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Journal of Modern Physics, 2013, 4, 127-150 http://dx.doi.org/10.4236/jmp.2013.44A013 Published Online April 2013 (http://www.scirp.org/journal/jmp) Predicting the Neutron and Proton Masses Based on Baryons which Are Yang-Mills Magnetic Monopoles and Koide Mass Triplets Jay R. Yablon Schenectady, New York, USA Email: jyablon@nycap.rr.com Received February 14, 2013; revised April 19, 2013; accepted April 26, 2013 Copyright © 2013 Jay R. Yablon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT We show how the Koide relationships and associated triplet mass matrices can be generalized to derive the observed sum of the free neutron and proton rest masses in terms of the up and down current quark masses and the Fermi vev to six parts in 10,000. This sum can then be solved for the separate neutron and proton masses using the neutron minus proton mass difference derived by the author in a recent, separate paper. The oppositely-signed charges of the up and down quarks are responsible for the appearance of a complex phase exp(iδ) and real rotation angle θ which leads on an independent basis to mass and mixing matrices similar to that of Cabibbo, Kobayashi and Maskawa (CKM). These can then be used to specify the neutron and proton mass relationships to unlimited accuracy using θ as a nucleon fitting an- gle deduced from empirical data. This fitting angle is then shown to be related to an invariant of the CKM mixing an- gles within experimental errors. Also developed is a master mass and mixing matrix which may help to interconnect all baryon and quark masses and mixing angles. The Koide generalizations developed here enable these neutron and proton mass relationships to be given a Lagrangian formulation based on neutron and proton field strength tensors that contain vacuum-amplified and current quark wavefunctions and masses. In the course of development, we also uncover new Koide relationships for the neutrinos, the up quarks, and the down quarks. Keywords: Proton Mass; Neutron Mass; Baryons; Magnetic Monopoles; Koide; CKM Mixing Angles; Current Quarks; Constituent Quarks 1. Introduction In an earlier paper [1] the author introduced the thesis that baryons are Yang-Mills magnetic monopoles. Using the t’Hooft magnetic monopole Lagrangian in (2.1) of [2] and a Gaussian ansatz for fermion wavefunctions from (14) of O’Hanian’s [3] to obtain energies according to 33 r dFFx gauge 1 dT 2 Ex L, it became possible in Equation (11.22) of [1] to predict the electron rest mass as a function of the up and down quark masses, specifically: 3 2 32π u m ed mm , (1.1) with the factor 3 2 2π emerging from three-dimensional Gaussian integration. Based on a “resonant cavity” analysis of the nucleons whereby the energies released or retained during nuclear binding are directly dependent upon the masses of the quarks contained within the nu- cleons, it was also predicted that latent, intrinsic binding energies of a neutron and proton, see (12.12) and (12.13) of [1], are given by: 3 2 2442π 7.640679 MeV, Pudd udu Bmmmmmm (1.2) 3 2 2442π 9.812358 MeV. Nduu udd Bmmm mmm (1.3) These predict a latent binding energy of 8.7625185 MeV per nucleon for a nucleus with an equal number of protons and neutrons, which is remarkably close to what is observed for all but the very lightest nuclides, as well as a total latent binding energy of 493.028394 MeV for 56Fe, in contrast to the empirical binding energy of C opyright © 2013 SciRes. JMP J. R. YABLON 128 492.253892 MeV. This is understood to mean that 99.8429093% of the available binding energy in 56Fe is applied to inter-nucleon binding, with the balance of 0.1570907% retained for the intra-nucleon quark con- finement. It was also noted that this percentage of energy released for inter-nucleon binding is higher in 56Fe than in any other nuclide, which further explains that although the quarks come closer to de-confinement in 56Fe than in any other nuclide (which also explains the “first EMC effect” [4]), they do always remain confined, as empha- sized by the decline in this percentage for elements with nuclear weights higher than 56Fe. In a second paper [5], the author showed how the thesis that baryons are Yang-Mills magnetic monopoles together with the foregoing “resonant cavity” analysis can be used to predict the binding energies of the 1s nu- clides, namely 2H, 3H, 3He and 4He to parts per hundred thousand for 3He and in all other cases to parts per mil- lion, and also to predict the difference between the neu- tron and proton masses according to: 3 2 3 2π du m32 NPu dμ MMm mmm . (1.4) This relationship, originally predicted in (7.2) of [5] to about seven parts per ten million in AMU, was later taken in (10.1) of [5] to be an exact relationship, and all of the other prior mass relationships which had been de- veloped were then nominally adjusted at the seventh decimal place to implement (1.4) as an exact relationship. The review of the solar fusion cycle in Section 9 of [5] served to emphasize how effectively this resonant cavity analysis can be used to accurately predict empirical binding energies, and suggested how applying gamma radiation with the right resonant harmonics to a store of hydrogen may well have a catalyzing effect for nuclear fusion. This relationship (1.4) will also play an important role in the development here. At the heart of these numeric calculations which accord so well with empirical data were the two outer products (4.9) and (4.10) in [5] for the neutron and the proton, with components given by (4.11) and related re- lationships developed throughout Sections 3 and 4 of [5]. In particular, the two matrices which stood at the center of these successful binding energy calculations were 3 × 3 Yang-Mills diagonalized matrices K of mass dimension 1 2 with components , ,diag N udd K mmm for the neutron and , ,diag P duu K mmm m m for the proton, where u is the “current” mass of the up quark and is the current mass of the down quark. d What is very intriguing about these K-matrices (which we designate with K to reference Koide), is that although they originate from the thesis that baryons are magnetic monopoles, they have a form very similar to matrices which may be used in the Koide mass formula [6] for the charged leptons, namely: 2 123 123 3 2 mm m Rmm m ,mmm m . (1.5) Above, when we take 12e and m3 = mτ to be the charged lepton masses, the ratio 32R 0.5109989280.000000011 MeV 105.65837150.0000035 MeV e m m gives a very precise relationship among these masses. Indeed, if we use the 2012 PDG data 1.500022828R and mτ = 1776.82 ± 0.16 MeV [7], we find using mean ex- perimental data that , very close to 3/2. Because the binding energies formulated in (1.2) and (1.3) are rooted in the thesis that baryons are Yang-Mills magnetic monopoles and specifically emerge from the calculation of energies via , see (11.7) of 3 dEx L [1] et seq., and because these binding energies can also be refashioned via Koide relationships as we shall show in the next Section, the author’s previous findings will provide us with the means to anchor the Koide relation- ships in a Lagrangian formulation. And, because Koide provides a generalization of the mass matrices derived by the author in [5], these matrices will provide us with the means to derive additional mass relationships as well, in particular, and especially, the free neutron and proton rest masses, which is the central goal of this paper. Specifically, after reviewing in Section 2 similarities between the author’s baryon/magnetic monopole matri- ces and the Koide matrices, we shall show in Section 3 how to reformulate the Koide relationships in terms of the statistical variance of Koide mass terms across three generations. This will yield some new Koide relation- ships for the neutrinos, the up quarks, and the down quarks. We then show in Section 4 how to recast these Koide relationships into a Lagrangian/energy formulation, which addresses the question as to underlying origins of these relationships, so that these relationships are not just curious coincidences, but can rooted in fundamental physics principles based on a Lagrangian. Most importantly, in this paper, we combine the au- thor’s previous work in [1,5] as well as [8], using the generalization provided by Koide triplet mass matrices of the form (2.1) below, to deduce the observed rest masses 938.272046 MeV and 939.565379 MeV of the free neu- tron and proton as a function of the up and down quark masses and electric charges and the Fermi vev. This mass derivation is presented in Sections 5 and 6. In Section 7 we connect the masses obtained in Section 6 to the em- pirically-observed Cabibbo, Kobayashi and Maskawa (CKM) quark mixing matrices. In Section 8 we examine Copyright © 2013 SciRes. JMP J. R. YABLON JMP 129 AB “constituent” and “vacuum-amplified” quark masses for the neutron and proton. Finally, in Section 9 we develop a Lagrangian formulation for these neutron and proton masses, which underscores that these relationships are not just close numerical coincidences, but originate from fundamental Lagrangian-based physics. author in [5] and those developed by Koide in [6] are highlighted if we define a Koide matrix K generally as: Copyright © 2013 SciRes. 