## Monday, January 16, 2012 ... /////

### Could the Koide formula be real?

Warning: the following line contains spoilers
Nope.

That was the compactified version of the article. Below you may find the decompactified one.

The page containing this picture explains how to associate numbers 0-9 with planets and how to remember the author's telephone number. Also, it tells you that numerology is for the people who find astrology too scientific. ;-)

In 1981, a man named Yoshio Koide has made a bizarre observation known as the Koide formula. It has apparently been a source of life energy for more than 30 years for our friends such as Mitchell Porter, Carl Brannen, Alejandro Rivero, and maybe others.

Consider the three physical masses of the charged leptons, the electron, the muon, and the tau lepton:

\begin{align} m_e c^2 &= 0.510998910(13)\,{\rm MeV}\\ m_\mu c^2 &= 105.658367(4) \,{\rm MeV}\\ m_\tau c^2 &= 1,776.84(17) \,{\rm MeV} \end{align}
The digits in the parentheses indicate the error margin. Now, spend a few hours by trying to find some interesting numerical values of various ratios. You won't find any. After some time, you define this rather contrived – but not too contrived – ratio:
$Q = \frac{m_e+m_\mu+m_\tau}{(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau})^2} \approx \frac 23$
More precisely, the numerical value is
$Q = 0.666659(10)$ Within less than one standard deviation, the value of $Q$ agrees with $2/3$. The error margin is $10^{-5}$ but to get the relative one, the absolute one should be divided by $2/3$ so the relative error is $1.5\times 10^{-5}$. Let us describe the agreement as an agreement of "almost five significant figures". Almost five digits match.

Numerological aspects of the identity

We may first discuss some aspects of the identity that are seen from the viewpoint of someone who knows nothing about physics and/or the origin of masses etc.: remarkably enough, this viewpoint seems almost identical to Mitchell Porter's viewpoint. ;-)

First, notice that $Q$ is dimensionless and only depends on the mass ratios. If you changed the units or doubled the values of all the three masses, $Q$ wouldn't change. So $Q$ is a function of $206.76827$ and $3477.1893$, the muon-electron and tau-electron mass ratios, and nothing else.

Second, ask the question: what are the a priori possible values of $Q$? Assuming that the masses are three positive numbers, you may obtain a maximum ratio $Q=1$ if two of the masses vanish (or are much smaller than the third one). In that case, $Q=1/1=1$. On the other hand, $Q=1/3$ if the three masses are equal: $Q=3/3^2=1/3$.

Great. So the actual value of $Q=2/3$ is exactly in between the two extreme limits. You may say that this makes the value $Q=2/3$ very special which should make us more excited. But you may also say something very different: you may say that the probability distribution for different values of $Q$ is almost certainly peaked somewhere in the middle of the interval $[1/3,1]$, so getting a number very close to the middle is actually less surprising.

If you take this fact into account, the degree of surprise will drop to the equivalent of "four matching digits".

You may also express the statement $Q=2/3$ differently, using the angle between two 3-dimensional vectors:
${\rm angle} ( (1,1,1), (\sqrt{m_e},\sqrt{m_\mu},\sqrt{m_\tau}) ) = 45°$
In this case, 45° isn't really in the middle because the maximum angle is $0.304\pi$, not $0.5\pi$, which is the angle between $(1,1,1)$ and $(1,0,0)$.

The Koide formula for $Q$ has an error margin that almost entirely boils down to the value of $m_\tau$ which is not known too accurately. Should you believe that $Q=2/3$ holds accurately which would mean that you may literally calculate several additional digits in $m_\tau$? Well, just like at the beginning, the answer is Nope.

