Introduction

Using photomeson reactions to study the structure of hadronic systems offers several significant advantages. First, the electromagnetic photon–nucleon interaction is well described within the framework of hadron electrodynamics. Second, at the large momentum transfers, characteristic of these reactions, virtual photons scan a nucleus at short distances, which allows one to obtain more complete information about nuclear interaction.

Photomeson reactions also enable the investigation of the structure of excited baryon states, particularly their excitation spectra and electromagnetic transition form factors. This bridges functional approaches in quantum chromodynamics with methods based on effective field theories and connects directly to a wealth of existing experimental studies1,2. Experiments particularly well suited for these purposes are the quasi-free production studies on light nuclei, primarily the deuteron. Due to its low binding energy and well-understood structure, the deuteron serves as an effective laboratory for investigating various aspects of meson-nucleon dynamics in electromagnetic reactions under controlled conditions.

Despite these advantages, the interpretation of the corresponding experimental results often depends on the theoretical model employed to describe the reaction. The main reason is that QCD, the fundamental theory of the strong interaction, cannot be treated perturbatively at low energies. This forces us to apply various approaches based on effective field theory, lattice QCD, chiral perturbation theory, non-relativistic scattering theory, and constituent quark model2,3,4,5,6. Being inherently phenomenological, these theoretical frameworks typically have limited predictive power. Consequently, different models often yield similar results only for a restricted set of observables, most notably the unpolarized differential cross section, while their predictions for other observables may differ significantly. Therefore, to assess the validity and applicability of various model approaches, it is essential to measure as broad a set of independent observables as possible.

In the present paper, we study the incoherent photoproduction of negative pions on the deuteron

$$\begin{aligned} \gamma + d \rightarrow \pi ^{-}+p+p. \end{aligned}$$
(1)

In the intermediate photon energy region (\(E_{\gamma }<1\) GeV), this reaction is well described within the impulse approximation, which leads to the so-called spectator model. In this framework, the pion is assumed to be produced on a single quasi-free (active) nucleon, while the second nucleon acts as a spectator and does not participate in the interaction. This model has been widely employed in the literature (see, e.g.,7,8,9 and references therein) and provides a reliable first approximation, capturing the dominant dynamical features of the reaction, at least in kinematic regions close to the quasi-free maximum.

The most important corrections to the quasi-free mechanism arise from final state interaction (FSI), specifically, NN and \(\pi N\) rescatterings. These additional mechanisms can significantly affect the energy and angular distributions of the outgoing particles and must therefore be included in any realistic description. Moreover, FSI in pion photoproduction is of intrinsic interest: the \(\pi NN\) system constitutes the simplest three-body system in which one of the particles, the pion, can be emitted or absorbed. In this sense, the dynamics of this system lies between that of nonrelativistic scattering theory and full quantum field theory. Consequently, the consistent inclusion of FSI effects in pion photoproduction on deuterons represents an important challenge in intermediate-energy pion-nuclear physics. This has motivated numerous theoretical studies aimed at identifying observables that are particularly sensitive to FSI, thereby allowing their effects to be clearly isolated and quantified.

One may expect FSI effects to be particularly significant in cases where the primary mechanism, photoproduction on quasi-free nucleons, is suppressed. This is precisely the situation in the kinematic region under study, where the average momenta p of the outgoing nucleons significantly exceed the characteristic momentum in the deuteron, \(p_0 = \sqrt{M\varepsilon _d} \approx 45\) MeV/c with \(\varepsilon _d\) denoting the deuteron binding energy and M the nucleon mass. The FSI mechanism enables the transferred momentum to be shared between both nucleons, thereby compensating for the imbalance between p and \(p_0\) and thus effectively reducing the aforementioned suppression.

In the case of polarization observables, FSI effects are further enhanced, as measurements involving polarized particles are generally more sensitive to various corrections than unpolarized cross sections. In particular, studies such as8,10 have shown that for the reaction \(\gamma d \rightarrow \pi ^{-}pp\), the tensor analyzing power components, \(T_{20}\), \(T_{21}\), and \(T_{22}\), exhibit significant sensitivity to NN and \(\pi N\) rescattering effects. Moreover, theoretical predictions indicate that the rescattering contribution increases with both the momentum transfer to the deuteron and the relative momentum of the final-state nucleon pair.

