arXiv:hep-ph/0505220v1 25 May 2005
The strange formula of Dr. Koide
and Andre Gsponer
February 2, 2008
We present a short historical and bibliographical review of the
lepton mass formula of Yoshio Koide, as well as some speculations on
its extensions to quark and neutrino masses, and its possible relations
to more recent theoretical developments.
A subjective slice of history
At the end of 1981 Yoshio Koide, working on some composite models of
quarks and leptons, had the go od or bad fortune of stumbling over a very sim-
ple relationship between the masses of the three charged leptons [1, Eq. (17)].
This resulted in a prediction of 1.777 GeV/c
for the mass of the tau lep-
ton. At that time, that prediction was more than two standard deviations
away from t he measured value, 1.7842 GeV/c
. So by January of 1983 Koide
sent to the Physical Review a decaﬀeinated presentation , holding a purely
phenomenological point of view, and intro ducing a correction term δ in order
to ﬁt the — then supposed — experimental value. Still, this paper paved
the way for a nascent research on democratic family mixing. This is because
Koide mass formula
Zaragoza University at Teruel, firstname.lastname@example.org
Independent Scientiﬁc Research I ns titute, Geneva, email@example.com
can be related to the eigenvectors of the well known democratic matrix
As Foot pointed out , Koide’s formula is equivalent to ask for an angle
of exactly π/4 between the eigenvector ( 1 , 1, 1) and the vector fo r med with
the square roots of the lepton masses. And the rotatio n around (1, 1, 1) is
of course determined by taking any basis of the nullspace of the democratic
mixing matrix. But the various uses of this doubly degenerated matrix are
not the theme of this review
We can only imagine the excitation when some years later the value o f the
mass of the tau lepton is revised... And the correction parameter δ becomes
plainly zero: The original prediction was right! Koide revives his formula
and builds new models [5, 8], but the impact is very small. In the electronic
archive arXiv.org, R. Foot [3 ] suggests a geometrical interpretation, and a
couple years later Esposito a nd Santorelli  revisited the formula, remark-
ing its stability under radiative corrections, at least below the electroweak-
breaking scale (and breaking at this scale would be, after all, a clue to the
origin of the relationship). Furthermore, it has not escaped that the down
quark-family masses are also approximated well enough by the formula, and
some eﬀort was made by Koide and others to see how the rest of the particles
could ﬁt in the picture. Contemporary descent s of this eﬀort now encompass
Meanwhile, t he increasing interest in seesaw models gave another oppor-
tunity to use the democratic matrix, as well as to generate a top quark mass
enhancement justifying a radiative origin for the lower generations. For some
years, seesaw ideas and Koide’s formula went hand in hand [9, 10, 11]. The
last incarnation we are aware of, e.g. from Refs. [11, 12], jumps to a justi-
ﬁcation based on discrete S(3 ) symmetry. Three-Higgs techniques, used by
several authors in the late nineties [13, 14] can b e given a role there.
This was only a little cut through the record (see the SPIRES database
for more), not the whole story, and for sure it is not over yet. After all, as
it is r emarked in , the current status o f the ﬁt against a theoretical unity
The matrix was ﬁrst used for glue-mediated mixing of mesons. We encourage the
reader towards any of the multiple articles fr om Harald Fritzsch. E.g.,  and references
Form leptons to quarks and neutrinos
Due to its success with leptons, it was quite obvious that Koide and others
would try to extend the mass formula to quarks. In order to see how this can
be done in a physically meaningful way, let us ﬁrst remove the square roots
from Eq. (1). Indeed, there is a large consensus that “mass” is at least in part
due to self-interaction processes which lead to expressions that are quadratic
in some underlying classical or quantum ﬁeld, and thus in the corresponding
coupling constant. For instance, in classical electrodynamics: ∆m = e
in general relativity: ∆m = Gm
/r, and in ﬁeld theory: ∆m = g
is also the case in pure ﬁeld theories such as Lanczos’s  and Weinberg’s
, where mass is entirely originating from t he self and mutual interactions
It is therefore natural to introduce a physical quantity that is more fun-
damental than the mass, which we pro pose to designate by the symbol ג, i.e.,
the third letter in the Hebrew alphabet,
and to deﬁne it as the gim of the
particle according to the identity m ≡ ג
. This enables to rewrite Koide’s
or, by combining the gims of the three lepton into the vector
ג = (ג
), as the more compact expression
where t he trace operation Tr acting o n a vector is to be interpreted as the
sum of its components.
In order to apply Eq. (3) to quarks, as well as to neglect the question of
radiative corrections already alluded to in the previous section, one has to
circumvent the problems that their masses are not directly measurable, and
that their estimated masses are obtained by methods which are diﬀerent for
each generation of them .
