Ratios of helicity amplitudes for exclusive \(\rho ^0\) electroproduction on transversely polarized protons
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Abstract
Exclusive \(\rho ^0\)meson electroproduction is studied by the HERMES experiment, using the 27.6 GeV longitudinally polarized electron/positron beam of HERA and a transversely polarized hydrogen target, in the kinematic region 1.0 GeV\(^2<Q^2<\) 7.0 GeV\(^2\), 3.0 GeV \(<W<\) 6.3 GeV, and \(t^\prime <\) 0.4 GeV\(^2\). Using an unbinned maximumlikelihood method, 25 parameters are extracted. These determine the real and imaginary parts of the ratios of several helicity amplitudes describing \(\rho ^{0}\)meson production by a virtual photon. The denominator of those ratios is the dominant amplitude, the nucleonhelicitynonflip amplitude \(F_{0\frac{1}{2}0\frac{1}{2}}\), which describes the production of a longitudinal \(\rho ^{0}\)meson by a longitudinal virtual photon. The ratios of nucleonhelicitynonflip amplitudes are found to be in good agreement with those from the previous HERMES analysis. The transverse target polarization allows for the first time the extraction of ratios of a number of nucleonhelicityflip amplitudes to \(F_{0\frac{1}{2}0\frac{1}{2}}\). Results obtained in a handbag approach based on generalized parton distributions taking into account the contribution from pion exchange are found to be in good agreement with these ratios. Within the model, the data favor a positive sign for the \(\pi \rho \) transition form factor. By also exploiting the longitudinal beam polarization, a total of 71 \(\rho ^0\) spindensity matrix elements is determined from the extracted 25 parameters, in contrast to only 53 elements as directly determined in earlier analyses.
1 Introduction
The formalism describing SDMEs of the produced vector meson was first presented in Ref. [4] for unpolarized targets only, and expressions of SDMEs in terms of helicity amplitudes were also established. The formalism was then extended to the case of polarized targets in Ref. [5]. An alternative, general formalism for the description of the process in Eq. (1) through SDMEs was presented in Ref. [6]. In the latter formalism, which is used throughout this paper, the SDMEs describing the production on an unpolarized target are denoted by \(u^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\), those describing the production on a longitudinally polarized target are denoted by \(l^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\), and those describing the production on a transversely polarized target are denoted by \(n^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\) and \(s^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\). Here, longitudinal and transverse polarization are defined with respect to the momentum direction of the virtual photon in the CM system of the process in Eq. (1).
The exact expressions for SDMEs [4, 5, 6], which are dimensionless quantities, can be rewritten in terms of ratios of helicity amplitudes. When fitting the experimental angular distribution of the finalstate particles, either the SDMEs or alternatively the amplitude ratios can be considered as independent free parameters. The first fit method is referred to as the “SDME method” in the rest of this paper, while the second one is referred to as the “amplitude method”.
Exclusive meson production in hard leptonnucleon scattering was shown to offer the possibility of constraining generalized parton distributions (GPDs), which provide correlated information on transversespatial and fractionallongitudinalmomentum distributions of partons in the nucleon (see Refs. [3, 7, 8, 9, 10, 11] and references therein). Vectormesonproduction amplitudes contain various linear combinations of GPDs for quarks of various flavors and for gluons. In particular, exclusive \(\rho ^{0}\) production on an unpolarized target is sensitive to the nucleonhelicitynonflip GPD H, while exclusive \(\rho ^{0}\) production on a transversely polarized target is sensitive to the nucleonhelicityflip GPD E, as well. Through the Ji relation [12, 13], the sum of both GPDs H and E is related to the parton total angular momentum. Access to GPDs relies on the factorization property of the process amplitude, i.e., the amplitude can be written as a convolution of GPDs and vectormeson distribution amplitudes, which are both nonperturbative quantities, and amplitudes of hard partonic subprocesses, which are calculable within the frameworks of perturbative quantum chromodynamics (pQCD) and quantum electrodynamics.
For spin1 particles, longitudinal (transverse) polarization is assigned by convention to the states with helicity \(\lambda =0\) (\(\lambda = \pm 1\)). The helicity amplitudes \(F_{0\frac{1}{2}0 \pm \frac{1}{2}}\) describe the transition of a longitudinally (L) polarized virtual photon to a longitudinally polarized vector meson, \(\gamma ^*_L \rightarrow V_L\), and dominate at large photon virtuality \(Q^2\). Although factorization was rigorously proven [14] only for these amplitudes, it was assumed in Refs. [15, 16] that factorization also holds for the amplitudes \(F_{1\frac{1}{2}1\pm \frac{1}{2}}\) and \(F_{0 \frac{1}{2}1\pm \frac{1}{2}}\), which describe the transition from a transversely polarized virtual photon to a transversely polarized meson, \(\gamma ^*_T \rightarrow V_T\), and a longitudinally polarized meson, \(\gamma ^*_T \rightarrow V_L\), respectively. The agreement found between certain calculated SDMEs and those extracted from HERMES [17], ZEUS [18], and H1 [19] data supports this assumption. In general, the differential and total cross sections for \(\rho ^0\)meson production by virtual photons are reasonably well described in the GPDbased approach of Refs. [15, 16], not only at the high energies of the HERA collider experiments [20, 21, 22, 23, 24], but also at intermediate energies covered by the fixedtarget experiments E665 [25] and HERMES [26].