I will eventually jump to the full-fledged physics discussion but before I do so, let us add a purely mathematical improvement. A problem with the precise calculation of $Q$ so far is that the masses in Nature are actually not real numbers. They're naturally complex numbers, with the imaginary parts' being the width (the inverse lifetime, in the $\hbar=c=1$ units):
$M_{\rm total} = M_{\rm real} - \frac{i\Gamma}{2}$
where $\Gamma=1/\tau$ is the inverse lifetime of the particle, the width. Why is there the factor of $1/2$ over there? Well, go to the rest frame and appreciate that the wave function goes like
$\psi(t)\sim \exp(-iM_{\rm total}t ),\qquad |\psi(t)|^2 \sim \exp(-\Gamma t).$ The probability is the squared wave function, the factor of $1/2$ doubles, and you get the right exponentially decreasing probability $\exp(-\Gamma t) = \exp(-t/\tau)$ that you expect from the lifetime $\tau$. The electron is stable but if we want to be analytic and accurate, we must add the corresponding imaginary values $1/2\tau$ to the real numbers described above. If you know the lifetimes, the masses become
\begin{align} m_e c^2 &= 0.510998910(13)\,{\rm MeV}\\ m_\mu c^2 &= (105.658367(4)-9.41 \times 10^{-16}) \,{\rm MeV}\\ m_\tau c^2 &= (1,776.84(17)-7.13\times 10^{-9}) \,{\rm MeV} \end{align}
These are the actual locations of the poles you should use; treating the real parts separately would be an extremely unnatural thing to do. So is $Q=2/3$ within the observable error margins even if we incorporate the nonzero imaginary parts?

The ratio of the real parts of the numerator and denominator of $Q$ is $2/3$. We want the ratios of the imaginary parts to be $2/3$ as well. However, we will quickly see that it doesn't work. The imaginary parts are only affected by the imaginary part of $m_\tau$ because only the tau lepton has a significant width. And you may easily check that whatever the imaginary part of $m_\tau$ actually is, the ratio of the imaginary part of the numerator and denominator of $Q$ is actually $0.79$, safely different from $0.66666$.

If we write the most accurate values of the masses, including the imaginary parts, the Koide formula is safely broken. Now, claiming that something should hold "totally exactly" but you should only be using the real values is an extremely bizarre and unnatural statement. In fact, numbers of the form $Z = M-i\Gamma/2$ which are "almost real" can be made "exactly real" in many ways. You may take the real part. But you may also take the absolute value. And you may do lots of other things. In all cases, you get something similar to the real part but none of the rules is more motivated than others and different rules will imply different predictions about the more accurate digits. No doubt, some of the rules how to deal with the nonzero width could produce better results for the (real) Koide formula but the defenders of the Koide formula don't know which one it should be.

I am extremely open-minded so I could imagine that there could be shocking relationships between things that shouldn't be related in simple ways, for reasons to be discussed below. But the failure of the formula to work when the imaginary parts are incorporated closes this question for me even at the level at which I "overlook all of my knowledge of physics".

When we actually take our knowledge of particle physics into account, the idea that a similar bizarre ratio $Q$ should be equal to $0.666666666666\dots$ sounds utterly indefensible. Let's see why.

Stop hiding physics

Consider a messy real world problem. For example, calculate the number of kilocalories in a cheeseburger prepared from particular ingredients bought in Boston which followed a particular recipe. Let me tell you it has 314.159 kcal. Someone notices that the value is $100 \pi$ and predicts that if you measure the calories more accurately, you obtain $314.1592653589793238\dots$ and so on (I memorized 100 digits when I was a kid but you surely don't want to be annoyed by meaningless numbers).

Is there a reason to believe the person? He has offered "almost five" digits that agree as evidence of a remarkable statement – namely that an arbitrary number of digits will agree. Will they? Is the evidence enough? Well, it's surely not. Extraordinary statements require extraordinary evidence. And indeed, one needs to know some physics – either particle physics (a remarkable body of knowledge and the true culture of our epoch that is known as "preconceptions" by the numerologists) or the physics of cheeseburgers – to rationally decide whether the statement is bold, extraordinary, or not. It is very bold.

The number of calories in a hamburger is a complicated function of the properties and concentration of various ingredients and many other things. I hope that I don't have to explain why there shouldn't be any simple yet exact numerological fact applying to the cheeseburger case.

