Note on OPE production amplitudes [#!aa!#]

Adam Szczepaniak, 08/25/99 pt

Charge exchange, OPE production amplitudes of $Y^0\pi ^+$ (Y is the isobar) via a₁, a₂, $\pi_2$ and $\hat\rho$.

$$\gamma p \to X^+ n \to Y^0 \pi^+ n (\to \pi^+\pi^-\pi^+ n)$$ (1)

$${{d^2\sigma }\over {dtdM_{\rho \pi }d\Omega_k}}= {1\over 2} ... ...\Omega_k,\lambda_\gamma,\lambda_N,\lambda'_N,\lambda_Y)\vert^2$$ (2)

Spin projections of the isobar, $\lambda_Y$ and resonances, $\lambda_X$, $M_{Y\pi}$, (see below) are defined in the GJ frame i.e. resonance at rest, photon momentum along the z axis, ( $\lambda_\gamma$=photon helicity). In contrast, nucleon will be parametrized by its helicity.

The amplitude A is written as a product of the production amplitude A_OPE (production as defined by Dennis) and the amplitude T for the resonance (assumed BW) to decay into $Y\pi$. (I think Dennis has that in his code, I'm giving it here to completeness),

$$A = \sum_{X,J^P,\lambda_X}A_{OPE}(s,t,\lambda_N,\lambda_\ga... ...,\lambda_X) T(M_{Y^0\pi^+},\Omega_k,J^P,\lambda_X,\lambda_Y)$$ (3)

$$A_{OPE}(s,t,\lambda_N,\lambda_\gamma,\lambda_N',J^P,\lambda_X ) \sqrt{2} g_{\pi NN} \left({s\over {s_0}}\right)^{\alpha_\pi(t)} {{\sqrt{t'} }\over (t - m_\pi^2)}\beta(t)\delta_{\lambda_N,-\lambda'_N}$$ =

$$\sum_{L_\gamma}\sqrt{{2L_\gamma+1}\over {4\pi}}\langle J,\lambda_... ...rangle \left( {{\omega}\over m_X}\right)^{L_\gamma} g^{L_\gamma}_{X\gamma\pi} .$$ x (4)

Here t'=|t-t_min|, $\omega=\lambda(m_X,0,t)$ is the breakup momentum of the resonance X with mass m_X decaying into photon and an off-shell pion with mass t, $g_{\pi NN}^2/(4\pi) \sim 14.4$, $s=2E_\gamma m_n + m_n^2$, and $\alpha_\pi=0.9\mbox{GeV}^{-2}(t-m_\pi^2)$. The form factor $\beta(t)$ was fitted to the $\gamma p \to a^+_2 n$ data from Ref. [#!a21!#], [#!a22!#] and is given by

$$\beta(t) = (a_1 e^{b_1 t } + a_2 e^{b_2 t})^{1/2}$$ (5)

with $b_1 \sim 35\mbox{GeV}^{-2}$, $b_2 \sim 3\mbox{GeV}^{-2}$, $a_1 \sim 30$, $a_2 \sim 1.5$ for $s_0 = 10\mbox{GeV}^2$.

$$T(M_{Y^0\pi^+},\Omega_k,J^P,\lambda_X,\lambda_Y) \sum_{L_{Y\pi}M_{Y\pi}} { {m_X g^{L_{Y\pi}}_{XY^0\pi^+}}\over {2(... ...ma_{X,J^P}}\over 2} ) } } \left( {{k_{Y\pi}(m_X)}\over {m_x}}\right)^{L_{Y\pi}}$$ =

$$\langle s_Y\lambda_Y,L_{Y\pi}M_{Y\pi}\vert J_X\lambda_X\rangle Y_{L_{Y\pi},M_{Y\pi}}(\Omega_k)$$ x (6)

The photocouplings $g^{L_\gamma}_{X\gamma\pi}$ are calculated from the radiative widths $\Gamma_{X\gamma\pi}$ using

$$\Gamma_{X\to \gamma\pi}(E) = m_X \sum_{L_\gamma,L'_\gamma} {{g^{L... ...gamma\pi}}\over {32\pi^2}} \left({{q}\over {m_x}}\right)^{L_\gamma+L'_\gamma+1}$$

$$\left[\delta_{L_\gamma,L'_\gamma}\left(\delta_{L_\gamma,J} + \del... ...'_\gamma,J+1}\delta_{L_\gamma,J-1}\right) {{\sqrt{J(J+1)}}\over {2J+1}}\right],$$ x

with $q=\lambda(m_X,0,m_\pi)$. Note that, if there is more then one partial amplitude, $L_\gamma$ contributing, $\Gamma_{X\gamma\pi}$ determines only a product of different couplings. In such a case, to completely determine all $g^{L_\gamma}_{X\gamma\pi}$'s we assumed that the amplitude ratios are the same as for $X\to \rho\pi$.

