# Mass and flavour eigenstates in SM

From what I understand, if we develop the terms in the Lagrangian of SM containing the $$SU(2)$$ quark doublets $$Q_{i,L}=\begin{pmatrix} u_{i,L}\\ d_{i,L} \end{pmatrix}$$ with $$u_i=u,c,t$$ and $$d_i=d,s,b$$, the terms that are responsible for the mass of the quarks mixes different quarks together. We then say this doesn't diagonalize the masses matrix, it's in the flavour eigenstates.

On the other hand, it's possible to do a linear combination of the quarks such that the mass terms don't mixes the quarks together anymore. We then say that it diagonalize the masses matrix and that it's in the mass eigenstates.

Here are my questions:

1. Is this explanation correct ?
2. Why do we call those two ways of writing the Lagrangian eigenstates? We don't talk about states (kets) here.
3. Apperently, the CKM matrix allow us to go from one point of view to the other, how exactly?

I've followed QFT and SM courses so you can use notions of those.

• Commented Oct 24, 2020 at 19:09

I should think your QFT or SM text, or WP describe the linear-algebraic unitary rotation of mixing adequately. States and quantum fields (e.g. acting on the vacuum) are equivalent descriptions for the same thing.

None of these does (/should) call what you call "flavor eigenstates" this way, and if they did, you should switch text.

Mass eigenstates are the ones that correspond to the eigenvalues of the mass operator, so they correspond to the u,c,t;d,s,b quarks. These states also define what we call flavor. An s quark is the state that propagates with the mass $$m_s$$, and has well-defined strangeness. These are what appear in generation charts, where the generation index increases with the mass of the respective quarks. (Such charts are conceptually arbitrary, and group particles in generations by mass; there is no extant deep reason you could not reassign the s to the first generation and the d to the second, perversely, provided you properly accounted for this change of basis--below.)

Now these states do not couple to the charged weak current simply: Every "up-like" quark (charge 2/3) couples to all three quarks d,s,b and a $$W^+$$. By convention, and nothing deeper, we denote the linear combination of d,s,b coupling to u as d'; to c, as s'; and to t as b'.

The respective coefficients comprise the rows of the CKM matrix, $$\begin{bmatrix} d^\prime \\ s^\prime \\ b^\prime \end{bmatrix} = \begin{bmatrix} V_{ud} & V_{us} & V_{ub} \\ V_{cd} & V_{cs} & V_{cb} \\ V_{td} & V_{ts} & V_{tb} \end{bmatrix} \begin{bmatrix} d \\ s \\ b \end{bmatrix}.$$ One sometimes calls these primed linear combination states "weak eigenstates" since they characterize the charged weak coupling. But they do not have a well-defined flavor, and thus generation. (This underlies the comparably oracular statement that "charged weak interactions violate flavor"...) However, above, it appears your are using $$d_i$$ for the down-like primed states, and not the mass eigenstates!

There is nothing special about the down-like quark mixing; it is a feature of convention: we could have kept the down-like mass eigenstates, and defined up-like primed combinations (weak eigenstates) by the hermitian conjugate of the above CKM matrix, without conceptual stress. (But don't ever do it...)

The majority component of these primed states is the homonymous unprimed one, so the biggest component of d' is d, etc... (but this cannot be done in the formally identical case of lepton mixing, which has often been the source of confusion). (I hope it is clear to you now why calling them, instead, "flavor eigenstates" is an invitation for confusion. A dangerous term to use. Flavor is a feature of masses. People have used the term "flavor basis" on occasion, but it is sadistically meant to send your head spinning, so reach for your holster/cudgel when you hear it used.)

For the neutral weak couplings, there are no such complications, and you simply use mass eigenstates, unprimed states.

Fine print/geeky. You might well wonder where this flavor-versus-mass nonsense started. It did so in the lepton sector, where elusive neutrinos are exclusively studied through their production and decay to charged leptons, so experimentally the primed states are characterized/identified by the flavor of the charged lepton they interact with, $$e,\mu,\tau$$. This is radically different usage than what you see for quarks above, and "neutrino flavor" means not mass eigenstate: it means a primed weak-coupling state. So "flavor oscillations" induced by neutrinos means mutation of the charged leptons involved in production versus decay of the neutrinos! From then on, one drifted to neutrino flavor states, the flavor basis, and, lunatically, to "flavor eigenstates", designed to confuse. The lunacy is receding with the PDG and SM chart makers supplanting mass eigenstates $$\nu_L,\nu_M,\nu_H$$, where they can.