1
A Model of Mesons
R Wayte
29 Audley Way, Ascot, Berkshire, SL5 8EE, England, UK.
e-mail: rwayte@googlemail.com
Research article. Submitted to vixra.org 19 September 2011
Abstract: Detailed models of mesons have been derived in terms of real structured
particles, in order to replace the formless quark/anti-quark singularities of standard
QCD theory. Pion design is related to the muonic mass, and a Yukawa potential is
calculated for the hadronic field. A charged pion is produced by adding a heavyelectron or positron in a tight orbit around the neutral core. Other mesons are found to
be ordered assemblies of pionic-size masses, travelling in bound epicyclical orbits,
with real intrinsic spin and angular momentum. These orbit dimensions are related to
the mean lifetimes of the mesons through action integrals. Decay products resemble
parts of their parent mesons, as expected for a relaxation process with traceability of
particles.
Key words: meson composite models
PACS Codes: 12.39.Pn, 12.60.Rc, 14.40.Be, 14.40.Df
2
1
Introduction
According to standard QCD theory, mesons have overall finite dimensions yet
consist of quark plus anti-quark singularities of infinite mass density. This is not
realistic and is of limited value in explaining the meson variations from type to type.
Of course, quantum theory is essential for describing interactions between particles,
yet it tells us little about particle structure, and there remains the need to elevate
quantum theory above mathematical inspiration or fantasy.
Comprehension of this analysis requires some unique concepts developed
previously for fermion models; see Wayte, Papers 1, 2, 3. Those models described an
isolated proton, electron or muon and were very successful at explaining the Yukawa
potential, the reality of spin and anomalous magnetic moment, and particle creation
mechanisms. On the other hand, the Standard Model of particle interactions has been
very successful at accounting for data from high energy collision experiments.
Consequently, the conceptual differences between these models can be explained if
particles in collisions reveal behaviour not immediately apparent in static models. To
link these models, the constituents of baryons and mesons need to behave like up,
down and strange quarks when in high energy collisions. It will be shown in
Appendix A how this happens.
When we consider that high energy collision experiments are theoretically
capable of producing a continuum of meson types, it must be significant that so few
types are somehow chosen to exist. Mesons will be described as real understandable
mechanisms with variability in their substructure for the different types. In collisions
and decay processes, historic traceability of products is considered to be very
important as a guide to design. Spin and angular momentum are always real structured
quantities. As for fermions, particle mass is simply localised energy travelling at the
velocity of light in bound orbits, so the Higgs postulate is unnecessary. This bound
energy has helicity which determines by whether the particle is matter or anti-matter.
A fundamental particle satisfies the Dirac equation but any concept of negative massenergy or time-reversal is excluded. Real particles are covered by their wavefunction:
Ψ = Ae ±i( Et −px ) / ,
(1.0)
where +i means right-handed helicity and -i means left-handed helicity of a circularly
polarised wavefunction; and E, t are real positive quantities only.
3
In this static meson model there are indeed two major pieces, but they do not
look or behave like QCD quarks, so they will be called quion q+ and anti-quion qaccording to their charge sign. These orbit the origin and can produce real spin
angular momentum, depending on their orientation and the orbit radius. Often, there is
also an extra complex particle located at the origin, which increases the mass and adds
variety to meson behaviour. We will start with the pion model as the basic design, and
then extrapolate to cover other mesons.
The lifetime of a free meson must depend upon its particular internal design.
This inference is based upon analogy with other physical systems; for example when
a charged capacitor C is connected to a resistor R, the discharge time constant is
determined by the hardware involved (CxR). Several separate batches of a given
meson type will decay so as to converge upon the same characteristic lifetime;
therefore the same exact mechanism must exist in all batches. The probability that a
single meson will decay in any given time increment is a constant, so no ageing
occurs. This implies that the smooth-running mechanism is perfect but subject to
spontaneous quantum fluctuations of the internal fields which can disturb it
catastrophically. Different types of mesons have characteristic lifetimes and
mechanisms but there are some common features.
The way that the meson's mean lifetime τ and decay width Γ cooperate during
its creation in a collision process is interesting. Given:
Γτ ≈
,
(1.1)
the real value of Γ must be established in the short creation period, whereas τ and the
decay probability appear realised after the creation is completed, over a much longer
period in some cases. This contrasts with an atomic emission-line in which the scatter
in energy depends upon the lifetime of the excited state before emission. Furthermore,
the Heisenberg uncertainty principle is written:
∆E∆t ≈
or ∆x∆p ≈ , ,
(1.2)
where ∆t implies incremental uncertainty in a larger macroscopic value of t; but τ in
Eq.(1.1) is the macroscopic value. Effectively, τ is established during the creation
stage as a coherence period of the controlling guidewave, (see Paper 1, Section 10.3).
The decay probability wavefunction follows as a consequence of this.
4
Section 2 will now concentrate on the detailed pion design. Section 3 will
cover the well-observed unflavoured mesons and Section 4 the remaining unflavoured
mesons. Section 5 describes the designs of strange mesons. Section 6 is the general
conclusion. Appendix A explains the compatibility of these designs with the Standard
Model for protons, neutrons and mesons.
All particle data have been taken from the Particle Data Group listing at
http://pdg.lbl.gov.
2
Pions
π o (135) : m = 134.9766MeV/c 2 , I G (J PC ) = 1− (0 − + )
π ± (140) : m = 139.57018MeV/c 2 , I G (J P ) = 1 − (0 − )
;.
In QCD theory, a pion is thought to consist simply of a quark and anti-quark
with net charge determined by the type of quarks. The pion effective charge radius
has been measured at around 0.65fm, see Amendolia et al. (1986) and Eschrich et al.
(2001).
Our pion model is illustrated in Figure 2.1, wherein a quion q+ and anti-quion
q−, each consisting of two smaller pearls, orbit the centre at radius roπ and velocity c to
constitute a πo neutral pion. These may then be orbited by a heavy-positron to make a
π+ (classed as matter), or a heavy-electron to make a π− (anti-matter). Analogous to
the proton and antiproton, the quion and anti-quion emit a radial pionic-type field, in
addition to possessing their own native electromagnetic charge, plus a gluonic strongfield running around their own circumferences. Overall angular momentum of the
pion is zero because the pair rotate about their own axes, counter to their orbital
motion, that is:
m q c′rq = m q croπ
,
where [c′ = c(π / 2)], [rq = roπ (2 / π)], and [roπ = 2e 2 / m πo c 2 ] .
(2.1)
5
Fig.(2.1) Pion component parts for matter π+ and anti-matter π−
2.1
Yukawa potential
The quion /anti-quion pair is proposed to emit an attractive nuclear-type
hadronic field similar to the proton; wherein the ‘field-mesons’ have reduced effective
masses [mπ' = mπo(h'/h) with h' << h] in order to produce a smooth copious field.
Published QCD calculations will here be considered unrealistic if the exchange field
particles are as massive as the source particle, see for example Gashi et al. (2001).
Nevertheless, the calculations may still be useful if aspects like field range can be reexpressed in terms of the mass mπ' .
The metric tensor component for the field is proposed to be analogous to the
proton field, (see Paper 1, Eq.(3.4)):
r
r − roπ
γ = 1 − oπ exp−
rπ
r
1/ 2
,
(2.2)
where (rπ = ħ/mπoc = 1.462fm) is the range factor, and (roπ = 2e2/mπoc2 = 2rπ /137) is
double the pion classical radius because of the quion /anti-quion pair; compare this
with positronium. The corresponding empirical potential is given by:
m πo c 2
,
Vc = (γ − 1)
a χπ
(2.3)
where aχπ represents the hadronic charge for pions, to be determined shortly. Then,
from Eqs.(3.8) (3.9) in Paper 1, the hadronic coupling constant for (π-π) is definable
as:
6
m c 2 r
χ π = π o o π
2
exp roπ
r
π
(1 + 2 / 137 ) .
/ c ≈
137
(2.4a)
This ‘mesonic-field’ is independent of any electromagnetic positive or negative
charge which orbits the hadronic pair. For pion-nucleon interactions it is probable
that the coupling constant will be of the order:
χ πN = (χ N χ π )1 / 2 ≈ (9 / 137 ) ≈ 0.065 ,
(2.4b)
where the nucleon-nucleon coupling constant is (χ N ≈ 1 / 3 ) in Paper 1.
Analogous to the proton derivation, hard core repulsion will be attributed to
rapid spinning of the quion/anti-quion field source which modulates the local field
and causes it to become repulsive. Here, the source frequency is (c /2πroπ) compared
with the Compton frequency of field-mesons (c/2πrπ); that is, (137/2) times greater.
Therefore, analogous to Eq.(4.1) of Paper 1, the full hard core metric tensor
component becomes:
1/ 2
r r r
r − roπ
r − roπ
exp −
γ hc = 1 − oπ 1 − π oπ exp −
r roπ r
rπ
roπ
15
10
MeV
5
0.5
0
Fermis
1
1.5
rπ
-5
-10
0.117fm
-15
Fig.(2.2) Hadronic potential energy function for the pion.
.
(2.5)
7
The empirical overall pion potential is given by:
m c2
Vhc = (γ hc − 1) πo ,
a χπ
(2.6)
where [aχπ = (χπħc)½]. This potential energy (aχπVhc) as a function of radius is
illustrated in Figure 2.2.
2.2
Pion mass
The quion /anti-quion masses may be related to electronic mass or to muonic
mass as was found for the pearls in a proton:
m πo = 2m q = 264.1426 m e = 134.9766 ± 0.0006 MeV/c2, (2.7)
and given muon mass (mµ = 105.658367MeV/c2) we have:
m µ
m πo = 2m q ≈ 2 × 2
3
1
1 − .
24
(2.8)
Here, we recall that a muonic mass could consist of 3 distinct packs of core-segments
(Paper 3, Eq.(4.2)); consequently each quion pearl here takes the mass of one such
pack (mµ /3), approximately. The small negative term in this equation will be taken to
express overall mass decrement due to binding energy of the attractive field between
pearls within the quion itself, plus the binding energy of the quion /anti-quion pair in
the pion circumference.
