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Since Quantum Chromo-Dynamics theory has been supposed to explain quarks confinement, has experimental search for free fractional electric charge elementary particles been put to an end? If not, what is the current best theoretical motivation to pursue this goal? Are there significant synergies with other experimental/theoretical endeavours?

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I was surprised to see that such searches have happened, over the years , in SLAC-PUB-8283 Oct, 1999

We have carried out a direct search in bulk matter for free fractional electric charge elementary particles using the largest mass single sample ever studied— about 17.4mg of silicone oil. The search used an improved and highly automated Millikan oil drop technique. No evidence for fractional charge particles was found. The concentration of particles with fractional charge more than 0.16 e (e being the magnitude of the electron charge) from the nearest integer charge is less than $4.71 * 10^{−22}$ particles per nucleon with 95% confidence level.

Surprised because of the million of events scanned and measured in particle phyisics there was never an excitement of a less than integer charge, which would stick out because of charge and energy conservation. Nobody thought of using this information to set a limit, I guess.

They have kept at it,

Summary as of January 2007.

Total mass throughput for all experiments- 351.4 milligrams of fluid

Total drops measured all experiments - 105.6 million

No evidence for fractional charge particles was found.

And searches are going on: here is a different recent search, APS april meeting 2018

Here, we present one such search for free particles with electrical charges less than the elementary charge, which are predicted by some extensions of the standard model. Such particles have not been observed and direct searches can restrict model parameter space. Using the results from our first year of physics data with the {\sc Demonstrator}, new direct-detection limits on the flux of lightly ionizing particles with charges as low as e/1000 can be set.

So the answer is, people are looking but not finding, yet?.

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