the hierarchy problem and naturalness
Imagine adding up a long list of huge positive and negative numbers — billions and trillions in size — and finding that the total comes out to a small, modest number like 100. You would suspect that is no accident: either you made an error, or the numbers were secretly arranged to cancel almost perfectly. The hierarchy problem is exactly this kind of suspicion, applied to the mass of the Higgs boson. The Higgs mass we measure is a comfortable 125 GeV, but the calculation that should produce it is a tug-of-war between enormous quantities, and they must cancel to staggering precision to leave such a small remainder.
Here is the physics behind it. In quantum field theory, a particle's mass gets contributions from its interactions with all the quantum fluctuations of every field it touches. For most particles, symmetries protect the mass from receiving giant corrections. The Higgs, being a spin-zero particle, has no such protection: its mass receives quantum corrections that grow with the highest energy scale where new physics might enter — potentially the scale of gravity, which is something like ten-thousand-trillion times the Higgs mass. For the final answer to land at a modest 125 GeV, the various huge contributions would have to cancel to roughly thirty decimal places. Nothing in the theory forces this cancellation; it just has to happen. That uncomfortable coincidence is the hierarchy problem.
Naturalness is the guiding intuition that such an extreme, unexplained cancellation is a sign that something is missing — that a deeper theory should make the small Higgs mass automatic rather than a fluke. This intuition motivated many proposed extensions of the Standard Model (supersymmetry being the most famous) that would tame the corrections. The honest state of affairs, and an important caveat: despite extensive searches, no such new physics has been found, which has led some physicists to question whether naturalness is the right guide at all. The hierarchy problem is real and unsolved, but it is a problem of aesthetics and expectation, not a contradiction with any experiment — the Standard Model works fine; it just looks finely tuned.
It is as if you measured a mountain's height as the difference between two distances each thousands of kilometers long, and the difference came out to be exactly one meter. Possible, but it cries out for an explanation. For the Higgs mass, the required cancellation is far more extreme — about one part in 10^30.
A modest leftover from a clash of giants — possible, but suspicious.
The hierarchy problem is not an experimental contradiction — the Standard Model still works. It is a problem of naturalness (it looks improbably fine-tuned), and the lack of any new physics found at the LHC has made some doubt whether naturalness is the right expectation.