The HERMES Collaboration

Spin Density Matrix Elements (SDMEs) describing the angular distribution of exclusive ρ 0 electroproduction and decay are determined in the HERMES experiment with 27.6 GeV beam energy and unpolarized hydrogen and deuterium targets. Eight (fifteen) SDMEs that are related (unrelated) to the longitudinal polarization of the beam are extracted in the kinematic region 1 GeV 2 < Q 2 < 7 GeV 2 , 3.0 GeV < W < 6.3 GeV, and −t < 0.4 GeV 2. Within the given experimental uncertainties, a hierarchy of relative sizes of helicity amplitudes is observed. Kinematic dependences of all SDMEs on Q 2 and t are presented, as well as the longitudinal-to-transverse ρ 0 electroproduction cross section ratio as a function of Q 2. A small but statistically significant deviation from the hypothesis of s-channel helicity conservation is observed. An indication is seen of a contribution of unnatural-parity-exchange amplitudes; these amplitudes are naturally generated with a quark-exchange mechanism.

mesons on nucleons offers a rich source of information on the mechanisms that produce these mesons, see e.g., Refs.[1,2].This process can be considered to consist of three subprocesses: i) the incident lepton emits a virtual photon γ * , which dissociates into a q q pair; ii) this pair interacts strongly with the nucleon; iii) from the scattered q q pair the observed vector meson is formed.

In Regge phenomenology, the interaction of the q q pair with the nucleon proceeds through the exchange of a pomeron or (a combination of) the exchanges of other reggeons (e.g., ρ, ω, π, ...).If the quantum numbers of the particle lying on the Regge trajectory are J P = 0 + , 1 − , ..., the process is denoted Natural Parity Exchange (NPE).Alternatively, the case of J P = 0 − , 1 + , ... is denoted Unnatural Parity Exchange (UPE).In perturbative quantum chromodynamics (pQCD), the interaction of the q q pair with the nucleon can proceed via two-gluon exchange or quark-antiquark exchange, where the former corresponds to the exchange of a pomeron and the latter to the exchange of a (combination of) reggeon(s).

Spin density matrix elements (SDMEs) describe the final spin states of the produced vector meson.In this work, SDME values will be determined and discussed in the formalism that was developed in Ref. [3] for the case of an unpolarized or longitudinally polarized beam and an unpolarized target.For completeness, we also present SDME values in the more general formalism of Ref. [4].The SDMEs can be expressed in terms of helicity amplitudes that describe the transi

ons from the
nitial helicity states of virtual photon and incoming nucleon to the final helicity states of the produced vector meson and the outgoing nucleon.The values of SDMEs will be used to establish a hierarchy of helicity amplitudes, to test the hypothesis of s-channel helicity conservation, to investigate UPE contributions, and to determine the longitudinal-totransverse cross-section ratio.

In the framework of pQCD, the nucleon struc ure can also be studied through hard exclusive meson production as the process amplitude contains Generalized Parton Distributions (GPDs) [5,6,7].For longitudinal virtual pho-tons, this amplitude is proven to factorize rigorously into a perturbatively calculable hard-scattering part and two soft parts (collinear factorization) [8,9].The soft parts of the convolution contain GPDs and a meson distribution amplitude.At leading twist, the chiral-even GPDs H f and E f are sufficient to describe exclusive vector-meson production on a spin-1/2 target such as a proton or a neutron, where f denotes a quark of flavor f or a gluon.These GPDs are of special interest as they are related to the total angular momentum car ied by quarks or gluons in the nucleon [10].