Similar to the proton model, we shall further assume that:
1 r
1 − ≡ 1 −
24 rq
,
(2.9)
where the real quion radius is (rq = roπ(2/π)), and the pearl radius r is thus:
r =
e2
=
24 m q c 2
rq
( 2 / π)
.
24
(2.10)
This implies that a pearl is dimensionally 24 times smaller than the quion, and that
there were 24 original pearl-seed particles which subsequently condensed into 2 equal
pearls of mass m to minimise action /energy, analogous to the 3 pearls in a proton’s
trineon. It is thought probable that a pearl consists of 24 gluon-loops, like a proton
pearl, but their constituent grains and mites are undefined.
8
The quion charge e was originally divided between the 24 pearl-seeds, thus:
e
mq
2.3
e / 24 e / 2
=
=
m s m
.
(2.11)
Charged pions
A charged pion is produced by adding a heavy-electron or positron to orbit the
central quion /anti-quion pair. This increases the total mass, as was found for the
neutron in Paper 1, viz:
π ± − π o = 4.5936 ± 0.0005MeV / c 2 = 8.9894 m e = m h
(2.12)
We shall assume that the mass increase is due solely to the orbiting heavy-electron (or
positron), with its compressed dimensions. Thus, the classical radius of this heavyelectron is to be equal to the orbit size:
rh = e 2 / m h c 2 = 0.31347fm .
(2.13)
At first sight this result appears arbitrary and does not explain why an electron should
attach itself to the hadronic core at all. However, by referring to the neutron analysis,
a proper physical explanation can be derived. The πo radius is given above
as (roπ = 2e 2 / m πo c 2 = 0.0213365fm ) , consequently (rh = 14.692roπ ) . Then this ratio
of radii must govern a special relationship because the neutral πo and heavy-electron
cooperate to produce a more stable charged pion,. Consequently, it is proposed that
spiralling electromagnetic feeler guidewaves are emitted by the charged quion and
anti-quion to communicate attractively and continuously with the heavy-electron (or
positron). An action equation for these guidewaves will be based upon the following
formula:
ln(rh / roπ ) = ln 14.692 ≈ π(e n / π) .
(2.14)
This may be differentiated and reduced to an electromagnetic action integral upon
multiplying through by (2e 2 / c = m πo croπ ) :
π
2 ×
en
2 πr
2π
h
δ(m πo )
δ( e 2 )
× ∫
dt ≈ ∫
croπ dθ .
2
2πroπ z
0
(2.15)
where δ may be around h΄/h, as used in Section 2.1. On the left is potential energy
action for the feeler guidewave spiralling out and back from the quion and anti-quion,
9
with (z = ct = 2πr) and including a contribution from the gluon energy, through factor
(π/en). On the right is kinetic energy action for the element of pion core material
which constitutes the guidewave energy.
Finally, the manner in which the free electron (or positron) is compressed onto
the πo core is interesting. Let the free electron spin-loop be first compressed down to
electron core radius roe , then further to (rhe = roe /2.843), as was found for the neutron
(see Paper 1). This is followed by compression by factor (rhe /rh = 3.1619) to get to
the final radius rh . Then (ln(rhe /rh) ≈ π/en) may be reduced to an action integral by
differentiation and applying Eq.(2.13):
2 πrh
2π
e2
1
m
− ∫
dt ≈ ∫ h crh dθ
z
2
en
2 πrhe
0
.
(2.16)
On the left is action due to potential energy of the collapsing electron charge loop
(z = 2πr), rotating at velocity c. The right side represents action of kinetic energy
around the loop for a second harmonic material helix.
2.4
Pion mean lifetimes
It has been shown previously that the lifetime of a neutron or muon may be
related to its internal period, by way of an action integral.
Similarly, the pion
lifetimes appear to be definite functions of internal periods, as follows:
(a)
Let the πo lifetime ( τ πo = 8.4 + .06 × 10-17s) represent a number N πq of
quion periods, (2πrq /c' = 1.81x10-25s):
(
)
N πq = τ πo / 2πrq / c' = 4.63 × 10 8 .
(2.17)
Then ( ln N πo ≈ 2π 2 ) will be taken to indicate that there is an action integral which
will describe the pion structure. Thus, after differentiating and multiplying through
by (e2/c = mπocroπ /2 = 2mqc'rq /2), we get:
N πo ( 2 πrq )
∫
( 2 πrq )
1 (e / 2)
2 z ′
2
2π
dt ≈ m c′rq dθ .
∫ 2
0
(2.18)
On the left, pearl charge is (e/2) and the integral represents classical potential energy
action required to create a pearl travelling around a quion loop, by assembly of charge
from the guidewave coherence distance Nπo(2πrq). In reality, the creation may be a
10
faster strong force process so Eq.(2.18) would represent theoretical dissipation of a
pearl. However, the classical viewpoint allows visualisation and ensures conservation
of energy and action. Distance (z' = c't) employs velocity [c' = c(π/2)]. The right side
represents kinetic energy action of a pearl as it travels at velocity c' over one quion
revolution 2πrq. Only mass (mℓ /2) is involved because half of the pearl mass is in the
external field which does not rotate. If Eq.(2.18) were multiplied by (δ = h'/h) as in
Eq.(2.15) then it would represent just the associated guidewave creation mechanism.
(b)
The π+ lifetime ( τ π + = 2.6033 × 10-8 s) may also represent a number of quion
periods,
(
)
N π+ = (τ π+ ) / 2πrq / c' = 1.437 × 1017 .
(2.19)
Thus (ln N π + ≈ 4π2) may be developed into an action integral similar to Eq.(2.18),
but with double the action on the right, which is now expressed in terms of the quion
kinetic energy action:
N π+ ( 2πrq )
∫
( 2 πrq )
2
1 (e / 2)
2 z ′
2π m
dt ≈ q
∫ 2
0
c' rq dθ .
(2.20)
However, the extended π+ lifetime relative to πo should probably be attributed to the
heavy-positron orbit period (2πrh /c), in some manner like the following. Let:
N P = (τ π+ ) / (2πrh / c ) = 3.963 × 1015 ;
(2.21a)
ln( N P ) ≈ 2π(137 / 24) .
(2.21b)
then,
Upon differentiating and multiplying through by (e2/c = mπocroπ /2), this expression
could represent an action integral, such as:
1
137
N p ( 2 πrh )
∫
2 πrh
2π
2
dθ
1 e
1 m πo
.
croπ
dt ≈ ∫
2
24 0 2
2 z
(2.22)
On the left is potential energy action required to establish one of the 137 pearls (see
Paper 2) within the orbiting heavy positron core, operating over guidewave coherence
distance NP(2πrh) at velocity c, in time
τ π + . On the right side is a quantity of kinetic
energy action due to one of the 24 pearl-seeds per quion over half a pion period. This
expression appears to relate the establishment of the orbiting positron's internal
11
mechanism to the pion's existing core mechanism, through some spiralling feeler
guidewave link. It is this physical linkage, plus that described in Eq.(2.15), which
could then govern the long decay lifetime until guidewave coherence is broken by
random peaks in the internal quantum fluctuations.
3
Various light unflavoured mesons
Design structures for a few light mesons will now be outlined, using concepts
developed for models of the pion, proton and neutron. The decay products can retain
some features from their parent and are simpler in design, as would be expected from
a self-controlled relaxation process.
3.1 Some J = 0 mesons
3.1a Eta-meson: η(548): m = 547.853MeV/c2, IG(JPC) = 0+(0− +).
The lowest η-meson has the mass of around 4 pions, and is thought to take the
basic design of a pion, see Figure 3.1a. Here the positive quion consists of 2 pearls of
approximately pionic mass each; likewise for the negative anti-quion. Analogous to
Eqs.(2.7)(2.8), we have:
m η = 2m q = 547.853 ± 0.12 MeV/c 2 ,
(3.1.1a)
and approximate quion mass,
1
m q ≈ 2m µ/ 1 −
,
37.7
(3.1.1b)
where (mµ' = 4mµ /3), which will be called a muonet like a miniaturised muon, as it
will be used frequently later. This reveals an overall binding mass decrement due to
an attractive field inside the meson. Particle core radius is given by:
roη = 2(e 2 / m η c 2 ) = 5.257 × 10 −3 fm ,
(3.1.2a)
and quion radius is,
rq = roη ( 2 / π) .
(3.1.2b)
Apparently, there were 37 original pearl-seed particles, and the pearl radius is
37.7 times less than the real quion radius (rq):
r = rq / 37.7 .
(3.1.3)
12
Although each pearl in a quion has roughly the mass of a pion, it is miniaturised and
cannot have identical design. The original 37 pearl-seeds in a quion are proposed to
have condensed into 2 pearls comprising 37 gluon-loops, each of mass mq /(2x37.7).
If the lifetime of an η-meson (τ η = / Γ = 5.063 × 10 −19 s) is related to its
core period (2πroη / c = 11.02 × 10 −26 s) , then:
N η = τ η /( 2πroη / c) = 4.59 × 10 6 ,
(3.1.4a)
and,
( )
ln N η = 15.34 = π 2 (π / 2) .
(3.1.4b)
After differentiation, this with Eqs.(3.1.2a), can be reduced to an integral representing
the action of creation (or dissipation):
Nη ( 2 πroη )
∫
2πroη
2
π
1e
dt ≈ ×
2
2 z′
2π
mq
croη dθ .
2
0
∫
(3.1.5)
On the left is the amount of potential energy action required to create (or dissipate) a
quion travelling around the spin-loop, by assembly of charge from the guidewave
coherence distance Nη(2πroη). Distance (z' = c't) may describe a spiral, and (c'τη)
represent a guidewave coherence length. The integral on the right is a quantity of
kinetic action for a quion as it travels at velocity c over one revolution (2πroη).
Coefficient (π/2) must be for weighting.
An η-meson may emit an attractive nuclear-type of field similar to the pion.
