diff --git "a/1tA0T4oBgHgl3EQfMv-f/content/tmp_files/load_file.txt" "b/1tA0T4oBgHgl3EQfMv-f/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/1tA0T4oBgHgl3EQfMv-f/content/tmp_files/load_file.txt" @@ -0,0 +1,993 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf,len=992 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='02137v1 [nucl-th] 5 Jan 2023 Reduced nuclear helicity amplitudes for deuteron electrodisintegration and other processes J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Flores and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Chabysheva Department of Physics, University of Idaho, Moscow ID 83844 USA J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Hiller Department of Physics, University of Idaho, Moscow ID 83844 USA and Department of Physics and Astronomy, University of Minnesota-Duluth, Duluth, Minnesota 55812 USA (Dated: January 6, 2023) 1 Abstract We extend the original idea of reduced nuclear amplitudes to capture individual helicity ampli- tudes and discuss various applications to exclusive processes involving the deuteron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Specifically, we consider deuteron form factors, structure functions, tensor polarization observables, photodisin- tegration, and electrodisintegration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The basic premise is that nuclear processes at high momentum transfer can be approximated by tree graphs for point-like nucleons supplemented by empirical form factors for each nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The latter represent the internal structure of the nucleon, and incorporate nonperturbative physics, which can allow for early onset of scaling behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The nucleon form factors are evaluated at the net momentum transfer experienced by the given nucleon, with use of GE for a no-flip contribution and GM for a helicity-flip contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Results are compared with data where available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The deuteron photodisintegration asymmetry Σ is obtained with a value of Σ(90◦) ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='06, which is much closer to experiment than the value of -1 originally expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The method also provides an estimate of the momentum transfer values required for scaling onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We find that the deuteron structure function B is a good place to look, above momentum transfers of 10 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' INTRODUCTION With the advent of the upgraded electron accelerator at the Thomas Jefferson National Accelerator Facility, scattering experiments with polarized beams and targets at high energy and high momentum transfer become possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In the regime of high momentum transfer to all relevant nucleons, quantum chromodynamics (QCD) implies that the internal structure of every nucleon is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Until ab initio QCD (lattice) calculations for nuclear scattering processes are available for more than very simple processes, one is led to consider models that can represent the basic physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' One such approach is the reduced nuclear amplitude (RNA) analysis pioneered by Brod- sky and Chertok [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In addition to their application to a generic deuteron form factor, the approach has been applied to deuteron disintegration [2], pion photoproduction [3], and photodisintegration of 3He [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' As originally developed, a nuclear process was modeled as a tree-level amplitude multiplied by a generic form factor for each nucleon, with each form factor evaluated at the net momentum transferred to that nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In order to model the behavior of polarization observables [5–15], we extend this approach to a reduced nuclear helicity amplitude (RNHA) method to combine a tree-level helicity amplitude for point-like nucleons with the appropriate form factor for each nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' When the nucleon does (not) flip its helicity, we use the electric (magnetic) form factor GEN (GMN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' As a check on the procedure, virtual photon absorption by a single nucleon in the RNHA approach is consistent with the definitions of GEN and GMN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' A caveat in applications of the RNA approach is that the normalization is not determined by the model and is fixed to data at infinite momentum transfer by the coefficient of the leading power-law behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' This means that the normalization cannot be determined in practice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' fitting to a data point at some intermediate kinematics will give the wrong nor- malization and the wrong magnitude at higher momentum transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Instead, ratios need to be considered, so that the normalization becomes irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The primary criterion for the asymptotic region is in the momentum transfer to each nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For every nucleon in the process, the momentum transfer must be above some common threshold, which is at least 1 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For example, for deuteron photodisintegration, 2 the momentum transfer to a nucleon is −tN = −(pN − p/2)2, where pN is the final four- momentum of the nucleon and p is the initial deuteron momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' When expressed in terms of the photon energy Eγ and the final nucleon angle θ, the constraint to be above 1 GeV2 becomes [16] mNEγ � 1 − � Eγ mN + Eγ | cos θ| � ≥ 1 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='1) This relationship is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Notice that away from 90◦, the lower limit is quite high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For electrodisintegration, only the most recent data [17, 18] begins to reach this θ (deg) 0 20 40 60 80 100 120 140 160 180 Eγ (GeV) 0 2 4 6 8 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Angular dependence of the scale for large momentum transfer in deuteron photodisinte- gration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Here we will focus on deuteron processes, including photodisintegration and electrodisin- tegration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For recent reviews of deuteron studies at high momentum transfer, see [16, 19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Elastic electron scattering data at high momentum transfer is presented in [21–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Recent