behavior of the electric (magnetic) theory crosses over from being dominated by the short distance fixed point to the long distance one (the typical mass scale of the theory). This raises a number of questions; it is not clear why $\Lambda$, $\bar\Lambda$ defined in such a way should satisfy a relation like  when the theories are only equivalent in the extreme infrared. Also, this definition leaves ambiguous a numerical factor, especially in theories with more than one scale.

One can think about eq.  and the scales appearing in it purely in the extreme IR theory. Since the scaling dimensions of various operators in the infrared are not the same as in the UV (see  and discussion below), one needs a dimensionful parameter to relate the UV operators to the IR ones. That dimensionful parameter, which can be defined for example through two point functions of such operators, is $\Lambda$. Similarly, in the magnetic theory one has $\bar\Lambda$, and additional parameters such as $\mu$ . The meaning of the scale matching relation is that $\bar\Lambda$ and $\mu$ must be chosen to obey  in order for the correlation functions of the two theories to agree including normalization. At any given point in 