Dynamical model for pion photo-production reactions on the deuteron and the extraction of neutron-target observables
Within the multiple scattering formulation, we predict the cross sections of by using a dynamical model of and reactions in the nucleon resonance region. The calculations include the impulse term and the final-state interaction (FSI) terms due to pion-exchange and nucleon-exchange. We show that the off-shell effects, calculated from the meson-exchange mechanisms, on the propagations of the exchanged nucleon and pion are significant in determining the reaction amplitudes. The FSI effects on the predicted cross sections are found to be important at energies near the (1232) resonance, and are still significant at higher energies. The results are in good agreement with most of the available data of and reactions. We use our predictions to examine a commonly used procedure for extracting the (un)polarized observables from the data. It is demonstrated that the extracted unpolarized cross sections still contain the FSI effects even after the kinematical cuts are applied to isolate the quasi-free events. However the same procedure works well for the polarization observables , , and . We also find that the FSI effects on the determinations of the cross sections of can be somewhat different from those of . We point out the importance of applying a cut on the final invariant mass to efficiently suppress a problematic Fermi smearing effect, thereby maintaining a good accuracy of the extraction. An experiment-theory iteration procedure for extracting the observables of from the data is suggested.
pacs:11.80.La, 13.60.Le, 13.88.+e, 14.20.Gk
LFTC-18-6/27, J-PARC TH-0129
The developments of dynamical models for and reactions were motivated by the success of the meson-exchange models machleidt () for the interactions and the earlier attempts donna () to relate the isobar models of nucleon resonances to the predictions of constituent quark models capstick (). Within the formulation given in Refs. sl96 (); msl07 (), the development of dynamical models of and reactions has two main objectives:
Develop interpretations of the nucleon resonances () within the framework that the excitations of the nucleon can be described in terms of bare baryon states and meson-exchange interactions. With the parameters determined by fitting the data of and reactions up to the invariant mass GeV, the poles and residues of nucleon resonances are extracted by performing analytic continuations of the partial-wave amplitudes predicted within the constructed model to the complex energy-plane.
Apply the constructed dynamical model to investigate the propagations of mesons and nucleon resonances in nuclei, which are crucial for analyzing data from experiments on nuclear targets in the nucleon resonance region, such as the recent experiments on the neutrino oscillations nu-exp ().
With the efforts reported in Refs. jlms07 (); jlmss08 (); djlss08 (); ssl09 (); kjlms09a (); jklmts09 (); kjlms09b (); sjklms10 (); knls10 (); ssl10 (); shkl11 (); knls13 (); knls16 (), we have achieved the first objective. In this paper, we will take a step toward reaching the second objective by applying the constructed dynamical coupled-channel model (ANL-Osaka model) presented in Ref. knls13 (); knls16 () to investigate pion photo-production on the simplest nucleus, the deuteron (). We will focus on the reactions for two reasons:
An uncertainty in the construction of the ANL-Osaka model was in the determination of the isospin structure of the nucleon resonances by fitting the available data of which were extracted from the data with some procedures to select the events. Furthermore, data included in our previous fit were rather scarce.111 For determining the isospin structure of nucleon resonances, data and either of or data are needed in the fit if the data are of very high accuracy. This is however not the case, thus the fits including both of the neutron data may be needed to reduce the uncertainty. We therefore need to examine whether our predictions of will be in agreement with the original data of and .
The extractions of cross sections from the data of and are important tasks at Jefferson Laboratory (JLab) jlab1 (); jlab2 (); clas1 (); clas2 () and Mainz mami3 (); mami1 (); mami4 (); mami5 (); mami2 (). It is therefore interesting to use our predictions to examine the extent to which their procedures are consistent with a dynamical model approach which has significant differences with the previous approaches tara (); tara-1 ().
