I am trying to understand the textbook derivation of the two-point function in CFT as spelled out in this CFT book by Francesco. The same derivation can also be found in this thread.
However, I don't understand the final step they do in their approach.
So the derivation goes something like this:
Consider two primary fields $\phi_1(x_1)$ and $\phi_2(x_2)$. Since we know how the primary fields transform we can write their two-point function as
\begin{equation} \langle\phi_1(x_1)\phi_2(x_2)\rangle =\left|\frac{\partial x'}{\partial x}\right|^{\Delta_1/d}_{x=x_1} \left|\frac{\partial x'}{\partial x}\right|^{\Delta_2/d}_{x=x_2} \langle\phi_1(x_1')\phi_2(x_2')\rangle. \end{equation}
Now, from translational and rotational symmetry we have that $$\langle\phi_1(x_1)\phi_2(x_2)\rangle = f(\left| x_1-x_2\right|).$$
By scaling $x$ i.e. letting $x\rightarrow \lambda x$ we get that
\begin{equation} f(\left| x_1-x_2\right|) = \lambda^{\Delta_1 +\Delta_2}f(\lambda \left| x_1-x_2\right|). \end{equation}
Now comes the "final" step that I fail to see or understand. Without too much explonation (maybe because it is straight forward, but I still don't se it) they say that this fixes the two point function up to an overall constant:
\begin{equation} \langle\phi_1(x_1)\phi_2(x_2)\rangle = \frac{C}{|x_1 - x_2|^{\Delta_1 + \Delta_2}}. \end{equation}
I don't see how letting
\begin{equation} \langle\phi_1(x_1)\phi_2(x_2)\rangle = f(\left| x_1-x_2\right|) = \lambda^{\Delta_1 +\Delta_2}f(\lambda \left| x_1-x_2\right|) \end{equation} tells us that we must have $f(x) \approx \frac{1}{x}$.
If I use this ansatz I get
\begin{equation} f(\left| x_1-x_2\right|) = \frac{1}{\left| x_1-x_2\right|}= \lambda^{\Delta_1 +\Delta_2} \frac{1}{\lambda \left| x_1-x_2\right|} = \lambda^{\Delta_1 +\Delta_2 -1} \frac{1}{ \left| x_1-x_2\right|}. \end{equation}
The only way I could see that this is true is if it was known that the sum of the scaling dimensions must always be one i.e. $\Delta_1 + \Delta_2 =1$, but I don't know if this is true?
Anyone who see my confusion?