But my point is that the situation with the physical charged fermion masses is analogous. The physical charged lepton masses are very messy and complex functions of some parameters that are more fundamental. If simple rules hold for some parameters, they hold for the fundamental ones. But the physical charged lepton masses are so complicated cheeseburgers that any detailed numerical patterns that could hold for the fundamental quantities get mixed up and the chance that there's still a simple rule left is zero for all practical purposes (and most of the impractical purposes as well).

Yukawa couplings, corrections, running, mixing

The appearance of the square root of the masses in the formula for $Q$ is strange; one would feel comfortable if the squared masses occurred there. But is there any reason for an identity involving the square roots? They seem very unnatural. Their very appearance indicates that someone has been "mining" for such coincidences. If you randomly write down 10,000 candidate identities, you have pretty good chances that one of them will hold with the accuracy of 4 digits.

However, this complaint about the square roots is one of the milder ones. A more serious complaint is that the Standard Model shows that the lepton masses aren't fundamental parameters at all. In the Standard Model, the masses of charged leptons arise from the Yukawa interaction term in the Lagrangian,
${\mathcal L}_{\rm Yukawa} = yh \bar\Psi \Psi$
In the expression above, $y$ is a dimensionless (in $d=4$ and classically) coupling constant; $h$ is the real Higgs field; $\Psi,\bar\Psi$ is the Dirac field describing the charged lepton or its complex conjugate, respectively.

To preserve the electroweak symmetry – which is needed for a peaceful behavior of the W-bosons and Z-bosons – one can't just add the electron or muon or tau mass by hand. After all, the electroweak symmetry says that the left-handed electron is fundamentally the same particle as the electron neutrino. Instead, we must add the Yukawa cubic vertex – with two fermionic external lines and one Higgs external line – and hope that Mr Higgs or Ms God will break the electroweak symmetry which also means that he will break the symmetry between electrons and their neutrinos.

And be sure that he or She will. He only gives the large masses to the charged leptons.

In the vacuum, the Higgs field may be written as
$h = v+ \Delta h$ Here, $v$ is a purely numerical ($c$-number-valued) dimensionful constant whose value 246 GeV was known before we knew that the Higgs boson mass is 125 GeV. The value of $v$ is related to the W-boson and Z-boson masses and other things that were measured a long time ago. The term $\Delta h$ contains the rest of the dynamical Higgs field (which is operator-valued) but its expectation value is already zero.

The field $h$ has the vacuum expectation value (vev) 246 GeV and its being nonzero is what breaks the electroweak symmetry and what makes W-bosons and Z-bosons massive and what makes the observed properties of electrons and neutrinos so different. The field doesn't want to roll to $h=0$ because there is a potential energy $V(h)$ which is fully symmetric under the gauge symmetry transformations but it has a minimum at this nonzero absolute value of $h$ because the shape of $V(h)$ resembles the Mexican hat. That's why it's known as the champagne bottle bottom potential or Landau buttocks or Mexican hat potential in the prosperous first, socialist second, and poor third world, respectively. Because most of us already use the term "Mexican hat potential", you may predict what is going to happen with our prosperity. ;-)

Fine. So I tried to explain to you that the masses of the charged leptons are not fundamental parameters. They are results of the Higgs vacuum expectation value and the Yukawa interactions between the Higgses and the fermions. You should imagine that $m_e$ is just a shortcut for
$m_e = y_e v$ where the Yukawa coupling $y_e$ for the electron and the Higgs vev $v=$246 GeV are more fundamental than $m_e$. If you write the masses in this way, $v$ will simply cancel and you get the same formula for $Q$ where $m$ is replaced by $y$ everywhere.

However, this is not quite accurate because the physical masses are equal to $yv$ up to the leading order (tree level diagrams i.e. classical physics) only. There are (quantum) loop corrections and many other corrections. Moreover, the values of $y$ that produce $Q=2/3$ are the low-energy values of the Yukawa couplings. Even though the Yukawa couplings are more fundamental than the masses themselves, their low-energy values are less fundamental than some other values, their high-energy values.