photodisintegration data can be found in [26–29], and for electrodisintegration data, in [17, 18, 30–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Other analyses of deuteron processes include hidden-color contributions to deuteron form factors [34], the hard rescattering mechanism [35], quark-gluon strings [36], the Moscow NN potential [37], and AdS/QCD models [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' One recent experiment [17] used the 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 GeV electron beam at JLab and the Hall C spectrometers to measure electron scattering from a liquid deuterium target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The final electron and the proton were detected, with the kinematics restricted to the exclusive process ed → e′pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' One spectrometer measured the final electron at a nominal 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2◦ degrees from the beam direction, with a momentum of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='1 GeV such that the recorded events had a distribution of momentum transfer squared reaching 5 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The events studied were taken from a bin of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5 GeV2 in the tail of the distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' however, the nominal transfer was 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 GeV2, because the majority of the events were in the lower half of the bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' A second spectrometer measured the proton momentum at a range of angles to the beam direction, tuned to select events where the (missing) neutron had an angle relative to the direction of the momentum transfer that fell within a chosen bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In the one-photon exchange approximation, which we assume, the momentum transferred is, of course, the photon momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The published neutron angles are binned at 35◦, 45◦ and 75◦, with the first two selected to minimize final-state interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For our purposes, the importance of 3 these two angles is that the momentum transferred to the neutron reaches 1 GeV in a zero- binding approximation, so that, rather than focus on the internal structure of the deuteron, we can consider the response to a large momentum transfer to all the nucleons involved and we can see that experiments may be approaching the threshold where our model can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The RNHA model is constructed in detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' II for two-nucleon processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In the remainder of the paper, we consider various processes for the deuteron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' III, the form factors,1 structure functions, and tensor polarization observables of elastic electron scattering from the deuteron are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Photodisintegration and electrodisintegration are analyzed in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' IV and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Within the zero-binding approximation, elastic scattering and photodisintegration live at edges of the kinematic range of electrodisintegration and are essentially special cases that provide introductory examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Section VI contains a summary of the results and suggestions for additional applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Many details of the electrodisintegration helicity amplitudes are left to an appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' CONSTRUCTION OF THE MODEL The basic process for a two-nucleon system to absorb a photon and exchange momentum between the nucleons is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' These diagrams are modeled on the primitive process of γ∗ff → ff, with f representing a point-like nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 The structure of each nucleon is then introduced by combining the Feynman amplitude for each diagram with the appropriate form factor for each nucleon, evaluated at the net momentum transfer for that nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For a deuteron process in the zero-binding limit, the initial nucleons share the initial deuteron momentum p equally, so that pp = pn = p/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We also neglect the nucleon mass difference, setting mp = mn ≡ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The distinction between different photon-absorption processes is then in the nature of the photon, being either real or virtual, and in the outcome for the final nucleons, bound as a deuteron or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The tree-level amplitudes for the four diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 2 are Mν a (λ′ p, λ′ n, λp, λn) = Aµν p (p/2 + q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' λ′ p, λp) 1 (p′ n − p/2)2Bnµ(λ′ n, λn), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='1) Mν b (λ′ p, λ′ n, λp, λn) = Aνµ p (p′ p − q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' λ′ p, λp) 1 (p′n − p/2)2Bnµ(λ′ n, λn), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2) Mν c (λ′ p, λ′ n, λp, λn) = Aµν n (p/2 + q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' λ′ n, λn) 1 (p′p − p/2)2Bpµ(λ′ p, λp), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='3) Mν d (λ′ p, λ′ n, λp, λn) = Aνµ n (p′ n − q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' λ′ n, λn) 1 (p′ p − p/2)2Bpµ(λ′ p, λp), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4) where Aµν N (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' λ′ N, λN) = ¯u′ Nγµ ̸p + m p2 − m2γνuN, Bµ N(λ′ N, λN) = ¯u′ NγµuN, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5) with uN (¯u′ N) the initial (final) spinor for the nucleon N with helicity λN (λ′ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The sub- amplitude AN represents the fermion line that absorbs the photon, and BN represents the 1 For discussion specifically in terms of perturbative QCD, see [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 2 In [2], the primitive process was γ∗q¯q → q¯q, with q corresponding to a point-like proton and ¯q to a point- like neutron, and direct interaction of the photon with the neutron was neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Here we amend and extend this, to retain information about helicity states of the fermions and include photon absorption by the neutron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 4 pp, λp pn, λn p′ p, λ′ p p′ n, λ′ n q, λγ pp, λp pn, λn p′ p, λ′ p p′ n, λ′ n q, λγ (a) (b) pp, λp pn, λn p′ p, λ′ p p′ n, λ′ n q, λγ pp, λp pn, λn p′ p, λ′ p p′ n, λ′ n q, λγ (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Tree