Anewempiricalapproachtoleptonandquarkmasses

Kevin Loch⇤ (Dated: 2017 February 27, updated November 11) 2 3 Aleptonratiowithasmalldimensionlessresidualke = mem⌧ /mµ and scaling factor ↵f =27me/m⌧ are used to construct empirical formulas for charged , left-handed and . The predicted 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 to explain fermion lifetimes and famously by with his charged 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 pole (physical) masses. This allows precise predictions me m⌧ = 27 . (5) of 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 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 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

me 2 k` = ke + k⌫ 8. (14) mc =2 1 1.69 GeV/c . (21) ' 3 2 ' ku ↵f If their sum is exactly 8 we can express the residuals with an angle ✓ke⌫ such that ku = 16 (22) 2 k⌫ = 8cos ✓ke⌫ , (15) These predicted pole masses are in excellent agreement with recent pole mass evaluations of mb =4.78 k = 8sin2✓ , ± e ke⌫ (16) 0.06 GeV and m =1.67 0.07 GeV [31]. c ± 3

For the light quarks perturbative QCD does not pro- pare them directly. The quark sector residuals can be vide reliable pole masses so we are left to extrapolate expressed in terms of an angle ✓ 19.47o such that kud ' formulas from the heavy quarks. Having proposed for- 2 mulas for mtop and mc we can extract from (18) kd = 144cos ✓kud , (26)

⇡2 2 2 mu = me 2.52 MeV/c . k = 144sin ✓ . (27) 2 ' u kud

While this value is close to the lattice theory derived MS Looking at different combinations of the lepton and quark mass commonly quoted m 2.2MeV/c2,wecaution residuals we find that u ' against comparing our predicted light quark pole masses kd to renormalized light quark masses directly. For the down k` = =8, (28) quark we assume a similar structure to our ku formula and a ratio m /m 0.5.Withthatcriteria u d ' our proposed formulas for the down and strange quarks ku are =2. (29) k` m =⇡2m 5.04 MeV/c2, (23) d e ' V. COMBINED VIEW 1 ⇡ 3 me 2 ms = 1 4 96.6MeV/c , (24) k 3 ↵ 3 ' While the preceding formulas in terms of me are conve- d f nient for calculating precision predicted values it is more interesting to view them together in terms of the Higgs k = 128. (25) vacuum expectation value v 246.22 GeV [31]. With the d ' top quark Yukawa coupling y = p2m /v 0.99779 we t top ' These are also close to the lattice theory MS masses can rewrite all of our proposed formulas in terms of yt though as mentioned before it is not appropriate to com- and v:

7 ytv 3 ytv 3 3 ytv 3 3 ytv m⌫ = 2⇡↵ ,me = 2⇡↵ ,md = 2⇡ ↵ ,mu = ⇡ ↵ , 1 f p2 f p2 f p2 f p2

7 4 5 7 3 3 3 2⇡↵f ytv 2⇡↵f ytv 2⇡ ↵f ytv ⇡↵f ytv m⌫2 =9 1 ,mµ =9 1 ,ms = 1 ,mc = 1 , 3 p2 0 3 1 p2 0 3 1 p2 3 p2 k⌫ ! ke kd ku ! @ A @ A

7 ytv 2 ytv 1 ytv ytv m⌫ = 27 2⇡↵ ,m⌧ = 27 2⇡↵ ,mb = 2⇡ 2 ↵f ,mtop = . (30) 3 f p2 f p2 p2 p2 ⇣ ⌘

There are many interesting patterns in these expressions, VI. CONCLUSION only some of which are enforced by the sector residual equations (2), (6) , (17) and (18). Despite these patterns In conclusion, we have presented a novel way to view there does not appear to be any relationship between fermion mass relationships with ↵f showing potential as them and electric charge or other properties besides gen- a scaling factor relating the widely differing magnitudes eration, sector and mass. The unexpected appearance of different masses across all sectors. While the predicted of the sector residuals on only the second generation is masses are all within experimental bounds many of those puzzling and may indicate these are just approximations bounds are relatively wide. Lower uncertainty of the ref- of more exact formulas yet to be found. The top quark erence masses would help greatly in validating and fur- Higgs Yukawa coupling appearing in every formula also ther developing this model. The most exciting area of ex- may be an approximation of slightly different values for perimental confirmation would be refinement of the neu- each fermion. In table I we provide a summary of the pre- trino oscillation parameters and especially measurement dicted masses and parameters derived from these equa- of the neutrino mass states directly. The KATRIN exper- tions. iment [35] will search for neutrino mass signatures in de- 4

Parameter THIS WORK Reference Value Ref.

me 0.5109989461(31) MeV CODATA 2014 [34] mµ 105.6583745(24) MeV CODATA 2014 [34] m⌧ 1776.86 0.12 MeV PDG 2017 [31] Model Inputs ± k` = ke + kv 8 (exact) ku 16 (exact) kd 128 (exact)

