Read the book of Feynman, QED - The strange theory of light and matter. And Scientific American of June 1980, Gerard 't Hooft, Gauge theories of the forces between elementary particles. In Four quarks in the shell knowledge of renormalization theory is necessary at that level.

A little knowledge about quaternions is necessary. The quaternion approach of strong force colors is worked out in the storyline QQD.

According to QED each quark A has a shell of quark-antiquark pairs and gluons around it. Although it is not really around it, one can say that the quarks in the shell have same size as quark A. However, let's still call it in the shell of quark A.

This page is a sequel of paragraph Massless coinciding at page 2 of EXPAN. The idea that is worked out is that gluons are made of a quark-antiquark pair, massless coinciding, that is taken out of the shell of a real existing quark like A.

For the process of massless coinciding it is necessary the quark absorbs from the Higgs field the same amount as the antiquark emits to the Higgs field, and that supposes the quark and the antiquark to exist as separate entities. The gluon then resembles more the old gluon from accepted QCD with its upper and lower color, like , and . The view of streams of color, as they are often represented in articles, agrees with colors keeping separated.

When a particle-antiparticle pair emerges, at what distance do they do that, initially? When the colors of all four quarks in the shell have to react with each other, this is possible: place them at the four corner points of a tetrahedron with sides of 0.9 fm. When some of the four quarks have to coincide massless, place them at most 10^-21 m apart (the tetrahedron shape is lost then).

In this page is shown how pairs of quarks, taken out of the shell of virtual particles that surrounds each quark, can form colored spin 1 and spin 0 gluons. When the gluon is emitted by a quark 1, follows its short route and is absorbed by another quark 2 nearby, the quark-antiquark pair that makes up the gluon is taken from the shell of the emitting quark 1 (without merging), follows its route without merging and then takes its place between the infinite other quark-antiquark pairs in the shell of quark 2, all without ever having merged. When a gluon is absorbed by the color of a target quark, the colors of the constituting quark and antiquark in the gluon are applied one after the other onto the target quark and that means multiplication of the colors of the quark and antiquark in the gluon one after the other with the color of the target quark.

 

Four quarks in the shell

Let's go back to the superposition of innumerable quark-antiquark pairs, that according to QED is shielding the naked color of the quark. The color coupling constant is about 1. Therefore 4 quarks appearing in 2 quark antiquark pairs, all within their time borders, all seeing each other, count with same importance as 2 quark antiquark pairs superposed to each other at same mutual distance. (The superposed two quark pairs don't see each other, don't react with each other.)              (a)

We just suppose the 4 quarks to appear within their time borders, at the Earth surface within 10^-19 m or 10^-4 fm (see EXPANSION OF THE UNIVERSE, page 2, paragraph The calculation of the time border). We assume the two pairs to emerge simultaneous within a length of time of 10^-23 sec. Quarks have maximum attraction at about 0.9 fm, so the 4 quarks hardly attract each other. They don't form pairs under strong force attraction.

Emerging pairs always consist of a quark and an antiquark with opposite taste, color and spin. Electric charge already is in the taste, e.g. when the taste is u then electric charge is +2/3 times the positron charge. When the taste is d then the electric charge is -1/3. When the tastes are opposite, the electric charges are so too.

Impulses don't have to be opposite. If one quark from the pair has a small impulse and the other quark has a large impulse in a different direction, then at higher quark velocity SR (special relativity) yields larger mass increase at one hand and lower pace of Higgs field absorption by time dilation on the other hand. Do these effects cancel out? If not then the mass absorption from the quark might not completely cancel the mass emission of the antiquark, making massless coinciding impossible.

Let's make things easier: for every pair of impulses there is one frame where both quarks have equal but opposite impulse. In that frame, when the two quarks coincide sufficiently, masslessness is achieved. The direction and the energy of the resulting lightspeed gluon is not immediately clear. Subsequently transforming to another frame goes by Lorentz transform.

A and D are quarks, B and C are antiquarks (the underlining means it is an antiquark), AB is one emerging pair, CD is the other emerging pair, = spin +1/2, = spin -1/2. Let's suppose the two emerging quark pairs have different color, independent from each other. The colors of the pairs can be or or .

