Low-energy -nucleon interaction studied with photoproduction off the deuteron
We develop a reaction model for photoproduction off the deuteron (), and study the reaction at a special kinematics, where the photon beam energy is GeV and the scattered proton is detected at , for the purpose of determining the -nucleon scattering length () and effective range (). In this kinematics, the -nucleon elastic rescattering is significantly enhanced while other background mechanisms are suppressed. We show that a ratio , the cross section divided by the cross section convoluted with the proton momentum distribution in the deuteron, has a very good resolving power of and . We conclude that the data with 5% error, binned in 1 MeV width of the -neutron invariant mass, can determine () at the precision of 0.1 fm (0.5 fm), significantly narrowing down the previously estimated ranges of the parameters. To arrive at the conclusion, it is essential to use the reaction model equipped with elementary amplitudes that are well constrained by and reaction data through a sophisticated coupled-channels analysis. This result strongly motivates the Research Center for Electron Photon Science (ELPH) at Tohoku University to measure .
The low-energy interaction between the meson and the nucleon () is, as with the interaction, a basic feature of the meson-baryon dynamics. It is characterized by the two complex parameters, the scattering length and effective range , defined through an effective-range expansion of the -wave scattering amplitude: with , where is the on-shell momentum in the center-of-mass (c.m.) frame and the phase shift. Because determines the attractive or repulsive nature of the interaction at , the existence of exotic -mesic nuclei, which have been actively searched for experimentally, hinges on its value etan8 (); etan9 (). Accurate values of and can also greatly help determine the pole position of the -wave resonance, the first spin- negative-parity excitation of the nucleon; the pole is known to be near the threshold but its accurate position is still uncertain pdg (). It is known that the -wave scattering parameters can well determine an -wave resonance pole near threshold pis (); knlskp2 ().
Despite its important role in nuclear and hadron physics, the low-energy interaction has not been well understood yet. This is attributed mainly to the fact that neither direct scattering experiments nor -ray measurements from -mesic atoms are possible due to the neutral and unstable nature of , and thus one has to rely on indirect information. Previous works have attempted to extract and by analyzing the and reaction data that have a sensitivity to the interaction through coupled-channels effects etan9 (). The reaction has also been analyzed to extract the interaction embedded in the strongly interacting system etan5 (). These analyses gave fairly convergent results for the imaginary parts of and , the values of which fall into –0.3 fm and –0 fm, respectively etan9 (); gw-2005 (). However, their real parts scatter in a rather wide range: –0.9 fm and to +1 fm. The large model-dependence of the previously extracted and originates from the difficulty of isolating the scattering amplitudes from other mechanisms involved in the reactions analyzed. Therefore, it is highly desirable to utilize reactions in which mechanisms associated with the elastic rescattering are significantly enhanced while other background mechanisms being suppressed.
To meet this demand, an photoproduction experiment plan () is planned at the Research Center for Electron Photon Science (ELPH), Tohoku University. In this experiment, a deuteron target is irradiated with a photon beam at the laboratory energy of GeV tagger (); bpm (), and the recoil proton from the reaction is detected at from the photon direction. At this chosen kinematics, an is likely to be produced almost at rest, being expected to strongly interact with the spectator neutron. Meanwhile, the struck proton goes away with a large momentum, leaving little chance to interact with the and neutron. This seems an ideal kinematical condition, to which we refer as the ELPH kinematics, to determine the low-energy scattering parameters. In this Rapid Communication, we show with a theoretical analysis that a combined cross-section data for and expected to be taken at the ELPH experiment would indeed lead to significant reduction of the uncertainty of and previously extracted, thereby providing crucial constraints on the existence of -mesic nuclei and the properties of .
The possibility of extracting from data was first explored by Sibirtsev et al. juelich2002a (), and a fairly large -dependence of angular and momentum distributions was shown. However, a subsequent work by Fix et al. mainz2004 () found a significantly less pronounced -dependence than those of Ref. juelich2002a (), leading to the conclusion that it is practically impossible to extract from data. Thus, until the present work, no practically useful connection has been made between data and . We note that these pioneering works juelich2002a (); mainz2004 () examined near the threshold ( 0.7 GeV) while we study the reaction in rather different kinematics ( GeV; ).
