Reaction and Models

Reaction with Resonances and Decay

The (photo-production) reaction is \( \gamma p \to \eta \pi^0 p\), its decay is described by an amplitude model with three possible resonances: \(a_2\), \(\Delta^+\), and \(N^*\).

Amplitude Models

The amplitude model is adapted from the Lecture 11 in STRONG2020 HaSP School, only the Breit-Wigner and Spherical harmonics terms are kept for doing PWA eventually, as shown in equation (1).

(1)\[\begin{split}\begin{eqnarray} A^{12} &=& \frac{\sum a_m Y_2^m (\Omega_1)}{s-m^2_{a_2}+im_{a_2} \Gamma_{a_2}} \\ A^{23} &=& \frac{\sum b_m Y_1^m (\Omega_2)}{s-m^2_{\Delta}+im_{\Delta} \Gamma_{\Delta}} \\ \ A^{31} &=& \frac{c_0}{s-m^2_{N^*}+im_{N^*} \Gamma_{N^*}} \end{eqnarray}\end{split}\]

where s is the Mandelstam variable, m is the mass, \(\Gamma\) is the width, \(Y^m_l\) is the spherical harmonics, \(\Omega_i\) is the decay angles (a pair of Euler angles), and \(a_i\), \(b_i\), and \(c_i\) are coefficients

Note

Mandelstam variables \(s_{ij}=(p_i+p_j)^2\), \(t_i=(p_a-p_i)^2\), and \(u_i=(p_b-p_i)^2\).

with intensity \(I\) and amplitude \(A\):

(2)\[\begin{split}\begin{eqnarray} I &=& |A|^2 \nonumber \\ A &=& A^{12} + A^{23} + A^{31} \\ \end{eqnarray}\end{split}\]

where \(\quad 1 \equiv \eta ; \quad 2 \equiv \pi^0 ; \quad 3 \equiv p\)

Note

The choice of the amplitude model (equations (1) and (2)) for PWA in this tutorial consists of three resonances, and each of them are formed by two terms: Breit-Wigner with Spherical harmonics (\(l = 2, 1, 0\)).

Important

The spin of \(\eta\) meson and \(\pi^0\) meson are all 0. But the spin of proton is not 0, it is spin-\(\frac{1}{2}\).

In this amplitude model spin of baryon is simplified (not realistic): \(\eta\), \(\pi^0\) and \(p\) are all treated as spin-0 particles.

This means that total intrinsic spin \(s\) is ignored in this model, the total angular momentum \(J\) of the system or any subsystems within this model will solely depend on the orbital angular momentum \(L\), characterized by quantum number \(l\). And this simplifies the use of spherical harmonics \(Y_l^m(\theta,\phi)\), since only the orbital angular momentum component is involved, and thus the combination of contribution is not considered (e.g. Clebsch-Gordan Coefficients).

In our case:

• \(A^{12}\) amplitude represents a d-wave interaction, as indicated by \(l=2\).

  • The possible \(m\) values are \(−2,−1,0,1,2\). Each of these values corresponds to different orientations of the d-wave. The wave type is solely determined by \(l\) and all these \(m\) values still describe d-wave characteristics.

• \(A^{23}\) amplitude represents a p-wave interaction, as indicated by \(l=1\).

  • The possible \(m\) values are \(−1,0,1\). Each of these values corresponds to different orientations of the p-wave. Similarly, these values are all p-wave orientations.

• \(A^{31}\) amplitude represents a s-wave interaction, as indicated by \(l=0\).

  • The only possible \(m\) value is 0, which is consistent with the spherical symmetry of s-waves.