The Koide formula explained by flavour mixing in a Weyl 2-spinor, Schroedinger ‘Zitterbewegung’ lepton

Above: Fig. 25.13 on p. 644 of the 2004 Cape edition of Penrose’s Road to Reality. Caption reads: ‘In the zigzag picture of a Dirac particle, the vertices may be viewed as interactions with the (constant) Higgs field.’ Because the mass of the particle is acquired as it interacts with the constant mass vacuum field quanta at the vertices (some kind of mass-producing field, not necessarily any particular speculative Higgs boson, which has never been observed), it follows that the ‘coupling constant’ for the interaction must be different where the resultant particles are different in mass: the coupling per vertice (there are two vertices needed for each complete cycle of de Broglie wave oscillation as the particle moves in the zigzag motion) is a square root factor which apparently explains the Koide formula for leptons, including Carl Brannen’s modification for neutrino masses. This results from the decomposition of Dirac’s spinor into a 2-spinor form by Weyl in 1929, where one component of the spinor is left handed and the other right handed. A truly massless neutrino would be an entirely left-handed spinor, like just the zig part of the zigzag motion of an electron. But if they have a small mass an can change flavour (as observed for solar neutrinos), neutrinos must occasionally interact with a massive vacuum field and therefore have a very small zag component.

Let the masses of the electron, muon and tauon be Me, Mm, and Mt, respectively. Koide’s formula in its usual presentation is then:

(Me1/2 + Mm1/2 + Mt1/2)2/(Me + Mm + Mt) = 3/2

Multiplying out the squared numerator, simplifying and rearranging gives (see appendix B at the end of this post for detailed steps):

Me + Mm + Mt = 4[(MeMm)1/2 + (MeMt)1/2 + (MmMt)1/2]

To further simplify the Koide formula, remember from the law of indices that any mass can be represented like: Me = (MeMe)1/2. The Koide formula can then be seen to consist entirely of a summation of terms of the form (MaMb)1/2, some positive and some negative.

Physically, the Koide formula in its original form presumably is high-energy averaging for all flavours of leptons (electron, muon and tauon) as suggested in a previous post (which considered a different explanation for the square roots, however): at high enough energy (in either collisions or simply in a strong enough static field very close to the real core of a fundamental particle), all three flavours of leptons can spontaneously occur as virtual fermions in the vacuum due to pair-production. The original Koide formula seems to apply to the averaging of the masses of leptons at high energy when they can all briefly arise as virtual particles formed from preon interations in the vacuum.


The Dirac spinor was first resolved into 2-spinors by Weyl in 1929, and the relevance here is that, as Penrose shows (Road to Reality, 2004, pp. 629-30):

‘The Dirac equation can then be written as an equation coupling these two spinors, each acting as a kind of “source” for the other, with a “coupling constant” 2-1/2M describing the strength of the “interaction” between the two. …

‘From the form of these equations, we see that the Dirac electron can be thought of as being composed of two ingredients … It is possible to obtain a kind of physical interpretation of these ingredients. We form a picture in which there are two “particles”, one described by alphaA and the other by betaA’, each of which is massless, and where each one is continually converting into the other one. Let us call these the “zig” particle and the “zag” particle … this is a realization of the phenomenon referred to as “zitterbewegung“, according to which, the electron’s instantaneous motion is always to be the speed of light, owing to the electron’s jiggling motion, even though the overall averaged motion of the electron is less than light speed. Each ingredient has a spin about its direction of motion, of magnitude h-bar/2, where the spin is left-handed in the case of the zig and right-handed for the zag. …

‘In this interpretation, the zig particle acts as the source for the zag particle and the zag particle as a source for the zig particle, the coupling strength being determined by M.’

This zig-zag motion of the electron is a kind of oscillation as it propagates, as Penrose explains on p. 630:

‘… we find that the average rate at which this [zig-zagging] happens is (reciprocally) related to the mass coupling parameter M; in fact, this rate is essentially the de Broglie frequency of the electron.’