2. Similarities between Baryon/Magnetic Monopole Matrices and Koide Matrices The similarities between the matrices developed by the 1 2 3 00 00 00 AB m Km m . (2.1) Then, the two latent binding energy relationships (1.2) and (1.3) may be represented as: 2 33 22 3 2 3 2 11 Tr Tr 2π2π 2442π7.640679 MeV 00 000000 1 0000Tr0000 2π 00 000000 PABBA BB ud dudu ddd d uuu u uuu u BKKK KK mm mmmm mmm m mmm m mmm m AA KK (2.2) T r 2 11 Tr Tr 00 00 00 u d d BKKKK KKK m m m 1d mm 33 22 3 2 3 2 2π2π 24 42π9.812358MeV 00 0000 1 Tr0000Tr00 2π 00 0000 NABBA AABB du uudd uu u dd d dd d mm mmm m mm m mm m mm m (2.3) where, starting with (2.1), in (2.2) we have set and and in (2.3) we have set 1u23u mmm mm and 23d. Again, these originate in the author’s thesis in [1] that baryons are Yang-Mills magnetic monopoles. Above, designates an outer matrix pro- duct. 12 , e mmm m mmm On the other hand, setting and 3 mm in (2.1), we may write: 2 123 Tr AB BAe K KKm mmmmm, (2.4) 2 123 Tr AA BBe KK KKmmmmmm 2 . (2.5) Then, using (2.4) and (2.5), Koide relationship (1.5) for charged leptons may be written as: 2 2 Tr 3 2 Tr eAA BB eABBA mm mKK KK Rm KKK mm . (2.6) Clearly then, the Koide matrices (2.1) provide a gen- eral form for organizing the study of both binding energy and fermion mass relationships which lead to very accu- rate empirical results. It thus becomes desirable to under- stand the physical origin of these Koide matrices and tie them to a Lagrangian formulation so that they are no longer just intriguing curiosities that yield tantaliz- ingly-accurate empirical results, but can also be rooted in fundamental physics principles based on a Lagrangian. And, it is desirable to see if these matrices can be ex- tended in their application to make additional mass pre- dictions and gain a deeper understanding of the particle J. R. YABLON 130 mass spectrum, especially the free neutron and proton masses to be explored here. We start in the next Section by showing how to refor- mulate the Koide relationships in terms of the statistical variance of the Koide terms across the three generations. 3. Statistical Reformulation of the Koide Mass Relationship We continue to examine the charged leptons by setting 12e,mmmm m and 3 m in (2.1). When we use the extremes of the experimental data ranges in [7], spe- cifically, the largest possible tau mass and the lowest possible mu mass, we obtain R = 1.5000024968. Al- though this is an order of magnitude closer to 3/2 than the ratio obtained from the mean data, is still outside of experimental errors. This means that while 32R is a very close relationship, it is still approximate even ac- counting for experimental error. For this to be within experimental errors, it would have to be possible to ob- tain some 32R for some combination of masses at the edges of the experimental ranges, and it is not. First, using (2.4), we write the average of masses i in a Koide mass triplet 123 , i.e., the “aver- age of the squares” of the matrix elements in (2.1), as: m,,mm m 22 12 Tr 3KK mm 3 3 3 AB BA i KK m m (3.1) Next, via (2.5), we write the “square of the average” of these matrix elements as: 2Tr KK 2 123 2 123 99 3 9 AA BB K K m m K mm mm (3.2) So, combining (3.1) and (3.2) in the form of (1.5) al- lows us for the charged leptons to write: 2 22 12 12 Tr 3Tr KKK KK mm mmm 2 3 3 3 2 AA BB AB BA KK KK m R (3.3) This allows us to extract the relationship: 2R22 1 32 K KK , (3.4) which naturally absorbs the 3 from the factor of 3/2. Now, we simply use (3.4) to form the statistical vari- ance K in the usual way, as: 22 22 2 2 3 11 3 31 1. 2 ii R K KK K K R mKKm R (3.5) The key relationship here, using first and last terms, is: K i m . (3.6) So the average i of the charged lepton masses is approximately (and very closely) equal to the statistical variance m K 1.500022828R of Koide matrix (2.1) when used for the charged leptons. This is a much simpler and more transparent way to express the Koide mass relationship (1.5), it completely absorbs the factor of 3/2, and it is entirely equivalent to (1.5). Of course, as noted at the outset of this Section, this is a very close, but still approximate relationship. The exact relationship, also extracted from (3.5), and using based on mean experimental data, is: 0.999969563 31iii K mmCm R , (3.7) where we have defined the statistical coefficient C and the inverse relationship for R as: 33 1; 1 CR RC . (3.8) Thus, we may rewrite the basic Koide relationship (1.5) more generally as: 2 123 123 3 1 mmm R mm mC . (3.9) In the circumstance where the statistical coefficient C = 1, i.e., where the average mass is exactly equal to the statistical variance, we have 32R 0.999969563 . So the statistical variance of the square roots of the three charged lepton masses is just a tiny touch less than the average of the three masses themselves. But the fac- tor of 3/2, which is somewhat mysterious in (1.5), is now more readily understood when we realize that it corre- sponds with C = 1 in (3.7). This means that the Koide relationship for any given triplet of numbers with mass dimension 1 2, may be al- ternatively characterized by the coefficient C. Thus, us- ing (3.7), the coefficient C for the charged lepton triplet is (we also include R for comparison): 0.999969563 1; 1.5000228 8322. R Ce e (3.10) Copyright © 2013 SciRes. JMP J. R. YABLON 131 So what about some other Koide triplets? For the neu- trinos, PDG in [9] provides upper limits e , and m for the neutrino masses. If we use these mass limits in a Koide triplet, we find that R = 1.202960231. But the significance of this is much more easily seen by using (3.8) to calculate: 2eVm 18 0.19 MeVm.2 MeV 1.4 1.20 e e R C 938480 ; 29602 32 365 (3.11) Here, we have another ratio very close to 3/2, but now it is the coefficient C rather than the coefficient R. So, for the upper neutrino mass limits, 32 K m . This in an interesting “coefficient migration” as between the charged and uncharged leptons, wherein for the charged leptons masses 32R to parts per 100,000, while for the neutrino lepton upper mass limits, 32C within about 0.4%. As we shall see, this is the start of a new Koide pattern. Turning to quark masses, we use u and d developed in (10.3) and (10.4) of [5] with the conversion 1u = 931.494061(21) MeV/c2. We also use , 2.223792405 647 eV 95 s m m 0335MeV 1.2750.025 G c m MeV 4.90m t and b from PDG’s [10]. For Koide triplets of a sin- gle electric charge type, we can then calculate that: 5MeV,173.5m 0.03 GeV .6.8 GeVm4.18 32688 ; 9134866 5 1.54 1.177 Cuct uRct (3.12) 1.18741 1.371483 Cdsb dsRb 6 5; 91115 11 (3.13) So we now see a distinctive pattern of coefficient mi- gration among (3.10) through (3.13). For the charged leptons in (3.10) which are the lower members of a weak isospin doublet, 32Re , as has long been known. For neutrinos which are the upper members of this dou- blet, 32 e C , which migrates the 3/2 from the R to the C coefficient. Then, for the up quarks, we find another coefficient migration such that 32Cuct , which is same as the C for the neutrinos. Both the up quarks and the neutrinos are the upper members of weak isospin doublets. Finally, we see that the 65Ruct coefficient for the up quarks, now migrates to Cdsb 65 for down quarks. So the migration is 32eC 32 e R for leptons, 32 e C 32Cuct provid- ing a “bridge” from “up” leptons to “up” quarks, and then 65Ruct 65Cdsb migrating from the up to the down quarks. The net upshot of this coefficient migration is that we now have Koide-style close relations for all four sets of fermions (and anti-fermions) of like-electric charge Q, namely: 2 6 05 e e mm m RQ mm m . (3.14) 2 3 12 e e mm m RQ mm m . (3.15) 2 26 35 uct uct mmm RQ mmm . (3.16) 2 115 311 dsb dsb mmm RQ mmm . (3.17) Each of these relationships takes twelve a priori inde- pendent fermion masses and reduces by 1, their mutual independence. So with (3.14) through (3.17), to first ap- proximation, we have now eight, rather than twelve in- dependent fermion masses. For some other commonly-studied Koide triplets we have: 0.6929012 ; 32 1.772105341 21 Cuds udsR (3.18) 1.00939 1; 1.4929941033 2 Cctb ctRb (3.19) 0.86795; 1.606042302Cusc uRsc , (3.20) 1.02783 1; 1.479416975 i2wth 3 s Ccsb cmsbR (3.21) 0.81520; 1.652718083Cdcs dRcs. (3.22) Cuds We note that the relationship (3.18) for 12 955MeVm 98.95303495 MeVm is accurate to within experimental errors. Spe- cifically, given the empirical s, (3.18) can be made into an exact relationship to ten digits (the accuracy of the up and down masses derived in [5]) if we set s . Of course, even the rela- tionship (3.15) for the charged leptons is a close but not exact relationship, see the discussion at the start of this Section, so we ought not expect (3.18) to be exactly 12Cuds. But, similarly to (1.5), see also (3.10), it may well make sense to regard this as a relationship ac- curate to the first three or four decimal places, which would improve our knowledge of the strange quark mass by four or five orders of magnitude. But this main point of the foregoing is not about the specific Koide relationships (though the set of relation- ships (3.14), (3.16) and (3.17) are important steps for- Copyright © 2013 SciRes. JMP J. R. YABLON Co2013 SciRes. JMP 132 If we generalize this to any three fermion wavefunc- tions ward in their own right), but about how the ratio pa- rameter R which for the charged lepton triplet is 12 3 ,, such that (4.1) represents the specific case 12 pyright © 32R , can be reformulated for any fermion triplet into the coefficient C in the statistical variance relationship i K Cm 1 C , which, for the charged leptons, is . And, as we see in (3.14) through (3.17), this can lead to additional rela- tionships via a