Setting up experiments to measure tensor asymmetries in photo-reactions on the deuteron requires a tensor polarized deuterium target. However, a high degree of tensor polarization can be achieved only with gaseous deuterium. This requirement severely limits the target thickness and thereby reduces the luminosity of experiments performed with an extracted electron beam, rendering their implementation practically unfeasible. For this reason, experimental studies of tensor asymmetries have never been included in the research programs of leading facilities such as ELSA11, MAMI12, JLab13, and ELPH14. Nevertheless, such experiments can be carried out without significant loss of luminosity by employing the internal target method. In this approach, an ultra-thin gaseous deuterium target is placed directly inside the accelerator ring, enabling operation without electron beam extraction and thus avoiding beam current loss. The high circulating beam current compensates for the reduced target density, allowing luminosities comparable to those achieved with solid targets. This technique was pioneered at the Budker Institute of Nuclear Physics in the mid-1960s with the development of the first electron storage rings and is currently implemented at the VEPP-3 accelerator facility15.

Over the past 30 years, a number of experiments have been carried out at VEPP-3 to measure tensor asymmetries for various photoreactions on a deuteron. In particular, accurate experimental values of the \(T_{20}\), \(T_{21}\), and \(T_{22}\) components of the tensor analyzing power have been obtained in the photodisintegration channel, \(\gamma d \rightarrow p n\), in Ref.16. In addition, the energy and angular dependencies of \(T_{20}\) for the reaction \(\gamma d \rightarrow \pi ^{0} d\) have been measured in17,18,19,20, where strong discrepancy between the experimental data and theoretical predictions was noted in the region of large momentum transfer21. As for study of the reaction (1) at VEPP-3, the first results for tensor asymmetry were obtained from the statistics accumulated in 199922 and 2002–200323. At the same time, due to the low statistical accuracy, these results allowed only a qualitative comparison with the existing theoretical predictions.

The first precise results for \(T_{20}\) in the reaction (1) were obtained in 2021 for the photon energies in the range \(300 \le E_{\gamma } \le 500\) MeV24. A comparative analysis based on sophisticated model calculations demonstrated that including NN and \(\pi N\) rescattering significantly improves the agreement between data and theory. At the same time, at higher energies (\(E_\gamma > 500\) MeV), where the typical momentum transfers increase, rescattering effects are expected to become even more pronounced.

Therefore, to achieve better conditions for studying the FSI effects, it is reasonable to move to higher energies. A possibility to measure the tensor asymmetry for the reaction \(\gamma d \rightarrow \pi ^{-} p p\) at \(E_{\gamma } > 500\) MeV appeared only in 2023 due to the commissioning of the photon tagging system at VEPP-3. The first preliminary results for \(T_{20}\) in the photon energy range 400–650 MeV were obtained in 2023 and published in25. Here, we present our final results and provide a detailed analysis of the role of rescattering in the large momentum transfer regime.

Research method

The basis of the polarization experiments at VEPP-3 is the measurement of the asymmetry in the reaction yield with respect to the sign reversal of the tensor polarization of the deuterium target. The experimental setup comprises a detection system for reaction products, a photon tagging system (PTS), an internal target cell, a source of polarized deuterium atoms, and a Low-Q Polarimeter (LQP). Except for the detection system and the PTS, all components have undergone only minor modifications over the past 20 years.

The design of the detection system is determined by the kinematic parameters and the types of particles to be detected. In particular, in 2012, a deuteron (or proton) in coincidence with one or two gamma quanta from the decay of a neutral pion was registered, enabling reconstruction of the kinematics of the two processes: \(\gamma d\rightarrow \pi ^{0} d\) and \(\gamma d \rightarrow \pi ^{0} p n\). During 2002–2003, coincidence data were accumulated for proton–neutron and proton–proton events, which allowed extraction of information on deuteron photodisintegration and incoherent pion photoproduction on the deuteron.