Hence, the electron (and leptons generally) have two components which drive one another cyclically with a mass coupling constant of 2-1/2. Each different lepton has a different mass, so the zig-zag amplitude will vary for electrons, muons, and tauons, each being a square root factor: the product of the two square root terms for the two components gives us the overall mass of the lepton of interest. The interaction of zigs and zags is a W-shape on a Feynman diagram, and according to Feynman’s rules (very nicely explained with clear examples in the 2008 book Quantum Field Theory Demystified) the amplitude for an interaction is simply the product of the various coupling constants and propagators involved, integrated in such a way as to conserve momentum for the interaction.

Now, at each zig and zag vertex in the W-shaped Feynman diagram for a lepton’s Zitterbewegung motion, the electron is actually interacting with a vacuum field particle which gives it mass, according to Figure 25.13 on page 644 of Penrose’s 2004 Road to Reality:

‘In the zig-zag picture of a Dirac particle, the vertices may be viewed as interactions with the (constant) Higgs field.’

This is how the mass of leptons mass arises, according to Sir Roger Penrose! My argument is that the square root products in the expanded Koide formula arise because the mass of a lepton is a composite of the product of two square roots of lepton masses (including neutrino masses, since Carl Brannen has shown that the Koide formula with a slight modification – a minus sign to the lightest neutrino mass square root term instead of a positive sign – fits the neutrino mass data very well); if mixing is allowed for massive particles as like electrons, muons and tauons, as occurs with other leptons (neutrinos) you get three distinct ways that the electron, muon and tauon mass square roots can mix:

1. electron-muon
2. muon-tauon
3. tauon-electron

We know from observations that some leptons at least, neutrinos, can mix varieties spontaneously while they are propagating: there are three flavours of neutrinos, only 33% of the flavour generated in the sun by fusion is detected here on earth, hence that neutrino has mixed flavours equally between all 3 flavours while on the way to the detector on the earth. In this way, the number of neutrinos of the original flavour has fallen by the factor 3 to just 33% as observed, and 67% are present as other flavours which the detector here (a larger underground swimming pool filled of carbon tetrachloride and a big scintillation photomultiplier) is simply unable to detect.

What happens with the electron, muon and tauon in Koide’s formula is simply that the sum of their masses is directly proportional to the sum of the products of all combinations of the square roots. The expanded Koide formula:

Me + Mm + Mt = 4[(MeMm)1/2 + (MeMt)1/2 + (MmMt)1/2]

shows that Me + Mm + Mt is equal to the sum of 4*3 = 12 separate terms, i.e., each of the masses is represented by 4 terms each. If we look at the right hand side of the equation above, the term (MmMt)1/2 is the biggest, (MeMt)1/2 is second biggest, and (MeMm)1/2 is the smallest.

It is probable that the Koide formula is a gross manifestation of something much deeper, because if we look at individual particle masses, we can’t get the electron mass by multiplying the square roots of two massive lepton masses together, unless maybe one of the masses is a neutrino mass and the other one is the muon or tauon mass.

The reason for bringing neutrinos in here is that Penrose states that neutrinos are equivalent to the ‘zig’ part of an electron, which makes them left handed (neutrinos are only left-handed in the Standard Model to make the weak interaction purely left-handed as experimentally observed; neutrinos interact with Z bosons in weak interactions). But because neutrinos have a very small mass, they can’t be entirely left-handed ‘zig’ particles and need to interact with a massive vacuum field occasionally in order to acquire mass so they can change flavour i.e. to ‘oscillate’ between electron neutrinos, muon neutrinos and tauon neutrinos. (Penrose, 2004, Figure 25.9.) Without mass, neutrinos would go at velocity c only and thus would be frozen and unable to oscillate; having a very small mass reduces their velocity to just under c and allows them to gradually change flavour on the 8.3 minute journey from the sun to the earth.

This zig neutrino idea is funny because it will force a change to the Standard Model: the neutrino is not 100% left-handed; it is very slightly right-handed in order to explain its mass. The small proportion of right-handedness presumably will be linked to the ratio by which matter outweighs antimatter in the universe; because it is the left-handed weak interaction which allows downquarks (in neutrons) to decay into upquarks (allowing neutrons to decay into protons), but not vice-versa (nobody has detected a proton decaying). The asymmetry of left-handedness of weak interactions is clearly tied into the asymmetry of matter over antimatter in the universe from the first fraction of a second onward.