The corresponding hard-core metric tensor component is like Eq (2.5), wherein roπ is
13
replaced by roη. Similarly in Eq (2.6), mπo is replaced by mη and aχπ by
(a χη = (χ η c)1/2 )
for (χη ≈ (1+0.5/137)/137) derived from Eq.(2.4a) after
replacements. Overall empirical potential energy (aχηVhc) as a function of radius is
illustrated in Figure 3.1b. It is 4 times deeper in the short range, than the pion
potential of Figure 2.2. For ηN interactions, the coupling constant will be:
χ ηN = (χ N χ η )1 / 2 ≈ (9 / 137) ≈ 0.065 .
(3.1.6)
15
10
MeV
5
0.5
Fermis
1
1.5
0
-5
-10
-15
-20
-25
-30
-35
-40
Fig.(3.1b) Hadronic potential energy function for η(548).
3.1b η'(958): m = 957.78MeV/c2, IG(JPC) = 0+(0− +)
This eta-meson has a mass of around 7 pions and decays predominantly into
η(548) plus π+π- or πoπo, which implies that it has a similar but more elaborate design
than η(548), see Figure 3.1c. In this case, the quions and anti-quion have 3 pearls
each, like a trineon in a proton. However, there is a further pearl at the centre with
mass that is less than the quion pearls, estimated as follows.
14
Analogous to Eq.(3.1.1b), let the quion mass be given by:
1
2
m q ≈ 3m µ/ 1 −
= 3 × 137.140922MeV / c ,
37.7
(3.1.7)
Factor 37.7 in the denominator means there were originally 37 pearl-seeds, and these
condensed into 3 pearls, each comprising 37 gluon-loops. Pearl radius is 37.7 times
less than quion radius. Now let the central pearl mass be less than a quion pearl mass,
in view of its central bound position:
1
m c ≈ m µ/ 1 − = 135.007913MeV / c 2 .
24
(3.1.8)
The total meson mass is therefore approximately:
m η′ ≈ 2m q + m c = 957.85MeV / c 2 .
(3.1.9)
Given that the quion's pearls consist of matter, and the anti-quion's pearls of antimatter, it appears that the central pearl must resemble a pion with its quion and antiquion components.
Particle core radius roη' is determined by the quion masses without the central
pearl:
roη′ = 2(e 2 / 2m q c 2 ) = 3.505 × 10 −3 fm ,
(3.1.10a)
and the quion radius is,
rq = roη′ (2 / π) .
(3.1.10b)
Lifetime is given by (τ η′ = / Γ = 3.26 × 10 −21 s) , and may be related to the
quion period (2πrq / c' = 2.97 × 10 −26 s) , thus:
N η′ = τ η′ /( 2πrq / c' ) = 1.096 × 10 5 ,
(3.1.11a)
15
and then,
( )
ln N η′ = 11.60 ≈ (7 / 6 )π 2 .
(3.1.11b)
After differentiation, this with Eqs.(3.1.10a,b), may be reduced to an integral for
action of creation (or dissipation) of the quion/anti-quion, plus a central pearl:
Nη′ ( 2 πrq )
∫
2πrq
2
7
1e
dt ≈ ×
6
2 z′
2π
mq
∫ 2
0
c′rq dθ .
(3.1.12)
On the left is potential energy action required to create a quion, rotating at velocity c' .
The guidewave coherence distance (z' = c't) may describe a spiral. On the right, the
integral covers the kinetic action for a quion (3 pearls) rotating at velocity c', plus the
action of half the central-pearl is included through coefficient (7/6).
3.1c a0(980): m = 980 MeV/c2, IG(JPC) = 1−(0++).
The a0(980) meson has zero spin and mass equal to 8 pions approximately.
Since the dominant decay mode is ηπ, it is proposed to have the basic form of η(548),
but now with binary pearls, see Figure 3.1d. This produces very strong binding energy
within the quion which has mass given by:
mq =
980MeV / c 2
(3 / 2)
≈ 4m µ/ 1 − 2
.
2
24
(3.1.13a)
Factor (3/2)/24 signifies binding energy, and (1/24) could be a preferred pearl size
relative to a quion. During creation there were probably 24 pearl-seeds, which
condensed into the two pearls per quion. If the pearls rotate parallel to their quion
rotation, it could increase the mass over the f0(980) with its anti-parallel rotation, say.
16
This would be one way of distinguishing the a0(980) from f0(980), as appears
necessary according to Scadron et al. (2003), Janssen et al. (1994), Baru et al. (2003,
2008), Wang & Yang (2005).
Core radius of the ao(980) is given by:
roa = 2(e 2 / m a c 2 ) ,
(3.1.13b)
and the quion radius is,
rq = roa (2 / π) .
(3.1.13c)
Lifetime ( τ a = / Γ = 8.8 × 10 −24 s) appears to be related to the core period
(2πroa /c = 6.13 × 10−26s) rather than the quion period:
N a = τ a /(2πroa / c) = 143 ,
(3.1.14a)
ln (N a ) = 4.955 ≈ π 2 / 2
(3.1.14b)
and then,
.
After differentiation, this with Eq.(3.1.13b) may be reduced to an action equation,
similar in part to Eq.(3.1.5):
Na ( 2 πroa )
∫
2 πroa
2π m
2
q
dθ
1e
croa
.
dt = ∫
2
2
2 z′
0
(3.1.15)
On the left is potential energy action required to create (or dissipate) a quion, where
distance (z' = c't) may describe a spiral. On the right, the integral covers the kinetic
action for a quion, travelling at velocity c around half the core circumference (πroa).
3.1d fo(980): m = 980 MeV/c2, IG(JPC) = 0+(0++).
The f0(980) meson probably has structure very similar to a0(980), but with the
pearls spinning anti-parallel to their quion rotation, to reduce the overall mass below
a0(980). The dominant decay (ππ) would exclude (η) because of this anti-parallel spin.
3.2 Some mesons with ( J = 1 )
3.2a Rho-meson ρ(770): m = 775.49 MeV/c2, IG(JPC) = 1+(1− −)
The ρ(770) meson is distinctly different from the π and η-mesons because of
its spin being J = 1
rather than zero. If, like other particles, only half its mass is
contained in the spin-loop and half is field energy which does not rotate, then:
17
J = ( m ρ / 2)crρ =
.
(3.2.1)
Spin-loop radius rρ is therefore:
rρ = 2( / m ρ c) = 137[2(e 2 / m ρ c 2 )] ,
(3.2.2)
which is 137 times the classical/theoretical radius for a quion/anti-quion pair in
rotation. The mass is around that of 6 pions and is thought to take the design of 3
pearls in the quion and 3 in the anti-quion, as shown in Figure 3.2a. Therefore:
mq =
1
775.49 ± 0.34MeV / c 2
≈ 3m πo 1 − ,
2
24
(3.2.3)
where m πo is the ''pionet'' mass like a miniaturised pion, rather than the muonet mass
used previously in Eq.(3.1.1b) etc. As found for the pion pearls in Eq.(2.10), these
pearls are smaller than the quion by 24 times. However, like trineons in a proton, the
quions are now very much smaller than the spin-loop:
rq = rρ / 137(2 / π) .
(3.2.4)
The electromagnetic lifetime given by (τρe = / Γee = 9.35 × 10 −20 s) and the
spin period (2πrρ /c = 1.0666 × 10-23s) may be related by:
N ρe = τ ρe /(2πrρ / c) = 8.77 × 10 3 ,
(3.2.5a)
and then,
(
)
ln N ρe = 9.079 ≈ (π / 2)(137 / 24) .
(3.2.5b)
18
This may be reduced to an action integral by differentiating and introducing
Eq.(3.2.2):
Nρe ( 2 πrρ )
∫
2 πrρ
2π
2
1e
1 1 m q
≈
dt
∫
2 z
24 2 0 2
dθ
crρ
.
2
(3.2.6)
On the left is the potential energy action required to create (or dissipate) a quion;
where (z = ct) describes a spiral over a guidewave coherence length. The integral on
the right side represents kinetic energy action of the quion travelling around half the
spin-loop. Coefficient (1/24) means that the quion originally comprised 24 pearlseeds, but the action of only one is involved here. These 24 pearl-seeds condensed
into 3 pearls, each containing 24 grains of reduced mass.
The full width
(Γρ = 149.4 ± 1.0 MeV) implies a strong lifetime (τρ =
4.406×10-24s), which is less than the spin period given above and may be related to
the period of the rotating quion (2πrq /c' = 7.783 x 10-26s):
N ρq = τ ρ /(2πrq / c′) = 56.6 ,
(3.2.7a)
ln N ρq = 4.036 = 0.41π 2 .
(3.2.7b)
and then,
(
)
Upon differentiating and applying Eqs.(3.2.2), (3.2.4), this reduces to an interesting
action integral:
Nρq ( 2 πrq )
∫
2 πrq
2π
2
1e
1 m q
≈
dt
∫
2 z′
2 0 2
dθ
c′rq
.
3
(3.2.8)
On the left is potential energy action required to create (or dissipate) a quion; where
(z' = c't) over the guidewave coherence length. The integral on the right side
represents kinetic energy action of a spinning quion over one third period (2πrq /3).
3.2b Omega-meson: ω(782): m = 782.65 MeV/c2, IG(JPC) = 0−(1− −)
The ω(782) meson design is similar to ρ(770), see Figure 3.2b, but with a
spin-loop radius slightly less:
rω = 2( / m ω c) = 137[ 2(e 2 / m ω c 2 )] ,
(3.2.9)
and a corresponding quion radius,
rq = rω / 137(2 / π) .
(3.2.10)
19
The mass is again around that of 6 pionets, although the binding energy within the
quions is a little less:
mq =
1
1
782.65 ± 0.12MeV / c 2
≈ 3m πo 1 −
. (3.2.11)
1 −
2
37.7 137
Electromagnetic lifetime ( τ ωe = / Γee = 1.10 × 10 −18 s) appears to be related
to the spin period (2πrω /c = 1.056 × 10-23s) by:
N ωe = τ ωe /( 2πrω / c) = 1.041 × 10 5 ;
(3.2.12a)
ln(N ωe ) = 11.55 ≈ π(137 / 37.7) .
