Predicting the Neutron and Proton Masses Based on Baryons which are Yang-Mills Magnetic Monopoles and Koide Mass Triplets 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 angle deduced from empirical data. This fitting angle is then shown to be related to an invariant of the CKM mixing angles 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.


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 3 3 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: 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 nucleons, 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:     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 493.028394MeV for 56 Fe, in contrast to the empirical binding energy of 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 developed 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 relationships 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  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:      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 relationships 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 matrices 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 relationships 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 author'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 938.272046MeV and 939.565379MeV of the free neutron 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 empirically-observed Cabibbo, Kobayashi and Maskawa (CKM) quark mixing matrices.In Section 8 we examine 0 0 Then, the two latent binding energy relationships (1.2) and (1.3) may be represented as: where, starting with (2.1), in (2.2) we have set  and and in (2.3) we have set 1 Again, these originate in the author's thesis in [1] that baryons are Yang-Mills magnetic monopoles.Above, designates an outer matrix pro-duct. Tr (2.5) Then, using (2.4) and (2.5), Koide relationship (1.5) for charged leptons may be written as: Clearly then, the Koide matrices (2.1) provide a general form for organizing the study of both binding energy and fermion mass relationships which lead to very accurate empirical results.It thus becomes desirable to understand 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 tantalizingly-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 extended in their application to make additional mass predictions and gain a deeper understanding of the particle mass spectrum, especially the free neutron and proton masses to be explored here.
We start in the next Section by showing how to reformulate the Koide relationships in terms of the statistical variance of the Koide terms across the three generations.

Statistical Reformulation of the Koide Mass Relationship
We continue to examine the charged leptons by setting ).When we use the extremes of the experimental data ranges in [7], specifically, the largest possible tau mass and the lowest possible mu mass, we obtain R = 1.5000024968.Although 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 3 2 R  is a very close relationship, it is still approximate even accounting for experimental error.For this to be within experimental errors, it would have to be possible to obtain some 3 2 R  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 1 2 3 , i.e., the "average of the squares" of the matrix elements in (2.1), as: Next, via (2.5), we write the "square of the average" of these matrix elements as: So, combining (3.1) and (3.2) in the form of (1.5) allows us for the charged leptons to write: This allows us to extract the relationship: in the usual way, as: The key relationship here, using first and last terms, is: 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: where we have defined the statistical coefficient C and the inverse relationship for R as: Thus, we may rewrite the basic Koide relationship (1.5) more generally as: In the circumstance where the statistical coefficient C = 1, i.e., where the average mass is exactly equal to the statistical variance, we have 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, This in an interesting "coefficient migration" as between the charged and uncharged leptons, wherein for the charged leptons masses to parts per 100,000, while for the neutrino lepton upper mass limits, 3 2 C  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/c 2 .We also use , for down quarks.
providing a "bridge" from "up" leptons to "up" quarks, and 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: Each of these relationships takes twelve a priori independent fermion masses and reduces by 1, their mutual independence.So with (3.14) through (3.17), to first approximation, we have now eight, rather than twelve independent fermion masses.
For some other commonly-studied Koide triplets we have: , (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 relationship (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

C uds 
. But, similarly to (1.5), see also (3.10), it may well make sense to regard this as a relationship accurate 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 relationships (3.14), (3.16) and (3.17) are important steps for-If we generalize this to any three fermion wavefunctions ward in their own right), but about how the ratio parameter R which for the charged lepton triplet is , 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 migration of coefficients.
Turning back to the neutron and proton triplets, For these triplets which all have a small 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 parameterization, we have C .

