Koide formula

The Koide formula is an unexplained empirical equation discovered by Yoshio Koide in 1981. In its original form, it is not fully empirical but a set of guesses for a model for masses of quarks and leptons, as well as CKM angles. From this model it survives the observation about the masses of the three charged leptons; later authors have extended the relation to neutrinos, quarks, and other families of particles.

Formula
The Koide formula is


 * $$Q = \frac{~ m_e + m_\mu + m_\tau ~}{\left(\ \sqrt{\ m_e\ } + \sqrt{\ m_\mu\ } + \sqrt{\ m_\tau\ }\ \right)^2} = 0.666661(7) \approx \frac{\ 2\ }{ 3 }\ ,$$

where the masses of the electron, muon, and tau are measured respectively as $m$$e$ = $0.511 MeV/c2$, $m$$$μ$$ = $105.658 MeV/c2$ , and $m$$$τ$$ = $1,776.86 MeV/c2$ ; the digits in parentheses are the uncertainties in the last digits. This gives $Q = 0.667$.

No matter what masses are chosen to stand in place of the electron, muon, and tau, the ratio $me$ is constrained to $mμ$. The upper bound follows from the fact that the square roots are necessarily positive, and the lower bound follows from the Cauchy–Bunyakovsky–Schwarz inequality. The experimentally determined value, $mτ$, lies at the center of the mathematically allowed range. But note that removing the requirement of positive roots, it is possible to fit an extra tuple in the quark sector (the one with strange, charm and bottom).

The mystery is in the physical value. Not only is the result peculiar, in that three ostensibly arbitrary numbers give a simple fraction, but also in that in the case of electron, muon, and tau, $Q$ is exactly halfway between the two extremes of all possible combinations: $Q$ (if the three masses were equal) and $1⁄3 ≤ Q < 1$ (if one mass dwarfs the other two). $Q$ is a dimensionless quantity, so the relation holds regardless of which unit is used to express the magnitudes of the masses.

Robert Foot also interpreted the Koide formula as a geometrical relation, in which the value $$\ \frac{ 1 }{\ 3\ Q\ }\ $$ is the squared cosine of the angle between the vector $$\ [\ \sqrt{\ m_e\ }, \sqrt{\ m_\mu\ }, \sqrt{\ m_\tau\ }\ ]\ $$ and the vector $$\ [\ 1, 1, 1\ ]\ $$ (see dot product). That angle is almost exactly 45 degrees: $$\ \theta = 45.000^\circ \pm 0.001^\circ ~.$$

When the formula is assumed to hold exactly ( $2⁄3$), it may be used to predict the tau mass from the (more precisely known) electron and muon masses; that prediction is $1⁄3$.

While the original formula arose in the context of preon models, other ways have been found to derive it (both by Sumino and by Koide – see references below). As a whole, however, understanding remains incomplete. Similar matches have been found for triplets of quarks depending on running masses. With alternating quarks, chaining Koide equations for consecutive triplets, it is possible to reach a result of $1$ for the mass of the top quark.

Speculative extension
Carl Brannen has proposed the lepton masses are given by the squares of the eigenvalues of a circulant matrix with real eigenvalues, corresponding to the relation


 * $$\sqrt{\,m_n\;} = \mu \left[\,1 + 2 \eta \cos\left( \delta + \frac{\,2\pi\,}{3}\cdot n \right) \,\right] ~,~$$ for $Q$ = 0, 1, 2, ...

which can be fit to experimental data with $n$$η$ = 0.500003(23) (corresponding to the Koide relation) and phase $2$ = 0.2222220(19), which is almost exactly $δ$. However, the experimental data are in conflict with simultaneous equality of η$2⁄9$ = $2$ and $1⁄2$ = $δ$.

This kind of relation has also been proposed for the quark families, with phases equal to low-energy values $2⁄9$ = $2⁄27$ × $2⁄9$ and $1⁄3$ = $4⁄27$ × $2⁄9$, hinting at a relation with the charge of the particle family ($2⁄3$ and $1⁄3$ for quarks vs. $2⁄3$ = 1 for the leptons, where &thinsp; $3⁄3$ × $1⁄3$ × $2⁄3$ ≈ $3⁄3$&thinsp;).