(3.2.12b)
and then,
By differentiating and introducing Eq.(3.2.9), this may be reduced to an action
integral:
Nωe ( 2πrω )
∫
2 πrω
2π
2
1 m q
1e
∫
dt ≈
37.7 0 2
2 z
dθ
crω
.
2
(3.2.13)
On the left is potential energy action required to create (or dissipate) a quion; where
(z = ct) could describe a spiral over the guidewave coherence length. The integral on
the right side represents kinetic energy action of the quion as it travels around half the
spin-loop. Factor 37.7 in the denominator means there were originally 37 pearl-seeds
and only one is being considered. These 37 pearl-seeds condensed into 3 pearls, each
containing 37 grains of reduced mass.
20
Full width (Γω = 8.49 ± 0.08 MeV) implies a strong lifetime (τω = 7.75 ×
10−23s), which may be related to one third of a quion's rotation period (2πrq/3c' = 2.57
x 10-26s):
N ωq / 3 = τ ω /(2πrq / 3c′) = 3016 .
(3.2.14a)
ln N ωq / 3 = 8.01 = 0.81π 2 ,
(3.2.14b)
Then,
(
)
and upon differentiating and introducing Eqs.(3.2.9) and (3.2.10), this reduces to an
interesting action integral:
Nωq / 3 ( 2 πrq / 3)
∫
( 2 πrq / 3)
2π m
2
q
1e
dt ≈ ∫
2
2 z′
0
dθ
c′rq
.
3
(3.2.15)
On the left is potential energy action required to create (or dissipate) a quion; where
(z' = c't) over the guidewave coherence length The integral on the right side represents
kinetic energy action of a spinning quion over one third of a revolution (2πrq /3).
Equation (3.2.14a) implies that a third harmonic guidewave is operating around the
quion.
3.2c Phi-meson φ(1020): m = 1019.455MeV/c2, IG(JPC) = 0−(1− −).
The φ(1020) meson has spin 1 given by:
J = (m φ / 2)crφ =
,
(3.2.16)
where spin-loop radius rφ is:
rφ = 137[2(e 2 / m φ c 2 )] ,
(3.2.17)
and quion radius,
rq = rφ / 137(2 / π) .
(3.2.18)
Mass is approximately that of 8 pionets and is to take the form of 4 pearls in the quion
and 4 in the antiquion, as shown in Figure 3.2c. During decay these usually convert to
separate kaons, although a rho + pi is also possible. Quion mass is given by:
mq =
1019.455MeV / c 2
(4 / 3)(3 / 2)
≈ 4m πo 1 −
37.7
2
.
(3.2.19)
The constituent pearls are smaller than the quion by 37.7 times. Factor (4/3) means
that three pearls are at the vertices of an equilateral triangle, and the fourth at the
centre, see Simo (1978). Again, the quions are 137(2/π) times smaller than the spin-
21
loop, like trineons in a proton. This ensures that a quion rotates 137 times at velocity
c' during one spin-loop orbit, which is a stable arrangement.
The electromagnetic lifetime given by (τ φe = / Γee = 5.183 × 10 −19 s) and the
spin period (2πrφ /c = 8.114 × 10−24s) may be related by:
N φ = τ φe /( 2πrφ / c) = 6.388 × 10 4 ,
(3.2.20a)
ln(N φe ) = 11.065 ≈ π(137 / 37.7) .
(3.2.20b)
and,
This can be differentiated and, with Eq.(3.2.17), reduced to an action integral similar
to Eq.(3.2.13):
Nφe ( 2 πrφ )
∫
2 πrφ
2π
2
1 m q
1e
≈
dt
∫
37.7 0 2
2 z
dθ
crφ
.
2
(3.2.21)
On the left is potential energy action required to create a quion, where (z = ct) along a
spiral over the guidewave coherence length. The integral on the right represents
kinetic energy action of the quion as it travels around half the spin-loop. Denominator
37.7 means that only one of the 37 pearl-seeds is being considered. These 37 pearlseeds condensed into 3 pearls, each containing 37 less-massive grains.
The full width (Γφ = 4.26 ± 0.04 MeV) implies a strong lifetime of (τφ = 1.55
×10-22s). This lifetime may be related to the period of the rotating quion (2πrq /c' =
5.921x10-26s):
22
N φ = τ φ /(2πrq / c′) = 2618 ,
(3.2.22a)
ln( N φ ) = 7.87 = 0.80π 2 .
(3.2.22b)
and then,
Upon differentiating and introducing Eqs.(3.2.17) and (3.2.18), this reduces to an
action integral:
Nφ ( 2 πrq )
∫
2 πrq
2π m
2
q
dθ
1e
c′rq
dt
.
≈
∫
2
3
2 z′
0
(3.2.23)
On the left is potential energy action expended to create (or dissipate) a quion; where
z' = c't over the guidewave coherence length. The integral on the right side represents
kinetic energy action of a spinning quion over one third revolution (2πrq /3).
4
General design of light unflavoured mesons
In the previous section, the very lightest mesons have been described in some
detail, but more massive mesons of each species have also been studied in order to
produce similar viable structures. Choice of design has been based upon the belief that
the decay process is a relaxation effect so that the products should be simpler, but
retain some of the parent features. Those decays accompanied by low levels of kinetic
energy are most likely to satisfy this criterion. Pearls are not created during a decay
process, so the number of pearls will either stay the same or decrease by annihilation.
It is easy for the pearl type to remain unchanged or to lose energy by changing from
muonet (mµ' = 140.9MeV/c2) to pionet (mπο = 135MeV/c2); but less easy for the
reverse process, except when enough free kinetic energy is accessible. These rules
restrict the use of mµ' to mesons with (C = +1, (π η a f)), and mπο to mesons with
(C = −1, (ρ ω φ b h)).
The mesons occupy 8 categories and have been listed with regard to their
properties in Table 4.1. They have 5 defining characteristics, IGJPC. Each class has a
particular parity P with a free choice of J value, but I, G and C are related through:
CG = 1 − 2I ,
(4.1)
which limits the number of classes to 8 only. The meson traditional nomenclature is:
(a) Pseudoscalar (JP = 0−). (b) Scalar (JP = 0+). (c) Pseudovector (JP = 1+). (d) Vector
(JP = 1−).
23
Table 4.1 Classification of the light unflavoured mesons.
PC
0− +
J
I=1
J −+
1− + 2− +
J ++
0 ++ 1++
2++
4++
J +−
1+ −
G = −1
a0(980)
π
a1(1260)
π(1300)
a2(1320)
π1(1400)
a0(1450)
π1(1600)
a4(2040)
π2(1670)
π(1800)
π2(1880)
G = +1
G = +1
b1(1235)
ρ(770)
ρ(1450)
ρ3(1690)
ρ(1700)
G = +1
η(548)
η'(958)
η(1295)
η(1405)
η(1475)
η2(1645)
G = −1
G = −1
h1(1170)
ω(782)
φ(1020)
ω(1420)
ω(1650)
ω3(1670)
φ(1680)
φ3(1850)
G = −1
ο
I=0
J −−
1− − 3− −
G = +1
f0(980)
f2(1270)
f1(1285)
f0(1370)
f1(1420)
f0(1500)
f2'(1525)
f0(1710)
f2(1950)
f2(2010)
f4(2050)
f2(2300)
f2(2340)
Scalar and pseudoscalar mesons with zero angular momentum appear to be
tight orbiting structures, consisting of discrete pearls of muonet-mass around
140.9MeV/c2, see Table 4.2. The quion and anti-quion rotate counter to their orbital
motion, in order to cancel angular momentum overall, as in Eq.(2.1) and Section 3.1.
All vector and pseudo-vector mesons (J ≥ 1) appear to be open structures, to generate
the spin, following Eq.(3.2.1). Some of these mesons also consist of mµ'- pearls, as
listed in Table 4.3. The others consist of pearls of pionet-mass (134.9776MeV/c2), and
decay into simpler pionic structures; see Table 4.4.
Mesons f1(1285), η(1295), η(1405), η(1475), and π(1800) can decay into
ao(980), so they have the same unusual twin-pearl structure, based on the inheritance
principle.
The mass decrements indicate the degree of binding overall and in some cases
the size of the pearls relative to their quion size, as in Section 2.2.
24
Table 4.2 Internal designs for scalar and pseudoscalar mesons, comprising pearls of
muonet-mass (mµ' = 140.877823MeV/c2), approximately. A mass analysis formula is
given, plus IG(JPC), full width Γ, and the main decay products Dy(…).
4µ'
η(548) / 547.853 ± 0.024 MeV
0+(0− +), Γ = 1.29keV, Dy(3π,2γ )
1
m ≈ 4m µ/ 1 −
37.7
= 548.56MeV
7µ'
η'(958) / 957.78 ± 0.06 MeV
0+(0− +), Γ = 0.202 MeV, Dy(π,η,ρ,ω ).
1
1
m ≈ 6m µ/ 1 −
+ m µ/ 1 −
24
37.7
= 957.85MeV
8µ'
a0(980) / 980 ± 20 MeV
1−(0+ +), Γ = 50-100 MeV, Dy(ηπ).
f0(980) / 980 ± 10 MeV
0+(0+ +), Γ = 40-100 MeV, Dy(ππ).
2(3 / 2 )
m ≈ 8m µ/ 1 −
= 986.1MeV
24
10µ'
η(1295) / 1294 ± 4 MeV
0+(0− +), Γ = 55 MeV, Dy(a0(980),η ).
(3 / 2)
2
m ≈ 8m µ/ 1 − + 2m µ/ 1 −
24
24
= 1297.2MeV
10µ'
π(1300) / 1300 ± 100 MeV
1−(0− +), Γ = 200-600 MeV, Dy(ρπ ).
(4 / 3)(3 / 2) + 2m / 1 − 1
m ≈ 8m µ/ 1 −
µ
24
24
= 1303.1MeV
25
10µ'
f0(1370) / 1350 ± 150 MeV
0+(0+ +), Γ = 200-500 MeV, Dy(ππ,ηη,KKc).