cascading migra- tion of coefficients. Turning back to the neutron and proton triplets, , ,, , , uu dd mm mm 92405 MeV, diag diag Pd Nu Km Km which were so central to obtaining accurate binding en- ergy predictions in [1,5], we find using the MeV equiva- lents of the mass values d obtained in (10.3) and (10.4) of [5] that: 2.2237 u m 6470335 MeV4.90m 2 Cp duu R pduu 0.0387876019; .8879821000 0298844997; 2.9129480061 3R uuu 0; (3.23) 0.Cn udd nuddR (3.24) For these triplets which all have a sm all variance in comparison to the earlier triplets which cross generations, the Koide ratio . In the circumstance where the variance is exactly zero because all three quarks have the same mass, for example, for the triplets and , using the Koide mass relationship for param- eterization, we haveC. ddd 3R 4. Lagrangian/Energy Reformulation of the Koide Mass Relationship The appearance of Koide triplets originating from the thesis that Baryons are Yang-Mills magnetic monopoles can be seen, for example, by considering Equation (11.2) of [1] for the field strength tensor of a Yang-Mills mag- netic monopole containing a triplet of colored quarks in the zero-perturbation limit, reproduced below: Tr '' '' '''' " RR RR GG GG Fi pm pm " BB BB pm (4.1) , R G and 3 B , , and, as we did prior to (11.19) of [1], if we consider the circumstance in which the interactions shown in Figure 1 at the start of Section 3 in [1] occur essentially at a point, then 0p , approaches an ordinary commu- tator, each of the , and the “quoted” denominator becomes an ordinary denominator, see (3.9) through (3.12) of [1] for further background. So also setting 12 R G mmm mand 3 B mm , (4.1) generalizes for a point interaction to a Koide-style field strength tensor: 11 1 2233 23 , Tr ,, Fi m mm (4.2) Then, we form a pure gauge field Lagrangian gauge 11 Tr Tr 22 F FFF L 2 Tr as in (11.7) of [1]. As discussed in Section 3 of [5], we consider both inner and outer products over the Yang- Mills indexes of F, i.e., we consider both F Tr AB BCAB BA F FFF and TrTr AB CD F FFF AA BB F F . Note carefully the different index structures in AB BA F F versus AA BB F F , and also contrast this to (2.2) through (2.5) in this paper, which we shall now seek to refashion into a Lagrangian formulation. To proceed, we use this Lagrangian gauge to calcu- late energies according to (11.7) of [1], also (1.8) of [5], which are reproduced below: L 33 gauge 1 dTr d 2 ExFFx L , . (4.3) In the case where 123du so that P F F represents the proton, then depending on whether we contact indexes using AB BA F F or AA BB F F , we obtain the inner and outer products in (3.6) of [5]. When 123ud , so N F F AA BB represents the neutron, we obtain the inner and outer products in (3.7) of [5]. Using (2.1), the Koide generali- zation of the outer products ( K K index summation) is: 3333 3 2 11 2 22 123 3 3 2 2 33 1111 dTrdTr dd 222 2π 00 00 11 Tr0000 2π2π 00 00 ABCDAABBAABB ExFFxFFxFFxKK mm mm mmm mm L (4.4) J. R. YABLON 133 while the Koide generalization of the inner products ( K AB BA K index summation) is: 123 1 33 33 11 22 33 3 22 2 33 11 1 dTr dTrdd 22 2 0000 11 Tr 0000 2π2π2π 00 00 AB BDAB BA AB BA ExFFxFFxFFx mm K Kmm mm L mmm (4.5) This means that is now becomes possible to express the Koide relationship (3.9) entirely in terms of energies E derived from the Lagrangian integration (4.3). Specifi- cally, combining (3.9) with (4.4) and (4.5) allows us to write: 3 23 3d 2 3 Trd 0 2 e Ex FF Fx LL 33 33 3 d d 3 1 AA BB AB BA 3 23 3 3 2 12 12 3 dTr dTr Tr d Tr d d d AA BB AB BA x FFx E E x FF x KK KK R C FF x Fx FF x FF x mmm mm m L L (4.6) This expresses the Koide mass relationship in multiple forms, in terms of an energy integral of the general La- grangian density form 1Tr 2 F FL, with general field strength (4.2). This means for any Ko- ide triplet of given empirical R, there is an energy R E 23 d 0 x RF x which vanishes under condition: 3 d Tr R ER FF LL (4.7) This is the Lagrangian/energy formulation of the Ko- ide relationship (3.9), and although different in appear- ance, it is entirely equivalent. So, for example, using the symbol as in Figure 1 and Table 3 of [8] to repre- sent the three generations of the fermions for any given charge, the four Koide relationships (3.14) through (3.17) for the pole (low probe energy) masses may be written as in the entirely equivalent, alternative form: 3 6d 5 6 Tr 5 Ex FF LL (4.9) 3 23 6d 5 6 Trd 0 5 u Ex FF Fx LL (4.10) 3 23 15 d 11 15 Trd 0 11 d Ex FFF x LL ,, (4.11) Whether these become exactly equal to zero for masses at high-probe energies, and whether there is an underlying action principle involved here, are questions beyond the scope of this paper which are worth consid- eration. What ties all of this together, is that we mod el the ra- dial behavior of each fermion in the triplet 123 using the Gaussian ansatz borrowed from Equation (14) of [3] and introduced in (9.9) of [1] which is reproduced below with an added label for each of the fermions and masses in (4.2): 1, 2,3i 23 d 0F x (4.8) 2 3 0 24 2 1 πexp 2 i ii i i rr rup m , (4.12) and that we also relate each reduced Compton wave- length i to its corresponding mass i via the De- Broglie relation ii mc , see [1] following (11.18). This is what makes it possible to precisely, analytically calculate the energy in integrals of the form (4.3), spe- cifically making use of the mathematical Gaussian rela- tionship (9.11) of [1]: 2 03 32 3 2 1expd 1 π rr x , (4.13) and variants thereof. It is (4.12) and (4.13) and 1m ii 1c (in units) which tie everything together at the “nuts and bolts” mathematical level when Copyright © 2013 SciRes. JMP J. R. YABLON 134 (4.2) is employed in (4.3) through (4.11). And this is what leads to accurate mass relationship (1.1) and bind- ing energy predictions (1.2) and (1.3), as well as the binding energy predictions for 2H, 3H, 3He and 4He and the proton-neutron mass difference (1.4) found in [5]. The final piece which also ties this together at nuts and bolts level, is the empirical normalization for fermion wavefunctions developed in (11.30) of [1], namely: 22 22 1 24 22 Em mm 24n 41 f Em Nn , (4.14) where f is the total number of fermions over three generations including three colors for each quark. Now, it is important to emphasize that the Gaussian ansatz (4.12) is not a theory, but rather, it is a modeling hypothesis that allows us to analytically perform the necessary integrations and calculate energies which for- tuitously turn out to correlate very well with empirical data. That is, explicitly in [1] and implicitly in [5], we hypothesized that the fermion wavefunctions can be modeled as Gaussians with specific Compton wave- lengths 1mii defined to match the current quark masses, we performed the integrations in (4.3), and we found that the energies predicted matched empirical binding data to—in most cases—parts per million. This, in turn, tells us that for the purpose of predicting binding energies, it is possible to model the current quarks as Gaussians (which means they act as free fermions), with masses and wavelengths based on their undressed, cur- rent quark masses, and to thereby obtain empirically- validated results. But, as also discussed at the end of Section 11 in [1], this use of a current quark mass does not apply when it comes predicting the short range of the nuclear interac- tion which we showed at the end of Section 10 in [1] is indeed short range with a standard deviation of 12 85.65 . For, if we use the current quark masses that work so well for binding energies, we find u F 41.04 and d F , and the predicted short range is still not short enough. If, however, we turn to the constituent quark masses which, at the end of Section 11 in [1], for estimation, we took to be 939 MeV/3 = 313 MeV, then we have 0.63 F and 04 1 2.5 F 432 , which tells us that the nuclear interaction virtually ceases at about F ,,mm m 2du mm . This is exactly what is observed. In both cases—for nuclear binding energies and for the nuclear interaction short range—we found that the Gaus- sian ansatz (4.12) does yield empirically-accurate results. But for binding energies, it was the undressed, current quark masses which gave us the right results, while for nuclear short range, it was the fully dressed, constituen t quarks masses that were needed to obtain the correct re- sult. Because we shall momentarily embark on a prediction of the fully dressed rest masses 938.272046 MeV and 939.565379 MeV of the free neutron and free proton, what we learn from this is that while we might also be able to approach the neutron and proton masses using a Gaussian ansatz for fermion wavefunctions, we will, however, need to be judicious in the fermion wavefunc- tions we choose and in the masses that we assign to the fermions. That is, the focus of our deliberations will be, not wh ether we can use the Gaussian ansatz, but on how to select the fermion wavefunctions and masses that we do use with the Gaussian ansatz, in order to obtain em- pirically accurate results. Now, with all of the foregoing as background, let us see how to predict the neutron and proton masses. 