The real parts of the amplitude ratios in \(\rho ^0\) and \(\phi \) meson electroproduction on the proton were first studied by the H1 experiment [19] at the HERA collider. In the HERMES experiment [27], \(\rho ^0\)meson production on unpolarized protons and deuterons was investigated. Both real and imaginary parts of the ratios of amplitudes without nucleon helicity flip were extracted at HERMES using a longitudinally polarized electron or positron beam. The results of the analysis of \(\rho ^0\)meson and \(\omega \)meson production on the unpolarized targets at HERMES using the SDME method were published in Refs. [17] and [28], respectively. The SDMEs for the electroproduction on transversely polarized protons were published in Ref. [29]. In this paper, the work of Ref. [29] is continued. Ratios of \(\rho ^{0}\) helicity amplitudes with respect to the amplitude \(F_{0\frac{1}{2}0\frac{1}{2}}\) are extracted from HERMES data collected with longitudinally polarized electron and positron beams scattered off transversely polarized protons. The amplitude ratios that require measurements with a transversely polarized target are reported for the first time in this paper.
At fixed \(Q^2\) and CM energy W in the \(\gamma ^{*}N\) system the cross section \({\text {d}}\sigma /{\text {d}}t\), which is differential in the Mandelstam variable t, contains the linear combination of squares of all helicity amplitudes. Including \({\text {d}}\sigma /{\text {d}}t\) in an amplitude analysis of all beam and targetpolarization states would allow the extraction of the moduli of all amplitudes and of the phase differences between them, while the common phase would remain undetermined.
The amplitude ratios measured at HERMES, as described in this paper, will also be compared to those evaluated within the GPDbased handbag approach by Goloskokov and Kroll [15, 16], hereafter referred to as “GK model”.
The paper is organized as follows. In Sect. 2, the theoretical formalism is introduced. Section 3 briefly describes the experimental setup and specifies the applied data selection. The extraction procedure of the amplitude ratios is treated in Sect. 4. The obtained results are discussed in Sect. 5. Summary and conclusions are given in Sect. 6.
2 Formalism
2.1 Kinematics
2.2 Definition of angles and coordinate systems
The angles used for the description of the process are defined in the same way as in Ref. [17], according to Ref. [30], and are presented in Fig. 1. According to Ref. [4], the righthanded “hadronic CM system” of coordinates XYZ of virtual photon and target nucleon is defined such that the Zaxis is aligned along the virtualphoton threemomentum \(\varvec{q}\) and the Yaxis is parallel to \(\varvec{q} \times \varvec{v}\), where \(\varvec{v}\) is the \(\rho ^0\)meson threemomentum. The angle \(\varPhi \) is the angle between the \(\rho ^0\)meson production plane (XZ plane, which coincides with the nucleon scattering plane) and the lepton scattering plane in the CM system. The angles \(\theta \) and \(\phi \) are defined in the righthanded xyz system of coordinates (see Fig. 1) that represents the \(\rho ^0\)meson rest frame. The y axis coincides with the Y axis. The angle \(\theta \) is the polar angle of the decay \(\pi ^+\)meson threemomentum with respect to the z axis, where the latter is aligned opposite to the direction of the momentum of the outgoing nucleon. The azimuthal angle of the \(\pi ^+\) momentum with respect to the \(\rho ^0\)meson production plane in the CM system is denoted \(\phi \). In the HERMES experiment, the vector \(\varvec{P}_T\) of the target polarization is orthogonal to the beam direction. The angle between the directions of the transverse part (with respect to the beam) of the scattered electron momentum and \(\varvec{P}_T\) is denoted by \(\varPsi \) and is defined in the target rest frame.
2.3 Natural and unnaturalparityexchange helicity amplitudes
 1.
The number of linearly independent NPE (UPE) amplitudes is equal to 10 (8);
 2.
No UPE amplitude exists for the transition \(\gamma _{L} \rightarrow \rho ^0_{L}\), so that in particular \(F_{0 \frac{1}{2} 0 \frac{1}{2} } = T_{0 \frac{1}{2} 0 \frac{1}{2} } \equiv T^{(1)}_{00}\);
 3.
2.4 Asymptotic behavior of amplitudes at small \(t'\)
2.5 Spindensity matrix of the virtual photon
2.6 Spindensity matrix of the initial nucleon
2.7 Spindensity matrix of the \(\rho ^{0}\) meson
2.8 SDMEs in the Diehl representation
2.9 Angular distribution of decay pions
It can easily be shown that the angular distribution \(\mathcal {W}(\varPhi , \varPsi ,\theta , \phi )\) cannot be negative for any set of values of the complex amplitudes \(F_{\lambda _{V}\mu _N\lambda _{\gamma }\lambda _N}\), even for unphysical ones. This property is of great importance for the fit procedure. It is worthwhile to note that using the SDME method one faces the problem of a possible negativity of \(\mathcal {W}(\varPhi , \varPsi ,\theta , \phi )\) for some angles when SDMEs assume unphysical values. As it is unknown in which region in the multidimensional space of SDMEs \(\mathcal {W}(\varPhi , \varPsi ,\theta , \phi )\) is not negative, serious problems may appear when applying the maximumlikelihood method. Hence the amplitude method is in that respect more reliable than the SDME method.
Altogether, Eqs. (46), (38–41), (29–31), and (32–37), with the substitutions \(T_{00}^{(1)}\rightarrow 1\) and for all other amplitudes \(T_{\lambda _{V}\lambda _{\gamma }}^{(j)}\rightarrow t_{\lambda _{V}\lambda _{\gamma }}^{(j)}\) and \(U_{\lambda _{V}\lambda _{\gamma }}^{(j)}\rightarrow u_{\lambda _{V}\lambda _{\gamma }}^{(j)}\), constitute a basis for the amplitude method, in which the extracted quantities are the helicityamplitude ratios.