The results presented in the present paper were obtained from the experimental statistics accumulated in 2023. The corresponding experimental setup is shown in Fig. 1. The detection system is designed to register proton–neutron and proton-proton coincidences. Protons were detected by the drift chambers (DC) and the scintillation spectrometers in the range of the polar emission angle \(50^{\circ }\)–\(90^{\circ }\) and the kinetic energy 55–170 MeV. Neutrons were recorded by sectional sandwich calorimeters consisting of ten layers of iron and scintillation strips. To extract \(\gamma d \rightarrow \pi ^{-} p p\) events, the statistics corresponding to proton–proton coincidences was used.

Fig. 1
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Experimental scheme.

For proton identification, the veto counter and proton scintillators are used in each arm. In the upper arm, identification is based on the \(\Delta E/E\) analysis, while in the lower arm, the TOF/E method is applied. In both detection arms, the kinetic energy of protons is reconstructed from the energy deposited in the proton scintillators. For energy calibration, GEANT4 simulations were employed. To select \(\gamma d \rightarrow \pi ^{-} p p\) events, the pion mass was reconstructed using the measured proton kinematic parameters together with the photon energy recorded by PTS. To validate the event selection procedure, the results of the data analysis were compared with those obtained from GEANT4 simulations using the GENBOS photoreaction generator26. The GENBOS generator was developed at JLab and is used for simulating photoreactions on the deuteron. GEANT4 simulations incorporating GENBOS were also employed to estimate the number of irreducible background events. A detailed description of the proton detection procedure, identification of the \(\gamma d\rightarrow \pi ^{-} p p\) reaction, and assessment of the background are presented in Ref.25.

The photon tagging system is a new component at VEPP-3, first commissioned in 2023. Its distinctive feature is that it must operate with the electron beam inside the acceleration chamber. This significantly complicates both the operational principle and the mechanical design of the PTS compared to those at JLab27 or MAMI28. The reason is that, unlike the single-dipole-magnet photon tagging systems employed at external-target facilities such as JLab and MAMI, the PTS at VEPP-3 requires the installation of three precisely aligned dipole magnets, a set of synchrotron radiation absorbers, and a compact tracking system within the limited space of the VEPP-3 straight section.

The PTS operation schematic is shown in Fig. 2. At the first stage, the electron beam is deflected from the initial orbit by the first dipole magnet D1 and directed onto the internal target. After passing through the target, the electrons enter the second dipole magnet, which, together with the GEM trackers, functions as a magnetic spectrometer, measuring the energy of scattered electrons in the range \(0.2E_0\) to \(0.5E_0\), where \(E_0\) is the beam energy. Thus, the energy range of the tagged photons is (0.5–0.8)\(E_0\). In particular, for an electron beam with an energy of 800 MeV, the tagged photons are registered in the range 400–650 MeV. Electrons that have not undergone an interaction with the target are returned to the initial orbit by the third dipole magnet D3. For a detailed description of the PTS at VEPP-3, we refer the reader to Ref.29.

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The photon tagging system, top view. D1, D2, D3: dipole magnets, GEM1, GEM2, GEM3: tracking detectors.

The tensor-polarized deuterium target is an elliptical storage cell built into the accelerating ring. During the experiment, gaseous tensor-polarized deuterium from the source of polarized atoms30 was continuously fed into the cell. At the output of the polarized atom source, the tensor polarization is close to 100%, and its sign is flipped every 30 s (from \(-2\) to \(+1\)).

Inside the storage cell, however, the tensor polarization is significantly reduced due to various depolarizing effects. Therefore, its degree within the storage cell is continuously monitored during data acquisition. For this purpose, the experimental setup is supplemented with an LQP, whose operating principle is based on measuring the asymmetry in elastic ed scattering at small momentum transfer.

To register the ed coincidences, an electron detector was mounted, and part of the main detector was used to detect the recoil deuterons. In the present work, the thin veto counters of the upper arm of the detection system were employed to detect the recoil deuterons (see Fig. 1). More information about the LQP can be found in Ref.31.

The tensor polarization \(P_{zz}\) of the deuteron target may be expressed in terms of the populations \(n_{0, \pm 1}\) of the deuteron states with the spin projections \(s_{z} = 0, \pm 1\) on the direction of the magnetic field as

$$\begin{aligned} P_{zz} = 1 - 3n_{0} = 3 (n_{+} + n_{-}) - 2. \end{aligned}$$
(2)

During the experiment, the deuteron target could be in the state with a positive (negative) tensor polarization \(P_{zz}^{+}\) (\(P_{zz}^{-}\)). According to the LQP data, the degree of tensor polarization averaged over the full duration of the experiment was \(P_{zz}^{+} = 0.39 \pm 0.025 \pm 0.009\) and \(P_{zz}^{-} = -0.66 \pm 0.043 \pm 0.015\), where the first and the second uncertainties are statistical and systematic, respectively.

