This section follows closely our publication in Ref. [Blin16a] .
See this publication for more detailed information and references.
Formalism
The two processes contributing to $\gamma\, p \to J/\psi p$ are shown in the figure. The top diagram represents the direct production of the $P_c(4450)$ resonance. The bottom diagram represent the background. The nonresonant background is expected to be dominated by the $t$-channel Pomeron exchange, and we saturate the $s$-channel by the $P_c(4450)$ resonance. In the following we consider only the most favored $J_r^P = 3/2^-$ and $5/2^+$ spin-parity assignments for the resonance. We adopt the usual normalization conventions, and express the differential cross section in terms of the helicity amplitudes $\bra{\lambda_{\psi}\lambda_{p^\prime}}T_r\ket{\lambda_\gamma \lambda_p}$, \begin{equation} \label{Edsigdcos} \frac{d\sigma}{d \cos\theta}= \frac{ 4\pi\alpha}{32 \pi s} \frac{p_f}{p_i} \frac{1}{4} \sum_{\lambda_\gamma,\lambda_p, \lambda_{\psi}, \lambda_{p'}} \left|\bra{\lambda_{\psi}\lambda_{p^\prime}} T \ket{\lambda_\gamma \lambda_p}\right|^2. \end{equation} Here, $p_i$ and $p_f$ are the incoming and outgoing center-of-mass frame momenta, respectively, $\theta$ is the center-of-mass scattering angle, and $W=\sqrt{s}$ is the total energy in the center-of-mass . Note that the electric charge $\sqrt{4\pi \alpha}$ is explicitly factored out from the matrix element. The contribution of the $P_c(4450)$ resonance is parametrized using the Breit-Wigner ansatz, \begin{align} \bra{\lambda_{\psi}\lambda_{p^\prime}}T_r\ket{\lambda_\gamma \lambda_p}= \frac {\bra{\lambda_{\psi}\lambda_{p^\prime}}T_{\text{dec}}\ket{\lambda_r}\bra{\lambda_r} T^\dagger_\text{em}\ket{\lambda_\gamma \lambda_p}} {M_r^2-W^2-\mathrm{i}\Gamma_rM_r}.\label{EBWamp} \end{align} The numerator is given by the product of photo-excitation and hadronic decay helicity amplitudes. The measured width is narrow enough to be approximated by a constant, $\Gamma_r=(39\pm 24)\mbox{ MeV}$. The angular momentum conservation restricts the sum over $\lambda_r$, the spin projection along the beam direction in the center of mass frame, to $\lambda_{r}=\lambda_{\gamma}-\lambda_{p}$. The hadronic helicity amplitude $T_\text{dec}$, which represents the decay of the resonance of spin $J$ to the $J/\psi$ state, is given by \begin{align} \bra{\lambda_{\psi}\lambda_{p'}}T_\text{dec}\ket{\lambda_r}= g_{\lambda_{\psi}\lambda_{p'}} d_{\lambda_r,\lambda_{\psi}-\lambda_{p'}}^{J}(\cos\theta), \end{align} where $g_{\lambda_{\psi}\lambda_{p'}}$ are the helicity couplings between the resonance and the final state. There are three independent couplings ($\lambda_{p^\prime} = \frac{1}{2}$, $\lambda_{\psi} = \pm 1,0$), the other three being related by parity. For simplicity, we assume all these couplings to be equal, i.e. $g_{\lambda_{\psi}\lambda'_p}\equiv g$. The helicity amplitudes and the partial decay width $\Gamma_{\psi p}$ are related by \begin{equation} \Gamma_{\psi p} = \mathcal{B}_{\psi p}\, \Gamma_r =\frac{\bar p_f}{32 \pi^2 M^2_{r}}\frac{1}{2J_{r}+1} \sum_{\lambda_{R}}\int d\Omega \,\vert\langle \lambda_{\psi}\lambda_{p'} \vert T_{dec} \vert \lambda_{R} \rangle |^2 = \frac{\bar p_f}{8\pi M_r^2} \frac{6g^2}{2J_r + 1} , \label{ampwidth} \end{equation} with $\mathcal{B}_{\psi p}$ being the branching ratio of $P_c \to J/\psi$ and $\bar p_f$ the momentum $p_f$ evaluated at the resonance peak. We assume that the $P_c(4450)$ decay is