I was reading these papers



They state that $m_s$ is proportional to $M_P/\sqrt{V} $ and that $m_{3/2}$ is proportional to $M_P/V$.

($m_s$:intermediated string scale, $m_{3/2}$ gravitino mass, $M_P$:Planck mass)

Then they claim that $V=10^{15} l_s^6$ is required to explain the weak/Planck hierarchy. I want to understand this statement, how is the weak scale related?

  • $\begingroup$ Here is somethig I think could help to find the answer: fuw.edu.pl/~tok/talksII/Conlon.pdf In page 17 "With $m_s = 1011GeV$, the weak hierarchy can be naturallygenerated through TeV-scale supersymmetry." $\endgroup$ – Anthonny Feb 5 '14 at 20:08
  • $\begingroup$ Sorry, the correct link is physics.rutgers.edu/het/video/conlon.pdf but the otherone seems useful also, there is a section titled weak scale. $\endgroup$ – Anthonny Feb 5 '14 at 20:41

If the supersymetry breaking scale is close to the weak scale, the problematic renormalization parameters in the scalar sector of the Standard Model are naturally limited. In supersymemtry (when all the fields are dynamic) there are powerful non-renormalization theorems. Therefore, perturbative effects affect the scalar sector only below the supersymetry breking scale $M_S$. They are therefore of order $$ \delta \propto \frac{1}{16 \pi^2} \left( \frac{M_S^2}{M_\mathrm{ew}^2}\right) $$ where $M_\mathrm{ew}$ is the electroweak scale. By having $M_s \sim 1$ TeV, $\delta \sim 1$ and the small values of parameters in the scalar sector are protected by supersymmetry.

Should the susy breaking scale (or generally the scale of new physics) be large i.e. comparable to the Planck scale, the corrections are $$ \delta \propto \frac{1}{16 \pi^2} \left( \frac{M_\mathrm{Pl}^2}{M_\mathrm{ew}^2} \right) \gg 1 $$ Then, there is no natural reason, why the parameters at the weak scale should be so small, if their value gets huge renormalization contributions.

A simile for this is investing one Billion dollars into a company with expected gains of one Billion and one dollar. You invest a lot to get a tiny value out. A miniscule deviation completely throws your theory off-balance.

One problem in this however is, that there is no specific reason why the SUSY breaking scale should be close to the weak scale. It would be nice as an explanation for the small parameters, but we would stil have to explain the value of the "small hierarchy" between the weak and SUSY breaking scales.

Therefore, mechanisms that create a small SUSY breaking scale from e.g. the Planck scale (which is waaay up there) are intersting to look at. The volume of extra dimensions reducing the SUSY breaking scale is one of such mechanisms.


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