One of the important features of the ANL-Osaka model is that the amplitudes are generated from an energy Hamiltonian which can be included in the conventional Hamiltonian formulation for developing many-body descriptions of nuclear reactions. This is achieved by using a unitary transformation method ut-osaka (); sl96 () to derive hadron-hadron interactions from relativistic quantum field theory with meson and baryon degrees of freedom. This feature allows us to use rigorously the well-established multiple-scattering formulation feshbach () to develop a reaction model for reactions. For our limited purpose here, we follow the previous investigations arenhover (); fix (); lev06 (); sch10 (); wsl15 (); tara (); tara-1 () to only include the single scattering (impulse) amplitudes and the double scattering amplitudes due to pion-exchange and nucleon-exchange mechanisms. The exchanges of unstable particles , and (1232), which can be generated from the ANL-Osaka model, are more difficult to calculate and are neglected in this work for simplicity.
Thus our task is to develop formula for calculating the amplitudes illustrated in Fig. 1. The and amplitudes for evaluating the pion re-scattering term of Fig. 1(c) can be generated from the ANL-Osaka model. The initial deuteron wave function and the amplitudes in Fig. 1(b) can be generated from any of the available high-precision potentials. To be consistent with the meson-exchange mechanisms of the ANL-Osaka model, we choose the CD-Bonn potential cdbonn (). There are two important issues in practical calculations of the matrix elements of these mechanisms, as discussed well in the earlier investigations thomas (); lee75 () of the multiple scattering of hadrons from nuclei within the Hamiltonian formulation. First, we need to define a Lorentz boost transformation to relate the , and amplitudes in Fig. 1 in the photon-deuteron laboratory frame to those in the two-body center of mass (CM) frame where the two-body amplitudes are generated from Hamiltonian by solving scattering equations in partial-wave representation. In particular, the spin rotations must be taken into account relativistically for investigating polarization observables. Here we follow the method of relativistic quantum mechanics, as detailed in Ref. polyzou (); polyzou-1 (); kei-pol (). The second issue is that the resulting FSI amplitudes will include loop-integrations over the off-energy-shell matrix elements of the two-body amplitudes in Fig. 1. Thus it is necessary to introduce an approximation to choose the collision energies in generating these two-body off-shell matrix elements. Here we will use an approach which accounts for the energy shared by the spectator nucleon and pion in Fig. 1.
A commonly used procedure of extracting cross sections for the neutron target from data on the deuteron is to apply a certain set of kinematical cuts to the deuteron data and assume that the selected events are from the quasi-free processes on a single nucleon. However, this procedure has been always with a concern that the rescattering effects, as illustrated in Fig. 1, can still be in the selected events and must be accounted for in the extraction. This has been addressed in recent years in Refs. jlab1 (); tara (); tara-1 () within a model using the two-body amplitudes generated from the SAID model said (). Here we will also investigate this problem by using our model with the mechanisms illustrated in Fig. 1. The mechanisms considered in Refs. jlab1 (); tara () for are the same as those illustrated in Fig. 1, while the pion-exchange mechanism [Fig. 1(c)] is not considered in Ref. tara-1 () for . Furthermore our approach has significant differences with Refs. jlab1 (); tara (); tara-1 () in the formulation of the reaction and the calculations. In particular, the off-shell propagations of pions and nucleons are treated rather differently.
Within our model, we develop explicit formulas, which can account for the kinematical cuts recently employed by the experimental groups, to predict cross sections. The predicted results are taken as ‘data’ to extract the observables which are then compared with those on a free neutron calculated from the ANL-Osaka model. The differences in the comparison determine the extent to which the FSI have been removed, or some artifacts have been introduced, by using the kinematical cuts to extract the observables. We will examine the differences with the investigations of Refs. tara (); tara-1 () in extracting the unpolarized differential cross sections. Meanwhile, no theoretical study with a photon-deuteron reaction model as ours has been done on the extraction of polarization observables for of the current interests clas1 (); mami2 (); we will conduct such a study for the first time.