One possible approach is to use some r easonable model, such as the
Nambu-Barut [18, 19] formula generalized by Gsponer and Hurni  which
gives a smooth and consistent ﬁt to both the lepton and quark data, except
Being the positional equivalent to “gamma”, the letter ג is called “gimel.” It means
“camel,” and is thus a nice word for the carrier of the mass.
for the top quark mass. Barut’s formula for leptons is
m(N) = m
is the mass of the electron and α ≈ 1/137 the electromagnetic ﬁne
structure constant. The masses of the electron, muon, and tau correspond
then to N = 0, 1, and 2, respectively. The masses of the quarks are also
given by this for mula, provided m
is replaced by m
/7.25, the mass
of the u-quark. The masses of the d, s, c, and b quarks ar e then given to an
excellent approximation by N = 1 , 2, 3 and 4, respectively, but the formula
fails completely for the t quark.
While Barut’s formula is very diﬀerent from Koide’s, it contains the 3/2
factor, and a fourth-power dependence on a quantum number N that is
typical of self-interaction in non-linear ﬁeld theories. There are several ways
of using Barut’s formula in relations to Koide’s:
First, in the generalized Barut formula the char ged leptons and three of
the quarks are r elated by a proportionality relation such that e corresponds
to u, µ to d, and τ t o s. Thus, if this model would be the correct underlying
theory of mass, Koide’s formula should equally well apply to the (e, µ, τ )
lepton-triplet, than to the (u, d, s) quark-triplet, i.e., the original Gell Mann –
Zweig SU(3) triplet. Indeed, using Barut’s masses for either of these triplets,
Koide’s fo r mula turns out to work at a precision of 2% for both. Similarly, if
one applies Koide’s for mula using Barut’s theoretical masses for the (d, s, b)
triplet (i.e., the d-type quark family) the precision is 2.7%, which is also
quite good. On the other hand for the (u, c, t) triplet (i.e., the u-type quark
family), the agreement is not as good, i.e., only about 28%. Therefore, while
Eq. (1) works rather well for the o riginal quark triplet and the d-type quark
family, it does not work so well for the u-type quark family — as was observed
in Ref. , and certainly by Koide and others earlier on.
A second way is to consider that while there a r e three massive leptons
forming some kind of a three-dimensional real vector, the six quarks could
correspond to the six real components of a complex 3-vector. This picture
has the advantage that this complex quark-gim vector could be put in re-
lation to the real lepton-gim vector introduced by Foot , and tha t bot h
vectors could have a sound ﬁeld theoretical interpretation in t erms of a real
or a complex gim vector ﬁeld. Conversely, if the lepton gim-vector is com-
plexiﬁed by assuming non-zero neutrino masses, one would have a six partons
generalization of Koide’s formula. In that perspective, if the masses of the
ﬁve ﬁrst quarks are taken from the generalized Barut formula , and the
Quark masses and gims
u 0.5 0.71
d 6 2.5
s 105 10
c 1250 35
b 4500 67
t 174000 417
total 180000 532
Ta ble 1: Average of the quark masses given by the Particle Data Book ,
and their correspo nding gims, i.e., ג ≡
mass of t he t from the data , i.e., 1 74 GeV/c
, the precision of the gen-
eralized Koide formula, Eq. (3) in which the sums of t he gims and of their
squares are taken over all six quarks, is about 9%.
Finally, a third and possibly the most interesting way is to repeat the
previous calculation by taking all six quark masses from the data . More
precisely, due to the theoretical uncertainties of extracting the quark masses
from the data, these six masses are taken as the averages between the extreme
values cited in Ref. , as listed in Table 1. Thus, as 532
/180000 = 1.572
instead of 1.5, Koide’s formula applies to a six-quark complex vector with a
precision of about 5%, which is quite good.
We can therefore conclude this section by stressing that Koide’s formula
applies reasonable well to quark masses, either if these masses are calculated
with Barut’s formula generalized to quarks, or if the masses are taken f r om
the Particle Data Book. Agreement is particularly good if we take for the
masses of the three lighter quarks those given by Barut’s formula, or if we
take all six quark masses from the data. This suggests that while Barut’s
formula seems to contain an important element of truth concerning the light
parton masses (where the factors 3/2 and α ≈ 1/137 appear to play some
fundamental role ), Koide’s formula seems to embody a similar element
of truth concerning the heavy partons (including the mass of the top quark
which does not ﬁt the Nambu-Barut formula). However, as is well known,
The ratio 283/180 being quite precisely equal to π/2 illustrates how easily one can get
a good agreement with a beautiful number. (Je an-Pierre Hurni, private communication.)
all phenomenological formulas which do not have a completely unambiguous
theoretical foundation should be used with caution. This is illustrated by
the less successful ﬁt provided by the u-quark triplet as compared to the
d-quark triplet, which implies that t he direct application of Koide’s formula
to neutrino masses, as is done in Ref. , would provide an important clue
if it were to b e supported by the data .
