If you open the imgur link it should be in high res. Anyways the reference is chapter 3, end of subchapter 3.3 of this pdfVanadium 50 said:I can't read light gray on a less light gray background. Can you use LaTex and maybe post your source?
Eq. (75) is just the definition of ##Z_1 = Z_g Z_2 Z_3^{\frac{1}{2}}##. Anyways thank you I understand now, the expansions of ##1/Z_2## , ##1/Z_3## are obviously at first order in ##\alpha_s##, which in turn means first order in the counterterms since they are proportional to ##\alpha_s## because they were computed at 1-loop. I guess the divergence in ##1/\epsilon## doesn't matter since we only take the limit ##\epsilon \to 0## in the end?vanhees71 said:The Dyson series is an expansion in the coupling constant ##g##. Obviously you are evaluating the quoted 2nd-order loop corrections to the quark and gluon self-energies as well as the gluon 3-vertex to get the counter terms to evaluate ##Z_1##, ##Z_2##, and ##Z_3##, from which ##Z_g## follows through a Slavnov-Taylor identity (which maybe is Eq. (75), which you didn't quote). Of course since you have the said ##Z##-factors only to order ##g^2## (or order ##\alpha_s=g^2/(4 \pi)##), you can determine only the counterterm contributing to ##Z_g## to this order, and thus you have to expand the expression for it also up to order ##\alpha_s##.
In the here obviously applied minimal-subtraction scheme you need to find the coefficients to ##1/\epsilon## order by order perturbation theory. This determines the counter terms order by order. At higher loop order you have to take care of the subdivergences by using the corresponding counter terms of subdiagrams.
"Expansion at first order in QCD counterterm" refers to a mathematical technique used in quantum chromodynamics (QCD) to calculate the behavior of quarks and gluons in high energy interactions. It involves expanding the equations of QCD in terms of a small parameter, known as the coupling constant, and then using counterterms to account for any divergences that arise in the calculations.
"Expansion at first order in QCD counterterm" is important because it allows scientists to make predictions about the behavior of quarks and gluons in high energy interactions, which are crucial for understanding the fundamental forces that govern the behavior of matter. It also helps to reconcile the theoretical predictions of QCD with experimental data.
"Expansion at first order in QCD counterterm" is different from other methods of calculation in QCD because it takes into account higher order corrections to the equations, which can significantly improve the accuracy of the predictions. It also allows for the inclusion of more complex interactions between quarks and gluons, making it a more comprehensive approach.
One limitation of "Expansion at first order in QCD counterterm" is that it can become increasingly complex and difficult to calculate at higher orders. Additionally, it does not take into account the effects of strong interactions, which can be important in certain high energy interactions. As such, it is often used in conjunction with other methods to improve the accuracy of predictions.
"Expansion at first order in QCD counterterm" is closely related to the concept of asymptotic freedom, which states that the strength of the strong nuclear force decreases at high energies. This allows for the use of perturbative techniques, such as "Expansion at first order in QCD counterterm", to make accurate predictions. However, at lower energies, the strong force becomes stronger and these techniques are no longer applicable.