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 magnetic monopole containing a triplet of colored quarks in the zero-perturbation limit, reproduced below: , 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 , approaches an ordinary commutator, 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  Then, we form a pure gauge field Lagrangian 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 , 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 calculate energies according to (11.7) of [1], also (1.8) of [5], which are reproduced below: In the case where represents the proton, then depending on whether we contact indexes using , we obtain the inner and outer products in (3.6) of [5].When represents the neutron, we obtain the inner and outer products in (3.7) of [5].Using (2.1), the Koide generalization of the outer products ( K K index summation) is: while the Koide generalization of the inner products ( K AB BA K index summation) is: 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).Specifically, combining (3.9) with (4.4) and (4.5) allows us to write: This expresses the Koide mass relationship in multiple forms, in terms of an energy integral of the general Lagrangian density form with general field strength (4.2).This means for any Koide triplet of given empirical R, there is an energy which vanishes under condition: This is the Lagrangian/energy formulation of the Koide relationship (3.9), and although different in appearance, it is entirely equivalent.So, for example, using the symbol  as in Figure 1 and Table 3 of [8] to represent 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: 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 consideration.
What ties all of this together, is that we model the radial behavior of each fermion in the triplet 1 2 3    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): and that we also relate each reduced Compton wavelength i to its corresponding mass i via the De- , see [1] following (11.18).This is what makes it possible to precisely, analytically calculate the energy in integrals of the form (4.3), specifically making use of the mathematical Gaussian relationship (9.11) of [1]: and variants thereof.It is (4.12) and (4.13) and units) which tie everything together at the "nuts and bolts" mathematical level when (4.2) is employed in (4.3) through (4.11).And this is what leads to accurate mass relationship (1.1) and binding energy predictions (1.2) and (1.3), as well as the binding energy predictions for 2 H, 3 H, 3 He and 4 He 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: 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 fortuitously 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 wavelengths 1 m   i i 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, current quark masses, and to thereby obtain empiricallyvalidated 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 interaction which we showed at the end of Section 10 in [1] is indeed short range with a standard deviation of , 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 , which tells us that the nuclear interaction virtually ceases at about . This is exactly what is observed.In both cases-for nuclear binding energies and for the nuclear interaction short range-we found that the Gaussian 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, constituent 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 938.272046MeV and 939.565379MeV 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 wavefunctions we choose and in the masses that we assign to the fermions.That is, the focus of our deliberations will be, not whether 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 empirically accurate results.
Now, with all of the foregoing as background, let us see how to predict the neutron and proton masses.

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 1 2 3 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.This means the "mass coverings" m (using a lowercase m) for the neutron and proton may be calculated to be: . (5.2) These mass coverings m represent the observed, fully-dressed neutron and proton masses M, less the sum u for the proton, and 1 2 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 v F = 246.219651GeV beyond what is shown in (5.4).Specifically, 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!" as weights of rather heavy "clothing" "covering" "bare" quarks.The sum of these two mass covers is: 1856.4466 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 derive the sum of these two masses, and then use (1.4) as a simultaneous equation to obtain each separate mass.The benefit of this approach using a sum, referring 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 symmetric under and interchange, because the sum contains three up quarks and three down quarks, as well as one neutron and one proton.Our empirical target, therefore is the mass sum But we can alternatively find this by finding the mass cover sum P N of (5.3) to which we can then readily add .These sums are what we now seek to predict.We now return to use the "clues" laid out in (3.6) through (3.8) of [8].We start in the simplest way possible by focusing our consideration on (3.8) of [8], reproduced below, but multiplied by a factor of 2 and separated into 4 F u v m and 4 F d v m in the second term, thus:  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 For the proton with a fermion triplet , the corresponding eigenvalue entries in (5.5) above are For the neutron and its triplet, the entries are 2 1 1 , , 3 3 3 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 we see that to match the mass dimension 1 2 of the terms with 4 vm u and 4 vm   , , m m m 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 for which we also take the fourth root to match (5.4).Thus, we use to define two new Koide triplets, one for the neutron and one for the proton, as follows: 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 coefficient In recent years, consideration has been given to having negative square root terms in Koide mass relations, see for example (3.21) 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 oppositelysigned charges of the up and down quarks.But for the moment, we ignore in the above and examine magnitudes only, and form and calculate the following Koide matrix product from (5.6) and (5.7) with excised: Comparing to (5.3) which tells us that we see that we have hit the target to within about 0.06%!That is: 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 relationship to hypothesize that a meaningful relationship is given by , then using the above with (5.3) to add the current quark masses u d  to this mass cover sum, we see that to within about 0.06%: (5.10) So it appears as though we have now discovered the correct coefficients for the "clue" in (5.4).These coefficients, 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 coincidental expression for the neutron plus proton mass sum, we see that this is symmetric under interchange, and that it is formed by taking the inner product

