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

  1. Feb 15, 2016 #1
    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?
     
  2. jcsd
  3. Feb 15, 2016 #2

    mfb

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    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.
     
  4. Feb 15, 2016 #3
  5. Feb 15, 2016 #4
    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?
     
  6. Feb 17, 2016 #5
  7. Feb 17, 2016 #6
    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.
     
  8. Feb 18, 2016 #7

    mfb

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    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.
     
  9. Feb 18, 2016 #8
    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?
     
  10. Feb 18, 2016 #9

    mfb

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    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.
    The pole mass of the Z, roughly 90 GeV.
     
  11. Feb 18, 2016 #10
    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.
     
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