Near-threshold $\pi^-$ photoproduction on the deuteron

TL;DR

This paper reports the first experimental measurement of the near-threshold cross section for incoherent $^-$ photoproduction on the deuteron, providing new data to test theoretical models of the elementary reaction and final-state interactions.

Contribution

It presents the first experimental measurement of the near-threshold $^-$ photoproduction cross section on the deuteron, using gamma-ray detection techniques.

Findings

Measured total cross section for $^-$ photoproduction near threshold.

Compared experimental data with theoretical models of elementary reactions.

Provided insights into pion-nucleon and nucleon-nucleon final-state interactions.

Abstract

The first experimental investigation of the near-threshold cross section for incoherent $\pi^-$ photoproduction on the deuteron $\gamma$ d -> $\pi^-$ pp is presented. The experimental technique involved detection of the ~131 MeV gamma ray resulting from the radiative capture of photoproduced $\pi^-$ in the target. The total cross section has been measured using an unpolarized tagged-photon beam, a liquid-deuterium target, and three very large NaI(Tl) spectrometers. The data are compared to theoretical models that give insight into the elementary reaction $\gamma$ n -> $\pi^-$ p and pion-nucleon and nucleon-nucleon final-state interactions.

Tables

Table 1. Table 1: Summary of the dominant systematic uncertainties. The right column indicates if the systematic uncertainty contributes to the standard deviation of the nine cross-section measurements.

quantity	source	magnitude	in std. dev.
$Y$	fit	2%–7%
	scattering	$\lesssim 5.5$%
	Kapton	1.5%
${\Omega }_{\mathrm{eff}}$	positioning	3%
	$E_{\mathrm{cut}}$	1.5%
	model	5%
$N_{\gamma }$	tagg. eff.	2%
${\kappa }_{\mathrm{eff}}$	geometry	1.2%
$P_c$	beam sim.	$\lesssim 1.6$%
	$\Delta E_{\gamma }$	$\lesssim 3.1$%
$B{R}_{\mathrm{rc}}$	measurement	1.5%

Table 2. Table 2: Measured total cross section for π − superscript 𝜋 \pi^{-} photoproduction on the deuteron with statistical and systematic uncertainties.

$E_{\gamma }$ [MeV]	$\sigma \pm {\mathrm{err}}_{\mathrm{stat}.}\pm {\mathrm{err}}_{\mathrm{sys}.}$ [$\mu b$]
147.0	$3.8\pm 0.2\left(5.3\%\right)\pm 1.1\left(28.9\%\right)$
149.7	$11.9\pm 0.3\left(2.5\%\right)\pm 1.8\left(15.1\%\right)$
152.3	$21.4\pm 0.3\left(1.4\%\right)\pm 2.5\left(11.7\%\right)$
154.9	$28.5\pm 0.5\left(1.8\%\right)\pm 3.0\left(10.5\%\right)$
157.6	$31.9\pm 0.4\left(1.3\%\right)\pm 3.0\left(9.4\%\right)$
159.8	$35.0\pm 0.5\left(1.4\%\right)\pm 3.4\left(9.7\%\right)$

Equations

$$\sigma =\frac{4\pi Y}{{\Omega }_{\mathrm{eff}}{N}_{\gamma }{\kappa }_{\mathrm{eff}}{P}_{\mathrm{c}}B{R}_{\mathrm{rc}}},$$

$${\Omega }_{\mathrm{eff}}=4\pi N_{E_{\det }\in [120,133]\mathrm{MeV}}/N_{\mathrm{tot}},$$

$${\kappa }_{\mathrm{eff}}=(8.14\pm 0.10)\cdot 10^{23}nuclei/c{\mathrm{m}}^2,$$

Full text

threshold pion photoproduction, final-state interactions, chiral perturbation theory, deuterium, radiative capture