Although there is no such rigorous proof for transverse virtual photons, phenomenological models use the modified perturbative approach [11] instead, which takes into account parton transverse momenta.The latter are included at subleading twist in the subprocess γ * f → Mf , where M denotes the meson, while the partons are still emitted and reabsorbed by the nucleon collinear to the nucleon momentum.By using this approach, the pQCDinspired phenomenological "GK model" can describe existing data on cross sections, SDMEs and spin asymmetries in exclusive vector-meson production for values of Bjorken-x, x B , below about 0.2 [12,13,14].It can also describe exclusive leptoproduction of pseudoscalar mesons by including the full contribution to the electromagnetic form factor from the pion, in contrast to earlier studies a leading-twist, which took into account only the relatively small perturbative contribution to this form factor (see Ref. [15] and references therein).The GK model also applies successfully to the description of deeply virtual Compton scattering [16].The results of the most recent variant of the GK model, in which the unnatural-parity contributions due to pion exchange are included to describe exclusive ω leptoproduction [17], will be compared in this paper to the HERMES proton data in terms of SDMEs and certain combinations of them.

Early papers on exclusive ω electroproduction are summarized in Ref. [18], which particularly contains results on SDMEs obtained at DESY for 0.3 GeV 2 < Q 2 < 1.4 GeV 2 and 0.3 GeV < W < 2.8 GeV.The symbol Q 2 represents the negative square of the virtual-photon four-m mentum and W is the invariant mass of the photon-nucleon system.

Recently, SDMEs in exclusive ω electroproduction were studied for 1.6 GeV 2 < Q 2 < 5.2 GeV 2 by CLAS [19] and it was found that the exchange of the pion Regge trajectory dominates exclusive ω production, even for Q 2 values as large as 5 GeV 2 .


Formalism


Spin Density Matrix Elements

The ω meson is produced in the following reaction:
e + p → e + p + ω,(1)
with a branching ratio Br = 89.1% for the ω decay:
ω → π + + π − + π 0 , π 0 → 2γ. (2)
The angular distribution of the three final-state pions depends on SDMEs.The first subprocess of vector-meson production, the emission of a virtual photon (e → e + γ * ), is described by the photon spin density matrix [3],
̺ U+L λγ λ ′ γ = ̺ U λγ λ ′ γ + P b ̺ L λγ λ ′ γ ,(3)
where U and L denote unpolarized and longitudinally polarized beam, respectively, and P b is the value of the beam polarization.The photon spin density matrix can be calculated in quantum electrodynamics.

The vector-meson spin density matrix ρ λV λ ′ V is expressed through helicity amplitudes F λV λ ′ N λγ λN .These amplitudes describe the transition of a virtual photon with helicity λ γ to a vector meson with helicity λ V , while λ N and λ ′ N are the helicities of the nucleon in the initial and final states, respectively.Helicity amplitudes depend on W , Q 2 , and t ′ = t − t min , where t is the Mandelstam variable and −t min represents the smallest kinematically allowed value of −t at fixed virtual-photon energy and Q 2 .The quantity √ −t ′ is approximately equal to the transverse momentum of the vector meson with respect to the direction of the virtual photon in the γ * N centre-of-mass (CM) syste

n 2φ + 2ǫ(1 − ǫ) sin
r 8 11 sin 2 Θ + r 8 00 cos 2 Θ − √ 2Re{r 8 10 } sin 2Θ cos φ − r 8 1−1 sin 2 Θ cos 2φ .(15)
Definitions of angles and reference frames are shown in Fig. 1.The directions of the axes of the hadronic CM system and of the ω-meson rest frame follow the directions of the axes of the helicity frame [3,20,21].

The angle Φ between the ω production and the lepton scattering plane in the hadronic CM system is given by
cos Φ = (q × v) • (k × k ′ ) |q × v| • |k × k ′ | , (16) sin Φ = [(q × v) × (k × k ′ )] • q |q × v| • |k × k ′ | • |q| . (17)
Here k, k ′ , q = k − k ′ , and v are the three-momenta of the incoming and outgoing leptons, virtual photon, and ω meson res cos φ = (q × p ′ ) • (p ′ × n) |q × p ′ | • |p ′ × n| ,(20)sin φ = − [(q × p ′ ) × p ′ ] • (n × p ′ ) |(q × p ′ ) × p ′ | • |n × p ′ | . (21)
3 Data Analysis