(4 / 3) + 2m / 1 − (3 / 2)
m ≈ 8m µ/ 1 −
µ
37.7
24
= 1351.3MeV
11µ'
η(1405) / 1409.8 ± 2.5 MeV
0+(0− +), Γ = 51 MeV, Dy(η,KKc,a0(980)).
2(3 / 2 )
2
m ≈ 8m µ/ 1 − + 3m µ/ 1 −
24
24
= 1402.9MeV
11µ'
a0(1450) / 1474 ± 19 MeV
1−(0+ +), Γ = 265MeV, Dy(η'(958),KKc)
(4 / 3) + 3m / 1 − 2
m ≈ 8m µ/ 1 −
µ
37.7
24
= 1474.6MeV
12µ'
η(1475) / 1476 ± 4 M eV
0+(0− +), Γ = 87 MeV, Dy(KKcπ,a0(980))
2( 3 / 2)
m ≈ 12m µ/ 1 −
24
= 1479.2MeV
12µ'
f0(1500) / 1505 ± 6 MeV
0+(0+ +), Γ = 109 MeV,
Dy(ππ,KKc,ηη,ηη'(958))
(5 / 3)(3 / 2) +
m ≈ 10m µ/ 1 −
24
2( 3 / 2)
+ 2m µ/ 1 −
= 1508.6MeV
24
13µ'
f0(1710) / 1720 ± 6 MeV
0+(0+ +), Γ = 140 MeV, Dy(KKc,ηη,ωω)
5/3
1
/
m ≈ 10m µ/ 1 −
+ 3m µ 1 −
24
24
= 1716.0MeV
26
14µ'
π(1800) / 1816 ± 14 MeV
1−(0− +), Γ = 207 MeV, Dy(a0(980),ηη'(958))
1
2
m ≈ 12m µ/ 1 − + 2m µ/ 1 −
24
24
= 1819.7MeV
Table 4.3 Internal designs for vector and pseudovector mesons, comprising pearls of
muonet-mass mµ' approximately.
9µ'
a1(1260) / 1230 ± 40 MeV
1−(1+ +), Γ = 250-600 MeV, Dy(ρπ).
1
1
m ≈ 6m µ/ 1 −
+ 3m µ/ 1 −
37.7
37.7
= 1234.3MeV
10µ'
f2(1270) / 1275.1 ± 1.2 MeV
0+(2+ +), Γ = 185 MeV, Dy(ππ,KKc)
(4 / 3)(3 / 2 )
m ≈ 8m µ/ 1 −
+
24
2(3 / 2)
2m µ/ 1 −
= 1279.6MeV
24
10µ'
f1(1285) / 1281.8± 0.6 MeV
0+(1+ +), Γ = 24.3 MeV, Dy(ao(980),KKc)
2( 3 / 2)
2
m ≈ 8m µ/ 1 − + 2m µ/ 1 −
24
24
= 1279.6MeV
10µ'
a2(1320) / 1318.3 ± 0.6 MeV
1−(2+ +), Γ = 107 MeV, Dy(ηπ,ωππ,KKc).
(4 / 3) + 2m / 1 − 2
m ≈ 8m µ/ 1 −
µ
24
24
= 1322.7MeV
27
10µ'
π1(1400) / 1354 ± 25 MeV
1−(1− +), Γ = 313 MeV, Dy(ηπ).
(4 / 3) + 2m / 1 − 1
m ≈ 8m µ/ 1 −
µ
37.7
24
= 1357.2MeV
11µ'
f1(1420) / 1426.4 ± 0.9 MeV
0+(1+ +), Γ = 54.9 MeV,
Dy(KKc*(892),φ)
(4 / 3)( 3 / 2)
m ≈ 8m µ/ 1 −
+
24
( 3 / 2)
3m µ/ 1 −
= 1429.3MeV
24
12µ'
f2/(1525) / 1525 ± 5 MeV
0+(2+ +), Γ = 73 MeV, Dy(KKc,ηη)
(5 / 3)(3 / 2)
m ≈ 10m µ/ 1 −
+
24
( 3 / 2)
2m µ/ 1 −
= 1526.2MeV
24
13µ'
π1(1600) / 1662 ± 15 MeV
1−(1− +), Γ = 234 MeV, Dy(η/(958))
(5 / 3)(3 / 2)
m ≈ 10m µ/ 1 −
+
24
(3 / 2) = 1658.2MeV
+ 3m µ/ 1 −
24
12µ'
η2(1645) / 1617 ± 5 MeV
0+(2− +), Γ = 181 MeV, Dy(a2(1320),KKc,ηπ)
(5 / 3) + 2m / 1 − 1
m ≈ 10m µ/ 1 −
µ
37.7
37.7
= 1620.8MeV
13µ'
π2(1670) / 1672.4 ± 3.2 MeV
1−(2− +), Γ = 259 MeV, Dy(f2(1270),ρ,KKc*)
( 6 / 3)
2
/
m ≈ 12m µ/ 1 −
+ mµ 1 −
24
24
= 1678.8MeV
28
15µ'
f2(1950) / 1944 ± 12 MeV
0+(2+ +), Γ = 472 MeV, Dy(K* Kc*(892),ηη)
( 3 / 2)
( 6 / 3)
/
m ≈ 12m µ/ 1 −
+ 3m µ 1 −
24
24
= 1945.9MeV
16µ'
f2(2010) / 2011 ± 60 MeV
0+(2+ +), Γ = 202 MeV, Dy(φφ,KKc)
2(4 / 3)
m ≈ 16m µ/ 1 −
24
= 2003.6MeV
16µ'
a4(2040) / 2001 ± 10 MeV
1−(4+ +), Γ = 313 MeV, Dy(KKc,ρω,f2(1270))
2(4 / 3)
m ≈ 16m µ/ 1 −
24
= 2003.6MeV
16µ'
f4(2050) / 2018 ± 11 MeV
0+(4+ +), Γ = 237 MeV, Dy(ππ, KKc,ηη)
(6 / 3)( 3 / 2)
m ≈ 12m µ/ 1 −
+
24
(4 / 3) = 2022.8MeV
4m µ/ 1 −
37.7
16µ'
18µ'
f2(2300) / 2297 ± 28 MeV
0+(2+ +), Γ = 149 MeV, Dy(φφ,KKc)
2(4 / 3)
1
m ≈ 16m µ/ 1 −
+ 2m µ/ 1 −
24
37.7
= 2278.0MeV
18µ'
f2(2340) / 2339 ± 60 MeV
0+(2+ +), Γ = 319 MeV, Dy(φφ,ηη)
2(4 / 3)
m ≈ 16m µ/ 1 −
+
37.7
2( 3 / 2)
+ 2m µ/ 1 −
= 2341.1MeV
24
29
Table 4.4 Internal designs for vector and pseudovector mesons, comprising pearls of
pionet-mass (mπο = 134.9776MeV), approximately.
6π
ρ(770) / 775.49 ± 0.34 MeV
1+(1− −), Γ = 149.4 MeV, Dy(ππ).
1
m ≈ 6m πo 1 −
24
= 776.1MeV
6π
ω(782) / 782.65 ± 0.12 MeV
0−(1− −), Γ = 8.49 MeV, Dy(πππ).
1
m ≈ 6m πo 1 −
37.7
= 788.4MeV
8π
φ(1020) / 1019.455 ± 0.020 MeV
0−(1− −), Γ = 4.26 MeV, Dy(KK,ρπ).
(4 / 3)(3 / 2)
m ≈ 8m πo 1 −
37.7
= 1022.5MeV
9π
h1(1170) / 1170 ± 20 MeV
0−(1+ −), Γ = 360 MeV, Dy(ρπ).
1
1
m ≈ 6m πo 1 − + 3m πo 1 −
24
37.7
= 1170.3MeV
10π
b1(1235) / 1229.5 ± 3.2 MeV
1+(1+ −), Γ = 142 MeV, Dy(ω,KK,φ ).
(4 / 3)(3 / 2) +
m ≈ 8m πo 1 −
24
2
+ 2m πo 1 − = 1237.3MeV
24
30
11π
ω(1420) / (1400-1450) MeV
0−(1− −), Γ = 180-250 MeV, Dy(b1(1235),ρ)
(4 / 3) + 3m 1 − 1
m ≈ 8m πo 1 −
πo
37.7
24
= 1429.7MeV
12π
ρ(1450) / 1465 ± 25 MeV
1+(1− −), Γ = 400 MeV, Dy(ππ)
(5 / 3)(3 / 2) +
m ≈ 10m πo 1 −
24
1
2m πo 1 − = 1467.9MeV
24
13π
ω(1650) / 1670 ± 30 MeV
0−(1− −), Γ = 315 MeV, Dy(ρπ,ωππ,ωη)
ω3(1670) / 1667± 4 MeV
0−(3− −), Γ = 168 MeV, Dy(ρ,ω)
(5 / 3) + 3m 1 − 1
m ≈ 10m πo 1 −
πo
24
37.7
= 1678.2MeV
13π
φ(1680) / 1680 ± 20 MeV
0−(1− −), Γ =150 MeV, Dy(KK*(892))
(5 / 3) + 3m 1 − 1
m ≈ 10m πo 1 −
πo
37.7
37.7
= 1684.3MeV
13π
ρ3(1690) / 1688.8 ± 2.1 MeV
1+(3− −), Γ =161 MeV, Dy(π,KK,ρ)
(5 / 3) + 3m 1 − 1
m ≈ 10m πo 1 −
πo
37.7
37.7
= 1684.3MeV
14µ'
ρ(1700) / 1720 ± 20 MeV
1+(1− −), Γ = 250 MeV, Dy(ρππ,ρρ,KKc)
(6 / 3) + 2m 1 − 2( 3 / 2)
m ≈ 12m πo 1 −
πo
24
24
= 1721.0MeV
31
15µ'
φ3(1850) / 1854 ± 7 MeV
0−(3− −), Γ = 87 MeV, Dy(KK*(892))
(6 / 3) + 3m 1 − 2
m ≈ 12m πo 1 −
πo
24
24
= 1855.9MeV
18π
φ(2170) / 2175 ± 15 MeV
0−(1− −), Γ = 61 MeV, Dy(KK,φ fo(980))
1
2(4 / 3 )
m ≈ 16m πo 1 −
+ 2m πo 1 −
24
24
= 2178.4MeV
Given all these meson designs, it is now possible to see how spin depends on
mass to some extent but not predictably. Measured spin is accurately known and the
number of pearls in each meson is known, but the binding energy varies greatly and
causes uncertainty. Figure 4.1 shows J vs M for these meson structures, overlaid by
lines of an average binding energy according to the expression:
M=
2.75 /
m µ (3J + n + 3) .
