# Why is the hierarchy problem a problem?

1. Nov 14, 2015

### Smattering

As I am not sure what is the most appropriate forum for this question, I am posting it here:

In another thread I came across a link to the Wikipedia article on the hierarchy problem:

https://en.wikipedia.org/wiki/Hierarchy_problem

Unfortunately, after reading the article several times, I am still not sure what the core of the problem actually is, and why it is a problem.

For one thing, there seem to be several different definitions of the problem that might be equivalent, but apparently I am lacking the required knowledge to understand why:

1. Why is the weak force $10^{32}$ times stronger than gravity?
2. Why is the fundamental value of some physical parameter vastly different from its effective value after renormalization?
3. Why is the Higgs boson so much lighter than the Planck mass?

Regarding 1: Why is this considered a problem?
Regarding 2: Renormalization is necessary in QFT, right? So to what QFT is this actually referring?
Regarding 3: What has the Planck mass to do with this?

Can someone help me to get a better idea what this is all about?

Best regards,
Robert

2. Nov 20, 2015

### Buzz Bloom

Hi Smattering:

I have been interested to see what the experts would say about this, so I am as disappointed as you are likely to be from getting no answers. I will give some thoughts about the questions in the hope that my foolish ideas might provoke a smart answer.

In general, "WHY" questions about physics are frequently unanswerable. To go along with the "Hierarchy Problem" how about the following:
1. Why is EM force about 1036 times stronger than gravity?
2. Why is EM force about 104 times stronger than the weak force?
3. Why doesn't either (1) or (2) deserve a problem name like the "Hierarchy Problem"? That is, why is the "Hierarchy Problem" more of a problem than (1) or (2)?

BTW, I did not understand your
Regarding 2: Renormalization is necessary in QFT, right? So to what QFT is this actually referring?​

Regards,
Buzz

3. Nov 20, 2015

### Smattering

I agree that why questions tend to be unphysical when referring to why nature behaves in a certain way. But in this case the why question does not refer to nature, but rather to the physicists who feel that there is a hierarchy problem. Physicists are people, and unlike nature, people are supposed to have motives.

Edit: Initially, I thought you were referring to my own why question from the thread title. But after re-reading your post, I now think that you were referring to the definition of the hierarchy problem which is itself a why question. Regarding this, I agree. I neither understand how a why question about the values of some natural values can even be physically meaningful.

Last edited: Nov 20, 2015
4. Nov 20, 2015

### Staff: Mentor

Maybe this will get more responses now that it's been moved to the particle physics forum.

5. Nov 20, 2015

### Smattering

Hi Buzz,

According to the Wikipedia article there is some physical parameter that has a fundamental value that is several magnitudes higher than its effective value after renormalization. I thought that this parameter must have something to do with gravitation, but renormalization is closely related to quantum field theory, and there is no generally accepted QFT of gravity.

Last edited: Nov 20, 2015
6. Nov 20, 2015

### Buzz Bloom

Hi Smattering:

Thanks for your answer. I confess my too quick look carelessly missed that discusion of renormalization in the Wikipedia article.

Now that I actually tried to read the article, I found it way over my head, especially
such quantum corrections are usually power-law divergent, which means that the shortest-distance physics are most important.​
I think I now get that the "Hierarchy Problem" is a problem because
Typically the renormalized value of parameters are close to their fundamental values​
and for the weak-gravity ratio, this is not the case, and apparently no one has an acceptable explanation for this anomoly.

Regards,
Buzz

7. Nov 23, 2015

### Staff: Mentor

Unfortunately that doesn't create an alert.

All those hierarchy problems are "just" things that look odd. Sure, a parameter can be exactly 1.000000000000000000000344, and the theory works, but without a deeper theory that predicts this value it looks odd. If the parameter does not have to be 1, and can be anything, why is it so close to 1 but not exactly 1? It is expected that some "more fundamental" theory will lead to some explanation of factors like that.

8. Nov 23, 2015

### Smattering

Hm ... but this sounds a bit like the layman's argument that the chance of winning a lottery with numbers "1 2 3 4 5 6" is less likely than winning it with more random looking numbers.

Having spent quite some time on statistical pattern recognition in university, I can certainly understand that fine tuning can be a problem due to the risk of overfitting your model on the existing observations such that it will not generalize well on new observations. But fine tuning (as I understand the term) does not refer to parameter values differing in magnitude. Rather it means that very small changes to a parameter's value lead to huge differences in result.

9. Nov 23, 2015

### Staff: Mentor

It is not, but if there is only one drawing ever and it gives 1 2 3 4 5 6 in that order, it is still a surprising result. It makes you wonder if the drawing was truly random or if someone simply coded "give me the smallest number not yet drawn" and ran that to generate the numbers.

