In Peskin and Schroeder Books Chapter 5 subsubection Bound State equation (5.43)

$\mathcal{M}(\uparrow \uparrow \rightarrow B)=\sqrt{2M}\int \frac{d^3k}{(2\pi)^3}\tilde{\psi^\ast}(\mathbf{k})\frac{1}{\sqrt{2m}}\frac{1}{\sqrt{2m}}\mathcal{M}(\uparrow \uparrow \rightarrow \mathbf{k} \uparrow, - \mathbf{k}\uparrow)$

My question : If i would like to calculate the decay width of pion decay, can anybody tell me how to use that equation.


You mixed up eq43 with eq44, you should replace $\left|B\right\rangle$ with $\mathcal{M}(\uparrow\uparrow\rightarrow B)$

But in any case, in the derivation of this expression an important assumption has been made, one which does not hold for pions!

The bound state must be non relativistic to a good approximation. This is true for heavy quark mesons, not for light quark mesons like the pion which is an intrinsically relativistic system (a good way to quickly understand why is to compare the binding energy with the rest energy of the constituents)

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  • $\begingroup$ thank you, I have edited my questions. Pion is composed by down quark and anti-up quark. So, because Pion is relativistic can I assume that down quark and anti-up quark as individual particle (not forming a bound state)? Then I use feynman diagram rules just like another case. $\endgroup$ – Panuluh Mar 7 '15 at 1:27
  • $\begingroup$ No they are still a bound state, but because they are relativistic you cannot use the non-relativistic schrodinger wave function description to represent them as in the given equation... So you need the full nonperturbative QFT apparatus $\endgroup$ – Ali Moh Mar 7 '15 at 1:29
  • $\begingroup$ So, you mean that for example we should use Bethe-Salpeter equation to solve this problem? Or is there another solution? $\endgroup$ – Panuluh Mar 7 '15 at 1:37
  • $\begingroup$ Except that the Bethe-Saltpeter equation can only be solved for relatively non relativistic systems... $\endgroup$ – Ali Moh Mar 7 '15 at 1:41
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    $\begingroup$ @Panuluh regarding p 264, you can see there that in the paradigm of effective field theory we write down the pion muon neutrino vertex based on known symmetries, combining all our (analytic) ignorance about the microscopic theory description of pion decay in one constant $f_\pi$. Then this is totally unknown, except you can estimate it using dimensional analysis (or precisely using the lattice as Melquiades noted). Then he goes on to calculate ratios which is the only thing you can predict, by canceling $f_\pi$ $\endgroup$ – Ali Moh Mar 8 '15 at 4:35

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