graphs for deuteron processes that absorb a photon of momentum q and helicity λγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The initial (final) nucleon momentum and helicity are pN (p′ N) and λN (λ′ N), with N = p or n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The two nucleons exchange momentum via a vector particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The four diagrams differ in the nature of the photon-absorbing nucleon and the order of this absorption and momentum transfer between nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' other fermion line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Calculation of these sub-amplitudes can be checked against the trace theorem for sums over helicities: � λN,λ′ N Aν′µ′∗ N (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' λ′ N, λN)Aµν N (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' λ′ N, λN) = Tr � γν′ ̸p + m p2 − m2γµ′(̸p′ N + m)γµ ̸p + m p2 − m2γν(̸pN + m) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6) � λN,λ′ N Bµ′∗ N (λ�� N, λN)Bµ N(λ′ N, λN) = Tr � γµ′(̸p′ N + m)γµ(̸pN + m) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='7) The full amplitude is constructed from the MX by combining them with form factors for each nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For a deuteron with initial helicity λd, we have Mν(λ′ p, λ′ n, λd) = � λp,λn Cλd λpλn \uf8ee \uf8f0 � X=a,b,c,d Mν X(λ′ p, λ′ n, λp, λn) \uf8f9 \uf8fb Gpλ′pλp(Q2 p)Gnλ′nλn(Q2 n), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8) 5 where Q2 N = −(p′ N − pN)2, Cλd λpλn = \uf8f1 \uf8f2 \uf8f3 δλp± 1 2δλn± 1 2, λd = ±1 1 √ 2 � δλp 1 2δλn− 1 2 + δλp− 1 2δλn+ 1 2 � , λd = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='9) and GNλ′λ = � GEN, λ′ = λ GMN, λ′ = −λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='10) The form factors GEN and GMN represent the internal structure of the nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' They can be represented by data or empirical fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For simplicity, we use the fits [40] GEp ≃ � 1 + Q2 N m2 0 �−2 , GMp ≃ µpGEp, GMn ≃ µnGEp, GEn ≃ − µnτ 1 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6τ GEp, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='11) where m2 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='71 GeV2, τ = Q2 N 4m2 , µp = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='79, and µn = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' To limit the analysis to a single mass scale, we take the parameter m0 to be proportional to the nuclear mass, with m2 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='80 m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We do not attempt to compute or assign an overall normalization to Mν, and the running of the strong coupling constant is not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The initial nucleon spinor, for a deuteron traveling along the negative z direction, is [41] uN = ̸p/2 + m � Ed/2 + m � φ(λN)(−ˆz) 0 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='12) with φ(1/2)(−ˆz) = � 0 1 � , φ(−1/2)(−ˆz) = � 1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='13) The final nucleon spinor is u′ N = ̸p′ N + m � E′ N + m � φ(λ′ N)(ˆp′ N) 0 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='14) with φ(1/2)(ˆp′ N) = � cos(θN/2) eiφN sin(θN/2) � , φ(−1/2)(ˆp′ N) = � −e−iφN sin(θN/2) cos(θN/2) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='15) where θN and φN are the polar and azimuthal angles of the outgoing momentum of the particular nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' As discussed in the Introduction, the overall normalization of the RNHA amplitude is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For comparison with data, we consider quantities which are themselves ratios or a ratio of the model to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' ELASTIC ELECTRON SCATTERING A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Form factors The three deuteron form factors, GC, GM, and GQ, are readily obtained from the hadronic helicity amplitudes of elastic electron-deuteron scattering in the Breit frame [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The kine- matics are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The photon four-momentum is q = (0, 0, 0, qz) and the initial 6 (final) deuteron four-momentum is p = (Ed, 0, 0, −qz/2) (p′ = (Ed, 0, 0, qz/2)), with q2 z = Q2 and Ed = � Q2/4 + m2 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In the zero-binding limit,3 md = 2m and the individual nucleon four-momenta are pp = pn = p/2 and p′ p = p′ n = p′/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The hadronic matrix elements are given by Gµ λ′ d,λd = � λ′p,λ′n C λ′ d λ′pλ′nMµ(λ′ p, λ′ n, λd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='1) The initial spinors are as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' the final spinors are specified by u′ N = ̸p′/2 + m � Ed/2 + m � φ(λ′ N)(ˆz) 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2) ⃗q, λγ −⃗q/2, λd ⃗q/2, λ′ d z ⃗pe, λe ⃗p ′ e, λ′ e FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Kinematics for elastic electron-deuteron scattering in the Breit frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The photon travels along the positive z direction, and the deuteron comes from the right, along the negative z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The three form factors are then extracted as [42, 43] GC = −1 2md √1 + η G+ 00 − 2G+ +− 3 , GM = 2 2md √1 + η Gx +0 √2η, GQ = −1 2md √1 + η G+ 00 + G+ +− 2η , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='3) with η ≡ Q2 4m2 d and the + superscript denoting the light-front sum of the 0 and z components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For the helicity matrix elements, the model yields the following Q2 dependence: G+ 00 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5588N m �m Q �9 � 1 + 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='1m2 Q2 + O(m4 Q4 ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4) G+ +− = −69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='85N m �m Q �11 � 1 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8m2 Q2 + O(m4 Q4 ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5) Gx +0 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='851N m �m Q �10 � 1 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8m2 Q2 + O(m4 Q4 ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6) with N the unknown normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The factor of m/Q associated with each helicity flip [44] is clearly evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For the form factors, we find GC = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5588 √1 + η N 12 �m Q �9 � 1 + 379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='1m2 Q2 + O(m4 Q4 ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='7) 3 The difference between the proton and neutron masses is neglected in