The underlying mechanism for the square roots of mass in the Koide formula is linked to the Weyl 2-spinor (left and right handed spinors) using Schroedinger’s ‘Zitterbewegung’ lepton as discussed by Penrose, Road to Reality, 2004. See Penrose’s Figure 25.13 for a Feynman diagram showing how two components of a lepton interact to produce the de Broglie oscillation of a moving particle. They acquire mass at the interaction vertex. The Zitterbewegung vertex coupling constant for the interaction is a square root factor, because you square the momentum integrated amplitude of coupling constants and propagators for a Feynman diagram to get the resultant probability or reaction rate.

Because Zitterbewegung involves interactions between two components, a zig and a zag, in a lepton, you need two interaction vertices in each complete oscillatory cycle of the particle as it propagates, and the coupling constants multiply together to give the amplitude according to Feynman’s rules; this provides the seed for an explanation of the square roots in the Koide formula. Since any lepton is acquiring mass from the same vacuum at vertices, it follows that to explain the variety of lepton masses that exist in nature, the Zitterbewegung vertex coupling constants must be proportional to the square root of the mass of the zig and zag components of a Zitterbewegung lepton. We know that neutrinos oscillate in flavour uniformly between three flavours while coming to us from the sun, which is why we detect just one third of the total (the third which have the flavour that our discriminate detector is searching for). If we extend this idea so that the zig and zag components of leptons in general, the masses of leptons will be represented by the sum of products of pairs of square roots of masses of different leptons. Although Koide’s original formula only applied to the masses of the electron, muon and tauon, Brannen has extended it with a modification to include neutrino masses.

The question now is to see if this mechanism for the Koide formula can be tied firmly into Carl Brannen’s application of the Koide formula to neutrino masses, mentioned in an earlier post.

One possibility is that the

Appendix A: History of Zitterbewegung (quoted from Wikipedia)

‘The existence of such motion was first proposed by Erwin Schrödinger in 1930 as a result of his analysis of the wave packet solutions of the Dirac equation for relativistic electrons in free space, in which an interference between positive and negative energy states produces what appears to be a fluctuation (at the speed of light) of the position of an electron around the median, with a circular frequency of , or approximately 1.6 × 1021 Hz.’

Appendix B: rewriting the Koide formula

Koide formula: (Me1/2 + Mm1/2 + Mt1/2)2/(Me + Mm + Mt) = 3/2.


2(Me1/2 + Mm1/2 + Mt1/2)2

= 3(Me + Mm + Mt).

Expand 2(Me1/2 + Mm1/2 + Mt1/2)2:

2(Me1/2 + Mm1/2 + Mt1/2)2

= 2[Me + Mm + Mt + 2{(MeMm)1/2 + (MeMt)1/2 + (MtMm)1/2}]

= 2Me + 2Mm + 2Mt + 4{(MeMm)1/2 + (MeMt)1/2 + (MtMm)1/2}

Which equals 3(Me + Mm + Mt).


Me + Mm + Mt = 4[(MeMm)1/2 + (MeMt)1/2 + (MmMt)1/2].

To further simplify the Koide formula, remember from the law of indices that for example Me = (MeMe)1/2, so Me + Mm + Mt = 4[(MeMm)1/2 + (MeMt)1/2 + (MmMt)1/2] can be written as:

(MeMe)1/2 + (MmMm)1/2 + (MtMt)1/2 = 4[(MeMm)1/2 + (MeMt)1/2 + (MmMt)1/2],

which may help.

Update (7 December 2011):

“Then Koide went some steps beyond and considered quarks and leptons with substructure, so that lepton mass quotients could predict the Cabibbo angle too, even if this is a mixing between quarks.”