††preprint: MAX-lab $\gamma d\rightarrow\pi^{-}pp$ Summary article for PRC

I Introduction

Incoherent pion photoproduction on the deuteron $\gamma d\rightarrow\pi\!N\!N$ provides information on the elementary reaction on the nucleon $\gamma N\rightarrow\pi N$ and on pion-nucleon ($\pi\!N$) and nucleon-nucleon ($N\!N$) final-state interactions (FSI). The near-threshold cross section for the elementary reaction is sensitive to the $E_{0+}$ amplitude, which has a long history of theoretical studies closely related to measurements of near-threshold pion photoproduction Drechsel and Tiator (1992). Partial-wave analysis (PWA) Arndt et al. (1990) of experimental data sets may be used to obtain values for this and other photoproduction amplitudes. These are vital inputs to low-energy descriptions of hadron physics based on dispersion relations Drechsel et al. (2007) or chiral perturbation theory ($\chi$PT) Hilt et al. (2013). The latter, which is also used for comparison with experimental data in this article, is an effective field theory of Quantum Chromodynamics (QCD), where hadrons, instead of quarks and gluons, act as relevant degrees of freedom. $\chi$PT emerges from the QCD Lagrangian in the chiral limit of vanishing up and down quark masses $(m_{u},\>m_{d}\rightarrow 0)$ and thus offers a way to investigate the fundamental symmetries and interactions of the strong force in an energy regime where QCD is non-perturbative.

Tagged-photon beams combined with improved detector technology have substantially increased the size of the global pion-production data set over the last decades. However, most measurements have focused on the $\pi^{0}$ channel Bergstrom et al. (1996, 1997); Bergstrom (1998); Hornidge et al. (2013), as the elementary amplitude for $\pi^{0}$ production vanishes in the chiral limit. Thus the $\pi^{0}$ data allow for direct probing of chiral symmetry breaking phenomena. The most recent of these experiments Hornidge et al. (2013) provided high-precision differential cross section and beam asymmetry data that have enabled stringent testing of $\chi$PT. Threshold measurements of charged pion photoproduction are scarce in comparison. While the threshold cross section for $\pi^{+}$ photoproduction was established in Ref. Booth et al. (1979), none of the $E_{\gamma}<200$ MeV $\pi^{-}$ measurements Rossi et al. (1973); Salomon et al. (1984); Wang (1992); Liu (1994) have probed the near-threshold region, with the lowest-energy data point at ${\sim}158$ MeV Liu (1994), more than 10 MeV above threshold. This article reports the pioneering measurement of the total cross section for $\pi^{-}$ photoproduction on the deuteron in the energy range 147 – 160 MeV. The well-understood radiative capture (RC) reaction on the deuteron $\pi^{-}d\rightarrow\gamma nn$ Gabioud et al. (1979) with an end-point photon-energy of 131.4 MeV is exploited in a novel way for the yield determination of the photoproduced $\pi^{-}$.

II Experimental setup

The experiment was performed at the Tagged-Photon Facility J.-O. Adler et al. (2013) of the MAX IV Laboratory Eriksson (2014) in Sweden. A tagged-photon beam with energies from ${140-160}$ MeV, created via the bremsstrahlung-tagging technique J.-O. Adler et al. (1990, 1997), was incident on a thin cylindrical Kapton vessel that contained liquid deuterium (LD2) with density ${\rho_{D}=(0.163\pm 0.001)}$ g/$\rm cm^{3}$. The Kapton vessel was a cylinder of 170 mm length and 68 mm diameter, aligned along the axis of the photon beam. The vessel walls were 120 $\mu$m thick. The tagged-photon energies $E_{\gamma}$ were determined by momentum analysis of the post-bremsstrahlung electrons using a dipole magnet together with a 64-channel focal-plane (FP) hodoscope J. M. Vogt et al. (1993). The tagged-photon energy resolution was ${\pm}0.3$ MeV. Electron arrival times at the hodoscope were digitized with multi-hit time-to-digital converters (TDCs). The post-bremsstrahlung electron counting rate (typically $0.1-1$ MHz per FP channel), necessary for the photon-flux determination, was measured by scalers, normalized to the counting time. Tagging efficiency, the fraction of bremsstrahlung photons which passed through the photon-beam collimation system en route to the target, was measured daily. The mean tagging efficiency was $({\sim}23\pm 2^{\rm sys.})$%.

