# Explorations of two empirical formulas for fermion masses

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## Abstract

Two empirical formulas for the lepton and quark masses (i.e. Kartavtsev’s extended Koide formulas), \(K_l=(\sum _l m_l)/(\sum _l\sqrt{m_l})^2=2/3\) and \(K_q=(\sum _q m_q)/(\sum _q\sqrt{m_q})^2=2/3\), are explored in this paper. For the lepton sector, we show that \(K_l=2/3\), only if the uncertainty of the tauon mass is relaxed to about \(2\sigma \) confidence level, and the neutrino masses can consequently be extracted with the current experimental data. For the quark sector, the extended Koide formula should only be applied to the running quark masses, and \(K_q\) is found to be rather insensitive to the renormalization effects in a large range of energy scales from GeV to \(10^{12}\) GeV. We find that \(K_q\) is always slightly larger than 2/3, but the discrepancy is merely about 5 %.

### Keywords

Energy Scale Quark Masse Neutrino Masse Charged Lepton Mass Hierarchy## 1 Introduction

Despite the glorious successes of the standard model of particle physics, the generations of fermion masses remain one of the most fundamental but unsolved problems therein. These masses are treated as free parameters in the standard model, which seem to be rather dispersed and unrelated and can only be determined experimentally. Therefore, it is reasonable to first seek some phenomenological relations of these masses in order to reduce the number of free parameters, and this will significantly help us for the future model buildings in and beyond the standard model.

This remarkable precision has aroused in both theorists [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] and phenomenologists [26, 27, 28, 29, 30, 31, 32, 33, 34] a longtime interest in the Koide formula, but unfortunately the underlying physics remains incomplete. The previous studies can be classified into three categories: (1) to explore the possible physical origin of the Koide formula (e.g. in a supersymmetric model [19] or in an effective field theory [20, 21, 22]); (2) to generalize the Koide formula from charged leptons to neutrinos and quarks (see Refs. [33] and [34] for example); (3) to examine the energy scale dependence of the Koide formula [18] (i.e. to check the stability of the Koide formula against radiative corrections, with the pole masses replaced by the running masses of fermions). In the present paper, we focus on the latter two aspects for two extended Koide formulas suggested by Kartavtsev in Ref. [35]. Before doing so, we briefly review the relevant work on the extensions of the Koide formula from charged leptons to neutrinos and quarks.

*u*,

*d*,

*s*) and heavy (

*c*,

*b*,

*t*) quarks, they separately introduced

These preliminary successes enlighten us to investigate Kartavtsev’s extensions of the Koide formula in detail, since the naive estimates in Ref. [35] were still very simple. We should first of all explore phenomenologically the validity of Eqs. (3) and (4) for both leptons and quarks, and these explorations will be greatly helpful for the relevant model building for the theoretical explanation of the extended Koide formulas. This is the purpose of our present paper.

The first 10 minima of \(|K_l-2/3|\) and the corresponding neutrino masses from the extended Koide formula in the normal mass hierarchy. The values of \(|K_l-2/3|\) are greatly larger than the precision of operation, meaning that the extended Koide formula for leptons is unsuccessful under the constraints from current experimental data.

\(|K_l-2/3|\) | \(m_1\) (eV) | \(m_2\) (eV) | \(m_3\) (eV) | \(m_e\) (MeV) | \(m_\mu \) (MeV) | \(m_\tau \) (MeV) |
---|---|---|---|---|---|---|

\(7.28894\times 10^{-6}\) | \(4.89844\times 10^{-8}\) | \(8.58877\times 10^{-3}\) | \(4.95356\times 10^{-2}\) | 0.510999 | 105.658 | 1776.98 |

\(7.29051\times 10^{-6}\) | \(1.85238\times 10^{-8}\) | \(8.58196\times 10^{-3}\) | \(4.95343\times 10^{-2}\) | 0.510999 | 105.658 | 1776.98 |

