Monday, December 16, 2013

Rare Top Quark Decays And Why There Are No Top Quark Hadrons

Fermilab produced about 150,000 top quarks in its entire run (the first one was observed in 1995); while the LHC has so far produced about 2,000,000 top quarks as of June 2012.

Top quarks almost always decay into a W+ boson and a bottom quark.  But, sometimes, they decay into a W+ boson and a strange quark, or into a W+ boson and a down quark.

Based on the current Particle Data Group CKM matrix values (based on a global fit of the four CKM matrix parameters), top quarks decayed into strange quarks about 240 times at Fermilab and about 3200 times at the LHC, while top quarks decayed into down quarks about 10-11 times at Fermilab and about 140 times at the LHC.  Of course, since the numbers are small there is considerable statistical sampling error in these values in addition to experimental error of other types (systemic error).

Top quarks have much shorter lifetimes than bottom or charm quarks (which have a mean lifetime of  a quadrillionth of a second), by a factor of about 10^11 or 10^12 (lighter quarks have even longer mean lifetimes).  Top quarks are still much longer lived than a hypothetical minimum unit of time called Planck time, by a factor of 10^20.

While the actual lifetime of any given bottom quark or top quark varies, there is virtually no overlap between the two in mean lifetimes.  Since the mean time necessary to form a meson (two quark composite particle) or baryon (three quark composite particle) is much longer than the mean lifetime of a top quark, and much shorter than the mean lifetime of all other quark types, top quarks generally do not form composite particles prior to decaying, while other quarks almost always form composite particles before decaying (i.e. they are confined).

What does that mean in terms that are understandable?

It means that if a bottom quark's mean lifetime were set equal to 70 years like a typical human, that a top quark's mean lifetime would be on the order of a hundredth of a second.  If the top quark's mean lifetime were actually a hundredth of a second, the longest lived top quark ever out of 2,150,000 top quarks ever made on Earth would probably be about 15 seconds.  Imagine that the mean time necessary to form a meson was about 2 and three-quarters hours in the example above.  The mean lifetime of a bottom or charm quark would be about a million times longer, while the mean lifetime a top quark would be about a million times shorter.

The number of non-top quarks that don't hadronize before decaying in that situation would be on the order of one per 2^9239 for bottom and charm quarks, and would be much less common for the three lighter flavors of quarks.  This is roughly one per 10^2781 charm and bottom quarks (and a much smaller share of up and down quarks).  Yet, there are only about 10^80 quarks is the entire universe and the vast, vast majority of them are up and down quarks.

Thus, while strictly speaking, a composite particle containing a top quark is not forbidden by the Standard Model, it is so vanishingly unlikely that there is no good reason to believe that such a composite particle will ever be observed, or that an unconfined quark of some other type will ever be observed.

The age of the universe is seconds is about 4*10^17, only a tiny proportion of the 10^80 quarks in the universe at any given time are top quarks (a number that should be roughly constant over time in the universe after its first few instants due to extreme baryon asymmetry and conservation of baryon number), and the probability of a top quark having a mean lifetime long enough to hadronize is vanishing low.  If that probability is less than 1 in 10^99 for example, it is an event that has almost surely never taken place even once in the entire universe, ever (in hypothetical baryongenesis which would last a few moments to a few hundreds of thousands of years, at most, the total number of quarks in the universe would be even smaller for the vast majority of that time although the proportion of top quarks would probably be quite high in that high energy environment). The real probability of a top quark living long enough to hadronize is probably closer to 10^2000 than it is to 10^99.

UPDATE 12/17/13:  The statements I made above about the mean lifetimes of the relevant quarks, the age of the universe, the number of quarks in the universe, Planck time, the number of top quarks every discovered, and the estimated number of rare top quark decays are all correct.  But, one number I used, and the inferences that flow from it, was misleading.

Going back to the scenario where a bottom quark mean lifetime is 70 years and a top quark mean lifetime is a hundreth of a second.  So far so good.  But, the average time to form a hadron is something like 1 second to one minute on that scale, not two and three-quarters hours.

This makes confinement of non-top quarks even more absolute than I suggested.  1 in 10^2781 charm or bottom quarks decaying before hadronizing was a gross overestimate of the likelihood of that happening.  In fact, it might be closer to 10^4000 or so.  These quarks are just not going to decay before hadronizing ever.

But, I have grossly overstated the rarity of a top quark living long enough to hadronize, in theory, anyway.  This is more like one in 2^4 to 2^7.  Thus, while top quark hadronization might be suppressed somewhere on the order of 90%-99.99% (the low end estimate assuming that two top quarks need to persist that long since no other quarks can get into their proximity that quickly) relative to all other quarks which hadronize 100% of the time, it shouldn't be suppressed on that basis alone so much that one would expect top quark mesons to have never formed in 2,150,000 cases where top quarks have been created.  One would expect something on the order of 200 top quark mesons to have been created.  (See a previous post discussing the possibility less rigorously here).

It may simply be that a t-t' meson decay is indistinguishable from the decay products of isolated decays of a t and a t' pair that never hadronize with each other, since the odds are overwhelming that the t will decay to a b and that the t' will decay to a b' in both cases and that this would happen very rapidly after hadronization (possibly leading to a brief intermediate period in which there is a top-bottom meson with a mass on the order of 178-180 GeV).  But, I would think that there would be at least some difference in experimental signature between the two cases.  Any t-t' decay is quite spectacular, with or without hadronization, as it is a 346 GeV+ event.  So, almost all such decays have probably been noted and studied in detail.  By comparison, the heaviest conceivable hadron with no top quarks (a spin-3/2 baryon, with electric charge -1, made of three b quarks), something that has not yet been definitively observed, would have a mass of more than 12.6 GeV, but probably less than 15 GeV.  A top-antitop meson would have a mass twenty-three times or more as great.

The characteristic time period required to hadronize is estimated, roughly speaking, on the assumption that two quarks moving at the speed of light relative to each other will hadronize if they are within 10^-15 meters of each other.  If that estimate is to brief, for some reason, then the actual time required for hadronization to take place could be longer than the 10^-23 seconds give or take a substantial margin of error that is often cited as an estimate, and that could explain the deficit of top quark mesons observed.

If this estimate were too short by a factor of ten, then the expected number of top quark mesons produced to date might be closer to 2 than 200 and this small number of outlier events might be too statistically insignificant to be noticed.

1 comment:

andrew said...

The expected chance of seeing just one flavor changing neutral current event from top decays in the Standard Model given the total number of top quarks observed to date is not more common than about one in a million.