A       C B       D       (1.1)

A       C B       D       (1.2)

 
The double pair in (1.1), and in (1.2) as well, form 2 gluons which can be investigated along 3 ways: AB CD, AD BC and AC BD (denoted as II column pairs, X crosswise pairs and = row pairs).

A and B have opposite spin and so do C and D.

II pairs in (1.1) and (1.2): quark spins +1/2 -1/2 = 0, the double pair can form two colorless spin 0 gluons. White gluons don't glue.           (b)

X pairs in (1.1) and (1.2): the pairs are a particle-particle pair and an antiparticle-antiparticle pair, wherein massless coinciding is not possible. So X-pairs in (1.1) and (1.2) will not form gluons.

= pairs in (1.1): the 4 quarks can form two colored spin 1 gluons, quark spins +1/2 +1/2 = +1 and -1/2 -1/2 = -1.              (c)

In fact in 18 from 25 cases the two spin 1 gluons in = have color and in 7 cases they are white-white or black-black, see table Color multiplication summary in the column at the right, and see paragraph And now for the chances below.

= pairs in (1.2): two colored spin 0 gluons can be formed. When trying to make up the vacuum from these gluons you need them in pairs, the color of one gluon canceling the color of the other gluon, but this is now a previous attempt, see page 2 of QG below the black bar.              (d)

The vacuum can be made of colorless spin 0 gluons directly. This is done at page 4 of EXPAN, especially (5.10) and (5.13) in the paragraph Remnant particles in the fermion explosions. The gluon is colorless but there is color inside: the constituting quarks have opposite colors or or . They don't glue, so they form a grid by a Bose condensate only. The grid property and the Bose condensate property of the vacuum are discussed in paragraph The fermion explosion of the vacuum particles at page 4 of EXPAN:

- there is no grid of cohering particles

- and an expanding cloud of particles should still be a Bose condensate.

Four quarks making up 2 gluons is the smallest number of quarks to appear within their time borders, in order to form colored spin 1 gluons.

The fourth axis

1 ------- -1

is not shown.

To avoid confusion, quarks depicted as actual colors are , , etc. Gluons depicted as actual colors are shown a little larger, like , , etc.

Let's regard the colors a little better. Let's go into quaternions. Eventually use the gluon table. Regard the 2 pairs again. Each pair consists of a color and an anticolor. For each pair that emerges, there are 4 possibilities: and and . And the fourth pair, is it ? The result per pair is to be taken as the application of both colors one after the other. One has to multiply the colors with each other and in quaternions multiplication order makes a difference. So which order is to be taken? Set e.g.:

A		i		C		-j
B	=	-i	and	D	=	j	(1.3)

 
II pairs:           (1.4)

= i * -i = = j * -j = 1 = , so the II-pairs form a pair of white gluons. The quark composition of the white gluon is or or . Multiplication order is not important. But how for the other combinations?

 
= pairs (form gluons) and X pairs (particle-particle or antiparticle-antiparticle, form no gluons):

= -j * i = k = = i * -j = -k =

= j * -i = k = = -i * j = -k =

= i * j = k = = j * i = -k =

= -i * -j = k = = -j * -i = -k =       (1.5)

We didn't need to worry. It are all combinations of i and j - with or without a minus sign in front - and the multiplication always will yield k, one with a minus sign in front and the other without. All possible arrangements and orders of the quarks of color i and j (with or without a minus sign in front) yield the gluon pair k -k. Two different pairs of color-anticolor that appear, always yield the third possible color-anticolor pair.

Quarks have color and antiquarks have anticolor           (1.6)

This is a result that should not be forgotten, see the alinea just after (5.2) in paragraph mesons at page 3 of QQD

The X pairs (3rd and 4th scheme, with i * j and -i * -j respectively) are particle-particle pairs or antiparticle-antiparticle pairs, in which massless coinciding is not possible. The X pairs don't yield gluons.           (1.7)

The only possibility to form colored gluons are the = pairs of the first two tables. It then can be spin 1 colored gluons or spin 0 colored gluons, depending on whether the spin distribution is like (1.1) or (1.2).           (1.8)

We can use (1.6) to settle another question that appears in the quark-antiquark representation of the gluon. When you have a gluon of quark composition then is suggested the application of the the two quarks * * (which yields ) has equal importance as * * (what is yielding ). Presenting the gluon as is the same as presenting it as . So if the gluon is applied when coupling to a colored quark, the superposition of the two outcomes can be expected, leading to a net color application of - = k -k = 0. (0 is not white, = 1).

