# How the resonance width can change depending on the different wave 1S, 2S, 3S etc ?

This is a picture of a Upsilon system. The thing I'm wondering is that energy width $\Gamma$ for forth one (4S) wave is very different than the first three.

I know there is a formula which explains the relation between cross-section and energy width $\Gamma$ but i don't understand how this graph can be explained using the formula:

$$\sigma (E) = \frac{2 J +1 }{(2S_1+1)(2S_2 +1)} \frac{4 \pi}{K^2} \frac{\Gamma^2 /4 }{(E-E_0)^2+ \Gamma^2 /4} B_{in} B_{out}$$

More precisely, How the resonance width can change depending on the different wave 1S, 2S, 3S etc ?

Any details answer will much appreciated.

• If you take a piece of paper and draw the full width at half maximum, you will see that increasing in masses have an increasing width, the last one since it has a smaller crossection is the more obvious. This would show in the Γ if you fitted a breit wigner for each resonance. – anna v Dec 3 '17 at 15:14
• correction:The last one since it has an extra decay channel decays faster ( the narrower the resonance the larger the lifetime – anna v Dec 3 '17 at 16:17
• I get it a bit bit but i want something more rigorous. Could you please elaborate a bit in mathematical and with Feynman diagram if possible? – Numerical Person Dec 3 '17 at 20:43
• the link you give above has the feynman diagrams. Have a look at this to see the complexity of what your are asking.arxiv.org/abs/0910.0967 – anna v Dec 4 '17 at 4:45