Anewempiricalapproachtoleptonandquarkmasses
Kevin Loch⇤ (Dated: 2017 February 27, updated November 11) 2 3 Aleptonratiowithasmalldimensionlessresidualke = 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 3500:1. If this is not just a coincidence we can attempt to find factors that cancel out with this formula to learn One outstanding problem in modern physics is the more about the underlying structure. We can then at- seemingly unrelated particle masses that exist as free tempt to apply those factors to search for similar struc- parameters in the standard model. Numerous attempts tures in other sectors. have been made over the years to derive a systematic Mac Gregor proposed using different exponents of the mathematical relationship between these masses, most QED coupling constant to explain fermion lifetimes and famously by Yoshio Koide with his charged lepton for- mass relationships using the zero energy scale ↵QED(0) mula [1, 2] [26, 27]. While we have not found convincing relation- ships with ↵QED(0) we do find m + m + m 2 e µ ⌧ m K = 2 . (1) e (pme + pmµ + pm⌧ ) ' 3 ↵f = 27 (3) m⌧ While this approximation works for charged leptons, and 1 interesting with a value ↵f 128.786. This is close appears to work for some combinations of quarks with ' 1 2 ↵ (M )= running masses, it does not work for neutrinos or at- to the effective QED coupling constant QED Z tempt to relate all fundamental fermions in a consistent 128.944 [28]. Despite the similarity in value to an ef- manner. It does however hint at the possibility of find- fective QED coupling constant we are assuming that ↵f ing other more fundamental relationships. Indeed, Koide is a separate parameter related only to mass generation and others [1–24] have used it as inspiration for extended and not electric charge. With (2) and (3) we can derive models to cover other fermions. lepton formulas While these extended Koide models are interesting we me present an alternative structure for charged leptons that mµ =9 1 2 , (4) k 3 ↵ 3 is easily applied to other sectors. we find these formu- e f las appear to relate all fundamental lepton and quark pole (physical) masses. This allows precise predictions me m⌧ = 27 . (5) of neutrino and quark pole masses. If correct this would ↵f significantly reduce the number of free parameters in the standard model (extended for neutrinos with mass). With our choice of definitions for ke and ↵f we have dis- covered an interesting and surprisingly simple charged lepton mass structure that could be used to look for sim- II. CHARGED LEPTONS ilar patterns in other sectors.
We begin by considering the small dimensionless resid- ual value in a charged lepton structure proposed by Ter- III. NEUTRINO SECTOR azawa [25] 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 [29]. Cosmological models also put con- straints on the sum of the three mass states [30]. By looking for equations of similar form to (4) and (5) and testing against the experimental neutrino constraints we ⇤ [email protected] can identify candidate neutrino mass formulas. Given 2
o that the experimental bounds on the neutrino mass states with ✓ke⌫ 24.42 . This is smaller than the weak mixing ' o ⇡ have them much closer together than the charged leptons angle ✓W 28.17 by approximately 48 radians. If the ' ⇡ we assume the same exponent of ↵f would need to ap- relationship is exactly ✓W = ✓ke⌫ + 48 then it would be 2 pear in each formula. By also using the same integer possible to calculate sin ✓W =0.222928(26) and mW = coefficients as the charged lepton formulas we find a sur- 80.3834(32) MeV,anorderofmagnitudemoreprecise prisingly close match to the experimental squared mass than previously known. differences when setting the neutrino residual k 6.64. ⌫ ' This results in left-handed neutrino sector formulas IV. QUARK SECTOR 2 m1m k = 3 6.64. (6) ⌫ m3 ' We will now attempt to apply the techniques we used 2 relating masses in the lepton sector to quarks. The quark sector residual formulas similar to (2) and (6) are 4 3 2 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 ku = 3 . (18) mc 4 2 2 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
2 5 2 4 have surprisingly found the best overall results using pole m 7.56 10 eV /c , (10) 21 ' ⇥ masses for the heavy quarks. The deconfined nature 2 3 2 4 m 2.51 10 eV /c , (11) of the top quark has enabled It’s mass to be measured 31 ' ⇥ 2 3 2 4 through decay products in particle collision experiments. m32 2.44 10 eV /c , (12) ' ⇥ The best fit we have found to the global average mtop and the sum of all three mass states pole mass using ↵f and simple coefficients is
2 2 me 2 ⌃m⌫ =6.09 10 eV/c . (13) mtop = 3 173.72 GeV/c . (19) '⇥ 2⇡↵f ' These predicted values are in excellent agreement with recent global analysis of oscillation experiments [29, 31] Several different analysis modes exists on data from the for normal mass ordering (m⌫1