HERMES Experiment

The data analyzed in this paper wer accumulated with the HERMES spectrometer during the running period of 1996 to 2007 using the 27.6 GeV long ometer, which is described in detail in Ref. [22], was built of two identical halves situated above and below the lepton beam pipe.It co sisted of a dipole magnet in conjunction with tracking and particle identifi when their polar angles were in the range ±170 mrad in the horizontal direction and ±(40 − 140) mrad in the vertical direction.The spectrometer permitted a precise measurement of charged-particle momenta, with a resolution of 1.5%.A separation of leptons was achieved with an average efficiency of 98% and a hadron contamination below 1%.


Selection of Exclusively Produced ω Mesons

The following requirements were applied to select exclusively produced ω mesons from reaction (1): i) Exactly two oppositely charged hadrons, which are assumed to be pions, and one lepton with the same charge as the beam lepton are ide the combined responses of the four particle-identification d ton invariant ma

to be in the inte
val 0.11 GeV < M (γγ) < 0.16 GeV.The distribution of M (γγ) is shown in Fig. 2.This distribution is centered at m π 0 = 134.69± 19.94 MeV, which agrees well with the PDG [24] value of the π 0 mass.

iii) The three-pion invariant mass is required to obey 0.71 GeV≤ M (π + π − π 0 ) ≤ 0.87 GeV.iv) The kinematic requirements for exclusive production of ω mesons are the following: a) The scattered-lepton momentum lies above 3.5 GeV.

b) The constraint −t ′ < 0.2 GeV 2 is used.c) For exclusive production the missing energy ∆E must vanish.Here, the missing energy is calculated both for proton and deuteron as ∆E =
M 2 X −M 2 p 2Mp
, with M p being the proton mass and M 2 X = (p + q − p π + -p π − -p π 0 ) 2 the missing mass squared, where p, q, p π + , p π − , and p π 0 are the four-momenta of target nucleon, virtual photon, and each of the three pions respect

ely.In this analysis, taking into account t
e spectrometer resolution, the missing energy has to lie in the interval −1.0 GeV < ∆E < 0.8 GeV, which is referred to as "exclusive region" in the following.d) The requirement Q 2 > 1.0 GeV 2 is applied in order to facilitate the application of pQCD.e) The requirement W > 3.0 GeV is applied in order to be outside of the resonance region, while an upper cut of W < 6.3 GeV is applied in order to define a clean kinematic phase space.

After application of all these constraints, the proton sample contains 2260 and the deuteron sample 1332 events of exclusively produced ω mesons.These data samples are referred to in the following as data in the "entire kinematic region".The invariant-mass distributions for exclusively produced ω mesons are shown in Fig. 3.Note the reasonable agreement of the fit result, m ω = 784.8± 55. 8 MeV for proton data and m ω = 784.6 ± 58.2 MeV for deuteron data, with the PDG [24] value of the ω mass.The dis ns of missing energy ∆E, shown in Fig. 4, exhibit clearly visible exclusive peaks.The shaded histograms represent semi-inclusive deep-inelastic scattering (SIDIS) background obtained ] Monte Carlo simulation that is normalized to data in the region 2 GeV < ∆E < 20 GeV.The simulation is used to determine the fraction of background under the exclusive peak, which is calculated as the ratio of number of background events to the total number of events.It amounts to about 20% for the entire kinematic region and increases from 16% to 26% with increasing −t ′ .


