Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Deacay and scattering amplitudes are not the same   We study the lowest lying $\pi^+ \pi^-$ resonance R, bound by the long range
Coulomb potential and destabilized by short range strong interactions mediating
the dominant decay into two neutral pions. Parametrizing the corresponding
partial decay width by $\Gamma (R \to 2 \pi^0) = 2/9 \alpha^3 \sqrt{m_{\pi^+}^2
- m_{\pi^0}^2} (\Delta a_{D} m_{\pi^+})^2$ the quantity $\Delta a_{D}$ depends
within QCD & QED on four basic parameters : $\Lambda_{QCD}, m_{u} + m_{d},
\alpha, m_{d}- m_{u}$. We are interested here only in the limit where the last
two parameters vanish $\Delta a_{D}^{0} = \Delta a_{D} (\Lambda_{QCD}, m_{u} +
m_{d}, \alpha = 0, m_{d} - m_{u} = 0)$.
  While isospin invariance implies $\Delta a_{D}^{0} = a_{D}^{I = 0} - a_{D}^{I
= 2}$ it is shown that beyond the first order expansion in the strong
interaction the identities $a_{D}^{I} = a^{I}$, where $a^{I}$ denote the strong
interaction scattering lengths, do not hold.

11
The categories of this paper are: hep-lat
14
The categories of this paper are: math-ph
13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th
18
The categories of this paper are: econ
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
15
The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin
27
The categories of this paper are: comp-gas
8
The categories of this paper are: hep-ex
1
The categories of this paper are: cs