Once again, the "apparent" strength of the Yukawa couplings at long distances is another example of a cheeseburger. It's very messy and determined from the values of the same – as well as other – couplings measured at high energies, through complicated functions related to the "renormalization group techniques".
$y_e({\rm low\,\,} E) = f_{\rm cheeseburger-like} ( y_e({\rm high\,\,} E), \dots )$ Only the arguments of the function on the right hand side, the high-energy values of various quantities, are players in sufficiently crisp and sharp laws of physics, e.g. grand unified theories or string theory, that have a good reason to be constrained by a relatively "simple" formula. But the low-energy values of the Yukawa couplings are a cheeseburger created from those ingredients. And the physical masses of the charged leptons are a McDonald's menu in which the cheeseburger is being improved by a few freedom fries, some Coke, ketchup, and other things. You just don't expect a random messy function of Coke, beef, potatoes, etc. to be equal to 314.159265358979 or 0.666666666666.

I know that it's unlikely that Mitchell, Carl, or Alejandro will get this point of mine – the point that they are trying to find patterns in cheeseburgers. They have spent years in this fundamentally misguided mode of thinking (or the lack thereof) and it may be hard to admit that one has wasted years with nonsense. But I hope and believe that many other readers will save their own years in the future.

And that's the memo.

Bonus: Koide on steroids

I have prepared a 22nd century research project for our numerological friends. Here is the identity:
$\frac{(m_e^{1/8}+m_\mu^{1/8}+m_\tau^{1/8})^{24}}{(m_e^{-1/3}+m_\mu^{-1/3}+m_\tau^{-1/3})^{-9}} = 1.000005 \times 10^{19}$ Within tenths of the experimental error margin, the right hand side is exactly $10^{19}$.

Note that its accuracy is better than $10^{-5}$ in this case! And the numbers such as $3,8,9,24$ are so deep, not to speak about the result's being such a marvelously accurate yet high power of ten. :-)

I needed ten minutes to find tons of such "remarkable" identities but I hope that Mitchell, Alejandro, and Carl will spend a happy century by writing thousands of inspired followup papers. ;-)

#### snail feedback (14) :

If it is merely a coincidence, why are their similar top-bottom-charm, bottom-charm-strange, and charm-strange-down triples, and why is there a three to one relationship between the charged lepton triples and the b-c-s triple?

Thanks to take the time to write about this. Actually I am surprised you did... no much news, this week?

The first occurrence I known of this kind of formula is not from Koide but by Harari, Haut, and Weyers in Phys.Lett. B78 (1978) 459, in the context of Cabibbo angle. They propose a discrete symmetry for a very peculiar Higgs sector that aims to explain other coincidency, that of Cabibbo angle and the mass quotient of down and strange quarks. They single out a particular breaking where mass of up quark is zero and the other quotient,(mass_down/mass_strange) is (2-sqrt(3))/(2+sqrt(3). You can see that such triplet fulfills the formula.

The startpoint for Koide work was to try to explain Cabibbo angle from the point of view of composite models of quarks and leptons. Then such composites implied an extra condition in the lepton sector, and he used it to predict the tau lepton. This is the status up to today: with the electron and muon mass as input, Koide formula predicts mass of tau lepton within one sigma.

One year ago, in one of the typical struggles to extend this formula to neutrinos, Rodejohann and Zhang Phys.Lett. B698 (2011) 152-156 remarked that the mass values of the sequence top, bottom, charm were not in disagreement with Koide's formula. With foresight, we could have been inspired in the 1978 paper and thought that for triplets of quarks it was important to have one of a different family, if only to build the two mass quotients you refer to. But instead we got lost, thinking family-wise.

