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We've seen several times that the strong interactions have special properties.

Quarks behave as almost as free particles at short distances inside the hadrons.

Nonetheless, free quarks have never been observed outside hadrons.

This indicates that the strong force confines them within bound states and

does not allow to separate them to large distances.

In this fourth video we’ll qualitatively discuss

these seemingly incompatible properties.

After following this video, you will know the main properties of strong

interactions, including its different vertices in Feynman diagrams,

color charge, gluons and their role in binding quarks together, and

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We also saw the immediate consequence to triple the cross section for

e+ e- annihilation into hadrons in video 4.5.

Color is the quark property responsible for their strong interaction.

The formal theory of this interaction is quantum chromodynamics, also called QCD.

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Color can take three different values, red, blue, and green for quarks.

We indicate it by a lower index when appropriate.

A quark can carry only one non zero color, antiquarks carry one anti-color.

The interaction between quarks proceeds through the exchange of color.

The intermediate vector bosons transmitting the strong force are the eight gluons, g.

They carry a color and an anti-color, and therefore are not color neutral.

This is in contrast to the photon which couples to the electromagnetic charge but

does not carry one itself.

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For strong interaction, there are eight gluons in total.

With three colors and

three anti-colors one would expect a total of nine combinations.

One of them, the fully symmetric combination, that is

red-antired plus green-antigreen plus blue-antiblue,

has no net color, it does not participate in strong interaction and

so cannot be produced.

The remaining eight gluons are vector bosons,

they are electromagnetically neutral and have zero mass.

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The basic vertex of the strong quark interactions changes

the color of the quark.

The coupling constant g appearing in the vertex

enters into the cross section through its square.

Alpha strong is equal to g^2/(4π),

in analogy to the electromagnetic fine structure constant alpha.

The interaction has the same strength for the three colors or

any of their superpositions.

So there is invariance under an overall rotation in color space.

According to Noether's theorem, this requires a conservation law for colors.

The vertex conserves color.

The corresponding amplitude is also independent of the flavors of

quarks and their electromagnetic charge, which are both ignored and

conserved by strong interactions.

If gluons carry themselves color,

they should be able to interact among themselves.

Because they carry even color and anti color, there are two additional vertices.

The color indices indicated here are only examples.

The three-gluonvertex is proportional to g and

has the same strength as the quark-gluon vertex.

We must, in each calculation, consider the fact that there are many more different

colors for gluons than for quarks.

The vertex with four gluons on the contrary is proportional to g^2

and thus disfavored with respect to the other two.

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has discovered pentaquark states consisting of five quarks.

However it's still unclear whether these are bound

baryon-meson states as shown in the picture on the left or

genuine compact multi-quark states as sketched on the right.

LHCb has also confirmed the existence of a tetra-quark state

first observed by the BELLE collaboration in Japan.

Again, the same reservations apply.

It could still be a meson-meson bound state,

rather than a compact four-quark state.

Mesons contain a superposition of quark-antiquark pais with

all colors in equal proportions.

A snapshot is shown here for the interior of a positive pion.

Gluons are constantly exchanged between quarks to maintain binding,

changing the color of quarks, as in this sketch,

while keeping the overall hadron white.

The same mechanism works inside baryons.

Between hadrons, for example in a nucleus, objects of neutral color

are exchanged to create binding, except at very short distances.

It follows that an ab initio description of the nuclear force stays very difficult,

although we dispose of a well-established quantum theory for

interaction between quarks and gluons, namely QCD.

At short distances, comparable to the hadron size,

the exchange of gluons produces a potential between quark and antiquark

which is similar to the one established by photons in a positronium bound state.

This potential varies with distance as 1/r like the Coulomb potential.

At large distances, there is a different behavior,

the potential becomes larger with distance.

We already saw a potential that fits the description of the spectra of J/Psi and

Upsilon states in video 5.3.

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The second term is created by the interaction between gluons.

In terms of a chromostatic language,

the additional force between gluons concentrates the color field

along a corridor that connects the two color charges.

Because of the particular shape of the field,

one also call this a string that connects the colors.

The potential proportional to

k times r does not allow color and anti-color to be separated,

but confines them to distances of the order of 1 fm.

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The potential inside hadrons ought to be generated dynamically

by the interaction between the quarks and gluons.

Obtaining results is complicated by difficulties which are both conceptual and

technical.

Both the confinement of color at large distance and

the relativistic and quasi-free movement of quarks must be incorporated.

One must treat states with multiple quarks and gluons.

The evolution of the strong coupling

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Calculation techniques are considerably simplified if one replaces

the space-time continuum by a mesh of equidistant discrete points,

similar to a crystal lattice with sides a.

One defines the quark fields on the sites of the lattice,

and gluon fields on the links.

The continuum of space-time is recovered in the limit of an infinitely large lattice

with a -> 0.

This discretization introduces a lower limit on momentum-transfer

of the order of 1/a,

and thus regularizes the divergences inherent to pertulative QCD.

Numerical calculations of this kind require tremendous computing resources.

They are performed on the most powerful supercomputers.

With this non-perturbative method, one arrives, fo

example, at results for the spectrum of light hadrons.

The calculated masses are the average for the different particle types.

For instance, the mass of nucleon is the average of the mass of the proton and

the neutron.

Parameters of the calculation are the strong coupling constant, alpha_strong,

and the masses of light quarks up and down, and the quark strange.

They are fixed using the measured mass of pions, kaons and

Xi as reference.

This results in a pretty impressive understanding of

the hadron spectrum, suggesting that QCD is indeed a correct theory for

strong interaction also at larger distances.

Distance laws for electromagnetic and

strong interaction are fundamentally different.

The electromagnetic potential decreases as 1/r as a function of distance.

The strong potential increases at long distances proportional to r.

To understand this difference,

we must consider quantum correction to the photon and to the gluon propagators.

A significant correction to the photon propagator

is the one which introduces an electron-positron loop.

This loop creates additional electric charges between the projectile and target.

We call this phenomenon vacuum polarization.

In electrostatic language we can say that these additional

charges screen the target charge.

Therefore the effective charge decreases with distance.

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That is, it increases with growing momentum transfer q^2.

This is, indeed, what we find experimentally.

For electrodynamics, this is a small effect.

To see a change of alpha of a few percent, we have to compare momentum

transfer at almost zero that is at very large distances,

where alpha is measured to be about 1/137

– see the example of the Penning trap in video 4.3 –

to the energy of the Large Electron Positron collider at CERN,

where the momentum transfer q^2 is around 10^4 GeV^2.

That is at very short distances.

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For strong interactions, the effect of vacuum polarisation is much more

pronounced. Because of the size of the strong coupling, the vacuum polarization

and the change in the net charge with distance are more important.

In addition, the sign of the effect is reversed.

The strong force becomes stronger with distance.

This is because there are two types of vacuum polarization graphs for gluons.

The analogue of electromagnetic polarization introduces quark-antiquark

loops in the gluon propagator which shield the colors charge.

But in addition, the coupling of gluons between them

introduces gluon loops which have the opposite sign.

And in the chromostatic language, we can say that the gluon loops

introduce additional color-anticolor charges which are attractive.

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Consequently, the color charge of the target increases with

distance proportional to the inverse of the momentum transfer.

Indeed one experimentally finds that this effect is

more significant than its electromagnetic counterpart.

In the next video we will see how these properties of strong

interaction keep us from seeing free quarks.

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