That flavor eigenstates may not  coincide with mass eigenstates is a property that must be included as input, i.e. at the ``classical'' level of field theory-then the dynamics leads to the possibility of studying how quantum corrections can affect it. This is what's known as flavor physics and there's a vast literature on the subject, available in summer school proceedings-cf. the arXiv-and is now presented in all textbooks on particle physics, whose editions have appeared in the last ten years, at least. 

The assertion about up and down quark matrices isn't correct.

I think it is a dream of many (most?) theorists to calculate such quantities. Unfortunately, renormalized perturbation theory requires you to specify equally many input parameters as there are parameters of this type to describe, leaving no predictions.

One may conjecture that the existing values correspond some fixpoint under renormalization group flow, and start calculating. I'm sure several have already tried, and that it didn't work out. If successful, it would certainly have been announced loudly. 

First of all, the Ward-Takahashi (resp. the Slavnov-Taylor) identities do constrain the observables, in general-and flavor observables in particular. This is the case, also, for the lattice calculations of flavor physics quantities. Renormalization group techniques are standard in this area, so they aren't singled out for particular mention. I think there's some misunderstanding here.

One review is: http://arxiv.org/abs/0907.5386 Cf. also http://twiki.ific.uv.es/twiki/bin/view/Flavianet/Reports

Incidentally, mathematical physics isn't particularly relevant here-in the sense that new mathematical techniques are developed, to address problems in physics. The mathematical techniques are known; computational physics techniques are of most relevance-that allow to address non-perturbative issues, once the perturbative aspects are under control.

Stam "...some misunderstanding..."

One must fix the renormalized mass and interaction parameters at some scale by choosing equally many bare parameters (under the given symmetry constraints).

But when one calculates the flow of these parameters under the renormalization group, when scaling towards the ultraviolet or infrared, they may flow towards fixpoints where their values are no longer independent. There is an example of such a  fixpoint (emergent symmetry) in the textbook by Peshkin and Schroeder.

This, in principle, opens for predictions. One problem is that you may have no control over the renormalization group equations under their full range of flow, if you can only calculate their coefficients perturbatively. And the value of the CKM matrix does not indicate that there should be any simple solution. (Although I think Cecilia Jarlskog found some interesting properties many years ago --- I have forgotten which ones.)

(Peshkin is a different physicist from Peskin :) )

The statements are true-indeed the values of the parameters of the quantum  theory, under the RG flow are NOT independent-they are related by the Schwinger-Dyson equations, which are, actually, identities. And they are, also, related by the W-T/S-T identities. 

The coefficients are, indeed, not calculated, only, perturbatively, but non-perturbatively, using lattice techniques. Cf. the review I linked to-it provides a fairly complete presentation. What does ``a simple solution'' mean?

Stam> "Cf. the review I linked to-it provides a fairly complete presentation."

Thank you, it look interesting. For some reason the links were not visible in the post I responded to. Is there a way to update previous post?

I gather the the collected results cannot be used to make working postdictions about the structure of the CKM matrix(?).

Stam>  "What does ``a simple solution'' mean?"

That is in the eye of the beholder, but patterns of symmetry would be simple. (I once had a student who calculated the value of the Weinberg angle to sin2 thetaW = 3/14, which we though was a simple solution, fitting experiments a that time very well. We were thrilled! Unfortunately it was all wrong, and had moreover been published before.) 

Thanks everyone for the great discussion. But, my point in the question was different. I am not asking about perturbative QFT or RG evolution. I am asking about diagonalising  the quark mass matrices using the matrix perturbation theory. The answers you give may explain the theoretical origin of the mass matrices where as I am interested in the form of the mass matrices that give the CKM matrix using matrix diagonalization using matrix perturbation theory. Only once you have the form om M_u and M_d that perturbatively give CKM matrix, one can think of explaining it from a flavour model.

Take a toy model: 2 fermions, ψ_1 and ψ_2. The indices label flavor. If the mass term is diagonal, i.e. of the form ψ_I m_I δ_IJ ψ_J, then flavor eigenstates are mass eigenstates. If it isn't, i.e. M_IJ=m_I δ_IJ + F^a σ^a_IJ, with σ^a the Pauli matrices, then flavor eigenstates aren't mass eigenstates. In this example 1 and 2 could be the flavors, u and d, for instance. This is a free theory, of course. 

Now add an interaction, either Yukawa, with a scalar field, or with gauge fields and it's possible to evaluate the corrections explicitly-perturbatively and non-perturbatively. 

It's at this point that something must be said about the symmetries one wants to impose-what are the properties of the scalar, in the Yukawa term, what's the gauge group and  how do the fermions transform under it.

Do a change of basis, that diagonalizes the mass matrix-this will induce mixing of the flavor eigenstates and interactions will induce flavor changing processes-that are experimentally constrained in the Standard Model, cf. the GIM mechanism, 

http://www.scholarpedia.org/article/Glashow-Iliopoulos-Maiani_mechanism

http://arxiv.org/abs/1303.6154

(Incidentally, are these links visible? For me they, and the previous ones, certainly are, so i don't know how to correct their appearence.)

