Open Phys. 2018; 16:427–429

Research Article

Piotr Żenczykowski* A Note on Koide’s Doubly Special Parametrization of https://doi.org/10.1515/phys-2018-0058 with kL being almost exactly 1. Received Mar 18, 2017; accepted May 10, 2018 Indeed, if one takes kL = 1 and uses the central values of experimental and masses (the Abstract: Three charged masses may be expressed pole masses me = 0.510998928(11) MeV and mµ = in terms of a Z3-symmetric parametrization relevant for 105.65836715(35) MeV [2]), the larger of the two solu- the discussion of Koide’s formula. After disregarding the tions of Eq. (1) is overall scale parameter, the observed pattern of lepton masses can be described extremely well if the remaining mτ = 1776.9689 MeV, (2) two parameters acquire the unexpectedly simple values of 1 and 2/9. We argue that an analogue of this doubly spe- while the experimental τ mτ is cial feature of the parametrization can also be seen in the mτ = 1776.82 ± 0.16 MeV. (3) quark sector provided that the mass of the is taken to be around 160 MeV,as might be expected in the The parameter kL entering into Eq. (1) may be defined low-energy regime. through the following Z3-symmetric parametrization [3] of m m m Keywords: Koide formula, quark and lepton masses three arbitrary masses ( 1, 2, 3): (︂ √ (︂ )︂)︂ √︀ √︁ 2πj PACS: 12.60.-i,12.15.Ff mj = Mf 1 + 2 kf cos + δf , (4) 3 (j = 1, 2, 3),

1 Introduction (with f = L) in terms of three parameters: the overall mass scale Mf , the average spread kf of the three masses (rela- Despite the general acceptance of the , a tive to the scale Mf ) and the pattern/phase parameter δf . i.e. truly successful resolution of the problem of mass ( the As δf is free, we assume that m1 ≤ m2 ≤ m3. The above actual prediction of the masses of fundamental fermions) parametrization is particularly suited to Koide’s formula, still eludes our understanding. Hopefully, the search for as the latter appears independent of parameter δf . Further- such an understanding could be guided by the analysis of more, from (4) one can derive a counterpart of Eq. (1) that the observed pattern of particle masses. The identified reg- depends on δL but is independent of kL [4]: ularities might then serve as the analogues of the Balmer √ √ √ 3 ( mµ − me) formula, thus helping us decipher the new physics. √ √ √ = tan δL . (5) 2 mτ − mµ − me 2 Koide’s parametrization 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, Koide discovered one of the most interesting regulari- one can use Eq.(5) and the experimental values of electron ties. Namely, it appears that the three charged lepton (L) and muon masses to predict the mass of the tauon: masses satisfy an empirical relation [1]:

2 mτ me + mµ + mτ 1 + k = 1776.9664 MeV. (6) √ √ √ = L , (1) ( me + mµ + mτ)2 3 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’. *Corresponding Author: Piotr Żenczykowski: Division of The- k δ oretical Physics, the Henryk Niewodniczański Institute of Nuclear While the origin of values L = 1 and L = 2/9 is unknown, Physics, Polish Academy of Sciences, Radzikowskiego 152, 31-342 the appearance of such numbers suggests the existence of Kraków, Poland; Email: [email protected] a fairly simple underlying algebraic framework.

Open Access. © 2018 P.Żenczykowski et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution- NonCommercial-NoDerivatives 4.0 License 428 Ë P. Żenczykowski et al.