3
(4.2)
Usually, the spin J falls below the main line (n = 0) and can increase with mass, but it
shows no obvious relationship.
In order to eliminate the confusion caused by variable binding energy, Figure
4.2 shows (J vs M) for theoretical meson structures with negligible binding energy,
according to the expression:
M = m µ/ (3J + n + 3) .
(4.3)
There now appears to be some order which may eventually be explicable. At the low
mass end, several vacancies still remain even after adding 4 strange mesons to fill
gaps. The ρ(770) stands out as unusual.
Missing meson resonances at (J = 0, n = 0, 2, 3, 6) correspond through
Eq.(4.1) to reduced masses 387MeV, 646MeV(f0(600)), 775MeV(κ(800)), 1162MeV.
These controversial states are currently supported by several investigators, for
example, Parganlija et al (2009).
32
n value
0
3
4
6
J, spin
3
9
2
1
0
0
250
500
750
1000
1250
1500
1750
2000
2250
2500
Mass, MeV
Fig.4.1
The actual relationship between spin and mass for light unflavoured
mesons, with average binding energy lines overlaid according to Eq.(4.2).
Factor n is given on the right ordinate. The solid points are for mesons in
Tables 4.2, 4.3, and hollow points are for the mesons in Table 4.4.
0
n value
3
4
6
J, spin
3
9
2
12
1
0
0
250
500
750
1000
1250
1500
1750
2000
2250
2500
2750
Mass, MeV
Fig.4.2
The theoretical relationship between spin and mass for light
unflavoured mesons, assuming negligible binding energy between pearls of
mass mµ' according to Eq.(4.3). Several mesons coincide in mass. Four strange
mesons (Ж) have been added to fill vacancies but several positions remain
vacant.
33
5.
Strange mesons
5.1
General features
When angular momentum is plotted against mass for strange mesons, it is
apparent that a linear relationship exists, even though many points are vacant, see
Figure 5.1. A reasonable fit exists for the empirical formula:
M K ≈ 2.875(J + n / 3)m πo + m K ±
,
≈ m πo (3J + n )(1 − 1 / 24) + m K ±
(5.1)
where n is an integer for the parallel lines as marked, and ( m πo = 134.9766MeV/c2),
(m K ± = 493.677MeV) are the pionet and kaon masses. This is like Eq.(4.1) and
suggests that individual pionets are added to increment meson mass. The mean
deviation of actual meson masses from Eq.(5.1) is 14MeV/c2, which is good
compared with 32MeV/c2 for a theoretical random mass distribution.
0
1
K-mesons
4
K*(2045) 16
3
4
3
J, spin
K3*(1780) 14
6
7
K2(1770)
K2*(1430) 11
2
14 K2(1820)
K1(1270)
10
K*(892) 7
1
11
K1(1400)
K*(1410)
K±(494) 4
0
500
'n'
value
14 K*(1680)
11 K0*(1430)
1000
1500
2000
Mass, MeV
Fig.(5.1) A plot of strange meson spin against mass, according to Eq.(5.1)
for the various values of ‘n’ given on the right ordinate. The number of
pionets in each strange meson is marked.
34
Equation (5.1) implies that a strange meson resonance can easily decay into a
single long-lived kaon plus pieces, albeit its own mean lifetime is very short.
Likewise, an unflavoured meson such as φ(1020) can produce a KK pair when it has
a quion and antiquion of sufficient mass.
Figure (5.2) represents our basic model for strange mesons, in which there are
(4 + n) neutral pionets bound by gluons in the compact core of radius rOK . At charge
radius r± there is a positron for K+ (matter) or an electron for K− (antimatter). At the
same radius there may also be a neutralising electron to produce a neutral kaon K0
(matter), or neutralising positron to produce a neutral K 0L (antimatter). It is this
neutralising electron which is emitted during semi-leptonic decay of the K0, and viceversa. These two neutral mesons are antiparticles and differ because the core pionets
have right-handed helicity within the original K+ or left-handed helicity within the
original K−, (just as a neutron differs from an anti-neutron).
e(rq)
spinloop (rs)
π π
+q
(+q)
π π
(-q)
(roK)
-q
(r±)
e+
Fig.(5.2)
Basic model schematic design for strange mesons, showing 4
pionets at the centre, orbited by a positron or electron, or both for a neutral
meson. Pionets may also be added to the core to increase the value of n. Spin
J is accomplished by adding pionets to the quion /antiquion pair in the spinloop.
35
The core always has zero net spin, but overall meson spin may be produced by
the quion /antiquion pair travelling in a larger spin-loop of radius rs at velocity c.
During hadronic decay, this spin-loop material plus n core pionets may convert
rapidly into free π, ρ, etc., eventually leaving the central kaon (2πo + 2πo) intact. This
spin-loop may only exist for less than one rotation period, although its creation must
have been completed more rapidly. For example, K*(892) has a decay full width (Г =
50.75 MeV) which corresponds to a lifetime of (τ = ħ/Γ = 1.3x10-23sec). Its spin-loop
period is given by (2πrs /c = 2.12x10-23sec), according to Eq.(5.4). In the case of
K4*(2045), its lifetime is only 0.33x10-23sec and its spin-loop period is also 2.12 x
10-23sec. Consequently, strange mesons with spin hardly come into existence before
decaying.
The spin of a strange meson is always given by:
J = (M s / 2)crs ,
(5.2)
where Ms is the quion+antiquion mass travelling at velocity c around the spin-loop at
radius rs . Given Figure (5.1), we will arbitrarily let the n pionets reside in the core, so
that 3 pionets must be added to the spin-loop to increase J by unity, then:
M S ≈ J × 2.875m πo ,
(5.3a)
and Eq.(5.1) becomes:
2.875
M K ≈ M S +
nm πo + m K ± .
3
(5.3b)
The spin-loop radius is independent of J at:
2
= 1.017 fm .
rS ≈
2
.
875
m
c
πo
(5.4)
For all values of J, pionets added to the spin-loop are bound by approximately the
same energy decrement, since the coefficient 2.875 implies:
2.875m πo = 3(1 − 1 / 24 )m πo .
(5.5)
The mass decrement ( m πo /24) is due mainly to the quion's or antiquion's self-binding
strong force, plus their mutual electromagnetic attraction.
36
5.2
Kaon mass structure
Kaons are denoted strange because of their long lifetime, which implies strong
binding of the component parts. Kaon mass m K + represents 4 strongly bound pionets
through the formula:
2
m K + = 493.677 MeV / c 2 ≈ 41 − m πo ,
24
(5.6)
where pionet mass is 134.9766MeV/c2. Here, the negative term represents binding
energy which keeps the pionets together by the strong force. The core radius r0K will
be given the classical value for 4 pionets arranged as a quion/antiquion pair like the
pion design:
rOK = 2 × (e 2 / 4m πo c 2 ) = 5.334 × 10 −3 fm .
(5.7a)
Then, by analogy with the electron and proton, the K+ charge radius is proposed to be:
r± = α-1r0K/2 = 0.3655fm,
(5.7b)
where (α = 1/137.036) is the fine structure constant. And a neutralising electronic
charge can be impressed into this same orbit to yield a neutral kaon K0 of mass (mK0 =
497.614MeV/c2).
Now we recall that for the neutron in Paper 1, the neutralising heavy-electron
around the proton had a mass determined roughly by its radius:
m ′he = e 2 /(c 2 rhe ) .
(5.8a)
Consequently, the mass of the neutralising electron here (at r±) might be simply:
m ′− = e 2 /(c 2 r± ) = 7.71m e = 3.94 MeV/c2 ,
(5.8b)
which would account for the measured difference in mass between a neutral and
charged kaon:
m K 0 − m K ± = 3.937 ± 0.028 MeV/c2 .
(5.9)
In addition, according to the neutron theory, the neutralising electron is also bound
and stabilised by its own self-interaction guidewave binding energy; see Paper 1,
Eq.(10.2.2). Therefore, the analogous expression for the binding energy here would
produce a heavy-electron of energy:
m − c 2 = 9m e c 2 −
e2
= 7.773m e c 2 = 3.97 MeV .
(2πr± )
(5.10)
This could take the form of 3 groups of 3 nominal electron masses; like 3 pearls in the
3 trineons of a proton.
37
The measured charge radius for the K+ is <r> = 0.560 ± 0.031fm, see PDG
(2010). This is an effective interaction size, to be compared with our real source size
(r± = 0.3655fm). Likewise, the effective/measured size of K0 is <r2> = -0.077 ±
0.010fm2, which implies that the negative and positive charges together at r± interact
differently with electrons in liquid hydrogen, so as to produce a net effective radius.
5.3
Mean lifetime
The long lifetime of a kaon will be attributed to the surrounding charge, with
due regard to its particular spin orientation. The basic K+ has a core structure (2πo +
2πo) which has not been observed to exist by itself without its positron. Decay of a
kaon occurs via the weak force, which is simply interpreted as repulsion due to direct
natural jostling between constituent pionets in their tight orbits.
K+. The kaons K+ have a central core period of (2πrOK /c = 1.12x10-25secs), so the
number of periods in its mean life (τ± = 1.2380x10-8secs) is:
N K = τ ± /(2πr0 K / c) = 1.11 × 1017
.