The review I linked to provides the material for understanding how the CKM matrix comes about and it would be useful  to go through it.

What fails to be appreciated, quite often,  is that, in a quantum field theory, quantities, that aren't protected by any symmetry, receive corrections and, conversely, if they are protected by a symmetry, their corrections need not vanish-they are, simply, constrained by the relations the symmetry implies with other quantities. So the mixing angle between the SU(2) and U(1)_hypercharge groups does receive corrections and these are sensitive, in fact, to physics beyond the Standard Model, that's one reason they're quite intensively studied. 

Conversely, take a diagonal mass matrix and add Yukawa or gauge interactions, that aren't diagonal in the flavors. Then the corrections, these will bring  to the mass matrix will lead to off-diagonal terms.

Even if the interaction respects the factorization of the flavors, it can be the case that the measure of the functional integral doesn't and this can lead to flavor-changing processes. 

The reason is that both flavors interact with the same scalar and gauge fields, thus, indirectly, with each other and it is this interaction that will mix them-unless they carry quantum numbers that ensure that the contributions cancel.

Also, one must include all terms consistent with the symmetry of the theory-if some are forgotten, they will be generated by quantum effects. So the symmetries of the theory are the starting point-in the Standard Model, they lead to the CKM matrix.

Of course, if the interactions and the mass terms are diagonal in the flavors, quantum corrections can't induce off-diagonal mass terms-as can be shown by writing the identities that stem from the symmetry.

Sanjeev> my point in the question was different.

You want to explain the higher order terms in \lambda in the Wolfenstein parametrization by the first order term? That could be an interesting approach, if you through this could reduce the number of independent parameters. Unfortunately, I don't see how this can possibly work, since the first order term only connect the lowest two generations. Any perturbation of that will preserve this property, hence you will have to introduce at least one more independent parameter, of order \lambda^2 (two parameters are still better than four). But perhaps you had something entirely different in mind? 

I am not aware of any work along the lines you indicate, but since I don't work actively in the field this does not amount to anything. For a serious check I would (i) investigate the works by Wolfenstein, and (ii) investigate works which cites Wolfensteins original article (where "investigate" includes filtering according to title and abstracts, to reduce the task to something of manageable size).

Stam> are these links visible?

Yes, no problems with that. I just think you had saved an early version of your post (without the links), which I read and responded to. I was not aware that one can update posts (remember my complaint on another thread of discovering piles of errors immediately after posting), but have learned to do that now.

I checked the (large) report on Flavor Physics in the Quark Sector, which you linked to. It is rather brief on lattice simulations, and I kind of doubt that the matter has been investigated sufficiently by lattice simulations yet. This is quite difficult due to the need for anomaly cancellation between the lepton and quark sectors in the Standard Model (after getting rid of the fermion doublers). Proposals of considering surface states of an 5-dimensional lattice does not seem accessible to lattice simulations yet.

This is a matter I currently have some professional interest in (contrary to discussions on other threads); hence I am grateful for any comments of insight.

The report indeed cannot and does not do true justice to the lattice simulations techniques themselves-I'll try to find one more focused on those issues. However I'd strongly recommend reading subsection 2.1 carefully, because it's there that the theoretical foundation is presented. In particular it's explained why the gauged chiral symmetry of the electroweak sector is, in fact, not relevant, so the leptonic partners to the quarks don't enter the picture. The chiral symmetry that's relevant is the (approximate) global chiral symmetry of the string interactions-the electroweak interactions enter in a much more subtle way and much effort has been devoted to the consistent definition of ``effective Hamiltonians'', their range of relevance and how  the operators can be renormalized ``non-perturbatively''-which is, another, subfield, itself, however. 

Already taking into account dynamical quarks is extremely challenging: if they're too heavy (like the c or b quarks) , they don't ``fit'' on currently available lattices-the lattice spacing isn't fine enough. If they're too light (like the u and d quarks, that are involved in  describing the chiral properties of the strong interactions), lattice simulations take much too long-it's only very recently that algorithms have been developed, that allow to describe pions with, almost, physical masses.

The review does provide a presentation of the theoretical basis that allow current simulations to obtain data that are relevant for flavor physics.

I'm interested in more theoretical matters, precisely, how to use domain wall fermions to simulate the effects of gauged chiral symmetry in different contexts, beyond QCD, so I would be interested in pursuing the discussion. My papers on the subject are available on the arXiv and (of course, through the profile here).    

Lattice simulations of 5D theories, for simulating 4D chiral fermions are possible and have been done-but it depends what one is after. For the phase diagram they're not too difficult; for studying global properties, relevant for QCD they're much harder: cf. here

https://www.princeton.edu/physics/graduate-program/theses/theses-from-2004/J.Laihothesis.pdf

for a review of one topic. 

For condensed matter applications, studies of 3D theories relevant for 2D chiral fermions should be much more feasible.