addition to the deuteron binding energy, the two being of the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 7 GM = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='851 � η(1 + η) N 2 √ 2 �m Q �10 � 1 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8m2 Q2 + O(m4 Q4 ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8) GQ = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5588 η√1 + η N 8 �m Q �9 � 1 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='086m2 Q2 + O(m4 Q4 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='9) The leading ± signs are as expected for large Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We have left the kinematic factor η = Q2/16m2 without substitution, because there can be three regimes for Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In addition to Q2 large or small, there can be an intermediate region where Q2 is large but η is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Such an intermediate regime does exist for GM and GQ, where the coefficients of the nonleading terms are small enough for this correction to be small while η is also small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For GC, this is not the case, because the coefficient of the nonleading term is large enough to require a Q2 value for which η is also large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In the intermediate regime, we obtain GM ∼ �m Q �11 , GQ ∼ �m Q �11 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='10) and for the large-η regime GC ∼ �m Q �10 , GM ∼ �m Q �12 , GQ ∼ �m Q �12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='11) Ratios of these form factors at very large Q2 can be compared with the tree-level ratios for a point-like spin-one particle, such as the W +, which are [43] GC GQ = 2 3η − 1, GM GQ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='12) Such behavior is immediately reproduced for form factors separated according to a Drell– Yan frame [45], with the assumption of strict G+ 00 dominance [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In terms of our hadronic matrix elements, we have GC GQ = 2 3η − 2η G+ +− G+ 00 + G+ +− , GM GQ = −2 � 2η Gx +0 G+ 00 + G+ +− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='13) As already observed in [43], these Breit-frame ratios cannot both be resolved by simply assuming G+ 00 dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' From our model, we obtain GC GQ = 2 3η + 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 + O(m2 Q2 ), GM GQ = −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 + O(m2 Q2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='14) The leading 2 3η is just kinematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The deviations of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 and -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 from -1 and -2, respec- tively, are due to nonleading contributions multiplied by powers of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Similar deviations will arise for calculations done in the Drell–Yan frame, because η factors again interfere with strict G+ 00 dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Plots of these ratios are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 8 �����V����e/����� V�� V�� � �� �� �� ��/M����0 � �� ��� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Ratios GC/GQ − 2η/3 (dashed) and GM/GQ (solid) for the model deuteron form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Structure functions Experiments designed to extract these form factors measure cross sections and polariza- tion observables in elastic electron-deuteron scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The unpolarized cross section dσ dΩ ∝ S, S ≡ A(Q2) + B(Q2) tan2(θe/2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='15) depends on the electron scattering angle θe and two structure functions A(Q2) ≡ G2 C + 8 9η2G2 Q + 2 3ηG2 M, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='16) B(Q2) ≡ 4 3η(1 + η)G2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='17) These have been measured at the highest Q2 yet attained at JLab [22–24], and A has been measured at comparable Q2 at SLAC [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' However, these do not yet reach the Q2 values needed for a definitive comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Figures 5 and 6 show plots of the data divided by the model, including an arbitrary normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In our model, expansions of these functions in inverse powers of Q2 are A(Q2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='1041N 2 �m Q �20 � 1 + 1246m2 Q2 + O(m4 Q4 ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='18) B(Q2) = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='06N 2 �m Q �20 � 1 + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6m2 Q2 + O(m4 Q4 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='19) Because the expansion for GC is valid only for large η, we have used the explicit form of η in constructing the expansion for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The function B is independent of η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' the leading factor of 9 � �l�70V����� 5 6 7 � � �7Ql���70 5 6 7 � � � � � FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Data for the deuteron structure function A(Q2) divided by the model function, including an arbitrary normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Experimental values are taken from [23] (circles) and [24] (squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' η(1+η) in its definition exactly cancels against factors in the relationship of GM to hadronic matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The expansion for B converges much faster than the expansion for A, and the leading Q2 behavior is dominant for Q2 ≫ 10 GeV2 only for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For A, one must wait until impossibly large Q2, which enters a regime where the collective quark substructure is important, including hidden-color effects [34], and the point-like approximation used in our model is invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In [22] the large Q2 behavior of B is quoted as being Q−24 from perturbative QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' This faster fall off compared to A is attributed to the extra suppression of the helicity flip involved in GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' However, there are other compensating factors, and, just as in our model, the behavior of B should be Q−20, which is the same as A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 7 we plot the ratio of B to A for a large range of Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' This ratio becomes constant at very large Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Although the plots begin at low Q2, there is nothing in the model that could reproduce diffractive minima, hence the smooth appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Tensor