We denote by \(N^{+}\) (\(N^{-}\)) the number of the reaction events corresponding to the tensor polarization \(P_{zz}^{+}\) (\(P_{zz}^{-}\)) of the deuteron target and registered in a given kinematic region. The experimental setup described above has allowed us to measure the asymmetry \(A^{T}=N^{+} - N^{-}\) of the reaction yield with respect to the change of the sign of the tensor polarization \(P_{zz}\). To suppress possible systematic error in measuring \(A^{T}\), the sign of \(P_{zz}\) was reversed every 30 s. In the general case, however, all the three components \(T_{20}\), \(T_{21}\), and \(T_{22}\) of the tensor analyzing power contribute to \(A^{T}\).

To extract an individual component \(T_{2M}\), one should use the general expression for the differential cross-section,

$$\begin{aligned} d\sigma = d\sigma _0 \left\{ 1 + \frac{1}{\sqrt{2}}\,P_{zz} \left[ d_{00}^2(\theta _H)\,T_{20} -d_{10}^2(\theta _H) \cos (\phi _H)\,T_{21}+d_{20}^2(\theta _H)\cos (2\phi _H)\, T_{22}\right] \right\} , \end{aligned}$$
(3)

where \(d\sigma\) (\(d\sigma _0\)) is the polarized (unpolarized) differential cross-section, \(d^j_{mm^\prime }(\theta _{H})\) are the Wigner rotation matrices defined as in32, and the angles \(\theta _{H}\) and \(\phi _{H}\) determine the orientation of the magnetic field in the coordinate system with the z axis directed along the photon momentum. Note, that Eq. (3) refers to the coplanar kinematics when the momenta of all the three final particles lie in the same plane in the laboratory system.

In the present experiment, the magnetic field was directed along the photon beam, so that the angle \(\theta _{H} = 0\). In this case, only the term with \(T_{20}\) survives in the square brackets in Eq. (3). Taking into account that the number of registered events \(N^{\pm }\) in a given kinematic region is proportional to the corresponding differential cross-section (3), one obtains

$$\begin{aligned} N^{\pm } = N_{0}\Big \{1+\frac{1}{\sqrt{2}}\,P_{zz}^{\pm }T_{20}\Big \}, \end{aligned}$$
(4)

where \(N_{0}\) is the number of reaction events in the unpolarized case. Equation (4) directly yields the expression for the \(T_{20}\) component in terms of the experimentally measured quantities:

$$\begin{aligned} T_{20}=\sqrt{2}\,\frac{N^{+}-N^{-}}{P_{zz}^{+}N^{-}-P_{zz}^{-}N^{+}}. \end{aligned}$$
(5)

Results

Our data are presented in Fig. 3. On the left panel, the dependence of \(T_{20}\) on the proton–proton invariant mass \(M_{pp} = (4 m_{p}^{2} + p_{r}^{2})^{1/2}\) is shown, where \(p_{r}\) is the magnitude of the relative pp momentum of the proton pair in the center-of-mass frame, and \(m_{p}\) is the proton mass. On the right panel, \(T_{20}\) is presented as a function of the momentum \(p_{s}\), defined as the smaller of the momenta of the two final protons in the laboratory frame. Statistical and systematic uncertainties are shown together with the experimental points. The dominant source of the systematic uncertainty is the uncertainty in the degree of the deuteron tensor polarization and the contribution from irreducible background events.

Fig. 3
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The tensor analyzing power component \(T_{20}\) for the reaction \(\gamma d \rightarrow \pi ^{-} p p\) as a function of the proton-proton invariant mass \(M_{pp}\) (left panel) and the momentum of the slow proton (right panel). The black filled circles represent the experimental results. Statistical and systematic uncertainties (red lines at the bottom) are shown together with the data points. The red squares (green triangles) correspond to the results of simulations based on the spectator model with (without) inclusion of \(\pi N\) and NN rescattering in the final state.