Here (1.6) comes to the rescue. We state that if the gluon couples to a quark with color, that after the coupling you still have a quark with a color (and not an anticolor). Likewise if the gluon couples to an antiquark with anticolor after the coupling you will still have an antiquark with an anticolor (and not an antiquark with a color). This means that all outcomes with minus signs in (1.5) are abolished, leaving only the i, j and k outcomes to be applied.

This seems to suggest that the gluon has a spatial extension, a spatial structure, such that in gluon-quark couplings the positive outcome of the two quarks in the gluon is permitted while the negative outcomes are forbidden. Compare (5.9) up to (5.10) in paragraph Building vacuum from gluons at page 2 of QG.

With gluon-gluon couplings it might be different. One can assign the gluons with color i, j, k as particles, while the gluons with anticolor -i, -j, -k then are antiparticles, but these assignments are in fact not of application. Gluons just like photons are supposed not to have a time arrow, none of the colored gluons are particles or antiparticles. So when a gluon with quark composition couples to a gluon with quark composition then indeed both applications * * = * j *-i = k = * and * * = * -i * j = -k = * superpose (and also * * = * and * * = * superpose) when the two gluons couple, leading to a net result of k -k = 0.

Gluons only couple to each other, gluons don't glue to each other           (1.9)

if I don't mistake.

 

What if the two pairs of quarks are the same?

A				C
B	=		and	D	=

= i * -i = 1 = = -i * i = 1 =

= i * i = -1 = = -i * -i = -1 =

          (2.1)

At first glance the = pairs as well as the II pairs yield white gluons (first scheme).           (2.2)

The X pairs (last scheme, with i * i and -i * -i) would yield black gluons only. However, it are a particle-particle pair and an antiparticle-antiparticle pair respectively, in which massless coinciding is not possible. The outcome of black gluons is not possible here.           (2.3)

So when AB and CD is the same pair of colors the result always seems to be a pair of white gluons.           (2.4)

The particle and antiparticle in a pair that emerges in the shell cancel each other out, that's why they can appear. + = + = + = + = i -i = j -j = k -k = 1 -1= 0, the colors and anticolors happen to add up to zero. But in quaternions, when we apply colors, we don't add them, we multiply them. Then * = * = * = i * -i = j * -j = k * -k = 1.           (2.5)

White and black are opposite colors, is the particle and is the antiparticle, they cancel each other out, so can emerge as a particle-antiparticle pair. The pair is a colorless color, that is no color at all, and so it is allowed to stay on its own: the particle will escape. Its range is infinite, as far as the color force is concerned.           (2.6)

Black and white multiply to * = * = 1 * -1 = -1 * 1 = -1 = . When this color -1 hits the color of a target quark and merge with it, the color of the target quark is multiplied with -1. When the quark is e.g. a d-quark it would yield an d-quark with anticolor but still with electric charge -1/3. Or when the color -1 meets a positron which has color +1, the positron obtains color -1 like the electron has but still with +1 electric charge. Both cases are forbidden in (1.6) and thus don't happen.           (2.7)

In paragraph Electrons as Baryons at page 4 of QG (yet to come in the TONE storyline) the black quark is recognized as the electron and the white quark is recognized as the positron. The photon, consisting of a color +1 quark and a color -1 quark, will never merge its color (that should be -1) with the color of the target particle and multiply it by -1. I mean: when the photon hits the target particle, its constituent color +1 quark and color -1 quark would apply their colors one after the other on the color of the target quark, thus one after the other multiplying it with their color *(+1) *(-1) = *(-1) and, as said, (1.6) forbids this. The photon doesn't react by color force. For the color of the photon we have to restrict to summing only.

