A Feynman loop diagrams and Dyson series for anomalous magnetic moment

[ohwilleke]

Second, α varies with distance. At a distance of about 1/40 of a femtometer, it's up to about 1/125. If you're predicting alpha, at what scale?

α varies with the momentum transfer in the interaction in question, commonly called "energy scale", and often identified with the variable "s", not with distance.

See Section 10.2.2 of "10. Electroweak Model and Constraints on New PhysicsRevised March 2022" by J. Erler and A. Freitas in R.L. Workmanet al.(Particle Data Group), Prog. Theor. Exp. Phys. 2022, 083C01 (August 11, 2022) at https://pdg.lbl.gov/2023/reviews/contents_sports.html

Specifically:

In most EW renormalization schemes, it is convenient to define a running α dependent on the energy scale of the process, with α⁻¹≈137.036 appropriate at very low energy, i.e. close to the Thomson limit. The OPAL and L3 collaborations at LEP could also observe the running directly in small and large angle Bhabha scattering, respectively. For scales above a few hundred MeV the low energy hadronic contribution to vacuum polarization introduces a theoretical uncertainty in α.

 

[Vanadium 50]

α varies with the momentum transfer in the interaction in question, commonly called "energy scale", and often identified with the variable "s", not with distance.

Wrong.

The source of the variation in alpha is charge screening by vacuum polarization. Shorter distance means less polarization. As a practical matter, short distance also means high momentum transfer (not s in general), which is more experimentally accessible, but the fundamental cause is that you get "inside" the region of charge screening by the vacuum.

 

[Adrian59]

As I said, the primary purpose of opening this thread was to was to explore the origin of claims made to the effect that α can be theoretically derived, and that its value when compared with experimentally derived values is the origin of the assertion that QED is one of the most accurate physical theories . This certainly is the claim made by many popular science books. It appears in more academic circles that there is no absolute derivation of α without some experimental value as an input.

Sure, but it's not a theoretical prediction of the value of α. It's a theoretical prediction of the electron's magnetic moment, making use of the best then current experimental value of α.

However, this is an indictment on academic physics that this is not made clear, and the myth of a pure theoretical derivation is allowed to permeate popular science books. Certainly twenty years ago that was what I thought, having read such claims.

You can look at the QED coupling constant beta function in this open access paper explaining how it is calculated.

The paper you linked here was an extremely good summary of the basics of quantum field theory (QFT) which the earlier sections were all very familiar to me having studied QFT. However, there it is again in table 1.1 a comparison of the theoretical and the experimental of α. Also, the reference is back to our old friend Aoyama, see my opening reference.

Then the first sentence of chapter 6, the conclusions explicitly makes the claim, 'quantum electrodynamics’ unparalleled agreement with experiment has made it one of the most successful theories in the physical sciences.'

There needs to be some clarity as to what is being claimed and by what method.

 

[renormalize]

As I said, the primary purpose of opening this thread was to was to explore the origin of claims made to the effect that α can be theoretically derived, and that its value when compared with experimentally derived values is the origin of the assertion that QED is one of the most accurate physical theories . This certainly is the claim made by many popular science books.

Can you cite some specific references from "popular science books" that claim $\alpha$ can be theoretically derived?

 

[PeterDonis]

the myth of a pure theoretical derivation is allowed to permeate popular science books

Pop science books are not good sources for learning actual science, so it would be no surprise to me (unfortunately) if they made claims about this that are not made in textbooks or peer-reviewed papers. This problem is certainly not limited to claims about theoretical derivations of $\alpha$.

The paper

It's not a paper, it's a thesis. The rules for those are somewhat different.

there it is again in table 1.1 a comparison of the theoretical and the experimental of α.

This table is based on the same kinds of derivations we have already discussed: they are not derivations of $\alpha$ from first principles, they are basically consistency checks on the perturbation models using the comparisons of model predictions of actual observables with the experimental values.

the conclusions explicitly makes the claim, 'quantum electrodynamics’ unparalleled agreement with experiment has made it one of the most successful theories in the physical sciences.'

There needs to be some clarity as to what is being claimed and by what method.

It seems to me that you are the only one who is (mistakenly) claiming that any of these papers are giving a derivation of $\alpha$ solely from first principles, or that "agreement with experiment" requires every single quantity in the theory to be derived solely from first principles. Nobody else seems to be misunderstanding them that way.

 

[Adrian59]

It seems to me that you are the only one who is (mistakenly) claiming that any of these papers are giving a derivation of $\alpha$ solely from first principles, or that "agreement with experiment" requires every single quantity in the theory to be derived solely from first principles. Nobody else seems to be misunderstanding them that way.