The strong couplings are computed from

$$\Gamma^{L_{Y\pi}}_{X,J^P} = m_X {{ (g^{L_{Y\pi}}_{XY\pi})^2 ... ...2\pi^2}} \left( {{k_{Y\pi}}\over {m_X}} \right)^{2L_{Y\pi}+1}.$$ (7)

Here $k_{Y\pi} = \lambda(m_X,m_Y,m_\pi)$ is the resonance breakup momentum. Photocouplings and strong couplings to selected $Y\pi$ channels are summarized in Table 2. These are calculated from Eqs. 7, 8 using, where known, the PDG values  for $\Gamma_{X\gamma\pi}$ and $\Gamma^{L_{Y\pi}}_{X,J^P}$ and ``reasonable guesses'' otherwise, (all listed in Table. 1).

Table I: Resonance parameters. $\rho \pi$ widths are taken from the PDG. $\Gamma$ is the total hadronic width, $\Gamma = \sum _{AB,L_{AB}}\Gamma ^{L_{AB}}_{XAB}$ and $\Gamma ^e$ is the total radiative width to $\gamma \pi$ Here I have assumed that resonances can be saturated by the (two body) decay channels listed above. This is close to reality, however in principle on can recalculate the couplings in Table 2 for the ``true'' PDG values''. (This assumption was made to make the theoretical model of Ref. [1] simpler).

Resonance	JPC	Mass[GeV]	Partial waves	$\Gamma^{L_{AB}}_{XAB}/ \Gamma$	$\Gamma$[MeV]	$\Gamma ^e$[keV]
decay channel			(LAB, $L_\gamma=L^{\rho\pi})$
a1	1++	1.26			400	640
$\;\;\;\;\;\;\rho\pi$			S	0.99
			D	0.01
a2	2++	1.32			110	295
$\;\;\;\;\;\;\rho\pi$			D	0.70
$\;\;\;\;\;\eta\pi$			D	0.30
$\pi_2$	2-+	1.67			258	300
$\;\;\;\;\;\;\rho\pi$			P	0.98*0.31
			F	0.02*0.31
$\;\;\;\;\;\;f_2\pi$			S	0.69
$\hat\rho$	1-+	1.6			400	170
$\;\;\;\;\;\;\rho\pi$			P	0.50
$\;\;\;\;\;\;b_1\pi$			S	0.50

Table II: Strong and electromagnetic couplings corresponding to (BW) widths given in Table 1. Strong couplings, $g^{L_{Y\pi }}_{XY^0\pi ^+}$ to the charged state $Y^0\pi ^+$ when Y⁰ is part of an isovector should be reduced by $\sqrt {2}$.

Resonance	Partial wave	${{(g^{L_{AB}}_{XAB}(k_{AB}(m_X)))^2}\over {4\pi}}$	${{(g^{L_\gamma}_{X\gamma\pi})^2}\over {4\pi}}$
decay channel
a1
$\;\;\;\;\;\;\rho\pi$	S	26.28	1.03
	D	32.54	1.04 x 10-2
a2
$\;\;\;\;\;\;\rho\pi$	D	442.4	2.37
$\;\;\;\;\;\;\eta\pi$	D	57.08	72.71
$\pi_2$
$\;\;\;\;\;\;\rho\pi$	P	20.01	1.74
	F	20.15	4.01 x 10-2
$\;\;\;\;\;\;f_2\pi$	S	13.72
$\hat\rho$
$\;\;\;\;\;\;\rho\pi$	P	24.59	1.51
$\;\;\;\;\;\;b_1\pi$	S	7.12

Footnotes

... values 

See caption for Table.1