A possible mistake in Equations (18.204)-(18.205) in Peskin & Schroeder

[MathematicalPhysicist]

They write the following on page 646:

Thus, according to the Altarelli-Parisi equations, the n-th moment of $f^{-}_f(x)$ obeys:
$$(18.204) \ \ \ \frac{d}{d\log Q^2}M^-_{fn} = \frac{\alpha_s(Q^2)}{8\pi}a_f^n \cdot M^-_{fn}.$$
To integrate this equation, we need the explicit form of $\alpha_s(Q^2)$.
Inserting expression (17.17), we find:
$$(18.205) \ \ \ \frac{d}{d\log Q^2} M^-_{fn}=\frac{a_f^n}{2b_0}\frac{1}{\log(Q^2/\Lambda^2)}M_{fn}^-$$

Now, equation (17.17) reads: $\alpha_s(Q) = \frac{2\pi}{b_0 \log(Q/\Lambda)}$, so if I plug $\alpha_s(Q^2) = \frac{2\pi}{b_0 \log(Q^2/\Lambda)}$ into Eq. (18.204) I get: $\frac{a^f_n}{4b_0}\frac{1}{\log(Q^2/\Lambda)}M^-_{fn}$.

Perhaps the $\alpha_s(Q^2)$ in equation (18.204) should be $\alpha_s(Q)$?

For this it works fine, I wonder how come it doesn't appear in the errata this typo?

 

[nrqed]

They write the following on page 646:Now, equation (17.17) reads: $\alpha_s(Q) = \frac{2\pi}{b_0 \log(Q/\Lambda)}$, so if I plug $\alpha_s(Q^2) = \frac{2\pi}{b_0 \log(Q^2/\Lambda)}$ into Eq. (18.204) I get: $\frac{a^f_n}{4b_0}\frac{1}{\log(Q^2/\Lambda)}M^-_{fn}$.

Perhaps the $\alpha_s(Q^2)$ in equation (18.204) should be $\alpha_s(Q)$?

For this it works fine, I wonder how come it doesn't appear in the errata this typo?

Q has the same dimension as \Lambda so they must appear with the same power in the log. If it works with $\alpha_s(Q)$ then that's correct. It does not appear just because they are not aware of it or they forgot to include it. No all typos are corrected in errata.

 

[Dr.AbeNikIanEdL]

At least in my copy, (17.17) reads

$\alpha_S(Q^2) = \frac{2\pi}{b_0\ln Q/\Lambda}$ ,

i.e., with a $Q^2$ instead of $Q$ as the argument.

 

[MathematicalPhysicist]

At least in my copy, (17.17) reads

$\alpha_S(Q^2) = \frac{2\pi}{b_0\ln Q/\Lambda}$ ,

i.e., with a $Q^2$ instead of $Q$ as the argument.

Which edition do you have?

 

[Dr.AbeNikIanEdL]

I am actually not sure, the front matter is not particularly helpful. Published by Perseus Books, so rather old if I understand correctly...

Edit: according to this https://www.slac.stanford.edu/~mpeskin/QFTold.html eq. (17.17) in the version I apparently have was corrected in later versions (to be consistent with (17.13)), possibly without changing later equations consistently. I think this is just a matter of convention what argument you prefer for $\alpha_S$, but in any case, the powers of $\Lambda$ and $Q$ should match of course, as nrqed said...

 

[vanhees71]

It's the running of the strong coupling constant. You find the one-loop calculation in my lecture notes

https://itp.uni-frankfurt.de/~hees/publ/lect.pdf

p. 255 ff. I hope I didn't make also some mistake or typo. Note that by definition $\alpha_s=g^2/(4 \pi)$. At least it's the same as in Schwartz, QFT and the Standard model (given that for SU(3) $C_A=3$ and that $\ln(\mu^2/\Lambda^2)=2 \ln(\mu/\lambda)$ ;-)).

So the correct formula is
$$\alpha_s=\frac{4 \pi}{\beta_0 \ln(\mu^2/\Lambda^2)}=\frac{2 \pi}{\beta_0 \ln(\mu/\Lambda)},$$
where $\beta_0=11 C_A-2 N_F$, where $C_A$ is the casimir of the adjoint representation and $N_F$ the number of quark flavors in the fundamental representation (as in QCD).

The two contributions come from the two gluon-loop diagrams and the quark-loop diagram, respectively, in the very elegant calculation via the gluon polarization function in background-field gauge provided in my manuscript, orignating from Abbott as cited in the manuscript).