Talk:Cabibbo–Kobayashi–Maskawa matrix

Kobayashi-Maskawa mechanism
There is no "Kobayashi-Maskawa mechanism" page. Should one be added to wikipedia? If not there should be a more concrete definition given here. — Preceding unsigned comment added by 174.4.38.199 (talk) 05:43, 6 May 2012 (UTC)

Matrix elements
Is there a reason why the matrix is not symmetric? --HappyCamper 02:46, 18 August 2006 (UTC)


 * There is no reason that it should be. The only requirement is that it be unitary. -- Josh Thompson 01:31, 19 November 2006 (UTC)

Doesn't appear in search?
A wikipedia search on 'CKM' doesn't find this page. Is there a way to fix that? -- Josh Thompson 01:32, 19 November 2006 (UTC)


 * I made this edit. Will it suffice? --HappyCamper 02:55, 19 November 2006 (UTC)
 * Works for me. Thanks. -- Josh Thompson 22:30, 20 November 2006 (UTC)

A minor issue
I just have a little quibble with the description of the vector on the left side of equation below as being a vector of strong force eigenstates, since the vector on the right side also is a strong force eigenstate vector. More appropriate would be, it seems, to call it a vector of mass eigenstates.


 * $$\begin{bmatrix} V_{ud} & V_{us} & V_{ub} \\ V_{cd} & V_{cs} & V_{cb} \\ V_{td} & V_{ts} & V_{tb} \end{bmatrix} \begin{bmatrix} \left| d \right \rangle \\ \left| s \right \rangle \\ \left| b \right \rangle \end{bmatrix} = \begin{bmatrix} \left| d' \right \rangle \\ \left| s' \right \rangle \\ \left| b' \right \rangle \end{bmatrix}$$ —The preceding unsigned comment was added by Dauto (talk • contribs) 19:38, 9 April 2007 (UTC).

strong force eigenstate
what's an eigenstate of the strong force? IMHO this is a misconception, and the article should read "mass eigenstate" instead. Please confirm. - Saibod 14:09, 20 September 2007 (UTC)


 * I don't know what, if anything I can add to this discussion, but here's a go at it. In his The Road to Reality (which is my only source on things in this area), Roger Penrose introduces a few of the concepts dealt with in this article (specifically the Cabibbo angle) through discussion of the K0 meson, which he describes as "an eigenstate of strong interactions" (his italics, not mine). If that applies here or not, I couldn't begin to say. 204.52.215.69 (talk) 15:40, 12 December 2007 (UTC)

Uncertainties
Why aren't uncertaities quoted in this article? Any particular reason besides "nobody's done it yet"? —Preceding unsigned comment added by Dauto (talk • contribs) 15:55, 5 February 2008 (UTC)

Unitary triangles.
Either condition $$\sum_k V_{ij}V^*_{ij}=0$$ doesn't seem to hold up or something is not explained as well as it should be.

Explicited, for i=u and j=c, this is

$$V_{ud}V^*_{cd}+V_{us}V^*_{cs}+V_{ub}V^*_{cb}=0.$$

Using the CKM values given, this yields

$$0.97419 \times 0.2256 + 0.2257 \times 0.97334 + 0.00359 \times 0.0415= ~0.4396, $$

which is evidently not zero.Headbomb {{{sup|ταλκ}}κοντριβς – WP Physics} 00:38, 18 December 2008 (UTC)


 * I think the problem is that whoever copied the values from the Particle Data Group only copied the absolute values – that is, the numerical values given there are really $$|V_{ud}|$$, $$|V_{cd}|$$ and so on – and didn't bother with the phases. Markus Poessel (talk) 10:08, 18 December 2008 (UTC)

That was my intuition too, but I wanted to make sure. Been a while since I've dealt with QM.Headbomb {{{sup|ταλκ}}κοντριβς – WP Physics} 10:15, 18 December 2008 (UTC)

uct primes
My reasoning is that what we call |1>,|2> and |3> in


 * $$\begin{bmatrix} \left| 1 \right \rangle \\ \left| 2 \right \rangle \\ \left| 3 \right \rangle \end{bmatrix} = \begin{bmatrix} V_{ud} & V_{us} & V_{ub} \\ V_{cd} & V_{cs} & V_{cb} \\ V_{td} & V_{ts} & V_{tb} \end{bmatrix} \begin{bmatrix} \left| d \right \rangle \\ \left| s \right \rangle \\ \left| b \right \rangle \end{bmatrix}$$

are given by expressions of the following form


 * $$\left| 1 \right \rangle = V_{ud} \left| d \right \rangle + V_{us} \left| s \right \rangle + V_{ub} \left| b \right \rangle.$$