Comparison of Data and Monte Carlo Events

Distributions of experimental data in some kinematic variables are compared to those simulated by PYTHIA.The comparison is shown in Fig. 5 and mostly demonstrates good agreement between experimental and simulated data.4 Extraction of ω Spin Density Matrix Elements


The Unbinned Maximum Likelihood Method

The SDMEs are extracted from data by fitting the angular distribution W U+L (Φ, φ, cos Θ) to the experimental angular distribution using an unbinned maximum likelihood method.The probability distribution function is W U+L (R; Φ, φ, cos Θ), where R represents the set of 23 SDMEs, i.e., the coefficients of the trigonometric functions in Eqs.(14,15).The negative log-likelihood function to be minimized reads
− ln L(R) = − N i=1 ln W U+L (R; Φ i , φ i , cos Θ i ) N (R) ,(22)
where the normalization factor
N (R) = NMC j=1 W U+L (R; Φ j , φ j , cos Θ j )(23)
is calculated numerically using events from a PYTHIA Monte Carlo generated according to an isotropic threedimensional angular distribution and passed through the same analytical process as experimental data.The numbers of data and Monte Carlo events are denoted by N and N MC , respectively.


Background Treatment

In order to account for the SIDIS background in the fit, first "SIDIS-background SDMEs" are obtained using Eqs.Events/1.4MeVFig. 5. Distributions of several kinematic variables from experimental data on exclusiv

ω-meson leptoproduction (black squares) in
comparison with simulated exclusive events from the PYTHIA generator (dashed areas).Simulated events are normalized to the experimental data.(22,23) for the PYTHIA SIDIS sample in the exclusive region.Then, SDMEs corrected for SIDIS background are obtained as follows [26]:
− ln L(R) = − N i=1 ln (1 − f bg ) * W U+L (R; Φ i , φ i , cos Θ i ) N (R, Ψ ) + f bg * W U+L (Ψ ; Φ i , φ i , cos Θ i ) N (R, Ψ ) .(24)
From now on, R denotes the set of SDMEs corrected for background, Ψ the set of the SIDIS-background SDMEs, and f bg is the fraction of SIDIS background.The normalization factor reads correspondingly
N (R, Ψ ) = NMC j=1 ( , φ j , cos Θ j ) . (25)

Syst ty on a given extracted SDME r is obtained by adding in quadrature the uncertainty from the background subtraction procedure, ∆r bg sys , and the one due to the extraction method, ∆r MC sys .The former uncertainty is assigned to be the difference between the SDME obtained with and without bac

round correction.This
conservative approach also covers the small uncertainty on the fraction of SIDIS background, f bg .The uncertainty statistics of the Monte Carlo data exceed those of the experimental data by about a factor of six.The generated events were passed through a realistic model of the HERMES apparatus using GEANT [27] a

tract the SDME set R MC .
n this way, effects from detector acceptance, efficiency, smearing, and misalignment are accounted for.Two uncertainties are considered to be responsible for the difference between input and output value of a given SDME r,
(r − r MC ) 2 = (∆r MC sys ) 2 + (∆r MC stat ) 2 ,(26)
where ∆r MC stat is the statistical uncertainty of r MC as obtained in the fitting procedure that uses MINUIT [28].From Eq. ( 26), ∆r MC sys is determined, using the convention that ∆r MC sys is set to zero if [(r − r MC ) 2 − (∆r MC stat ) 2 ] is negative.


Results

The results on SDMEs in the Schilling-Wolf [3] representation are given in Tables 1-5 in Appendix B and in the Diehl [4] representation in Table 6 in the same Appendix.The SDMEs for the entire kinematic region are discussed in Sect.5.1, while their dependences on Q 2 and −t ′ are discussed in Sect.5.3.


SDMEs for the Entire Kinematic Region

The SDMEs of the ω meson for the entire kinematic region ( Q 2 = 2.42 GeV 2 , W = 4.8 GeV, and −t ′ = 0.080 GeV 2 ) are presented in Fig. 6.These SDMEs are divided into five classes corresponding to different helicity transitions.The main terms in the expressions of class-A SD tual photons to longitudinal vector mesons, γ * L → V L , and from transverse virtual photons to transverse vector mesons, γ * T → V T .The dominant terms of class B correspond to the interference of these two transitions.The main terms of class-C, class-D,

