# Finding a suitable form factor for given conditions

This is basically a physics problem but I will try my best to highlight the mathematics behind it. Suppose I have two functions:

$$T(z,B)=\frac{\text{z}^3 e^{-3 A(\text{z})-B^2 \text{z}^2}}{4 \pi \int_0^{\text{z}} \xi ^3 e^{-3 A(\xi )-B^2 \xi ^2} \, d\xi },$$ $$\phi(z,B)=\int_0^z \sqrt{-\frac{2 \left(3 x A''(x)-3 x A'(x)^2+6 A'(x)+2 B^4 x^3+2 B^2 x\right)}{x}} \, dx$$ where $$z \in \mathbb{R^+}$$ and $$B \in [0,1]$$

and I want to find a function $$A(z)$$, which is known as the form factor in literature, such that the plot of the function $$T(z,B)$$ v/s $$z$$ has one minima along with the condition that $$T(z,B)\rightarrow\infty$$ when $$z\rightarrow0$$, also $$\phi(z)$$ is real valued. When I take the ansatz $$A(z)=-a z^2$$, I am able to satisfy the above condition for $$B\in[0,0.6]$$ and get plots like: Now for a different model, I need to use such an ansatz for $$A(z)$$ such that I may be able to satisfy the real valued-ness of $$\phi(z)$$ and get minima as well as a maxima for the plot of $$T(z,B)$$ v/s $$z$$ with the condition that $$T(z,B)\rightarrow\infty$$ when $$z\rightarrow0$$ and $$T(z,B)\rightarrow0$$ when $$z\rightarrow\infty$$ to get plots like these: If possible I want to keep $$B\in[0,1]$$. The constant $$a$$ has to be used in the form factor somehow, whose value is 0.15. The form factor can also be written in terms of $$A(z,B)$$. Is there a way to use Mathematical analysis to come up with such a form factor? Any help in this regard would be truly beneficial!