Color ph = color e-  +  color e+ = -1 +1 = 0           (2.8)

The net color of the photon is zero, see (2.1) and (2.2) in paragraph The color of the photon at page 5 of NET FORCE IN QED.

In paragraph Two gluons of opposite sign do not react at page 3 of QG, below the black bar at that page, (7.2) says: Two gluons of opposite sign will pass each other by or stay put to each other and form a composition, as if they don't react with each other, because the time border is between them. This also prevents * from application.

Is there another way to construct a black gluon? In accordance with (1.10), quark combinations that yield -1 are = i * i = = j * j = = k * k = = -i * -i = = -j * -j = = -k * -k = -1, but these are all particle-particle pairs or antiparticle-antiparticle pairs in which massless coincidence doesn't take place. These pairs of quarks will never yield a gluon. *a)

Here the text Fig. (3.9) in QUATERNIONS up to Fig 3.10, in paragraph Other colorshifts and mesons at page 3 of QCD comes to the resque. Two gluons of color i that meet and merge yield

i * i = = = -1, a gluon of color -1, a black gluon. The possible quark compositions of the gluon i are

-k * j = = k * -j = = i =           (2.9)

But as said, there is no composition of a quark and an antiquark yielding color -1. There is the colorless quark, the white quark that is the e+. And there is the colorless antiquark, the black quark that is the e-. These together yield a black gluon, but as said, that is no gluon, that is the photon.

I conclude that the two gluons when yielding a black gluon do not merge, but instead yield a gluon that consists of four quarks.

The black gluon consisting of 2 colorless quarks is the photon       (2.10)

And there are no white photons. Well, that is, maybe at (2.16).

The black gluon made of colored quarks consists of 4 quarks       (2.11)

Possible states for the 4 quarks in the black gluon:

Black gluon type 1. All four quarks on top of each other, massless coinciding within distance 10^-21 m, where the vacuum cannot distinguish different locations anymore. They stay on top of each other because at little larger distances the vacuum does distinguish the quark locations and then the quarks have to be supplied with their mass, for which the energy is usually not available.

This 4-quark configuration is massless and I guess it is stable.

Black gluon type 2. The two gluons and circle around each other as is suggested in GluNons, page 7 of QCD: The gluons orbit each other with impulse momentum ..., -2, -1, 0, 1, 2, ....

At what distance do they orbit each other? 10^-21 m? 0.9 * 10^-15 m? Something in between? Can this configuration be massive? It doesn't look very stable either.

Black gluon type 3. When energy is available the 4 quarks can arrange at the cornerpoints of a tetrahedron with mutual distances of 0.9 fm where the color force is at maximum, see Table 1 and (1.10) up to and including (1.13) in paragraph The proton at page 4 of QCD. The 4 quarks then are no longer massless coinciding. The needed energy for this is the mass of all four quarks minus the condensation energy that is freed when the quarks settle at 0.9 fm mutual distance, when coming from zero distance (coinciding).

Two of the quarks must have opposite spin, the other two quark spins align to spin +1 or spin -1. This configuration certainly is massive, but it is in fact no longer a gluon. It rather resembles two mesons put together in one particle. Mesons are unstable - how about this particle?

The black gluon type 3, consisting of 4 colored quarks at the cornerpoints of a tetrahedron with sides of 0.9 fm, is my one and only candidate for Dark Matter       (2.12)

Dark Matter, mind the capitals, is the matter that keeps too fast stars within their galaxies and too fast galaxies within their clusters. In this website dark matter (without capitals) is antimatter, matter that goes backward in time. *b)

In Fig 3.9 and Fig 3.10 at page 3 of QCD the two gluons merge into Dark Matter.

Maybe there is a problem with the photon picture of (2.8) and (2.10). If ph = + then the color application of two photons at the same spacetime point would yield

* * * = -1 * -1 = 1           (2.13)

Two photons, simultaneously at the same place, could together react by their color force with a color.

