Analysis of the differential cross section and photon beam asymmetry data for $\gamma p\to\eta^{\prime}p$

The study of nucleon resonances ($N^{\ast}$s) has been of great interest in hadron physics, since a deeper understanding of $N^{\ast}$s can help us get insight into the nonperturbative regime of quantum chromodynamics. Apart from the $\pi N$ scattering or $\pi$ photoproduction reactions where we get most of knowledge about $N^{\ast}$s, the production processes of heavier mesons such as $\eta$, $\eta^{\prime}$, $K$, $K^{\ast}$, $\omega$, and $\phi$ have been gaining increasing attention, especially in study of the resonances that couple weakly to $\pi N$ but strongly to other meson-baryon channels. In the present work, we concentrate on the $\eta^{\prime}$ photoproduction reaction. As $\eta^{\prime}N$ has a much higher threshold than $\pi N$, this reaction is more suitable than pion production reactions to investigate the $N^{\ast}$s in the less-explored higher-energy region. Furthermore, the $\eta^{\prime}$ photoproduction acts as an “isospin filter,” isolating the $N^{\ast}$s with isospin $I=1/2$.

Experimentally, increasing amounts of data on differential cross sections and photon beam asymmetries for $\gamma p\to\eta^{\prime}p$ became available in recent years. In 2009, the CLAS Collaboration at the Thomas Jefferson National Accelerator Facility reported the high-precision differential cross section data in the center-of-mass energy range $W\approx 1.925\sim 2.795$ GeV CLAS09 , and the CBELSA/TAPS Collaboration released the differential cross section data in the energy range $W\approx 1.934\sim 2.35$ GeV CBELSA09 . In 2017, the A2 Collaboration at MAMI reported the differential cross section data from threshold, $W\approx 1.896$ GeV, up to $W=1.956$ GeV MAMI . The first photon beam asymmetry data at two center-of-mass energies, 1.903 and 1.912 GeV, were published by the GRALL Collaboration in 2015 beam2015 . Later in 2017, the CLAS Collaboration reported the photon beam asymmetry data for this reaction in a much wider energy range, $W\approx 1.9\sim 2.1$ GeV beam2017 . In 2019, the GlueX Collaboration reported the photon beam asymmetry data for this reaction at $W\approx 4.172$ GeV GlueX2019 . These photon beam asymmetry data provide much stronger constraints than the differential cross section data on constructing the reaction amplitudes for $\gamma p\to\eta^{\prime}p$.

Theoretically, before the availability of the photon beam asymmetry data beam2015 ; beam2017 ; GlueX2019 , there were several works that aimed to describe the differential cross section data for $\gamma p\to\eta^{\prime}p$, e.g., Refs. NH06 ; Zhong:2011 ; Huang:2013 . Nevertheless, none of them predicted correctly the experimentally observed behavior of the photon beam asymmetries for this reaction. Very recently, the photon beam asymmetry data from the GRALL and the CLAS Collaborations together with the differential cross section data for $\gamma p\to\eta^{\prime}p$ have been simultaneously described in partial-wave analysis performed by the BnGa group Anisovich2017 ; Anisovich2018 and in the updated $\eta$MAID model etaMAID2018 . The partial-wave analysis of BnGa group concluded that the contributions from the $N(1895)1/2^{-}$, $N(2100)1/2^{+}$, $N(2120)3/2^{-}$, and $N(1900)3/2^{+}$ resonances are significant in the $\eta^{\prime}p$ photoproduction reaction Anisovich2017 ; Anisovich2018 , while in the updated $\eta$MAID model it was claimed that the $N(1895)1/2^{-}$, $N(1880)1/2^{+}$, $N(1860)5/2^{+}$, and $N(1990)7/2^{+}$ resonances are of significance in the $\gamma p\to\eta^{\prime}p$ photoproduction process etaMAID2018 .

In the present work, we investigate the $\gamma p\to\eta^{\prime}p$ reaction within an effective Lagrangian approach at the tree-level approximation. In addition to the $t$-channel $\rho$ and $\omega$ exchanges, $s$- and $u$-channel nucleon ($N$) exchanges, and generalized contact current, we consider as few as possible $N$ resonances in the $s$ channel in constructing the reaction amplitudes to describe the available differential cross section data and photon beam asymmetry data from the CLAS, A2, and GRALL Collaborations CLAS09 ; MAMI ; beam2015 ; beam2017 . The differential cross section data from the CBELSA/TAPS Collaboration CBELSA09 are not included in the fit as these data cover a narrower energy range and have less precision than the data from the CLAS Collaboration CLAS09 . The beam asymmetry data from the GlueX Collaboration GlueX2019 are not included in the present work, either. These data are measured at $W\approx 4.172$ GeV. In such a high energy region, a pure Regge model JPAC2017 might be needed rather than a Feynman model as employed in the present work.