3 Experiment and data selection
3.1 Experiment
A detailed description of the HERMES experiment can be found in Ref. [34]. The data analyzed in this paper were collected between the years 2002 and 2005. A longitudinally polarized positron or electron beam of 27.6 GeV was scattered from a pure gaseous, transversely polarized hydrogen target internal to the HERA lepton storage ring. The helicity of the beam was typically reversed every 2 months. The beam polarization was continuously measured by two Compton polarimeters [35, 36]. The average value of the beam polarization for the events used in the analysis is about \({\pm } 0.30\) with a relative uncertainty of 2%. The target polarization was reversed every 60 to 180 s [37]. The measured mean value of the target polarization is \(\langle P_{T}\rangle =0.72 \pm 0.06\) [38, 39].
Mean values for the kinematic variables W, \(Q^2\), \(t'\), and \(f_{bg}\) under the exclusive peak in each of the (\(Q^2\), \(t'\)) cells
Cell limits  \(\langle W \rangle \), GeV  \(\langle Q^2 \rangle \), GeV\(^2\)  \( \langle t^\prime \rangle \), GeV\(^2\)  \(f_{bg}\) 

1.0 GeV\(^2<Q^2 <1.4\) GeV\(^2\)  4.70  1.19  0.128  0.065 
1.4 GeV\(^2<Q^2 <2.0\) GeV\(^2\)  4.75  1.67  0.128  0.073 
2.0 GeV\(^2<Q^2 <7.0\) GeV\(^2\)  4.80  3.06  0.136  0.122 
0.00 GeV\(^2<t' <0.05\) GeV\(^2\)  4.75  1.89  0.023  0.064 
0.05 GeV\(^2<t' <0.10\) GeV\(^2\)  4.75  1.92  0.074  0.085 
0.10 GeV\(^2<t' <0.20\) GeV\(^2\)  4.71  1.94  0.145  0.108 
0.20 GeV\(^2<t' <0.40\) GeV\(^2\)  4.72  2.00  0.281  0.147 
3.2 Event selection
 1.
The longitudinal beam polarization is restricted to the interval \(15\%<  P_B  < 80\%\).
 2.
Events with exactly two oppositely charged hadrons and one lepton with the same charge as the beam lepton are selected. All tracks are required to originate from the same vertex.
 3.
The scattered lepton has to have an energy larger than 3.5 GeV in order to not introduce effects from varying trigger thresholds.
 4.
The twohadron invariant mass is required to lie around the \(\rho ^{0}\) mass, i.e., it is required to obey 0.6 GeV \(<M(\pi ^+\pi ^)<1.0\) GeV.
 5.
The photon virtuality is required to obey 1 GeV\(^2<Q^2< 7\) GeV\(^2\). The lower limit is a minimum requirement for the application of pQCD, while the upper one delimits a well defined kinematic phase space.
 6.
The \(t'\) variable is restricted to \(t' \le 0.4\) GeV\(^2\) in order to reduce nonexclusive background of the reaction under study.
 7.
The invariant mass W is required to obey 3 GeV \(< W \le 6.3\) GeV. The requirement \(W>3\) GeV is imposed in order to be outside of the resonance region. The upper constraint delimits a well defined kinematic phase space.
 8.
For exclusive \(\rho ^{0}\)meson production, \(\varDelta E\) as defined in Eq. (8) must vanish. In the present analysis, taking into consideration the spectrometer resolution, the missing energy has to be in the region \({}1.0\) GeV \(< \varDelta E < 0.6\) GeV. This region is referred to as “exclusive region” in the following.
4 Extraction of amplitude ratios
4.1 Fit of the angular distribution
4.2 Background corrections
One of the main sources of background contamination to exclusive \(\rho ^0\)meson electroproduction in deepinelastic scattering originates from SIDIS. A PYTHIA MC using GEANT3 [43] to simulate the HERMES apparatus and tuned to the kinematics of the HERMES experiment [44], hereafter referred to as “SIDIS MC”, is used for the estimation of this background contribution. The same kinematic and geometrical requirements are imposed on both simulated and real data samples. The normalization of the MC data to the experimental data is performed in the region 2 GeV \(< \varDelta E < 20\) GeV (see Fig. 3 in Ref. [29]), and the number of background events in the exclusive region is estimated. The fraction of SIDIS background \(f_{bg}\) as estimated from the SIDIS MC is shown in the fifth column in Table 1.
It is assumed that the angular distribution of the SIDIS background events is reasonably well reproduced by the SIDIS MC simulation. The fit of the angular distribution of the SIDIS MC events under the exclusive peak for each (\(Q^2,t'\)) cell is performed using Eq. (49) in which the substitutions \(\mathcal {W} \rightarrow \mathcal {W}_{bg}\), \(N_i \rightarrow N^{bg}_i\), and \(\mathcal {R} \rightarrow \mathcal {S}\) must be performed, and the sum runs over all background MC events. The set of free parameters \(\mathcal {S}\) represents the complete set of 15 “unpolarized” \(u^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\) SDMEs describing the background. The normalization factor for the background, \(N_{i}^{bg}\), is determined in an analogous way as done for the signal events, but in the present analysis the background angular distribution is considered to be independent of the beam and target polarizations, i.e., \(P_B\) and \(P_T\) are set to zero. After the fit of the SIDIS MC events, the angular distribution of the background is considered to be fixed, hence \(\mathcal {W}_{bg}(\mathcal {S},\varPhi _i, \varPsi _i,\theta _i, \phi _i)\), \(N^{bg}_i\), and \(f_{bg}\) do not contain any free parameters.