Three large NaI(Tl) spectrometers, named Boston University Sodium Iodide (BUNI) J. P. Miller et al. (1988), Compton and Two Photon Spectrometer (CATS) Hünger et al. (1997) and Detector Of Iodine And Sodium (DIANA) L. S. Myers (2010), were placed at laboratory angles $\theta=60^{\circ},\>120^{\circ}$ and $150^{\circ}$ to detect RC photons originating from the LD2 target. The positioning of the detectors relative to the beam and the target is depicted in Fig. 1. Each spectrometer consisted of a cylindrical core crystal surrounded by an annulus of optically isolated crystal segments. The segments were in turn surrounded by plastic scintillators. Scintillation light was read out by photomultiplier tubes (PMTs) attached to the rear faces of the scintillators. Analog signals from the PMTs were recorded by charge-integrating analog-to-digital converters (ADCs).

Data were recorded on an event-by-event basis. The data-acquisition and data-analysis software were based on ROOT Brun and Rademakers (1997) and RooFit Verkerke and Kirkby (2003) frameworks. The data acquisition was triggered by an energy deposition greater than ${\sim}50$ MeV in any NaI(Tl), which initiated the readout of the ADCs and started the TDCs. The TDC stop signals came from the post-bremsstrahlung electrons striking the FP channels. The ADC information was used to reconstruct detected photon energies, whereas the FP TDC information established the coincidence between the post-bremsstrahlung recoil electrons and the particles detected with the spectrometers. The data were collected over three run periods in 2011 and 2015.

III Analysis

Calibration

Each NaI(Tl) detector was calibrated from its in-beam response to a low-intensity tagged-photon beam. Cosmic-ray muons that traversed the detectors during data taking were identified with the annulus scintillators by requiring coincident signals in opposing annular segments. Selection of the cosmic-ray events is illustrated in Fig. 2. Shifts in the pulse-height distributions of selected cosmic-ray muon events were used to correct for PMT gain instabilities. After calibration, the NaI(Tl) detectors had a resolution of ${\sim}2$% (full width at half maximum) for the incident photon energies. The absolute calibration of the tagged-photon energies and the NaI(Tl) detectors was determined with an accuracy of $\pm 0.4$ MeV by reconstructing the 131.4 MeV photon-energy end-point from the RC reaction $\pi^{-}d\rightarrow\gamma nn$ Gabioud et al. (1979).

Signal identification

Photons from the RC reaction were also used to determine the yield of photoproduced $\pi^{-}$. The near-threshold $\pi^{-}$ had low kinetic energies and most were instantaneously captured inside the LD2 target. The two dominant capture channels are non-radiative capture (NRC) $\pi^{-}d\rightarrow nn$ (absolute branching ratio ${B\!R_{\rm nrc}=0.739\pm 0.010}$) and RC (${B\!R_{\rm rc}=0.261\pm 0.004}$) Highland et al. (1981). Using these branching ratios and the energy spectrum of the RC photons Gabioud et al. (1979); Gibbs et al. (1975); de Téramond (1977), the $\pi^{-}$ photoproduction yield was obtained. Figure 3 depicts the simulated energy spectra of the dominant background reactions of deuteron photodisintegration (np sim), $\pi^{0}$ photoproduction ($\pi^{0}$ sim) and $\pi^{-}$ NRC (nn sim), alongside the theoretical RC spectrum ($\gamma nn$ th) Gibbs et al. (1975); Gibbs and Gibson , the simulated RC spectrum ($\gamma nn$ sim) and the measured energy spectrum (exp. data). Simulations were based on GEANT4 Allison et al. (2016). The photoproduced $\pi^{+}$ did not constitute a significant background, as the muons from the dominant subsequent decay ${\pi^{+}\rightarrow\mu^{+}\nu_{\mu}}$ did not deposit more than ${\sim}50$ MeV in any of the NaI(Tl) detectors. Positrons from the decay ${\mu^{+}\rightarrow e^{+}\nu_{e}\bar{\nu}_{\mu}}$ were almost always outside the timing coincidence window with respect to the post-bremsstrahlung electron. The simulated RC spectrum was obtained by first matching the Monte Carlo in-beam data to the experimental in-beam data Strandberg et al. (2018); Myers et al. (2015). Then, photons with energies sampled from the theoretical RC spectrum and an isotropic angular distribution were generated in the LD2 target into $4\pi$ solid angle. Energy deposited by the photons in the NaI(Tl) detectors was smeared to account for the previously determined resolution effects, which led to the simulated RC spectrum (Fig. 3). The simulation, which is in excellent agreement with the data, indicated that the dominant background reactions could be removed by selecting the detected energy $E_{\rm det}\in[120,\>133]$ MeV. Background from elastic $\gamma d\rightarrow\gamma d$ and inelastic $\gamma d\rightarrow\gamma np$ Compton scattering could not be separated. Contamination from Compton scattering channels was angle- and energy-dependent, but at the present energies, the scattering cross section is only a few percent of the charged-pion photoproduction cross section. The cross-section data from Refs. Strandberg et al. (2018); Kossert et al. (2002) were extrapolated to produce conservative scattering-contamination estimates. These indicated that the effect on the extracted $\pi^{-}$ cross section was typically $\pm 3$% (maximum of 5.5% at lowest $E_{\gamma}$). This effect was accounted for in the systematic uncertainty analysis discussed below.