Despite my best efforts you have repeatedly got the wrong end of the stick. I am in no way mistaken on this issue as I have said repeatedly in this thread. If you doubt this then you need to re-read the whole thread. In fact I have agreed with you consistently through the thread. My stated aim was to explore the origin of such a claim which I have no doubt has been made by OTHERS as I have referenced.

The only issue may be that the terminology is poorly applied, so that when an author says theoretically derived they do not mean completely theoretically derived right from first principles. However, my point is that these authors should be more careful with their wording.

 

[PeterDonis]

In fact I have agreed with you consistently through the thread.

Not when you make claims like this:

this is an indictment on academic physics

There needs to be some clarity as to what is being claimed and by what method.

my point is that these authors should be more careful with their wording.

These are the kinds of claims I am pushing back on, because I do not agree with them, and I don't see where there is any basis for them in any of the references you have given.

 

[PeterDonis]

when an author says theoretically derived they do not mean completely theoretically derived right from first principles

We do not have anything in physics that is "completely theoretically derived right from first principles". Every single physical theory we have and have ever had has quantities in it that have no first principles derivation but must be input from experimental results. Every physicist knows this, and expects anyone reading what they write to know it too, so they don't have to laboriously restate it every time they write a paper.

 

[Adrian59]

We do not have anything in physics that is "completely theoretically derived right from first principles". Every single physical theory we have and have ever had has quantities in it that have no first principles derivation but must be input from experimental results. Every physicist knows this, and expects anyone reading what they write to know it too, so they don't have to laboriously restate it every time they write a paper.

So we do actually agree, as I thought all along. I will have to re-read the Aoyama et al paper to try and find what they are really up to. I can see if it is purely a numerical derivation how it works; but as I have said, and something you agreed on in #7 their terminology does appear rather confusing.

 

[Meir Achuz]

Whether were not its size can be derived (It has not yet been derived, although I have tried.), its size is remarkable. If it were larger or smaller by enough, life as we know it would be impossible.

 

[PeterDonis]

we do actually agree

If we do, then why did you post the things I quoted in post #37? Those are all direct quotes from you, and as I said in that post, I disagree with all of them. Are you now retracting those statements?

 

[PeterDonis]

its size

The size of what? Did you possibly intend this post for a different thread?

 

[PeterDonis]

something you agreed on in #7 their terminology does appear rather confusing

It's confusing with regard to their use of the term "analytic" as opposed to "numerical", as I described in that post. It's not confusing (at least not to me and others in this thread besides you) about whether it claims to be a complete first principles derivation of $\alpha$ with no experimental input. It's not, and it doesn't claim to be.

 

[Meir Achuz]

The size of what? Did you possibly intend this post for a different thread?

The FSC=1/137.

 

[Meir Achuz]

?

 

[PeterDonis]

?

?

 

[Meir Achuz]

The FSC=1/137.

I was trying to ask what was skeptical about

The FSC=1/137.

 

[PeterDonis]

@Meir Achuz I don't have any idea what your actual question is. Nobody is "skeptical" about the value of the fine structure constant.

 

[vanhees71]

Well, the discussion is about the question, whether there's a theoretical explanation for the value of the fine structure constant. This has been asked since Sommerfeld introduced it when treating the fine structure of the hydrogen spectrum using Bohr-Sommerfeld quantization. It's indeed an ironic coincidence that he gets the fine structure right without introducing spin (which is pretty impossible within Bohr-Sommerfeld quantization anyway) by just solving the relativistic equations of motion of a classical point particle in a Coulomb field.

Now the fine structure constant is, in the SI, simply
$$\alpha=\frac{e^2}{4 \pi \epsilon_0 \hbar c}.$$
With the new SI the question about the status of its understanding is very easily answered: in the above formula $e$ ("elementary electric charge", $\hbar=h/(2 \pi)$ ("modified Planck's action quantum"), and $c$ ("speed of light in vacuo") are all defined constants fixing our system of units, but $\epsilon_0$ has to be measured, i.e., it cannot be derived somehow from any theory. So the value of $\alpha$ is an empirical input in our contemporary best theory we have (in this case the Standard Model of elementary particle physics).

 

[Meir Achuz]

@Meir Achuz I don't have any idea what your actual question is. Nobody is "skeptical" about the value of the fine structure constant.

I thought that someone put a 'skeptical' imogi on my original post.

 

[Meir Achuz]

Well, the discussion is about the question, whether there's a theoretical explanation for the value of the fine structure constant. This has been asked since Sommerfeld introduced it when treating the fine structure of the hydrogen spectrum using Bohr-Sommerfeld quantization. It's indeed an ironic coincidence that he gets the fine structure right without introducing spin (which is pretty impossible within Bohr-Sommerfeld quantization anyway) by just solving the relativistic equations of motion of a classical point particle in a Coulomb field.

