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CAF123
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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!
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!