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BACKGROUND

So far I understood that the hierarchy problem was the large difference between the gravitational scale, $M_{pl}\sim 10^{18}\; [GeV]$, compared with the electroweak scale, $M_{ew}\sim 10^3\;[GeV]$.

However, I heard that the hierarchy problem is due to the existence of quadratic divergences in the scalar sector of the Standard Model.

QUESTION

Can someone explain with ease the hierarchy problem?

Additionally, Is it possible to relate both of the above points of view?

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    $\begingroup$ Possible duplicate physics.stackexchange.com/q/43303 $\endgroup$ – DJBunk Nov 6 '12 at 19:43
  • $\begingroup$ @DJBunk: Thank you for pointing out that post. There exist an overlap between the questions! Cheers $\endgroup$ – Dox Nov 7 '12 at 13:38
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The hierarchy problem is not only about big numbers, such as $M_{pl}/M_{EW}$, per se. In fact in QCD there is no hierarchy problem associated to the ratio $M_{pl}/\Lambda_{QCD}$.

The problem is actually about the quantum numbers of certain operators in a Wilsonian EFT. The point is that we understand the SM as an effective low-energy description of the dynamics associated to relatively light degrees of freedom. Because of QM, the heavy degrees of freedom that one has integrated out actually leak into the effective description by changing the couplings of the local operators of the EFT.

It's pretty simple, by means of dimensional analysis, to see which operators are strongly affected by the UV degrees of freedom that live at the scale $\Lambda$ or above:

$\delta\mathcal{L}=\sum_{\mathcal{O}}c_\mathcal{O}\Lambda^{4-\Delta_{\mathcal{O}}}\mathcal{O}$,

where $\Delta_{\mathcal{O}}$ is the scaling dimension of the operator $\mathcal{O}$. It's thus clear that relevant operators, i.e. with $\Delta<4$, are very sensitive to the scale of UV physics. Marginal ($\Delta=4$) or almost marginal ($\Delta\simeq 4$) are pretty much insensitive to the scale of UV physics whereas irrelevant operators ($\Delta>4$) are suppressed by large powers of $\Lambda$.

Notice that in the SM the smallness of neutrino masses, and the conservation of B and L quantum numbers, follow from the irrelevance of the operators associated with those operations.

However, in the SM, the operator $|H|^2$ is relevant and one would expect its coefficient to scale with $\Lambda^2$: a light Higgs and a hierarchically small vev (especially if compared to $\Lambda\sim M_{pl}$), are hard to accommodate without finely tuning some cancellation in the UV.

One could solve the hierarchy problem by introducing new degrees of freedom that enforce such a cancellation as a symmetry requirement (rather than accidentally) which would suppress the couplings of the relevant operator. For example: Supersymmetry (SUSY).

By the way, QCD doesn't have the hierarchy problem (apart from the strong CP-problem...). This is true for two reasons:

1) the QCD gauge coupling is almost marginal so that the actual scale $\Lambda_{QCD}$ is generated only when the coupling has run for a very long time (that is, for very a large energy range) to get into a strong coupling regime which allows for strong bound states to form; and

2) there are no relevant operators that are not forbidden by symmetry.

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The great explainer, Richard Feynman, touched on this issue briefly in lay man's terms in his messenger lectures series, lecture 1, around the 48:20 time mark.

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