Partial-wave analyses of and using a multichannel framework
This paper presents results from partial-wave analyses of the photoproduction reactions and . World data for the observables , , , , , and were analyzed as part of this work. The dominant amplitude in the fitting range from threshold to a c.m. energy of 1900 MeV was found to be in both reactions, consistent with results of other groups. At c.m. energies above 1600 MeV, our solution deviates from published results, with this work finding higher-order partial waves becoming significant. Data off the proton suggest that the higher-order terms contributing to the reaction include , , and . The final results also hint that is needed to fit double-polarization observables above 1900 MeV. Data off the neutron show a contribution from , as well as strong contributions from and .
In recent years, a wealth of new high-precision experimental data has been measured at various facilities including JLab, MAMI, LEPS, SLAC, and GRAAL for a number of observables with the goal of better understanding the spectrum of and resonances. Despite past efforts, there are still predicted resonances that have not been found, known as the problem of the “missing resonances”, and other resonances whose properties are not well determined. Two possible explanations for this are that (1) the missing resonances do not exist or (2) they couple mainly to reactions not yet analyzed. This work investigates the second possibility. Knowledge gained from this and future work is expected to guide theorists trying to understand the fundamental features of Quantum Chromodynamics (QCD) or the theory of the quarks and gluons that bind matter into hadrons.
It is well known that at least eight measured observables are needed to perform a complete experiment Tabakin . The database analyzed in this work for and include significant amounts of data for five of the eight needed observables. These five are , , , , and measured at various c.m. energies from threshold to 1900 MeV. Also analyzed were seven and 12 data points. Data for the helicity-dependent cross section were analyzed as and data. Table 1 tabulates the number of data available for each observable and shows that while there are a wealth of differential cross-section data, the polarization measurements are still limited. Because there are still insufficient data for a complete experiment, information from other reactions, including and , was used to constrain the fits.
This work analyzed the world data of eta photoproduction off the nucleon in the c.m. energy range from threshold up to almost 2000 MeV. The final generated energy-dependent solutions were then used in the KSU multichannel framework to improve knowledge about the and resonance parameters. Section II outlines the basic formalism used throughout this work including sign conventions for the different spin observables. Section III describes the general procedure that we used to obtain the results. Section IV describes results of the analyses for the reactions and . Comparisons to results from BnGa (2016) Sarantsev and Jülich (2015) Julich are also shown. Fits to the data are shown in Appendix A.
Four helicity amplitudes are needed to describe the photoproduction of a pseudoscalar () meson and a baryon off of a nucleon target BDS1 . Each of the four helicity amplitudes can be expanded in terms of electric and magnetic multipoles and , respectively, where is the orbital angular momentum of the final-state hadrons and is the total angular momentum. Each multipole is a complex function of energy, which makes the helicity amplitudes complex functions of both energy and scattering angle:
The naming convention for the four helicity amplitudes above follows that of the SAID group SAID1990 . All 16 single- and double-polarization observables can be written in terms of these four helicity amplitudes; however, in the literature, different sign conventions are used in their definitions. The definitions for each of the observables included in this work are given in Table 2.
The literature also mentions measurements of and , which are the helicity-dependent cross sections Witthauer17 . They are related to the and observables by
and are then pure helicity- amplitudes, while and are pure helicity- amplitudes. The full differential cross section is recovered by the relationship
Iii Fitting Procedure
We began our analysis by performing single-energy partial-wave analyses. The starting point was to assemble all data within specified small c.m. energy ranges into individual bins. Observables within a single bin were then approximated as functions of just the scattering angle. It was determined that 5-MeV wide bins were needed near threshold where the amplitude dominates due to the rapid rise in the cross section near the (1535) resonance. At c.m. energies above MeV, a trade-off between small bin sizes and keeping sufficient polarization data within the energy bin meant that larger bin sizes of 15 to 20 MeV were needed to constrain the fits.
Once the data were binned, the global phase was initially determined by first fitting the data with a purely real amplitude to determine its magnitude. Then an energy-dependent fit, similar to those of Shrestha and Manley ShresthaED , was used to determine its phase through unitarity constraints. With the amplitude fully determined, initial values for the higher-order amplitudes could then be determined.
Due to complexities that arise from interference effects, an iterative procedure was needed to obtain good quality fits to the data. The procedure involved two main steps that were iterated as many times as necessary to obtain convergence. The first step (single-energy fits) was to allow a subset of the partial-wave amplitudes to vary in each energy bin. This generated a discrete solution for each of the varied partial-wave amplitudes. These single-energy results were then used as input to energy-dependent fits (the second step) that were also used to determine the resonance parameters. In this second step, resonance parameters were adjusted to generate a smooth energy-dependent solution of the single-energy amplitudes. Finally, the output of the second step was used as input to the first step. This iterative procedure was continued until reached a global minimum for this and all other analyzed reactions in the energy-dependent analysis, which included and as well as , , , and . This means the final binned solutions found as part of this work are within a few percent, but not at, the exact minimum for the reaction.