(5.11a)
This very large ratio is reminiscent of the pion Eq.(2.19), then:
ln(1.11 × 1017 ) ≈ 4π 2
(5.11b)
is probably to do with guidewave action and coherence length. For example, after
differentiating this and multiplying through by (e2/c = 4mπocr0K /2 = 2mqc'rq /2), we
get:
N K ( 2 πr0 K )
∫
2 πr0 K
2π m
2
q
1 (e / 2)
dt
≈
∫
2 cr0 K dθ .
2 z ′
0
(5.11c)
On the left, pearl charge is (e/2) and the integral represents potential energy action
required to create the pearl travelling around the core. The right side represents kinetic
energy action of a quion as it travels at velocity c during one core revolution 2πr0K .
K 0L .
The extended lifetime of this kaon (τ0L = 5.116x10-8s) implies that the
neutralising heavy-electron must interact constructively with the co-rotating core.
This lifetime represents a number of periods for the heavy-electron at r± :
N K 0 L = τ 0 L /(2πr± / c) = 6.68 × 1015 .
(5.12a)
38
Again this ratio is interesting because of its interpretation in terms of a guidewave's
coherence time and action, through the formula:
ln( N K 0 L ) ≈ π(137 / 12) ,
(5.12b)
which, after differentiating may be reduced with Eq.(5.7) to:
1
137
N ( 2 πr± )
∫
2 πr±
2π
2
1e
1 m q
=
dt
∫
2 z
24 0 2
crOK dϑ .
(5.12c)
This expression accounts for the long lifetime by coordinating action in the
neutralising heavy-electron and the core mechanism. On the left, the integral is
potential energy action required to create the electron with its spiralling
electromagnetic guidewave, which communicates continuously with the core to
stabilise it. Weighting coefficient (1/137) records that the electron core consists of
137 pearls, (see Paper 2). The integral on the right is kinetic energy action for a quion
running around the core, at radius rOK. Coefficient (1/24) confirms there were 24
original pearl-seeds in each pionet.
K 0S . The greatly reduced lifetime of this kaon (τ0S = 0.8956x10-10s) implies that the
neutralising heavy-electron with the native positron are not very successful at
stabilising a counter-rotating core. However, the core by itself would not exist at all,
so some stabilisation must be occurring. The lifetime may again represent a number of
core periods (2πrOK /c = 1.12x10-25secs):
N K 0S = τ 0S /(2πrOK / c) = 8.01 × 1014
(5.13a)
This ratio may be interpreted in terms of guidewave action and coherence through the
formula:
ln( N K 0S ) ≈ 4π 2 (e n / π)
,
(5.13b)
which after differentiating may be reduced to an expression like Eq.(5.11c):
π
en
N K ( 2 πr0 K )
∫
2 πr0 K
2π m
2
q
1 (e / 2)
cr0 K dθ .
dt ≈ ∫
2
2 z ′
0
(5.13c)
On the left, pearl charge is (e/2) and the integral covers action required to create the
pearl and stabilising guidewave travelling around a quion loop. Factor (π/en) implies
the participation of gluons in the process. The right side represents kinetic action of a
quion as it travels at velocity c during one core revolution 2πr0K . This expression
39
coordinates the heavy-electron and core mechanisms, to produce some stability of the
core for a short while.
The fact that K 0L and K S0 are produced in equal quantity will be attributed to
random orientation of the core spin relative to the angular momentum of orbiting
charge. Energy difference of 3.491x10-12MeV between the states is then comparable
with the hyperfine splitting of interstellar hydrogen (5.874x10-12 MeV). The higher
energy state K 0L is expected to be for parallel spins, which is evidently more stable.
Regeneration of K S0 during interaction of K 0L with matter is understandable in terms
of spin inversion. Earlier, the extended lifetime of the K 0L relative to that of K± was
proposed to be due to the stabilising effect of the neutralising charge on the corotating core, through the spiralling guidewaves. This is analogous to the charged pion
being more stable than the neutral pion.
According to observations, neutral kaons K 0L oscillate between the matter and
anti-matter state while propagating. In quark theory, this has been explained (see
Perkins, 2000), as being due to a second-order weak interaction which also causes K 0L
and K S0 particles to have the different masses mentioned above:
∆m = m KL − m KS = 3.491x10 −12 MeV / c 2 ,
(5.14)
and the oscillation period is (τ0 = h/∆mc2 = 1.18786x10-9 s). Herein, the K 0L and K 0L
must have equal status and such oscillations could be attributed to a change in helicity
of the core from right-handed for K 0L to left-handed for K 0L , maybe due to prompting
from the orbiting charges.
Figure (5.3) illustrates our model for the 4 possible
particles. It is only anti-parallel spin of the core, not helicity, which causes the shorter
lifetime for K S0 and K S0 .
Clearly, all mean lifetimes must be determined by definite internal
mechanisms and may actually be related. For example, the long lifetime of K 0L may
be related to the above oscillation period through (τ0L/τ0 ≈ 137/π). And for some
reason, two other lifetimes (K± and K S0 ) are in the ratio (τ± /τ0S ≈ 137).
40
Fig.(5.3) Schematic diagram of the K 0L and K SO neutral kaons with their
anti-kaons K 0L and K S0 .
5.4
General designs
According to Figure 5.1 strange mesons consist of (4 + n) bound pionets in a
central core, plus a quion /antiquion pair in the spin-loop, as drawn in Figure 5.2.
Three pionets must be added to the spin-loop to increase J by unity. By considering
the decay products and kinetic energy, it is possible to derive a design for each one.
Thus the decay process is to be regarded as relaxation, in which component parts
separate ergonomically, preserving some features, especially when the free energy is
low. Table 5.1 illustrates some simple tentative designs for strange mesons, and the
formulae satisfy their mass distribution fairly well. The grey circles are pearls of pionet
mass, and the small ovals have half the pionet mass.
41
Table 5.1 Tentative design of strange mesons, based upon ergonomic agreement with their
decay products. The pearls shown as small ovals have half the pionet mass, only.
7π
K*(892) / 891.66 ± 0.26 MeV
½(1−), Γ = 50.8 MeV, Dy(Kπ)
3/ 2
1
m ≈ 4m πo 1 −
+ 3m πo 1 −
24
24
= 894.2MeV
10π
K1(1270) / 1272 ± 7 MeV
½(1+), Γ = 90 MeV, Dy(Kρ,K*(892))
1
3/ 2
m ≈ 7m πo 1 −
+ 3m πo 1 −
24
24
= 1273.9MeV
11π
K1(1400) / 1403 ± 7 MeV
½(1+), Γ = 174 MeV, Dy(K*(892)π,Kρ)
1
3/ 2
m ≈ 8m πo 1 −
+ 3m πo 1 −
24
24
= 1400.4MeV
11π
K*(1410) / 1414 ± 15 MeV
½(1−), Γ = 232 MeV, Dy(K*(892)π,Kρ)
1
1
m ≈ 8m πo 1 − + 3m πo 1 −
24
24
= 1422.9MeV
11π
K0*(1430) / 1425 ± 50 MeV
½(0+), Γ = 270 MeV, Dy(Kπ)
1
m ≈ 11m πo 1 −
24
= 1422.9MeV
11π
K2*(1430) / 1425.6 ± 1.5 MeV
½(2+), Γ = 98.5 MeV, Dy(K*(892),Kρ)
1
1
m ≈ 5m πo 1 − + 6m πo 1 −
24
24
= 1422.9MeV
42
14π
K*(1680) / 1717 ± 27 MeV
½(1−), Γ = 322 MeV, Dy(Kπ,Kρ,K*(892))
1
2(3 / 2)
m ≈ 11m πo 1 −
+ 3m πo 1 −
24
24
= 1687.2MeV
14π
K2(1770) / 1773 ± 8 MeV
½(2−),Γ = 186MeV, Dy(K2*(1430), K*(892))
1
2
m ≈ 8m πo 1 − + 6m πo 1 −
24
24
= 1765.9MeV
14π
K3*(1780) / 1776 ± 7 MeV
½(3−), Γ = 159 MeV, Dy(Kρ,K*(892)π,Kη)
1
2
m ≈ 5m πo 1 − + 9m πo 1 −
24
24
= 1782.8MeV
14π
K2(1820) / 1816 ± 13 MeV
½(2−), Γ = 276MeV, Dy(K2*(1430),f2(1270))
1
1
m ≈ 8m πo 1 − + 6m πo 1 −
24
24
= 1810.9MeV
16π
K4*(2045) / 2045 ± 9 MeV
½(4+), Γ = 198 MeV, Dy(φK*(892),ρKπ)
2
1
m ≈ 4m πo 1 − + 12m πo 1 −
24
24
= 2047.1MeV
Meson K0*(1430) apparently has no net spin-loop yet accommodates 11
pionets. It has a very short lifetime (τ = 0.22x10-23 s) and could consist of a quion +
antiquion pair, which closely orbit the inner core of 7 pionets in counter-rotation.
43
5.5 Charge neutralisation of K*(892) and K2*(1430)
These two strange mesons have been measured in their neutral and charged
states, well enough to analyze like the kaon:
K * (892) o − K * (892) ± = 4.28 ± 0.34MeV ≈ 8.38m e c 2 ,
(5.15)
K 2 * (1430) o − K 2 * (1430) ± = 6.8 ± 2.0MeV ≈ 13.3m e c 2 .
(5.16)
As for the kaon, the increased mass of the neutralised meson will be attributed to
adding a heavy-electron, without any change in the quion /anti-quion spin-loop or
mass. In addition, an explanation of the particular heavy-electron mass value is
desirable.
(a) K*(892). If, following Eq.(5.8a), the heavy-electron radius were smaller than
the free electron radius at roe /8.38 = 0.336fm, it would be less than (r± = 0.3655fm) in
the central kaon. This could be an unstable orbit, in view of the earlier kaon analysis.