If the two photons are identical, their spin would add to +1 +1 = +2 or -1 -1 = -2.           (2.14)

If the two photons are not identical, then one photon can have spin 0 (the two constituting quarks have opposite spin) while the other photon has spin 1 (the two constituting quarks align to spin +1 or spin -1) resulting in a two-photon net spin of +1 or -1, as is required for gluons.           (2.15)

Maybe the two coinciding photons have to be identical to form a white photon: same energy, direction, spin and phase. Anyway, let's conjecture:

The white photon consists of 2 black quarks and 2 white quarks       (2.16)

Some other reactions.

A				C
B	=		and	D	=

= i * -1 = nop = -i * 1 = -i = a)

= j * -1 = nop = -j * 1 = -j = b)

= k * -1 = nop = -k * 1 = -k = c)                 (2.17)

Nop means no operation, or not possible.

X pairs are particle-particle or antiparticle-antiparticle and thus form no gluons.

II pairs: AB forms a white gluon and CD is a photon.

The = pairs are more interesting: in a) the is a red quark; the is an electron. (2.7) shows that the electron reacting with a quark or antiquark by color force yields forbidden states. Moreover, the masses of the electron e- and the quark don't match, so when trying to coincide massless they at best partially succeed. If the particle would form, it would not reach out far.

Ditto for * = -i * 1 = -i = . The color multiplication * is allowed, the 10^-23 sec color coupling *) of the positron color with the cyan antiquark color . But their masses don't match either. Efforts to gain masslessness can only partially succeed.

*) coupling, not gluing, colorless colors don't glue.

See also 2) in paragraph Electrons as Baryons at page 4 of QG.

Likewise for b) and c) in (2.17).

A				C
B	=		and	D	=

= 1 * -1 = -1 = = -1 * 1 = -1 =     (2.18)

X pairs are particle-particle or antiparticle-antiparticle and thus form no photons. The color coupling however is allowed:

= 1 * 1 = 1 = and = -1 * -1 = 1 =

One would expect this to occur. But two e+ and likewise two e-, repel each other by EM (electromagnetic force) and so don't come close enough to each other for a 10^-23 sec colorless color coupling, couplings that don't glue.

S HIS   ORRECT?

= pairs as well as II pairs yield photons, spin 1 photons or spin 0 photons according to (1.1) and (1.2).

As you see, the white photon as outcome doesn't occur.       (2.19)

It's all multiplication of colors. That means coupling, merging of the preceding two colors into a merger color, see (1.2), (1.3) and (1.4) at page 3 of QQD. From the merger color the information is lost what precisely were the colors of the quarks just before merging.

 

Taste

And last but not least there is taste. In the 1st generation there are two tastes available for quarks: u and d. There u has always +2/3 electron charge, d has always -1/3, anti-u = u has -2/3 and anti-d = d has +1/3 electric charge. As to speak, electric charge is in the taste. There are 4 possible u and d distributions for the quark pairs AB and CD that emerge in the shell. Keep in mind we assigned spin +1/2 to A and C, and spin -1/2 to B and D.

A

C

u

u

d

d

u

d

d

u

B

D

u

u

d

d

u

d

d

u

(d)

In the first, and similar in the second pair of pairs, II-pairs as well as =-pairs are possible, there the tastes cancel out. But X would form electric charged gluons and that are no gluons. Moreover, X are particle-particle pairs or antiparticle-antiparticle pairs, in which massless coincidance is not possible.              (e)

In the third and fourth pair of pairs, the 3rd and 4th scheme in (d), situation is even worse: only II can form gluons (colorless spin 0 gluons only) and X and = would yield electric charged gluons.

The = pairs in the 1st and 2nd scheme are the only possibility to form colored spin 1 gluons, the gluons that mediate the strong force.

Mark the = pairs in the 3rd and 4th scheme are interesting: they form pairs of spin 1 particles of unit charge and the particles can have color but may be white or black too. IF u and d have different mass - and I think they have because of the difference in electric charge - THEN the Higgs field absorption of the one do not cancel the Higgs field emission of the other precisely. The quarks cannot coincide massless, the particle will have some remnant mass.