Yield determination

The total cross section for $\pi^{-}$ photoproduction on the deuteron was determined according to

[TABLE]

where $Y$ is the yield of RC photons, $\Omega_{\rm eff}$ is the detector acceptance, $N_{\gamma}$ is the tagged-photon flux incident on the target, $\kappa_{\rm eff}$ is the effective target thickness, $P_{\rm c}$ is the $\pi^{-}$ capture probability inside the LD2 target and $B\!R_{\rm rc}$ is the branching ratio for RC. The factor $4\pi$ originates from the assumption that RC photons are emitted isotropically. For the yield determination, timing-coincidence spectra with respect to the post-bremsstrahlung recoil electrons were filled for events inside the cut $E_{\rm det}\in[120,\>133]$ MeV. The FP channels were grouped in eight ${\sim}2.5$ MeV wide bins, resulting in eight spectra per detector. The resulting spectra had a coincidence peak superimposed upon events that were in random coincidence. As the dominant background reactions were removed by the cut on $E_{\rm det}$, $\pi^{-}$ capture yields could be determined directly from fits to the coincidence spectra (Fig. 3 inset). The signal peak was represented by a Gaussian. The background from random coincidences had a time structure due to a time modulation of the electron-beam intensity related to the pulse-stretching and beam-extraction apparatus R. Al Jebali (2013). The first two FP energy bins were below $\pi^{-}$/$\pi^{+}$ threshold. Thus, the coincidence spectra for these bins were completely dominated by random coincidences, which allowed estimation of the random-background shape. The background shape visible in the inset of Fig. 3 (black line) was obtained from the sub-pion-threshold data and employed in the fit of the super-pion-threshold data. Tools in the RooFit package enable creation of a fit shape from any histogram, which circumvents the difficulty of defining an analytical form for the non-trivial shape of the random background coincidences. The fit was moderately dependent on the width of the fitted window around the coincidence peak, which led to a systematic uncertainty of ${\sim}2$% (7% at lowest $E_{\gamma}$). Systematic uncertainty due to contamination from $\pi^{-}$ produced in the thin-walled Kapton vessel was estimated to be ${\sim}1.5$% by taking into account the chemical composition of Kapton, the thickness of the endcaps of the vessel and assuming conservatively that the $\pi^{-}$ photoproduction cross section on $\rm{}^{12}C$ and $\rm{}^{16}O$ scales linearly with the number of neutrons per atom.

