I think that the unperturbed masses, ie the value (1/3 sum(sqrt(M))^2, should have some role in this kind of sceneries. 313 MeV is an interesting value.
Also, for the t,u,c mass, other scenary is interesting: where the initial mass, before cosines, of u and c is zero, while the one of the top is nonzero. And another one, with u and c equal but nonzero, and t huge.
Um, you mean scenario, Alejandro. English is mean, heh? By the way, I don't know where your last comment ended up ... you seem to like posting on old posts.
Good points. Remember that this mass sum depends only on the $r$ parameter, since summing a triplet of either cosines or cosine squares gives a constant. We can create many types of Koide triplet. To get two equal non zero eigenvalues, we just need to set the phase to zero. To get two zero eigenvalues, set $r = 2$.
My point is that we have got a well known quantity of the standard model, 313 MeV, the SU(3) self-mass of coloured particles. But for the top another well known quantity should appear naturally, the electroweak vacuum, 246 GeV. Remember that mass(top)= 0.99 * 246 GeV / sqrt(2) in the standard model.
Hmm, to my mind the square roots are associated to the categorical (matrix) dimension $2$, so I guess this formula makes sense for any $n$, but I'm not sure it is physically useful to think about $n$ generations this way.
I think that the unperturbed masses, ie the value (1/3 sum(sqrt(M))^2, should have some role in this kind of sceneries. 313 MeV is an interesting value.
ReplyDeleteAlso, for the t,u,c mass, other scenary is interesting: where the initial mass, before cosines, of u and c is zero, while the one of the top is nonzero. And another one, with u and c equal but nonzero, and t huge.
Um, you mean scenario, Alejandro. English is mean, heh? By the way, I don't know where your last comment ended up ... you seem to like posting on old posts.
ReplyDeleteGood points. Remember that this mass sum depends only on the $r$ parameter, since summing a triplet of either cosines or cosine squares gives a constant. We can create many types of Koide triplet. To get two equal non zero eigenvalues, we just need to set the phase to zero. To get two zero eigenvalues, set $r = 2$.
Actually, the automatic corrector is even worse than English... Yes, I like to add extra documentation to old posts.
ReplyDeleteI am not sure of how good is my point. In the charged leptons, it is very natural:
sqrt(M_i)=sqrt(313 Mev) (1 + sqrt(2) cos(phase_i))
For the top quark, with r=2, it should be something as
sqrt(172 GeV)/3 ( 1 + 2 cos(phase_i))
But we really could also set the 1 to 0, ie put for the up and charm
sqrt(?) (0 + r cos (phase_i))
and let for the top
sqrt(?) (1 + r cos (phase_i))
My point is that we have got a well known quantity of the standard model, 313 MeV, the SU(3) self-mass of coloured particles. But for the top another well known quantity should appear naturally, the electroweak vacuum, 246 GeV. Remember that mass(top)= 0.99 * 246 GeV / sqrt(2) in the standard model.
ReplyDeleteAlejandro, recall that I use a Koide scale factor of $\mu = 22.80$ GeV for the up quarks. This roughly satisfies the rule
ReplyDelete$\sqrt{\mu / 246} = 1/3$
... is that not what you want?
More or less. I was thinking to put explicitly (0 + r cos) for up and charm, but (1 + r cos) for top. At the end it makes not a big difference.
ReplyDeleteGoing to another subtopic, on the same theme. Could it be sensible to write Koide's as
square(average(root(m)))=1/2 average(m)
It allows for a generalisation to n generations, but I am not sure even if it is the right generalisation.
Hmm, to my mind the square roots are associated to the categorical (matrix) dimension $2$, so I guess this formula makes sense for any $n$, but I'm not sure it is physically useful to think about $n$ generations this way.
ReplyDelete