\(7.29142\times 10^{-6}\) | \(9.25236\times 10^{-8}\) | \(8.59159\times 10^{-3}\) | \(4.95360\times 10^{-2}\) | 0.510999 | 105.658 | 1776.98 |

\(7.30195\times 10^{-6}\) | \(2.81761\times 10^{-7}\) | \(8.62488\times 10^{-3}\) | \(4.95420\times 10^{-2}\) | 0.510999 | 105.658 | 1776.98 |

\(7.31602\times 10^{-6}\) | \(6.75381\times 10^{-8}\) | \(8.60216\times 10^{-3}\) | \(4.99676\times 10^{-2}\) | 0.510999 | 105.658 | 1776.98 |

\(7.32170\times 10^{-6}\) | \(3.37241\times 10^{-10}\) | \(8.57321\times 10^{-3}\) | \(5.02470\times 10^{-2}\) | 0.510999 | 105.658 | 1776.98 |

\(7.35571\times 10^{-6}\) | \(6.34196\times 10^{-6}\) | \(8.58244\times 10^{-3}\) | \(4.97131\times 10^{-2}\) | 0.510999 | 105.658 | 1776.98 |

\(8.32900\times 10^{-6}\) | \(1.67259\times 10^{-3}\) | \(8.73485\times 10^{-3}\) | \(4.95610\times 10^{-2}\) | 0.510999 | 105.658 | 1776.98 |

\(1.00832\times 10^{-5}\) | \(6.54993\times 10^{-3}\) | \(1.07890\times 10^{-2}\) | \(4.99830\times 10^{-2}\) | 0.510999 | 105.658 | 1776.97 |

\(1.71168\times 10^{-5}\) | \(4.87567\times 10^{-2}\) | \(4.95407\times 10^{-2}\) | \(6.95290\times 10^{-2}\) | 0.510999 | 105.658 | 1776.98 |

## 2 Extended Koide formula for leptons

In this section, we examine the extended Koide formula for leptons. However, this can only be performed with the masses of all the six leptons known. Unfortunately, the lack of the absolute masses of neutrinos makes this examination impossible. But on the other hand, thanks to the more and more precise experiments of neutrino oscillations, we have already had two firm constraints of the mass-squared differences of the neutrino mass eigenstates. Therefore, if we assume the validity of the extended Koide formula for leptons (i.e., \(K_l=2/3\) exactly) and regard it as the third constraint, the three neutrino masses may thus be extracted.

From Table 1, we clearly observe that although the values of \(|K_l-2/3|\) are already very small \(\sim \mathcal {O}(10^{-6})\), they are still greatly beyond the computing accuracy that we adopt. Since we have set the precision of operation to 30 significant figures, \(|K_l-2/3|\) should be \(\sim \mathcal {O}(10^{-30})\), if the masses \(m_1\), \(m_2\), \(m_3\) make \(|K_l-2/3|\) converge to zero absolutely. As a result, we conclude that there is no solution for neutrino masses from Eq. (5) under the current experimental constraints in Eqs. (6)–(10). In other words, Kartavtsev’s extended Koide formula for leptons in Eq. (5) is unsuccessful. This situation is the same for the inverted mass hierarchy.

The running quark masses and the running parameter \(K_q(\mu )\) in the extended Koide formula at some typical energy scales. The mass of the Higgs boson is taken as 125 GeV, and the cutoff scale for vacuum stability is \(4\times 10^{12}\) GeV (data from Table 1 of Ref. [38]). The running quark masses are found to decrease monotonically, but \(K_q(\mu )\) is almost stable in a sizable range of energy scales from GeV to \(10^{12}\) GeV. Moreover, \(K_q(\mu )>2/3\) at all energy scales, but the deviations are only about 5 %.

