A problem of hierarchy

One of the many puzzles (a.k.a. Mysteries of Life) faced by modern theoretical physics is the so-called hierarchy problem: when one compares [1] the relative strength of the four fundamental forces, two widely separated scales are evident:

Interaction Coupling constant
Strong 1
Electromagnetic 1/137
Weak 1/10^6
Gravitational 1/10^39

Or, as Lisa Randall puts it in this interview:

The gist of it is that the universe seems to have two entirely different mass scales, and we don’t understand why they are so different. There’s what’s called the Planck scale, which is associated with gravitational interactions. It’s a huge mass scale, but because gravitational forces are proportional to one over the mass squared, that means gravity is a very weak interaction. In units of GeV, which is how we measure masses, the Planck scale is 10 to the 19th GeV. Then there’s the electroweak scale, which sets the masses for the W and Z bosons. These are particles that are similar to the photons of electromagnetism and which we have observed and studied well. They have a mass of about 100 GeV. So the hierarchy problem, in its simplest manifestation, is how can you have these particles be so light when the other scale is so big.

As you probably know, Randall’s response to this conundrum implies a long detour through multiple dimensions, as recently reviewed over at Backreaction, which was predated by a proposal by Arkani-Hamed, Dimopoulos and Dvali, nicely explained for the rest of us in this Physics Today article. (As a warmup for higher-dimensional physics, you may find entertaining this recent pedagogical review of Kaluza-Klein theories.)

An alternative solution has been put forward by the supersymmetry proponents. As explained (hyped?) in this beatiful review of particle physics:

According to supersymmetry, every “ordinary” particle has a companion particle — differing in spin by half a unit, but with otherwise identical properties. Furthermore, the strengths of the interactions of the superpartners are identical to those of the corresponding ordinary particle. Supersymmetry so simplifies the mathematics of quantum field theory and String Theory that it allows theoriests to obtain solutions that would otherwise be far beyond their calculating ability.

For reasons too complex to explain here (even if i really understood them: see here and here for some of the nitty-gritty details), supersymmetry is claimed to lead to a unification of fundamental forces at very high energies (some 10^28K, or 10^{-39} seconds after the Big Bang), somehow making natural the wild differences in scale of the (energy-dependent) coupling constants in our current universe. As mentioned, the theory also makes easier to define renormalizable QFTs (due to some magical cancellations), and has become one of the main ingredients of String theory, although there is at least another extension to the standard model of particle physics that seem to share these magic cancellation virtues, solving in the process the hierarchy problem: this Physics World article gives an introduction to this so-called ‘little Higgs’ theory.

Personally, i find all these untestable super-theories and multiple dimensions rather unconvincing, and would prefer some old good 4-dimensional solution. Alas, no one seems to be avaialable… maybe it’s time to embrace Compactified Dementia and be done with that.


[1] The excellent little comparison of coupling constants for the fundamental forces pointed to by the above link is part of a nifty site called HyperPhysics, an amusing experiment combining HyperCards, Javascript experiments and similar online tricks with well written contents. Visit it for fun, hierarchy problem or not.

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8 Responses to “A problem of hierarchy”

  1. Luis Sanchez Says:

    Actually I initially shared your opinion that all this multidimensional hoopla was out of control (remember the heterotic string with it chiral asymmetry implying the excitations of the string had 26 dimensions counter clockwise and 10 clockwise).

    Despite I would also like a 4 dimensional solution, not only because of the obvious fact we only observe 4 dimensions but also because:

    1. In “classical” quantum field theory you only get a meaningful (renormalizable) theory with d less or equal to 4.

    2. Hadrons (like the proton) are bound states of quarks and gluons, but it is not possible to confine quarks and gluons for more than 4 dimensions. We have loads of experimental evidence showing that confinement is an essential part of any theory describing hadrons. It should be noted that calculations of confinement in QCD are amazingly difficult (the pertubative approach fails) but evidence from lattice QCD has convinced mos of us that confinement is indeed a part of QCD.

    3. Getting D=4 out of the M Theory D=11 (or D=10 of a strings-only theory) requires compactification (of course the Randall Sundrum models are the alternative) which ruins all the predictive power of the theory as there are an infinity of ways to do it.

    Anyway it should be said that there is (a rather small chance) to test one of the Randall Sundrum models (the one with an additional brane) because a graviton might actually go to that additional brane which will produce apparent violations of conservation of energy.

    There is an additional way to test the RS models, they predict a very peculiar pattern for gravitational waves. The current experiments (mainly LIGO) are unable to detect this pattern (LIGO can only detect gravitational radiation from really violent events like black hole mergers, and even that hasn’t been achieved yet), but there is a chance that PLANCK can detect gravitational waves in the CMB, I dont really know if PLANCK is sensible enough to detect the pattern predicted by the RS models, but it might be an interesting possibility.

  2. Fabio Says:

    The closer you look at these “beautiful” solutions to the heirarchy problem, the less beautiful they become. They still require alot of coincidences and fine tunings to work right, just not as severe as before.

  3. Mimath Says:

    I think that we have got so entangled that we are looking in all directions which only lead inward. Having been a Martial Artist for many years I use and understand Duality, I see no mystery. Look at what is ‘out there’ for real and define it for what it is and not for it ought to be in Bra and Ket equations. As an analogy, Dolphins and Sharks have been called the most streamline creatures in the sea but the point is they must both obey the ‘law of the sea’ (whatever that might be). If one saw a single dorsal fin from a distance it might be one or the other yet the choice is between two diffent types of animal. One beats the water up and down the other from side to side. Evolution made the Dolphin breathe air so why should it live in an environment denying that need. Like a virtual particle, it pops up for air every now and then. Maybe ‘effective theory’ needs an ‘effective theory’ to explain it and…whoops, going inward again.
    I’m just an interested layman

  4. Mimath Says:

    Sorry, left an ‘m’ out of my e-mail address

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