dominated by the lowest partial wave, with angular momentum $\ell=0$ for $J_r^P=3/2^-$ and $\ell=1$ for $J_r^P=5/2^+$. We recall that the following near-threshold behavior of the helicity amplitudes holds: $g \propto p_f^\ell$. The helicity matrix elements of $T_\text{em}$ are usually parametrized in terms of two independent coupling constants, $A_{1/2}$ and $A_{3/2}$, which are related to the matrix elements with $\lambda_r=1/2, 3/2$, respectively. The other two helicities $-1/2$ and $-3/2$ are constrained by parity. Using the standard normalization convention, in which the helicity couplings $A_{\lambda_{R}}$ have units of $\text{GeV}^{-1/2}$ and are proportional to the unit electromagnetic charge, \begin{equation} \label{a12a32sec} \langle \lambda_{\gamma}\lambda_{p} \left| T_\text{em} \right| \lambda_{R} \rangle =\frac{W}{M_{r}}\sqrt{\frac{8M_{N}M_{r} \bar p_{i}}{4\pi\alpha}} \sqrt{\frac{\bar p_i}{p_i}}A_{\lambda_{R}}, \end{equation} with $\bar{p}_i$ the momentum $p_i$ evaluated at the resonance peak. The electromagnetic decay width $\Gamma_\gamma$ is given by \begin{equation} \label{a12a32width1} \Gamma_{\gamma}= \frac{\bar p_i^2} {\pi}\frac{2M_{N}}{(2J_{r}+1)M_{r}} \left[ \left | A_{1/2} \right |^{2}+\left | A_{3/2} \right|^{2} \right]. \end{equation}
Model
The photon helicity amplitudes for a pentaquark are not known. To rely on data as much as possible, we start by following Ref. [Kar16a] and assume a vector-meson dominance (VMD) relation for the transverse vector-meson helicity amplitudes \begin{equation} \bra{\lambda_{\gamma}\lambda_p}T_\text{em} \ket{\lambda_r} = \frac{\sqrt{4\pi \alpha} f_\psi}{M_\psi} \bra{\lambda_{\psi}=\lambda_\gamma ,\lambda_p}T_\text{dec} \ket{\lambda_r}, \label{vmd} \end{equation} with $f_\psi$ being the $J/\psi$ decay constant which is proportional to the electromagnetic current matrix elements, $\bra{0} J^\mu_\text{em} (0)\ket{J/\psi( p,\lambda)} = \sqrt{4\pi \alpha} f_\psi M_\psi \epsilon^\mu(p,\lambda).$ The decay constant is related to the $J/\psi$ wave function via the Van~Royen-Weisskopf relation, and can be estimated from the leptonic decay width of the $J/\psi \to l^+l^-$, yielding $f_\psi=280$ MeV. Finally, the VMD hypothesis leads to \begin{align} \Gamma_\gamma=4\pi \alpha \,\Gamma_{\psi p} \left(\frac{f_{\psi}}{M_{\psi}}\right)^2 \left(\frac{\bar p_i}{\bar p_f}\right)^{2\ell + 1} \times \frac{4}{6}, \label{eqvecdom} \end{align} with the factor $4/6$ due to the fact that in Eq. \eqref{vmd} only the transverse polarizations of the $J/\psi$ contribute. Again, we use $\ell=0$ for $J_r^P=3/2^-$ and $\ell=1$ for $J_r^P=5/2^+$. With the help of Eqs. \eqref{a12a32width1} and \eqref{eqvecdom}, one can constrain the size of the photocouplings. The background in the resonance region is assumed to be dominated by diffractive production, which we parametrize by an effective, helicity-conserving, Pomeron exchange model: \begin{align} \langle \lambda_\psi\lambda_{p^\prime}|T_{P} |\lambda_\gamma \lambda_p\rangle = iA~\left(\frac{s-s_t}{s_0}\right)^{\alpha(t)}~e^{b_0 t'} \delta_{\lambda_p\lambda_{p'}}\delta_{\lambda_{\psi}\lambda_\gamma}. \label{EqPom} \end{align} Here $s_0=1\mbox{ GeV}^2$ is fixed and $t' = t-t_\text{min}$. The Pomeron trajectory is given by $\alpha(t)=\alpha_0 + \alpha' \: t $, where $\alpha_0$ and $\alpha'$ are parameters to be determined, as well as the normalization $A$, the effective threshold parameter $s_t$, and the $t$-slope parameter $b_0$.