polarization observables Experiments can also extract tensor polarization observables [20, 25] t20 ≡ − 1 √ 2S �8 3ηGCGQ + 8 9η2G2 Q + 1 3η � 1 + 2(1 + η) tan2(θe/2) � G2 M � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='20) t21 ≡ 2η √ 3S cos(θe/2) � η + η2 sin2(θe/2)GMGQ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='21) t22 ≡ − η 2 √ 3S G2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='22) 10 � �l��-G����� 1V � V V � 3 3 � � � � ��Ql����- 3 3 � � � � � FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Data for the deuteron structure function B(Q2) divided by the model function, including an arbitrary normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Experimental values are taken from [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' � �8��15�8��1 � �� �� �� �� ��� ��� ��� ��Q8����1 ��� ��� ��� ��� ��� ��� ��� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Ratio of B to A for the model deuteron structure functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The highest Q2 measurements of these were also done at JLab [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' When η is held explicit, expansions in m/Q are t20 = − √ 2 + 1064[1 + 2(1 + η) tan2(θe/2)] �m Q �2 + O(m4 Q4 ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='23) 11 t21 = 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 sec(θe/2) � η + sin2(θe/2)m Q (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='24) + sec(θe/2) � η + sin2(θe/2)[48606 + 77869(1 + η) tan2(θe/2)] �m Q �3 + O(m4 Q4 ), t22 = −434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5 �m Q �2 + [544703 + 872133(1 + η) tan2(θe/2)] �m Q �4 + O(m6 Q6 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='25) While at very large Q2, they are t20 = − √ 2 + 133 tan2(θe/2) + 1064[1 + 2 tan2(θe/2)] �m Q �2 + O(m4 Q4 ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='26) t21 = 1217 sec(θe/2) sin(θe/2)[tan2(θe/2) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='007972] + [] �m Q �2 + O(m4 Q4 ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='27) t22 = −[434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5 − 54508 tan2(θe/2)] �m Q �2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='28) +[544703 + 872133 tan2(θe/2)] �m Q �4 + O(m6 Q6 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The coefficients of nonleading terms are quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Thus, very large Q2 is required for the leading term to be dominant, well beyond any available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The limit of − √ 2 for t20 at θe = 0◦ was an early prediction of perturbative QCD [44, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' However, as argued elsewhere [43], this value is obtained only at very large Q2, and the value is quite different for small η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Figures 8 and 9 show plots of these observables at angles of 0◦ and 30◦, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We also compare with data [25] in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 10, 11, and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' At these ‘small’ values of Q2, only t22 is consistent with data, something which is likely accidental with both data and model values near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' PHOTODISINTEGRATION In the photodisintegration of a deuteron, a real photon is absorbed and the two constituent nucleons emitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' This process is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The initial deuteron and photon four- momenta in the center-of-mass (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=') frame are p = (Ed, 0, 0, −qz) and q = (qz, 0, 0, qz), where the incident photon is taken along the positive z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The final proton and neutron four-momenta are p′ p = (E′ p, ⃗p ′ p) and p′ n = (E′ n, ⃗p ′ n), with θp and φp the polar and azimuthal angles of the final proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' By ignoring the nucleon mass difference, we have E′ p = E′ n, because momentum conservation guarantees ⃗p ′ n = −⃗p ′ p in the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In terms of the Mandelstam variable s, the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' energies and momenta are Ed = (s + 4m2)/(2√s), qz = (s − 4m2)/(2√s), E′ p = √s/2, |⃗p ′ p| = √ s − 4m2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='1) The photon energy in the lab frame is Eγ = (s − 4m2)/4m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We will work at large s, so that momentum transfers are large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The standard definition of helicity amplitudes for photodisintegration is [47] Fi± ≡ ǫν(λγ)Mν(λ′ p, λ′ n, λd) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2) 12 � ��5e0���e0��� V�)� V�)� V�)� V� V5)� V5)� V5)� V5)� 5 ��08����Q �55 �5� �5� �5� �5� �5� �5� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Deuteron tensor polarization observables t20 (solid), t21 (dashed), and t22 (dotted) as computed in the model at an angle of θe = 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The asymptotic value of t20(0◦) is − √ 2, as predicted by perturbative QCD [44, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' with ǫ the polarization vector for a photon with helicity λγ and Mν given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The index i is associated with particular helicity combinations as follows: F1± = ǫν(1)Mν(±1 2, ±1 2, 1), F2± = ǫν(1)Mν(±1 2, ±1 2, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='3) F3± = ǫν(1)Mν(±1 2, ±1 2, −1), F4± = ǫν(1)Mν(±1 2, ∓1 2, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4) F5± = ǫν(1)Mν(±1 2, ∓1 2, 0), F6± = ǫν(1)Mν(±1 2, ∓1 2, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5) The other helicity combinations are related to these by parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The helicity amplitudes can be used to compute various polarization observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The recoil-proton polarization Py measures the asymmetry parallel/antiparallel to the normal ˆy ∝ ⃗q × ⃗p ′ p to the scattering plane: Py = 2Im 3 � i=1 [F † i+Fi+3,− + F † i+3,+Fi−]/f(θ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6) where f(θ) = �6 i=1[|Fi+|2 + |Fi−|2] is the sum of all the helicity amplitudes squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The transferred polarizations Cx′ and Cz′ measure asymmetries parallel/antiparallel to the ˆx′ ∝ ⃗p ′ p × ˆy and ˆz′ = ˆp ′ p directions: Cx′ = 2Re 3 � i=1 [F † i+Fi+3,− + F † i+3,+Fi−]/f(θ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='7) Cz′ = 6 � i=1 [|Fi+|2 − |Fi−|2]/f(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8) 13 � ���e0���e0��� V�)� V�)� V�)� V�)� V�)� � ��03����Q ��� ��� ��� ��� ��� ��� ��� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 8 but for an angle of θe = 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The