5. Predicting the Neutron plus Proton Mass Sum to within about 6 Parts in 10,000 Because we can connect any Koide matrix products to a Lagrangian via (4.4) and (4.5), let us work directly with the Koide matrix (2.1) to determine how to assign the masses 123 so as to predict the neutron and proton masses. Then at the end (in Section 9), we can backtrack using the development in Section 4 to connect these masses to their associated Lagrangian. In other words, we will first fit the empirical mass data, then we will backtrack to the underlying Lagrangian. Each of the neutron and proton contains three quarks. The sum of the current quark masses is for the neutron and 12.0367331 MeV2ud mm 9.35405514 MeV 2.223792405 MeVm for the proton, using u and d earlier introduced before (3.12) as developed in (10.3) and (10.4) of [5]. For a free neutron and proton, none of this rest mass is released as binding energy, and so these quark mass sums are fully included in N 4.906470335 MeVm 939.565379 MeVM and P respectively, where we use an uppercase M to denote these fully-dressed, observed masses. As demonstrated in Sections 11 and 12 of [1] and throughout [5], these rest masses are reduced when the neutron and proton fuse with other nucleons. But for free protons and neutrons, the entire rest mass is retained and all of the latent bind- ing energy is used to confine quarks. 938.272046 MeVM 928.91799152M eV PP ud mM mm This means the “mass coverings” m (using a lowercase m) for the neutron and proton may be calculated to be: 927.52864572M eV NN ud mM mm , (5.1) . (5.2) 12 3AB BA These mass coverings m represent the observed, fully-dressed neutron and proton masses M, less the sum K Kmmm of the current quark masses, Copyright © 2013 SciRes. JMP J. R. YABLON 13 SciRes. JMP 135 123d m mm 3 mm Copyright © 20 ,mm , u mmm with u for the proton, and 12 d for the neutron, see (2.4). One may think of P release energies of all the 1s nuclides with very close precision. We shall wish to add to this toolkit here, and in particular, will wish to refine our use of the Fermi vev vF = 246.219651 GeV beyond what is shown in (5.4). Spe- cifically, as noted after (3.8) of [8], we need to put (5.4) “and like expressions into the right context and obtain the right coefficients. And where do such coefficients come from? The generators of a GUT!” m and N m 37 3 MeV 3 P u d m m as weights of rather heavy “clothing” “covering” “bare” quarks. The sum of these two mass covers is: 1856.4466 NP N mmMM (5.3) Now, at the end of Section 10 of [5], after deriving the neutron minus proton mass difference (1.4), we noted that the individual masses for the neutron and proton could now be obtained by deriving some independent expression related to the sum of their masses, and then solving these two simultaneous equations—sum equation and difference equation—for the two target masses, namely, those of the neutron and proton. We shall do exactly that here. In particular, it will be our goal to de- rive the sum N P M M nud of these two masses, and then use (1.4) as a simultaneous equation to obtain each sepa- rate mass. The benefit of this approach using a sum, re- ferring to the so-called mass “toolbox” in (4.11) of [5] and also the discussion of the alpha nuclide following (5.4) of [5], is that in selecting mass terms to consider, we can eliminate any candidates not absolutely symmet- ric under and interchange, because the sum p N P M M contains three up quarks and three down quarks, as well as one neutron and one proton. Our em- pirical target, therefore is the mass sum NP MM 1856.4466 37 But we can alternatively find this by finding the mass cover sum PN of (5.3) to which we can then readily add . These sums are what we now seek to predict. 1877.837425 MeV 33 ud mm MeV mm We now return to use the “clues” laid out in (3.6) through (3.8) of [8]. We start in the simplest way possi- ble by focusing our consideration on (3.8) of [8], repro- duced below, but multiplied by a factor of 2 and sepa- rated into 4 F u vm and 4 F d vm in the second term, thus: 2 4 2 44 22 MeV1803.6 70518 F udFu FdFud mvmvm vmd vm (5.4) Here, vF = 246.219651 GeV is the Fermi vev. Because this is about 3% smaller than P N in (5.3) and is closer to mm P N than either (3.6) or (3.7) of [8], and also is symmetric under interchange, we shall see if (5.4) can be used, by itself, to provide the founda- tion for hitting the mass target (5.3). As we shall, it can be so used! mm mm ud .446637 MeV 1856.446637 MeV 1856 PN Now it is time to “cash in” on the GUT we developed in [8] to obtain the coefficients needed to bring (5.4) closer to the target mass of in (5.3). Because the vev that seems based on (5.4) to bring us into the correct “ballpark” is the Fermi vev, we focus on electroweak symmetry breaking which occurs at the Fermi vev, and which, in (8.2) of [8], is specified by breaking electroweak symmetry using electric charge generator Q via: In (4.11) of [5], we developed a “toolkit” of masses which we used for calculating the binding and fusion diag diag 211 122 0,,,,1,,,diag 333333 i FiF FF T vvQ ,,duu (5.5) For the proton with a fermion triplet , the corresponding eigenvalue entries in (5.5) above are 122 ,, 33 3 FFF vvv ,,udd . For the neutron and its triplet, the entries are 211 ,, 333 FFF vvv . We now wish to use these to establish Koide triplet matrices for the neutron and proton which can then be used to generate the sum of their masses. Looking at these vacuum triplets 122 ,, 33 3 FFF vvv and 211 ,, 333 FFF vvv , we see that to match the mass dimension 1 2 of the terms with 4vmu and 4vm ,,mmm d in (5.4) and use these as Koide triplets, we will need to take the fourth roots of these vacuum triplets. So we do exactly that, and pair these triplets with the mass triplets duu and ,,mmm udd for which we also take the fourth root to match (5.4). Thus, we use 0.5 444 1221 2 2 ,,, , 3333 33 F FFFdFuFu v v vivmvmvm J. R. YABLON 136 and 0.5 444 211 21 ,, , 333 33 F F FFuFd vvv vmivm 0.5 1 , 3 Fd ivm to define two new Koide triplets, one for the neutron and one for the proton, as follows: 4 0.5 4 2 3 1 3 00 Fu ABF d vm KNi vm 0.5 4 00 00 1 3Fd iv m (5. 6) 0.5 4 4 1 3 2 0 3 00 Fd ABF u 4 00 0 2 3 F u m vm ivm KP v (5.7) What we have done here is simply develop (5.6) and (5.7) to match the mass dimensionalities in (5.4) while bringing in the coefficients from (5.5) which reflect the electric charges of the up and down quarks. We see that because of the negatively-signed (-) charge for the down quark, of which we have taken the fourth root, each of these triplets contains components with the complex co- efficient 0.5 4 2 11 1 ii . In recent years, consideration has been given to having negative square root terms in Koide mass relations, see for example (3.21) in which one uses s m to derive a close relation for the csb 0.5 i 0.5 i triplet (see Rivero’s original finding of this in [11]). The above, (5.6) and (5.7) take this a step further, because they raise the specter of Koide triplets with complex square root coefficients! In the next Section we explore the profound implications of these complex coefficients, which arise from the oppositely- signed charges of the up and down quarks. But for the moment, we ignore in the above and examine mag- nitudes only, and form and calculate the following Koide matrix product from (5.6) and (5.7) with excised: 44 44 44 2 4 12 00 00 1857.570635 Me 33 21 Tr V 0000 33 21 00 00 33 2 39 Fd Fu ABBAF uF d Fu Fd Fud vm vm K PK Nvmvm vm vm vmm (5.8) Observed 1856.446637 MeV PN mm Comparing to (5.3) which tells us that we see that we have hit the target to within about 0.06%! That is: Observed 1857.570635 MeV 1856.446637 MeV 1.000605457! BA NP KPKN mm AB (5.9) This is extremely close, and in particular, we now see that the sum of the neutron and proton mass coverings may be expressed solely as a function of the up and down quark masses and charges and the Fermi vev to within about 6 parts in 10,000! So if we use this close relation- ship to hypothesize that a meaningful relationship is given by mmKPKN 33mm NP ABBA, then using the above with (5.3) to add the current quark masses ud to this mass cover sum, we see that to within about 0.06%: 2 42 3333 333 9 M N PNPu dABBAu dFudu d Mmmm mKPKNm mvmmmm. (5.10) So it appears as though we have now discovered the correct coefficients for the “clue” in (5.4). These coeffi- cients, which are based on none other than the electric charges of the quarks, yield the neutron plus proton mass sum to 6 parts in 10,000! Further qualifying (5.10) as a proper and not merely Copyright © 2013 SciRes. JMP J. R. YABLON 137 coincidental expression for the neutron plus proton mass sum, we see that this is symmetric under inter- change, and that it is formed by taking the inner product AB BA ud K PKN of the Koide proton matrix K P and the Koide neutron matrix K N, which product is symmetric under interchange. Further, both of these fully embed the electric charges and mass magni- tudes of the current quarks as well as the Fermi vev. So in sum, (5.10) makes sense on multiple bases: it yields an empirical match to within 6 parts in 10,000; it is the product of a proton matrix with a neutron matrix; the proton matrix contains the masses and charges of two up quarks and one down quark while the neutron matrix contains the masses and charges of two down quarks and one up quark; and it is fully symmetric under both and interchange. p pn n ud Furthermore, if we divide (5.8) by 2, we see that: 2 4 2 32 29 928.785 ABBA Fud KPKN vmm3174 MeV 179915 MeV (5.11) This actually falls between and N from (5.1) and (5.2), so (5.10) clearly appears to be a correct expression for the leading terms in the neutron and proton masses. Based on this close concurrence and “threading the needle” be- tween the neutron and proton masses with (5.11) and all of the appropriate symmetries