At what scale is the charge of an electron -2e?

[utesfan100]

The charge of an electron is -e in energy scales well into the atomic scale. At infinitesimal scales it becomes infinite. This relation must be continuous for re-normalization to work, thus the intermediate value theorem asserts that it attains all values between at some energy level. I want to determine the scale at which the charge is observed to be -2e.

This should only involve a few highest order terms. Where can I find the highest order perturbation terms for the charge of an electron as the energy scale increases/length scale decreases?

 

[mfb]

At infinitesimal scales it becomes infinite.

Where did you read that, and what does that mean?

The coupling strength, not the charge, is scale-dependent, but it does not get infinite below the Planck scale. And we know our physics doesn't work beyond that.

 

[utesfan100]

https://www.physicsforums.com/threads/infinite-charged-electron-in-qft.477144/

 

[utesfan100]

To contextualize your answer I am picturing something like

bare charge = coupling strength * e

in the limit where the length scale goes to infinity.

At what scale, then, is the coupling strength 2?

 

[nikkkom]

You can calculate it yourself.

https://www.ippp.dur.ac.uk/~krauss/Lectures/QuarksLeptons/QCD/AsymptoticFreedom_1.html

On that page, there is a formula for fine structure constant (alpha) running with energy scale (Q). At Q=0, alpha=1/137. At Q~=90 GeV, alpha=1/128. Find a Q at which alpha = 2/137. Then convert Q to distance scale.

 

[utesfan100]

Thank you for the link, but I am having difficulties using the formula provided. When I plug the values you give I don't get the values you give. The formula given is:

α1=α/[1-α/(3π)*log(Q^2/me^2)]

In particular, the log of 0 diverges to -infinity, so at 0 the formula goes to positive 0. That said, it gives a value of 1/138.3 at the energy scale of the CMBR, so it diverges very slowly, and exactly 1/137 at the energy scale of the rest mass of an electron times c^2.

At Q=90GeV≈180,000*me*c^2, I don't get 1/128.

log(180,000^2)=10.51
a/(3π)*10.51=0.00814
1-0.00814=0.99186
1/137/0.99186=1/135.9 ≠ 1/128

Even interpreting the log as ln, as some web references occasionally use, only gets me to 1/134.4.

 

[mfb]

The 128 looks like an error. Compare it with this plot, which agrees with 1/134.4=0.00744, but is clearly inconsistent with 1/128=0.00781.

Natural logarithm.

 

[utesfan100]

Thank you. So then, solving for Q in terms of α1/α I get:

Q=me*e^[3π/α*(α1/α-1)]≈me*e^[645.6*(α1/α-1)]

For α1/α=2 I get 2E+280me. This appears to answer my question. :)

Before leaving I have two quick follow ups.
1) This is significantly larger than the plank energy. Would I be wrong to think that higher order terms certainly appear before then?
2) What is a Z-pole?

 

[mfb]

1) This is significantly larger than the plank energy. Would I be wrong to think that higher order terms certainly appear before then?

They should follow the square, cube, ... of α/(3π)*log(Q^2/me^2) with some different numerical prefactor. Below the Planck scale, this term is much smaller than one, so higher orders should be smaller.

2) What is a Z-pole?

The pole mass of the Z, roughly 90 GeV.

 

[nikkkom]

Would I be wrong to think that higher order terms certainly appear before then?

QED essentially stops being a useful description of the interaction when energy scale gets significantly larger than Higgs vacuum energy. At those scales, SU(2)xU(1) weak isospin/weak hypercharge is a better description, and you need to concern yourself with their constants and their running, not fine structure constant's running.

Your question, thus, was a theoretical one, about the imaginary Universe where QED is the actual interaction, not a low-energy limit of weak force.