Consider the real emission correction to the tree level process ##e^+ e^- \rightarrow q \bar q## involving a final state gluon emitted from either the outgoing quark or antiquark line. The differential cross section for producing a quark with fractional energy ##x_1## in the ##q \bar q g## final state is $$\frac{1}{\sigma_0} \frac{d \sigma}{d x_1} = \frac{2 \alpha_s}{3 \pi} \int dx_2 \frac{x_1^2 + x_2^2} {(1-x_1)(1-x_2)},$$ where ##\sigma_0## is the tree level cross section for ##e^+ e^- \rightarrow q \bar q##. The singularities present for ##x_i \rightarrow 1## are associated with soft and collinear divergences which are removed upon consideration of virtual corrections to the tree level process for ##q \bar q ## production. Thus, $$\int_0^1 dx_1 \frac{d \sigma_R^{(1)}}{d x_1} + \sigma_V^{(1)} = \frac{\alpha_s}{\pi} \sigma_0 = \text{finite}$$ This may be rewritten as $$ \int_0^1 dx_1 \left( \frac{d \sigma_R^{(1)}}{d x_1} + \left( \sigma_V^{(1)} -\frac{\alpha_s}{\pi} \sigma_0 \right) \delta(1-x) \right) = \text{finite}$$ Now, define $$F(x)_+ = \text{lim}_{\beta \rightarrow 0} \left( F(x) \theta(1-x-\beta) - \delta(1-x-\beta) \int_0^{1-\beta} F(x')dx' \right),$$ where ##\beta## acts as a regulator. We can then write $$ \frac{1}{\sigma_0} \frac{d \sigma^{(1)}}{dx_1} = \frac{1}{\sigma_0} \left(\frac{d \sigma^{(1)}}{dx_1}\right)_+ +\alpha_s R \delta(1-x)\,\,\,\,\,(1)$$ where ##R = \sigma_0(1+\alpha_s/\pi)\,\,\,\,(2)## and $$\int_0^1 dx_1 \left(\frac{d \sigma^{(1)}}{dx_1}\right)_+ = 0\,\,\,\,\,(3)$$ and so $$\left(\frac{d \sigma^{(1)}}{dx_1}\right)_+ = \frac{\alpha_2}{2\pi} P_{q \rightarrow qg}(x_1) \cdot L + \alpha_s f(L),\,\,\,\,\,(4)$$ with ##L## the logarithmically divergent piece and ##f(L)## a left over function depending on how the z integration was regularised.(adsbygoogle = window.adsbygoogle || []).push({});

Most of this was cited from https://www.ippp.dur.ac.uk/~krauss/Lectures/QuarksLeptons/QCD/DGLAP_2.html. I'm just wondering if someone could explain where equations (1)-(4) come from? I think (3) is clear from the definition of the plus distribution but I am not sure how (4) comes from the earlier equations and how (1) and (2) are initially obtained.

Thanks!

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# I Derivation of splitting function from cross section

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