In addition to the experimental points, Fig. 3 presents the results of Monte Carlo simulations. The reaction amplitude used in the simulation algorithm was calculated within the model of Ref.33, which includes the contributions of the three diagrams shown in Fig. 4. Among the FSI mechanisms, NN rescattering turns out to be particularly important, primarily due to the significantly higher intensity of the NN interaction compared to \(\pi N\) (see also the discussion in Refs.7,8,9). Furthermore, in our experiment both final protons have relatively high kinetic energies, and the NN rescattering mechanism efficiently redistributes energy between the fast active nucleon and the spectator. In contrast, the intermediate pion cannot transfer a sufficient amount of kinetic energy between the two nucleons because of its small mass. The Monte Carlo simulation was performed using the algorithm described in detail in Ref.34. The energy of the incident photons was sampled according to the Dalitz spectrum35.

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The mechanisms of the incoherent pion photoproduction on a deuteron: (a) quasi-free pion photoproduction on a single nucleon, (b) NN rescattering, (c) \(\pi N\) rescattering.

As may be seen from Fig. 3, the tensor analyzing power \(T_{20}\) is rather small in the kinematic region under consideration. It is also evident that the experimental results cannot be satisfactorily described, if a pure spectator model that neglects FSI is used. The discrepancy increases with the relative momentum of the proton pair. In contrast, inclusion of rescattering effects significantly improves the agreement with the data, particularly in the region of large relative proton–proton momentum. The overall agreement between experiment and simulation is fairly good (\(\chi ^2_{tot}/n<1\)).

One may anticipate that a possible future improvement of experimental data may necessitate a refinement of the theory, in particular, the incorporation of additional mechanisms into the model. Such mechanisms might include, for example, the interaction between a single nucleon and a nucleon resonance in an intermediate state, or the contribution of the \(\Delta \Delta\) component of the deuteron wave function36,37. Higher-order multiple-scattering diagrams38 may also play a role in the kinematic region of the present experiment.

In this regard, we would like to highlight an aspect of the \(\gamma d\rightarrow \pi NN\) processes that is generally little discussed in the literature. Namely, in the first and second resonance regions, a rather strong discrepancy persists between theory and experiment8, especially in the \(\pi ^0 np\) channel, the origin of which is still not very well understood. One commonly considered explanation is that certain additional FSI mechanisms are still missing from current models. This may be particularly relevant in the second resonance region, where various inelastic channels (\(K\Lambda\), \(\pi \pi N\), etc.) start to open. However, our analysis strongly suggests that the source of the present discrepancy apparently lies elsewhere, possibly in the very mechanism of pion photoproduction on a quasi-free nucleon itself.

Conclusion

We have obtained new, relatively precise data for the tensor analyzing power \(T_{20}\) in the reaction \(\gamma d \rightarrow \pi ^{-} p p\) in the photon energy range from 400 to 650 MeV and for proton momenta \(p_{1,2} > 400\) MeV/c. The kinematic region considered in the present analysis is of particular interest due to its enhanced sensitivity to final state interaction effects. The measured values of \(T_{20}\) were compared with the results of Monte Carlo simulations performed within the framework of a spectator model that also includes FSI mechanisms (NN and \(\pi N\) rescattering in the final state).

The dependence of \(T_{20}\) on the final particle energies shown in Fig. 3 constitutes the most informative aspect of our measurement and, within the context of the presented analysis, represents our main result. By comparing the Monte Carlo calculations with the experimental data, we conclude that the overall structure of the \(T_{20}\) energy spectra is reasonably well described, indicating that our interpretation of the reaction mechanism is essentially correct. The agreement between the calculated and measured values is rather good, especially considering the high sensitivity of polarization observables to the details of the wave function and the NN and \(\pi N\) scattering amplitudes. The calculations successfully reproduce the deviation of the experimental \(T_{20}\) values from the predictions of the pure spectator model, demonstrating that this deviation is primarily attributable to FSI.

Overall, our results clearly demonstrate that FSI effects become appreciable in the kinematic region under study and tend to grow with increasing final proton momentum as well as with increasing invariant mass of the final pp-system. Their inclusion significantly improves the agreement between theory and experiment.