The present paper is organized as follows. In Sec. II, we briefly introduce the framework of our theoretical model, including the treatment of gauge invariance, the effective interaction Lagrangians, the resonance propagators, and the phenomenological form factors employed in the present work. In Sec. III, we present our theoretical results of the differential cross sections and the photon beam asymmetries, where discussions about the reaction mechanisms are also made and a prediction of the target asymmetries for $\gamma p\to\eta^{\prime}p$ is given. Finally, we give a brief summary in Sec. IV.

Following a field theoretical approach of Refs. Haberzettl:1997 ; Haberzettl:2006 , the full photoproduction amplitudes for $\gamma N\rightarrow\eta^{\prime}N$ can be expressed as

Here the first three terms $M^{\mu}_{s}$, $M^{\mu}_{t}$, and $M^{\mu}_{u}$ stand for the $s$-, $t$-, and $u$-channel pole diagrams, respectively, with $s$, $t$, and $u$ being the Mandelstam variables of the internally exchanged particles. They arise from the photon attaching to the external particles in the underlying $NN\eta^{\prime}$ interaction vertex. The last term, $M^{\mu}_{\rm int}$, stands for the interaction current that arises from the photon attaching to the internal structure of the $NN\eta^{\prime}$ interaction vertex. All the four terms in Eq. (1) are diagrammatically depicted in Fig. 1.

In the present work, the following contributions, as shown in Fig. 1, are considered in constructing the $s$-, $t$-, and $u$-channel amplitudes: (i) $N$ and $N^{\ast}$ exchanges in the $s$ channel, (ii) $\rho$ and $\omega$ exchanges in the $t$ channel, and (iii) $N$ exchange in the $u$ channel. Using an effective Lagrangian approach, one can, in principle, obtain explicit expressions for these amplitudes by evaluating the corresponding Feynman diagrams. However, the exact calculation of the interaction current $M^{\mu}_{\rm int}$ is impractical, as it obeys a highly nonlinear equation and contains diagrams with very complicated interaction dynamics. Following Refs. Haberzettl:2006 ; Huang:2012 ; Huang:2013 , we model the interaction current by a generalized contact current,

Here $\mu$ is the Lorentz index for $\gamma$, and $\Gamma_{NN\eta^{\prime}}$ is the vertex function of $NN\eta^{\prime}$ coupling given by the Lagrangian of Eq. (11),

$C^{\mu}$ is an auxiliary current, which is nonsingular, introduced to ensure that the full photoproduction amplitudes of Eq. (1) are fully gauge invariant. Following Refs. Haberzettl:2006 ; Huang:2012 , we choose $C^{\mu}$ for $\gamma N\rightarrow\eta^{\prime}N$ as

with

Here $p$, $p^{\prime}$, and $k$ are four momenta of the incoming $N$, outgoing $N$, and incoming photon, respectively; $Q_{N}$ is the electric charge of $N$; $f_{s}$ and $f_{u}$ are the phenomenological form factors for the $s$-channel $N$ exchange and $u$-channel $N$ exchange, respectively; $\hat{h}$ is an arbitrary function of $s$, $u$, and $t$, and it goes to unity in high-energy limit to prevent the violation of scaling sd1972 . In the present work, as usual Haberzettl:2006 ; Huang:2012 , we treat $\hat{h}$ as a fit parameter. It was found that the fitted value of $\hat{h}$ is very close to $1$. Thus, to reduce the number of fitting parameters, we simply fix $\hat{h}=1$ instead of treating it as a fit constant. Actually, as we shall see in Sec. III, the contributions from the interaction current [cf. Eq. (2)] are rather small in the present work.

Note that the $u$-channel $N^{\ast}$ exchanges are neglected in the present work. These contributions are expected to be rather small, as can be seen from the data on differential cross sections in the high-energy backward angles. We mention that neglecting the $u$-channel $N^{\ast}$ exchanges will not affect the gauge invariance of the full reaction amplitudes, since the transition Lagrangians for $N^{\ast}\to N\gamma$ are transverse [cf. Eqs. (14)-(17)].

In the rest of this section, we present the effective Lagrangians, resonance propagators, phenomenological form factors, and Reggeized treatment of the $t$-channel $\rho$ and $\omega$ exchanges employed in the present work.