The organization of the rest of this paper is as follows. In Sec. II, we present formulas for calculating the amplitudes of the impulse term [Fig. 1.(a)], nucleon re-scattering (-exchange) term [Fig. 1.(b)], and pion re-scattering (-exchange) term [Fig. 1.(c)]. The results for comparing our predictions with the available data of and , and for examining the importance of FSI and off-shell effects will be given in Sec. III. In Sec. IV, we examine within our model the extent to which the procedure using kinematical cuts to extract from the data of can be justified. The discussions on future possible developments are given in Sec. V.
In this section, we describe the theoretical formulation used in our investigation of reactions. All of the formulas for calculations are presented with the normalizations: for plane-wave states and for bound states.
ii.1 Model Hamiltonian with meson and baryon degrees of freedom
Our starting point is the following many-body Hamiltonian
where is the free Hamiltonian for all particles in the considered processes, is a nucleon-nucleon potential between the nucleons and , and
is the interaction Hamiltonian of the ANL-Osaka (AO) model. The channels included are and with resonant , and components. The energy independent meson-exchange potentials are derived from phenomenological Lagrangians by using the unitary transformation method sl96 (); ut-osaka (). The vertex interaction defines the formation of a bare state from a channel . The parameters of the Hamiltonian have been determined in Refs. knls13 (); knls16 () by fitting about 26,000 data points of the and reactions up to the invariant mass GeV.
Here we mention that the model Hamiltonian defined above is a very significant extension of the Hamiltonian developed in Refs. lee83 (); mitzutani () where only , , and degrees of freedom are included. It can be used to investigate , , and two-pion production reactions on nuclei in the nucleon resonance region. The first attempt in this direction was made to investigate the reaction in the context of extracting the low-energy scattering parameters etaN (). Here we focus on the pion photo-production on the deuteron to test our predictions and to also address the issues concerning the extractions of the observables of the from the data of reactions.
ii.2 Scattering amplitudes for reactions
Starting with the scattering -matrix given by the Hamiltonian Eq. (1), it is straightforward to follow the well-developed procedures feshbach () to obtain a multiple scattering formulation of the reactions. Following the previous investigations arenhover (); wsl15 (); tara (); tara-1 (), we will only keep the single-scattering (impulse) terms and the double scattering terms with intermediate states. The -matrix operator for the reactions can then be written as
where is the total energy of the system, is a scattering operator associated with the -th nucleon in the deuteron. Note that the matrix elements of are determined by the total Hamiltonian Eq. (1) and cannot be calculated exactly. Therefore, we adopt the spectator approximation thomas (); lee75 () as
where is the two-body scattering operator in free space, and is the energy of the spectator of the two-body scattering in the or three-particle states, as seen in Fig. 1. The resulting -matrix operator is consistent with that of the Faddeev framework up to and including the double scattering terms, as has been also discussed in Ref. etaN ().
To proceed further, we need to define the matrix elements of the scattering operator with an invariant mass . Within the ANL-Osaka model, these matrix elements are generated in the two-body CM frame by solving the following coupled-channel equations in each partial wave:
where , and are the momenta in the CM frame, is the energy of a particle in a channel with the mass , are the considered channels, and channel is included perturbatively; is the self-energy for the unstable channels , and is zero for the other stable channels. The driving term is
where is the bare mass of an excited nucleon state ; is a particle-exchange Z-diagram in which a channel is included. Note that the initial and final states of each matrix element in Eq. (5) can be on-energy-shell  or off-energy-shell  within the Hamiltonian formulation.
