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  Interpretation of transverse momentum as the logarithm of renormalization scale in Altarelli and Parisi's derivation of DGLAP equation

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I'm reading Altarelli and Parisi's 1977 paper "Asymptotic freedom in parton language" in which they derive their own equations for the evolution of the parton distribution functions.

In page 309, they compare the cross sections of the processes (in massless QCD)

  • $A+D\to C+f$, in which the initial states $A$ and $D$ interact through an virtual intermediate particle $B$ whose momentum is "nearly" on the mass shell,
  • $B+D\to f$, treating $B$ as an initial state on its own.

The state $D$ as well as the final state $f$ are not specified; it is assumed that the process takes place with $A$ splitting in two almost-collinear particles $B$ and $C$, such that their transverse momentum $p_\perp$ with respect to the direction of motion of $A$ is small.

Regarding the longitudinal components of momentum, $B$ is given a fraction $z$ of $A$'s longitudinal momentum, and $C$ the remaining part.

After factorizing the first process as a "composition" of the separate processes $A\to B+C$ and $B+D\to f$ and calculating the terms relative to the former, they compare the result to a previous equation to extract the probability that $A$ contains a $B$-type particle with a fraction $z$ of the longitudinal momentum.

The cross section $\sigma_{A+D\to C+f}$ contains an integral on $p_\perp^2$, and we can change integration variables to $\mathrm{\ln p_\perp^2}$. At this point, they say

for a virtual mass $-Q^2$ for particle $D$, the integral in $p_\perp^2$ has an upper limit of order $Q^2$, so that, at the leading logarithmic approximation, $\mathrm{d}\ln p_\perp^2$ can be directly interpreted as $\mathrm{d}t$,

where $t=\ln\frac{Q^2}{Q_0^2}$, $Q_0$ being a suitable renormalization point.

I don't get how this interpretation of $p_\perp^2$ is explained. Why does $D$ have a virtual mass, when $D$ is an external state, and why should it limit $p_\perp^2$ from above?

And from where does the $Q_0$ term arise in all this? An analogous calculation but in a QED context where $B$ is an electron, as done by Peskin and Schroeder in their Intro to QFT book, shows that, since the denominator of $B$'s propagator has been approximated in the limit of vanishing mass, if we restore $B$'s mass then the square of the transverse momentum should actually be cut off from below by $m_B^2$. The result, though, shows $Q_0^2$, not the square of $B$'s mass, which anyway sometimes may be zero, i.e. if $B$ is a gluon.

asked Apr 9 in Theoretical Physics by yellon (5 points) [ no revision ]

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