That is related to the hierarchy problem(s). Changing the parameter 1.000000000000000000000344 to 1.000000000000000000000484 (made-up numbers) could have a huge effect.

10. Nov 23, 2015

### Smattering

Having *your* numbers drawn in a lottery is always a surprising result, isn't it? If you told me to bet on "23 41 17 34 3 8", I would find it equally suprising if these numbers were drawn in exactly the sequence you predicted.

I can understand why this would be an issue. But the Wikipedia article implied to me that the hierarchy problem is not so much about the fine tuning of single parameters, but rather the differing value scales of two or more parameters.

Last edited: Nov 23, 2015
11. Nov 23, 2015

### Staff: Mentor

If you are the one running the lottery (something that makes you unique - we have only one universe to observe) it would be surprising if you win your own lottery, independently of the numbers. Sure, it can happen by chance, but manipulation is certainly a relevant alternative hypothesis.

Those concepts are related. If a mass value can be anything from 0 to the Planck scale, and has to be subtracted from a value that should be around the Planck scale, it is surprising if the difference is orders of magnitude below the Planck scale. That's the factor 1.00000000000000000000463 I mentioned earlier (again, random digits).

12. Nov 23, 2015

### Smattering

Sorry, but I do not understand what you are referring to. Can you please explain this in more detail?

13. Nov 23, 2015

### Staff: Mentor

The Higgs mass is the sum (or difference, depending on sign conventions) of two unrelated terms:
- its bare mass, which can take any value
- radiative corrections, which (in the absence of new physics below the Planck scale) should be of the order of the Planck mass

The Higgs is 17 orders of magnitude lighter than the Planck mass, so in the Standard Model the two terms have to be very close together to create such a huge difference between Planck scale and Higgs mass.

Supersymmetry and other models lead to smaller radiative corrections, so the necessary amount of fine-tuning goes down.

14. Nov 23, 2015

Staff Emeritus
Maybe it's worth putting some numbers in (courtesy of Michael Dine): m(H)2 = 36,127,890,984,789,307,394,520,932,878,928,933,023 - 36,127,890,984,789,307,394,520,932,878,928,917,398 GeV2.

It is entirely possible that the two numbers come from completely unrelated processes and their closeness is purely coincidental. Just like it's possible to walk into a room and find all the pencils are pefectly balanced on their points. But does that seem likely to you?

15. Nov 23, 2015

### ohwilleke

I have to say that I am not at all impressed with the presumptuous premise of the hierarchy problem and a number of other "problems" of modern physics such as the problem of matter-antimatter asymmetry in the universe, the problem that the cosmological constant has the value that it does, and the "problem" that the strong force Lagrangian doesn't have a CP violating term even though a generalized version of the equation has a very obvious place to put one. Nature is what it is and there is no particular reason that its fundamental constants should have any particular value, which is what "fundamental" means.

If a physical constant value present in Nature looks unnatural, in my mind, this is evidence that your looking at the situation in the wrong way. But, it isn't necessarily a hint that you need to devise news laws of Nature that make physical constant values seem "natural" by hand.

Supersymmetry is a particularly brute force solution to the hierarchy problem that could probably be answered with additional laws of Nature (e.g. the sum of the square of the masses of the fundamental fermions equals the sum of the square of the masses of the fundamental bosons, which is true experimentally to within all applicable margins of error) that are more subtle and do not require a host of new particles that have not been observed.

16. Nov 23, 2015

### nikkkom

I tried to find an example from the history of physics where an apparent "mysterious" fine tuning of free parameters of a theory was resolved by a newer theory, but I'm not that good with history of science. Anyone?

17. Nov 24, 2015

### haushofer

18. Nov 24, 2015

### Staff: Mentor

Not physics, but evolution explained why so many different species exist, all "fine-tuned" to their specific environment, and all sortable into groups of very similar species.

If you think of atomic energy levels as independent free parameters, then quantum mechanics explained their relation (e. g. 1/n^2 for hydrogen-like atoms).
We don't see them as independent parameters today as we found a theory predicting fixed relations between them.

The orbits of the planets all follow Kepler's law - could look like fine-tuning, but Newton's theory of gravity gave a simple explanation for it.

19. Nov 24, 2015

### nikkkom

This is not quite a type of example I was looking for. There was no "deferent and epicycle" atomic theory which was predicting atomic energy levels, before we've got our current one.

20. Nov 24, 2015

### Smattering

But the explanation that evolution can offer here is not any sophisticated mechanism, but simply selection bias.