asymmetry Σ for linearly polarized photons is given by Σ = −2Re �� ± (F † 1±F3∓ − F † 4±F6∓) − F † 2+F2− + F † 5+F5− � /f(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='9) Each observable is formed as a ratio, which sets aside questions of normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Because we only need to consider photons with helicity +1, the polarization vector is always ǫ = − 1 √ 2(0, 1, i, 0), relative to the momentum in the positive z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The final Dirac spinors are u′ N = ̸p′ N + m � E′ N + m � φ(λ′ N)(ˆp′ N) 0 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='10) with θn = π − θp, φn = φp + π = π, and φ(1/2)(ˆp′ N) = � cos(θN/2) eiφN sin(θN/2) � , φ(−1/2)(ˆp′ N) = � −e−iφN sin(θN/2) cos(θN/2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='11) With these spinors as input, the amplitudes ǫνMν X can be evaluated in terms of Dirac matrix and spinor products and then combined to construct the predefined amplitudes Fi±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' At large s, these RNHA predictions for the helicity amplitudes reduce to F1+ ∼ 4 √ 2 √scsc2(θp 2 )GEn(θp)GEp(θp), F1− ∼ 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='12) F2+ ∼ 2m s cot3(θp 2 )[GEn(θp)GMp(θp) − GMn(θp)GEp(θp)], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='13) F2− ∼ 2m s cot(θp 2 )[GMn(θp)GEp(θp) − GEn(θp)GMp(θp)], F3+ ∼ 0, F3− ∼ 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='14) 14 � ��� 1��� 1��� 1��� 1��� � ��� ��� ��� ��� ��0(����G ��� ��� ��� ��� ��� ��� ��� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Plots of the tensor polarization observable t20 of the deuteron from both data [25] (circles) and the model (squares) considered in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The angle θe varies and is as follows in order of increasing Q2: 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6◦, 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4◦, 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8◦, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='3◦, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0◦, and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' F4+ ∼ −4 √ 2m s cot(θp 2 )GMn(θp)GEp(θp), F4− ∼ 4 √ 2m s cot(θp 2 )GEn(θp)GMp(θp), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='15) F5+ ∼ 2 √s cot2(θp 2 )GEn(θp)GEp(θp), F5− ∼ 2 √sGEn(θp)GEp(θp), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='16) F6+ ∼ 4 √ 2m s cot3(θp 2 )GEn(θp)GMp(θp), F6− ∼ −4 √ 2m s cot(θp 2 )GMn(θp)GEp(θp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='17) From these we can calculate the various observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Plots of the results and recent data [48– 50] are given in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 14, 15, 16, and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Because the tree-level amplitudes are real, Py is automatically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' That Cx′ is of order m/√s, rather than zero, is a correction to hadron helicity conservation [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Also, we find the asymmetry Σ(90◦) to be approximately -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='06, rather than the nominal expectation [52] of -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In general, the trends with photon energy seem to be modestly consistent with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' ELECTRODISINTEGRATION The kinematics of the electrodisintegration process are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The initial (final) momentum and helicity of the electron are pe (p′ e) and λe (λ′ e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The intermediate photon carries four-momentum q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The azimuthal angle φp of the proton measures the rotation of the hadronic reaction plane relative to the electron scattering plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In the lab frame, with the z axis taken along the photon three-momentum and the electron mass neglected, the initial and final electron four-momenta are pe = (Ee, Ee sin θe, 0, Ee cos θe), p′ e = (E′ e, E′ e sin θ′ e, 0, E′ e cos θ′ e), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='1) 15 � ��� V�)� V�)� � �)� �)� �)� �)� �)� �)� ��1Q���� �)� �)� � �)� �)� �)� �)� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 10 but for t21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' � ��� 8�4�� 8�4� 8�4�� 8�4� 8�4�� � �4�� �4� �4�� ��-G����e �4� �4� � �4� �4� �4� �4� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 10 but for t22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' with E′ e and ˜θ = θ′ e − θe, the angle of the scattered electron to the beam direction, being measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The photon four-momentum q = (Eγ, 0, 0, qz) is just pe − p′ e, which yields Q2 ≡ −q2 = 2EeE′ e(1 − cos ˜θ), Eγ = Ee − E′ e, qz = � E2γ + Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2) The deuteron four-momentum is p = (md = 2m, 0, 0, 0), and in the zero-binding limit, the initial proton and neutron four-momenta are pp = pn = (m, 0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The final nucleon 16 ⃗q, λγ ⃗p, λd ⃗p ′ p, λ′ p ⃗p ′ n, λ′ n θn = π − θp θp z FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Kinematics for deuteron photodisintegration in the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' frame, with ⃗q the photon mo- mentum and ⃗p = −⃗q the deuteron momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The final proton and neutron momenta are ⃗p ′ p and ⃗p ′ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The λ’s are helicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Coordinates are chosen such that the photon enters along the positive z direction and the azimuthal angle φp of the proton is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Eγ (GeV) 0 1 2 3 4 5 6 Py (90 deg) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 θ (deg) 0 20 40 60 80 100 120 140 160 180 Py (Eγ=2 GeV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Recoil proton polarization Py as a function of (a) photon energy Eγ and (b) proton angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For the latter, the photon energy is 2 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The solid line is the RNHA prediction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' the data points are from [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' four-momenta are p′ p = (E′ p = � ⃗p ′2 p + m2, |⃗p ′ p| sin θp cos φp, |⃗p ′ p| sin θp sin φp, |⃗p ′ p| cos θp), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='3) p′ n = (E′ n = � ⃗p ′2 n + m2, −|⃗p ′ n| sin θn cos φp, −|⃗p ′ n| sin θn sin φp, |⃗p ′ n| cos θn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content="4) Within the one-photon-exchange approximation, the scattering amplitude is proportional 17 Eγ (GeV) 0 1 2 3 4 5 6 Cx' (90 deg) 