↵f 0.00776481(52) 1 ↵f 128.7862(87) Model Parameters ke 1.36777(18) k⌫ =8 ke 6.63223(18) 2 sin ✓ke⌫ 0.170972(23) 3 m1 1.85756(50) 10 eV ⇥ 3 m2 8.8981(23) 10 eV ⇥ 2 m3 5.0154(14) 10 eV 2 ⇥ 5 2 5 2 Neutrinos m21 7.5626(39) 10 eV 7.53 0.18 10 eV PDG 2017 [31] 2 ⇥ 3 2 ±+0.038 ⇥ 3 2 m31 2.5120(14) 10 eV 2.524 0.041 10 eV Esteban 2016 [29] 2 ⇥ 3 2 ⇥ 3 2 m32 2.4363(13) 10 eV 2.45 0.05 10 eV PDG 2017 [31] ⇥ 2 ± ⇥2 m⌫ 6.0910(16) 10 eV < 9.68 10 eV Giusarma 2016 [30] ⇥ ⇥ +0.5 Pmd 5.043357447(31) MeV 4.7 0.4 MeV (MS) PDG 2017 [31] +8 Down-type ms 96.5812(87) MeV 96 4 MeV (MS) PDG 2017 [31] mb 4.78172(65) GeV 4.78 0.06 GeV PDG 2017 [31] ± +0.6 mu 2.521678724(15) MeV 2.2 0.4 MeV (MS) PDG 2017 [31] mc 1.68173(23) GeV 1.67 0.07 GeV PDG 2017 [31] Up-type ± mtop 173.719(35) GeV 173.5 1.1GeV PDG 2017 [31] ± yt 0.99779(20) 2 md/me ⇡ (exact) 2 mu/me ⇡ /2 (exact) mu/md 0.5 (exact) 0.38 0.58 PDG 2017 [31] Ratios +0.7 m =(mu + md)/23.782518086(23) MeV 3.5 0.3 MeV PDG 2017 [31] ms/md 19.1502(17) 17-22 PDG 2017 [31] ms/m 25.5336(23) 27.3 0.7 PDG 2017 [31] ± 2 ⇡ sin ✓W (if ✓W = ✓ke⌫ + 48 ) 0.222928(26) 0.22290(33) PDG 2017 [31] ⇡ Electroweak mW /mZ (if ✓W = ✓ke⌫ + 48 ) 0.881517(14) 0.88153(17) PDG 2017 [31] ⇡ mW (if ✓W = ✓k + ) 80.3834(32) GeV 80.385 0.015 GeV PDG 2017 [31] e⌫ 48 ±

Table I. Predicted parameters, pole masses and derived values.

cay from tritium -decay but has a minimum sensitivity m1 and 25 times higher than our predicted m3.Fur- threshold of 0.2eV,100timeshigherthanourpredicted ther refinement of the heavy quark experimental masses is another important confirmation opportunity.

[1] Y. Koide, Nuovo Cim. Lett. 34, 201 (1982). doi:10.1143/PTP.97.459, arXiv:hep-ph/9612322. [2] Y. Koide, A Fermion- Composite Model of quarks [7] Nan Li, Bo-Qiang Ma, Estimate of neutrino masses and leptons, Phys. Lett. B 120 (1983) 161. from Koide’s relation, Phys.Lett. B609 (2005) 309-316, [3] Y. Koide, Phys. Rev. D 28 (1983) 252. arXiv:hep-ph/0505028. [4] R. Foot (1994), A note on Koide’s lepton mass relation, [8] A. Rivero, A. Gsponer, The strange for- arXiv:hep-ph/9402242. mula of Dr. Koide, arXiv:hep-ph/0505220. [5] Y. Koide and M. Tanimoto, U(3)-Family Nonet Higgs doi:10.1016/j.physletb.2005.01.066, Boson and its Phenomenology, Z.Phys.C72:333-344,1996, [9] W. Krolikowski (2005), Excellent approximate solution doi:10.1007/s002880050253, arXiv:hep-ph/9505333. to the mysterious mass equation by Koide, arXiv:hep- [6] Y. Koide and H. Fusaoka, Analytical Expressions ph/0508039. of Masses and Mixings in a Democratic Seesaw [10] J. M. Gerard, F. Goffinet, and M. Herquet, A New Mass Matrix Model, Prog.Theor.Phys.97:459-478,1997, Look at an Old Mass Relation, Phys.Lett. B633 (2006) 5

