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Open Physics

formerly Central European Journal of Physics

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Volume 16, Issue 1


Volume 13 (2015)

A Note on Koide’s Doubly Special Parametrization of Quark Masses

Piotr Żenczykowski
  • Corresponding author
  • Division of Theoretical Physics, the Henryk Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences, Radzikowskiego 152, 31-342, Kraków, Poland
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Published Online: 2018-07-17 | DOI: https://doi.org/10.1515/phys-2018-0058


Three charged lepton masses may be expressed in terms of a Z3-symmetric parametrization relevant for the discussion of Koide’s formula. After disregarding the overall scale parameter, the observed pattern of lepton masses can be described extremely well if the remaining two parameters acquire the unexpectedly simple values of 1 and 2/9. We argue that an analogue of this doubly special feature of the parametrization can also be seen in the quark sector provided that the mass of the strange quark is taken to be around 160 MeV, as might be expected in the low-energy regime.

Keywords: Koide formula; quark and lepton masses

PACS: 12.60.-i; 12.15.Ff

1 Introduction

Despite the general acceptance of the Higgs mechanism, a truly successful resolution of the problem of mass (i.e. the actual prediction of the masses of fundamental fermions) still eludes our understanding. Hopefully, the search for such an understanding could be guided by the analysis of the observed pattern of particle masses. The identified regularities might then serve as the analogues of the Balmer formula, thus helping us decipher the new physics.

2 Koide’s parametrization

Koide discovered one of the most interesting regularities. Namely, it appears that the three charged lepton (L) masses satisfy an empirical relation [1]:


with kL being almost exactly 1.

Indeed, if one takes kL = 1 and uses the central values of experimental electron and muon masses (the pole masses me = 0.510998928(11) MeV and mμ = 105.65836715(35) MeV [2]), the larger of the two solutions of Eq. (1) is


while the experimental τ mass mτ is


The parameter kL entering into Eq. (1) may be defined through the following Z3-symmetric parametrization [3] of three arbitrary masses (m1, m2, m3):


(with f = L) in terms of three parameters: the overall mass scale Mf, the average spread kf of the three masses (relative to the scale Mf) and the pattern/phase parameter δf. As δf is free, we assume that m1m2m3. The above parametrization is particularly suited to Koide’s formula, as the latter appears independent of parameter δf. Furthermore, from (4) one can derive a counterpart of Eq. (1) that depends on δL but is independent of kL [4]:


It was observed by Brannen and Rosen [5, 6] that the three charged lepton masses satisfy Eq.(5) with δL being almost exactly 2/9. Indeed, if one assumes that δL is 2/9, one can use Eq.(5) and the experimental values of electron and muon masses to predict the mass of the tauon:


This prediction of the tauon mass is as good as that given in Eq.(2) by Koide’s original formula. For this reason Koide’s parametrization may be rightly called ’doubly special’. While the origin of values kL = 1 and δL = 2/9 is unknown, the appearance of such numbers suggests the existence of a fairly simple underlying algebraic framework.

Although the two predictions of Eqs (2) and (6) are incompatible with each other, the degree of their incompatibility is extremely small. Thus, once the overall scale ML is properly adjusted, the choice of kL = 1 and δL = 2/9 gives an almost perfect description of the charged lepton masses. The observed tiny deviations from the (kL = 1, δL = 2/9) case could then be attributed to some higher order corrections of the underlying scheme.

3 Modification of parametrization

A peculiar feature of Koide’s and Brannen-Rosen’s observations is that if instead of the charged lepton pole masses one takes their running masses as evaluated at a higher mass scale μ, the left-hand sides of Eqs (1) and (5) correspond to kL and δL deviating more from their ’perfect’ values of 1 and 2/9 (at the mass scale of MZ the deviations are 0.2 % and 0.5% respectively [7, 8]). This suggests that a deeper understanding of the observed regularity should be sought at the low (and not high) energy scale. Another peculiarity is the actual value of the phase parameter δL which is 2/9 radians (and not a simple fraction times π that might be expected). This value of δL suggests a less geometric and more algebraic origin of the regularities observed in the lepton spectrum. One may then wonder if replacing the spread parameter kf by its logarithm, i.e.


