What is new with Koide sum rules?

[fzero]

The idea seems to be that the Koide relation holds exactly at high energies, and it also holds for the pole masses, because the corrections due to the family gauge bosons cancel the QED corrections for each charged lepton, at its own mass scale. Above that scale, the mass will just run normally as in the SM, until the scale where electroweak unifies with the family force (100s or 1000s of TEVs), at which point the Koide relation becomes manifest again.

But I'm just telling you how I think it's supposed to work, I'm still getting my head around the details.

Let's denote the pole masses by m_i(m_i). The Koide result is that

\frac{ \sqrt{m_e(m_e)} + \sqrt{m_\mu(m_\mu)} +\sqrt{m_\tau(m_\tau)}}{\sqrt{m_e(m_e) + m_\mu(m_\mu) +m_\tau(m_\tau) }} = \sqrt{\frac{3}{2} } \pm 10^{-5} .

Now m_i(E) definitely runs with energy and we know this because it's been measured. What I understood is that, in Sumino's model, the one-loop corrections to

r(E) = \frac{ \sqrt{m_e(E)} + \sqrt{m_\mu(E)} +\sqrt{m_\tau(E)}}{\sqrt{m_e(E) + m_\mu(E) +m_\tau(E) }}

cancel.

However, we know from running the pole masses in the first relation that r(E) differs from \sqrt{3/2} by one part in 10^{-3} (stated below eq (2) in Sumino). Now, since m_e(m_e)/m_\tau(m_\tau) \sim 3 \cdot 10^{-4}, at this level of precision, we might as well just drop the terms with m_e from r(E). The Koide relation really doesn't convincingly extend to the electron and is just some numerology involving \mu and \tau. The situation for the up quarks is even worse since m_u/m_t is much, much smaller than the experimental uncertainty in the top mass.

 

[mitchell porter]

Now m_i(E) definitely runs with energy and we know this because it's been measured.

I think the point is that in a theory with "Koide symmetry" (i.e. whatever it is that produces the Koide relation) but not "Sumino family symmetry", the m_i(m_i) start out at a set of values which don't satisfy Koide symmetry. The additional Sumino family symmetry adjusts the RG trajectory so that the pole masses do satisfy Koide symmetry. But that doesn't mean that Koide symmetry is exact for the running masses at low energies; it only becomes exact above the family-symmetry unification scale. The masses do not satisfy the symmetry at any single value of E below that scale; but the relation happens to hold for the pole masses at their different scales.

I still haven't verified this! But I believe this is how it's supposed to work.

 

[arivero]

corrections between m_e and m_W

By the way, according for instance table IV of http://arxiv.org/pdf/hep-ph/0601031v2 or section F of http://prd.aps.org/abstract/PRD/v46/i9/p3945_1 (where, note, the wrong measured value for tau is still used), all the damage to Koide relation for leptons is done already when moving electron and muon up to the GeV scale. From 1 GeV up to any high energy (without GUT), the mismatch keeps about 1.0017 - 1.0019, i.e. about the 0.2% of "error".

I wonder, is there some context where pole masses are more relevant that running masses? For instance, when we compare two masses to decide if the particle A can decay to particle B, are we supposed to compare pole masses, or to run the mass of B to the A scale, or run the mass of A to the B scale? I'd expect the two later procedures to be equivalent, but given that stability is about the total balance of energy, perhaps the former procedure is more relevant.

 

[fzero]

I think the point is that in a theory with "Koide symmetry" (i.e. whatever it is that produces the Koide relation) but not "Sumino family symmetry", the m_i(m_i) start out at a set of values which don't satisfy Koide symmetry. The additional Sumino family symmetry adjusts the RG trajectory so that the pole masses do satisfy Koide symmetry. But that doesn't mean that Koide symmetry is exact for the running masses at low energies; it only becomes exact above the family-symmetry unification scale. The masses do not satisfy the symmetry at any single value of E below that scale; but the relation happens to hold for the pole masses at their different scales.

I still haven't verified this! But I believe this is how it's supposed to work.

After reading a bit more, I see that the paper is saying that he can engineer r(\Lambda) to have the right value and this is treated as an initial condition for the EFT. He makes remarks saying that the running of r(\mu) is formally protected but notes that the physical argument breaks down below \Lambda.

The troubling part is that he seems to be pushing for some running of masses below this scale that is not at all like what actually happens. We know that the ratios of the pole masses are not the same as the ratios of the running masses at observable energies. There's no calculation in the paper that uses real physics to explain how to get from r(\mu) to the corresponding ratio of pole masses. It still seems that he wants to fix the masses in r(\Lambda) to match the ratios between the pole masses. This is precisely what I'm saying is completely unphysical.

By the way, according for instance table IV of http://arxiv.org/pdf/hep-ph/0601031v2 or section F of http://prd.aps.org/abstract/PRD/v46/i9/p3945_1 (where, note, the wrong measured value for tau is still used), all the damage to Koide relation for leptons is done already when moving electron and muon up to the GeV scale. From 1 GeV up to any high energy (without GUT), the mismatch keeps about 1.0017 - 1.0019, i.e. about the 0.2% of "error".

Assuming that the calculations in the paper are correct, this is very useful to illustrate my point. However the authors obviously reach the wrong conclusions. They seem to think that k-1 being different from zero at a larger degree than the ratio m_1/m_3 or the experimental uncertainty \Delta m_3/m_3 still implies that "Koide's relation is a universal result." This is not a scientific conclusion, we require a higher standard.

I wonder, is there some context where pole masses are more relevant that running masses? For instance, when we compare two masses to decide if the particle A can decay to particle B, are we supposed to compare pole masses, or to run the mass of B to the A scale, or run the mass of A to the B scale? I'd expect the two later procedures to be equivalent, but given that stability is about the total balance of energy, perhaps the former procedure is more relevant.