There seems to be a rapid decrease of the cross section in the threshold region and the shift parameter $s_t$ is introduced to enable a smooth connection between the high energy, $W \sim 100\mbox{ GeV}$, and the threshold.
The result is presented below for the model of Table I in Ref. [Blin16a] .
From left to right, the plots are:
- The differential cross section in the forward direction as a function of the photon energy $\frac{d\sigma}{dE_\gamma}(t' = 0)$.
- The total cross section as a function of the photon energy $\sigma_\text{tot}(E_\gamma)$.
- The angular distribution at the peak $\frac{d\sigma}{dt}(s = M_r^2)$.
References
[Blin16a]
A. N. Hiller Blin et al. (JPAC collaboration),
``Studying the $P_c$(4450) resonance in $J/\psi$ photoproduction off protons,''
Phys. Rev. D 94, 034002 (2016),
[Kar16a]
M. Karliner and J. L. Rosner,
``Photoproduction of Exotic Baryon Resonances,''
Phys. Lett. B752, 329 (2016).
Resources
- Publication: [Blin16a]
- Material:
- Web.tar.gz, compressed file with the codes. See below for the description of the files,
- C++ codes in a zip file,
- Formalism in a zip file
- Contact person: Astrid H. Blin
- Last update: December 2017
Description of the files in the compressed archive Web.tar.gz:
There are four folders:
- Cpp : include the C/C++ code and its input/output files.
The code computes our model with standard value of the parameters.
The parameters can be changed in the beginning of the file.
The kinematic are determined from 4-vectors, read from a file. - Formalism : include the latex files.
- fortran : include the fortran files.
Compile them with the command 'gfortran readplot.f -o nameyouwant.out'.
It reads the file 'test.txt' in the Input folder. - gnuplot : include the gnuplot files .gnu to create the plots.
The plots are created in the folder Plots. The fortran code generate the output files in the folder Data.
They can be group in the following categories:
- Parameters 1-8 are real numbers. They are the Pomeron and resonance parameters: $\alpha_0, \alpha', A, s_t, b_0, B_R, M_r, \Gamma_r$.
- Parameters 9 and 10 are strings, one gives the choice of spin (32 or 52), the other of smearing (0,60 or 120 MeV).
- Parameters 11 and 12 are integers of photocoupling choice:
either both are 0 0, or both have the same size 1 1, or one of them is 0: 1 0 or 0 1. - Finally there is the choice of the observable the user wants to get plotted:
1 for diff. cross section vs photon energy, 2 for total cross section, 3 for diff. cross section vs cos(angle) at the resonance peak.
Simulation
We invite the users to reproduce our plots with their own set of parameters.
In order to do so, the possible choices are for:
- The fitting parameteres of the paper $\alpha_0$, $\alpha'$, $A$, $s_t$, $b_0$ and $\mathcal{B}_{\psi p}$;
Choose $\mathcal{B}_{\psi p}=0$ in order to reproduce only the Pomeron background; - The physical mass and width of the $P_c(4450)$, $M_r$ and $\Gamma_r$, to enable varying them within their errors;
- The spin assignment $3/2$ or $5/2$;
- The size of the smearing due to the experimental resolution;
- The photocoupling assignment: $r_{1/2} = A_{1/2}/(|A_{1/2}|^2+|A_{3/2}|^2)^{1/2}$.
The norm $|A_{1/2}|^2+|A_{3/2}|^2$ is fixed.
- Finally, the observable to plot:
- The differential cross section in the forward direction as a function of the photon energy,
- The total cross section as a function of the photon energy,
- The angular behaviour of the differential cross section at the resonance peak,
- The asymmetry $K_{LL}$ in the forward direction as a function of the photon energy,
- The angular behaviour of the asymmetry $K_{LL}$ at the resonance peak,
- The spin-density matrix element (SDME) $\rho_{00}$ in the forward direction as a function of the photon energ
- The angular behaviour of the SDME $\rho_{00}$ at the resonance peak,
- The SDME $\rho_{10}$ in the forward direction as a function of the photon energy,
- The angular behaviour of the SDME $\rho_{10}$ at the resonance peak,
- The SDME $\rho_{1-1}$ in the forward direction as a function of the photon energy,
- The angular behaviour of the SDME $\rho_{1-1}$ at the resonance peak.
- If you choose the differential cross section or any of the SDMEs in the forward direction,
you can choose the maximal energy to display, $E_\text{max}$.