What is new with Koide sum rules?

arivero

I am not sure. Consider a decay muon to electron plus a pair neutrino antineutrinos, as usual. As it is possible that the electron is left in the same rest frame that the initial muon, I could say that the energy available for the neutrino pair is the difference of pole masses of muon and electron, not the muon pole mass minus the renormalised electron mass at muon scale. I think I should had put more care when I attended to the undergraduate lectures, twenty years ago.

Of course it is irrelevant for the experimental results, the running of electron fro .511 to 105 is surely negligible.

fzero

Thinking a bit more, I think the best way to look at the issue is the most straightforward. If you compute the decay using the bare masses, then a proper treatment of loop corrections automatically takes into account running of the masses and coupling constants. Determining what bare parameters to use amounts to choosing a renormalization scheme and then extracting the pole mass by finding the pole in the full propagator.

I think that any simplification (like just using effective mass parameters) probably leaves too much physics out that is of the same degree of importance.

arivero

I agree, bare plus corrections seems the best approach, and in fact it is the usual approach to calculate the decay width. But I am intrigued really about the size of phase space, and more particularly about which is the maximum energy that the neutrino pair can carry. In principle is is a measurable quantity. Is it [itex]105.6583668 - 0.510998910[/itex], i.e, [itex]m_\mu(m_\mu) - m_e(m_e)[/itex] (?), or is it [itex]m_\mu(m_\mu) - m_e(m_\mu)[/itex]? I think that the solutions to the RG running make the electron mass to _decrease_ when the scale goes up, so the second answer would extract energy from magic. And the first answer is then an example of a physical comparision of lepton mass at different scales. So I'd conclude that the need of comparing masses at the same scale is just a rule of thumb, not a general axiom.

ohwilleke

The assumption that Lubos makes that pole masses are necessarily more fundamental the the rest masses we know and love isn't necessary right. This is particularly true if decay width, rather than being a truly independent parameter of a particle is actually a function of some other property or properties of that particle according to a function whose form is not currently known.

By analogy, while it is often more helpful to use the expected decay time of a particle adjusted by a Lorentz transformation to reflect its kinetic energy (we could say that this quantity runs with the energy level of the particle), that doesn't necessarily mean that the Lorentz transformed decay time from the perspective of an observer watching the particle wizz by him is really more fundamental than the decay time of the particle from an observer in the particle's rest frame that does not run.

Likewise, until we understand the underlying mechanism by which Koide's formula arises, there is no particularly good reason to conclude that all of the masses in the formula must be computed at the same energy scale as arivero notes in #37.

mitchell porter

I still haven't fully worked through Sumino's paper, but I want to highlight another curious fact, that the family symmetry group which he proposes is U(3) x SU(2) (later he embeds this in bigger groups). Since that is the SM gauge group, I've been wondering whether his mechanism can be realized by some form of dimensional deconstruction.

mitchell porter

What do you get if you minimize this expression:

[itex]V = \sum\limits_{1 ≤ i,j,k ≤ 6; \text{ } i,j,k \text{ different}}(\frac{x_i^2 + x_j^2 + x_k^2}{(x_i+x_j+x_k)^2} - \frac{2}{3})^2[/itex]

Do you get something like a descending chain of Koide triplets from the squares? (For some ordering of the "x"s.)

arivero

http://arxiv.org/abs/1205.4068
Neutrino masses from lepton and quark mass relations and neutrino oscillations,
by Fu-Guang Cao,
suggests the use of Koide-like sums for all the six leptons.

arivero

Hmm, rumours of fermiophobic Higgs!

It is even better than leptophobic; it implies that the Higgs has not role in the mass of the bcsdu quarks. It is agnostic about top, because a 125 GeV Higgs obviously can not decay into top quarks.

hareyvo

Precisely, it is possible that the fine structure constant has a role in the calculation of the mass.
With α the fine structure constant, e the charge of electron, me its mass, re its length, q the charge of Planck, m its mass, r its length, according to
http://en.wikipedia.org/wiki/Planck_units,
we have
q^2 = 4πc(hbar)ε_0 = 4πmr(c^2)ε_0 = mr.10^7
αq^2 = e^2 = αmr.10^7 ≡ me.re.10^7
Write α = yz and αq^2 = αmr.10^7 = ym.zr.10^7
With ym = me = 9.1093829100.10^-31 kg,
y = me/m = 4.1853163597.10^-23
With zr = re = 2.8179403250.10^-15 m,
z = re/r = 1,7435592744.10^20 = (4.1755948971.10^10)^2
Then y = [(10α)^ 1/3]/(9, 98451148382.10^21)
and z = [(10α)^2/3].(9.9845040300.10^20)
from which
me = ym ≈ m(10^-22)[(10α)^ 1/3] =m(α/10^65)^ 1/3
and
re = zr ≈ 10r(α.10^31)^2/3

arivero

In order to correct a bit the distorsion introduced by hareyvo (please, guys, do your homework and read the old threads before posting. Ah, and use your blog part if you do not aim for general discussion), let me stress again what the fermiophobic higgs should mean for ALL the low-energy approach to masses: basically that the field becomes open, because we should have experimental evidence of the nullity of the yukawa coupling for particles lighter than the Higgs itself.

