Zee QFT – Part I.6

I.6.1

Putting n = 1 in (3), we get:

\displaystyle M^2_{TG} = \frac{M^2_{Pl}}{[M_{TG}R]^1}

\displaystyle M^2_{TG} = \frac{M^2_{Pl}}{M_{TG}R}

\displaystyle R = \frac{M^2_{Pl}}{M_{TG}^3}

If we use M_{TG} = 10 TeV as the large dimensions energy scale, we get:

\displaystyle R \approx \frac{(2\cdot10^{18}\,\mathrm{GeV})^2}{(10^4\,\mathrm{GeV})^3} = 4\cdot10^{24}\,\mathrm{GeV}^{-1}.

To convert from this units to standard length units we must use the speed of light (c) and the (reduced) Planck’s constant (\hbar):

\displaystyle R \approx 4\cdot10^{24}\,\mathrm{GeV}^{-1}\cdot10^{-9}\,\mathrm{GeV}\cdot\mathrm{eV}^{-1}\cdot7\cdot10^{-16}\,\mathrm{eV}\cdot \mathrm{s}\cdot3\cdot10^8\,\mathrm{m}\cdot\mathrm{s}^{-1}

\displaystyle R \approx 84\cdot10^{7}\,\mathrm{m} = 8.4\cdot10^5\,\mathrm{km},

more than twice the distance to the Moon.

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