noted in the previous paragraph, we now regard (5.10) as a meaningful (rather than coincidental) close expression for 928.9 P m 6457 MeV927.528m P N M M to 0.06%. It will simplify and clarify the calculations from here to use an uppercase M notation to define what we shall hereafter refer to as “vacuum-amplified” up and down quark masses according to: 604.1 2 3 uFu Mvm751345 MeV, (5.12) 634.5 1 3 dFd Mvm784463 MeV. (5.13) Consequently: 2 4619.1 2 9 ud Fud MM vmm902116MeV. (5.14) With these definitions, the neutron plus proton mass sum (5.10) may be rewritten more transparently as: 33 3 N PNPu d udu d M Mmmm m M Mmm (5.15) 0.5 00 00 00 d AB u u iM KP M M while the Koide mass matrices (5.6) and (5.7) for the neutron and proton become: , (5.16) 0.5 0.5 00 00 00 u AB d d M KN iM iM . (5.17) These matrices now restore the 0.5 11 2 ii coef- ficient that we excised to calculate (5.8). Thus, as in (5.8), but including this complex factor, we now take: 0.5 0.5 0.5 0.5 00 Tr 00 00 00 00 11857.570635 0 MV1 2 0 3e AB BA d u u u d d ud KPKN iM M M M iM iM iMM i (5.18) Having found a very close magnitude, we could make use of a 2 factor and continue to match the empirical data by writing 2Re ABBAPN K PKNmm . But this just sidesteps understanding the meaning of this complex coefficient and it does not help us past the 0.06% difference that still remains between the predicted and the empirical data. We now need to find a more fundamental way to un- derstand this complex factor, as well as how to close the remaining 0.06% gap between the predicted and the ob- served neutron plus proton mass sum. That will be the subject of the next two Sections. 6. Exact Characterization of the Neutron and Proton Masses via a Mixing Angle θ and Phase Angle δ The complex factor 0.5 11 2 ii which arises from the oppositely-signed up and down quark charges, as we shall now see, is actually like the subtle clue in a good detective story which, when pulled like a small thread and pursued to its logical end, eventually cracks the en- tire mystery. So, let us start to pull on this thread and see where it leads us. Copyright © 2013 SciRes. JMP J. R. YABLON Copyright © 20 JMP 138 0.5 111 11 exp0000 0cossin010 0sincos 001 AB ii U 13 SciRes. We first represent this factor 11 2 in te 0.5 iirms of a phase angle defined such that π4 , so that: . (6.5) 0.5 1exp 1 2cos siniii i . (6.1) Then, we brme So (6.5) sandwich-multiplied by (6.4) simply general- iefly rena K K and use this phase to rewrite (5.18) as: e00 0 00 00 0 e u u i d N P KN M M M M M M iM Mmm (6.2) expii Tr 0 0e 00 3exp AB BA i d u i d ud KP with 0.5 in separate matrices (5.16), (5.17) en we use thalso. is to rewrite mass sum (5.15) with 0.5 expii Th restored as: 3 3exp NPu d udu d MM m3 NP mm m M Mimm (6.3) where we have also brd iefly rename M M and ,, P NPN mm , all with π4 . is importaNow, (nt, because it gives us an oppor- tu atrix 6.3) nity to define a new Koide matrix AB which we shall refer to as the “electron generation m” as such: 400MM 30 0 00 ud AB u d m m . (6.4) Then, making note ose exp i f the pha which mul- tiplies ud M M in (6.3) and keepind how the Kobayaaskawa mixing matrices are formed for three generations, we introduce a new angle 1 ing in m shi and M such that 10 and form a unitary matrix 1 U with i e : izes the appearance of the term 0.5 ud iMM in (5.18). But now let us permit both and to rotate freely, , 0 . Then, using (6.4) and (6.5), we may form the neutron plus proton mass sun according to Equation (6.6) at the bottom of the page. and For the special case where π4 , (6.6) precisely reproduces (6.3). But in (6.6) we have removed the approximation sign that was in (6.3), because we are now going to define the an- gles , , so as to precisely match up with the empi rica l values of the neutron and proton masses. That is, just as (1.4) is an exact formula for the proton-neutron mass difference, we shall now regard (6.6) as an exact formula for the neutron plus proton mass sum, with the numerical values of defined by empirical data so as to make this an exact fit. Now before we proceed, let us pause to make clear, the cascading detective work we have just done: We have used the matrix 0.5 diag ,1,1Ui implicit in (6.3) and explicit in (6.5) as a hint that there exists a matrix diagexp,1,1Ui with π4 . Then we use diagexp,1,1Ui as a further hint that there exists a matrix (6.5). Then we allow both of these angles to freely rotate to form (6.6) which generalizes (6.3). Fol- lowing all of this, we will use these freely rotated angles to permit the otherwise close relationship (6.3) to be fit- ted exactly by empirically choosing these angles so as to yield an exact fit. But before we do this, however, there is a final, deep cascade to this hint, which is to recognize that (6.5) with angles free to rotate is one of the three matrices used to define the CKM matrices used for electroweak genera- tion mixing, see (7.11) in [8], and in particular, is the matrix that is use to introduce the phase angle response- ble for CP violation. We also see that (6.4) is strictly a function of the first (electron generation) quark masses and the Fermi vev which makes its upper left component 4ud M M containing the “vacuum-enhanced” quark 4 4 111 11 11 1 11 00 00 exp0 0 3Tr000cossin00 0sincos 00 00 0 0 3Tr0cossin 3expcosco 0sincos ud ud NPABBC CAuu d d ud uud udud ud d MM MM i MM Umm mm MM mmmMMimm mm m 1 s (6.6) exp i J. R. YABLON 139 masses substantially larger than its middle and lower righ onen168,758 MeV 2 3 tt Mvm , (6.10) t compts u and md m. KM mixing has two more matrices and also mixes two more generations, let us now form two more and analogous to (6.4) for the muon and tan generation of quarks, following the pattern for mi the original parameterization of Kobayashi and Ma. Thus, we put the largents Because C 18,522 Me 3V 1 bb Mvm, (6.11) which yields the higher-generation analogues to (5.14): matrices uo xing in skawa compone4cs M M and 6356 MeV cs MM , (6.12) 4tb M M into the And, as a atter of convention, we keep thic charge = n andn matrices as: lower right positions. e up (electrm +2/3) series of mass terms in the middle position. Thus we define the muo tauon generatio 4 4 00 300 ; 00 00 300 . 00 s AB c cs b AB t tb m m MM m m MM (6.7) 55,908MeV tb MM . (6.13) These values are calculated from the laid out prior to (3.12), rounded to the nearest MeV (recognizing substantial experimental un We also define two more matrices an s in [8]: 22 222 cos sin sin cos AB U , analogously to (6.6), for the second and third generations, respectively, we form: At the same time, analogously to (5.12) and (5.13), we define the vacuum-enhanced higher-generation quark masses: 14,467MeV, 3 2 cc Mvm (6.8) PDG data [10] certainties). alogous to (6.5) for the second and third generations in same manner as i used to form the CKM mixing matrices, again see (7.11) 33 00 1 cossin 0 (6.14) 33 3 sincos0. 001 AB U Then 0 0 ; 2792 MeVm, (6.9) 1 3 ss Mv 22 2 cossin 0 0 3cosco ssc c scs cs mmm MMmm MM 22 2 3Trsin cos 00 AB BCCAs cc Um mm 2 s , (6.15) 33 333 cos 0 3Trsin cos0 0 bb b tt mm Ummm 33 3cos cos t btb tb MMmm MM sin t m 0 AB BC CA iply all three of (6.6), (6.15) and (6.16) together in the same manner that the Cabibbo mixing matrices are formed, again see (7.11) in [8], to obtain a master “mass and mixing matrix” with mass dimen- sion +3, defined as: . (6.16) Then, we mult 213 123 123 1 23 23 123 123 ee 27 uscbtusct udsct b ii udsbudsbt uct uc bt t s UUU mmmmmcss mmmmcscmmmmMM ss MM mmccMM mmmcs mmmccc mm mmccsmm mMMsc m 2 12 23 23 e eud c b i i udcbt ud scb udc sbt MM m mmss MM mmmsc mm MM mms 13 13u cst smmMMmsc 1dd cs tb mMMMMc (6.17) Copyright © 2013 SciRes. JMP J. R. YABLON 140 This master matrix contains all six of the quark masses in all three generations, all three of the real mixing an- gles and the one phase angle that appears when the three generations are mixed, and implied in the vacuum-en- hanced mass terms, the Fermi vev and the electric charges of all of these quarks. If all of the masses are set to equal 1, this reduces to the usual generational mixing matrix in the original parameterization of Kobayashi and Maskawa, seen in, e.g., (7.11) in [8]. In the circumstance where 23 0, 0ss , this reduces to: 11 11 e0 0 270 cossin 0sincos i udsb uctud ctb udc stdc st b MM mm mmmmm mMM mmMMmm MMMM . (6.18) and in the further circumstance where all of the second andrd generation masses are set to 1, this further reduces to 9 times the matrix shown in (6.6): thi e0 0 i ud MM 11 11 27 0 cossin 0sincos uu d ud d mm m mm m . (6.19) neutron plus proton mass sum of (6.6): So in this particular special case, (6.17) even contains the 1Tr 3expcoscos 9ud 11udNP M MimmMM ! (6.20) So this neutron plus proton mass sum now is a special case of (6.17) which includes all the generation mixing a shion from the simple hint of a matrix with 0.5 diag,1,1Ui in the neutron plus proton mass for- mula (6.3), with the 0.5 i itself having emerged from the simple f d angles and all the quark masses and their electric charges and the Fermi vev! Consequently, one expects that (6.17) can be used to signe gin substantial new insights into fermion and baryon asses generally. And all of this emerges in cascade m fa act that up and down quarks have oppositely- charges which led to terms containing 41 when Such is t we formed Koide matrices to represent masses. he nature of this detective mystery! igression of (6.7) eturn to solve (1.4) and (6.6) as simultaneous