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content="0 θ (deg) 0 20 40 60 80 100 120 140 160 180 Cx' (Eγ=2 GeV) 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 14 but for the transferred polarization Cx′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=" Eγ (GeV) 0 1 2 3 4 5 6 Cz' (90 deg) 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content="0 θ (deg) 0 20 40 60 80 100 120 140 160 180 Cz' (Eγ=2 GeV) 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 15 but for Cz′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' to Med(λ′ p, λ′ n, λ′ e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' λd, λe) = ¯u′ eγµue Dµν q2 Mν(λ′ p, λ′ n, λd), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5) with ue (u′ e) the initial (final) spinor of the electron and Mν given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The numerator of the photon progator is the sum over photon polarizations Dµν = 1 � λ=−1 (−1)λǫ∗ µ(λ)ǫν(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6) 18 Eγ (GeV) 0 1 2 3 4 5 6 Σ (90 deg) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 θ (deg) 0 20 40 60 80 100 120 140 160 180 Σ (Eγ=2 GeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='10 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 14 but for the asymmetry Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The data points are from [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The polarization four-vectors are4 ǫ(±1) = ∓ 1 √ 2(0, 1, ±i, 0), ǫ(0) = (qz/Q, 0, 0, Eγ/Q) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='7) relative to the photon four-momentum q = (Eγ, 0, 0, qz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Polarization observables [5–15] can then be computed from these helicity amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' ⃗pe, λe ⃗p ′ e, λ′ e x ⃗q ⃗p ′ p, λ′ p ⃗p ′ n, λ′ n y′ x′ z, z′ y θp φp θe θ′ e z θn ˜θ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Kinematics for deuteron electrodisintegration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The unprimed axes are defined relative to the electron scattering plane, and the primed axes relative to the final nucleon momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The final proton momentum has polar angle θp and azimuthal angle φp relative to the unprimed frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 4 In the hadronic c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' frame, the longitudinal polarization vector is ǫ(0) = (q′ z/Q, 0, 0, E′ γ/Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 19 In keeping with the notation of [6, 7] and [15], the differential cross section for elec- trodisintegration, summed over the final electron and neutron helicities in the lab frame, is [14, 15]5 dσ ≡ dσ5 dE′dΩ′ edΩ′ p (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8) = mpmn|⃗p ′ p| 16π3md σMott frec [νLRL + νT RT + νTTRTT + νLTRLT + 2λeνLT ′TLT ′ + 2λeνT ′RT ′] , where Ω′ e (Ω′ p) is the solid angle of the scattered electron (proton), σMott is the Mott cross section, frec = |1 + (Eγ|⃗p ′ p| − E′ pqz cos θp)/(md|⃗p ′ p|)| is the lab recoil factor, νL = Q4 q4z , νT = Q2 2q2z + tan2 ˜θ 2, νTT = Q2 2q2z , νLT = Q2 √ 2q2 z � � � �Q2 q2z + tan2 ˜θ 2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='9) νLT ′ = − Q2 √ 2q2z tan ˜θ 2, νT ′ = tan ˜θ 2 � � � �Q2 q2 z + tan2 ˜θ 2, and ˜θ = θ′ e − θe is the angle between the incoming and outgoing electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The response functions RX depend upon the hadronic helicity amplitudes and the azimuthal angle φp of the hadronic scattering plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The subscripts refer to the polarization of the intermediate photon, which enters on substitution of the polarization expansion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6) for the numerator of the photon propagator in the hadronic amplitude (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The amplitude then decomposes into separate leptonic and hadronic factors Med(λ′ p, λ′ n, λ′ e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' λd, λe) = − 1 � λ=−1 ¯u′ e̸ ǫ∗(λ)ue (−1)λ Q2 ǫν(λ)Mν(λ′ p, λ′ n, λd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='10) The leptonic factors give rise to the νX coefficients, and the hadronic factors to the response functions in the square of the amplitude used to construct the cross section [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The subscript L(T) indicates a purely longitudinal (transverse) contribution, while LT is a cross term between longitudinal and transverse photon helicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The TT subscript marks a cross term between different transverse helicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' A prime indicates a different combination of transverse helicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The response functions are computed from components of the hadronic tensor wλ′,λ = 2 3 � λ′′p,λ′p,λ′n,λ′′ d,λd ǫ∗ ν(λ′)Mν∗(λ′′ p, λ′ n, λ′′ d)ρp λ′′p,λ′pǫµ(λ)Mµ(λ′ p, λ′ n, λd)ρd λ′′ d,λd, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='11) with ρp(ρd) the density matrix for the proton (deuteron) helicity state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We construct these in the xyz coordinate system of the electron scattering plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The particular components are [6] RL = w0,0, RT = w1,1 + w−1,−1, RT ′ = w1,1 − w−1,−1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='12) RTT = 2Rew1,−1, RLT = −2Re [w0,1 − w0,−1] , RLT ′ = −2Re [w0,1 + w0,−1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 5 In [6], h is 2λe but in [15], h is just λe, which leads to additional factors of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 20 For an unpolarized target, the deuteron density matrix is proportional to the identity, ρd = 1 3I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' similarly, if the proton helicity is not detected, ρp = 1 2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We then have the unpolarized cross section [6] dσunpol = mpmn|⃗p ′ p| 16π3md σMott frec σ0, σ0 ≡ νLRU L + νTRU T + νTTRU TT + νLTRU LT , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='13) where the RU X are computed with the simple density matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' These are then computable in our model, with the basic computation being the evaluation of ǫ(λγ)µMµ, which differs from the photodisintegration calculation in only two ways: Q2 is not zero and λγ ranges over all three possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The unpolarized response functions RU LT ′ and RU T ′ are identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' With ρd replaced by 1 3I and the form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='7) of the polarization vectors taken into account, Rew(0, 1) is just the negative of Rew(0, −1), and w1,1 is equal to w−1,−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Thus, the inputs to RU LT ′ and RU T ′, as