- #1
CAF123
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Consider the leading-order (LO) DGLAP ((Dokshitzer–Gribov–Lipatov–Altarelli–Parisi) equation $$x \mu^2 \frac{d xg(x,\mu^2)}{d\mu^2}= \alpha_s \int_x^1 dz P_{gg}(z) \frac{x}{z} g(\frac{x}{z}, \mu^2) + \dots\,\,\,\,\,\,\,\,\,\,\,\,\,(1) $$
Define an unintegrated parton density function (PDF) by $$F(x,\mu^2) \equiv x \frac{dg(x,\mu^2)}{d\mu^2},$$ with the initial condition $$\mu^2 F^{(0)}(x,\mu^2) = \theta(1-x) \theta \left(\frac{\mu^2}{Q_o^2}-1\right),\,\,\,\,\,\,\,\,\,\,(2)$$ where ##Q_o## is some non perturbative scale. The first order contribution to ##F## is then given by inserting (2) into (1) so that $$\mu^2 F^{(1)}(x,\mu^2) \approx \int_x^1 dz P_{gg}(z) \int_{Q_o^2}^{\mu^2} dk^2 \alpha_s(k^2) F^{(0)}(\frac{x}{z}, k^2)$$
My questions are
1)From the definition of ##F## it follows that $$g(x,\mu^2) = x^{-1} \int^{\mu^2} dk^2 F(x,k^2)$$ so I can see how the integral in the last display arises but what I don't see is why does this correspond to the first order contribution to ##F##? Doesn't equation (1) tell us that the evolution of the LO gluon PDF is given by the integral convolution of the LO splitting function kernels with the LO gluon PDF itself? So where does next-leading order (NLO) effects, i.e that suggested by 'first order in ##F##' arise?
2)What are the physical meanings of the ##F^{(i)}##? Are these defined through some expansion and, if so, what is this expansion in?
Define an unintegrated parton density function (PDF) by $$F(x,\mu^2) \equiv x \frac{dg(x,\mu^2)}{d\mu^2},$$ with the initial condition $$\mu^2 F^{(0)}(x,\mu^2) = \theta(1-x) \theta \left(\frac{\mu^2}{Q_o^2}-1\right),\,\,\,\,\,\,\,\,\,\,(2)$$ where ##Q_o## is some non perturbative scale. The first order contribution to ##F## is then given by inserting (2) into (1) so that $$\mu^2 F^{(1)}(x,\mu^2) \approx \int_x^1 dz P_{gg}(z) \int_{Q_o^2}^{\mu^2} dk^2 \alpha_s(k^2) F^{(0)}(\frac{x}{z}, k^2)$$
My questions are
1)From the definition of ##F## it follows that $$g(x,\mu^2) = x^{-1} \int^{\mu^2} dk^2 F(x,k^2)$$ so I can see how the integral in the last display arises but what I don't see is why does this correspond to the first order contribution to ##F##? Doesn't equation (1) tell us that the evolution of the LO gluon PDF is given by the integral convolution of the LO splitting function kernels with the LO gluon PDF itself? So where does next-leading order (NLO) effects, i.e that suggested by 'first order in ##F##' arise?
2)What are the physical meanings of the ##F^{(i)}##? Are these defined through some expansion and, if so, what is this expansion in?