Let's assume

An amount of 1/3 electric charge is the only fundamental difference between the u- and the d-quark         (3.1)

The quarks are within their time borders *c), they will not separate because the presence within the time border gives some mass reduction. Separation means the quarks have to be supplied by their complete mass; that thus might be huge. We know little about the quark mass, see paragraph Quark mass at the previous page. This is the binding mechanism that succesfully works against the mutual electric repulsion of u and d, or u and d.       (3.2)

And there also is the mechanism of binding by aligned spins, isn't it? The spins +1/2 and +1/2 of the u and d are aligned, otherwise they wouldn't yield the spin +1 of W+.       (3.3)

Admitted, the particle has some resemblance to the W+ W- particle. But well, why W has mass 80385 MeV while the meson consisting of the same quarks then, has mass 140 MeV?

 

The conversion between -meson and W

For the photon we assumed it consists of an electron and a positron massless coinciding, and the electron and positron then are assumed to be a quark of color -1 and +1 respectively, the colorless colors -1 and +1. In accepted QCD the coupling constant of W+ W- and the coupling constant of the photon are nearly the same.

Assume that the -meson as well as the W+ W- consist of quarks with color. When the quark and the antiquark are very near to each other so that the partial mass cancelation can work, then it is the W+ or W-. At such small distances the strong force is virtually zero, that's why the quark and the antiquark don't bind by color then. And when the quark and the antiquark are about 0.9 fm apart then it is the -meson. The quark and the antiquark then are too far apart for the mass reduction mechanism to work. Instead the quarks are bound by color. Color works the best at a distance of 0.9 fm, there the strong force is at its maximum.       (4.4)

The strong binding causes a large mass defect. The electromagnetic repulsion reduces this mass defect a tiny little.       (4.5)

Mass u + mass d - color force mass defect + EM force mass surplus = 140 MeV -meson mass.       (4.6)

A meson can spontaneously convert to W+ W- and vice versa. When the quark and the antiquark change from very near to 0.9 fm, or the other way around, the conversion between a -meson and the W+ W- takes place.When you convert the -meson into a W+ or W- then you first have to supply the binding energy, not to free the quark and the antiquark from each other into the outside, but to free them from each other into the inside, where the strong force is (nearly) zero too. They are allowed to do so during the time t given in the uncertainty relation

ΔE * Δt h/2

wherein E = the binding energy.       (4.7)

If they can approach each other in that time, within the minimum distance that the vacuum can distinguish (within their time borders), about 10^-21 m that is, then the mass reduction from (1.17) takes place. For this conversion finally the mass W+ or W- minus mass -meson = 80385 MeV - 140 MeV = 80245 MeV has to be supplied. Therefore the transition from to W will not happen often.       (4.8)

When converting from W+ W- to , according to the same mechanism in reversed time order, 80245 MeV should be freed. Where does this energy go? In the kinetic energy of the -meson?

A previous attempt

Another possibility is that the u d and d u pairs that constitute W+ and W- respectively, must be quarks that don't glue: u- and d-quarks of color +1 (white) and u- and d-antiquarks of color -1 (black), according to (5.2) and (5.9) in paragraph Mesons at page 3 of QQD. It then couples by the strong force (but doesn't glue) at its typical pace of 10^-23 times per second; and it interacts by the electromagnetic force at the typical pace of the electromagnetic force. The W+ W- reacts as a kind of massive photon. This then is the W+ and W-.       (2.1)

A meson can spontaneously convert to W+ W- and vice versa. Which means that the particle pair colors i  -i and j  -j and k  -k, and the particle pair colors 1  -1 in principle can convert into each other, in doing so making the conversion take place. *d)       (2.2)

A question is, do such quarks exist, quarks with fractional electric charges +1/3, -1/3 and +2/3, -2/3 but with color +1 and -1? So far we only had quarks with color i, j and k and electric charge +2/3 and -1/3. And we had antiquarks with color -i, -j and -k and electric charge -2/3 and +1/3. We had also quarks with color +1 and electric charge +1, which in this website is recognized as positrons. And we had quarks with color -1 and electric charge -1, which are electrons.       (2.3)

I see no reason to forbid colorless quarks with fractional electric charge, but it would be something new. Maybe it is allowed as long as it cannot be observed as such. The rule seems to be that only integer electric charge can be observed.       (2.4)

When the u d and d u pairs have color, then it is the -meson.

