The Large Hadron Collider (LHC) has been smashing particles for a long time and sometimes people say that it has found new dimensions. How is it even possible for a particle accelerator to find new dimensions?
First, no evidence for other dimensions has been found.
However, there are ways for particle colliders to detect other dimensions. One of the main ones is to see if any energy "disappears" under very certain circumstances...then it could've possibly gone into another dimension.
Another way is to look for particles that can only exist if there are other dimensions. These particles would be around 100 times the mass of the W and Z bosons (these carry the electroweak force). Particles of this size could really only be detected by the LHC.
Other ways include examining the evaporation of mini black holes and seeing what particles are produced and looking for gravitons - gravitons would quickly disappear into extra dimensions, leaving an apparent loss of energy, which could be detected.
I'll be updating this as I find more information, though these are some of the main ways. Hope this helps!
This website has more information.
It is important to understand how experiments work. With very few very very basic exceptions, all experiments and their measurements involve a theoretical framework.
Fact is, we almost never measure things explicitly. For crude examples consider:
Temperature: a mercury thermometer measures length (that of the column of mercury). An electric thermometer measures either voltage, current, or resistance.
Speed: we usually measure position and time, and calculate the speed. The speedometer in your car very likely measures current.
With modern cutting-edge experiments, there is a lot more theory behind. The measures themselves are just numbers that the machines output (and again, the actual sensors are probably just measuring electric current, magnetic charge, etc., and for this they use the accepted electro-magnetism theories, say, which are very well established but they are theories, not facts). This data is processed by software designed following the principles of the accepted theory. Said like this, it doesn't sound very exciting, but it definitely has its merits.
In any case, experiments either confirm the numbers predicted by the theory, in which case they contribute to the theory's standing, or they contradict the theory, and then the theorists need to work on understanding what is wrong with the theory.
The bottom line is that experimental gadgets almost never measure directly the effects they intend to measure. They will measure some consequences of the effects, which are then studied according the accepted theories.