# How does the Super-Kamiokande experiment falsify SU(5)?

In his book "The Trouble With Physics", Lee Smolin writes that he is still stunned by the falsification of the $SU(5)$ Georgi-Glashow model by the null results of proton decay experiments.

I should like a simple but, if possible, quantitative explanation of how $SU(5)$ is falsified by the null results of attempts like Super-Kamiokande to witness proton decay? Obviously this theory foretells proton decay and we haven't seen it, but have we waited long enough or looked hard enough to say at a reasonable confidence that the null result falsifies the theory?

I am not a particle physicist, and you might see my outsiders answer to "What is the significance of Lie groups SO(3) and SU(2) to particle physics?" to gauge my level of competence.

Here is my reasoning so far, so, unless it is wrong (if so, please correct me), please use the following to help structure an answer.

1. The Georgi-Glashow model postulates that the theory of the primordial universe was a gauge theory with $SU(5)$ structure group. So this symmetry of physical laws is exact in the absence of some symmetry breaking mechanism which has taken hold today. I would like to know if there is a succinct summary of what this symmetry breaking mechanism might be.

2. So, at high enough energies - much greater than the potential drop that physical systems today get from "falling down the potential hill" begotten by the symmetry breaking mechanism in (1) - particles of the standard model should behave according the ancient $SU(5)$-symmetric laws.

3. Quarks and leptons would show themselves not to be fundamental but to be superpositions of the particles of the $SU(5)$ theory.

4. So, quarks and leptons are indeed coupled and, at today's everyday energy levels, there should be a tiny, but nonzero rate of "jumping over the unification energy barrier" - quantum tunnelling - so that protons should slowly spontaneously become other superpositions of $SU(5)$ model particles - i.e. protons should "decay".

So, the rate of decay is related to the size of the unification energy.

Presumably, if $SU(5)$ can be deemed falsified, we have a reasonable confidence in an upper bound to the unification energy such that the very slow proton decay rates ($<10^{-34} \mathrm{year}^{-1}$) consistent with the Super-Kamiokande null result imply a unification energy well above this upper bound. Usually quantum tunnelling rates are exponentially dependent on barrier sizes, so that small errors in energy barriers mean huge errors in tunnelling rates, so I'd be intrigued to see an analysis of the sensitivity of results to observational uncertainties.

So, in summary, here are my questions:

1. How in detail are unification energies related to implied decay rates?

2. How do we know what the unification energies are, or plausibly could be? How can we be sure that the bounds on these energies imply we should be seeing proton decay?

3. Conversely, what could be the symmetry breaking mechanism for $SU(5)$?

Since these things are probably well known to the relevant people, references instead of detailed answers would certianly be acceptable to me.

• Excellent questions! I don't really have time for an answer now, but may come back to one. In the meantime check out this relevant Q&A. PDG review on GUTs is here (pdf). See especially the sections on gauge coupling unification starting on pg. 4, Figure 15.1 (which is already enough to rule out the classic SU(5)) and nucleon decay starting pg. 9. Zee has nice introductory chapters in his QFT book. And there are many other reviews out there... – Michael Brown Sep 30 '13 at 15:05
• @DImension10AbhimanyuPS Why, do you think he's a bit loopy?!! Actually I liked the book from the autobiographical perspective, and at least Smolin seems to be one of the few criticizing string theory (which I have no grasp of BTW) from a standpoint of having gotten the expertise in the field himself. I thought he kind of gainsaid himself on one of the main themes - the supposed lack of falsifiability of ST - because he also seems to say do not underestimate the craft of the astronomers and experimentalists who come up with ingenious and unforeseen ways to test the seemingly untestable... – Selene Routley Oct 1 '13 at 0:25
• @DImension10AbhimanyuPS ... all the time. So, whilst the lack of falsifiability so far seems to be a problem, it would seem a bit too soon to say there never will be one - that's from an outsider's point of view. That experimental discovery slows down relative to the work put in is not surprising. Another experimentalist who seems to know her stuff on this site also seems to be of the view that the string theorists will turn some symmetry up that will further "organizes" the experimental results. – Selene Routley Oct 1 '13 at 0:30
• @DImension10AbhimanyuPS Smolin is a deliberate provocateur with a minority viewpoint, and so gets put on the banned list for a certain kind of person. But any time there is a dominating theoretical program without any direct experimental support it is helpful to have such different, non-crackpotty, voices around. Of course take what he says with a healthy grain of salt and verify everything you can for yourself, but that applies for every author. ;) ... – Michael Brown Oct 1 '13 at 7:35
• Some comments on your points 1-4. 1. The simplest way to break the SU(5) symmetry is with a second Higgs field, in addition to the one which subsequently breaks SU(2)xU(1) to U(1). 2. Yes. 3. Quarks and leptons would still be fundamental, but they are classified into larger and fewer multiplets (5- and 10-dimensional reps of SU(5)) than in the SM. 4. The new ingredient is that a quark can become a lepton (or vice versa), thanks to the extra SU(5) bosons (X and Y particles, or leptoquark bosons). It's rare because they are heavy. The proton disintegrates e.g. into a pion and a lepton. – Mitchell Porter Oct 2 '13 at 10:32

