Virtual Compton scattering and the generalized polarizabilities of the proton at ${Q}^{2}=0.92$ and 1.76 GeV${}^{2}$

Virtual Compton scattering and the generalized polarizabilities of the proton at $Q^{2} = 0.92$ and 1.76 GeV $^{2}$

H. Fonvieille et al. (The Jefferson Lab Hall A Collaboration)

Phys. Rev. C 86, 015210 – Published 25 July 2012

Abstract

Virtual Compton scattering (VCS) on the proton has been studied at the Jefferson Laboratory using the exclusive photon electroproduction reaction $e p \to e p γ$ . This paper gives a detailed account of the analysis which has led to the determination of the structure functions $P_{L L} - P_{T T} / ε$ and $P_{L T}$ and the electric and magnetic generalized polarizabilities (GPs) $α_{E} (Q^{2})$ and $β_{M} (Q^{2})$ at values of the four-momentum transfer squared $Q^{2} = 0.92$ and 1.76 GeV $^{2}$ . These data, together with the results of VCS experiments at lower momenta, help building a coherent picture of the electric and magnetic GPs of the proton over the full measured $Q^{2}$ range and point to their nontrivial behavior.

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Received 18 May 2012

DOI:https://doi.org/10.1103/PhysRevC.86.015210

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Vol. 86, Iss. 1 — July 2012

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Figure 1
Feynman graphs of photon electroproduction.
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Figure 2
$(e p \to e p γ)$ kinematics; four-momentum vectors notation and Compton angles $(θ_{c . m .}, ϕ)$ in the $γ p$ center of mass.
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Figure 3
The range in $q_{c . m .}^{'}$ , or $W$ , covered by the various data sets for the $e p \to e p γ$ events. The vertical lines show the upper limit applied in the analyses: the pion threshold (dotted line at $W = 1.073$ GeV) for the LEX analyses and $W = 1.28$ GeV (dashed line) for the DR analyses. $W$ and $q_{c . m .}^{'}$ are related by $W = q_{c . m .}^{'} + \sqrt{q_{c . m .}^{' 2} + m_{p}^{2}}$ .
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Figure 4
The 17 HRS-H settings for the proton detection in data set I-a. Each setting is represented by a box in momentum and angle. The closed curves correspond to in-plane $e p \to e p γ$ kinematics at fixed values of $q_{c . m .}^{'}$ : 45 MeV/ $c$ (inner curve) and 105 MeV/ $c$ (outer curve). The $e p \to e p$ elastic line is also drawn at a beam energy of 4.045 GeV.
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Figure 5
Coincidence time spectrum of data set I-a. The central peak is 0.5 ns wide in rms.
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Figure 6
The fitted offset in beam energy, $Δ E_{beam}$ , versus the setting number (time ordered). There is one point per setting. The various data sets are delimited by the vertical lines. The horizontal line at $Δ E_{beam} = 0$ corresponds to the nominal beam energy from the accelerator, $E_{beam} = 4.045$ GeV.
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Figure 7
A sample of data set II: the experimental spectrum of the missing mass squared at various levels of cuts, added successively and labeled from 1 to 5. (a) The raw coincidences (1) and adding the $R$ -function cut (2). (b) Adding the conditions $W > 0.96$ GeV (3), $Y_{dif} < - 0.012$ m (4), and $| x_{dif} | < 3$ mm (5) (see the text for the description of the variables). Inset (c) shows histogram 1 in full scale abscissa.
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Figure 8
A sample of data set II: the experimental $W$ spectrum for the coincidence events after the $R$ -function cut. The insets show the proton impact in the HRS-H collimator calculated “elastically” (see text). Events in insets (a) and (b) correspond to the two hatched zones of the histogram: the $e p$ elastic peak [ $W < 0.96$ GeV (a)] and a typical VCS region [ $1.0 < W < 1.073$ (b)]. In inset (a) a sketch of the tungsten collimator is drawn. The upper inset (r) shows the $e p$ elastic events before applying the $R$ -function cut.