\(\mu \) | \(m_u(\mu )\) (MeV) | \(m_d(\mu )\) (MeV) | \(m_s(\mu )\) (MeV) | \(m_c(\mu )\) (GeV) | \(m_b(\mu )\) (GeV) | \(m_t(\mu )\) (GeV) | \(K_q(\mu )\) |
---|---|---|---|---|---|---|---|

\(m_c(m_c)\) | \(2.79^{+0.83}_{-0.82}\) | \(5.69^{+0.96}_{-0.95}\) | \(116^{+36}_{-24}\) | \(1.29^{+0.05}_{-0.11}\) | \(5.95^{+0.37}_{-0.15}\) | \(385.7^{+8.1}_{-7.8}\) | \(0.701^{+0.010}_{-0.011}\) |

2 GeV | \(2.4^{+0.7}_{-0.7}\) | \(4.9^{+0.8}_{-0.8}\) | \(100^{+30}_{-20}\) | \(1.11^{+0.07}_{-0.14}\) | \(5.06^{+0.29}_{-0.11}\) | \(322.2^{+5.0}_{-4.9}\) | \(0.698^{+0.010}_{-0.011}\) |

\(m_b(m_b)\) | \(2.02^{+0.60}_{-0.60}\) | \(4.12^{+0.69}_{-0.68}\) | \(84^{+26}_{-17}\) | \(0.934^{+0.058}_{-0.120}\) | \(4.19^{+0.18}_{-0.16}\) | \(261.8^{+3.0}_{-2.9}\) | \(0.696^{+0.011}_{-0.009}\) |

\(m_W\) | \(1.39^{+0.42}_{-0.41}\) | \(2.85^{+0.49}_{-0.48}\) | \(58^{+18}_{-12}\) | \(0.645^{+0.043}_{-0.085}\) | \(2.90^{+0.16}_{-0.06}\) | \(174.2^{+1.2}_{-1.2}\) | \(0.691^{+0.010}_{-0.010}\) |

\(m_Z\) | \(1.38^{+0.42}_{-0.41}\) | \(2.82^{+0.48}_{-0.48}\) | \(57^{+18}_{-12}\) | \(0.638^{+0.043}_{-0.084}\) | \(2.86^{+0.16}_{-0.06}\) | \(172.1^{+1.2}_{-1.2}\) | \(0.692^{+0.010}_{-0.010}\) |

\(m_H\) | \(1.34^{+0.40}_{-0.40}\) | \(2.74^{+0.47}_{-0.47}\) | \(56^{+17}_{-12}\) | \(0.621^{+0.041}_{-0.082}\) | \(2.79^{+0.15}_{-0.06}\) | \(167.0^{+1.2}_{-1.2}\) | \(0.691^{+0.010}_{-0.010}\) |

\(m_t(m_t)\) | \(1.31^{+0.40}_{-0.39}\) | \(2.68^{+0.46}_{-0.46}\) | \(55^{+17}_{-11}\) | \(0.608^{+0.041}_{-0.080}\) | \(2.73^{+0.15}_{-0.06}\) | \(163.3^{+1.1}_{-1.1}\) | \(0.691^{+0.010}_{-0.010}\) |

1 TeV | \(1.17^{+0.35}_{-0.35}\) | \(2.40^{+0.42}_{-0.41}\) | \(49^{+15}_{-10}\) | \(0.543^{+0.037}_{-0.072}\) | \(2.41^{+0.14}_{-0.05}\) | \(148.1^{+1.3}_{-1.3}\) | \(0.693^{+0.010}_{-0.010}\) |

\(\Lambda _\mathrm{VS}\) | \(0.61^{+0.19}_{-0.18}\) | \(1.27^{+0.22}_{-0.22}\) | \(26^{+8}_{-5}\) | \(0.281^{+0.02}_{-0.04}\) | \(1.16^{+0.07}_{-0.02}\) | \(82.6^{+1.4}_{-1.4}\) | \(0.705^{+0.011}_{-0.011}\) |

Last, we should stress that we only focus on the pole masses of leptons in this section, but not their running masses. Since the lepton mass ratios are rather insensitive to radiative corrections [18], this is not a severe problem.