Our observations tell us that baryon and lepton number are conserved, within the accuracies of our experiments and observations. This means we have chosen as a standard model SU(3)xSU(2)xU(1) because in the group structure of the possible representations of all the quantum numbers assigned to the particles and resonances we know, there is no exchange particle that could change a proton to something else and no calculable Feynman diagram like:

This means that within the standard model the probability for a proton to decay is zero with any precision in the perturbative expansions which we calculate crossections with.

Extending the standard model, i.e.stating that it could be embedded in a higher group's representations, as an SU(5) model, with the standard model embedded, introduces new particles that can be exchanged, which carry lepton and quark numbers such that their exchange allows a proton decay. In this specific model these diagrams have been calculated and give a lifetime of order of 10^36 years.

The experimental limits of the half life are close to 10^34 years.

So it is not that

Quarks and leptons would show themselves not to be fundamental but to be superpositions of the particles of the SU(5) theory.

They are fundamental in SU(5). It is the group structure that is extended, and new exchange paths that allow Feynman diagrams that lead the (non fundamental) proton to decay. This larger structure also introduces new particles, like leptoquarks. To discover leptoquarks is an objective for the new lepton collider under consideration.

This paragraph summarizes :

Some beyond-the-Standard Model grand unified theories (GUTs) explicitly break the baryon number symmetry, allowing protons to decay via the Higgs particle, magnetic monopoles or new X bosons. Proton decay is one of the few observable effects of the various proposed GUTs. To date, all attempts to observe these events have failed.

This paragraph explains the "stunned":

Early grand unification theories, such as the Georgi–Glashow model, which were the first consistent theories to suggest proton decay postulated that the proton's half-life would be at least 10^31 years. As further experiments and calculations were performed in the 1990s, it became clear that the proton half-life could not lie below 10^32 years.

I suppose the book was written before the later calculations that recalculated the limit to 10^36years.

The reason we are not satisfied with the standard model and are trying extensions to higher groups and symmetries comes from the experimental observation that the coupling constants of the three interactions in the standard model tend towards the same region as the energies get larger. The theoretical urge to unite gravity to the other three forces which is supported by the cosmological observations to date are another strong impetus. But this is another stunning story :).

It is string theories that are trying to change our concept of "particle with specific quantum numbers" to "fundamental vibrating strings with specific vibrational levels". The future will tell.

• Thanks Anna. I cleared a few misconceptions up - a great summary. – Selene Routley Apr 2 '15 at 7:20
• Is the loopy connector distinct from if u, u and X met at a triangular intersection? – gen-ℤ ready to perish Jan 12 at 3:03
• @gen-zreadytoperish sorry I do not understand your question. – anna v Jan 12 at 5:37
• The up quarks in the diagram are connect by a horseshoe shape. The positron and antidown quark are connected by an angle. Do these two types of connectors intentionally reflect a distinction? – gen-ℤ ready to perish Jan 12 at 5:39
• It implies how quantum numbers are taken by the X, it carries away 2/3 baryon number .generates a -1 electron number and a -1/3 baryon number which implies the interaction cannot exist within the standard model, a new force carrier. is necessary – anna v Jan 12 at 5:45