Because not all measured observables are available in all energy bins, a penalty term was added to the standard term to constrain the bin-to-bin solution. The explicit form for the penalty term was
where is the starting value found in step two of the iterative process and are the amplitudes that were varied to improve the fits further. This biased results to single-energy solutions that were somewhat similar to the current energy-dependent solution. was a parameter that was chosen to control the strength of the constraints, with smaller values more tightly constraining the fits. To minimize bias from the penalty term, multiple starting solutions were used to determine which produced the best fit.
Once the solution converged to a minimum, final uncertainties on the amplitudes in the single-energy solutions from step one were obtained by fixing the phases of each partial-wave amplitude at the values from our energy-dependent solution, and then allowing only their moduli to vary. This phase constraint was needed to fix both the global phase of the solution and to constrain the final results due to lack of all spin observables at all energies. An additional penalty constraint was used as well, but kept small enough that the penalty contribution to was less than 10. The resulting single-energy solutions, projected into real and imaginary parts, were then used as input to a final energy-dependent fit in which all parameters were free to vary, to determine final uncertainties in the and resonance parameters. Our final fits included all amplitudes up to for both and , although was the highest amplitude necessary for good fits.
This section presents final results for the partial-wave analyses of both and . It compares results with those of other groups and shows the quality of agreement for the integrated cross-section data that were not directly fitted.
For the reaction , the fits of the observables , , , and were very good over the entire energy range; however, the fits of the beam asymmetry showed minor problems at backward angles in the c.m. energy range 1650 to 1800 MeV. Table 3 shows the contribution from each individual observable for the different works as well as the total over all observables. We note that the contribution from the differential cross section obtained in this work is significantly smaller than that by other groups with minimal impact to the spin observables. This is in part because we included in our fits new high-precision data for some observables that were unavailable to the other groups at their time of analyses. In order to provide a good description of the data, the amplitudes and were needed starting at threshold and , , and waves were important above 1600 MeV. At energies above MeV, appears important, but the lack of spin observables at these energies made it difficult to make any definitive conclusions. The magnitude of partial-wave amplitude above 1600 MeV was found to differ from the results of BnGa Sarantsev , which found saturating the integrated cross section over almost the entire energy range analyzed in this work.
|Observable||KSU||BnGa (2016)||Jülich (2015b)|
The integrated cross section was obtained by integrating the differential cross section over the full angular range. The helicity-1/2 and 3/2 cross sections were generated by integrating the helicity-dependent differential cross-section data as defined in Eqs. (2) and (3). Figures 1, 2, and 3 show the full integrated cross section as well as the helicity-1/2 and 3/2 integrated cross sections. The curves generated through the integrated cross section points were obtained by fitting the differential cross section data and extracting the integrated cross section from the individual partial-wave amplitudes. Overall, the curves describe the data well through the entire energy region. The curve for slightly overshoots the data near 1600 MeV but our fits to data were found to be in good agreement. The helicity-1/2 and helicity-3/2 plots also showed good agreement within experimental uncertainty and the scatter in the points.
Figure 4 compares the partial-wave amplitudes from this work with results from BnGa Sarantsev and Jülich Julich . For this reaction, the only amplitude that is in agreement between all the groups is . Higher partial waves all exhibit major discrepancies with at least one of the groups. This lack of agreement indicates that additional data are needed from unmeasured double polarization observables to obtain a unique solution.
To obtain further progress towards the goal of a single solution for photoproduction off the nucleon, Fig. 5 shows what observables show the most difference between the three groups compared in this work. A measurement of both and would be ideal at c.m. energies between 1600 and 1800 MeV while above 1800, the predictions for these two observables actually converge and a better measurement would be either or .
For the reaction , our fits to the published observables , , and are overall very good. Fits to the observable are overall quite good with just a few bins showing minor local problems. Using wide binning showed that the data varied significantly from bin-to-bin, which prevented further improvements to the fits. The amplitude dominates the reaction from threshold up to MeV, with and showing significant contributions in the region of the narrow structure near MeV. Table 4 lists shows the contributions for this work and BnGa (2016) Sarantsev . Again, individual and total contributions are shown. This work does a slightly better job at describing a few of the observables while BnGa (2016) does better at others. It is noted that BnGa did not have available the data for the observables , , and .