Since the quantities involved are related to u more than they are to d, it makes sense to call |1> --> |u'> rather than |d'>.Headbomb {{{sup|ταλκ}}κοντριβς – WP Physics} 10:37, 18 December 2008 (UTC)


 * As noted elsewhere, I strongly disagree. The linear combination of down type fields is still a down type field. (It has the same Tz eigen value.) If you would denote the weak interaction eigen state by u' the weak interaction term becomes (proportional to) W+μu γμu'. That is odd. In fact in some sense what is currently denoted d' should may be actually be called d, since it is what in the context of the PMNS matrix for neutrinos would be called the flavor eigenstate. What we currently are calling d, the mass eigenstate can only be identified with the down flavour because the CKM matrix is relatively small flavour mixing.


 * On a slightly unrelated note. What is the ket notation supposed to convey in this context? It seems to imply that we are viewing the quark fields as elements of some Hilbert space. But I just can't see what Hilbert space this should be. (TimothyRias (talk) 12:20, 18 December 2008 (UTC))


 * I agree with TimothyRias - in 1, for instance, there is no admixture whatsoever of the u state, the flavor quantum numbers all show that it's a down state, the magnitudes of the CKM matrix show that 1 is d with small additional components. Just to make sure, I've looked through a few text-books and reviews, and where there are new primed variables for the weak eigenstates, it's always d', s', b'. This includes the book by Hughes that's currently cited as a reference for this particular paragraph – look at section 12.8, for instance, and you'll find that, yes, Hughes calls the Cabibbo-mixed states d' and s'. Also, I agree that the kets should be dropped - I don't see why they should be there. Markus Poessel (talk) 14:32, 18 December 2008 (UTC)

Alright I'll expand my position a bit (I'm not saying that's what should go in the article, just the reasons why I feel this way).


 * Concerning the bra-ket notation, all I gather from this is that the |1>, |2>, |3> weak eigenstate is a linear combination of what we call the |d>, |s>, |b> quark states because the |d>, |s>, |b> states are not orthogonal to each other since they mix. The CKM matrix is the transformation for going from the non-orthogonal basis (|d>, |s>, |b>) to the orthogonal one (|1>, |2>, |3>). And that's all there is to it (and seems to jive with what I know of Hilbert spaces which is, admittedly, not a lot). The MNS article uses ket notation as well to convey the same ideas, so I would be surprised if using kets here is some kind of heresy. Hence:
 * $$\begin{bmatrix} \left| 1 \right \rangle \\ \left| 2 \right \rangle \\ \left| 3 \right \rangle \end{bmatrix} = \begin{bmatrix} V_{ud} & V_{us} & V_{ub} \\ V_{cd} & V_{cs} & V_{cb} \\ V_{td} & V_{ts} & V_{tb} \end{bmatrix} \begin{bmatrix} \left| d \right \rangle \\ \left| s \right \rangle \\ \left| b \right \rangle \end{bmatrix}$$
 * Now for the name of the |1>,|2>,|3> states, I know that the notation seems to be dominantly |d'>, |s'>, |b'>, and that the reason given for the choice of these names is that the dominant state in |d'>, |s'>, |b'> are |d>, |s>, |b> respectively. But one thing I hate is a notation that is based on arbitrary things, such as numerical values. What if the CKM matrix values were these instead?

\begin{bmatrix} |V_{ud}| & |V_{us}| & |V_{ub}| \\ |V_{cd}| & |V_{cs}| & |V_{cb}| \\ |V_{td}| & |V_{ts}| & |V_{tb}| \end{bmatrix} = \frac{1}{\sqrt{3}} \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}.$$
 * Would the choice of the names |d'>, |s'>, |b'> make sense then? Hence my preference for |u'>, |c'>, |t'> : The |1> state reflects the coupling of |d>,|s>,|b> quarks states to the |u> quark state, and so on. The weak isospin doublets would then be written in this fashion:
 * $$\begin{pmatrix} |u \rangle \\ |u' \rangle \end{pmatrix},\begin{pmatrix} |c \rangle \\ |c' \rangle \end{pmatrix}, \begin{pmatrix} |t \rangle \\ |t' \rangle \end{pmatrix}$$
 * where the prime denotes the "down typeness" of the field, and the symbol denotes to what up-type field/state that particular down-type field/state is the weak isospin partner of. It's also interesting to note that one could've just-as-well chosen to construct the CKM matrix in this manner instead:
 * $$\begin{bmatrix} \left| d' \right \rangle \\ \left| s' \right \rangle \\ \left| b' \right \rangle \end{bmatrix} = \begin{bmatrix} V_{du} & V_{dc} & V_{dt} \\ V_{su} & V_{sc} & V_{st} \\ V_{bu} & V_{bc} & V_{bt} \end{bmatrix} \begin{bmatrix} \left| u \right \rangle \\ \left| c \right \rangle \\ \left| t \right \rangle \end{bmatrix}$$
 * leading to this for doublets instead
 * $$\begin{pmatrix} |d' \rangle \\ |d \rangle \end{pmatrix},\begin{pmatrix} |s' \rangle \\ |s \rangle \end{pmatrix}, \begin{pmatrix} |b' \rangle \\ |b \rangle \end{pmatrix}$$

