Is it possible for an x86 processor to match an ARM processor in terms of performance per watt?

Arm processors have been getting increasingly complex - so its an apple to orange comparison - arm has only had a 64 bit varient for about 2 years, and even within the same generation power use varies. Its probably fairer to consider contemporary ARM processors and their atom counterparts as anandtech have done here.

Most of the power use isn't the processor either - the anandtech article I linked tends to suggest that maybe 1/4 of overall power use is the processor, and that a clovertrail atom uses slightly over half of the ARM processor in a surface. It also shows an interesting alternate viewpoint - to optimise systems per component for power use (as an aside - its entirely possible that you can get pretty significant power savings doing this off a standard desktop platform as well). Something as simple as avoiding inefficient power conversions can do a fair bit. On most phones, for example, your display is probably taking up close to half your total power use.

As you can see from the numbers on that article - Atom processors designed for mobile devices already match ARM processors on the power efficiency front - so its probably worth wondering why they arn't more common.

Arm doesn't make their own silicon - they design and test it, and sell the IP to make processors. For companies that need to design their own processors, or to tweak it, this means significant savings in R&D without needing to develop everything from scratch (tricky) or to buy processors from another company (with x86, we have Via, AMD and Intel, but only intel seem really interested in the mobile space, and I have no clue what via is up to).

The idea that arm processors are more efficient is a bit of a myth - they've made a different set of tradeoffs (power efficiency over raw speed) and are moving in a different direction and a different set of tradeoffs in an attempt to go after the server market. Intel on the other hand, is effectively segmenting modern atom designs into server parts, desktop parts (like new pentium models) and phones, to go after the low end. Neither design is inherently better at everything than the other.