Consequently, it is proposed that the heavy-electron is split into 3 loops of mass
around 2.79me each, but carrying only one electronic charge in total. The radius of
this heavy-electron assembly is then rhe ≈ roe /2.79 = 1.009fm: just inside the quion
/anti-quion orbit at 1.017fm. If rhe is now substituted into Eq.(5.10) in place of r± ,
then the resultant heavy-electron mass is (m − c 2 ≈ 8.56m e c 2 = 4.37MeV) , which is
acceptable. It is thought that the heavy-electron actually travels around the spin-loop,
with the gluons emitted by the quion and anti-quion. This extra mass would decrease
the spin-loop radius slightly.
The compression sequence of the heavy-electron here follows that of the
neutron, and by analogy in the final stage it is quantisable in terms of action because
[ln(roe /rhe ) ≈ ln(2.79) ≈ π/3], which leads to an action integral by differentiating then
multiplying by (e2 = mecroe):
2 πrhe
−
∫
2 πroe
e2
1
dt ≈ ×
z
3
2π
me
croe dθ .
2
0
∫
(5.17)
On the left is the integral for potential energy action done in compressing the
electron, and on the right is one third of the standard kinetic energy action of an
electron core.
(b) K2*(1430) The accuracy of this data is not enough for certainty but a model
similar to K*(892) will be proposed with a heavy-electron mass of approximately
44
14me. If the heavy-electron radius were actually smaller than the free electron, at (rhe
≈ roe /14 = 0.20fm), it would be less than r± in the central kaon. This is again
unsatisfactory so let the heavy-electron be split into 5 loops of mass 2.8me each, but
carrying only one electronic charge in total. The radius of this heavy-electron
assembly is again coincidental with the spin-loop as for K*(892). It follows that the
heavy-electron mass of 5 loops is around (5/3)(4.37MeV/c2) ≈ 7.2MeV/c2. And the
compression sequence of the heavy-electron follows that of the K*(892).
5.6 Mean lifetimes of K-mesons
(a) K*(892). The full widths of the charged and neutral mesons are similar, so their
lifetimes are probably governed by their spin-loop or quion periods independent of the
charge. The full width ( ΓΚ∗(892) ~ 50 MeV) implies a lifetime (τΚ = 1.30×10-23s),
which is less than the spin-loop period (2πrS /c = 2.13×10-23s from Eq.(5.4)). It may
instead be related to the period of the rotating quion (2πrq /c' = 1.55 x 10-25s):
N Kq = τ K /(2πrq / c′) = 83.9 ,
(5.18)
and then,
(
)
ln N Kq = 4.43 = 0.448π 2 .
(5.19)
Upon differentiating and multiplying by the general format [e 2 / c = (2 / π) 2 m q c' rq ] ,
this reduces to an action integral like Eq.(3.2.8):
N Kq ( 2 πrq )
∫
2 πrq
2π
2
1e
1 m q
dt ≈ ∫
2 z′
2 0 2
dθ
c′rq
3
.
(5.20)
On the left is potential energy action required to create (or dissipate) a quion; where
(z' = c't) over the guidewave coherence length. The integral on the right side
represents kinetic energy action of a spinning quion over one third period (2πrq /3).
(b) K1*(1270) through to K4*(2045). The other K-mesons all have lifetimes less
than K*(892), with their actions in the range 60% -100% of Eq.(5.20).
6. Conclusion
The quark/anti-quark singularity design of meson QCD theory has been
replaced entirely by very well defined real particles. Particle mass represents
45
organised, localised energy, so the Higgs mechanism is not required. Detailed models
of mesons have been derived in terms of structured components. First, pion design
was derived by relating it to the muonic mass. A Yukawa potential was calculated for
the hadronic field, analogous to the proton's field. By adding a heavy-electron or
positron in a tight orbit around the hadronic core, a charged pion was produced. Other
mesons were found to be ordered collections of muonet or pionet masses, travelling in
bound epicyclical orbits. Periods of these orbits were then related to the mean
lifetimes of their mesons through specific action integrals. Decay products were
descended from existing components within parent mesons, as expected for a
relaxation process. This provided some traceability of particles and increased
confidence in the analysis. The design of strange mesons with their relatively massive
core was distinctly different from the flavourless mesons.
Appendix A: Compatibility with Standard Model
The models for an isolated proton, electron or muon given in Papers 1, 2, 3,
were very successful at explaining the Yukawa potential, the reality of spin and
anomalous magnetic moment, and particle creation mechanisms. On the other hand,
the Standard Model of particle interactions has been very successful at accounting for
data from high energy collision experiments. Consequently, the conceptual
differences between these models can be explained if particles in collisions generate
aspects not immediately apparent in static models. To link these models, the trineons
in the proton and quions in mesons need to behave like up, down and strange quarks
when in high energy collisions. It will be found that quark masses are specific for
each particle type and not related necessarily to other particles. On average over many
collisions, anti-quarks may even appear to be mixed with quarks in deep inelastic
lepton-nucleon scattering experiments.
A.1 Proton and Neutron
Consider Figure A.1 wherein the proton of Paper 1 is depicted as 3 trineons
travelling around the spin-loop at the velocity of light. Each trineon has a charge (+e)
but only emits an electromagnetic field due to (+e/3) into the exterior space, so the
proton's total external charge is (+e) as observed. Trineons also emit an e.m field
around the spin-loop, equivalent to (+2e/3) each.
46
Fig.A.1 A schematic proton consisting of 3 trineons in the spin-loop with
external and internal electromagnetic fields due to charge (e/3) and (2e/3),
as experienced by an incident charge D.
Consequently, an energetic incident charge D is able to approach any one of
the 3 trineons closely and will experience an interaction which depends upon the
position and direction of that trineon within the spin-loop. For example, D on A will
vary as [e/3 + (2e/3)cos(θ)], whereas D on B will vary as [e/3 + (2e/3)cos(θ+120o)],
and D on C will vary as [e/3 + (2e/3)cos(θ+240o)]. These 3 possibilities for interaction
of particle D with a proton are shown overlaid in Figure A.2. Clearly the effective
interaction charge for a trineon can vary from (3e/3) to (-e/3).
e
Trineon interaction charge
A
2e/3
C
e/3
0
0
−e/3
30
B
60
90
120 150 180 210 240 270 300 330 360
Phase angle
Fig.A.2 Variation of interaction charge for trineons A,B,C.
47
For correspondence with the Standard Model, we require A(+2e/3), B(-e/3),
and C(+2e/3), which occur at (θ = 60o) where the squared values are nearest to each
other: A(4e2/9), B(e2/9), C(4e2/9). It happens that the average of [e/3 + (2e/3)cos(θ)]2
over one spin-loop cycle is e2/3, which is also the average of quark charges-squared
(4e2/9 + 4e2/9 + e2/9)/3. The A,B,C, nominations are interchangeable at (θ = 120o,
240o).
Thus, the appearance of a negative interaction charge (-e/3) within a positive
proton is remarkable. This only happens for inelastic collision processes where a
trineon reacts according to its internal mechanisms and direction of travel. Trineons
are tightly confined by strong force gluons within a proton, so any collision of an
incident particle with a single trineon might appear to involve a quark of spin-(1/2).
For the neutron model in Paper 1, a heavy-electron closely orbits the proton to
neutralise its positive charge. Then if this heavy-electron joins with trineon A say, in
opposing incident particle D, the effective interaction quark charges would be A(-e/3).
B(-e/3), C(2e/3) as required. This combining-process for a neutron as proposed will
also be required for the meson interactions below.
A.2 Mesons
For the neutral pion model described in Section 2, the quion requires a total
charge (+e) according to creation equation (2.18). So, analogous to a proton's trineon,
this charge appears to be distributed as (+e/3) for an external field and (+2e/3) for an
internal field,. Consequently, a πo has the immediate appearance of a d d quark pair.
However, the quion with its internal charge (+2e/3) is travelling around the meson
circumference (2πroπ) at the velocity of light and could interact with an incident
particle just like a trineon in the proton of Figure A.1. Thus, it could behave like an up
or down quark, and the corresponding anti-quark could interact like an anti-up or antidown quark. On average therefore, the πo can interact like a mixture of u d u d quarks.
The π+ has an orbiting heavy-positron as in Figure 2.1. In a collision process,
this positron combines with the anti-quion (-e/3) to interact like an up quark, so the π+
could be viewed as an ud quark pair. Similarly, the π− would interact like an ud quark
pair. Obviously, these meson quark assignments describe the charges only, and do not
represent masses, which are less than in the proton. Masses of other unflavoured
48
mesons are multiples of the pionet mass, but their quion and anti-quion charges are
the same as for the pions, as if this is the ground state.
Mesons with zero spin must generate spin for their quions as necessary during
collisions, but mesons with spin can be considered to possess spin-1/2 quarks in
collisions.
A.3 Strange quarks
Strange quarks were introduced to account for long lifetimes of some particles,
and they also add more variety to the types of particles. In fact, the more massive
strange particles decay rapidly into the long-lived lowest form, so that a strange quark
does not extend the lifetime of its original particle. For example, heavy K-mesons
described in Section 5 simply have a strongly-bound core which survives the initial
rapid decay and converts to a kaon of long lifetime. Allocation of charge (-e/3) to a
strange quark makes K± analogous to π±, and it also makes K 0 the exact anti-particle
of K0 by way of helicity.
References
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Baru V et al 2003 arXiv:hep-ph/0308129
Eschrich et al 2001 Phys Lett B522 233-239
Gashi et al 2002 Nucl Phys A699 732-742
Janssen G et al 1994 arXiv:nucl-th/9411021
Parganlija D, Giacosa F, Rischke DH 2009 arXiv:nucl-th/0911.3996
Perkins DH 2000 Intro to High Energy Physics 4th Ed Cambridge Univ Press
Scadron MD et al 2003 arXiv:hep-ph/0309109
Simo C 1978 Celestial Mechanics 18 165-184
Wang Z G & Yang W M 2005 The European physical journal C42 89-92
Wayte R (Paper 1) 2010 A Model of the Proton www.vixra.org/abs/1008.0049
Wayte R (Paper 2) 2010 A Model of the Electron www.vixra.org/abs/1007.0055
Wayte R (Paper 3) 2010 A Model of the Muon www.vixra.org/abs/1008.0048
RICHARD WAYTE