QCD, show that a Quark and an anti-quark of the same colour attract

[Chain]

This isn't really a homework question but it's from a past paper for revision. It is a short question so it shouldn't require lengthy mathematics. It asks you to show that a quark and an anti-quark of the same colour will attract.



The problem gives the 8 generator matrices of SU(3) and the Feynman rule for the exchange of a single gluon: $ig_sT^a \gamma ^\mu$



I'm not really sure how to find the force between them. To my understanding the Feynman rules give the probabilities of certain exchanges occurring, I don't know how this would be used to show the direction of the force between the quarks.

Thank you for your time :)

 

[fzero]

One way to do this is to recall that, for electromagnetism, the Coulomb law arises by computing the Fourier transform of the momentum-space amplitude for single photon exchange. Attraction and repulsion are deduced from the sign of the interaction. By analogy, here we want the amplitude for single gluon exchange.

 

[cmcraes]

Anti quarks have both opposite electrical charge and colour charge. So a positive red up quarks' antiparticle will be a negative anti-red(cyan) anti-up quark. A boson pair of quarks (a meson) consists of a quark and its corrisponding antiquark. The fact that these mesons annihilate so quickly is proof that they are attracted to each other. Also the fact that opposite charges will attract and quarks are only found where the colour charges produce a 'white' charge. So they have to attract other wise these mesons would not exist and then decay so quickly!

 

[Chain]

Ah okay, thanks for the responses guys (:

 

[paranormal]

As a scientist, it is important to understand the fundamental forces and interactions between particles. In the case of quarks, they are bound together by the strong nuclear force, described by Quantum Chromodynamics (QCD). In QCD, quarks interact through the exchange of gluons, which are the force carriers of the strong force.

To show that a quark and an anti-quark of the same color attract, we need to look at the interaction between them through the exchange of gluons. In the Feynman rules, the term "ig_sT^a" represents the strength of the interaction between the particles, where "g_s" is the strong coupling constant and "T^a" is the generator matrix for the color charge of the particles.

The Feynman rule also includes the term "\gamma^\mu", which represents the spin of the particles. Spin is important in determining the direction of the interaction between particles.

In the case of a quark and an anti-quark of the same color, the exchange of a gluon will result in an attractive force between them. This is because the gluon carries a color charge that is opposite to the color charge of the particles, causing them to attract each other.

To further understand this, we can look at the SU(3) generator matrices, which represent the different possible color charges of particles. In the case of a quark and an anti-quark of the same color, both particles will have the same color charge, represented by the same generator matrix. This means that when a gluon is exchanged between them, it will carry a color charge opposite to that of the particles, resulting in an attractive force.

In conclusion, the Feynman rules and the SU(3) generator matrices can be used to show that a quark and an anti-quark of the same color will attract each other through the exchange of gluons. This is a fundamental aspect of QCD and helps us understand the strong nuclear force that binds quarks together.