where the sub-indices and stand for the final state , and the initial state , respectively, is the photon energy, and for the laboratory frame,
The exchange terms in Eq. (LABEL:eq:amp_decomp) can be obtained from Eqs. (8)-(10) by flipping the overall sign and interchanging all subscripts 1 and 2 for the nucleons in the intermediate and final states. Here, the deuteron state with spin projection is denoted as ; the nucleon state with momentum and spin and isospin projections and ; the photon state with momentum and polarization ; the pion state with momentum and the isospin projection . The total energy in the laboratory frame, , is given by where is the deuteron mass. The invariant masses for the matrix elements of the two-body subprocesses are calculated according to the momentum variables specified in Fig. 1 and of the spectator approximation defined by Eq. (4):
To be consistent with the chosen CD-Bonn potential cdbonn () for generating the scattering amplitudes and the deuteron bound state, the two-nucleon energy in the propagator of the nucleon re-scattering amplitude in Eq. (9), is calculated with a non-relativistic approximation
The amplitudes of the pion photoproduction and of the pion-nucleon scattering in Eqs. (8)-(10) are first generated from the ANL-Osaka model by solving the coupled-channel equation of Eq. (5) in the two-body CM frame. The resulting matrix elements are then boosted to the considered -deuteron frame. Here we follow the approach of Ref. polyzou () based on the instant form of relativistic quantum mechanics kei-pol (). The same frame-transformation procedure is also needed to calculate the matrix element of scattering in Eq. (9). The formulas for calculating these matrix elements in the - deuteron frame are given in Appendix A.
Here we note an important difference between our dynamical model approach and the approach of Refs. tara (); tara-1 (). In the loop-integrations of Eqs. (9) and (10), the two-body matrix elements can be off-energy-shell and are calculated exactly from the ANL-Osaka model and the CD-Bonn potential as explained above. The equations for calculating FSI in Refs. tara (); tara-1 () are similar to our expressions, but the two-body matrix elements are taken as their on-shell values and are taken out of the loop-integrations. For , they include a monopole form factor of Ref. lev06 () to account for the off-shell effect. These simplifications greatly reduce the computation task. We will examine the on-shell approximation within our formulation in Sec. III.
ii.3 Cross section formula
The reaction cross section for is defined by
where is the relative velocity of the initial - system, and the Lorentz invariant amplitude has been defined by Eqs. (LABEL:eq:amp_decomp)-(10). The unpolarized differential cross section with respect to the pion emission angle in the laboratory frame () derived from Eq. (15) can be written as
where is the photon polarization, is the -component of the deuteron spin, and
Here is the total energy in the laboratory frame, the momentum of in the CM frame, and the invariant mass of the two nucleons in the final state. The magnitudes of the momenta are simply denoted, for example, by , and . To evaluate the Lorentz invariant amplitude using Eqs. (LABEL:eq:amp_decomp)-(10), we need to express each outgoing momentum in Fig. 1 in terms of , and . This can be done by using the standard Lorentz transformation. For a given , , and the angles , we can calculate them by
We will also present results of the differential cross section in the the CM frame of the - system. We find that
where and can be obtained from and in the laboratory frame by the Lorentz transformation with . The momentum variables for the calculations of the invariant amplitudes using Eqs. (LABEL:eq:amp_decomp)-(10) can be obtained from the same Eqs. (19)-(24), but setting .
We will also present results for three polarization observables of current interest. We choose the -axis along the incident photon direction, , and the -axis along the vector product, . Then, the photon asymmetry for the reaction is defined by
where denotes either of differential cross sections of Eqs. (16) or (25), but the initial photon polarization is fixed to instead of the average. The components for the linear photon polarization in Eq. (28) are
The asymmetry of the circularly polarized photons on a deuteron target polarized along the -axis () is defined by
The asymmetry of the linearly polarized photons on a polarized deuteron is defined by
where the components for the photon polarization are
In our calculations of the amplitudes Eqs. (8)-(10), the partial-wave amplitudes up to and including (: the orbital angular momentum) generated from the ANL-Osaka model of Refs. knls13 (); knls16 () are taken into account. To calculate the loop-integrations in Eqs. (9) and (10) for the - and -exchange mechanisms, respectively, the off-energy-shell two-body matrix elements are generated from solving the scattering equation Eq. (5) for the meson-nucleon scattering and a similar Lippmann-Schwinger equation of Eq. (LABEL:eq:LS) for the scattering. The partial wave amplitudes up to and including (: the total angular momentum) generated from the CD-Bonn potential are included in the calculations.
