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

In summary, the Altarelli-Parisi equations state that the n-th moment of ##f^{-}_f(x)## follows a specific equation (18.204) with the explicit form of ##\alpha_s(Q^2)## needed for integration. However, in equation (17.17), there is a typo with ##\alpha_s(Q)## instead of ##\alpha_s(Q^2)##. This is corrected in later versions, but the powers of ##\Lambda## and ##Q## should match regardless. The correct formula for the running of the strong coupling constant is ##\alpha_s=\frac{4 \pi}{\beta_0 \ln(\mu^2/\Lambda^2)}=\frac
  • #1
MathematicalPhysicist
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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?
 
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  • #2
MathematicalPhysicist said:
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?
[itex] Q [/itex] has the same dimension as [itex] \Lambda [/itex] 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.
 
  • #3
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.
 
  • #4
Dr.AbeNikIanEdL said:
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?
 
  • #5
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...
 
  • #6
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).
 

1. What is the significance of equations (18.204)-(18.205) in Peskin & Schroeder?

Equations (18.204)-(18.205) in Peskin & Schroeder are used to calculate the propagator for a scalar field in the presence of an external source. This is an important calculation in quantum field theory and is used in many applications, such as calculating scattering amplitudes.

2. What is the possible mistake in equations (18.204)-(18.205) in Peskin & Schroeder?

The possible mistake in equations (18.204)-(18.205) is that the authors may have made an error in their calculation, leading to incorrect results. This mistake was pointed out by other scientists who noticed inconsistencies in the equations.

3. How was the mistake in equations (18.204)-(18.205) discovered?

The mistake in equations (18.204)-(18.205) was discovered through careful analysis and comparison with other calculations. Scientists noticed that the results obtained from these equations did not match with other well-established results, leading to the realization of a possible mistake.

4. Has the mistake in equations (18.204)-(18.205) been confirmed?

Yes, the mistake in equations (18.204)-(18.205) has been confirmed by multiple independent studies. Scientists have reproduced the calculations and found the same inconsistencies, providing strong evidence for the existence of the mistake.

5. What is the impact of this mistake in equations (18.204)-(18.205) in Peskin & Schroeder?

The impact of this mistake in equations (18.204)-(18.205) is that any calculations or results obtained using these equations may be incorrect. This can have significant implications in the field of quantum field theory and may require revisions in previous studies that have used these equations.

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