Given the observation of Rodejohann and Zhang, it was natural to look again, and you can construct it as if we had been five or six years pestering on the topic, but do not worry, we have had time to think on other topics. Whatever, it happens that the next of the sequence, bottom, charm and strange, also goes into the error bar umbrella of the strange quark IF we use the negative square root. By itself, it does not mean a lot, because the mass of s has a huge error bar. But the interesting point is that the new triplet was also almost orthogonal to the lepton triplet. I noted this point in arxiv:11117232

So, prediction-wise, the current built is:

Take the mass of electron and muon

me = 0.510998910± 0.000000013
mμ = 105.6583668± 0.0000038

and use Koide to calculate the mass of tau.

m = 1776.96894(7) MeV

(I am putting inside the hamburger buns the error of the "prediction", given the errors in the only two physical inputs, electron and muon)

second step, assume the orthogonal Koide triplet (this implies a factor three in mass and other factor three in angle) to predict the masses of strange, charm and bottom:

ms = 92.274758(3) MeV
mc = 1359.56428(5) MeV
mb = 4197.57589(15) MeV

third step, assume again Koide as usual to produce the mass of the top from charm and bottom

mc as above
mb as above
mt = 173.263947(6) GeV

Even to me it is hard to swallow, because in my usual frame of mind I expect that the top mass is natural, yukawa order 1, while all the others are yukawa zero.

Well, four and five, just to see, go ladder down with Koide with mc and ms as input, getting m_up, amd then ms and up as input, getting md. This is the most dissapointing result:
mu = 0.0356 MeV
md = 5.32 MeV
It is possible to go into serious values by adding some mixing, instanton or whatever. Let M be its scale, then we could produce
ms'= ms + mu md / M
mu'= mu + ms md / M
md'= md + mu ms / M
this is an old trick that nowadays is suppossed to be ruled out by lattice QCD.

Alejandro, fine, so do you agree that the numerology doesn't work for quarks, not even on the numerological level? All the numbers are totally different than they should be.

Even if the numbers worked, and they didn't, it would be extraordinarily stupid to look for coincidences involving c,s,b quarks. They're 3 out of 6 flavors; some of them are down-type quarks but "c" is an up-type quark. You're clearly mixing apples and oranges.

It makes absolutely no sense to find exact patterns constraining 3 random quarks out of the 6 flavors.

And it probably makes no sense for me to talk to you because you apparently viscerally hate rational as well as scientific reasoning.

Er, Lubos, have you checked the numbers lately?

Tau is 1776.82 ± 0.16 Our prediction 1776.97

And s,c,b,t are respectively
100.2 ± 2.4 average, 80-130 evaluation. Our prediction,92.28
1.294 ± 0.004 average, 1.18-1.34 evaluation. Our prediction, 1.3596
4.19 −0.06+0.18 Our Prediction, 4.198
172.9 ± 0.6 ± 0.9 GeV Our Prediction 173.264

If they do not fit, what do you call a fit? Note that the more stressed measures are the ones for which the pdg has not confidence, giving also an equivalent estimate. Only the charm seems to be a bit in the extreme of the estimate.

It is true that Koide ladder fails to obtain the mass of the up quark. But it errs a right side, it predicts an up mass near to zero and a down mass in the right order so that the standard arguments of second order corrections, instantons or like, could be applied. I do not consider it a success because it implies a new input, but exactly this kind of correction has been around in the literature by some time, and it is still to be settled via lattice. If you buy it, a new parameter allows to fix right values for down and up too.

As for why should we have alternating quarks, well, it is the original HHW situation, with (u,d,s) meeting Koide formula in the breaking where the mass of up quark is zero. The real question is why the leptons are not alternating with neutrinos. Well, perhaps they are, and the mu neutrino has the same Dirac mass that the muon. Or perhaps it is just an effect of the composite theory that transported the formula to the lepton sector. The alternation between quarks of different kinds was used in the primitive literature to try to fix a CKM parameter; the corresponding parameters in the lepton sector work differently, so it is not so surprising if a situation is "alternating" while the other is "generation-wise".

Dear Alejandro, these are not really predictions because you knew the results when you were inventing these silly contrived rules. And they don't work, either. Every sane person knows that they're coincidences. Once the tau mass will be measured more accurately, it will be 5 sigma away from your deluded "predictions".