The pole masses are obviously the right ones in processes such as particle production at threshold. For decay processes, I believe the right criterion is that the process has to make sense in the rest frame of the decaying particle. Therefore the pole mass of A is the right one to use and any running of the B mass is a small contribution to the kinematics of the final state.

 

[arivero]

The pole masses are obviously the right ones in processes such as particle production at threshold. For decay processes, I believe the right criterion is that the process has to make sense in the rest frame of the decaying particle. Therefore the pole mass of A is the right one to use and any running of the B mass is a small contribution to the kinematics of the final state.

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]

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.

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 105.6583668 - 0.510998910, i.e, m_\mu(m_\mu) - m_e(m_e) (?), or is it m_\mu(m_\mu) - m_e(m_\mu)? 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:

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

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]

Today Bentov and Zee have a paper about the second scenario.

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]

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

 

[arivero]

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)

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 let's 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.

        | prediction          |  (pdg 2012)
========+=====================+===================
tau     | 1776.96894(7) MeV   | 1776.82 ± 0.16 GeV
strange | 92.274758(3) MeV    | 95 ± 5 MeV ev ([URL="http://pdglive.lbl.gov/ideograms/Q123SM.png"]ideogram[/URL] 94.3±1.2)
charm   | 1359.56428(5) MeV   | 1.275 ± 0.025 GeV   
bottom  | 4197.57589(15) MeV  | 4.18 ± 0.03 GeV
top     | 173.263947(6) GeV   | 173.5 ± 0.6 ± 0.8 GeV | 173.18±0.94 GeV (Tevatron arxiv:1207.1069)

 

[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:

t:174.10 GeV--> b:3.64 GeV---> c:1.698 GeV --> s:121.95 MeV ---> u:0 ---> d:8.75 KeV

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.

 

[MTd2]

The up quark is massless?

 

[arivero]

The up quark is massless?

Indeed

There are, with some variants, two main arguments here.

- You can consider it pragmatically, that we are just looking for solutions of the Waterfall that happen to produce a small mass for the quark up, say 3 KeV, starting from 175 GeV. That is six orders of magnitude, and it is a convenient device to count the zero mass solutions and then, barring catastrophes, do all the numerical search from it. It should work the same for the down quark in this case.

- Or, you can take seriously the requirement in order to avoid the theta problem of QCD, and claim that the mass of the up is really zero and its measured mass is of a secondary nature, so that really for s,d,u the masses are modified following
m'_x=m_x + {m_y m_z \over M} + ...
with M coming from QCD. Note that if the up is really massless at the point where you apply the formula, then s and d are not modifyed in first order.

 

[MTd2]

What do you mean by " its measured mass is of a secondary nature"? I don't understand how you hid its mass.

 

[mitchell porter]

I have been looking into "quark-hadron duality" for an approach to the Koide waterfall that I'm not yet ready to explain. But I have to point out something I just found in the literature - in "The origins of quark-hadron duality" by Close and Isgur: that one manifestation of this duality, is that the same formula can be expressed as the square of a sum or as a sum of squares - see page 4. Doesn't that sound like the Koide formula? - with the "square roots of the masses" as the basic quantities that you sum or that you square.

 

[arivero]

What do you mean by " its measured mass is of a secondary nature"? I don't understand how you hid its mass.

Where is the problem, exactly? I put the mass of the up equal to zero and then I use an expansion to produce a final mass, this is a very usual recipe. The problem is that M is an interaction scale which comes from the chiral scale of QCD, so it is not fundamental in the Koide waterfall, hence the name of "secondary"

BTW, I mean MeV, no KeV, of course.

 

[arivero]

I have been looking into "quark-hadron duality" for an approach to the Koide waterfall that I'm not yet ready to explain. But I have to point out something I just found in the literature - in "The origins of quark-hadron duality" by Close and Isgur: that one manifestation of this duality, is that the same formula can be expressed as the square of a sum or as a sum of squares - see page 4. Doesn't that sound like the Koide formula? - with the "square roots of the masses" as the basic quantities that you sum or that you square.

I am not sure. Koide seems about finite sequences, mostly triples, and duality is about sums over all the states.

 

[arivero]

t:174.10 GeV--> b:3.64 GeV---> c:1.698 GeV --> s:121.95 MeV ---> u:0 ---> d:8.75 MeV

Just for the record, this is the only waterfall with a sensible value of t/b for five steps to zero. If we aim for six steps, there is (only) other solution, rather more peculiar:

t:174.10 GeV-->c:1.859 GeV-->b:3.401 GeV-->s:132.23 MeV-->d:9.49 MeV --->u:0

Really it uses the same triples uds, scb and cbt, but instead of csu to land into the zero, it calls for bsd.

There is not monotonic solution of six steps compatible with t/b "desert" and crossing zero. And this two are really the only six step series with such compatibility. Of these, the former survives better when going to experimental values.

 

[MTd2]

Where is the problem, exactly?

Suppose the meson rho0 or omega0, where both are u anti u, where does it get its mass, from gluons only?

 

[arivero]

Suppose the meson rho0 or omega0, where both are u anti u, where does it get its mass, from gluons only?

Indeed, and this is true for most of the low mass pions. I am not conversant in QCD, but if you open a thread on the topic in the SM subforum, I will try to follow it.

The main use of a massless up quark is to solve the strong CP problem (again, a topic where someone in the SM subforum can be more conversant than me) Here you can see
http://arxiv.org/abs/hep-ph/9403203
to Banks, Nir, and Seiberg telling that they do not believe that the controversy has solved. Since then, data from lattice show up quark with a mass different of zero, but again it is not clear if they are already accounting for some QCD trick (such as the one I told above).