equations, that is, we now solve the simultan With the important contextual d through (6.20) as backdrop, we now r eous equation set: 11 3 2 cos 2π d m3expco 32 3 PN udu NPu dμd MMMMi m MMmmmm m s u (6.21) We now need no more than elementary algebra to determe that the neutron and proton masses, separate given by: in ly, are each 3 323 dudμdu mmmmmm 2 3 1 2π 13expcos 2 u Pud udu MM Mimmm (6.22) These can be made into exact theoretical expressions fo 1 13expcos 2 Nud MMMim 2 3232π dμdu mmmm r the neutron and proton mass by solving for 1, , to find their will need to form the square modulus magnitude empirical values based on the empirical neu- and proton masses. Let’s now do so. on Because each of (6.22) contains a complex phase, we 2 M MM of these masses. So first we deduce: tr Copyright © 2013 SciRes. JMP J. R. YABLON 141 3 22 1 2 3 2 1 3 2 49 6cos3cos3232π 3cos3 232π; 3 2π Nudud ududμdu ud udμdu du MMMMM mmmmmmm mm mmmmm m m m (6.23) 1 2 3232π dudμdu mmmmm Now we solve these as simultaneous equations for 1 1 49 6cos 3cos 3cos3 2 Pu dud u ud udμ MM MMM m mm mmm 2 3 2 and . First we restructure (6.2 terms of 3) in to arrive at: 2 3 2 3 2 2 3 2 3 493cos3232π cos 63cos3232π 493cos3232π cos Nudududμdu udu dudμdu Pud ududμdu MMMmmmmmmm MMm mmmmmm MMM mmmmmmm m m (6.24) We now set tese two cos 2 1 1 2 1 2 1 63 cos 3232π ud u dudμdu MM m mmmm h equal to one another to eliminate and solve for . It will be easier to see the underlyinge of these equations as well as solve them if we write (6.24) above as: structur 2 1 1 2 1 1 cos cos cos cos A BA A BA (6.25) using the following substitution of variables: cos NB C PB 2 2 3 49; 49 32 3 3;6 Nud Pud ud du ud ud NM MM PM MM Ammmm m BmmC MM (6.26) Next, we reduce the second and third terms of (6.25) successively in five steps as follows: 2 2π; 22 222 2 111 1 22 2222 11 22 222 2 1 1 coscos cos cos cos cos 4) :coscoscos B A PBABABA BAA BABA NBA ABA ABA 22 11 5) :02coscos2ABBNPANPA In the final step, we arrive at a quadratic for 1 cos 11 1 2 2 111 1 1) :coscoscos 2) :coscoscos cos 3) : NBAA PB NBABA BA NB AP A 1 cosBA 1 1 cosAPB 3 (6.27) , and so obtain a solution via the quadratic equation. Then, we use the variables (6.26) including the empirica l masses of the neutron and proton, to calculate that: 224 1 0.94745412 82 os 4 2 c 4 NPNPANP A AB (6.28) Additionally, 0.3198sin9167 1 . In the above, we use the negative root, because this yields 1 1cos 1 . This means the empirically-determined value of 1 is: 10.32561515rad 18.65637386 π9.64817715 (6.29) We shall refer to 10.947454co 2s124 in (6.28) used to precisely fit (6.22) to the observed neutron and Copyright © 2013 SciRes. JMP J. R. YABLON 142 proton masses as the “nucleon fitting angle”. In the next Section we shall show how to tie this angle to the ob- served CKM mixing angles, so it is not a “new” angle, but is related to other known mixing data. Now, we use (6.28) in (6.25) to solve for , and cal- culate to find that: 2 224 224 2 224 24 824 cos 824 824 1 24 NNPNPANPA AA CNPNPANP AAA PNPNPANPAAA NP AAA (6.30) This numerical calculation reveals that cos 1 28CNP NPA , ex- actly, to all decimal places, so the phase factor 0 . This means that when the variables in (6.26) are tuted into (6.30), the extremely unwieldy-looking result- inill reduce to 1 ident substi- g expression wically! So to the ex- tent that may be a CP-violating phaset 0 , and given tha is a deduced result for the neutron and proton mthis deductively tells us that there are no asses (6.22), CP-violating effects associatn and proton. in the circum- st 24 ed with neutro This is validated by empirical data which shows the mass of the antiproton is equal to that of the proton, and the mass of the antineutron is equal to that of the neutron, see, e.g., [12,13]. So, we take (6.22) to be exact formula- tions of the neutron and proton masses, ance where empirically-determined angle 10.947454cos12 and CP-violating phase 0 . So we now return to (6.22), set 0 , and so obtain our final expressions for the neutron and proton masses: 3 2 1 2 1 13cos3232π 2 13cos 3232π 2 dududμdu dududμdu MMMmmmmmmm MMMmmmmmmm which are exact relations with thrical substitution 10.947454 2s124 3 (6.31) Nu Pu e empi . able us to back to the masses (nuclear weights) for the 1s nuclides predicted in [5] to high accuracy and rewrite (8.6), (8.1), co These relationships (6.31), in turn, now en go (8.3) and (8.5) of [5], respectively, as: 2 11 3cos P Nuudud u M MMm MMmm , (6.32) m 3 32 1 3 1 242 2π 19cos 7 2 PNu μd udu du MM Mmmm MMmmm 3) 2 232π μdu mm m 3 d m (6.3 3 2 3 1 22 19cos 523232π 2 PN uud ududuuddμdu MMMmmm MMmmmmmmmmm (6.34) 2 3 42 2 1 22661010 162π2 6cos 2 PN u dduudud du du d MM M mmmmmmmm M M Now, 0, AA ZPNZ BZM NMM which is binding energy B for any given nuclide with Z protons and N neutrons hence A = Z + N nucleons, thus 2NZAZ , may also be rewritten generally in relation to nuclear 3 2 1010162π udd uud mmmmmm (6.35) um mmm 0 Copyright © 2013 SciRes. JMP J. R. YABLON 143 weights using (6.31), in the form: 01 32 13cos 2 22π dμ AA ZZud udu mm BMAMMmm AZm 3 2 3 du mm (6.36) One final exploratory exercise of interest is to return to he master mass and mig matrix in (6.17) and set 23 0 txin while using 10.947454co 2s124 found in (6.28). In this circumstance, (6.17) reduces to: 1 1 00 27 0cos0 cos udsb uct tb MM mm mmm MM 00 dc s mM M . (6.37) Th root, an ated with the neutron plus proton mass sum) to get mass nuthat should be related to ind is is in dimensions of mass3. If we take the cubed d divide by 2 (because we know that this origi- n mbers ividual baryons, we find 3 1diag MeV939.72,1163 MeV ,17 MeV73 2 (and we also get a coefficient 32723 2, back to e neutron mass expected a priori, Koide!). This first entry is very close to th 939.565379 MeV which would not be ut this is because b630 MeV sb mm which is not too far from 619 MeV ud MM . Perhaps this is yet an- other close relationship among fermn masses!? The io ecowould become smaller snd entry at 116 23 0, 3 MeV, which 0 when ss of the 0 d readily be com , is only ab 1115.68uds pensated by out 4% larger than the 3 MeV baryon, which non-zero 23 , ma coul angles charm and top quark eV, is perhaps sugges- ply pointed out in an ted that in (6.17) is just one representation of a mass/mixing matrix and that one can also vary the way in which o Koide triplets (6.4) and (6.7), so as to be able to obtain this as a tiv explo well as experime sses. The final e e (6.37) rela ratory spirit, an ntal errors in the ntry at 1773 M tionships are sim d it is to be m e of the 1672.45 MeVsss baryon mass, however, contra, there are no omitted angles and some- where we should expect to come across a baryon with a third generation quark. Thes no ne sets up the matrix in several different representations. uld be clear that the master matrix (6.17) and like matrices that can be simi- larly constructed are an exceedingly useful tool for trying to develop and fit mutual relationships among mixing angles, CP violating phases, and quark an 7. Relation of the Nucleon Fitting Angle θ to the CKM Mixing Angles - cleon fitting angle 10.947454co 2s124 Whatever the correct fits may turn out to be with various higher-generation baryons, it sho d baryon masses. Following the development in the last Section, the nu found in (6.28) is a new empirical parameter that enables us to precisely formulate the neut (6.31). While this is an imp standing the neutron and proton masses, it would be even better if this angle could be related in some way to the wn CKM quark mixing ron and proton masses using ortant step forward in under- empirically-kno angles, which could then relate the neutron and proton masses them- selves to the CKM angles. This is highly preferable to having 1 cos be a new, separate parameter. 12 12 23 12 2312 0.974270.000150.2253 ud us ub cdcscb tb V cc VVVV scc V ssc Toward this end, we first write the CKM matrix with the “standard choice” of angles and its empirical values from PDG’s [14] as: 13121313 1223 1312231223 1323 13 23 1312231223 132313 0. e ee ee 4 0.000650.00351 i ii ii td ts VVsc s ssccssssc VVcscsscs cc 0.00015 00014 0.0011 0.0005 0.0011 0.000021 0.0005 0.000046 4 0.000160.0412 404 0.999146 . (7.1) e loangles are between 0 and 0.00029 0.00031 0.22520 0.00065 0.9734 0.00867 0.0 (We use a negative sign for the threwer-left empirical entries to match the negative values in the terms which the standard CKM matrix takes on when the π2.) Now, 1 cos 0.9474541242 does not fit any particular one of these elements. But what is of interest is the determinant Copyright © 2013 SciRes. JMP J. R. YABLON 144 Vwhich may be calculated from the CKM mixing and phase angles ij and to be: 1 udcs tbus cb tdub cdts ub cs tdus cdtbudcb ts VVVVVVV VVV VVVVVV VVV (7.2) and which contains invariant expressions of interest (See also [15] which cleverly connects this determinant, when real as in the standard angle choice (7.1), to the Jarlskog determinant). Specifically, if we employ the mean ex- perimental values in (7.1), we find that sum of the three positively-signed (+) terms in the determinant, denoted Vning all nine matrix rminant,” is determined from the empirical data in (7.1) to be: elements, and which we shall refer to as the “major de- , which is an invariant contai te 0.947535 udcstbus cb VVVVVV tdub cdts VVVV (7.3) This major determinant is to 1 cos very close 0.947454 , truncated to the kn of nown precisioV . In fact we find 1 cos 0.000 perimental 0.947192 262V if we use the lower bounds of all the ex