d class-
SDMEs are proportional to small amplitudes describing γ * T → V L , γ * L → V T , and γ * T → V −T transitions respectively.The SDMEs for the proton and deuteron data are found to be consistent with each other within their quadratically combined total uncertainties, with a χ 2 per degrees of freedom of 28

class-A SDMEs r 1  1−1 and I
{r 2 1−1 } can be violated only by the quadratic contributions of the doublehelicity-flip amplitudes
T 1± 1 2 −1 1 2 and U 1± 1 2 −1 1 2 with |λ V − λ γ | = 2.
The observed validity of SCHC means that their possible contributions are sm rtainties.Also for class-B SDMEs, to which the same small double-helic rly, no SCHC violation is observed.In addition, class-B SDMEs contain the contribution of the two small products
T 0± 1 2 1 1 2 T * 1± 1 2 0 1 2 (U 0± 1 2 1 1 2 U * 1± 1 2 0 1 2
).As the SCHC hypothesis is fulfilled, all these contributions are concluded to be negligibly small compared to the experimental uncertainties.This validates the assumption made in Sect.2.2 that the double-helicity-flip amplitudes can be neglected.

All SDMEs of class C to E have to be zero in the case of SCHC.The class-C SDME r 5 00 deviates from zero by about three standard deviations for the proton and two standard deviations for the deuteron (see Fig. 6).Since the numerator of the equation for r 5 00 [20],
r 5 00 = Re T 0− 1 2 1 1 2 T * 0− 1 2 0 1 2 + T 0 1 2 1 1 2 T * 0 1 2 0 1 2 N ,(27)
contains two amplitude without an amplitude analysis of the presented data it cannot be concluded which contribution to r 5 00 dominates.Both amplitudes T 0− 1 2 1 1 2 and T 0 1 2 1 1 2 have to be zero if the SCHC hypothesis holds.

Figure 6 shows that out of the six SDMEs of class D three, i.e., r 5  11 , r 5 1−1 , and Im{r 6 1−1 }, slightly differ from ze e calculated linear combination of these three SDMEs, r 5  11 + r 5 1−1 − Im{r 6 1−1 }, is −0.14 ± 0.03 ± 0.04 for the proton and −0.10 ± 0.03 ± 0.03 for the deuteron.These values differ from zero by about three standard deviations of the total uncert inty for the proton.This, together with the experimental information on measured class-C and class-D SDMEs, indicates a violation of the SCHC hypothesis in exclusive ω production.


Dependences of SDMEs on Q 2 and −t ′ and Comparison to a Phenomenological Model

In th ations of them are presented and interpreted wherever possible.In particular, the proton data presented in this paper are compared to the calculations of the phenomenological GK model described in Sect. 1.In each case, model calculations are shown with and without inclusion of the pion-pole contribution.In order to stay in the framework of handbag factorization and to avoid large 1/Q 2 corrections, model calculations are only shown for Q 2 > 2 GeV 2 , which leaves for the Q 2 dependence only two data points that can be compared to the model calculation.This paucity of comparable points makes it sometimes difficult to draw useful conclusi matic dependences of SDMEs on Q 2 and −t ′ are presented in three bins of Q 2 with Q 2 = 1.28 GeV 2 , Q 2 = 2.00 GeV 2 , Q 2 = 4.00 GeV 2 , and t ′ with −t ′ = 0.021 GeV 2 , −t ′ = 0.072 GeV 2 , −t ′ = 0.137 GeV 2 .Table 7 shows the average value of Q 2 and −t ′ for bins in −t ′ and Q 2 , respectively.

The Q 2 and −t ′ dependences of class-A SDMEs are shown and compared to the model calculations in Fig. 7.All three SDMEs clearly s

the sign of δ N .