{(sqrt(M_e)+sqrt(M_mu)+sqrt(M_tau))^2} /( M_e + M_mu +M_tau) = 2/3

The key factor of 2/3 in the Koide relationship is the fractional electric charge of the up/charm/truth quarks, which arises from a mixing effect. It’s the 2/3 electric charge of up/charm/truth quarks that’s so interesting. The -1/3 charge of the down/strange/bottom quarks is very easily predicted by analysis of vacuum polarization for the case of the omega minus baryon (Fig. 31 in ). It appears that the square root of the product of two very different masses gives rise to an intermediate mass (see for the maths) that the Koide relationship implies a bootstrap model of fundamental particles (akin to the bootstrap concept Geoffrey Chew was trying to develop to explain the S-matrix in the 1960s before quarks were discovered). The square root of the product of the masses of a neutrino and a massive weak boson may give an electron mass, for instance. This seems to be the deeper significance of the Koide formula, from my perspective for what it’s worth. All fundamental particles are connected by various offshell field quanta exchanges, so their “charges” are dependent on other charges around them. This means that the ordinary approach of analysis fails, because of the reductionist fallacy. If your mathematical model of rope is the same for 100 one-foot lengths as for a single 100 foot length, it leads to customer complaints when you automatically send a sailor the former, not the latter. It’s no good patiently explaining to the sailor that mathematically they are identical, and the universe is mathematical. If the Koide formula is correct, then it points to an extension of the square root nature of the Dirac equation. Dirac made the error of ignoring Maxwell’s 1861 paper on magnetic force mechanisms: the chiral handedness of magnetism (the magnetic field curls left-handed around the direction of propagation of an electron) is explained in Maxwell’s theory by the spin of “field quanta” (Maxwell had gear cogs, but in QFT it’s just the spin angular momentum of field quanta). Maxwell’s theory makes EM an SU(2) Yang-Mills theory, throwing a different light on the Dirac’s spinor. It just so happens that the Yang-Mills equations automatically reduce to Maxwell’s if the field quanta are massless, because of the infinite self-inductance of electrically charged field quanta, so SU(2) Maxwellian electromagnetism in practice looks indistinguishable from Abelian U(1), explaining the delusions in modern physics.

The very interesting results Alejandro Rivero’s gives are from equation 4 on page 3 of his paper, which solves the Koide formula by writing one mass in terms of the two lepton other generation masses. Koide’s formula also implies (my 2009 post):

Me + Mm + Mt = 4 * [(Me * Mm)^1/2 + (Me * Mt)^1/2 + (Mm * Mt)^1/2]

where Me = electron mass, Mm = muon mass, Mt = tauon mass. I.e., the simple sum of lepton masses equals four times the sum of square roots of the products of all combinations of the masses, making it seem that if Koide’s formula is physically meaningful, then Geoffrey Chew’s bootstrap theory of particle democracy must apply to masses (gravitational charge) in 4-d. At high energy, early in the universe, tauons, muons and electrons were all represented and we only see an excess of electrons today because the other generations have decayed, although some of the other masses may actually exist as dark matter, and thus still undergoes the interaction of graviton exchange, which determines the Koide mass spectrum today (this dark matter is analogous to right-handed neutrinos). The basic physics of the Koide formula seems to be the Chew bootstrap applied to gravitation (Chew applied it to the strong force, pre-QCD):

“By the end of the 1950s, [Geoffrey] Chew was calling this [analytic development of Heisenberg’s empirical scattering or S-matrix] the bootstrap philosophy. Because of analyticity, each particle’s interactions with all others would somehow determine its own basic properties and … the whole theory would somehow ‘pull itself up by its own bootstraps’.” – Peter Woit, Not Even Wrong, Jonathan Cape, London, 2006, p148. (Emphasis added.)

The S-matrix went out when the SM was developed (although S-matrix results were used to help determine the Feynman rules), but at some stage a Chew-type bootstrap mechanism for Koide’s mass formula may be needed to further develop a physical understanding for the underlying theory of mass mixing, leading to a full theory of mixing angles for both gravitation (mass) and weak SU(2) interactions of leptons and quarks.


3 thoughts on “The Koide formula explained by flavour mixing in a Weyl 2-spinor, Schroedinger ‘Zitterbewegung’ lepton

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