## 3 Extended Koide formula for quarks

*u*,

*d*,

*s*) are estimated as the current quark masses, the masses of relatively heavy quarks (

*c*,

*b*) mean the running quark masses, and the mass of the heaviest

*t*quark is measured as its pole mass. Therefore, it is meaningless to calculate \(K_q\) as a combination of all these six masses without distinction.

Different from leptons, quarks are confined inside hadrons and cannot be observed as physical particles in experiments, so their masses cannot be measured directly. Therefore, concerning quark masses, we should first make clear their quantitative definitions and meanings. For example, the masses of light quarks in chiral perturbation theory always mean the current quark masses. While, in a particular non-relativistic hadron model, we mean the quark masses by the constituent quark masses. Moreover, the quark masses computed directly from lattice quantum chromodynamics (QCD) are the bare quark masses. Whereas, in the Koide formula, the charged lepton masses are the pole (physical) masses, which correspond to the positions of divergence in their propagators in the on-shell renormalization scheme. However, the pole masses of quarks can only be defined in perturbation theory and are not reliable at low energies because of the non-perturbative infrared effects in QCD. Hence, the pole masses of quarks are not well defined, so the extension of the Koide formula for quarks should only refer to their running masses.

In Table 2, we list the running quark masses taken from Table 1 of Ref. [38]. These masses were calculated in the standard model at a number of typical energy scales: for example, \(m_c\) evaluated at the scale equal to its mass, 2 GeV where light quark masses are often quoted in the \(\overline{\mathrm{MS}}\) scheme, the Higgs mass \(m_H\approx 125\) GeV, until the cutoff scale \(\Lambda _\mathrm{VS}\approx 4\times 10^{12}\) GeV, where the vacuum stability in the standard model is lost due to a relatively small Higgs mass. We clearly see that the running quark masses monotonously decrease at large energy scales. The running parameter \(K_q(\mu )\) in the extended Koide formula is also listed in the last column in Table 2.

*t*quark mass. Besides, we do not find that \(K_q(\mu )\) crosses 2/3 at some particular energy scale \(2~\mathrm{GeV}<\mu <m_Z\), as claimed by Kartavtsev in Ref. [35], since this cross was naively estimated from the data in Eq. (11). But as we have explained above, these values of quark masses in Eq. (11) have different definitions and cannot be consulted simultaneously. The stability of \(K_q(\mu )\) against the running effects is also illustrated in Fig. 2.

## 4 Conclusions and discussions

The standard model of particle physics has achieved triumphant successes in the last five decades. However, one of the most crucial shortcomings therein is a large number of free parameters, including 12 fermion masses. Any reduction of this number will pave a way for our comprehension of the underlying flavor physics. The Koide formula is one of the appealing attempts in this direction. Unfortunately, this formula only associates the masses of three charged leptons, but not of all the 12 flavors of fermions. However, charged leptons should not be particular in fermions, so the idea to extend the original Koide formula, including all leptons and quarks on an equal footing, is thus very natural and desirable. Kartavtsev’s extensions [35] in Eqs. (3) and (4) treated six leptons and quarks in a totally democratic manner, with a maximal *S*(6) permutation symmetry, and a preliminary estimate indicated a certain plausibility of these extensions.