We study relevant to the ELPH experiment with a model based on the impulse and the first-order rescattering mechanisms as depicted in Fig. 1. The -exchange mechanism [Fig. 1(b)] contains the subprocess we are interested in, while the other mechanisms (the impulse [Fig. 1(a)], -exchange [Fig. 1(c)], and -rescattering [Fig. 1(d)] mechanisms) are backgrounds for our purpose. With the momenta defined in Fig. 1, the amplitudes for (impulse), (-exchange), (-exchange), and (-rescattering), are explicitly written in the laboratory frame as
plus the exchange terms obtained from Eqs. (1)–(3) by flipping the overall sign and interchanging all subscripts 1 and 2 such as . The elementary (off-shell) amplitudes for photoproduction, meson-baryon, and rescattering are denoted by , , and , respectively. Here, is the deuteron state at rest with spin projection ; the nucleon state with momentum and spin and isospin projections and ; the photon state with momentum and polarization ; and (=) the pseudoscalar meson state with momentum . The total scattering energy of the system in the laboratory frame is given by the sum of the photon laboratory energy and the deuteron rest mass, , and the invariant masses of the two-body subprocesses in the above equations are defined to be , , and , where with being the mass of a particle .
The above definition on would call for an explanation, because other choices of have also been seen in the literature w-choice (). We calculate the mechanisms [Fig. 1(a)–1(d)] in a manner consistent with the well-established Faddeev framework up to the truncated higher order terms. The Faddeev framework uniquely specifies the energy (and thus ) of an interacting two-body subsystem in an intermediate state. A requirement is to combine the equation with elementary (off-shell) amplitudes calculated consistently with the Faddeev framework. Our elementary amplitudes are, as discussed shortly, calculated with meson-nucleon and nucleon-nucleon potentials that perfectly fit the Faddeev framework. Meanwhile, another prescription of corresponds to another three-dimensional scattering equation that should work with its own consistent elementary amplitudes but not with ours. Therefore, it does not make sense for us to use the other choices of . However, if one uses dynamical inputs that are not consistent with any of the scattering frameworks, as has been the case in most of the past works, there is no principle to determine , and thus various choices need to be considered.
We now specify our reaction model to evaluate Eqs. (1)–(3). The model must be built with reliable amplitudes for elementary , , and processes with =, as well as with a realistic deuteron wave function, so that we can reliably isolate the amplitude for the subprocess from data with well-predicted contributions from all the other background mechanisms. Regarding and amplitudes, we employ those generated with a dynamical coupled-channels (DCC) model knls13 (); knls16 (). The DCC model is a multichannel unitary model for the and reactions in the nucleon resonance region. It was constructed fitting data points, and successfully describes knls13 (); knls16 (); pipin () and reactions over the energy region from the thresholds up to GeV. As an example, we present the differential cross sections calculated with the DCC model of Ref. knls16 () in Fig. 2. The figure shows a very good agreement between the model and data gp-etap-data () over the energy region relevant to the following calculations of . This verifies that the most important amplitudes out of the elementary amplitudes for describing are well constrained by the data. This DCC model predicts the scattering parameters to be fm and fm, which are consistent with the previously estimated ranges. As for the deuteron wave function and the scattering amplitudes, we employ those generated with the CD-Bonn potential cdbonn ().
Previous models mainz1997 (); juelich2002a (); mainz2004 (); mainz2015 () also took account of the mechanisms shown in Fig. 1; the -exchange mechanism was considered only in Ref. mainz2015 (). However, comparing the elementary amplitudes implemented in the previous models, the DCC model possesses unique and sound features such as: (i) the model describes all the meson-baryon and photoproduction amplitudes relevant to in a unified manner; (ii) the model generates, by construction, off-shell amplitudes that are well-suited for working with the Faddeev framework. We also note that a simple model including only the -excitation mechanism mainz1997 (); juelich2002a () is not enough for practically describing at the ELPH kinematics because the amplitudes of –1.7 GeV give a large contribution.
