About hierarchy problem of higgs mass

[beacon]

Hi, I have a puzzle about the hierarchy problem of higgs mass.
I don't understand why delta m is much larger than m.
If we use MS bar scheme, the huge correction can be simply dropped and the higgs mass only get a small correction.
You can also view the SM as an effecive field theory. In this way, there are infinite higher order nonrenormalizable coupling terms in the Langrangian. At low energy, these terms are suppressed by (E/Lambda)^n where E is the typical energy at low scale and Lambda is the Planck Scale. If we do the momentum integral in Higgs mass correction up to cutoff Lambda, the contributions from the higher order terms become O(1) order and are not suppressed. If we sum all these contributions, they will probably give a small correction to the Higgs mass.
So I don't see why there is a hierarchy problem.

 

[Haelfix]

It depends what you mean by the 'Hierarchy problem'

In GUT theories, the problem is already manifest at tree level, eg fine tuning of the parameters through 26 orders of magnitude.

In regular SDM, even if you fix the mass of the higgs doublet to be small at tree level, you get huge 'quadratically divergent' quantum corrections to the mass squared ~ lambda ^2. In otherwords, you have to fix the bare mass through 26 orders.

No matter how you attempt to mask it by tailoring your renormalization scheme, it will come back and haunt you in the next order. Try it to next order in MsBar if your still not convinced =)

 

[beacon]

By "Hierarchy problem" I mean exactly what you called the huge quadratically divergent mass correction of order 26.
Once talking about Lambda, you are using cutoff regularization. But if I use dimensional regularization, the term of order 26 will become
1/epsilon and is simply dropped. So there seems no need to do fine tuning in the bare mass.
Even if I calculate to the next order, still the divergence can be dropped in such a way.

 

[Haelfix]

It doesn't matter what scheme you do, since perturbation series is independent of scheme up to the next order.

Fn - Fm ~ F(n+1) for some small epsilon going to zero.

I'm staring atm at my notes from a few years ago, where I wrote this calculation out explicitly. Using Pauli-Villars regularization, its perfectly apparent from where I am sitting.

I suspect you might be doing a silly mistake, but its hard to spot in such a forum.

Remember the key thing is to substitute for the cutoff independent quantity lambda, the only natural mass scale available. Say Mgut.. To have the physical mass (squared) come out to the electroweak scale would require an extreme cancelation between m^2 and dm^2

 

[Yustas]

No, beacon is right -- there are no power divergencies in the dimensional regularization with mass-independent subtraction like MSbar. That's why it's used in effective field theory calculations (chrial PT, etc), as it does not spoil power counting by introducing unwanted mass scales.

The problem with Higgs in the SM is that its mass is not protected by any symmetry (like fermion masses -- chiral symmetry, gauge bosons -- gauge symmetries). So it can be anything, not necessarily small as required for the theory to be weakly coupled.

 

[Haelfix]

Ok!

You're right, but since we expect the SDM to break down its natural to include an appropriate physical cutoff scale (the formula I gave above assumes the same regularization procedure). If you do such a thing in dimensional regularization you're guarenteed to be left with the same problem (I imagine that is a hard and unnatural computation, but I suspect it has been done)...

Its just hidden naively b/c of the assumptions in the scheme (1/epsilon divergences etc) gobbling up the physical quantities of interest. eg Its only in pure SDM valid at all scales where the above claim is true.. there really isn't a hierarchy problem perse unless you assume *something else*.

Assume new physics at some higher energy scheme, nothing protects us from Yukawa couplings in fermion terms.

The lack of a good symetry to protect the Higgs, incidentaly motivated SUSY.