PROTON DECAY IN STRING THEORY Edward Witten Sulakfest Boston University October 22, 2005

Three reasons: I.Instantons in the standard model △ B = 3 Why do we believe Baryon number is violated? q q q q q q q q l l l q

II.Black holes violate Baryon number BLACK HOLE RADIATION OUT MATTER IN

III. Unification seems to require it …Quarks and leptons in one representation of a gauge group – in four or more dimensions – leads to proton decay

But it takes a more specific framework to motivate a proton lifetime near what we might observe… Original motivation for the pioneering IMB experiment – the original SU(5)- like GUT’s plus the famous “running of the three couplings”

The proton, sadly, didn’t decay at IMB and Kamioka … However, the same motivation survives in a slightly modified form with “Supersymmetry” … raises the proton lifetime above our current bounds and also, in the context of GUT’s, gives an excellent fit to sin 2 θ W

SUSY GUT’s give a relatively model- independent mechanism for proton decay T p = 2x10 36 years x (M GUT /2 x GeV) 4 Too long! But close enough that any enhancement would be interesting, even a factor of 2π in the amplitude

But supersymmetry also gives proton decay mechanisms that are more model dependent – dimension five operators that actually are a bit embarrassing Potentially even dimension four operators, this being part of a larger problem than includes flavor changing processes, CP violation, maybe even the Higgs being a little heavy

The drawbacks seem large, but so are the virtues -- let us not forget the neutrino masses, whose order of magnitude has turned out to be close to what one would guess from GUT’s – hinting that this picture is on the right track

“Split supersymmetry” is a recent attempt to have our cake and eat it too – keeping only the virtues of SUSY – and whatever its merits, it is potentially interesting for proton decay

As my title indicates, I am focusing today on proton decay in string theory, but I am really going to narrow it down a lot to consider only GUT-like models derived from strings. Apart from the general attractions of this framework, the reason is that otherwise there is no reason for the proton lifetime to be near what we might observe.

To me, a model is “GUT-like” if it naturally leads to the same fermion representations as the usual GUT’s, and the same prediction for sin 2 θ W I’ll be talking a little about proton decay in GUT-like string models – but I have no miracles to offer and the result may be an anticlimax!

Now part of what made string theory popular 20 years ago …is how neatly GUT-like phenomenology CAN be achieved in string theory originally via the heterotic string on a Calabi-Yau manifold More recently … many related models

¯ In the heterotic string context, one just starts with the numbers 3,4, or 5 And out pop the right gauge groups and chiral fermions Input Gauge group Fermion Rep 3 … E … SO(10) 16 5… SU(5)

Nowadays there is a much larger zoo of possibilities to get GUT-like models from strings … Type I superstrings Strongly coupled heterotic string Intersecting D-branes M-theory on a manifold of G 2 holonomy ….. Often dual to each other

Now one important fact about all these models is that they satisfy my definition of a GUT- like model but they are not GUT’s in the strict sense of four-dimensional unified gauge theories …

They lead to similar results for quantum numbers of chiral fermions – hopefully, the light quarks and leptons we know – and the same input values for the strong, weak, and electromagnetic couplings. But, because there isn’t four- dimensional unification, other things are different.

The most dramatic difference is that the usual quantization of electric charge does not hold – typically, though not always, GUT-like models derived from strings have unconfined fractional electric charges at a very high scale. … Maybe even a dark matter candidate, if inflation didn’t get in the way!

The usual proton decay amplitude has a key factor g 2 M x 2 where g is the unified gauge coupling and M X is the mass of the X boson, which is the GUT partner of the ordinary SU(3) x SU(2) x U(1) gauge bosons.

In the string models, there is no four- dimensional unification, and so there is no X or Y boson. Instead there is an infinite tower of Kaluza- Klein states, or string states, that have the same quantum numbers and mediate proton decay.

So instead of a simple X or Y boson propagator g 2 M x 2 there is a whole infinite sum ci2ci2 Mx2Mx2 i g st 2 Σ i

In the original models, based on the perturbative heterotic string, the sum is over Kaluza-Klein harmonics. The sum converges, meaning that you can do the calculation in ten- dimensional field theory… No need for recourse to the full string theory

The answer is qualitatively the same as in four-dimensional GUT’s … But this is a problem where that isn’t precise enough … A factor of 2π in the amplitude would make all the difference…

There is no hope of getting a real answer right now, because even if string theory is correct and even if nature is based on one of its GUT-like realizations, there are far too many possibilities. A couple years ago, Igor Klebanov and I did an illustrative calculation (following work with T. Friedmann in a related case)

We wanted a GUT-like string model in which we could calculate all of the factors relevant to proton decay, in a way that would not be too complicated, and would apply to a whole class of models – without depending too many details. The framework we picked involved “intersecting D-branes” Further advantage: the relation of the Planck and GUT scales comes out more or less correctly in this class of models

5 5 5 Q ¯ M ¯ 10

Now there is an interesting difference in this case from the perturbative heterotic string: One can again try to calculate the proton decay amplitude via a sum over Kaluza- Klein harmonics

But this time the sum over Kaluza-Klein harmonics diverges! This means that one cannot do the calculation in field theory – not even higher dimensional field theory – one has to use the full string theory to get the result for the proton decay amplitude.

It is this divergence that causes the result to be independent of many details of the model … The amplitude is dominated by the behavior near the points where a 10 of SU(5) is inserted. One might say that in this kind of model, proton decay is a stringy effect, not just a GUT effect

There is even a consequence that is observable in principle … To the extent that the divergent term dominates, in the decay of a proton to a e+ or µ+, the final state lepton will be left-handed ¯ 5 (The divergence only affects the 10 of SU(5), not the.)

In the case of the µ +, this possibly could be observed in a next generation experiment after proton decay is discovered… But this reminds me to tell you that I cannot say what fraction of the proton decays do go to µ + rather than e + or τ + as this does depend on a lot more details

We had some fun with this computation because of the enhancement coming from the way the field theory divergence is cut off… There also is another nice factor in these models coming from the GUT scale “threshold corrections.” These are calculable (TF and EW) and often give a nice enhancement.

For a while, it seemed we might get a prediction for proton decay by dimension six operators that would be significantly more favorable than the usual case. But sadly, another factor (coming from ten- dimensional kinematics) ultimately intruded to spoil the fun, as well as the punchline of this talk.

Our eventual result for these models was surprisingly close to the usual answer computed in four dimensions T p = 2x10 36 years x (M GUT /2 x GeV) 4 though any of three non 4d factors (amplitude, threshold correction, kinematics) would individually make a very significant difference

I am sure that in this picture

there are models that would make us happier… I can only conclude by hoping that nature is based on one of them … and wishing “Happy Birthday, Larry”