4.3 Choice of free parameters
As explained in Sect. 1, the angular distribution of the detected particles depends on the amplitude ratios. The total number of linearly independent helicityamplitude ratios defined by Eqs. (25) and (26) is 17, which means that 34 real functions of \(Q^2\), \(t'\), and W determine all SDMEs and angular distributions. As established in Ref. [27], the large amplitudes at \(t' \le 0.4\) GeV\(^2\) and \(Q^2 \ge 1\) GeV\(^2\) are \(T^{(1)}_{00}\), \(T^{(1)}_{11}\), \(U^{(1)}_{11}\), and \(T^{(1)}_{01}\). For the ratios of large amplitudes, the parameterization of the \(Q^2\) and \(t'\) dependences is chosen as in Ref. [27]. For the small amplitude ratios, only the \(t'\) dependence following from angular momentum conservation (see for instance Ref. [6]) is taken into account, while averaging over the kinematic range in \(Q^2\). If all other amplitudes are expected to be significantly smaller than the large amplitudes, a possibility to extract the small amplitudes exists only if they are multiplied by large amplitudes. This means that they contribute linearly to the angular distribution.
The easiest way to interpret the extractability of the various helicityamplitude ratios is through their contribution to the SDMEs, as detailed in Ref. [6]. For the transversely polarized target, the SDMEs \(n^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\) and \(s^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\) contribute [6], while the contribution of the SDMEs \(l^{\lambda _V \lambda '_{V}}_{\lambda _{\gamma }\lambda '_{\gamma }}\) is neglected in this analysis, since the latter are multiplied by the longitudinal component of the target polarization \(\hat{P}_Z\). Indeed, \(\hat{P}_Z\) is proportional to \(\sin \theta _{\gamma }\) (see Eq. (36)), which in turn is proportional to \(Q/\nu \) (according to Eq. (33)). At HERMES kinematics Q is much smaller than \(\nu \), with \(Q/\nu \) of the order of 0.1. The helicityamplitude ratios \(t^{(2)}_{\lambda _V \lambda _{\gamma }}\) and \(u^{(2)}_{\lambda _V \lambda _{\gamma }}\) can be extracted from data collected with a transversely polarized target, as they contribute linearly to the SDMEs \(n^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\) and \(s^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\), respectively. Contributions of squares of moduli of the helicityamplitude ratios \(u^{(1)}_{01}\), \(u^{(1)}_{10}\), \(u^{(1)}_{11}\) to \(u^{\lambda _V \lambda '_{V}}_{\lambda _{\gamma }\lambda '_{\gamma }}\) are much smaller than the contribution of \(u^{(1)}_{11}^2\) according to the hierarchy of amplitudes established in Refs. [17] and [27]. The small helicity amplitude ratios \(u^{(1)}_{\lambda _V \lambda _{\gamma }}\) with \(\lambda _V \ne \lambda _{\gamma }\) contribute linearly to the SDMEs \(l^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\). The latter are multiplied by the small factor \(\hat{P}_Z \sqrt{1\epsilon }\) and cannot be extracted from the angular distributions of finalstate pions. Therefore the helicityamplitude ratios \(u^{(1)}_{\lambda _V \lambda _{\gamma }}\), with the exception of \(u^{(1)}_{11}\), are set equal to zero in the fit.
For an unpolarized target, only the SDMEs \(u^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\) contribute to the angular distribution. As follows from the previous analysis at HERMES [27] the ratios \(t^{(1)}_{\lambda _V \lambda _{\gamma }}\) and \(u^{(1)}_{11}^2+u^{(2)}_{11}^2\) can be reliably extracted from data collected with an unpolarized target. The value of \(u^{(1)}_{11}\) can be extracted from the unpolarized data, since the numerators of some SDMEs \(u^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\) contain \(u^{(1)}_{11}^2+u^{(2)}_{11}^2\). However, the phase \(\delta _u\) of \(u^{(1)}_{11}\) cannot be obtained reliably given the limited statistics in the present analysis. Another function that cannot be reliably extracted from the present data is \(\mathrm{{Im}}\{t^{(1)}_{11}\}\). The reason is that it contributes mainly to the imaginary parts of the SDMEs \(u^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\), which are multiplied by the small factor \(P_B\sqrt{1\epsilon }\). This factor is smaller than 0.15, since \(\epsilon \) is about 0.8 and the mean value of \(P_B\) is about 0.3.