We first compare our predictions with the available data of and reactions. We then examine whether our results can be improved if the employed model of Ref. knls16 () is adjusted to also fit the recent Mainz data mami1 (); mami2 () of . We also examine the roles of the off-energy-shell and FSI effects in explaining the data. The differences with the approach of Refs. tara (); tara-1 () will also be discussed.
iii.1 Comparisons with data of reactions
We have found that our predictions agree reasonably well with the available data of and reactions. In Figs. 2 and 3, we show some typical comparisons of the data for unpolarized differential cross sections and our results (blue dotted curves) using the amplitudes generated from the ANL-Osaka model. Clearly there are some discrepancies with the data, in particular for in the 500–750 MeV region ( is the photon energy in the laboratory frame). One possible source of the discrepancies is that, when determining the parameters for the ANL-Osaka model, we used rather scarce dataset that are extracted from data. To see whether this is the case, we have extended the fit of Ref. knls16 () to also include recently extracted data for from Mainz mami1 (); mami2 (), and also recent data for from JLab clas1 (); clas2 (). However, the amount of these new data are much less than the total number of the world data for included in the fit knls13 (); knls16 () for the ANL-Osaka model. We thus do not expect that the resonance parameters and the hadronic parameters determined in Ref. knls16 () will be changed significantly by including the recent data, and therefore vary only the parameters for the bare couplings in the new fit.
To be consistent with the procedure in constructing the ANL-Osaka model, the calculations for the fits include partial waves. In addition to the data of and included in the fits in Ref. knls16 (), the recent data mami1 (); mami2 (); clas1 (); clas2 () of the unpolarized differential cross sections and for are included in the new fits. In Fig. 4, we show the comparisons of our results with the data of the differential cross sections. The fits to the data which are not shown in Fig. 4 are of similar quality and hence are omitted to simplify the presentation. Clearly, the results from this new fit (red solid curves) and those from the ANL-Osaka model (blue dashed curves) are very similar. On the other hand, we see in Fig. 5 some large differences between the fits and the data for the polarization observable . Further improvements are possible only if we also vary the non-resonant parameters around the values determined in Ref. knls16 (). This will be worthwhile to pursue when the data for other spin observables, such as and , become available. For our present purposes, the quality of the fit shown in Figs. 4-5 is sufficiently good. We note here that the higher partial waves (), which include only non-resonant Born amplitudes, just slightly change the shape of the angular distributions for in the forward pion kinematics, and has little effects on . These weak higher partial wave amplitudes are therefore neglected in all of our calculations for the reactions.
With the parameters from the new fits, the calculated differential cross sections for and are the solid curves in Figs. 2 and 3. Comparing with the blue dotted curves from using the ANL-Osaka amplitudes, we see that the agreements with the data for in the region MeV are improved, but some significant discrepancies with the data remain. It could be due to the neglect of higher order terms in our multiple scattering calculations such as -, -, and -exchange terms. On the other hand, we must examine critically the procedures used in extracting the cross sections from the data of . This is the subject we will address in Sec. IV. The results presented in the rest of this paper are from using the parameters from the new fit.
iii.2 Off-shell effects
In Sec. II, we show that the two-body matrix elements in the calculations of amplitudes can be off-energy-shell within the Hamiltonian formulation where all particles are on their mass-shell. In Fig. 6, we see that, if we set all two-body matrix elements in Eqs. (8)–(10) to their on-shell values, the full results (solid curves) are changed to the blue dashed curves. In general, the effects are more significant at low energies and less at higher energies, as illustrated at energies near 300 and 700 MeV in Fig. 6. The differences between the solid and blue dashed curves suggest that the extracted using the on-shell calculation results of Refs. tara () are reasonable, but could be modified if the off-shell effects are included in their formulation. The off-shell effects are however model dependent and one must use well-established physics to determine them. In our approach, the off-shell effects are determined by the meson-exchange mechanisms in the ANL-Osaka model and CD-Bonn potential. Similar considerations must also be taken in other approaches.