Although the two predictions of Eqs (2) and (6) are in- 4 Analysis compatible with each other, the degree of their incompat- ibility is extremely small. Thus, once the overall scale ML Equations (1) and (5) work well for lepton pole masses (and is properly adjusted, the choice of kL = 1 and δL = 2/9 not for running masses) to such a high degree of accu- gives an almost perfect description of the charged lepton racy that it seems appropriate to consider the counterparts masses. The observed tiny deviations from the (kL = 1, of these masses in the quark sector as well. This suggests δL = 2/9) case could then be attributed to some higher that the pole masses should be used for the heavy (b, c, t) order corrections of the underlying scheme. 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 3 Modification of parametriza- on mass ratios (z1 = m1/m3, z2 = m2/m3) alone. In or- tion der to study the quark sector in detail it is therefore suffi- cient to estimate the relevant mass ratios in the light and A peculiar feature of Koide’s and Brannen-Rosen’s obser- heavy sectors as well as the relative mass scale of the two vations is that if instead of the charged lepton pole masses sectors. Now, the relative mass ratios can be estimated in- one takes their running masses as evaluated at a higher dependently in the light and heavy sectors. mass scale µ, the left-hand sides of Eqs (1) and (5) corre- In the light quark sector the results of lattice calcula- spond to kL and δL deviating more from their ’perfect’ val- tions (supplemented with the phenomenological studies ues of 1 and 2/9 (at the mass scale of MZ the deviations of isospin breaking effects) give (in the MS scheme at the are 0.2 % and 0.5% respectively [7, 8]). This suggests that renormalization scale µ = 2 GeV) [10]: a deeper understanding of the observed regularity should mu /md ≈ 0.46(5) and ms /md ≈ 19.9(2). (8) be sought at the low (and not high) energy scale. Another peculiarity is the actual value of the phase parameter δL These numbers agree very well with Weinberg’s old low- which is 2/9 radians (and not a simple fraction times π energy estimates based on the ratios of pion and kaon that might be expected). This value of δL suggests a less masses [11]: geometric and more algebraic origin of the regularities ob- served in the lepton spectrum. One may then wonder if re- mu /md ≈ 0.56, ms /md ≈ 20.2. (9) placing the spread parameter kf by its logarithm, i.e. We may therefore safely assume that mu /md = 0.50±0.05 ms m λf = ln(kf ) (7) and that / d = 20.0. In the sector of heavy quarks the perturbatively cal- - which introduces a completely parallel treatment of the culated pole masses mq are related to the running mq(µ) spread (λf ) and pattern (δf ) parameters (via the depen- masses via [10]: dence of the r.h.s. of (4) on a single complex argument [︂ ]︂ 4αs(mq) λf + iδf ) - could provide a welcome translation to the al- mq = mq(mq) 1 + + ... , (10) 3π gebraic level suggested by the peculiar value of δL. After all, the ‘simple’ value of kL = 1 corresponds to the equally where the dots symbolize substantial higher order cor- ‘simple’ value for λL, i.e. 0. Obviously, the replacement of rections [10]. As these corrections are quite uncertain we kf by λf may be productive only if it leads to a novel ob- treat the relevant independent ratios of heavy quark pole servation elsewhere. Now, it should be noted that the phe- masses (i.e. mc /mt, mb /mt) in two ways. nomenological analysis performed in [4] suggested the set First, we approximate these ratios by mc(mc)/mt(mt) of values δL = 2/9, δD = 4/27, δU = 2/27, and δν = 0 for and mb(mb)/mt(mt), where mq(mq) are the central values charged , quarks (f = D, U), and . With of the estimates given in [12] (in GeV): λ L = 0 belonging to this set, it seems that a replacement of +0.05 mc(mc) = 1.29−0.11, (11) kL by λL (a counterpart of δf ’s) might be a good idea. We m m +0.18 m m +1.1 need to investigate if other values of λf also belong to this b( b) = 4.19−0.16 and t( t) = 163.3−1. . set. In order to analyse this point we turn to quarks where With mc(mc)/mt(mt) and mb(mb)/mt(mt) being pure the values of kf are known to deviate significantly from 1 numbers and mt, at around 160 − 170 GeV, setting the ab- if the running quark masses are used. Indeed, at the mass solute scale of heavy quark masses, the relative scale of scale µ = MZ one obtains: for the down quarks kD ≈ 1.1, the light and heavy quark masses may be parametrized by and for the up quarks kU ≈ 1.3 [7, 9]. ms. The choice of ms fixes then the values of the four mass A Note on Koide’s Doubly Special Parametrization of Quark Masses Ë 429

ratios z1 and z2 (in the up and sectors). In a References previous paper [4] it was argued that the pattern of phases [1] Koide Y., Fermion-boson two-body model of quarks and lep- δf is particularly simple for ms ≈ 160 MeV (i.e. not for the tons and Cabibbo mixing, Lett. Nuovo Cim., 1982, 34, 201; A value of ms of the order of 90 − 100 MeV which is appro- fermion-boson composite model of quarks and leptons, Phys. µ GeV priate for = 2 ). It is therefore interesting to see how Lett., 1983, B120, 161-165; A new view of quark and lepton mass the values of λD and λU change when ms varies from 90 to hierarchy, Phys. Rev., 1983, D28, 252. 160 MeV or so. The upper part of Table 1 shows the rele- [2] Olive K.A., et al. (), Review of Particle vant ms-dependence of λD and λU obtained using (in the Physics, Chin. Phys., 2014, C38, 090001. [3] Koide Y., Quark and lepton mass matrices with a cyclic per- estimates of mc /mt, ms /mb, etc.) the values given in Eq. mutation invariant form, hep-ph/0005137 (unpublished); Tribi- (11). maximal mixing and a relation between neutrino and charged lepton-mass spectra, J. Phys., 2007, G34, 1653-1664.

Table 1: Values of λD and λU [4] Żenczykowski P., Remark on Koide’s Z3 symmetric parametriza- tion of quark masses, Phys. Rev., 2012, D86, 117303. [5] Brannen C., The lepton masses, http://brannenworks.com/ ms (MeV) 90 100 130 160 170 MASSES2.pdf, 2006. λ +0.094 D +0.081 +0.046 +0.016 +0.007 [6] Rosen G., Heuristic development of a Dirac-Goldhaber model for λU +0.214 +0.213 +0.212 +0.212 +0.212 lepton and quark structure, Mod. Phys. Lett., 2007, A22, 283- λD +0.112 +0.100 +0.068 +0.039 +0.031 288. [7] Xing Z.-Z., Zhang H., On the Koide-like relations for the running λU +0.194 +0.194 +0.193 +0.193 +0.192 masses of charged leptons, neutrinos and quarks, Phys. Lett., 2006, B635, 107. [8] Żenczykowski P., Koide’s Z3-symmetric parametrization, quark Second, instead of employing Eq. (11), we directly masses, and mixings, Phys.Rev., 2013, D87, 077302. use the estimates of pole masses given in [12] (i.e. [9] Rodejohann W., Zhang H., Extension of an empirical charged (mc , mb , mt) = (1.84, 4.92, 172.9)) (in GeV). The cor- lepton mass relation to the neutrino sector, Phys. Lett., 2011, B698, 152-156. responding variations of λD and λU are presented in the [10] Manohar A.V., Sachrajda C.T., Quark masses, in [2]. lower part of Table 1. We view the differences between the [11] Weinberg S., The problem of mass, Trans. N.Y. Acad. Sci., 1977, upper and lower parts of Table 1 as providing an estimate 38, 185. of the errors involved. [12] Xing Z.-Z., Zhang H., Zhou S., Impacts of the Higgs mass on vac- We observe that for ms ≈ 160 − 170 MeV the values uum stability, running fermion masses and two-body Higgs de- cays, Phys. Rev., 2012, D86, 013013. 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).