Parametrization of the helicityamplitude ratios and parameter values extracted from the fit. The combinations of the helicityamplitude ratios \(\xi \) and \(\zeta \) are defined in Eq. (58). An additional scale uncertainty of 8% originating from the uncertainty on the target polarization is present for the ratios \(t^{(2)}_{\lambda _V \lambda _{\gamma }}\), \(u^{(2)}_{\lambda _V\lambda _{\gamma }}\), \(\xi \) and \(\zeta \), but not shown. An extra scale uncertainty of 2% originating from the uncertainty on the beam polarization is present for the ratios \(\mathrm{{Im}}\{t^{(1)}_{\lambda _V\lambda _{\gamma }}\}\), \(\mathrm{{Re}}\{t^{(2)}_{\lambda _V\lambda _{\gamma }}\}\), \(\mathrm{{Re}}\{u^{(2)}_{\lambda _V\lambda _{\gamma }}\}\), and \(\zeta \), but also not shown.The correlations between the 25 parameters are listed in Table 3 in Appendix C
Parametrization  Value of parameter  Statistical uncertainty  Total uncertainty 

\(\mathrm{{Re}}\{t^{(1)}_{11}\}=b_1/Q\)  \(b_1=1.145\) GeV  0.033 GeV  0.081 GeV 
\(u^{(1)}_{11}=b_2\)  \(b_2=0.333\)  0.016  0.088 
\(\mathrm{{Re}}\{u^{(2)}_{11}\}=b_3\)  \(b_3=0.074\)  0.036  0.054 
\(\mathrm{{Im}}\{u^{(2)}_{11}\}=b_4\)  \(b_4=0.080\)  0.022  0.037 
\(\xi =b_5\)  \(b_5=0.055\)  0.027  0.029 
\(\zeta =b_6\)  \(b_6=0.013\)  0.033  0.044 
\(\mathrm{{Im}}\{t^{(2)}_{00}\}=b_7\)  \(b_7=0.040\)  0.025  0.030 
\(\mathrm{{Re}}\{t^{(1)}_{01}\}=b_8\sqrt{t'}\)  \(b_8=0.471\) GeV\(^{1}\)  0.033 GeV\(^{1}\)  0.075 GeV\(^{1}\) 
\(\mathrm{{Im}}\{t^{(1)}_{01}\}=b_9\frac{\sqrt{t'}}{Q}\)  \(b_9=0.307 \)  0.148  0.354 
\(\mathrm{{Re}}\{t^{(2)}_{01}\}=b_{10}\)  \(b_{10}=0.074\)  0.060  0.080 
\(\mathrm{{Im}}\{t^{(2)}_{01}\}=b_{11}\)  \(b_{11}=0.067\)  0.026  0.036 
\(\mathrm{{Re}}\{u^{(2)}_{01}\}=b_{12}\)  \(b_{12}=0.032\)  0.060  0.072 
\(\mathrm{{Im}}\{u^{(2)}_{01}\}=b_{13}\)  \(b_{13}=0.030\)  0.026  0.033 
\(\mathrm{{Re}}\{t^{(1)}_{10}\}=b_{14}\sqrt{t'}\)  \(b_{14}=0.025\) GeV\(^{1}\)  0.034 GeV\(^{1}\)  0.063 GeV\(^{1}\) 
\(\mathrm{{Im}}\{t^{(1)}_{10}\}=b_{15}\sqrt{t'}\)  \(b_{15}=0.080\) GeV\(^{1}\)  0.063 GeV\(^{1}\)  0.118 GeV\(^{1}\) 
\(\mathrm{{Re}}\{t^{(2)}_{10}\}=b_{16}\)  \(b_{16}=0.038\)  0.026  0.030 
\(\mathrm{{Im}}\{t^{(2)}_{10}\}=b_{17}\)  \(b_{17}=0.012\)  0.018  0.019 
\(\mathrm{{Re}}\{u^{(2)}_{10}\}=b_{18}\)  \(b_{18}=0.023\)  0.030  0.039 
\(\mathrm{{Im}}\{u^{(2)}_{10}\}=b_{19}\)  \(b_{19}=0.045\)  0.018  0.026 
\(\mathrm{{Re}}\{t^{(1)}_{11}\}=b_{20}\frac{(t')}{Q}\)  \(b_{20}=0.008\) GeV\(^{1}\)  0.096 GeV\(^{1}\)  0.212 GeV\(^{1}\) 
\(\mathrm{{Im}}\{t^{(1)}_{11}\}=b_{21}\frac{(t')}{Q}\)  \(b_{21}=0.577\) GeV\(^{1}\)  0.196 GeV\(^{1}\)  0.428 GeV\(^{1}\) 
\(\mathrm{{Re}}\{t^{(2)}_{11}\}=b_{22}\)  \(b_{22}=0.059\)  0.036  0.047 
\(\mathrm{{Im}}\{t^{(2)}_{11}\}=b_{23}\)  \(b_{23}=0.020\)  0.022  0.026 
\(\mathrm{{Re}}\{u^{(2)}_{11}\}=b_{24}\)  \(b_{24}=0.047\)  0.035  0.039 
\(\mathrm{{Im}}\{u^{(2)}_{11}\}=b_{25}\)  \(b_{25}=0.007\)  0.022  0.029 
4.4 Systematic uncertainties
In this subsection, the sources of systematic uncertainties and their effect on the extracted amplitude ratios are discussed. All systematic uncertainties except the one due to the uncertainty on the target and beam polarization measurements are added in quadrature to calculate the total systematic uncertainty. The statistical uncertainty and the total systematic uncertainty are added in quadrature to form the total uncertainty.
4.4.1 Systematic uncertainties due to beam and target polarization uncertainties
The measured mean value of the target polarization is \(\langle P_{T}\rangle =0.72 \pm 0.06\) [38, 39], i.e., the fractional uncertainty of the target polarization amounts to 0.08. The ratios \(t^{(2)}_{\lambda _V \lambda _{\gamma }}\) and \(u^{(2)}_{\lambda _V \lambda _{\gamma }}\) have a corresponding scale uncertainty of 8%, since through their linear contribution to the “transverse” SDMEs \(n^{\lambda _V\lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\) and \(s^{\lambda _V\lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\), they are multiplied by \(\langle P_{T}\rangle \). It was checked that the amplitude ratios \(t^{(1)}_{11}\), \(t^{(1)}_{10}\), \(t^{(1)}_{11}\), \(t^{(1)}_{01}\), and \(u^{(1)}_{11}\), which can be extracted from data taken with an unpolarized target (see Ref. [27]), are effectively insensitive to the uncertainty on the target polarization.