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Figure 9
A sample of data set II: the experimental $Y_{dif}$ spectrum (see text) for coincidence events surviving the application of the three following cuts simultaneously: $R$ function $> 0$ , $- 5000 < M_{X}^{2} < 5000$ MeV $^{2}$ and $W > 0.96$ GeV (solid histogram). The insets show the “elastically calculated” proton impact in the HRS-H collimator (see text) for $Y_{dif} < - 0.012$ m [clean events (a)] and for $Y_{dif} > - 0.012$ m [punch-through protons (b)]. The dashed histogram corresponds to the VCS simulation with the same three cuts.
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Figure 10
Data sets I-a (top) and II (bottom) after all cuts: comparison of experiment (solid histogram) and simulation (dotted histogram). [(a) and (c)] The missing mass squared in the VCS region; the peak full width at half maximum (FWHM) is about 1650 MeV $^{2}$ . [(b) and (d)] The geometrical variable $x_{dif}$ (see text); the peak FWHM is about 1.9 mm.
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Figure 11
Accepted phase space in ( $θ_{c . m .}^{'}, ϕ_{c . m .}^{'})$ for data set I-a. The two crosses denote the position of the BH peaks and the horizontal line corresponds to in-plane kinematics ( $θ_{c . m .}^{'} = 90^{\circ}$ or $ϕ = 0^{\circ}$ and $180^{\circ}$ ).
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Figure 12
Data set I-a below the pion threshold at $q_{c . m .}^{'} = 105$ MeV/ $c$ . The $(e p \to e p γ)$ cross section is shown in-plane [ $θ_{c . m .}^{'} = 90^{\circ}$ , (a)] and out-of-plane [ $θ_{c . m .}^{'} = 40^{\circ}$ , (b)]. The dotted curve is the BH+Born calculation. The solid curve includes the first-order GP effect calculated using our measured structure functions. The errors on the points are statistical only, as well as in the six next figures. The upper plot (c) shows the in-plane BH+Born cross section with a full-scale ordinate and the more traditional abscissa running between $ϕ_{c . m .}^{'} = - 180^{\circ}$ and $ϕ_{c . m .}^{'} = + 180^{\circ}$ .
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Figure 13
The ratio $(d σ^{EXP} - d σ^{BH + Born}) / d σ^{BH + Born}$ for the data points of the previous figure. The solid curve shows the first-order GP effect calculated using our measured structure functions.
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Figure 14
The $(e p \to e p γ)$ cross section for data sets I-a (a) and II (b) at $q_{c . m .}^{'} = 215$ MeV/ $c$ , in-plane ( $θ_{c . m .}^{'} = 90^{\circ}$ ) as a function of $ϕ_{c . m .}^{'}$ . The dashed curve is the DR model calculation, with parameter values as fitted in the experiment. The dotted (respectively, solid) curve is the BH+Born cross section (respectively, plus a first-order GP effect).
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Figure 15
Data set I-a. The $(e p \to e p γ)$ cross section at the lowest $q_{c . m .}^{'}$ of 45 MeV/ $c$ , for in-plane (a) and out-of-plane (b) kinematics. The solid curve is the (BH+Born+first-order GP) cross section. The right plot (c) shows the reduced $χ^{2}$ of the normalization test.
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Figure 16
The $q_{c . m .}^{'}$ dependence of the quantity $Δ M$ [see Eq. (9)] in each angular bin. Example of data set I-a at $θ_{c . m .}^{'} = 40^{\circ}$ (out-of-plane). The value of $ϕ_{c . m .}^{'}$ is written in each plot. The shaded band is the result of the LEX fit, “ $Δ M = const$ ,” within $\pm 1 σ$ error (statistical). The fit is performed on the three points below the pion threshold (solid circles). The other points (open circles) are above the pion threshold and do not participate to the LEX fit. The dashed curve shows the calculation of $Δ M$ by use of the DR model, using the results of our DR fit (see Sec. 5b2) which is performed on all points (solid+open circles).