As mentioned in Sec. I, before the photon beam asymmetry data for $\gamma p\to\eta^{\prime}p$ from the GRALL and CLAS Collaborations became available in 2015 and 2017 beam2015 ; beam2017 , the theoretical works on $\gamma p\to\eta^{\prime}p$ could describe only the differential cross section data. After the availability of the photon beam asymmetry data, simultaneous descriptions of the differential cross section data and the photon beam asymmetry data for $\gamma p\to\eta^{\prime}p$ have only been performed by the BnGa group Anisovich2017 ; Anisovich2018 and in the updated $\eta$MAID model etaMAID2018 . The resonance contents reported from these theoretical works are quite different.

In the present work, we analyze the available differential cross section data and the photon beam asymmetry data for $\gamma p\to\eta^{\prime}p$ from the CLAS, A2, and GRALL Collaborations CLAS09 ; MAMI ; beam2015 ; beam2017 within an effective Lagrangian approach in the tree-level approximation, with the purpose being to understand the reaction mechanisms and to extract the resonance contents and their associated parameters in this reaction. Apart from the $t$-channel $\rho$ and $\omega$ exchanges, $s$- and $u$-channel $N$ exchanges, and generalized contact current, we consider a minimum number of nucleon resonances in the $s$ channel in constructing the reaction amplitudes to describe the available data. We test different number of nucleon resonances and various combinations of them. We treat the resonance’s mass, width, and hadronic and electromagnetic coupling constants as fit parameters. Actually, only the products of the resonance electromagnetic and hadronic coupling constants are relevant to the reaction amplitudes, which are what we really fit in practice.

We have in total $877$ data in the fit, i.e., $681$ differential cross section data from CLAS CLAS09 , $120$ differential cross section data from A2 MAMI , $14$ photon beam asymmetry data from GRALL beam2015 , and $62$ photon beam asymmetry data from CLAS beam2017 . After numerous trials with the inclusion of different number of nucleon resonances and different combinations of them, we found that if only one resonance is taken into account in fitting the data, the $\chi^{2}$ per data point, $\chi^{2}/$N, will be larger than $4.4$ and results in a rather poor fitting quality. We then considered the exchanges of two nucleon resonances in the $s$ channel. We found that the available cross section data from the A2 and CLAS Collaborations and the beam asymmetry data from the GRALL and CLAS Collaborations can be well described by including the $N(1875){3/2}^{-}$ and $N(2040){3/2}^{+}$ resonances, which are rated as three-star and one-star resonances in PDG PDG , respectively. The resulting $\chi^{2}/{\rm N}$ is $1.8$ for beam asymmetry data, $3.2$ for differential cross section data, and $3.1$ for all data. The fitted results will be shown and discussed in detail below. With the inclusion of other two resonances, the best fit will be the one with the $N(1875){3\textfractionsolidus 2}^{-}$ and $N(1860){5\textfractionsolidus 2}^{+}$ resonances, which results in $\chi^{2}/{\rm N}=2.3$ for beam asymmetry data, $4.1$ for differential cross section data, and $3.9$ for all data. It is seen that the differential cross sections at high-energy forward angles are considerably underestimated in this fit. The other fits with two resonances have even worse fitting qualities and thus are not considered as acceptable. If one more resonance apart from the $N(1875)3/2^{-}$ and $N(2040)3/2^{+}$ resonances is further taken into account, the value of $\chi^{2}/{\rm N}$ reduces a little bit for the differential cross section data, but it remains almost the same or gets even worse for the beam asymmetry data. Moreover, in this case, there will be many solutions with similar fitting qualities, and thus no conclusive conclusion can be drawn about the resonance contents and parameters extracted from the available limited data. Therefore, we leave the fits with three or more nucleon resonances to future work when more data on spin observables for this reaction become available.