error ranges in (7.1), and 1 e upper bounds. So this is within experimental errors. Therefore, using 10.94cos 7454 cos 0.0 0.947854 00400V if we us as the baseline against which to compare V , we find that: 0.000400 0.000400 0.1 0002620.000262 0.947454cosV . (7.4) This means that the nucleon mixing angle 1 cos is ated to the invariant scalar relV according to: 1 cos VVV udcs tb us cb tdub cdts V VVV VVV (7.5) which is well within experimental errors! If we now take this to be a meaningful relationship given that it falls well within experimental errors, this means that we can go back to (6.31) and use (7.5) to rewrite the neutron and proton masses completely in terms of the CKM matrix elements, and specifically in terms of the major determi- nant V, according to: 3 2 2 32 3 2π 13 23 2π dμdu dμdu 3 2 3 Pu dudu 13 2 Nududu M MMVm mm mm mm MMMVmmm mm nnects the proton and neutron ma mm This now cosses to the major determinant (7.6) V which is an invariant of the CKM mixing matrix V the 0.06% difference of (5.18) between the predicted and the em- pirical neutron and proton masses using1 cos . This not only closes , but it connects 1 cos to the CKM mixing angles so that (7.6) now specifies the exact masses of the free neutron and proton as a function of the up and down masses and charges and the Fermi vev and the CKM quark mixing angles without introducing any new physical parameters to do so! Because 10.947454co 2s124 is known with better ecision than pr0.947535V, w 1 cos e then use as the basis for specifying V, i.e., we now set: 10.947s 4541242 , (7.7) which is then a further ingredient used to tighten the em- pirical data in (7.1). Further, because coV V injects into the proton and neu- tron masses an imaginary term with a Jarlskog deter- minant 2 13 1223 12 1323sin CKM Jcccsss culated using the angles in (7.1) withCKM (which may be cal- ), and if we wish to maintain the proton and neutron masses to be entirely real based on cos 1 (the “nucleon phase angle” CKM ) deduced in (6.30), then we can achieve this by restoring the phase to the vacuum-en- as in (6.21), ihanced mass term.e., by restoring exp ud ud M MMMi and then choo,sing in sin ud iMM to absorb the terms with t dethe Ja ne s f V ... when the whole determinant is made real” as it is in (7.2). Specifically, referring to (7.6), this mean set he Jarlskog erminant, again see [15] which shows how trl- skog determinant is “the imaginary part of any oele- ment among the six componentof determinant o s that one would sin Im0iMMVmm udu d to maintain CP symmetry for the neutron and proton. Given that Im 3VJ , this means that: 2 13 12231213 sin 3 3 ud ud ud mm JMM mm cccss23 s in CKM ud s M M (7.8) will define a very tiny phase in the term exp ud M Mi in the proton and neutron masses such that these masses remain real and thus maintain CP symmetry. While beyond the scope of this paper, this could provide additional insight into the so-called CP problem. Finally, as regards fermion masses, if we write each elementary fermion mass “strong f m in terms of the Fermi vev using a dimensionless coupling f G as 2 f fF mGv, see, e.g., (15.32) of [16], then use these relationships in (6.17) for or a similarly-formed matrix in a CKM representation (such as1)), we find that the matrix entries will contain terms of the form 33 34 , (7. f FfF GvGv and Copyright © 2013 SciRes. JMP J. R. YABLON opyright © 201 JMP 145 depending on representation, 35 f F Gv . This may help us gain further insight into fermion masses as well as high-orderangian vacuum terms 345 ,, which specify how much of the observed neutron and proton masses arise from each of th and te quarksheir in much does each down masses? In ot , for Lagr . All of this mystery cracking is the result of the detec- tive work embarked upon at the start of Section 6, of pulling on the tiny thread of the complex factor teractions with the vacuum. The question we now ask, referring to the neutron and proton mass formulas (6.31), is how much does each up quark contribute, and how quark contribute, to these total her words, what are the “constituent” 0.5 11 2 ii which fourth root arises from at em ide matrices ( 604 MeV ses whic on ma taking the anates fro sitelye up in order to form 5.6) a lues al ese “vacu h qua the neutron pl masses of the up quarks and down quarks in each of the neutron and proton, as opposed to their bare “current” masses? Referring to the neutron and proton masses (6.31) of the minus (−) sign thm the oppo- -signed electric charges of th and down quarks, nd (5.7). the Ko l between rk c us prot the square root terms ud M M and μd mm , w not directly segregate the up quark mass contri m that of the down e can- bution froquark. In these square root terms, 8. Vacuum-Amplified and Constituent Quark Massesthe up and down are coequal mass contributors. So we shall allocate instead. For the term 3ud M M in the neutron mass, we allocate a 1ud M M contribution to the one up quark and a total 2ud M M contribution to the two down quarks. For the proton, we allocate 1ud M M to the one down quark and 2ud M M to the two up quarks. We similarly allocate the μd mm terms. But as to terms which contain u m alone, o ctly to the up and down quark constituent masses, respectively. Thus, we identically rewrite each of (6.31) while defin- ing respective constituent quark mass sums 2 In (5.12) through (5.14) we defined three very helpful mass vaand 635 MeV. It is natural therefore to inquire whether thum-am- plified” quark masses might be related to the so-called “constituent” quark mash specify how much mass eacontributes to total mass of a nucleon or baryon, as opposed to the bare “current” quark masses. Specifically, recalling that these were the ingredients in ss sum, we note 2 u M r d m alone, we segregate these and apply them dire N N UD 302.0875673MeV, 317.2892232MeV2 d M in (5.12) ich is about 1/3 of the neutron and proton and (5.13), wh masses. This suggests that (5.12) to (5.14) may be related to the constituen t masses of the up and down quarks C3 SciRes. and 2 P P UD, as: 1 u u 33 22 133 22 23 32π2π 12 43 23cos 32π2π μdu ud 3c os NN μdd udd mm m MM mm 2 N M UD mm m MMm , (8.1) 133 22 33 22 43 23cos 32π2π 2 23 32π2π μdu ud uu 1 1 2 3c os P PP μdd mm m MM mm ud d M UD mm m , (8.2) butions respectively sp re arate contribu tions emanating from up and down quarks. We then separate out the constituent quark masses and calculate them using 10.947454co 2s124 MM m with the up and down quark contri ecified in the upper and lower lines of each of (8.1) and (8.2). That is, the abovpresent a deconstruction of the neutron and proton masses intoe sep- e, as follows: th 133 22 3 π2π 314.0092987 MeV μdu u mm m , (8.3) 2 13c os 232 Nu d u UM Mmm J. R. YABLON 146 1 2 13 cos 22 32π μ Nudd m DMMm 33 2 eV d m , 2 22π 312.7780400 M 3d m (8.4) 1 2 13 cos 22 3 Pu du u UM Mmm 33 22 π22π 310.0274283 MeV . (8 ) 3 μdu mm 2 m .5 133 22 2 3cos 232π μ Pudd mm DMMm 318.2171900 M 3eV d m . ) 1d (8.6 The first expr8.3) for 2π ession ( N U is the constituent contribution of the uark to the mass of the neutron. The second expressi(8.4) for up q on N D is the constituent contribution of eac e two down quarks to the mass of neutron. h of th the P U(8.5) is the constituent contribu- each of the p quarks to the mass of the pro- . Finally, in two ution of ton P D in (8.6) is the constituent contribution e down quarke mass of the proton. One can of th to th verify that 2 N NN M UD and 2 P PP M UD, numerically and analytically. It is important to observe that N P UU and N P DD, which is to say that the constituent contribution of each quark to the mass of a nucleon is not the same for different nucleons, but rather is dependent upon the particular nucleon in question, in this case, a proton or a neutron. So the lone up quark in e neutron makes a slightly greater contribution to the This sort of context-dependent variable behavior de- pending upon nuclide is to be expected based not only on what we uncovered throughout [5], but more generally based on the fact that when nucleons bind together, they release binding energy, so that different nuclides have different weights per nucleon, and indeed, different nu- cleons within a given nuclide should be expected to have different weights from one another based on their shell haracterization. Constituent mass Equations (8.3) through along these same lines, that the constituent mass contributions from each quark will differ depending upon the particular nuclide in question, and indeed, upon the particular nucleon with which a quark is associated ithin that nuclide. The above, (8.3) through (8.6), make the point that this type of variable mass behavior of indi- vidual quarks already starts to appear even as between the free neutron and proton. We also see that the “vacuum-amplified” quark masses (5.12) through (5.14), are not synonymous with con- stituent quark masses. These vacuum-amplified masses are ingredients which are used as part of the calculation of the constituent quark masse quark masses vary from one nucleon and nuclide and nucleon within a nuclide to the next, the vacuum-ampli- fied quark masses do not vary. They are mass constants (to the same degree that current quark masses are con- stants, recognizing mass screening) which do not change from one nucleon or nuclide to the next, and which are used as ingredients for calculating the quark masses, as we see in (8.3) for calculating neutron and proton masses (6.31) and nuclear weights (6.32) through (6.36). ert to the start of Section 5, where we noted that we can connect any Koide matrix products to a La- grangian via (4.4) and (4.5). Now that we have obtained a theoretical expression for the neutron and proton masses, it is time to backtrack usin Section 4 to connect these masses to their associated La- grangian expression. This is simply to put all of the foregoing into a more formal physics context so that this is understood as going beyond si numbers to make them numerically fit an equation with opaque origins. We shall develop such a Lagrangian formulation for the neutron plus proton mass sum (6.6), recognizing that a Lagrangian connection for the separate masses of the neutron and proton ca using Yang-Mills matrix expressions such as (5.3), (5.4), (6.3) and (7.4) of [5] to also develop a Lagrangian for- mulation of neutron minus proton mass difference (1.4). Using the Pauli spin matrix 2 T, a unitary rotation ma- trix may of course be formed using: th overall neutron mass than each of the two down quarks, and the lone down quark in the proton makes a slightly greater contribution to the proton mass than each of the two up quarks. 