Headbomb {{{sup|ταλκ}}κοντριβς – WP Physics} 03:03, 19 December 2008 (UTC)


 * And if you think this drives me nuts, don't get me started on hadron nomenclature. :P Headbomb {{{sup|ταλκ}}κοντριβς – WP Physics} 03:11, 19 December 2008 (UTC)


 * Well in the case you mention (i.e. if the CKM matrix had maximal mixing, which is almost the case for the PMNS matrix!) the mass eigen states would no longer be referred to as up, charm and top, but would simply call them the first, second and third mass eigenstate! This is basically what people do for the neutrinos. And I on the issue of the kets. Yes, I have noticed that other other wiki article (like the PMNS matrix one) use them as well, but I don't think I've ever seen it elsewhere, and I really don't see the point of it.(TimothyRias (talk) 07:01, 19 December 2008 (UTC))


 * Kets are useful here to distinguish states from common variables. Otherwise one might think that d (or d' or whatever) are simply numbers (d=4 doesn't make a whole lot of sense). I've reverted that. I've reverted the non-unitarity about the cabibbo angle treatment as well. I don't know why you say it'S OR nonsense. It follows rather straightfowardly from the fact that if probabilities don't add up to one, there's something missing. Headbomb {{{sup|ταλκ}}κοντριβς – WP Physics} 08:48, 19 December 2008 (UTC)


 * No the kets are not really useful, the imply something that isn't there. As for the unitarity remark see the quark talk page, it is nonsense because the numbers you say don't add up to 1 actual do add up to 1 easily within the margin of error. (the sum is 1.0000 +/- 0.0006) This was a plain case of WP:SYNTH. (TimothyRias (talk) 08:57, 19 December 2008 (UTC))


 * The fact that the t decays into c and u requires that they don't add to 1. There's no OR here. Headbomb {{{sup|ταλκ}}κοντριβς – WP Physics} 09:03, 19 December 2008 (UTC)
 * Yes but there is no way to deduce to opposite conclusion from the experimental data, which is what you are claiming. The fact is that the the mixing with the third generation is very small. In fact it is smaller than the error on the other mixing coefficients. So the claim you are making is nonsense. So even with today's data it would be impossible to predict the existence of a third generation just from measuring quark mixing angles. Nor, is it likely that we would be able to in the near future. (TimothyRias (talk) 09:38, 19 December 2008 (UTC))
 * The errors are linked, you can't simply add them up like you did. Individual they are |V_ud|2=(0.97419±0.00022)2 and |V_us|2=(0.2257±0.0010)2. Collectively |V_ud|2+|V_us|2 = 1–(0.00359±0.00016)2=0.9999871119±0.0000011488.Headbomb {{{sup|ταλκ}}κοντριβς – WP Physics} 09:56, 19 December 2008 (UTC)
 * Well in the case of numbers quoted they are, but that is because they were fitted with the assumption that the total is 1. If you take the actual independent direct measurements of |V_ud|2 and |V_us|2 from PDG review they actual have even bigger errors (and they are mostly independent) and add up to one with an even larger error margin. (TimothyRias (talk) 10:23, 19 December 2008 (UTC))
 * Actually, (0.97419 ± 0.00022)2 = 0.94905 ± 0.00043 and (0.2257 ± 0.0010)2 = 0.05094 ± 0.00045. Assuming the errors are indipendent, the total error is the root of the sum of the squared errors, and 0.94905 ± 0.00043 + 0.05094 ± 0.00045 = 0.99999 ± 0.00062. -- Army1987 – Deeds, not words. 20:29, 22 December 2008 (UTC)
 * And what do the ket implies that isn't there?Headbomb {{{sup|ταλκ}}κοντριβς – WP Physics} 09:05, 19 December 2008 (UTC)

Any notation that writes down, strange and bottom quarks effectively out of the picture, and denotes a mixture of clearly down-like quarks (all have the same electric charge, after all!) using the name of up-type quarks only (all with the same different electric charge) is very confusing, to say the least. As for the coupling: There is no rule that like only couples with like in the standard model. Neutrinos couple to heavy leptons, quarks of different color charges couple to each other. That's the way it is, so there's not a particularly good case for changing names to accommodate coupling, especially when doing so obscures important differences.

