This page concerns the Finite Energy Sum Rules for pion photoproduction
from the publication [Mat18b] .
A brief description of the formalism is given in the Formalisms section.
Please see the publication [Mat18b] for further details.
Both side of the sum rules are compared interactively in the Simulation section.
The user can also display the observables at high energies.
The codes can be downloaded in Resources section.

Formalism

The photoproduction of a single pion is a function of 2 variables. We are using the the momentum transfered squared $t$ and the crossing variable $\nu$ defined by $$\begin{align} \nu = \frac{s-u}{4M} = E_\gamma^\text{lab} + \frac{t-\mu^2}{4M}. \end{align}$$ The pion mass is $\mu$ and the proton mass is $M$.
The Mandelstam variables, $s = (k+p)^2$, $t = (k-q)^2$ and $u= (k-p')^2$, satisfy $s+t+u= 2M^2+\mu^2$.

The scattering amplitude $\left(A _{ji}^{a}\right) _{\lambda', \lambda \lambda_\gamma}$ for the reaction $\gamma (k, \lambda_\gamma) N_i (p, \lambda) \to \pi^a (q) N_j (p',\lambda')$ is decomposed into the CGLN scalar amplitudes . Assuming isospin symmetry, we can attach to each scalar amplitude an isospin index: $$\begin{align} \left(A _{ji}^{a}\right) _{\lambda', \lambda \lambda_\gamma} & = \sum_{k = 1}^4 \bar u(p',\lambda')\ (A_{ji}^{a})_k(s,t) M_k(s,t,\lambda_\gamma)\ u(p, \lambda). \label{eq:T} \\ (A_{ji}^{a})_k & = (\tau)_{ji}^{a} A_k^{(0)} + \delta_{ji}\delta^{a3} A_k^{(+)} + \frac{1}{2}\left[\tau^{a},\tau^{3} \right]_{ji} A_k^{(-)} \end{align}$$ The indices $a$ and $(i,j)$ are isospin indices for $I=1$ (pion) and $I=\frac{1}{2}$ (nucleon).
The $\tau$ matrices are the Pauli matrices and the structures $M_k(s,t,\lambda_\gamma)$ are defined in [Mat15a] .
The scalar functions have good isospin and $G$-parity in the $t$-channel. Their properties are summarized in the following table:

The signature of the exchange, $\tau = (-1)^J$, provides the crossing properties of the scalar function,
i.e. $A_k^{(\sigma)}(-\nu-i\epsilon, t) = \tau A_k^{(\sigma)}(\nu+i\epsilon, t)$.

The relation to scalar function for given reaction is $$\begin{align*} A\left(\gamma p \to \pi^0 p \right)&= A^{(+)} + A^{(0)}, & A\left( \gamma p \to \pi^+ n \right)&= \sqrt{2} \left(A^{(0)} +A^{(-)} \right), \\ A\left(\gamma n \to \pi^0 n \right)&= A^{(+)} - A^{(0)} & A\left(\gamma n \to \pi^- p\right)&= \sqrt{2} \left(A^{(0)} -A^{(-)} \right),\\ \end{align*}$$

If, for an amplitude $A_i^{(\sigma)}$, we assume a sum of Regge poles of the form: $$\begin{align} \text{Im } R_{i}^{(\sigma)}(\nu,t) = \sum_{j} \beta^{(\sigma)}_{ij}(t) (r \nu)^{\alpha_j(t)-1}. \end{align}$$ The scale factor is chosen to be $r = 1$ GeV$^{-1}$.
The finite energy sum rules are $$\begin{align} B_k^{(\sigma)}\frac{\pi}{\Lambda} \left(\frac{\nu_N}{\Lambda}\right)^k + \int_{\nu_0}^{\Lambda} \text{Im } A_i^{(\sigma)}(\nu,t) \left(\frac{\nu}{\Lambda}\right)^k \frac{d \nu}{\Lambda} = \sum_j \beta^{(\sigma)}_{ij}(t) \frac{(r \Lambda)^{\alpha_j(t)-1}}{\alpha_j(t)+k} \label{eq:FESR} \end{align}$$ The moment $k$ is even (odd) for an even (odd) amplitudes, i.e. $\tau (-1)^k = +1$
The Born term are given in the table:

Model

At high energy $E_\gamma^\text{lab}>3.0$ GeV, we use a Regge parametrization for the invariant amplitudes.
The residue of the Regge pole $j$ in the amplitude $i$ is of the form (omitting the indices) $$\begin{align} \beta(t) & = \left[\alpha(t)\right]^\kappa\ t^\delta \times \beta e^{b t} (1-\gamma_1 t) (1-\gamma_2 t) \end{align}$$ We use linear trajectories $\alpha(t) = \alpha(0) + \alpha' t$.

The left-hand side (LHS) of the sum rules can be computed from any low energy model given in term of multipoles.
Below the user could generate the LHS of the FESR with the models: SAID, MAID
Other models will be included soon. Contact Vincent Mathieu (mathieuv.at.indiana.edu) if you want your models to be included here.

The integration in the FESR Eq. $\eqref{eq:FESR}$ is performed at fixed $t$.
For $t\neq t_{\text{min}}$, the integral involve an unphysical region.
The physical region ($|z = \cos\theta| \le 1$) is the grey area in the figure.
The solid (dashed) red line represent the $\pi N$ threshold (nucleon pole).
The extrapolation outside the physical region is performed by evaluating the Legendre polynomials for $\cos\theta \le -1$.
High partial wave might compromise the numerical stability of the amplitudes in the unphysical region.
For this reason, we limit the number of waves to $L_\text{max} = 5$ for the models in the resonance region.

References

[Mat18b]
V. Mathieu, J. Nys, C. Fernandez-Ramirez, A. N. Hiller Blin, A. Jackura, A. Pilloni, A. P. Szczepaniak and G. Fox
``Structure of Pion Photoproduction Amplitudes,'' arXiv:1806.08414 [hep-ph], Phys. Rev. D 98, 014041 (2018)

[Mat15a]
V. Mathieu, G. Fox and A. P. Szczepaniak
``Neutral Pion Photoproduction in a Regge Model,'' arXiv:1505.02321 [hep-ph], Phys. Rev. D 92, 074013 (2015)

Resources

• Publications: [Mat18b]
• Multipoles: SAID, MAID
• C/C++: FESR-PiPhot-Low.c, FESR-PiPhot-Regge.c
• Input file: multipoles-model.txt, Simulation-param.txt, Trajectories.txt, Residues.txt.
• Contact person: Vincent Mathieu (mathieuv.at.indiana.edu)
• Last update: June 2018

Simulation