Longitudinal-to-
ransverse Cross-section Ratio

Usually, the longitudinal-to-transverse virtual-photon differential cross-section ra om the measured SDME r 04 00 using the approximated equation
[20] R ≈ 1 ǫ r 04 00 1 − r 04 00 . (43)
This relation is ex dence for the ρ 0 meson [20] is shown.For ω mesons produced in the entire kinematic region, it is found that R = 0.25 ± 0.03 ± 0.07 for the proton and R = 0.24 ± 0.04 ± 0.07 for the deuteron data.Compared to the case of exclusive ρ 0 production, this ratio is about four times smaller, and for the ω meson this ratio is almost independent of Q 2 .The R Fi . 14.The Q 2 (left) and −t ′ (right) dependences of the longitudinal-to-transverse virtual-photon differential cross-section ratio for exclusive ω and ρ 0 electroproduction at H RMES, where the −t ′ bin covers the interval [0.0-0.2]GeV 2 for ω production and [0.0-0.4]GeV 2 for ρ 0 entire kinematic region.Otherwise as for Fig. 7.

t ′ dependence of R is shown in the right panel of Fig. 14.

The comparison of the proton data to the GK model

alculations with and without inclusion of the p
on-pole contribution demonstrates the clear need to include the pion pole.The data are w follow the t ′ dependence suggested by the model when the pion-pole contribution is included.This i plies that transverse and longitudinal virtual-photon cross sections have different t ′ dependences.Hence the usual high-energy assumption that their ratio can be identified with the corresponding ratio of the integrated cross sections does not hold in exclusive ω electroproduction at HERMES kinematics, due to the pion-pole contribution.The GK model appears to fully account for the unnatural-parity contribution to R and shows rather good agreement with the data.


The UPE-to-NPE Asymmetry of the Transverse Cross Section

The UPE-to-NPE asymmetry of the transverse differe tial cross section is defined as [29]
P = dσ N T − dσ U T dσ N T + dσ U T ≡ dσ N T /dσ U T − 1 dσ N T /dσ U T + 1 = (1 + ǫR)(2r 1 1−1 − r 1 00 ), (44)
where σ N T and σ U T denote the part of the cross section due to NPE and UPE, respectively.Substituting Eq. (43) in Eq. (44) leads to the approximate relation
P ≈ 2r 1 1−1 − r 1 00 1 − r 04 00 . (45)
The value of P obtained in the entire kinematic region is -0.42 ± 0.06 ± 0.08 and -0.64 ± 0.07 ± 0.12 for proton and deuteron, respectively.This means that a large part of the transverse cross section is due to UPE.In Fig. 15, the Q 2 and t ′ dependences of the UPE-to-NPE asymmetry of the transverse differential cross section for exclusive ω production are presented.Again, the GK model calculation appears to fully account for the unnatural-par

y contribution and shows very good agreement with the dat
both in shape and magnitude.


Hierarchy of Amplitudes

In order to develop a hierarchy esulting hierarchy is given in Eqs. ( 62) and (64) below.


U 10 versus U 11

From Eqs. ( 35) and (37), the relation
2(u 2 2 + u 2 3 ) u 1 ≈ |U 11 U * 10 | |U 11 ǫ|U 10 /U 11 | 2 (46)
is obtained.Using the measured values of those SDMEs that determine u 1 , u 2 , and u 3 , the following amplitude ratio is estimated:
|U 10 | |U 11 | ≈ 2(u 2 2 + u 2 3 ) u 1 ≈ 0.2. (47)
In order to reach the best possible accuracy for such estimates, the mean values of SDMEs for the proton and deuteron are used and preference will be given to quantities that do not contain polarized SDMEs, which have much less experimental accuracy than the unpolarized SDMEs.The relatively large value for the ratio |U 10 /U 11 |

due to the large measur
d value of u 3 .However, as this value is compatible with zero within about one standard deviation of the total uncertainty, the contribution of u 3 in Eq. (47) can be neglected, which leads to the valu

< W < 6.3 GeV, and −t ′ < 0.2 GeV
.The average kinematic values are Q 2 = 2.42 GeV first time, 8 polarized spin density matrix elements are extracted.The kinematic dependences of all 23 SDMEs are presented for proton and deuteron data.No significant differences between proton and deuteron results are seen.

