Musings

High Energy Supersymmetry

I’ve really gotta stop posting about the Landscape. Posts on the subject rot your teeth and attract flies.

Still, it helps to sort out one’s thinking about anthropic ideas, which definitely clash with the sort of “explanation” we have become used-to in field theory. In any scientific framework, one needs to understand what’s just given — input data, if you will — what needs to be “explained” and (most importantly) what counts as an explanation. There’s a temptation to mix and match: to envoke the anthropic principle to explain some things, and “technical naturalness” to explain others. But that is simply inconsistent; a statistical distribution in the space of couplings does not favour technically-natural ones over others.

Consider the question: why is the QCD scale so much lower than the Planck scale (or the GUT scale)?

We are accustomed to saying that this large hierarchy is natural because it arises from renormalization-group running. The QCD coupling starts out moderately small at the GUT scale ($\alpha_{\text{GUT}}\sim 1/{25}$), and increases only logarithmically as we go down in energy.

But, in the Landscape, there’s a probability distribution for values of $\alpha_{\text{GUT}}$, which might just as easily be $1/{10}$, or $1/{150}$. What sounded like a virtue now sounds like a vice. The ratio $\Lambda_{\text{QCD}}/M_{\text{GUT}}$ depends exponentially on $\alpha_{\text{GUT}}$, and so is an exquisitely sensitive function of the moduli — exactly the sort of thing about which it is hard to make statistical predictions.

Instead, there’s an anthropic explanation for the value of $\Lambda_{\text{QCD}}$. Namely, the proton mass (which is essentially determined by the QCD scale) is tightly constrained. Vary $m_p/M_{\text{Pl}}$ by a factor of a few, and stars cease to exist. Hence $\alpha_{\text{GUT}}$ must be pretty close to $1/{25}$, otherwise, we aren’t here.

Similarly, point out Arkani-Hamed and Dimopoulos, the electroweak scale cannot be vastly different from $\Lambda_{\text{QCD}}$. For the ratio enters into the neutron-proton mass difference. If the neutron were lighter than the proton, there would be no atoms at all. If it were much heavier, all heavy elements would be unstable to beta decay, and there would be only hydrogen. Either way, we would not exist.

If the electroweak scale is anthropically-determined, is there any reason to expect any beyond-the-Standard-Model particles below the GUT scale? We don’t need low-energy supersymmetry to make $M_{\text{E.W.}}/M_{\text{Pl}}\ll 1$ natural. Arkani-Hamed and Dimopoulos posit a scenario where supersymmetry is broken at a high scale, with squarks and sleptons having masses in the $10^9$ GeV range (more on that below), whereas the “'inos” (the higgsino, the gluino, the wino, zino and photino) survive down to low energies.

Light fermions are, of course, technically natural. But there’s no reason to expect the theory to have approximate chiral symmetries. So technical naturalness is not, in this context an explanation for the light fermions. Instead, Arkani-Hamed and Dimopoulos argue that low-energy supersymmetry does have one great virtue — it ensured the unification of couplings around $10^{16}$ GeV. The “'inos” contribute to the $\beta$-function at 1-loop, so the 1-loop running in this model is exactly as in the MSSM. The squarks and sleptons contribute at 2-loops (as they come in complete $SU(5)$ multiplets, their 1-loop contribution does not affect the unification of couplings), and removing them from low energies actually improves the fit somewhat.

Arguing for coupling constant unification sounds equally bogus until you turn the argument on its head (thanks to Aaron Bergman for helping me see the light). Assume that at short distances one has grand unification. Then one needs light “'inos” so that the 3-2-1 couplings flow to their anthropically-allowed values at long distances.

Once we’ve abandoned low-energy, SUSY breaking, why not let the SUSY breaking scale be all the way up at the GUT scale? The reason is, again, anthropic. The gluino is a light colour-octet fermion, and hence very long-lived (it decays only via gravitino exchange). If you push the SUSY breaking scale up too high, the long-lived gluino creates problems for cosmology. Arkani-Hamed and Dimopoulos favour a SUSY-breaking scale, $M_S\sim 10^9$ GeV.

This gives a big improvement over low-energy SUSY in the context of the landscape. Flavour-changing neutral currents are no longer a problem. And it ameliorates, but does not really solve the problem of proton decay.

Proton decay via squark exchange, with two R-parity violating vertices.

Recall that there’s no reason for the generic vacuum on the Landscape to respect R-parity. R-parity is respected only on very high codimension subvarieties of the moduli space (if it’s present at all). So, generically, one expects R-parity violating terms in the superpotential to be unsuppressed. Since the squarks are much lighter than $M_{\text{GUT}}$, the dominant contribution to proton decay comes from squark exchange and the proton lifetime is roughly $$T \sim \left(\frac{M_S}{\lambda M_{\text{GUT}}} \right)^4 \times 10^{32}\text{years}$$ where $\lambda$ is the strength of the R-parity violating Yukawa couplings.

For TeV-mass squarks, the anthropic bound on the proton lifetime gives $\lambda \lt 10^{-9}$, whereas the observational bound is $\lambda \lt 10^{-13}$. Pushing the squark masses up to $10^9$ GeV, the bound on $\lambda$ is no longer so absurdly small. The anthropic bound is $\lambda \lt 10^{-3}$, and the observational bound is $\lambda \lt 10^{-7}$, but there is still a 4-orders of magnitude discrepancy which needs explaining.

I think that’s still a serious challenge for the anthropic principle. Why are the R-parity violating Yukawa couplings 4 orders of magnitude smaller than required by the anthropic bound?

A possible way out was suggest to me by Nima in the course of our email conversation. The lightest superpartner (one of the neutralinos) decays as well through R-parity violating interactions (a similar diagram to the one which led to proton decay, but with one R-parity violating and one R-parity preserving vertex, instead of two R-parity violating vertices. If we want the lightest superpartner to furnish a candidate for the dark matter (leading to structure formation and hence to us) we need its lifetime to be at least comparable to the age of the universe. For $M_{\text{lsp}}\sim$ a few hundred GeV, to get a lifetime of $10^{17}$ seconds, one ends up requiring $\lambda\sim 10^{-7}$.

Perhaps it is the existence of dark matter that “explains” the nearly exact R-parity in our universe. I’m still pretty sceptical, but I’m keeping an open mind. So, there may well be more posts on this subject in the future …