What is the Hierarchy Problem & How Can it be Solved?

[jdstokes]

Hi,

In my research I've heard of the Hierarchy problem expressed under at least two different guises:

1) Hugeness of the Planck mass compared to the electroweak/Higgs mass, resolved by e.g., warped compactifications.

2) Some kind of fine-tuning of the electroweak scale to magically offset quantum corrections, resolved by e.g., supersymmetry.

I'm clear aobut the first issue but very vague about the second. Can anyone suggest any references to help me understand what the actual problem is, and how approaches such as supersymmetry can resolve it?

 

[jdstokes]

On a related note. I would also like to know how many accepted Hierarchies are known to exist in the Standard Model, and which if any, are related. Off the top of my head I can think of the following (unrelated?) hierarchies

- Planck/Higgs mass hierarchy.
- Gauge coupling hierarchy between the electroweak and strong sectors.
- If one thinks of gravity as a gauge theory, is there also a hierarchy between the gravitational coupling constant and the SU(3) x SU(2) x U(1) gauge couplings?
- Fermion mass hierarchies, e.g., between up and top quarks, electron and electron neutrino.

Any others?

 

[nrqed]

Hi,

In my research I've heard of the Hierarchy problem expressed under at least two different guises:

1) Hugeness of the Planck mass compared to the electroweak/Higgs mass, resolved by e.g., warped compactifications.

2) Some kind of fine-tuning of the electroweak scale to magically offset quantum corrections, resolved by e.g., supersymmetry.

I'm clear aobut the first issue but very vague about the second. Can anyone suggest any references to help me understand what the actual problem is, and how approaches such as supersymmetry can resolve it?

The connection is this. If we compute the one-loop correction to a scalar particle like the Higgs, we find a quadratic divergence (as opposed to the usual logarithmic divergences.). This means that to get a "low" mass (relative to the Planck mass which is, presumably, the natural scale for the cutoff) one needs a fine tuning to an extraordinary precision. Logarithmic divergences do not require such a high level of fine tuning since a log grows so slowly.

Supersymmetry takes care of this because the quadratic divergences introduced by the scalar loops are canceled by the quadratic divergences produced by fermion loops. There rae no quadratic divergences at all in SUSY theories. In fact, almost all SUSY calculations are finite. There is only one class of logarithmically divergent graphs that are present and these can all be taken care of by a wavefunction renormalization.

Hope this helps

 

[jdstokes]

Thanks for the explanation. That helps a lot.

Unfortunately I haven't done QFT calculations above tree level. So I'm quite ignorant about ultra-violet divergences. What would you say is a good introduction to the topic, preferably with mention of the Hierarchy problem?

 

[nrqed]

Thanks for the explanation. That helps a lot.

Unfortunately I haven't done QFT calculations above tree level. So I'm quite ignorant about ultra-violet divergences. What would you say is a good introduction to the topic, preferably with mention of the Hierarchy problem?

You are welcome!

The most pedagogical example showing explicit cancellations of quadratic divergences in SUSY can be found in Aitchison's book on supersymmetry, pages 77-87. The calculation is done in such detail that everything will be clear even if you haven't done loops before. He discusses the hierarchy problem in the first chapter and at other points in the book. This is an excellent book to learn about susy, by the way.