 
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 magnitudes 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.
Furthermore, if we divide (5.8) by 2, we see that: 928.785 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" between 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 928.9With these definitions, the neutron plus proton mass sum (5.10) may be rewritten more transparently as: while the Koide mass matrices (5.6) and (5.7) for the neutron and proton become: . (5.17) These matrices now restore the ficient that we excised to calculate (5.8).Thus, as in (5.8), but including this complex factor, we now take: Having found a very close magnitude, we could make use of a 2 factor and continue to match the empirical data by writing . 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 understand this complex factor, as well as how to close the remaining 0.06% gap between the predicted and the observed neutron plus proton mass sum.That will be the subject of the next two Sections.

Exact Characterization of the Neutron and Proton Masses via a Mixing Angle θ and Phase Angle δ
The complex factor 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 entire mystery.So, let us start to pull on this thread and see where it leads us.
We first represent this factor Then, we br me So (6.5) sandwich-multiplied by (6.4) simply general- and use this phase to rewrite (5.18) as:  in separate matrices (5.16), (5.17) en we use th also.
is to rewrite mass sum (5.15) with where we have also br d  But now let us permit both  and  to rotate freely, . 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  , (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 angles ,   , so as to precisely match up with the empirical 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 3) and explicit in (6.5) as a hint that there exists a matrix . Then we use 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).Following all of this, we will use these freely rotated angles to permit the otherwise close relationship (6.3) to be fitted 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 generation mixing, see (7.11) in [8], and in particular, is the matrix that is use to introduce the phase angle responseble 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 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]: , 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: PDG data [10] certainties).alogous to (6.5) for the second and third generations in same manner as i used to form CKM mixing matrices, again see (7.11) 2792 MeV m  , (6.9) 1 3   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 dimension +3, defined as: Then, we mult

M M m m m s s M M m m m s c m m M M m m s
This master matrix contains all six of the quark masses in all three generations, all three of the real mixing angles and the one phase angle that appears when the three generations are mixed, and implied in the vacuum-enhanced 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 2 3 0, 0 s s   , this reduces to: .18)and in the further circumstance where all of the second and rd generation masses are set to 1, this further reduces to 9 times the matrix shown in (6.6): neutron plus proton mass sum of (6.6): So in this particular special case, (6.17) even contains the 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 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 g in 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 oppositelycharges which led to terms containing 4 1  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: We now need no more than elementary algebra to determ e that the neutron and proton masses, separate given by: These can be made into exact theoretical expressions fo 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 neuand proton masses.Let's now do so.on Because each of (6.22) contains a complex phase, we Now we solve these as simultaneous equations for  and .First we restructure (6.2 terms of 3) in   to arrive at: We now set t ese two cos h  equal to one another to eliminate  and solve for  .It will be easier to see the underlying e of these equations as well as solve them if we write (6.24) above as: Next, we reduce the second and third terms of (6.25) successively in five steps as follows: In the final step, we arrive at a quadratic for and so obtain a solution via the quadratic equation.Then, we use the variables (6.26) including the empirical masses of the neutron and proton, to calculate that:   in (6.28) used to precisely fit (6.22) to the observed neutron and proton masses as the "nucleon fitting angle".In the next Section we shall show how to tie this angle to the observed 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 calculate to find that:

N N P N P A N P A A A C N P N P A N P A A A P N P N P A N P
This numerical calculation reveals that cos 1 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 resultin ill reduce to 1 ident substig expression w ically!So to the extent that  may be a CP-violating phase t 0 , and given tha   is a deduced result for the neutron and proton m this deductively tells us that there are no asses (6.22), CP-violating effects associat n and proton.