In the present paper, we explore Kartavtsev’s extended Koide formulas for both leptons and quarks at length. For the lepton sector, it proves that \(K_l\) cannot be equal to 2/3 exactly with the current experimental data of the charged lepton masses and the mass-squared differences of neutrinos within \(1\sigma \) confidence level. Then our strategy is to assume the rigorous validity of the extended Koide formula for leptons and relax the uncertainty of the most inaccurate tauon mass \(m_\tau \). By this means, the neutrino masses can be extracted from the extended Koide formula, if the uncertainty of \(m_\tau \) is relaxed to about \(2\sigma \) confidence level in both the normal and the inverted mass hierarchies. The central values for three neutrino masses read: \(m_1=2.06\times 10^{-4}\) eV, \(m_2=8.68\times 10^{-3}\) eV, \(m_3=5.02\times 10^{-2}\) eV (normal hierarchy), and \(m_3=2.00\times 10^{-4}\) eV, \(m_1=4.94\times 10^{-2}\) eV, \(m_2=5.02\times 10^{-2}\) eV (inverted hierarchy). These results are consistent with the most stringent upper bound on the sum of neutrino masses from the measurements of the cosmic microwave background temperature spectra from the WMAP and Planck satellite experiments: \(m_1+m_2+m_3<0.66\) eV (95 % confidence level) [39], and also from the data combined with the baryon acoustic oscillations: \(m_1+m_2+m_3<0.23\) eV (95 % confidence level) [39]. It is interesting to note that, even if the neutrino masses increase near this cosmological bound (e.g. \(m_1+m_2+m_3\approx 0.23\) eV), with the constraints in Eqs. (6) and (7), the discrepancies between \(K_l\) and 2 / 3 almost remain unchanged: \(2.0\times 10^{-5}<|K_l-2/3|<3.8\times 10^{-5}\) for the normal mass hierarchy and \(2.0\times 10^{-5}<|K_l-2/3|<3.9\times 10^{-5}\) for the inverted mass hierarchy. This is understandable, as the uncertainty of \(K_l\) mainly comes from the uncertainties of charged leptons.

For the quark sector, the various definitions of quark masses greatly complicate the situation. The pole masses in the Koide formula become ill-defined for the light quarks, due to the non-perturbative effects in QCD at low energies. Therefore, the exploration of the extended Koide formula should only be implemented for the running quark masses. We find that the running parameter \(K_q(\mu )\) is almost stable in a very large range of energy scales from GeV to \(10^{12}\) GeV, mainly as a result of the large mass hierarchy in the quark sector. However, \(K_q(\mu )\) is always slightly larger than 2 / 3, meaning the invalidity of the extended Koide formula for the running quark masses, but this deviation is merely about 5 %. We omit the discussion of the running behavior of the extended Koide formula in the lepton sector, as the running effects are negligibly tiny because the lepton mass ratios are rather insensitive to radiative corrections [18].

Below, we give some general discussions on the Koide formula. The mystery of the Koide formula is twofold. The first is its surprising simplicity and accuracy, but only for charged leptons. The inclusion of neutrinos is a reasonable balance between the charged and uncharged leptons, but this inclusion is meaningful only if the generation mechanism of neutrino masses is the same as that of charged lepton masses (i.e. neutrinos are of Dirac type). On the contrary, if the tiny neutrino masses are generated from the seesaw mechanism [40, 41, 42], Kartavtsev’s extension of the Koide formula will be pointless due to the Majorana mass term. The second is that the Koide formula consists of the pole masses of fermions, which are the low energy quantities and are defined at different energy scales. This is extremely counterintuitive, since we always expect simple formulas at high energy scales, where some symmetries are restored. Hence, the renormalization effects will not allow the Koide-like formulas for both the pole and the running fermion masses simultaneously.

Finally, we should point out that Kartavtsev’s extension of the Koide formula [35] and our corresponding detailed explorations are still at the phenomenological level. A similar work (the most general extension of the Koide formula), taking all the 12 fermions into account, i.e. \((\sum _f m_f)/(\sum _f\sqrt{m_f})^2=2/3\), is also not quite successful. Therefore, a possible direction for further extensions of the Koide-like formulas is to seek the theoretical basis of these empirical relations, as Koide originally did in a composite model or an extended technicolor-like model [1, 2, 3]. We should incorporate in the extended Koide formulas the elements and the mixing and phase angles in the lepton and quark mixing matrices [43, 44, 45, 46, 47], and maybe also the fermion charges. This will be the topic for our future research.

We are very grateful to Prof. Jean-Marc Gérard for his stimulating idea and also to Fengjiao Chen for fruitful discussions. This work is supported by the Fundamental Research Funds for the Central Universities of China (No. N140504008).

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