Scaling laws in the dilute limit

By virtue of Eq. (11.10) this is identical to a ratio of renormalized cumulants, and invoking the theorem of renormalizability we find 

To find the general scaling law we now fix the renormalized length scale by the condition 

Clearly corresponding laws hold for all properly normalized quantities. 

To summarize the renormalization group proves two parameter scaling. The two parameters J q3 z however show a more complicated temperature dependence than assumed in the naive two-parameter scheme. The latter is correct only close to the 0-point. Furthermore the scaling functions take two different forms, representing the weak or the strong coupling branch. 

In Eqs. (11.49) (11,50) we indicated the branch by the index ( , ). In the sequel we will omit that index, the twobranched structure being understood throughout. 

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