The Bjorken-x depicitive

[blue2script]

Hello!

Sorry to bother you with all these questions coming to my mind in preparing for my oral exam in QCD. I was just wondering about the nature of the Bjorken x. You see, it is defined as:

$$x = \frac{Q^2}{2M\nu} = \frac{Q^2}{2P\cdot q}$$

where $$Q = k - k'$$ (k is the momentum of the electron), M the mass of the proton and $$\nu = \epsilon - \epsilon'$$ is the energy difference of the electron (in the rest frame of the nucleon).

So, what we find is that we can measure the Bjorken-x solely from the initial and final state of the electron. On the other side, we know

$$0 = p'^2 = \left(p+q\right)^2 = 2p\cdot q + q^2 = 2p\cdot q - Q^2$$

thus

$$1 = \frac{Q^2}{2p\cdot q} = \frac{Q^2}{2\xi P\cdot q} = \frac{x}{\xi}$$

$$\Rightarrow x = \xi$$

where p is the momentum of a single quark, P the momentum of the nucleous and $$\xi$$ is the momentum fraction of the quark, meaning $$p = \xi P$$. So, what we find is that the Bjorken x is the momentum fraction of the quark. However, we can calculate x and therefore the momentum fraction solely from the electron. That means there is a very deep relation between the momentum transfer through the photon and the momentum fraction of the quark.

My question is then: Where does this deep connection comes from? Is there any depictive way to make this argument, this connection, more visual, transparent?

Thanks for all your comments! With best regards,
Blue2script

PS: Another question of the same kind is: The connection above binds Q^2 to x. But then, how can we treat Q^2 and x independent in the naive parton model?

 

[AndyW]

Dear Blue2script,

Thank you for your questions about the nature of Bjorken-x and its relation to the momentum fraction of quarks. This is a very interesting topic in QCD and I am happy to provide some insights.

To start, let's look at the definition of Bjorken-x, which you have correctly stated as x = Q^2/2Mν = Q^2/2P·q. This quantity represents the fraction of the nucleon's momentum carried by the struck quark in deep inelastic scattering (DIS). In other words, it tells us how much of the nucleon's momentum is transferred to the quark during the scattering process.

Now, let's consider the second equation you have presented, which shows the connection between x and the momentum fraction of the quark, ξ. This equation arises from the fact that in the naive parton model, we assume that the nucleon is made up of point-like quarks, each carrying a fraction of the nucleon's momentum (ξ). This is a simplified picture, but it is useful in understanding the basic principles of DIS.

So, where does this connection between x and ξ come from? It is a consequence of the underlying theory of QCD. In QCD, the momentum of the nucleon is carried by its constituent quarks and gluons, which are constantly interacting and exchanging momentum. In DIS, the virtual photon interacts with a quark inside the nucleon, and the resulting scattering process reveals information about the quark's momentum fraction (ξ) and the momentum transfer (Q^2). This is why we can calculate x and ξ solely from the initial and final state of the electron.

As for your question about treating Q^2 and x as independent in the naive parton model, this is because the parton model is a simplified picture and does not take into account the complexities of QCD. In reality, Q^2 and x are not completely independent, as we can see from the equations you have presented. However, in the parton model, we treat them as independent variables for simplicity.

I hope this helps to clarify the deep connection between Q^2 and x, and how it arises from the underlying theory of QCD. Unfortunately, there is no visual or intuitive way to understand this connection, as it is a fundamental aspect of the theory. But I hope this explanation has provided some insight into the nature of Bjork

 

[sir-pinski]

Hi Blue2script,

First of all, don't apologize for asking questions - it's always good to have a clear understanding of the concepts before an exam!

The Bjorken-x is a dimensionless quantity that represents the fraction of momentum carried by a parton (quark or gluon) in a nucleon. It is related to the momentum transfer Q^2 in deep inelastic scattering (DIS) experiments, where a high-energy electron is scattered off a nucleon and the structure of the nucleon is probed.

In the equation x = Q^2/(2M\nu), Q^2 is the square of the four-momentum transfer between the initial and final states of the electron, M is the mass of the nucleon, and \nu is the energy difference of the electron. This equation can also be written as x = Q^2/(2P\cdot q), where P is the four-momentum of the nucleon and q is the four-momentum of the photon.

Now, in the naive parton model, we assume that the nucleon is made up of point-like partons (quarks and gluons) that carry a fraction of the nucleon's momentum. This means that the momentum transfer Q^2 is shared between the parton and the nucleon, and we can write Q^2 = \xi P\cdot q, where \xi is the momentum fraction of the parton. Since x = Q^2/(2P\cdot q), we can see that x = \xi, which means that the Bjorken-x is equal to the momentum fraction of the parton.

But why is this connection between Q^2 and x so deep? It is because the momentum transfer Q^2 is directly related to the virtuality of the exchanged photon in DIS experiments. The higher the virtuality, the smaller the Bjorken-x, which means that we are probing deeper into the nucleon and seeing the structure of the partons at smaller scales. This is why the Bjorken-x is often referred to as the "Bjorken scaling variable" - it scales with the energy of the scattering process.

As for a more visual or depictive way to understand this connection, one way is to think of the nucleon as a target and the parton as a bullet hitting the target. The momentum fraction x can then be thought of as the ratio of the momentum of the bullet to the total