-type quark, and is the right-handed down-type quark. If you’re confused about why the left-handed and right-handed particles have different letters, take a break and read a bit about chiral models (it’s important).
The indices and label generations: for example refers to the charm quark. To keep things simple, I’ve left out the hermitian conjugate terms. See note  below if you’re suspicious about the SU(2) indices.
The quarks obtain their masses through electroweak symmetry breaking. This just meas that the Higgs doublet obtains a vacuum expectation value which we may choose to be:
Inserting this into the Yukawa couplings, we see that the ground state of the theory now contains mass terms for the quarks:
I’ve explicitly labelled the chirality of the quarks so we can keep track of those that came from the left-handed doublet and the right handed singlets $u,d$. (If you’re worried about how SU(2) indices worked out, this is because of the `technical detail’ mentioned in note .)
Where the mass matrices are:
The first glaring thing is that these matrices are not, in general, flavor-diagonal. In order to find the physical states of a theory (up to perturbations), one needs to diagonalize the mass terms. The mass matrices can be diagonalized by unitary transformations on the fields:
For as appropriate. We have at our disposal the freedom to rotate our fields in flavour-space, so it seems like we can do this trivially by defining the ‘physical’ fields:
(Recall from linear algebra that we can diagonalize complex matrices with unitary transformations in this way.)
But alas! We’ve made a mistake. We do not actually have this much freedom. Recall that the left-handed up- and down-type quarks actually belong to the same SU(2) doublet! This means that the SU(2) symmetry of the Standard Model forces and to transform with the same unitary flavor matrix .
Said in another way, the flavor symmetry of the quark sector is . We only have the freedom to make three rotations, not the four required to diagonalize the two (up and down) mass matrices!
By convention, we go ahead and diagonalize the up-type quarks. But this means the down-type mass matrix is not flavour diagonal, and the physical down-type quarks mix flavours. The mixing matrix is called the CKM matrix (after the folks mentioned above) and is, explicitly:
Where is the matrix that would have diagonalized the down-type mass matrix.
Next time, we’ll start discussing how the CKM matrix affects your everyday life (kinda). See note  for furthe reading suggestions.
The unitarity of the CKM matrix allows you to write down some nice relations that can be represented graphically as the `unitarity triangle.’ Bee and Stefan over at Backreaction wrote a very informative post about the unitarity triangle as part of the Backreaction Advent calendar.
I’ve been planning a few posts on the CKM matrix for some time now, but they got to it first. Perhaps noteably, this wasn’t the last time I’d be scooped on Backreaction’s Advent calendar.🙂
Notes and References
 For those taking bets on Nobel prizes, by the way, the CKM trio is long overdue.
 To be completely accurate, should be , since this is in the representation of SU(2). But this is a `technical detail’ for those not familiar with representation theory.
 Further References:
- Yosef Nir’s lectures from the 2007 CERN-Fermilab Hadron Collider Summer School
- Flavor Physics and Beyond the SM; Part 1, Part 2, Part 3, Part 4
- Review article: Buras, Flavour Physics and CP Violation (hep-ph/0505175); See also some other reviews on the subject by Buras: (The big one: hep-ph/9806471, Erice School: hep-ph/0101336)
- Gilman and Nir, `Quark Mixing: the CKM Picture‘
- Chapter 12 and 13 of Cheng and Li, Gauge Theory of Elementary Particle Physics.
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