The fractional uncertainty on the beam polarization amounts to 2% [47]. This results in an additional scale uncertainty on \(\mathrm{{Im}}\{u^{\lambda _V\lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\}\), \(\mathrm{{Re}}\{n^{\lambda _V\lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\}\), and \(\mathrm{{Re}}\{s^{\lambda _V\lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\}\) of 2%, since these SDMEs enter the expression of the angular distribution of finalstate particles multiplied by the beam polarization [6]. From the expression of SDMEs in terms of helicityamplitude ratios, it follows that there is an additional scale uncertainty of \(2\%\) for \(\mathrm{{Im}}\{t^{(1)}_{\lambda _V\lambda _{\gamma }}\}\), \(\mathrm{{Re}}\{t^{(2)}_{\lambda _V\lambda _{\gamma }}\}\), and \(\mathrm{{Re}}\{u^{(2)}_{\lambda _V\lambda _{\gamma }}\}\), while the influence of the uncertainty on the beam polarization can be neglected for \(\mathrm{{Re}}\{t^{(1)}_{\lambda _V\lambda _{\gamma }}\}\), \(\mathrm{{Im}}\{t^{(2)}_{\lambda _V\lambda _{\gamma }}\}\), and \(\mathrm{{Im}}\{u^{(2)}_{\lambda _V\lambda _{\gamma }}\}\). The scale uncertainty arising from the uncertainty on the beam and target polarizations is not shown in the figures but quoted separately.
4.4.2 Systematic uncertainty due to the extraction method
4.4.3 Systematic uncertainty due to the background contribution
The helicityamplitude ratios are extracted from the experimental data once taking into account the background contribution (see Eq. (55)) and once neglecting this contribution (see Eq. (49)). The systematic uncertainty from the background contribution of each amplitude ratio is computed as the modulus of the difference of the amplitude ratios obtained for these two cases. This conservative approach is used, since the background correction is estimated from MC data instead of experimental data. The SDMEs \(n^{\lambda _V \lambda '_V}_{\lambda _{\gamma } \lambda '_{\gamma }}\) and \(s^{\lambda _V \lambda '_V}_{\lambda _{\gamma } \lambda '_{\gamma }}\) are, as shown in Ref. [29], much less sensitive to the background contribution than the SDME \(u^{\lambda _V \lambda '_V}_{\lambda _{\gamma } \lambda '_{\gamma }}\), since they enter the formula for the angular distribution multiplied by the target polarization. As the amplitude ratios \(t^{(2)}_{\lambda '_N \lambda _N}\) and \(u^{(2)}_{\lambda '_N \lambda _N}\) contribute linearly to these SDMEs, they are expected to be less sensitive to the background contribution than the amplitudes relevant for scattering off an unpolarized target. It was checked that this is indeed the case. The small influence of the background correction to the nucleonhelicityflip amplitude ratios \(t^{(2)}_{\lambda '_N \lambda _N}\) and \(u^{(2)}_{\lambda '_N \lambda _N}\) can be explained by the statistical correlations between these amplitude ratios and \(t^{(1)}_{\lambda '_N \lambda _N}\), \(u^{(1)}_{11}\).
4.4.4 Systematic uncertainty due to the omission of inaccessible amplitude ratios
4.4.5 Systematic uncertainty due to the experimental uncertainty of \(A_1^{\rho }\)
The uncertainty on the asymmetry \(A_1^{\rho }\), as explained in Appendix A, leads to a range of \(\delta _{u}\) between \({}26.2^{\circ }\) and \(51.1^{\circ }\), corresponding to \(A_1^{\rho }=0.24+0.14\) and \(A_1^{\rho }=0.240.14\).
4.4.6 Systematic uncertainty due to the experimental uncertainty of \(\mathrm{{Im}}\{t^{(1)}_{11}\}\)
5 Results on the amplitude ratios
5.1 Discussion of results on the amplitude ratios
The result obtained from the 25parameter fit is presented in Fig. 2. The results for the large amplitudes are calculated at \(t'=0.132\) GeV\(^2\) and \(Q^2=1.93\) GeV\(^2\), while integrating W over the entire kinematic region. The results for the small amplitudes are calculated at \(t'=0.132\) GeV\(^2\), while integrating \(Q^2\) and W over the entire kinematic region. Here, the values \(t'=0.132\) GeV\(^2\) and \(Q^2=1.93\) GeV\(^2\) are the mean values of the kinematic variables over the entire kinematic region, 0.0 GeV\(^2 \le t' \le 0.40\) GeV\(^2\), 1.0 GeV\(^2 \le Q^2 \le 7.0\) GeV\(^2\), and 3.0 GeV \(\le W \le 6.3\) GeV. The mean value of W over the entire kinematic region is 4.73 GeV. The NPE amplitude ratio without nucleonhelicity flip, \(t_{11}^{(1)}\), is the dominant amplitude ratio. Its real and imaginary parts differ from zero by more than five standard deviations. As also already known from the previous analysis [27], \(\mathrm{{Re}}\{t_{01}^{(1)}\}\) is significantly nonzero. In this analysis, the UPE amplitude ratios without nucleonhelicity flip, \(\mathrm{{Re}}\{u^{(1)}_{11}\}\) and \(\mathrm{{Im}}\{u^{(1)}_{11}\}\), are individually extracted and found to be nonzero with a significance of about four standard deviations of the total uncertainty. The values of \(u^{(1)}_{11}\) and \(u^{(2)}_{11}\), with \(\sqrt{u^{(1)}_{11}^2+u^{(2)}_{11}^2}=0.35\pm 0.06\), agree with the result \(\sqrt{u^{(1)}_{11}^2+u^{(2)}_{11}^2} \approx 0.40 \pm 0.02\) obtained in the previous HERMES analysis [27]. The extracted values of the amplitude ratios show that the main contribution to the term \(\sqrt{u^{(1)}_{11}^2+u^{(2)}_{11}^2}\) comes from the amplitude \(U^{(1)}_{11}\) without nucleonhelicity flip, and in particular they show that \(U^{(1)}_{11}^2 \gg U^{(2)}_{11}^2\). The amplitude ratios \(\mathrm{{Im}}\{t^{(2)}_{01}\}\), \(\mathrm{{Im}}\{u^{(2)}_{11}\}\), and \(\mathrm{{Im}}\{u^{(2)}_{10}\}\) deviate from zero by about two standard deviations, while the other extracted amplitude ratios with nucleonhelicity flip are consistent with zero within two standard deviations. The amplitude ratios \(\mathrm{{Im}}\{t^{(2)}_{01}\}\) and \(\mathrm{{Im}}\{u^{(2)}_{10}\}\) are part of those ratios in Eq. (24), which can be nonzero at \(t'=0\). Among the amplitude ratios that can be zero at \(t'=0\), only the amplitude ratio \(\mathrm{{Im}}\{u^{(2)}_{11}\}\), which is proportional to \(\sqrt{t'}\) at \(t' \rightarrow 0\), differs from zero by about two standard deviations of the total uncertainty.
