Friday, 13 April 2012

The Hierarchy Problem

I've got several drafts in progress, but none of them look like getting finished soon and I wanted to get something substantial out today.  So I thought I'd talk briefly about an important concept in theoretical particle physics, the hierarchy problem.

One question that it is worth thinking about is how we judge a model of the Universe to be more or less fundamental.  After all, if you have ever studied string theory, quantum field theory or even quantum mechanics, you will know that those theories are not simple.  Indeed, quite the opposite: more fundamental theories seem to be more complex.

Part of the answer is that the more convoluted theories are also more constrained.  For example, in quantum mechanics the thing that defines the model is the potential, which can take almost any form.  In quantum field theory, you instead have a finite number of interactions, each of which has a single number (the coupling) associated with it.1  And string theory only has one input, which is probably unmeasurable.2  What we can't do in quantum field theories is predict the values of these couplings.  We can only assume that they are set by some more fundamental theory, such as string theory, at high energies we can not yet probe.

However, it's not quite true that we have no idea about what these couplings should be.  Each coupling gets two contributions: the fundamental value, and quantum corrections.  We know how to calculate the quantum corrections to couplings.  We can be confident in this because the corrections depend on the energy scale at which we make our measurement.  While we can't predict the value of one measurement, we can predict how two would be related.  We have done this for the electric and strong nuclear interactions, and they are powerful confirmations of theory.

Now, for most couplings in the Standard Model of particle physics, quantum corrections are small.  The electric coupling goes from approximately 1/137 at low energies, to approximately 1/128 at the highest energies we have measured it.  Or consider the electron mass, another `coupling': the total correction all the way up to the Planck scale is less than 1% of the mass.3  There is one big exception, however.  The mass of the Higgs boson receives corrections proportional to the energy scale of our fundamental theory.  We know from indirect measurements that the Higgs cannot be heavier than about 1000 times the mass of the proton, and probably less.  The Planck mass is 1019 times the proton mass.

That's not actually inconsistent.  It's possible that the unpredictable, fundamental value for the Higgs mass just happens to cancel the quantum corrections to one part in ten thousand million million.

Yeah, doesn't sound likely.  This is the hierarchy, the large discrepancy between the Planck scale and the Higgs mass (electroweak scale).

There are two alternatives.  Either the scale at which we replace quantum field theory with something better is much lower than the Planck scale; or our calculation of the correction is wrong.  The only way it can be wrong is if particles we have not yet discovered modify things.  These particles are, after the Higgs, the main discovery target of the LHC; unfortunately, they are not guaranteed to be discovered.  But we are all hopeful.

1 I'm cheating slightly and assuming renormalisability.  That's a topic for another post, though.
2 Again, neglecting the huge number of dynamic parameters that are not strictly inputs, but might as well be.  That's not only the topic for another post, but one I'm currently writing.
3 The Planck scale is the energy scale at which quantum gravity must kick in, and so quantum field theory must be replaced by something more fundamental.  It is possible that this could happen at lower energies, but it cannot be deferred to higher ones.

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