Detector acceptances

The detector acceptance $\Omega_{\rm eff}$ was determined from the simulated RC spectrum described previously. The detector acceptance was determined by

[TABLE]

where $N_{\rm tot}$ is the total number of Monte-Carlo photons simulated inside the target, with energies sampled from the theoretical RC spectrum and directions sampled from a phase-space distribution over $4\pi$ solid angle. The numerator is the number of events in a detector within the energy cut $E_{\rm det}\in[120,\>133]$ MeV. The acceptances of the detectors at $60^{\circ}$, $120^{\circ}$ and $150^{\circ}$ were ${\sim}46$ msr, ${\sim}30$ msr and ${\sim}26$ msr, respectively. The dominant systematic uncertainty of 5% originated from the uncertainty in the theoretical model for RC Gibbs and Gibson . Systematic uncertainty from the positioning accuracy of the detectors and the target was estimated to be ${\sim}3$% by varying the detector and target positions in the simulation within realistic limits. The $\pm 0.4$ MeV uncertainty in the overall energy calibration of the detectors propagated into the acceptance calculation and was estimated to have an effect of ${\sim}1.5$% by varying the energy cut by the uncertainty in the simulation and recording the effect on the acceptance.

Tagged-photon flux & target thickness

The tagged-photon flux $N_{\gamma}$ was established by multiplying the FP hodoscope counts by the measured tagging efficiencies (${\sim}2$% systematic uncertainty from tagging efficiency). The effective target thickness was

[TABLE]

with a ${\sim}1.2$% systematic uncertainty originating from the geometry of the target. Further details about $N_{\gamma}$ and $\kappa_{\rm eff}$ can be found in Ref. Strandberg et al. (2018).

Pion-capture probability

The capture probability of photoproduced $\pi^{-}$ $P_{\rm c}$ was estimated from a GEANT4 simulation, where $\pi^{-}$ were simulated inside the LD2 target. The X-Y coordinates of the vertices were sampled from a simulated intensity distribution of the photon beam determined by the geometry of the beam line, and the Z-coordinates (along the beam axis) were distributed uniformly over the length of the target. In sampling the momenta of the $\pi^{-}$, the Fermi momentum of the bound neutron in the deuteron Lacombe et al. (1981), the energy of the incident photon and the angular distribution of the pions in the elementary photoproduction reaction web were taken into account. The dominant systematic uncertainty of ${\lesssim}$3.1% originated from the ${\pm}0.4$ MeV uncertainty in the tagged-photon energies. The effect of uncertainty in the beam profile was estimated to be ${\lesssim}1.6$% by changing the beam radius by $\pm 10$% in the simulation and recording the effect on $P_{\rm c}$. The simulated radius of the photon beam spot at the target center, $r_{\rm beam}\sim 20$ mm, was in good agreement with a beam photograph at that location and was substantially smaller than the $r_{\rm vessel}=34$ mm radius of the Kapton vessel. Additionally, the $\pi^{-}$ escape from the target occurred predominantly from the downstream endcap, which explains the relatively weak dependence on the radius of the beam. Figure 4 depicts the dependence of $P_{\rm c}$ on the incident photon energy $E_{\gamma}$ with systematic uncertainties.

Results

The cross section for threshold $\pi^{-}$ photoproduction at each energy was determined as a statistically weighted average of nine measurements (three detectors and three run periods). The standard deviation of the nine measurements was used to estimate the combined systematic uncertainty. Sources of systematic uncertainties are summarized in Table 1. The right column of Table 1 specifies whether or not a given systematic uncertainty contributed to the standard deviation of the nine measurements. Typically, the uncertainty estimated from the standard deviation was of similar magnitude compared to the uncertainty estimated from adding the contributing sources in quadrature. The non-contributing sources were then added to the standard deviation in quadrature to produce the final systematic uncertainties (Table 2, Fig. 5). Of the non-contributing uncertainties, only the capture efficiency affected the shape of the cross-section curve. Others affected the scale of the results. The angle- and energy-dependent uncertainties from scattering channels are accounted for in the standard deviation of the combined result, as they contributed to the observed spread of the nine measurements. A full account of the analysis of the experimental data is available in Ref. B Strandberg (2017).

