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
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
FL,
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
vmm3174 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
Mvm751345 MeV, (5.12)
634.5
1
3
dFd
Mvm784463 MeV. (5.13)
Consequently:
2
4619.1
2
9
ud Fud
MM vmm902116MeV. (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. Yablon, “Why Baryons Are Yang-Mills
Monopoles,” Hadronic Journal, Vol. 35, No. 4, 2012, pp.
399-467.
http://www.hadronicpress.com/issues/HJ
.pdf
[2] G. t’Hooft, “Magnetic Monopoles in Unified Gauge
Theories,” Nuclear Physics B, Vol. 79, No. 2
276-284. doi:10.1016/0550-3213(74)90486
n masses using the neutron minus pro-
ton 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 matri-
ces 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 nu-
cleon fitting angle deduced in (6.28) from empirical data.
This fitting angle is then sho
invariant of the CKM mixing angles within experi-
mental errors. Also of interest is a master mass and mix-
veloped in (6.17) which may help to inter-
connect all baryon and q masses and mixing angles.
The Koide generalizations developed here enable these
n and proton mass relationships to be given a La-
grangian formulatibased on neutron and proton field
strength tensors that contain vacuum-amplified and cur-
rent quark wavefunctions and masses, as shown in Sec-
tions 8 and 9. In the cof development, we also un-
cover new Koide relatio
r the neutrinos, the up quarks, and the down quarks.
Magnetic
/VOL35/HJ-35-4
, 1974, pp.
-6
[3] H. C. Ohanian, “What Is Spin?” American Journal of
Physics, Vol. 54, No. 6, 1986, pp. 500-505.
doi:10.1119/1.14580
[4] http://www.tau.ac.il/~elicomay/emc.html
[5] J. R. Yablon, “Predicting the Binding Energies of the 1s
Nuclides with High Precision, Based on Baryons which
Are Yang-Mills Magnetic Monopoles,” Journal of Mod-
ern Physics, Vol. 4 No. 4A, 2013, pp. 70-93.
doi:10.4236/jmp.2013.44A010.
[6] Y. Koide, “Fermion-Boson Two-Body Model of Quarks
and Leptons and Cabibbo Mixing,” Lettere al Nuovo
Cimento, Vol. 34, No. 8, 1982, pp. 201-205.
doi:10.1007/BF02817096
[7] http://pdg.lbl.gov/2012/tables/rpp2012-sum-leptons.pdf
[8] J. R. Yablon, “Grand Unified SU(8) Gauge Theory Based
on Baryons which Are Yang-Mills Magnetic Mono-
poles,” Journal of Modern Physics, Vol. 4 No. 4A, 2013,
pp. 94-120. doi:10.4236/jmp.2013.44A011
[9] http://pdg.lbl.gov/2012/listings/rpp2012-list-neutrino-pro
p.pdf
[10] http://pdg.lb.gov/2012/tables/rpp2012-sum-quarks.pdf
[11] A. Rivero, “A New Koide Tuple: Strange-Charm-Bot-
tom,” 2011. http://arxiv.org/abs/1111.7232
[12] http://cerncourier.com/cws/article/cern/29651
[13] M. Cresti, G. Pasquali, L. Peruzzo, C. Pinori and G. Sar-
tori, “Measurement of the Antineutron Mass,” Physics
Letters B, Vol. 177, No. 2, 1986, pp. 206-210.
doi:10.1016/0370-2693(86)91058-0
[14] http://pdg.lbl.gov/2012/reviews/rpp2012-rev-ckm-matrix.
pdf
[15] J. E. Kim and M.-S. 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