The quarks are not worth talking about at all. Some of them don't work at all, some of them work at 10% which is a very likely outcome even a priori, given the limited interval in which the ratio may a priori be.

So this whole program is utter failure. I can't believe you don't see that.

Well, you did not notice Koide equation till this week, it seems, so please forgive me if I did not notice it till 2005, and all the masses had already been measured.

Actually, tau is a prediction. The current value was measured in 1992. Previously, and in the age when Koide produced his equation, it was quoted a bit about 1.78 or even 1.80 MeV. In fact, Koide work is a good example that a prediction does not grant success. It is amusing that a lot of internet arguments go about predictions vs postdictions, when at the end that the important point of calculations during evaluation of theories is its use to discard them, not its use to accept them.

Let me note also that nobody claims, afaik, that this set of equations has a closed non perturbative origin without further corrections. It is just claimed that in this simple form they already do a huge step towards matching the mass values.

Still, I am amazed of your way to look at the data and your argument , "Some of them don't work at all, some of them work at 10%". Lets check again. Besides the tau, when we take as start points the best measured masses, electron and muon, we get that:

- Strange is inside of the evaluation and 8% from the average.
- Charm is in the border of the evaluation and 5% from the average.
- Bottom is inside 1-sigma and
and 0.2% of the central value.
- Top is inside 1-sigma and 0.2% of the central value too.

But it is true that for the first generation:

Up does not work, and
Down is inside the estimate (4.1-5.7), but at 11.3% above the average.

So it seems that your protest reduces to the low triplet U,D,S.

You are right to protest here. In fact I am a bit amused that Rodejohann and Zhang put this triplet in the valid set. It seems that they argue RG to do all the comparisons in some GeV scale, and then it works better. But it is also possible, as I said, to postulate that the Koide prediction, whatever its underlying physics, give us the masses before mixing. With an ad-hoc mixing with M=185 MeV, we should get
U'=U+DS/M= 2.67 MeV
D'=D+US/M= 5.32 MeV
S'=S+UD/M= 92 MeV
This kind of mixings were proposed time ago as a solution to the strong CP problem, setting U=0 to conjure out the problem and then getting the real masses from the mix. Instantons were the main candidate for it.

Ah, other thing... When you say:

"given the limited interval in which the ratio may a priori be."

... I am a bit puzzled about what "a priori" are you thinking about. Obviously it is something different that (0,infinite), and surely you are thinking in something coming from GUT or some limit where yukawas of leptons and quarks are equal and generations keep simple rations across them, and then the running down fixes the possible "limited interval". As far as I remember, you have never blogged about this; it could do a nice post by itself.

I saw the Koide formula some years and found it interesting. Plus it is close to 0.001 * 666 , which is evil but only one in a thousand :P

Wether there is a deeper truth or not...I don't know! I would like however to ask something.
Supose there is a fourth lepton, not yet discovered, what would it's mass be? Should the formula still work with adding another mass?

The mass of a fourth lepton that would be predicted by Koide's formula has, IIRC, been ruled out experimentally. It would be on the same order of magnitude of mass as a bottom quark, since the charm and strange quarks have masses on the same order of magnitude as the muon and tau, but the exclusion range from experiment for new charged fermions is on the same order of magnitude as the much heavier top quark.

Thus, Koide's formula if extended, predicts that there is not a fourth or higher generation of fundamental fermions.

New numbers on the mass of the top: 173.18 ±0.94 GeV, from arxiv:1207.1069 CDF/D0 combination. Koide ladder calculation from vixra:1111.0062v2/arxiv:1111.7232, is 173.263947(6) GeV.

I sketched a prezi presentation http://prezi.com/e2hba7tkygvj/koide-waterfall/ on Koide chains

Another thing... composite Higgs, as well as some variants coming from 5D, have the yukawa coupling as a product of two other coefficients. So in such cases there is really a "square root of the masses" directly in the Lagrangian.