IV Theoretical Analysis

The experimental data for the $\gamma d\rightarrow\pi^{-}pp$ reaction are now compared with model predictions. Compared to the elementary reaction $\gamma n\rightarrow\pi^{-}p$, the additional final-state proton introduces additional FSI that have a non-negligible effect on the cross section. The model is a version of that described in Ref. Tarasov et al. (2011), simplified for the near-threshold region. It is calculated from the four diagrams displayed in Fig. 6, where $M_{a}$ is the Impulse Approximation (IA) term, $M_{b}$ and $M_{c}$ are the $N\!N$ and $\pi\!N$ FSI terms, and $M_{d}$ is the $N\!N$-FSI term with pion rescattering in the intermediate state (the ‘two-loop’ term). The ingredients and the approximations for the computation of the four terms are given in the following list.

The elementary reaction is described by the $s$-wave amplitude, which is determined by the $E_{0+}$ multipole. The value of $E_{0+}$ as extracted by various analyses has been very stable over the last decades and here ${E_{0+}\!=-31.9}$ from Ref. Drechsel and Tiator (1992) is used. Here and elsewhere in the article the $E_{0+}$ amplitude is expressed in the conventional units of $10^{-3}/m_{\pi^{+}}$. Further, in diagrams $M_{c}$ and $M_{d}$ of Fig. 6, only charged intermediate pions are included as the neutral-pion photoproduction amplitude is much smaller than the charged-pion photoproduction amplitude in the near-threshold region. In this approximation, the cross section is proportional to $\lvert E_{0+}\rvert^{2}$. 2. 2.

The $s$-wave $pp$-scattering amplitude includes Coulomb effects and is taken in the Effective-Range Approximation Landau and Lifshits (1965), using the values $a_{pp}=-7.8$ fm for the $pp$ scattering length and $r_{pp}=2.8$ fm for the effective range. Off-shell effects are included as in Refs. Levchuk et al. (2006); Tarasov et al. (2011). 3. 3.

For $\pi\!N$ scattering, the $s$-wave $\pi\!N$ amplitude ${f_{\pi^{-}p}=b_{0}-b_{1}}$ is used, fixed by the isospin scattering lengths $b_{0}=-28$ and $b_{1}=-881$ in units of $10^{-4}/m_{\pi}$ Döring et al. (2004). 4. 4.

The deuteron wave function (DWF) derived from the Bonn potential is used in the parameterized form from Ref. Machleidt (2001). The IA diagram $M_{a}$ includes both the $s$- and $d$-wave part of the DWF, with the $d$-wave having only a small effect on the cross section at energies close to threshold. The inclusion of the $d$-wave in other diagrams is expected to have a negligible effect and is neglected to simplify calculations.

The cross-section model is compared with the experimental data in Fig. 5. The dashed curve indicates the IA ($M_{a}$ in Fig. 6), whereas the solid curve indicates the full model (all terms in Fig. 6). The dominant correction to the IA term $M_{a}$ originates from the $N\!N$-FSI amplitude $M_{b}$, whereas the combined contribution from the $\pi\!N$-FSI ($M_{c}$) and the two-loop term ($M_{d}$) is typically $\lesssim 10$%. Considering terms $M_{c}$ and $M_{d}$, the relative contribution of $M_{c}$ to the combined result of $M_{a}$ and $M_{b}$ is stable at around ${\sim}4$%, while the effect of $M_{d}$ reduces from ${\sim}8$% to ${\sim}2$% as $E_{\gamma}$ increases from threshold to ${\sim}160$ MeV. While the model and the experimental data agree within uncertainties in the energy region 147 – 157 MeV, it overestimates the data above 157 MeV, due to two dominant factors:

The model does not account for the energy dependence of $E_{0+}$. Since $E_{0+}$ decreases with energy, this causes the theoretical model to overestimate the cross section as $E_{\gamma}$ increases from threshold. 2. 2.

The model uses only the $s$-wave amplitude for the elementary reaction $\gamma n\rightarrow\pi^{-}p$ and for $N\!N$-FSI, which is expected to contribute to the divergence as higher partial waves become significant at energies $\gtrsim 10$ MeV above threshold.

