- #1
MathematicalPhysicist
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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?
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?