Now the fine structure constant is, in the SI, simply
$$\alpha=\frac{e^2}{4 \pi \epsilon_0 \hbar c}.$$
With the new SI the question about the status of its understanding is very easily answered: in the above formula $e$ ("elementary electric charge", $\hbar=h/(2 \pi)$ ("modified Planck's action quantum"), and $c$ ("speed of light in vacuo") are all defined constants fixing our system of units, but $\epsilon_0$ has to be measured, i.e., it cannot be derived somehow from any theory. So the value of $\alpha$ is an empirical input in our contemporary best theory we have (in this case the Standard Model of elementary particle physics).

But, it is $\alpha$, not $\epsilon_0$, that is actually measured.

 

[PeterDonis]

I thought that someone put a 'skeptical' imogi on my original post.

Probably because they had no idea what you were trying to say in that post. Neither did I.

 

[Adrian59]

Maybe, but the papers reporting these calculations generally explain, at a conceptual level, all of the terms that go into those calculations. Like any Feynman diagram calculation, it includes all possible paths from the starting point to the ending point and usually sorted by the number of loops involved.

So, since we are agreed that α cannot be derived from first principles, it is only the coefficients of the power series that connects the anomalous magnetic moment to the fine structure constant (α) that are derived. I can see how one could do this by a straightforward numerical method. However, it appears that these authors are using Feynman diagrams to derive these coefficients, and depending on the order being calculated, one can use either 1, 7, 72, 891, or 12 672 for the lowest order terms. Even at order 5 there are far too many loops to do this without a computer.

This table is based on the same kinds of derivations we have already discussed: they are not derivations of α from first principles, they are basically consistency checks on the perturbation models using the comparisons of model predictions of actual observables with the experimental values.

So, I have two remaining questions:
1) how does one explicitly derive this from a single Feynman loop, especially what values are ascribed to the momenta carried by the loop;
2) how are the more complex numbers of loops dealt with numerically.

 

[vanhees71]

But, it is $\alpha$, not $\epsilon_0$, that is actually measured.

True, but $\epsilon_0$ then follows through simple algebra from that measured value.

 

[vanhees71]

So, since we are agreed that α cannot be derived from first principles, it is only the coefficients of the power series that connects the anomalous magnetic moment to the fine structure constant (α) that are derived. I can see how one could do this by a straightforward numerical method. However, it appears that these authors are using Feynman diagrams to derive these coefficients, and depending on the order being calculated, one can use either 1, 7, 72, 891, or 12 672 for the lowest order terms. Even at order 5 there are far too many loops to do this without a computer.

It can not (yet?) derived from first principles but has to be measured (see my posting above).

So, I have two remaining questions:
1) how does one explicitly derive this from a single Feynman loop, especially what values are ascribed to the momenta carried by the loop;
2) how are the more complex numbers of loops dealt with numerically.

I've no idea to which calculations you refer to, i.e., Feynman diagrams for which processes you are talking about.

 

[Adrian59]

An example is from the Aoyama et al paper referenced in the OP. Equation 31 appears to be written from a loop diagram. Later on the paper the authors derive the 4th order coefficient with α to the power 8.

 

[vanhees71]

Yes, it's a two-loop contribution to the electron self-energy. What has this to do with a derivation of the numerical value of $\alpha$? In QED $\alpha$ is an input parameter, not something that's calculated.

 

[Vanadium 50]

Let.s step back.

The fine structure constant is the electric charge (squared). It is the one element of its definition present in all systems of units, and if you were to somehow define a sert of units without an e², in those units it would still be 4x bigger when discussing a (hypothetical) Q=+2 proton and Q=-2 electron atom.

The mathematical structure of QED permits any value of elementary charge.

Therefore, QED alone cannot predict α. No matter what you do. Sure, it is possible that some larger theory can predict it, but QED cannot. Because the mathematical structure of QED permits any value of elementary charge.

The answer tp the question posed in this thread is "no". If something does allow it to be predicted in the future, that something will not be QED.

 

[Adrian59]

Yes, it's a two-loop contribution to the electron self-energy. What has this to do with a derivation of the numerical value of α? In QED α is an input parameter, not something that's calculated.

That is my point entirely. So more specifically, what are Aoyama et al up to when they appear to start from such a loop diagram in section 4 of the referenced paper!

 

[vanhees71]

They are up to what's said in the title of the paper, i.e., to review the theory of the anomalous magnetic moment of the electron, i.e., the electron's Lande factor including higher-order loop corrections.