We study Koide equation sequentially in a chain of mass triplets. We notice at least a not previously published triplet whose existence allows to built a quark mass hierarchy from fixed yukawas for top and up quarks. Also, the new triplets are used to build mass predictions either descending from the experimental values or top and bottom, or ascending from the original triplet of charged leptons.

1. Koide equations for quark mass triplets.
Alejandro Rivero∗
Institute for Biocomputation and Physics of Complex Systems (BIFI),
University of Zaragoza,Mariano Esquillor, Edificio I + D , 50018 Zaragoza ,Spain
(Dated: February 6, 2013)
We study Koide equation sequentially in a chain of mass triplets. We notice at least a not
previously published triplet whose existence allows to built a quark mass hierarchy from fixed
yukawas for top and up quarks. Also, the new triplets are used to build mass predictions either
descending from the experimental values or top and bottom, or ascending from the original triplet
of charged leptons.
PACS numbers: 12.15Ff
I. ORIGIN OF KOIDE FORMULA
An early [3, 8] intuition of the mass of the d quark as
the result of Cabibbo mixing with s quark, md ∼ θ2
c ms,
guided the efforts of modellers in the seventies [4, 23, 24]
to derive either this relationship or a close one, namely
tan θc =
md
ms
(1)
Particularly, Harari, Haut and Weyers [10] used a dis-
crete symmetry to produce not only the above equation
but also two exact equations
mu = 0;
md
ms
=
2 −
√
3
2 +
√
3
(2)
His model was a sequential symmetry breaking, using
the trivial and bidimensional irreducible representations
of the permutation group S3 The work was promptly cri-
tiqued [5] because at the end it was unclear how do they
get the mass eigenvalues; they rotate the mass matrix to
select a particular representation and then they disregard
non diagonal terms. Besides they did not provide an ex-
plicit representation for the Higgs sector (As for mu = 0,
it was not really a problem: other QCD mechanism could
happen later, to give it a mass of order mdms/MΛ). Still,
the result contributed to sustain an interest on the use
of discrete symmetry to predict structures in the mass
matrix.
Following explicitly this topic, in 1981 Koide [12, 13]
attempted another approach, incorporating the symme-
try in the context of a model of preons. Given that such
preons were components both of quarks and leptons, the
result was a formula for Cabibbo angle using the mass
of charged leptons and, as a by-product, a formula (eq.
(17) of [14]) linking their masses:
(
√
m1 +
√
m2 +
√
m3)2
m1 + m2 + m3
=
3
2
(3)
∗ arivero@unizar.es
The formula predicts a tau mass of 1776.96894(7)
MeV, perfectly matching the experimental [1] value of
1776.82 ± 0.16 MeV. In 1981, the measurement was still
1783 ± 4 MeV, so in some sense Koide’s work was a real
prediction, even if its composite character was ruled out
by experiments.
This formula (3), considered independently of a un-
derlying model, is referred as “Koide formula” or “Koide
equation”, and a tuple of three masses fulfilling it is called
a “Koide triplet” or “Koide tuple”
The masses (2) are a Koide triplet.
II. MODERN RESEARCH ON KOIDE
TRIPLETS
The use of a composite model can be substituted by
an ad-hoc Higgs sector with discrete symmetries. Such
models of ”flavons” have been revisited by Koide in later
work [16, 17]. In some proposals [21, 22], more symmetry
is added in order to protect Koide equation from renor-
malisation running.
Only with the SM, the equation is not protected, and
its running under the renormalisation group has been
studied for leptons as well as for some quark triplets. A
detailed study is for instance [25]. Generically, it is no-
ticed that the running up to GUT scale corrects the equa-
tion (3) by less than a ten per cent in the most extreme
case, and that in the case of charged leptons the equation
works better for pole masses. The reason for the stability
of the formula is that we are considering mass quotients
and the correction becomes proportional, for the case of
leptons, to mµ/mτ , and similarly for triplets of quarks
with same charge. For some quark triplets studied pre-
viously [18, 25], the general results indicate too that the
match is slightly better in the zone of low energy, and
that the discrepancy remains stable along the running.
