Anewempiricalapproachtoleptonandquarkmasses
Kevin Loch⇤ (Dated: 2017 February 27, updated November 11) A novel alternative to the Koide formula and it’s extensions is presented. A new lepton ratio with 2 3 asmalldimensionlessresidualke = mem⌧ /mµ and mass scaling factor ↵f =27me/m⌧ are used to construct empirical formulas for charged leptons, left-handed neutrinos and quarks. The predicted masses are in excellent agreement with known experimental values and constraints.
PACS numbers: 14.60.Pq, 12.15.Ff, 12.10.Kt
I. INTRODUCTION to find factors that cancel out with this formula to learn more about the underlying structure. We can then at- One outstanding problem in modern physics is the tempt to apply those factors to search for similar struc- seemingly unrelated particle masses that exist as free tures in other sectors. parameters in the standard model. Numerous attempts Mac Gregor proposed using different exponents of the have been made over the years to derive a systematic QED coupling constant to explain fermion lifetimes and mathematical relationship between these masses, most mass relationships using the zero energy scale ↵QED(0) famously by Yoshio Koide with his charged lepton for- [25, 26]. While we have not found convincing relation- mula [1, 2] ships with ↵QED(0) we do find
me me + mµ + m⌧ 2 ↵f = 27 (3) K = 2 . (1) m⌧ (pme + pmµ + pm⌧ ) ' 3 1 interesting with a value ↵f 128.786. This is close While this approximation works for charged leptons, and ' 1 2 to the effective QED coupling constant ↵ (M )= appears to work for some combinations of quarks with QED Z running masses, it does not work for neutrinos or at- 128.944 [27]. Despite the similarity in value to an ef- tempt to relate all fundamental fermions in a consistent fective QED coupling constant we are assuming that ↵f manner. It does however hint at the possibility of find- is a separate parameter related only to mass generation ing other more fundamental relationships. Indeed, Koide and not electric charge. With (2) and (3) we can derive and others [1–24] have used it as inspiration for extended lepton formulas models to cover other fermions. m m =9 e , While these extended Koide models are interesting we µ 1 2 (4) k 3 ↵ 3 present an alternative structure for charged leptons that e f is easily applied to other sectors. we find these formu- las appear to relate all fundamental lepton and quark me m⌧ = 27 . (5) pole (physical) masses. This allows precise predictions ↵f of neutrino and quark pole masses. If correct this would significantly reduce the number of free parameters in the With our choice of definitions for ke and ↵f we have dis- standard model (extended for neutrinos with mass). covered an interesting and surprisingly simple charged lepton mass structure that could be used to look for sim- ilar patterns in other sectors. II. CHARGED LEPTONS
We begin by introducing a charged lepton formula with III. NEUTRINO SECTOR asmalldimensionlessresidualvalue We will now use ↵f and a sector structure formula similar to (2) to look for formulas for left handed neu- 2 mem⌧ trinos and predict mass state values for them. While ke = 3 1.37. (2) mµ ' neutrino mass states have not been directly measured yet numerous neutrino oscillation experiments have es- As with the Koide formula most of the mass differences tablished increasingly refined limits on neutrino squared cancel despite the input masses having a range of over mass differences [28]. Cosmological models also put con- 3500:1. If this is not just a coincidence we can attempt straints on the sum of the three mass states [29]. By looking for equations of similar form to (4) and (5) and testing against the experimental neutrino constraints we can identify candidate neutrino mass formulas. Given ⇤ [email protected] that the experimental bounds on the neutrino mass states 2
o have them much closer together than the charged leptons with ✓ke⌫ 24.42 . This is smaller than the weak mixing ' o ⇡ we assume the same exponent of ↵f would need to ap- angle ✓W 28.17 by approximately 48 radians. If the ' ⇡ pear in each formula. By also using the same integer relationship is exactly ✓W = ✓ke⌫ + 48 then it would be 2 coefficients as the charged lepton formulas we find a sur- possible to calculate sin ✓W =0.222928(26) and mW = prisingly close match to the experimental squared mass 80.3834(32) MeV,anorderofmagnitudemoreprecise differences when setting the neutrino residual k⌫ 6.64. than previously known. ' This results in left-handed neutrino sector formulas
IV. QUARK SECTOR 2 m1m3 k⌫ = 3 6.64. (6) m2 ' We will now attempt to apply the techniques we used relating masses in the lepton sector to quarks. The quark 4 3 2 sector residual formulas are m1 = ↵f me 1.86 10 eV/c , (7) ' ⇥ 2 mdmb kd = 3 , (17) 4 ms ↵f 3 2 m2 =9 1 me 8.90 10 eV/c , (8) 3 ' ⇥ k⌫ 2 mumtop k = . (18) u m 3 4 2 2 c m3 = 27↵f me 5.02 10 eV/c . (9) ' ⇥ While most attempts to find relationships between lep- These proposed masses give squared mass differences of tons and quarks use renormalized running masses we have surprisingly found the best overall results using pole 2 5 2 4 m 7.56 10 eV /c , (10) masses for the heavy quarks. The deconfined nature 21 ' ⇥ 2 3 2 4 of the top quark has enabled It’s mass to be measured m 2.51 10 eV /c , (11) 31 ' ⇥ through decay products in particle collision experiments. 2 3 2 4 m 2.44 10 eV /c , (12) m 32 ' ⇥ The best fit we have found to the global average top pole mass using ↵f and simple coefficients is and the sum of all three mass states me 2 2 2 mtop = 3 173.72 GeV/c . (19) ⌃m =6.09 10 eV/c . (13) 2⇡↵ ' ⌫ '⇥ f These predicted values are in excellent agreement with Several different analysis modes exists on data from the recent global analysis of oscillation experiments [28, 30] ongoing top quark experiments at the LHC with a com- prehensive summary in [31]. Our predicted pole mass is for normal mass ordering (m⌫1