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Figure 17
A graphical representation of the LEX fit for data sets I-a (a) and II (b). Each point in $v_{1} / v_{2}$ corresponds to a different bin in $(θ_{c . m .}^{'}, ϕ_{c . m .}^{'})$ . The solid circles correspond to out-of-plane data. The inset in plots (a) and (b) is a close-up of the in-plane data (triangles), all concentrated at small values of $v_{1} / v_{2}$ . The straight line refers to the fit performed on all data points (in-plane+out-of-plane).
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Figure 18
Data set I-b. The $(e p \to e p γ)$ cross section at fixed $\cos θ_{c . m .} = - 0.975$ and six bins in $ϕ$ (from Table 17). The curves show the DR model calculation for parameter values $(Λ_{α}, Λ_{β}) = (0.70, 0.63)$ GeV (solid line), (1.20, 0.63) GeV (dashed line), (0.70, 1.00) GeV (dotted line).
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Figure 19
The structure functions obtained by the two methods (LEX and DR) at $Q^{2} = 0.92$ GeV $^{2}$ and 1.76 GeV $^{2}$ . Each $Q^{2}$ corresponds to a solid-line polygon as marked. For each point, the inner ellipse is the contour at ( $χ_{\min}^{2}$ +1), yielding the standard deviation on each structure function independently. The outer ellipse at ( $χ_{\min}^{2}$ +2.3) corresponds to a probability of 68 $%$ that both structure functions are inside the contour simultaneously. The statistical errors quoted in Tables 5 and 7 are given by the boundaries of the inner contour. Dotted crosses give the size of the systematic error.
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Figure 20
The structure functions $P_{L L} - P_{T T} / ε$ (a) and $P_{L T}$ (b) measured at Bates [8], MAMI [1, 3], and JLab [5] (this experiment). The RCS point deduced from Ref. [17] is also included. The insets are a close-up the $Q^{2}$ region of this experiment. Some points are slightly shifted in the abscissa for visibility. The inner (outer) error bar on the points is statistical (total). The thin solid curve is the HBChPT calculation [27]. The other curves show DR calculations [32] performed with various sets of parameters. (a) $Λ_{α} = 0.7$ GeV (thick solid line) and $Λ_{α} = 1.79$ GeV (dashed line). (b) $Λ_{β} = 0.7$ GeV (thick solid line) and $Λ_{β} = 0.5$ GeV (dashed line). $ε = 0.9$ is chosen to draw the curves for $P_{L L} - P_{T T} / ε$ . The dotted curve in the insets is the spin part as given by the DR model: $- P_{T T} / 0.9$ (upper) and $P_{LTspin}$ (bottom).
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Figure 21
Theoretical predictions for the spin part of the measured structure functions. The DR curves labeled 1a and 1b are calculated with the MAID 2000 and MAID 2003 multipoles, respectively. The HBChPT curves 2a and 2b are obtained at $O (p^{3})$ [27] and $O (p^{4})$ [62], respectively. They are drawn up to an arbitrary value of $Q_{\max}^{2} = 0.5$ GeV $^{2}$ .
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Figure 22
The world data on the electric GP (a) and the magnetic GP (b), with statistical (inner) and total (outer) error bar. The solid curve is the DR calculation drawn for typical parameter values obtained in our experiment: $Λ_{α} = 0.70$ GeV (left) and $Λ_{β} = 0.70$ GeV (right). The short-dashed curves show the two separate contributions to this calculation: the pion-nucleon intermediate states (curve labeled “ $π N$ ”) and the [asymptotic+beyond $π N$ ] contribution (curve labeled “asy”). The long-dashed curve is the full DR calculation for other parameter values: $Λ_{α} = 1.79$ GeV (left) and $Λ_{β} = 0.51$ GeV (right).
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Figure 23
The angular systems to measure the polar and azimuthal angles of the momentum vector $\vec{q}_{c . m .}^{'}$ in the ( $γ p$ ) center of mass.
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