Koide formula has had two relevant rewritings. First,
Foot [7] suggested an exact angle between the vector of
square roots of the masses and the permutation invariant
tuple:
(
√
m1,
√
m2,
√
m3)∠(1, 1, 1) = 45◦
(4)
More recently, when generalised to neutrinos [2, 6], it

2. 2
was seen that some fits with negative signs of the square
root could be need, and so nowadays a presentation more
agnostic about such signs is preferred,
√
mk = M
1
2
0 1 +
√
2 cos
2π
3
k + δ0 (5)
For the charged lepton, we have then three masses fit-
ted with two parameters
Meµτ = 313.8MeV, δeµτ = 0.222 (6)
and as we move δ0, we can produce zero and negative
values for some square root.
D. Look and M. D. Sheppeard [19] proposed (5) in a
generalised way, allowing for any real value λ0 instead of√
2. Sheppeard noticed [20] that the angles δ0 have some
extra regularity, compared to its value for leptons, when
the match is applied to the up type (uct) and the down
type (dbs) quarks: δeµτ = 3δuct and δdsb = 2δuct.
This insight was revised by Zenczykowski [27]. He
notes that the value of δ0 in (5) can be extracted directly
from
1
√
2
(
√
m2 −
√
m1) = 3M0 sin δ0 (7)
1
√
6
(2
√
m3 −
√
m2 −
√
m1) = 3M0 cos δ0 (8)
and then he studies its experimental value independently
of the whole formula, confirming
δeµτ = 3δuct (9)
The above equations are related to the initial research
on mixing angles, as Koide used the combination of
(7),(8) to produce Cabibbo angle, while a third condi-
tion
1
√
3
(
√
m3 +
√
m2 +
√
m1) = 3M0 (10)
joined to the previous two, implies Koide formula from a
pythagorean condition on the LHS of the three equations
(7)2
+ (8)2
= (10)2
(11)
The above equations relate to the doublet and singlet
irreps of S3, where δ0 can be understood a rotation of
the basis for the bidimensional irrep.
Generalisations of Koide equation to neutrinos have
been addressed in the literature but without a conclusive
result. We do not address them in this paper. Last, ex-
tensions to six-quark equations where done by Goffinet[9,
p. 78] and more recently by Kartavsev[11], but a definite
method to build tuples with an arbitrary number of par-
ticles has not been built.
III. RECENT EMPIRICAL FINDINGS
Two years ago, Rodejohann and Zhang [18] noticed
that if one forgets the prejudice of using equal-charge
quarks, then the triplet (c, b, t) matches Koide equation
accurately.
We can inquire if this success is repeatable and we
can keep connecting triplets of quarks with alternating
isospin. Note that already the first solution of Harari et
al involved quarks of different charge.
To explore this possibility, we solve (3) to get one of
the masses as a function of the other two
m3 = (
√
m1 +
√
m2) 2 − 3 + 6
√
m1m2
(
√
m1 +
√
m2)2
2
(12)
And then iterate a waterfall from experimental values
of top and bottom [1]:
mt = 173.5 GeV
mb = 4.18 GeV
mc(173.5, 4.18) = 1.374 GeV
ms(4.18, 1.374) = 94 MeV
mq(1.374, 0.094) = 0.030 MeV
mq (0.094, 0.000030) = 5.5 MeV
(13)
Experimental mc and ms are, respectively, 1.275 ±
0.025 GeV and 95 ± 5 MeV. So a new Koide triplet, not
detected previously, is apparent: (s, c, b). The reason of
the miss, besides the need of alternate isospin, is that the
solution to in this case has a negative sign for
√
ms.
This has also the collateral effect that
−
√
ms,
√
mc,
√
mb is almost orthogonal to the lep-
tonic tuple
√
mτ ,
√
mµ,
√
me. From the view of Foot’s
cone, they are in opposite generatrices.