Figures 6, 7, and 8 show integrated cross-section results. Both Figs. 6 and 7 reveal what appears to be a narrow structure less than 100 MeV wide near 1680 MeV. Much has been written about this structure. Some researchers have concluded that the bump must be from either a resonance or due to an interference effect between two resonances Kuznetsov ; Anisovich1685 because the structure only appears in the helicity- data (see Figs. 7 and 8). The argument has been made that the helicity- cross section contains contributions from and , while the helicity- cross section does not, so the bump must be due to these two amplitudes.
The present work provides an alternative interpretation of the bump as a complicated structure generated by a number of resonances, specifically , , and the tail of the . As mentioned above, this would generate a bump in the helicity-3/2 cross section. Our predictions of the helicity cross sections (Figs. 7 and 8) show that the data allow and even hint at a small bump within the size of the error bars and scatter of the points. While the fits to the integrated cross section seem to overshoot the data at c.m. energies near 1650 MeV, the energy resolution in the region around the bump is 30 MeV (and wider at higher energies), despite the cross section points being roughly 10 MeV apart Werthmuller .
Plots comparing the partial-wave amplitudes determined in this work with BnGa results Sarantsev are shown in Figure 9. The amplitude is similar except in the c.m. energy region near 1680 MeV where the BnGa group explains the bump as an interference and what looks like a cusp effect, possibly from the opening of the channel near 1680 MeV. The only other amplitude that is similar between the two groups is the amplitude.
In our fits, there appears to be more structure than that found in the BnGa solution. Resonance peaks are clearly seen in multiple amplitudes with the imaginary part of the amplitude forming a peak when the corresponding real part approaches zero, as expected for a Breit-Wigner resonance.
At this point, additional measurements of any of the 16 observables would be useful to constrain the fits further and confirm previous measurements. As such, no predictions are shown at this point.
V Summary and Conclusions
Results from a partial-wave analysis of available data for were presented. , , , and amplitudes were found to be important for the the reaction in the energy range from threshold to 2000 MeV. , , , and were important for . This is consistent with the Moorhouse selection rule Moorhouse , which predicts that the (1675) resonance may couple to but not to .
Also, despite the wealth of new data, measurements of additional double polarization measurements are still needed to obtain agreement between the different partial-wave analyses.
Acknowledgements.The authors would like to thank Igor Strakovsky for supplying much of the database. This work was supported in part by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics Research Division, under Awards No. DE-FG02-01ER41194 and DE-SC0014323, and by the Department of Physics at Kent State University.
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Appendix A Final Fits to Experimental Data
Figures A - A show fits to and data for the observables , , , , and . The partial-wave amplitudes used to generate the curves are available in the form of data files Datafile . Also shown are the fits from BnGa 2016 Sarantsev and Jülich 2015b Julich .
Data sources shown in the plots are: HEUSCH 1966 Heusch66 , PREPOST 1967 Prepost67 , BACCI 1968 Bacci68 , BLOOM 1968 Bloom68 , ERBE 1968 Erbe68 , DELCOURT 1969 Delcourt69 , HOLT 1969 Holt69 , HEUSCH 1970 Heusch70 , HONGOH 1971 Hongoh71 , CHRIST 1973 Christ73 , BOOTH 1974 Booth74 , ABRAMIAN 1977 Abramian77 , VARTAPETYAN 1980 Vartapetyan80 , HOMMA 1988 Homma88 , KRUSHE 1995 Krushe95 , PRICE 1995 Price95 , BOCK 1998 Bock98 , AJAKA 1998 Ajaka98 , KOUZNETSOV 1998 Kouznetsov98 , DUGGER 2002 Dugger02 , AHRENS 2003 Ahrens03 , CREDE 2005 Crede05 , NAKABAYASHI 2006 Nakbayashi06 , BARTALINI 2007 Bartalini07 , ELSNER 2007 Elsner07 , SUMIHAMA 2009 Sumihama09 , WILLIAMS 2009 Williams09 , CREDE 2009 Crede09 , MCNICOLL 2010 Mcnicoll10 , SIKORA 2010 Sikora10 , AKONDI 2014 Akondi14 , SENDEROVICH 2014 Senderovich14 , WITTHAUER 2017 Witthauer17 , and KASHEVAROV 2016 Kashevarov17 .
figureFits to data for at = 1790 to 1990 MeV. See text for references.
figureFits to data for at = 1490 to 1710 MeV. See text for references.
figureFits to data for at = 1490 to 1850 MeV. See text for references.
figureFits to data for at = 1490 to 1850 MeV. See text for references.
figureFits to data for at = 1595 to 1735 MeV. See text for references.
figureFits to data for at = 1510 to 1885 MeV. See text for references.