Origins
The original derivation postulates $$\ m_{e_i} \propto\ (z_0 + z_i)^2\ $$ with the conditions
 * $$\ z_1 + z_2 + z_3 = 0\ $$
 * $$\ \tfrac{\ 1\ }{ 3 }\ (z_1^2+z_2^2+z_3^2) = z_0^2\ $$

from which the formula follows. Besides, masses for neutrinos and down quarks were postulated to be proportional to $$\ z_i^2\ $$ while masses for up quarks were postulated to be $$\ \propto\ ( z_0 + 2 z_i )^2 ~.$$

The published model justifies the first condition as part of a symmetry breaking scheme, and the second one as a "flavor charge" for preons in the interaction that causes this symmetry breaking.

Note that in matrix form with $$ M = A\ A^\dagger $$ and $$ A = Z_0 + Z $$ the equations are simply $$ \operatorname{tr} Z = 0 $$ and $$ \operatorname{tr} Z_0^2 = \operatorname{tr} Z^2 ~.$$

Similar formulae
There are similar formulae which relate other masses. Quark masses depend on the energy scale used to measure them, which makes an analysis more complicated.

Taking the heaviest three quarks, charm (1.275 ± 0.03 GeV), bottom (4.180 ± 0.04 GeV) and top (173.0 ± 0.40 GeV), regardless of their uncertainties, one arrives at the value cited by F. G. Cao (2012):


 * $$Q_\text{heavy} = \frac{m_c + m_b + m_t}{\big(\sqrt{m_c} + \sqrt{m_b} + \sqrt{m_t}\big)^2} \approx 0.669 \approx \frac{2}{3}.$$

This was noticed by Rodejohann and Zhang in the first version of their 2011 article, but the observation was removed in the published version, so the first published mention is in 2012 from Cao.

Similarly, the masses of the lightest quarks, up (2.2 ± 0.4 MeV), down (4.7 ± 0.3 MeV), and strange (95.0 ± 4.0 MeV), without using their experimental uncertainties, yield


 * $$Q_\text{light} = \frac{m_u + m_d + m_s}{\big(\sqrt{m_u} + \sqrt{m_d} + \sqrt{m_s}\big)^2} \approx 0.57 \approx \frac{5}{9},$$

a value also cited by Cao in the same article.

Note that an older article, H. Harari, et al., calculates theoretical values for up, down and strange quarks, coincidentally matching the later Koide formula, albeit with a massless up-quark.


 * $$Q_\text{light} = \frac{0 + m_d + m_s}{\big(\sqrt{0} + \sqrt{m_d} + \sqrt{m_s}\big)^2} $$

Running of particle masses
In quantum field theory, quantities like coupling constant and mass "run" with the energy scale. That is, their value depends on the energy scale at which the observation occurs, in a way described by a renormalization group equation (RGE). One usually expects relationships between such quantities to be simple at high energies (where some symmetry is unbroken) but not at low energies, where the RG flow will have produced complicated deviations from the high-energy relation. The Koide relation is exact (within experimental error) for the pole masses, which are low-energy quantities defined at different energy scales. For this reason, many physicists regard the relation as "numerology".

However, the Japanese physicist Yukinari Sumino has proposed mechanisms to explain origins of the charged lepton spectrum as well as the Koide formula, e.g., by constructing an effective field theory with a new gauge symmetry that causes the pole masses to exactly satisfy the relation. Koide has published his opinions concerning Sumino's model. François Goffinet's doctoral thesis gives a discussion on pole masses and how the Koide formula can be reformulated to avoid using square roots for the masses.

As solutions to a cubic equation
A cubic equation usually arises in symmetry breaking when solving for the Higgs vacuum, and is a natural object when considering three generations of particles. This involves finding the eigenvalues of a 3×3 mass matrix.

For this example, consider a characteristic polynomial
 * $$\ 4\ m^3 - 24\ n^2\ m^2 + 9\ n\ (n^3 - 4)\ m - 9\ $$

with roots $$\ m_j\ :\ j = 1, 2, 3\ ,$$ that must be real and positive.