That we can choose such negative signs in the search
for Koide triplets can be inferred for the rewritting (5)
above, as they are produced forcefully for some values
of the angle. Furthermore, the formalism of equations
(7),(8) and (10) allows to justify, in a model independent
way, that the sign of the square root can be arbitrary
with the only condition of keeping the same choosing in
the three equations. Formally, their LHS can be seen as
a row sum of the product matrix Q,
Q =



0 1√
2
− 1√
2
2√
6
− 1√
6
− 1√
6
1√
3
1√
3
1√
3





√
m3 0 0
0
√
m2 0
0 0
√
m1

 (14)
The matrix Q being constructed so that QQ+
(and
Q+
Q too) is a mass matrix having m1, m2, m3 as eigen-
values, independently of the choosing of signs (or phase)
in the square root of the diagonal matrix. Of course, the
argument to impose the pythagorean equation in the sum
of each row will be model dependent.
In the obtained chain of equations, we infer that q and
q correspond respectively to the masses of up and down

3. 3
quarks, and so the other two triplets are (usc) and (dus)
A zero mass for the up quark allows to suspect that we are
seeing some symmetry which is only slightly displaced.
Plus, a situation where the up quark is massless at some
stage of the breaking constitutes an escape from the QCD
θ problem.
We can question if there is some other lost triplet be-
tween the 20 possible combinations. From the empir-
ical masses in [1] we can check that only two triplets
meet Koide equation (for either sign) with a 1% of toler-
ance: (cbt) and (scb). Next, (usc), (dsb) and (dsc) can fit
within a 10%, and the rest of the triplets go worse. If we
use instead the GUT-level masses of [26], the five better
matches, within a 10% of tolerance, are still the same but
now neither of them are in the 1% accuracy, again sug-
gesting that the formula works better in the electroweak
scale than at GUT energies.
Comparing the “waterfall” chain with this exploration,
the disagreement comes only with the last step, (dus),
obviously because of the non zero mass of the up quark
in the measured data. But the alternatives really only
match Koide equation within a ten percent and break
the pattern of chaining, that here was proceeding reg-
ularly (generation-wise and isospin-wise). We will com-
ment briefly on these alternatives later; their main differ-
ence comes in the way they manage to derive the down
quark mass, and the two next sections do not involve it.
IV. A PREDICTION UPSTREAM
In the extra symmetry revised by Zenczykowski [27],
the angle δ0 of charged leptons was three times the an-
gle for up-type quarks. A similar proportionality is also
happening here. For the triplet bottom-charm-strange,
the fitted parameters are
Mscb = 941.2MeV, δscb = 0.671 (15)
Here not only the angle, but also the mass seems re-
lated by a factor three.
Were we to accept the relationships Mscb = 3Meµτ
and δscb = 3δeµτ , the set of Koide equations would be
the most predictive semi-empirical formula for standard
model masses: taking electron and muon as input, we
can build the parameters of the lepton triplet and then
seed them to produce the (scb) triplet. The chain pre-
dicts good values for down, strange, charm, bottom, and
top quarks, as well as for the tau lepton. For the top
quark, with usual values of me and mµ as input, the pre-
dicted mass is 173.264 GeV, right in the target of current
averages, from Tevatron and LHC, of the pole mass.
For reference, the complete comparison between exper-
iment and prediction is:
exp. pred.
t 173.5 ± 1.0 173.26
b 4.18 ± 0.03 4.197
c 1.275 ± 0.025 1.359
τ 1.77682(16) 1.776968
s 95 ± 5 92.275
d ∼ 4.8 5.32
u ∼ 2.3 .0356
(16)
Where we do not quote error widths in the prediction
side because the input data, electron and muon mass,
is known with a precision higher than the rest of the
measurements, so all shown digits are exact. The table
can be taken more as an illustration of the power of Koide
triplets than as a working prediction; the experimental
quark masses are not the the pole masses, except for
the top quark, and the relationships Mscb = 3Meµτ and
δscb = 3δeµτ are ad-hoc as explained.
V. RESULTS WITH SIMPLE INPUTS
To understand the factors of three in the last section,
it is instructive to examine a case without experimental
inputs, considering instead that the Yukawa coupling of
the top quark is exactly one and the Yuwaka coupling
of the up quark is exactly zero. The point of asking for
yu = 0 is that there is the possibility of having some
extra symmetry that becomes restored in this limit.