Actually, it is a bit of complex, as it also means that Higgs production has smaller rates than the SM. And the current scenario does not tell anything about the top yukawa coupling, as it is negligible as a channel for observation (if the Higgs is at 125 GeV) and surely (can someone confirm?) also as a production channel -we need to produce a top and then collide it again-.

mitchell porter

A reminder of why the Koide relation, and its generalizations reported at the start of this thread, are challenging: a short paper from India lists the fermion masses at M_Z scale and at GUT scale in various theories (SM, SM + extra higgs doublet, MSSM). Of course, the masses are different at GUT scale, often very different, and yet that is supposed to be where symmetries are more manifest.

The world of QFT (and strings) contains many unexpected equivalencies between different-looking pictures of the same physics. It may be that Koide relations won't really be understood without switching to a "UV/IR-dual" picture in which the IR looks simple and is somehow the starting point for the theory. Since you drop degrees of freedom in the RG flow from high energies to low, that sounds unlikely - the IR just doesn't have the information needed to reconstruct the UV.

But in string theory we already have various constraining relationships between IR and UV properties. So perhaps for the right sort of theory, we can find a new picture, in which the UV can be completely reconstructed from IR + "something else". Somehow, we want the new heavy degrees of freedom to enter at higher energies, yet the way in which they do so is constrained or foreshadowed or otherwise allows deep and nonaccidental relationships between IR quantities.

mitchell porter

While considering how to produce the "cascade" or "waterfall" of Koide relations for quark masses that Alejandro discovered, I have thought in terms of a range of possible multihiggs models. At one extreme, you have many higgses and they all contribute to the masses of all the fermions (Adler's circulant models, mentioned in comment #14 in this thread), and the cascade structure comes from a complicated inter-scalar potential. At the other extreme, you basically have one distinct higgs per fermion - again, the cascade must come from the potential, but you would also have to have the scalars and fermions appropriately charged under special discrete symmetries, to enforce "one higgs" or "few higgses" per fermion.

Today Bentov and Zee have a paper about the second scenario. I hasten to add that they don't talk about Koide at all, but you can see from their work something of how it would go. In the standard model, there's just one higgs VEV, and the fermion mass spectrum comes from the yukawas - O(1) for the top, much smaller for the other fermions. In these so-called "Private Higgs" models (every fermion has its own private higgs), all the yukawas are O(1), and it's the VEVs which produce the cascade!

My guess is that a model like this can produce the full Koide cascade at tree level, but that loop corrections should spoil the exactness of the Koide relation. But we won't know for sure, until someone tries to make it work...

edit: Other papers to read: Private Higgs, Private Higgs for leptons, and paper by Ernest Ma which explains Koide using the same discrete symmetry group.

arivero

Zee is the last of the big phenomenologists.

Let me remember that the first paper which actually brought a Koide equality (albeit with one of the three masses equal a zero), Harari Haut Weyers, was critiquized because its permuting of exchanging left and right quarks across generatations was really a way to present a complicated Higgs structure. Surely the same criticism applies to any other multiple Higgs ideas, but the escape comes if, as it happens in the Koide waterfall, it is always the same kind of step along all the ladder.

By the way, I have noticed that pdg has moved again their evaluation of the mass of the top, now it has the central value at 173.5 ± 0.6 ± 0.8 GeV, so near of the postdiction of the ascending waterfall of vixra:1111.0062v2/arxiv:1111.7232, which is 173.263947(6)

Edit: if we think of an unperturbed Koide, the main problem is that setting electron to zero but keeping the "QCD" mass to 313 GeV gives a slightly higher value for the top, namely 180 GeV. Of course we could scale everything down and set top to the electroweak vacuum, then the unperturbed levels should be 174.10 GeV (top), 3.64 GeV (bottom), 1.70 GeV (charm,tau), 121.9 MeV (strange, muon), 0 eV (up, electron), 8.75 MeV (down). I am not sure that I like it, but it has the merit of using a single input, the Fermi constant - to produce the initial seed of 174.10-. On other hand, Koide triples are a lot of quadratic equations, and surely there are more solutions also producing the 0 eV up quark; this one must impose also the extra condition of being monotonic, always descending, from top to up.

arivero

But their Higgses are proportional to the square root of the mass of each fermion. Perhaps the authors have got the wind.

arivero

And now the final evaluation of Tevatron moves it to 173.18 GeV! So since the upload of the paper, the difference has evolved from .36 to .24 and now to .08. The personal combination from Déliot for TeV-LHC is a bit lower, down to 173.1, but lets see how it evolves towards the pdg.

It is also intriguing that the only really wrong mass is the one of the charm quark, where they are finding some stress against the standard model (in CP violating decays).

To be sure, let me quote the table from the preprint, adding the current known (MS scheme) values of quark masses. Reminder, the only inputs are me = 0.510998910 and mu= 105.6583668 and only assumptions are Mq = 3Ml and q = 3l (quasi-orthogonality quarks/leptons). All the rest is to repeat Koide for each triple.

arivero

I have set up a prezi presentation in the Koide ladder (or waterfall)

http://prezi.com/e2hba7tkygvj/koide-waterfall/

feel free to disseminate

arivero

It is possible to use only the mass of the top, or the electoweak vacuum, and ask for a Koide waterfall chaining solutions until we arrive to a mass of the top equal to zero. There are five such chains, only three of them are actually "falls", and of those only one uses always the same solution of the Koide equation (see my paper, or this thread above). The waterfall is:

Note that the last triplet is even older than Koide, from Harari et al.

This descent uses only one input, Fermi scale, and the mases of c and s are even near of tau and muon that in the descent with two inputs. It supports then the idea of an unperturbed spectrum, where charged leptons are degenerated with some quarks, and then a perturbations that somehow commutes with the cause of Koide.