Threshold photoproduction of ${\eta }_c$ and ${\eta }_b$ using holographic QCD

We discuss the possibility that threshold photoproduction of ${\eta }_{c,b}$ may be sensitive to the pseudovector $1^{+-}$ glueball exchange. We use the holographic construction to identify the pseudovector glueball with the Kalb-Ramond field, minimally coupled to bulk Dirac fermions. We derive the holographic C-odd form factor and its respective charge radius. Using the pertinent Witten diagrams, we derive and analyze the differential photoproduction cross section for ${\eta }_{c,b}$ in the threshold regime, including the interference from the dual bulk photon exchange with manifest vector dominance. The possibility of measuring this process at current and future electron facilities is discussed.

Figure 1

Threshold photoproduction of ${\eta }_c$ through (a) Kalb-Ramond exchange and (b) P-wave photon exchange.

Figure 2

Witten diagrams for threshold production of ${\eta }_{c,b}$ through (a) Odderon and (b) photon exchange.

Figure 3

C-even and C-odd nucleon form factors in the approximation ${\kappa }_N={\kappa }_{\gamma }$ with the normalization fixed by the charge, magnetic moment and (3.22).

Figure 4

(a) Holographic differential cross section for threshold photoproduction of ${\eta }_c$ at $W=4.3\mathrm{GeV}$, with the P-wave photon exchange (solid green, Dirac; dotted green, Pauli), the Odderon exchange (solid red with $g_{B\psi }=\{1,0.5\}$) and their coherent sum (solid black with $g_{B\psi }=\{1,0.5\}$). (b) Holographic differential cross section as in (a), compared to the Primakoff photon exchange estimate (open blue circles) in [16].

Figure 5

(a) Holographic differential cross section for threshold photoproduction of ${\eta }_c$ at $W=10\mathrm{GeV}$, with the P-wave photon exchange (solid green, Dirac; dotted green, Pauli), the Odderon exchange (solid red with $g_{B\psi }=\{1,0.5\}$) and their coherent sum (solid black with $g_{B\psi }=\{1,0.5\}$). (b) Holographic differential cross section as in (a), compared to the eikonalized dipole approximation in [14] with the Primakoff photon exchange (green diamonds), the Odderon exchange (red triangles) and their sum (black diamonds).

Figure 6

(a) Holographic differential cross section for threshold photoproduction of ${\eta }_c$ at $W=50\mathrm{GeV}$, with the P-wave photon exchange (solid green, Dirac; dotted green, Pauli), the Odderon exchange (solid red with $g_{B\psi }=\{1,0.5\}$) and their coherent sum (solid black with $g_{B\psi }=\{1,0.5\}$). (b) Holographic differential cross section as in (a), but with the photon contribution summed.

Figure 7

(a) Holographic differential cross section for threshold photoproduction of ${\eta }_c$ at $W=300\mathrm{GeV}$, with the P-wave photon exchange (solid green, Dirac; dotted green, Pauli), the Odderon exchange (solid red with $g_{B\psi }=\{1,0.5\}$) and their coherent sum (solid black with $g_{B\psi }=\{1,0.5\}$). (b) Holographic differential cross section as in (a), compared to the Primakoff photon-exchange (blue open circles) from [16], the Odderon model (orange diamonds) from [27] and the Odderon model (green triangles) from [26].

Figure 8

(a) Differential cross section for threshold photoproduction of ${\eta }_b$ at $W=11\mathrm{GeV}$, with the P-wave photon exchange (solid green, Dirac; dotted-green, Pauli), the Odderon exchange (solid red with $g_{B\psi }=\{1,0.5\}$) and their coherent sum (solid black with $g_{B\psi }=\{1,0.5\}$). (b) Holographic differential cross section as in (a), for the unseparated contributions.

Figure 9

Holographic differential and total cross sections for threshold photoproduction of $J/\mathrm{\Psi }$ (blue-shaded) from [9], and the present results for ${\eta }_c$ (black-shaded with $g_{B\psi }=\{1,0.5\}$). (a) $W=4.58\mathrm{GeV}$, (b) $W=4.30\mathrm{GeV}$, (c) $W=10\mathrm{GeV}$, (d) $W=50\mathrm{GeV}$, (e) $W=300\mathrm{GeV}$; and holographic total cross section for threshold photoproduction of $J/\mathrm{\Psi }$ (f). The black data points are from GlueX [28]. The magenta and green data points are from SLAC [29] and Cornell [30], respectively.

Figure 10

Minimal and maximal transverse momentum transfer $t_{\min }$, $t_{\max }$ in the physical region for ${\eta }_c$ (a) and ${\eta }_b$ (b) versus $W=\sqrt{s}$. The photon momentum is taken to be at the optical point $q^2=-Q^2=0$, and the hadron masses are given by $M_N=0.938\mathrm{GeV}$, $m_{{\eta }_c}=2.984\mathrm{GeV}$ and $M_{{\eta }_b}=9.399\mathrm{GeV}$.