I think this is a misconception. Quarks u with color +1 and d with color -1 are just a positron and an electron respectively, albeit positrons and electrons with electric charge +1/3 and +2/3. And electrons and positrons form photons, not W+ or W-. I regard this as a previous attempt. - June 2026

End of previous attempt

We now conclude the reasoning of four quarks in the shell leading to real gluon pairs with color (as well as photons). Finally one of the colored spin 1 gluons formed from the 4 quarks in the shell, can be absorbed by the real quark in which shield all this is happening, changing its real color. The other color then can be reabsorbed too (yielding no change at all) OR escape to another real quark. This can happen in mesons as well as in baryons.

Also possible is that both gluons leave the real quark where they are born, each of them going to a different quark - in baryons only, baryons have 3 quarks. The mother quark then doesn't change color, but the other two do.

Gluons might be CTL's (Closed Time Loops), see CTL considerations at page 8 of SR.

 

And now for the chances

(   TILL O EVISE   )

The 4 quarks in the shell emerge in 2 particle-antiparticle pairs AB and CD. From (a) there is a chance of 1 out of 2 for emerging 4 quarks all seeing each other within their time borders.              (chance 1 = 1/2)

The remaining chance of 1 out of 2 is for the emerging pairs just to superpose, despite they appear at precisely the same spots.              (chance 2 = 1/2)

A

C

u

u

d

d

u

d

d

u

B

D

u

u

d

d

u

d

d

u

(d)

 
The best way to proceed now is to start with the taste. We are in chance 1. When written down as in (d) there is a chance of 1 out of 2 for the first or second pair-of-pairs to form.              (chance 3 = 1/2)

The remaining chance of 1 out of 2 is for the third or fourth pair-of-pairs to be formed, yielding colorless spin 0 gluon pairs only.              (chance 4 = 1/2)

(For convenience we assumed the u is as likely to appear as d, which might be wrong)

When in chance 3 (1st and 2nd pair-of-pairs in d), there is a chance of 1 out of 2 for II-pairs, yielding sole colorless spin 0 gluons.              (chance 5 = 1/2)

And a chance of 1 out of 2 for = pairs to be formed.              (chance 6 = 1/2)

And no chance for X-pairs to be formed.

We now shift to spin and color. Set A at spin +1/2 and B at spin -1/2. When in chance 6, there is a chance of 1 out of 2 for C spin +1/2 and D spin -1/2.              (chance 7 = 1/2)

And a chance of 1 out of 2 for C spin -1/2 and D spin +1/2.              (chance 8 = 1/2)

As long as we have no reason to assume otherwise, we assign an equal chance to the 5 possible color-anticolor pairs to emerge. In the scheme at the right we see 25 possible combinations, 18 have color and glue, and 7 are white-white or black-black pairs from which the gluons don't glue.

18/7 = 2.57, 7/18 = 0.39, 25/18 = 1.39, 18/25 = 0.72, 25/7 = 3.57, 7/25 = 0.28.

In chance 7 there is a chance of 18 out of 25 for a pair of colored spin 1 gluons, the particles that make up the strong force.              (chance 9 = 18/25)

When in chance 8, there is a chance of 7 out of 25 for a pair of colorless spin 1 gluons (black-black or white-white).              (chance 10 = 7/25)

When 4 quarks in 2 pairs emerge within their time borders, the chance for two colored spin 1 gluons (two opposite colored spin 1 gluons) is:

Chance 1 x chance 3 x chance 6 x chance 7 x chance 9

= 1/2 * 1/2 * 1/2 * 1/2 * 18/25 = 18/400 = 0.045 or about 5 percent.

This percentage is making up the entire strong force. For spin 1 colored gluons, what we interpreted so far as one reaction in one cycle of time, up until now taken as 10^-23 s, must be about 1/0.045 = 22 reactions, 22 cycles of time - from which only one of them is yielding colored spin 1 gluons. One strong force cycle of time then must be rather 0.5 * 10^-24 s. In that time a light speed gluon covers only 0.15 fm.

 

References

Scientific American june 1980, Gerard 't Hooft, Gauge theories of the forces between elementary particles, for the model used in Four quarks in the shell.