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 formulations of the neutron and proton masses, ance where empirically-determined angle 1 0.947454 1 2
So we now return to (6.22), set 0   , and so obtain our final expressions for the neutron and proton masses: which are exact relations with th rical substitution 1 0.947454 2 s 1 2 4 3 (6.31) 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: which is binding energy B for any given nuclide with Z protons and N    , may also be rewritten generally in relation to nuclear One final exploratory exercise of interest is to return to he master mass and mi g matrix  in (6.17) and set    found in (6.28).In this circumstance, (6.17) reduces to:    angles charm and top quark eV, is perhaps suggesply 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 uld be clear that the master matrix (6.17) and like matrices that can be similarly constructed are an exceedingly useful tool for trying to develop and fit mutual relationships among mixing angles, CP violating phases, and quark an Whatever the correct fits may turn out to be with various higher-generation baryons, it sho d baryon masses.

Relation of the Nucleon Fitting Angle θ to the CKM Mixing Angles
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 underempirically-kno angles, which could then relate the neutron and proton masses themselves to the CKM angles.This is highly preferable to having 1 cos be a new, separate parameter.
Toward this end, we first write the CKM matrix with the "standard choice" of angles and its empirical values from PDG's [14] as: e lo angles are between 0 and (We use a negative sign for the thre wer-left empirical entries to match the negative values in the terms which the standard CKM matrix takes on when the π 2 .)Now, 1 cos  0.9474541242 does not fit any particular one of these elements.But what is of interest is the determinant V which may be calculated from the CKM mixing and phase angles ij  and  to be: 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 experimental values in (7.1), we find that sum of the three positively-signed (+) terms in the determinant, denoted V ning 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 as the baseline against which to compare V  , we find that: V  according to: 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 determinant V  , according to: nnects the proton and neutron ma This now co sses 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 empirical neutron and proton masses using 1 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 ), and if we wish to maintain the proton and neutron masses to be entirely real based on cos    to absorb the terms with t det he 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 t rlskog determinant is "the imaginary part of any o element among the six component of determinant o s that one would CP symmetry for the neutron and proton.Given that Im 3 V J    , this means that: , 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 as 1)), we find that the matrix entries will contain terms of the form 3 3 3 4  , (7. Copyright © 2013 SciRes.JMP J. R. YABLON opyright © 201 JMP 145 depending on representation, 3 5   f F G v .This may help us gain further insight into fermion masses as well as high-order angian vacuum terms 3 4 5 , , which specify how much of the observed neutron and proton masses arise from each of th and t e quarks heir in much does each down masses?In ot , for Lagr    .All of this mystery cracking is the result of the detective 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"   m m 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 defining respective constituent quark mass sums 2 In (5.12) through (5.14) we defined three very helpful mass va and 635 MeV.It is natural therefore to inquire whether th um-amplified" quark masses might be related to the so-called "constituent" quark mas h specify how much mass eac ontributes 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  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 constituent masses of the up and down quarks C 3 SciRes.3 cos 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 abov present a deconstruction of the neutron and proton masses into e sep -e , as follows: The , 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 depending 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 nucleons 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 individual 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 constituent 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-amplified quark masses do not vary.They are mass constants (to the same degree that current quark masses are constants, 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 Lagrangian 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 Lagrangian 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 formulation of neutron minus proton mass difference (1.4).
Using the Pauli spin matrix 2 T , a unitary rotation matrix 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.

The Lagrangian Formulation of the Neutron plus Proton Mass Sum
Consequently, the square root of this rotation matrix is: 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   1 in combination with a rotated "electron generation matrix" and an adjoint matrix defined via right-multiplication with 1 U as: Copyright © 2013 SciRes.JMP In the above, .Consequently, we may use (9.4) and (9.5) to write the mass sum 3) in a Lagrangian formulation, using these rotated Koide matrices, via (4.4) and (4.5) as: y introducing new field strength tensors defined in the manner of (4.2) as: , , , Tr where the "vacuum-amplified" masses 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 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 and xing matrix  as specified in (6.17), see (6.20)  So, referring back to the discussion at the end of Section 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 , u d 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 waveer 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 current 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 current quarks and their binding energies, the current quarks can be modeled as free fermions to obtain empiriay b ling vacuum-enhanced e whole p us 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 wavelengths 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 m e obtained by mode quarks in combination with current quarks as free fermions, in the manner outlined above.
Th oint of the disc sion in this Section has been to make clear that the neutron plus pro m sum (and thus the individual neutron and proton masses) developed in this paper is not just the result of developing formulas which fit the empirical data   x .This puts the neutron and proton mplication via  as specified in (6.17), masses as well) into the context of fundamental, 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.