The SDMEs are presented in five classes corresponding to different helicity transitions between the virtual photon and the ω meson.While the values of class- y conservation, the class-C SDME r 5 00 indicates a violation of this hypothesis.The values of those class-D SDMEs that correspond to the transition γ * L → ω T also indicate a small violation of the hypo the UPE transition γ * T → ω T is larger than the NPE amplitude for the same transition, i.e., |U 11 | 2 > |T 11 | 2 .The importance of UPE transitions is also shown by a combination of SDMEs denoted u 1 .This suggests that at HER-MES energies in exclusive ω electroproduction the quarkexchange mechanism, or π 0 , a 1 ... exchanges in Regge phenomenology, play a significant role.

The phase shift between those UPE amplitudes that describe transverse ω production by transverse and longitudinal virtual photons, U 11 for γ * T → ω T nd U 10 for γ * L → ω T , respectively, as well as the magnitude of the phase difference between the NPE amplitudes T 11 and T 00 is determined for the first time.

The ratio R between the differential longitudinal and transverse virtual-photon cross-sections is determined to be R = 0.25 ± 0.03 ± 0.07 for the ω meson, which is about four times sma try of the transverse virtual-photon cross section is determined to be P = −0.42± 0.06 ± 0.08 and P = −0.64 ± 0.07 ± 0.12 for the proton and deuteron data, respectively.

From the extracted SDMEs, two slightly different hierarchies of helicity amplitudes can be derived, which remain indistinguishable for the given experimental accuracy of the presented data.Both hierarchies consistently mean that the UPE amplitude describing the γ * T → ω T transition dominates over the two NPE amplitudes describing the γ

L → ω L
and γ * T → ω T transitions, with the latter two being of similar magnitude.

Good agreement between the presented proton data and results of a pQCD-inspired phenomenological model is found only when including pion-pole contributions, which are of unnatural parity.The distinct t ′ dependence of the pion-pole contribution leads to a t ′ dependence of R.This invalidates for exclusive ω production at HERMES energies the common high-energy assumption of identifying R with the ratio of the integrated longitudinal and transverse cross sections.

Fig. 1 .
1
Fig.1.Definition of angles in the process eN → eN ω, where ω → π + π − π 0 .Here, Φ is the angle between the ω production plane and the lepton scattering plane in the center-of-mass system of the virtual photon and the target nucleon.The variables Θ and φ are respectively the polar and azimuthal angles of the unit vector normal to the decay plane in the ω-meson rest frame.


Fig. 2 .
2
Fig.2.Two-photon invariant mass distribution after application of all criteria to select exclusively produced ω mesons.The Breit-Wigner fit to the mass distribution is shown as a continuous line and the dashed line indicates the PDG value of the π 0 mass.


Fig. 3 .
3
Fig.3.Breit-Wigner fit (solid line) of π + π − π 0 invariant mass distributions after application of all criteria to select ω mesons produced exclusively from proton (top) and from deuteron (bottom).The dashed line represents the PDG value of the ω mass.


2 Fig. 4 .
24
Fig.4.The ∆E distributions of ω mesons produced in the entire kinematic region and in three kine atic bins in −t ′ are compared with SIDIS ∆E distributions from PYTHIA (shaded area).The vertical dashed line denotes the upper limit of the exclusive region.


Fig. 6 .
6
Fig. 6.The 23 SDMEs for exclusive ω electroproduction extracted in the entire HERMES kinematic region with Q 2 = 2.42 GeV 2 , W = 4.8 GeV, −t ′ = 0.080 GeV 2 .Proton data are denoted by squares and deuteron data by circles.The inner error bars represent the statistical uncertainties, while the outer ones indicate the statistical and systematic uncertainties added in quadrature.Unpolarized (polarized) SDMEs are displayed in the unshaded (shaded) areas.


)Fig. 7 .