If we further keep only the on-shell pole terms of the propagators by setting
in the loop integrations in Eqs. (9) and (10), we obtain the green dotted curves. Clearly, they are significantly different from the solid curves of the full calculations including the off-shell effects, in particular, at 305 MeV for , and in the forward angles () at 700 MeV for . Here we mention that Eq. (34) is sometimes used to reduce the computation effort for the loop-integration. The differences between the solid and dotted curves in Fig. 6 give some estimates on the accuracy of such a simplified approach which neglects the off-shell effects completely.
iii.3 Final state interaction effects
In all of the previous investigations of reaction, the -exchange term illustrated in Fig. 1(b) is found to dominant the FSI. This can be examined by using Eq. (27) to calculate the dependence of the differential cross sections on the invariant mass . In the upper half of Fig. 7, we see at a small angle of that the main contributions to the differential cross sections of at MeV are in the region close to the threshold where the scattering is dominated by the partial wave which is orthogonal to the deuteron bound state. Consequently, the FSI greatly reduces the cross section of the impulse () term (dashed curve) to that of the terms (dotted curve). When the pion-exchange term () is included, we obtain the solid curve which is almost indistinguishable from the dotted curve. As the scattering angle increases to , the peak position of the spectrum is shifted to the larger invariant mass region where the reduction due to the FSI is still large, but much less than that in the forward angles. The FSI effects become weak at large angles and since the large portion of the spectrum is in the region far away from the threshold.
At a higher photon energy of MeV (lower half of Fig. 7), the reduction due to the FSI is similarly dependent on how close the spectrum peak is to the threshold. We also note that while the FSI effect is much smaller than the FSI effect overall, it can give a main FSI correction at large pion emission angles; for example, see the results at in the lower half of Fig. 7. Integrating over , the FSI effects on the differential cross sections of are shown in Fig. 8. The large reduction due to the FSI is clear. Clearly, the large reduction due to FSI is essential in obtaining a reasonable agreement with the data in Fig. 2. (Note that the results in Fig. 2 is given in a different frame chosen in Ref. gd-pi0pn_data () to present the data). This is similar to the previous findings arenhover (); fix (); tara-1 (). The FSI effects at higher energies are weaker, but still significant in the forward angle, .
The FSI effects on are shown in Fig. 9. Here, we see that the FSI effect is also clearly visible in the small angle region where the spectrum peak is close to the threshold and hence the FSI going to the partial wave gives a dominant portion of the FSI effect. Contrary to the case, its interference with the impulse term is to enhance the cross section from the dashed curves to the dotted curves. The FSI effect is also weak here, as can be seen in the negligible difference between the solid and dotted curves. As the angle increases, the FSI effect is getting discernible but overall the FSI effect quickly becomes much smaller. Therefore, the FSI effect on the differential cross sections are very weak, as seen in Fig. 10.
We have also investigated the FSI effects on the polarization observables , , and defined in Eqs. (28)-(32) which are of current interests. The results are shown in Figs. 11 and 12. Clearly, the FSI effects do not play an important role here. It will be interesting to compare our predictions with the data in near future.
Iv Extraction of cross sections from data
We now turn to investigating the extractions of the unpolarized cross sections and polarization observables for the from the data of reactions.
iv.1 Unpolarized cross sections
For extracting cross sections of from those of , we need a formula that gives a relation between them. For deriving it, we first consider an ideal situation that the incident photon interacts with only one of the nucleons inside the deuteron [Fig. 1(a)]. The cross section for this ‘quasi-free’ process can be calculated from keeping only the first term of Eq. (LABEL:eq:amp_decomp) in our calculations; the FSI terms and exchange terms are neglected. In this simplified situation, it is straightforward to derive from Eq. (15) that the cross section () is related to the cross section () by