As discussed above, the most recent differential cross section data and photon beam asymmetry data for $\gamma p\to\eta^{\prime}p$ from the A2, CLAS, and GRALL Collaborations CLAS09 ; MAMI ; beam2015 ; beam2017 can be satisfactorily described with the inclusion of the three-star resonance $N(1875)3/2^{-}$ and the one-star resonance $N(2040)3/2^{+}$. The model parameters determined by the fit with these two resonances are listed in Tables 1 and 2. In Table 1, the asterisks below resonance names denote the overall status of these resonances evaluated by PDG PDG . The decay branching ratios (in $\%$) in bold font are values fixed in the calculation. For the $N(1875)3/2^{-}$ resonance, they are fixed to the centroid values of the dominant decay modes quoted by PDG. For the $N(2040)3/2^{+}$ resonance, its electromagnetic branching ratio is fixed to be $0.2\%$ as there is no information from PDG for the decay of this resonance. The other adjustable parameters are determined by a fit to the available data. The quantities in square brackets below the resonance mass $M_{R}$ and width $\Gamma_{R}$ of $N(1875)3/2^{-}$ are the corresponding ranges quoted in PDG. One sees that our fitted mass of $N(1875)3/2^{-}$ is within the range of the corresponding PDG value, but the fitted width lies outside the corresponding range given by PDG. Nevertheless, our fitted width $392$ MeV is not too far away from the value $321\pm 21$ MeV given by Hunt and Manley in their partial wave analysis Hunt19 . Note that for $s$- and $u$-channel $N$ exchange, the coupling constant $g_{NN\eta^{\prime}}$ is treated as a fit parameter, while the cutoff is set to be $900$ MeV in order to reduce the number of free parameters of the model. In Tables 1 and 2, the uncertainties of the values of fit parameters are estimated from the uncertainties (error bars) associated with the fitted experimental differential cross section and photon beam asymmetry data.

The theoretical results of the differential cross sections for $\gamma p\to\eta^{\prime}p$ obtained with the parameters listed in Tables 1 and 2 are shown in Fig. 2. There, the black solid lines represent the results from the full calculation. The blue dashed, red dash-dotted, green dash-double-doted, and magenta dot-double-dashed lines represent the individual contributions from the $s$-channel $N(1875)3/2^{-}$ exchange, $s$-channel $N(2040)3/2^{+}$ exchange, $t$-channel $\omega$ exchange, and $u$-channel $N$ exchange, respectively. The contributions from the interaction current, $t$-channel $\rho$ exchange, and $s$-channel $N$ exchange are not presented, as these contributions are too small to be clearly seen with the scale used in this figure. The green stars and red full circles represent the data from the A2 Collaboration MAMI and CLAS Collaboration CLAS09 , respectively. The blue empty squares denote the data from the CBELSA/TAPS Collaboration CBELSA09 which are not included in the fit. The numbers in parentheses denote the centroid values of the photon laboratory incident energy (left number) and the corresponding total center-of-mass energy of the system (right number), in MeV.

One sees from Fig. 2 that overall our calculated differential cross sections agree quite well with the corresponding data from the A2 and CLAS Collaborations in the entire energy region considered. Small discrepancies are also seen in a comparison of our theoretical results with the CBELSA/TAPS data, although these data have much larger error bars, which might show that the data from the CBELSA/TAPS and CLAS Collaborations are not compatible with each other. In Ref. Anisovich2017 , the theoretical curves were multiplied by overall scaling factors $0.9$ for differential cross sections and $0.94$ for beam asymmetries to compare with the corresponding CLAS data. In the present work, if we multiply a scaling factor $0.9$ to the CBELSA/TAPS differential cross section data, the incompatibility between the CBELSA/TAPS and CLAS datasets will become less obvious, but at high energies the CBELSA/TAPS data still tend to be larger than the CLAS data, as has already been observed in Ref. Anisovich2017 . From Fig. 2 one also sees that, at high-energy forward angles, the differential cross sections are dominated by the $t$-channel $\omega$ exchange. The $u$-channel $N$ exchange provides small contributions at high-energy backward angles. Actually, the parameters for $t$-channel and $u$-channel interactions are determined mainly by the high-energy data at forward angles and backward angles, respectively. Then, at lower energies, one has to introduce the resonance exchanges to describe the experimental cross sections. The resonance $N(1875)3/2^{-}$ contributes significantly in the energy range from threshold up to $W\sim 2.3$ GeV. The resonance $N(2040)3/2^{+}$ provides smaller but considerable contributions in the energy range $2.0$ GeV $<W<2.3$ GeV.

Figure 3 shows our theoretical results for the photon beam asymmetries compared with the data. There, the solid lines represent the results from the full amplitudes. The blue dashed and red dash-dotted lines represent the results obtained by switching off the contributions from the $N(1875)3/2^{-}$ and $N(2040)3/2^{+}$ resonances, respectively, from the full model. The blue stars and red circles represent the data from the GRALL Collaboration beam2015 and CLAS Collaboration beam2017 , respectively. One sees that our theoretical results for the photon beam asymmetries are in good agreement with the corresponding data. When the contributions from the $N(1875)3/2^{-}$ resonance exchange are switched off, the theoretical photon beam asymmetries deviate significantly from the data in the whole energy range considered. When the contributions from the $N(2040)3/2^{+}$ are switched off, considerable deviations of the theoretical beam asymmetries with the data are only seen above $W\sim 2$ GeV. This coincides with the observations from the differential cross sections as discussed above; i.e., the resonance $N(2040)3/2^{+}$ provides considerable contributions in the energy range $2.0$ GeV $<W<2.3$ GeV, while the resonance $N(1875)3/2^{-}$ provides significant contributions in the energy range from threshold up to $W\sim 2.3$ GeV.