Alternate origins of Koide’s relation
Asymmetric Weyl spinors. In the early age of single-electron and pure
electrodynamics, textbooks (such as Berestetskii-Lifschitz-Ditaevskii) told
us not to worry about diﬀerent m a nd m
masses when combining two Weyl
spinors int o o ne Dirac spinor, because their quotient can a lways be hidden
inside a redeﬁnition of the spinors
. With the advent of generations, a new
scheme was required to control the new freedom of rotation between equally
charged particles: The “CKM” and “MNS” mass-mixing matrices in the
quark and lepton sectors. But, is it still true that we can hide from reality
any diﬀerence in their eigenva lues?
Suppose M is a degenerated diagona l matrix λ1
, and that M
still a diag onal nondegenerated matrix. The corresponding Dirac equation
will have a mass squared matrix MM
composed of eigenvalues λm
its real masses will be pro portional to
. If one is able to impose some
symmetry in this scheme, Koide’s formula will follow, at the cost o f three
Non-Commutative Geometry. Models based on non-commutative
geometry usually have the potential of the Higgs sector to be determined
by the lepton mass matrix, and sometimes by its square and its trace. For
instance, in order to get a non-tr ivial vacuum, the early electroweak Connes-
Lott model imposed the condition
6= 0. (5)
Could this technique b e related to Koide’s ﬁndings?
Thus any lack of self-adjointness of m was only illusory, wasn’t it?
Switching couplings oﬀ. It has b een remarked and footnoted
of times that the muon and the electron (or equivalently the chiral and elec-
tromagnetic breaking scales) are separated by a factor o f order 1/α. It is
less often mentioned [2 3], but similarly intriguing, that the tau and the elec-
troweak vacuum (or equivalently the SU(3) and electroweak scales) are like-
wise separated by the same Sommerfeld’s constant.
We can mentally visualize that if we shut down this constant , the mass of
the electron is pushed towa r ds zero, while the Fermi scale is pushed towards
inﬁnity. In this scenario, Foot’s hint of an additional symmetry invites to use
Koide’s formula as a constraint: The net result is that the muon mass should
increase by a few MeV/c
, while that of the tau should decrease a little in
order to keep the angle. And depending of your views of chiral perturbation
theory, the pion should either go to zero mass or just to lose a few MeV/c
of its mass due to the component quarks, thus becoming mass degenerated
with the muon!
Moreover, the switching oﬀ of the weak coupling constant will aﬀect the
link between the bottom and the top quarks, causing the former to lose some
weight. In the limit of a very small α, we have a bunch of almost massless
particles, another one of hugely massive bosons (completed by the top quark),
and then some surviving elements making use of the SU(3) gap: The muon
and the strange quark courting themselves and perhaps the pion; the tau,
the charmed and bottom quarks dancing ar ound the nucleon and its glue
(see Fig. 1). The mass ratio between tau and muon becomes exactly (
i.e., the mass ratio predicted by Koide’s formula when one of the masses
) approaches zero. The three in this expression is coming from the
number of generations, not from the number of colors, and its numerical value
(≈ 14) is very close to the measured coupling constant of the pseudoscalar
pion-nucleon theory of strong interactions (see, e.g., [24, p.450]).
One of the puzzling mysteries of Nature is why the massive leptons, which
are colorless, are wandering just there. A hint o f universality could come if
we go further, switching oﬀ the SU(3) coupling! Even without contribution
from strings and glue, we can build mesons starting at the pion scale, because
of the strange quark; and baryons at the scale of the nucleon, because of the
other two massive quarks. Family masses conspire to save the gap.
Mostly in the seventies, but there are many earlier references, e.g., [18, 22] and others
cited in .
d s b
Figure 1: Masses of the elementary particles in logarithmic plot. The lengths
of the two upper horizontal lines, and that of the lower shorter one, are the
numbers 1/α ≈ 137 and (
≈ 14, respectively.
Cabibbo angle. From the very beginning , Koide’s formula has been
associated to another one for the Cabibbo angle, involving square roots of
the three generations. Of course when one of the masses is driven to zero, the
two extant ones form a square root of a mass quotient, similar to the kind
of expressions that nowadays are popular folklore in the phenomenology of
mixing. If one has followed the gedakene xercise of the previous paragraph,
one will not be surprised that the Cabibbo angle can be obtained both from
leptons and from quarks. This possibility has also been noticed recently
by Carl Brannen  in a variant, previously used by Koide , of the
democratic mixing, namely,
1 0 0
0 1 0
0 0 1
which happens to have eigenvalues proport io nal to our
when θ is Cabibbo’s
angle. In  this solution is avoided, p erhaps intentionally, in order to ob-
tain a complementary ﬁt in the quark sector.
Other. The ﬁrst author maintains on internet a permanent quest f or
phenomenologically inspired relationships related to the Standard Model.
Please check the wiki page .
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