Conclusion
In conclusion, we have shown how the Koide relationships and associated triplet mass matrices can be generalized to de e 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 a oto nd 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 n masses using the neutron minus proton mass difference (1.4) earlier derived in [5], as shown in (6.22).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), see (6.5) and (6.14).These can then be used to specify the neutron and proton mass relationships to unlimited accuracy as shown in (6.31) using θ as a nucleon fitting angle deduced in (6.28) from empirical data.This fitting angle is then sho invariant of the CKM mixing angles within experimental errors.Also of interest is a master mass and mixveloped in (6.17) which may help to interconnect all baryon and q masses and mixing angles.The Koide generalizations developed here enable these n and proton mass relationships to be given a Lagrangian formulati based on neutron and proton field strength tensors that contain vacuum-amplified and current quark wavefunctions and masses, as shown in Sections 8 and 9.In the c of development, we also uncover new Koide relatio r the neutrinos, the up quarks, and the down quarks.
where u is the "current" mass of the up quark and is the current mass of the down quark. d relationship among these masses.Indeed, if we use the 2012 PDG data and m τ = 1776.82± 0.16 MeV [7], we find using mean experimental 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 d E x m m m ,(2.4) .4) which naturally absorbs the 3 from the factor of 3/2.Now, we simply use (3.4) to form the statistical vari-

. 6 )
So the averagei of the charged lepton masses is approximately (and very closely) equal to the statistical variance m

. 2 ,
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 factor of 3/2, which is somewhat mysterious in (1.5), is now more readily understood when we realize that it corresponds with C = 1 in (3.7).This means that the Koide relationship for any given triplet of numbers with mass dimension 1 may be alternatively characterized by the coefficient C. Thus, using (3.7), the coefficient C for the charged lepton triplet is (we also include R for comparison): some other Koide triplets?For the neutrinos, PDG in [9] provides upper limits e 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:

13 ),
PDG's[10].For Koide triplets of a single electric charge type, we can then calculate that:So we now see a distinctive pattern of coefficient migration among (3.10) through(3.13).For the charged leptons in (3.10) which are the lower members of a weak isospin doublet, 3 2 R e   , as has long been known.For neutrinos which are the upper members of this douwhich migrates the 3/2 from the R to the C coefficient.Then, for the up quarks, we find another coefficient migration such that 3 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 6 5 R uct  coefficient for the up quarks, now migrates to   reformulated for any fermion triplet into the coefficient C in the statistical variance relationship i 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:

and form a unitary matrix 1 U
 with i e   : izes the appearance of the term 0


of these masses.So first we deduce: tr 6.31), in the form: however, contra, there are no omitted angles and somewhere we should expect to come across a baryon with a third generation quark.Thes no ne sets up the  matrix in several different representations.


as the basis for specifying V  , i.e., we now set: a further ingredient used to tighten the empirical data in (7.1).Further, because co V   V  injects into the proton and neutron masses an imaginary term with a Jarlskog determinant 2 13 12 23 12 13 23 sin CKM J c c c s s s   culated using the angles in (7.1) with CKM (which may be cal-   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 f F m G v 

5
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 th m the oppo--signed electric charges of th and down quarks, nd (not directly segregate the up quark mass contri m that of the down e canbution fro quark.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 3 two down quarks.For the proton, we allocate 1 quarks.We similarly allocate the μ d exp 1 i  , and write the neutron plus proton mass sum us in root rotation matrix as: the square root mass u d strength tensor (4.2) and as just noted, 1 m   in the Koide matrix (2.1), then followed the remaining development of Section 4 with the E the entirety of a three-space volume element3  d in which one uses