Recall from the standard model that when yt = 1, the
mass of the top is 174.1 GeV, via mq = yq v /
√
2; we are
going to use this proportionality instead of quoting the
adimensional value of the yukawa coupling.
We should consider the four possible solutions in each
step of equation (12). A fast exploration with a computer
shows that it is possible to try different combinations of
triplets to get the goal yu = 0, but that the best matching
with data is the case of our sequential chain, and that it
is also the case with the greatest gap between top and
bottom. Thus we are going to use this fact as a postulate
to select the solution, in two steps:
First, we ask the difference between the top and the
next quark to be the greatest possible within Koide equa-
tion. This amounts to set the angle δcbt = 0. The masses
of bottom and charm are then the same, 2.56 GeV. The
next triplet has an angle, δscb = 60◦
and predicts a mass
for the strange quark of 150 MeV. And then the next
triplet has an angle of 12.62◦
and predicts a small but
non zero mass for the next quark: 0.810 MeV. All these
steps are unambiguous.
So this case with a single input, yt = 1, already pro-
vides a good match with the standard model patterns.
Now we proceed to smoothly move the parameters in or-
der to approach this yu to zero, fixng it at the cost of

4. 4
losing exact angles in the upper mass triplets. We get:
mt 174.1 GeV
mb 3.64 GeV
mτ mc 1.698 GeV
mµ ms 121.95 MeV
me mu 0 MeV
md 8.75 MeV
(17)
In the above table, we have chosen by hand a position for
lepton masses. This has some value: contemplate simply
the lower part of the table, written now as
mb 4Mscb
mτ mc
2+
√
3
2 Mscb
mµ ms
2−
√
3
2 Mscb
me mu 0
(18)
It can be seen that in this case Meµτ = Musc = Mscb/3,
giving some ground to the ad-hoc use of this factor in the
previous section; and also that the angles are one of them
three times the other: δeµτ = 15◦
and δscb = 45◦
, then
agreeing both with our observation and Zenczykowski’s.
VI. DISCUSSION
A question about the validity of Koide triplets is if they
are equations for the pole masses or for running masses,
and if the later, for which scale should they be applied.
Looking at the running, it can be argued that they can
be used by modellers in any situation, as they are good at
electroweak scale and still approximate enough at GUT
scale, but that their better agreement seems to happen
with pole masses (or, for quarks, for mq(mq) values).
This was already known for the lepton triplet, and here
we have provided an ”upstream prediction” going from
the e, µ pole mass as input to the experimental top mass,
which is also a pole mass.
Consider (s, c, b). For MS masses in current data [1],
the quotient LHS/RHS of (3) is 0.986. The value con-
temporary to [26] was 0.974. Using running masses from
[26], the quotient at MZ is 0.949, and at GUT scale it is
0.947.
Pole masses can be more natural on models where mix-
ing is the origin of Koide equation, while running masses
are appropriate for models where a symmetry breaking
at some scale produces the mass pattern.
There is the question of the model itself. We do not aim
in this paper to propose such; there are some attempts
in the literature to include more symmetry. Now, we can
say some words about the fact that we seem to have not
one, but four equations in the quark sector.
We believe that S4 = V4 S3 is an interesting candi-
date for a bigger group. It is the group of rotations of a
cube, and in fact we can classify our triplets by drawing
such cube with quarks in the faces, such that u, c, t meet
in a vertex, with respective opposite faces b, d, s. The
vertexes, arranged by oppositeness across the diagonal,
are
bds usc scb cbt
uct btd tdb dus
(19)
These four diagonal axis, have associated S3 sub-
groups, and each axis has at least one of the triplets
we are interested on. It can be noticed that an axis is
“overloaded” with the triplet dus, and this can become
an argument to substitute it by bds. Some other substi-
tutions are possible, and while we can not concrete the
option without resorting to an explicit model, the point
of being able to select here, from the 20 possible triplets,
the ones having a best matching with data, is an argu-
ment to use S4 at least for an initial organisation.