9. The Lagrangian Formulation of the Neutron plus Proton Mass Sum Now we rev c (8.6) tell us w s. While the constituent varying constituent through (8.6), as well as g the development in mply playing with mass n then be developed Copyright © 2013 SciRes. JMP J. R. YABLON 147 234 222 2 2 23 23 4 4 2 111 exp 1 2! 3!4! 10 000 0 11 1 010 2! 3!4! 1 3 iTiTiTiTiT (9.1) 24 3 00 11 12! 4! 11 1 3! 2! Consequently, the square root of this rotation matrix is: 3 4 0 cos sin ! 1 sin c 4! os 22 pexpiT iT 1 2 ex 1 sin 2 11 cos 22 cos 1 2sin . (9.2) ing the phase With this in mind we start with the expression (6.6) incl exp i which we later found in (6.30) is ud exp 1i , and write the neutron plus proton mass sum us in root rotation matrix as: g a square 2 3Tr 0 0 3exp c ud ud u iM Mm 2 11 1 11 0 0 11 cos sin 22 11 sincos 22 cos uu dd d mm mm m 11 1 NPABBC CAABBC MMUU 1 4exp os CD U MM i DAABBA (9.3) in combination with a rotated “electron generation matrix” defined via left multiplication with 1 U as: 41 exp 0 2 30 co 0s ud AB u MMi m m 11 1 0 1 sin 2 d CB AC mU 11 1 cos 2 d m 4 11 11 1 s 2 1 in 2 1 exp00 200 11 30cossin00 22 00 11 0sincos 22 u ud u d i MM m m (9.4) and an adjoint matrix defined via right-multiplication with 1 U as: Copyright © 2013 SciRes. JMP J. R. YABLON 148 4 11 1 4 1 exp0 0 2 11 30cossin 22 11 0sin 22 1 exp0 0 2 00 11 000cos 22 00 11 0sinc 22 ud ABu u dd ud u d MM i mm mm i MM m m 1 11 11 cos sin os AC CB U (9.5) .947454 2s124 In the above, 10co mber found in (6.28), is the empirical nuand 0 is identically true as fAB ound in (6.30). The above, and AB , are just the Koi the electron generation ltiplying from the left and fr de triplet matrix AB tated into primed state by m for uro om the right via 1CB AC U and 1AC CB U. ow from (4.4) and (4.5) that as soon as we But we kn a Koide matrix, we can backtrack into a Lagrangian formulation. In this case, in (2.1) for a generalized Koide matrix AB K have , we are setting 12 , udu mMMmm and 3d mm , and the only new feature is that we are then rotating this matrix both from the left and the right via K U K and K KU. Consequently, we may use (9.4) and (9.5) to write the mass sum N P M M in (9.3) in a Lagrangian formulation, using these rotated Koide matrices, via (4.4) and (4.5) as: 33 3 33 3 22 2 2 4 33 211 1 11 2πd2πTrd 2πTr d 22 1 exp 0 2 111 2πd3Tr0 cossin 222 11 0sincos 22 3expcos c NP AB BD ud ABBAABBAuu dd MM xxx MMi xmm mm iMMm m 1udud 0 LEE EE EE os MM (9.6) y introducing new field strength tensors defined in the manner of (4.2) as: 1NP b ,,, Tr ududuudd ud ud imm MM E, (9.7) ,,, Tr ududuu dd ud ud imm MM E, where the “vacuum-amplified” masses (9.8) M and d M u as well as the square root mass ud M M are defined as in 5.14), and where the Koide mass matrices are (5.12) to ( formed for E using left-multiplication (9.4) and for using right-multiplication (9.5). Referring back to Sections 2 and 4, this means that here we have set 123 ,, udu d in the field strength tensor (4.2) and as just noted, 1ud mMM, 23 , ud mmmm in the Koide matrix (2.1), then fol- lowed the remaining development of Section 4 with the E Copyright © 2013 SciRes. JMP J. R. YABLON 149 only addition being that we now are also employing the rotations (9.4) and (9.5) on these Koide triplet matrices. We also now have the knowledge which can be exploited for further future development, that (9.3) for the neutron plus proton mass sum specifies a special case of the very general master mass andxing matrix as specified in (6.17), see (6.20). So thes us a hook into a Lagran- gian formulation for othegenerations of fermion, and therefore, for formulating er charmed, strange, top and bottom-containing baryo As a consequence of toregoing, the unrotated fer- mion eigenstates used to fm (9.7) and (9.8) are a triplet ,, ud u mi is giv r oth ns. he f or d consistin a wavefunction for a vac- uum-enhrmion (using upper case Greek), together the ory fermion wavefunctions , g of ud dinar anced fe with ud foe up and d current quarks (lower case Gree ud function th nderan tron r thown wave u k). It is the at is responsible for generating the vast prepoce of the constituent mass contributions to the neuplus proton mass sum, see Section 8, while ,d are responsible for the cur- rent mass contributions. Lastly, as in (4.12) through (4.14), at the nuts and bolts level, we apply theansatz (he Gaussian 4.12), in tform: 2 3 0 24 2 1 πexp2 uu rr ru , (9.9) u 2 0 2 1 2 dd d rr rd , (9.10) 3 24 πexp 2 0 2 1 2 ud ud ud rr rV , (9.11) and for the reduced Comn wavelengths, converting to 1c units, we specify: 3 24 πexp pto 1 uu u mc m, (9.12) 1 dd d mc m, (9.13) 1 ududu d M Mc MM. (9.14) So, referring back to the discussion at the end of Sec- tion 4, as was the case with the short range of the nuclear interaction, we can indeed use the Gaussian ansatz to model fermion wavefunctions as Gaussians and obtain e fully-dressed neutron and proton masses. But to do so, in the above we are using the undressed “current” quarks , ud th which yielded binding energies in [1,5], together in the same Koide triplet with a vacuum-amplified quark avefunction ud and associated masses and wave- er obtain a precise concurrence with empirical data. So, insofar as fully covered protons and neutrons are concerned, it looks as if the vacuum-amplified quarks in combination with the curren t quarks, are behaving as free fermions, as specified in detail in all of the foregoing. This underscores the role of the Gaussian ansatz as a modeling tool used to derive effective concurrence with empirical data, rather than as a part of the theory per se. The theory is centered on bary magnetic monopoles, and nucleons releasing or retaining binding energies based on their resonant properties which in turn depend upon the current quark content of those nucleons. For calculations which involve the components and emissions of protons and neutrons such as their cur- rent quarks and their binding energies, the current quarks can be modeled as free fermions to obtain empiri- ay bling vacuum-enhanced e whole pus ton ass but have unclear, opaque origins in the way that the Koide relations have also had unclear origins. Rather, as shown in (9.6) this mass sum can be formulated as the energy w lengths. So here too, it is not a question of wheth we can use a Gaussian ansatz, but rather, it is a question of which wavefunctions with which masses and wave- lengths we need to use in the Gaussian ansatz, in order to ons being Yang-Mills cally-accurate results. For other calculations which in volve the bulk behavior of protons and neutrons, accurate - results me obtained by mode quarks in combination with current quarks as free fer- mions, in the manner outlined above. Thoint of the discsion in this Section has been to make clear that the neutron plus pro m sum (and thus the individual neutron and proton masses) de- veloped in this paper is not just the result of developing formulas which fit the empirical data 33 2 33 2 2πd 12πTr d 2 NP MM x x L EE arising from integrating a Lagrangian density 1 2 LEE over the entirety of a three-space vol- ume element 3 d x . This puts the neutron and proton mplication via as specified in (6.17), masses as well) into the context of funda- mental, Lagrangian-based physics, and shows how these mass formulas (as well as those of Koide) are not just coincidental numeric coincid but truly are real physics relationships with a Lagrangian foundation. 10. Conclusion In conclusion, we have shown how the Koide relation- ships and associated triplet mass matrices can be gener- alized to dee neutron and pr fo masses (and by i other baryon ences of unexplained origin, rive the observed sum of the fre n rest masses in terms of the up aotond down current quark masses and the Fermi vev to six parts in 10,000, see (5.18). This sum can then be solved r the separate Copyright © 2013 SciRes. JMP J. R. YABLON Copyright © 2013 SciRes. JMP 150 neutron and proto wn in (7.5) to be related to an ing matrix de uark neutro on ourse nships (3.14), (3.16) and (3.17) fo REFERENCES [1] J. R. 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Seo, “A Simple Expression of the Jarlskog Determinant,” 2012. http://arxiv.org/abs/1201.3005 [16] F. Halzen and A. D. Martin, “Quarks and Leptons: An Introductory Course in Modern Particle Phy Wiley & Sons, Hoboken, 1984. sics,” John |