563-566, doi:10.1016/j.physletb.2005.12.054, arXiv:hep- ph]. ph/0510289 [24] Yoshio Koide, Hiroyuki Nishiura, Flavon VEV Scales in [11] C. Brannen, The Lepton Masses, U(3)×U(3)￿ Model, arXiv:1701.06287 [hep-ph] http://brannenworks.com/MASSES2.pdf, (2006). [25] H.Terazawa and M.Yasu‘e, Phys.Lett. B307, 383(1993), [12] Yoshio Koide, Tribimaximal Neutrino Mixing and a doi:10.1016/0370-2693(93)90237-C Relation Between Neutrino- and Charged Lepton-Mass [26] M. H. Mac Gregor, generation of leptons Spectra, J.Phys.G34:1653-1664,2007, doi:10.1088/0954- and hadrons with reciprocal a-quantized lifetimes 3899/34/7/006, arXiv:hep-ph/0605074. and masses, Int.J.Mod.Phys. A20 (2005) 719- [13] Y. Koide, Charged Lepton Mass Relations in a Supersym- 798, doi:10.1142/S0217751X05021117, arXiv:hep- metric Yukawaon Model, Phys.Rev.D79:033009,2009, ph/0506033. doi:10.1103/PhysRevD.79.033009, arXiv:0811.3470 [hep- [27] M. H. Mac Gregor, A " Mass Tree" with alpha- ph]. quantized Lepton, Quark and Hadron Masses, arXiv:hep- [14] Yukinari Sumino, Family Gauge Symmetry and ph/0607233 Koide’s Mass Formula, Phys.Lett.B671:477-480,2009, [28] K. Hagiwara, R. Liao, A.D. Martin, Daisuke No- doi:10.1016/j.physletb.2008.12.060, arXiv:0812.2090 mura, T. Teubner, (g-2)_mu and alpha(M_Z^2) [hep-ph] re-evaluated using new precise data, J. Phys. G [15] Y. Sumino, Family Gauge Symmetry as an Ori- 38 (2011) 085003, doi:10.1088/0954-3899/38/8/085003, gin of Koide’s Mass Formula and Charged Lep- arXiv:1105.3149 [hep-ph]. ton Spectrum, JHEP 0905:075,2009, doi:10.1088/1126- [29] Ivan Esteban, M. C. Gonzalez-Garcia, Michele Maltoni, 6708/2009/05/075, arXiv:0812.2103 [hep-ph]. Ivan Martinez-Soler, Thomas Schwetz, Updated fit to [16] Y. Sumino, Family Gauge Symmetry as an Origin of three neutrino mixing: exploring the accelerator-reactor Koide’s Mass Formula and Charged Lepton Spectrum, complementarity, IFT-UAM/CSIC-16-114, YITP-SB-16- arXiv:0903.3640 [hep-ph]. 45, arXiv:1611.01514 [hep-ph]. [17] Yoshio Koide, Yukawaon Approach to the Sumino Re- [30] Elena Giusarma, Martina Gerbino, Olga Mena, Sunny lation for Charged Lepton Masses, Phys.Lett.B687:219- Vagnozzi, Shirley Ho, Katherine Freese, On the improve- 224,2010, doi:10.1016/j.physletb.2010.03.019, ment of cosmological neutrino mass bounds, Phys. Rev. arXiv:1001.4877 [hep-ph]. D94,083522(2016),doi:10.1103/PhysRevD.94.083522, [18] A. Kartavtsev, A remark on the Koide relation for arXiv:1605.04320 [astro-ph.CO]. quarks, arXiv:1111.0480 [hep-ph]. [31] C. Patrignani et al. (), Chin. Phys. [19] Werner Rodejohann, He Zhang, Extended Empirical C, 40, 100001 (2016). Fermion Mass Relation, Phys.Lett.B698:152-156,2011, [32] Giorgio Cortiana, Top-quark mass measurements: review doi:10.1016/j.physletb.2011.03.007, arXiv:1101.5525 and perspectives, arXiv:1510.04483 [hep-ex]. [hep-ph]. [33] G. Aad, et al.(ATLASCollaboration),Determina- [20] Jerzy Kocik, The Koide Lepton Mass Formula and Ge- tion of the top-quark pole mass using $t \bar{t}+1$- ometry of Circles, arXiv:1201.2067 [physics.gen-ph]. jet events collected with the ATLAS experiment [21] Guan-Hua Gao, Nan Li, Explorations of two empirical in 7 TeV $pp$ collisions, JHEP 10 (2015) 121, formulas for fermion masses, Eur. Phys. J. C (2016) doi:10.1007/JHEP10(2015)121, arXiv:1507.01769 [hep- 76:140, doi:10.1140/epjc/s10052-016-3990-3. ex]. [22] Yoshio Koide, Hiroyuki Nishiura, Quark and Lepton [34] Peter J. Mohr, David B. Newell, Barry N. Mass Matrix Model with Only Six Family-Independent Taylor, CODATA Recommended Values of Parameters, Phys. Rev. D 92, 111301 (2015), the Fundamental Physical Constants: 2014, doi:10.1103/PhysRevD.92.111301, arXiv:1510.05370 doi:10.1103/RevModPhys.88.035009, arXiv:1507.07956 [hep-ph]. [physics.atom-ph]. [23] Yoshio Koide, Hiroyuki Nishiura, Quark and Lepton [35] R. G. H. Robertson, for the KATRIN Collaboration, Mass Matrices Described by Charged Lepton Masses, J.Phys.Conf.Ser.120:052028,2008, doi:10.1088/1742- doi:10.1142/S021773231650125X, arXiv:1512.08386 [hep- 6596/120/5/052028, arXiv:0712.3893 [nucl-ex].