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Is the quadratic divergence of the Higgs mass really bad?

  1. Oct 4, 2008 #1
    I just happened to read two papers that pretend that the quadratic divergence of the Higgs mass is not a problem.

    The first is "Vacuum energy: Quantum Hydrodynamics vs Quantum Gravity" http://arxiv.org/abs/gr-qc/0505104 (Update: this is now the correct paper from arxiv) where Volovik says that the quadratic divergences has the same origin as the quartic divergence of the vacuum energy. He says, since the vacuum energy is experimentally zero, the quartic divergence argument is wrong, as thus also the quadratic argument about the Higgs mass. Volovik concludes that the quadratic divergence of the Higgs mass needs simply to be ignored.

    The second is "On Naturalness of Scalar Fields and Standard Model" http://arxiv.org/abs/0712.0402, published in PRD, which makes a similar claim. It says in its conclusions: "On the other hand, if some unknown mechanism provides for small mass of scalar particles, perturbation theory is quite able to explain relative stability of the scalar mass against small variations in fundamental parameters. We demonstrated that there is no fine tuning problem in the theory of quantum scalar field, ... "

    Does this mean that the quadratic divergence issue is not as bad as has been said for the last 30 years?

    Last edited: Oct 5, 2008
  2. jcsd
  3. Oct 6, 2008 #2


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    This certainly sounds wrong, but I don't have the time right now to read these papers more carefully. But let me say a few words about the hierarchy problem and quadratic divergences:

    First of all, if you really have a thing about quadratic divergences, why not just use dimensional regularization?! There, no quadratic divergences, and everything is done. Divergences are not physical (there are no infinite masses!) so what's the big deal?!

    The problem is more serious than quadratic (or quartic!) divergences. The problem is UV sensitivity. If you set a coupling to zero, quantum corrections will regenerate it with the size of the largest masses available. So for example: the Higgs mass is not a hierarchy problem IF we assume that there is no physics above the weak scale. This is ludicrous, since we already know for a FACT that there is new physics at the Planck scale (whatever Quantum Gravity is), so we EXPECT that the mass of the higgs scalar should be the Planck mass! The fact that it's not means that (1) there is some symmetry that causes these things to cancel, or (2) there is a God who is tuning knobs to cancel things just right to make things work. Needless to say, we physicists are not fond of (2), so this leads us to consider things like supersymmetry, technicolor, little higgs, extra dimensions, etc.

    The same argument goes for the cosmological constant: these power divergences are sensitive to the highest scales of the theory, and for us, that's the Planck scale. THAT is why there is a problem, not because there is a divergence. QFT has divergences galore! It's what the divergences are telling you that matters, not the fact that they are there.

    Your summaries of these two papers make them sound vacuous. In the first paper, saying that the EXPERIMENTAL c.c. is zero means there's no problem is crazy: we calculate the c.c. to be HUGE, so it sounds like it's a GIGANTIC problem! To the second statement: "if some unknown mechanism provides for small mass of scalar particles" - then we have solved the hierarchy problem! This is the WHOLE POINT!!
  4. Oct 7, 2008 #3
    Thank you for the feedback!

    Can I try to summarize? Experiment says that the cosmological constant (or the Higgs mass) is small, and theory says it is about 10^120 times larger (10^17 times for the Higgs mass). It is obvious to everybody that the theory is wrong. There are two options: (1) the theory needs to be modified (with susy, technicolor, etc.) or (2) the calculation was done incorrectly.

    Volovik seems to argue for point (2), whereas most people, like you explain, argue for option (1).

    Could there be any way in which the calculation is wrong? Integrating up to Planck scale is, after all, a tricky thing to do: we do not know what happens there. There might be non-perturbative effects that make the problem disappear.

    I once read that Dirac, just before he died, wanted to change QED because the calculations yielded infinities. Everybody now says that the calculation is ok, it was simply wrongly interpreted. Are we repeating Dirac's mistake, asking to change the theory whereas in fact we only need to be careful with the calculation?

  5. Oct 7, 2008 #4


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    The problem is the following with regards to finetuning.

    Lets say you have a difference D between two quantities A and B such that
    D = A - B

    Lets also say that both A and B are 'expected', to be both on the order of some scale S

    Naively you would then expect D to be a number like 1 or 2, or 3.0036 or something like that (assuming you normalize your units that describe S to unity).

    Instead, what we measure is more like D = .000000000000000000...4234.

    In other words, A and B conspire to cancel to fantastic accuracy. Such a numerical coincidence is very hard to believe, even though its possible in principle. It begs the question, maybe there is some mathematical reason why they are so close, but not identically the same. Such a mathematical reason does not exist in the standard model (or at least no one can see it), but it can exist in extensions (for instance supersymmetry makes such a cancellation natural in the case of the Higgs mass)

    Otoh, the calculation which is done in the standard model for both the CC and the hierarchy problem is very robust and actually rather simple, so I'd be very skeptical about any paper claiming otherwise.
  6. Oct 7, 2008 #5


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    But don't you see: this is PRECISELY what (1) is suggesting: that there is some new physics at or slightly above the weak scale, that make the problem go away (SUSY, etc)! I agree entirely that it shouldn't be that we can take our theory up to the Planck scale unmodified, and that such a thing is very naive. But that's why so many physicists believe in physics beyond the SM!

    Put another way: IF we assume that there is nothing above the weak scale except gravity, then there IS a hierarchy problem - this is a robust statement, as Haelfix says. But you have it right that assuming there's nothing above the weak scale is very naive and unbelievable. The question is: what is it?

    I'm not sure what you mean by this. We now understand what these infinities are telling us, and they're telling us that there is something new we haven't seen yet. So now we're out to look for it.
  7. Oct 7, 2008 #6
    I see. So in fact the problem is where the mass of the particles comes from. If one had a solution for that, it could still be that the standard model were valid up to Planck energies (even though this is naive and unbelievable).

  8. Oct 8, 2008 #7


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    The real problem is the **Higgs** mass. If we understood THAT, then we'd be in great shape! :wink:
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