As for the contrafactual CKM matrix: yes, if the laws and parameters of physics were different, our nomenclature would be very different, too. But in this particular universe, calling the mixed states d', s' and b' is useful. If we ever introduce inter-multiverse wikilinks, I'll be glad to reconsider.

Returning to the more narrow focus of what is appropriate for Wikipedia: All the text-books that I've found which address the issue, and in particular the reference currently quoted in support for this very paragraph, use this particular notation. The actual values of the CKM matrix and the up-type/down-type distinction are a good reason for doing so. I've yet to come across a single article that shares your point of view on this. In view of all this, I think the choice is pretty clear. Markus Poessel (talk) 09:19, 19 December 2008 (UTC)


 * Like I said, the uct prime notation doesn't have to end up in the article. Consensus seems against that, so it'll stay out.Headbomb {{{sup|ταλκ}}κοντριβς – WP Physics} 09:31, 19 December 2008 (UTC)


 * OK, good. Markus Poessel (talk) 15:58, 19 December 2008 (UTC)

Parametrization.
It would be nice if someone calculated the values of the parameters using the PDG values for the CKM matrix.Headbomb {{{sup|ταλκ}}κοντριβς – WP Physics} 03:30, 19 December 2008 (UTC)


 * The values of which parameters? (The values of A, λ, ρ and η are in the document linked) (TimothyRias (talk) 22:14, 22 December 2008 (UTC))


 * I've added the Wolfenstein params after I made the remarks, but the "standard" parameters are still value-less.Headbomb {{{sup|ταλκ}}κοντριβς – WP Physics} 05:33, 23 December 2008 (UTC)


 * Standard parameters:
 * θ12 = 13.04° ± 0.05°
 * θ13 = 0.20° ± 0.01°
 * θ23 = 2.38° ± 0.06°
 * δ = 1.20 ± 0.08
 * (obtained by inverting relations for Wolfenstein parameters in the PDG paper. Results are somewhat approximate, may update later with more careful values.) (TimothyRias (talk) 08:20, 23 December 2008 (UTC))
 * More careful calculation yields the same values. Anyway this should be precise enough for wikipedia. (TimothyRias (talk) 08:49, 23 December 2008 (UTC))


 * If you have time, could you do the same but for the KM parameters? Headbomb {{{sup|ταλκ}}κοντριβς – WP Physics} 10:41, 23 December 2008 (UTC)

Could someone confirm that I've placed the angles at the right place? Headbomb {{{sup|ταλκ}}κοντριβς – WP Physics} 23:51, 17 September 2009 (UTC)


 * Poke. Headbomb {talk / contribs / physics / books} 04:20, 4 September 2010 (UTC)

En-dashed name
The title of this article and the references to the matrix all use an en-dash between the names Cabibbo, Kobayashi, and Maskawa. An en-dash is not the appropriate punctuation. One could argue that a hyphen should replace the en-dash; but I would suggest simply a space: According to Brian Garner's widely used (and highly respected) usage guide, proper nouns used in a compound adjective should not be hyphenated --- Cabibbo Kobayashi Maskawa matrix. — Preceding unsigned comment added by 138.246.2.239 (talk) 11:56, 14 October 2014 (UTC)
 * This is the proper usage see WP:ENDASH.TR 15:04, 14 October 2014 (UTC)

Units
Units are multiplicative, and values with uncertainties should be stated with brackets. Alternatively, both value and uncertainty can be written with the corresponding unit. In this article, only the uncertainties in the mixing angles are stated with units. (I have not learned yet how to use or amend the scripted expressions to state values.) 146.200.139.78 (talk) 09:15, 28 May 2021 (UTC)


 * New comments belong to the bottom of the page, where I moved this. The matrix elements are, of course, unitless pure numbers. The usage is that of the PDG, the professional instrument you might take your gripe too. Cuzkatzimhut (talk) 13:06, 28 May 2021 (UTC)