# Has B-L some role in the mass matrix?

1. Apr 19, 2015

### arivero

So, B-L is a U(1) generator extracted out of some unified theories of leptons and quarks and in such theories it is traceless, with B=1/3 and L=1, and the trace taken over a "four coloured" multiplet, namely a lepton and three colored quarks.

Now, I am amazed that there is another Matrix that happens to be traceless, and it is the product of B-L times Mass, when taken only with a single colour of light quarks, I mean we have

Tr (L x M )=sum of lepton masses = $1882.98 \pm 0.16$ MeV
Tr (B x M) = 1/3 sum of udscb masses= $1852.37 \pm 13.13$ MeV

Moreover, there is at least one lattice team out of the consensus for charm quark,
http://arxiv.org/abs/1311.2793 and with the quark values of this paper, we would have Tr(B x M) = 1882.23 It is also true that the same group has another paper with a slightly higher b mass, but with a compatible error anyway.

So it seems that discarding the top quark, the matrix (B - L) x M is compatible with tracelessness when looking only to one colour.

The question is, are there GUT theories using this product matrix? or topcolor or technicolor theories? Georgi-Jarlskog?

2. Apr 19, 2015

### ChrisVer

Wouldn't then the Tr[ (B-L) x M ] = Tr[B x M ] - Tr[L x M] != 0 ?

3. Apr 19, 2015

### arivero

Yep, that is my point; it is intriguing that discarding the top quark, (B-L) times Yukawas happens to be a traceless matrix. I was not putting explicitly this way because one could argue that we still need to account for three colours. On the other hand, colour seems to commute with the yukawas.

Also putting explicitly one sees that the sum in each sector is about two times the nucleon mass, or if you wish about six times the "current quark mass". I think this also is a hint connecting the mass mechanism to colour.

4. Apr 19, 2015

### ChrisVer

I meant not-equal to zero, so it won't be a traceless matrix...
$1852.37-1882.98 =- 30.61 \pm 13.13$

5. Apr 19, 2015

### arivero

Ah, sorry. It is equal to zero with paper 1311.2793 and from reading this paper and the alternative lattice calculations I was considering that perhaps the charm mass is undervaluated. Note that Erler keeps this paper listed in the most current review, surely because its calculation of the strong constant is more coincident with the electroweak fit than the other calculations. But even if not valid -or if other discussions of scale invariance and renormalization apply- it is still very near of zero, and then naturalness a la 't Hoft could be invoked: if a quantity is very near of zero, we should look for a symmetry protecting it.

Last edited: Apr 19, 2015
6. Apr 19, 2015

### The_Duck

There are probably some subtleties here: quark masses depend on the renormalization scale. Lattice calculations are usually renormalized around 2-3 GeV, pretty low compared to the GUT scale. Your equality may not hold at the GUT scale.

7. Apr 19, 2015

### arivero

And lepton masses too.

8. May 7, 2015

### ohwilleke

Very interesting. I wonder why the outlier paper has a charm mass that is 75 MeV heavier than is typical? What assumption is different? Is it because it is Nf=2+1+1 rather than Nf=6?

It is also worth noting that Tr (B x M) = 1/3 sum of udscb masses= 1852.37±13.13 MeV has a value that is identical within the MOE to 1/3 sum of scb masses, a relationship that has been known for some time now. And, the 1/3 sum of scb masses is comparing triple to triple and so doesn't have to concern itself with the unexplained omission of the t mass from the trace.