Lets conclude by reviewing what we have found in this
paper: we have done a complete exploration of Koide
quark triplets. We have found at least a new exact tuple
for Koide equation, for quarks s, c, b, and two more, ap-
proximate, for u, s, c and d, u, s; even if the later is not
really new in the literature and we could be in doubt
about substituting it for d, s, b.
We noticed that these tuples allow to build a chain of
equations, starting from c, b, t from [18]; we have verified
its empirical validity and we have suggested yet two other
methods to reproduce the mass scales of the Standard
Model fermions: either to fix the yukawas of top and up
quark to the natural values 1 and 0 or, independently,
to propose a proportionality between the parameters of
the charged leptons tuple and those of the s, c, b tuple.
Both methods have some virtue: if the last method is
applied using as input the experimental values of two
lepton masses, all the quark masses, except the up quark,
coincide with the experimental measurements within one
or two sigmas at most, and a very good prediction of
the top quark pole mass is reached. And in the former
method, with fixed yt = 1, and yu = 0, we could tell
that the only input from experiment is Fermi constant
and the number of generations, and that Koide triplets
provide all the necessary levels for mass.
ACKNOWLEDGMENTS
The author wants to acknowledge C. Brannen, M.
Porter and J. Yablon by their support and comments.
[1] J. Beringer et al. (Particle Data Group), Phys. Rev. D86,
010001 (2012)
[2] C. A. Brannen, “The Lepton Masses”, preprint (2006)
[3] N. Cabibbo and L. Maiani, Phys. Lett. B 28, 131 (1968).

5. 5
[4] A. Ebrahim, Phys. Lett. B 73, 181 (1978).
[5] J. M. Frere, Phys. Lett. B 80, 369 (1979).
[6] H. Fritzsch, arXiv:hep-ph/0207279.
[7] R. Foot, arXiv:hep-ph/9402242.
[8] R. Gatto, G. Sartori and M. Tonin, Phys. Lett. B 28,
128 (1968).
[9] F. Goffinet, “A Bottom-up approach to fermion masses”,
Th`ese Univ. Cath. de Louvain, (2008)
[10] H. Harari, H. Haut and J. Weyers, Phys. Lett. B 78, 459
(1978).
[11] A. Kartavtsev, arXiv:1111.0480 [hep-ph].
[12] Y. Koide, Phys. Rev. Lett. 47 (1981) 1241.
[13] Y. Koide, “Quark And Lepton Masses Speculated From
A Subquark Model,” preprint (1981)
[14] Y. Koide, Phys. Lett. B 120, 161 (1983).
[15] Y. Koide, Phys. Rev. D 28 (1983) 252.
[16] Y. Koide, Mod. Phys. Lett. A 5, 2319 (1990).
[17] Y. Koide, Phys. Rev. D 79, 033009 (2009)
[arXiv:0811.3470 [hep-ph]].
[18] W. Rodejohann and H. Zhang, Phys. Lett. B 698 (2011)
152 [arXiv:1101.5525 [hep-ph]].
[19] M. D. Sheppeard, blog entry, “Update 19”, (2010)
http://pseudomonad.blogspot.com.es/2010/11/
theory-update-19.html
[20] M. D. Sheppeard, blog entry, “Lesson 342”, (2010)
http://pseudomonad.blogspot.com.es/2010/07/
m-theory-lesson-342.html
[21] Y. Sumino, Phys. Lett. B 671, 477 (2009)
[arXiv:0812.2090 [hep-ph]].
[22] Y. Sumino, JHEP 0905, 075 (2009) [arXiv:0812.2103
[hep-ph]].
[23] S. Weinberg, Trans. New York Acad. Sci. 38, 185 (1977).
[24] F. Wilczek and A. Zee, Phys. Lett. B 70, 418 (1977)
[Erratum-ibid. 72B, 504 (1978)].
[25] Z.Z. Xing and H. Zhang, Phys. Lett. B 635, 107 (2006)
[hep-ph/0602134].
[26] Z.Z. Xing, H. Zhang and S. Zhou, Phys. Rev. D 77,
113016 (2008) [arXiv:0712.1419 [hep-ph]].
[27] P. Zenczykowski, Phys. Rev. D 86, 117303 (2012)
[arXiv:1210.4125 [hep-ph]].