Calculating Hadron Scattering Processes from the Strong Force

[RedX]

Conceptually, how does one calculate hadron scattering processes from the strong force, if one has the amplitudes for 2x2 scattering of quarks and gluons? For example, the strong force between a proton and a proton, both made from uud quarks - is this therefore a 6x6 scattering?

I went through a lot of tedious math, manipulating a bunch of twistors, to get the amplitude for 2x2 elastic scattering of: 4 quarks; 2 quarks and 2 gluons; and 4 gluons; but the book stops there and doesn't talk about how to calculate hadron scattering processes. I don't even know why I calculated 4 gluon scattering and 2 quarks 2 gluons scattering - are there free gluons inside the hadrons? How many gluons are there inside? So a proton is really made of uud quarks and more than 1 gluon? So proton-proton scattering by the strong force is greater than 6x6 if you include the gluons?

The energy scale in order to apply perturbation theory to strong processes is said to be the mass of the Z-boson. Is this the reason that QED is sometimes renormalized at the Z-boson mass, so that one can compare the effects of the strong force to the electromagnetic?

Also twistors seem kind of tedious. Are QCD calculations without using the twistor method easier?

 

[humanino]

Conceptually, how does one calculate hadron scattering processes from the strong force, if one has the amplitudes for 2x2 scattering of quarks and gluons?

The amplitude is usually split in two parts convoluted together. What you calculated is called a "hard part". It involves only partons. The "soft part" contains the non-perturbative physics nobody knows how to calculate from QCD (strictly), except on by brute force on a lattice, and basically encodes the hadronization process (where confinement sets in). This is called factorisation. The reason factorisation is useful is because the soft part depends only on the (final state) hadrons involved, not on the particular process at hand. So we can measure the soft part in one experiment, and use it to predict what we will get in another experiment. The soft part is sometime called "structure function", and this universal piece is as close as it gets to a measure of the "wavefunction" of the hadrons in a Fock tower of partons. Indeed, the soft part scales, it changes with scale, because as you look ever closer to the quarks, they radiate ever more gluons which themselves tend to fluctuate in quark-antiquark pairs. Alternatively, you can think of long-distance partons as propagating with a "cloud" of virtual partons giving them an effective constituent mass.

Factorization theorems hold mathematically in the so-called "Bjorken limit". There are simple criteria the amplitude fullfills in this limit, so we can check whether in a given experiment, the limit was not reached yet (of course, we can never know for certain that the limit was indeed reached, but when we get scaling we have no reason to doubt).

Also twistors seem kind of tedious. Are QCD calculations without using the twistor method easier?

Try using only spinors. It is often worse, so people use twistors more and more often.

 

[Tingish]

Sorry for not contributing to the discussion but it should be chromodynamics instead of chronodynamics. Chromo- relates to colour while chrono- relates to time. And dynamics usually refers to the time evolution of physical processes, so quantum chronodynamics would be the quantum description of the time evolution of time. Definitely interesting, but a bit different than QCD.

 

[humanino]

Sorry for not contributing to the discussion but it should be chromodynamics instead of chronodynamics. Chromo- relates to colour while chrono- relates to time. And dynamics usually refers to the time evolution of physical processes, so quantum chronodynamics would be the quantum description of the time evolution of time. Definitely interesting, but a bit different than QCD.

I did not notice the typo :rofl:
The concept of "chronodynamics" is the funniest physics joke I ever heard.

 

[DrFaustus]

RedX -> gluons, unlike photons, are self-interacting particles. This means that, say, in a proton when a gluon is exchanged between two of the three quarks and another gluon is exchanged between another pair of quarks, the two gluons can interact.

 

[RedX]

RedX -> gluons, unlike photons, are self-interacting particles. This means that, say, in a proton when a gluon is exchanged between two of the three quarks and another gluon is exchanged between another pair of quarks, the two gluons can interact.

Are you talking about two protons, or one proton? So if you're talking about one proton, then the first pair, 12, exchanges a gluon. Then another pair, either 13 or 23, exchanges a gluon? So the process would look like normal 2x2 scattering of quarks, but one of the outgoing external quark lines of the 2x2 scattering would hook up with a gluon and exchange this with another quark, so this would be like a 3x3 scattering instead? Why is this particular diagram important?

Are the gluons inside the baryons external lines and therefore on-shell? Normally it doesn't matter if the external lines are on-shell, because there's nothing special about that: once you calculate the diagrams you just set the momentum of the external lines to be on-shell by amputating the diagrams and setting the momentum to be on-shell. However, the twistor method requires that the gluons be on-shell (since 4-momentum must be light-like in order to represent the polarization vector as a product of spinors - or a twistor). The amplitudes calculated for gluon-gluon scattering is written in terms of twistors, so one cannot use the results from those diagrams to build bigger diagrams since that will make the gluons internal.

humanino: The reason factorisation is useful is because the soft part depends only on the (final state) hadrons involved, not on the particular process at hand.

So it doesn't depend on the initial state hadrons? Usually amplitudes depend on the initial and final state, and you add all processes. I thought maybe the confinement part tells you how the momentum is distributed inside a baryon, given the total 4-momentum of the baryon. Once you have that information for the initial and final hardrons, you can figure out the 4-momentum of initial and final quarks, and calculate using SU(3) gauge theory. So this is too naive?

humanino: The soft part is sometime called "structure function", and this universal piece is as close as it gets to a measure of the "wavefunction" of the hadrons in a Fock tower of partons.

So this seems to say that the soft part is determining the Fockspace wavefunction, which is the same as determining the distribution of momentum within a hadron. So does this Fockspace includes free gluons, so that instead of a baryon being made of 3 quarks, it has some gluons too? In fact a whole cloud of them, along with quarks and anti-quarks - or are these all virtual and the only thing real are the 3 quarks that make the baryon?

Tingish: it should be chromodynamics instead of chronodynamics.

I should have known that. One of my favorite video games long ago was a game called chrono cross/trigger which was about time travel to save the future. My general relativity is bad, so I still retain fond memories of the game.

 

[humanino]

So it doesn't depend on the initial state hadrons? Usually amplitudes depend on the initial and final state, and you add all processes.

It does depend on the initial and final states (reaction), but one on the particular process.

I thought maybe the confinement part tells you how the momentum is distributed inside a baryon, given the total 4-momentum of the baryon. Once you have that information for the initial and final hardrons, you can figure out the 4-momentum of initial and final quarks, and calculate using SU(3) gauge theory. So this is too naive?

You need interplay between position and momentum if you are interested in the full energy-momentum densities and (two-body) correlations. One can even invent higher order correlations. The modern approaches are more along the line of density matrices and Wigner operator than Fockspace wavefunctions.

So this seems to say that the soft part is determining the Fockspace wavefunction, which is the same as determining the distribution of momentum within a hadron. So does this Fockspace includes free gluons, so that instead of a baryon being made of 3 quarks, it has some gluons too? In fact a whole cloud of them, along with quarks and anti-quarks - or are these all virtual and the only thing real are the 3 quarks that make the baryon?

The soft part contains the entire Fockspace, valence and sea. Distinguishing real and virtual in general is a tricky question. Only heavy valence quarks are close to their MSbar mass shell (in heavy hadrons), everything else is pretty much always virtual. Alternatively, you can give up MSbar (light) quarks and go with constituent quark picture (and it's not restricted to good ol'(non) relativistic constituent quark models. Dyson-Schwinger models give very consistent results).