1. The Bell Reasoning and the Aspect Experiment. 

The reasoning of Bell and the experimental investigation of his theoretical results by Aspect were a most interesting attempt to find experimental proof of the existence of a particlelike objective reality underlying quantum mechanics.  Bell’s theorem essentially says that (1) if particles exist and (2) if one presumes that a measurement on one particle cannot affect the results of a measurement on a second, distant particle, then under certain circumstances quantum mechanics predicts an incorrect result.  The Aspect experiment showed that quantum mechanics gives the correct result. 

2. What Conclusions are to Be Drawn? 

Because the first “if” is usually implicitly assumed to hold (rather than being explicitly stated), it is normally assumed that the second “if” does not hold.  If that is assumed, the implied long-range, instantaneous “interaction”  between the two particles, needed to explain the influence of one particle on the second, distant particle, seems mysterious indeed.  But if one assumes there are no particles, or more generally, no objective reality, then it is the first “if” that does not hold.  In that case, since there are just wave functions (rather than wave functions plus mysteriously coupled particles), the mystery in the Bell-Aspect results completely disappears; the experiment simply confirms the veracity of quantum mechanics (which may not always agree with our local classical expectations because, under certain circumstances, it predicts correlations between widely separated events).

Thus the most economical interpretation—because there is no need to have an underlying particle theory that must incorporate instantaneous action-at-a-distance—of the Bell-Aspect results is that there are no particles.  Only the wave function exists.

3. Note on Bohm and Bell. 

Bell's no-influence-at-a-distance condition is taken from our everyday experience.  But perhaps that is not the best guide.  In Bohm’s 1952 model (IIIB2), the "particle" trajectories are defined in terms of the wave function.  Because the wave function allows nonlocal effects, Bohm’s hidden variable model allows effects at a distance, as Bohm explicitly remarks.  And in fact, a little thought shows that, if there are underlying particles, we expect their trajectories to be affected by the wave function.

The no-influence-at-a-distance condition imposed in deriving Bell’s theorem, however, excludes Bohmlike models.  So while Bell's work and the related experiments certainly demonstrate that the nonlocal reality implied by quantum mechanics is odd, they are not particularly relevant to particles or particle theories.

References.

J. S. Bell, Physics 1, 195 (1964).

A. Aspect, P. Grangier, and G. Rogers, Phys. Rev. Lett. 47, 460 (1981).