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so John Baez has this nice list of what he enumerates as the 26 Fundamental Universal Physical Constants and they're dimensionless, as they should be to be meaningfully fundamental. one if his grad students worked out the numerical values of those of the 26 that are known.

my question is, where is the low-energy fine-structure constant $ \alpha = \frac{e^2}{(4 \pi \epsilon_0) \hbar c} $ in all of this? i see where the high-energy $\alpha$ is related to the gauge coupling constants, $g_{U(1)}$ and $g_{SU(2)}$:

$$ \alpha(m_Z) = \frac{1}{4 \pi} \frac{g_{U(1)}^2 \cdot g_{SU(2)}^2}{g_{U(1)}^2 + g_{SU(2)}^2} = \frac{1}{128.16} $$

but that is not the familiar $ \alpha = \frac{e^2}{(4 \pi \epsilon_0) \hbar c} = (137.0359991)^{-1} $ value. can someone please explain where $\alpha$ fits into this list of 26 fundamental constants of nature? how does $\alpha(m_Z)$ get related to $\alpha$?

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  • $\begingroup$ Did you see/read the reference Black used to compile the information on $\alpha(m_Z)$? $\endgroup$ – Kyle Kanos Nov 11 '14 at 3:45
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    $\begingroup$ Have you forgotten that coupling constants run? i.e. are functions of energy? the familiar ~ 1/137 becomes ~1/128 , as measured experimentally see www2.ph.ed.ac.uk/~rhorsley/SII09-10_mqft/lec16_2.pdf for example $\endgroup$ – anna v Nov 11 '14 at 5:23
  • $\begingroup$ it's not that i've forgotten it. not being a physicist (i'm an electrical engineer into signal processing), i hadn't learned this but i understand, conceptually, that some of these numbers depend on the various energies of different particles. but i still see no relationship that relates some subset of those 26 constants to the fine-structure constant, approx $(137.036)^{-1}$, yet we understand $\alpha$ to be a universal fundamental constant. $\endgroup$ – robert bristow-johnson Nov 11 '14 at 13:35
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The fine structure constant is the electromagnetic coupling constant. In the article he says

Instead of the electromagnetic coupling constant together with the masses of the W, Z, and Higgs, we could have used 4 other constants: the U(1) coupling constant, the SU(2) coupling constant, the mass of the Higgs, and the expectation value of the Higgs field.

and all four of those are in the list. I don't see where he shows the relationship between the four he lists and $\alpha$. The linked Black paper shows the relationship.

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  • $\begingroup$ i do not see a relationship between any set of numbers in the linked Black paper and some number that comes out as approximately $(137.036)^{-1}$. what, specifically, is that relationship? $\endgroup$ – robert bristow-johnson Nov 11 '14 at 13:31
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    $\begingroup$ There is a note that $\alpha(m_Z)=\dfrac 1{4 \pi}\dfrac {g \cdot g'}{g^2+g'^2} \approx \dfrac 1{128}$ and that the unfamiliar value is because of the energy level. $\endgroup$ – Ross Millikan Nov 11 '14 at 14:12
  • $\begingroup$ i know that! again, my question is, if the 26 values in Baez's list are all of the fundamental constants, that is they are the numbers that cannot be derived from other numbers, they just are; then how is the common value of the fine-structure derived from the fundamental constants of nature? $\endgroup$ – robert bristow-johnson Nov 11 '14 at 15:27
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I think your problem starts here:

...yet we understand $\alpha$ to be a universal fundamental constant.

This isn't actually true, we do not understand $\alpha=e^2/hc$ to be a fundamental constant, instead the coupling constants are the fundamental constants.

