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A New Empirical Approach to Lepton and Quark Masses

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A New Empirical Approach to Lepton and Quark Masses 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 <m⌫2 <m⌫3 )andthe sum is just below cosmological model limits [29]. A com- an excellent match for recent analysis including the 2016 PDG pole mass average of m = 173.5 1.1GeV/c2 parison of our predicted values to experiments is shown top ± [30]. We also have a particularly close match to the cen- in table I. Remarkably, these equations have the identical +1.0 ter value of Mt = 173.7 1.5 1.4 0.5 reported in [32]. integer coefficients (1, 9, 27) as our charged lepton formu- ± ± − las and also satisfy a left handed neutrino sector struc- For the bottom and charm quarks we target the pole ture equation similar to (2) for charged leptons. Right masses derived from perturbative QCD MS mass analysis handed sterile neutrinos, if the exist do not yet have reli- from collider experiments. The best formulas we have able constraints established from which we could search found using ↵f and relatively simple coefficients for the for similar relationships. bottom and charm quarks are At this point we must ask why the sector residual fac- m m = e 4.78 GeV/c2, k k b 1 2 (20) tors e and ⌫ only appear as terms in the second gener- ⇡ 2 ↵ ' ation and if their values are related? While we do not yet f have an answer to the first question, the second may be answered by looking at the sum of the lepton residuals me 2 mc =2 1 1.69 GeV/c . (21) 3 2 ' ku ↵ k = k + k 8. (14) f ` e ⌫ ' If their sum is exactly 8 we can express the residuals with ku = 16 (22) an angle ✓ke⌫ such that These predicted pole masses are in excellent agreement 2 k = 8cos ✓ , (15) with recent pole mass evaluations of mb =4.78 ⌫ ke⌫ ± 0.06 GeV and m =1.67 0.07 GeV [30]. c ± For the light quarks perturbative QCD does not pro- k = 8sin2✓ , e ke⌫ (16) vide reliable pole masses so we are left to extrapolate 3 formulas from the heavy quarks. Having proposed for- pare them directly. The quark sector residuals can be mulas for m and m we can extract from (18) expressed in terms of an angle ✓ 19.47o such that top c kud ' 2 2 k = 144cos ✓ , ⇡ d kud (26) m = m 2.52 MeV/c2. u 2 e ' 2 ku = 144sin ✓kud . (27) While this value is close to the lattice theory derived MS mass commonly quoted m 2.2MeV/c2,wecaution Looking at different combinations of the lepton and quark u ' against comparing our predicted light quark pole masses residuals we find that to renormalized light quark masses directly. For the down kd quark we assume a similar structure to our up quark k` = =8, (28) formula and a ratio m /m 0.5.Withthatcriteria ku u d ' our proposed formulas for the down and strange quarks are ku =2. (29) k` m =⇡2m 5.04 MeV/c2, (23) d e ' 1 V. COMBINED VIEW ⇡ 3 me 2 ms = 1 4 96.6MeV/c , (24) 3 3 ' kd ↵f While the preceding formulas in terms of me are conve- nient for calculating precision predicted values it is more interesting to view them together in terms of the Higgs kd = 128. (25) vacuum expectation value v 246.22 GeV [30].
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