5.2 Comparison of calculated SDMEs with directly extracted SDMEs
A comparison of the SDMEs obtained from the SDME method in Refs. [17] and [29] to those calculated from the amplitude ratios extracted in the present analysis is presented in Figs. 3, 4 and 5. The SDMEs are calculated in each individual bin using the average kinematics in the parameterizations obtained for the amplitude ratios. Furthermore, their mean value is then determined by weighting the SDME value calculated in a given bin by the number of events in this bin. The correlation matrix for the 25 parameters is taken into account for the calculation of the statistical uncertainties of the SDMEs \(u^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\), \(n^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\), and \(s^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\) obtained in the amplitude method. As already mentioned in Sect. 1, the SDMEs \(n^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\) and \(s^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma \ }}\), presented in Figs. 4 and 5, can only be extracted from measurements with a transversely polarized target so that the helicityflip amplitude ratios \(t^{(2)}_{\lambda _V \lambda _{\gamma }}\) and \(u^{(2)}_{\lambda _V \lambda _{\gamma }}\) are extracted in this paper for the first time. The systematic uncertainties of the SDMEs from the amplitude method are determined in an analogous way as for the amplitude ratios by varying the relevant parameters, as explained in Sects. 4.4.2–4.4.6, and recalculating the corresponding SDMEs. The total uncertainty is the sum in quadrature of the statistical and the total systematic uncertainties.
Those SDMEs that can be extracted only from data taken with a longitudinally polarized lepton beam are shown in shaded areas. Figure 3 shows that for each SDME \(u^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\) determined from our present results, there exists an SDME \(u^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\) published in Ref. [17]. However, Figs. 4 and 5 show that for some of the SDMEs \(n^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\) and \(s^{\lambda _V \lambda '_V}_{\lambda _{\gamma }\lambda '_{\gamma }}\) determined in this analysis no published results from Ref. [29] exist, because the beam polarization was not exploited in the analyses presented in Ref. [29]. While in Refs. [17] and [29] a total of 53 SDMEs could be extracted, the amplitude method presented here allows for the calculation of 71 SDMEs based on the extraction of 25 parameters.
As seen from the figures, there is reasonable agreement between SDMEs obtained with the SDME method and those from the amplitude method. It is possible that the values of the SDMEs obtained in these two methods do not coincide, because the parameter space for SDMEs in the SDME method is different from that in the amplitude method. Indeed, the SDMEs should belong to a special region in the 71dimensional real space to give a nonnegative angular distribution. However, at present the equations determining the boundaries of this region are unknown. The physical SDMEs can be represented in terms of 17 helicityamplitude ratios. This restricts the region in the 71dimensional space. This requirement is not taken into account in the SDME method, but it suppresses statistical fluctuations especially when a SDME value is close to the boundary of the allowed region. Note that the positivity requirement on the angular distribution is inherent to the amplitude method, while it is not to the SDME method, where it is usually imposed artificially.
5.3 Comparison to amplitudes calculated in a GPDbased handbag model
Details of the calculations of the amplitudes as well as the parametrization of the GPDs, the meson wavefunction and the \(\pi \rho \) transition form factor can be found in the original papers [16, 51]. The evaluation of the amplitudes represent an intermediate step of the calculation of the observables discussed in these papers. These amplitudes are divided by \(F_{0 \frac{1}{2} 0 \frac{1}{2}}=T^{(1)}_{00}\) in order to obtain the amplitude ratios that can be compared to the ones discussed above. The phase convention from Eq. (9) is taken into account.
 \(t_{11}^{(1)}\):

Contributions come from GPDs \(H\xi ^2/(1\xi ^2) E \simeq H\). Good agreement is observed for the real part, which is by far the largest amplitude ratio. The calculated imaginary part appears to be too small. Note that a part of this difference is due to the known underestimation of the relative phase between the \(\gamma ^*_T\rightarrow V_{T}\) and \(\gamma ^*_L\rightarrow V_{L}\) amplitudes in the GK model [17].
 \(u_{11}^{(1)}\):

Contributions come from GPDs \(\widetilde{H}\) and the pion pole. The GK calculations underestimate the unnaturalparity contribution to the \(\gamma _T^*\rightarrow V_T\) amplitude, which is related to the small unnaturalparity cross section used in the GK model [51]. It may be traced back to the neglect of the nonpole contribution of the GPD \(\widetilde{E}\) or to a too small value for the \(\pi \rho \) transition form factor in the GK model.
 \(t_{11}^{(2)}\):

Contributions come from GPDs E. The calculated imaginary part agrees with the measurement.