To derive the Koide relation, let $$\ m \equiv x^2\ $$ and the resulting polynomial can be factored into
 * $$\ (\ 2\ x^3 - 6\ n\ x^2 + 3\ n^2x - 3\ )(\ 2\ x^3 + 6\ n\ x^2 + 3\ n^2\ x + 3\ )\ $$

or
 * $$\ 4\ (\ x^3 - 3\ n\ x^2 + \tfrac{ 3 }{\ 2\ }\ n^2x - \tfrac{ 3 }{\ 2\ }\ )(\ x^3 + 3\ n\ x^2 + \tfrac{ 3 }{\ 2\ }\ n^2\ x + \tfrac{ 3 }{\ 2\ }\ )\ $$

The elementary symmetric polynomials of the roots must reproduce the corresponding coefficients from the polynomial that they solve, so $$ x_1 + x_2 + x_3 = \pm 3\ n $$ and $$ x_1 x_2 + x_2 x_3 + x_3 x_1 = + \tfrac{ 3 }{\ 2\ }\ n^2 ~.$$ Taking the ratio of these symmetric polynomials, but squaring the first so we divide out the unknown parameter $$\ n\ ,$$ we get a Koide-type formula: Regardless of the value of $$\ n\ ,$$ the solutions to the cubic equation for $$\ x\ $$ must satisfy
 * $$\ \frac{\ 2\ ( x_1 x_2 + x_2 x_3 + x_3 x_1)\ }{~ ( x_1 + x_2 + x_3 )^2\ } = \frac{\ (3\ n^2)\ }{~ (\pm 3\ n)^2\ } = \frac{\ 1\ }{ 3 }\ $$

so
 * $$\ 1 - \frac{\ 2\ x_1 x_2 + 2\ x_2 x_3 + 2\ x_3 x_1\ }{~ ( x_1 + x_2 + x_3 )^2\ } = 1 - \frac{\ 1\ }{ 3 } = \frac{\ 2\ }{ 3 }\ ~.$$

and
 * $$\ 1 - \frac{\ 2\ x_1 x_2 + 2\ x_2 x_3 + 2\ x_3 x_1\ }{~ ( x_1 + x_2 + x_3 )^2\ } = \frac{\ ( x_1 + x_2 + x_3 )^2 - 2\ x_1 x_2 - 2\ x_2 x_3 - 2\ x_3 x_1\ }{~ ( x_1 + x_2 + x_3 )^2\ } = \frac{\ x_1^2 + x_2^2 + x_3^2\ }{~ ( x_1 + x_2 + x_3 )^2\ } ~.$$

Converting back to $$\ \sqrt{\ m\ } = x\ $$
 * $$\frac{\ m_1 + m_2 + m_3\ }{\ \left(\ \sqrt{\ m_1\ } + \sqrt{\ m_2\ } + \sqrt{\ m_3\ }\ \right)^2\ } = \frac{\ 2\ }{ 3 }\ ~.$$

For the relativistic case, Goffinet's dissertation presented a similar method to build a polynomial with only even powers of $$\ m ~.$$

Higgs mechanism
Koide proposed that an explanation for the formula could be a Higgs particle with $$U(3)$$ flavour charge $$\Phi^{a\overline{b}}$$ given by:


 * $$\ V(\Phi) = \left[\ 2\ \left[tr(\Phi)\right]^2 - 3\ tr(\Phi^2)\ \right]^2\ $$

with the charged lepton mass terms given by $$\ \overline{\psi}\ \Phi^2\ \psi ~.$$ Such a potential is minimised when the masses fit the Koide formula. Minimising does not give the mass scale, which would have to be given by additional terms of the potential, so the Koide formula might indicate existence of additional scalar particles beyond the Standard Model's Higgs boson.

In fact one such Higgs potential would be precisely $$V(\Phi) = \det[(\Phi-\sqrt{m_e})]^2 + \det[(\Phi-\sqrt{m_\mu})]^2 + \det[(\Phi-\sqrt{m_\tau})]^2$$ which when expanded out the determinant in terms of traces would simplify using the Koide relations.