The setup described above allows us to make a parameter-free prediction for the cross sections. We thus confront our model predictions with existing data, thereby assessing the validity of the model. In Fig. 3, we show the angular distribution at and 775 MeV from our DCC-based model with and without the rescattering contributions along with the data. Our parameter-free prediction is found to be in an excellent agreement with the data. A slight enhancement in the backward direction due to the rescattering is important for this agreement. Fix et al. mainz2015 () also have done a comparable calculation, and found a rather minor role of the rescattering mechanism in the angular distribution at these energies. The slight underestimation of their results at backward angles (Fig. 5 of Ref. mainz2015 ()) is likely to be ascribable to the different scattering lengths; fm in our model and fm in Ref. mainz2015 (). Regarding the cross sections with the impulse mechanism only, our result is close to that of Ref. mainz2015 () while significantly smaller than that of Ref. juelich2002a (). See Ref. mainz2004 () for a detailed discussion on the difference with Ref. juelich2002a ().
Now let us consider the reaction at the ELPH kinematics with GeV and . In Fig. 4(top), our model predictions for the threefold differential cross section, , are presented as a function of . We find that the dominant contribution is from the impulse mechanism [Fig. 1(a)] that contains the amplitudes, while the amplitudes negligibly contribute. The -exchange mechanism [Fig. 1(b)] has a substantial contribution to the cross section, which changes the impulse result by 40 to +20% [difference between the dashed and dotted curves in Fig. 4(bottom)]. Meanwhile, the -exchange [Fig. 1(c)] contribution is smaller, and suppresses the cross sections by 9% (difference between the dashed and dash-dotted curves). The rescattering [Fig. 1(d)] contribution (deviation of the dash-dotted curve from 1) is very small for GeV. This feature is what we expect to find in this special kinematics. The -exchange mechanism is strongly suppressed even though the elementary amplitude is significantly larger than that of at the considered energies. This is the exchanged pions have rather large momenta near their on-shell, picking up high-momentum components with very small probabilities in the deuteron wave function. The -rescattering mechanism is hindered by the same kinematical reason, and also by the rather weak scattering at this kinematics where the relative momentum is large.
We have shown that the in the ELPH kinematics for GeV are described with the impulse and -exchange mechanisms and with the smaller (almost negligible) correction from the -exchange (-rescattering) mechanism. This indicates that the proton is well separated from interacting with the system, and thus multiple rescatterings beyond the first-order rescattering [Figs. 1(b)–1(d)] should be safely neglected in this kinematical region. We have also confirmed that an off-shell momentum effect associated with the scattering amplitude is very small and that partial wave amplitudes higher than the wave give negligibly small contributions. These facts allow us to modify the full model by replacing the scattering amplitude with the -wave one parametrized with and , and then to determine these parameters through analyzing the forthcoming ELPH data. To make contact with the ELPH data, we need to take one more step because the data are actually given in a form of the ratio, denoted by , of the measured cross sections for divided by those for convoluted with the proton momentum distribution in the deuteron. This is for removing systematic uncertainties of the acceptance from the detector coverage. Thus, from the theoretical side, the corresponding quantity to calculate is:
where () is calculated with the modified full model (the impulse term only). The remaining questions to address are how sensitively changes as and are varied, and how well with a certain error can determine and .
First we vary over 0.2 – 1.0 fm, with fixed values of fm and fm. At the ELPH kinematics and GeV, the obtained cross sections are mostly within the red striped region shown in Fig. 5(top). The corresponding variation of is shown in Fig. 5(bottom) where the sensitivity to the variation of is more clearly seen. As the striped bands show, the cross section and thus changes by 25% at the quasi-free (QF) peak position at GeV. Meanwhile, the green solid bands, which are covered when is varied by fm from 0.6 fm, have the widths of 5% at the QF peak. The result indicates that data of 5% error per MeV bin, which is achievable in the planned ELPH experiment plan (), can determine at the precision of fm, significantly narrowing down the current uncertainty.