 \(u_{11}^{(2)}\):

In GK calculations only the pion pole contributes since \(\widetilde{E}\) is neglected, so that the GK result is mirror symmetric upon sign change of the \(\pi \rho \) transition form factor. Good agreement with the data is seen for the positive sign.
 \(t_{00}^{(2)}\):

Contributions come from GPDs E. Agreement is observed with the measurement.
 \(t_{01}^{(1)}\):

Contributions come from GPDs \(\bar{E}_T\). Agreement is observed with the measurement.
 \(t_{01}^{(2)}\):

Contributions from GPDs \(H_T\). There is no pionpole contribution to this ratio, hence data cannot decide on the sign of the form factor. The measured imaginary part seems to be lower than the GK calculation.
 \(u_{01}^{(2)}\):

Contributions from GPDs \(H_T\). Since these GPDs have no specific parity, \(u_{01}^{(2)}\) is equal to \(t_{01}^{(2)}\) in the GK calculation.
 \(u_{10}^{(2)}\):

Contributions come from the pion pole only, so that the GK result is mirror symmetric upon sign change of the \(\pi \rho \) transition form factor. The positive sign is favored by the data.
The \(\gamma ^*_T\rightarrow V_{T}\) amplitudes, corresponding to the amplitude ratios \(t_{11}^{(1)}\), \(t_{11}^{(2)}\), and \(u_{11}^{(2)}\), are neglected in the GK model. This is seen to be in reasonable agreement with the data. Only gluon transversity GPDs could contribute and the contribution from the pion pole is suppressed by \(1/Q^3\) as compared to the longitudinal amplitudes. Both are neglected in the GK model.
As discussed in Sect. 4.3, the ratios \(u_{01}^{(1)}\), \(u_{10}^{(1)}\) and \(u_{11}^{(1)}\) cannot be determined experimentally in the present analysis and are hence put equal to zero. In the GK model, \(u_{01}^{(1)}\) and \(u_{11}^{(1)}\) are also set equal to zero, while \(u_{10}^{(1)}\) is nonzero due to a contribution from the pion pole, but small. Apart from the \(\gamma ^*_T\rightarrow V_{T}\) amplitudes, \(u_{01}^{(1)}\) and \(u_{11}^{(1)}\), also \(t_{10}^{(1)}\) and \(t_{10}^{(2)}\) are set equal to zero. This is consistent with what is extracted from the data.
As the unnaturalparity amplitudes depend on the sign of the \(\pi \rho \) transition form factor, a conclusion on the sign of the latter can be drawn when comparing the calculated GK amplitude ratios to the data. Only the amplitude ratios \(u_{11}^{(2)}\) and \(u_{10}^{(2)}\) appear sensitive to the sign of the form factor and are hence used to calculate the \(\chi ^2\) per degree of freedom, i.e., \(ndf=4\). For the positive sign \(\chi ^2/ndf=1.8/4\) is obtained and for the negative sign \(\chi ^2/ndf=30.3/4\). Hence the positive sign of this form factor is clearly favored.
6 Summary and conclusions
Within the total experimental uncertainty, all determined amplitude ratios with nucleonhelicity flip are consistent with zero. The extracted values of the amplitude ratios show that the main contribution to the quantity \(\frac{\sqrt{U_{1\frac{1}{2}1\frac{1}{2}}^2+U_{1\frac{1}{2}1\frac{1}{2}}^2}}{T_{0\frac{1}{2}0\frac{1}{2}}}\) obtained in Ref. [27] originates from the unnaturalparityexchange amplitude \(U_{1\frac{1}{2}1\frac{1}{2}}\) and that \(U_{1\frac{1}{2}1\frac{1}{2}}^2 \gg U_{1\frac{1}{2}1\frac{1}{2}}^2\). Furthermore, it is shown that the 53 SDMEs extracted in Refs. [17, 29] can be described with good accuracy using the 25 amplitude ratios obtained in the present analysis. By also exploiting the longitudinal beam polarization, 18 additional \(\rho ^0\) SDMEs are determined from the extracted 25 parameters for the first time.
The unnaturalparity amplitudes depend on the sign of the \(\pi \rho \) transition form factor, so that the comparison of certain amplitude ratios to calculations within a GPDbased handbag model taking into account the contribution from pion exchange allows the conclusion that the positive sign of this form factor is favored.
Together with precise data on the unpolarized differential cross section \({\text {d}}\sigma /{\text {d}}t\) of exclusive \(\rho ^0\) production in deepinelastic scattering, the extracted amplitude ratios could be used to obtain the amplitude \(T_{0\frac{1}{2}0\frac{1}{2}}\), for which the factorization property is proven.
Notes
Acknowledgements
We gratefully acknowledge the DESY management for its support and the staff at DESY as well as the collaborating institutions for their significant effort. This work was supported by the State Committee of Science of the Republic of Armenia, Grant No. 15T1C401; the FWOFlanders and IWT, Belgium; the Natural Sciences and Engineering Research Council of Canada; the Alexander von Humboldt Stiftung, the German Bundesministerium für Bildung und Forschung (BMBF), and the Deutsche Forschungsgemeinschaft (DFG); the Italian Istituto Nazionale di Fisica Nucleare (INFN); the MEXT, JSPS, and GCOE of Japan; the Dutch Foundation for Fundamenteel Onderzoek der Materie (FOM); the Russian Academy of Science and the Russian Federal Agency for Science and Innovations; the Basque Foundation for Science (IKERBASQUE) and MINECO (Juan de la Cierva), Spain; the U.K. Engineering and Physical Sciences Research Council, the Science and Technology Facilities Council, and the Scottish Universities Physics Alliance; as well as the U.S. Department of Energy (DOE) and the National Science Foundation (NSF).
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