The measured cross section for $\gamma d\rightarrow\pi^{-}pp$ is also compared to a previous $\chi$PT prediction for the isospin-partner channel $\gamma d\rightarrow\pi^{+}nn$ Lensky et al. (2005). Comparison of the $\pi^{-}$ experimental data with the $\pi^{+}$ prediction is insightful as, compared to Ref. Tarasov et al. (2011), the $\chi$PT calculation uses higher-order partial waves both for the elementary reaction ${\gamma p\rightarrow\pi^{+}n}$ and for the $N\!N$-FSI. It also accounts for the energy dependence of $E_{0+}$. In the leading order of the chiral expansion, the elementary amplitudes $\gamma n\rightarrow\pi^{-}p$ and $\gamma p\rightarrow\pi^{+}n$ are equal. The most important difference between the elementary $\pi^{+}$ and $\pi^{-}$ photoproduction reactions is the proton recoil in the latter, which increases the dipole moment of the final $\pi\!N$ system. Due to the absence of proton recoil in the $\pi^{+}$ reaction, the absolute value of $E_{0+}$ is approximately 12% smaller for $\gamma p\rightarrow\pi^{+}n$ compared to $\gamma n\ \rightarrow\ \pi^{-}p$. For this calculation $E_{0+}(\pi^{+}n)=28.2$ from Ref. Bernard et al. (1996) was used. This effect suppresses the cross section for $\gamma d\rightarrow\pi^{+}nn$ compared to $\gamma d\rightarrow\pi^{-}pp$. On the other hand, there is no Coulomb FSI in $\gamma d\rightarrow\pi^{+}nn$, which leads to a relative increase in the cross section compared to $\gamma d\rightarrow\pi^{-}pp$. These two effects are expected to cancel partially. The $\chi$PT calculation for $\gamma d\rightarrow\pi^{+}nn$ with theoretical uncertainties (see Ref. Lensky et al. (2005) for details of the uncertainty calculation) is depicted as a gray band in Fig. 5. The starting $E_{\gamma}$ value of the theoretical curve has been shifted to $145.8$ MeV to account for the difference in the reaction threshold compared to the $\pi^{-}$ channel. The calculation has been performed at the order ${\chi}^{5/2}$ of the chiral expansion parameter $\chi=m_{\pi}/m_{N}$, where $m_{\pi}$ ($m_{N}$) stands for the generic pion (nucleon) mass. The experimental data and the $\gamma d\rightarrow\pi^{+}nn$ model agree within uncertainties, suggesting that the differences between the $\pi^{+}$ and $\pi^{-}$ channels indeed tend largely to cancel. The good agreement between the models and the experimental data at energies $E_{\gamma}<157$ MeV suggests that in the immediate vicinity of the threshold, the dominant processes that contribute to the cross section are relatively well understood.

V Summary

In summary, the first measurement of the near-threshold cross section for $\pi^{-}$ photoproduction on the deuteron has been presented along with model predictions. The models and the experimental data are in good agreement in the vicinity of the threshold and provide new insight into the FSI behavior in this energy regime. The behavior further away from the threshold could be investigated by a dedicated $\chi$PT calculation for the measured $\gamma d\rightarrow\pi^{-}pp$ reaction. Further insight into the discrepancies between experimental data and the models at energies $\gtrsim 10$ MeV above threshold could be gained from differential cross-section measurements for $\gamma d\rightarrow\pi^{-}pp$, which would allow for a more detailed study of the effects of various partial waves.

Acknowledgements.

The authors would like to thank W. R. Gibbs, B. F. Gibson and G. F. de Téramond for constructive discussions. The authors acknowledge support from the staff of the MAX IV Laboratory. This research has been supported by the Swedish Research Council (Contracts No. 40324901 and No. 80410001), the Crafoord Foundation (Grant No. 20060749), the Scottish Universities Physics Alliance (SUPA), the SUPA Prize Studentship, UK STFC Grants No. 57071/1 and No. 50727/1, US NSF Grant No. PHY1309130, US DOE Grants No. DE–SC0016581, No. DE–SC0016583 and No. DE–FG02–06ER41422, the NFS/IRES Award No. 0553467, RFBR Grants No. 16–02–00767 and 16–02–00767a, and the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Center, SFB 1044.