Baez says this about the use of the gauge coupling constants instead of the fine structure constant (all emphasis mine):

I should warn you here: there are different ways of slicing the pie. Instead of the electromagnetic coupling constant together with the masses of the W, Z, and Higgs, we could have used 4 other constants: the U(1) coupling constant, the SU(2) coupling constant, the mass of the Higgs, and the expectation value of the Higgs field. These are the numbers that actually show up in the fundamental equations of the Standard Model. The idea is that the photon, the W and the Z are described by an U(1) × SU(2) gauge theory, which involves two coupling constants. The beautiful symmetry of this theory is hidden by the way it interacts with the Higgs particle. The details of this involve two further constants - the Higgs mass and the expectation value of the Higgs field - for a total of 4. If we know these 4 numbers we can calculate the numbers that are easier to measure in experiments: the masses of the W and Z, the electromagnetic coupling constant, and the mass of the Higgs. In practice, we go back backwards and use the constants that are easy to measure to determine the theoretically more basic ones

So Baez chooses to use the gauge coupling constants, rather than the fine structure constant, because U(1)$\times$SU(2) appears to be more fundamental because the electric charge is not a natural quantity in the unified theories, it's calculated from the coupling constants: $$ e=\frac{g_{U(1)}\cdot g_{SU(2)}}{\sqrt{g_{U(1)}^2+g_{SU(2)}^2}} $$ which is in Black's document, on the last page.

Black's document also shows how to obtain $\alpha(m_z)$ from $g_{U(1)}$ and $g_{SU(2)}$, which you represent in your question. Note that here, $m_z\sim90$ GeV whereas the electromagnetic coupling is at an energy of 511 keV. In this post, Lubos notes that,

The fine-structure constant is essentially constant below the mass of the electron - the lightest charged particle - which is 511,000 eV or so. You are extrapolating the running of the electromagnetic fine-structure constant $\alpha=1/137.03599$ all the way up to the Planck scale, about 10,000,000,000,000,000,000,000,000,000 eV. I chose to avoid the scientific notation to make it more explicit how far you have extrapolate

While you aren't quite running it up that far, you are trying to extrapolate up to 91,000,000,000 eV from 511,000 eV, so the principle remains: the fine-structure isn't as fundamental as you are led to believe.

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  • $\begingroup$ yes, i realize that if $\alpha$ itself is not "fundamental", then it (the $(137.036)^{-1}$ value) should be able to be calculated from the fundamental constants and i haven't yet seen such a calculation. because this $$ e = \frac{g_{U(1)}\cdot g_{SU(2)}}{\sqrt{g_{U(1)}^2+g_{SU(2)}^2}}$$ does not come out to be $$ e = \sqrt{4 \pi \alpha} = 0.30282212 $$, then i don't understand where the connection is. $\endgroup$ – robert bristow-johnson Nov 11 '14 at 18:13
  • $\begingroup$ Because, as I stated in my answer, you're way too high of an energy. The values you are using are at $m_z\sim90$ GeV whereas the fine-structure constant is $\alpha(511\,{\rm keV})$. $\endgroup$ – Kyle Kanos Nov 11 '14 at 18:21
  • $\begingroup$ Also, you are using $\alpha\sim1/137$ in that second relation when you should be using $\alpha\sim1/128$. $\endgroup$ – Kyle Kanos Nov 11 '14 at 19:45
  • $\begingroup$ i am trying to find out how, from the subset of the 26 Fundamental Physical Constants that are known depicted in the Black paper, what mathematical expression with some subset of those numbers yields $(137.036)^{-1}$. it is not $$ \frac{1}{4 \pi} \frac{g_{U(1)}^2 \cdot g_{SU(2)}^2}{g_{U(1)}^2 + g_{SU(2)}^2} $$ ... then what is it? $\endgroup$ – robert bristow-johnson Nov 12 '14 at 2:41
  • $\begingroup$ Well you'd have to determine $g(\mu)$ for energy scale $\mu$. The case of Black's paper is $\mu=m_z=91\,{\rm GeV}$; the case you want is $\mu=0$. Thus it is the renormalization group that you desire (see also the Beta function). $\endgroup$ – Kyle Kanos Nov 12 '14 at 3:59

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