Figure 4 shows our predicted total cross sections (black solid lines) together with individual contributions from the $s$-channel $N(1875)3/2^{-}$ exchange (blue dashed line), $s$-channel $N(2040)3/2^{+}$ exchange (red dash-dotted line), $t$-channel $\omega$ exchange (green dash-double-dotted line), and $u$-channel $N$ exchange (magenta dot-double-dashed lines) obtained by integrating the corresponding results for differential cross sections. The contributions from the interaction current, $t$-channel $\rho$ exchange, and $s$-channel $N$ exchange are not shown, as these contributions are rather small. The data are from the CBELSA/TAPS Collaboration CBELSA09 but not included in the fit. One sees that our theoretical total cross sections are considerably lower than the data from the CBELSA/TAPS Collaboration, which is not a surprise as we have already seen from Fig. 2 that our theoretical differential cross sections do not agree well with the CBELSA/TAPS data although they are in good agreement with the data from the A2 and CLAS Collaborations. From Fig. 4 one sees that it is the $t$-channel $\omega$ exchange that dominates the background contributions of the reaction $\gamma p\to\eta^{\prime}p$, while the contributions from the other nonresonant terms are negligible. The resonance contributions from both $N(1875)3/2^{-}$ and $N(2040)3/2^{+}$ are significant. In particular, the $N(1875)3/2^{-}$ resonance dominates the sharp rise of the total cross sections near threshold and provides a bump around $W\sim 1.95$ GeV, while the $N(2040)3/2^{+}$ resonance is responsible for the little bump structure around $W\sim 2.1$ GeV.

In Fig. 5, we show the predictions of the target nucleon asymmetries ($T$) at two selected energy points from the present model. We hope that this observable can be measured in experiments in the near future, which can help us to further constrain the theoretical model and thus to get a better understanding of the reaction mechanisms and the associated resonance contents and parameters in $\gamma p\to\eta^{\prime}p$.

In the present work, we employ an effective Lagrangian approach at the tree-level approximation to analyze the available differential cross section and photon beam asymmetry data for the photoproduction reaction $\gamma p\to\eta^{\prime}p$. We consider the exchanges of a minimum number of nucleon resonances in the $s$ channel, in addition to the $t$-channel $\rho$ and $\omega$ exchanges, $s$- and $u$-channel $N$ exchanges, and generalized interaction current, in constructing the reaction amplitudes to describe the data. The $s$-, $u$-, and $t$-channel amplitudes are obtained by evaluating the corresponding Feynman diagrams, and the generalized contact current is constructed in such a way that the full photoproduction amplitudes are fully gauge invariant. It is found that the available differential cross section data and photon beam asymmetry data for $\gamma p\to\eta^{\prime}p$ from the A2, CLAS, and GRALL Collaborations CLAS09 ; MAMI ; beam2015 ; beam2017 can be well reproduced by introducing the $N(1875)3/2^{-}$ and $N(2040)3/2^{+}$ resonances in the $s$-channel interactions. Further analysis shows that the $t$-channel $\omega$ exchange dominates the high-energy forward angle cross sections, and the $u$-channel $N$ exchange contributes considerably to the high-energy backward angle cross sections. The resonance $N(1875)3/2^{-}$ contributes significantly to both differential cross sections and photon beam asymmetries in the energy range from threshold up to $W\sim 2.3$ GeV. The resonance $N(2040)3/2^{+}$ contributes considerably in the energy range $2.0$ GeV $<W<2.2$ GeV to both differential cross sections and photon beam asymmetries. The contributions from the interaction current, $s$-channel $N$ exchange, and $t$-channel $\rho$ exchange are found to be negligible. The predictions of the target nucleon asymmetry for $\gamma p\to\eta^{\prime}p$ from the present model are also presented. Experimental information on this observable and on other spin observables are called for to further constrain the theoretical model and thus to get a better understanding of the reaction mechanisms for this reaction. In this respect, the data for electroproduction reactions from CLAS12 in the relevant range of $W$ and up to $Q^{2}$ of $12$ GeV² will hopefully provide an opportunity for better extractions of resonance parameters and couplings Carman20 .