# I Deep inelastic scattering and the Q^2 large limit

1. Oct 24, 2016

### CAF123

I am reading through Bailin and Love's argument (see P.151-152 of 'Introduction to Gauge Field Theory') that as $Q^2 \rightarrow \infty$, we probe the product of the two electromagnetic currents appearing in the hadronic tensor for DIS on the lightcone. I will write out the argument here and point out my questions as I go along.

The Bjorken limit is defined as the limit in which $Q^2 \rightarrow \infty, p \cdot q \rightarrow \infty$ with Bjorken $x$ fixed. As we will now show, the Bjorken limit corresponds to studying the light cone in coordinate space. To see this, it is convenient to work in the frame in which $p = (m_N, 0, 0, 0 )$ and $q = (q^0, 0, 0, q^0)$ with the z axis chosen along the direction of momentum of the virtual photon. (1st question: This parametrisation of momenta does not allow for a virtual photon - $q^2 = 0$ no? Perhaps there is a typo in his parametrisation because then the rest of the argument has no foundation I think)

In lightcone variables, $q^2 = q_+ q_-$ and $p \cdot q = (m_N/2)(q_+ + q_-)$ and therefore $$x = \frac{-q_+ q_-}{m_N(q_+ + q_-)}$$The Bjorken limit is thus the limit $q_+ \rightarrow \infty$ with $q_-$ fixed (and negative). Consequently, $x = -q_-/m_N$. (2nd question: why the Bjorken limit corresponds to $q_+ \rightarrow \infty$? $x$ as written is symmetric in $q_+ \leftrightarrow q_-$ so singling out $q_+$ as the one that gets large seems incorrect, no?)

Expanding the exponential appearing in the hadronic tensor in terms of lightcone components: $$e^{i qx} = \exp \left(\frac{i}{2} (q_+ x_- + q_- x_+) \right)$$ The exponential oscillates rapidly as $q^+ \rightarrow \infty$ so that the only contribution to the integral comes from the region $x_- = 0$. Then $$x^2 = x_+ x_- - \mathbf x^2 \leq 0$$ The strict inequality can't hold because of causality therefore we must have $x^2 = 0$, as required. (3rd question: $x^2 < 0$ would have implied that the currents were spacelike separated - why is this a bad thing? Is it because they would be in different regions of the lightcone diagram and hence never 'talk' to each other? )

Thanks!

2. Oct 29, 2016

### Greg Bernhardt

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

3. Oct 31, 2016

### MathematicalPhysicist

What seems to be the problem? $q^2 = q_0^2-q_0^2=0$, didn't you notice it?

4. Nov 1, 2016

### CAF123

Yes, that is exactly the problem. The whole point is to have a hard scale $q^2 \neq 0 \gg \Lambda_{\text{QCD}}$ so that perturbative QCD can be applied to this process.I think his parametrisation is just a typo and I thought about 3) more so only 2) is the remaining question I have.

5. Nov 3, 2016

### MathematicalPhysicist

I am not expert in physics, but perhaps there's something mentioned in the 2-volume of Aitchinson's about this problem. (don't know, perhaps the same problem arises there as well).