given in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='12), immediately cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The recent ed → e′pn experiment at JLab [17] does not include polarization but does begin to reach momentum transfers sufficient to consider the RNHA approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Once po- larization data is available, the expressions developed here and in the Appendix can be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' SUMMARY We have extended the reduced nuclear amplitude approach [1, 2] to helicity amplitudes and applied this model to analysis of elastic electron-deuteron scattering, deuteron photo- disintegration, and deuteron electrodisintegration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' These are just examples of the approach, which is generally applicable to exclusive nuclear processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The primary limitation is that, for any process, the net momentum transfer to every nucleon must be large;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' therefore, as the number of nucleons increases, the required beam energy can increase dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The primary gain is precocious scaling in the dependence on momentum transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' What the model (or the original RNA approach) does not provide, though, is an overall normalization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' comparisons must be made in terms of ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' By considering helicity amplitudes, many more quantities can be studied, including po- larization dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' All three of the deuteron’s electromagnetic form factors can be cal- culated and from there various elastic scattering observables can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' III we considered the standard structure functions A and B as well as the tensor polarizations t2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Generally, the model implies the need for momentum transfers larger than one would have hoped for seeing simple perturbative QCD scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' However, our results do imply that the deuteron structure function B is a good place to look, above a transfer of 10 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The RNHA results for polarization observables in deuteron photodisintegration, consid- ered in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' IV, are somewhat consistent with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In particular, our result for the asymmetry Σ, with a value of Σ(90◦) ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='06, is much better than the value of -1 originally expected [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Higher photon energies would, of course, be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We have also constructed the RNHA framework for analysis of deuteron electrodisinte- gration, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' This stands ready for comparison with experiment when data is available at sufficient energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' One aspect that does remain is to consider polarization of the outgoing proton, in addition to polarization of the beam and target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Other processes that one might consider include deeply virtual Compton scattering on the deuteron, pion photoproduction on the deuteron [3], and photodisintegration of 3He [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 21 In each case, our approach can provide not only information about helicity amplitudes but also an analysis of nonleading momentum transfer dependence with respect to the onset of perturbative QCD scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We look forward to experiments at larger momentum transfers for all of these processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work began in conversations with S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Brodsky and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Hwang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Some calculations were checked by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Miller and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Salveson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Diagrams were drawn with use of JaxoDraw [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Appendix A: Electrodisintegration with polarization If we consider polarization for the beam and the target,6 the proton density matrix is still just ρp = 1 2I, but the deuteron density matrix in the xyz frame is [6] ρd = 1 3 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 1 + � 3 2T10 + 1 √ 2T20 − � 3 2(T ∗ 11 + T ∗ 21) √ 3T ∗ 22 − � 3 2(T11 + T21) 1 − √ 2T20 − � 3 2(T ∗ 11 − T ∗ 21) √ 3T22 − � 3 2(T11 − T21) 1 − � 3 2T10 + 1 √ 2T20 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (A1) For a target polarization defined relative to the beam direction, rather than the xyz system used above, the tensor polarization coefficients TJM are related to the coefficients ˜TJM defined relative to the beam [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' If only ˜T10 and ˜T20 are nonzero,7 the nonzero TJM are T10 = cos ˜θ ˜T10, T11 = − 1 √ 2 sin ˜θ ˜T10, (A2) T20 = 1 4(1 + 3 cos 2˜θ) ˜T20, T21 = − � 3 8 sin 2˜θ ˜T20, T22 = � 3 32(1 − cos 2˜θ) ˜T20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The density matrix can then be written as ρd = �1 3I + ˜T10ρdV + ˜T20ρdT � , (A3) where ρdV = 1 3 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed � 3 2 cos ˜θ √ 3 2 sin ˜θ 0 √ 3 2 sin ˜θ 0 √ 3 2 sin ˜θ 0 √ 3 2 sin ˜θ − � 3 2 cos ˜θ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (A4) and ρdT = 1 3 \uf8eb \uf8ec \uf8ec \uf8ed 1 4 √ 2(1 + 3 cos 2˜θ) 3 4 sin 2˜θ 3 √ 32(1 − cos 2˜θ) 3 4 sin 2˜θ − 1 2 √ 2(1 + 3 cos 2˜θ) −3 4 sin 2˜θ 3 √ 32(1 − cos 2˜θ) −3 4 sin 2˜θ 1 4 √ 2(1 + 3 cos 2˜θ) \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (A5) 6 For discussion of a polarized outgoing proton, see [7] and [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 7 The spherical tensor moments are related to the Cartesian tensor moments as ˜T10 = � 3 2Pz and ˜T20 = 1 √ 2Pzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 22 The response functions can then be separated into unpolarized, vector, and tensor contri- butions as RX = RU X + ˜T10RV X + ˜T20RT X, with RU X, RV X, and RT X computed with ρd replaced by 1 3I, ρdV , and ρdT , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' With dσunpol defined as the unpolarized cross section, given in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='13), the full cross section can be written as dσ = � 1 + ˜T10 � AV d + 2λeAV ed � + ˜T20 � AT d + 2λeAT ed �� dσunpol, (A6) in terms of the single and double asymmetries AV 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