Next we vary over a rather broad range of the current estimates, 6 – 0 fm; the scattering length is fixed at fm, the value from the latest DCC analysis knls16 (); fm. The corresponding changes of the cross section and cover the red striped region in Fig. 6.
Because plays no role very close to the threshold, its effect starts to be visible at 5 MeV above the threshold. The red striped (green solid) band of shows that at GeV changes by 30% (5%) when is varied over 6 – 0 fm (3.5 to 2.5 fm). Therefore, data of 5% error per MeV bin can also determine at the precision of 0.5 fm, significantly improved precision over the current estimates.
Regarding the imaginary part, we vary in the range of 0.2 – 0.3 fm, the currently estimated range, and with fm and fm. The cross sections and change at most 5%. When varying over the currently estimated range, 1 – 0 fm, with fm and fm being fixed, we found a similar situation.
We argue that theoretical uncertainties hardly affect the above results. A major part of the uncertainty of the cross section is from the elementary amplitudes that take over errors (5%) from data fitted. However, what we need in analyzing the ELPH data is not the cross section itself but in which theoretical uncertainty in the cross section is largely canceled out. We have confirmed that is very stable (0.1%) even when the overall magnitude of the amplitudes is varied over %. Another possible source of the uncertainty is the subthreshold amplitudes which are not well-constrained by the data. However, at the ELPH kinematics, the cross sections (and thus ) are found to hardly change (0.1% at the QF peak; 1% for GeV) even when the subthreshold contributions are omitted. We have also studied the model dependence of the deuteron wave function. We used those of the CD-Bonn cdbonn (), Nijmegen I nij (), and Reid93 nij () models, and found a rather good stability ( 0.5% at the QF peak; 1% at GeV) of .
Finally, we make clear what we have advanced from the previous investigations juelich2002a (); mainz2004 () on extracting from data. For this purpose, it would be illustrative to compare our main result (Fig. 5) with Fig. 6 (bottom) of Ref. mainz2004 () that also shows the -dependence of differential cross sections at a fixed proton angle. Despite the similarity, the authors of Ref. mainz2004 () were concerned with the cross section shape while we utilize the absolute values of that has a significantly better sensitivity to the scattering parameters. What enables us to utilize the values is our very well-controlled calculation as follows. At the kinematics chosen in Ref. mainz2004 () ( MeV, , ), according to our model, we found: (i) the elementary amplitudes give a contribution comparable to that from the amplitudes; (ii) the subthreshold amplitudes give a 30% contribution; (iii) the rescattering contribution is not well suppressed (10% contribution) and thus, considering the precision in question, a contribution from multiple rescatterings beyond the first-order rescattering would be nonnegligible. On the other hand, our result obtained at the ELPH kinematics is essentially free from these contributions (i)-(iii) that are currently difficult to control with a high precision. Another benefit of utilizing the ELPH kinematics is that the cross sections are fairly large near the QF peak, making a precise measurement possible. Indeed, our cross sections at the QF peak in Fig. 5 are 20 times larger than those shown in Fig. 6 (bottom) of Ref. mainz2004 (). One more advancement is that we proposed to use the ratio, Eq. (4), to cancel out the 5% uncertainty inherent in any elementary amplitudes. The advancements described above lead us to a conclusion that it will be possible to significantly improve the precision of the scattering parameters using the ELPH data.
In conclusion, we have analyzed the reaction at GeV and , and found that, once data of 5% error binned in 1 MeV width are given, () can be determined at the precision of fm ( fm), which is significantly better than the currently estimated uncertainty. We emphasize that, for reliably extracting the scattering parameters from the data, it is prerequisite to control all the relevant subprocess in with a sophisticated model like the DCC model knls13 (); knls16 ().
Acknowledgements.We thank T. Sato for a useful discussion. This work was supported in part by JSPS KAKENHI Grant Nos. JP25105010, JP25800149, and JP26400287, and Fundação de Amparo à Pesquisa do Estado de São Paulo-FAPESP, Process No. 2016/15618-8.
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