- which introduces a completely parallel treatment of the spread (λf) and pattern (δf) parameters (via the dependence of the r.h.s. of (4) on a single complex argument λf+iδf) - could provide a welcome translation to the algebraic level suggested by the peculiar value of δL. After all, the ‘simple’ value of kL = 1 corresponds to the equally ‘simple’ value for λL, i.e. 0. Obviously, the replacement of kf by λf may be productive only if it leads to a novel observation elsewhere. Now, it should be noted that the phenomenological analysis performed in [4] suggested the set of values δL = 2/9, δD = 4/27, δU = 2/27, and δν = 0 for charged leptons, quarks (f = D, U), and neutrinos. With λL = 0 belonging to this set, it seems that a replacement of kL by λL (a counterpart of δf’s) might be a good idea. We need to investigate if other values of λf also belong to this set. In order to analyse this point we turn to quarks where the values of kf are known to deviate significantly from 1 if the running quark masses are used. Indeed, at the mass scale μ = MZ one obtains: for the down quarks kD ≈ 1.1, and for the up quarks kU ≈ 1.3 [7, 9].

4 Analysis

Equations (1) and (5) work well for lepton pole masses (and not for running masses) to such a high degree of accuracy that it seems appropriate to consider the counterparts of these masses in the quark sector as well. This suggests that the pole masses should be used for the heavy (b, c, t) quarks and the low-energy scale masses should be utilised for the light (u, d, s) quarks.

Furthermore, we observe that Eqs (1) and (5) depend on mass ratios (z1 = m1/m3, z2 = m2/m3) alone. In order to study the quark sector in detail it is therefore sufficient to estimate the relevant mass ratios in the light and heavy sectors as well as the relative mass scale of the two sectors. Now, the relative mass ratios can be estimated independently in the light and heavy sectors.

In the light quark sector the results of lattice calculations (supplemented with the phenomenological studies of isospin breaking effects) give (in the MS scheme at the renormalization scale μ = 2 GeV) [10]:


These numbers agree very well with Weinberg’s old low-energy estimates based on the ratios of pion and kaon masses [11]:


We may therefore safely assume that mu/md = 0.50 ± 0.05 and that ms/md = 20.0.

In the sector of heavy quarks the perturbatively calculated pole masses mq are related to the running mq(μ) masses via [10]:


where the dots symbolize substantial higher order corrections [10]. As these corrections are quite uncertain we treat the relevant independent ratios of heavy quark pole masses (i.e. mc/mt, mb/mt) in two ways.

First, we approximate these ratios by mc(mc)/mt(mt) and mb(mb)/mt(mt), where mq(mq) are the central values of the estimates given in [12] (in GeV):


With mc(mc)/mt(mt) and mb(mb)/mt(mt) being pure numbers and mt, at around 160 – 170 GeV, setting the absolute scale of heavy quark masses, the relative scale of the light and heavy quark masses may be parametrized by ms. The choice of ms fixes then the values of the four mass ratios z1 and z2 (in the up and down quark sectors). In a previous paper [4] it was argued that the pattern of phases δf is particularly simple for ms ≈ 160 MeV (i.e. not for the value of ms of the order of 90 – 100 MeV which is appropriate for μ = 2 GeV). It is therefore interesting to see how the values of λD and λU change when ms varies from 90 to 160 MeV or so. The upper part of Table 1 shows the relevant ms-dependence of λD and λU obtained using (in the estimates of mc/mt, ms/mb, etc.) the values given in Eq. (11).

Table 1

Values of λD and λU

Second, instead of employing Eq. (11), we directly use the estimates of pole masses given in [12] (i.e. (mc,mb,mt) = (1.84, 4.92, 172.9)) (in GeV). The corresponding variations of λD and λU are presented in the lower part of Table 1. We view the differences between the upper and lower parts of Table 1 as providing an estimate of the errors involved.

We observe that for ms ≈ 160 – 170 MeV the values of λU,D in the quark sector are not far from λL = 0 and δL = 2/9. Deviations from these values may be tentatively assigned (as in the case of charged leptons) to some higher order corrections (which, on account of strong interactions being involved, could be larger than in the lepton case). Note that the regularities in question (for λU,D here and for δU,D in [4]) are observed at approximately the same value of ms ≈ 160 MeV.

5 Conclusions

In conclusion it seems that there is a numerical hint that the doubly special character of Koide’s parametrization can also be seen in the quark sector (with both λD and λU belonging to the set (0, 2/27, 4/27, 2/9)) (provided ms is taken to be around 160 MeV).


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About the article

Received: 2017-03-18

Accepted: 2018-05-10

Published Online: 2018-07-17

Citation Information: Open Physics, Volume 16, Issue 1, Pages 427–429, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0058.

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© 2018 P. Żenczykowski et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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