# Why is the hierarchy problem a problem?

Smattering
Both dark energy (another fine tuning problem) and the absence so far of new physics beyond the HIggs boson at the LHC are prompting some soul searching about whether naturalness is really a reliable criterion for evaluating success in physical theories. Could Wilson have steered us wrong?"

Is naturalness anything else than a pure aesthetic argument? Why should we expect nature to be elegant?

Is naturalness anything else than a pure aesthetic argument?

No, it is not. We have examples of theories with fine tuning being superseded by theories without one.
Give me a case where the opposite happened, if you know one.

Is naturalness anything else than a pure aesthetic argument? Why should we expect nature to be elegant?

There is some aesthetics to it, but not the one you wrongly believe motivates it. The aesthetics is that we assume that our theory is not the final theory. Within that framework, naturalness can be technically phrased. See slide 8 of http://www.slac.stanford.edu/econf/C040802/lec_notes/Lykken/Lykken_web.pdf.

Mentor
Well, we know that the Standard Model is not complete. It does not include gravity, its options to account for dark matter are at best questionable, it tells us nothing about dark energy or inflation, and even if we ignore gravity we would have the Landau pole as problem at even higher energies.

atyy
Gold Member
One example of concerns about fine tuning leading to fruitful scientific theories would be the anomalous magnetic dipole moment of the electron aka "g-2" (i.e. why is the magnetic dipole moment of the electron, "g", not exactly 2, but instead, some tiny but very exactly measured small amount greater than two).

It turns out that this slight discrepancy arises in QED from interactions with virtual photons, and that if your theory doesn't allow for virtual photons (and other odd assumptions of path integrals like inclusion of photon paths at slightly more and slight less than the speed of light "c" even though those paths are highly suppressed) that you get an answer different from the physical one. The notion would be that fine tuning if it is observed must exist because we are missing something of the same sort of mathematical character as the inclusion of virtual photons in our theory which is why our expectations are so off. The search for why g-2 was fine tuned produced theoretical progress. Now, I can't say that the intellectual history of that discovery really establishes that fine tuning was the insight that really made the difference in figuring out that virtual loops needed to be considered in QED (and the rest of the Standard Model as well), but it is a historical example that captures the notion.

haushofer and mfb
Anomalous magnetic moment in QED is not an example of fine tuning.

An example of fine tuning would be a theory where two large free parameters interact (subtracted, divided, etc) to give a vastly smaller number.

Before QED, no theory at all explained electron's anomalous magnetic moment. (I'm not sure we even had a theory or any kind which was predicting the value of electron's magnetic moment, anomalous or not).

atyy
Mentor
Dirac's relativistic quantum mechanics "predicted" a value of exactly 2. If you modify the theory to get a free parameter, this parameter seems to be fine-tuned to nearly agree with Dirac's prediction.

Staff Emeritus
2021 Award
I don't think the magnetic moment is a good example of fine tuning. As said above, it does not arise by a near cancellation. The "if you replace it with a free parameter" argument can be applied to absolutely every constant and every measurement, so it provides no real insight.

atyy
A classic example would be the self energy of the electron in classical EM.
$$M_{observed}=M_{bare}+\frac{e^{2}}{4\pi \epsilon r_{e}}$$
Experiment sets a limit on the the size r so that the self energy was greater than 10 GeV and thus the first bare term must be chosen to cancel the second self energy term with a finetuning at about the O(.001) level
.511Mev=-9999.489+10000.000 Mev
Of course quantum mechanics comes to the rescue to 'explain' this finetuning, by adding the positron and the related quantum electrodynamic symmetry. This sort of picture is sort of the conceptual equivalent to explaining the Higgs mass by something like Technicolor.

$$\kappa [\gamma_{\mu},\gamma_{\nu}]F^{\mu \nu} \psi$$
can be added to the Dirac equation for arbitrary $\kappa$, which makes the magnetic moment an adjustable parameter while satisfying all necessary symmetries. It's actually Wilson who saves us here: this term is non-renormalizable, so it's irrelevant at low energies, giving us Dirac's universal result. It's not a near-cancellation scenario like the cosmological constant or Higgs mass, but Dirac did "fine-tune" $\kappa$ to zero.